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J. Chem. Phys. 153, 184307 (2020); https://doi.org/10.1063/5.0030808 153, 184307 © 2020 Author(s).Stereodynamics of ultracold rotationally inelastic collisions Cite as: J. Chem. Phys. 153, 184307 (2020); https://doi.org/10.1063/5.0030808 Submitted: 25 September 2020 . Accepted: 22 October 2020 . Published Online: 11 November 2020 Masato Morita , and Naduvalath Balakrishnan COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical Chemistry ARTICLES YOU MAY BE INTERESTED IN Stereodynamics of rotationally inelastic scattering in cold He + HD collisions The Journal of Chemical Physics 153, 091101 (2020); https://doi.org/10.1063/5.0022190 QCT calculations of O 2 + O collisions: Comparison to molecular beam experiments The Journal of Chemical Physics 153, 184302 (2020); https://doi.org/10.1063/5.0024870 The effect of collisions on the rotational angular momentum of diatomic molecules studied using polarized light The Journal of Chemical Physics 153, 184310 (2020); https://doi.org/10.1063/5.0024380The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Stereodynamics of ultracold rotationally inelastic collisions Cite as: J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 Submitted: 25 September 2020 •Accepted: 22 October 2020 • Published Online: 11 November 2020 Masato Moritaa)and Naduvalath Balakrishnanb) AFFILIATIONS Department of Chemistry and Biochemistry, University of Nevada, Las Vegas, Nevada 89154, USA Note: This paper is part of the JCP Special Collection in Honor of Women in Chemical Physics and Physical Chemistry. a)Electronic mail: masatomorita2013@gmail.com b)Author to whom correspondence should be addressed: naduvala@unlv.nevada.edu ABSTRACT Recent experiments on rotational quenching of HD in the v= 1, j= 2 rovibrational state in collisions with H 2, D 2, and He near 1 K have revealed strong stereodynamic preference stemming from isolated shape resonances. So far, the experiments and subsequent theoretical analyses have considered the initial HD rotational state in an orientation specified by the projection quantum number mor a coherent superposition of different mstates. However, it is known that such stereodynamic control is generally not effective in the ultracold energy regime due to the dominance of the incoming s-wave ( l= 0, partial wave). Here, we provide a detailed analysis of the stereodynamics of rotational quenching of HD by He with both mandm′resolution, where m′refers to the inelastically scattered HD. We show the exis- tence of a significant mdependence in the m′-resolved differential and integral cross sections even in the ultracold s-wave regime with a factor greater than 60 for j= 2→j′= 1 and a factor greater than 1300 for j= 3→j′= 2 transitions. In the helicity frame, how- ever, the integral cross section has no initial orientation ( k) dependence in the ultracold energy regime, even resolving with respect to the final orientation ( k′). The distribution of final rotational state orientations ( k′) is found to be statistical (uniform), regardless of the initial orientation. Published under license by AIP Publishing. https://doi.org/10.1063/5.0030808 .,s I. INTRODUCTION Cold and ultracold molecules are rapidly transforming our understanding of chemical reaction dynamics and energy transfer phenomena in the deep quantum regime. The ability to control their properties and interaction through external electric, mag- netic, and optical fields has led to new applications in emerg- ing areas of quantum information processing, quantum comput- ing, quantum sensing, and precision spectroscopy.1–11Ultracold chemistry experiments can offer much insight into reaction inter- mediates and energy dispersal, as recently demonstrated for the benchmark KRb + KRb →K2+ Rb 2reaction that was shown to occur through a four-center mechanism and a transient K 2Rb2 intermediate complex.12,13Even for closed-shell molecules with- out an electric dipole moment (or molecules with a weak dipole moment like HD) or a hyperfine structure, selective preparation of the initial molecular rotational state in an orientation specifiedby projection quantum number m(or a superposition of m-states) allows considerable control of angular distribution of the inelasti- cally scattered molecule, as demonstrated in recent experiments of Perreault et al.14–18 There is a long history of molecular collisions exploring the dependence of inelastic and elastic (polarization) cross sections on the projection quantum numbers mand m′of initial and final molecular rotational states.19–31In particular, the search for propen- sitiesΔm=m′−myielded new insights into collisional reorienta- tion, providing ideas for developing approximate calculation meth- ods such as the jzconserving coupled-states/centrifugal-sudden (CS) approximation as well as the infinite-order sudden (IOS) approxi- mation.23–25,29Analogous efforts to relate dynamical outcomes to the reagent approach geometry in chemical reactions (dynamical stere- ochemistry) have also been developed.32–34Nowadays, such efforts have been broadly termed stereodynamics in which the correla- tion between various vector properties in collisions and reactions is J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp examined, drawing extensive research interest to deepen the physical insight into the underlying dynamics.35–45 An experimental scheme recently proposed by Perreault et al.14–16has proved to be a powerful tool for probing stereodynamics of molecular collisions when a small number of partial waves control the collision outcome. In their intrabeam scheme, a co-expansion of colliding species in a supersonic molecular beam effectively reduces the relative collision energy to around 1 K (provided that the col- lision partners have comparable initial velocity distribution). The initial rovibrational state ( v,j) as well as the orientation of the molec- ular rotational state specified by the projection quantum number m onto the initial relative velocity vector between the collision partners is prepared by the Stark-induced Adiabatic Raman Passage (SARP) method.17,18This approach eliminates the requirement of external electromagnetic fields to confine and control molecules during the collision process and enables collisional studies of molecules that are not amenable to the external field control. Perreault et al. showed that rotational quenching of HD ( v= 1 and j= 2) in collisions with H 2, D 2, and He depends strongly on whether the initial HD molecular bond axis is preferentially aligned horizontal (H-SARP) or vertical (V-SARP) to the initial relative velocity vector. For more details of the intrabeam method for cold and ultracold collisions, we refer to recent works of Amarasinghe et al.46,47 The pioneering experiments of Perreault et al. motivated theo- retical studies48–51that revealed rotational quenching of HD by H 2, and He is governed by isolated shape resonances near a collision energy of∼1 K. Thus, the stereodynamic control of the resonance features provides a direct handle to influence the collision outcome. As shown in a recent theoretical study, such control is not limited to isolated shape resonances but also in the presence of overlap- ping shape resonances, as demonstrated for the H 2+ HCl52sys- tem that is characterized by a stronger anisotropic interaction than H2+ HD. While these successful demonstrations are promising, it implies that stereodynamic control is effective only in the energy region where one or more shape resonances exist. In other words, it is dif- ficult to control collision outcomes utilizing the m-dependence of cross sections (stereodynamic preference) in the ultracold energy regime (<1 mK) dominated by an incoming s-wave ( l= 0 partial wave). Furthermore, it has been shown that the stereodynamic con- trol of the collisional rotational transition and chemical reaction is not possible at the level of the integral cross section (ICS) for the l= 0 partial wave.53Nonetheless, here we show that there exists com- plex stereodynamics in the ultracold energy regime and the control of the outcomes by preparing initial mis still possible by a selec- tive measurement of inelastically scattered molecules in an oriented rotational state designated by m′. We demonstrate that both m- and m′-resolved measurements can reveal a significant stereodynamic effect over a wide range of energies, regardless of the presence of a shape resonance. In this paper, we examine He + HD collisions as an illustra- tive case as it has already been studied by the SARP experiment and is a benchmark system for both experiment and theory. The use of He as the collision partner also avoids the difficulty of preparing H 2 in a single rotational level of ortho (nuclear spin I= 1, odd rota- tional levels) or para (nuclear spin I= 0, even rotational levels) states due to the thermal population of rotational states.14,15Furthermore, highly accurate potential energy surfaces (PESs) with spectroscopicaccuracy are available for the HeH 2system.54–56Indeed, our recent study51of rotational quenching of HD ( v= 1,j= 2→v′= 1,j= 0) in cold collisions with He yielded good agreement with experimental results of Perreault et al.16for the angular distribution of inelastically scattered HD. The paper is organized as follows: Section II provides a brief outline of the theory and computational details. The results for rota- tional quenching processes v= 1,j= 2→v′= 1,j′= 1 (Sec. III A) andv= 1,j= 3→v′= 1,j′= 2 (Sec. III B) are presented in Sec. III, and a summary of our findings is given in Sec. IV. II. THEORY AND COMPUTATION A. Scattering formalism The quantum mechanical scattering problem of collisions between He (1S) and HD (1Σ) is numerically solved using the MOLSCAT (v.14) code.57It implements the solution of the close- coupling (CC) equations58derived from the time-independent Schrödinger equation using the total angular momentum represen- tation in the space-fixed (SF) coordinate frame. The Hamiltonian for the collision complex of He + HD may be written (̵h= 1) in the Jacobi coordinate as ˆH=−1 2μRd2 dR2R+ˆl2 2μR2+ˆhHD+Vint(R,r,γ), (1) whereμ(=1.721 871 434 amu) is the reduced mass of He and HD, ˆl is the orbital angular momentum operator for the relative motion of He and HD, and ˆhHDis the rovibrational Hamiltonian for the iso- lated HD molecule. The interaction potential Vint(R,r,γ) between He and HD in the electronic ground state is generated from the BSP3 PES for the HeH 2system.54 The total wavefunction for a given value of the total angular momentum Jof the collision complex, its projection component M onto the SF z-axis, and the inversion parity εI= (−1)j+l(all three quantities are conserved during the collision) are expanded as ΨJMεI=1 R∑ vjlFJMεI vjl(R)χj v(r) r∣JMεI(lj)⟩, (2) where FJMεI vjl(R)are the radial expansion coefficients in R,χj v(r) denotes the radial part of the rovibrational eigenfunctions of HD specified by quantum numbers of vand j, and | JMεI(lj)⟩denotes the basis functions for the angular degrees of freedom in the total angular momentum representation for a given parity εI. The coefficients FJMεI vjl(R)satisfy the CC equations obtained by substituting Eq. (2) into the time-independent Schrödinger equa- tion, [1 2μd2 dR2−l(l+ 1) 2μR2+EC]FJεI vjl(R) =∑ v′j′l′FJεI v′j′l′(R)∫∞ 0⟨JεI(lj)∣χj v(r)Vint(R,r,γ)χj′ v′(r)∣JεI(l′j′)⟩dr, (3) J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where Mis omitted since the CC equations are independent of M. The collision energy ECis given by EC=E−Ev,j, where Edenotes the total energy and Ev,jdenote the rovibrational energies of the HD molecule obtained by solving the eigenvalue problem, [−1 2μHDd2 dr2+j(j+ 1) 2μHDr2+VHD(r)]χj v(r)=Ev,jχj v(r), (4) whereμHD(=0.671 699 9 amu) is the reduced mass of HD. For the potential energy curve VHD(r) in the electronic ground state of the HD molecule, we adopt a modified version of Schwenke’s H 2 potential ( X1Σ+ g) reported by Boothroyd et al.59The modified log- derivative propagation method of Manolopoulos60is employed to numerically solve the CC equations from R= 2.0 Å to 50.0 Å with a propagation interval of ΔR= 0.05 Å. As discussed in our recent work,51the BSP3 PES54used in our calculations is the most accurate ab initio PES for the HeH 2system, and it yields line-shape parameters of H 2and HD immersed in He in excellent (subpercent) agreement with highly accurate experimental results.55,56The uncertainty of the BSP3 PES in the van der Waals well region is estimated to be less than 0.04 K, sufficiently below the mean collision energies in the experiments of Perreault et al.16As illustrated in our previous work,51other available He–H 2PESs59,61 also yield very similar results as the BSP3 PES. B. Rotational quenching cross section The solution of the CC equations yields the scattering S-matrix from which scattering amplitudes for various state-to-state transi- tions specified by quantum numbers ( v,j,m) and ( v′,j′,m′) can be constructed. Since the S-matrix is evaluated in the total angular momentum representation in the SF coordinate frame, its elements are specified by the quantum numbers ( v,j,l) and ( v′,j′,l′). It is nec- essary to transform the basis set to obtain the scattering amplitudes specified by ( v,j,m). Here, we focus on pure rotational transitions within a vibrational manifold and omit vand the parity designation εIfor simplicity. Cross sections for vibrational transitions are 5 to 6 orders of magnitude smaller than pure rotational transitions as in H2+ HD48and H 2+ HCl52collisions. The scattering amplitude for rotational transition between oriented ( m- and m′-specified) HD rotational states is given by22,51,58,62,63 fjm→j′m′(θ,ϕ,E) =√π(−1)j+j′ ∑ J=0J+j ∑ l=∣J−j∣J+j′ ∑ l′=∣J−j′∣il′−l(2J+ 1)√ 2l+ 1 ×(j l J m0−m)(j′l′J m′m−m′−m)TJ jl,j′l′(E)Yl′m−m′(ˆR), (5) whereθandϕare the scattering polar and azimuthal angles, T(E) is the T-matrix obtained from the S-matrix as T(E) = 1−S(E), and Y denotes a spherical harmonic as a function of ˆR=(θ,ϕ). The corresponding differential cross section (DCS) is given by the square of the modulus of the scattering amplitude, dσjm→j′m′ dΩ=∣fjm→j′m′(θ,ϕ,E)∣2 k2 C, (6)where kC=√ 2μECis the magnitude of the wave vector in the incident channel. In the experiment of Perreault et al .16theϕ- dependence is not observed and the θ-dependence of the DCS is obtained after averaging over ϕ,51 dσjm→j′m′ dθ=2πsinθ∣fjm→j′m′(θ,E)∣2 k2 C, (7) where fjm→j′m′(θ,E) is related to the full scattering amplitude of Eq. (5) by the relation fjm→j′m′(θ,ϕ,E) =fjm→j′m′(θ,E)exp{−i(m −m′)ϕ}. The ICS ( σjm→j′m′) is obtained by taking an integral of the DCS overθfrom 0 toπ. In our previous studies,51we examined the quenching from v= 1,j= 2 to v′= 1,j′= 0 for comparison with the experimental results of Perreault et al.16Here, we consider the stereodynamic effect of the quenching to v′= 1, j′= 1 from the same initial rovibrational state with a focus on the ultracold regime. We show that a significant stereodynamic effect can be observed in the ultracold energy regime by selective measurement of the inelastically scattered HD ( v′= 1, j′= 1) in an orientation specified by the value of m′. A similar effect is reported for quenching from v= 1,j= 3 to v′= 1,j′= 2. III. RESULTS AND DISCUSSIONS A.v=1,j=2→v′=1,j′=1 Figure 1 shows the calculated ICS for the rotational quenching of initially oriented HD ( v= 1,j= 2,m) by collisions with He to j′= 0 (Δj=−2) and j′= 1 (Δj=−1) within the v= 1 manifold. We note that |m| in the legend of this figure means that the resultant cross section is independent of the sign of m; thus, the cross section is invariant even if we prepare the molecule in a state described by a coherent superposition of | j,−m⟩and | j,m⟩with any (normalized) expan- sion coefficient. The effects of relative phases in the initial coherent superposition of different mstates is not observed unless we observe the DCS with ϕresolution. The results for j= 2→j′= 0 are the same as those reported previously.51As discussed in our previous paper, the primary peak centered around a collision energy of 0.2 cm−1and FIG. 1 . Integral cross sections for rotational quenching of initially prepared HD (v= 1, j= 2, m) to ( v′= 1, j′= 0) and ( v′= 1, j′= 1) in collisions with He. The initial m-dependence is displayed in red ( m= 0), green (| m| = 1), and blue (| m| = 2). The non-polarized case without the preparation of mis displayed in black (isotropic). J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the shoulder around 2 cm−1are due to shape resonances from the incoming p-wave ( l= 1) and d-wave ( l= 2), respectively. Except for the overall magnitude, the ICSs for both j′= 0 and j′= 1 feature sim- ilar energy dependence and m-dependence. The higher magnitude for |Δj| = 1 reflects the leading anisotropic term V1(R,r) in the angu- lar dependence of the interaction potential [ Vint(R,r,γ) =∑λVλ(R, r)Pλ(cosγ), where Pλis a Legendre polynomial] that drives the | Δj| = 1 transition. Note that unlike He–H 2, the leading Vλ(R,r) terms of odd order in λare non-zero for He-HD. Them-dependence in the ICSs is pronounced only around the resonances for both transitions consistent with previous theoretical results for other systems.49,50,52Thus, it is expected that the stereody- namic control of collision outcomes by preparing the initial molec- ular rotational state in an orientation specified by the value of mis most effective in the region of a shape resonance. It has been shown rigorously that there is no m-dependence for the ICS in s-wave col- lisions even if the molecule has internal angular momenta such as the spin or electronic orbital angular momentum in addition to the rotational angular momentum.53These findings raise the follow- ing questions: (1) Is stereodynamics relevant in the ultracold energy regime? (2) Does the similar qualitative m-dependence of the ICS between Δj=−1 andΔj=−2 imply that these processes are driven by similar dynamic effects and are not influenced by stereodynamic preparation? To address the above questions, for j= 2→j′= 1, we decompose the ICS into the three contributions designated by the m′values ( m′ =−1, 0, 1 for j′= 1) although such m′selective detection has not been realized in the experiments of Perreault et al. However, we note that Perreault et al.14indirectly extracted the m,m′dependence for HD + D2collisions by fitting the observed angular distributions of inelasti- cally scattered HD to an expansion of relevant outgoing partial waves present in the experiment. Recently, Sharples et al.31reported four vector correlations between relative velocities and rotational angu- lar momenta in the initial and final states, namely, the k–j–k′–j′ correlation ( k: relative momentum) for collisions of electronically excited NO(A2Σ+) with Ne by observing the scattering angle depen- dence of ionization probabilities using a circularly polarized light at a collision energy of about 680 cm−1. In Figs. 2(a) and 2(b), we show the m-dependence of the ICS form′= 0 and m′= 1, respectively, for the j′= 1 state. In (a), we see a significant m-dependence for the cross section in the entire energy region including the ultracold energy regime dominated by the Wigner threshold behavior. The largest cross section corre- sponds to m= 0→m′= 0 that conserves m. On the other hand, more than an order of magnitude suppression is observed in the quenching from | m| = 2. We note that the ICS with any type of initial rotational state preparation, including the HD bond axis alignment performed previously in the SARP experiment, is given as the sum of these cross sections in (a) with positive valued weighting factors;50,51 thus, the upper and lower limits of the control range with the initial HD preparation is given by the red ( m= 0→m′= 0) and blue (| m| = 2→m′= 0) curves, respectively. In Fig. 2(b), the ICS for m= 2 →m′= 1 (blue solid curve) is the largest, and the m-conserving pro- cessm= 1→m′= 1 (green solid curve) becomes largest only in the vicinity of the l= 1 resonance (0.2 cm−1), indicating that the stereo- dynamic preference may largely depend on the partial wave ( l) and collision energy. On the other hand, the efficiencies for processes that involve a change in the sign of m(dashed lines, | Δm| = 2 and3) are largely suppressed. Including the blue curve in (a) for | m| = 2 (|Δm| = 2), significant suppression of the ICS for these processes in the ultracold energy ( s-wave) regime correlates with the restriction on the number of outgoing partial waves. For the initial rotational sate of j= 2, the total angular momentum quantum number for the collision complex with the incoming s-wave ( l= 0) is J= 2; thus, due to the conservation of the total angular momentum and the parity [J=J′and (−1)j+l= (−1)j′+l′], the possible outgoing partial waves for the final rotational state of j′= 1 are l′= 1 and 3. Furthermore, due to the conservation of the projection of J, the relation M=m +ml=m′+ml′is satisfied for the sum of the projections of rota- tional and orbital angular momenta yielding ml′=m−m′for the case of l= 0 ( ml= 0); thus, the l′= 1 component cannot exist for the processes | Δm|≥2. A suppression of ICS with increasing | Δm| was also reported by Krems and Dalgarno27for the reorientation of the electronic angular momentum in O(3Pj=2,m=2) + He collisions at ultracold temperatures. The non-polarized cross sections without preparing the initial orientation (black solid curves) show very similar behavior in (a) and (b) and that m′selective measurement without preparing the initial orientation ( m) is not useful in gaining insights into the stereody- namics in the ultracold energy regime. In all, the hidden information on stereodynamics in the ultracold energy regime is revealed only when the ICS is examined with both mandm′resolution. While them-dependence for j= 2→j′= 1 and j= 2→j′= 0 in Fig. 1 is similar when m′is not resolved, stereodynamics of j= 2→j′= 1 reveals a much more intricate picture when m′is specified. The quenching to the rotational ground state ( j′= 0) does not exhibit the m-dependence because there is only a single m′= 0 component (no m′resolution). We omit the results for m′=−1 because they can be obtained by inverting the sign of min the legend, as shown in Fig. 2(b). Next, we point out that some qualitative features in stereody- namic preference (order of the magnitude of ICS with respect to mandm′) are independent of the system in the ultracold energy regime. In Eq. (5), for the incoming s-wave ( l= 0), there are two possible terms related to the l′= 1 and 3 outgoing partial waves forJ= 2, j= 2, and j′= 1, as discussed above. For each l′, the rel- ative m,m′dependence for the partial ICS is determined by the product of two 3-j symbols since the T-matrix element is indepen- dent of mandm′, and the integral of the square of the modulus of spherical harmonic Yl′,m−m′overθyields unity in evaluating ICS. As discussed in our previous work,51form′= 0 in Fig. 2(a), the ratio of the partial ICS for l′= 1 is 1:0.75 for m= 0 and |1| (see the Appendix for more details) and zero for | m| = 2. On the other hand, for l′= 3, the ratio of the partial ICS with m′= 0 is 1:0.89:0.56 for m= 0, |1|, and |2|, respectively. Thus, we can con- clude that for m′= 0, the stereodynamic preference is m= 0>|m| = 1>|m| = 2, regardless of the value of the T-matrix elements which are system dependent. This trend is clearly displayed in Fig. 2(a). It is important to emphasize that the above trend for the stereodynamic preference comes from the purely algebraic part of the scattering amplitude. An important goal of stereodynamics is to gain insights into how the anisotropy of the interaction potential (intermolecular force) controls the collision outcome. However, in the s-wave regime, the overall trend in stereodynamic preference may not reflect details of the interaction potential. In such cases, system J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Integral cross sections for rotational quenching of initially prepared HD ( v= 1, j= 2, m) to ( v′= 1, j′= 1, m′) in collisions with He: (a) m′= 0 and (b) m′= 1. Results form′=−1 are identical to (b) by inverting the sign of min the legend. Results for initial kand final k′helicity dependences are displayed in (c) and (d) for k′= 0 and k′= 1, respectively. Results for k′=−1 are identical to (d) by inverting the sign of kin the legend. dependent information coded in the T-matrix (S-matrix) elements is revealed only by quantitative measurements of m,m′- resolved ICS. On the other hand, in the case of m′= 1 [Fig. 2(b)], we can- not predict even a qualitative trend of ICS with respect to mwithout the information of T-matrix since the 3-j symbols give rise to a dif- ferent trend of stereodynamic preference between l′= 1 and l′= 3 (see the Appendix). These aspects underscore the difficulty in pin- pointing specific stereodynamic preference in the ultracold regime. However, it is worth emphasizing that some qualitative trends in stereodynamic preference are independent of the interaction poten- tial for purely s-wave collisions. At higher energies ( >1 cm−1) due to contributions from multiple partial waves, it is difficult to ana- lytically discuss the feature of the m,m′dependence. However, the large m-dependence even around 10 cm−1implies that higher partial waves result in mechanisms similar to that of the s-wave. The orientations ( m,m′) of the initial and final HD rota- tional states are specified in the SF coordinate frame whose z-axis isparallel to the initial relative velocity for the collision. In the SARP experiments, due to the co-expansion of colliding species, the ini- tial relative velocity is also parallel to the laboratory fixed molecular beam as well as the direction of the time-of-flight axis, simplify- ing the relation between experimental data and scattering properties defined in the SF frame.14,15However, to investigate more details of the dynamics as well as hidden propensity rules, it would be worth exploring stereodynamics in other frames. Here, we consider the ini- tial and final helicity64,65dependence of the ICS in Figs. 2(c) and 2(d). The initial projection (helicity) component kis obtained by taking the projection of the molecular rotational angular momentum onto the axis parallel to the incident relative momentum. On the other hand, the final helicity ( k′) for the inelastically scattered HD (here, v′= 1 and j′= 1) is defined by the projection onto the direction of the final (recoil) relative momentum. The scattering amplitude in the helicity frame is expressed using the scattering amplitude in SF frame [Eq. (5)] and the Wigner D-matrix as22,25 J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp fjk(=m)→j′k′(θ,ϕ,E)=j′ ∑ m′=−j′Dj′∗ k′m′(ϕ,θ, 0)fjm→j′m′(θ,ϕ,E), (8) where kis equal to msince the quantization z-axis for the initial helicity component is parallel to the initial relative velocity vector. Similar to the transformation between the SF frame and body-fixed (BF) frame, the third Euler angle in the D-matrix is not unique unless we define it. Here, we set the angle to be zero adopting the conven- tion of Alexander.22An explicit form of the θ-dependent scattering amplitude in the helicity frame similar to fjm→j′m′(θ,E) in Eq. (7) is convenient in the following discussion:65,66 fjk→j′k′(θ,E)=(−1)j+j′ 2∑ J=0(2J+ 1)dJ k′,k(θ)J+j ∑ l=∣J−j∣J+j′ ∑ l′=∣J−j′∣il′−l+1 ×√ 2l+ 1√ 2l′+ 1(j J l k−k0)(j′J l′ k′−k′0)TJ jl,j′l′(E). (9) We note that the T-matrix in the right hand side is specified by the (j,l) index. Since kandmare the same, the k-dependence of the ICSs is identical to the m-dependence in Fig. 1 as long as we sum over the contributions from all k′(=−1, 0, 1) components. On the other hand, once we specify k′, the k-dependence [Figs. 2(c) and 2(d)] is dras- tically different from the m-dependence in (a) and (b). Evidently, thek-dependence is significant only around the resonance regions, as seen in (c) and (d). Furthermore, the kdependence fades with a decrease in the collision energy, and it vanishes in the ultracold limit. This particular kindependence of the ICS is due to the dominance of the s-wave ( l= 0) and the resultant dominance of the J=jcompo- nent. For J=jandl= 0, the first 3-j symbol in the right hand side of Eq. (9) results in (−1)J−k/√2J+ 1(=(−1)j−k/√ 2j+ 1);67,68thus, k appears only as a phase factor. The remaining kdependence is cap- tured in the Wigner small d-matrix dJ k′,k(θ). While the behavior of thed-matrix is reflected in the DCS, it has no effect on the magnitude of the ICS once integrated over θfrom 0 toπ. Thus, as a univer- sal feature, there is no k-dependence in the ICS in the ultracold energy regime even with k′-resolution. This feature applies regard- less of the collision energy for the l= 0 partial wave. On the otherhand, the absolute value of the ( k-independent) ICS is system depen- dent and is determined by the T-matrix elements associated with l′= 1 and 3. So far, we have focused on the m/k-dependence in a selected m′/k′component. Now, we shall discuss the final m′/k′distribution with an initial preparation specified by the value of m/k. To obtain them′distribution, we need to compare the curves with the same color in (a) and (b). At the collision energy of 10−4cm−1, the m′ = 0 to m′= 1 ratios of the ICSs (branching ratio) are 3.81, 23.2, 1.02, 0.333, and 0.0113 with initial orientations m= 0,−1, 1,−2, and 2, respectively. Here, we clearly observe the m-dependence in ICS [Figs. 2(a) and 2(b)] and predict branching ratios by resolving the final orientation m′. In the same way, we consider the final k′distri- butions given by the k′= 0 to k′= 1 ratio of the ICSs at 10−4cm−1. As we can expect from Figs. 2(c) and 2(d), the ratio is independent of the initial kvalue, yielding 1.46 with all possible kvalues ( k= 0, −1, 1,−2, and 2), implying that the branching ratio is similar to a statistical (uniform) distribution. We note that the branching ratio is not universal because it is determined by the T-matrix elements associated with l′= 1 and 3. We described the same stereodynamics in two different frames (SF vs helicity); thus, the different final distributions of the (projec- tion) components are not necessarily important. Previous studies of the m,m′-dependence at higher collision energies ( ∼10 cm−1 to∼1000 cm−1) have addressed the frame dependence to explore theΔmpropensity and develop associated approximation meth- ods.22,25,26,28,30Indeed, a statistical distribution22,25as well as asym- metric distribution with respect to the sign of m′/k′were observed in some cases. However, to our knowledge, a significant m′- dependence similar to the one presented here was not reported. Previous studies analyzed the frame dependence based on the dif- ference in the interaction potentials in the short-range repulsive and long-rang attractive regions and the resultant scattering directions. It is not obvious whether such a criterion is effective also in the ultracold energy regime where both the long-range and short-range forces strongly influence the scattering dynamics and the collision is dominated by a single incoming partial wave. Furthermore, as noted above, the qualitative orientation dependence in the cross section is not necessarily related to the system dependent properties such as the interaction potential. There might exist a frame exhibiting a FIG. 3 . Differential cross sections for rotational quenching of initially prepared HD ( v= 1, j= 2, m) to ( v′= 1, j′= 1, m′) in collisions with He at a collision energy of 10−4cm−1. (a) The initial orientation ( m) dependence; (b) the nitial orientation ( m) dependence with m′= 0; and (c) the initial orientation ( k) dependence with k′= 0 in the helicity frame. J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp specific propensity rule in ultracold collisions, but an elaborate sys- tematic search for finding such a frame is beyond the scope of this work. Next, we examine the effect of m′/k′selection on the m/k- dependence for the differential cross section (DCS). Such a fully resolved DCS corresponds to the k-j-k′-j′correlation and has recently been reported for collisions of NO with rare-gas atoms at around 500 cm−1–700 cm−1to analyze irregular diffraction pat- terns for NO and identify propensity rules.30Unlike ICS, DCS can exhibit m-dependence even at ultracold energies without m′reso- lution.53However, as shown in Fig. 3(a), the m-dependence for the DCS is actually very weak at a collision energy of 10−4cm−1if we sum over all m′. We note that the m-dependence in (a) is equal to thek-dependence as long as we sum over all k′components since mis equal to k, as discussed above. For non-polarized initial HD (black), the DCS exhibits a sinusoidal behavior arising from the sin θ term in Eq. (7) indicating the isotropic nature of ∣fjm→j′m′(θ,E)∣2 for the s-wave, as discussed by Aldegunde et al.53Thus, without m′resolution, it is very difficult to detect the m-dependence of the DCS in the ultracold energy regime. On the other hand, the m′-resolved DCS [(b) m′= 0] exhibits a large m-dependence both in the magni- tude and oscillatory behavior ( θ-dependence). Compared to (a), it is easy to observe the m-dependence even without fine resolution in θ and the absolute magnitude. The results in the helicity frame in (c) exhibit distinct oscillations in θfor different k; thus, k′resolution has a substantial impact on the DCS in the ultracold energy regime in the helicity frame unlike ICS [Figs. 2(c) and 2(d)]. In other words, the integrals of the DCSs in Fig. 3(c) over θresult in essentially identical ICS values regardless of the value k. B.v=1,j=3→v′=1,j′=2 Finally, in Fig. 4, we present the ICS for the rotational quench- ing of initially oriented HD ( v= 1, j= 3, m) by collisions with He toj′= 2 within the v= 1 manifold. Similar to the j= 2 results in FIG. 4 . Integral cross sections for rotational quenching of initially prepared HD ( v= 1, j= 3, m) to ( v′= 1, j′= 2, m′) in collisions with He: (a) m′= 0 and (b) m′= 1. Results form′=−1 are identical to (b) by inverting the sign of min the legend. Results for initial kand final k′helicity dependences are displayed in (c) and (d) for k′= 0 and k′= 1, respectively. Results for k′=−1 are identical to (d) by inverting the sign of kin the legend. J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Fig. 1, the ICS exhibits a primary peak due to the l= 1 shape res- onance near 0.2 cm−1and a shoulder due to the l= 2 partial wave in the 1 cm−1–3 cm−1regime. We observe a strong m-dependence of the ICS with m′resolution, as shown in Figs. 4(a)–4(c). Com- pared to Figs. 2(a) and 2(b), the basic trend in the m-dependence prevails except for the number of possible values of m. Again, Δm transitions that involve a sign change of mare significantly sup- pressed. The quenching rate generally decreases with the increase in |Δm| with some exceptions. For example, the | Δm| propensity is reversed between | m| = 2→m′= 0 and | m| = 3→m′= 0 in (a). We see that the order of the magnitude of the control range of the ICS with respect to mis larger than that shown in Figs. 2(a) and 2(b) forj= 2. Thus, it may be possible to expand the range of control for higher initial rotational states. In Fig. 4, (d) shows results obtained by adding the contributions from all possible m′(=−2,−1, 0, 1, 2) (simi- lar to Fig. 1), indicating that the conventional stereodynamic control of the collision outcomes is not effective except in the resonance region. The m′distribution for a given mat a collision energy of 10−4cm−1is as follows: the m′= 0 to m′= 1 ratios of the ICSs are 3.03, 39.5, 0.759, 0.026, 0.0006, 27.7, and 1.01 for m= 0,−1, 1,−2, 2,−3, and 3, respectively, and the m′= 0 to m′= 2 ratios are 47.1, 59.3, 4.95, 0.50, 0.0013, 11.1, and 0.016 for m= 0,−1, 1,−2, 2,−3, and 3, respectively. Again, this leads to a highly asymmetric final dis- tribution with respect to the sign and magnitude of m′. We omit the corresponding results in the helicity frame as the k-dependence is limited to the resonance regions even with k′resolution, as shown in Figs. 2(c) and 2(d). IV. SUMMARY AND CONCLUSIONS We have carried out rigorous quantum scattering calculations ofmandm′resolved cross sections in He + HD ( v= 1,j,m)→He + HD ( v′= 1,j′,m′) collisions for the j= 2 to j′= 1 and j= 3 to j′= 2 transitions in an energy range of 10−4cm−1–10 cm−1spanning the ultracold and cold regimes. For both transitions, the integral cross sections display a l= 1 shape resonance near 0.2 cm−1and a shoulder feature from a l= 2 partial wave in the 1 cm−1–3 cm−1regime similar to the previously studied j= 2 to j′= 0 transition. Key findings of this work are summarized below: ●For a given initial rotational level, the stereodynamic prefer- ence for the different final rotational levels is found to be similar even in the resonance region if the orientation of the final rotational state is not specified by resolving the m′ quantum number. However, this result is not universal and may depend on the molecular system. ●There exists intricate stereodynamics in the ultracold regime that is revealed only by resolving the cross sections in both mandm′. ●We do not observe any initial helicity [ k(=m)]-dependence for the integral cross section even by detecting inelastically scattered HD in a given helicity ( k′) component. ●Stereodynamic preference is frame-dependent in the ultra- cold regime. However, this vanishes for the integral cross section when summed over all possible m′/k′components. ●The m-dependence of each m′component of ICS is partly accounted for without details of the T-matrix elements (and,in turn, the details of the interaction potential). A similar universal trend has been discussed in our previous study51 in the region of a l= 1 partial wave shape resonance. Our findings indicate that the stereodynamic control of colli- sion outcomes in the ultracold energy regime governed by the l= 0 partial wave is possible if selective measurements of the orientation of final rotational states are carried out. As discussed before, the m′- dependence of the differential cross section for rotationally inelastic collisions of HD by D 2has been extracted in the experiment of Per- reault et al.14through a partial wave analysis and a non-linear fit to the experimental data. However, their analysis includes l= 0 and a few higher-order partial waves, and the explicit nature of the stere- odynamics for the purely s-wave regime was not revealed. It is our expectation that sensitive detection schemes may allow direct m′- resolved measurements and even attain the l= 0 regime for collisions involving light species with similar masses (e.g., intrabeam collisions of3He and HD) or merged beam techniques. The analysis of the scattering amplitudes in the helicity frame and the k,k′dependence of the resultant integral cross section may provide additional insights into the dynamics and sensitive compar- isons between theory and experiment. Our findings indicate that the m,m′analyses similar to those performed in earlier studies for dis- cussing the propensities in various frames may also be effective in the ultracold energy regime. The results presented here are for collisions in the absence of external fields and are motivated by the SARP experiments that use co-expansion of colliding species in a co-propagating molec- ular beam. It may also apply to buffer-gas cooled molecules or merged beam techniques, which are applicable to a broader class of molecules. Many techniques for cooling, trapping, and controlling cold molecule collisions involve external electric and/or magnetic fields and open shell molecules. The degeneracy of the rotational levels will be lifted in an external field and this would affect the preparation of the initial state, especially when a linear combination of initial m-states is involved, as in the case of V-SARP. Also, for collisions in confined geometries, stereodynamics may play an even greater role in ultracold collisions.39,69 DEDICATION This paper is dedicated to Kate Kirby for her outstanding con- tributions in Chemical Physics and AMO Physics as well as her leadership role in serving the physics community. ACKNOWLEDGMENTS This work was supported, in part, by NSF [Grant No. PHY- 1806334 (N.B.)] and ARO MURI [Grant No. W911NF-19-1-0283 (N.B.)]. We thank Nandini Mukherjee for helpful discussions. APPENDIX: ANALYSIS OF THE m, m′DEPENDENCE OF THE ICS IN THE s-WAVE REGIME As discussed in the main text, the product of two 3-j symbols in the term associated with the incoming s-wave ( l= 0) and an outgoing partial wave l′in the scattering amplitude [Eq. (5)] determines the relative m,m′dependence of the partial ICS in the ultracold energy J. Chem. Phys. 153, 184307 (2020); doi: 10.1063/5.0030808 153, 184307-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp regime. Since the cross section is related to the square of the modulus of the scattering amplitude, the key quantity is ∣(j l J m0−m)(j′l′J m′m−m′−m)∣2 . (A1) For the quenching of HD ( j= 2,m→j′= 1,m′) in the s-wave regime, outgoing partial waves l′= 1 and 3 contribute. For l′= 1, the m- dependence of the ICS for m′= 0 [Fig. 2(a)] is determined by ∣(2 0 2 m0−m)(1 1 2 0m−m)∣2 . (A2) Evaluating the 3-j symbol yields,67,68for the ratio of the ICS, 1:0.75 form= 0 and |1| and zero for | m| = 2, as discussed in the main text. We note that the first 3-j symbol results in (−1)−m/√ 5 in which themvalue determines only the sign; thus, the second 3-j symbol controls the ratio of the ICS with respect to m. For the l′= 3 term, ∣(2 0 2 m0−m)(1 3 2 0m−m)∣2 , (A3) the ratio is 1:0.89:0.56 for m= 0, |1|, and |2| for the ICS. From these results, we can conclude that the ratio of the ICS in the ultracold energy regime is m= 0>|m| = 1>|m| = 2 for m′= 0, as shown in Fig. 2(a). On the other hand, the relative contribution between l′= 1 and 3 is system and energy dependent because it is determined by the values of the respective T-matrix elements in Eq. (5). Thus the ratio of the total (sum of the l′= 1 and 3 contributions) ICSs with respect to min Fig. 2(a) reflects the system dependent information such as the interaction potential. Form′= 1 [Fig. 2(b)], we can perform similar analysis for l′= 1 andl′= 3. However, as shown below, the resultant relative ratios for the partial ICS with respect to mshow a different trend between l′= 1 and l′= 3. Thus, for an accurate prediction of the trend in the magnitude of total ICSs, the T-matrix information also needs to be considered. For the l′= 1 component, ∣(2 0 2 m0−m)(1 1 2 1m−1−m)∣2 (A4) yields the ratio of the partial ICS to be 0.17:0.5:1 for m= 0, 1, and 2, respectively. Here, m=−1 and−2 result in 0, as discussed in the main text. For the l′= 3 component, ∣(2 0 2 m0−m)(1 3 2 1m−1−m)∣2 (A5) yields 1:0.5:0.17:1.7:2.5 for m= 0, 1, 2,−1, and−2 for the ICS ratio; thus, m=−2 shows the largest partial ICS. The ratio of Eq. (A4) with m= 2 ( l′= 1) to Eq. (A5) with m=−2 (l′ = 3) is 1.4. Therefore, the significant difference between m= 2 and m=−2 contributions in Fig. 2(b), is attributed to the cor- responding T-matrix elements (in fact, the actual ratio of the square of the modulus of the T-matrix elements of l′= 1 and l′= 3 is 21.3 at 10−4cm−1). DATA AVAILABILITY The data that support the findings of this study are available within the article.REFERENCES 1K. M. Jones, E. Tiesinga, P. D. Lett, and P. S. Julienne, “Ultracold photoassocia- tion spectroscopy: Long-range molecules and atomic scattering,” Rev. Mod. 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5.0025173.pdf

J. Chem. Phys. 153, 184112 (2020); https://doi.org/10.1063/5.0025173 153, 184112 © 2020 Author(s).A way of resolving the order-of-limit problem of Tao–Mo semilocal functional Cite as: J. Chem. Phys. 153, 184112 (2020); https://doi.org/10.1063/5.0025173 Submitted: 13 August 2020 . Accepted: 22 October 2020 . Published Online: 11 November 2020 Abhilash Patra , Subrata Jana , and Prasanjit Samal ARTICLES YOU MAY BE INTERESTED IN An improved seminumerical Coulomb and exchange algorithm for properties and excited states in modern density functional theory The Journal of Chemical Physics 153, 184115 (2020); https://doi.org/10.1063/5.0022755 r2SCAN-D4: Dispersion corrected meta-generalized gradient approximation for general chemical applications The Journal of Chemical Physics 154, 061101 (2021); https://doi.org/10.1063/5.0041008 r2SCAN-3c: A “Swiss army knife” composite electronic-structure method The Journal of Chemical Physics 154, 064103 (2021); https://doi.org/10.1063/5.0040021The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A way of resolving the order-of-limit problem of Tao–Mo semilocal functional Cite as: J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 Submitted: 13 August 2020 •Accepted: 22 October 2020 • Published Online: 11 November 2020 Abhilash Patra,a) Subrata Jana,b) and Prasanjit Samalc) AFFILIATIONS School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India a)Electronic mail: abhilashpatra@niser.ac.in b)Author to whom correspondence should be addressed: subrata.jana@niser.ac.in and subrata.niser@gmail.com c)Electronic mail: psamal@niser.ac.in ABSTRACT It has been recently shown that the Tao–Mo (TM) [J. Tao and Y. Mo, Phys. Rev. Lett. 117, 073001 (2016)] semilocal exchange–correlation energy functional suffers from the order-of-limit problem, which affects the functional performance in phase transition pressures [Furness et al. , J. Chem. Phys. 152, 244112 (2020)]. The root of the order-of-limit problem of the TM functional is inherent within the interpolation function, which acts as a switch between the compact density and the slowly varying density. This paper proposes a different switch function that avoids the order-of-limit problem and correctly interpolates the compact density and the slowly varying fourth-order density correction. By circumventing the order-of-limit problem, the proposed form enhances the applicability of the original TM functional on the diverse nature of solid-state properties. Our conclusion is ensured by examining the functional in predicting properties related to general-purpose solids, quantum chemistry, and phase transition pressure. Besides, we discuss the connection between the order-of-limit problem, phase transition pressure, and bandgap of solids. Published under license by AIP Publishing. https://doi.org/10.1063/5.0025173 .,s I. INTRODUCTION The Kohn–Sham (KS) formalism of the density functional the- ory (DFT)1,2is the de facto standard for performing the electronic structure calculation of atoms, molecules, solids, and clusters. While the theory is exact, the accuracy of DFT depends on the approxi- mations of the exchange–correlation (XC) functionals having all the many-body effects. The development of efficient yet accurate XC functionals has been an emerging topic in DFT for the last couple of decades and continues to be the same. Constructing approximations have been proposed in different levels, such as local density approx- imation (LDA),3generalized gradient approximation (GGA),4–17 meta -GGA,18–38the (screened-)hybrid density functional,39–42and double-hybrid density functionals,43to improve the electronic struc- ture calculations of solids and quantum chemical systems. How- ever, within different levels of semilocal approximations (i.e., LDA, GGA, and meta -GGA), meta -GGA functionals are the advanced and (more) accurate ones for the diverse nature of solid-state and chem- ical properties.44–56In general, a meta -GGA XC energy functional is written asExc[ρ↑,ρ↓]=∫d3rρ(r)ϵLDA xFxc(ρ↑,ρ↓,∇ρ↑,∇ρ↓,τ↑,τ↓), (1) whereϵLDA x=−3 4π(3π2ρ)1/3is the LDA exchange energy per particle. In Eq. (1), the total charge density ρ=ρ↑+ρ↓, andτσ=1 2∑i∣∇ψiσ∣2 is the KS kinetic energy density. Due to the dependence of the KS kinetic energy density and other built-in ingredients, meta -GGA functionals recognize single, overlap, and slowly varying density regions in a much better way than GGA does.54However, in DFT, it is important to construct more accurate semilocal XC functionals by getting rid of the deficiencies of the proposed XC functionals. Recent advances in the development of semilocal functionals demonstrate that more accurate density functionals can be proposed by satisfying quantum mechanical constraints. One of the most important developments came through the strongly constrained and appropriately normed (SCAN)26meta -GGA functional, which is an accurate functional for the diverse nature of solid-state and molec- ular properties.57Several modifications of the SCAN functional are also proposed.58,59Besides the SCAN functional, the recently proposed Tao–Mo (TM)28functional is also showing promising J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp performance for finite and extended systems.45,48,51,52The intrinsic TM exchange hole is employed to construct range-separated hybrid functionals.60–63In addition, a revision of TM functionals (revTM) has been proposed very recently.36Notably, TM based function- als correctly satisfy two important paradigms: (i) they are accurate for one- and two-electron systems, which is essential for quantum chemistry, and (ii) they recover the correct gradient expansion for slowly varying densities up to fourth-order, which is relevant to con- densed matter physics. However, both TM and revTM functionals suffer from the order-of-limit problem anomaly, which is a critical limitation and degrades their performance in predicting the transi- tion pressure of solids.64The order-of-limit problem of meta -GGA functionals arises from the iso-orbital indicator z=τW/τ= 1/(1 + (3/5)(α/p)), whereα= (τ−τW)/τunifis another iso-orbital indi- cator known as the Pauli kinetic energy enhancement factor. Here, p(=s2=∣∇ρ∣2/[4(3π2)2/3ρ8/3]is the square of the reduced den- sity gradient s,τW= |∇ρ|2/(8ρ) is the von Weizsäcker kinetic energy density, and τunif=(3/10)(3π2)2/3ρ5/3is the Thomas–Fermi kinetic energy density. In the limit of vanishing αandp, the order-of-limit problem arises in zas27,65 lim p→0⎡⎢⎢⎢⎢⎣lim α→01 1 +3 5α p⎤⎥⎥⎥⎥⎦=1, (2) while lim α→0⎡⎢⎢⎢⎢⎣lim p→01 1 +3 5α p⎤⎥⎥⎥⎥⎦=0. (3) In the main paper of the TM functional,28the problem has been overlooked with the statement “this only happens near a nucleus.” However, it was shown recently32that this problem deteriorates the functional performance for the transition pressure of solids. Although the order-of-limit problem does not seem to be an impor- tant restriction for non-covalent ( α≫1) and slowly varying density regions (α≈1), it appears to be a significant limitation for single- bond regions, where both small αand small p(around the bond crit- ical points) may occur.27In addition, during the change of the phases of two solids, the formation of bonding and energy differences are important. Hence, during the meta -GGA functional construc- tion, the order-of-limit problem must be taken into account. Note that both Tao–Perdew–Staroverov–Scuseria (TPSS)22and its revised version (revTPSS)23also suffer from the order-of-limit anomaly, which is resolved by regularizing the functionals known as the regTPSS functional.27Recent meta -GGAs like meta -GGA made sim- ple (MS134and MS234),meta -GGA made very simple (MVS66), and SCAN26functionals do not suffer from the order-of-limit. To find a way to resolve the order-of-limit problem of the TM functional [which we named as the regularized Tao–Mo (regTM) functional], in this paper, we propose a slightly different form of the iso-orbital indicator z. Our resolution is described by arrang- ing the paper as follows: In Sec. II, we will in brief describe the TM functional and its order-of-limit problem. A possible way to resolve the order-of-limit issue of the TM functional is also discussed in Sec. II. Following that, we demonstrate the functional performance by assessing it for general purpose solids, molecules, and transition pressure problems.II. THEORY We start with the functional form of the Tao–Mo (TM) func- tional and the underlying problems associated with it. The exchange enhancement factor of the TM functional is given by28 FTM x(p,z,α)=wFDME x +(1−w)Fsc x, (4) where the enhancement factor derived using the density matrix expansion (DME) is FDME x(p,α)=1 f2+7R 9f4, (5) with R=1 + 595(2λ−1)2p 54−[z3−3(λ2−λ+ 1/2)(z3−1−z2/9)] andf=[1 + 10(70y/27)+βy2]1/10. Here, z2=τW/τunif= 5p/3, z3=τ/τunif=z2+α,y= (2λ−1)2p, andτunif=3 10(3π2)2/3ρ5/3.λand βare the fitted parameters. The slowly varying correction part of the enhancement factor is given as28 Fsc x(p,α)={1 + 10[(10 81+50 729p)p+146 2025q2 −73 405q3z 5(1−z)]}1/10 . (6) Here, z=τW/τ= 5p/(5p+ 3α), and w=(z2+ 3z3)/(1 +z3)2. The order-of-limit problem of the TM exchange enhancement fac- tor arises from the order of two limiting conditions p→0 andα→0. The discontinuity in the enhancement factor is observed with lim α→0[lim p→0[FTM x(p,α)]]=1.013 72 (7) and lim p→0[lim α→0[FTM x(p,α)]]=1.1132. (8) In addition, it is observed that none of FDME x(p,α)andFsc x(p,α)face any order-of-limit problem as lim p→0[lim α→0[FDME x(p,α)]]=lim α→0[lim p→0[FDME x(p,α)]]=1.1132 (9) and lim p→0[lim α→0[Fsc x(p,α)]]=lim α→0[lim p→0[Fsc x(p,α)]]=1.013 72. (10) Hence, the issue arises from the weight factor wused in the TM func- tional [Eq. (4)]. Moreover, the order-of-limit problem of the weight factor is attributed to the iso-orbital indicator z. In search of an appropriate weight factor, we modify the used form of the iso-orbital indicator ztoz′as z′=1 1 +(3 5)[α p+f(α,p)], (11) where the function f(α,p)=(1−α)3 (1+(dα)2)3/2e−cpis adopted from Ref. 27 with d= 1.475 and c= 3.0. This choice of the iso-orbital indicator lifts the order-of-limit problem as lim p→0[lim α→0[z′]]=lim α→0[lim p→0[z′]]=1. (12) J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Note that any small positive definite quantity or real number (greater than zero) instead of f(α,p) can remove the order-of-limit problem ofz′. However, the present choice of z′keeps the exchange enhance- ment factor close to that of the TM functional [except p→0 and α→0 (orα→0 and p→0)]. In Ref. 27, f(α,p) acts as an interpolat- ing function between α= 0 and ordinary αvalues. However, in the present case, f(α,p) is added to pso that at p→0, and for finite α,α p becomes a finite number. We keep the original values of candd, as mentioned in Ref. 27, since the order-of-limit is important only in the limit p→0 andα→0. With the modified iso-orbital indicator, the weight factor of the TM functional becomes w′=z′2+ 3z′3 (1 +z′3)2. (13) With this choice of w′, the exchange enhancement factor of the TM functional [named as regularized TM (regTM)] becomes FregTM x(p,z,z′,α)=w′FDME x +(1−w′)Fsc x. (14) It is noteworthy to mention that the change from ztoz′is only applied to the functional form of the interpolation function w(now w′), not in Fsc xorFDME x. This is because Fsc xandFDME x do not suffer from any order-of-limit problem as discussed before. For the one or two-electron singlet state, α= 0, which implies z′= 1,w′= 1, and FregTM x=FDME x. For the slowly varying density region,α≈1,f(α,p)≈0, and w′is small. Hence, Fsc xdominates in the slowly varying density region, essential for solids. For non- covalent bonding, α≫1, and f(α,p) is small except for small s. For example, at α= 10, f(10, p) is zero for s>1.6. The present form of f(α,p) ensures a close matching between FregTM x andFTM x for different αandpvalues except at limiting conditions. It is nec- essary because FTM xis a quite good functional for weakly bonded systems and strongly bound solids, including non-covalent inter- actions and layered materials. Other forms of w′can be proposed based on different iso-orbital indicators like αandβ,32but those may make the functional behavior quite different from FTM x, especially for largerα.54,67Forα= 0, both FTM xandFregTM x reduce to FDME x. Now, the regTM enhancement factor Eq. (14) eludes the order-of-limit problem as lim p→0[lim α→0[FregTM x(p,α)]]=lim α→0[lim p→0[FregTM x(p,α)]]=1.1132. (15) For comparison, in Fig. 1, we plot wandw′of TM and regTM functionals for the Kr atom along with the radial distance ( r) from the nucleus. The proposed function w′matches very closely with win the near nucleus and the density tail region ( α≈1). In the middle of the inter-shell region, only a slight difference is observed between these two curves, indicating consistency of both wandw′ by construction. In Fig. 2, we compare the variation in the enhancement fac- tors of both TM and regTM exchange functionals with αfor a particular value of s. The order-of-limit correction is clear at α = 0 for s= 0 curves. For other values of αand s, the regTM exchange enhancement factor matches very closely to that of the TM functional. FIG. 1 . Shown is the wof TM (w′of regTM) for the Kr atom along the radial distance from the nucleus, r. Regarding correlation, we consider the regTPSS correlation energy functional.27The regTPSS correlation functional is also uti- lized with MS1, MS2, and MVS exchange energy functionals, and it seems to be a more suitable correlation functional for exchange func- tionals proposed by removing the order of limit problem. The use of the TM or TPSS correlation energy functional does not work well with the regTM functional as the obtained error for AE6 atomization energies becomes ≈8.0 kcal/mol. The only price the regTPSS cor- relation pays is not taking care of the one-electron self-correlation, which is vital for molecules having many H atoms, such as water clusters. However, as stated in Ref. 27, molecular reaction ener- gies are not influenced much by the atomic energy errors, and it depends on the error cancellation effects obtained from exchange and correlation. It is also important to note that in the low-density or strong-interaction limit, like TPSS and revTPSS functionals, the TM XC energy functional remains independent of ζ=ρ↑−ρ↓ ρ↑+ρ↓for FIG. 2 . Difference in the enhancement factor of TM and regTM, with the variation inαfor particular values of sbeing shown. For s= 0, the drastic difference at α= 0 is clearly from the correction of the order-of-limit problem. J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Exchange energies (in Ha), correlation energy [in mHa/number of electrons (N e)], and total XC energy (in Ha) for several atoms as obtained from TM and regTM functionals. The results are obtained using the Hartree–Fock (HF) orbitals and densities.68The reference HF exchange energy and correlation values are taken from Refs. 28, 65, and 69. Ex Ec Exc=Ex+Ec Atoms HF TM regTM Ref.aTM regTM Ref.bTM regTM H−0.313−0.312−0.312 0.0 0.0 −3.2−0.313−0.312−0.319 He−1.026−1.032−1.032−21.0−14.5−22.0−1.068−1.061−1.076 Li−1.781−1.782−1.782−15.1−11.6−18.0−1.826−1.817−1.836 Ne−12.109−12.154−12.164−39.1−31.5−35.8−12.500−12.469−12.522 Ar−30.190−30.109−30.147−40.1−36.2−39.9−30.912−30.761−30.866 Kr−93.892−93.204−93.277−57.4−46.8−49.6−95.958−94.892−95.063 aSee Table 1 of Ref. 69 and all references therein. bAdding HF energy of column 2 and correlation energy of column 5. 0≤|ζ|<0.7 (see Fig. 2 of Ref. 28). In fact, this is a constraint important for atomization energies.26,65 Next, we calculate the exchange and correlation energies of the TM and regTM functionals for a few atoms (e.g., H, He, Li, Ne, Ar, and Kr). The results are summarized in Table I. The exchange ener- gies for regTM are close to TM exchange energies, indicating the likeness of wandw′. The deficiency of one-electron self-interaction free correlation energy of the regTM functional for the H-atom is clear from the mentioned table. In addition, the combined XC ener- gies of the regTM functional are more close to the reference values than TM for atoms with a higher atomic number. III. RESULTS AND DISCUSSIONS A. Molecular assessment To assess the regTM functional performance against the TM functional, we consider general-purpose quantum chemical and solid-state test sets. For quantum chemistry, the Minnesota 2.070is considered except G2/148 (atomization energies of 148 molecules),71S22 (22 non-covalent interactions),72WATER27 test sets,73,74and the dihydrogen bond complex dataset.75The G2/148 test set is considered from Ref. 71, whereas geometries of S22 and WATER27 are taken from GMTKN55,76and the geome- tries of the dihydrogen bond complex test set are collected from Ref. 75. Overall, all the test sets are divided into main group thermochemistry, barrier heights, bond length, and non-covalent interactions. The thermochemistry group consists of (1) AE6— atomization energies of six molecules,77(2) G2/148—atomization energies of 148 molecules,71(3) EA13—13 electron affinities,70 (4) IP13—13 ionization potentials,70(5) PA8—8 proton affini- ties,70and (6) DC9—9 difficult cases.78The barrier height test set consists of (1) BH6—6 barrier heights,77(2) HTBH38—38 hydrogen barrier heights,79and (3) NHTBH38—38 non-hydrogen barrier heights.79For bond lengths of molecules, we consider (1) MGHBL9—9 main group hydrogenic bond lengths7and (2) MGNHBL11—11 main group non-hydrogenic bond lengths7,78 For the non-covalent group, we consider (1) the HB6—6 hydro- gen bond test set,80(2) DI6—6 dipole interactions,80(3) CT7— 7 charge transfer molecules,80(4) PPS5—binding energies of fiveπ–πstacking complexes,80(5) WI7—7 weekly interaction com- plexes,80and (4) S22—22 non-covalent interaction molecules including H-bond, dispersion interaction, and mixed bonds,72,81(5) WATER27—27 water cluster binding energies,73,74and (6) dihydro- gen bond complexes.75 The performance of the TM and regTM functionals for the molecular test cases is summarized in Table II. Both the function- als behave equivalently in main group thermochemistry except PA8, for which the regTM is inferior to TM. For nine difficult cases, the MAE of regTM is slightly better than TM. A similar behavior is also observed for main group hydrogenic bond lengths. However, the TM functional is marginally better than regTM for the main group non-hydrogenic bond lengths. In the case of barrier height, regTM performs slightly better than TM for hydrogen and non-hydrogen transfer barrier heights. Notably, the resolution of the order-of- limit problem seems important for the barrier height.27Exciting results are also observed for H-bond and non-covalent interactions. Regarding the performance, regTM produces a slightly better result than TM for H-bonded dipole interactions, charge transfer, and π–πstacking complexes. The improvement of H-bonds for the regTM functional is also reflected in the performance of the S22 test set, where regTM is overall better than the TM functional for H-bonded and mixed complexes. We observe that for non-covalent interactions, especially for H-bonded systems, regTM binding ener- gies are slightly lower than the TM functional, making binding energies closer to the experimental values. However, this moder- ate underestimation results in slightly worse performance of regTM for 27 water cluster binding energies than the TM functional. This behavior of the regTM functional for non-covalent interaction may happen because FregTM x is slightly enhanced than FTM x[coming from thef(α,p) function] for α≫1, important for the non-covalent inter- action. In addition, one cannot rule out the lack of one-electron self- interaction free correlation in regTM. The dihydrogen bond test set representing H–H interaction in 32 molecular complexes is a chal- lenge for semilocal functionals.75In this case, the regTM functional possesses a smaller error than the TM functional in calculating the interaction energies. In addition to TM and regTM functionals, we calculate the errors using the SCAN functional. SCAN is also quite an accurate J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Tabulated are the mean absolute errors (MAE) of SCAN, TM, and regTM functionals for molecular test cases (main group thermochemistry, bond lengths, bar- rier heights, and non-covalent interactions) and solid-state properties of 29 bulk solids [equilibrium lattice constants (LC29), bulk moduli (BM29), and cohesive energies (COH29)], as compiled in Ref. 24. Best values within TM and regTM are marked with a bold style. The zero-point an-harmonic expansion (ZPAE) corrected reference values for lattice constants, bulk moduli, and cohesive energies are taken from Refs. 36, 46, and 49. Of the 29 solids, a test set is extracted following Refs. 27 and 49 that contains lattice constants, bulk moduli, and cohesive energies of 20 materials called LC20, BM20, and COH20, respectively. SCAN TM regTM Molecular tests Main group thermochemistry (kcal/mol) AE6 3.4 4.5 4.4 G2/148 3.7 6.5 5.7 IP13 4.43 3.17 3.76 EA13 3.22 3.79 3.25 PA8 1.41 2.13 5.13 DC9/12 11.13 26.69 22.40 Bond length (Å) MGHBL9 0.002 0.007 0.007 MGNHBL11 0.006 0.004 0.006 Barrier heights (kcal/mol) BH6 7.68 7.59 5.47 HTBH38 7.31 7.25 7.17 NHTBH38 7.88 8.86 8.29 Non-covalent interactions (kcal/mol) HB6 0.76 0.23 0.10 DI6 0.53 0.40 0.30 CT7 2.99 2.87 2.67 PPS5 0.72 0.74 0.62 WI7 0.07 0.04 0.04 S22 0.92 0.61 0.55 WATER27 7.44 1.44 1.53 Dihydrogen bond 0.92 0.43 0.32 Solid-state tests Lattice constants (Å) Simple metals 0.031 0.051 0.044 Transition metals 0.030 0.024 0.026 Ionic solids 0.036 0.039 0.037 Semiconductors and insulators 0.013 0.015 0.028 Total MAE 0.027 0.033 0.034 LC20 0.025 0.032 0.033 Bulk moduli (GPa) Simple metals 1.0 1.5 1.1 Transition metals 9.3 9.9 9.9 Ionic solids 4.8 4.2 4.4 Semiconductors and insulators 5.3 6.4 6.1 Total MAE 5.1 5.5 5.3 BM20 4.1 4.2 3.8TABLE II . (Continued. ) SCAN TM regTM Cohesive energies (eV/atom) Simple metals 0.120 0.274 0.155 Transition metals 0.364 0.751 0.570 Ionic solids 0.210 0.069 0.095 Semiconductors and insulators 0.066 0.122 0.078 Total MAE 0.190 0.304 0.224 COH20 0.159 0.282 0.216 functional for main group thermochemistry. Especially for AE6, PA8, and DC9 test sets, the SCAN functional performs better than TM and regTM. In addition, it is a convincing functional for the main group hydrogenic bond lengths. However, the SCAN func- tional is not as precise as TM and regTM for the non-covalent inter- action test sets, especially for the HB6, DI6, CT7, WATER27, and dihydrogen bonds. The performance of the SCAN functional for non-covalent interactions is discussed further in Refs. 67 and 82. B. Solid-state assessment Before applying the regTM functional on solids, we plot the variation in f(α,p),z,z′,w, andw′in Fig. 3. Considering the bulk Li (simple metal), the behavior of f(α,p),z, and z′is shown in the left panel of the mentioned figure, and in the right panel, we show the variation in wandw′. Far away from the nucleus, both w andw′decay to zero, indicating the likeness of both interpolation parameters. However, the inset representing the zoomed-in view suggests that w′possesses a small oscillatory pattern in the valence region. The oscillatory pattern is inherited from the behavior of z′ andf(α,p). We also observe a similar resemblance between wand w′for LiCl (ionic solid) and Si (semiconductor) (shown in Fig. 4). The similar behavior of wandw′indicates the judicious choice ofw′. Overall, it correctly recovers wthroughout the range of the slowly varying bulk solids and rapidly or moderately varying atomic densities. Next, we turn to the solid-state performance of TM and regTM XC functionals. For solids, recovery of the slowly varying fourth- order gradient approximation of exchange is essential. By construc- tion, both TM and regTM functionals respect the slowly varying fourth-order gradient approximation. In Table II, the MAE in lat- tice constants, bulk moduli, and cohesive energies of 29 bulk solids (compiled in Ref. 24) is compared, as obtained from SCAN, TM, and regTM functionals. For the experimental reference values of lattice constants, we consider the zero-point an-harmonic expan- sion (ZPAE) corrected values taken from various literature stud- ies.36,46,49,64In predicting lattice constants, except semiconducting materials, both TM and regTM underestimate equivalently. For semiconductors, the TM functional predicted results are closer to the experimental values. Note that solids like Li, Na, and K are known as “soft matter.” In these cases, the lattice constants are influenced by the short-range part of the vdW interaction.84Although semilocal density functionals such as MS1, MS2, MVS, and SCAN function- als include some short-range part of the vdW interaction,26,54it is J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Shown are the different meta -GGA ingredients and interpolation functions for Li from the atom at (0, 0, 0) to (1/2, 1/2, 1/2). The left panel shows f(α,p),z, and z′. The right panel shows wandw′. The position of atoms is indicated in dark circles. unclear whether the TM and regTM functionals incorporate some amount of the short-range vdW interactions or not. However, in Refs. 85–88, it is claimed that inclusion of the short- as well as intermediate-range vdW interactions leads to an improvement of the lattice constants for TM functionals. Concerning the impact of the correlation energy on the per- formance of semilocal functionals, the main improvement for the solids comes from the slowly varying density limit of the correla- tion and satisfaction of the local density linear response criterion of the XC functional. In this respect, the TM functional correctly recovers the slowly varying density limit of the correlation. The regTPSS correlation functional used to construct the regTM func- tional respects the correct formal properties of the correlation energy functional, important for solids. Nevertheless, the performance of the regTM functional may be further improved by constructing ameta -GGA ingredient dependent correlation energy functional, compatible with the present exchange. Overall, for the LC29 test set,the MAE of the regTM functional is 0.036 Å whereas the MAE of the TM functional is 0.033 Å. Next, we focus on the bulk moduli of the solids. The bulk mod- uli are obtained from the equation of state fitting of the energy vs volume curve of the unit cell with the third-order Birch–Murnaghan isothermal equation of state. The unit cell volume is varied in the range V0±5%, where V0is the equilibrium volume. This test depends on the accuracy of the geometries as predicted by the semilocal functional. As both the TM and regTM are quite good at predicting the geometries, the overall mean absolute errors obtained from both the functionals are about ≤5.5 GPa. For comparison, we also show the BM20 results, which can be directly compared with the results of other functionals presented in Refs. 27 and 49. The cohesive energy is also an important property for solids, and it is related to the thermodynamics of solids. However, better accuracy in lattice constants does not guarantee better accuracy in cohesive energies. For example, the performance of PBEsol in the FIG. 4 . Interpolation functions wandw′of LiCl (left panel) and Si diamond (right panel) are shown. For LiCl, the behavior of both wandw′is plotted from the Li atom at (0, 0, 0) to the Cl atom at (1/2, 1/2, 1/2), and for the silicon diamond, it is from the Si atom at (1/8, 1/8, 1/8) to the center of unit cell. The position of atoms is indicated in dark circles. J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp cohesive energies is not as efficient as that in the lattice constants.27,52 In most cases, meta -GGA functionals are better than GGA function- als in calculating cohesive energies because of their dependence on the additional ingredient, i.e., KS kinetic energy density.52A simul- taneous good behavior in valence densities (which is moderately and rapidly varying) of atoms and in the bulk densities (which is slowly varying) is required to get accurate cohesive energies. The cohesive energy per/atom is calculated as Ecoh=1 N{∑ atomsEatom−Ebulk}, (16) where Ebulkis the total energy of the bulk unit cell having Natoms andEatom is the energy of the constituent atoms. It is observed that alkali metals’ and transition metals’ cohe- sive energies are the most challenging for meta -GGA, where GGA Perdew-Burke-Ernzerhof (PBE) generally gives better results.52,89,90 From Table II, we observe that in almost all cases except ionic solids, regTM has a lower error than TM. The improved performance of regTM is probably due to the slightly improved atomization energies and oscillatory nature of w′in the valance band of solids and the core of the isolated atoms. Overall, regTM performs somewhat better for the slowly varying bulk density and rapidly or moderately varying atomic densities. Note that the SCAN functional also shows similar accuracy (or more accurate) for the 29 test set, which is presented in Table II. To understand the behavior of both TM and regTM function- als in dispersion dominated materials, we plot the binding energy curves for Ar 2dimer and bi-layer graphene in Fig. 5. These two systems are often relevant to assess the quality of a functional in non-covalently interacting molecular and solid-state systems. The regTM binding energy curve is slightly up-shifted compared to the TM functional for both the Ar 2dimer and bi-layer graphene. This is probably due to the lack of the one-electron self-interaction free correlation. Nevertheless, regTM performs very closely to the TM functional for these systems, indicating the reliability and closeness ofwandw′. C. Structural phase transition Accurate prediction of the pressure-induced structural phase transition of solids from low-pressure phases (LP) to high-pressure phases (HP) has practical implications.94–96Semilocal or higher- order wave function methods are very successful in predicting the phase transition pressure.83,97–101The accurate phase transi- tion pressure depends on the precise prediction of both equi- librium geometries and energy differences of the LP and HP. Although LDA is useful for structural properties, it underestimates the energy difference of two phases.83,97Out of various semilocal approaches, the PBE GGA overestimates the volume and performs better than LDA for energy differences and phase transition pres- sure.83,97Recent advances in the development of semilocal func- tionals show that meta -GGA functionals like MS1, MS2, MVS, and SCAN perform better than PBE in predicting the phase transition pressure.83,98,101 Formeta -GGA functionals, the phase transition pressure is related to the artifact of the order-of-limit problem,27,64which is associated with the wrong energy difference between LP and HP phases. The meta -GGA having an order-of-limit problem mostly FIG. 5 . Shown are the (a) Ar 2dimer binding energy curve and (b) bilayer graphene binding energy curve, as obtained from TM and regTM functionals. The CCSD(T) values for the Ar 2curve are collected from Refs. 91 and 92, and the random phase approximation (RPA) binding energy value for bilayer graphene is taken from Ref. 93. underestimates the energy difference and mispredicts the phase transition pressure.27The TM functional, which also possesses the order-of-limit problem, underestimates the phase transition pres- sure.64As stated in Ref. 27, the order-of-limit problem is more severe for covalent molecules and insulating solids, where α≈0 and p≈0 are often encountered around the critical bond point. To illustrate the improvement in the phase transition pres- sure from the regTM functional, in Table III, the structural phase transition quantities as obtained from the TM and the regTM func- tional are compared. We observe that regTM yields phase tran- sition pressure and energy differences closer to the experimental one than yielded by the TM functional. For the Si phase transi- tion, the regTM functional improves the energy difference of the two phases markedly. Compared with the SCAN and ACSOSEX (adiabatic-connection second-order screened exchange) values from Ref. 83, regTM is more close to SCAN, indicating its improvement over TM upon eliminating the order-of-limit problem. A very simi- lar tendency is observed for other structural phase transitions, where regTM improves considerably over the TM functional. The regTM J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Tabulated are the phase transition pressure ( Pt) (in GPa) and energy difference ( ΔEe) (in eV/functional) of highly symmetric phases using TM and regTM functionals. For reference comparison, ΔEeis compared with SCAN/ACSOSEX from Ref. 83. All values are without temperature corrections. Pt ΔEe Solids Expt.aSCAN TM regTM SCAN/ACSOSEXaTM regTM Si 12.0 14.5 3.9 14.5 0.417 /0.328 0.246 0.415 Ge 10.6 11.3 6.7 7.5 0.265 /0.280 0.365 0.394 SiC 100.0 74.1 52.5 66.9 1.631 /1.599 1.227 1.548 GaAs 15.0 17.1 8.2 14.9 0.825 /0.978 0.659 0.728 Pb 14.0 16.4 10.0 15.08 0.015 /0.027 0.007 0.024 C 3.7 4.6 2.17 4.81 0.088 /0.130 0.040 0.086 BN 5.0 2.8 −1.2 1.11 0.105 /0.048 0.044 0.042 aSee Ref. 83 and all references therein. phase transition pressure of Si, GaAs, and Pb becomes very close to the experiment, where TM underestimates substantially. For those solids, the energy differences between the two phases are also close to that of the SCAN/ASCOSEX values.101From cubic to the hexagonal phase transition of BN, the agreement of phase transition pressure for the TM functional is inferior. It shows negative phase transition pressure, where regTM improves impressively over TM. However, we do not include the temperature corrections in our calculations, which considerably improves the BN phase transition pressure.83 It is noteworthy to mention that accurate prediction of the phase transition pressure depends on both the energy difference (ΔEe) and the volume difference ( ΔV0). While both functionals perform almost similarly for geometries, regTM performs better for energy differences. Therefore, eliminating the order-of-limit problem is essential for improving the structural phase transition properties from the semiconductor to metallic phases. D. Connection to the bandgap On eliminating the order-of-limit problem, the regTM func- tional also improves the semiconductor bandgaps. In Table IV, we show the bandgaps of a few selective solids for which a clear improvement of regTM over TM is evident. For diamond C, the regTM bandgap increases by almost 1 eV. Similarly, for Si, Ge, SiC, BP, and GaAs, we observe an improvement in the bandgap. The improvement is related to the slope of the exchange enhance- ment factor with respect to α, which is discussed in Refs. 35 and TABLE IV . Selective semiconductor band gaps (in eV) as obtained from regTM and TM functionals. Solids Expt. SCAN TM regTM C 5.50 4.56 4.08 5.08 Si 1.17 0.83 0.56 1.26 Ge 0.74 0.22 0.32 0.40 SiC 2.42 1.70 1.29 1.75 BP 2.40 1.53 1.20 1.94 GaAs 1.52 0.86 0.91 1.0155. For regTM, the slope ∂Fx/∂αis more negative, which includes a more derivative discontinuity ( Δxc) in the generalized KS scheme.35 This also happens for functionals like MS1, MS2, MVS, SCAN, meta -GGA using cuspless hydrogen model (MGGAC38), and Thilo Aschebrock-Stephan Kümmel (TASK35) behavior. However, for regTM, this happens only for α≈0 and s≈0 regions. For other regions, the regTM exchange enhancement factor matches closely to that of the TM functional. Next, we consider the relationship between improvement to the phase transition pressure and the bandgap, which is discussed in Refs. 102 and 103. In Ref. 102, it is argued that the underestima- tion of the phase transition pressure for Si for LDA and GGA may be related to the bandgap of solids. Higher-order methods like GW and the screened hybrid functional HSE06 enhance the density of state (DOS) near the Fermi level and describe the covalent bond- ing more conveniently, which may be responsible for the improved phase transition pressure for those methods.102Contrary to Refs. 102 and 103, it is stated that the impact of the fundamental bandgap may not be that important for the improvement in the phase tran- sition pressure. However, in this case, the bandgap improvement for regTM for semiconductors, especially for Si, indicates that the regTM transition pressures may be related to the improvement in the bandgap in Si, as suggested in Ref. 102. IV. CONCLUSIONS The order-of-limit problem is an important limitation of the TM functional to predict the phase transition pressure, as shown in Ref. 64. In this paper, a modified interpolation function of the Tao– Mo (TM) functional is proposed, which resolves the order-of-limit of the TM functional. Using the modified interpolation function, the proposed functional correctly retains the accuracy of the parent functional for the one- or two-electron limit, slowly varying density correction, and non-covalent interacting systems. This is important because, along with resolving the order-of-limit problem, we retain the main functional accuracy for thermochemistry and solid-state physics, which are simultaneously important. It is shown that the phase transition pressure of the proposed functional is improved corresponding to the TM functional. As claimed in Ref. 64, redesign- ing the interpolation function as a function of αmay also resolve J. Chem. Phys. 153, 184112 (2020); doi: 10.1063/5.0025173 153, 184112-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the order-of-limit problem of the TM functional. Still, in that case, one has to ensure the good accuracy of the parent functional simul- taneously. In these prospects, the present modification of the TM functional to make it free from the order-of-limit problem seems quite suitable. Finally, we conclude that the present modification of the iso- orbital indicator zis simple and quite useful as it can be used further to construct meta -GGA functional development. Along this line of construction, a one-electron self-interaction free correlation with the regTM exchange functional may also enhance its performance for several thermochemical and solid-state structural properties. However, this can be further revisited in future publications. A. Computational details The molecular calculations of the functionals are performed using the developed version of the Q-CHEM code104with the def2- QZVP basis set (for IP13 and WATER27, the def2-QZVPD basis set is used) with a 99 point radial grid and 590 point angular Lebedev grid. Note that like TM, the regTM functionals are not sensitive to the choice of the grid. Using the 350 point radial grid and 590 point angular Lebedev grid, we do not see any difference in the potential energy curves for non-bonded interactions. Hence, the choice of a denser grid is not required, and the present choice of the grid is quite adequate for energy convergence All solid-state calculations are performed in the plane wave suite code Vienna ab initio simulation package (VASP).105–111The lattice constants are calculated by relaxing the volume and internal co-ordinates using the conjugate gradient algorithm. For bulk calcu- lations, we used 20 ×20×20Γ3centered kpoints with 800 eV energy cutoff. The spin polarized atomic calculation for cohesive energies are performed with an orthorhombic box of size 23 ×24×25 Å3. To calculate the bulk moduli and phase transition pressure, the third order Birch–Murnaghan equation of state112is used. To plot Figs. 3 and 4, we use the density and kinetic energy density as obtained from PBE XC functional using the all-electron WIEN2k113code. SUPPLEMENTARY MATERIAL See the supplementary material for the details of the pseu- dopotentials and solid-state calculation results of TM and regTM functionals. ACKNOWLEDGMENTS S.J. would like to acknowledge and thank Dr. Lucian Con- stantin for the Hartree–Fock density of atoms shown in Fig. 1 and Table I. S.J. is also grateful to the NISER for partial financial sup- port. Q-CHEM and VASP simulations have been performed on the KALINGA andNISERDFT high performance computing facility at the NISER, Bhubaneswar. P.S. thanks Q-CHEM Inc. and developers for providing the source code. 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5.0006075.pdf

Appl. Phys. Rev. 7, 031308 (2020); https://doi.org/10.1063/5.0006075 7, 031308 © 2020 Author(s).Material platforms for defect qubits and single-photon emitters Cite as: Appl. Phys. Rev. 7, 031308 (2020); https://doi.org/10.1063/5.0006075 Submitted: 27 February 2020 . Accepted: 13 August 2020 . Published Online: 21 September 2020 Gang Zhang , Yuan Cheng , Jyh-Pin Chou , and Adam Gali COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Surface-enhanced Raman spectroscopy for chemical and biological sensing using nanoplasmonics: The relevance of interparticle spacing and surface morphology Applied Physics Reviews 7, 031307 (2020); https://doi.org/10.1063/5.0015246 Elastocaloric switching effect induced by reentrant martensitic transformation Applied Physics Reviews 7, 031406 (2020); https://doi.org/10.1063/5.0007753 Developing silicon carbide for quantum spintronics Applied Physics Letters 116, 190501 (2020); https://doi.org/10.1063/5.0004454Material platforms for defect qubits and single-photon emitters Cite as: Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 Submitted: 27 February 2020 .Accepted: 13 August 2020 . Published Online: 21 September 2020 Gang Zhang,1 Yuan Cheng,1 Jyh-Pin Chou,2,3,a) and Adam Gali4,5,a) AFFILIATIONS 1Institute of High Performance Computing, A /C3STAR, 1 Fusionopolis Way, Singapore 138632, Singapore 2Department of Physics, National Changhua University of Education, Changhua 50007, Taiwan 3Department of Mechanical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Hong Kong 999077, Hong Kong 4Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, P.O. Box 49, H-1525 Budapest, Hungary 5Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki /C19ut 8, H-1111 Budapest, Hungary a)Authors to whom correspondence should be addressed: jpchou@cc.ncue.edu.tw andagali@eik.bme.hu ABSTRACT Quantum technology has grown out of quantum information theory and now provides a valuable tool that researchers from numerous fields can add to their toolbox of research methods. To date, various systems have been exploited to promote the application of quantum information processing. The systems that can be used for quantum technology include superconducting circuits, ultracold atoms, trappedions, semiconductor quantum dots, and solid-state spins and emitters. In this review, we will discuss the state-of-the-art of material platformsfor spin-based quantum technology, with a focus on the progress in solid-state spins and emitters in several leading host materials, includingdiamond, silicon carbide, boron nitride, silicon, two-dimensional semiconductors, and other materials. We will highlight how first-principles calculations can serve as an exceptionally robust tool for finding novel defect qubits and single-photon emitters in solids, through detailed predictions of electronic, magnetic, and optical 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/10.1063/5.0006075 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. DEFECT QUBITS WITH VARIOUS MAGNETO- OPTICAL PARAMETERS . . . . . . . . . . . . . . . . . . . . . . . . . . 4 A. Basic first-principles methods in a nutshell and theoretical spectroscopy . . . . . . . . . . . . . . . . . . . 4 B. List of defect qubits in semiconductors and wide bandgap materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 III. DIAMOND-HOSTED QUBITS. . . . . . . . . . . . . . . . . . . . . 8 A. NV center. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 B. Silicon-vacancy center . . . . . . . . . . . . . . . . . . . . . . . . 9C. Other types of split-vacancy complex color centers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 D. Other selected single color centers. . . . . . . . . . . . . . 12 IV. DEFECT CENTERS IN SIC. . . . . . . . . . . . . . . . . . . . . . . . 12 V. DEFECT CENTERS HOSTED IN 2D MATERIALS. . . . 15 A. Hexagonal boron nitride . . . . . . . . . . . . . . . . . . . . . . 15 B. Transition-metal dichalcogenides . . . . . . . . . . . . . . . 17 VI. OTHER MATERIALS AS HOSTS . . . . . . . . . . . . . . . . . . 17A. Other wide bandgap materials . . . . . . . . . . . . . . . . . 17 1. Zinc oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172. Zinc sulfide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173. Titanium oxide . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4. Gallium nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5. Cubic BN and hexagonal BN nanotube . . . . . . 186. Wurtzite aluminum nitride. . . . . . . . . . . . . . . . . 187. Rare-earth ions in garnets, silicate, and vanadate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 8. Lithium fluoride . . . . . . . . . . . . . . . . . . . . . . . . . . 19 B. Potential moderate/small bandgap semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1. Carbon nanotube . . . . . . . . . . . . . . . . . . . . . . . . . 192. Silicon: Kane quantum computer . . . . . . . . . . . 20 VII. SUMMARY AND OUTLOOK. . . . . . . . . . . . . . . . . . . . . 22 I. INTRODUCTION Quantum technology is a platform that exploits the basic rules of quantum mechanics to realize certain techniques and functionality Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-1 VCAuthor(s) 2020Applied Physics Reviews REVIEW scitation.org/journal/aremore efficiently than classic technologies.1The development of quan- tum technology opens new arenas of previously impossible technolo- gies due to the nature of quantum mechanics. In recent years, the interest in quantum technology has grown rapidly.2Quantum technol- ogy has grown out of quantum information theory, which relies oncontrollable quantum bits (qubits) as the most elementary unit of quantum information. 3Quantum information theory is predicted to solve such complex problems by algorithms based on qubits that are otherwise intractable by classic digital computers3–8as well as led to the foundation of inherently secure communication based on the no-clone theorem of quantum states. 9The latter is the fundamental of quantum communication applications that are now available commer- cially, in particular, by the use of quantum key distribution.10The sim- ulation of quantum systems and quantum computation is still in its infancy, but quantum computers11are now publicly available and pro- ducing the first important results.12The key for the development of reliable quantum devices is to realize physical qubits that have suffi-ciently long coherence times for performing quantum operations on them with high fidelities. In practice, the fidelity of the operation with physical qubits is not perfect and quantum error correction is neces- sary; this correction can be realized by multiple physical qubits for each logical qubit of the quantum algorithm. An exception would be the physical realization of the Majorana qubit, 13–16which is intrinsi- cally a logical qubit. One may distinguish two types of physical qubits in terms of their role in quantum information processing: local qubits are the units for quantum operations at the quantum nodes, whereas the quantum information is delivered by flying qubits between the distant quantum nodes. A natural physical realization of the flying qubits isthe photons that can connect the nodes. However, photons are not able to store quantum information at a given site, and thus, local qubits are typically realized on the basis of materials with sufficiently long coherence times of the quantum state. In this review, we focus on the physical realization of material qubits. We note that the quantumnodes may be connected to each other by realizing an interface between the material qubit and the photon. The material platforms of local physical qubits are numerous, such as atoms in optical lattices, 17trapped ions,18,19nuclear spin,20–22 superconducting circuits,23–28spins in semiconductor quantum dots (QDs),29–35and atomic sized defects in solids36–38(see Fig. 1 ). Innovative areas of device application will arise once the physical properties of these materials are fully understood. From a physical per- spective, materials with highly controllable quantum states will help to improve the key performance regarding the dynamical properties of quantum devices in general. Practical quantum devices will likely exploit different real materials for certain purposes. For example, asolid-state spin qubit 36may be used in a quantum memory because of the long coherence time, while a superconducting Josephson junc- tion23–25has an advantage as a computational qubit due to the fast processing capability. Since quantum technology is a subject of interest to a broad audience, the literature on this topic is now sizable.Obviously, given the breadth of the topics touched by quantum tech- nology and defects in solids, not all technical details can be provided in one article. Interested readers are directed to relevant references. For a brief overview of quantum simulators, they can refer to Ref. 39; for specialized reviews on trapped atoms and ions, they can refer to Refs. 40–43 . For reviews on superconducting circuits, they should referto Ref. 26. This review attempts to provide a self-contained description of the current status of research on materials development for solid- state defect quantum technologies. A key representative of solid-state defect qubits is the nitrogen- vacancy (NV) center in diamond, which is a paramagnetic color cen- ter. 44,45The electron spin of single defect centers can be initialized and read out through the spin-dependent luminescence measurement by optical pumping (see Sec. III A). Importantly, the electron spin state can be coherently manipulated with an alternating magnetic field. In addition, the electron spin of the defect center exhibits long coherence times ( /C24ms) even at room temperature,46,47which makes it promising for room temperature quantum information processing. The single defect manipulation can be achieved by using a confocal microscopy setup for excitation and optical readout of the spin state in such dia- mond samples where the defect concentration is sufficiently low to excite a single defect within the confocal spot. The NV center then actsas a single-photon emitter, which can be the source of quantum com- munication protocols or may be applied as a fluorescent agent in bio- markers and other applications. We note that the nuclear spin of the nitrogen and 13C isotopes around the NV defect may be applied as qubits to realize quantum memory47–49and quantum error correction protocols50and bear great potential in quantum simulation and quan- tum computation applications. The magneto-optical parameters and the electron spin coherence time of the NV center in diamond are rela- tively sensitive to the environment inside the diamond lattice or exter- nal to diamond, in particular, for NV centers residing close to the diamond surface. The sensitive dependence of spin on strain,51mag- netic fields,52,53electric fields,54and temperature55–57combined with its atomic-scale resolution makes the defect center in a solid a versatilequantum sensor. These excellent and versatile properties make the defect in a solid a promising candidate for use in a wide range of appli- cations, including quantum information processing and quantum sensing of biological systems 58–64(seeFig. 2 ). One advantage in the solid-state qubit is its capability of being integrated into the traditional microelectronic structures that have rev- olutionized our world once already. Diamond has favorable opticalproperties with room temperature defect qubits, but it is difficult to grow diamond at the wafer scale and fabricate a semiconductor device from diamond. This was one motivation to seek alternative materials and potential atomic-scale defects that are both robust and easily con- trolled, similar to the favorable coherence and readout properties of the diamond NV center. These conditions pose restrictions on the choice of host materials and defect states, as explained in detail in Ref.61. We briefly emphasize here that optical readout and the need of well-confined atomic-scale defect states favor wide bandgap materials as hosts; nevertheless, moderate or small bandgap materials can also have great potential, in particular, with low-temperature operation. The list of possible applications of defect qubits in Fig. 2 already hints that a single defect qubit cannot cover the diverging criteria of various quantum applications. This is another reason to search for alternative solid-state defect qubits. We briefly demonstrate this issue on the most successful defect qubit again. The diamond NV center has very broad phonon-assisted fluorescence in the region of 637–900 nm under green laser illumination, where the contribution of the zero- phonon-line (ZPL) emission to the total emission is only /C243%, which is called the Debye–Waller (DW) factor. The electron spin resonance (ESR) occurs typically in the microwave region ( /C243G H z ) w h e n n oApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-2 VCAuthor(s) 2020external magnetic field is applied, i.e., ESR resonance at the zero field splitting (ZFS).76Biological studies need room temperature operation that the diamond NV center satisfies. On the other hand, in vivo bio- logical applications also favor near-infrared (NIR) optical excitation and emission where the absorption and the auto-fluorescence of the living cells are minimal (see the definition of biological windows inFig. 9 in Sec. VII). Furthermore, it is favorable to avoid the use of intense microwave spin excitation at 2–3 GHz in biological systems because the latter can heat the organism. As a consequence, alternativeroom temperature defect qubits with these magneto-optical propertiesare in the focus of intense research. On the other hand, room tempera- ture operation for quantum communication application is not demanding, but rather, the optical properties should suit to the techni-calities of the present communication technologies. The dominant and bright emission should come at the ZPL with coherent photons, and the ZPL should be stable against the stray electric fields to avoid spec-tral diffusion. 77Furthermore, the ZPL wavelength should ideally be in the region of 1300–1550 nm to be compatible with the maximum transmission of current optical fibers available nowadays (see Fig. 9 ). Thus, it is evident that new solid-state defect qubits should be foundwith these favorable magneto-optical properties. FIG. 1. The illustration of five pursuable quantum technologies and the corre-sponding manufacture materials, includingquantum dots of Si, 65InSb,66and GaAs;67 trapped ions/ultracold atoms of9Beþ68 and40Caþ;69topological qubits of Bi2Te315and FeTe 0.55Se0.4516with realiz- ing Majorana bound states; superconduct- ing circuits of Al70and C;71and solid-state spins/defects of diamond (C),72SiC,73 ZnO,74and BN.75These materials can be applied to quantum computers, quantum communication, and quantum sensing. This review focuses on (spin) defects insolids. The famous Chinese symbol in thecenter, “Tai Chi,” is represented by a circle divided into light and dark, or “Yang” and “Yin,” just as is the case with the compo-nents of qubits of S¼61/2. Applica/g415ons of defects in solidSingle photon sources qubits Quantum sensingQuantum LED Quantum compu/g415ng Quantum ElectrometerQuantum radiometer Quantum Key Distribu/g415on Biomedical imaging High-resolu/g415on Bio-marker Quantum memories Quantum Piezometer Quantum Thermometer Quantum MagnetometerQuantum registers FIG. 2. The major applications of defects in solids include single-photon sources, qubits, and quantum sensors due to the coupling of spin to strain, magnetic fields,electric fields, and temperature. For each application, diverse magneto-opticalparameters are expected for the defect center. For details, refer to the main text.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-3 VCAuthor(s) 2020These issues drive the field in searching for new solid-state qubits that operate akin to the NV center in terms of initialization and read- out61,78but have the aforementioned magneto-optical properties for quantum sensing and communication. It is very difficult to experi-mentally perform systematic studies on various point defects embed- ded in numerous host materials. The search for new candidates can be seriously accelerated by highly predictive ab initio studies that can first identify the target defect that can be realized experimentally.Alternatively, experimentation might find a new, prospective quantum emitter by chance, but the interpretation of the signals and improve- ment on the quantum protocols are hindered by the fact that themicroscopic origin of the quantum emitters is unknown. Often, it isvery difficult to obtain knowledge about the nature of the quantum emitters by experimental methods, in particular, for such emitters that rarely occur in the host material. Again, ab initio modeling can be extremely powerful in the identification of quantum emitters and qubits. As a consequence, this review summarizes the recent effort and results from ab initio atomistic simulations, i.e., theoretical spectros- copy, and experiments hand-in-hand. We will provide a brief overviewabout the ab initio modeling and show the physical concept of the key magneto-optical parameters of solid-state defect qubits in Sec. II. We here review the most fundamental magneto-optical proper- ties of the observed defect qubits and single-photon sources, with thefocus on wide bandgap materials that can host room temperature qubits, but moderate or small bandgap materials are also briefly dis- cussed in Sec. VI B. We attempt to provide a full coverage of the already identified qubits or single-photon sources. We note that otherreview papers have been appeared related to this topic, and probably numerous review papers have been written parallel to this work or are now being in the preparation phase. We realized that previous reviewpapers are either specific to a given material such as diamond, 79,80sili- con carbide (SiC),81,82or hexagonal boron nitride ( h-BN)83,84or only focusing on the issues of specific quantum applications of selectedcolor centers (e.g., Ref. 85). Our review is not limited either to specific materials or applications. The full list of the fundamental magneto- optical properties of the already observed single-photon sources in various three-dimensional and low dimensional materials orients thereaders to pick up those qubits or single-photon emitters in theirresearch that might suit the target quantum application. We will briefly discuss the ongoing research associated with the given qubits or the host materials in this regard. Accordingly, we show the physi-cal concept of the key magneto-optical parameters of solid-state defect qubits and their intimate connection to theoretical spectros- copy from ab initio modeling in Sec. II. In Secs. III–VI , we will briefly discuss the defect qubits and single-photon emitters in diamond,SiC, cubic BN, gallium nitride (GaN), aluminum nitride (AlN), zinc oxide (ZnO), zinc sulfide (ZnS), titanium dioxide (TiO 2), silicon (Si), wide bandgap materials hosting rare-earth ions (REIs), and two-dimensional materials such as h-BN and low-dimensional materials including carbon nanotubes. Finally, we conclude this review paper in Sec. VII. II. DEFECT QUBITS WITH VARIOUS MAGNETO- OPTICAL PARAMETERS In the example of diamond NV centers, we already listed the key magneto-optical parameters of qubits in Sec. I. These parameters and their sensitivity on external effects (electric and magnetic fields, strain,temperature, etc.) determine their applicability for a given quantum application. The physical concept of these parameters will be discussedin Sec. II A, also in the context of ab initio calculations. In a recent review, 45a full coverage of ab initio methods was provided for the dia- mond NV center. These ab initio methods heavily rely on the applica- tion of Kohn–Sham density functional theory (DFT) plane wave supercell calculations to defects in solids where the basic equationsand advanced functionals were summarized in an earlier general review paper 86and other recent review papers focused on specific defect qubits.87We, therefore, describe the basic first-principles meth- ods very briefly in Sec. II A, in order to make a connection between the theoretical and experimental spectroscopy without extensively deferring the readers from the context of the present review. We thendescribe how these parameters are determined in the experiments in Sec.II Band provide a very comprehensive list of the magneto-optical properties of the known single defect qubits in various wide bandgapmaterials in Table I , which is a central point of this review paper. A. Basic first-principles methods in a nutshell and theoretical spectroscopy Characterization of point defects in solids is the key task for iden- tification candidates for qubits. Computation of the electronic struc- ture of the point defects is the first inevitable step to this end. Havingthe electronic structure in hand, the key magneto-optical properties can be calculated, which may be called theoretical spectroscopy of point defects in solids. We briefly summarize the typical ab initio methods employed to this end. Born–Oppenheimer approximation is usually applied for the electron–nuclei system of point defects, in which the ions are treated as charged particles with atomic masses, and the electrons move fast in the adiabatic potential created by the ions. The total energy of the elec-tronic states should be calculated as a function of coordinates of ions of the system so that the adiabatic potential energy surface (APES) of the system can be mapped. The global energy minimum of the APES for a given electronic configuration can be found by minimizing the quantum mechanical forces acting on the ions. The quasi-harmonicvibration modes can also be calculated by interpolating a parabola around the global minimum by moving the ions out of the equilibrium positions and solving the Hessian equation. To this end, the totalenergy of the electronic system should be determined. Two widespread basic methods can be used: (1) quantum chemistry codes with linear combination of atomic orbitals (LCAOs) and Gaussian-type orbital(GTO)-based molecular clusters and (2) density functional theory (DFT) supercell methods. Quantum chemistry codes with coupled cluster or configuration interaction (CI) methods can treat highly cor-related orbitals, but only small systems can be calculated, which is problematic when the band edges converge slowly to the infinite clus- ter’s values because of the quantum confinement effect, which opens the gap between empty and occupied defect levels. DFT supercell methods are often applied together with a plane wave basis set, as it isa natural basis for periodic boundary conditions (nevertheless, it can also be a GTO basis), which requires pseudopotentials 88or projected augmented wave (PAW)89methods. The charged defects in the super- cell methods can be calculated by neutralizing the supercell with an opposite charge delocalized in the entire supercell, which results in artificial interaction between the periodic images of these charges. Thiscan be usually compensated for by Lany–Zunger correction 90basedApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-4 VCAuthor(s) 2020TABLE I. Summary of the key parameters in experiment, such as zero phonon line (ZPL), zero field splitting (ZFS), and the Debye–Waller (DW) factor, for emerging quantum- coherent materials. The ZPL weight, DW factor, and the electron–phonon coupling parameter, the Huang–Rhys (HR) factor, are deduced from the relatio nship w¼e/C0S, where wis the DW factor and Sis the HR factor. DAP labels the donor–acceptor pair defect and REI for rare-earth ions. V(4 þ) represents vanadium impurities, substituting Si in the SiC lattice that should not be read as a vacancy. Most of the listed defects were observed as single-photon emitters; PbV( /C0) in diamond, Cr(4 þ) in 4H SiC and GaN, V B(/C0)i n h-BN, all the defects in c-BN, and all the defects in ZnO except for V Znwere observed as ensembles. ZPL (eV) ZFS (GHz) DW Diamond NV(0) 2.156109 NV(/C0) 1.945110NV(/C0)2A2, 2.88,763E, 1.43118NV(0) 0.14121 SiV(0) 1.310111SiV(0) 1.00119NV(/C0) 0.0462 SiV(/C0) 1.681112SiV(/C0)2Eg, 50,120 2Eu, 260120SiV(0) 0.90 60.10122 GeV( /C0) 2.06113GeV( /C0)2Eg, 181,113 2Eu, 1120113SiV(/C0) 0.75–0.79123 SnV( /C0) 2.00114SnV( /C0)2Eg, 850,114 2Eu, 3000114GeV( /C0) 0.61113 NiN 41.546115SnV( /C0) 0.41114 PbV( /C0) 2.385116 ST1 2.255117ST1 1.274 60.139117ST1<0.1117 4H-SiC V Si–VC(0) 1.094–1.150124VSi–VC(0)3A2, 1.22–1.34124VSi–VC(0)/C240.05132 PL5 divacancy 1.190125PL5 divacancy 1.353125 PL6 divacancy 1.193125PL6 divacancy 1.365125 VSi–NC(/C0) 0.99–1.06126–128VSi–NC(/C0)3A2, 1.282–1.331128 VSi(V1) 1.438129VSi(V1)4A2, 0.004129VSi(V1) 0.08133 VSi(V2) 1.352129VSi(V2)4A2, 0.035129VSi(V2) 0.09133 Cr(4þ) 1.158, 1.190130Cr(4þ)3A2, 1–6.46130Cr(4þ) 0.75130 V(4þ) 0.970, 0.929131V(4þ)2E, 529, 43131V(4þ) 0.25, 0.5131 3C-SiC V Si–VC(0) 1.121132VSi–VC(0)3A2, 1.336132VSi–VC(0)/C240.05132 VSi–NC(/C0) 0.845134VSi–NC(/C0)3A2, 1.303134 1.27135,a 6H-SiC V Si–VC(0) 1.094–1.134136VSi–VC(0)3A2, 1.236–1.347124 VSi–NC(/C0) 0.960–1.000127 VSi(V1) 1.433129VSi(V1)4A2, 0.028137 VSi(V2) 1.398129VSi(V2)4A2, 0.128137 VSi(V3) 1.366129VSi(V3)4A2, 0.028137 V(4þ) 0.948, 0.917, 0.893131V(4þ)2E, 524, 25, 16131V(4þ)<0.5131 h-BN nanotube 1.941138 2.172 and 2.179139 h-BN /C242.0b, 4.1c, 5.3d0.82,750.59141 VB(/C0)/C241.6140,eVB(/C0)3A2g, 3.4140/C240.030e WSe 2 0.936142 WS 2 VS1.72 and 1.981430.30 and 0.50143 2H-MoS 2 1.174144 c-BN O N–VB/C241.63145 Bi3.3146 Bi–VN3.57146Bi–VN0.017148 CN4.09147 Si impurity 4.94148Si impurity 0.007148 ZnO V Zn2.331149 CuZn2.859150CuZn0.0015153 VZn–Cl O2.365151 DAP 3.333152DAP 0.996154 GaN 0.855155,a0.71156 3.33f, 2.594g, 1.82b0.63g Cr(4þ) 1.193130Cr(4þ) 0.73130Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-5 VCAuthor(s) 2020on the Makov–Payne theory91or Freysoldt correction92in 3D materi- als. For 2D materials, new schemes that should be applied have beendeveloped, 93,94but doing so can be painstaking. The application of charge correction or donor–acceptor pair (DAP) models is highlyimportant, for the value of the charge correction could go beyond 1 eV in 2D materials. 94We mention another method to treat charged 2D systems. We note here that Kaviani and co-workerstransferred this troublesome problem of charged slab supercell cal-culation into another but readily solvable problem: 95a neutral dia- mond surface model is used for the negatively charged NV defect, NV(/C0), at the expense of a substitutional, positively charged nitro- gen defect N Sentering the diamond slab. Due to the valence elec- tron of a nitrogen atom being one more than that of a carbon atom, this N Sdonor will naturally donate its additional electron to the neutral NV acceptor defect by creating a pair comprising an NVcenter and a positively charged N S.I ft h i sd e f e c tp a i ri sp l a c e di n the same layer of the slab and the slab has a cubic-like shape, then the dipole–dipole interaction between the periodic images of the defect pairs can be minimized (also see Ref. 96). In most of the defect calculations, the charged defects are embed- ded in the supercell models. By using the correction energy ( Ecorr)f o r charged defective supercells, one can calculate the formation energy ðEf qd½/C138Þof a given point defect ( d)i nac h a r g e ds t a t e q,w h i c hp r o v i d e s thermodynamic properties. For a binary compound material (XY),this can be written as 97 Ef qd½/C138¼Etotd½/C138/C0nXlX/C0nYlY/C0ndldþqE FþEV ðÞ þEcorr;(1) where liis the chemical potential of ion ( i), with ninumber in the supercell, and EFis the Fermi-energy with respect to the calculated valence band maximum EV, which is usually aligned to zero energy in the corresponding band diagrams, and then EFvaries between zero and the fundamental bandgap ( Eg) of the host material. The adiabatic charge transition levels between the qandqþ1 charged states can be calculated asEqjqþ1 ðÞ ¼ Eqþ1 totd½/C138þEqþ1 corr/C16/C17 /C0Eq totd½/C138þEq corr/C0/C1 þEV;(2) which defines the position of the Fermi-level where the defect adiabati- cally switches between the qand qþ1 charge states with respect to EV. The calculated charge transition levels provide the ionization thresholds of the defect qubit, as explained in Fig. 3(a) . These quanti- ties are very important to understand the photostability of the qubit. Photoexcitation of the defect above the ionization threshold leads totemporary or complete loss of the qubit, which is often called the dark state in the literature (no fluorescence from the defect qubit). If the qubit defect was photoionized, then the qubit state may be restored byphotoexcitation of the dark state. The photoionization threshold needed to restore the qubit can be larger than the ionization threshold of the qubit (see divacancy defect qubits in 4H-SiC 98,99). We note that the given charge and electronic configuration should be calculated as a function of the spin state too, and the lowestenergy spin configuration belongs to the ground state spin of the sys- tem at a given charge state. This is highly important, for the qubit state is often associated with the high-spin ground state. The spin levelsmay split due to spin–orbit or dipolar spin–spin interactions at zero magnetic field that was previously called ZFS, which can be calculated from first principles (see Ref. 100). The ZFS can be measured by either alternating magnetic fields (microwave transition to rotate the spin state) or resonant optical transitions [see Fig. 3(c) ]. The calculation of the key optical quantities, such as the ZPL and DW factors, requires the computation of the optically allowed excited state and the vibration modes. In the Huang–Rhys (HR) approxima-tion of the optical transition, it is sufficient to calculate the vibration modes only in the ground state. 101Nevertheless, the calculation of the ZPL energy requires computing the excited state and finding the globalminimum of the APES in this electronic configuration, i.e., computa- tion of the quantum mechanical forces acting on the ions. The calcula- tion of the total energy in the excited state is not trivial forKohn–Sham DFT. Nevertheless, it may work by the constraintTABLE I. (Continued. ) ZPL (eV) ZFS (GHz) DW Silicon G-center C iCs0.969157G-center C iCs0.30159 Er(3þ) 0.805158,h REI YAG:Pr(3 þ) 4.122160 Ce(3þ) 2.536161YAG:Ce(3 þ) 0.002165 YSO: Er(3 þ) 0.807162 YVO:Yb(3 þ) 1.280163 LaFe 3:Pr(3þ) 2.594164 aThe possible candidates are C NONHi,166CN-Hi,166or C N-SiGa.167 bTheoretical study predicted C B–VN173and experimental study proposed N B–VN.75 cDAP174and C N175were originally proposed, but a recent theoretical study predicts the C N–CBdimer defect.75 dTheoretical study predicted O N176and experimental studies proposed V NB177and DAP.178 eAs the reported measurements140were performed at room temperature with no observable ZPL energy, we show the estimated value from ab initio calculations for ZPL and DW factor.179 fTheoretical studies predicted that the possible candidates are C N168or other complexes,169such as V Ga-3H or V GaON-2H. Experimental studies proposed that the possible candi- dates are C N–ON170,171and C N-SiGa.171 gPoint defects near cubic inclusions within the hexagonal lattice of GaN were proposed.172 hCo-doping oxygen with erbium in silicon has been reported to enhance emission and improve luminescence at high temperature.180Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-6 VCAuthor(s) 2020occupation of Kohn–Sham states, also known as the DSCF method (see the application to the NV center in diamond in Ref. 102). We will discuss below situations when this simple formalism fails and the pos- sible methods to calculate this quantity. By combining these calcula- tions, the DW factor ( w) can be derived as S¼/C0ln w,w h e r e Sis the Huang–Rhys factor obtained in the ab initio calculations103,104[see Fig. 3(b) ]. B. List of defect qubits in semiconductors and wide bandgap materials Over the past few decades, qubits have been implemented in a wide range of materials, including atoms,105superconductors,106semi- conductors,36and rare-earth materials.107Among these possible can- didates, point defects in semiconductors and wide bandgap materials, also called “color centers,” show a rich spin and optoelectronic physics that can be exploited to fabricate quantum information devices, including qubits for quantum computing and quantum sensing tech- nology and single-photon emitters in both quantum communication and quantum computation. Color centers are point defects in a semi- conductor or insulator that bind electrons to an extremely localized region on the scale of A ˚. Thus, color centers behave as atomsembedded in the host crystal. These atom-like states make the color centers ideal candidates for solid-state qubits, for many deep centers exhibit a non-zero spin magnetic moment in the ground state and might be optically polarized, manipulated into energetically excitedstates by illumination and radiation of microwave fields. At a low concentration of point defects in a solid, the photoexci- tation source can be focused on the target point defect with a simulta-neously induced ESR transition by sweeping alternating magnetic fields. If the fluorescence intensity is spin dependent, then the electron spin resonance can be observed by monitoring the fluorescence inten-sity as a function of the microwave frequency. This technique is calledthe optically detected magnetic resonance (ODMR) measurement. 108 The spin resonance may be detected at zero magnetic field for non-isotropic defects where the low symmetry crystal field will split thespin levels known as ZFS. Here, we list these most basic key optical (ZPL, DW) and magnetic properties (ZFS in the ground and excited states at the given spin state) for the materials of prominent qubits andsingle-photon emitters that have been experimentally identified to date, including three-dimensional materials such as diamond, SiC pol- ymorphs (4H, 3C, and 6H), ZnO, GaN, cubic BN ( c-BN), 2H-MoS 2, yttrium aluminum garnet (Y 3Al5O12or YAG), yttrium orthosilicate (Y2SiO 5or YSO), and yttrium orthovanadate (YVO 4or YVO); and FIG. 3. Illustrations of key magneto-optical parameters. (a) Ionization threshold. The defect has a deep acceptor level in the bandgap. Assuming that the ne gatively charged defect is the qubit, one can see that E1energy is needed to neutralize the defect by photoexcitation of the electron from the defect to the conduction band edge, whereas E2 energy is needed to restore the qubit by ionizing the defect from the valence band. (b) Theory of emission spectrum. Adiabatic potential energy surfac e (APES) of the ground state and excited state along the configuration coordinate. In the Huang–Rhys approximation, it is assumed that the APES of the excited state has the sa me shape as that of the ground state. The thin lines in the APES represent the energy levels of the effective phonon. In reality, many phonons may contribute to the optical transition, which may result in a broad phonon sideband. (c) Illustration of the typical optically detected magnetic resonance (ODMR) spectrum. The intensity of the emiss ion changes as microwave frequency sweeps, which is the signature of electron spin transition. At B ¼0 zero magnetic field, the position of the dip is the ZFS parameter.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-7 VCAuthor(s) 2020two-dimensional materials such as hexagonal BN ( h-BN) and transition-metal dichalcogenides (TMDs) (WSe 2and WS 2). In Secs. III–VI , we will discuss these materials in detail. III. DIAMOND-HOSTED QUBITS Diamond has a bandgap of about 5.5 eV; thus, it can host numer- ous color centers with emission wavelengths in a wide region from NIR through visible to UV.181Diamond has strong and short car- bon–carbon bonds, which makes it a dense and hard material. This may be considered as disadvantageous from the doping point of view but also results in high frequency phonon modes. It is worth empha- sizing that phonons play critically important roles in quantum infor- mation.182,183The phonon, the quantized excitation of lattice vibrational modes in crystals, can carry heat and information, which has broad applications for heat energy control and management. Unlike in bulk materials, in low-dimensional nanostructures, the con- fined phonon mode changes the nature of phonon transport, leading to anomalous thermal conductivity and heat energy diffusion.184,185In addition to the key role in the thermal conductivity of solid-state sys- tems,186–188the interactions of phonons with other carriers, such as electrons and photons, also have important influences on the perfor- mance of nanostructures,189optoelectronic devices,190and thermo- electric devices.191In particular, phonons significantly affect the coherence time of defect spins. For example, the spin coherence time of an NV center can be about a few seconds at cryogenic temperatures but is remarkably reduced to 1.8 ms at room temperature.46,47,192We note here that many magneto-optical properties of NV centers depend on the phonon density of states, which highlights the advantage of dia- mond as a host material, i.e., its high Debye temperature of /C242200 K. Hence, room temperature can be considered as a relatively low tem- perature for diamond, which results in a relatively low phonon popula- tion even under ambient conditions that generally suppresses the possibility of spin–lattice relaxation. Therefore, other materials with high Debye temperatures may also be promising for hosting defects with long spin–lattice relaxation times, such as boron-based materials.193,194 A. NV center Among the many color centers in diamond, the NV center stands out because of its desirable spin coherence and readout proper- ties.100,195,196The NV center is particularly attractive due to its ability to interface with a variety of external degrees of freedom. While differ- ent charge states of the NV defect have been reported, the negatively charged NV defect [NV( /C0) center] has the most attractive quantum properties and is usually denoted as the NV center. Structurally, the NV center is a point defect in diamond in which a substitutional nitro- gen atom is adjacent to a vacancy [ Fig. 4(a) ], which can capture an extra electron to form an NV( /C0) defect. A simplified schematic of the electronic structure of an NV center is shown in Fig. 4(a) .I ti sc l e a r that the ground state is a spin triplet (3A2), the excited state is also a spin triplet state (3E), and two metastable singlet spin states lie between them (1A1and1E). The working mechanism of a diamond NV center as a qubit is shown in Fig. 4(a) . First, a spin-conserving optical excitation is applied between the spin triplet ground state (3A2) and spin triplet excited state (3E), with an excitation energy of 1.945 eV (637 nm) when no phonons participate in the optical transition.100,195,196At low temperatures(<20 K), a fine structure emerges in the3Emanifold because of spi- n–orbit and spin–spin interactions (not shown). At room temperature, electron–phonon interactions will rapidly mix the orbital levels, lead-ing to an effective orbital singlet with a fine structure that stronglyresembles the ground state, 100,195,196as shown in Fig. 4(a) .T h er a d i a - tive decay from the triplet excited state to the ground state is spin-conserving. In addition, an efficient non-radiative decay pathway viaintermediate dark states 1A1and1Eexists between the mS¼61s u b l e - vels of the triplet excited state and the mS¼0 sublevel of the triplet ground state, respectively, which is called intersystem crossing. Rogerset al. 197applied uniaxial stress to demonstrate the ZPL of the infrared transition between these two singlet states that was found at 1.19 eV.The energy gap between 3Eand1A1was estimated at around 0.4 eV from the simulation and measurement of the intersystem crossing rate.198,199The combination of radiative and spin selective non- radiative pathways makes the NV center optically polarized into theground state upon illumination. Once initialized, the ground state spincan then be coherently manipulated by microwave radiation, and itsstate can be read out by the observed fluorescence intensity uponillumination. The electronic structure of the NV center is shown in Fig. 4(b) . The bound states of this deep center are multiparticle states composed of six electrons, which come from the four dangling bonds surrounding the vacancy (five electrons) and one electron from the environment.The double degenerate e(e x,ey) states are completely occupied in the spin majority channel and are empty in the spin minority channel,whereas the a 1orbital is fully occupied in the electronic ground state. The spin-conserving optical excitation from the3A2to the3Eexcited triplet can be described as promoting an electron from the a1state to the empty ex/eystate in the defect level diagram.200We note that the 1A1state is a strongly correlated state that can only be described by multireference methods beyond the Kohn–Sham DFT methods.201–203 At room temperature, extremely long spin coherence times of up to 1.8 ms are reported for the ground state of the NV center,46which Carbon Vacancy |0⟩|±1⟩ Ground state (3A2)Excited state (3E) 1A1|0⟩|±1⟩ (a) NV(0) NV(−)6 5 4 3 2 1 0Energy level (eV) a1ex,y a1exa1ex,y ex a1eyey(b) 1E FIG. 4. (a) Structure of an NV center in diamond and its energy level diagram in an optical polarization cycle. The green arrows represent the spin-conserving absorp-tion and emission, and the gray dotted and red straight arrows represent the weak and efficient intersystem crossings, respectively, and the cyan dotted arrow repre- sents a weak near-infrared emission competing with the direct non-radiative decay.The excited state level diagram is simplified to the case of elevated temperatures.(b) The ground state of single particle NV defect energy levels in the fundamental bandgap of diamond. Filled and empty arrows depict the occupied (electron) and unoccupied (hole) states, respectively, as found in spin-polarized density functionaltheory calculations.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-8 VCAuthor(s) 2020is close to the regime needed for quantum error correction. To realize the long electron spin coherence time, the deep center should reside in a wide bandgap host material that allows large energy spacing between adjacent bound states, and it should have small spin–orbit coupling. The remarkable features of the NV center are largely afforded by thehost material, diamond. Therefore, the electronic bound states of the NV center can reside deep within the bandgap, resulting in weak inter- action with states in the valence and conduction bands. This may imply that wide bandgap materials are suitable hosts for color centers akin to the NV center in diamond. It is worth noting that due to thequantum confinement effect, the semiconductor bandgap increases with reductions in the dimensions of the crystals. 204,205Therefore, for some narrow bandgap semiconductors, despite these semiconductors not being ideal to host deep centers, their nano-sized counterparts may become promising candidates for elevated temperature operation. The temperature dependence of the optical transition linewidth and excited state relaxation of a single NV center were associated with strong electron–phonon coupling.206It was shown that, at low temper- ature, the electron–phonon coupling caused by the dynamic Jahn–Teller effect in the3Eexcited state is the dominant mechanism for the optical dephasing. The Jahn–Teller effect was first reported in 1937 by Hermann Arthur Jahn and Edward Teller.207It states that a system with a spatially degenerate electronic state will undergo a spon- taneous geometrical distortion, resulting in symmetry breaking, because the removal of degeneracy can lower the overall energy of the system. Using a combination of first-principles calculations and the Jahn–Teller type of electron–phonon interaction model analysis, Abtew et al. studied the dynamic Jahn–Teller effect in the3Eexcited state of a diamond NV( /C0)c e n t e r .208To understand the Jahn–Teller effect, the APES was obtained by carrying out constrained DFT calcu- lations. For the NV center, there are six symmetrized displacements with two A1and two Evibrations induced by the motion of the imme- diate neighbor atoms of the vacancy. The totally symmetric A1vibra- tional modes do not lower the symmetry, but the Evibrational modes will lower the symmetry and split the defect levels. The combination of theoretical modeling208and experimental measurement206clearly suggested that the dephasing of the ZPL of the NV center is dominated by the coupling between the double degenerate Estate and the Epho- non modes. This results in a T5dependence of the ZPL linewidth. First-principles calculations from Thiering and Gali have shown that the strong electron–phonon coupling, i.e., the dynamic Jahn–Tellereffect, with higher energy phonons plays a crucial role in determining the fine structure of the 3Eexcited state at cryogenic temperatures and affects the intersystem crossing rate and the shape of the PL spectrum.209,210 Recent advances in quantum technology make it possible to cre- ate proximate qubits that form a quantum network in solids. Via elec- tron spin–spin interaction, the proximate qubits might interact with each other. It was found that in diamond with extremely high concen- trations of NV centers (about 45 ppm), the electron spin coherence time was significantly reduced to about 67 ls,211in striking contrast to t h e1 . 8 m sc o h e r e n c et i m eo ft h ei s o l a t e dN Vc e n t e r s .46The decoher- ence is caused by the charge fluctuation between neighboring NV( /C0) and NV(0) defects. To obtain deep knowledge about the underlying physical mechanism, very recently, Chou et al. , using first-principles calculations, explored the tunneling-mediated charge diffusion between point defects in diamond.212Based on the quantummechanical tunneling of the electron of NV( /C0) and a proximate acceptor defect in diamond, they proposed a physical model for thesource of decoherence in NV qubits. Although the wave functionsdecay fast and the electron probability is only about 0.1% at a distanceof 1 nm away from the center of the defect, strong interaction and charge transfer still exist even when the distance between neighboring NV(/C0) and NV(0) centers is up to 4.4 nm. Importantly, Chou et al. theoretically suggested that, to maintain the coherence time ( /C251m s ) of the isolated NV qubit, a distance of 9 nm between the NV sensorand the acceptor defect is required. This is of high importance for both quantum network and quantum sensor applications of the solid-state qubits. 85,213,214 The most promising area of application of the NV center is the ultrasensitive nanoscale sensors.95,215,216The spin levels and coherence time of the NV center are sensitive to external fields, such as an elec-tromagnetic field, pressure, and temperature, making it an atomic-scale quantum sensor capable of detecting changes in the surroundingenvironment, and, in particular, it is very attractive for biological or biomolecule sensing applications. In sensor applications, the NV cen- ters should be engineered relatively close to the surface of the dia-mond. However, compared to that of deeply buried NV centers indiamond, the coherence time of near-surface NV centers is signifi-cantly reduced. One possible approach to improvement is proper sur-face functionalization with different molecule/atom groups. 95,216,217So far, not all the possible sources of the noise causing this effect are fully understood, and the foundation of a precise strategy to keep the longcoherence time as well as high sensitivity is still under intenseresearch. We note that the diamond NV center has great potential to real- ize room temperature quantum computing, in striking contrast to thepresent ultralow-temperature operation of superconductor-basedquantum computers. It has been recently shown that single diamond NV center can coherently address 27 13C nuclear spins.218 Furthermore, a high fidelity of >99% has been demonstrated for the gate operation for single- (99.99%) and two-qubit gates (99.2%).50,219 By combination of present advances in the formation and activation ofNV centers by high precision ion implantation (e.g., Ref. 220and references therein), the photocurrent-based spin readout technique with going below the diffraction limit in the readout process, 221,222an array of NV centers can be envisioned with addressing /C2430 qubits around each NV center with sufficiently close distance between NVcenters to produce a scalable room temperature quantum processorunit. Ab initio theory can provide a great asset to locate the individual 13C spins around the NV center,223,224but further theoretical effort is needed to explore the complex spin–spin interactions between thecluster of 13C spins and the central electron spin, in terms of optimiz- ing the quantum control and maintaining the favorable coherencetime of the NV electron spin. B. Silicon-vacancy center Recently, the silicon-vacancy (SiV) defect in diamond, which forms a structure with inversion symmetry, has been of great inter- est. 123The inversion symmetry of defects is one strategy to suppress the influence of the local electric field noise to signal, i.e., the spectraldiffusion. In particular, near transform-limited photons could be gen-erated by the negatively charged SiV( /C0) centers because of the strong ZPL emission. 225–227The SiV defect consists of an interstitial siliconApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-9 VCAuthor(s) 2020atom in a split-vacancy configuration belonging to the symmetry group of D3d, which gives rise to an electronic level structure con- sisting of ground (2Eg) and excited (2Eu)s t a t e st h a tb o t hh a v ea double orbital degeneracy.120,228,229In the SiV( /C0) center of dia- mond, these degenerate orbitals are occupied by a single hole with spin S ¼1/2, leading to both orbital and spin degrees of freedom. Thus, spin projection and orbital angular momentum are goodquantum numbers for this system. The energy spacing between thespin levels in the electronic ground and excited states is caused bythe spin–orbit coupling reduced by the electron–phonon interac-tion, 79which are about 50 GHz and 250 GHz, respectively. The interaction of the external magnetic field and strain to the spin andelectronic orbitals was developed based on group theory princi-ples 120and later combined with the extended theory of the Jahn–Teller effect from first-principles calculations.79 By employing group theory and an advanced ab initio method with a realistic bandgap of bulk diamond, Gali and Maze determinedthe electronic structure of the SiV defect in diamond. 229In this struc- ture, the Si atom goes automatically into a bond center position between two adjacent vacancies, i.e., split-vacancy configuration, which is the inversion center of the diamond lattice. Thus, this defectis also called the V–Si–V defect in the literature, but the SiV label iscommon in the quantum optics community, and we will use this labelin this context. Since carbon atoms are more electronegative than sili-con atoms, the charge is transferred from the silicon atom to thenearby carbon atoms. The Si-related four sp 3states should form a1g, eu,a n d a2uorbitals, while the carbon dangling bonds form a1g,a2u,eu, andegorbitals, and all of them form the defect states in diamond. For the neutral SiV defect, only the egorbital occurs in the bandgap, with 0.3 eV above the valence-band maximum (VBM). Unlike the posi-tively charged state, the neutral and negatively charged states are foundto be stable in a large range of Fermi levels. In the SiV( /C0) center, the ground state electron configuration is e 4 ue3g, where the excited state can be described as promoting an electron from the euto the eglevel. The calculated ZPL energy is 1.72 eV without spin–orbit interaction, whichagrees well with the experimental data [1.682 eV (Ref. 230)]. It is worth noting that the ground ( 2Eg) and excited (2Eu) states have similar charge densities; therefore, only small charge redistribution occursupon optical excitation of the SiV( /C0) center, explaining the relatively large Debye–Waller factor of about 0.7 or 70%. This is quite differentf r o mt h ec a s eo ft h eN V ( /C0) center, where the charge densities of the ground state and excited state significantly differ and results in a large motion of ions upon illumination. 102The relatively sharp and stable ZPL emission with tiny spectral diffusion made it possible to demon-strate indistinguishable photon production from two single SiV( /C0) centers without any external tuning. 231 We note that both the ground2Egand excited2Eustates are sub- ject to the Jahn–Teller effect. As a consequence, the electron–phononinteractions have a significant impact on the photon emission. Figure 5 shows the full width at half maximum linewidths for the sin- gle SiV( /C0)c e n t e r . 227It is obvious that the linewidth increases with temperature, but with different scaling laws at different ranges of tem-perature. From 70 K to 350 K, the linewidth scales as the cube of thetemperature. However, at low temperature ( <20 K), there is linear dependence. Temperature can introduce vibronic coupling between different orbitals due to the dynamic Jahn–Teller effect, resulting inthe population mixture. Jahnke et al. developed a microscopic model of the electron–pho- non processes within the ground and excited electronic levels todescribe the observed temperature dependent optical linewidth of the SiV(/C0)c e n t e r . 227In this model, the electron–phonon coupling is a consequence of the linear Jahn–Teller interaction between the E-sym- metric electronic states and corresponding acoustic phonon modes. Atlow temperature, the first-order transition between the orbital statesinvolves the absorption or emission of a single phonon. The frequency of the phonon is resonant with the spin–orbit splitting energy /C22hD. Within the frame of time-dependent perturbation, the transition ratesare approximately described as c/C25 2p /C22hvqD2kBT; (3) where vis the interaction frequency for phonons, qis the proportion- ality constant, and Tis the temperature. Hence, the single phonon mediated transition results in the relaxation of the population betweenthe ground and excited states, as well as the dephasing of the states. Atlow temperature, it leads to a linear dependence. 800 600 400 200 Line width (MHz) 468101214164 3 2 1 056Line width (THz)Temperature (K) Temperature (K)0 50 100 150 200 250 300 35018 Lifetime limit (a) (b) FIG. 5. (a) The SiV center consists of a silicon atom centered between two neighboring vacant lattice sites (red small balls). Si ion sits in the inversion cent er of the diamond lattice, which makes the optical transition less dependent against the stray electric fields than that for the NV center in diamond. (b) Linewidth of th e SiV center for different tem- peratures. The green curve corresponds to cubic scaling in the high temperature range ( >70 K). At low temperatures ( <20 K), the pink line represents linear scaling, as shown in the inset. Reproduced with permission from Jahnke et al. , New J. Phys. 17, 043011 (2015). Copyright 2015 Authors, license under a Creative Commons Attribution 3.0 (CC BY 3.0) International License.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-10 VCAuthor(s) 2020Conversely, at high temperature, the line broadening deviates from the linear relationship with temperature, suggesting the involve-ment of higher order phonon coupling in the relaxation process. It isworth emphasizing that for NV( /C0) centers, the inelastic Raman-type scattering process dominates; however, for SiV( /C0) centers, the inelas- tic Raman scattering is suppressed and the elastic Raman-type scatter-ing process is dominated. Thus, the two-phonon elastic scatteringdominated transition rate is c/C25 2p3 3/C22hv2q2D2k3 BT3: (4) Therefore, the two-phonon scattering dominates the dephasing of the orbital states at high temperature, which agrees well with theexperimental results. Understanding the electronic structure and coherence properties of the SiV( /C0) center contributed to the development of a coherent population trapping experiment, 225,226wherein the spin state of the center could be optically addressed by applying resonant lasers at cryo-genic temperatures. However, SiV( /C0) has short coherence times of about 100 ns at cryogenic temperatures because of the inelastic Ramanprocesses. 232By cooling the system down to the millikelvin region, the phonon density of states can be reduced and millisecond coherencetimes can be achieved. 233,234In special diamond structures at millikel- vin temperatures, coupling of identical SiV( /C0) spins has been achieved235,236and coherent manipulation of a single SiV( /C0)c e n t e r spin by acoustic phonons has been recently demonstrated.237 We note a recent breakthrough on creating stable and individual neutral SiV defects122in which the zero-phonon line is at 946 nm (see Table I ) and the spin state is S ¼1. Unlike in the SiV( /C0) center, the ground state spin does not strongly couple to phonons similar to the case of the NV( /C0) ground state, and so it has a long spin-relaxation time and presumably long coherence times at cryogenic temperatures.Optical spin polarization of SiV(0) was achieved, 122which could be partially understood from the known electronic structure of thedefect. 111,119,229,238–241Very recently, ODMR signal from single SiV(0) has been reported,242which is, in particular, spectacular because it has been achieved via excitation of the Rydberg states. The Rydbergexcited states have been analyzed by the combination of group theoryandab initio calculations, 242where Rydberg excited states are bound exciton states in which the valence band hole is attracted by Coulombforces of the negatively charged defect. Interestingly, similar chargecorrection is required for the description of the Rydberg excited statessimilar to that for the fully ionized defect up to a certain cutoff radius from the center of the defect that is associated with the Bohr radius of the Rydberg excited state. 242The underlying mechanism of the ODMR contrast has not yet been fully understood, and furtherresearch is required for optimizing the corresponding signal.Nevertheless, SiV(0) certainly opens a novel avenue for quantum tele-communication applications. C. Other types of split-vacancy complex color centers The inversion symmetry-based group-IV vacancy color centers XV(/C0), where X ¼Si, Ge, Sn, and Pb, in diamond are fast emerging qubits that can be harnessed in quantum communication and sensorapplications. 79As mentioned above, SiV( /C0) possesses several advan- tages, such as a narrow inhomogeneous linewidth,243negligible spectradiffusion,235and immunity to electric field noise to first order;235how- ever, it suffers from low quantum efficiencies of /C2410% (Ref. 244)a n d short coherence times at cryogenic temperatures.232These limitations urged researchers to seek alternative quantum emitters that potentially have larger spin–orbit splitting that may raise the spin-coherence times. The heavier group-IV vacancy centers of GeV, SnV, and PbV are characterized by similar geometric structures and optical properties but have increased energy spacing between the spin levels because ofthe expected larger spin–orbit coupling. Indeed, these centers could be formed in diamond. The GeV( /C0) center has a strong photolumines- cence band with a zero-phonon line at 602 nm. 113,245–247Siyushev et al. used two optical excitation wavelengths with resonant matching with the different electronic transitions between the ground and excited states, and they observed coherent population trapping.247 This demonstrates that the GeV color center in diamond has excellentoptical spectral stability and controllable spin states. The SnV( /C0)c e n - ter revealed a narrow emission linewidth of /C24232 MHz (Ref. 248)a n d a 17-fold greater ground state splitting of /C24850 GHz, 79implying that i th a sp o t e n t i a lf o ral o n gs p i nc o h e r e n c et i m e .I th a sa l s ob e e nf o u n d to have a large quantum efficiency ( /C2480%) and a long ZPL wavelength of 2.0 eV.114Novel color centers have been associated with PbV cen- ters. Cryogenic photoluminescence measurements revealed several transitions, including a prominent doublet near 520 nm, and the ground state splitting of 5.7 THz far exceeds that reported for othergroup-IV split-vacancy centers. 116,249We emphasize that the degener- ate orbital in the ground and excited states results in strong Jahn–Teller interaction for these defects, i.e., a strong electron–phonon coupling well beyond the Born–Oppenheimer approximation;79there- fore, the electron–phonon or vibronic spectrum should be determinedinab initio calculations. It is spectacular that theory predicted a similar order of magnitude of strength of electron–phonon coupling and spi- n–orbit coupling for the PbV( /C0) center, which results in an effective phonon-spin coupling. 79Although a similar effect has been demon- strated for the SiV( /C0) center at ultralow temperatures with the use of complex materials engineering to produce acoustic waves in dia- mond,237the significantly enlarged energy spacing between the spin sublevel of the PbV( /C0) center should enable us to demonstrate this coupling at elevated temperatures with laser excitation of high energy acoustic phonons; thus, the complex material processing can be avoided. Pb is a huge ion; thus, it may create many unwanted defects after ion implantation. Presently, the optical250and spin proper- ties248,251of the SnV( /C0) center are rather under intense research to apply as a convenient alternative to the SiV( /C0)c e n t e ri nt e r m so f operation temperature, where high temperature annealing or chemical vapor deposition (CVD) after-growth of diamond after Sn implanta- tion resulted in high-quality diamond.250,252Very recently, GeV( /C0) and SnV( /C0) centers have been grown into diamond via microwave plasma CVD process253as alternative methods for smooth introduc- tion of dopants. This has the advantage of high-quality diamond crys- tal but the positioning of the individual color centers is not controlled. We note that the sister defects of SiV(0) were studied by first- principles calculations,240where the complex nature of the triplet excited states, i.e., a product of the Jahn–Teller effect, was studied indetail. We note here that the three triplet excited states are strongly coupled by phonons much well beyond the Born–Oppenheimer approximation. The three triplet excited states are separated due to the exchange interaction between electrons, which cannot be accuratelyApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-11 VCAuthor(s) 2020described by Kohn–Sham DFT, and multireference electronic struc- ture calculations are required. Nevertheless, deep insight into the nature of the exchange interaction makes it feasible to have a fair esti- mate for the energy splitting between the triplet excited states by using Kohn–Sham DFT.240It is predicted that GeV(0), SnV(0), and PbV(0) have ZPL transitions at around 1.80 eV, 1.82 eV, and 2.21 eV, respec- tively.240The corresponding optical centers have not yet been reported, although it is likely that they could appear in diamond besides their negatively charged counterparts because they do not require high p-type doping, in contrast to the case of SiV(0) defects.79 It is likely that GeV(0) and SnV(0) centers will be observed in dia-mond in the near future that can be prospective as visible emitter alter- native to the NIR emitter SiV(0). D. Other selected single color centers Two other color centers in diamond were selected and are listed inTable I , which were observed as quantum emitters. The NiN 4defect has NIR emission with inversion symmetry; therefore, it is an interest- ing alternative to other color centers in diamond. However, after the early observation of its quantum emission,115no progress has been reported so far. Further exploration of this color center may be difficult because its controlled formation (clustering of nitrogen atoms around Ni impurity) is a very challenging task. The ST1 color center was also picked up because it has a unique property: the optical emission occurs between the singlet states, whereas the qubit state is a metastable S ¼1 state.117The defect exhibits a high room temperature readout contrast at 45%. The electron spin could be coherently coupled to proximate 13C nuclear spins. It was demonstrated that the nuclei coupled to sin- gle metastable electron spins are useful quantum systems with long memory times, in spite of electronic relaxation processes.117This qubit has not yet fully exploited as the microscopic origin is yet unknown, which makes its controlled formation difficult. Ab initio calculations are needed for the identification of this solid-state defect qubit, which may boost the quantum memory application of this qubit. Identification of the microscopic structure would help to perform symmetry analysis on the excited state and metastable state, which can, principally, lead to the optimization of the quantum optics protocol. IV. DEFECT CENTERS IN SIC Inspired by the color centers in diamond, deep color centers have been considered in other host crystals.78,254Weber et al. proposed a list of physical criteria that a candidate color center and its host mate- rial should meet,61and silicon carbide (SiC) was identified as a poten- tial candidate because of its large bandgap and low concentration of nuclear spins, which was in line with a previous theoretical study that proposed, in particular, a divacancy defect in SiC as an alternative tothe NV center in diamond. 254SiC is a wide bandgap semiconductor with extensive applications. High-quality 4-in. wafers are available, and this availability contributes to the success of SiC in industrial applications. Defect centers in SiC have emerged as promising candi- dates for quantum technology applications due to their lower cost, the availability of mature microfabrication technologies, and their favor- able optical properties. SiC is composed of silicon and carbon atoms in a hexagonal lattice arrangement in the plane, which can be stacked as quasi-hexagonal ( h) or quasi-cubic ( k) Si–C bilayers in the sequence. Resulting from the stacking of these bilayers, over 200 polytypes existin SiC, including 3C, 4H, and 6H phases in the Ramsdell notation with bandgaps of 2.4 eV, 3.2 eV, and 3.0 eV, respectively, where C and H refer to cubic and hexagonal crystals, respectively, and the number represents the instances of Si–C bilayers in the unit cell. Among these polytypes or polymorphs of SiC, 4H-SiC is commonly used as every- day semiconductors for electronic devices, which makes this material a unique platform for the integration of classical semiconductor technol- ogy with quantum technology. In 4H-SiC, handkbilayers alter each other, creating two distinct configurations for monovacancy defectsand four different configurations for pair-like defects such as two adja- cent vacancies, called divacancies. This multiplies the complexity of the identification of defects in this material but also provides opportu- nities to produce multiple qubits with distinct but similar properties at the same time. As a compound semiconductor, SiC contains intrinsic defects of carbon vacancies (V C), silicon vacancies (V Si) ,a n t i - s i t et y p ed e f e c t s( S i C and C Si), the carbon antisite-vacancy pair defect (C Si–VC), and diva- cancy (V Si–VC). The identification of the microscopic configuration of point defects is a key step in the advance of quantum information mate- rials. First-principles calculations combined with magneto-optical spec- tra were widely employed to study the electronic and spin properties of these defect centers in SiC (e.g., recent ab initio results about native d e f e c t si n4 H - S i Ci nR e f . 255and a previous result in Ref. 256). It was expected that the optically active native point defects might act as single color centers and qubits in well-engineered SiC materials. Indeed, one of the brightest solid-state quantum emitters in the visible region (ZPLs around 600 nm) was reported in 4H-SiC, and it was identified by ab initio calculations as configurations of the posi- tively charged C Si–VCdefect, which has an S ¼1/2 electron spin.257 The manipulation of the spin state of these emitters has not yet been reported. Using optical and microwave techniques similar to those used with diamond qubits, Koehl et al. demonstrated that several point defects in 4H-SiC are optically active and coherently controlled with a range of temperature from 20 K to room temperature.125They were mostly identified as divacancy configurations,256which they called PL1-4 centers. Neutral divacancies in 4H-SiC have S ¼1 ground state spin and a ZPL of around 1.1 eV with a DW factor of about 0.03. The coherence times of V Si–VCare 1.2 ms with an ODMR readout contrast o f1 5 %i n4 H - S i Ca n d0 . 9m si n3 C - S i C .132Compared to the NV cen- ter in diamond, the divacancy with correlated states in SiC can provide competitive advantages, such as that the emission wavelength in the NIR region would suit biological studies because it can effectively pen-etrate organic tissue, the ODMR and ZPL signals can be resolved at room temperature, and ZFS is at lower frequencies ( /C241.3 GHz) than that of the NV center in diamond, which is also preferential for biolog- ical systems. 258–260Besides PL1-4 centers, other ODMR centers with similar ZPL and ZFS energies were also found and called PL5-6 centers. Very recently, these centers have been identified as neutral divacancies inside the stacking faults of 4H-SiC,260which shows the principle that stacking faults or short polytype inclusions may generate atomic-scale defect qubits distinct from the configurations in the bulk counterpart. These defect spins could be fabricated and manipulated a tt h es i n g l ed e f e c tl e v e l .132,261Recently, the ability to control the spe- cific charge states of divacancy spin defects in 4H-SiC has been real- ized experimentally, providing enhanced spin-dependent readout and long-term charge stability.99Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-12 VCAuthor(s) 2020The divacancy spin levels and coherence times are sensitive to the presence of strain,73,262,263electric fields,73magnetic field,264and temperature136and can potentially be harnessed in sensor applications or used to drive the electron spins mechanically265or electrically.259,266 The variety of divacancy qubits and the Stark-shift tuning of the opti-cal and spin levels make possible the production of interference between different types of divacancy qubits in a controlled fashion when engineered into SiC electronic devices. 267It was experimentally proven that the spin polarization can be efficiently transferred fromthe electron spin toward nearby nuclear spins in the SiC lattice, 136 which was understood by the combined ab initio and effective spin Hamiltonian study,268with the prediction of the flipping of nuclear spins by optical means and small constant magnetic fields.269 Silicon-vacancy related ODMR centers were well known in hex- agonal SiC polytypes and even in a rhombohedral SiC polytype. Thesecenters were labeled as V1 and V2 in 4H-SiC 270,271and as V1, V2, and V3 in 6H-SiC,272where the corresponding emission comes at the near-infrared region (see Table I ) with a broad phonon sideband. Very recently, the DW factor was found to be around 6%.133,273The corre- sponding electron paramagnetic resonance (EPR) signals showed S¼3/2 spin states with a relatively small ZFS in the ground state between the sublevels caused by the crystal field of the hexagonal crys-tal, which slightly deviates from the quasi-tetrahedral symmetry andresults in the C 3vsymmetry.272The origin of these ODMR emitters was debated in the literature. Soltamov and co-workers interpreted the observed hyperfine signatures for a V2 center so that the V2 centershould be a complex of V Sinear a second or farther neighbor V Calong the crystal axis.274Using first-principles calculations and electron spin resonance measurements, Iv /C19adyet al. have studied the isolated silicon vacancies, the negatively charged silicon-vacancy, and a proximateneutral carbon vacancy [V Si(/C0)þVC(0)], as shown in Fig. 6 .271The calculated DFT energies indicated that the pair vacancy [VSi(/C0)þVC(0)] is a metastable configuration. Furthermore, ZPL energy and hyperfine tensor were calculated via theHeyd–Scuseria–Ernzerhof (HSE06) hybrid exchange–correlation func-tional, 275,276and the zero-field splitting calculations were conductedusing the Perdew–Burke–Ernzerhof (PBE) functional.277These high- precision first-principles calculations showed that the pair vacancy [VSi(/C0)þVC(0)] has a spin-1/2 ground state without any zero-field splitting. By theoretical simulations and high-resolution EPR measure-ments, Iv /C19ady et al. demonstrated that the isolated silicon-vacancy [V Si(/C0)] accounts for the majority of the experimentally observed magneto-optical properties. The molecular orbitals of the isolated silicon-vacancy are constructed from symmetry-adapted linear combi-nations of the three equivalent sp 3-orbitals from the basal-plane car- bons and the sp3-orbital belonging to the carbon atom on the crystalline c-axis;278thus, it is not a part of a nearby carbon vacancy, as proposed by Soltamov et al. The conclusion is that these ODMR centers are the configurations of isolated V Si(/C0) defects, where V1( h) and V2( k) in 4H-SiC and V1( h), V2( k2), V3( k1) in 6H-SiC were iden- tified by first-principles calculations,278–280which can be engineered as single photon emitters.458An individual V Si(/C0) qubit at room temper- ature was coherently controlled with a coherence time of about1.5 ms. 281The ODMR contrast does not exceed 1% for any configura- tion. V Si(/C0) qubits could be engineered into SiC electronic devices, where the charge state of single defect qubits could be stabilized282and electroluminescence from ensemble V Si(/C0) qubits could be observed283with a potential to produce masers.284 Group theory considerations together with first-principles calcu- lations have already revealed the basic electronic structure of V Si(/C0) qubits and the corresponding selection rules in the optical excitation of these defects:285,286the four carbon dangling bonds [see Fig. 6(a) ] create an a1andt2orbital in the a1(2)t2(3)electronic configuration in the negatively charged state in 3C SiC, where the t2orbital splits to a1 andeorbitals in the hexagonal SiC polytypes. The lowest excited state may be described as promoting an electron from the lower a1level to the upper a1level with forming4A2ground and excited states. Soykal et al. further elaborated the group theory by revealing the fine spin level structures and states for both the optically active spin quartet states and the dark spin doublet states.278First-principles calculations revealed that a strong Jahn–Teller effect appears in the excited state,causing the appearance of a polaronic V1 0level in the PL spectrum FIG. 6. Models of silicon-vacancy related qubits in 4H-SiC. (a) Isolated Si-vacancy model and the alignment of V1–V2 centers to different silicon-vacancy c onfigurations in 4H- SiC; (b) vacancy pair model of the V2 center in 4H-SiC. Here, the red lines highlight the stacking of the SiC double layers to identify the different confi gurations of silicon and carbon vacancies. The highlighted orange lobes demonstrate the spin density of the defects. It can be seen that, based on the calculations shown [Repr oduced with permission from Iv /C19adyet al. , Phys. Rev. B 96, 161114 (2017). Copyright 2017 American Physical Society], the isolated vacancy model in (b) is assigned to the V1 and V2 centers in 4H- SiC. Reprinted figure with permission from Iv /C19adyet al. , Phys. Rev. B 96, 161114 (2017). Copyright 2017 American Physical Society.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-13 VCAuthor(s) 2020and the temperature dependence of the spin dephasing.133In addition, the V1 center shows negligible spectral diffusion287because of the small change in the charge density between the quartet lowest energy excited state and the ground state.288The favorable magneto-optical properties of V Si(/C0) qubits could be harnessed to realize the spin- controlled generation of indistinguishable and distinguishable photons from silicon-vacancy centers in 4H-SiC.289 Nitrogen-vacancy pair defects (N CVSi) have recently been observed in N-doped 3C, 4H, and 6H-SiC polytype crystals.128,290The negatively charged N CVSishow broad photoluminescence spectra and NIR emission arising from the transition between the3Eexcited and 3A2ground states, both of which exhibit S ¼1 spin states.291,292The ODMR measurement of the single nitrogen-vacancy centers in 4H- SiC at room temperature has been very recently achieved in N-doped SiC,293which presumably shows similar properties to their NV coun- terpart in diamond. The photoluminescence signals from transition- metal point defects were also observed in SiC, e.g., chromium, vana- dium, niobium, and molybdenum, which possess relatively high DW factors. Chromium defects in hexagonal SiC have about 0.73 DW fac- tor and S ¼1 spin ground state, which could be spin polarized upon illumination at the ensemble level.130The emission wavelength is at 1.158 eV with long optical decay rates of /C24100ls, where the latter is due to the intra-configurational spin–flip transition between the spin singlet excited and spin triplet ground states.294The neutral vanadium defects have two configurations in 4H-SiC with optical emission at 0.969 ( a) and 0.929 ( b)e V ,295and three centers with similar NIR emission were found in 6H-SiC.296Individual vanadium centers cre- ated by vanadium implantation were observed in 4H- and 6H-SiC, in which the observed coherence time was roughly 1 ls at cryogenic tem- peratures.131The high-quality molybdenum doped SiC (ZPL is at 1.106 eV and 1.152 eV in 6H and 4H polytypes, respectively) resulted in a coherence time of 0.32 lsf o rt h eS ¼1=2electron spin at cryogenic temperatures, behaving as doublets with a highly anisotropic Land /C19eg- factor297largely deviated from the value of the free electron, which implies a contribution of the spin–orbit interaction and should result in a relatively large zero-field splitting. In these defects, the dorbitals play a crucial role by carrying angular momentum and can interact with the external magnetic fields. However, these systems are also Jahn–Teller active, and the electron–phonon coupling will significantly affect the strength of interaction of the system with external magneticfields, explaining the observations on vanadium and molybdenum qubits.298 Overall, SiC has obvious advantages in the applications of quantum technology. A high-quality 4-in. 4H-SiC wafer is widely exploited in the semiconductor industry with well controlled doping technologies, which is promising to reduce the production cost of the future SiC-based quantum devices. Indeed, single divacancy256 and Si-vacancy spins299have been integrated and controlled in SiC diode structures (see Fig. 7 ). Furthermore, single SiC spins could be integrated into photonics structures, which is a promising route toward future quantum optoelectronics devices.2994H-SiC has a smaller than diamond’s but still considerably large bandgap at 3.3 eV, which can host mostly NIR quantum emitters. NIR emission is generally favorable for quantum sensors for biology and quantum communication. On the other hand, the relatively low small energy spacing between the defect levels and about twice smaller Debye temperature of 4H-SiC (about 1000 K)300than diamond’s make the defect qubits’ key magneto-optical properties sensitive to tempera- ture. For instance, the ODMR readout contrast of single divacancy spins upon off-resonant excitation is well above 10% close to cryo- genic temperatures but significantly reduces at room temperature. We refrain here that the room temperature ODMR readout contrast of the Si-vacancy V2 center is around 1%. Compared to the /C2430% room temperature ODMR readout contrast of the diamond NV cen- ter, the low ODMR contrast of 4H-SiC spins is disappointing as the high readout contrast is an important factor in the overall sensitivity and temporal resolution of quantum sensors. Very recently, a break- through has been achieved in 4H-SiC,301where an /C2420% off- resonant room temperature ODMR readout contrast is demon- strated for single PL6 spins. The theory implies that the ODMR readout contrast starts to decline at about 150 K for divacancy defect qubits in 4H-SiC,301in contrast to diamond NV center’s data at around 600 K (Ref. 302) because of the smaller energy spacing between the defect levels in 4H-SiC with respect to those of NV dia- mond.301Nevertheless, PL6 divacancy qubit’s room temperature ODMR contrast still persists at relatively high value. This opens the door for SiC quantum technology in the field of room temperature quantum nuclear magnetic resonance (NMR) measurements and other biology applications. The tight control of the surface and inter- face of SiC and silicon dioxide (SiO 2) is far from being solved (e.g., FIG. 7. As discussed here, SiC offers cer- tain advantages in applications in quantumtechnology. A high-quality 4-in. 4H-SiC wafer is widely exploited in the semicon- ductor industry with well controlled dopingtechnologies, which can reduce the pro-duction cost of future SiC-based quantum devices. Single divacancy and Si-vacancy spins have been integrated and controlledin SiC diode structures. (a) Si-vacancyand divacancy structures in 4H-SiC. (b) Defect qubits integrated into SiC diodes. (c) Defect qubits integrated into SiC pho-tonics structures.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-14 VCAuthor(s) 2020Ref.303), which will be the next important task in the development of sensitive quantum sensors from SiC. V. DEFECT CENTERS HOSTED IN 2D MATERIALS Two-dimensional (2D) materials have been attracting extensive research interest in the past few decades. Recently, room temperature quantum emitters have been reported in two-dimensional (2D) wide bandgap materials,75,304–307creating great interest on quantum emit- ters and potential quantum bits in 2D materials. The big advantage of 2D materials is that the creation of the defects can be well controlled with the present experimental techniques with the almost determinis- tic localization of the defects that is needed for scaling-up quantum bits for quantum computers. In addition, strain and electric fields can be engineered on these 2D materials, and optical cavities can be formed around them.308Another playground for 2D materials is the stacking of different 2D materials on top of each other, and one might imagine sandwiching an atom or molecule between these 2D layers as qubits. Besides quantum computing and quantum communication purposes, it is noted that 2D materials are surfaces per se, and so they can be good hosts for quantum sensing. The single-layer MoS 2layers, delta-doped diamond slabs, and Si thin slabs were predicted as prom- ising hosts for spin qubits.309Isotopic purification is much more effec- tive in 2D materials that lead to an exceptionally long spin coherence time of the order of 30 ms. Apparently, there is great potential for using 2D materials for quantum technology. However, the first quan- tum emitters found in hexagonal boron nitride ( h-BN) have not been unambiguously identified, and the origin of these emitters is still under intense research.75,148,173,176,306,310–315For instance, some quantum emitters have been tentatively assigned to specific native point defectsinh-BN. 176If these quantum emitters are identified and then engi- neered into a nanocavity (e.g., for WSe 2in Ref. 316), then room tem- perature photon blockades, which are an ultimate demonstration of non-linearity at a single-photon level, might be realized for the first time at room temperature. A. Hexagonal boron nitride Hexagonal boron nitride ( h-BN) is one representative 2D mate- rial. It is a wide bandgap ( /C246 eV) 2D material with the potential to host color centers that are promising candidates for quantum technol- ogy. Many emitters in h-BN are bright with narrow linewidths, are tunable, and have high stability.75,305,317,318In 2016, room tempera- ture, polarized, and ultrabright single-photon emission from color cen- ters in 2D h-BN was first experimentally demonstrated.75These advantages have sparked strong research interest in h-BN defects.75,173,305,311,312,317–323 Inh-BN, a narrow emission band has been observed experimen- tally with a ZPL transition at a range of 1.6 eV–2.2 eV,75,305buth-BN has also shown luminescence with a ZPL at 4.1 eV (Ref. 174)a n d 5.3 eV.177Tran et al.75experimentally demonstrated room tempera- ture and ultrabright emission from h-BN with a ZPL energy of 1.95 eV. A plasma-treatment of h-BN removes this emitter and produ- ces another emitter with ZPL emission at /C241.7 eV that was associated with the presence of oxygen in the quantum emitters.313Observations suggest that some emitters may absorb at and emit from distinct elec- tronic states.318The origin of these emitters is still unclear. A broad spectral distribution, spanning /C241e Vo ft h eg r o u po f2 - e VZ P Le m i t - ters also observed in h-BN, implies that a number of distinct defectstructures may exist, and they are tentatively attributed to the struc- tural composition variations,306the defect charge state uncertainty,324 and the local strain and dielectric environment variations.307The broad window of ZPLs that are generated by numerous color centers makes the identification of these quantum emitters very challenging.Indeed, it has been shown that the Stark shift on the order of 10 nm can be observed upon a 1 GV/m electric field on the ZPL energy of sin- gle emitters in h-BN, 325which demonstrates the strong coupling of these emitters to external fields. Inspired by the similar phenomenon in NV centers in diamond, it is naturally suggested that the origin of these quantum emitters is the combination of substitutional and vacancy defects. h-BN has a wide variety of possible intrinsic defects, antisite defects, and uninten- tional impurities (such as H, O, C, and Si) in its lattice structure. Several DFT computational works have attributed these emissions tocharge neutral native and substitutional defects with deep bandgap states. 173,176,311,312,321The plausible defect structures proposed by theo- retical studies and experimental observations are shown in Fig. 8 , including boron vacancy V B;140,326nitrogen vacancy V N;326carbon substitutional C Band C N;327,328oxygen substituting for nitrogen ON;327silicon substituting for boron Si B;329av a c a n c yn e x tt oas u b s t i - tutional atom, e.g., nitrogen V NNB;75,312oxygen V NOB;325carbon VNCB;312,322and C NVB;321boron vacancy passivated by oxygen VB2O313and hydrogen V B3H;176,310and a double substitutional car- bon defect C NCB:330 The intrinsic defects V Band V Nin a freestanding h-BN have been observed in a TEM experiment.326The defect V Npossesses D3h symmetry, which has two defect orbitals in the bandgap that enable optical excitation below the bandgap energy. The negatively charged VNshows a closed-shell singlet ground state, which is excluded as the source of the observed EPR signal.321Room temperature EPR and ODMR observations demonstrated a triplet ground state in electronirradiated and annealed h-BN crystal and exfoliated flakes, 140which was identified as the negatively charged V B:179,312Substitutional impu- rity defects such as O N,CN,CB, and C NCBcarbon pairs were directly observed by annular dark-field imaging in a STEM;327that study dem- onstrated that substitutional carbon atoms may occur mostly in pair- form. The 4.1 and 5.3 eV ZPLs were plausibly attributed to the deep donor C Band deep acceptor C N;328which has not yet been confirmed either from experiment or theory, though these defects have indeed low formation energies in h-BN.176Recent experiments have recorded ODMR signals on single or a few defects at room temperature for color centers in h-BN140,331that were tentatively assigned to C N defects based on the detected hyperfine signatures compared to the calculated ones.321On the other hand, the calculated ZPL energy of CNdefects does not agree with the observed ones;176thus, further investigations are needed to identify the origin of the close-to-single ODMR centers. McDougall et al.310combined a x-ray absorption near-edge structures (XANES) experiment and DFT calculation on the shift of the core levels of the host atoms to investigate the nature of point defects in h-BN. They were able to identify the presence of O N and V B3H defects, where the latter was a boron vacancy with three hydrogen atoms saturating the dangling bonds. As the h-BN samples exhibited a 4.1 eV emission, this was tentatively assigned to the V B3H defect based on DFT calculations that underestimated the bandgap by about 1.5 eV. Recent theoretical investigations indicated that, based on the calculated ZPL energy, the lifetime of the excited state and DWApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-15 VCAuthor(s) 2020factor332of carbon pair defect gives rise to a relatively narrow lumines- cence band, which can be associated with the 4.1-eV emitter. Turiansky et al.333found, by first-principles calculations, that the internal transition from the boron dangling bonds to a boron pzstate has a ZPL at 2.06 eV and emission with a Huang–Rhys factor of 2.3,which was suggested for the origin of the group of 2-eV quantum emitters in h-BN. In this model, the boron dangling bond was accom- panied by hydrogen atoms saturating the other dangling bonds of the vacancies, 333but the presence of hydrogen has not yet been proven in experiments. In addition, the calculated ionization threshold of the negatively charged dangling bond was smaller in energy than the cal- culated ZPL energy at 2.06 eV,333which makes this proposition tenta- tive at the moment. Advanced first-principles calculations in combination with group theory analyses would be a powerful approach to analyze the defects’ magneto-optical properties, as shown in Ref. 312.A b d i et al. predicted that V B(/C0) can exist and produce an S ¼1 ground state,312and this phenomenon was later observed by ODMR.140On the other hand, the prediction of the excited states from DFT calculations has limitations for the highly correlated orbitals or multireference states that typically appear in vacancy-type defects. As an example, Cheng et al. predicted that a V NCBdefect has an S ¼1 ground state in the neutral charge state, which is stable in a relatively wide range of Fermi levels.311The estimated ZPL energy between the triplet states is about 1.6 eV, where emitters with such ZPL energy are often observed in h-BN samples. Incontrast to this result, Sajid and co-authors found that the ground state of V NCBis1A1singlet with the C 2vsymmetry. They predicted 2.08 eV ZPL energy for the emission, where the phonon sideband should bebroad. 173This may explain the observed 1.95 eV emitters in h-BN.75If the intersystem crossing is very fast from the singlet excited state toward the triplet state, then the predicted ZPL energy is about 1.58 eV between the triplets,321which basically agrees with the previous DFT result in Ref. 311. A multireference character of the closed-shell and open-shell ground states of the defect-induced strong electron correlation effect would be another challenge in computational methods to accurately describe the energy levels of point defects.334As an example of V NCB defects in h-BN, hybrid functionals such as HSE06 work very well for excitations within the triplet manifold of the defects; however, they sig- nificantly underestimate the triplet-state energies by about 1 eV. Theestimation of the inaccuracy was based on an h-BN flake cluster model of a few tens of atoms terminated by hydrogen atoms by comparing the DFT results with Hartree–Fock-based multireference methods. The esti- mated DFT error might be overestimated in Ref. 334because of the lim- itations of the small models. Recently, a multireference method wasapplied on a supercell model of about 200 atoms for the calculation of the V B(/C0)s t a t e si n h-BN based on a density matrix renormalization group algorithm, and the error in HSE06 DFT was not so severe.179 It is worth noting that phonons play a critical role in determining the magneto-optical properties of the quantum emitters in h-BN. Energy (eV)Intensity (a. u.) Energy (eV)).u .a( ytisnetnI(b)(a) 1.5 0.5 0.01.0 0 /g415meg2(t)(c) kVB VN VNCBVB3H CNCB CNVBCN CBH ON C BONVB2OVNNB VNOB FIG. 8. Atomic defects in h-BN. (a) High-resolution transmission electron microscope image showing lattice defects in h-BN such as single vacancies and larger vacancies, respectively. These can be seen as triangle shapes with the same orientation. The scalebar in this figure is 1 nm. The figure has been reprinted with permi ssion from Jin et al. , Phys. Rev. Lett. 102, 195505 (2009)326Copyright 2009 American Physical Society. (b) Typical visible and ultraviolet PL spectra from quantum emitters. The photon correlation spectrum ( g2) reveals the quantum nature of the emitters. (c) Geometry of various possible defects in h-BN responsible for the quantum emitters, including V B,VN,O N,CB, CN,VNNB,VNOB,VNCB,CNVB,VB3H, V B2O, and C NCBdefects.Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-16 VCAuthor(s) 2020Strong electron–phonon interaction activates the emission of V NNB defects,335which also causes a strong coupling of strain to the ZPL emission. In this particular case, the out-of-plane or membrane pho- non modes are coupled strongly to the defect, which moves the atomsout of the plane. Experiments combined with first-principles calcula- tions imply that this motion of ions can explain the polarization of the emitted photons from such emitters. 325The strong electron–phonon coupling, manifested as the Jahn–Teller effect in the excited state, is responsible for the weakly allowed optical transition in V B(/C0) states in h-BN, and membrane phonons may play a crucial role in the intersys- tem crossing responsible for the ODMR contrast.179 B. Transition-metal dichalcogenides The family of two-dimensional materials, including graphene, h- BN, transition-metal dichalcogenides (TMDs) (e.g., MoS 2,W S 2, etc.), and phosphorene, have attracted much attention due to their extraor-dinary physical, chemical, and mechanical properties, as well as their promising applications in nanoelectronics, thermoelectric devices, wearable electronics, flexible displays, and smart health diagnos- tics. 336–345In addition to h-BN, TMDs have semiconducting charac- teristics, with direct bandgaps in their monolayer structures. For aMoS 2monolayer, there exist several types of intrinsic defects.346,347 The defect-based single-photon sources were realized in WSe 2.348–356 The potential advantage of a 2D single-photon source is obvious, for it offers the possibility of integrating single-photon sources with van der Waals heterostructures. The most observed optical emission from WSe 2in the band between 1.63 eV and 1.72 eV, with narrow line- widths of 120–130 leV, and strong photon anti-bunching were reported, unambiguously establishing the single-photon nature.348–351 This feature is attributed to spatially localized exciton states and theelectron–hole interaction in the presence of anisotropy. 352,353The pho- ton emission properties can be controlled via the application of exter- nal electric and magnetic fields. The decay time of these emissions is around 10 ns, which is 10-fold longer than that of the broad localized excitons. This reveals that the exciton bound to deep-level defects isthe underlying mechanism for the observed single-photon emis- sions. 354In 2017, large-scale deterministic creation of quantum emit- ters was demonstrated experimentally.355,356Although research on defect centers in TMDs is currently intensifying, this is a new field that addresses many open questions, such as the major defects dominatingthe single-photon emission, their atomic configurations, and electronic structures, which still need to be determined. Manipulating desired point defects at a predetermined set of locations lie at the heart of 2D quantum sensing engineering, and it can be technically realized. 357 VI. OTHER MATERIALS AS HOSTS A. Other wide bandgap materials Similar to the aforementioned host materials, other wide bandgap materials also host optically active defects that can principally emit sin- gle photons. Compared to the NV center in diamond, the research of color centers in these wide bandgap materials is still in its infancy.Preliminary results show a great advance in these materials; however, further effort is still required to obtain a comprehensive understand- ing, such as the correlation between defect orbitals with the pristine orbitals of host material, the effects of strain, and phonon coupling on electron coherence. For the completeness of this review article, here,we briefly introduce the basic properties, performance, and openquestions of these wide bandgap semiconductors for application in quantum information technology. 1. Zinc oxide Point defects in ZnO were explored for potential application as single-photon sources.149,358,359Numerous theoretical and experimen- tal studies have investigated the electrical and optical properties ofZnO with a bandgap of around 3.4 eV, and the role of defects, includ-ing oxygen vacancies (V O), zinc vacancies (V Zn), the zinc interstitials (Zn i), and the oxygen interstitials (O i).360Using hybrid density func- tional calculations, the native vacancies, interstitials, and danglingbonds in ZnO have been investigated. 361The oxygen vacancy (V O) was found to be a neutral defect, and the highest-energy photolumi-nescence peak associated with V Ois at 0.62 eV. Under realistic growth conditions, the zinc vacancy (V Zn) is the lowest-energy native defect in n-type ZnO, acting as an acceptor. V Zngives rise to multiple transition levels and emission between 1 and 2 eV. Compared with isolated V Zn, hydrogen atoms form highly stable complexes with V Zn,s h i f t i n gt h e acceptor levels closer to the valence band edge. Hydrogenated V Znhas optical transitions similar to those of isolated V Zn, both in good agree- ment with recent experimental results, supporting them as the sourceof photon emission. It is suggested that Zn dangling bond-relatedemission may be an intrinsic source of green luminescence in ZnO.The first report on the room temperature quantum emitters withbroad phonon sideband (see Table I ) was associated with the different charge states of the Zn-vacancy. 149Room temperature single-photon emission was also found in ZnO nanoparticles.362,363The application of ZnO in quantum technology would benefit from the detailedknowledge of its optical properties that should be further explored.We note that the exchange interaction between the bands of Zn dorbi- tals and O porbitals makes the accurate DFT calculation of pristine ZnO and defective ZnO complicated with respect to the case of othersemiconductor materials. Hybrid DFT functionals resulted in a greatadvance as listed above; however, further effort is required to study thelocalized and correlated orbitals of the vacancies and related defects. 2. Zinc sulfide A room temperature quantum emitter was also observed in cubic ZnS particles of /C24100 nm in diameter.364T h ez i n cb l e n d eZ n Se x h i b i t s a room temperature bandgap of /C243.6 eV, which is somewhat larger than that of ZnO. The fluorescence intensity of the quantum emitteremerges at 600 nm; it has a maximum at around 640 nm, and it decaysto zero at around 750 nm upon a 532-nm excitation. The ZPL peak isnot visible at room temperature. The observed lifetime is at /C242.2 ns. Since the emitter was observed without any treatment, it was tenta-tively associated with an intrinsic defect, the Zn-vacancy. Ab initio pre- dicts for a ZPL energy of about 2.4 eV (516 nm) for the Zn-vacancy, 365 which is a bit far from the observed shortest wavelength of the emis-sion. Further efforts are needed for the identification of this single-photon emitter. A high-quality ZnS material with large bandgap has agood potential in hosting visible quantum emitters. 3. Titanium oxide Recently, defects in TiO 2thin films and nanopowders exhibiting single-photon emission have been found.366T h ee x c i t e ds t a t ea n dApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-17 VCAuthor(s) 2020non-radiative lifetimes were found to be within the range of several nanoseconds and tens of nanoseconds, respectively. The fluorescenceoccurred in the red emission band. The three types of room tempera-ture quantum emitters show relatively broad luminescence with somepeaks between 600 and 700 nm. Low-temperature measurements for optical characterization and ab initio calculations may reveal the origin of these single-photon emitters in this prospective material with abandgap at /C243.0 eV. 4. Gallium nitride Room temperature quantum emitters in gallium nitride (GaN) were reported to exhibit narrowband luminescence.172In semiconduc- tor quantum dots, because of the high densities of defects and chargetraps, the rapid charging/de-charging can result in an obvious broad-ening of the emission linewidth. To overcome this issue, interface fluc- tuation GaN quantum dots have been developed. 367In the interface- fluctuation quantum dots, because the formation of charge localizationcenters is located at positions of thickness fluctuation in quantumw e l l s ,t h e ye m i ti nt h en e a ru l t r a v i o l e ta taw a v e l e n g t ho f /C24350 nm and exhibit narrow linewidths as compared to typical QDs. 367–369 Recently, near-infrared emitters in GaN have been found; they exhibit both excellent photon purity and a record-high brightness exceeding 106counts/s.155The origin of the emitter remains unknown. The sub- stitutional chromium (Cr4þ) impurity is a qubit candidate in GaN that possesses small phonon sidebands, so most of the fluorescence can beharnessed for quantum communication. 130 5. Cubic BN and hexagonal BN nanotube In addition to h-BN 2D crystals, the zinc blende allotropic form of boron nitride, the so-called cubic boron nitride ( c-BN), has shown important application as a wide bandgap semiconductor. Recently, Tararan et al.148provided a literature survey and systematically inves- tigated the optical properties of c-BN, including the optical gap and the luminescence of intragap defects. Using EELS, a large optical gapexceeding 10 eV was reported. A number of defect luminescence cen-ters were demonstrated in the visible and UV spectral ranges. In theUV spectral range, possible single-photon emission was observed, 148 which may motivate further investigation on this material. Another possible form of BN is the hexagonal boron nitride nanotubes ( h-BNNT). Unlike carbon nanotubes, the polarized B–N bonds create a large bandgap of /C246 eV independent from the chirality of the nanotube. The first room temperature quantum emitters werereported in about 5-nm-diameter h-BNNT. 138Recently, two similar room temperature quantum emitters have been observed in 50-nm diameter h-BNNT.139The peaks for the two emitters are found at 571 nm and 569 nm, respectively, presumably attributed to ZPL of thedefects. Both emitters have broad features and asymmetric line shapesin the spectra. Similar quantum emitters in the visible wavelengthregion were reported in 2D h-BN. 75For comparison, they also con- ducted experiments on BNNT samples with an average diameter of about 5 nm under the same condition, on different substrates.139It is much more difficult to find a quantum emitter, and the emissions arevery unstable under the 532 nm illumination. Most of them bleachwithin 10 s. This is consistent with former results 138and is likely due to a much larger curvature in 5-nm-diameter BNNTs.6. Wurtzite aluminum nitride The wurtzite phase of aluminum nitride (w-AlN) has C6vsym- metry. It is a wide bandgap (6.03 /C06.12 eV) semiconductor that has many applications.370The very large bandgap can suppress coupling between bandgap levels and bulk states. Moreover, the small spin–or- bit splitting of 19 meV was reported,371which can enhance the qubit- state lifetime. These two unique advantages make AlN an attractive host material for scalable solid-state qubits analogous to diamond and SiC. Furthermore, the growth of high-quality crystal w-AlN has been reported.372The spin states of neutral nitrogen vacancy in AlN have been experimentally detected using electron paramagnetic resonance.373 AlN has a large bandgap; this meets the criteria for a host mate- rial for the addressable single-photon emitter. However, in AlN, the defect levels induced by anion vacancy are too close to the band edge, resulting in strong resonance with the bulk band edges. This hinders the application of AlN in scalable quantum technologies. To overcome this issue, Varley et al. proposed that alloying AlN with transition- metal dopants can push the defect levels deeper into the bandgap.374 For example, Ti and Zr atoms substitute on the Al site can lead to theformation of the desired electronic and spin states. Seoet al. proposed an alternative by applying strain. 375It was found that in the stress-free negatively charged nitrogen vacancy(V /C0 N), the S ¼1 state is slightly higher in energy than the S ¼0 state, revealing that the two spin states are approximately degenerate in energy. Conversely, these two spin states are associated with two dis- tinct Jahn–Teller distortion configurations. This suggests that these two spin states can be separated in energy by applying strain. TheirDFT calculation results demonstrate this expectation, i.e., even at a small compressive strain of /C03% along the [1120] direction, the S ¼1 state is significantly lower by about 250 meV than the S ¼0 state. In addition to the difference between the energy of these two spin states, the hyperfine tensors are also sensitive to loading strain. Very recently, Xue et al. have observed numerous room tempera- ture single-photon emitters in w-AlN films. 376At low temperatures, the PL spectra exhibit relatively narrow and strong ZPL peaks with positions varying from the visible (543 nm) till the NIR ( /C24980 nm) region. The full PL spectra and, thus, the corresponding Debye–Wallerfactors have not reported. They also applied DFT calculations with such functionals that do not well reproduce the bandgap or not well tested, and only the vertical excitation energies were determined. 376 Based on these calculations, they concluded that some NIR quantumemitters in the AlN film originate from the antisite nitrogen vacancycomplexes (N AlVN) and divacancy complexes (V AlVN). Further char- acterization is required at both the theoretical and experimental fronts in this highly prospective host material. 7. Rare-earth ions in garnets, silicate, and vanadate Rare-earth ions (REIs) embedded into the solid-state matrix are prospective building blocks for quantum memory and quantum com-munication applications. 377In particular, it has been demonstrated for an ensemble of europium doped into Y 2SiO 5or YSO that the quantum information can be stored for six hours.378In REI systems, the 4felec- trons split in the low symmetry crystal field, and the optical transition basically originates from these split atomic states. The atomic-likestates have the advantage of producing narrow optical emission lines,Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-18 VCAuthor(s) 2020and the large ion has a relatively large hyperfine fine structure in the ground state with the long coherence time. However, the disadvantage is that the long optical lifetimes because of the optical transition dipolebetween the atomic states are often forbidden in the first order. This limitation has been partly circumvented in the case of Pr 3þdoped Y3Al5O12or YAG by using an upconversion process that has the mutual benefits of accessing a short-lived excited state and providing background-free optical images,160which was the first demonstration of REI quantum emission at He flow temperatures. Quantum emission of Pr3þdoped LaF 3at 1.5 K was also demonstrated in a following study.164Later, spin-to-photon interface and single spin readout were observed in Ce3þdoped YAG,161,379where the optical transition occurs between the 5 dexcited state and 4 fground state. 4 fstates that the spin is highly mixed with the orbital momentum. This enables spin-flip optical transitions between the 4 fand the 5 dlevels with an optical lifetime of /C2460 ns. Under dynamic decoupling, spin coherence lifetime reaches T2¼2 ms and is almost limited by the measured spin–lattice relaxation time T1¼4.5 ms. Another possibility to circum- vent the intrinsically low emission of REI is to engineer them near high-quality optical resonator, which can significantly increase the rate of spontaneous emission selectively at the resonator mode by harness-ing the Purcell effect. This has been recently achieved in Er 3þdoped YSO380and ytterbium doped YVO 4or YVO381that made possible to detect single-photon emission from these ions. Furthermore, by care-ful choice of the direction of an external constant magnetic field, the enhancement of the emission has been tuned toward the spin- conserving excitation only with well-distinguishable optical transitions for the different spin states. 380,381If the laser and the cavity modes were tuned only in one of the spin-conserving excitation energies,then emission is expected only for the resonant bright spin state and no emission is expected for the off-resonant dark spin state. During the optical cycles, a spontaneous decay from the bright spin state to the dark state can occur with some low probability that remains dark. Thus, the observation of the emission is correlated with the spin stateof the REI in its ground state. Aside from/in addition to the optical readout of the REI spin state, spin dephasing and the spin dephasing limits in YVO have been observed. 381 Theory here faces several challenges: highly correlated atomic like orbitals, large spin–orbit interaction, and possibly non-negligible electron–phonon interaction for the case of quasi-degenerate orbi-tals. One obvious choice is to apply post Hartree–Fock-based meth- ods where the correlation energy between the orbitals can be systematically taken into account by raising the complexity of the approach, including relativistic effects (cf. a review in Ref. 382). The crystals are modeled by finite small clusters (tens of atoms like ultra-small quantum dots) in the post Hartree–Fock-based methods. To the best of our knowledge, no systematic theoretical study has been conducted on how the size of the clusters would affect the results. Alternatively, the Kohn–Sham DFT theory may be used but it may fail due to the complex correlation effects of the atomic like orbitals.Recently, a computationally tractable solution has been developed that combines the HSE hybrid density functional theory with the orbital-dependent exchange interaction that was demonstrated forcerium dioxide. 383Another solution can be to embed CI methods into DFT203or to very efficiently calculate the CI method in medium sized clusters with periodic boundary conditions by means of density matrix renormalization group (DMRG) algorithms.1798. Lithium fluoride Using time-dependent density functional theory embedded- cluster simulations, the electronic and optical properties of two adjacent singlet-coupled F color centers (anion-vacancy) in lithiumfluoride (LiF) were investigated, 384where LiF has an extremely large bandgap of over 10 eV. It was found that to accurately simulate the entangled-defect system, it is necessary to consider the dynamicalcorrelations between the defect electrons and the adjacent ionic lattice. To the best of our knowledge, no single-photon emitter has been observed so far in this insulator. B. Potential moderate/small bandgap semiconductors Diamond is already proven to be a promising host material for quantum information technology that builds up from carbon atoms. The graphite, including the single sheet of graphite, i.e., graphene, is azero-gap semiconductor; therefore, it cannot host color centers. However, some forms of carbon nanotubes with a relatively small diameter can introduce small bandgap, which opens the door for host-ing NIR emitters. By considering small bandgap materials for hosting qubits, silicon crystal is an obvious choice, which is an elementary semiconductor. Silicon is the most used semiconductor for integratedcircuits, which has a bandgap at 1.17 eV at cryogenic temperatures. Besides using dopant atoms for realizing qubits, i.e., Kane quantum computer (see below), NIR color centers have been introduced forquantum optics studies (see Table I ). Considering its ideal interface compatibility with conventional Si-based technology, the application of silicon in quantum information technology has attracted a great attention in recent years. In this sub-section, we discuss the character- istic, performance, and perspective of these two promising materialsfor quantum information technology. 1. Carbon nanotube The development of a solid-state photon source based on carbon-related nanomaterials has received a lot of attention.385,386The photoluminescence of semiconducting single-walled carbon nanotubes was first observed in 2002.387The bandgap of carbon nanotubes can vary from 0 to 2 eV, depending significantly on the tube chirality. The advantages of carbon nanotubes include the straightforward tuning of the emission wavelength, between 850 nm and 2 lm, by changing the chiral species. Unfortunately, in principle, single-photon emission requires a quantum mechanical quasi-two-level system, yet the one- dimensional (1D) band structure of carbon nanotubes conflicts withthis major requirement. One scheme toward a single-photon source is the strong exciton–exciton interaction in 1D systems. 388Via this strong excito- n–exciton interaction, the multiple excitons created in a carbon nano- tube can be annihilated until only a single exciton remains to emit asingle photon, namely, the exciton–exciton annihilation effect. As it does not rely on exciton localization, room temperature single-photon generation is possible. In 2015, room temperature partial single- photon emission with a purity of about 50% was observed. 389 Another strategy for obtaining reliable single-photon emission from carbon nanotubes relies on exciton localization. Deep potential trapping (much greater than kBT) is required for localization at room temperature. However, because the trapping potential in a pristineApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-19 VCAuthor(s) 2020carbon nanotube is very shallow, single-photon emission in a pristine carbon nanotube is limited to cryogenic temperatures. Chemical func- tionalization of carbon nanotubes provides one way to strengthen the exciton localization and generate localized excitons with well depths of 100 meV or greater, which are higher than kBTat room tempera- ture.390–393The physical mechanism is the strong exciton trapping at the sp3defect centers.391,394,395The zero-dimensional nature of trapped excitons also brings novel physical phenomenon, such as the interaction of 1D excitons with the 1D phonon modes in carbon nano- tubes.394,395Furthermore, exciton localization can lead to significantly longer decay times, and with diameter decreases of (7, 5)–(5, 4) of the nanotubes, the decay time increases from 75 ps to 600 ps, demonstrat- ing strong chirality dependence.396 2. Silicon: Kane quantum computer A completely different approach toward building quantum com- puters was proposed by Kane in 1998,397which is based on the combi- nation of classical semiconductor technology and electron-nuclear magnetic resonance techniques. As the most successful material applied in the semiconductor device technology, silicon (Si) has an indirect bandgap of about 1.17 eV at cryogenic temperatures that reduces to about 1.12 eV at room temperature. The isotopically pure 28Si has a nuclear spin of 0 that can be a perfect host for defect qubit spins. When the silicon substrate is doped with isotopically pure31P (phosphorus), with providing a donor electron with an electron spin and also a nuclear spin of1=2, the nuclear spin of the phosphorous donors can realize the function to encode qubits with extremely long coherence times and low error rates because the donors are extremely separated from the environment. This design combining the advantage of tunable quantum well for localizing the donor electron using gate elec- trodes—called quantum dots398—and nuclear magnetic resonance has the advantage of scalability.397In this system, also named the Kane quan- tum computer, the nuclear spins of the phosphorous donors perform single-qubit operations. Two nuclear spins can interact mediated by the extended electron wave packet. As the electron wave packet is sensitive to the external electric field, the nuclear spin interaction can be con- trolled by gate voltage, which is essential for realizing the operation of quantum computation. With an array of such donors embedded beneath the surface of a pure silicon wafer, a silicon-based nuclear spin quantum computer was expected. Indeed, using electron- nuclear double resonance technique, the electron spin could be suc- cessfully encoded into the nuclear spin of31P with an overall store- readout fidelity of 90%. The coherence lifetime of the quantum memory element at 5.5 K exceeded 1 s in28Si enriched Si.399 The idea of a Kane quantum computer faces at least two chal- lenges: (i) efficient readout of single qubits and (ii) precise alignment of the donor atoms in Si. By engineering the P donors between twoelectrodes, i.e., field-effect transistor (FET) configuration, the electrical detection of single P donor electron spins was demonstrated via spin- selective recombination between a deep paramagnetic defect acting as a trap for the P donor electron at the silicon and SiO 2interface. This result was published4006 years after Kane’s publication.397The mecha- nism can be described as a spin-to-charge conversion, where the high purity of the Si material makes possible to detect single electrons in the FET device. With using the pulsed electrical detected magnetic reso- nance, the Rabi nutation of single P donor electrons (the coherentstate) was observed in the Si FET device with using the same mecha- nism.401The disadvantage of this method is that the spin-to-charge conversion probability between the states of the deep defect (presum- ably a Si dangling bond at the Si/SiO 2interface) and the P donor heavily depends on their locations, and it is extremely complicated tocontrol the formation and location of the deep defect. A breakthrough in the single-shot readout of P donors was achieved in a single electron transistor (SET) architecture with using a 1.5 T magnetic field for effi- cient Zeeman splitting at 40 mK temperature, in order to achieve a spin-selective tunneling of the P donor electron from the SET regiontoward the electrodes. 402,403Morello et al. observed a spin lifetime of 6 s at a magnetic field of 1.5 T and achieved a spin readout fidelity bet- ter than 90%.403Although this method operates at ultralow tempera- tures compared to the spin-to-charge readout mechanism, here, only the location of the P donor should be precisely controlled. Later, a top- gated nanostructure, fabricated on an isotopically engineered28Si sub- strate, was used to increase the electron spin readout contrast to about97% and 31P nuclear spin readout contrast to 99.995%.404Although the control fidelity approached 99.99%,404the coherence time of the 31P nuclear spin can be much extended by removing the donor elec- tron spin by illumination, i.e., charging the donor. It was demonstrated that the coherence time of the31P nuclear spin is 39 min at room tem- perature and 3 h at cryogenic temperatures.405We note that the con- trol and readout of the qubit still occur at low temperatures. Thesefavorable qubit properties make feasible to study the scalability of the qubits and their interaction. 406To this end, the tight control on the formation and placement of the donors is required. A breakthrough in the precise alignment of P donors was achieved by a combination of scanning tunneling microscopy (STM) andhydrogen-resist lithography. 407Simmons group demonstrated a single- atom transistor in which an individual phosphorus dopant atom was deterministically placed with a spatial accuracy of one lattice site in SET. The SET operated at liquid helium temperatures, and millikelvin electron transport measurements confirmed the presence of discrete quantumlevels in the energy spectrum of the P atom. 407Furthermore, the coher- ent control of single P donor qubit was demonstrated.408This technology has been recently applied to fabricate the two-qubit exchange gate between phosphorus donor electron spin qubits in silicon using indepen- dent single-shot spin readout with a readout fidelity of about 94%, which operates very fast (about 800 ps).409In the two-qubit exchange mecha- nism, the precise alignment of the two P donors was inevitable. This is acrucial step with respect to the previous breakthroughs achieved in the demonstration of various two-qubit gates in Si. 409–413 In addition to phosphorous dopant, arsenic in silicon is another attractive platform for quantum computing since arsenic dopants have many advantages over phosphorus, including a higher solid solubilityin bulk silicon and a lower diffusivity than phosphorus. Moreover, the atomic spin /C0orbit interaction strength of arsenic is twice that of phos- phorus and triple times of the nuclear spin value. 414The high nuclear spin value presents opportunities for simplifications in physical imple- mentations of quantum gate structures. Very recently, using a scan-ning tunneling microscope tip, Stock et al. reported the successful fabrication of the atomic-precision of arsenic in silicon. 415Obviously, further studies on the quantum properties of the arsenic in silicon are desired. Arsenic has an advantage of the relatively large quadrupole of the nuclear spins that can be harnessed to control the nuclear spin states of the ionized arsenic donors via strain.416Obviously, furtherApplied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-20 VCAuthor(s) 2020studies on the qubit properties of the arsenic in silicon are desired. A very interesting donor alternative in Si is bismuth (Bi). Although Bi has a much lower solubility than P in Si,209Bi offers a 20-dimensional Hilbert space rather than a four-dimensional Hilbert space of31P, which has a great potential in quantum operations. It was experimen- tally demonstrated that the Bi donor has similar coherence times like P donor’s, and the nuclear spins can be coherently manipulated.417 Notably, group-III shallow acceptors in silicon, e.g., boron, which have strong spin–orbit coupling, exhibit ultralong coherence times of 10 ms that can rival the best electron-spin qubits.418The large spin–orbit coupling may help to manipulate the qubit with electric fields insteadof magnetic fields, which is technologically much friendlier. We note that it was a common belief that the relatively shallow donors and acceptors in Si can be well understood by using the effec-tive mass theory with some fitting parameters for the central cell cor- rection and related empirical tight binding theory. 419,420In particular, the electrostatic tuning of the hyperfine interaction between the elec- t r o ns p i na n dt h ePd o n o rn u c l e a rs p i ni na nF E T421or the large valley-orbit splitting in a silicon gated nanowire device422could be well understood with this picture. Ab initio theory predicted that the former effect can be even significantly greater in a silicon nanowire where the physical dimension of the silicon nanowire is reduced to thequantum confinement regime, 423w h e r et h es t r a i na c t i n go nt h eq u b i t depends on the diameter and surface termination of the Si nanowire. Recently, strain was applied to various donors in Si to tune the hyper- fine coupling between the corresponding nuclear spin and the donor spin.424Although the empirical tight binding theory seemed to mostly reproduce the experimental data, the ab initio theory may be neededfor accurate prediction, in particular, for the large donor ions. The challenge for ab initio periodic cluster calculations is to accommodate the defect (wavefunction) in a sufficiently large supercell or to apply the appropriate scaling method to approach the isolated donor limit. The spin-to-photon interface to create flying qubits in Si may be realized by deep donors like chalcogen donors.425For the prototypical 77Seþdonor, lower bounds on the transition dipole moment and excited-state lifetime were measured with long-lived spin at cryogenic temperatures, enabling access to the strong coupling limit of cavity quantum electrodynamics using the known silicon photonic resonator technology and integrated silicon photonics.426,427 A n o t h e ra p p r o a c hi st ou s ed e e pfl u o r e s c e n tc e n t e r si nS it or e a l - ize quantum emitters. Erbium in Si emits light in the telecom wave- length, which is an REI impurity in Si with optical transition between the split 4forbitals (e.g., Ref. 107). A hybrid approach was demon- strated for the readout of the electron spin in which optical excitation is used to change the charge state (conditional on its spin state) of an erbium defect center in a SET, and this change is then detected electri- cally.428Recently, ensembles of G-centers and G-center-related single- photon emitters were created in silicon by carbon implantation and annealing.159,459A G-center mostly emits at 1280 nm (0.97 eV) wave- length in the near-infrared region, as shown in Fig. 9 ,m a k i n gi tv e r y compelling for quantum telecommunication.429The G-center has a known ODMR signal with an S ¼1 metastable state with singlet ground and excited states (see Ref. 430and references therein); thus, this quantum emitter may act as a qubit or a quantum memory. A combined deep level transient spectroscopy, PL, EPR, and ODMR spectroscopy study strongly argued that G-center is a bistable form of FIG. 9. The plots of quantum-coherent materials vs zero-phonon line emission (unit in nm). In the region of near-infrared light (700–2500 nm), there are two w indows presented as pink color (first biological window: 650–950 nm) and gray color (second biological window: 1000–1350 nm), which are important for in vivo imaging applications.436The three telecommunication operating wavelengths for fiber optic communication, namely, O-band (original), E-band (extended), and S-band (short), are pre sented as green, blue, and orange colors, respectively. The ZPL values with unit in eV are shown in Table I . The color centers with unknown origin are not labeled. The symbol codes are the followings: 1D color center—purple square; 2D color center—purple triangle; 3D color center—yellow circle; coherent ensemble spin control—circle with blue ar rows; and coherent single spin control—circle with a single green arrow. In silicon, an oxygen-related center, C-center, has a ZPL at 1570 nm and the corresponding DW factor is 0 .100435besides a self- interstitial-related color center, W-center, which has a ZPL energy at 1220 nm and the corresponding DW factor is /C240.40.157The C-center and W-center in silicon were reported in neutron irradiated and annealed samples as ensembles157and, thus, are not listed in Table I .Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-21 VCAuthor(s) 2020two carbon atoms sharing one Si site in the lattice;430however, previ- ous and recent ab initio calculations did not reach any satisfying con- sensus and conclusion about the microscopic origin of the center.431–434Additional ab initio studies are required for the unam- biguous identification and further characterization of the G-center insilicon. We note that the ZPL wavelength of the C-center and W-center in silicon 435is also very promising for quantum communication purposes (see Fig. 9 ). Further theoretical and experimental effort is needed to harness these color centers in quantum technologyapplications. VII. SUMMARY AND OUTLOOK In this review, we have focused on color centers in solids, the emerging material platform for quantum information technology withdistinctly promising properties. A summary of the key parameters inthe experimental measurement for these promising materials, e.g., dia-mond, 4H-SiC, 3C-SiC, 6H-SiC, ZnO, GaN, c-BN, and 2H-MoS 2,a n d for 2D materials, e.g., h-BN, WSe 2, and WS 2, is given in Table I .O v e r the past decade, scientists have tried various materials, such as dia- mond, h-BN, SiC, and other wide bandgap semiconductors, for con- structing the basic elements for quantum information processingdevices. 437Some startup companies have already begun to commer- cialize solid-state quantum systems, such as diamond NV centers, forquantum sensing. 438Diamond NV centers have great potential to real- ize quantum computers operating at elevated temperatures when thetechnology is developed to imprint the qubits in a scalable 2D arrayarrangement. However, obviously, many challenges for industry-level applications remain in any platform. Within the next few decades, one may expect that traditional and quantum computers will coexist, foreach type has its own advantages, 439as predicted by Turing Award laureate Andrew Chi-Chih Yao. In 2016, The Quantum Manifesto440 called upon Member States and the European Commission to launchae1/C210 9Flagship-scale Initiative in Quantum Technology. Two years later, the National Science Foundation (USA) invested funds ofup to $25 /C210 6in a new program, called Enabling Quantum Leap: Convergent Accelerated Discovery Foundries for Quantum MaterialsScience, Engineering, and Information (Q-AMASE-i), to establish foundries with “mid-scale infrastructure for rapid prototyping and development of quantum materials and devices,” according to the pro-gram solicitation. 441Obviously, the above shows that the quantum computer will have a prominent part in the future. Although quantumcomputers may not supersede traditional computers in tasks for whichtraditional computers are already proven to be highly efficient, quan-tum computers could excel in many tasks, such as the design of newmaterials for health, energy storage and production, high temperaturesuperconductors, and new catalysts. We briefly mentioned the challenges in this road in the paper that we perpetuate here. At limited temperature, the excitation of a phononwill induce atoms in the solids displacing from their equilibrium posi-tions. As a result, the electron/spin energy level has significant broad-ening because of the fluctuation of the atoms, leading to a finiterelaxation time for spin states. For example, a significant reduction inthe spin coherence time was reported in the isolated NV center atroom temperature with respect to low temperature. 46,195Furthermore, the excitation of phonons may also break the time-reversal symmetry and result in a net magnetization in the non-magnetic system.442In addition to its influence on the spin coherence time, physicists alsofigured out how quantum computers can operate using coherent pho- nons.443–445In light of the aforementioned facts, therefore, we suggest that better knowledge of spin-phonon coupling is highly important not only for understanding the fundamental physics but also for the appli- cation of quantum technologies. The complex interplay between thephonons and spins can be monitored in such defect qubits in which the electron–phonon coupling and spin–orbit coupling are in the same order of magnitude. 79The theory predicts that SnV and PbV defects in diamond are ideal platforms to this end.79 Besides quantum computation, the development of quantum sensing and quantum communication devices is also a driving force in the field. In quantum sensor applications, the room temperature oper- ation is a must for in vivo biological and medical studies; thus, the search for room temperature defect qubits is of high importance in this context. Furthermore, the temperature dependence of materialproperties, such as high-temperature superconductivity, can only be monitored by such quantum sensors that can operate in a broad range of temperatures. The use of light either in the manipulation or readout of the room temperature qubits poses restrictions on the wavelength region for biological studies. Two NIR wavelength regions are defined in the context where the absorption is minimal by typical biological systems (see NIR-I and NIR-II in Fig. 9 and Ref. 436). Ideally, the color centers should be photoexcited and emit in the NIR-I region or rather in the NIR-II region to this end. We note that the phonon sideband of the emission may fall to the desired region, which has a longer wave- length than the marked ZPL wavelength in Fig. 9 ,a st h eo p t i c a lr e a d - out of many color centers with a low DW factor is carried out in the phonon sideband. The quest for the design of the optical emission is also evident for quantum communication applications. The efficiencyof the quantum communication can be significantly increased if the critical ZPL emission of the color centers falls into the present telecom- munication bands of the optical fibers (see Fig. 9 ), and so it does not require any wavelength conversion that always leads to a loss of the signal intensity. The list of the presently observed single-photon emit-ters and qubits in the context of technologically relevant wavelength regions is summarized in Fig. 9 . It is obvious that the search for qubits with the given fluorescence properties is required for the optimization and efficient implementation of quantum technologies. We note that the search of point defects for a given ZPL wave- length is an insufficient condition in the optimization of defect qubits. The interaction and coupling between different particles and quasi- particles in solids, including electrons, phonons, photons, and spin, are the important physical process in nature and have direct impacts on the performance of quantum devices. Considering the complex multi- particle interaction, it is still challenging in the computational design of new material platform for quantum technology. Recently, with bigdata generated by theory and experiments, high-throughput calcula- tion, screening, and machine learning methodologies have been adopted to material genome initiatives and material informatics. 446–448 These methodologies have been exploited in the discovery of variousfunctional materials, including valleytronics materials, 449thermoelec- tric materials,450thin film solar cells,451organic–inorganic perov- skites,452and topological electronic materials.453Very recently, candidate materials for hosting defect qubits have been identified with the principles of constituting of the crystal by low-spin isotopes for securing long coherence times and of possessing >2 eV bandgap for robust optical control by means of machine learning techniques.460Applied Physics Reviews REVIEW scitation.org/journal/are Appl. Phys. Rev. 7, 031308 (2020); doi: 10.1063/5.0006075 7, 031308-22 VCAuthor(s) 2020However, quantum-coherent materials are realized by point defects in the host that should be also identified. The first steps were taken to conduct this type of research to seek solid-state defect qubit candi- dates,454which is based on the computation of the key magneto- optical properties of the defects, as listed in Table I , i.e., the database of defect properties based on the theoretical spectroscopy. Therefore, it isalso expected that machine learning methodology can provide new insights into and facilitate the development of new material platforms for quantum technology. A crucial issue that may stem to employ machine learning methodology to this end is the accuracy of the com- putation methods, in particular, for the excited states, which is critical in the credibility of the resulting database. The combination of the advantages of the density functional theory and configuration interac-tion seems to provide a good balance in terms of the tractable system size and the relatively good accuracy (about 0.1 eV) for highly corre- lated excited states. 203 It is exciting to note that, due to the enormous potential of quan- tum computers in the design of new materials and the importance of new material platforms for building scalable quantum computer net-works with much better performance, the birth of the usable quantum computer will definitely speed up the development of new material plat- forms, resulting in “self-accelerating” progress in quantum information technology. The initial step has been recently taken in this direction. 455 In this context, the theoretical simulation of experimental characteriza-tion is expected to play a more important role in future research. Finally, we allude here to an important point from Ref. 45.W e note that the full description of a quantum bit cannot be separated from the description of the environment. The environment often is a source of noise for the qubits causing decoherence. This is clearly dis- advantageous for quantum computer application but can be harnessed in quantum sensor protocols. We briefly mentioned some key quanti- ties of the qubits such as readout contrasts, the longitudinal spin relax- ation time, and the coherence time, which are dependent on the environment such as strain, electric and magnetic fields, and tempera-ture. By computing the coupling of the defect qubit properties to these key quantities, the sensitivity of quantum sensing protocols and opti- mization of quantum control can be designed and may guide future experimental studies. Only this comprehensive approach makes the ab initio search for alternative solid-state defect quantum bits reliable and powerful that might be superior for a given quantum technology appli- cation. One possible route along this direction is to combine the ab ini- tiocalculations and effective spin Hamiltonian approach to compute the readout contrasts, the longitudinal spin relaxation time, the coher- ence time, and other key quantities. A recent study has taken the first steps into this direction, 456which has been successfully applied to understand the optical response of qubits in a real environment.457 Again, the usable quantum computer may result in a self-acceleratingprocess by directly simulating the evolution of spin states of the defect qubit in the bath of external spins. All-in-all, the future of color center qubits looks bright in the joint effort of ab initio simulations—theoretical spectroscopy and qubit control—and experiments. ACKNOWLEDGMENTS A.G. acknowledges the support from the National Office of Research, Development, and Innovation in Hungary for the QuantumTechnology Program (Grant No. 2017-1.2.1-NKP-2017-00001) and the National Excellence Program (Grant No. KKP129866), and fromthe European Commission for the QuanTELCO project (Grant No. 862721). G.Z. and J.-P.C. thank Dr. M. Sun for his discussion and data collection. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982). 2A. Ac /C19ın, I. Bloch, H. Buhrman, T. Calarco, C. Eichler, J. Eisert, D. Esteve, N. Gisin, S. J. Glaser, F. Jelezko, S. Kuhr, M. Lewenstein, M. F. Riedel, P. O.Schmidt, R. Thew, A. Wallraff, I. Walmsley, and F. K. Wilhelm, New J. Phys. 20, 080201 (2018). 3D. P. DiVincenzo, Science 270, 255 (1995). 4S. Lloyd, Science 273, 1073 (1996). 5J. I. Cirac and P. Zoller, Science 301, 176 (2003). 6L.-M. 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5.0022150.pdf

APL Mater. 8, 111102 (2020); https://doi.org/10.1063/5.0022150 8, 111102 © 2020 Author(s).Photoemission electron microscopy of magneto-ionic effects in La0.7Sr0.3MnO3 Cite as: APL Mater. 8, 111102 (2020); https://doi.org/10.1063/5.0022150 Submitted: 17 July 2020 . Accepted: 04 October 2020 . Published Online: 04 November 2020 Marek Wilhelm , Margret Giesen , Tomáš Duchoň , Marco Moors , David N. Mueller , Johanna Hackl , Christoph Baeumer , Mai Hussein Hamed , Lei Cao , Hengbo Zhang , Oleg Petracic , Maria Glöß , Stefan Cramm , Slavomír Nemšák , Carsten Wiemann , Regina Dittmann , Claus M. Schneider , and Martina Müller ARTICLES YOU MAY BE INTERESTED IN The influence of lattice defects, recombination, and clustering on thermal transport in single crystal thorium dioxide APL Materials 8, 111103 (2020); https://doi.org/10.1063/5.0025384 Effect of antiferromagnetic order on topological electronic structure in Eu-substituted Bi2Se3 single crystals APL Materials 8, 111108 (2020); https://doi.org/10.1063/5.0027947 Oxygen vacancies: The (in)visible friend of oxide electronics Applied Physics Letters 116, 120505 (2020); https://doi.org/10.1063/1.5143309APL Materials ARTICLE scitation.org/journal/apm Photoemission electron microscopy of magneto-ionic effects in La 0.7Sr0.3MnO 3 Cite as: APL Mater. 8, 111102 (2020); doi: 10.1063/5.0022150 Submitted: 17 July 2020 •Accepted: 4 October 2020 • Published Online: 4 November 2020 Marek Wilhelm,1,a) Margret Giesen,1 Tomáš Ducho ˇn,1 Marco Moors,1,2 David N. Mueller,1 Johanna Hackl,1Christoph Baeumer,1 Mai Hussein Hamed,1 Lei Cao,3,4 Hengbo Zhang,4 Oleg Petracic,4 Maria Glöß,1,2,5 Stefan Cramm,1Slavomír Nemšák,1,6 Carsten Wiemann,1 Regina Dittmann,1,5 Claus M. Schneider,1,7,8 and Martina Müller1,9 AFFILIATIONS 1Forschungszentrum Jülich GmbH, Peter-Grünberg-Institute, 52425 Jülich, Germany 2Leibniz-Institute of Surface Engineering (IOM), Department of Functional Surfaces, 04318 Leipzig, Germany 3Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany 4Jülich Centre for Neutron Science (JCNS-2), Peter-Grünberg-Institute (PGI-4), JARA-FIT, Forschungszentrum Jülich GmbH, 52425 Jülich, Germany 5JARA-FIT, RWTH Aachen University, 52056 Aachen, Germany 6Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA 7Faculty of Physics, University of Duisburg-Essen, 47057 Duisburg, Germany 8Department of Physics, UC Davis, Davis, California 95616, USA 9Faculty of Physics, University of Konstanz, 78457 Konstanz, Germany Note: This paper is part of the Special Topic on Magnetoelectric Materials, Phenomena, and Devices. a)Author to whom correspondence should be addressed: m.wilhelm@fz-juelich.de ABSTRACT Magneto-ionic control of magnetism is a promising route toward the realization of non-volatile memory and memristive devices. Magneto- ionic oxides are particularly interesting for this purpose, exhibiting magnetic switching coupled to resistive switching, with the latter emerging as a perturbation of the oxygen vacancy concentration. Here, we report on electric-field-induced magnetic switching in a La 0.7Sr0.3MnO 3 (LSMO) thin film. Correlating magnetic and chemical information via photoemission electron microscopy, we show that applying a positive voltage perpendicular to the film surface of LSMO results in the change in the valence of the Mn ions accompanied by a metal-to-insulator transition and a loss of magnetic ordering. Importantly, we demonstrate that the voltage amplitude provides granular control of the phenom- ena, enabling fine-tuning of the surface electronic structure. Our study provides valuable insight into the switching capabilities of LSMO that can be utilized in magneto-ionic devices. ©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/10.1063/5.0022150 .,s I. INTRODUCTION Transition metal oxides (TMOs) reveal a multitude of chemical, structural, and physical properties, such as insulating and metal- lic states, piezoelectricity, superconductivity, and different magnetic ordering. They can be altered by subtle distortions of the coor- dination environments, i.e., changes in composition, temperature, and external electric fields.1–3The variety inherent in TMOs stemsfrom the complex relation between the cationic d-electron config- uration and their corresponding crystal structure, which provide a multitude of electronic configurations with small energy differ- ences.4,5Moreover, transition metals can appear in multiple oxi- dation states, which enable an electron transfer between those dif- ferent cationic valencies and may stabilize a variety of magnetic states.6 APL Mater. 8, 111102 (2020); doi: 10.1063/5.0022150 8, 111102-1 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm Special attention is paid to mixed-valence La 0.7Sr0.3MnO 3−δ (LSMO), a half-metallic magnetic oxide, which exhibits a large spin polarization ( ∼95%) and the highest Curie temperature of T C = 370 K known for the group of manganites.7,8Furthermore, a magnetoresistance (MR) larger than ∼1800%, which has been reported,8,9has been widely used in magnetic oxide tunnel junction- based devices with effective spin injection behavior. As an exten- sively investigated magnetoelectric (ME) material, LSMO has been the subject of numerous studies revealing ME coupling via strain,10,11charge carrier doping,12and interfacial exchange processes.13 The strong correlation between the physical properties and the electronic- and atomic structure leads to a considerable sensi- tivity to ionic crystal defects and particularly oxygen vacancies.14 The formation and dynamics of oxygen vacancies can be substan- tially controlled by strain,15temperature,16and external electric fields,17providing reversible manipulation between two or multiple states,18which has attracted enormous scientific and technological interest. Recently, the field of magneto-ionics received increased atten- tion as the voltage-driven chemical intercalation of ionic species opened a novel route toward the control of magnetism in bulk mate- rials. One of the first studies reported about the migration of oxygen vacancies in magnetic Fe–O/Fe layers in contact with ionic liquid electrolytes, which modulated near-surface and bulk magnetism.19 In the scope of magnetic oxides, electrolyte-gated LSMO devices showed a substantial formation and annihilation of oxygen vacan- cies affecting magnetic bulk properties (20 nm)20and providing switching capabilities for large sample areas with large spatial homo- geneity and high switching speed.21Beyond ionic electrolyte gating also, solid–solid devices exhibit promising magneto-ionic effects. It has been shown that using strongly reducing films as a capping layer on LSMO induces significant oxygen migration and enables the con- trol of magnetic properties. A subsequent long-time exposure to atmospheric oxygen can restore the pristine state of LSMO, and oxy- gen vacancies as induced by reducing capping layers are refilled with oxygen atoms.22Recently, a combined resistive switching experi- ment with a structural monitoring utilizing in situ transmission elec- tron microscopy (TEM) revealed a reversible phase transition from a half-metallic and perovskite (PV) structure into an insulating and brownmillerite (BM) state.23Moreover, an annealing study in vac- uum demonstrated a topotactic phase transition from BM to PV accompanied with a magnetic transition from antiferromagnetic to a ferromagnetic ordering and an insulator-to-metal transition.24 The prevalent multi-phase transition capabilities in LSMO open up valuable potential for future non-volatile memristive devices. Although, there are many reports about electric field- induced chemical and physical changes in LSMO, the driving mech- anism is not completely understood and needs further investigations utilizing advanced combined probing techniques. In this study, we employ local conductivity atomic force microscopy (AFM) and x-ray photoemission electron microscopy (PEEM) to investigate the correlation of magnetic and chemical information of electrically modified areas of LSMO thin films on a Nb:STO (100) substrate. Our data demonstrate a voltage-driven change in the Mn ion valency accompanied by a metal-to-insulator transition and a loss of magnetic ordering. The strength of the underlying microscale redox-processes can be tuned by the voltageamplitude and is reflected by significant spectral variations in the Mn L 3,2and the O K absorption edges. II. EXPERIMENTAL DETAILS 10 nm thick La 0.7Sr0.3MnO 3films were grown epitaxially on TiO 2-terminated Nb (0.5%):SrTiO 3(100) substrates by means of pulsed laser deposition (PLD) using a 50 W KrF excimer laser at a repetition rate of 5 Hz and an energy density of 3.3 J cm−2. For the film growth, the substrate was heated to a deposition temperature of 700○C (ramping heat 50○C/min) in an oxygen environment of pO2= 0.2 mbar. The distance between the substrate and target was set to 6 cm. In order to remove surface contaminations, the target was pre-ablated with 1800 pulses at a repetition rate of 10 Hz. After deposition, the chamber was filled with oxygen (p O2= 500 mbar) and the heater was immediately turned off. The specimen cooled down to room temperature within 1.5 h. The LSMO growth pro- cess was monitored by an in situ high-pressure reflection high- energy electron diffraction (RHEED) system, which confirmed a two dimensional layer-by-layer growth. Subsequently, the morphology and the crystal structure were characterized by atomic force microscopy (AFM) and x-ray diffrac- tion (XRD), revealing an epitaxial, single crystalline film and an atomically flat surface. Vibrating sample magnetometry (VSM) was used to investigate the saturation magnetization moment along the magnetic (110) easy axis, which amounts to M Sat= 3.4 ±0.34 μB/f.u. In preliminary experiments, we verified that the resistivity of LSMO can be changed by the application of a voltage to the sur- face, as shown in detail in Fig. S1. We find high (low) resistive states (HRS/LRS) in those areas where positive (negative) voltages were applied. These pre-studies allowed us to optimize the voltage ranges for electrical modification of the samples investigated by PEEM in the following. The electrical treatment of the pre-characterized LSMO sample was realized by utilizing local conductivity atomic force microscopy (LC-AFM) of a commercial variable temperature scanning probe microscope (VT-SPM, Scienta Omicron) equipped with a silicon ultrananocrystalline diamond (UNCD) cantilever (AppNano). The cantilever is coated with a boron-doped diamond layer and exhibits a tip radius of R Tip<150 nm. In order to remove any adsorbates, the sample was heated to 190○C for about 30 min in an oxygen envi- ronment of 0.1 mbar prior to the LC-AFM measurements. The base pressure during the switching experiment was set to 3 ⋅10−9mbar. The electrical modification of the LSMO sample was realized by scanning over 2 ×2μm2sized areas with constantly applied volt- ages, while the substrate was set to ground (Fig. 1). The tip scan started at one corner of the area, and the tip was moved back and forth using a total of 500 lines to cover the entire area. The scan speed was set to 5 μm/s. We wrote areas with different polarities between −3 V and +5 V. Hereinafter, we label these areas by referring to the respective voltages applied. In order to prevent unintentional modi- fications of the resistive states after the writing process, no read out measurements were performed afterward. After the LC-AFM writing procedure, the samples were exposed to air and brought to the UE56-1_SGM LEEM/PEEM beamline at BESSY II, where we performed soft x-ray PEEM. All presented linearly and circularly polarized XAS-PEEM images and APL Mater. 8, 111102 (2020); doi: 10.1063/5.0022150 8, 111102-2 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 1 . Setup of the experiment in UHV: Composition stack consisting of a con- ducting 5 ×5 mm2Nb:STO (100) substrate and a 10 nm thick La 0.7Sr0.3MnO 3 thin film. 2 ×2μm2sized squares are written by a scanning LC-AFM tip electrode under applied constant voltages. spectra were recorded in the total electron yield (TEY) mode and in the same field of view (20 μm). X-ray magnetic circular dichroism (XMCD) measurements were performed at room temperature and across the Mn L 3,2-edge (630 eV–670 eV) at a fixed angle of θ= 20○ between the beam axis and the surface plane of the specimen. As no external magnetic field was applied during the spectromicroscop- ical analysis, the XMCD spectra reflect a remanent ferromagnetic response. We did not observe any magnetic inhomogeneities for the LSMO sample. Hence, the pristine LSMO sample is assumed to be in a single domain state or in an anisotropic multidomain state with a sufficiently large net magnetization and a domain size below the resolution of our PEEM measurement ( ∼100 nm). Therefore, we conclude that the magnetic contrast as observed in our experiments is related to the electrical treatment. III. RESULTS AND DISCUSSION A. Photoemission electron microscopy (PEEM) To study the micro-scale redox processes present in the elec- trically modified areas of the LSMO thin film, we performed x-ray photoemission electron microscopy (PEEM). PEEM provides the capability of measuring all voltage-treated areas at the same time. This allows us to directly compare magnetic and chemical prop- erties with the electrical pre-treatment and set them into relation, since one does not have to deal with homogeneity issues and sam- ple preparation reproducibility. The respective images and spectra were recorded for the Mn L 3,2and O K-edges. Subsequently, we performed principle component analysis (PCA)25,26on the PEEM data. In PCA, the original dataset is described in a new orthogonal coordinate system, which is spanned along new principal compo- nent (PC) axes. The PCs are oriented along the largest, uncorrelated variances in the data. Spectral information is covered by the largest data variances, whereas smaller variances are generally related to noise. Hence, only a small number of PCs is sufficient to describe the relevant spectral features, and one can reconstruct the original data using merely the small set of relevant PCs, thereby improving the signal-to-noise ratio substantially. As has been demonstrated in Refs. 27 and 28, PCA can reveal small signals mostly camouflaged by noise. A further advantage of PCA is that the data variances can be related to certain pixels in the original PEEM image, thereby provid- ing a high spatial chemical resolution. A detailed description of theprincipal components and the respective PEEM data reconstructions for various photon energies of the absorption edges can be found in Figs. S2–S4. Figure 2(a) shows the PCA reconstruction of the original PEEM image for the Mn L-edge ( hν= 642.2 eV). The contrast of the areas modified with positive voltages raises with increasing ampli- tude and shows a significant chemical transition between 3 V and 4 V. From the previous resistive switching experiment shown in Fig. S1, we conclude that there is also a metal-to-insulator transi- tion between 3 V and 4 V. Thus, there is an obvious correlation between the enhanced chemical contrast and the transition from pristine low resistivity to high resistivity. The areas written with a negative voltage reveal only weak contrast levels without any obvious trend. For the cluster image and the corresponding spectra shown in Figs. 2(b) and 3–5, respectively, we made use of a cluster analysis, which is based on ortho -non-negative matrix factorization ( o-NMF) and group classification of similar Euclidian distance, as described in Ref. 29. In the work presented here, o-NMF identifies pixel groups where PEEM spectra are similar: chemical inhomogeneities cause spectral changes that lead to mean-square deviations between spec- tra. The o-NMF identifies those deviations and sorts spectra into dif- ferent groups, which are visualized in Fig. 2(b) as differently colored pixels. In the case of voltage-treated LSMO, four clusters with suffi- cient spectral resolution and the most physical sense (gray, blue, red, and green) are identified for the Mn L and O K-edges. The gray clus- ter corresponds to untreated sample regions and represents the pris- tine state. Isolated, scattered blue pixels seem to be located mainly within areas treated with negative voltage. While the gray cluster can be ascribed to a fully stoichiometric La 0.7Sr0.3MnO 3thin film, the blue cluster indicates a partially reduced state. Between −5 V and −3 V, increasing the voltage leads to a slight increase in the contri- bution of the blue cluster. However, this increase is hardly above the noise level. It has been shown that LSMO thin films initially tend FIG. 2 . Real space imaging of the Mn L 3-edge. (a) Reconstruction of the original PEEM image by the main principle components. The chemical contrast of the mod- ified regions at the Mn L 3-edge ( hν= 642.2 eV) reveals a significantly enhanced intensity for 4 V and 5 V (HRS). (b) False-color representation of the PCA PEEM image in (a) indicating pixels of high spectral similarity as obtained by orthogonal non-negative matrix factorization ( o-NMF). The mean spectra determined from the average over all spectra within the colored areas (gray, blue, red, and green) are plotted in Figs. 3 and 4 (top) using the same color code. APL Mater. 8, 111102 (2020); doi: 10.1063/5.0022150 8, 111102-3 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 3 . Top panel: x-ray absorption spectra of the Mn L 3,2-edge averaged over the cluster regions, as shown in Fig. 2(b) using the same color code. The areas modified with a positive tip polarization show an increasing Mn2+contribution with increasing voltage. Bottom panel: reference spectra indicating Mn2+, Mn3+, and Mn4+. Adapted from Ref. 31. to show a certain Mn2+contribution at the surface30due to sym- metry breaking effects, which could explain the observed scattered blue pixels. Hence, we conclude that the application of a negative tip bias does not significantly affect the oxidation state, since the pristine state is already in a conductive and an oxygen saturated state. FIG. 4 . Top panel: x-ray absorption spectra of the O K-edge averaged over the cluster regions, as shown in Fig. 2(b) using the same color code. The areas mod- ified with a positive voltage reveal a decreased O 2 p–Mn 3 dhybridization peak with increasing voltage. Bottom panel: reference spectra indicating Mn2+, Mn3+, and Mn4+. Adapted from Ref. 31. FIG. 5 . XMCD spectra of the Mn L 3,2-edge extracted and averaged over the color- coded cluster regions defined by the o-NMF analysis (inset). The XMCD signal decreases with increasing positive voltage and reveals a significant ferromagnetic contrast between 3 V and 4 V. The chemical contrast significantly changes between 2 V and 5 V. The area treated with 3 V shows a homogeneously distributed contribution of the blue cluster. The 4 V area consists of three clus- ters, which form a concentrical alignment of rectangles. From the outermost region to the center, the clusters are encoded in blue, red, and green, respectively. In the case of 5 V, we observe a similar concentrical arrangement of clusters, but with a significantly larger contribution of green in the center. Figure 3 shows a direct comparison of the according spectra recorded at the Mn L 3,2-edge, where each color is assigned to the respective cluster as defined in the o-NMF analysis [Fig. 2(b)]. We observe an emerging low energy peak at h ν= 642.2 eV from gray (stoichiometric La 0.7Sr0.3MnO 3) to green. A significant increase in the peak is obtained between the blue and the red spectrum, notic- ing that this accompanies the metal-to-insulator transition. The low energy feature that is separated by 1 eV from the main peak can be ascribed to the appearance of Mn2+cations.30,32We also note that the main peak of the red spectrum and green spectrum, located at hν= 644 eV, reveals an apparent shift of the maximum intensity Imaxtoward lower photon energy [see the inset of Fig. 3]. The Mn L-edges of MnO, Mn 2O3, and MnO 2reference samples are shown in Fig. 3. They indicate the three valence states of Mn2+, Mn3+, and Mn4+31and illustrate the possible components, which might form the Mn L 3,2-edge spectra of a LSMO thin film after voltage treatment. In particular, the MnO reference spectrum illustrates the origin of the low energy peak of the measured spectra at h ν= 642.2 eV, which is assigned to Mn2+. The arising Mn2+contribution is a consequence of oxygen vacancies, whose concentration increases with increasing positive voltage. Applying a positive voltage induces a migration of the negatively charged oxygen ions to the surface. Subsequently, a surface exchange process leads to the release of the oxygen, which is driven by the gradient of the chemical potential of oxygen across the LSMO/vacuum interface.33The exchange process of the oxygen leads to a significant valence change from a Mn3+/4+to a Mn2+/3+ dominated state. The concentrical alignment of the clusters, shown in Fig. 2(b), is likely caused by the gradient of the applied field, the scanning APL Mater. 8, 111102 (2020); doi: 10.1063/5.0022150 8, 111102-4 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm pattern, and the scanning speed. The difference in exposure to the electric field is known to influence magneto-ionic effects.34 A variation of the oxygen vacancy concentration should also impact the O K-edge spectra (Fig. 4), particularly on the peak located at hν= 532.3 eV, which is assigned to the hybridization state between the O 2 pand the Mn 3 dstates.35,36As it is obvious from the reference spectra in Fig. 4 (bottom), a decrease in the oxidation state of the Mn leads to a decrease in the O 2 p–Mn 3 dhybridization peak. The gray spectrum reveals the largest intensity as it represents the oxygen saturated pristine LSMO thin film. The peak inten- sity decreases from blue to green, indicating less unoccupied elec- tronic states of the manganese and corroborating the increasing oxy- gen vacancy concentration for increasing positive voltages. In line with the metal-to-insulator transition and the results from the Mn L-edge, we observe a considerable modification of the spectral inten- sities between the blue spectrum and the red spectrum. Our obser- vations of significant changes in the manganese valence state, the incorporation of oxygen vacancies, and the metal-to-insulator tran- sition already strongly suggest that the magnetic properties may very likely be affected due to a diminished double exchange coupling. B. X-ray magnetic circular dichroism In order to gain a deeper understanding of the electrically induced change in the magnetic properties and altered exchange interactions, spin-sensitive and element-selective XMCD-PEEM measurements of the Mn L 3,2-edge will scrutinize the interplay between valence change, resistivity, and magnetic ordering. The XMCD signals of each cluster exhibit the characteristic ferromagnetic signature of perovskite LSMO thin films at room tem- perature,37,38which originates from the double exchange interac- tion between Mn3+and Mn4+. The gray XMCD spectrum reveals the largest XMCD signal and represents the pristine ferromagnetic state. The strength of the XMCD signal decreases with increasing positive voltage. From this, we deduce a clear dependence of the fer- romagnetic properties on the oxygen vacancy concentration and the corresponding Mn valence state. A straightforward explanation for this phenomenon is a disturbed double exchange interaction, which has two reasons: In LSMO, with a Sr doping concentration of 30%, oxygen vacancies act as electron donors (causing a valence change in order to preserve charge neutrality), weakening the double exchange interaction and promoting antiferromagnetic coupling. Moreover, the delocalization of the e gelectron between the Mn3+and Mn4+ cations is crucial in order to stabilize a parallel and ferromagnetic alignment of the respective t 2gelectron spin states. Since the delo- calization of the e gelectron is mediated by an interlinking oxygen anion between Mn3+and Mn4+, the exchange process is perturbated due to a missing oxygen anion. Thus, an increasing concentration of oxygen vacancies leads to a gradual transition from a ferromagnetic through antiferromagnetic to a nonmagnetic state. Besides a perturbation of the driving double exchange mech- anism, there could be other explanations for the decreasing ferro- magnetic signal. For example, recent studies on LSMO reported on a structural phase transition from a perovskite (PV) ABO 3to a brown- millerite (BM) ABO 2.5structure driven by the application of an elec- tric field to the thin film surface23and by an annealing procedure in vacuum.24The latter study also revealed a concomitant magnetic phase transition from a ferromagnetic (PV) to an antiferromagneticordering (BM) with a Néel temperature of T N= 30 K and a para- magnetic state at room temperature. Thus, we performed XAS also on a BM reference sample, which is discussed in the supplementary material (see Fig. S8). While the XMCD results could support the picture of an existing BM phase in the HRS, the absorption spectra are not evidential. The green and red spectra at the Mn L 3,2-edge (Fig. 3) would corroborate a BM phase. The spectra we find for the green and red cluster at the O K-edge (Fig. 4), however, are not consistent with the reference spectra for BM (Fig. S8), rendering it unlikely that the HRS state corresponds to the BM phase. For clari- fication, a complementary analysis of the modified regions by trans- mission electron microscopy may unveil possible BM structures and will be part of our future investigation. Under well-chosen experi- mental conditions (proper thickness of the cut lamella and probing the right crystal orientation), prevalent BM phases embedded in a perovskite matrix could provide sufficient transmission contrasts in order to identify respective filaments.23 IV. CONCLUSION We performed a combined element selective and spatially resolved XAS and XMCD study of electrically modified areas in La0.7Sr0.3MnO 3thin films. Our results demonstrate the direct inter- play between resistivity, chemical composition, and magnetic order- ing driven by an oxygen exchange process across the LSMO film surface. Significant chemical modifications are observed in the high resistive state, where the incorporation of oxygen vacancies leads to a distinct valence change from Mn3+/4+to Mn2+/3+. Furthermore, a direct correlation between oxygen deficiency and a degradation of the ferromagnetic properties is found. In this light, these results provide novel insight into the vacancy-driven magneto-ionic control of magnetoelectric oxides. Moreover, they open up novel routes toward a multiphase-control of physical and chemical properties in complex oxides and novel ionotronic devices. SUPPLEMENTARY MATERIAL See the supplementary material for a discussion of the elec- tronic properties, a detailed description of the principal component analysis, information about the XMCD analysis, and a comparison of the brownmillerite and perovskite x-ray absorption fingerprints. ACKNOWLEDGMENTS This work was funded by the Deutsche Forschungsgemein- schaft (DFG) within the Grant No. SFB 917. M. Glöß acknowledges for the support from the Emmy Noether program of the DFG. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1S. V. Kalinin and N. A. Spaldin, “Functional ion defects in transition metal oxides,” Science 341, 858–859 (2013). 2M. A. Peña and J. L. G. Fierro, “Chemical structures and performance of perovskite oxides,” Chem. Rev. 101, 1981–2018 (2001). APL Mater. 8, 111102 (2020); doi: 10.1063/5.0022150 8, 111102-5 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm 3S. 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5.0019438.pdf

AIP Conference Proceedings 2265 , 030432 (2020); https://doi.org/10.1063/5.0019438 2265 , 030432 © 2020 Author(s).Influence of carrier concentration and mobility tuned magneto resistance of Bi1−xSbx nanocrystals Cite as: AIP Conference Proceedings 2265 , 030432 (2020); https://doi.org/10.1063/5.0019438 Published Online: 05 November 2020 Hasan Afzal , R. Venkatesh , and V. Ganesan ARTICLES YOU MAY BE INTERESTED IN Negative differential resistance and magnetotransport in Fe 3O4/SiO 2/Si heterostructures Applied Physics Letters 114, 242402 (2019); https://doi.org/10.1063/1.5092872 Magnetic field induced resistivity upturn and plateau in antimony crystal AIP Conference Proceedings 2265 , 030437 (2020); https://doi.org/10.1063/5.0017445 Ab-initio study of thermoelectric properties of Co 2XGa (X = Mn, Mo, Pt) AIP Conference Proceedings 2265 , 030449 (2020); https://doi.org/10.1063/5.0016802Influence of Carrier Concentration and Mobility Tuned Magneto Resistance of Bi 1−xSbx Nanocrystals Hasan Afzala), R. Venkatesh and V. Ganesan. Low Temperature Laboratory, UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore 452 001, M.P, India a)Corresponding author: hasana@csr.res.in Abstract: In the present work, influence of Sb doping on magneto transpo rt properties of the topological insulator Bi 1−xSbxin its nano crystalline form, prepared by microwave assisted so lvothermal synthesis, is investigated. At a fixed temperature of 4K, for x=0.06, a posit ive linear magneto resistance (PLMR) at low magnetic fields was observed, which trans forms towards a parabolic class ical MR curve with a further increase in Sb molar ratio x. The slope of MR was found to increase with x, the devi ation from linearity was found to be attributable to a monotonic decrease in average ca rrier mobility as well as a systematic increase in the carrier concentration (electron- hole) asymmetry ratio. This elucidates on a possible mechanism for tuning of MR by varying carrier concentration as well as mobility and hence is likely to be instrumental in deve lopment of MR dependent devices. INTRODUCTION Bi 1−xSbxalloys arewell known narrow band gap semiconductors, However, s tudies on topological phases in condensed matter physics have shown Bi 1−xSbxto behaveas a “topological insulator” belonging to a nontrivial Z2 topological class for (0.07< x<0.22) [1] . It has been confirmed by ARPES measurements. Its ambipolar natu re owing to the presence of both electrons and holes as charge carriers is also reported [2]. Topological insulators (TI) are a class of materials having ins ulating states in the bulk but robust metallic surface states protected by Spin Orbit Coupling and Time Revers al Symmetry [3], resulting in these states being non-dissipative. Also such materials are known to exhibit posit ive linear magneto resistance PLMR even up to high magnetic fields [4]. Consequently, these materials are pre dicted to be useful for applications in spintronics. However, the surface state conduction is likely to be influence d by bulk conduction carriers, the effect of which is expected to be reduced by lowering sample dimensions so that the bulk band gap is enhanced to the extent that the fermi level does not reach the conduction band. Thin f ilms and nano structuring are two viable options to bring about reduction in dimension. Owing to the reduction i n crystallite size due to nano structuring, surface state conduction is likely to be enhanced as compared to bulk s ingle crystals. Consequently, in this work, the influence of the variati o n o f S b m o l a r r a t i o x o n m a g n e t o r e s i s t a n c e (MR) of Bi 1−xSbxnano particles has been pursued, with the intention of studying the effect of change in x on the nature of MR and the underlying mechanism. The nano particles w ere synthesized using micro-wave assisted solvo thermal method as it is quick and effective in production of nano particles as well as tuning of particle size. SAMPLE PREPARATION AND CHARACTERIZATION Bi1-xSbx nanocrystals having different values of relative molar ratio o f Bi and Sb ranging from (x= 0.00 to 1.00) were synthesized by microw ave assisted solvo thermal meth od using a single-mode variable-power 300 Watts, CEM microwave oven, taking BiCl 3 and SbCl 3 as precursors and NaOH and Oleylamine (OAM) as reducing agents with ethylene glycol as the solvent. The mixtur e was at first, subjected to vigorous stirring as well as heating for 1 hour to remove oxygen and moisture respec tively, followed by heating at 195OC for one hour in a closed chamber microwave oven. The cleaning of the pr ecipitate obtained was done using ethanol, acetone and DI water for a multiple number of times. The sample s with varied Sb molar ratio mentioned in this paper are labelled as “BS6” for un doped Bi 0.94Sb0.06, “BS9” for Bi 0.91Sb0.09and"BS38" for Bi 0.62Sb0.38 for convenience. DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030432-1–030432-4; https://doi.org/10.1063/5.0019438 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030432-1The structural and morphology of n anocrystals were characterize d by XRD and FESEM respectively. The crystal structures were determined from powder X-ray Diffractio n XRD, performed using Bruker D8 Advance X-ray diffractometer (CuK-α) and morphology has been characteri zed using a FEI made “Nova NanoSEM 450” field-emission scanning electron microscopy (FESEM) XRD images showed single phase formation of Bi 1-xSbx with rhombohedral R-3mH structure. FESEM images showed hexagona l shaped nanocrystals having size of approximately 1µm (while EDX measurements showed Sb inclusion p ercentage in BS6, BS9, BS38 as 6, 9 and 38%respectively. The nanocrystal size and shape as well as morp hology is not seen to vary much in the reported samples as seen from (Fig.1b) H ence, the contribution of nanopa rticle size, shape and morphology in the variation of magneto transport properties of the reported un do ped and Sb and Co doped samples can be neglected. RESULTS AND DISCUSSION Resistivity measurements were p erformed on different samples o f B i 1−xSbxfor (x=0.06, 0.09, 0.38) as a f u n c t i o n o f t e m p e r a t u r e a n d m a g n e t i c f i e l d d o w n t o 2 K u s i n g a PPMS system from QD, USA. From the resistivity versus magnetic field (R vs B) data corresponding t o a particular temperature of 4K, the magneto resistance MR,given by (R(B)–R(0) / R(0) x 100%), where R(B) an d R(0) are the resistances at non zero and zeromagnetic field respectively, w as calculated for the three s amples BS6, BS9 and BS38. The corresponding MR vs B plots of the three samples are depicted in the fig 2a. Also, the plots were fitted using a power law equation to ascertain the B dependence of the slope of MR (fig 2a). The MR corresponding to BS6(Fig. 3), shows, a positive linear magneto resistance (PLMR) at low magne tic fields (fig 2a). With further increase in Sb molar ratio x, the slope of MR deviates from linearity (shown i n fig 2a, also displayed in table 1). In order to explore the mechanism behind the deviation from lin earity, the MR vs B plots were fitted using the two band model [5] for two types of charge carriers (electr ons and holes) owing to the known existence of oppositely charged carriers in the sample [2]. In such a situat ion, the MR is given by ܴܯ ൌሺஜା ஜ ሻమାሺஜା ஜ ሻሺஜା ஜ ሻஜஜ మ ሺஜା ஜ ሻమାሺିሻమஜమஜ మమെ1 ( 1) 0210420630 098196294 20 40 60 80099198297x=0.06 x=0.09 x=0.38intensity (arbitrary units) 2 theta0210420630 098196294 27.0 27.2 27.4 27.6 27.8 28.0099198297x=0.06 x=0.09 x=0.38intensity (arbitrary units) 2 theta FIGURE 1a.) (top left) XRD pattern of BS6, BS9 and BS38 (top to bottom) b)(top above) FESEM images of BS6, BS9 and BS38 (left to right) FIGURE 2a.) (top left) MR vs B of BS6, BS9 and BS 38 along with power law fit b)(top right) MR vs B of BS6, BS9 and BS38 along with two carrier model fit 030432-2where μ,μ ,݊,݊,are the electron and hole mobility and concentration respective ly, were used as parameters while MR and B were the variables. To simplify the a bove equation, the assumption ( μൌμ ሻ was employed in eqn 1 as suggested and used by Yang et al [5]. The simplified version [5] of equation 1 can be written as ܴܯ ൌଵାஜమమ ଵାమஜమమെ1 (2) Where µ is the average mobility of electrons and holes, assumi ng the mobility to be equal for the two types of carriers and k is the electr on hole concentration asymmetry ratio given by ݇ൌ|ି| |ା| (3) The plots fitted with eqn 2 are shown in fig 2b. The values of µ and k obtained from the fits corresponding to the samples BS6, BS9 and BS38 are displayed in table 1. Also th eir values were plotted against the deviation of magneto resistance MR from linear behaviour given by the relati on D (= n-1) where n is the index of the power law fit to the MR vs B plot. The value of 1.0 represents linear behaviour. The results found were interesting. The deviation from linearity D was found to undergo a monotonic decrease with increase in average carrier mobility µ(fig 3a) and as well as a systematic decrease with in crease in the carrier concen tration (electron-hole) asymmetry ratio k (fig 3b) over the range of Sb molar ratio x f rom 6 % right up to 38 %. Also k and µ were found to be inversely proportiona l to each other (fig 3c). Thus positive linear magneto resistance PLMR (power n=1 and deviation from linearity D =0) corresponds to highest c arrier mobility and lowest carrier concentration asymmetry ratio k. Thus single type of charge carriers appears to facilitate linear MR as well as high carrier mobility whereas the presence of two types of charge carriers s hows reduced mobility as well as directs MR from linearity towards a classi cal parabolic nature. Thus this outcome highlights the inter relationship between MR, carrier mobility and carrier concentration and hence provid es a possible route for inventing a mechanism for tuning of MR by varying carrier mobility and carrier concen tration as well as and is thus likely to be instrumental in development of MR dependent devices. 0.82 0.83 0.84 0.850.00.10.20.30.40.50.60.7D=n-1 carrier conc asymmetry ratio KBS6BS9BS38 0 . 81 . 01 . 21 . 41 . 61 . 82 . 02 . 20.00.10.20.30.40.50.60.7D=n-1BS38 BS9 BS6 average carri er mobility ( 104cm2V-1s-10 . 81 . 01 . 21 . 41 . 61 . 82 . 02 . 20.8150.8200.8250.8300.8350.8400.8450.8500.855 (104cm2V-1s-1BS6BS9BS38 K TABLE 1. n, D, k and µ values of BS6, BS9 and BS38 SAMPLE CODE PARAMETERS P O W E R O F MR(n) D=n-1 K µ(104cm2V-1s-1) BS6 1.03 0.03 0.8213 2.02057 BS9 1.31 0.31 0.8377 1.41895 BS38 1.67 0.67 0.8523 0.88176 CONCLUSIONS In the present work, influence of variation of Sb doping on mag neto transport properties of the topological insulator Bi 1−xSbxin its nano crystalline form, prepared by microwave assisted so lvothermal synthesis, is investigated. At a fixed temperature of 4K, for x=0.06, a posit ive linear magneto resistance (PLMR) at low magnetic fields was observed, which gets modified into a parabo lic nature by a further increase in Sb molar ratio x. The power of MR w.r.t. magnetic field was found to increase with x and deviate from linearity, the deviation from linearity was found to be attributable to a monotonic decr ease in average carrier mobility as well as a systematic increase in the carri er concentration (electron-hole ) asymmetry ratio. This outcome guides to a FIGURE 3a.) (top left) D vs k of BS6, BS9 and BS38 b)(top right) D vs µ of BS6, BS9 and BS38 c)(top right) k vsµ of BS6, BS9 and BS38 030432-3possible mechanism for tuning of MR by varying carrier concentr ation as well as mobility and hence is likely to be instrumental in development of MR dependent devices. ACKNOWLEDGMENTS Authors thank Director, Dr. A.K. Sinha, UGC-DAE CSR Indore, Dr. D.M.Phase, Mr. P. Saravanan, other members of LTL, & DST for their support. REFERENCES 1. F.Nakamura et al,Phys.Rev.B 84, 235308,1-8 (2011). 2. Jin et al. Appl. Phys. Lett 101, 053904 (2012) 3. H. Zhang, C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. C. Zhang , Nature physics, 5 , 438-442(2009). 4. Wang, et al Phys. Rev. Lett. 108 , 266806 (2012) 5. Yang et al. SciRep 6, 26903, DOI: 10.1038/srep26903 030432-4

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AIP Advances 10, 095018 (2020); https://doi.org/10.1063/5.0022525 10, 095018 © 2020 Author(s).Origin of band inversion in topological Bi2Se3 Cite as: AIP Advances 10, 095018 (2020); https://doi.org/10.1063/5.0022525 Submitted: 22 July 2020 . Accepted: 04 September 2020 . Published Online: 17 September 2020 Stephen Chege , Patrick Ning’i , James Sifuna , and George O. Amolo 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 Catalog of magnetic topological semimetals AIP Advances 10, 095222 (2020); https://doi.org/10.1063/5.0020096 Enhancement of spin signals by thermal annealing in silicon-based lateral spin valves AIP Advances 10, 095021 (2020); https://doi.org/10.1063/5.0022160 Topological thermoelectrics APL Materials 8, 040913 (2020); https://doi.org/10.1063/5.0005481AIP Advances ARTICLE scitation.org/journal/adv Origin of band inversion in topological Bi 2Se3 Cite as: AIP Advances 10, 095018 (2020); doi: 10.1063/5.0022525 Submitted: 22 July 2020 •Accepted: 4 September 2020 • Published Online: 17 September 2020 Stephen Chege,1,a) Patrick Ning’i,1 James Sifuna,1,2,b) and George O. Amolo1 AFFILIATIONS 1Materials Modeling Group, School of Physics and Earth Science, The Technical University of Kenya, 52428-00200 Nairobi, Kenya 2Department of Natural Science, The Catholic University of Eastern Africa, 62157-00200 Nairobi, Kenya a)voshtilimbugua@gmail.com b)Author to whom correspondence should be addressed: jsifuna@cuea.edu ABSTRACT Topological materials and more so insulators have become ideal candidates for spintronics and other novel applications. These materials portray band inversion that is considered to be a key signature of topology. It is not yet clear what drives band inversion in these materials and the basic inferences when band inversion is observed. We employed a state-of-the-art ab initio method to demonstrate band inversion in topological bulk Bi 2Se3and subsequently provided a reason explaining why the inversion occurred. From our work, a topological surface state for Bi 2Se3was described by a single gap-less Dirac cone at⃗k= 0, which was essentially at the Γpoint in the surface Brilloiun zone. We realized that band inversion in Bi 2Se3was not entirely dependent on spin–orbit coupling as proposed in many studies but also occurred as a result of both scalar relativistic effects and lattice distortions. Spin–orbit coupling was seen to drive gap opening, but it was not important in obtaining a band inversion. Our calculations reveal that Bi 2Se3has an energy gap of about 0.28 eV, which, in principle, agrees well with the experimental gap of ≈0.20 eV–0.30 eV. This work contributes to the understanding of the not so common field of spintronics, eventually aiding in the engineering of materials in different phases in a non-volatile manner. ©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/10.1063/5.0022525 .,s I. INTRODUCTION We appreciate the fact that topological materials form a broad class of materials ranging from insulators,1superconductors,2and semi-metals.3To date, the topological idea in materials is still scanty and perhaps the best understood topological materials are topolog- ical insulators,4which form the basis of our work herein. Topolog- ical insulators in bulk have insulating properties but often exhibit conducting edge states on the boundary protected by time rever- sal symmetry.5An important aspect of a topological insulator is band inversion, which results from the crossing of the valence band and conduction band of different parities and a gap opening due to spin–orbit coupling.6 As we focus on novel topological insulators, it is important to shed light on their intrinsic character. These insulators are char- acterized by a Hamiltonian that is not, in any way, adiabatically connected to the atomic limit. This implies that when tuning exter- nal parameters to change the Hamiltonian, the process is extremely slow such that the material remains in the ground state throughout.7 A typical example of the atomic limit may be a case of diamond.Imagine that we pull the carbon atoms extremely far apart, yielding isolated carbon atoms.8If we use diamond, it is possible to do this process without closing the gap implying that diamond is adiabati- cally connected to the atomic limit. This outcome is expected since diamond is a normal insulator.9 The same process can be applied to Bi 2Se3, which in crystalline form is a topological insulator with a gap of 0.27 eV–0.30 eV.10–12 If the previous procedure is repeated such that we pull the system apart to reach the atomic limit, it is impossible to do it without closing the bulk gap. This implies that Bi 2Se3is not adiabatically con- nected to the atomic limit and thus suits to be called a topological insulator.11 The big question here is as follows: “What is the origin of such anomaly in topological insulators?” It is important to note that the electron wave function twists as you cross the Brillouin zone in a topological insulator.13This is, however, not true in an atomic limit, and so the twist needs to be undone when getting to such a limit. This is what is achieved by gap closure. Equally important is the fact that these twists can be labeled by topological invariants14whose mathematical form is highly dependent on the type of topology but AIP Advances 10, 095018 (2020); doi: 10.1063/5.0022525 10, 095018-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv are related to the Berry phase-like quantities15that determine the evolution of the wave function as one crosses the Brillouin zone. In topological three-dimensional insulators, the topological invariant is usually a set of four numbers16that take one of the two val- ues (Zclassification) and can be calculated by the Wannier charge center evolution across the Brillouin zone, as described in Refs. 5 and 17–19. Alternatively, if a system possesses inversion symme- try, the topological invariant is obtained by calculating the parity of eigenstates at special points in the Brillouin zone, as explained in Ref. 18. Apart from the topological insulators, there is also another interesting class of insulators known as Chern insulators.20These are two-dimensional materials with topological and magnetic orders, as explained by Garrity and Vanderbilt.20In this class of materials, it is worth noting that the topological invariant is the Chern number attained by an integration of the Berry curvature over the Brillouin zone. First-principles calculations have been employed in calculat- ing topological invariants21–23by involving Berry phase-like quan- tities. Many studies have employed Wannier functions and pack- ages that implement such calculations and have interfaces to some density functional theory (DFT) packages such as Wannier Tools.24 There are also numerous databases such as the Topological Materi- als Database25that can aid in the topological classification of many materials based on semi-local DFT. If one accesses such a database, it forms a perfect starting point to figure out the topological order of a material. Caution has to be employed if higher levels of theory are applied such as hybrid functionals and GW. This application may easily lead to different results from those in the existing databases subsequently necessitating further analysis. Having known that band inversion is a key ingredient of any topologically non-trivial material,26we will, therefore, focus on the fundamental definition of band inversion, how to recognize it in a band structure, and making inferences if band inversion is observed in a material. Our work will mainly focus on the topological insu- lator, Bi 2Se3,and we show that indeed spin–orbit coupling is not a causative agent of band inversion. This paper is organized as follows: In Sec. II, we account for the technicalities used in our calculations that may be critical for future reproducibility. In Sec. III, we present finer details of Bi 2Se3with and without spin–orbit coupling (SOC). We present the structural properties of Bi 2Se3in comparison to other works (Sec. III A), the various orbital contributions in Bi 2Se3and their specific locations in the band-structure (Sec. III C), the band-structure of Bi 2Se3(Sec. III B), the demonstration of band inversion in Bi 2Se3with and without spin–orbit (Sec. III D), and the calculation of topological invariants (Sec. III E). Conclusion and future perspectives of this article are highlighted in Sec. IV. II. CALCULATION DETAILS We carried out scalar relativistic and fully relativistic first- principles calculations employing the density functional theory27,28 formalism as implemented in the SIESTA method.29We used a single- ζbasis set for the semi-core states and double- ζplus polariza- tion for the valence states of all the two atoms. Exchange and correlation functionals were treated within the generalized gradi- ent approximation.30Core electrons were replaced by ab initionorm-conserving fully separable,31Troullier–Martin pseudopoten- tials.32InSIESTA , the one-electron eigenstates are expanded in a set of numerical atomic orbitals. A Fermi–Dirac distribution with a tem- perature of 0.075 eV was used to smear the occupancy of the one- particle electronic eigenstates. To get a converged system, a two-step procedure was performed: First, to relax the atomic structure and the one-particle density matrix with a sensible number of k-points (9 ×9 ×5 Monkhorst–Pack33k-point mesh); second, freezing the relaxed structure and density matrix, a non-self consistent band structure calculation was performed with a much denser sampling of 60 ×60 ×60 in the real space integrations, and a uniform grid with an equivalent plane-wave cutoff of 600 Ry was used. In this system, we equally computed the fat bands ( Fi,n,σ,⃗k), which are defined as the periodic equivalent of the Mulliken population, Fi,n,σ,⃗k=ΣjCi,n,σ,⃗kCj,n,σ,⃗kSi,j,⃗k, (1) where Ci,n,σ,⃗kand Si,j,⃗kare the orbital coefficients and the overlap matrix elements in that order, respectively. The indices iandjdenote basis functions, nis the band index, σis the spin index, and⃗kis a reciprocal vector in the Brillouin zone. Since Bi 2Se3has inversion symmetry,6we calculated the topo- logical invariants from the symmetry of Bloch functions at the Brillouin zone points by employing the newly developed interface between SIESTA and WANNIER 90 .34 In this calculation, all the atomic coordinates were relaxed until the forces were smaller than 0.01 eV/Å, and the stress tensor com- ponents were below 0.0001 eV/Å3. For spin–orbit coupling (SOC) in SIESTA , we employed the recipe described in Refs. 35 and 36. III. RESULTS AND DISCUSSIONS In Fig. 1, we illustrate Bi 2Se3with a rhombohedral crystal struc- ture. In addition, it has a space group D5 3d(R3m). Usually, it is dis- played in quintuple layers with an Se 1–Bi 1–Se 2–Bi 1′–Se 1′arrange- ment. Bi 2Se3portrays layered structures with a triangular lattice within its single layer. It is important to note that this material con- sists of five-atom layers arranged along the z-direction, known as quintuple layers (QLs). A QL consists of five atoms with two equiv- alent Se atoms, two equivalent Bi atoms, and a third Se atom. Some scholars11denote the two Se atoms as Se 1and Se 1′, the two Bi atoms as Bi 1and Bi 1′, while the third Se as Se 2. A. Structural properties of Bi 2Se3 It is usually important to perform geometrical relaxations in any first-principles calculation to avoid errors and for an accurate prediction of other dependent quantities such as the electronic prop- erties that are extremely sensitive. Table I presents the fully relaxed lattice parameters of Bi 2Se3with a rhombohedral crystal. Our cal- culations, with and without taking into account the effects of spin– orbit coupling, are in good agreement with the available theoretical and experimental results. It is clear that our lattices are slightly over- estimated. This is a well-known occurrence when employing GGA, as described in Ref. 37. When using bare GGA, we found that the lattices aand c were overestimated by 1.47% and 0.73%, respectively, in comparison to the experimental values. On employing spin–orbit coupling, we AIP Advances 10, 095018 (2020); doi: 10.1063/5.0022525 10, 095018-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . The crystal structure of Bi 2Se3.24The green (purple) balls represent Se atoms (Bi atoms). The bond angles are α=β= 90○andγ= 120○. realized that it increased the error in lattice parameters. This kind of discrepancy can easily be corrected by using van der Waals (vdW) corrections, as employed in Ref. 38. The inclusion of vdW in such calculations will definitely give matching experimental results as far as the lattices are concerned. B. Electronic band structure of Bi 2Se3 We calculated the band structure of Bi 2Se3along high- symmetry points Γ-Z-F-Γ-L in the first Brillouin zone. In Fig. 2(a), we set the Fermi level at 0 eV, and it is represented by the red dot- ted line. For a plain GGA calculation, an energy separation at Γof 0.05 eV is observed between the top of the valence band and the bot- tom of the conduction band. This is also in agreement with 0.08 eV obtained by Aguilera and co-workers40when using LDA.41This hints to the fact that Bi 2Se3is a direct bandgap material. This energy gap was severely underestimated due to the DFT limitation that arises from the approximations used in the exchange and correlation functional. When we considered the effect between the orbital motion and electron spin as shown in Fig. 2(b), we found the energy gap to be of TABLE I . Lattice parameters of Bi 2Se3in Å. Reference a c This work (GGA) 4.18 28.85 This work (GGA + SOC) 4.14 28.96 Theory114.07 29.83 Experiment394.12 28.64 FIG. 2 . Band structure of Bi 2Se3. (a) shows almost gap closure, while (b) shows gap opening, as a result of SOC. The Fermi has been set to zero in both cases. As expected, it is very clear that band inversion occurs at the Γpoint, as shown in Fig. 5. the order of 0.28 eV but no longer direct as also reported in the liter- ature.40The gap was found between ZandF. This is consistent with the previous DFT and experimental findings, as shown in Table II. From Fig. 2(b), we show that spin–orbit coupling has a significant impact on the band structure and alters the bandgap at the Γpoint. Since we no longer have a direct gap, it is noted that SOC induces gap opening between the conduction and the valence band, as shown in Fig. 4 and elaborated in Fig. 5. The inversion seen illustrates non- trivial topology in Bi 2Se3due to the band inversion in momentum space. Another exciting feature in this material is that the topological surface states for this crystal are seen to be simple and are described by a single gap-less Dirac cone at the⃗k= 0,Γpoint in the surface Brilloiun zone.40 One important feature in Fig. 2(b) is the lifting of the degen- eracy of one-electron energy levels in the band structure. This has been seen to be significantly dominant at the Γpoint. Without the employment of spin–orbit coupling as depicted in Fig. 2(a), we found doubly degenerated bands. This type of degeneracy is usually broken by a spin-dependent term in the Hamiltonian, as described in Refs. 35 and 36, of the SIESTA method. Apart from the increased bandgap, the bandwidth also increased. TABLE II . Bandgap of Bi 2Se3in eV in relation to other studies. Reference E gap(eV) This work (GGA + SOC) 0.28 Theory110.30 Experiment100.30 Experiment120.27 AIP Advances 10, 095018 (2020); doi: 10.1063/5.0022525 10, 095018-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv C. Active orbitals in Bi 2Se3 An investigation of the total and partial densities of states always helps to clarify the nature of the bandgaps. In principle, we get information about the origin of various orbitals in the band structure. From Fig. 3, it can clearly be seen that the s-orbitals of Bi and Se contribute majorly in the core states, while the p-orbitals of Se are seen to dominate the valence states. A close look at the conduction states shows domination by the p-orbitals of Bi. At the Fermi level, the p-orbitals of Se and Bi are fully responsible for the properties seen herein. A similar orbital arrangement in Bi 2Se3was reported in Ref. 38. There is also a very strong hybridization in the Sepand Bi pstates. A second look at Fig. 3 shows that the spin–orbit coupling sup- presses the density of states. The total density of states from ≈30 states/eV in the scalar relativistic scenario decreases to ≈3 states/eV in a fully relativistic case. This should not raise an alarm since similar observations were seen in Ref. 42. One common effect of spin–orbit coupling is that it always tends to lower the energy of a system.43 D. Band inversion in Bi 2Se3 1. Band inversion with spin–orbit coupling It is well known that in Bi 2Se3, the topological index distin- guishing ordinary insulating from topological insulating behavior is controlled by band inversion at the Γpoint,44as shown in Fig. 5. The goal of this work was to demonstrate band inversion in Bi 2Se3and, thereafter, seek to answer the question of the origin of band inver- sion. The usual strategy to get a topological insulator, and already used by Kane and Mele,16is to induce the wave function twist using spin–orbit coupling (SOC). This is known for Bi 2Se3but may be present in other materials. Without spin–orbit coupling (SOC), as shown in Fig. 5(a), the “conduction band” in green is majorly made of Bi p zorbitals with FIG. 3 . Total and partial densities of states of Bi 2Se3: (a) without SOC and (b) with SOC. The Fermi level, in both cases, has been set to zero. In both (a) and (b), we see that the active orbitals are pfor Se and Bi. The sorbitals for Se and Bi are slightly deep in energy, and it is expected that they will have very little contribution in the conduction band.a little admixture of Se p zand the “valence band”, shown in red, is made of Se p zorbitals. However, due to DFT underestimation, the bands tend to overlap, giving the system a gap of 0.05 eV, as explained in Sec. III B. When spin–orbit coupling (SOC) is included in Fig. 5(b), a gap opens at Γpoint, and now, we have a proper con- duction band that has contributions from the band that made the valence band originally (red) and vice versa. This is called a band inversion. A band inversion like this may suggest that the material has a topological order, but the only way to confirm it is by calculat- ing the topological invariant and later to calculate the Weyl chirality to prove the topological nature. The actual cause of band inver- sion in materials is still a subject of discussion as far as scholars are concerned. 2. Band inversion without spin–orbit coupling Following the recommendations in Ref. 26, it is argued that band inversion is not necessarily induced by spin–orbit coupling (SOC) but may also occur when the strength of some other external parameter such as structural distortion increases. This is demon- strated in Fig. 6(b). We found out that after slight distortions of the lattice ( aby 0.47% and cby 2.36%), band inversion occurred at F andΓpoints. Unfortunately, the bands did not open as portrayed in Fig. 5(b), confirming that spin–orbit is necessary for the open- ing of the gap in our case. With the lattice distortion in the system, we also noted an increase in the bandwidth. The lifting of degen- eracy around the Γpoint was also seen just as it was predicted in Fig. 5(b), giving a notion that degeneracy can be lifted using lattice distortions. The authors herein argue that while what has been illustrated in Fig. 6(b) is not enough to have a topological insulator, it is still vital to open the bandgap to get to a situation similar to the one depicted in Fig. 4 or Fig. 5(b), which usually still needs spin–orbit coupling (SOC). Often, it is argued that spin–orbit is responsible for band inver- sion in topological Bi 2Se3. This is not true at all based on our lattice FIG. 4 . A sketch to illustrate band inversion at the Γpoint in Bi 2Se3. A detailed fat band description has been depicted in Fig. 5. AIP Advances 10, 095018 (2020); doi: 10.1063/5.0022525 10, 095018-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 5 . Fat bands of Bi 2Se3illustrating band inversion at the Γpoint. The Fermi level has been set to zero in both cases: without spin–orbit coupling (a) and with spin–orbit coupling (SOC) (b). As expected, it is very clear that band inversion occurs between Bi 6 p(green) and Se 4 p(red). strained band inversion demonstrated in Fig. 6(b). Since we per- formed a fully relativistic optimization, an elongation of the clattice was observed, while the alattice shrank. We had to repeat a scalar relativistic calculation with a distorted lattice and similar effects were FIG. 6 . Orbital distribution in the band-structure of Bi 2Se3illustrating band inver- sion at the Γpoint as a result of lattice distortions. The Fermi level has been set to zero in both cases:equilibrium lattice (a) and with distorted lattice (b). As expected, it is very clear that band inversion occurs between Bi 6 p(green) and Se 4 p(red) at F and Γpoints.seen. The point herein is that it is possible to witness band inver- sion (i) in materials whose spin–orbit coupling energy is too low, as reported in Ref. 26, and (ii) in a strained lattice similar to the case of Fig. 6(b). It is therefore prudent to note that band inversion is caused by a couple of factors: (i) scalar relativistic effects such as mass velocity and Darwin terms and (ii) lattice distortion in the crystal. Spin–orbit coupling in Bi 2Se3is only vital in opening the gap and making it topological. The mass velocity and Darwin terms in this case can be written as −P4/8m3c2and−̵h2/4m2c2⋅dV/dr⋅∂/∂⃗r, respectively, as elaborated in Ref. 45. We note herein that even when no spin– orbit is considered, the scalar relativistic terms are included in our scheme since we have a heavy element such as Bi. Such contributions will not split bands, but they are rather dependent on the angular momentum behavior of the band, particularly for pstates.46 E. Topological invariants in Bi 2Se3 We note that the response of Bi 2Se3toward a given perturba- tion is characterized by topological invariants.18,19This means that a given topological invariant describes the fundamental properties of the band structure at preferred k-points. Usually, it is used to describe strong topological insulators and if present, the system will have topologically protected bands at the surface. Since we have inversion symmetry in Bi 2Se3, we obtained the topological invari- ants from the symmetry of the Bloch function at six special Brillouin zone points, Γi=(n1,n2,n3)=1 2(n1b1+n2b2+n3b3), (2) where the reciprocal lattice vectors are bi, while ni= 0, 1. If we con- siderψi,nto be the nth occupied Bloch function at γi, we defined the symmetry function as δi=Πn√ ⟨ψi,n∣Θ∣ψi,n⟩, (3) where Θis the inversion operator. Since we know the symmetry functions in Bi 2Se3, it follows that the strong topological invariant νis given by (−1)ν0=Π8 i=1δi. (4) There also exist three weak invariants ν1,ν2, andν3, given by the products of four δi’s for which Γireside in the same plane, (−1)νk=Πnk=1,nj≠k=0, 1δi=(n1,n2,n3). (5) Usually, the weak invariants are not so much robust toward atomic disorder. We found out that the Z2topological index in Bi 2Se3was [1, 0, 0, 0], a signature of a strong topological material. IV. CONCLUSION In this work, we have demonstrated the fundamental band inversion in topological Bi 2Se3, which is a result of both scalar rela- tivistic and fully relativistic effects. Our work indicates a critical role played by spin–orbit coupling in such materials on the opening of the band structure. Our bandgap energy of 0.28 eV was also in a per- fect agreement with the existing scholarly work. One strong finding AIP Advances 10, 095018 (2020); doi: 10.1063/5.0022525 10, 095018-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv in this work is the fact that spin–orbit coupling is not the driving force for band inversion in Bi 2Se3, but scalar relativistic terms and lattice distortions do. It is also important to note that the basic fea- tures of topological materials can equally be probed and understood by the use of simple tight-binding models using the newly devel- oped interface between SIESTA and WANNIER 90, as illustrated in Ref. 47. This will give us room to deal with complex dynamics including electron–phonon and electron–electron interactions that will permit us to simulate polaron charge transport or excitonic phenomena in Bi2Se3. It would be interesting to see if band inversion can still be observed at the Γ-point if higher theories such as hybrid functionals andGW are employed. ACKNOWLEDGMENTS The authors gratefully acknowledge the computer resources, technical expertise, and assistance provided by the Centre for High Performance Computing (CHPC - MATS862 and MATS0712), Cape Town, South Africa. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. E. Moore, Nature 464, 194 (2010). 2M. Sato and Y. Ando, Rep. Prog. Phys. 80, 076501 (2017). 3A. A. Burkov, Nat. Mater. 15, 1145 (2016). 4N. Batra and G. Sheet, Resonance 25, 765 (2020). 5M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 6J. Betancourt, S. Li, X. Dang, J. D. Burton, E. Y. Tsymbal, and J. P. Velev, J. Phys.: Condens. Matter 28, 395501 (2016). 7D. Gosset, B. M. Terhal, and A. Vershynina, Phys. Rev. Lett. 114, 140501 (2015). 8A. Skurativska, T. Neupert, and M. H. Fischer, Phys. Rev. Res. 2, 013064 (2020). 9W. Gajewski, P. Achatz, O. A. Williams, K. Haenen, E. Bustarret, M. Stutzmann, and J. A. 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Appl. Phys. Lett. 117, 132404 (2020); https://doi.org/10.1063/5.0020199 117, 132404 © 2020 Author(s).A half-metallic ferrimagnet of CeCu3Cr4O12 with 4f itinerant electron Cite as: Appl. Phys. Lett. 117, 132404 (2020); https://doi.org/10.1063/5.0020199 Submitted: 29 June 2020 . Accepted: 15 September 2020 . Published Online: 29 September 2020 Hongping Li , Zhizhong Ge , An Sun , Zhipeng Zhu , Yi Tian , and Weiyu Song 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 Spin-reorientation transition induced magnetic skyrmion in Nd 2Fe14B magnet Applied Physics Letters 117, 132402 (2020); https://doi.org/10.1063/5.0022270 Observation of surface dominated topological transport in strained semimetallic ErPdBi thin films Applied Physics Letters 117, 132406 (2020); https://doi.org/10.1063/5.0023286A half-metallic ferrimagnet of CeCu 3Cr4O12 with 4f itinerant electron Cite as: Appl. Phys. Lett. 117, 132404 (2020); doi: 10.1063/5.0020199 Submitted: 29 June 2020 .Accepted: 15 September 2020 . Published Online: 29 September 2020 Hongping Li,1 Zhizhong Ge,1AnSun,1Zhipeng Zhu,1YiTian,2,a)and Weiyu Song3 AFFILIATIONS 1Institute for Advanced Materials, School of Materials Science and Engineering, Jiangsu University, Zhenjiang 212013, People’s Republic of China 2Institute for Energy Research, Jiangsu University, Zhenjiang 212013, People’s Republic of China 3State Key Laboratory of Heavy Oil Processing, College of Science, China University of Petroleum-Beijing, Beijing 102249,People’s Republic of China a)Author to whom correspondence should be addressed: tianyi@ujs.edu.cn ABSTRACT Half-metals have drawn extensive interest due to their unique electronic structure and wide application in spintronics. We report an A-site- ordered quadruple perovskite CeCu 3Cr4O12with half-metallic behaviors using first-principles calculations. Our calculations demonstrate that CeCu 3Cr4O12is a ferrimagnet with a saturated magnetic moment of 7.00 lBf.u.–1. Effective ferrimagnetic interactions are generated from the antiparallel spin arrangement between the A0-site Cu and B-site Cr. The electronic structure analyses reveal that CeCu 3Cr4O12 exhibits half-metal performance, which can be attributed to the mixed-valence of B-site Cr. More importantly, a small amount of 4f itinerantelectrons are located on the A-site Ce, i.e., both itinerant electron magnetism and localized magnetic moments are observed in the theoreticalcalculations. The charge distribution in this system is confirmed to be Ce (4/C0d)þCu2þ 3Cr(3.5þd)þ 4O2/C0 12. The physical properties of the AA03B4O12-type perovskite CeCu 3Cr4O12revealed in this study show that this class of materials shows promise in applications of half-metals. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020199 Magnetic half-metals have gained considerable attention over the past few years due to their unique electronic structure. Charge carriers near the Fermi level are entirely spin polarized in half-metallic materi- als, which generates a wide variety of intriguing spin-electronic prop- erties.1,2The first half-metal was reported in NiMnSb by de Groot et al.3Since then, several compounds have been proposed with half- metallic behaviors both experimentally and theoretically. These studies focused primarily on Heusler alloys (NiMnSb,4CoMnSb,5and Co2FeSi6), metallic oxides (rutile CrO 27and spinel Fe 3O48), lantha- num strontium manganite (La 0.7Sr0.3MnO 39), and zinc-blende compounds (CrAs,10CrSb,11and MnAs12). The high spin polariza- tion and unique electronic structures provide the potential for this class of materials to be applied in spintronics. This differs from tra-ditional electronics because, in addition to their charge state, elec- tron spins are exploited as an additional degree of freedom, which has implications in the efficiency of data storage and transfer. Typical applications include magnetic tunnel transistors, storage media, nonvolatile magnetic random access memory (MRAM), and magnetic sensors.Recent investigations have focused on perovskite-type com- pounds to search for half-metal materials with magnetic transitions above room temperature (RT). The A-site-ordered quadruple perov- skite AA 03B4O12is a typical perovskite-type structure, where the A site is generally occupied by nonmagnetic cations and the B/B0sites are usually filled with different kinds of transition-metals (TMs). Multiplemagnetic interactions occur in these compounds, which generate intriguing physical properties such as disproportionation, 13multifer- roic behavior,14low-field magnetoresistance,15giant dielectric con- stant,16and half metallicity.17Therefore, the research of half-metallic AA03B4O12-type perovskites has attracted vast attention both experi- mentally and theoretically. For example, half-metal properties have been successfully predicted in LaCu 3Mn 4O12,18,19where metallic behaviors are obtained for the spin-up channel and insulator proper- ties with a 0.3 eV bandgap are seen for the spin-down channel. CaCu 3Ni4O12was also predicted to be a half-metallic ferromagnet with a Cu-Ni parallel spin arrangement.20The strong electron correla- tions and a mixed valence of Ni3.25þlead to its half-metal perfor- mance. Moreover, it has been experimentally confirmed that Appl. Phys. Lett. 117, 132404 (2020); doi: 10.1063/5.0020199 117, 132404-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplBiCu 3Mn 4O12with Cu–Mn ferromagnetic coupling exhibits a half- metallic performance,17which indicates that AA03B4O12-type perov- skites are potential materials with potential half-metallic characteristics. The oxidation state of A-site cations plays a significant role in determining the internal charge combination along with the magnetic and electronic transport properties of AA03B4O12-type perovskites. Although ACu 3B4O12with divalent or trivalent ions at the A-site (Ca2þ,L a3þ,B i3þ, etc.) has been widely investigated, few researchers focused on higher valent A-site ions, such as Ce4þ. To date, few com- pounds noted as CeCu 3B4O12(B¼TMs) have been reported, but they show unique physical properties. The CeCu 3Mn 4O12(CCMO) dis- plays a charge-disproportionation of 65% Mn4þand 35% Mn3þ.21 The CeCu 3Fe4O12(CCFO) also exhibits a charge-disproportionation of 4Fe3.5þ!3Fe3þþFe5þ, while the antiferromagnetic ordering of Fe ions is predominant in the charge-disproportionatedCe 4þCu2þ 3Fe3.5þ 4O2/C0 12phase.22Recently, CeCu 3Cr4O12(CCCO) was synthesized under high-pressure and high-temperature condi- tions.23The material showed a ferrimagnetic transition of TC¼333 K and an experimentally observed semiconductor behavior. The þ4o x i - dation state of A-site Ce was assumed because the Ce–O bond lengths of CCCO (2.525 A ˚at 290 K) were very close to those of CCMO (2.524 A ˚at RT) and CCFO (2.555 A ˚at RT). However, the interesting results obtained in our calculations differ slightly from the experimen- tal results. To disclose the intrinsic electrical properties and charge combination of CCCO, we investigated the magnetic coupling and electronic transport properties using first-principles calculations. The results indicate that CCCO is a ferrimagnetic half-metal with a Ce(")–Cu(#)–Cr(") magnetic configuration. The A-site Ce approaches b u ti sl e s st h a nt h e þ4 oxidation state, while itinerant electron magne- tism (Ce 4f) and localized magnetic moments (Cu and Cr 3d) were found. There were no inter-site charge transfers or charge dispropor- tionation transitions in the CCCO. The Vienna ab initio simulation package (VASP)24,25based on spin-polarized first-principles calculations was used in our calculations. The generalized gradient approximation in the form of the Perdew–Burke–Ernzerhof exchange-correlation functional (GGA-PBE) 26was implemented in our calculations. The valence electron configurations o fC e ,C u ,C r ,a n dOw e r e4 f15d16s2,3 d104s1,3 d54s1,a n d2 s22p4, respectively. The wave functions were implemented assuming the plane waves with a kinetic cutoff energy of 450 eV and a Brillouin zone (BZ) over a 7 /C27/C27 k-mesh. The geometric structure was completely relaxed until the energy convergence dropped below 10/C05eV/atom, and the remaining force was less than 10/C02eV/A˚. The electron correla- tion effect was considered using the effective Hubbard parameter Ueff (Ueff¼U/C0J,w h e r e Uis the Hubbard parameter and Jis the exchange parameter) to accurately describe the strongly correlation of the Cu/Cr 3d and Ce 4f electrons. For simplicity, UCe,UCu,a n d UCrare used here- inafter to represent the effective parameter Uefffor Ce, Cu, and Cr, respectively. As shown in Fig. 1 ,t h eA A03B4O12-type perovskite CCCO crys- tallizes in the cubic phase with the Im 3 space group. The structure was fully optimized, and the lattice parameters are tabulated in Table I . The lattice constant obtained from the GGA is slightly smaller than that from the GGA þU(7.323 A ˚vs 7.362 A ˚), but both of them are similar to the experimentally measured value (7.313 A ˚). The bond dis- tances of Ce–O, Cu–O, and Cr–O within the GGA þU(GGA) methodare 2.549 (2.537) A ˚, 1.965 (1.975) A ˚, and 1.954 (1.936) A ˚, correspond- ing to the experimental results of 2.530 A ˚, 1.971 A ˚, and 1.934 A ˚, respectively. It is found that our optimized crystal structure andparameters agree well with the experimentally determined values. In addition, the calculated bond angles are slightly reduced to relieve the distortion of the CrO 6octahedral, making the crystal structure rela- tively more stable in the theoretical study. The chemical valence of A-site Ce is critical to determine the internal electron distribution in CCCO, which can be reflected by the Ce–O bond distances. As a comparison, the structures of the perov- skite compounds CCMO and CCFO were also relaxed. Similar Ce–Obond lengths were observed in CCFO (theoretical 2.559 A ˚vs experi- mental 2.555 A ˚) 22and CCMO (theoretical 2.550 A ˚vs experimental 2.524 A ˚),21in which the corresponding Ce oxidation state is proven to beþ4a n d þ3.38, respectively. However, the Ce–O bond lengths are noticeably longer in CeMn 7O12(2.656 A ˚)27and CeCuMn 6O12 (2.612 A ˚),28where the Ce is in a þ3 state. The results from the lattice parameters suggest that the Ce in CCCO could be the þ4o x i d a t i o n state or close to it. In addition, our calculations demonstrate that all the Ce–O and Cr–O bond distances are the same in CCCO,which verifies the absence of inter-site charge transfer or charge disproportionation. It is noted that the A 0–B interactions always play the predomi- nant role in AA03B4O12-type perovskites when both the A0site and B site are occupied by TMs. Thus, two magnetic configurations between FIG. 1. Crystal structure of the A-site-ordered quadruple perovskite CeCu 3Cr4O12. TABLE I. Experimental and theoretical lattice parameters a(A˚), bond distances (A ˚), and bond angle (/C14) for CeCu 3Cr4O12. GGA GGA þU Experimental a(A˚) 7.323 7.367 7.305 Ce–O (A ˚) 2.537 2.550 2.525 Cu–O (A ˚) 1.975 1.965 1.974 Cr–O (A ˚) 1.936 1.956 1.931 /Ce–O–Cu (/C14) 107.8 108.6 108.0 /Ce–O–Cr (/C14) 89.2 89.1 89.4 /Cu–O–Cr (/C14) 108.3 108.9 108.2Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132404 (2020); doi: 10.1063/5.0020199 117, 132404-2 Published under license by AIP Publishingthe Cu and Cr sublattices were designed: (i) Cu( ")–Cr(")f e r r o m a g n e - tism (FM) and (ii) Cu( #)–Cr(") ferrimagnetism (FiM). The “ þU” method was carried out to improve the description of Cu and Cr 3d electrons. A series of Uvalues were tested for Cu and Cr ( UCufrom 0.0 to 7.0 eV and UCrfrom 0.0 to 4.0 eV). The corresponding results are summarized in Table II . The results show that the FiM magnetic con- figuration becomes the ground magnetic configuration when Uis included due to the minimum theoretical total energy. As Uincreases, the bandgap for the spin-down channel gradually increases from 1.23 to 1.67 eV, whereas a metallic conducting behavior is found for the spin-up channel, regardless of the combination of UCuand UCr.T h i s remarkable increase in the bandgap is accompanied by the enhance- ment of the spin magnetic moment of Cr from 2.18 to 2.69 lBand that of Cu from /C00.43 to /C00.66lBwhen Uis included. It is worth noting that a non-negligible magnetic moment is located at the A-site Ce in both FM and FiM configurations (shown in Table II )a l t h o u g h the magnetic properties have been amended with different Ucombina- tions. The same phenomenon occurs in our calculated results of the CCMO system with Ce3.4þ, suggesting that the A-site Ce in CCCO similarly approaches but is less than the þ4 oxidation state. To verify our hypothesis, the more sophisticated functional HSE06 was consid- ered for comparison. The magnetic moment of Ce slightly decreases but still exists (0.13 lB), suggesting that a small quantity of 4f itinerant electrons are located at the A-site and contribute to the weak magneticmoment. Therefore, the Ce 4f electrons are not completely lost, leading to a residual magnetic moment for Ce in CCMO. It is expected that a small amount of 4f electrons are located on the A-site Ce in CCCO.Thus, as shown in Fig. 2 , four additional magnetic configurations involving magnetic Ce were constructed: (i) FiM-1: Ce(")–Cu(#)–Cr(")[Fig. 2(a) ]; (ii) FiM-2: Ce( #)–Cu(#)–Cr(") [Fig. 2(b) ]; (iii) FiM-3: Ce( #)–Cu(")–Cr(")[Fig. 2(c) ]; and (iv) FM-4: Ce( ")–Cu(")–Cr(")[Fig. 2(d) ].TABLE II. Calculated energy differences DE(meV per formula unit) (with FiM as a reference), bandgap at the spin-down ( Eg/C0) and spin-up channel ( Egþ), and magnetic moments for CeCu 3Cr4O12with an increasing Ucomposition. More sophisticated functionals (HSE06) were considered as a comparison. UCu/UCr(eV) DE(meV f.u./C01) Eg/C0(eV) Egþ(eV)M(lB) Ce Cu Cr Total FM 0.0/0.0 /C021 0.24 0.00 0.28 /C00.44 2.20 6.77 4.0/2.0 518 1.09 0.00 0.22 0.58 2.57 12.24.0/3.0 370 0.00 0.80 0.24 /C00.12 2.85 10.8 5.0/2.0 693 1.85 0.00 0.28 0.66 2.57 12.7 5.0/3.0 497 1.84 0.00 0.27 0.50 2.84 12.7 6.0/3.0 567 1.84 0.00 0.22 0.68 2.67 12.87.0/3.0 494 1.85 0.00 0.24 0.72 2.65 12.87.0/4.0 268 1.83 0.00 0.24 0.58 2.91 12.9HSE06 489 1.89 0.00 0.11 0.85 2.80 12.9 FiM 0.0/0.0 0 0.00 0.00 0.33 /C00.43 2.18 6.77 4.0/2.0 0 1.23 0.00 0.21 /C00.57 2.46 6.90 4.0/3.0 0 1.38 0.00 0.20 /C00.58 2.58 7.01 5.0/2.0 0 1.50 0.00 0.26 /C00.59 2.48 6.92 5.0/3.0 0 1.57 0.00 0.22 /C00.60 2.59 7.01 6.0/3.0 0 1.59 0.00 0.22 /C00.63 2.58 7.00 7.0/3.0 0 1.62 0.00 0.23 /C00.66 2.59 6.99 7.0/4.0 0 1.67 0.00 0.19 /C00.66 2.69 7.08 HSE06 0 1.72 0.00 0.13 /C00.79 2.80 6.99 FIG. 2. Magnetic coupling of CeCu 3Cr4O12considering Ce, Cu, and Cr as magnetic ions: (a) FiM-1, (b) FiM-2, (c) FiM-3, and (d) FM-4.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132404 (2020); doi: 10.1063/5.0020199 117, 132404-3 Published under license by AIP PublishingSlater and Pauling reported that the magnetic moments of 3d ele- ments and their binary compounds can be described using the meannumber of valence electrons per atom. This is the famousSlater–Pauling rule, which is used to describe the magnetic propertiesof Heusler compounds. Perovskite-type oxides still obey the Slater–Pauling rule, indicating that an integer magnetic moment should exist under first-principles calculations. As shown in Table II , the total magnetic moment is an integer of 7.00 l Bf.u./C01asUCu/UCr ¼6.0/3.0 eV. Thus, the Uvalue for Ce, Cu, and Cr ions was chosen to beUCe¼5.0 eV, which has been used in CeCrO 3,29,30UCu¼6.0 eV, and UCr¼3.0 eV for further calculations. The corresponding details are given in Table III . Obviously, the total energies for the FiM-3 and FM-4 configurations with a parallel alignment electron arrangementbetween the Cu and Cr are larger than those in FiM-1 and FiM-2. Theenergy costs of inter-atomic exchange interactions for Ce in the crys-tals indicate that the total energy of FiM-2 is slightly larger than that of FiM-1. This implies that the FiM-1 [Ce( ")–Cu(#)–Cr(")] is the most favorable magnetic configuration in CCCO. In addition, the saturationmoment in FiM-1 is the same as that in the FiM state (7.00 l Bf.u./C01 vs 7.00 lBf.u./C01), suggesting that the calculated results with UCe/UCu/ UCr¼5.0/6.0/3.0 eV are reasonable, and further electronic structures are based on this Ucombination. As shown in Table II ,c o m p a r e dw i t ht h eH S E 0 6a p p r o a c h ,t h e calculated results (at least the magnetic properties) of GGA þUare more accurate and reasonable for the ferrimagnetic half-metal CCCOdue to the integer total magnetic moment (7.00 l Bf.u./C01vs 6.99 lB f.u./C01). The energy bandgaps for both approaches are similar. For half- metallic materials, this difference is unimportant and could beaccepted. Therefore, we prefer using GGA þUi n s t e a do fh y b r i df u n c - tionals where a reasonable Uremedies the GGA shortcomings. These effects can seem similar to the HSE06 in our system. Overall, CCCO isa ferrimagnet with a saturation moment of /C247.00l Bf.u./C01, while the Cu(#)–Cr(") antiferromagnetic coupling plays an important role in determining the magnetic properties of the system. The magneticmoments for Ce, Cu, and Cr are 0.17 l B,/C00.63 lB,a n d2 . 5 9 lB, respectively. In particular, the O ions also carry a /C00.17lBmagnetic moment, which is presumably due to interactions between the itiner-ant Ce and the localized Cu and Cr states. Figure 3 presents the band structures and density of states (DOSs) within the FiM-1 magnetic configuration based on the GGAand GGA þUmethods. For GGA calculations [shown in Fig. 3(a) ], the circuitous band lines in the spin-up channel are across the Fermi level (E F), whereas a relatively small bandgap (nearly 0 eV) appears in the spin-down channel. Figure 3(b) shows the results obtained from theGGAþUmethod with UCe¼5.0 eV, UCu¼6.0 eV, and UCr¼3.0 eV. The effective Coulomb-repulsion Upushes the unoccupied states to a higher energy region, which creates a wider energy bandgap in thespin-down channel. In contrast to the spin-asymmetric insulating band structure of CCCO, the up-spin conduction bands across the Fermi level consist primarily of Cr 3d and O 2p with negligible contri-butions from Cu 3d. The spin-down channel has a direct bandgap of0.62 eV with both the conduction band minimum and the valence band maximum located at the X point. We further analyze the partial DOSs (PDOSs) of Ce, Cr, Cu, and O as obtained from the GGA þUmethod. As shown in Fig. 4(a) , both spin-channels for the Ce 4f orbitals are mostly empty, and relativelyfew 4f electrons are scattered around the Ce ion range, which suggests aC e (4/C0d)þwith the 4fdelectronic configuration. Figure 4(b) indicates that most of the Cr 3d xy,3 d yz,a n d3 d xzorbitals in the spin-up channel are occupied and are divided by the EF. This suggests that the elec- tronic charge of Cr is larger than þ2 but not quite þ3. All of the Cu 3d orbitals are filled except for the 3d xyspin-up channel [ Fig. 4(c) ], which confirms the 3d9electronic configuration of Cu2þ. The chemical valence of the O anion should be around /C02d u e to the nearly occupied 2p orbitals [ Fig. 4(d) ]. To verify the chemical valence state in CCCO, a quantitative analysis of the electronic occu-pation was performed using the LOBSTER code. The Cr 3d electronic occupation is (d xy)0.81(dyz)0.82(dxz)0.84(dz2)0.38(dx2 /C0y2)0.38for theTABLE III. Energy differences DE(in eV per formula unit) within different magnetic configurations (with FiM-1 as a reference) for CeCu 3Cr4O12withUCe¼5.0 eV, UCu ¼6.0 eV, and UCr¼3.0 eV. Magnetic coupling DE(meV f.u./C01)M(lB) Ce Mn Cr Ce Cu Cr Total FiM-1 "#" 0 0.17 /C00.63 2.59 7.00 FiM-2 ##" 10 0.17 /C00.63 2.60 7.00 FiM-3 #"" 557 0.18 0.68 2.70 12.8 FM-4 """ 576 0.18 0.68 2.70 12.8 FIG. 3. Band structures and DOSs of CeCu 3Cr4O12within the FiM-1 configuration obtained using (a) GGA and (b) GGA þU.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132404 (2020); doi: 10.1063/5.0020199 117, 132404-4 Published under license by AIP Publishingspin-up channel and (d xy)0.10(dyz)0.11(dxz)0.11(dz2)0.21(dx2/C0y2)0.22for the spin-down channel. The total electronic occupation for Cr is Cr3.52þ (3d2.48þ) (sum of the difference between the spin-up and spin-down channels), which is consistent with the experimental bond valencesum (BVS) analysis (Cr 3.52þvs Cr3.5þ). It is noted that the absolute values, which are somewhat dependent on the chosen atomic spheres, should not be overinterpreted. The calculated results of the mixedchemical valence of Cr agree well with predictions based on charge conservation. In addition, all Cr ions per formula unit possess the same chemical valence, which verifies that there is no inter-site charge transfer or charge disproportionation in CCCO. As a result, negligible Ce 4f itinerant electrons were found in the system, and the chemicalvalence of Ce approaches þ4, which suggests that the electronic distri- bution is Ce (4/C0d)þCu2þ 3Cr(3.5þd)þ 4O2/C0 12. No inter-site charge trans- fer or charge disproportionation was found in CCCO, which is different from CCFO and CCMO. In summary, we systematically investigated the crystal structure, magnetic configuration, and electronic properties of quadruple perov- skite CCCO using first-principles calculations. Ferrimagnetic couplingwas found in CCCO and a strong Cu( #)–Cr(") antiparallel spin arrangement was predominant when determining its magnetic prop- erties. The electronic structure analyses revealed that the CCCO exhib- its half-metal characteristics with an integer magnetic moment of 7.00 l Bf.u./C01. No inter-site charge transfer or charge disproportionation transitions are found in CCCO, which is in sharp contrast to analogCCFO and CCMO. The charge combination is confirmed as Ce(4/C0d)þCu2þ 3Cr(3.5þd)þ 4O2/C0 12with minimal Ce 4f itinerant elec- trons. The Ce in the itinerant state is of interest in CeCu 3B4O12 (B¼TM) systems and needs further experimental verification, which was outside the scope of this study. This work was supported by the National Natural Science Foundation of China (Grant Nos. 21301075 and 21901086) and theSenior Talent Foundation of Jiangsu University (Grant Nos. 12JDG096 and 19JDG017). DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1I./C20Zutic´, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). 2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. V. Molnar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). 3R. A. Groot, F. M. Mueller, P. G. Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983). 4Z. C. Wen, Z. Y. Qiu, S. T €olle, C. Gorini, T. Seki, D. Hou, T. Kubota, U. 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5.0020804.pdf

Appl. Phys. Lett. 117, 161601 (2020); https://doi.org/10.1063/5.0020804 117, 161601 © 2020 Author(s).Molecular beam epitaxy growth and strain- induced bandgap of monolayer 1T′-WTe2 on SrTiO3(001) Cite as: Appl. Phys. Lett. 117, 161601 (2020); https://doi.org/10.1063/5.0020804 Submitted: 04 July 2020 . Accepted: 08 October 2020 . Published Online: 20 October 2020 Huifang Li , Aixi Chen , Li Wang , Wei Ren , Shuai Lu , Bingjie Yang , Ye-Ping Jiang , and Fang-Sen Li ARTICLES YOU MAY BE INTERESTED IN Epitaxial strain dependent electrocatalytic activity in CaRuO 3 thin films Applied Physics Letters 117, 163906 (2020); https://doi.org/10.1063/5.0020934 Rotated angular modulated electronic and optical properties of bilayer phosphorene: A first- principles study Applied Physics Letters 117, 163102 (2020); https://doi.org/10.1063/5.0023296 Open-circuit voltage photodetector architecture for infrared imagers Applied Physics Letters 117, 163503 (2020); https://doi.org/10.1063/5.0020000Molecular beam epitaxy growth and strain- induced bandgap of monolayer 1T0-WTe 2 on SrTiO 3(001) Cite as: Appl. Phys. Lett. 117, 161601 (2020); doi: 10.1063/5.0020804 Submitted: 4 July 2020 .Accepted: 8 October 2020 . Published Online: 20 October 2020 Huifang Li,1,2Aixi Chen,2LiWang,2Wei Ren,2,3Shuai Lu,2Bingjie Yang,2Ye-Ping Jiang,1,a)and Fang-Sen Li2,3,a) AFFILIATIONS 1Key Laboratory of Polar Materials and Devices (MOE), Department of Electronic, School of Physics and Electronic Science, East China Normal University, Shanghai 200241, China 2Vacuum Interconnected Nanotech Workstation (Nano-X), Suzhou Institute of Nano-Tech and Nano-Bionics (SINANO),Chinese Academy of Sciences (CAS), Suzhou 215123, China 3School of Nano-Tech and Nano-Bionics, University of Science and Technology of China, Hefei 230026, China a)Authors to whom correspondence should be addressed: ypjiang@clpm.ecnu.edu.cn andfsli2015@sinano.ac.cn ABSTRACT A monolayer 1T0-WTe 2film is grown on SrTiO 3(001) with in-plane tensile strain. A height of /C240.7 nm, obvious charge transfer, and incom- mensurate charge fluctuations in 1T0-WTe 2suggest strong coupling to the STO substrate. Scanning tunneling spectroscopy on the surface reveals that a large energy gap opens at the Fermi level with nearly zero conductance. The opened energy gap decreases with the increase in the WTe 2island size. The lack of the metallic edge state on monolayer 1T0-WTe 2/SrTiO 3(001) indicates the absence of the quantum spin Hall (QSH) state. Our study here demonstrates that the energy gap of monolayer 1T0-WTe 2can be tuned by lattice strain and illustrates the importance of interface coupling to realize the metallic edge state and QSH in monolayer 1T0-WTe 2. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020804 Tungsten ditelluride (WTe 2) has become a hot material due to its interesting phenomena, such as unsaturable magnetoresistance,1,2 superconductivity,3,4topological nontrivial type-II Weyl semimetal,5,6 and quantum spin Hall (QSH) effect.7–10The QSH state, characterized as the metallic edge state and an energy gap opened by strong spin- orbital coupling (SOC), has been observed on monolayer 1T0-WTe 2 on bilayer graphene/6H–SiC(0001) (BLG/SiC),8,11cleaved 1T0-WTe 2 monolayer,9and bulk WTe 2surface.10Bulk WTe 2can be recognized as unstressed monolayer WTe 2strongly coupled to bulk. The MBE growth of few layer 1T0-WTe 2exhibits high quality and thickness in control.8,11,12Tang et al.8found a topological bandgap of /C2450 meV from ARPES and STS measurements, while Song et al.11observed the appearance of the electron–electron-induced Coulomb gap. Such a Coulomb gap was symmetric, pinned at the Fermi level, and the den- sity of states (DOS) shows a linear dependence on energy near the Fermi level. Much effort has been adopted to tune the QSH state in mono- layer 1T0-WTe 2theoretically13–15and experimentally.16–18Qian et al.13calculated the response of the energy gap from biaxial strain and found a tunable bandgap. Xiang et al.14predicted a phasetransition from a semimetal to a QSH insulator under the uniaxial strains. When cutting 1T0-WTe 2into nanoribbons perpendicular (par- allel) to the W–W chains,15the system would undergo a semimetal to semiconductor (metal) transition. The external electric field can tune QSH insulator-metal-superconductor switching.18However, experi- mental investigation about strain-induced band engineering on mono- layer WTe 2has seldom been done. Here, we try to grow strained monolayer 1T0-WTe 2on STO and investigate how the electronic state changes under substrate-induced strain. We find that monolayer 1T0-WTe 2strongly couples to the STO substrate with tensile strain. The differential d I/dVconductance shows a large bandgap on monolayer WTe 2.W eh a v en o to b s e r v e dt h e metallic edge state, which indicates the absence of the QSH state. Our experiments were carried out in a low temperature STM sys- tem (Unisoku 1300) with a preparation chamber. Monolayer WTe 2 fi l m sw e r eg r o w no nN b - d o p e dS r T i O 3(001) (STO, Nb:0.5 wt. %, KMT) by co-depositing high purity W and Te (99.9999%) from an e- beam evaporator and a Knudsen cell, respectively. Before STM/STS measurements, the WTe 2films were annealed at 300/C14Cf o r1 h .T h e STM/STS were obtained at 4.8 K, unless otherwise specified. The Appl. Phys. Lett. 117, 161601 (2020); doi: 10.1063/5.0020804 117, 161601-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldifferential d I/dVspectra were acquired using a standard lock-in tech- nique with a modulation voltage of /C2410 mV and at a frequency of 991 Hz at 4.8 K. The STM topographic images were processed withWSXM software. 19 Monolayer 1T0-WTe 2hosts distorted the octahedral structure with one-dimensional W–W chains, as shown in Fig. 1(a) . The top Te atoms are not in plane. Previous MBE growth of monolayer WTe 2on BLG/SiC exhibits step-diffusion limited behavior with random edgegeometries. 8,12Here on the STO substrate, the edge geometry of 1T0- WTe 2is also random with a smaller size. Figure 1(b) shows such a typ- ical morphology with /C2485% coverage. The optimal growth tempera- ture is rather narrow from 230/C14Ct o2 5 0/C14C. High growth or annealing temperature results in a needle-shape structure (see the sup- plementary material , Fig. S1). We seldom observed second-layer WTe 2but clusters when the coverage is larger than 1 ML. The line profile in Fig. 1(c) shows that the step height is /C240.70 nm, nearly equal to the step height (0.701 nm) of bulk 1T0-WTe 2,20indicating much stronger coupling than those on BLG/SiC. Figure 1(d) shows the tunneling current image of the striped 1T0phase with different orien- tations. Statistics of the orientations show that domains with an angleof 90 /C14dominate. It should conform to the twofold symmetry of WTe 2 and fourfold symmetry of the STO substrate. Figure 1(f) shows the high resolution image of monolayer 1T0- WTe 2taken at 78 K. The corresponding fast Fourier transform image yields in-plane lattice constants of a/C240.36 nm and b/C240.66 nm. An /C243/C14angle distortion is observed here, consistent with previous mea- surements.12,21,22The enlarged lattice than bulk values ( a¼0.348 nmand b¼0.628 nm) suggests tensile strains: 3.4% and /C245% along [100] and [010] directions, respectively. The smaller strain along the [100]direction agrees well with the previous calculation that atom motionsare less sensitive along W–W chains. 23,24It is worth mentioning that the lattice constant scatters on islands with different sizes, ranging from/C240.63 nm to /C240.68 nm, indicating varying strains. We probed the electronic state of monolayer 1T0-WTe 2/STO by differential d I/dVconductance. Figure 2(a) shows a typical d I/dVspec- trum at the center of the monolayer 1T0-WTe 2/STO island. The over- all curve resembles measurements on 1T0-WTe 2/BLG/SiC,8,12such as obvious upwarp at negative bias voltage and a round peak near 0.5 V. We label edges of valence bands and conduction bands as V 2,V3and C2,C3inFig. 2(a) . Compared with 1T0-WTe 2/BLG/SiC,8,11,12here we find that the energy levels shift downward by /C240.25 eV, indicating electron doping from the STO substrate. The most striking feature of the d I/dVspectrum in Fig. 2(a) is the largely suppressed density of states near the Fermi level. A fundamental bandgap of /C24112 meV can be clearly identified [see the small-scale STS in Fig. 2(b) ]. The conduc- tance goes to near-zero within the gap, contrast to the semimetal prop- erty of bulk 1T0-WTe 2. The gap is highly asymmetrical with respect to the Fermi level, ruling out the possibility of the superconducting gap. To illustrate the nature of such a large bandgap, we performed line spectroscopy across monolayer 1T0-WTe 2islands. The STM image in Fig. 2(c) shows an island with an area of /C2485 nm2, and a black arrow marks the location where the spectra were taken. The color plot of the line spectroscopy in Fig. 2(d) shows that electronic states are not uniform. However, the bandgap near the Fermi level can FIG. 1. MBE growth of monolayer 1T0-WTe 2films on SrTiO 3(001). (a) Crystal structure of monolayer 1T0-WTe 2, showing the distorted structure and octahedral lattice. (b) Surface morphology of monolayer WTe 2grown at Ts/C24235/C14C. Scanning parameters: V¼1.5 V, I¼100 pA, and 100 nm /C2100 nm. (c) Line profile of the yellow arrow in (b), showing that the height of the 1T0-WTe 2monolayer is /C240.7 nm. (d) The tunneling current image, showing orientation of WTe 2islands. Scanning parameters: V¼1.0 V, I¼100 pA, and 30 nm /C230 nm. (e) Histogram of orientations in (d). (f) Atomic-resolved STM image obtained at 78 K. The inset shows its fast Fourier transform image and black circles mark the reciprocal lattice. Scanning parameters: V¼0.12 V, I¼2 nA, and 4 nm /C24 nm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 161601 (2020); doi: 10.1063/5.0020804 117, 161601-2 Published under license by AIP Publishingbe identified on the whole island (also see the zoom-in line spectroscopy in the supplementary material ,F i g .S 2 ) .A tt h es t e pe d g e ,w eo b s e r v e d an enlarged bandgap rather than a conductive edge state different fromthe metallic edge state on nearly free-standing monolayer 1T 0-WTe 2on BLG/SiC.8,9In addition to edges with random geometries, we also have not found the robust conductive state at the domain boundary along the adirection (see the supplementary material ,F i g .S 3 ) . We turn to figure out the origin of the observed bandgap in monolayer 1T0-WTe 2on STO. When monolayer WTe 2was grown on BLG/SiC,8,11,12a topological bandgap /C2450 meV was observed due to strong spin–orbit coupling (SOC)-induced band inversion. Here on t h eS T Os u r f a c e ,t h el a c ko ft h ec o n d u c t i v ee d g es t a t ec a nh e l pu st oexclude the possibility of the QSH state in monolayer WTe 2.T h e much larger bandgap of monolayer WTe 2/STO suggests that in addi- tion to strong SOC, other factors should be taken into account, such aslattice strain, considering the enlarged lattice constant. Previous calcu-lations 13–15point out that adding strain into the WTe 2monolayer can tune the size of the bandgap and even induce topological non-trivial totrivial phase transition. Another evidence for the strain-induced effect is the incommen- surate charge fluctuation, as shown in Fig. 3(a) . The line profile along the black arrow (see the supplementary material , Fig. S4) shows that the period of this fluctuation is 0.76–1.45 nm (2–4 unit cells) with aheight of 20–30 pm, suggesting its incommensurate property. Jiaet al. 12reported a weak charge ordering (CDW) in 1T0-WTe 2/BLG/ SiC and claimed that CDW induced the observed small gap. However,recently they concluded that such a soft gap was a Coulomb gap dueto the electron-electron interaction. 11Here, we observed a large U- shape bandgap rather than a linear V-sharp soft gap,11indicating that it is not a Coulomb gap on monolayer 1T0-WTe 2/STO. Because of the incommensurate charge fluctuation just along the adirection, it is unlikely to be attributed to electron scattering from random edgegeometries or the moir /C19e pattern between the film and the substrate. Zhang et al. 25have observed a similar charge fluctuation in monolayer 1T0-WSe 2grown on the STO substrate, where in-plane compressive strain was developed. Therefore, we conclude such a charge fluctuationis caused by in-plane tensile strain. We have measured the bandgap on WTe 2islands with different areas, as shown in Fig. 3(b) . As a whole, the bandgap decreases with the increase in the area of WTe 2. Based on previous calculations,13the bandgap of monolayer WTe 2increases with strain monotonically. It agrees with our observation here. The bandgap even goes to/C24200 meV with an area of /C2412 nm 2.Q i a n et al.13predicted an /C24180 meV when biaxial strain goes to /C248%. However, such size- dependent band evolution also recalls the quantum size effect (QSE),which has been noticed since the 1960s. 26Within the regime of the QSE, due to the confinement of the movement of electrons in reduceddimensions, the space of energy level DE Fnear the Fermi surface is proportional to 1/ L,w h e r e Lis the size of the reduced system. We have measured the evolution of the energy level of V 2inFig. 3(c) , which moves to lower energy with the increase in area. It is inconsis-tent with the QSE. Thus, in-plane tensile strain tunes the bandgap of FIG. 2. (a) Typical d I/dVspectrum on monolayer WTe 2. The labels V 2,V3, and C 2, C3here mark the edges of some valence bands and conduction bands. (b) The zoomed-in d I/dVcurves (blue), clearly indicating the features of the bulk bandgap. The red curve is the corresponding log (d I/dV) plot to extract the opened bandgap of/C24112 meV. (c) A STM image of the WTe 2island with an area of /C2485 nm2. The black arrow marks the position where line spectra in (d) were taken. Yellow circlesmark the positions for spectra at the center and edge of WTe 2islands, respectively. Scanning parameters: V¼700 mV, I¼140 pA, and 15 nm /C215 nm. (d) Color plot of the line d I/dVspectroscopy. The d I/dVcurves at the center and edge were dis- played. The obvious dark border near the Fermi level is the opened gap. FIG. 3. The charge fluctuation and size dependent gap of monolayer 1T0-WTe 2on STO. (a) Zoom-in STM image, showing spatial incommensurate charge fluctuations along theadirection. Scanning parameters: V¼330 mV, I¼550 pA, and 5 nm /C25 nm. (b) Size of the bandgap as a function of the area of monolayer WTe 2islands on STO. The “$” mark indicates the gap size of monolayer WTe 2grown on weak-coupled bilayer graphene/SiC in Ref. 8. (c) The energy level of the valence band V 2as a function of the area of monolayer WTe 2islands. The inset shows the energy level with a minimum slope for determination of V 2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 161601 (2020); doi: 10.1063/5.0020804 117, 161601-3 Published under license by AIP Publishingmonolayer 1T0-WTe 2. If we need to further increase the bandgap, the nanoribbon with a smaller width will be one of the strategies. InFig. 3(b) , we also list some bandgaps obtained on monolayer 1T0-WTe 2/BLG/SiC from Ref. 8, in which the intrinsic electronic structure was expected. We find that on islands with the same area, the bandgap on STO ( /C2420 meV) is smaller than those on BLG/SiC (/C2450 meV). It is strange because according to the theoretical calcula- tion,13the bandgap was positively related to the strains, and a larger bandgap was expected on STO. To explore the reason and investigate the distribution of the bandgap, we have performed STS mapping on monolayer 1T0-WTe 2/STO under various energies (see the supple- mentary material , Fig. S5). The STM image in Fig. S5(a) shows a small WTe 2island with an area of /C2442 nm2. Incommensurate charge fluctu- ations can be also identified. Figure S5(b) shows the spatial zero- energy mapping at the box area of Fig. S5(a), indicating a uniform low-density state at the Fermi level and bandgap developed on thewhole islands. However, STS mapping under different energies in Fig. S5(c) shows irregular and unpredictable distribution of the electronic state, emphasizing the nonuniform interface coupling between WTe 2 and STO. Such nonuniform interface coupling should be the reason for bandgap suppression and absence of the conductive edge state on monolayer WTe 2/STO. In conclusion, we have grown monolayer 1 T0-WTe 2films with tensile strain on SrTiO 3(001) substrates. Obvious electron transfer from the STO substrate and charge fluctuations are identified. An asymmetrical full energy gap is observed. Such an energy gap can be tuned by lattice strain on WTe 2islands with different sizes. We have not observed the metallic state at the step edge of WTe 2islands, indi- cating the absence of the QSH state. Strong nonuniform coupling between WTe 2and STO should induce such topological to non- topological phase transition. Our study emphasizes the importance of interface interaction to tune the QSH gap and hold the metallic edgestate. See the supplementary material for additional descriptions of (1) morphologies of the monolayer WTe 2grown at different tempera- tures, (2) small range of line spectroscopy, (3) incommensurate charge ordering with varying distances, and (4) surface STS mapping. AUTHORS’ CONTRIBUTIONS H.L. and A.C. contributed equally to this work. We acknowledge support from the National Natural Science Foundation of China (Grants Nos. 11604366 and 11634007) and the National Natural Science Foundation of Jiangsu Province (Grant No. BK 20160397). F.-S.L. acknowledges support from the Youth Innovation Promotion Association of Chinese Academy of Sciences (No. 2017370). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1M. N. Ali, J. Xiong, S. Flynn, J. Tao, Q. D. Gibson, L. M. Schoop, T. Liang, N. Haldolaarachchige, M. Hirschberger, N. P. Ong, and R. J. 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5.0023590.pdf

J. Chem. Phys. 153, 144118 (2020); https://doi.org/10.1063/5.0023590 153, 144118 © 2020 Author(s).Effects of perturbation order and basis set on alchemical predictions Cite as: J. Chem. Phys. 153, 144118 (2020); https://doi.org/10.1063/5.0023590 Submitted: 30 July 2020 . Accepted: 27 September 2020 . Published Online: 14 October 2020 Giorgio Domenichini , Guido Falk von Rudorff , and O. Anatole von Lilienfeld ARTICLES YOU MAY BE INTERESTED IN Theory and implementation of a novel stochastic approach to coupled cluster The Journal of Chemical Physics 153, 144117 (2020); https://doi.org/10.1063/5.0026513 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Effects of perturbation order and basis set on alchemical predictions Cite as: J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 Submitted: 30 July 2020 •Accepted: 27 September 2020 • Published Online: 14 October 2020 Giorgio Domenichini, Guido Falk von Rudorff, and O. Anatole von Lilienfelda) AFFILIATIONS Institute of Physical Chemistry and National Center for Computational Design and Discovery of Novel Materials (MARVEL), Department of Chemistry, University of Basel, 4056 Basel, Switzerland a)Author to whom correspondence should be addressed: anatole.vonlilienfeld@unibas.ch ABSTRACT Alchemical perturbation density functional theory has been shown to be an efficient and computationally inexpensive way to explore chemical compound space. We investigate approximations made, in terms of atomic basis sets and the perturbation order, introduce an electron-density based estimate of errors of the alchemical prediction, and propose a correction for effects due to basis set incompleteness. Our numerical analysis of potential energy estimates, and resulting binding curves, is based on coupled-cluster single double (CCSD) reference results and is limited to all neutral diatomics with 14 electrons (AlH ⋯NN). The method predicts binding energy, equilibrium distance, and vibrational frequencies of neighboring out-of-sample diatomics with near CCSD quality using perturbations up to the fifth order. We also discuss simultaneous alchemical mutations at multiple sites in benzene. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023590 .,s I. INTRODUCTION Chemical space, the ensemble of all possible molecules that constitute matter, is unfathomably large. The number of molecules in chemical space is estimated to be higher than 1060just consid- ering small organic molecules.1,2The size of this problem makes it impossible to enumerate and assess a significant portion of chemical space with standard quantum chemistry methods alone. Therefore, modern approaches in materials design have the need for computa- tionally less demanding yet sufficiently accurate methods.3Promis- ing results were obtained, for instance, through quantum machine learning models.4–9Provided that the machine has been trained on a sufficiently large number of molecules, quantum machine mod- els can provide fast yet accurate predictions. For example, in 2017, machine learning models were trained to predict multiple electronic properties for thousands of organic molecules reaching the den- sity functional theory (DFT) accuracy in a milli-second prediction time.10,11 Alchemical perturbation density functional theory (APDFT) aims at a similar goal, namely, to obtain a rapid screening of chemical space. The approach is fundamentally different though:While machine learning models require thousands or millions of training instances in order to interpolate toward similar systems, APDFT relies on a single explicit calculation on a reference sys- tem to give approximate but still accurate predictions for properties of a multitude of isoelectronic and similar target compounds.12–16 APDFT yields consistent predictions of both energies and electron densities.17Its reliability has been shown in several applications such as energies in BN-doped aromatic systems and non- covalent interactions thereof,17decomposition of energy contri- butions,18or estimation of deprotonation energies.19,20Through the use of pseudopotentials, quantum alchemy can be a power- ful tool in solid state chemistry,12,13,21–23with possible applica- tions in the catalyst design.24–26In recent applications, it has been shown that APDFT—if applied to high level coupled-cluster sin- gle double (CCSD)27–29calculations for the reference molecule— can outperform17widely used methods in computational chem- istry [e.g., Hartree–Fock (HF),30,31MP2,32and DFT33–36]. This approach holds the promise to shift the computational cost from many medium-quality calculations throughout chemical compound space to a few select high-quality calculations that serve as a subsequent basis to obtain alchemical estimates of electronic J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp observables for a combinatorially larger number of molecules in one shot. APDFT is straight-forward: The non-degenerate electronic ground state in the Born–Oppenheimer approximation E0(RI,ZI,Ne) is a continuous function of (i) the positions of the nuclei RI, (ii) the nuclear charges ZI, and (iii) the number of electrons Neof a molecule. Partial derivatives of observables such as E0with respect to the nuclear charges (constants RIand Ne) are called alchemi- cal derivatives and can be used to perturb a reference molecule and obtain efficient estimates of solutions for multiple target molecules via the Taylor series expansion. As such, isoelectronic alchemical changes take place within the same Hilbert space populated by all the solutions to Schrödinger’s equation for any combination of real values for { ZI,RI}. Obviously, a comparison to experimental real- izations is only meaningful when all nuclear charges assume integer positive values. Due to the Hellmann–Feynman theorem, the first alchemi- cal derivative can be obtained analytically and typically at neg- ligible additional cost.37,38The second derivatives of the ground state energy with respect to Ne,RI, and ZIand their mixed partial derivatives constitute a unified Hessian matrix,39which is a gen- eralization of the “geometrical” Hessian matrix in which only the derivatives with respect to the nuclear coordinates are included and which can be diagonalized in order to generate a complete basis in which chemical compound space can be expanded. The unified Hes- sian matrix also contains mixed derivatives with respect to nuclear charges and the total number of electrons∂2E ∂ZI∂Ne. If included in the Taylor expansion, these terms can be used to further increase the number of possible targets (or properties) from one reference calculation.39–43 While varying electron numbers are worthwhile and of inter- est, in principle, for this article, we limit ourselves to isoelectronic transmutations only for the following reasons: (i) The well-known derivative discontinuity, including fractional electron numbers, rep- resents a challenging quantity and carries its own additional com- plexity if the proper spin-state and multi-reference character is to be accounted for, and (ii) the vastness of chemical compound space does not originate in different numbers of electrons but rather in the combination of chemical elements and their molecular spatial configurations. As such, we consider the exploration of all com- pounds within a given fixed electron number to be the more pressing item. Rigorously rooted in Rayleigh–Schrödinger-perturbation the- ory, APDFT is an approximate method in practice. While other (uncontrolled) approximations are common in DFT, the approxi- mate nature of APDFT has not yet been explored in full. As such, it is highly desirable to obtain at least an estimate of the error associ- ated with alchemical estimates for given reference calculations and given perturbation orders. The control of prediction errors repre- sent an important goal for computational material design efforts, as meaningful trade-offs between a calculation’s cost and its accuracy can be established. Note that while, in principle, the concept of alchemical predic- tion is applicable to all electronic properties,17here we focus on the ground state energy E0. More specifically, we give a comprehensive discussion of the sources of errors from a practical point of view, i.e., also including those sources of errors that are not necessarily inherent to APDFT per se but rather result from restrictions inpresent quantum chemistry codes. We explain the relevance of the various sources of errors; in particular, we show that a substantial portion of the error results from neglecting the derivatives of the basis set coefficients in the APDFT Taylor expansion. We refer to this effect as “basis set error.” While universal basis sets exist, many of the more popular contracted Gaussian basis sets have different atomic orbitals for reference and target molecules and thus suffer from a substantial basis set error that can vary depending on the basis set size and the optimization technique used in its construction. Based on a set of diatomics, we calibrate a measure for the error of quantum alchemy and propose a correction for the basis set error. We vali- dated the results obtained for diatomics on pyridine, bipyridine, and triazine. Finally, we demonstrate that APDFT calculations of vibra- tional frequencies in dimers can be nearly as good as the reference method employed (CCSD) and more accurate than common DFT approximations. II. METHODS A. Alchemical perturbation density functional theory Following the earlier work17where details of the derivation are given, we define the alchemical coordinate 0 ≤λ≤1 as a linear trans- formation of the nuclear charges from the reference vector of nuclear charges ZRto the target ZT. For atom I, ZI(λ)≡ZR I+λ(ZT I−ZR I)=ZR I+λΔZI. (1) For isoelectronic changes, the total electronic ground state energy of the molecule is continuous and differentiable in λ.12The function E(λ) can be expanded as the Taylor series centered at λ= 0, i.e., at the reference molecule. With the energy of the reference molecule ER=E(λ= 0) and its derivatives, we can approximate the energy of the target molecule ET=E(λ= 1), ET=∞ ∑ n=01 n!∂nE(λ) ∂λn∣ λ=0=ER+∞ ∑ n=11 n!∂nE(λ) ∂λn∣ λ=0. (2) The first derivative can be evaluated via the Hellmann–Feynman theorem,44 ∂E ∂λ∣ λ=0=⟨ψR∣ˆHT−ˆHR∣ψR⟩, (3) which can be evaluated analytically for any wavefunction-based or density-based quantum chemistry method. Given the electron density, one finds12 ∂E ∂λ=∫Ωdr(vT(r)−vR(r))ρ(r,λ), (4) with the external potentials vRandvTcorresponding to reference and target systems, respectively. From this it is clear that Eq. (4) can be written in terms of higher order perturbations of the electron density alone,17 ∂n+1E ∂λn+1=∫Ωdr(vT(r)−vR(r))∂nρ(r,λ) ∂λn. (5) J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Combining Eqs. (5) and (2), one obtains ET=ER+∫Ωdr(vT(r)−vR(r))˜ρ(r) (6) —an orbital free density functional that is exact provided that (i) ER is exact and that (ii) the Taylor expansion converges and where the averaged density ˜ρis given by ˜ρ(r)=∑∞ n=11 n!∂n−1ρ(r,λ) ∂λn−1∣λ=0. The integrals 4 and 5 can be obtained from numerical integra- tion by projecting the one particle electron density on an integration grid or analytically by contracting the one particle density matrix with the nuclear attraction operator in the atomic orbital basis. Both methods are equivalent provided a suitable integration grid is used. In this work, we evaluate the derivatives of the electron density via finite differences on an evenly spaced five point stencil with Δλ = 0.05. Based on previous results, this stencil yields a good numer- ical stability.17The stencil coefficients have been calculated45such that the leading error term is O(Δλ5). In practice, the infinite sum from the Taylor expansion is truncated after the perturbation order n. Estimates made using all terms up to order nare denoted by APDFT n. B. Sources of prediction errors Calculating the alchemical derivatives via finite differences requires the evaluation of the density derivatives in the basis set of the reference. Consequently, APDFT derivatives actually build up the energy of the target molecule in the basis set of the reference molecule. Nearly all atom centered orbital basis sets have different coeffi- cients for the target and reference systems, with the atomic orbitals being optimized for each element. This results in a potentially noticeable error in the target energy, as pointed out earlier.46 The energy of the target molecule with the basis set of the refer- ence ET[R]is, in general, different and usually higher than the energy of the target with its optimized basis set ET[T]. In this paper, we define the energy error due to the use of the reference basis set ΔEBSas the difference between these energies, ΔEBS≡ET[R]−ET[T]. (7) In contrast to this error, the truncation of the Taylor series to a certain number of leading terms introduces other errors. This error contribution is equal to the difference between the predicted energy EAPDFTand the energy of the target molecule ET[R]evaluated with the basis set of the reference molecule, ΔEtrunc≡EAPDFT−ET[R]. (8) Besides these higher order terms, practical implementation details commonly introduce two more sources of error: (i) the error in the estimation of derivatives due to the finite difference scheme and (ii) the error from the numerical integration of the Coulomb interaction with the perturbed densities. These two errors can potentially be minimized by choosing a more numerically stable integration grid or a more accurate finite difference stencil.As such, we view the intrinsic error of APDFT nas the sum of two dominating contributions: one due to the truncation of the Tay- lor series at order nand the other one due to using the basis set of the reference molecule, ΔEAPDFT=EAPDFT−ET[T] =EAPDFT−ET[R]+ET[R]−ET[T] =ΔEtrunc +ΔEBS. (9) While the truncation order can be controlled through the increase in n(as long as the Taylor expansion converges), basis set effects have been studied less. C. Correction for atom-centered orbital basis sets For small to medium atomic basis sets, the error contributions ΔEBSthat arise from representing the density derivatives in the basis set of the reference molecule dominate the total deviation from a self-consistent evaluation of the target molecule. While the error seems to be reduced toward the complete basis set limit, a correction constitutes a worthwhile alterna- tive, enabling accurate absolute energy estimates while remaining cost-effective. Technically, these basis set errors can be avoided by including alchemical derivatives with respect to basis set coefficients.46This, however, would substantially increase the complexity of the prob- lem and the computational cost as we go to higher orders in the perturbation. Without any loss of generality, we note the alternatives that can be used to mitigate the problem of different basis sets for ref- erence and target molecules in the context of quantum alchemy. One possibility is to concatenate the basis sets of the reference and target.46Concatenating the basis sets produces overcomplete basis sets often with linear dependencies. From a technical perspective, the number of basis functions doubles, which is expensive but also hard to converge due to the linear dependencies. Conceptually, the atomic orbitals have become dependent on the reference system. Alternatively, one can also use the same basis for all the chemical elements, as demonstrated for or the Universal Gaussian Basis Set (UGBS),47a non-contracted Gaussian basis set in which the expo- nents are shared among all the elements. This is not only conceptu- ally appealing but also quite common for quantum calculations in the condensed phase using plane wave pseudo-potential simulations in periodic boundary conditions.48,49The drawback of the universal basis functions is that they need to be very large, and, in the case of an atomic basis, the lack of polarization functions worsens the convergence. In this article, we present a general way to correct for ΔEBS through a correction that can be implemented readily for less exotic basis sets and quantum chemistry methods. To motivate the correction, we observe that the largest contri- bution to the total electrostatic energy of the molecule is given by the core orbitals; thus, we can also assume that the main contribution in ΔEBSis due to the change in the core orbitals’ coefficients. This is both because the core orbitals have a larger contribution to the total energy than valence orbitals and because they are less flexible than their valence counterparts. J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IfΔEBSis predominantly unaffected by the chemical environ- ment, we can approximately estimate it using the full dissociation limit, i.e., the individual free atoms. For each atom I, an atomic contribution ΔEBS,Iis collected, the sum of which amounts to our approximate correction of ΔEBS, ΔEBS≈∑ IET[R] I−ET[T] I≡∑ IΔEBS,I. (10) For every atom, EBS,Iis the difference between the free atom energy with the basis set of the reference element for that site minus the energy of the free atom with its proper basis set. This correction has no effect on the shape of the potential energy surface, and it does improve total energies with limited and constant additional computational effort, i.e., a single free atom cal- culation for each target element is required. For example, in order to study BN-doping in sp2-hybridized carbon sites, four such elemental calculations are necessary, regardless of the number of target com- pounds: EB[B],EB[C],EN[N],EN[C]. Here, all free atom calculations have been performed in their lowest electronic spin state. However, certainly of interest, the study of alchemical changes involving other spin-state combinations is so substantial that it warrants a separate publication. III. RESULTS As the test case for this work, we consider all pairwise APDFT predictions that can be made within the series of neutral diatomic molecules with 14 electrons (HAl, HeMg, LiNa, BeNe, BF, CO, and NN). This set not only covers the elements most relevant to organic chemistry but also allows us to assess APDFT predictions between many different elements. All of these molecules are iso- electronic, therefore the energy of any molecule in this set can be estimated via alchemical transformation from any other molecule in this set. To account for contributions from non-equilibrium geometries, we consider 20 bond lengths from 1.3 bohrs to 3.2 bohrs. Since the accuracy of APDFT depends significantly on the basis set, we performed all calculations for major basis set families. Within each family, we chose two representatives: one smaller and another more expanded basis set. This allows us to compare the performance of APDFT both within and between basis set families. The employed basis sets are STO-3G, STO-6G,50Pople’s split valence (3-21G and 6- 31G∗),51,52Dunning’s correlation consistent (cc-pVTZ and aug-cc- pVQZ),53,54and the Karlsruhe (def2-TZVP and def2-QZVPP)55,56 basis sets. The dataset for the prediction up to APDFT5 within the 14 electron diatomic series is made available under open access.57 All calculations employ CCSD in the frozen core approxima- tion using a RHF reference. For changes in covalent bonding in vicinity to the equilibrium and for the closed-shell systems stud- ied, we assume that CCSD is sufficiently accurate. CCSD is also used for its desirable properties of being size-consistent, account- ing for electronic correlation, and representing the first step in the cluster expansion. In order to systematically reach higher accu- racy, e.g., to model van der Waals interactions, within converged APDFT estimates, perturbations of coupled-cluster single double triple (CCSDT) based densities will be required.For comparison in Sec. III E, DFT calculations with B3LYP and PBE functionals were performed. Unless otherwise noted, we used the quantum chemistry software MRCC.58,59 The calculations of the response density matrix in the atomic basis for benzene in Sec. III F were implemented using PySCF.60 A. Error from the reference basis set For all alchemical absolute energy estimates within the diatomic isoelectronic series ( Ne= 14) for which the nuclear charges of the reference and target molecule differ at most by 2, Fig. 1 shows the comparison between the basis set error ΔEBSand the total error for all the different contributions for APDFT perturbation orders ranging from 2 to 5. Each panel corresponds to a different basis set. For all basis sets and perturbation orders, we see substantially larger errors for ΔZ=±2 than for ΔZ=±1. This is not surprising since the former also constitutes a much larger perturbation than the latter. In view of the scales and the correlation between the over- all APDFT error and the basis set contribution thereto, this suggests that the basis set contribution constitutes the dominating source of error. Moreover, the results indicate that the contribution is highly systematic, especially when it comes to the smaller perturbation (ΔZ=±1). For some basis sets (3-21G, 6-31G∗, and aug-cc-pVQZ), we observe clusters of errors due to ΔEBSbeing mostly constant for neighboring elements. In the case of aug-cc-pVQZ, the cluster is subdivided, as the contribution for ΔEBSfor elements within the second period is smaller than the contribution for those in the third period. Also as | ΔZ| increases, we see less correlation between the total error and the basis set contribution. This is predom- inantly caused by the truncation of the Taylor series as shown by the higher order predictions being much more consistently off. In rare cases, those two contributions become comparable in magnitude. The convergence with expansion orders is slower with the min- imal STO or Pople basis sets. Interestingly, one consequence is that higher order energies are not always an improvement over lower orders. We attribute this to the representability of the alchemical density derivatives in those basis sets. Consequently, the overall magnitude of the errors in these basis sets is significantly higher than for the Karlsruhe basis sets, and we note that due to such erratic behavior, the use of APDFT in small basis set calculations is notto be recommended. In the cases of aug-cc-pVQZ, def2-TZVP, and def2-QZVPP, we obtained some negative basis set errors. Those correspond to the alchemical transformations BeNe →HeMg, BeNe →LiNa, and BF →LiNa. In these transformations, the total number of core electrons decreases from the target to reference. Since we used frozen core CCSD, a reduction of the correlation energy is observed, consequently leading to a total basis set error of negative sign. For some noticeable outliers in Fig. 1, HeMg →BeNe and HAl →LiNa and even fourth order contributions for the largest basis set still yield significant residual differences. Note that in all these cases, an atom from the first period is used as a reference for an atom from the second period with a substantially increased electronic extend. Consequently, we must be cautious against the use of APDFT across periods when using nuclear charges only. J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . The scatter plot shows the correlation between ΔEBSandΔEAPDFT for alchemical transmutations with | ΔZ| = 1 (solid markers) and | ΔZ| = 2 (hollow mark- ers), respectively. Every pair of reference-target molecules within the 14 electron diatomic series is represented by a point for each APDFT order. The error shown is the median of all considered geometries. B. Quantifying the basis set derivative error Since for some dimer pairs in this work the number of core electrons is not constant under alchemical transformation that can lead to an additional error for frozen-core CCSD, we divided our dimer set into two subsets: HAl and HeMg as well as BeNe, BF, CO, and NN. Within each subset, the number of core electrons remains constant under alchemical transformation between any set element.All sources of error depend on the basis set chosen for the reference calculation. By and large, the basis set error tends to be smaller for more expanded basis sets, approaching zero toward the complete basis set limit. Increasing the number of contracted Gaus- sians per atomic orbital has a little effect in reducing ΔEBS, as shown in Fig. 2, for the STO-3G/STO-6G pair. Similarly, the inclusion of polarization functions has little effect for these linear molecules; passing from the 3-21G basis set to the 6-31G∗basis set improves the quality of the results, in general, but ΔEBSremains comparably large. Adding more valence functions, in general, lowers ΔEBS: we observed that for the pairs cc-pVTZ/aug-cc-pVQZ and def2- TZVP/def2-QZVPP, going from triple to quadruple zeta basis sets leads to a substantial reduction of ΔEBS. Using large basis sets might be advised in any case since not only APDFT but also the quality of the underlying quantum method might be better. In the end, the basis set choice should also depend on the systems analyzed and on the quantum chemistry method used, e.g., for good description of hydrogen atoms in a molecule diffuse functions are generally required. As shown in Fig. 2 and Table I B, we find the Karlsruhe def2 basis set family to be the most reliable in the context of alchemical changes as they outperform any other basis set family, in particular, considering their number of basis functions. The Karlsruhe basis set def2-TZVP has a basis set contraction scheme (11 s, 6p, 2d, 1f)→[5s, 3p, 2d, 1f] similar to the cc-pVTZ basis set (10 s, 5p, 2d, 1f)→[4s, 3p, 2d, 1f]. Even though they differ only by one sbase function, the error for second row elements in cc-pVTZ is fivefold higher than in def2-TZVP. If we look at the quadruple-zeta basis set, def2-QZVPP with a second period contraction scheme of (15 s, 8p, 3d, 2f, 1g)→[7s, 4p, 3d, 2f, 1g] is a substantially smaller basis set than aug-cc-pVQZ with a contraction scheme of (16 s, 10p, 6d, 4f, 2g)→[9s, 8p, 6d, 4f, 2g]. Despite def2-QZVPP being smaller in the number of basis functions, it performs better than aug-cc-pVQZ for APDFT, with typically half FIG. 2 . Dependence of ΔEBSon the basis set type and size. For alchemical trans- mutations within diatomics containing elements of the second period (BeNe, BF, CO, and NN) grouped by | ΔZ| are shown the median, the 10th, and 90th percentile ofΔEBS. J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Median of ΔEBS[mHa] for second period diatomics with ΔZ=±1, 2 and for the pair HAl and HeMg ( ΔZ=±1). Basis set ΔZ=±1 ΔZ=±2 HAl ↔HeMg STO-3G 2215 8519 1684 STO-6G 2238 8460 1687 3-21G 1855 6956 1775 6-31G∗1862 6947 1762 cc-pVTZ 553 2382 1260 aug-cc-pVQZ 23 190 1144 def2-TZVP 105 552 1081 def2-QZVPP 12 89 100 the error for ΔEBS. The reasons for this may be found in the opti- mization procedures of these basis sets. For example, the first order energy derivative for vertical changes by virtue of the chain rule can be written as ∂E ∂λ=∑ I∂E ∂ZI∂ZI ∂λ+∑ i∂E ∂ci∂ci ∂λ, (11) where the second sum represents the contribution of the basis set coefficients cias they change from the reference to the target molecules. By construction,55,56the first order terms for HF are zero for the def2 basis set family, which might contribute to the good performance in the context of APDFT. As discussed above, we treated the alchemical transmutation HAl↔HeMg independently. In that case, the best results were obtained with the def2-QZVPP basis set. In this case, the basis set error is 100 mHa, one order of magnitude smaller than the error for any other basis set. FIG. 3 . Systematic reduction in signed error ET[T]−ET[R]due to the basis set correction we suggest, here shown for def2-QZVPP and the diatomic pairs of the second period. For every alchemical transmutation (labeled reference/target in the figure), different interatomic distances lead to slightly different errors shown in box- plots. The red line is the median of the distribution, and the arrows indicate the shift due to the correction outlined in Sec. II C.C. Correcting the basis set derivative error As shown above, choosing appropriate atomic orbital basis sets can significantly reduce the overall error in APDFT. Still, even in our best case (def2-QZVPP), such error is on the order of tens of milli-Hartree, while for other smaller basis sets, the error is up to two orders of magnitude larger. With the simple correction (see Sec. II C), however, this error can be substantially reduced for all basis sets, as shown in Fig. 3. Due to the inclusion of several interatomic distances for each dimer in our dataset, we always have a distribution of errors that covers the thermally accessible range. Since the correction is based on free atoms, it is independent of the interatomic distances. Con- sequently, the correction always moves the whole error distribu- tion. As shown in Fig. 3, the signed error is consistently improved since the largest error source is approximately corrected for. It is remarkable that for the cases with ΔZ=±1, the basis set error can be reduced to few milli-Hartrees for our large span of interatomic distances. It is worth noting that errors in Fig. 3 differ between a pre- diction in one direction and its inverse. With the exception of the BeNe/BF pair, errors are smaller for the predictions that lead to the most similar diatomic (e.g., the error in the prediction CO →N2 is lower than the one of the prediction N 2→CO). For the trunca- tion error, a similar effect has already been observed, discussed, and explained.39These two effects are independent since the error in the Taylor expansion is related to the shape of the function E(Z), while the reasons for the basis set error are rooted in the construction and the optimization of the basis set coefficients and is outside the scope of our article. Future comprehensive work regarding the detailed impact of individual basis functions on this asymmetry in alchemi- cal basis set errors might provide further insights into this trend. With the exception of the case BeNe →BF, the correction generally overestimates the error. In molecules, the orbitals placed on different atoms can overlap (superposition). Due to this, the degrees of freedom in the description of the electronic structure of each atom are more complete in the molecule than as an isolated atom. This effect leads to the well-known basis set superposition error (BSSE61) in predicting dissociation energies. In our case, this becomes visible as ΔEBSis lower for molecular species than for iso- lated atoms and the correction then tends to slightly overestimate the error. D. Truncation error As with any truncated series expansion, APDFT suffers from truncation errors, i.e., the total contribution from higher order terms in the series expansion ΔEtrunc=EAPDFT−ET[R]. (12) Ana priori estimation of the truncation error would be highly valu- able to judge the accuracy of APDFT estimations and, consequently, to define a trust region similar to a convergence radius where the extent of the region is derived from the required accuracy with regard to the reference method. J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Based on the earlier work for crystal systems,13we propose to use the integrated absolute electron density difference Δρ≡ ρT(r)−ρR(r) between the reference and target as a proxy of the error, ∥Δρ∥≡∫Ωdr∣Δρ(r)∣. (13) Since alchemical energy derivatives are evaluated through the alchemical perturbations of the electron density, two systems are close to each other if and only if they possess similar electron densi- ties. In this way, a small density change in the alchemical path results in a small error. In the limit of ∥Δρ∥=0, the reference and tar- get are identical and the error in alchemical energy prediction is zero. Note that this integral is only weakly sensitive to changes in the level of theory or a basis set, as the corresponding differences in electron density are minute. This allows for simple evaluations in practice where the electron density of a low level of theory or even atomic densities can serve as a substitute of the self-consistent electron density for the purpose of error estimation. Figure 4 shows that the error of APDFT correlates with ∥Δρ∥for APDFT2 to APDFT5 for representative basis sets. The FIG. 4 . Correlation between ΔEtruncand the electron density displacement ∥Δρ∥. Each panel shows the median of the error for one basis set for the different trunca- tion orders in APDFT. For comparison, horizontal lines denote the mean absolute errors of Hartree–Fock and MP2 calculations, respectively.consistent trends observed in that figure allow us to give an empir- ical formula for the unsigned truncation error. We see that this error increases exponentially with the integrated absolute charge difference for diatomic molecules. Different expansion orders yield different steepness of this relation only. To put the focus of the empirical error estimation on small truncation errors, we propose to fit a linear function to the logarith- mized data, log(∣ΔEtrunc∣)≈α+β∥Δρ∥, (14) where the coefficients αandβare found via a fit for each APDFT order and basis set. Table II shows the resulting fits. With this empir- ical fit, we can estimate the expected error for APDFT numbers without any additional DFT calculations. It is of great value, in prac- tice, to have an expected uncertainty associated with every APDFT energy. For more complex molecules, we can consider the size- consistency of APDFT and its error. If more than one transmuta- tion is done on non-interacting independent sites, the total error is decomposed in a sum of site-defined contributions. Due to the symmetry of the dimers,17the error is similar for the second and third order as well as for the fourth and fifth order. For small ∥Δρ∥, higher order APDFT performs substantially bet- ter than lower order ones, while for large ∥Δρ∥, the error obtained with different APDFT orders becomes comparable. We attribute this to the finite precision of the density derivatives as obtained from finite differences where for larger changes in ΔZ, numerical noise is amplified. In Fig. 4, the comparison of the truncation error between dif- ferent basis sets gives results similar to those obtained for the basis set error earlier. In fact, more expanded basis sets yield lower ΔEtrunc than smaller ones and the triple and quadruple zeta basis sets give very good results: for APDFT5 with ∥Δρ∥<5e, the error is less than 10 mHa. The best performance again was obtained with the def2 basis set: for def2-QZVPP APDFT performs better than MP2 up to ∥Δρ∥ =6e(indicative of ΔZ≤2) and performs better than HF up to ∥Δρ∥ =8e(ΔZ= 3). TABLE II . For the fitting parameters αandβthat describe the total truncation error for APDFT of order naccording to Eq. (14), the data are extracted from the binned values, as shown in Fig. 4 and in Fig. S1. Basis set nα[log(Ha)] β[log(Ha) e−1] R2 STO-6G 5 −3.161 0.522 0.95 3-21G 5 −3.434 0.480 0.98 6-31G∗5 −3.572 0.480 0.97 cc-pVTZ 5 −3.634 0.458 0.92 def2-TZVP 5 −3.828 0.460 0.96 aug-cc-pVQZ 5 −3.743 0.416 0.97 def2-QZVPP 2 −2.355 0.273 0.84 def2-QZVPP 3 −2.806 0.314 0.98 def2-QZVPP 4 −3.643 0.388 0.94 def2-QZVPP 5 −3.973 0.442 0.92 J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp For transmutations that do not involve a large electron dis- placement, we note that the most important term for error estima- tion in Eq. (14) is the leading term αthat indicates the intercept in the regression line. From Table II, we can compare the performance of the most relevant basis sets taken in analysis. There is a continuous improvement in accuracy passing from a minimal basis set to a double ζbasis set to a triple and a quadruple ζ basis set. A split valence basis set can represent molecules with frac- tional nuclear charges better as the additional basis set flexibility is associated with a smoother density response and consequently gives a more accurate energy expansion. The effect of basis sets on the accuracy is also influenced by the type of systems investigated. For instance, the effects of polarization functions (passing from the 3-21G basis set to the 6-31G∗basis set), as shown in Table II, are visible but might be more stark for a non- linear system. An extended version of Table II and Fig. 4 for the other basis sets considered can be found in the supplementary material. E. Vibrational frequencies Accurate estimates of potential energy surfaces open up access to the application in vibrational spectroscopy. Even more so, if the residual errors from either higher order terms or basis set intrica- cies are systematic around the minimum geometry configuration.62 Starting from a scan of interatomic distances for one molecule, we have tested this application for APDFT energies at the same inter- atomic distance of a different isoelectronic molecule from our set. We then interpolated these points with a cubic spline to obtain both the curvature and minimum of the target molecule. From the curvature, we calculated vibrational frequencies in the harmonic approximation. We used CCSD as a reference method, and we expect our results to be most accurate in thermally accessible regions of the bond dissociation curve, assuming good description of properties around the minimum of the PES. For a more accurate description of the whole dissociation curve, especially toward the dissociation limit, a multi reference method would likely be required. To the best of our knowl- edge, APDFT has not been extended to multi reference methods yet. Figure 5 shows the error made by APDFT in comparison with the result that can be obtained with other computational meth- ods. The alchemical prediction is accurate up to 2 kcal/mol for the dissociation energies, while the equilibrium bond distance is correct up to 0.01 bohr. Consequently, the dissociation profiles of the alchemical estimations overlay with the self-consistent CCSD profiles. In part, this is achieved; thanks to the basis set correc- tion described in Sec. II C that shifts the profile by a fixed energy, leaving minimal bond distances and the vibrational frequencies unchanged. The dissociation profiles for other methods do not agree well with CCSD results. In particular, MP2 and PBE yield errors for dis- sociation energies of up to 20 kcal/mol compared to CCSD. In the right part of Fig. 5, we can see that vibrational frequencies from APDFT are closer to the CCSD reference calculations than other established methods. For both energies and vibrational frequencies, we observe that APDFT using perturbations of a higher level of FIG. 5 . Homolytic bond dissociation energy EBDand harmonic vibrational frequen- ciesνharmof CO, NN, and BF calculated at the CCSD, MP2, HF, PBE, B3LYP, and APDFT5 level of theory. For APDFT, the basis set correction from Sec. II C is included. All data are shown for the def2-QZVPP basis set. theory can consistently outperform self-consistent results of lower levels of theory, as shown in Table III. This is particularly remark- able for vibrational frequencies that are much less affected by rel- ative errors then total energies. Among the APDFT predictions, those are more accurate when the reference and target have similar electronic structures, e.g., the predictions from CO to N 2and vice versa are better than the predictions from CO to BF and from BF to CO. F. APDFT derivatives on benzene Finally, we show how the results from the dimer case can be transferred to larger molecules. To this end, we investigate three targets: pyridine, pyrimidine, and triazine. With the reference molecule benzene being of the D 6hsym- metry, and all sites being equivalent, we can reduce the number of derivatives that need to be calculated. For a APDFT3, only three derivatives of the electron density are needed to predict all targets, namely, the first∂ρ ∂Z1, the second∂2ρ ∂Z2 1, and the second mixed∂2ρ ∂Z1∂Z2, all of which were obtained by finite differences. The basis set correction described in Sec. II C was obtained subtracting the CCSD energy of a nitrogen atom with its proper basis functions from the CCSD energy of an isolated nitrogen atom with the basis set of a carbon atom and a ghost hydrogen, J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Comparison of the dissociation profile in terms of homolytic bond disso- ciation energy [kcal/mol], bond length [Bohr], and vibrational frequencies [cm−1] for different APDFT orders, HF, MP2, CCSD, and the DFT functionals, PBE and B3LYP. Method Bond energy Bond length νharm CO→NN APDFT1 1827.97 1.90 3118 APDFT2 −210.76 2.07 2425 APDFT3 −219.71 2.08 2388 APDFT4 −214.22 2.07 2445 APDFT5 −214.29 2.07 2444 HF −117.32 2.01 2728 MP2 −235.28 2.10 2197 PBE −243.22 2.08 2347 B3LYP −228.07 2.06 2453 CCSD −213.43 2.07 2439 NN→CO APDFT1 1805.18 1.93 2954 APDFT2 −250.73 2.12 2270 APDFT3 −250.73 2.12 2270 APDFT4 −248.22 2.13 2222 APDFT5 −248.22 2.13 2222 HF −175.58 2.08 2427 MP2 −269.41 2.14 2133 PBE −268.94 2.14 2134 B3LYP −254.52 2.12 2211 CCSD −247.87 2.12 2234 BF→CO APDFT1 1781.15 1.97 2800 APDFT2 −232.08 2.10 2358 APDFT3 −252.25 2.13 2247 APDFT4 −250.84 2.13 2218 APDFT5 −249.58 2.13 2228 CO→BF APDFT1 1893.64 2.08 2250 APDFT2 −195.06 2.44 1314 APDFT3 −183.34 2.42 1340 APDFT4 −174.49 2.38 1441 APDFT5 −174.34 2.38 1420 HF −137.70 2.35 1508 MP2 −188.87 2.39 1417 PBE −186.79 2.41 1350 B3LYP −179.79 2.38 1399 CCSD −176.17 2.39 1421 ΔEcorr=E[N][C-H]−E[N][N]. (15) The value of ΔEcorrwas subtracted once for every CH to N transmutation in the alchemical transformation. Based on the results in Sec. III B, we chose the def2-TZVP basis set as a compromise between cost and accuracy. Figure 6 and Table IV show errors and corrections for the predictions of pyridine, bipyridine, and triazine. FIG. 6 . Errors in the prediction of the total energies of pyridine, pyrimidine, and tri- azine: basis set effect ΔEBSand the error in alchemical prediction ΔEAPDFT3 before and after the correction proposed in this work. For comparison, unsigned errors are shown as well. APDFT3 energies are calculated from a benzene reference using CCSD/def2-TZVP. TABLE IV . Errors and corrections in the APDFT3 predictions of pyridine, pyrimidine, and triazine from benzene using CCSD/def2-TZVP, as shown in Fig. 6. Pyridine Bipyridine Triazine ΔEtrunc −8.73 −17.98 −27.74 ΔEBS 55.09 110.37 165.90 ΔEAPDFT 46.35 92.38 138.16 Correction −65.64 −131.28 −196.92 Error after correction −19.28 −38.90 −58.76 From the results, we can see that ΔEBSis roughly six times larger than ΔEtrunc and constitutes the dominating source of error, and both errors increase with an approximately linear trend with the number of transmuted atoms. The basis set correction obtained overestimates the basis set error by a sixth, as observed earlier in Sec. III C. The truncation error is negative in sign, therefore after the correction, it cancels to some degree with the error in the correction; nevertheless, the corrected prediction is still improved by a factor of three. If we consider the total difference in the energy of benzene and pyridine of 16.019 Ha, it is remarkable that the error with APDFT3 of 19.28 mHa constitutes 0.12% of the total energy difference. IV. CONCLUSION In this work, we analyzed prediction errors in APDFT12,17in terms of atomic basis set effects as well as the perturbation order in the context of its application to energetics and vibrational frequen- cies in isoelectronic diatomics, as well as for mutating benzene to pyridine, pyrimidine, and triazine. Our numerical results indicate that absolute energy estimates are dominated by two main sources of error: truncation of the Tay- lor expansion and differences in basis sets between the reference and target compound. The error due to the basis set is an issue for all the element-specific basis sets: for small Gaussian contracted basis sets, this error can be substantial, while parametrically optimized basis sets such as the Karlsruhe basis sets yield reasonably accurate alchemical derivatives. J. Chem. Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We showed that this error is reduced with the increase in basis set size and we expect it to vanish in the limit of a complete basis set. Furthermore, as noted before, if reference and target molecules have identical basis functions, this error is zero. For the use of the more commonly used and well tested con- tracted Gaussian orbital basis sets, we proposed a single atom cor- rection to the error that is easily implemented at a very limited computational cost. The error due to the Taylor expansion truncation has been shown to be related to the total displacement of electronic charge between the reference and target molecules. We provide linear fits as an empirical relation between the two, which can help to estimate the expected accuracy of an APDFT energy up to order n= 5. We show that this error for small || Δρ|| and for APDFT5 can be as small as 1 kcal/mol for the largest basis set considered. Regarding the prediction of vibrational frequencies, we have observed that APDFT based on CCSD calculations can reproduce the shape of dissociation curves better than generalized gradient approximation (GGA) (at the second order for ΔZ= 1) and B3LYP or MP2 (at the fifth order). Consequently, quantities derived from these curves, such as equilibrium distances or vibrational frequen- cies, are also predicted more accurately using APDFT than using DFT. While obtaining the alchemical derivatives from finite differ- ences is expensive for higher orders, we showed in that good predic- tions can regularly be achieved already at the APDFT3 level—even for vibrational frequencies. In this work, we use n+ 1 additional sin- gle points for all APDFT terms up to and including the n-th order. APDFT becomes cost effective for problems with a large number N of transmutation sites. APDFT3 requires N2+N+ 1 single point calculations without any symmetry, while the number of possible targets increases in a combinatorial manner ( ≃2N). Symmetry can be used to further reduce the required number of reference QM calcula- tions.17This scaling behavior renders APDFT suitable for exploring large chemical spaces from much fewer reference calculations than potential targets. SUPPLEMENTARY MATERIAL The complete set of parameters from fitting the truncation error for all basis sets considered (extending Table II) as well as the equivalent of Fig. 4 for all other basis sets are given in the supplementary material. ACKNOWLEDGMENTS O.A.v.L. acknowledges funding from the Swiss National Sci- ence Foundation (Grant No. 200021_175747) and from the Euro- pean Research Council (ERC-CoG Grant QML). This work was partly supported by the NCCR MARVEL, funded by the Swiss National Science Foundation. 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Phys. 153, 144118 (2020); doi: 10.1063/5.0023590 153, 144118-11 Published under license by AIP Publishing

5.0023992.pdf

Appl. Phys. Lett. 117, 142405 (2020); https://doi.org/10.1063/5.0023992 117, 142405 © 2020 Author(s).Quantification of interfacial spin-charge conversion in hybrid devices with a metal/ insulator interface Cite as: Appl. Phys. Lett. 117, 142405 (2020); https://doi.org/10.1063/5.0023992 Submitted: 03 August 2020 . Accepted: 11 September 2020 . Published Online: 06 October 2020 Cristina Sanz-Fernández , Van Tuong Pham , Edurne Sagasta , Luis E. Hueso , Ilya V. Tokatly , Fèlix Casanova , and F. 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Sebasti /C19anBergeret1,5,a) AFFILIATIONS 1Centro de F /C19ısica de Materiales (CFM-MPC), Centro Mixto CSIC-UPV/EHU, 20018 Donostia-San Sebasti /C19an, Basque Country, Spain 2CIC nanoGUNE, 20018 Donostia-San Sebasti /C19an, Basque Country, Spain 3IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Basque Country, Spain 4Nano-Bio Spectroscopy Group, Departamento de F /C19ısica de Materiales, Universidad del Pa /C19ıs Vasco (UPV/EHU), 20018 Donostia-San Sebasti /C19an, Basque Country, Spain 5Donostia International Physics Center (DIPC), 20018 Donostia-San Sebasti /C19an, Basque Country, Spain a)Authors to whom correspondence should be addressed: cristina_sanz001@ehu.eus ;v.pham@nanogune.eu ;ilya.tokatly@ehu.es ; f.casanova@nanogune.eu ; and fs.bergeret@csic.es ABSTRACT We present and experimentally verify a universal theoretical framework for the description of spin-charge interconversion in non-magnetic metal/insulator structures with interfacial spin–orbit coupling (ISOC). Our formulation is based on drift-diffusion equations supplemented with generalized boundary conditions. The latter encode the effects of ISOC and relate the electronic transport in such systems to spin loss and spin-charge interconversion at the interface. We demonstrate that the conversion efficiency depends solely on these interfacial parame-ters. We apply our formalism to two typical spintronic devices that exploit ISOC: a lateral spin valve and a multilayer Hall bar, for which wecalculate the non-local resistance and the spin Hall magnetoresistance, respectively. Finally, we perform measurements on these two devices with a BiO x/Cu interface and verify that transport properties related to the ISOC are quantified by the same set of interfacial parameters. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023992 A thorough understanding of charge and spin transport in sys- tems with spin–orbit coupling (SOC) is crucial for the electric control of spin currents.1,2The latter leads to the widely studied spin Hall effect (SHE)3–5and Edelstein (EE) effect,6–9which are at the basis of spin–orbit torque memories10–12and spin-based logic devices.13,14 Of particular interest are systems with spin-charge interconver- sion (SCI) at the interface between an insulator (I) with a heavy ele- ment and a normal metal (N) with negligible SOC and long spin relaxation length as, for example, BiO x/Cu bilayers. In these systems, the SCI occurs at the hybrid interface via an interfacial spin–orbit cou- pling (ISOC).15,16Whereas the electronic transport in N is well described by customary drift-diffusion equations, the interfacial effectsoccur at atomic scales near the interface and, hence, their inclusion is more subtle. Some works use an intuitive picture based on an idealized 2DEG with Rashba SOC at the interface, 16–18in which the intercon- version takes place via the EE and its inverse (IEE). Such a descriptionis clearly valid for conductive surface states in (e.g., topological) insula- tors19,20or 2DEGs.21,22However, in metallic systems, it requires addi- tional microscopic parameters to model the coupling between interface states and the diffusive motion of electrons in the metal. Moreover, thevery existence of a well-defined two-dimensional interface band and its relevance for the electronic transport in systems such as BiO x/Cu is not obvious as realistic structures are frequently polycrystalline and disor-dered. Moreover, one can contemplate other microscopic scenarios to describe the SCI. For example, at the BiO x/Cu interface, Bi atoms could diffuse into Cu inducing an effective extrinsic SHE in a thin layer nearthe interface. 23Alternatively, a SCI can be generated via an interfacial spin-dependent scattering of the bulk Bloch states.24–26Each of these scenarios will invoke different sets of microscopic parameters to beinferred from macroscopic transport measurements. In this Letter, we approach the problem from a different angle and propose a universal theoretical framework, which is independent Appl. Phys. Lett. 117, 142405 (2020); doi: 10.1063/5.0023992 117, 142405-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplof microscopic details. We combine the drift-diffusion theory with effective boundary conditions (BCs)27to account for ISOC. Such BCs describe two types of interfacial processes: SCI and spin-losses, quanti- fied, respectively, by the interfacial spin-to-charge/charge-to-spin con-ductivities, r sc=cs, and the spin-loss conductances Gk=?for spins polarized parallel/perpendicular to the interface. The SCI efficiency is determined by the ratio between the strengths of these two processes. This ratio coincides with the widely used conversion efficiency and theinverse Edelstein length k IEEsuch that kIEE¼rsc=Gk.F u r t h e r m o r e ,w e apply our theory to describe two typical experimental setups: non-localresistance measurement in a Permalloy/copper (Py/Cu) lateral spin valve (LSV) with a middle BiO x/Cu wire, Fig. 2(a) ,a n dm e a s u r e m e n t of the spin Hall magnetoresistance (SMR) in a BiO x/Cu/YIG trilayer Hall bar, Fig. 4(a) . From the fitting of our theory to the experimental results, we show that both experiments are described by similar valuesof the ISOC parameters. This confirms that the SCI only depends on the intrinsic properties of the BiO x/Cu interface. Moreover, we demon- strate that rsc¼rcs, in accordance with Onsager reciprocity. We start considering the I/N structure depicted in Fig. 1 .I nt h e N layer, spin and charge transport is described by the diffusionequations, r 2^l¼^l k2 N; (1) r2l¼0: (2) Here, ^l¼ðlx;ly;lzÞandlare the spin and charge electrochemical potentials (ECP), where the symbol ^:indicates spin pseudovector. It isassumed that N has inversion symmetry with an isotropic spin relaxa- tion described by the spin diffusion length kN.28The diffusive charge and spin currents are defined as e^j¼/C0rNr^land ej¼/C0rNrl, respectively, with e¼/C0 j ejandrNthe conductivity of N. Equations (1)and(2)are complemented by BCs at the interfaces. At the interface with vacuum, one imposes a zero current condition,whereas at the I/N interface with ISOC, the BCs for the spin and charge densities read: 27 /C0rNðr /C1 nÞ^lj0¼G?^l?j0þGk^lkj0þrcsn/C2rðÞ lj0;(3) /C0rNðr /C1 nÞlj0¼rscn/C2rðÞ ^lj0: (4) Here, nis the unitary vector normal to the interface, see Fig. 1(a) . The last term in the rhs of Eq. (3)describes the charge-to-spin con- version quantified by the conductivity rcs. This term couples an e f f e c t i v ee l e c t r i cfi e l da n dt h e( o u t g o i n g )s p i nc u r r e n td e n s i t ya tthe interface 27,29–31and can be interpreted as an interfacial SHE. Alternatively, it can be interpreted as if the electric field induces a h o m o g e n e o u ss p i nE C Pv i aa ni n t e r f a c i a lE E ,w h i c hi nt u r nd i f - fuses into N. Both interpretations are fully compatible within thepresent formalism. The second type of process taking place at theinterface are spin-losses [first two terms in the rhs of Eq. (3)], quantified by the spin-loss conductances per area G ?=kfor spins perpendicular/parallel ( ^l?/^lk)t ot h ei n t e r f a c e . The charge is obviously conserved and, therefore, the rhs of Eq.(4)only contains the spin-to-charge conversion term. The latter is the reciprocal of the last term in Eq. (3)32and can be interpreted as an interfacial inverse SHE but, again, an alternative interpretation is pos-sible: from the conservation of the charge current at the interface, wecan relate the bulk charge current to the divergence of an interfacialcurrent j IasrNðr /C1nÞlj0¼/C0 er/C1jI. Comparing the latter with Eq.(4),w ed e fi n e jIas33 ejI¼/C0rscn/C2^lðÞ j0: (5) Written in this way, Eq. (4)describes the conversion of a non- equilibrium spin into an interfacial charge current, which correspondsto an interfacial IEE, see Fig. 1(b) .T h i si n t e r p r e t a t i o na l l o w su st o introduce the commonly used conversion length k IEE,d e fi n e da st h e ratio between the amplitude of the induced interfacial charge current density, jI, and the amplitude of the spin current injected from the bulk, rNðr /C1nÞ^lj0. According to Eq. (5), the effect is finite only if the spin current is polarized in a direction parallel to the interface. UsingEqs.(3)and(5),w eo b t a i n k IEE¼rsc Gk: (6) This is a remarkable result that follows straightforwardly from our description of hybrid systems with ISOC and for which the spin-charge interconversion occurs only at the interface. k IEEis purely determined by interfacial parameters and it is indeed a quantification of the conversion efficiency: it is the ratio between the spin-to-chargeconversion and the spin-loss at the interface. Both parameters, r scand Gk, depend on the microscopic properties of the interface, which are intrinsic for each material combination, and may depend on temperature. From an experimental perspective, the spin-to-charge conversion is usually detected electrically, by measuring a voltage drop [see FIG. 1. Sketch of the non-magnetic insulator ( z>0)/metal ( z<0) system under study. ISOC is finite at the interface with normal vector n. (a) Charge-to-spin con- version: a charge current Icinduces at the interface an out-of-plane spin current density ^jdiffperpendicularly polarized. (b) Spin-to-charge conversion: a ninjected spin current density ^jdiffinduces at the interface a voltage drop perpendicular to the polarization of ^jdiff.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 142405 (2020); doi: 10.1063/5.0023992 117, 142405-2 Published under license by AIP PublishingFigs. 2(a) and4(a)]. For concreteness, we consider the generic setup of Fig. 1(b) : a spin current polarized in the xdirection flows towards the interface, and a voltage difference is generated in the transverse y direction according to Eq. (5). The averaged voltage drop between the points y¼6Ly=2 is given by (see supplementary material Note S1) Vsc¼rsc erNANððLy 2 /C0Ly 2n/C2^lj0/C0/C1/C1eydxdy; (7) where eyis a unitary vector in the ydirection and AN¼tNwNis the wire cross section, with tNandwNbeing its thickness and width, over which the voltage drop is averaged. According to Eq. (7), the voltage drop between two points is proportional to the spin accumulation between them created via the ISOC. Next, we calculate the voltagedrop associated with SCI in two different devices with an I/Ninterface. We start analyzing the double Py/Cu LSV shown in Fig. 2(a) (see supplementary material Note S2 for experimental details). A charge current I cis injected from the ferromagnetic injector F2 into the Cu wire. F2 forms a LSV either with the detector F1 or F3. We use the F1–F2 LSV as a reference device. In the F2–F3 LSV, there is an additional middle Cu wire covered by a BiO xlayer, resulting in an I/N interface with ISOC, in which part of the spin current is absorbed andconverted to a transverse charge current. Quantitative description of electronic transport in LSVs has been widely studied in the literature. 34,35In our Cu wires, kN/C29tN;wN, allowing us to simplify the ECPs diffusion to a one-dimensional prob-lem, 34–36seeFig. 2(b) and supplementary material Note S3. At the BiO x/Cu wire, the z-integration using Eq. (3)leads to a renormaliza- tion of kN,kNk¼kNffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þGkk2 N rNtNs : (8) At the node, x¼0i n Fig. 2(b) , we use Kirchhoff’s law for the spin currents (see supplementary material Note S3), /C0ANrN@x^lkj0þ 0/C0¼/C0 GNk^lkjx¼0/C0Aeff nrcsejc rN^ex: (9) Here, GNk¼tNrNAeff n k2 Nkis the effective spin (bulk) conductance of the BiO x/Cu wire, with Aeff n¼wNðwNþ2kNkÞ. The latter is the effective area of the BiO x/Cu interface that absorbs (injects) spin current. Indeed, the rhs of this equation corresponds to Eq. (3)with an effective spin-loss conductance counting for both the interfacial and bulk spin-losses at the middle wire. The last term in Eq. (9)corresponds to the last term in Eq. (3)and it is proportional to the total injected charge current I calong the middle wire oriented in the ydirection. If we assume an homogeneously distribution of the current, then jc¼jc^ey, with jc¼Ic AN. The Cu/Py interfaces are described by the following BC:34,37,38 /C0ANrN@x^lk/C12/C12/C12x¼/C0L/C0x 2 x¼/C0Lþ x 2¼/C0 AFr/C3 F@y^lkF2jy¼0þpFejc/C16/C17 ; /C0ANrN@x^lk/C12/C12/C12x¼L/C0x 2 x¼Lþ x 2¼/C0 AFr/C3 F@y^lkF3jy¼0; (10) where ^lkF2=F3is the spin ECP at F2/F3, pFthe spin polarization, and r/C3 F¼rFð1/C0p2 FÞthe effective conductivity of Py. Lxis the distance FIG. 2. (a) SEM image of the two Py/Cu LSVs, the reference one between ferromagnets F1–F2 and the one with a middle BiO x/Cu wire (light red covering) between F2–F3. Non-local voltages Vref nl(blue circuit) and Vabs nl(red circuit) are measured applying an external magnetic field ( B) along the yaxis. The spin-to-charge conversion voltage Vsc (red circuit) is detected applying Balong the xaxis. (b) Effective one-dimensional model of the device. (c) Geometry and mesh of the 3D finite element method model. The BiO x/Cu interface is simulated as a thin layer (yellow) on top of the transverse Cu wire (purple).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 142405 (2020); doi: 10.1063/5.0023992 117, 142405-3 Published under license by AIP Publishingbetween consecutive ferromagnetic wires and AFthe Py/Cu junction area.34F o rt h er e f e r e n c eL S V ,w es u b s t i t u t eF 3b yF 1i nE q . (10). Because the Py/Cu interfaces are electrically transparent, we assume the continuity of ^lk. This condition, together with the one- dimensional version of Eq. (1)and the BCs (9)and(10), determines the full spatial dependence of ^lk. Specifically, we need ^lkat the detector F1/F3 to determine the non-local voltage Vnl¼e/C01pF^lkF1=F3jy¼034,39[see Fig. 2(b) ]a n d the corresponding non-local resistance, Rnl¼Vnl=Ic,w h e r e Icis the current injected from F2. Rnlchanges sign when the magnetic configuration of the ferromagnetic injector and detector changes from parallel, RP nl, to antiparallel, RAP nl.T h i sa l l o w su st or e m o v e any baseline resistance coming from non-spin related effects by taking DRnl¼RP nl/C0RAP nl[see Fig. 3(a) ]. Comparing the resistancemeasured at F3, DRabs nl, with the one measured at F1, DRref nl,w e determine the magnitude of the spin absorption and, therefore, the value of the spin-loss conductance, Gk. For this, we compute the ratioDRabs nl=DRref nl¼^lkF3=^lkF1jy¼0by solving the full boundary problem, DRabs nl DRref nl¼1þGNk 2GNðGFþ2GNÞ/C0GFe/C0Lx kN ðGFþ2GNÞþGFe/C0Lx kN2 43 5/C01 : (11) Here, Gi¼riAi kiare the bare Cu ( i¼N) and Py ( i¼F) wires spin con- ductances. The form of Eq. (11)agrees with the one obtained in previ- ous works.35,38,40However, our expression is more general since it distinguishes via GNkbetween interfacial and bulk losses at the BiO x/ Cu wire. Consequently, we can ensure that our calculation of Gkand, therefore, kIEE, is only related to interfacial effects [see Eqs. (6),(8), and(9)]. Figure 3(b) shows a weak temperature dependence of the absorp- tion ratio, DRabs nl=DRref nl/C250:5, revealing that about half of the spin cur- rent is absorbed at the BiO x/Cu middle wire. The temperature dependence of rNis measured ( supplementary material Note S4), with tN¼wN¼80 nm and Lx¼570 nm. The specific properties of the Py and Cu wires ( qFand pFtemperature dependencies and con- stant spin resistivities kF=rF¼0:91 fXm2andkN=rN¼18:3fXm2) are well characterized from our previous work.41Thereupon, by insert- ing these experimental values into Eq. (11) for different temperatures, we obtain the Gkdependence shown in Fig. 3(b) . A slight decrease in Gkcan be observed with increasing temperature, which seems to arise from the Cu conductivity. A linear relation between GkandrN(see supplementary material Note S5A) suggests a Dyakonov–Perel mecha- nism of the spin-loss, expected for a Rashba interface and in agree- ment with Ref. 42. We can also determine rsc=csin the same device. By injecting a charge current Icfrom F2, an x-polarized spin current is created and reaches the BiO x/Cu wire, where a conversion to a transverse charge current occurs via Eq. (5). This is detected as a non-local voltage Vsc along the BiO x/Cu wire, determining the non-local resistance RLSV sc ¼Vsc=Icas a function of an in-plane magnetic field Bx.B yr e v e r s i n g the orientation of the magnetic field, the opposite RLSV scis obtained. The difference of the opposite values of RLSV sc;2DRLSV scinFig. 3(c) , allows us to remove any baseline resistance. By swapping the voltage FIG. 3. (a) Non-local resistances as a function of By(trace and retrace) measured atIc¼70lA and 10 K for the reference LSV (blue squares) and the BiO x/Cu LSV (red circles). DRref nlandDRabs nlare tagged. (b) Temperature dependence of the spin absorption ratio (upper panel) and the corresponding spin-loss conductance (lowerpanel). (c) and (d) Reciprocal SCI non-local resistances as a function of B x(trace in red and retrace in black), from which we extract the spin-to-charge (2 DRLSV sc) from an average of seven sweeps and charge-to-spin (2 DRLSV cs) signals from an average of four sweeps, respectively. Measurements are performed at 10 K and Ic¼70lA (c) and Ic¼150lA (d). FIG. 4. (a) Measurement configuration of the TADMR in the BiO x/Cu Hall-cross device on YIG. An in-plane B-field (100 mT) is rotated an angle ( a) with respect to the applied current ( Ic¼5 mA) direction. (b) Double SCI at the BiO x/Cu interface. (c) Transverse resistance measured as a function of a(black squares). The solid red curve corresponds to a fit to Eq. (14).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 142405 (2020); doi: 10.1063/5.0023992 117, 142405-4 Published under license by AIP Publishingand current probes, the reciprocal charge-to-spin conversion signal, RLSV cs¼Vcs=Ic, can be determined. Theoretically, from the full spatial dependence of ^lk,w ec o m p u t e Vscfrom Eq. (7)andVcsfrom Vcs¼e/C01pFlx kF2jy¼0, yielding DRLSV sc=cs¼6rsc=cs rNAeff n ANpFeLx 2kN GF1/C0GNk 2GN/C18/C19 þeLx kNðGFþ2GNÞ1þGNk 2GN/C18/C19 : (12) Experimentally, Figs. 3(c) and 3(d) confirm the reciprocity between both measurements, DRLSV cs¼DRLSV sc. The broken time reversal sym- metry, due to the magnetic contacts, leads to the opposite sign forreciprocal measurements. Contrasting this with the result of Eq. (12), one confirms that r sc¼rcs.43The experimental value, 2 DRLSV sc /C251563lXat 10 K, yields rsc=cs/C254469X/C01cm/C01and kIEE /C250:1660:03 nm. The latter value is of the same order of magnitude but somewhat smaller than the previously reported results obtained byspin pumping experiments, k IEE/C250:2/C00:7n m ,15,18,42and LSV experiments, kIEE/C250:5/C01 nm.36This discrepancy might be due to a different quality of the BiO x/Cu interface: ex situ deposition in our experiment and in situ deposition in other works. The temperature dependence of the different parameters is presented in Note S5B in thesupplementary material . One observes a decreasing trend of r scby increasing the temperature, which translates in a decrease of kIEE,i n agreement with the previous literature.42 The accuracy of our 1D model is checked by performing a 3D finite element method simulation detailed in supplementary material Note S6. Figure 2(c) shows the geometry of the simulated device and the mesh of the finite elements. The ISOC is simulated as a thinlayer of finite thickness t int, spin diffusion length kint, and a spin Hall angle heff int. From the definition kIEE¼1 2heff inttint,44we obtain kIEE¼0:1060:02 nm, in good agreement with our 1D model. To verify that both ISOC parameters, Gkandrsc, are interface specific, we carry out another experiment involving a BiO x/Cu inter- face. Namely, we measure the SMR in a Cu layer sandwiched betweenBiO x(atz¼0) and Y 3Fe5O12(YIG) insulating layers (at z¼/C0 tN), shaped as a Hall bar, see Fig. 4(a) andsupplementary material Note S2 for the experimental details. In this setup, a double SCI takes place assketched in Fig. 4(b) . A charge current I cin the xdirection induces an out-of-plane y-polarized spin current density via Eq. (3).T h i ss p i n current propagates towards the Cu/YIG interface where it is partlyreflected with mixed xandypolarizations. 45–47The reflected spin cur- rent diffuses back to the BiO x/Cu interface, where its x-polarized con- tribution is reciprocally converted to an interfacial charge current. Theoverall effect is then proportional to r csrsc¼r2 sc. The electron spin reflection at the Cu/YIG interface depends on the direction of magnetization mof ferrimagnetic YIG. The effective BC describing this interface is known and reads48,49 /C0rNðr /C1 nÞ^lj/C0tN¼Gs^lj/C0tNþGrm/C2^l/C2mðÞ j/C0tN þGim/C2^l ðÞ j/C0tN: (13) Here, Gsis the so-called spin-sink conductance and Gr;iare the real and imaginary parts of the spin-mixing conductance (per area),G "#¼GrþiGi.I nY I G , Gi/C28Grand, hence, Giis neglected.47,50–52 We measure the transverse angular dependent magnetoresistance (TADMR) in the BiO x/Cu/YIG Hall bar of Fig. 4(a) . The transversevoltage, VT, depends on the direction of the in-plane applied magnetic field, parameterized by the angle a. The TADMR measurements are shown in Fig. 4(c) . Theoretically, we calculate the spatial dependence of ^lby solving the boundary problem of Eqs. (1),(3),a n d (13) by assuming transla- tional invariance in the x–yplane. We then determine VTfrom Eq. (7) and obtain for RT¼VT=Ic, RT/C25r2 sc 2r2 Nt2 NGr ðGkþGsÞðGkþGsþGrÞsinð2aÞ¼DRTsinð2aÞ: (14) Here,DRTis the amplitude of the modulation and we assume that kN/C29tN(seesupplementary material Note S7). The parameters of the Cu/YIG interface, Gr;s, add to the spin-loss at the BiO x/Cu interface Gk. We identify by comparison of Eqs. (3)and(13)two effective spin- loss conductances, Gx¼ðGkþGsÞand Gy¼ðGkþGsþGrÞ,f o r spins polarized in the xand ydirections, respectively. The amplitude of the SMR signal, Eq. (14), is then proportional to Gx/C0Gy. From Fig. 4(c) , we estimate DRT/C250:03 mXatT¼130 K. At this temperature, from the LSV measurements, we obtain Gk/C251:5 /C21013X/C01m/C02andrsc=cs/C2511:3X/C01cm/C01,a ss h o w ni n Figs. 3(b) and S3b, respectively. The spin conductances GsandGrin light metal/ YIG interfaces have been estimated in evaporated Cu53and Al.54 Whereas Gs¼3:6/C21012X/C01m/C02for Cu/YIG53is a consistent value in the literature,54,55the reported Gris very low,53as generally observed in evaporated metals on YIG.54,56By substituting Gk;Gs, andrsc=csvalues in Eq. (14), we obtain Gr/C256:1/C21013X/C01m/C02.T h i s value for sputtered Cu on YIG is much larger than that estimated in evaporated Cu on YIG, in agreement with the reported difference between sputtered and evaporated Pt.56Importantly, the obtained Gr satisfies the required condition Gs<Gr,47,55which confirms the valid- ity of our estimation. In summary, we present a complete theoretical framework based on the drift-diffusion equations to accurately describe electronic trans- port in systems with ISOC at non-magnetic metal/insulator interfaces. Within our model, the interface is described by two types of processes: spin-losses, parameterized by the interfacial conductances Gk=?,a n d SCI, quantified by rscandrcs. These parameters are material specific. The efficiency of the spin-to-charge conversion is quantified by the ratio rsc=Gk, which coincides with the commonly used Edelstein length kIEE. The Onsager reciprocity57–59is directly captured by rsc¼rcs, as demonstrated by comparing our theoretical and experi- mental results. Our theory is an effective tool for an accurate quantifi- cation of SCI phenomena at interfaces, which is of paramount importance in many spintronic devices. It is important to emphasize that the present formulation of our theory is valid for interfaces between non-magnetic materials. In principle, one could go beyond our theory and address the problem of magnetic moment transfer at a metal/magnetic insulator interface by including interfacial exchangeinteraction and magnon dynamics into the model. 47 See the supplementary material for additional details on the deri- vation of the spin-to-charge averaged voltage, Eq. (7), and the renor- malized spin diffusion length and node boundary condition for the LSV, Eqs. (8)and(9), respectively; measured temperature dependence of the Cu resistivity and analysis on the temperature dependence ofApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 142405 (2020); doi: 10.1063/5.0023992 117, 142405-5 Published under license by AIP Publishingthe ISOC parameters in the LSV; a brief explanation of the 3D simula- tion and the relation between the simulation and ISOC parameters; theoretical result for the transverse resistance measured in the multi- layer Hall bar, i.e., which leads to Eq. (14); and the experimental details of the nanofabrication and measurements of the LSV and multilayerHall bar devices. AUTHORS’ CONTRIBUTIONS C.S-F. and V.T.P. contributed equally to this work. C.S-F., F.S.B., and I.V.T. acknowledge funding from the Spanish Ministerio de Ciencia, Innovaci /C19on y Universidades (MICINN) (Project Nos. FIS2016–79464-P and FIS2017–82804-P)and Grupos Consolidados UPV/EHU del Gobierno Vasco (GrantNo. IT1249-19). The work of F.S.B. is partially funded by EU’s Horizon 2020 research and innovation program under Grant Agreement No. 800923 (SUPERTED). The work at nanoGUNE issupported by Intel Corporation through the SemiconductorResearch Corporation under MSR-INTEL TASK No. 2017-IN-2744and the “FEINMAN” Intel Science Technology Center and by the Spanish MICINN under the Maria de Maeztu Units of Excellence Programme (No. MDM-2016–0618) and under Project Nos.MAT2015–65159-R and RTI2018–094861-B-100. V.T.P.acknowledges postdoctoral fellowship support from the “Juan de laCierva–Formaci /C19on” program by the Spanish MICINN (Grant No. FJCI-2017-34494). E.S. thanks the Spanish MECD for a Ph.D. fellowship (Grant No. FPU14/03102). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1I./C20Zutic´, J. Fabian, and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004). 2G. Vignale and J. Supercond, Nov. Magn. 23, 3 (2010). 3J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). 4S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006). 5T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, 156601 (2007). 6A. Aronov and Y. B. Lyanda-Geller, JETP Lett. 50, 431 (1989), available at http://www.jetpletters.ac.ru/ps/1132/article_17140.shtml . 7V. M. Edelstein, Solid State Commun. 73, 233 (1990). 8Y. Ando and M. Shiraishi, J. Phys. 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5.0021125.pdf

AIP Advances 10, 115202 (2020); https://doi.org/10.1063/5.0021125 10, 115202 © 2020 Author(s).Temperature-dependent growth of topological insulator Bi2Se3 for nanoscale fabrication Cite as: AIP Advances 10, 115202 (2020); https://doi.org/10.1063/5.0021125 Submitted: 08 July 2020 . Accepted: 08 October 2020 . Published Online: 02 November 2020 Muhammad Naveed , Zixiu Cai , Haijun Bu , Fucong Fei , Syed Adil Shah , Bo Chen , Azizur Rahman , Kangkang Zhang , Faji Xie , and Fengqi Song ARTICLES YOU MAY BE INTERESTED IN 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 The concept of spin ice graphs and a field theory for their charges AIP Advances 10, 115102 (2020); https://doi.org/10.1063/5.0010079 Large magnetoresistance in topological insulator candidate TaSe 3 AIP Advances 10, 095314 (2020); https://doi.org/10.1063/5.0015490AIP Advances ARTICLE scitation.org/journal/adv Temperature-dependent growth of topological insulator Bi 2Se3for nanoscale fabrication Cite as: AIP Advances 10, 115202 (2020); doi: 10.1063/5.0021125 Submitted: 8 July 2020 •Accepted: 8 October 2020 • Published Online: 2 November 2020 Muhammad Naveed,1Zixiu Cai,2Haijun Bu,1Fucong Fei,1,a)Syed Adil Shah,1Bo Chen,1Azizur Rahman,3 Kangkang Zhang,1 Faji Xie,1and Fengqi Song1,a) AFFILIATIONS 1National Laboratory of Solid-State Microstructures, Collaborative Innovation Center of Advanced Microstructures, and College of Physics, Nanjing University, Nanjing 210093, China 2School of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA 3High Magnetic Field Laboratory, Chinese Academy of Science, Hefei, 230031 Anhui, China a)Authors to whom correspondence should be addressed: feifucong@nju.edu.cn and songfengqi@nju.edu.cn ABSTRACT Topological insulators and their characteristics are among the most highly studied areas in condensed matter physics. Bi 2Se3nanocrystals were synthesized via chemical vapor deposition at different temperatures on a silicon substrate with a gold catalyst. The effects of tem- perature on the obtained Bi 2Se3nanocrystals were systematically investigated. The size and length of Bi 2Se3nanocrystals change when the temperature increases from 500○C to 600○C. We found that the crystallization quality of the Bi 2Se3nanocrystals synthesized at 560○C is optimal. At this temperature, we can get the desired thickness and length of the nanocrystals, which is quite suitable for nanoscale fabrication. ©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/10.1063/5.0021125 I. INTRODUCTION The materials with a non-trivial topological order that act as an insulator in its interior but whose surface possesses conducting states are declared as topological insulators (TIs).1The conduct- ing states are not unique topological materials because the nor- mal band insulators also support the conducting states. The novel part of the topological insulator is that these conducting states are symmetry protected by time-reversal symmetry.2–4Bi2Se3is a bulk three dimensional (3D) TI containing conducting states with non- degenerate spins.5,6Bi2Se3is a fascinating TI because the simple surface states make it an ideal candidate to realize the magneto- electric effect, and its large bulk bandgap specifies great potential for possible high-temperature spintronic applications.7However, in the transport measurements of Bi 2Se3, the bulk involvement is very high as compared to the surface contribution because of the shifting of Fermi energy caused by the selenium vacancy defects, and this is the point where both fundamental research and practical applications confront challenges regarding sample fabrication.8,9Previous stud- ies revealed that Bi 2Se3is a layered rhombohedral crystal structurewith a space group of D5 3d(R3m), and each layer has five atomic sheets consisting of Se–Bi–Se–Bi–Se.5,10The unit cell of Bi 2Se3is given in Fig. 1(a). For the layered properties of Bi 2Se3, one can pos- sibly obtain thin film samples from the bulk crystals or bottom-up growth thin flake samples. In order to obtain a benefit from the thickness of the samples, the surface-to-volume ratio is high and the Fermi energy is possibly tunable by applying the gate voltage, so the contribution from the surface states can be enhanced. There are mul- tiple techniques for the synthesis of Bi 2Se3thin-films, but most of them are expensive and less handy.11The widely used method for the growth of nanostructures is chemical vapor deposition (CVD). This method is highly adjustable and low cost and can achieve high- quality Bi 2Se3nanostructures.12A silicon substrate with a gold (Au) catalyst is ideal for CVD in order to get the one-dimensional (1D) nanostructure. The catalyst-free substrate greatly reduces the 1D nanostructures, and the majority of nanostructures are nanowires in the case of catalyst. The Au catalyst induces the growth of Bi 2Se3 nanostructures. The silicon substrate with the Au catalyst is able to give high-quality 1D Bi 2Se3nanostructures for fabrication.13The details of the different growth techniques are given in Table I. AIP Advances 10, 115202 (2020); doi: 10.1063/5.0021125 10, 115202-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv There are some reasons due to which Bi 2Se3has gained great interest. For example, the band structure of Bi 2Se3creates a nearly ideal Dirac cone compared to TIs with much more complicated structures, Bi 2Se3is a stoichiometric material, and Bi 2Se3has a bulk bandgap of 0.3 eV. This made it unique, having almost the largest bulk bandgap, compared to the other known TIs. The chalcogenide materials’ growth and physical properties have been studied for a long time because of their interesting thermoelectric properties. However, research interest on this family of compounds increased dramatically because Bi 2Se3, Bi 2Te3, and Sb 2Te3were discovered to be 3D TIs, a new quantum state with conductive massless Dirac fermions on their surfaces.35–37Most of the success of semicon- ductor technology can be attributed to the transport properties of semiconductors that are readily tunable with an electrical gate. In the past several decades, the field-effect approach has also played an important role in the discovery of the quantum Hall effect, the quantum spin Hall effect, and several other breakthroughs in basic research.38–40 In this article, we studied the synthesis of Bi 2Se3nanocrystals by CVD on a silicon substrate with Au catalyst at different growth temperatures. We found that the morphology of Bi 2Se3was greatly affected by the growth temperature, which can be referred to as different growth dynamics at different temperatures. The results show that the nanostructures of Bi 2Se3deposited at 560○C areoptimal, and at this temperature, at least we can get the desired length and thickness of the nanocrystals, which are suitable for nanoscale device fabrication. II. EXPERIMENTAL DETAILS The Bi 2Se3nanocrystals were grown by the CVD method, and the growth of Bi 2Se3nanostructures was carried out inside a 12-in. quartz tube placed inside a one zone furnace. Ultra-pure Bi 2Se3pow- der (99.999%, Alfa Aesar) was placed in a quartz boat as a source material; then, the quartz boat was placed inside the quartz tube. The silicon substrates coated with the Au catalyst were placed in the downstream of the quartz tube at a distance ranging from 9 cm to 12 cm away from the source material. Argon (Ar) gas was used as a carrier to transform the vapor into the downstream of the furnace. For the synthesis, the tube was initially pumped down to the base pressure of 100 mTorr and flushed with Ar gas repeatedly several times in order to remove the residual oxygen. The source material was placed at the center of the horizontal tube furnace equipped with a 12-in. quartz tube. After deposition, the system was naturally cooled down to room temperature. In order to study the influence of the growth temperature on the morphology of Bi 2Se3nanostructures, we set the growth tem- perature from 500○C to 600○C including a growth time of 2 h. The FIG. 1. Crystal structure and characterization of Bi 2Se3. (a) Unit cell of Bi 2Se6. (b) The EDS spectra of the synthesized Bi 2Se3crystals at different temperatures. (c) XRD pattern of Bi 2Se3. AIP Advances 10, 115202 (2020); doi: 10.1063/5.0021125 10, 115202-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Different growth techniques. Techniques Advantages Disadvantages References (1) CVDHigh growth rates High process temperature 5,14–18Low-cost equipmentThickness of the samples cannot be exactly controlled Good reproducibility Low vacuum (compared with MBE) Nanostructures with various morphologies Samples are easy to be contaminated (nanoplates, nanoribbons, and nanowires) (hydrogen incorporation and oxidization) (2) Confined thinCheap with expensive instruments, Nanostructures’ growth is not 19film meltingprecursors, or catalyst uniform on flat insulating and 2D material substratesOne step synthesis that is favorable to the device fabrication Tunability of the density of nanostructures and various shapes (3) MBEMild growth temperature Expensive and complicated equipment 20–24Large scale of uniform thin films Extremely high vacuum is needed High quality of the synthesized samplesGrowth rate is low and not suitable for mass production The thickness of the films can be atomically controlled Suitable for producing complicated doping with the desired atomic ratio (4) Melting methodSimple and low-cost techniqueNeed an additional exfoliation 25–28process to fabricate mesoscopic transport devices Large bulk crystals instead of Defects such as vacancies and nanostructures are produced substitutions are unavoidable Choosing the appropriate container can produce crystals of pre-assigned diameter (5) Solvothermal/Low growth temperature is neededGrowth parameters are complex and 29–34hydrothermal method(normally less than 250○C)large amount of trials are needed to optimize the growth process Various morphologies of the synthesizedThe scales of the synthesized crystalssamples (nanoplates, nanoflowers,are normally smallnanoribbons, and nanowires) Lower cost equipment (even cheaper Sometimes alkaline environment than the CVD method) is needed Easy to scale up and suitableHigh defect concentrationfor mass production Low carrier mobility of the synthesized samples energy-dispersive x ray spectroscopy (EDS) and XRD confirm the successful growth of Bi 2Se3with a perfect atomic ratio and excel- lent crystallinity. The morphology of Bi 2Se3nanowires, ribbons, and plates was studied by scanning electron microscopy (S-4500 SEM), the thickness was studied by atomic force microscopy (AFM) [Asy- lum Research Scanning Probe Microscope (SPM) operated in the AC mode], electron beam evaporation (EBE) was used to deposit the gold layer of 5 nm–10 nm on the silicon substrate, and the electron beam of 7.5 kV was used. Photolithography equipment(URE-2000/25) was used to draw the pattern, and for comparatively small size crystals, electron beam lithography (S-4500 EBL) was used to draw the designs. III. RESULTS AND DISCUSSION A. First growth on temperature (530○C) The first growth was done at 530○C when the silicon wafer was removed from the system, and it was observed that the source AIP Advances 10, 115202 (2020); doi: 10.1063/5.0021125 10, 115202-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 2. Typical SEM images of Bi 2Se3from the first growth. [(a)–(d)] Nanowires, nanoribbons, and nanoplate crystals with the size slowly increasing with respect to temperature, which were grown on the silicon substrate at the growth temperature of 530○C. material had entirely been consumed and there was a thick layer of growth on the hot edge of the wafer. The EDS spectrum at 560○C confirms the exact stoichiometry [Fig. 1(b-ii)]. The XRD pattern of Bi 2Se3with excellent crystallinity is given in Fig. 1(c). Figures 2(a)–2(d) show the typical SEM images of the Bi 2Se3crys- tal grown on a silicon substrate at a growth temperature of 530 ○C. It can be seen that when the temperature is 530○C, which is a common temperature used for the growth of the Bi 2Se3nanostruc- tures, the size of the crystals gradually increased, and we obtained very clear nanowires, nanoribbons, and nanoplates, and the lay- ered hexagonal structure and flat surfaces can be easily observed.41 The average size of the crystals at this growth temperature was 10 μm, and by comparing with the growth results on other higher temperatures, which will be explained below, one can see that the size of the crystals slowly increased. Due to the low energy of the source material, evaporation of the source material is not large enough, resulting in small size of nanocrystals reported at this temperature. B. Second growth on temperature (560○C) The average size of the crystals at 560○C was 20 μm, which is quite enough for device fabrication, and size matters a lot during nanoscale fabrication, and comparing with the crystals at 530○C, we observed similarities in the crystal structure, but the size of the crystals was different, and as the temperature increased, the evap- oration rate of the source material also increased, but there were different binding energies on different surfaces of Bi 2Se3, which led to different growth rates.42Figures 3(a)–3(c) present typical SEM images of the nanocrystals, comparatively bigger in size, and we also clearly observed the gold ball on the edge of the nanowire in the zoomed-in view of the image [Fig. 3(d)], which verifies that the growth mechanism is dominated by the vapor–liquid–solid (VLS) mechanism. When we repeatedly studied the crystal structures at temperatures lower and higher than 560○C, we found that the depo- sition of Bi 2Se3nanostructures at 560○C was optimal, and at this FIG. 3. Second growth SEM images of Bi 2Se3. [(a)–(c)] Nanostructures with a comparatively bigger size. (d) Gold ball on the edge of the nanowire can be seen clearly, which verifies the VLS mechanism, and the growth temperature for the second growth was 560○C. temperature, we obtained the suitable size of the crystals for nanoscale devices. C. Third growth on temperature (600○C) When the growth temperature was further increased, the size of the crystals decreased, as shown in Figs. 4(a)–4(d). The differ- ent morphologies observed at different growth temperatures can be attributed to the different growth dynamics at different temper- atures. The average size of the nanocrystals at 600○C was 9 μm, and the main disadvantage of the higher temperature is the pro- duction of the selenium vacancies. When the temperature is fur- ther increased, the higher energy of the selenium atom led to the formation of selenium vacancies and decrease in the selenium ele- ment content in the sample. On the other hand, the addition of more selenium powder ensures that there is sufficient selenium in FIG. 4. Typical SEM images of Bi 2Se3from the third growth. [(a)–(d)] Nanocrystals with increased size as the temperature increases compared to the first and second growth, and this growth was carried out at the growth temperature of 600○C. AIP Advances 10, 115202 (2020); doi: 10.1063/5.0021125 10, 115202-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv the growth process, which can minimize the formation of sele- nium vacancies.43In order to get high purity crystal products, the deposition temperature should be at 600○C because it is very easy to form selenium vacancies at high temperatures. We kept the pressure and airflow constant during the growth at different temperatures. D. Atomic force microscopy (AFM) The AFM results in Fig. 5 present the surface morphology and different thicknesses of the crystals, and the thickness of the nanocrystals is very important for nanoscale device fabrication. The sample with comparably small in thickness can give desirable results, and as we knew from the different growth temperatures, the density of the crystals was quite high, and it was difficult to select a suit- able crystal for fabrication because the crystals were closely peaked. Hence, we transferred the crystals to oxidized silicon in order to select a suitable crystal. We measured four different samples and found that the thickness varies from 14 nm to 200 nm, as shown in Figs. 5(a)–5(d). E. Nanoscale fabrication The existence of strong spin–orbit coupling in TIs leads to certain topological invariants in the bulk, which is quite different from their values in a vacuum. The abrupt change in invariants at the interface results in metallic, time-reversal invariant surface states whose properties are beneficial for applications in spintronicand quantum computation. Despite that, one of the major prob- lems is to fabricate these materials on the nanoscale suitable for devices and probe the surface. We successfully fabricated some devices by photolithography and EBL, and there were many sim- ilarities between photolithography and EBL; in the sample prepa- ration, instead of using the mask, a beam of the electron is used in EBL to write the designs, and this method is able to produce features analogously smaller. The main objective of lithography is to make contact between the interesting nanostructures and the metal leads, which were then be able to easily connect with the measurement equipment.44Figures 6(a)–6(c) clearly described the six- and eight-probe devices, where the first one was designed by photolithography and the other two by EBL. In order to check the devices, the temperature-dependent resistance has been measured. The temperature-dependent longitudinal resistance with zero mag- netic fields shows metallic behavior [Fig. 6(d)]. It is better to mention that this metallic behavior is from the bulk contribution, not from the metallic surface states, because the details of the metallic sur- face state cannot be simply extracted from the R(T) results because of the selenium vacancies discussed above, which cannot be fully suppressed, and the Fermi level of Bi 2Se3samples grown by CVD normally present in the conduction bulk band.45,46Thus, the trans- port that forms the bulk also shows metallic behavior. The bulk and surface transport characteristics exhibited different thicknesses and angle dependences, and thickness-dependent or angle-dependent magneto-transport studies can provide insight into bulk and surface states.45,47 FIG. 5. Topographic images of the Bi 2Se3crystals by AFM. [(a)–(d)] Crystals with enough thickness varying from 14 nm to 200 nm, which were quite suitable for nanoscale device fabrication. AIP Advances 10, 115202 (2020); doi: 10.1063/5.0021125 10, 115202-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 6. Device fabrication: [(a)–(c)] Six- and eight-probe nanoscale devices were fabricated by photolithography and EBL. (d) Temperature-dependent resistance. IV. CONCLUSION We investigated the CVD growth of Bi 2Se3at different tem- peratures on a silicon substrate with a gold catalyst. The mor- phology of Bi 2Se3was greatly affected by the growth tempera- ture due to different growth dynamics at different temperatures, and we found that when the temperature increases from 500○C to 600○C, the size and length of the nanocrystals changed. In order to get high purity products, the deposition temperature should be at 600○C, and the results have shown that the nanocrystals of Bi2Se3deposited at 560○C is optimal, and at this temperature, crys- tals with suitable size and thickness for nanoscale fabrication were observed. ACKNOWLEDGMENTS The authors gratefully acknowledge the financial support from the National Key R & D Program of China (Grant No. 2017YFA0303203), the National Natural Science Foundation of China (Grant Nos. 91622115, 11522432, 11574217, U1732273, U1732159, 61822403, 11874203, 11904165, and 11904166), the Natural Science Foundation of Jiangsu Province (Grant Nos. BK20160659 and BK20190286), the Fundamental Research Funds for the Central Universities (Grant Nos. 020414380150, 020414380151, and 020414380152), and the Opening Project of Wuhan National High Magnetic Field Center. 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5.0028943.pdf

J. Chem. Phys. 153, 224104 (2020); https://doi.org/10.1063/5.0028943 153, 224104 © 2020 Author(s).Geometry relaxation-mediated localization and delocalization of excitons in organic semiconductors: A quantum chemical study Cite as: J. Chem. Phys. 153, 224104 (2020); https://doi.org/10.1063/5.0028943 Submitted: 09 September 2020 . Accepted: 13 November 2020 . Published Online: 08 December 2020 M. Deutsch , S. Wirsing , D. Kaiser , R. F. Fink , P. Tegeder , and B. Engels COLLECTIONS Paper published as part of the special topic on Excitons: Energetics and Spatio-temporal Dynamics ARTICLES YOU MAY BE INTERESTED IN Vibronic and excitonic dynamics in perylenediimide dimers and tetramer The Journal of Chemical Physics 153, 224101 (2020); https://doi.org/10.1063/5.0024530 Photoexcitation dynamics in perylene diimide dimers The Journal of Chemical Physics 153, 244117 (2020); https://doi.org/10.1063/5.0031485 On the quantum origin of few response properties The Journal of Chemical Physics 153, 221101 (2020); https://doi.org/10.1063/5.0027545The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Geometry relaxation-mediated localization and delocalization of excitons in organic semiconductors: A quantum chemical study Cite as: J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 Submitted: 9 September 2020 •Accepted: 13 November 2020 • Published Online: 8 December 2020 M. Deutsch,1 S. Wirsing,1 D. Kaiser,1R. F. Fink,2 P. Tegeder,3and B. Engels1,a) AFFILIATIONS 1Institut für Physikalische und Theoretische Chemie, Universität Würzburg„ Emil-Fischer-Str. 42, D-97074 Würzburg, Germany 2Institut für Physikalische und Theoretische Chemie, Universität Tübingen, Auf der Morgenstelle 18, 72076 Tübingen, Germany 3Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 253, 69120 Heidelberg, Germany Note: This paper is part of the JCP Special Topic on Excitons: Energetics and Spatio-temporal Dynamics. a)Author to whom correspondence should be addressed: bernd.engels@uni-wuerzburg.de ABSTRACT Photo-induced relaxation processes leading to excimer formations or other traps are in the focus of many investigations of optoelectronic materials because they severely affect the efficiencies of corresponding devices. Such relaxation effects comprise inter-monomer distortions in which the orientations of the monomer change with respect to each other, whereas intra-monomer distortions are variations in the geom- etry of single monomers. Such distortions are generally neglected in quantum chemical investigations of organic dye aggregates due to the accompanied high computational costs. In the present study, we investigate their relevance using perylene-bisimide dimers and diindenop- erylene tetramers as model systems. Our calculations underline the importance of intra-monomer distortions on the shape of the potential energy surfaces as a function of the coupling between the monomers. The latter is shown to depend strongly on the electronic state under consideration. In particular, it differs between the first and second excited state of the aggregate. Additionally, the magnitude of the geo- metrical relaxation decreases if the exciton is delocalized over an increasing number of monomers. For the interpretation of the vibronic coupling model, pseudo-Jahn–Teller or Marcus theory can be employed. In the first part of this paper, we establish the accuracy of density functional theory-based approaches for the prediction of vibrationally resolved absorption spectra of organic semiconductors. These inves- tigations underline the accuracy of those approaches although shortcomings become obvious as well. These calculations also indicate the strength of intra-monomer relaxation effects. Published under license by AIP Publishing. https://doi.org/10.1063/5.0028943 .,s INTRODUCTION Functionalized polycyclic aromatic molecules are in the focus of a multitude of experimental and theoretical investigations because they are promising materials in the field of organic semiconduc- tors.1–5In this context, perylene-based dyes have been frequently investigated as they exhibit favorable properties.1In particular, 3,4,9,10-perylene tetracarboxylic acid bisimides (PBIs) have been explored as a replacement for fullerenes in organic photovoltaics as they possess higher electron mobilities6,7and high extinction coefficients in the visible region.8Furthermore, they are relatively inexpensive and remarkably stable toward light as well as air.1Eventually, their electronic properties and the packing geometries can be tuned by varying the substitution pattern.9,10However, despite all these advantages, PBI containing photovoltaic devices show reduced efficiencies in comparison to fullerene analogs.11 Reasons for these shortcomings were attributed to morphological issues12and to trapping processes, which arise due to the forma- tion of excimer states.13,14For PBI thin films, a fast relaxation of excitons leading to long-lived immobile states was identified.14For α-perylene, the formation of excimers induced by motions of two monomers with respect to each other was indicated by Raman spec- troscopy.15Recent investigations on excimer formation by Hoche and co-workers,16as well as Kennehan and co-workers,17also J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp indicated a similar mechanism. For amorphous systems, additional traps are due to the strongly varying excitation energies due to the relative orientation of the molecules with respect to each other.18,19 Actually, it was shown that the energetic disorder is mostly caused by these orientation effects and only to a smaller degree by envi- ronmental effects. In this investigation, the energetical disorder pre- dicted by using the Bässler model20,21could only be reproduced if a delocalization of excimers is assumed.22 A detailed understanding of possible relaxation pathways is very important for the rational design of materials with improved functionality.1,23However, while the mentioned studies agree with the fact that exciton trapping occurs rather frequently, the details of the underlying processes are still under debate. Within H-aggregates, trapping can occur because the exciton is initially excited to the higher lying Frenkel state (in a dimer system, this is the S 2state). If the relaxation to the lowest Frenkel state (S 1) is faster than the competing hopping process, the exciton gets trapped because hopping to the higher lying S 2state is no longer possible due to the energy loss accompanied with the relaxation to the S 1state. Furthermore, hopping between the lower Frenkel states—which is energetically feasible—is very inefficient due to the vanishing transition dipoles of these states. Non-negligible contributions of charge transfer (CT) states result in an even more complex situation. Using the dimer M–M (M≡Monomer) as the most simple aggregate, Frenkel configura- tions arise from linear combinations of the locally excited M∗M and MM∗determinants, while the CT-type configurations stem from linear combinations of the electron exchanging determinants M+M−and M−M+. Due to significant couplings between locally excited- and CT-configurations, the adiabatic states determined by common quantum-chemistry excited state calculations are generally strong mixtures of these configurations. The ratio of Frenkel and CT characters depends on a subtle interplay of the energy differ- ence and the electronic coupling between the diabatic states.21,24–26 Hence, a rugged potential energy surface (PES) with various con- ical intersections results, which may provide very efficient decay pathways to lower lying states.27–30Going to larger aggregates, the number of CT states increases considerably (trimer: 3 Frenkel, 6 CT; tetramer: 4 Frenkel, 12 CT), resulting in a decreased energy spacing between the states. Consequently, transitions into lower lying states become even more efficient. Note that due to the cou- plings between states, CT states can influence the position of coni- cal intersections even if they are higher in energy than the Frenkel states. Photo-induced relaxation processes in PBI aggregates repre- sent one example for the complicated interplay of various electronic states and geometrical relaxations and distortions.29–32Model cal- culations predict that excimer formation is the reason for the mea- sured strong red shift in the emission spectra of PBI dimers in the minimum structure of the ground state. This red shift is obtained by twisting one of the two monomers of an eclipsed arrangement by 30○about the symmetry axis parallel to the molecular plane [RL= R T= 0 Å, R z= 3.4 Å, and φ= 30○according to the dimer structure parameters in Fig. 3(a)].33According to these model cal- culations, the excimer formation starts with an exciton relaxation from the initially populated S 2Frenkel state to the lower lying S 1 Frenkel state. However, this first step is not a direct transfer but is mediated by a CT state, which crosses the initially populated S 2state, resulting in a fast population transfer. The transition between the transiently populated CT state and the S 1Frenkel state is non- radiative and also very efficient due to the strong coupling between both states. The final emission then takes place from the S 1state.29–32 The computations show that intra-monomer (e.g., variations of interatomic distances) and inter-monomer (e.g., the torsional angle between both monomers) relaxation effects have to be taken into account. The relaxation effects not only strongly influence the effi- ciencies of the relaxation processes but also induce strong red shifts in the emission. On the basis of these processes, it was possible to assign the known spectra of PBI aggregates as well as femtosecond time-resolved experiments.30,31,33Recently, the model was also suc- cessfully employed to explain the ambient-stable, bright, steady-state photoluminescence from long-lived excitons of H-aggregated PBI crystals.34Another important example for localization effects is due to intra-chain dynamics.35,36 While intra-monomer relaxations leading to the CT geometry were included in the description of the PBI trapping processes, possi- ble geometry relaxations of the Frenkel states themselves were omit- ted. Their importance became apparent through our monomer com- putations, which indicated relaxation effects of about 0.3 eV–0.4 eV. This is in the range of the Davydov splittings found for aggregates of these organic semiconductors. The pronounced impact of geom- etry relaxation raises the question of how their explicit consideration for Frenkel states may change predictions for aggregate clusters. The inclusion of such relaxation effects might not only induce a change in the energetic position of the excited states but may also affect the individual character of the electronic states. For example, the nuclear relaxation of a single monomer in an aggregate could lead to a local- ization of the exciton on that monomer. The energetics associated with this situation is sketched in Fig. 1 using a dimer as the sim- plest model for a molecular aggregate. For clarity, we neglect possible influences of CT states. Figure 1 depicts the coupling between the two localized S 1states of the monomers if both monomers adopt the same geometry (blue) or if one monomer adopts a structure leading to a lower energy of its localized S 1state (red). If both monomers adopt the same geometry (blue), the exciton is completely delocal- ized over both monomers due to simple symmetry considerations. Going to the case given in red, one monomer is relaxed to the FIG. 1 . Comparison of different relaxations. Blue: Both monomers remain in the geometry of the ground state of the monomer (R 0). Red: One monomer is relaxed to the geometry of the excited state of the monomer (R 1) while the second one remains in the ground state geometry of the monomer (R 0). Green: Both monomers relaxed by the same amount (R d= optimal geometry for S 1-state of dimer). J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp optimal geometry of its localized S 1state (R 1), while the other remains in the geometry of the ground state (R 0). Because the energy of the localized S 1state in the optimal S 1geometry is lower, both interacting localized states differ not only in their geometry but also in their associated energies. Due to these inequalities, the splitting decreases, and the exciton starts to localize on the monomer in the S1geometry for the lower Frenkel state. For the exciton in a dimer, it could also be that both monomers relax by the same amount (green situation, R dis the optimal geometry for the S 1state of the dimer). Then, the energy of the exciton would also decrease, but the exci- ton remains delocalized over the whole dimer. Which situation is most appropriate to characterize the molecular aggregate depends on the relative energies of the resulting Frenkel states. It might also be that the exciton becomes localized for the lowest Frenkel state but stays delocalized for the second Frenkel state. Since CT states are always delocalized, the degree of delocalization of an exciton may vary strongly during complex relaxation processes involving several electronic states. Localization effects in self-assembled perylene helices were investigated by Segalina and co-workers.37Their study offers valu- able insights into the behavior of such aggregates, but localization effects resulting from geometrical changes in monomers were omit- ted. Such localization effects were investigated by Talipov and co- workers.38They investigated relaxation effects in polychromophoric assemblies using covalently linked cofacially arrayed polyfluorenes as model systems. Their computations show that for these systems, excitons always localize on dimers irrespective of the number of chromophores. This behavior is explained by very strong geome- try relaxation effects of the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO) in this specific case. Thus, it remains unclear whether these systems can serve as real model systems. Possible models were already worked out by Fulton and Gouterman who successfully described these excitonic effects with vibronic coupling theory.39,40They also showed that this theory is formally equivalent with the pseudo Jahn–Teller (pJT) framework and similar to the potential energy model used in Marcus theory. The vibronic coupling theory was later employed by Diehl and co- workers41to assign the spectra for a homologous series of oligo( p- phenylene) bridged PBI dimers with intermolecular center-to-center distances ranging from 1.3 nm to 2.6 nm. For these dimer systems, the reorganization effects are in the range of 0.3 eV–0.4 eV, while the coupling between monomers is small by a factor of 5 or even more. Consequently, a double minimum potential arises in which the exci- ton is either localized on one or the other monomer. In simulations, the monomer on which the exciton remains adopts the S 1equilib- rium geometry, while the other monomer stays in the S 0structure. For the fully delocalized situation, both monomers were assumed to adopt the S 0geometry (blue case in Fig. 1). Dreuw and co-workers employed the pJT framework to model relaxation effects in CO and in the benzene dimer.42,43In both cases, they varied inter-monomer geometries of the dimers to mod- ify the coupling between both monomers. They showed that for larger monomer distances, the coupling is weaker than the relax- ation, which results in a double-minimum potential. For smaller distances, a parabolic shape is found because the coupling is stronger than the relaxation. More details in relation to our study will be given below. FIG. 2 . Lewis structures of the investigated monomer systems. (1) 3,4,9,10- perylene tetracarboxylic acid bisimides (PBIs), (2) diindenoperylene (DIP), and (3) dicyanoperylene-bis(dicarboximide) (PDIR-CN 2). In the present study, we will extend these previous investiga- tions of the interplay of geometry relaxation and exciton localization. Whether an exciton is delocalized or localized depends on the sub- tle interplay between the coupling strength between the units of the aggregate and the decrease in the associated energy due to the intra- monomer relaxation of one unit into its S 1geometry. While the former favors a delocalization of the exciton, the latter induces its localization. We start the investigations with perylene dimers [Figs. 2 and 3(a)] and vary the inter-monomer geometries (e.g., the longi- tudinal shift or the distance) and the intra-monomer geometry of one monomer. In a second step, we investigate the diindenopery- lene (DIP) tetramer as an example for larger clusters. Variations in the electronic character are monitored by computing the localization of the exciton. This paper is organized as follows: After briefly describing the details of the used theoretical approaches, we first establish expected error bars for different quantum chemical approaches by compar- ing computed vibrationally resolved excitation energies of the PBI monomer (1) (Fig. 2) with measurements of Klebe et al.44and Wewer and Stienkemeier.45,46For DIP (2) and PDIR-CN 2(3), we extend these investigations to solvent effects. After establishing the accuracies of our approaches, we turn to the dimer and trimer com- putations to investigate the interplay of exciton localization and delocalization effects. TECHNICAL DETAILS Multi-reference approaches would yield ideal reference results for benchmarks because they provide very accurate results for J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . (a). Description of the varied inter-monomer coordinates for the computed PBI dimer. The coordinates include transversal (R T) and longitudinal (R L) shifts, the distance between both monomers R Z, and the rotation of the upper monomer around φ. (b) Enlarged sketch of the geometry change between the equilibrium structures of the S 0to the S 1states of the PBI monomer. The sum of the variations is <0.005 Å. (c) Arrangement of the DIP tetramer. excited states,47–50potential energy surfaces,51,52as well as other molecular properties.53For the size of the present system, however, they are computationally too expensive. Hence, we test the accuracy of time-dependent density functional theory (TD-DFT) calculations against SCS-CC254,55in combination with the SVP (split valence plus polarization)56,57and the TZVPP basis sets.56,58We compared the TD-DFT-based approaches with respect to SCS-CC2 to keep the computational costs feasible. SCS-CC2 predicted adiabatic excita- tion energies with a mean absolute error (MAE) of 0.05 eV and a standard deviation of 0.06 eV for a test set of 0–0 transitions in medium-sized and large organic molecules.59In this work, SCS-CC2 was shown to perform slightly better than its parent approach CC2 (MAE = 0.09 eV). Furthermore, it improves the description of CT states.60For the “low-cost” TD-DFT approach, we mainly used the modern range-separated hybrid ωB97X-D functional61in combi- nation with SVP, TZVP, cc-pVDZ, cc-pVTZ, and def2-SVP basis sets.56,58,62,63This method is sufficiently efficient to compute aggre- gates up to tetramers.32,64,65For the benchmark, we optimized the ground state and the first excited states within the given approxi- mation unless stated otherwise. Vibrational effects were computed using the Franck–Condon approximation alone or in combination with the Herzberg–Teller correction as implemented in the Gaus- sian16 program package.66For the vibrational effects, we used not only the standard time-independent approach67but also the time- dependent formalism.68For excited state calculations in solution, we used the equilibrium and non-equilibrium options in the IEF-PCM (Integral Equation Formalism version of Polarizable Continuum Model) approach implemented in the Gaussian program suite.69–71 In the non-equilibrium option, only ultrafast solvent processes (e.g., polarization of the electron cloud of the solvent) are considered.72 In the equilibrium option, also slower effects (e.g., the reorienta- tion of solvent molecules) are included.73An equilibrium calcula- tion describes a situation where the solvent has had time to fully respond to the solute. A non-equilibrium calculation is appropriate for processes that are too rapid for the solvent to have time to fully respond, e.g., for the energy of the 0–0 excitation in an absorption spectrum. The vibrational propagation had to be estimated based on frequencies in the equilibrium model due to technical limitations. To mimic intra-monomer relaxation effects, we modulated the monomer geometry linearly from the monomer ground stategeometry [ ⃗R(S0)] to the monomer geometry in its S 1state [⃗R(S1)] using ⃗Rχ=⃗R(S0)+χΔ⃗R, withΔ⃗R=⃗R(S0)−⃗R(S1). (1) We varied χfrom −0.5 to 1.5 in 0.5 steps for each monomer of the dimer. According to Eq. (1), χ= 0.0 gives ⃗R(S0), while⃗R(S1) is obtained with χ= 1.0. An enlarged description of the difference between ⃗R(S0)and⃗R(S1)for PBI is given in Fig. 3(b). The result- ing 2D surfaces for energy and properties were obtained by spline interpolation between the resulting 25 points. We only include relax- ations to the equilibrium geometry of the S 1state of the monomer because higher electronic states of the monomer units are nor- mally not relevant for the photo-induced behavior of crystals, thin films, or amorphous systems.18,19,74To investigate the interplay between inter-monomer and intra-monomer changes, we computed the intra-monomer variations for the longitudinal shifts (R L= 0.0 Å, 0.5 Å, 1.0 Å, 1.4 Å, 1.7 Å, and 2.5 Å) and the distances between both monomers (R Z= 3.1 Å, 3.31 Å, 3.5 Å, 4.0 Å, 5.0 Å, and 10.0 Å). Beside the energies, we also computed the oscillator strengths, the PRvalue, and the CT value. The P Rvalues give the delocalization of the exciton, i.e., P R= 2 denotes the situation in which the exciton is completely delocalized between the two monomers in a dimer, while PR= 1 characterizes an exciton that solely resides on one monomer. The CT value gives the percentage of CT character, i.e., CT = 0 corresponds to a pure Frenkel state, and CT = 1 corresponds to a pure CT state. To determine the P Rand CT values, we employed the TheoDORE program package.75,76 Benchmark for vibrationally resolved absorption spectra of perylene-based organic semi-conductors In order to computationally model the relevant processes in organic semiconductor thin films, sufficiently large molecular clus- ters have to be used to mimic the high density of states, their energy shifts, and the important mixing of CT and Frenkel states. Addition- ally, the experimental absorption and emission spectra exhibit dis- tinct vibrational progressions, i.e., vibrational effects should also be considered. Finally, the influence of the environment on the energy position of the different states can also be important because the J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp position of conical intersections or even the energetic order of states might change due to polarizable environments. Obviously, the com- putational modeling of large molecular clusters and very accurate electronic descriptions exclude each other. Computations of such cluster models are only possible with DFT or even simpler methods because the system size excludes costly high-level multi-reference approaches.77–79The inclusion of vibrational effects—even solely on the harmonic level—necessitates geometry optimizations and the determination of the Hessian of the excited state, which is generally too costly for more accurate approaches. Table I estimates the error bars arising from the use of TD- DFT and neglect of the vibrational effects by comparing the com- puted results for the PBI monomer (1) with measurements of Klebe et al.44as well as Wever and Stienkemeier.45Klebe et al. noted that the first band peak of PBI monomers dispersed in, e.g., polystyrene appears at 530 nm (2.339 eV), while Wever and Stienkemeier mea- sured the 0–0 transition of N–N-dimethyl PBI in He-droplets at 486 nm (2.553 eV). The difference of about 0.2 eV within the exper- imental spectra may result from the different temperatures, solvent effects, or the influence of substituents. Because we performed the computations of single molecules without considering environmen- tal (solvent) effects, we chose the values of Wever and Stienkemeier as our reference, i.e., a 0–0 transition at 2.553 eV. In the following, we focus on the S 0→S1transition because the vertical energy of the S 2state is about 1 eV higher. Furthermore, the corresponding transition moment vanishes. From our experience, also the range-separated ωB97X-D func- tional should give reliable results for perylene systems.74,80–82In contrast, the B3LYP functional that is very often used for the com- putations of excited states was less accurate and tends to underes- timate the excitation energies of CT states.28,81,83Comparing com- puted vertical excitation energies obtained for vacuum with mea- sured 0–0 energies in He-droplets, B3LYP/TZVP deviates by only −0.122 eV. The discrepancies for SCS-CC2/SVP (+0.418 eV) and ωB97X-D/cc-pVDZ (+0.287 eV) are considerably larger. The pic- ture changes if vibrational effects are included and the transitionsbetween the lowest vibrational states of ground and excited states (0–0 transition) are compared. While the discrepancies between the SCS-CC2/SVP and ωB97X-D/cc-pVDZ values and their experi- mental counterparts decrease to +0.143 eV and +0.008 eV, respec- tively, the values obtained with B3LYP/TZVP are too low by about 0.3 eV. The error found for SCS-CC2 decreases to −0.010 eV if the larger TZVPP basis is employed, i.e., SCS-CC2 shows a much improved description with larger basis sets. In contrast, for ωB97X- D/cc-pVDZ, we find an error compensation because the agreement to the experimental result of 2.553 eV slightly deteriorates if the basis sets are enlarged (Table II). Comparing the results obtained with the cc-pVDZ basis with those computed with the improved aug-cc-pVDZ basis sets, the 0–0 excitation energy decreases to 2.49 eV, i.e., the deviation from the experimental result increases to about 0.06 eV. The same trend is also found for other larger basis sets. Hence, at least for PBI, ωB97X-D/cc-pVDZ represents an excel- lent choice for the description of the electronic states. The agreement is much better than expected because various benchmarks indi- cate larger error bars for the computations of excitation energies of organic molecules.84,85 The variations going from vertical to vibrationally-resolved excited state estimations mainly result from the relaxation ener- gies of the first excited state [ES1(RS1)−ES1(RS0)], which amount to 0.14 eV–0.21 eV depending on the employed method. This explains the distinct vibrational progression found in experimental absorp- tion and emission spectra. Figure 3(b) gives an enlarged description of the geometrical structure difference between the equilibrium geometry of the S 0state to that of the S 1state. Note that the sum of all variations is less than 0.005 Å. Table I clearly shows that the small deviations between theoretical B3LYP/TZVP verti- cal energies and experimental 0–0 energies stem from error com- pensation. In comparison to the more accurate approaches, B3LYP underestimates the excitation energies by about 0.4 eV–0.5 eV. This error is partly compensated by the neglect of the relaxation energy of the excited state (0.15 eV–0.2 eV). For SCS-CC2/SVP, the TABLE I . Comparison of computed and measured excitation energies for a PBI monomer. B3LYP/ ωB97X-D/ SCS-CC2/ SCS-CC2/ TZVP cc-pVDZ SVP TZVPP Reference 44 Reference 45 Evert(eV)a2.431 2.840 2.971 2.810b Eadi(eV)c2.289 2.631 2.766 2.613b Erelax(eV)d0.143 0.209 0.205 0.197 ZPE (eV)e0.030 0.070 0–0 (eV)f2.259 2.561 2.696g2.543g2.339 2.553 Ecorr(eV)h0.172 0.279 0.275 aVertical excitation energy. bSingle point computation on the SCS-CC2/SVP geometrical structure. cAdiabatic excitation energy. dRelaxation energy of the excited state (E vert–Eadi). eEnergy correction due to zero-point vibrational effects. f(0–0) excitation energy. gUsing the ZPE of ωB97X-D/cc-pVDZ. hTotal correction (E vert–E0–0). J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Influence of functionals and basis sets on the computed excitation energies for the S 1state of a PBI monomer. For more explanations, see Table I and the text. ωB97X-D/ ωB97X-D/ ωB97X-D/ ωB97X-D/ ωB97X-D/ cc-pVDZ aug-cc-pVDZ 6–311++G(d,p) def2-TZVP def2-TZVPP Evert(eV)a2.840 2.763 2.801 2.814 2.811 Eadi(eV)b2.631 2.554 2.587 2.596 2.593 0–0 (eV)c2.561 2.494 2.511 2.510 2.512 B3LYP/ CAM-B3LYP/ M06-2X/ LC- ωHPBE/ cc-pVDZ cc-pVDZ cc-pVDZ cc-pVDZ Evert(eV)a2.414 2.803 2.820 3.160 Eadi(eV)b2.297 2.602 2.620 2.878 0–0 (eV)c2.226 2.531 2.546 2.803 B3LYP/ CAM-B3LYP/ M06-2X/ LC- ωHPBE/ def2-TZVPP def2-TZVPP def2-TZVPP def2-TZVPP Evert(eV)a2.383 2.766 2.789 3.118 Eadi(eV)b2.262 2.556 2.580 2.831 0–0 (eV)c2.174 2.468 2.489 2.744 aVertical excitation energy. bAdiabatic excitation energy. c(0–0) excitation energy. vertical excitation energies do not coincide with the experimental 0–0 transition because no error compensation takes place. Remain- ing deficiencies in the basis set (SVP vs TZVPP) even lead to an overestimation of the vertical energies by about 0.1 eV–0.2 eV. Assuming SCS-CC2/TZVPP results as a reference value, the ωB97X- D functional overestimates the vertical excitation energies by less than +0.1 eV. These estimates are in good agreement with previ- ous investigations of perylenetetracarboxylic dianhydride (PTCDA) single crystals.29,32,64 Table S1 shows the results of extended computations employ- ing additional functionals and basis sets, and Table II summarizes the influence of the basis set size for ωB97X-D and the influ- ence of the functionals for the smallest (cc-pVDZ) and the largest basis sets (def2-TZVPP). Comparing the vertical excitation ener- gies obtained for the def2-TZVPP basis sets, B3LYP (2.383 eV) gives the lowest value. The B3LYP excitation energy differs by about 0.74 eV from the excited state energy obtained for LC- ωHPBE (3.118 eV). The ωB97X-D functional, which seemed to be quite accurate according to the comparisons given in Table I, predicts a value of 2.811 eV, which is 0.045 eV higher than that of CAM-B3LYP (2.766 eV), the long-range corrected version of B3LYP. The M06- 2X functional lies in between (2.789 eV). Comparing the 0–0 transitions with the corresponding experimental values (Table I, 2.553 eV), ωB97X-D/def2-TZVPP shows the best agreement (−0.041 eV deviation from the experimental value of Ref. 45). CAM-B3LYP/def2-TZVPP and M06-2X/def2-TZVPP give com- parable deviations ( −0.085 eV and −0.064 eV, respectively). As already discussed above, B3LYP/def2-TZVPP considerablyunderestimates the experimental value by 0.379 eV, and PBE0 behaves rather similar, while LC- ωHPBE/def2-TZVPP overesti- mates the experimental value by 0.191 eV. Tables II and S1 also show interesting trends with respect to basis sets’ selection. Going from larger basis sets to the double-zeta cc-pVDZ basis set, which represent the smallest employed basis sets, the computed 0–0 excitation energies increase irrespective of the used functionals. However, the increments are less than 0.1 eV in all cases. This shows that the neglect of vibrational effects is a more severe approximation than the use of smaller basis sets. The aug-cc- pVDZ basis sets are well adapted to the description of electronically excited states, and indeed, the corresponding 0–0 excitation ener- gies are slightly lower than the ones obtained with the def2-TZVPP basis sets. However, for the most reliable functionals ( ωB97X-D, CAM-B3LYP, and M06-2X), the predicted values become too low in comparison to the experimental value of 2.553 eV. In summary, the best agreement is found for M06-2X/cc-pVDZ, which only devi- ates by 56 cm−1(0.007 eV). ωB97X-D/cc-pVDZ underestimates the experimental value by 65 cm−1(0.008 eV). Note that while ωB97X-D/cc-pVDZ predicts the 0–0 exci- tation energy rather accurately, a comparison with the SCS-CC2 benchmark values indicates that it seems to overestimate the relax- ation energy by about 4%. As indicated by the even larger relax- ation energies obtained with larger basis sets (Table II), the choice of the relatively small cc-pVDZ basis set again tends to improve the performance of the method. Figure 4 depicts a comparison of computationally modeled and experimental absorption spectra. We present the spectrum obtained J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Comparison of the experimental (red full lines) and computed (black dashed line) S 0–S1absorption spectra of the PBI monomer. The computed spectra were shifted by−117 cm−1and 275 cm−1, respectively, so that the 0–0 lines of experiment and theory coincide. The computed intensities are adjusted with respect to the experimental 0–0 transition intensity; however, the intensities cannot be compared due to experimental reasons. For more information, see the text. within the ωB97X-D/cc-pVDZ ansatz as an example. To simplify the comparison, the computed spectra were shifted so that the 0–0 lines of experiment and theory coincide. Furthermore, the computed intensities were adjusted with respect to the experimental 0–0 tran- sition intensity. The overall agreement in the energetic positions of the quantized absorption peaks is excellent. As mentioned by Wewer and Stienkemeier,45the intensities in their laser induced flu- orescence spectra do not agree with absorption spectra, which are simulated in our theoretical approach. It is obvious that the sizes of the employed AO basis sets do not have a relevant influence on the accuracy of the computational prediction. Table III and Fig. 5 extend our investigations to the DIP molecule. Table III summarizes the computed data and compares the 0–0 band of the S 0→S1transition with the energy of the lowest band of the experimental absorption spectrum, measured in ace- tone at room temperature.87Figure 5 compares the stick spectrum and a convoluted version of the computed vibrational transitions with the measured absorption spectrum. The computed data are shifted by 0.04 eV to match the maximum of the energetically lowest peak in the absorption spectrum. More information is given in the supplementary material. For the DIP monomer, the S 2state is only slightly higher in energy than the S 1state. Nevertheless, it can be omitted in regard to the assignment of the absorption spectrum as the S 0→S2transition is symmetry forbidden.88S3will also not con- tribute because it is too high in energy and the computed transition dipole moment vanishes.88We combine the ωB97X-D/cc-pVTZ approach with a (non-equilibrium) continuum-solvation approach to account for solvent effects (acetone) in the absorption spectrum. We predict the 0–0 band of the S 0–S1excitation at 2.38 eV, which is in excellent agreement with the energetically lowest peak of the experimental absorption spectrum of the monomer (2.35 eV). The corresponding vertical energies are predicted at about 2.7 eV, under- lining that the neglect of vibrational effects leads to a blue shift inthe absorption spectrum of 0.32 eV in vacuum and 0.34 eV in ace- tone. Table III shows that for vacuum and acetone, the strongest contributions to the difference between vertical excitation energies and the 0–0 peak result from the geometrical relaxation into the S1geometry (0.24 eV–0.25 eV). Variations in the zero-point ener- gies (ZPEs) of both electronic states are about 0.1 eV, i.e., they fur- ther decrease the excitation energy. All other contributions to the TABLE III . Monomer calculations on the S 0→S1transition of DIP. All energies are given in eV. For more explanations, see Table I and the text. Acetone Solvent Vacuum Eq. Non-eq. Method ωB97X-D SCS-CC2 ωB97X-D Basis sets cc-pVDZ cc-pVTZ Experiment Evert(eV)a2.80 2.77 2.72 2.72 Eadi(eV)b2.59 2.53 2.29 2.47 Erelax(eV)c0.21 0.24 0.42 0.24 ZPE (eV)d0.11 0.10 0.09 0–0 (eV)e2.48 2.43 2.20 2.38f2.35 Ecorr(eV)g0.32 0.34 0.51 0.34 aVertical excitation energy. bAdiabatic excitation energy. cRelaxation energy of the excited state (E vert–Eadi). dEnergy correction due to zero-point vibrational effects. e(0–0) excitation energy. fCalculated with the equilibrium ZPE since frequency calculations are not feasible in non-equilibrium solvation. gTotal correction (E vert–E0–0). J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The monomer S 0→S1absorption spectrum of DIP ( ωB97X-D/cc-pVTZ): The black full line shows the experimental data (in acetone).86The computational estimate in acetone is given in red (stick spectrum and its Gaussian convolution, FWHM = 500 cm−1). For a better comparison, these theoretical data were shifted by 0.04 eV so that the energetically lowest peak of the theoretical and experimental spectra coincides. The corresponding data in blue give the computational results in vacuum. These datapoints are shifted by 0.14 eV. 0–0 energies are smaller. The reduction in the basis set’s size from valence triple-zeta (cc-pVTZ) to valence double-zeta (cc-pVDZ) seems to introduce an error of 0.06 eV. In contrast to PBI, an increase in the basis sets slightly improves the agreement with the experimental values for DIP. Using the wave function based SCS-CC2 approach in combination with the cc-pVTZ basis set, the computed excitation energies change by only about 0.05 eV. In sum- mary, as for PBI, the explicit consideration of the photo-induced relaxation of the geometry (vertical excitation vs adiabatic exci- tation) introduces the strongest shift in the computed excitation energies. Figure 5 shows the vibrational structure of the theoretical absorption spectra in acetone (red) or vacuum (blue) at T = 0 K and the experimental spectrum.86For the corresponding convolu- tion with a Gaussian, we used a FWHM of 500 cm−1because it seemed to reflect the overall shape of the bands better than, for example, a FWHM of 270 cm−1, which is the default option in the Gaussian16 program. Decreasing the FWHM introduces additional features in the envelope of the first band, which are not found in the experimental spectra (Figs. S1 and S2). However, while the com- puted energy spacings between the first two bands and their widths agree quite nicely with the experimental results, the intensity ratio between the two lowest energy (highest intensity) bands is predicted less well (Fig. 5). The experimental measurements find a decrease in intensity with an increase in energy. In contrast, the lowest band has less intensity than the next higher one in the computed spectrum. This erratic intensity ratio is found for the theoretical spectra in vac- uum and even stronger in acetone. Thus, while ωB97X-D/cc-pVTZ reproduces the excitation energy quite accurately, it overestimates the relaxation of the DIP molecule due to excitation in the S 1state.The amount of which the relaxation of the excited state is overesti- mated can be assessed by approximating the potentials of the ground end excited states by shifted harmonics. Then, the intensity Inof the vibrational line nis given by a Poisson distribution (In=Sn n!e−S), with the Huang–Rys parameter S=1 2(Δu)2. Here,Δuis the dif- ference between the equilibrium distances in dimensionless coordi- nates.89With the observed intensity patterns, it results that ωB97X- D/cc-pVTZ overestimates the structural change in the excited state (Δu) by about 20%. This seems sufficiently small to employ this model for the qualitative analysis of the excited states shown below. Furthermore, as discussed in the supplementary material, the shape of the theoretical spectrum changes by modifying the FWHM of the Gaussian convolution or by including temperature effects (Fig. S2), while the Herzberg–Teller correction is hardly significant (Fig. S3). The corresponding data for the PDIR-CN 2molecule are given in the supplementary material (Table S2, Fig. S4). We find the same trends as already discussed for PBI and DIP. Similar perfor- mance of the quantum chemical models for these molecules could be expected since the electronic excitations are dominated by the perylene core.28,65However, in a recent study, we also find compara- ble relaxation effects for tetracene, which represents a considerably larger molecule.90 Intra-monomer relaxation effects in aggregate structures The above-mentioned monomer relaxation effects of 0.2 eV –0.4 eV raise the question how distortions of the monomer geometry (intra-monomer relaxation) influence the photophysical behavior of organic semiconductors. The Davidov splitting in such molecu- lar aggregates is often in a similar range. Previous computational investigations often neglect intra-monomer relaxation effects since full optimizations of aggregates are too time-consuming if suffi- ciently accurate approaches are employed. Hence, to make the com- putations feasible, the frozen-monomer approximation has often been used. In this approximation, the geometries of the monomers are fixed to their S 0equilibrium structures. Using these frozen monomers, the potential energy surfaces are computed as a func- tion of the inter-monomer geometrical parameters to model inter- monomer relaxation effects. The inter-monomer parameters are shown in Fig. 3(a). To investigate how inclusion of intra-monomer effects changes the figure, we utilized the PBI-dimer for model cal- culations. For this model, we took various relative orientations of the monomers in the dimer and characterized the intra-monomer relax- ation by distorting each monomer independently from the equi- librium structure of the ground state of the monomer ⃗R(S0)to the equilibrium structure of the first excited state of the monomer ⃗R(S1). We performed the computations for the vacuum case, an almost nonpolar environment with a high refractive index ( ε= 5 andεinf= 4.45) and water ( ε= 78.355 and εinf= 1.776) to inves- tigate the influences of polarizable environments. However, since the comparison of the corresponding PESs (Fig. S5) shows very small changes when changing the polarity of the environment, we focus the discussion on the values obtained for the almost nonpolar medium. Figure 6 summarizes the PESs of the various states as a func- tion of distortions of the monomers. To differentiate between the J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp electronic dimer and monomer states, the dimer states are abbrevi- ated as D 0–D 3, where D 0represents the singlet ground state of the dimer, while D nstands for its n-th excited singlet state. The white lines are given for a better overview. Within the figures, the abscissa and ordinate depict the distortion of the monomers. Moving along the lowest horizontal white line of the PESs, one monomer remains fixed at its ground state geometry, while the other monomer varies its geometry from ⃗R−0.5=⃗R(S0)+(−0.5)Δ⃗R to⃗R1.5=⃗R(S0)+ 1.5Δ⃗R [Eq. (1)]. In Eq. (1), Δ⃗R =⃗R(S0)−⃗R(S1)represents the difference between the equilibrium structures of the monomer in its ground and first excited singlet states. Consequently, ⃗R(S0)is related to χ= 0.0, while ⃗R(S1)corresponds to χ= 1.0. Along the left vertical white line, the structure of the other monomer changes. The PES must be symmetric due to the symmetry of the system [Fig. 3(a)]. Along the diagonal, both monomers are equally distorted so that the exciton must be completely delocalized over the whole dimer. In the outer diagonal parts, both monomers are differently distorted so that the exciton can localize on one monomer. Single points on the PESs are designated as [ ⃗RA;⃗RB] where the structure of the first monomer is ⃗R(S0)+AΔ⃗R, while the second monomer’s geometry is ⃗R(S0)+BΔ⃗R.To make the small variations visible, we choose contour lines with an energy difference of only 0.02 eV. All energies are given with respect to the equilibrium geometry of the ground state, and all dimer structures have a vertical (molecular plane to plane) dis- tance of R Z= 3.31 Å [Fig. 3(a)]. All calculations were performed with theωB97X-D/def2-TZVP approach. Previous investigations show that the dimer is additionally stabilized by small transver- sal shifts (R L) or alternatively by a torsion of both monomers with respect to each other.13,27Nevertheless, in the scope of the present work, we only vary the longitudinal shift and the distance between the two monomers. Figures 6 and 7 compare the energies and the properties of the various dimer states at their optimal R Lorien- tations. Note that the optimal R Zvalue is 3.31 Å for all states. This comparison allows us to differentiate between intra- and inter- monomer relaxation effects. Figures 8 and 9 focus on the differences in the behavior of the D 1and D 2states for the same inter-monomer orientations (the same R Land R Zvalues). As expected, the minimum of the D 0surface (Fig. 6, top left) is found if both monomers adopt essentially the monomer ground state geometry. The small elongation of the equilibrium bond distance in the direction of the excited state indicates that the FIG. 6 . PES as a function of the monomer distortions computed for a monomer distance of 3.31 Å and the optimal longitudinal shift R Lfor the respective electronic state. The energy color code is given on the left side of each PES. The energy difference between two contour lines is 0.02 eV. R(S 0) and R(S 1) denote equilibrium geometries for the monomer in its ground (S 0) and first excited states (S 1), respectively. The white lines are given for a better overview. For more information, see the text. J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Variation in the oscillator strength f osc(a.u.) as a function of the monomer distortions computed for a monomer distance of 3.31 Å and the optimal longitudinal shift R L for the given electronic state. The color code for f oscis given on the left side of each surface. R(S 0) and R(S 1) denote equilibrium structures for the monomer in its ground and first excited states, respectively. The white lines are given for a better overview. For more information, see the text. interaction of the two monomers tends to promote electrons from the HOMO of one monomer into the LUMO of the other one and vice versa. Thus, both monomers are partially excited due to the interaction with the other one, which introduces the observed departure of the dimer equilibrium geometry from the monomer values. However, the energy lowering with respect to the monomer structure amounts to only about 0.02 eV, which is hardly signif- icant. For the D 1state, our computations predict different inter- and intra-monomer geometries compared to the ground state. The optimal inter-monomer orientation for the ground state is found at RL= 1.4 Å, whereas R L= 1.0 Å gives the lowest computed energy for the first excited state. In the D 1state, the optimal intra-monomer geometry is at [ ⃗R0.7;⃗R0.7]. This indicates that an exciton that is delo- calized over a dimer influences the geometry of the single monomers less than it is found for the S 1state of the monomer. This is in line with the work of Harbach and Dreuw on the CO dimer.42In the CO dimer, the CO distance of the dimer D 1state is 1.1875 Å, which lies between the distance found for S 0(1.126 Å) and S 1(1.222 Å) of the monomer. In terms of the χ-parameter, they obtain a value of 0.65, which is surprisingly close to our value of 0.7. Figure 6 also reveals that the D 1-PES shows a curved valley ranging from[⃗R0.7;⃗R0.3] over [⃗R0.7;⃗R0.7] to [⃗R0.3;⃗R0.7]. In this range, the energy varies by less than 0.02 eV, i.e., all structures are virtually isoener- getic. For the vertical energy of the D 1state at the structure [ ⃗R0.0; ⃗R0.0] and R L= 1.0 Å, we compute an excitation energy of 2.452 eV. Intermolecular relaxation along the longitudinal axis to R L= 1.0 Å decreases the excitation energy to 2.360 eV (Einter relax = 0.09 eV). Adding the intra-monomer relaxation ([ ⃗R0.0;⃗R0.0] to [⃗R0.7;⃗R0.7]) for RL= 1.0 Å, the excitation energy drops to 2.225 eV, i.e., Eintra relax amounts to 0.135 eV. Hence, the intra-monomer relaxations are slightly larger than the inter-monomer ones. However, for the for- mer, we only included one degree of freedom, while all possible relaxations are included in the latter. Note that Eintra relax computed for the dimer (0.135 eV) is smaller than the variation found for the monomer (Table I; 0.201 eV). For the D 2state, inter- and intra-monomer effects differ from those computed for the D 1state. The optimal R Lvalue for D 0and D2is equal, i.e., inter-monomer relaxation effects vanish. Further- more, the minimum of the D 2-PES lies along the diagonal, i.e., structures with equally distorted monomers are favored with respect to those with unequally distorted monomers. The intra-monomer relaxation effects for the D 2state are computed to be 0.13 eV. As J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Comparison of the PESs obtained for the longitudinal shifts R L= 0.5 and 2.5 Å. The horizontal distance between both monomers is always 3.31 Å. The energy color code is given on the left side of each PES. The energy difference between two contour lines is 0.02 eV. R(S 0) and R(S 1) denote equilibrium structures for the monomer in its ground and first excited states, respectively. The white lines are given for a better overview. For more information, see the text. for the D 1state, the monomers do not fully relax to the S 1geome- try of the monomer. The minimum ranges from about [ ⃗R0.5;⃗R0.5] to [⃗R0.7;⃗R0.7] instead. The shape of the intra-monomer PES of the D 3state seems to be a mixture of shapes of the D 1and D 2states. However, while D 1and D2possess mainly Frenkel character, the D 3represents a CT state. Hence, to describe its intra-monomer relaxation effects, we would have to distort the monomers to the anionic and cationic structures rather than to the S 1structures of the monomers. Since these addi- tional variations are out of the scope of the present work, we will focus in the following on D 1and D 2states of the dimer. The PES of the D 1states possesses a broad minimum, which comprises unequally distorted monomers and equally distorted monomers. For the latter, the exciton must be completely delocal- ized due to symmetry reasons. Since unequally distorted monomers can induce exciton localization, it is of interest to investigate the variation in the P Rvalue obtained from the TheoDORE pro- gram. The P Rvalue differentiates between completely delocalized (PR= 2.0) and localized (P R= 1.0) excitons. Hence, for equally dis- torted monomers, P R= 2 is obtained due to the symmetry. The P R value for [ ⃗R0.3;⃗R0.7] is about 1.9, showing that excitons remainsmainly delocalized in the range of the curved valley of the D 1 state. For the D 2state, the minimum runs along the diagonal, and therefore, the excitons are also delocalized. Another property that differentiates between localized and delocalized excitons is the oscillator strength f osc. For a completely delocalized exciton, f oscshould vary roughly between twice the monomer value and zero. For completely localized excitons, all oscil- lator strengths should resemble the monomer values. For example, for an H-aggregate, the D 1state should be dark (f osc= 0) if the exciton is delocalized. If the exciton starts to localize, the oscillator strength of D 1should increase to the limit of the oscillator strength of the S 1monomer state. The corresponding data are summarized in Fig. 7. Indeed, for the D 1state, f oscvanishes for equally dis- torted monomers. For unequally distorted ones, the values smoothly increase. For the D 2state, the opposite is found. The variations found for D 3resemble the behavior expected for a CT state. In the vibronic coupling theory, the relative size of the Davy- dov splitting with respect to the monomer relaxation energy deter- mines whether an asymmetrical distortion of both monomers will take place or not. The monomer relaxation energy within the pJT framework is related to the size of the vibronic interactions.42,43 J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . Comparison of the PESs obtained for the longitudinal shifts of R L= 0.5 and 2.5 Å. The horizontal distance between both monomers is always 5.0 Å. The energy color code is given on the left side of each PES. The energy difference between two contour lines is 0.02 eV. R(S 0) and R(S 1) denote equilibrium structures for the monomer in its ground and first excited states, respectively. For more information, see the text. Figures 8 and 9 investigate these interplays. We compare the sit- uations obtained for R L= 0.5 Å and R L= 2.5 Å in Figs. 8 and 9. The horizontal distance between both monomers (R Z) is 3.31 Å in Fig. 8. Thus, the coupling is stronger but should vary due to the variation in R L. The coupling J is given as half of the energy differ- ence between D 1and D 2(ΔE = 2 J) for equally distorted monomers. Using this relationship, for R Z= 3.31 Å and R L= 0.5 Å, our com- putations predict a coupling of 0.43 eV for [ ⃗R0.0;⃗R0.0] and 0.39 eV [⃗R1.0;⃗R1.0]. We compute a coupling of 0.12 eV for both monomer relaxations for R Z= 3.31 Å and R L= 2.5 Å, i.e., the coupling is indeed smaller. Figure 8 shows that this variation in the coupling influences the shapes of the PESs, but the overall energy variations are small. Figure S6 gives the f oscvalues. They show that even for [⃗R0.5;⃗R1.0], foscbehaves as expected for an H-aggregate (R L= 0.5 Å) or a J-aggregate (R L= 2.5 Å), showing the excitons to be still delocalized. For R Z= 5.0 Å, the coupling is only 0.04 eV–0.05 eV for RL= 0.5 Å and 0.03 eV for R L= 2.5 Å. The considerably smaller cou- plings are reflected in the PES shapes, as given in Fig. 9. The PESs of the D 1state possess two distinct minima at [ ⃗R1.0;⃗R0.0] and [⃗R0.0;⃗R1.0] for which one monomer remains at the ground state structure,while the other relaxes to the S 1monomer structure. The exciton is strongly localized in these positions at the relaxed monomer as shown by the computed P Rvalue of 1.08. Hence, the system repre- sents a localized exciton on the relaxed monomer, which is weakly perturbed by the non-excited PBI monomer at the distance of 5 Å for both minima. For R L= 0.5 Å, the intra-monomer relax- ation energy (E relax) is 0.204 eV, while 0.216 eV is computed for RL= 2.5 eV. We included environmental effects for these calcula- tions. The corresponding vacuum environment data are 0.143 eV for RL= 0.5 Å and 0.155 eV for R L= 2.5 Å. Both values are smaller than Erelax calculated for the monomer in vacuum (0.218 eV in Table II; ωB97X-D/def2-TZVP). This implies that the disturbed, non-excited monomer slightly decreases the relaxation energy, while a polariz- able environment leads to a small increase. Our computations show the well-known double-well potential of the Marcus theory or of a pJT-surface with a strong vibronic coupling for the PESs of the D 1 state. The computations also show that an estimate of the barrier height based on the unrelaxed geometry [ ⃗R0.0;⃗R0.0] leads to an over- estimation thereof. The barrier height is estimated at 0.20 eV for RL= 0.5 Å if [ ⃗R0.0;⃗R0.0] is taken as top of the barrier. The value reduces to only about 0.09 eV if the optimal arrangement ( ≈[⃗R0.7; J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ⃗R0.7]) is used. The corresponding values for R L= 2.5 Å are 0.22 eV (via [⃗R0.0;⃗R0.0]) and 0.08 eV (via [ ⃗R0.7;⃗R0.7]). The shapes of the PESs of D 1and D 2differ considerably for R Z = 3.31 Å and 5.0 Å. Especially the computed shapes for 5.0 Å nicely reflect the 1D potential curves given by Marcus theory for ground and excited states of weakly interacting monomers or that of a pJT effect with strong vibronic coupling. In Fig. 9, the D 1state exhibits the double-well potential if both minima are connected, while the shape of D 2along this coordinate is harmonic. A very similar poten- tial was obtained by Garcia-Fernandez et al. for the benzene dimer in D6hsymmetry at a distance of 5 Å, emphasizing the generality of our results.43As in the Marcus theory picture, the difference between D1and D 2in our model results from the required orthogonality between two solutions of a Hermitian operator such as the elec- tronic Hamilton operator. The exciton is localized on the relaxed monomer for the minimum of the D 1state at [ ⃗R0.0;⃗R1.0]. Due to the required orthogonality, a localized exciton would reside on the unre- laxed monomer for D 2. If it is localized on the relaxed monomer, it would populate the S 2state of the monomer, which is considerably higher in energy. However, if the exciton localized on the unrelaxed monomer, both monomers exhibit unfavorable geometries for this localization. Hence, the symmetrical relaxation of both monomers leading to a delocalized exciton represents the energetically most stable situation for D 2. For our investigations on a larger cluster, we selected DIP tetramers in which the monomers take the position of the crys- tal structure as a first example. While the monomer positions were taken from the experimental crystal structure (CCDC code 642482),91,92the intra-monomer coordinates were determined by full optimizations for the S 0and S 1states of the DIP monomer. As for the PBI dimer, the intra-monomer geometries were varied by changing the monomer structures linearly from ⃗R(S0)to⃗R(S1). The energy variations obtained for the four lowest electronic states of the tetramer (T 1–T4) as a function of varying monomer geometries are collected in Fig. 10. The abscissa gives the distortion of the respective monomers toward ⃗R(S1). To simplify the abbreviations, we intro- duce a different nomenclature for the tetramers. If all monomers are equally distorted, the distortion is indicated only once. For example, “1” denotes the structure in which all monomers are in the S 1geom- etry, while “0.25” abbreviates the structure in which all monomers are distorted to ⃗R0.25([⃗R0.25;⃗R0.25;⃗R0.25;⃗R0.25] in dimer notation). For differently distorted monomers, the abbreviation denotes which one is distorted (Fig. 3). For the [0.5 0 0 0.5] monomer, 1 and 4 are distorted to ⃗R0.5, while 1000 and 0001 abbreviate the clusters in which monomer 1 or 4 adopts the S 1geometry. For the cluster computations, these geometries have different energies because the corresponding monomers are not transferred into each other by a symmetry operation. Symmetrically equivalent are only monomers 1 and 2 as well as 3 and 4. As expected, the global minimum of the tetramer ground state T0is obtained for “0.” Distortions of the monomers toward ⃗R(S1) increase the ground state energies. For example, for “0.25,” the energy slightly increases by 0.05 eV, while an energy increase of 0.81 eV is obtained for “1.” For “1000,” the energy raises by 0.2 eV. The minimum of the first electronically excited tetramer state T 1is found for the structure “0001,” i.e., if monomer 4 (Fig. 3) adopts the S1geometry, while all other monomers remain in the ground state geometry. The excitation energy with respect to the energy of the FIG. 10 . Energies of the four lowest excited states of the DIP tetramer (T 1in red, T2in blue, T 3in green, and T 4in purple) as a function of the monomer geometries (for details see text). All energies are given with respect to the energy of the equi- librium geometry of the ground state.∗indicates states with considerable oscillator strengths. A full∗denotes f osc>0.5, while an empty one gives f osc>0.2. equilibrium geometry of T 0(Eadi) is 2.46 eV, and the corresponding value for “1000” is 2.50 eV. The computed P Rvalues of 1.04 and 1.13 indicate that the unequal distortions lead to a nearly complete local- ization of the exciton on the distorted monomer. If all monomers are equally distorted by a quarter of the difference between S 0and S 1 (“0.25”), E adiincreases to 2.55 eV. For a tetramer, P Rvaries between 4 (exciton delocalized over all 4 monomers) and 1 (exciton localized on one monomer). For “0.25,” the P Rvalue is 3.97. The P Rvalue dif- fers from 4 because not all monomers are symmetrically equivalent. The comparison of the “1000,” “0001,” and “0.25” structures shows that for the T 1state of the DIP tetramer, the localized structures are about 0.1 eV lower in energy than the delocalized ones. The partially localized structures “0.5 0.5 0 0” (E adi= 2.58 eV) and “0.5 0 0.5 0” (Eadi= 2.52 eV) are similar in energy to the delocalized structures “0.25.” For “0.5 0 0.5 0,” we compute a P Rvalue of 2.09, while 3.17 was computed for “0.5 0.5 0 0.” The difference arises because the exciton is delocalized over two neighbors for “0.5 0 0.5 0” but not for “0.5 0.5 0 0.” The structure “0.25” is slightly lower in energy than “0.5” (E adi= 2.59 eV). If the distortion increases to “0.75” and “1,” the Eadiincreases to 2.73 eV and 2.97 eV. In all cases, the P Rvalues are near to 4. As for the PBI dimer, the T 2state of the DIP tetramer adopts equally distorted structures. The same holds for the T 3and T 4states. For the T 2state, the adiabatic excitation energies of the equally dis- torted structure “0.25” and of the partially unequally distorted struc- ture “0.5 0.5 0 0” are 2.61 eV. However, the corresponding P Rvalues are only around 2, indicating that for “0.25,” the exciton is not com- pletely delocalized as found for the corresponding structure of the T1state. A closer inspection, indeed, showed that for the T 2state, the exciton is only delocalized over the monomers 3 and 4. For T 3, it is delocalized over the monomers 1 and 2. Such a localization J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp is possible because only 1 and 2 as well as 3 and 4 are symmetri- cally equivalent. For T 4, the exciton is again completely delocalized for “0.25.” Note that the unit cell of the DIP crystal contains two monomers, i.e., also for the crystal, not all monomers are identical. The P Rvalue computed for the “0.5 0.5 0 0” structure of the T 2is also lower than the corresponding value obtained for T 1. For T 3and T 4, the “0.25” structures represent the respective minima (2.67 eV and 2.85 eV). For the T 3state, this equally distorted structure lies about 0.2 eV below the “1000” or “0001” structures and still 0.1 eV below the structures in which only two monomers are distorted. Even the “0.5” structure is still lower than the unequally distorted structures. For the T 4state, energy variations are smaller. The “1000” structure is in the same energy range (E adi= 2.88 eV) as the equally distorted structure “0.25” (2.85 eV), while the partially unequally distorted structures “0.5 0.5 0 0” and “0.5 0 0.5 0” lie slightly higher in energy (Eadi= 2.93). For the T 1state of the DIP tetramer, the unsymmetrical dis- tortion is clearly favored over the symmetrical one, while for the D 1 state of the PBI dimer, both structures were isoenergetic. This can be explained by the couplings for the DIP tetramer, which are expected to be smaller because the molecular planes of the monomers are tilted with respect to each other. To check if differences in the elec- tronic structure of both molecules also contribute, we performed test computations for DIP dimers in which both monomers lie on top of each other. Based on previous investigations of the minimum structures of such dimers, we choose R L= 1.5 Å and R Z= 3.6 Å.93 For this dimer, the symmetrically distorted [ ⃗R0.5;⃗R0.5] structure (Eadi= 2.47 eV) and unsymmetrically distorted [ ⃗R0.0;⃗R1.0] (E adi = 2.46 eV) are again virtually identical. The corresponding val- ues for the PBI dimer with R L= 1.4 Å and R Z= 3.5 Å are 2.41 eV ([ ⃗R0.5;⃗R0.5]) and 2.35 eV [ ⃗R1.0;⃗R0.0], i.e., the variations are similar. Other structures like [ ⃗R0.0;⃗R0.0] and [⃗R1.0;⃗R1.0] are higher in energy for DIP (E adi= 2.60 eV and 2.55 eV) and for PBI (E adi= 2.53 eV and 2.45 eV). In both dimers, symmetri- cally distorted structures are favored with respect to unsymmet- rical ones in the D 2state (DIP: [ ⃗R0.5;⃗R0.5] E adi= 2.65 eV; [ ⃗R0.0; ⃗R1.0] E adi= 2.75 eV; PBI: [ ⃗R0.5;⃗R0.5] E adi= 2.51 eV; [ ⃗R0.0;⃗R1.0] Eadi= 2.69 eV). This comparison indicates that the favored unsym- metrically distorted monomers for the DIP tetramer mainly result from the different orientations of monomers, while differences between the electronic structure of DIP and PBI are of minor importance. Comparing the optimized geometries of the first excited states of monomer, dimer, and tetramer, respectively, shows that the dis- tortion relative to the ground state structure decreases. Taking the factor of 1.0 for the difference found for the monomer, for PBI, DIP, and the CO-dimer, the delocalized exciton leads to smaller distor- tions of 0.7, 0.65, and about 0.5. For the DIP tetramer, the structure in which each monomer is distorted by a factor of “0.25” is already lower in energy than the “0.5” geometry. This trend indicates that for the first excited states, the size of the photo-induced geome- try distortions decreases with an increase in delocalization of the exciton. Indeed, the trend is supported by test calculations for a clus- ter consisting of 14 tetracene. In this system, the vertical excitation energy (factor 0.0) is already lower than the excitation energy to the geometry in which all monomers are equally distorted by a factor of 1/14. Computations to investigate if this is a general trend are under way.CONCLUSIONS In the first part of the present publication, we show that DFT- based descriptions of vibrationally resolved absorption spectra of perylene-based organic semiconductor monomers such as PBI, DIP, and PDIR-CN 2are very accurate. The computations show that photo-induced geometrical relaxation effects are quite important and have to be included for an accurate description of the spec- tra. The ZPE also contributes but is of minor influence. For solvent effects, the non-equilibrium option is important. The most inten- sive band mainly represents the 0–0 transition. Small errors in the ratio of the intensities of the various bands indicate that the photo- induced relaxation effects are slightly overestimated. The Franck– Condon and harmonic approximations are essentially appropri- ate, but the inclusion of temperature (hot bands) is in some case significant. The size of the photo-induced relaxation effects found for monomers suggested that the inclusion of intra-monomer relaxation effects is also important for the description of spectra of aggregates. Our investigations of the PBI dimer, indeed, underline their impor- tance. The D 1state shows the expected double well shape for larger distances for which the coupling is weaker than the intra-monomer relaxation. Therefore, the exciton localizes on one monomer. How- ever, even in this situation, the D 2PES exhibits the harmonic form, i.e., the exciton remains localized. The strong dependence on the state under consideration results because the wavefunctions of both states have to be orthogonal to each other for each geometry. For the optimal distance between both monomers (R Z= 3.31 Å), the coupling is sufficiently strong that the excitons remain mainly delo- calized for the D 1as well as for D 2states, but the differences in the shapes of both PES are obvious. Additionally, the optimal longitudi- nal shifts (R L) of both states also differ. The results can be modeled by the vibronic coupling, the Marcus theory or by pJT-effect. All models represent a two-state system in which the eigenvalues are determined by an interplay between energy difference of the diag- onal elements and size of the outer-diagonal elements. The pJT is closer to the physical nature of the effect. In the delocalized situation for the dimer, both monomers adopt the same structures, which lie somewhere between the S 0and S 1structures of the monomer. The relaxation to a localized exciton is then induced by a force, which drives one monomer toward the S 1geometry and the other toward the S 0geometry. Obviously, this force leads to the antisymmetric motion, which is also obtained from symmetry considerations. The Marcus theory also allows for a description of these effects; however, the underlying forces remain more abstract as only one coordinate is considered. For the DIP tetramer, the coupling strength is smaller so that for the lowest electronically excited state T 1, the localized structure is the global minimum. In this structure, one monomer adopts the equilibrium geometry of the S 1state of the monomer, while all other monomers remain undisturbed. Structures in which all monomers are equally distorted are more stable for the T 2–T4states. However, even for these states, the amount of delocalization varies. For T 2 and T 3, the exciton is only delocalized over two monomers, while it is completely delocalized for T 4. This finding emphasizes that intra-monomer relaxation effects strongly depend not only on the ratio of intra-monomer relaxation and coupling but also on the electronic state under consideration. Note that the photo-induced J. Chem. Phys. 153, 224104 (2020); doi: 10.1063/5.0028943 153, 224104-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp geometry distortions decrease if the exciton is delocalized over more and more monomers. For dimers, the distortion is smaller by a factor of 0.5–0.7 than the distortion found for the monomer. The distortions shrink to about ¼if the exciton is delocalized over a tetramer. AUTHORS’ CONTRIBUTIONS M.D., S.W., and D.K. performed all computations and con- tributed equally to this work. B.E. supervised the calculations. P.T., R.F.F., and B.E. wrote this manuscript. B.E., S.W., and P.T. thank the DFG (B.E. and S.W. in the framework of the GRK2112, P.T. for the project TE479/6-1) and D.K. thanks the Fonds der chemischen Industrie for funding. SUPPLEMENTARY MATERIAL See the supplementary material for further information. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1F. Würthner, C. R. Saha-Möller, B. Fimmel, S. Ogi, P. Leowanawat, and D. Schmidt, “Perylene bisimide dye assemblies as archetype functional supramolecular materials,” Chem. Rev. 116, 962–1052 (2016). 2H. E. Katz, A. J. 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J. Vac. Sci. Technol. A 38, 063205 (2020); https://doi.org/10.1116/6.0000402 38, 063205 © 2020 Author(s).First-principles theoretical analysis and electron energy loss spectroscopy of vacancy defects in bulk and nonpolar ( ) surface of GaN Cite as: J. Vac. Sci. Technol. A 38, 063205 (2020); https://doi.org/10.1116/6.0000402 Submitted: 18 June 2020 . Accepted: 11 September 2020 . Published Online: 30 September 2020 Sanjay Nayak , Mit H. Naik , Manish Jain , Umesh V. Waghmare , and Sonnada M. Shivaprasad ARTICLES YOU MAY BE INTERESTED IN Structural breakdown in high power GaN-on-GaN p-n diode devices stressed to failure Journal of Vacuum Science & Technology A 38, 063402 (2020); https:// doi.org/10.1116/6.0000488 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 Band alignment at β-Ga2O3/III-N (III = Al, Ga) interfaces through hybrid functional calculations Applied Physics Letters 117, 102103 (2020); https://doi.org/10.1063/5.0020442First-principles theoretical analysis and electron energy loss spectroscopy of vacancy defects in bulk and nonpolar (10 10) surface of GaN Cite as: J. Vac. Sci. Technol. A 38, 063205 (2020); doi: 10.1116/6.0000402 View Online Export Citation CrossMar k Submitted: 18 June 2020 · Accepted: 11 September 2020 · Published Online: 30 September 2020 Sanjay Nayak,1,a) Mit H. Naik,2Manish Jain,2 Umesh V. Waghmare,3and Sonnada M. Shivaprasad1,b) AFFILIATIONS 1Chemistry and Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore 560064, India 2Department of Physics, Centre for Condensed Matter Theory, Indian Institute of Science, Bangalore 560012, India 3Theoretical Sciences Unit, Jawaharlal Nehru Centre for Advanced Scientific Research (JNCASR), Bangalore 560064, India a)Electronic mail: sanjaynayak.physu@gmail.com b)Electronic mail: smsprasad@jncasr.ac.in ABSTRACT We determine atomic structure, electronic structure, formation energies, magnetic properties of native point defects, such as gallium (Ga) and nitrogen (N) vacancies, in bulk and at the nonpolar (10 10) surface of wurtzite gallium nitride ( w-GaN) using first-principles density functional theory (DFT) based calculations. In bulk and at the (10 10) surface of GaN, N vacancies are significantly more stable than Ga vacancies under both Ga-rich and N-rich conditions. We show that within DFT-local density approximated N vacancies formspontaneously at the (10 10) surface of GaN when doped to raise the Fermi level up to /C251:0 eV above valence band maximum (VBM) while with valence band edge correction it is 1.79 eV above VBM. We provide experimental evidence for occurrence of N vacancies with electron energy loss spectroscopy measurements, which further hints the N vacancies at surface to the source of auto-doping which may explainhigh electrical conductivity of GaN nanowall network grown with molecular beam epitaxy. Published under license by AVS. https://doi.org/10.1116/6.0000402 I. INTRODUCTION Group III-nitride based semiconductors are important to applications in opto-electronic devices 1,2such as light emitting diodes (LEDs) and lasers because of their direct and tunable bandgap (0.7 –6.0 eV). In addition, GaN has emerged as a strong candidate for dilute magnetic semiconductors,3,4high power and high frequency devices,5,6and high electron mobility transistor applications.7As the group III-nitride semiconductors commonly crystallize in the polar wurtzite structure, they have a large internal piezoelectric field8(/difference106Vcm/C01) along the (0001) direction, which suppresses the radiative recombination. To avoid sucheffects of the piezoelectric field on their electronic structure, GaNbased heterostructures are grown along nonpolar (10 10)9and (1120)10directions. During the growth process, defects such as vacancies and dislocations, nucleate naturally and are commonly observed.First-principles calculations have been quite effective in under- standing the electronic properties of pristine w-GaN as well as its defects.11–14While atomic and electronic structures of Ga vacancies in GaN have been studied extensively, the structure and associatedproperties of N vacancies remain elusive. Numerous attempts havebeen made to introduce magnetism in GaN by incorporating mag-netic impurities 15,16for spintronic applications. Several authors proposed that the cation vacancies in w-GaN may lead to ferro- magnetism,17,18while w-GaN with anion vacancies exhibits para- magnetic behavior.19This is linked with the fact that cation vacancies acts as e/C0acceptors, whereas anion vacancies act as shallow donor.20 Overcoming the unintentional n-type doping of GaN has been a great challenge for its semiconductor technologies and the origin of such auto-doping is controversial. Van de Walle andNeugebauer 20eliminated N vacancies as a possible cause ofARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-1 Published under license by A VS.auto-doping arguing that its formation energy is too high and suggested that oxygen and/or carbon impurity may explain the observed high conductivity in films grown with metal organicchemical vapor deposition, where organometallic precursors areused and could be one of the sources of these impurities. However,unintentional n-type doping and high conductivity are also observed in GaN grown with molecular beam epitaxy (MBE). 21As MBE uses pure metal and ultrapure gas as the sources, the films aregrown in ultrahigh vacuum conditions, the cause of auto-doping inthis case may be attributed to point defects instead of carbon andoxygen impurities. Calculations carried out by Boguslawski et al. 22 showed that N vacancy introduces a shallow donor state and may be relevant to auto-doping in GaN. There is a lot of variation in estimates of formation energy of N vacancies in bulk w-GaN under N-rich conditions in reports based on first-principles calculations which range from 1.1 to 5.08 eV.12–14,22–25Second, the structure and associated properties of vacancies at the (10 10) surface have not been studied in depth.26 Defects can form more easily in lower dimensional structures, forexample, at the surface of thin films and nanowires due to the factthat the surface itself is a planar defect with its atom having lower coordination number and unsaturated dangling bonds. InRefs. 21 and 27, relatively high electrical conductivity was observed in the GaN nanowall network (NWN) grown on the sap-phire substrate, and the surface electronic structure was proposed to be at the origin of high conductivity of the material. Atomistic investigation of the structure shows that majorly the structure has(10 10) facets.28To investigate the role of the (10 10) surface in yielding a high conductivity of the material, we determined theatomic and electronic structure of the (10 10) surface of w-GaN containing vacancies defects using first-principles density func- tional theory (DFT) calculations. For completeness and a consistentcomparison, we present analysis of the atomic structure and elec-tronic properties of Ga and N vacancies in bulk w-GaN. II. METHODS A. Experimental details The faceted GaN nanostructure has been synthesized using a plasma assisted molecular beam epitaxy system on a (0001) plane ofsapphire. The detailed procedure of substrate preparation and growth parameters can be found elsewhere. 28,29Structural quality of the films is determined by a high-resolution x-ray diffractometer (Discover D8Bruker) with a Cu K αx-ray source of wavelength of 1.5406 Å. The surface stoichiometry is determined by x-ray photo-electron spectro-scopy (XPS) using Al K α(1486.7 eV) as excitation source. Transmission electron microscopy (TEM) and spectroscopy experiments were performed in an aberration, corrected a high-resolution TEM (FEI, TITAN 80-300). The electron accelerationvoltage was set at 300 kV. To avoid beam induced damages, theexposure time of the measurements was kept in the order of a second or less. Electronic bandgap and valence band electronic structure were determined at the nanoscale region using high-resolution electron energy loss spectroscopy (HREELS) with abeam energy of 300 keV. A gun monochromator was used in our study which provides a spectral resolution better than 180 meV. We removed the zero loss peaks from the HREELS spectra by fitted thelogarithm tail method. We employ the Kramers –Kronig analysis to calculate the optical absorption coefficient. Focused ion beam tech- nique is used for TEM sample preparation. B. Computational details Our calculations are based on density functional theory as implemented in “Spanish Initiative of Electronic System with Thousands of Atoms ”(SIESTA) code. 30Local density approximated functional parametrized by Ceperley and Alder31was used in treat- ing electronic exchange and correlation energy. Norm-conserving pseudopotentials generated by the scheme of Troulleir and Martin32 in the Kleinman and Bylander33form were used to model ionic cores of Ga and N with valence electronic configurations of3d 104s24p1and 2 s22p3, respectively. Interaction between core and valence electrons was included through the nonlinear core correc- tion.34The valence electron wave functions were expanded by using a combination of single zeta and double zeta with polarization function(seeTable IV in the Appendix). Hartree potential and charge density were computed on a uniformly spaced grid with a maximum kinetic energy cutoff of 200 Ry. Integration over the Brillouin zone of w-GaN were sampled on a Γ- centered 5 /C25/C23 mesh of k points in the unit cell of reciprocal space. 35Positions of all the atoms were allowed to relax using the conjugate gradient technique to minimize energy until force on each atom was less than 0.02 eV/Å. For simulation of Ga vacancy in bulk GaN, we used a 4 /C24/C22 supercell (128 atoms) amounting to a vacancy concentration of 0.78%. To simulate Nvacancies in bulk GaN, we used three different supercells, 2 /C22/C22, 3/C23/C22, and 4 /C24/C22, which model vacancy concentrations of 3.125%, 1.38%, and 0.78%, respectively. In simulation of stoichiomet- ric (10 10) surface of GaN, we used a symmetric slab of 32 atoms. A vacuum space of /difference12 Å was used to keep the interaction between the periodic images of the slab weak. We used a 2 /C22 periodic-cell in the plane of the slab to simulate surface vacancies (Ga and N). Formation energy of defects (for neutral state) in bulk as well as at surfaces was calculated using Zhang –Northrup scheme,36 Ef¼Etot(VNo rG a )/C0Etot(pristine )þΣniμi(No rG a ) þq(EFþEVBM/C0ΔV0=b)þEcorr q, where Etot(VNo rG a )a n d Etot(pristine ) are the total energies of the supercells containing a neutral vacancy each and the reference pris- tine supercell, respectively. niandμirepresent the number of vacan- cies and chemical potential of the ith species, respectively. In this work, we have calculated the defect formation energy under bothGa- and N-rich conditions. Under N-rich conditions, μ Nis the energy of N atom [obtained from the total energy Etot(N2)o f N2 molecule, i.e., μN¼1 2Etot(N2)], and the chemical potential of Ga is calculated assuming the thermodynamic equilibrium, i.e.,μ GaþμN¼EGaN[bulk], where EGaN[bulk]i st h eb u l kt o t a le n e r g yo f one formula unit of w-GaN. Similarly, for Ga-rich conditions, μGa is the chemical potential of Ga atoms in bulk α-Ga (i.e., μGa¼μGa[bulk]), and chemical potential of N atom ( μN)i sc a l c u l a t e d from the thermal equilibrium condition. Under Ga-rich conditions,the defect formation energies of Ga and N vacancies are shifted by /C0ΔH fandΔHf, respectively, with respec t to the N-rich conditions. Here, ΔHfis the formation enthalpy of w-GaN and is estimated byARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-2 Published under license by A VS.the formula, ΔHf¼EGaN[bulk]/C0μGa[bulk]/C01 2Etot(N2). The esti- mated value of ΔHf[GaN ]i s/C01.9 eV, which is in good agreement with earlier calculations based on DFT-LDA.12,13EFandEVBMare the Fermi energy and energy of bulk valence band maximum (VBM) oft h ep r i s t i n eG a N ,r e s p e c t i v e l y . ΔV 0=bis used for aligning the electro- static potentials of bulk and the supercells with neutral defects and is obtained by comparing of electrostatic potentials in the bulk like region far from the neutral defect an d in the pristine bulk calculation. We have used the electrostatic correction term Ecorr qfor a charged defect in the supercell by using the method proposed by Freysoldtet al., 37as implemented in CoFFEE code.38We have used the static dielectric constant ( εs) of GaN as 9.06 obtained from Ref. 39 in esti- mation of Ecorr q. The thermodynamic transition level ε(q1=q2) is estimated by the relation ε(q1=q2)¼Ef(q1;EF¼0)/C0Ef(q2;EF¼0) q2/C0q1, where Ef(q;EF¼0) is the formation energy of defect X in charge state qwhen the Fermi energy is at the VBM. Surface energy ( σ) is estimated using σ¼1 2A(Eslab/C0n/C2EGaN[bulk]), where Eslabis the total energy of slab, “n”is the number of formula units of GaN in the slab, and Ais the the area of surface unit cell in slab. In simulations of vacancies at the surface,corresponding atoms were removed from the sites at both surfaces of the slab. III. EXPERIMENTAL HRTEM AND HREELS RESULTS A dark-field TEM image of the GaN NWN grown on sapphire [Fig. 1(a) ], and HRTEM images of the sidewall of the GaN NWN [Figs. 1(b) and1(c)] confirm that the sidewalls are made of (10 10) surfaces of GaN. Formation of atomic steps in the structure can be clearly seen in Fig. 1(c) [shown by red dots, representing Ga atoms, inFig. 1(c) ]. We have obtained both Valence EELS and core-loss EELS of the (10 10) surface, where the beam direction is along [1010], and the spectra are shown in Figs. 1(d) and 1(e), respec- tively. Figure 1(d) shows the energy loss function and the imagi- nary part of the dielectric function [Im ε(ω)], obtained from Kramers –Kronig analysis. We have recorded a transition at 2.2 eV along with dielectric profile starting from /C253.0 eV, which suggest that defects in GaN nanowall have a combination of both deep and shallow natures in the bandgap region. An extensive XPS study on a similar GaN nanowall structure reveled Ga-rich surfaces with N:Ga = 1:1.57. We corroborate a relatively high electrical conductivityobserved with an n-type carrier density 21,27of 1019cm/C03of non- stoichiometric surface using first-principles DFT simulations, and discussed thoroughly in Sec. IV .IV. THEORETICAL ANALYSIS AND DISCUSSION A. Atomic and electronic structure of bulk w-GaN To establish the credibility in the numerical parameters used in our work here, we compared results obtained from our simula- tion with the earlier publishes ones (see Table I ). Our obtained results are in good agreement with the published work in the litera-ture. The equilibrium lattice parameters ( aand c), internal parame- ter ( u), and the bandgap (E g) of bulk w-GaN estimated with different methods and their experimental values are listed inTable I . Our estimates of lattice parameters agree well with earlier calculations and are within a deviation of /C250:5% from experimen- tal values. 40Our estimate of the bandgap of bulk w-GaN is 2.06 eV atΓpoint [see Fig. 11(a) ], which agrees well with earlier calcula- tions41,42based on plane wave (PW) basis. This is underestimated with respect to experimental value of 3.40 eV at RT, as is typical ofthat DFT-LDA. 42,43A detailed comparison of electronic structure is provided in Appendix. 1. Ga vacancies in bulk w-GaN Due to a neutral cation (Ga) vacancy at concentration of 0.78%, four neighboring N atoms move away from their positions causing a contraction in their bonds with the other Ga neighborsby/C251:9%–2.3%. This is smaller but agrees qualitatively with earlier estimates of the change in bond lengths of /C253:5%–3.7% reported by Neugebauer and Van de Walle, using a PW pseudopotential based calculations within LDA ( Ref. 14 ) and by Carter and Stampfl 45using GGA with SIESTA code (2.9% –3.7%). For Ga vacancies in charge states of /C01,/C02, and /C03, the contraction in bond length is 2.5% –2.6%, 3.3% –3.4%, and 4.1% –4.3%. Our esti- mate of the formation energy of isolated neutral Ga vacancy obtained under N-rich conditions is 6.90 eV, which is in good agreement with earlier DFT calculations (see Table II ). The forma- tion energies of Ga vacancies in /C01,/C02, and /C03 charge state at p-type growth conditions are 7.17, 8.29, and 10.18 eV, respectively. The defect formation energies versus Fermi energy is presented in Fig. 2(a) . We find the thermodynamic transition levels (0/ /C0), (/C0/2/C0), and (2 /C0/3/C0)o f VGaare at 0.27, 1.11, and 1.89 eV above the VBM. Our estimate of the correction term ( Ecorr qþqΔV0=b)i s 0.12, 0.65, and 1.56 eV for /C01,/C02, and /C03e charge states of the defect, respectively. With p-type growth conditions ( EF¼EVBM), neutral Ga vacancies are more stable, while with n-type growthconditions ( E F¼ECBM), Ga vacancies with /C03e charge state are more stable. The thermodynamic transition levels estimated hereagree qualitatively with the calculation where a finite size correc- tions was adopted. 50In the electronic structure of GaN with Ga vacancy concentration of 0.78% [see Fig. 3(a) ], we identify defect bands (denoted as D) by visualizing the spatial distribution of wavefunctions at high symmetry k points such as Γ, A, and M. The elec- tronic structure supports that the neutral Ga vacancies in bulk GaN are triple acceptors and the associated states are spin polarized. The three states are located at 0.59, 0.63 and 0.63 eV above the VBM atΓ-point. As is seen from the spin dependent DOS and PDOS of GaN with vacancy concentration of 0.78% [see Fig. 3(b) ], acceptor states evidently have the character of 2 porbitals of N atoms thatARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-3 Published under license by A VS.coordinate the Ga vacancy site, displaying a local magnetic moment of /C253:0μBconsistent with earlier report.17The estimated spin polarization energy ( Espincolonun /C0polarized /C0Espincolonpolarized )i s /C250:2 eV, which suggests that the magnetic state may be realizedwell above the room temperature. However, according to Dev et al. ,17the coupling between spins of neutral Ga vacancies in bulk GaN has been found to be antiferro magnetic in nature. As theneutral Ga vacancies in bulk w-GaN act as p-type dopants along FIG. 1. TEM image of GaN NWN (a). (b) and (c) are high-resolution TEM images of the sidewall surface of nanowall. The surface is identified to (10 10). Circular (red) dots are placed over the atomic sites to ease the visualization. (d) and (e) are the valence EELS and nitrogen core-loss EELS spectra, respectively. TABLE I. Optimized lattice parameters ( aand c in Å), internal parameter ( u), and bandgap ( Egin eV) of bulk w-GaN. SIESTA LDA ( Ref. 44 ) SIESTA GGA ( Ref. 45 ) VASP LDA ( Ref. 41 ) VASP LDA ( Ref. 42 ) Expt. ( Ref. 40 ) Present LDA a 3.23 3.28 3.155 3.160 3.189 3.17 c 5.19 5.31 5.145 5.150 5.186 5.16u — 0.378 0.3764 0.3765 0.377 0.377 E g 2.37 1.44 2.12 2.10 3.4 2.06ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-4 Published under license by A VS.with weaker dispersion and band width ( /C250:4–0.7 eV) of bands, we conclude that observed higher conductivity of GaN is unlikely to arise from neutral Ga vacancies. 2. N vacancies in bulk w-GaN To simulate N vacancies at various concentrations in bulk w-GaN, we use three supercells, i.e., 2 /C22/C22( 3 2 a t o m s ) ,3 /C23/C22 (72 atoms), and 4 /C24/C22 (128 atoms) and removed one N atom cre- ating vacancy concentrations of 3.125%, 1.38%, and 0.78%, respec-tively. As stated earlier, reports of defect structure and formationenergy of N vacancies under N-rich conditions exhibit a widespread, 12–14,22–25partly due to two different relaxation mechanisms proposed by different groups. Neugebauer and Van de Walle51 reported, using plane wave DFT (LDA) (32 atom supercell i.e., vacancy concentration of 3.125%), that the neighboring Ga atomsmove away from the N-vacancy site and found that the subsequentcontraction of Ga –N bonds near the vacancy is /C254%. This is consis- tent with the calculations of Gulans et al. , 52where the contraction of Ga–N bond length is by /C253:7%–3.9% (96 atom supercell) based on a DFT-GGA and atomic orbital basis for the expansion of Kohn –Sham states. Furthermore, Carter and Stampfl45reported a similar trend by u s i n gt h eG G Ac a l c u l a t i o n sw i t hS I E S T Ac o d eb u tt h e i re s t i m a t e so f bond length contraction is much weaker [ /C250:2–0.3% (96 atom super- cell)] compared to the results discussed earlier. They concluded thatsuch low values of outward relaxation may arise from the inclusion ofGa 3d electrons in the valence. In contrast to these results, Gorczycaet al. 13reported inward relaxation of Ga atoms toward the N vacancy site in 32 atoms supercell of bulk cubic-GaN (c-GaN) using the LMTO (linear muffin-tin orbital) method (LDA) and an elongationof Ga –N bond by /C252%. Another report by Chao et al. 53based on calculation with a plane wave basis code with GGA also showed aninward relaxation of Ga atom causing the elongation of Ga –N bonds by/C251:9%–3.4% using a 16 atom supercell in w-GaN. To present a clear picture of ionic relaxation due to N vacan- cies in w-GaN, we have considered three different concentrations of N vacancies: 3.125%, 1.38%, and 0.78%. During ionic relaxation,structural changes in these configurations of neutral N-vacancy concentrations are similar, and the Ga atoms coordinating N vacancy site toward the vacancy site causing elongation of theirbonds with other N neighbors. The extent of elongation of bondlength varies with N-vacancy concentration. In the case of vacancyconcentrations of 3.125%, 1.38%, and 0.78%, the Ga –N bond stretches by /C251:23–2.5%, 0.8 –3.6%, and 0.6 –0.7%, respectively, similar to the theoretical predictions of Gorczyca et al. 13and Chao et al.53A recent work based on HSE (Heyd –Scuseria –Ernzerhof) methodology, which is computationally much more expensive, pre-dicts similar results. 54Similar ionic relaxation has also been seen in the case of N vacancies in indium nitride.55For N vacancies with charged configuration of +1, +2, and +3, on the other hand, thenearest neighbor Ga atoms displaced away from the N-vacancy sitecausing reduction in Ga –N bond length to 0.10% –0.35%. Under N-rich conditions, formation energy of N vacancies estimated as a function of concentrations is /C254:73, 4.75, and 5.21 eV for 2 /C22/C22, 3/C23/C22, and 4 /C24/C22 supercells, respec- tively. The estimated formation energy of V Nin GaN grown under p-type condition with charge states of +1, +2, and +3 is 3.10, 3.75, and 4.90 eV, respectively. The thermodynamic transition level of VNsuch as (+/0), (2+/3+), and (3+/+) are present 2.10 eV above VBM (i.e., inside the conduction band), 0.7 and 0.9 eV belowVBM, respectively. Our estimation to the correction term(E corr qþqΔV0=b) is 0.17, 0.74, and 1.70 eV for the defect with +1, +2, and +3 charge states, respectively. We find VNwith +1 charge state is the most stable defect within DFT-LDA [see Fig. 2(b) ]. TABLE II. Vacancy formation energies for Ga and N vacancies in bulk w-GaN and at the (10 10) surface of w-GaN under both Ga- and N-rich conditions. All the values are given in electron volt (eV) unit. Present calculationReference-1 (Ref. 46 )Reference-2 ( Ref. 47 ) Reference-3 ( Ref. 48 ) Reference-4 ( Ref. 49 ) Configuration Charge state Ga-rich N-rich Ga-rich N-rich Ga-rich Ga-rich Ga-rich VBulk Ga 0 8.80 6.90 7.02 6.58 8.40 8.84 9.06 −1 9.07 7.17 8.90 8.46 8.83 9.23 9.31 −2 10.19 8.29 10.56 9.60 9.60 9.98 9.95 −3 12.08 10.18 14.14 13.70 10.67 11.14 11.05 VBulk N 0 3.31 5.21 2.59 3.03 3.16 —— +1 1.20 3.10 −0.58 −0.14 0.82 — 0.10 +2 1.85 3.75 —— 0.95 —— +3 3.00 4.90 −1.95 −1.51 0.89 — −1.08 VSurf Ga 0 2.07 3.97 —— — — — −1 2.45 4.35 —— — — — −2 2.72 4.62 —— — — — −3 6.02 7.92 —— — — — VSurf N 0 1.27 3.17 —— — — — +1 −1.86 0.04 —— — — — +2 −3.21 −1.31 —— — — — +3 −4.76 −2.86 —— — — —ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-5 Published under license by A VS.N vacancy in the bulk w-GaN is a triple donor and introduces four defect states in the electronic structure. Out of these defectstates as suggested by Neugebauer and Van de Walle, 14one fully occupied state lies below the conduction band and three other defect states lie inside the conduction band manifold. However, thework of Carter and Stampfl 45suggested presence of three singlet states near the CBM, two of which are below the CBM by 0.1 and 0.9 eV, respectively, and third one is above the CBM by 0.3 eV. In addition, there is a state about 0.5 eV below the VBM. To find theenergies of defect states, as a function of concentration of Nvacancy in GaN, we examined their electronic structure and DOS(see Fig. 4 ). At all concentrations of N vacancies considered here, one fully occupied state is present below the VBM and the other three states are present near the CBM. We have visualized thebands at different high symmetric k points and identified the defectbands [shown in Figs. 4(d) –4(g)] of N-vacancy concentration at 0.78%. The energy level of fully occupied band lies below the bulk VBM by 0.6, 0.5, and 0.5 eV for 3.125%, 1.38%, and 0.78%, respec- tively. The other three states occur in the gap below the CBM. To analyze magnetic properties of N vacancies in bulk w-GaN, we have obtained spin-polarized DOS within the LDA. At all the defect concentrations studied here, these do not give rise to any net magnetic moment. Our results are consistent with the results of Xiong et al. , 56while they contradicted other works19,23 reporting a net magnetic moment of 1.0 μBper N vacancy. To further-check our results, we performed “fixed-spin ”calculation with a fixed net magnetic moment of 1.0 μBand found out the N vacancies with vanishing magnetic moment is energetically lowerthan with a net magnetic moment of 1.0 μBby 0.23 eV at N-vacancy concentration of 3.125%. Although, N vacancy is atriple donor, the presence of fully occupied state below the VBM makes it effectively a semiconductor with one electron per single N vacancy for conduction. Higher defect formation energy and pres-ence of the fully occupied state just below the VBM indicate thatthe cause of auto-doping is not the N vacancies in bulk w-GaN. Second, very weak dispersion and band widths of the defect bands indicate that N vacancy states are localized and would not contrib-ute to its mobility and observed electrical conductivity. B. Atomic and electronic structure of the (10 10) surface of w-GaN 1. Pristine (10 10) surface The (10 10) surface of w-GaN can have two inequivalent con- figurations of surface termination: one with a dangling bond per atom is energetically more stable than the other with two dangling bonds per atom at the surface.57This prompted its use in the present calculations, where a slab geometry is used to model thepristine nonpolar (10 10) surface of GaN. In the relaxed (10 10) surface of w-GaN [shown in Fig. 5(b) ], Ga atoms at the surface of the slab move into the bulk, whereas N atoms move out into the vacuum, causing a vertical separation of /C250:4 Å along ,1010. between Ga and N atoms and buckling of surface Ga –Nb o n db y 14:2/C14, which is slightly overestimated than 7/C14–11/C14estimated in calculations based on PW basis.42,58Due to relaxation of the surface, bond length of Ga –Na tt h e( 1 0 10) surface reduces to FIG. 2. Defect formation energies of Ga vacancies (a) and N vacancies (b) as a function of DFT-LDA Fermi level (the bottom X axis) and valence band edge shifted Fermi level (the top X axis) in bulk of w-GaN. The white region indicates the LDA estimated bandgap.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-6 Published under license by A VS.1.83 Å contracted by /C256%w.r.t that in the bulk. This agrees well with earlier works,42,58–60where it was suggested that the struc- tural relaxation of the (10 10) surface involves electronic rehybrid- ization of surface Ga and N atoms resulting in sp2and sp3 configurations, respectively,59which is also evident in our analy- sis. Our estimates of the surface energy ( σ)o ft h e( 1 0 10) surface is 168 meV/Å2.From the surface electronic structure [shown in Figs. 5(c) and 5(d)], a fundamental direct bandgap of the (10 10) surface slab is 1.86 eV, which is 0.2 eV lower than the calculated bulkbandgap (see Table III ). This is because of the fact that the two surface states present within the bandgap of bulk w-GaN. The occupied surface N state ( S N), which mainly originates from the 2p orbital of the N atoms, from the first surface layer and thesecond subsurface layer with a small contribution from the firstsubsurface layer. In the electronic structure, it appears near tovalence band and has a weaker dispersion and band width of 0.44 eV reflecting its confinement to the surface. Unoccupied Ga surface state ( S Ga), which originates mainly from 4s orbital of the Ga atoms, from the first layer with a weaker contribution fromthe second layer of the slab. Unlike S N, electronic band of SGa exhibits stronger dispersion and band width of 2.2 eV, appearing near to the conduction band in the electronic structure. Layer resolved DOS [see Fig. 5(e) ] clearly shows the origin of surface states, where a significant difference can be seen between DOS ofthe bulklike layer and the first layer of the slab. There is an opendebate regarding the precise position of Ga-derived surface state with respect to GaN bulk band edges. 60Our finding based on the method used in Refs. 60 and 61, the unoccupied Ga-derived surface state is 0.05 eV below the bulk CBM edge at Γpoint. The estimated value is quite small in comparison to the estimates by using much advanced methodology in Ref. 60 and references therein. This discrepancy in the separation of Ga-derived surfacestate and bulk CBM is due to the under-estimation of bulk andsurface bandgaps by the methodology used here. From the spin-polarized DOS, it is seen that the dangling bonds at the (10 10) surface do not necessarily show spontaneous spin polarization. A st h ep r i s t i n e( 1 0 10) surface of w-GaN is insulating in nature and the estimated effective mass of electrons in Ga-derivedsurface states is /C250:2m esimilar to bulk, high electrical conduc- tivity may not arise from electrons in these surface states. 2. Ga vacancies at the (10 10) surface In simulations of Ga vacancies at the (10 10) surface, we used a 2/C22 in-plane supercell (128 atoms) and removed one Ga atom from each surface of the slab, creating a surface Ga vacancy con- centration of 12.5%. In the relaxed structure of Ga vacancies at the FIG. 3. Electronic band structure of Ga vacancies in bulk w-GaN with vacancy concentration of 0.78% (a). Defect states are designated with D. Total density of states (DOS) and projected density of states (PDOS) are presented in (b).TABLE III. Calculated change in bond lengths ( Δb) w.r.t. bulk, vertical separation between surface Ga and N atoms ( Δz), buckling angle ( ωb), surface energy ( σ) and bandgap ( Eg) of the (10 10) surface of w-GaN. Present calculation LDAPWPP LDA (Ref. 59 )VASP LDA (Ref. 42 )VASP GGA (Ref. 42 )PWPP LDA (Ref. 58 ) Δb(in %) 6 6 7.23 7.51 6 Δz(in Å) 0.44 0.22 —— 0.36 ωb(in°) 14.03 7 7.5 8.183 11.5 σ(in meV/Å2)168 118 123 97.70 — Eg(in eV) 1.86 — 1.815 1.534 —ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-7 Published under license by A VS.(1010) surface, the neighboring N atoms displace away from the vacancy site causing a /C251:5%–2% contraction of Ga –N bond length in the neighborhood of vacancies compared to the ideal(10 10) truncated surface. The contraction of Ga –N bond for the surface Ga vacancy in charge states of /C01,/C02, and /C03 are 1.5 – 1.7%, 2.0 –4.81%, and 3.4 –6.5%, respectively, compared to the ideal (1010) truncated surface. Contraction by /C252.3%–3.5% ( Ref. 62 ) and/C252:9%–3.7% ( Ref. 45 )o fG a –N bonds has been observed at the sidewall surface of a nanowire structure due to neutral Ga vacancies. The estimated defect formation energy, under N-richconditions, of Ga vacancy at the (10 10) surface is 3.97 eV, which is 2.93 eV less than in the bulk w-GaN, and is a consequence of lower coordination of atoms at the surface. The estimated defect forma- tion energy of surface Ga vacancies in charge states /C01,/C02, and /C03 are 4.35, 4.62, and 7.92 eV, respectively. Under p-type condi- tions, neutral Ga vacancies are found to be stable while undern-type growth conditions, Ga vacancies with /C02e charge state is more stable, with a thermodynamic charge transition level of (0/2/C0) at 0.32 eV above surface VBM (see Fig. 6 ). In estimation of formation energies of point defects in various charged configurations at the surface, we adopt the schemeproposed by Komsa and Pasquarello. 63The isolated vacancy in its different charge state has a very localized defect state, which is rep- resented by a Gaussian charge distribution. The width of the Gaussian charge distribution is determined from the charge densityplot of the defect states obtained from DFT calculation [forexample, see Fig. 7(a) ]. As both sides of the slab in our simulations contains defect, we used a double Gaussian charge distribution. Furthermore, such charge distribution is placed in a dielectricmedium. Since in the slab model, the systems show a spatially varying dielectric constant along one direction, we modeled thedielectric profile of the system as shown in Fig. 7(b) . Within the slab, the dielectric constant is fixed with the value used in bulk cal- culations, whereas in the vacuum it is 1.0. The electrostatic poten- tials obtained from DFT calculation and from the model chargedistribution agree well far from the defect site (in the vacuumregion), as shown in Fig. 7(c) , indicating that the model is con- structed appropriately. For estimation of E iso, we appropriately extrapolate the value of electrostatic energy (E per) to infinitely large supercell. In Fig. 8 , we show the uncorrected and corrected forma- tion energies of V Gaat the surface of the slab in /C01 charge state with variable supercell sizes and different vacuum thickness. These corrections lead to the similar estimates of the formation energies independent of the size of the supercell. Such convergence in for-mation energy with supercell dimensions indicates that the correc-tion scheme is appropriately chosen for the system. Similarapproaches were used in estimation of formation of N vacancy at the surface. In the electronic structure of the (10 10) surface with Ga vacancy concentration of 12.5%, three vacancy-related states(designated as D) are evident near the valence band just above the Nsurface state (see Fig. 9 ). Thus, neutral Ga vacancies at the (10 10) surface act as a p-type dopant though its band width is small, and it is unlikely that the Ga vacancies are the source of high electrical con-ductivity in GaN NWN as reported in Refs. 21 and27. Furthermore, PDOS analysis [ Fig. 9(b) ] shows that these hole states originate from 2s/2p orbitals of the N atoms that coordinate with the Ga vacancy and are spin polarized. Due to lower site symmetry at the surface, FIG. 4. (a)–(c) Show the electronic structure of N vacancies in bulk w-GaN with vacancy concentration of 3.125%, 1.38%, and 0.78%, respectively. All identified defect bands are designated as D. The wave function of defect bands lie below VBM and below the CBM are visualized and charge distribution of the bands are shown in (d) – (g) for the vacancy concentration of 0.78%. The isosurface was fixed at 5 :0/C210/C02e=A/C143.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-8 Published under license by A VS.FIG. 5. (a) and (b) Show the ideal and relaxed model of the slab for (10 10) surface, respectively. (c) Shows the electronic band structure of the pristine (10 10) surface. The band on top of the VB is a N derived (green colored) while that on the bottom of the CB is a Ga derived (red colored) surface state. DOS along with PDOS of N (gr een) and Ga (red) atoms are presented in (d). Layer resolved DOS (in black lines) of the slab along with atom projected DOS of N (in green) and Ga (red) are shown in ( e). FIG. 6. Defect formation energies of Ga vacancies (a) and N vacancies (b) at the (10 10) surface of w-GaN vs the DFT-LDA surface Fermi level (the bottom X axis) and valence band edge shifted Fermi level (the top X axis). The white region indicate the LDA estimated bandgap.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-9 Published under license by A VS.the three N atoms contribute asymmetrically to the net magnetic moment with individual magnetic moments of 1.462, 0.427, and 0.427 μB. Estimated spin polarization energy of 1.1 eV is relatively higher than that of bulk GaN. Thus, magnetic moments due to Gavacancies at the surface are more stable than that in the bulk w-GaN. 3. N vacancies at the (10 10) surface To simulate of N vacancies at the (10 10) surface, we removed one N atom from each surface of the 2 /C22 in-plane supercell of slab (i.e., a surface N vacancy concentration of 12.5%). Ga atoms coordinating the N vacancy at the (10 10) surface displace toward the vacancy site resulting an elongation of Ga –N bonds by 2.6%–6% relative to the bond length at the pristine (10 10) surface. Elongation of Ga –N bonds near surface Ga vacancy in charge states +1, +2, and +3 are 0.64% –1.56%, 0.67% –1.12%, and 0.48% –0.51%, respectively, relative to the ideal (10 10) truncatedsurface. Similar structural changes have been reported by Carter and Stampfl45in their work on N vacancies at the surface of a GaN nanowires. Because of N vacancies at the (10 10) surface, the basal plane Ga atom of N vacancy site get displaced toward the vacuum, while the remaining surface Ga atoms move into the bulk. Our esti- mate of formation energy of N vacancy at (10 10) surface of w-GaN is 3.17 eV, which is 2.04 eV less than that of bulk w-GaN. The esti- mated defect formation energy of surface N vacancies in chargestates +1, +2, and +3 are 0.04, /C01.31, and /C02.86 eV, respectively. A comparison on energetics of defects is made at Table II . The for- mation energy versus the Fermi level are shown in Fig. 6 . The ther- modynamic transition levels of V Non (10 10) surfaces such as (0/+) and (3+/+) are 3.14 and 1.45 eV above VBM, respectively. It is clear that the formation energy of the N vacancy both in the bulk and the (10 10) surface is significantly less than that of Ga vacancy, sug- gesting that the concentration of N vacancies will dominate duringcrystal growth. Further from Fig. 6 it can be deduce that N vacan- cies can form spontaneously up to the Fermi level of /C251:0e V above surface VBM. With p-type growth conditions, N vacancy in +3 charge state is most stable while the charge state +1 is moreenergetically preferable under n-type conditions. The electronic structure of the (10 10) surface containing N vacancies (see Fig. 10 ) reveals that N vacancies at the surface also act as n-type donors and donate only one electron per N vacancy for conduction. A half-occupied band appears in the fundamentalbandgap about 0.33 eV below the CBM at the Γpoint, while remain- ing defect bands appear overlapping with the conduction andvalence bands. Band width of the spin-polarized half-occupied band is/C250:25 eV, and the defect state forms from hybridized 4s and 4p orbitals of Ga atoms [see Fig. 10(b) ]. In contrast to N vacancies in bulk w-GaN, a net magnetic moment of /C251:0μ Barises per surface N vacancy at the surface. The magnetization is largely localized at Ga atoms that coordinate with the surface N vacancy site. The Ga atom in the basal plane which has two dangling bonds (db) due to FIG. 7. (a) Shows charge densities profile of defect states obtained from DFT calculation (dashed lines) and model profile used for the estimation of E corr. (b) Shows dielectric profile used for same. (c) Shows the Hatree potential obtained from DFT calculation (dashed lines) and with model charge distribution (solid lines). FIG. 8. Shows formation energy of gallium vacancy with /C01 charge state with various supercell sizes. Note that for these calculation, ionic relaxations were not performed.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-10 Published under license by A VS.the surface N vacancy contributes 0.415 μB, whereas the other two Ga atoms present in the first subsurface (with 1 db), each contributes0.1μ Bto the total magnetic moment. Estimated spin polarization energy is /difference0:11 eV, which is relatively lower than that of surface Ga vacancy. Thus, the stability of magnetic moment at the surface N vacancies is relatively weaker than that of surface Ga vacancies.It should be noted that though LDA has limitations in cap- turing the physics of defect electronic states, and its estimates offormation energy are relatively overestimated relative to the HSEand/or GW, observed qualitative trends are quite useful in under-standing the defect physics in this system. Several authors 46,50,64 proposed that a qualitative understanding in defect formation FIG. 9. Spin un-polarized electronic band structure (a) and spin-polarized DOS and PDOS (b) of surface Ga vacancies at the (10 10) surface with a surface vacancy concentration of 12.5%. The defect states are designated as D. FIG. 10. Spin un-polarized electronic band structure (a) and spin-polarized DOS and PDOS (b) of the N vacancies at the (10 10) surface of GaN with a surface vacancy concentration of 12.5%. The identified defect states are denoted as D.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-11 Published under license by A VS.energies and thermodynamic transition levels can be obtained between the computed local or semilocal functionals and thevalues estimated with HSE06 functional. Though hybrid function- als are not employed in SIESTA, we have used the correction method proposed by Freysoldt et al. 64where the valence band of the [10 10] surface is shifted by /C00.79 eV (see Fig. 6 ). With this correction, we now estimate that the (+/0) transition level ofnitrogen vacancy in bulk is now (2 :10þ0:79¼) 2.89 eV above VBM. Miceli and Pasquarello 46estimate it to be 3.17 eV while a self-consistent HSE calculation yields 3.25 eV.50Similarly, HSE calculation reveals the (2 /C0/3/C0) transition level of gallium vacancy is at 2.80 eV above VBM while with shifted VB edge we obtained a value of 2.68 eV. The HSE functional estimates of the formation energy of neutral nitrogen vacancy in bulk GaN is 3.1 eV while weestimate it to be 3.31 eV with DFT-LDA, suggesting that for deepdefect transition levels, our results and previous HSE predictionsare in fair qualitative agreement. With this correction now the Fermi level pins at (1 :50þ0:79¼) 2.29 eV above VBM which is close to our experimental value [(E F–EVBM)=2.45 eV]. Furthermore, the N vacancies will form spontaneously up to aFermi level of 1.79 eV. The (3+/+) thermodynamic transition levelof surface nitrogen vacancy is (1 :45þ0:79¼) 2.24 eV close to the intense transition of EELS at 2.2 eV. Occurrence of shallow donor states caused by N vacancies are again in agreement with the earlyrise in the intensity of dielectric profiles ( /C253:0 eV) in HREELS data. The observed reduction in electrical conductivity of GaNNWN after chemical etching treatment 27also suggests that the higher conductivity of GaN NWN can be attributed to defect states at the surface. Thus, based on our EELS data and first-principles simulation, we infer that high carrier densities in GaNnanowall network is caused by N vacancies present on the (10 – 10) sidewall surfaces which also results in a high electrical con- ductivity as observed experimentally.V. SUMMARY In summary, we have calculated the atomic and electronic structure, the formation energy, the stability, and the magneticground state of native point defects in the bulk w-GaN and at the (10 10) surface using first-principles DFT-based calculations. Both in bulk and at (10 10) surfaces of N vacancies in GaN act as n-type donors and donate 1e/vacancy for conduction. Within DFT-LDA while in bulk w-GaN, N vacancies in /C01 charge state is most stable throughout the Fermi level, at the (10 10) surface it stabilizes in +3 and +1 charge states under p and n-type growth conditions, respec-tively. Most importantly, formation energies of the N vacancy at the surface is notably more favorable than in the bulk giving to native n-type character of the (10 10) surface. Experimental evidence for the FIG. 11. Obtained electronic band structure of w-GaN (a) and total electronic DOS and PDOS (b) in its bulk phase. TABLE IV . Cutoff radii of zeta functions that are used in construction of the basis set in our simulations. Orbital Basis sizeCutoff radii of first Zeta function (in Bohr)Cutoff radii second Zeta function (in Bohr) Ga 4s Double Zeta 5.16 3.13 Ga 3d Double Zeta 5.03 2.694Ga 4p Single Zeta 4.12 — N 2s Double Zeta with a polarization function3.68 2.87 N 2p Double Zeta with a polarization function4.28 2.94ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000402 38,063205-12 Published under license by A VS.presence of N vacancies was found by EELS measurements with a deep transition level of 2.2 eV in agreement with a deep level (3+/+) transition of N vacancy at (10 10). The presence of deep as well as shallow defect electronic states and its low formation energy confirmabundance of N vacancies in GaN which results into higher electricalconductivity in faceted GaN nanostructures. ACKNOWLEDGMENTS S.N. acknowledges JNCASR for a fellowship. U.V.W. thanks SERB-DST for support through a JC Bose National fellowship andSheikh Saqr Fellowship. APPENDIX 1. Cutoff radii of zeta functions 2. Electronic structure of pristine w-GaN PDOS [ Fig. 11(b) ] shows that the valence band is primarily composed of 2p orbitals of N along with a small contribution from 4p orbitals of Ga while the conduction band is composed of 4s and 4p orbitals of Ga and a weak contribution from 2s and 2p orbitalsof N. Our estimates of the crystal field splitting ( Δ CF) (in the absence of spin –orbit coupling) at Γpoint is /C2550 meV, which is a bit weaker than the earlier theoretical estimates of 72 meV by Suzuki et al.65and larger than 42 meV estimated by Wei and Zunger,43where both calculations were carried out by using LDA. The experimental estimates of ΔCFreported in different works spanned from 10 to 25 meV.66–69The theoretical estimates of ΔCF have been higher than the experimental values. 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5.0017901.pdf

J. Appl. Phys. 128, 114902 (2020); https://doi.org/10.1063/5.0017901 128, 114902 © 2020 Author(s).Prediction of large magnetic anisotropy for non-rare-earth based permanent magnet of Fe16−x MnxN2 alloys Cite as: J. Appl. Phys. 128, 114902 (2020); https://doi.org/10.1063/5.0017901 Submitted: 10 June 2020 . Accepted: 26 August 2020 . Published Online: 16 September 2020 Riyajul Islam , and J. P. Borah Prediction of large magnetic anisotropy for non-rare-earth based permanent magnet of Fe 16−xMnxN2alloys Cite as: J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 View Online Export Citation CrossMar k Submitted: 10 June 2020 · Accepted: 26 August 2020 · Published Online: 16 September 2020 Riyajul Islam and J. P. Boraha) AFFILIATIONS Department of Physics, National Institute of Technology Nagaland, Dimapur, Nagaland 797103, India a)Author to whom correspondence should be addressed: jpborah@rediffmail.com ABSTRACT Exploring the metastable magnetic nanostructures of Mn substituted α00-Fe16N2with large saturation magnetization μ0MS, high Curie temperature TCand giant magnetic anisotropy are of technological merit as promising candidates for non-rare-earth based permanent magnets. Here, we present in-depth analysis for the structural and magnetic properties of Fe 16−xMn xN2using first-principles calculations. We predict a large magnetocrystalline anisotropy energy (MAE) constant of K1= 2.02 MJ/m3for the Fe 14Mn 2N2alloy, which is more than twice that of pristine Fe 16N2. The underlying mechanism associated with boosting K1is attributed to the local distortion of orbitals induced by Mn substitution. The MAE is also carefully analyzed in terms of reciprocal space analysis by employing the magnetic force theorem,revealing the regions in the Brillouin zone that are prominent for giving rise to MAE. Published under license by AIP Publishing. https://doi.org/10.1063/5.0017901 I. INTRODUCTION The pursuit of novel high energy rare-earth (RE) free permanent magnets (PMs) is of great interest in recent years. 1 Owing to their significance in the field of many energy-savingtechnologies, PMs persist in playing a crucial role in the new energy economies. The decisive attributes of the quality of PM materials are the operating range of temperature and, particularly,the maximum energy product, ( BH) max. Currently, RE based Nd2Fe14B and SmCo 5are the most extensively used PMs due to their large ( BH)max and high Curie temperatures. However, given the recent availability and escalating price as well as poor geographical diversity of the RE elements, there has been growinginterest in developing RE free PM materials. 2–4 The energy product that mainly determines the efficiency of the magnetic material is relatively a complex quantity. In addition to the microstructure of the material (size, shape, and orientation of crystalline grains), the energy product and eventually theefficiency of PMs rely on the intrinsic magnetic properties like satu-ration magnetization μ 0MS, Curie temperature TC, and magneto- crystalline anisotropy energy (MAE). So far, remarkable efforts have been made in improving the physical and magnetic properties of RE free PMs. Several RE free materials having highmagnetization show promising aspects in replacing RE based PMs.5–9Among them, the tetragonal metastable α00-Fe16N2phase of iron nitride with a distorted body-centered tetragonal (bct)structure, which was first prepared by Jack, 10has attracted substan- tial attention due to its low-cost Fe and high magnetic moment, ms of 3.0 μB/Fe.11Previous studies reported a wide variation in msof α00-Fe16N2extending from 2.4 to 3.5 μB/Fe.9,12–18These widespread msvalues may indicate the formation of secondary phases. The key issue in α00-Fe16N2is its low thermal stability, which decomposes intoα-Fe and Fe 4N at a temperature close to 500 K.19 Several theoretical and experimental works have reported that alloying α00-Fe16N2with certain transition metals (TMs), such as Ti, Mn, Co, etc., can significantly enhance the thermal stability of α00-Fe16N2.20–23However, such inclusions of TMs at higher concen- trations can lead to reduced μ0MS, which is unfavorable for most practical applications. Presently, no experimental study is available forTCof any alloyed α00-Fe16N2samples stabilized at higher temperatures. Recently, Bhattacharjee and Lee7investigated the magnetic exchange interactions and TCof pure α00-Fe16N2and vanadium doped α00-Fe16N2using the first-principles method. In contrast, the obtained TCusing mean-field approximation (MFA) was much higher than the TCobtained by Sugita et al .12byJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-1 Published under license by AIP Publishing.extrapolating the data obtained at low temperatures. Many theoreti- cal studies have been devoted to exploring the origin and enhanc- ing of the MAE in α00-Fe16N2by alloying.9,14,18Although there have been quite a few reported experimental results, regardless ofthem being diverse and inconclusive, extending from 4.8 × 10 4to 1.6 × 106J/m3. Specifically, Sugita et al.12attained an in-plane MAE with easy axis along the (100) direction, while Takahashi et al.,13 Kita et al.,24and Ji et al.16obtained uniaxial MAE with easy axis along the (001) direction. Consequently, a better grasp is necessaryto further study the origin and the fundamental factors enhancingthe MAE. In this paper, we systematically investigate the saturation mag- netization μ 0MS, Curie temperature TC, and magnetocrystalline anisotropy energy (MAE) of Fe 16−xMn xN2alloys using density functional theory (DFT) calculations. To the best of our knowledge,there have been a few theoretical investigations of the exchange interactions and Curie temperature T Cin Mn-doped Fe 16N2alloys. Our study shows a possible way of improving the magneticproperties ensuing high magnetization and enhanced MAE for PMapplications. II. COMPUTATIONAL DETAILS α 00-Fe16N2crystallizes in the bct type structure ( Fig. 1 ) with space group I4/mmm (No. 139). The unit cell of α00-Fe16N2can be built by considering a 2 × 2 × 2 supercell of bcc α-Fe and inserting two N atoms into the interstitial octahedral sites, which leads to the bct structure with trivial distortion in the Fe sites. Thus, theα 00-Fe16N2unit cell contains 16 Fe atoms in the three inequivalent Fe sites specified as four 4 esites, eight 8 hsites, and four 4 dsites. To investigate the effect of Mn substitution, a 1 × 1 × 1 supercell was built on the α00-Fe16N2unit cell. Initially, Fe was replaced by Mn on 4 esite as Mn prefers to occupy the 4 esite to construct two low concentration Mn substituted Fe 14Mn 2N2and Fe 12Mn 4N2 alloys,22and subsequently, Fe was replaced completely both in the4eand 4 dsites to construct one high concentration Mn substituted Fe8Mn 8N2alloy. Density functional theory (DFT) calculations of Fe 16N2 and Mn substituted Fe 16N2were carried out within full-potential linearized augmented plane wave method (FP-LAPW) + localorbitals (lo) method, 25implemented in the WIEN2k code.26,27The exchange-correlation potential was treated with the generalized gra- dient approximation (GGA) in the Perdew, Burke, and Ernzerhof(PBE) form. 28We employed GGA + U approach to treat the strong correlation interactions within d-electrons with the self-interaction correction (SIC) method29for the correction of double counting. The on-site Coulomb interaction Uwas parametrized by Ueffective =U−J, where Uand Jare the Hubbard parameter and approximation of Stoner exchange parameter, respectively. EffectiveU effective of 0.08 Ry, 0.10 Ry, and 0.29 Ry were used for the three inequivalent Wyckoff sites 4 e,8h, and 4 d, respectively, of Fe 16N2 as adopted by previously reported data obtained using the embed- ded cluster method.30,31The Kohn –Sham wave functions were expanded up to RMTKmax= 7.0 for better convergence of basis set, where RMTis the smallest muffin-tin radius of the atomic spheres and Kmaxis the maximum reciprocal lattice vector used in the expansion of plane wave. The partial wave functions in the muffin- tin atomic spheres were expanded using lmax= 10. The charge den- sities were Fourier expanded in the interstitial region up to cutoffvector G max=1 2R y1/2. Reciprocal space integration was performed with the help of tetrahedron method using a grid of 5000 k-points in the irreducible Brillouin zone (BZ) for structural relaxationalong with a convergence criterion 10 −4e and 10−4Ry. The dynamical stabilities of the Fe 16−xMn xN2alloys were also ensured using the supercell approach by the PHONOPY code.32 The optimized structure obtained from WIEN2K calculations were used to estimate the exchange coupling parameters usingthe Munich spin-polarized relativistic Korringa –Kohn –Rostoker (SPR-KKR) package 33,34by employing a real-space approach for- mulated by Liechtenstein et al.35and by mapping the total energy of the full system to a Heisenberg-Hamiltonian expressed as H¼/C0X i=jJijsi:sj, where Jijdenotes the exchange coupling parameters between sites i andjand sidenotes the unit vectors pointing along the direction of the local magnetic moment along the ith site. 30 energy points were used to execute the integration over the Green ’s functions and basis functions were expanded up to l= 3. The Curie temperature within mean-field approximation (MFA) was calculated by solving the following coupled equation for a multi-sublattice system: 3 2kBTCeμhi¼X vJμv 0evhi, where heμiis the average z-component of the unit vector heμ rialong the direction of the magnetic moment in sublattice μwith Jμv 0¼P r=0Jμv 0r. The Curie temperature corresponds to the largest eigenvalue of the Jμv 0matrix.36,37The r-summation was performed to a radius of |r|/a= 5.0, where ais the in-plane lattice constant. FIG. 1. Unit cell of α00-Fe 16N2crystal structure denoting three inequivalent Fe Wyckoff sites.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-2 Published under license by AIP Publishing.For MAE calculations using WIEN2k code, the relativistic spin–orbit coupling (SOC) effects were incorporated using the second variational approach.38The MAE constant was estimated using the following force method:39K1¼P 0,k[ε(n,k)a/C0ε(n,k)c], where ε(n,k)aandε(n,k)care the energy eigenvalues of the occu- pied states along a-a n d c-directions of magnetization, respectively, obtained in the presence of SOC. A denser grid of 10 000 k-points was used to ensure a well converged MAE. III. RESULTS AND DISCUSSIONS A. Structural and stability analysis Our estimated tetragonal distortion c/aand volume for the unit cell of pure Fe 16N2and Mn substituted Fe 16−xMn xN2as a function of the Mn concentration are displayed in Fig. 2(a) along with the optimized lattice parameters [the inset of Fig. 2(a) ]. Our optimized lattice parameters for Fe 16N2are a= 5.696 Å and c= 6.185 Å, which are slightly smaller than the experimentallyreported value a= 5.71 Å and c= 6.29 Å.40Mn substitution causes a decrease in lattice distortion c/afor x = 2 and 4, while there is a sudden rise in c/aat the Mn composition of Fe 8Mn 8N2. However, the variation of volume with increasing Mn content is relativelysmall in terms of magnitude as shown in Fig. 2(a) . Zhao et al. 41 proposed that the sudden changes in c/aand volume can be closely ascribed to the magnetic effects in the crystal systems. Huang et al.42and Benea et al.43showed that the volume and atomic radii of the TMs of Fe 16−xMxN2alloys with M = Ti, Cr, Mn, Co, and Ni are directly related, whereas an anisotropic variation was obtainedbetween a- and c-axes. Szymanski et al. 23have also demonstrated that an increase or a decrease of a-axis was associated with the atomic radius of the TMs, whereas the c-axis was found to be cor- related with the magnetic coupling of the TMs. To demonstrate thedynamical stability of these alloys, we investigated the phonondensity of states (DOS) by considering a 2 × 2 × 2 supercell. The structures were fully relaxed until the force per atom was less than 0.001 eV Å −1. The phonon DOS of the alloys, shown in Fig. 2(b) , exhibits no negative frequency modes, indicating the structures aredynamically stable. B. Electronic structure and magnetic moments Figure 3 represents the site-resolved spin-polarized DOS for the studied compounds around the Fermi level, E Fcalculated within GGA + U approach. All the DOS plots exhibit a typicalexchange splitting behavior expected for ferromagnetic materials. The DOS of Fe 16N2is in accordance with the previously reported studies in the literature.42–44The basic features of the DOS results are equivalent for all the compounds. Strong hybridizationobtained around −7 eV and −6 eV for the majority and minority spins states, respectively, shows the covalency between TM 3 dand N2 pstates displayed in Fig. 3 . However, such hybridization occurred substantially only for the 4 eand 8 hsite atoms, which are directly coordinated to the N atoms and weaker in the case of 4 d atoms. In the majority spin 4 eand 8 hbands, the tail of the major peak extends above the E F. Conversely, the partial DOS of 4 dsite atoms is below the EFin the majority state. This can be attributed to the 3 d-band narrowing of the 4 dsite atoms, as a result of weak hybridization with other 4 e,8h, and N atoms. While in the minor- ity spin state, 4 dsite atoms have unoccupied 3 dorbitals above the EF, showing greater exchange splitting than TM atoms on other sites. Most of the occupied states in both spin directions compriseitinerant d-electrons involved in metallic bonding between the TM atoms at the 4 dsite and its neighboring 4 eand 8 hsite TM atoms, indicating the presence of both itinerant and localized electron configuration within α 00-Fe16N2.15 Table I shows the spin magnetic moments computed using GGA + U approach for all Fe 16−xMn xN2alloys. The results for spin magnetic moments obtained within GGA approach can befound in Table SI of the supplementary material . The site decom- posed magnetic moments of Fe for pure Fe 16N2at 4e,8h, and 4 d sites are found to be 2.308, 2.507, and 3.234 μB, respectively, with an average magnetic moment of 2.643 μB/Fe, considerably higher than the results obtained using the GGA approach. Our results are in close agreement with those previously reported in theoretical DFT results43,45and experiments.46,47Such variations in magnetic FIG. 2. (a) The c/a and volume per unit cell variation as a function of Mn content, x, where the inset shows the optimized lattice parameters a and c and (b) phonon DOS of the Fe 16−xMxN2structures.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-3 Published under license by AIP Publishing.moments among all the Fe sites are because of the different bonding environments. The Fe atoms at the 4 eand 8 hsite atoms have lower magnitude of magnetic moments ( Table I ), being closer to the N atoms exhibiting strong hybridization with the 2 porbitals of N atoms, which leads to a substantial amount of electron locali- zation. However, the increased magnetic moment of the 4 dsite Fe atoms can be attributed to the reduction of occupancy in theminority spin state as well as the 3 d-band narrowing due to its greater bond length with N atoms and surrounding Fe atoms at the 4eand 8 hsites leading to a metallic environment resulting in a charge transfer among different Fe sites analogous to our DOSresults shown in Fig. 3(a) . 15,48–50However, the magnetic moments of the Fe atoms are still greater than those of bulk α-Fe (2.2 μB) and can be ascribed to the expanded lattice for Fe 16N2resulting from the inclusion of interstitial N atoms. Although a high satura-tion magnetization μ 0MS= 2.434 T was obtained for pure Fe 16N2,i t is still lower than the recently reported magnetization of 3.1 T in thin film grown on MgO (001) substrate.51Moreover, investigations by several other groups on Fe 16N2, mostly in the form of thin films grown on different substrates with a variety of techniques, led to contradictory results with μ0MSaround 2.6 T.11,52Still there is no report of obtaining such high magnetization of Fe 16N2in the bulk FIG. 3. Spin-polarized site-projected density of states of (a) Fe 16N2, (b) Fe 14Mn2N2, (c) Fe 12Mn4N2, and (d) Fe 8Mn8N2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-4 Published under license by AIP Publishing.form. The computed magnetic moments for Fe 16−xMn xN2alloys tabulated in Table I show a minor reduction of the average magnetic moments with increasing Mn concentrations; however, the totalmagnetic moment, m tot, of the alloys are still high (varying frommaximum 41.899 μBfor x = 0 to 38.675 μBfor x = 8). With the inclu- sion of Mn, local magnetic moments of surrounding Fe atoms at the 4eand 8 hsites decrease, while a slight increase in Fe magnetic moments at the 4 dsites are predicted. Prediction of such high and ferromagnetically coupled magnetic moments of Mn in Fe –Mn–N alloys is in close agreement with previous theoretical reports.18,42 Furthermore, previous results also suggest conflicting outcomes regarding the nature of the coupling between Fe –Mn (ferromagnetic or antiferromagnetic).53,54Recent analysis by Szymanski et al.23con- cluded that magnetic couplings in these alloys are dependent on theconcentration of Mn impurities. In addition, magnetic moments of the N atoms are relatively very small. C. Calculations of J ijand estimations of T C For the purpose of practical use, high TCis imperative to maintain thermal stability and have the potential of being used as aTABLE I. Site decomposed and total spin magnetic moments (in μB)o f Fe16−xMnxN2alloys. x (Mn)Magnetic moment ( μB) 4e(Mn/Fe)8h (Fe)4d (Mn/Fe) 2 a(N)ms (μB/TM) mtot 0 —/2.308 2.507 —/3.234 −0.027 2.643 41.899 2 2.078/2.259 2.449 —/3.252 −0.051 2.580 40.979 4 1.741/ — 2.399 —/3.268 −0.062 2.452 39.106 8 1.608/ — 2.208 3.675/ —−0.059 2.425 38.675 FIG. 4. Exchange coupling constants J ijfor (a) Fe 16N2, (b) Fe 14Mn2N2, (c) Fe 12Mn4N2, and (d) Fe 8Mn8N2as a function of atomic distance r/a.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-5 Published under license by AIP Publishing.PM material. So, first, we estimate the magnetic exchange parame- ters for the considered compounds via SPR-KKR package and are plotted in Fig. 4 . Our estimated results for Fe 16N2displayed in Fig. 4(a) exhibit some significant differences in contrast to the pre- viously reported data.7,18The magnetic exchange interactions of Jij= 38.69 meV between nearest-neighbor 4 e–4esites at a distance of 2.35 Å is the largest among all other interactions, which corre- sponds to the interaction between 4 esite Fe atoms of two neighbor- ing octahedrons, while for Bhattacharjee and Lee,7the largest was between the 8 h–4dsites. However, Ke et al.18obtained two contra- dicting results employing different methods, where the largest was between 8 h–4esites obtained using atomic sphere approximation (ASA) and quasiparticle self-consistent GW approximation(QSGW), whereas when using local density approximation (LDA),the largest magnetic interactions were obtained between 4 e–4esites. Interestingly, the large coupling between 4 e–4esites having J ij= 23.75 meV at a distance of 2.60 Å obtained by Ke et al.18using LDA increased up to Jij= 36.55 meV at a distance of 2.36 Å after considering the experimental atomic coordinates that agree wellwith our results. This is followed by the next strongest interactionsbetween 8 h–4dsites ( J ij= 29.67 meV) at a distance of 2.53 Å and 8h–4esites ( Jij= 21.01 meV) at a distance 2.33 Å both along the [111] direction. As seen in Fig. 4(a) , the nearest-neighbor magnetic interaction at the 8 h–8hsites with Jij= 6.72 meV at a distance of 3.09 Å is between Fe atoms of two neighboring octahedrons is ferromagnetic, while the strongest 8 h–8hmagnetic interaction at a distance of 4.02 Å is antiferromagnetic in nature withJ ij=−7.67 meV, corresponding to the 1800Fe–N–Fe superexchange interactions. Moreover, the strongest 4 d–4dmagnetic interactions Jij= 6.72 meV is at a distance of 3.09 Å. Among all the sublattice interactions, 4 e–4dferromagnetic interactions are the weakest with Jij= 3.48 meV at the nearest-neighbor distance of 2.87 Å, which is still ferromagnetic, despite being very weak. From Fig. 4 , it is clear that the magnetic exchange interactions between nearest-neighbor4e–4esites are significant for all the compounds, except for Fe 12Mn 4N2, where the nearest-neighbor 8 h–4dsites have stronger magnetic exchange interactions. For Fe 14Mn 2N2, the strongest mag- netic exchange interaction with Jij= 37.50 meV is between the two nearest-neighbor 4 e(Fe)-4 e(Mn) atoms at a distance of 2.34 Å, as displayed in Fig. 4(b) . Except for the interactions between 4 e(Mn) with other Fe atoms, the Fe –Fe interactions are reasonably similar to that of the parent compound [ Fig. 4(a) ]. However, for Fe12Mn 4N2and Fe 8Mn 8N2, as shown in Figs. 4(c) –4(d), the mag- netic exchange interactions shows weak ferromagnetic nature having the strongest interactions of Jij= 23.90 meV between 8 h (Fe)–4d(Fe) atoms and Jij= 14.00 meV between nearest-neighbor 4e–4esite Mn atoms, while the strongest antiferromagnetic interac- tions are between 8 h(Fe)–8h(Fe) with Jij=−5.80 meV and between 4e(Mn) –4d(Mn) with Jij=−11.18 meV, respectively. More specifi- cally, as the exchange interactions tend to align the neighbor spins on the atomic scale, so such reduction in exchange interactions canbe attributed to the reduced spin magnetic moments with theincrease of Mn concentrations. Figure 5 shows the normalized exchange coupling parameter J 0of all the compounds with respect to J0of Fe 16N2as well as the mean-field Curie temperature, TC. It is evident that with increasing Mn content, the exchange coupling parameter decreasesmonotonically because of the weak interaction of Mn with the sur- rounding atoms. TCestimated from MFA for Fe 16N2is 1450 K, much higher than the previously reported experimentally estimatedT Cof 813 K12and also than the bcc-Fe ( ∼1023 K). Such discrep- ancy between theoretical and experimental TCmay arise from the exchange interactions estimated from the first-principles methodthat are long-ranged and oscillatory and also due to the exclusionof thermal effect or the experimental estimation of T C, whose precise value is still unknown for single-phase Fe 16N2due to its low thermal stability. Ke et al .18suggested that as magnetization in Fe16N2increases compared to the pure bcc-Fe, thus TCin highly pure Fe 16N2must also be larger than 813 K and pure bcc-Fe. Moreover, it is also observed that TCfollows the same trend as the exchange coupling parameter J0with increasing the Mn content x (Fig. 5 ). Nevertheless, Fe 8Mn 8N2alloy does not fulfill the funda- mental requirement of high-efficient PMs, having TC= 538 K less than the required TC> 550 K, as proposed by Coey.55 D. MAE and the mechanism The variation of calculated MAE constant K1for the Fe16−xMn xN2alloys as a function of Mn content, x, is displayed inFig. 6(a) . Within GGA + U + SOC, we obtain a uniaxial MAE for Fe 16N2having K1= 0.81 MJ/m3, slightly higher than the result obtained using GGA + SOC approach (Fig. S1 in the supplementary material ). Albeit this value is quite comparable to a number of formerly reported experimental values,16,52but consid- erably smaller than what has been recently reported by Li et al.56 with K1= 1.9 MJ/m3. Such discrepancy may occur most likely due to the different lattice parameters leading to different lattice distor- tion c/aand magnetic moments of TMs, as these are sensitive parameters for MAE. The increase of MAE for Fe 16N2with regard to bcc-Fe is caused by the tetragonal distortion by breaking the cubic symmetry with the inclusion of N atoms, which causes second-order anisotropy terms to appear. Accordingly, this effect FIG. 5. Normalized exchange coupling parameter J 0with respect to Fe 16N2 and the Curie temperature T Cas a function of the Mn content, x.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-6 Published under license by AIP Publishing.was previously confirmed for α0-Fe8Nxwith the N content of 0≤x≤1 in thin films.57,58Based on the metastable alloys obtained after mixing Mn with Fe, we found a significant enhancement ofMAE with K 1= 2.02 MJ/m3in the metastable structure Fe 14Mn 2N2, however, for Fe 12Mn 4N2alloy the MAE constant K1is relatively small, whereas Fe 8Mn 8N2alloy possesses an in-plane anisotropy not desirable for PM application. The enhancement of MAE forFe 14Mn 2N2, despite having weak SOC, can be attributed to the large exchange interactions between nearest-neighbor Mn and Featoms at the 4 esites as shown in Fig. 4(b) . Furthermore, the DOS results in Fig. 3(b) also demonstrates the overlapping of E Fby the 4esite Mn 3 dorbitals other than 4 eand 8 hsite Fe atoms similar to Fe16N2. Although in the case of Fe 12Mn 4N2and Fe 8Mn 8N2, sub- stantial overlapping of EFby the 4 esite Mn 3 dorbitals is being observed, but due to the lower exchange interactions, the MAE is significantly reduced. Mori et al.59demonstrated that the dominantcontributions to the MAE comes from the doubly degenerate bands intersecting EF, split by the SO interactions. Daalderop et al.60and Strange et al.61suggested that even a ±0.01 electron filling may be sufficient to switch the sign of MAE and also concluded that merely10 −2or fewer electron at the EFcan contribute to the MAE. Furthermore, Sakuma et al.62confirmed that the sign of MAE depends on the number of valence electrons. They showed that a decrease in the number of valence electrons alters the sign of K1. So, the source of in-plane K1for x = 8 may be ascribed to the reduction in the number of valence electrons due to the substitutions of Mn. In order to gain further insights into the origin of MAE, we now analyze the anisotropy of orbital magnetic moment, Δml, which is often closely linked to the MAE as63 MAE ¼P iξi 4(m(Mkc) l/C0m(Mka) l)/C25Δml, where ξiis the SOC cons- tant. The equation is valid only if the spin conserved terms are con-sidered and spin-flip terms are neglected. Figure 6(b) demonstrates the total as well as atomic site-resolved orbital moment anisotropy, Δm las a function of Mn doping concentration, x. One can see that the major contribution to the total Δmlcomes from the anisotropy of the 4 eand 4 dsites, while the contribution from the 8 hsite is small. Moreover, the sign of total Δmlis opposite to that of MAE values, indicating that the assumptions in Bruno ’s formula break down in these alloys, which is in agreement with the previousstudy. 9The DOS results in Fig. 3 show the presence of non- negligible unoccupied majority spin states above the EF. Therefore, the spin-flip terms must also be considered in addition to the spin conserved terms in calculating MAE of Fe 16N2and its alloys. In 3 dbased itinerant systems with weak SOC, where SOC constant ξis substantially smaller compared to the bandwidth (in the range of 10 –100 meV/atom), it is reasonable to illustrate SOC Hamiltonian in terms of second-order perturbation theory. As MAE of a magnetic nanostructure originates from the SOC, Wanget al. 64interpreted MAE as a function of the angular momentum operator components along the x-axis ( Lx) and z-axis ( Lz), MAE ¼ξ2X uα,oβ(2δαβ/C01)uαhjLzjoβi2/C0uαhjLxjoβi2 Euα/C0Eoβ/C20/C21 , where Euαand Eoβare the unoccupied and occupied states of energy levels with spins αand β, respectively. Consequently, for the d-states contributions to the MAE, the non-zero matrix elements of the Lxand LzarehxyjLxjxzi¼1,hx2/C0y2jLxjyzi¼1, hz2jLxjyzi¼ffiffiffi 3p ,hxzjLzjyzi¼1, and hx2/C0y2jLzjxyi¼2. Accordingly, the positive contributions toward MAE mainly origi-nate from the two L zmatrix elements for the SOC coupling between the same spin channel, while Lxmatrix elements provide negative contributions. Conversely, it is the opposite for SOC cou-pling between different spin channels. So as to reveal the intrinsicmechanism enhancing the MAE in Fe 14Mn 2N2, the d-orbital pro- jected DOS (PDOS) specifying m=0 ( dz2),m=1 ( dxz/dyz), and m=2 ( dxy/dx2 −y2) of Fe-4 eatoms compared with that of the pristine Fe16N2is shown in Fig. 7 . As Mn was substituted in the 4 esite, the changes in the PDOS of Fe-4 eatoms are more significant in con- trast to the PDOS of 8 hand 4 dsite atoms (not presented here). In Fig. 7(b) ,m= 2 peaks are heightened in both the occupied majority spin state and unoccupied minority spin state compared with that FIG. 6. Variation of (a) K 1and (b) Δmlas a function of Mn content, x in Fe16−xMnxN2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-7 Published under license by AIP Publishing.of the pristine Fe 16N2inFig. 7(a) . Moreover, the m= 2 peaks of Fe-4eatoms for Fe 14Mn 2N2are shifted toward the EFin the major- ity spin occupied state, while in the minority spin unoccupied state,the peaks shifted further away from the E Fcompared with the parent compound. Furthermore, we obtained an increase in the majority spin occupied state by the peak of m= 1, whereas it is reduced in the minority spin occupied state near the EF. The reduc- tion in the minority spin occupied m= 1 states coupling with unoc- cupied m= 0 and m= 2 states for the same spin channel suppresses the negative contribution from hz2jLxjyzi,hxyjLxjxzi, and hx2/C0y2jLxjyzi, which eventually enhances the MAE. Additionally, the coupling of the different spin channels between the majorityspin occupied m= 1 states and minority spin unoccupied m=2 states also significantly enhances the MAE with positive contribu- tion from hx 2/C0y2jLxjyzi. The enhancement of MAE can also be correlated to the reduced energy separation between the orbitalslying in-plane d x2 −y2and dxycoupled via the matrix element hx2/C0y2jLzjxyidue to the reduced crystal symmetry.63 To gain deeper insight into the MAE origin, the band struc- tures of Fe 16N2and Fe 14Mn 2N2, obtained with the inclusion of SOC along (001) and (100) magnetization directions are plotted inFig. 8 , together with the MAE contributions per k-point estimated using the magnetic force theorem (MFT). Spin –orbit splitting leads to different band structures for different quantization direction and as the SOC constant for 3 dTMs is below 100 meV/atom,consequently, spin –orbit splitting as displayed in Fig. 8 , does not exceed this value. The substantial amount of band present withinfew electron volts near the Fermi surface, and due to hybridization of the band, makes the analysis of the MAE on the basis of band structure difficult. As the MAE is positive in both the consideredcases, the overall positive MAE contribution per k-point is antici- pated to dominate the band structures. As can be seen fromFig. 8(a) , there is a strong MAE contribution from the high sym- metry Г-point and there is a rather weak negative MAE contribu- tion is seen around the HandN-points, which is expected to be canceled out in the entire Brillouin zone integration. On the otherhand, for Fe 14Mn 2N2inFig. 8(b) , the overall MAE contribution perk-point remains positive over the entire Brillouin zone, with the most positive contributions to the MAE originates along the path N-Σand, particularly, weak contribution from the Г-point. Moreover, it is also evident that for the assessment of MAE varia-tion per k-point, a very accurate modeling of the electronic band structure and very dense k-mesh are significant. IV. CONCLUSIONS In conclusion, a comprehensive first-principle study has been performed for the Fe 16−xMn xN2bct phase compounds with the purpose of exploring the intrinsic magnetic properties, such as sat- uration magnetization μ0MS, Curie temperature TC, and MAE for PM applications. Although the average magnetic moments decrease FIG. 7. The d-orbital PDOS of 4e atoms for (a) Fe 16N2and (b) Fe 14Mn2N2. FIG. 8. Band structure of (a) Fe 16N2and (b) Fe 14Mn2N2incorporating SOC along quantization directions 001 (red dashed line) and 100 (black dashedlines), together with the MAE contribution per k-point (blue line) as estimated byMFT .Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 114902 (2020); doi: 10.1063/5.0017901 128, 114902-8 Published under license by AIP Publishing.with increasing Mn concentrations, a large perpendicular MAE of 2.02 MJ/m3has been achieved along with high TC≈1200 K for the structure Fe 14Mn 2N2. Analyses of the orbital resolved PDOS and the matrix elements of the SOC Hamiltonian reveal the underlyingmechanism for the enhancement of MAE and can be predomi-nantly associated with the local lattice distortion induced by Mn substitution, which altered the SOC Hamiltonian and hence the MAE. Such a substantial improvement of the magnetic propertiespredicted for the Fe 16−xMn xN2alloys provides the basis of model- ing a promising candidate for RE free PM. Further research andexperimentations are needed to enhance thermal stability in order to determine the potential use of Fe 16−xMn xN2alloys as PM materials. SUPPLEMENTARY MATERIAL See the supplementary material for the results of spin magnetic moment and MAE constant K1obtained within the GGA approach. ACKNOWLEDGMENTS The authors have no conflicts of interest. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1J. Cui, M. Kramer, L. Zhou, F. Liu, A. Gabay, G. 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5.0026161.pdf

J. Appl. Phys. 128, 155703 (2020); https://doi.org/10.1063/5.0026161 128, 155703 © 2020 Author(s).Ag and Ag–Cu interactions in Si Cite as: J. Appl. Phys. 128, 155703 (2020); https://doi.org/10.1063/5.0026161 Submitted: 24 August 2020 . Accepted: 02 October 2020 . Published Online: 19 October 2020 T. M. Vincent , and S. K. Estreicher ARTICLES YOU MAY BE INTERESTED IN The effect of vacancy-impurity complexes in silicon on the current–voltage characteristics of p–n junctions Journal of Applied Physics 128, 155702 (2020); https://doi.org/10.1063/5.0023411 Wigner functions in optoelectronics: Wave-packet phase-space Monte Carlo solver for waveguide-ring coupling Journal of Applied Physics 128, 153102 (2020); https://doi.org/10.1063/5.0021642 Crack kinking in h-BN monolayer predicted by energy dissipation Journal of Applied Physics 128, 154302 (2020); https://doi.org/10.1063/5.0020130Ag and Ag –Cu interactions in Si Cite as: J. Appl. Phys. 128, 155703 (2020); doi: 10.1063/5.0026161 View Online Export Citation CrossMar k Submitted: 24 August 2020 · Accepted: 2 October 2020 · Published Online: 19 October 2020 T. M. Vincent and S. K. Estreichera) AFFILIATIONS Physics Department, Texas Tech University, Lubbock, Texas 79409-1051, USA a)Author to whom correspondence should be addressed: stefan.estreicher@ttu.edu ABSTRACT Noble metals are often used for contacts on Si. A considerable amount of research has been done on Cu- and Au-related defects, but much less is known about Ag. Silver is a common contaminant in metallic copper and the *Cu 0photoluminescence defect has been shown to contain one Ag atom. In this study, we predict the properties of isolated interstitial (Ag i) and substitutional (Ag s) silver. The calculated migration barrier of Ag iis 0.53 eV, less than half the value extracted from the high-temperature solubility data. Ag ihas a donor level high in the gap and is in the positive charge state for most positions of the Fermi level. When interacting with a pre-existing vacancy, Ag ibecomes Agswith a gain in energy slightly higher than in the case of Cu but still less than the formation energy of the vacancy calculated at the same level of theory. The calculated donor and acceptor levels of Ag sare close to the measured ones, and we predict a double-acceptor level that matches a Ag-related (but otherwise unidentified) level reported in the literature. The Ag sCuipair is more stable than the Cu sAgipair. Ag s can trap several Cu is and form Ag s1Cuincomplexes (n = 1 –4) that are similar to the Cu s1Cuinones. When needed, their calculated binding energies are corrected to account for a change of the charge state following the formation of the complex. This correction is Fermi leveldependent. We tentatively assign the *Cu 0defect to Ag s1Cui3even though the single-donor level associated with *Cu 0does not match the calculated one. Published under license by AIP Publishing. https://doi.org/10.1063/5.0026161 I. INTRODUCTION The studies1of noble metal related defects in Si have over- whelmingly focused on Cu and Au, which are commonly used for metal contacts in Si devices. Even though silver paste is a standardback contact of solar cells, there are fewer studies of Ag in Si, but itis expected to behave similarly to Cu. Since a considerable amountof information is available on Cu and Cu –Cu interactions in Si, we begin this introduction with a review of Cu, then of the “Cu 0”and “*Cu 0”photoluminescence (PL) centers, and finally discuss the much less-known Ag in Si. For most positions of the Fermi level, isolated copper is the interstitial Cu i+. It hops from one interstitial tetrahedral (T) site to the next with an activation energy of 0.18 ± 0.02 eV.2The low migra- tion barrier was originally predicted by ab initio Hartree –Fock calcu- lations in clusters3and later confirmed experimentally2and with density-functional type calculations in periodic supercells.4,5 If copper is introduced into Si at high temperatures and the sample cooled down to room temperature (RT), Cu out-diffuses and precipitates at internal defects and sometimes in the defect-freematerial. It also forms small complexes, which are electricallyactive. 6–12PL and deep-level transient spectroscopy (DLTS)studies13show the presence of two closely related defects originally labeled Cu 0and *Cu 0, respectively. The Cu 0defect is now known as Cu PL, but since the present study focuses on its sibling *Cu 0,w e stick to the old notation. Cu0is isoelectronic and exhibits trigonal symmetry under uni- axial stress.14–16Its PL band14,17has a zero-phonon line at 1014 meV with sharp anti-Stokes sidebands separated by 7.05 meV,indicative of a pseudolocal vibrational mode (PLVM) at 57 cm −1. However, the details of the spectra are complicated as additionalPLVMs are present. The PL band has been correlated 18with a DLTS center at E v+ 0.10 eV. The dissociation enthalpy19of the defect in the temperature range 333 –417 K is 1.02 ± 0.07 eV. Since the formation dynamics of Cu 0implies that it incorporates the fast- diffusing Cu i+, the binding enthalpy in that temperature range is 0.84 ± 0.09 eV. The annealing of the Cu 0defect leaves Cu s, showing that Cu sis the core of the defect.20,21 This center was first thought to be the Cu sCuipair13and theory confirmed that this pair exhibits most of its key features.22However, PL studies in isotopically pure28Si samples23,24have shown that Cu 0 contains (at least) four Cu atoms. Shirai et al .25proposed the Cus1Cui3trigonal defect consisting of one substitutional and three interstitial copper atoms. The properties of this defect and a plausibleJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155703 (2020); doi: 10.1063/5.0026161 128, 155703-1 Published under license by AIP Publishing.formation process that could explain why Cu s1Cui1and Cu s1Cui2 should be short-lived have been calculated by Carvalho et al.26A recent experimental and theoretical study27of the family of defects CusmCuin(m = 1 –2; n = 1 –6) details the formation of successive Cus1Cuindefects. The properties of the intermediate complexes match the observed27ones, and a mechanism for the growth of the defects into larger Cu precipitates in defect-free Si is outlined. The identification of Cu 0as Cu s1Cui3is not certain. Indeed, the calculated binding energies of the Cu s1Cuin0complexes are 0.79, 0.77, 0.69, and 0.61 eV for n = 1, 2, 3, and 4, respectively.27 However, experiments show a higher binding energy for Cu s1Cui3 than for Cu s1Cui2and Cu s1Cui1. The calculated gap levels of all the defects match the measured ones except for the donor level ofCu s1Cui3, which is much higher (E v+ 0.30 eV) than observed (Ev+ 0.10 eV). However, a double-donor level calculated at Ev+ 0.13 eV is quite close to it. Thus, the donor levels and binding energy of Cu s1Cui3are still unresolved. The *Cu 0and Cu 0defects sometimes coexist, but less is known about *Cu 0. Its PL band13,28,29has a zero-phonon line at 944 meV assigned to the excitonic recombination of an isoelec-tronic center. Phonon replicas separated by 6.42 meV indicate a PLVM at 52 cm −1, but additional (weaker) structures are seen. The 944 meV band correlates with a DLTS level at E v+ 0.185 eV.21,30 The *Cu 0defect is long-lived at RT. Uniaxial stress and Zeeman splitting studies28of *Cu 0suggest T-to-A transitions in T dsymme- try, but these low-stress data are not fully understood.31This defect has first been assigned32to a metastable configuration of the CusCuipair, but PL studies23,24in isotopically pure28Si samples have demonstrated that *Cu 0contains (at least) one Ag and three Cu atoms. Since Ag is a common impurity in metallic Cu, in-diffusing Cu at high temperatures into Si can introduce small amounts of Ag as well. These PL studies23,24have also shown that the 780 and 867 meV bands are associated with Ag 4and Cu 2Ag2 complexes, respectively. The study of thermal ionization rates33,34have shown the pres- ence of Ag-related donor (E v+ 0.395 eV) and acceptor (E c−0.578 eV) levels. The MOS technique has revealed35Ag-related gap levels at Ev+0 . 3 6e Va n d E c−0.33 eV. DLTS studies have shown the presence of a donor level at E v+0 . 4 8e V i n R e f . 36and donor (E v+0 . 3 7e V ) and acceptor (E c−0.55 eV) levels in Ref. 37. These levels have been associated with isolated Ag s.Al e v e la tE c−0.55 eV has been assigned t oaA g –Ag pair.37Aliet al.38have used DLTS in irradiated n-Si samples in-diffused with Ag and reported a Ag-related level atE c−0.35 eV, which transforms into E c−0.21 eV upon annealing. The concentration of these centers matches that of the E c−0.54 eV level associated with isolated Ag s. Thus, the consensus is that Ag shas a donor level near E v+ 0.4 eV and an acceptor level near midgap. The levels reported at E c−0.33 eV35and E c−0.35 eV38have not been assigned to specific defects. High-temperature solubility studies39have shown that, in the range of 1300 –1600 K, Ag diffuses with a migration barrier 1.15 eV and the authors assumed that this was Ag i. However, the nature of the diffusing species at these high temperatures has not been firmlyestablished, and it is not clear that the data can be extrapolated all the way down to RT. Electron-paramagnetic resonance (EPR) studies 40,41of Ag centers have produced a number of spectra. In particular, theSi-NL42 spin-1/2 center with T dsymmetry has been assigned to isolated Ag i. Finally, the PL band at 779 meV42–44has been assigned to a trigonal Ag-related donor level at E v+ 0.34 eV. This PL band is now known23,24to belong to Ag 4. A first-principles calculation45in the Si 64periodic supercell predicted that Ag iis stable at the T site and in the +1 charge state for most positions of the Fermi level. We report here the results of first-principles calculations of isolated Ag and of Ag –Cu interactions in Si. The methodology is discussed in Sec. II. The results (Sec. III) for isolated silver include Agi(gap levels and migration barrier), its interaction with a pre- existing vacancy, and the gap levels of Ag s. Next, we discuss Ag –Cu interactions, starting with the energetics of Cu sAgivs Cu iAgs.W e then consider the series of Ag sCuincomplexes (n = 1 –4) and compare their electrical properties to those of the Cu sCuincom- plexes. Some Ag sCuincomplexes change charge state upon capture of Cu i+. The small correction to the energy involved in this electron or hole capture is evaluated for various positions of the Fermi level.A summary of the key points and a discussion are in Sec. IV. II. METHODOLOGY Our spin-density-functional calculations are based on the SIESTA method 46,47in Si 216periodic supercells. This approach has been used successfully for Ti,4,48V,49Fe,49–52Ni,53Cu,26,27and Co.54The lattice constant of the impurity-free cell is optimized in each charge state. The change of lattice constant with charge state is very small (virtually negligible) for a supercell of that size. The defect geometries are then obtained with a conjugate-gradientalgorithm. A 3 × 3 × 3 Monkhorst –Pack 55mesh samples the Brillouin zone. The electronic core regions are removed from the calculations using Troullier –Martins norm-conserving pseudopotentials56opti- mized for SIESTA.57 The Cu and Ag pseudopotential includes semicore (3s, 3p) and (4s, 4p) states, respectively. The valence regions are treated with spin-density-functional theory within the revised generalized gradient approximation for the exchange-correlation potential.58 This potential leads to the prediction of accurate activation energiesfor diffusion of impurities in Si. 59The charge density is projected on a real-space grid with an equivalent cutoff of 350 Ry to calculate the exchange-correlation and Hartree potentials. The basis sets for the valence states are linear combinations of numerical atomicorbitals: 60,61double-zeta for the first two rows of the Periodic Table with a set of polarizations functions (five 3d ’s) for Si. The basis set of Cu and Ag include two sets of s and d orbitals and one set of p ’s. The structures containing Ag and/or Cu were optimized in each charge and spin state using a conjugate-gradient algorithmuntil the maximum force component dropped below 0.003 eV/Å.Our notation for the spin and charge of a defect X is spinXcharge. The activation energies are calculated using the nudged elastic band (NEB) method62with a 2 × 2 × 2 Monkhorst –Pack mesh but in the smaller Si 64supercell because of the substantial amount of computer time required in NEB calculations. This cell size and k-point sampling have been shown to predict very accurate migra- tion barriers for isolated impurities in Si.59Our implementationJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155703 (2020); doi: 10.1063/5.0026161 128, 155703-2 Published under license by AIP Publishing.uses the climbing image method63for finding the saddle points. Local tangents are estimated using the improved-tangent formal- ism.64The images are connected with springs with a constant of 0.1 eV/Å. The converged diffusion path is the one for which themaximum force component perpendicular to the path at eachimage is less than 0.04 eV/Å. The gap levels are evaluated using the marker method. 65,66For Cu-related defects, the gap levels calculated with this method orusing plots of the formation energies vs chemical potential werefound to be within 0.01 eV of each other. 27Our universal marker is the perfect crystal: the reference for the donor and acceptor levels are the top of the valence band and the bottom of the conduction band, respectively. We use the low-T Si bandgap of 1.17 eV whenplotting the levels. This implementation of the marker methodworks well for a wide range of defects provided that the defectgeometries and the lattice constant of the supercell are optimized in each charge state with a 3 × 3 × 3 k-point sampling. This pro- duces converged energies for the supercell size used here. III. RESULTS A. Isolated Ag in Si Interstitial Ag i+behaves like a small closed-shell 4d105s0ion with little chemical activity. In the neutral charge state, theunpaired electron is delocalized and minimally affects the localinteractions, including the calculated migration barrier (below). Ag iis stable at the T site, and the Ag i−Si internuclear distance (in the 0 and + charge states) is 2.53 Å as compared to 2.45 Å inthe case of Cu i−Si and a T −Si distance of 2.35 Å in the impurity- free Si 216supercell. The calculated donor and acceptor levels of Ag iare at Ev+ 0.96 eV and E c−0.08 eV, respectively. The latter number is close to the accepted error bar of the marker method. Interstitialsilver is 0Agi+for most position of the Fermi level and1/2Agi0in heavily n-type materials. Thus, the NL-Si42 EPR defect observed in p-Si40,41cannot be the isolated interstitial since0Agi+is not paramagnetic. The NEB activation energy for migration from T site to T site is 0.53 eV in the + and 0 charge states. This is almost three timeslarger than that of Cu i+(0.18 eV), but still low enough to allow Agito hop around at RT. Indeed, the migration barriers of HBC+(0.48 eV67) and Fe i+(0.69 eV59) are comparable and both inter- stitials are known to be mobile at RT. When Ag iencounters a pre-existing vacancy, it becomes Ag s with a gain in energy larger than that for copper, but still less than the formation energy of the vacancy calculated at the same level of theory (3.85 eV26):1/2Agi0+0V0→1/2Ags0+ 3.57 eV (copper: 3.12 eV). Note that3/2Ags0is more stable by just 0.02 eV, a negligi- ble energy difference well within the error bar of our calculations.The internuclear distance between Ag sand one of its four Si neigh- bors is 2.41 Å, slightly longer than that calculated for Cu s(2.29 Å). In the positive charge state, the binding energy is slightly lower: 0Agi++0V0→0Ags++ 2.97 eV (copper: 2.51 eV).1Agi+is negligibly higher in energy, 0.02 eV. Thus, isolated Ag in defect-free Si is mostly interstitial but more likely to become substitutional than isolated Cu.The calculated donor level of Ag sis at E v+ 0.36 eV, the single acceptor level is at E c−0.61 eV, and we find a double-acceptor level at E c−0.38 eV. Thus,1/2Ags0is a plausible candidate for the NL-Si42 EPR center observed in p-Si.40,41Our calculated donor and single acceptor levels are close to those reported in the litera-ture (Sec. I), but the double-acceptor level of Ag shas not been dis- cussed. Since the single acceptor level of Ag sis near midgap, we are not surprised to find a double-acceptor level in the upper half ofthe gap. Furthermore, Cu sis similar and it has a double-acceptor level at E c−0.16 eV.27As discussed in the introduction, Ag-related levels have been reported at E c−0.30 eV,37Ec−0.33 eV,35and Ec−0.35 eV.38They are all close to our double-acceptor level at Ec−0.38 eV. B. Ag –Cu interactions The probability of Ag i–Cuiinteractions is low because both are in the +1 charge state for most positions of the Fermi level. The range of Fermi levels for which they are in the 0 charge state isnarrow. We also ignore the interactions between Ag sand Cu ssince they are not mobile at RT. Therefore, we focus here on the interac- tions between Cu i+and Ag s−or Cu s−and Ag i+(Fermi level midgap). Since the binding energy of Ag to the vacancy is larger than that ofCu, we expect Ag sCuito be more stable than Cu sAgi.I n d e e d ,s t a r t i n g with the interstitial near a T site adjacent to the substitutional, weget 0Agi++0Cus−→0Ags−+0Cui+with a net gain of 0.25 eV. The energy barrier involved in this reaction is 0.58 eV (NEB method, Fig. 1 ). Thus, Ag iwill kick-out and replace substitutional Cu sat RT. C. The Ag sCuincomplexes We have calculated the stable configurations of the Ag sCuin complexes (n = 1 –4). Their gap levels are compared to the corre- sponding Cu-only complexes27inFig. 2 (the numerical values are inTable I ). We have little confidence in the levels located very close to a band, such as the acceptor levels of Ag sCui3or Ag 1Cui4. The error bar for such shallow levels is unknown, and they may or may not exist. FIG. 1. Potential energy surface for the reaction0Agi++0Cus−→0Ags−+0Cui+. The dots are the seven images, the dashed line is a fourth-order polynomial fit,and Q is a generalized coordinate.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155703 (2020); doi: 10.1063/5.0026161 128, 155703-3 Published under license by AIP Publishing.The binding energies are obtained by comparing the total energies of the separated species (in different supercells) to those of the final complex plus the perfect supercell. In such comparisons, the spin and charge states must match on both sides of the reac-tion. However, depending on the position of the Fermi level (E F), the charge state of the final complex may be wrong, and a correc-tion is needed to reflect the gain in energy associated with the trap- ping of an electron or hole. This correction has been ignored in previous calculations. Since PL experiments are often performed inhigh-resistivity samples to reduce free-carrier absorption, we pickas an example the Fermi level to be midgap (0.585 ≈0.59). Figure 2 shows that if E Fis midgap, then Cu iand Ag sare in the +1 and −1 charge states, respectively, while the final complex Ag sCui1is in the 0 charge state. The binding energy of 0AgsCui10involves comparing total energies in the reaction 0Si216Cui++0Si215Ags−→0Si215AgsCui10+0Si2160. The energy gain is 0.61 eV, and0AgsCui10happens to be in the correct charge state. No correction is required. The next reaction is0Si216Cui++0Si215AgsCui10→0Si215 AgsCui2++0Si2160at a gain of 0.56 eV. However, if E Fis midgap, the AgsCui2c o m p l e xs h o u l db ei nt h e0c h a r g es t a t e :i tt r a p sa ne l e c t r o n a tag a i ni ne n e r g ye q u a lt ot h ee n e r g yd i f f e r e n c eb e t w e e nE Fand the donor level: 0.59 −0.39 = 0.20 eV. The corrected binding energy of 1/2Si215AgsCui20is 0.76 eV.The next reaction is0Si216Cui++1/2Si215AgsCui20→1/2Si215Ags Cui3++0Si2160at a gain of 0.53 eV. Here again, the Ag sCui3complex should be in the 0 charge state. It traps an electron at a gain in energy of 0.59 −0.42 = 0.17 eV. Thus, the binding energy 0Si215AgsCui30is 0.70 eV. Finally,0Si216Cui++0Si215AgsCui30→0Si215AgsCui4++0Si2160at a gain of 0.58 eV, but the Ag sCui4complex should be in the +2 charge state. It traps a hole at a gain in energy equal to 0.80−0.59 = 0.21 eV, and the binding energy is 0.79 eV. Thus, calculated binding energies have two terms. The first one is the total energy difference between the complex and the fully separated components. However, if the complex is in the wrong charge state for the given E F, a correction to the energy asso- ciated with the trapping of an electron or hole from the appropriategap level must be added. This implies that the measured binding energies should exhibit a small dependence on the position of the Fermi level. This could explain the occasional scatter in the experi- mental data. Table II shows the corrected binding energies for E F midgap, E F=Ev+ 0.42 eV (slightly p-type) and E F=Ec−0.42 eV (slightly n-type). When needed, the correction was estimated usingthe calculated gap levels. Table II also includes the corrected binding energies for the Cu-only complexes. Their gap levels can also be read from Fig. 2 (red dashed lines), but the numerical values are in Ref. 27. These involve just three Fermi level positions, and the numbers will change in strongly p- and n-type Si. IV. DISCUSSION We have calculated the properties of interstitial and substitu- tional Ag in Si, as well as the interactions between Ag and Cu. Both of them are noble metals with the (atomic) electronic structureTABLE II. Binding energies (eV) corrected for the Fermi level dependence associated with a change of charge state upon trapping of Cu i+(see text). defect E F=Ev+ 0.42 eV E Fmidgap E F=Ec−0.42 eV AgsCui1 0.62 0.61 0.69 AgsCui2 0.59 0.76 0.77 AgsCui3 0.53 0.70 0.68 AgsCui4 0.39 0.79 0.63 CusCui1 0.79 0.86 1.02 CusCui2 0.79 0.83 0.99 CusCui3 0.73 0.69 0.69 CusCui4 0.99 0.81 0.65 FIG. 2. Calculated gap levels of Ag i,A g s,A g sCui1,A g sCui2,A g sCui3, and AgsCui4. The levels calculated for the corresponding copper-only defects27are the red dashed lines. The thin black arrows show how the lines shift from Ag toCu (unless the shift is obvious and very small). The low-T bandgap of Si is 1.17 eV . TABLE I. Calculated gap levels (eV) for Ag-related complexes. The donor (acceptor) levels are given relative to the valence (conduction) band. The levels of t he corresponding Cu defects27are in parenthesis. Defect (+/+ +) (0/+) ( −/0) ( −/−) Agi Ev+ 0.96 (0.85) Ec −0.08 (0.14) Ags Ev+ 0.36 (0.20) Ec −0.61 (0.72) Ec −0.38 (0.16) AgsCui1 Ev+ 0.35 (0.23) Ec −0.50 (0.65) Ec −0.15 (0.20) AgsCui2 Ev+ 0.39 (0.27) Ec −0.48 (0.64) AgsCui3 Ev+ 0.17 (0.13) E v+ 0.42 (0.30) Ec −0.03 (0.03) AgsCui4 Ev+ 0.80 (0.79) E v+ 0.98 (0.94) Ec −0.06 (0.06)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155703 (2020); doi: 10.1063/5.0026161 128, 155703-4 Published under license by AIP Publishing.nd10(n + 1)s1, with n = 3 for Cu and n = 4 for Ag. Many similarities between them are expected when they are isolated and when they interact with each other. Isolated interstitial silver is at the T site, and the Ag i-Si inter- nuclear distance is slightly longer than the Cu i-Si one, reflecting the fact that Ag is a larger ion. Cu iand Ag iare in the +1 charge state for most positions of the Fermi level. We predict the donor level to be very high in the gap (E v+ 0.96 eV for Ag iand Ev+ 0.85 eV for Cu i) with an acceptor level very close to the con- duction band (E c−0.08 eV for Ag iand E c−0.14 eV for Cu i). These levels have not been observed experimentally. The Si-NL42 EPR center that has been associated with Ag iin p-type Si cannot be 1/2Agi0, but it could be1/2Ags0. The migration barrier of Ag ibetween adjacent T sites is 0.53 eV, larger than that for Cu i(0.18 eV) but low enough for Ag ito move around at RT. When encountering a pre-existing vacancy, Ag ibecomes Ag s with a gain of energy of 3.57 eV in the 0 charge state and 2.96 eVin the + charge state. These energy gains are larger than for copper(3.13 eV and 2.51 eV, respectively) but still smaller than the forma-tion energy of the vacancy calculated at the same level of theory(3.85 eV in the 0 charge state). Thus, we expect isolated silver in defect-free Si to be mostly Ag i, but with a larger fraction becoming Agsthan in the case of Cu. The calculated donor (E v+ 0.36 eV) and acceptor (E c−0.61 eV) levels of Ag sare close to the measured ones, and we predict a double-acceptor level (E c−0.38 eV) close to a Ag-related level reported by several authors. The interactions of Ag iwith Cu sresults in the formation of the Ag sCuipair with a gain in energy of 0.25 eV. The barrier involved in the kick-out of Cu sis only 0.58 eV. Since Ag sis in the −1 charge state when E Fis near midgap, it attracts the fast-diffusing Cu i+, which initiates the formation of AgsCuincomplexes. Some of the calculated binding energies must be corrected if a change of charge state of the complex is required.We estimate this correction to be the energy difference between theFermi level and the appropriate donor or acceptor level in the complex. Thus, this charge state correction varies with the Fermi level. The corrected binding energies of the Ag sCuinand Cu sCuin complexes are given in Table II for three values of E F. The gap levels of all the Ag sCuincomplexes are compared with those recently obtained27for the Cu sCuincomplexes. As expected, there are clear similarities, but the levels of Ag sCuin (n = 1 –4) are shifted upward relative to the corresponding Cu sCuin complexes ( Fig. 2 ). The precipitation of Cu i+’s stops at n = 4 because of the Coulomb repulsion with Ag sCui4++. The Ag sCui3complex has many of the features of the *Cu 0PL center, just like Cu sCui3resembles the Cu 0=C u PLone. However, *Cu 0 is associated with a single-donor level at E v+0 . 1 8 5e V21,30while we find a double-donor level at E v+ 0.17 eV. This situation is similar to CusCui3, which has a calculated double-donor level at E v+0 . 1 3e V instead of a single-donor level at E v+ 0.10 eV. In both cases, the calcu- lated single-donor level is much too high in the gap to explain the data. 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Artacho, D. Sánchez-Portal, P. Ordejón, A. García, and J. M. Soler, Phys. Status Solidi B 215, 809 (1999). 48D. J. Backlund and S. K. Estreicher, Phys. Rev. B 82, 155208 (2010). 49D. J. Backlund, T. M. Gibbons, and S. K. Estreicher, Phys. Rev. B 94, 195210 (2016). 50M. Sanati, N. Gonzalez Szwacki, and S. K. Estreicher, Phys. Rev. B 76, 125204 (2007).51S. K. Estreicher, M. Sanati, and N. Gonzalez Szwacki, Phys. Rev. B 77, 125214 (2008). 52N. Gonzalez Szwacki, M. Sanati, and S. K. Estreicher, Phys. Rev. B 78, 113202 (2008). 53J. Lindroos, D. P. Fenning, D. Backlund, E. Verlage, A. Gorgulla, S. K. Estreicher, H. Savin, and T. Buonassisi, J. Appl. Phys. 113, 204906 (2013). 54T. M. Gibbons, D. J. Backlund, and S. K. Estreicher, J. Appl. Phys. 121, 045704 (2017). 55H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 (1976). 56N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 (1991). 57P. Rivero, V. M. García-Suárez, D. Pereñiguez, K. Utt, Y. Yang, L. Bellaiche, K. Park, J. Ferrer, and S. Barraza-Lopez, Comput. Mater. Sci. 98, 372 (2015). 58B. Hammer, L. Hansen, and J. K. Nørskov, Phys. Rev. B 59, 7413 (1999). 59S. K. Estreicher, D. J. Backlund, C. Carbogno, and M. Scheffler, Angew. Chem. 50, 10221 (2011). 60O. F. Sankey and D. J. Niklevski, Phys. Rev. B 40, 3979 (1989). 61O. F. Sankey, D. J. Niklevski, D. A. Drabold, and J. D. Dow, Phys. Rev. B 41, 12750 (1990). 62G. Mills and H. Jonsson, Phys. Rev. Lett. 72, 1124 (1994); H. Jonsson, G. Mills, and K. W. Jacobsen, in Classical and Quantum Dynamics in Condensed Phase Simulations, edited by B. J. Berne, G. Ciccotti, and D. F. Coker (World Scientific, Singapore, 1998), p. 385. 63G. Henkelman, B. P. Uberuaga, and H. Jonsson, J. Chem. Phys. 113, 9901 (2000). 64G. Henkelman and H. Jonsson, J. Chem. Phys. 113, 9978 (2000). 65A. Resende, R. Jones, S. Öberg, and P. R. Briddon, Phys. Rev. Lett. 82, 2111 (1999). 66J. P. Goss, M. J. Shaw, and P. R. Briddon, in Theory of Defects in Semiconductors , edited by D. A. Drabold and S. K. Estreicher (Springer, Berlin, 2007), p. 69. 67S. K. Estreicher, M. Stavola, and J. Weber, “Hydrogen in Si and Ge, ”inSilicon, Germanium, and Their Alloys Growth, Defects, Impurities, and Nanocrystals , edited by G. Kissinger and S. Pizzini (CRC, Boca Raton, 2015), pp. 217 –254.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155703 (2020); doi: 10.1063/5.0026161 128, 155703-6 Published under license by AIP Publishing.

5.0017571.pdf

AIP Conference Proceedings 2265 , 030297 (2020); https://doi.org/10.1063/5.0017571 2265 , 030297 © 2020 Author(s).Investigation of magnetoelectricityin La2NiMnO6 thin film deposited by pulsed laser deposition Cite as: AIP Conference Proceedings 2265 , 030297 (2020); https://doi.org/10.1063/5.0017571 Published Online: 05 November 2020 Sweta Tiwary , S. Kuila , Sourav Chowdhury , M. R. Sahoo , A. Barik , R. Ghosh , R. J. Choudhary , Uday Deshpade , P. D. Babu , and P. N. Vishwakarma ARTICLES YOU MAY BE INTERESTED IN Magnetoelectricity in La 2NiMnO 6 and its PVDF impregnated derivative Journal of Applied Physics 124, 044101 (2018); https://doi.org/10.1063/1.5037736 Magnetoresistance in CoFe 2O4/BiFeO 3 core-shell nanoparticles near room temperature Journal of Applied Physics 124, 154104 (2018); https://doi.org/10.1063/1.5031170 Effect of crystal symmetries and phase boundaries on the magnetoelectricity of La 2NiMnO 6 prepared under ambient conditions Journal of Applied Physics 127, 214101 (2020); https://doi.org/10.1063/5.0003395Investigation of Magnetoelectricity in La 2NiMnO 6 Thin Film Deposited by Pulsed Laser Deposition Sweta Tiwary1, S. Kuila1,Sourav Chowdh ury2, M. R. Sahoo1, A. Barik1, R. Ghosh1, R. J . Choudhary2, Uday Deshpade2, P. D. Babu3,and P. N. Vishwakarma1, a) 1Department of Physics & Astron omy, National Institute of Technology, Rourkela-769 008, Odisha INDIA 2UGC-DAE Co nsortium for Scientific Research, Indore 452001, India 3UGC DAE Consortium for Scientific Research, Mumba i Centre, BARC, Mumbai 400 085, India a)Corresponding au thor: prakashn@nitrkl.ac. in and pnviisc@gmail. com Abstract. Polycrystalline thin film of La 2NiMnO 6 (LNM) is deposited successfully on Pt/TiO x/SiO 2/Si substrates using pulsed laser deposition technique. The X-ray diffraction data and the Raman spectra confirm the formation of a single phase with P2 1/n space group and alsothe indication of Mn4+obtains in X-ray photoelectron spectroscopyhints towards the formation of ordered double perovskites. Significant magnetoele ctric voltage is observed in the temperature range of 138 to 400 K, which may be due to the huge strain present in the sample andthe short range ordering (upto 400 K) of the sample.The resistivity vs. temperature measurement reveals the metallic behavior of the thin film. INTRODUCTION The quest for magnetoelectric materials (ME)1has attracted a lot of attention from the perspective of desig ning new devices. Recently, double perovskites LNM has received considerable attention because it exhib its a ferromagnetic order near room temp erature (Tc ~ 275 K)2 and short range ordering up to 400 K3. The structural studies reveal that the long range ordering of Ni and Mn correspon ds to P21/n space group at low temperature whereas R-3 space group at high temp erature4. Strikingly, thin film of LNM shows different behavior depending upon the growth process and the mismatch b etween the lattice parameters of sample and substrate,which introd uces epitaxial strain that cancau se structural and magn etic anisotropy which can considerab ly influ ence the magnetic and magneto electric properties5. In our previous study, it was observed that the bulk samples of LNM are found to be dominated by magnetoloss and true magnetoelectricity is observ ed after impregnating the bulk sample with PVDF polymer2. So LNM in the fo rm of thin film can also be one of th e best ideas to red uce the conductivity and to get the true magnetoelectric coupling mediated by the strain. In order to realize this in practice and to get better physical properties, it is very crucial to make the thin film of LNM unde roptimized deposition condition s. This proceed ing presents the results of X-ray diffractio n (XRD), Raman spectroscopy, X-ray photoelectron spectroscopy (XPS) and Magnetoelectric measurements of LNM thin film grown by pulsed laser deposition techniques. EXPERIMENTAL DETAILS LNM film was grown on (100) Pt/TiO x/SiO 2/Si by pulsed laser deposition techniques. A polycry stalline stoich iometric LNM target was synthesized and the details of its synthesis conditions are reported elsewhere2. The target were ablated by KrFex cimer laser (λ= 248 nm) with a repetition rate of 5 Hz. During depo sition, the substrate were kept in th e 800°C temperature with the co ntinuous flow of oxygen (oxygen pressure is ~ 500 mTorr). After the film deposition, the films were annealed within the deposition chamber for 1 hour at the same temperature (800°C), an d then the film was co oled to room temperature with the rate of 1 0°C/min. DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030297-1–030297-4; https://doi.org/10.1063/5.0017571 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030297-1Synthesized films were characterized by X-ray diffraction (XRD) , Raman spectroscopy and X-ray photoelectron spectroscopy (XPS). The direct magn etoelectric measurements were done using Lock – In Amplifier, as a function of temperature (125K to 300K) at a frequency of 210Hz. The magnetoelectric measurements are done in the homemade setup6. RESULTS AND D ISCUSSIONS Thecrystalline quality and phase of the LNM thin film was analy zed by using XRD measurement. The XRD patterns of both Bulk and thin film of LNM is shown in Fig. 1 (a). The major peak of the LNM sample ~ at 2.72 is also observed in the thin film. The index ed peak suggests the P21/n phase of the deposited samp le on the substrate. No diffraction peaks from any other secondary phases or randomly oriented grains were observed. In the Raman spectra two well separated peaks were observed in the spectral range of 530-534 cm-1 and ~ at 668 cm-1for antisymmetric stretching (AS) and symmetric stretching (S) modes of vibrations respectively. The high frequency peak ~ at 6 68 cm-1 has b een reported as the stretchin g “breathin g” vib rations of the Ni/MnO 6octahedra of the ordered (P2 1/n) phase of LNM. On the other hand the peak near 530 cm-1 is in a frequency range where, for the P21/n stru cture, it is attributed as the antistretchings or/and bendings of the Ni/MnO 6octahed ra3. Two unknown peaks are also observed ~418 an d 570 cm-1, which may be fro m the substrate. Hence, the XRD and Raman spectra clearly demon strate the format ion of single phase LNM deposite d on the Pt/TiO x/SiO 2/Si substrates using pulsed laser deposition technique. 2.5 2.6 2.7 2.8 2.9 3. 0Intensity (arb. units) d-Spacing / Bulk Sample Thin film Å(020) (112) -(112) (200) 200 300 400 500 600 700 800 900 100 0(530)Intensity (arb. units) Raman shift [cm-1](668) (a) (b) FIGURE 1 . XRD patterns of (a) bulk LNM samp le (b) t hin film of LNM in the 2 θ range of 31-35° and (c) Raman spectra of LNM thin film m easured at room te mperature. This observation is further confirmed with XPS, which shows the presence of Mn4+ in the thin film, as presented in Fig. 2. The spectra have 2p3/2 and 2p1/2 spin-o rbit doublet peaks located ~ at 642 eV and 654 eV, respectively and also spectra of Mn 3p in which peak is centered at ~ 50 eV (shown in the inset of Fig 2)and one broad peak located around 664 eV which is the signature peak of 4+ state o f Mn. 030297-2635 640 645 650 655 660 665 670Mn 2p3/2Intensity (arb.units) Binding Energy (eV)Mn 2p1/2 42 49 56Intensity (arb.units) Binding Energy (eV)Mn 3p FIGURE 2: Mn 2p and Mn 3p (inset) core-level XPS spectra of LNM thin film . The temperature depen dent ME voltage of LNM thin film in the temperature range of 138-400K is shown in Fig.3 (a).The schematic of the layers of the substrate (Pt/TiO x/SiO 2/Si) and the dep osited sample along with the measuring probe and the direction o f the field is shown in the lower inset of Fig. 3(a) The ME voltag e is increasin g monotonically with increase in temperature up to 370K and then a saturating behavior is observed . The same behavior is observ ed in the high temp erature (300 – 400 K) resistance vs temperature measurement (shown in the inset of Fig.3 (a)), in which the resistance is increasing with increase of temperature and beco mes saturating near 400K. And the resistance value is very high even at room temperature which indicates that, the lossy natu re in the bulk samples of LNM due to conductiv ity is highly redu ced in th in film. 150 200 250 300 350 4000.750.800.850.900.951.001.051.10 R (M) T (K)V (mV) H (kOe)B300 350 400560580600620 -10 -5 0 5 100.840.850.860.870.880.890.900.91 V (mV) H (kOe)V (mV) H (kOe)(300K) (b) -10 -5 0 5 101.0081.0101.0121.0141.016(400K) ( a) (b) FIGURE 3 (a) Variation of magnetoelectric vol tage with temperature of LNM th in film, the upper inset s hows R vs T plot in the temperature range of 300 – 400 K and the lower inset shows theS chematical depiction of the layers of the substrate and depositedsample (b) ΔV vs. H, a t 300 K and 400 K ( shown in th e inset) is shown. 030297-3In the plot of ΔV vs H (Fig.3 (b)), the ME voltage ~ 400 K is a lso linearly varies with magnetic field (see inset of Fig.3 (b) ). The linear behavior is observed in the LNM thin film, is due to the reversing nature of ME v oltage on rever sal of the magnetic field. Interestingly, this linear beha vior is the signature of true ME coupling in the sample w hich is extended much above th e Tc ( 275K) of the sample. So it should be emphasized th at the observed ME voltage upto 4 00 K is may be due to the presence of huge strain and short range ordering in the sample, which is reported upto 400K in LNM thin film3 as well as in LNM bulk samples7. CONCLUSION In conclusion, crystalline LNM thin film with an ordered perovs kite structure was grown by pulsed laser deposition. XRD and Raman spectra reveal the successful formation of single phase LNM thin film. The XPS result indicates that the Mn is in its +4 state which reconfirms the growth of ordered perovskite structure. Significant magnetoelectricity is observed up to 400 K, up to which the system exhibits short range ordering. But the resistance behavior with temperature is seems to be contradictable, as it does not follow the semiconducting behavior of LNM. Further investigation is in progress to have in-depth understanding of the observed behavior of the sample. ACKNOWLEDGMENTS Th e authors are thankful t o DST, New Delhi and UGC DAE CSR, Mum bai for their support in the form of projects “EMR/2014/000 341” and “CRS-M-223/2016/724” respectively. Dr. V. Siruguri is also acknowledged for his support and time to time valuable suggestions. The author Sweta Tiwary would also like to thank CSIR, India, for CSIR-SRF fellowship (09/983(0021)/2k18-EMR-I) and financial assistance. R EFERENCES 1. B. Paudel, I. Vasiliev, M. Hammo uri, D. Karpov, A. Chen, V. Lau ter, and E. Fohtung, RSC Advances 9, 13033 (2019) 2. S. Tiwary, S. Kuila, M. Sahoo, A. Barik, P. Babu, and P. Vishwakarma, Journal of Applied Physics 124, 044101 (2018) 3. M. N. Iliev, H. Guo, and A . Gupta, Appl. Phys. Lett. 90, 151914 (2007). 4. C. L. Bull, D. Gleeson, and K. S. Knight, J. Phys.: Condens. Ma tter 15, 4927 (2003). 5. S. Kazan, F. A. Mikailzade, M. Özdemir, B. Aktaş, B. Rameev, A. Intepe, and A. Gupta, Appl. Phys. Lett. 97, 072511 (2010). 6. S. Kuila, S. Tiwary, M. Sahoo, A. Barik, and P. Vishwakarma, Jo urnal of Alloys and Compounds 709, 158 (2017). 7. S. Zhou, L. Shi, H. Yang, and J. Zhao., Appl. Phys. Lett. 91, 172505 (2007) 030297-4

5.0023593.pdf

J. Chem. Phys. 153, 154302 (2020); https://doi.org/10.1063/5.0023593 153, 154302 © 2020 Author(s).Factors determining formation efficiencies of one-electron-reduced species of redox photosensitizers Cite as: J. Chem. Phys. 153, 154302 (2020); https://doi.org/10.1063/5.0023593 Submitted: 01 August 2020 . Accepted: 29 September 2020 . Published Online: 16 October 2020 Kyohei Ozawa , Yusuke Tamaki , Kei Kamogawa , Kazuhide Koike , and Osamu Ishitani COLLECTIONS Paper published as part of the special topic on 65 Years of Electron Transfer ARTICLES YOU MAY BE INTERESTED IN Potential and pitfalls: On the use of transient absorption spectroscopy for in situ and operando studies of photoelectrodes The Journal of Chemical Physics 153, 150901 (2020); https://doi.org/10.1063/5.0022138 Use the force! Reduced variance estimators for densities, radial distribution functions, and local mobilities in molecular simulations The Journal of Chemical Physics 153, 150902 (2020); https://doi.org/10.1063/5.0029113 A minimum quantum chemistry CCSD(T)/CBS dataset of dimeric interaction energies for small organic functional groups The Journal of Chemical Physics 153, 154301 (2020); https://doi.org/10.1063/5.0019392The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Factors determining formation efficiencies of one-electron-reduced species of redox photosensitizers Cite as: J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 Submitted: 1 August 2020 •Accepted: 29 September 2020 • Published Online: 16 October 2020 Kyohei Ozawa,1Yusuke Tamaki,1Kei Kamogawa,1Kazuhide Koike,2,a) and Osamu Ishitani1,a) AFFILIATIONS 1Department of Chemistry, School of Science, Tokyo Institute of Technology, 2-12-1-NE-1 O-okayama, Meguro-ku, Tokyo 152-8550, Japan 2National Institute of Advanced Industrial Science and Technology, Onogawa 16-1, Tsukuba 305-8569, Japan Note: This paper is part of the JCP Special Topic on 65 Years of Electron Transfer. a)Authors to whom correspondence should be addressed: k-koike@aist.go.jp and ishitani@chem.titech.ac.jp ABSTRACT Improvement in the photochemical formation efficiency of one-electron-reduced species (OERS) of a photoredox photosensitizer (a redox catalyst) is directly linked to the improvement in efficiencies of the various photocatalytic reactions themselves. We investigated the primary processes of a photochemical reduction of two series [Ru(diimine) 3]2+and [Os(diimine) 3]2+as frequently used redox photosensitizers (PS2+), by 1,3-dimethyl-2-phenyl-2,3-dihydro-1 H-benzo[ d]imidazole (BIH) as a typical reductant in detail using steady-irradiation and time- resolved spectroscopies. The rate constants of all elementary processes of the photochemical reduction of PS2+by BIH to give the free PS•+ were obtained or estimated. The most important process for determining the formation efficiency of the free PS•+was the escape yield from the solvated ion pair [ PS•+–BIH•+], which was strongly dependent on both the central metal ion and the ligands. In cases with the same central metal ion, the system with larger −ΔGbet, which is the free energy change in the back-electron transfer from the OERS of PS•+to BIH•+, tended to lower the escape yield of the free OERS of PS2+. On the other hand, different central metal ions drastically affected the escape yield even in cases with similar −ΔGbet; the escape yield in the case of RuH2+(−ΔGbet= 1.68 eV) was 5–11 times higher compared to those of OsH2+(−ΔGbet= 1.60 eV) and OsMe2+(−ΔGbet= 1.71 eV). The back-electron transfer process from the free PS•+to the free BIH •+ could not compete against the further reaction of the free BIH •+, which is the deprotonation process giving BI •, in DMA for all examples. The produced BI •gave one electron to PS2+in the ground state to give another PS•+, quantitatively. Based on these findings and investigations, it is clarified that the photochemical formation efficiency of the free PS•+should be affected not only by −ΔGbetbut also by the heavy-atom effect of the central metal ion, and/or the oxidation power of the excited PS2+, which should determine the distance between the excited PS and BIH at the moment of the electron transfer. ©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/10.1063/5.0023593 .,s INTRODUCTION Photochemical redox reactions have been intensely investi- gated for a long time. In artificial photosynthesis, for example, H 2 evolution1and CO 2reduction2,3can be initiated by a photoinduced electron transfer using a redox photosensitizer from an electron donor to a catalyst. Ru(II) trisdiimine complexes are most frequently used as redox photosensitizers owing to their stability in both theexcited and the one-electron-reduced states.4,5Since CO 2reduc- tion and H 2evolution require a two-electron reduction, the redox photosensitizer requires a catalyst that can accept electrons from the reduced redox photosensitizer to give the product(s). Recently, Os(II) trisdiimine complexes have been used as redox photosen- sitizers, as they use light at longer wavelengths compared to the corresponding Ru(II) complexes owing to their much stronger singlet–triplet absorption.6In many organic reactions, photoredox J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-1 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp SCHEME 1 . Photochemical formation processes for the one-electron reduced species of the redox photosensitizer ( PS•+). catalysts, another name for redox photosensitizers, play an impor- tant role.7In many reports, Ru(II) trisdiimine complexes have been used for various photocatalytic reactions, such as the reduc- tion of olefins,8,9carbonyl compounds,10,11and nitrogen functional compounds,12,13reductive dehalogenation reactions,14,15reductive radical cyclization,16reductive opening reactions of epoxides and aziridines,17,18and many C–C coupling reactions.19 Such photochemical redox reactions include the photochem- ical electron transfer process. Electron transfer has been thor- oughly investigated and the Marcus theory has been established; however, the photochemical formation of a one-electron reduced species (OERS) from the redox photosensitizer includes not only theelectron transfer processes from a reductant to the excited redox photosensitizer but also other processes as shown in Scheme 1 (the abbreviations PS2+andDare used for the redox photosensitizer and the reductant, respectively). For example, back-electron trans- fer from the produced OERS of the redox photosensitizer ( PS•+) to the oxidized reductant ( D•+) in a solvated ion pair competes with the separation process of the ion pair to free PS•+andD•+through the diffusive collision of free PS•+andD•+species. These processes lower the quantum yield for the formation of PS•+in the reaction solution. In many of the reported photocatalytic redox reactions, the electron transfer from Dto the excited redox photosensitizer (∗PS2+) giving free PS•+initiates the reaction.2,7The efficiency of the photochemical formation of PS•+via the electron transfer is important because it strongly affects the efficiency of the whole pho- tocatalytic redox reaction, i.e., the quantum yields for the formation of the products and the turnover frequency of the photocatalytic system. It is crucial in developing efficient photocatalytic redox reac- tions to clarify how to improve the quantum yield for the formation ofPS•+. To the best of our knowledge, however, the formation of PS•+has not been systematically investigated. Herein, the quantum yields for the formation of OERSs for both [Ru(X 2bpy) 3]2+(RuX2+) and [Os(X 2bpy) 3]2+(OsX2+) series (Chart 1, X 2bpy = 4,4′-X2-2,2′-bipyridine) are investi- gated. The PS2+, which have the same charge and a similar molecular size and shape, are used in photochemical reactions with a typical reductant, 1,3-dimethyl-2-phenyl-2,3-dihydro-1 H- benzo[ d]imidazole (BIH),18,20–22and each elementary process in the photochemical formation was systematically investigated under both steady-irradiation and laser-flash photolysis. RESULTS Photophysical and electrochemical properties Figure 1 shows the UV–Vis absorption spectra of RuX2+and OsX2+in an N,N-dimethylacetamide (DMA) solution. Although, in CHART 1 . Structures of RuX2+andOsX2+. J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-2 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . UV–Vis absorption spectra of RuX2+and OsX2+measured in N, N-dimethylacetamide. FIG. 2 . Emission spectra of RuX2+and OsX2+measured in N,N-dimethylacetamide at room temperature. both series, absorption bands at λmax=∼300 nm and λmax= 450 nm– 500 nm attributed to1π–π∗and1MLCT excitations were observed, only spectra of OsX2+showed a relatively strong broad absorption band at λmax= 650 nm–750 nm. These are attributed to singlet to triplet (S-T) absorption bands, i.e., direct excitation to the3MLCT excited state of OsX2+, which is a forbidden transition in principleof the quantum chemistry but is partially allowed owing to the strong heavy-atom effect of the central Os(II). All of RuX2+andOsX2+were emissive at room temperature in DMA solutions, and these emission spectra were broad as shown in Fig. 2. These are attributed to the emission from the3MLCT excited state of the complexes. On the other hand, the emission measured at a lower temperature had vibrational structures (Fig. 3), which are used for determining the excitation energy ( E00) of these com- plexes by coupling with the Franck–Condon analysis. The emission decays of all the complexes measured by the single-photon counting method could be fitted using a single exponential decay (Fig. S1). The photophysical properties of RuX2+andOsX2+are summarized in Table I. Comparing the series of RuX2+andOsX2+, a similar tendency was obtained. The shortest wavelengths of the3MLCT emission maxima were observed for the complexes in which X = H. In the complexes with either strong electron-withdrawing (X = CO 2Me) or strong electron-donating (X = OMe) groups, the1MLCT absorption and3MLCT emission maxima were observed at longer wavelengths. RuCO 2Me2+has the lowest energy of the π∗orbital (LUMO) for the diimine ligand in the series of the Ru(II) complexes and this could be the reason for the low excitation energy. This is supported by the non-radiative decay rate constant ( knr) of the3MLCT excited state ofRuCO 2Me2+being the lowest for the Ru(II) complexes (Table I) even though the excitation energy is lower than RuH2+andRuMe2+ and the energy-gap low should work. One of the main pathways for non-radiative decay of the3MLCT excited state proceeds through a thermal transition to the3d–d excited state and the low energy of the π∗orbital of the diimine ligand should suppress this thermal transition in the case of RuCO 2Me2+. This is the case for the Os(II) complexes as shown in Table I. On the other hand, the high energy of theπ∗orbital for the (MeO) 2bpy ligands should increase the energy level of the t 2gorbital (HOMO) centered on the Ru(II) owing to a weak π-back donating ability resulting in a lower excitation energy forRuOMe2+compared to RuH2+andRuMe2+. Although these tendencies were already reported in previous papers about Ru(II) FIG. 3 . Emission spectra of RuX2+andOsX2+measured in N,N-dimethylacetamide at 77 K (red) and fitting results using the Franck–Condon analysis (green). J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-3 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Photophysical properties of RuX2+andOsX2+measured in N,N-dimethylacetamide. λabs(nm) [ε(103M−1cm−1)]a Complex π–π∗ 1MLCT3MLCT λema(nm) Φemaτema(ns) kr×105s−1knr×105s−1 RuCO 2Me2+310 (58.4) 473 (23.8) ... 653 0.112 1050 1.1 9 RuH2+<290 456 (14.4) ... 631 0.143 905 1.6 10 RuMe2+<290 462 (15.0) ... 641 0.150 758 2.1 11 RuOMe2+<290 482 (12.8) ... 680 0.032 190 1.7 50 OsCO 2Me2+314 (69.1) 504 (20.8) 705 (4.92) 787 0.016 76 2.1 130 449 (27.9) OsH2+293 (87.1) 482 (14.1) 590 (3.96) 756 0.004 42 1.0 240 OsMe2+292 (75.9) 490 (14.5) 600 (4.14) 778 0.005 24 2.1 410 aMeasured at room temperature. polypyridine complexes,4,5it remains important for investigating the photochemical reduction of not only the Ru complexes but also the Os complexes, which have the same charge and a similar molecular shape, and both series of Ru and Os complexes have the same low- est excited state, i.e.,3MLCT excited state, and similar photophysical tendencies. All cyclic voltammograms (CVs) of RuX2+andOsX2+showed three reversible reduction waves, which are attributed to the one- electron reduction of each diimine ligand, in a scan of negative potential (Fig. 4). Table II summarizes the redox potentials, which have a good linear relationship in the Hammett plots (Fig. S2). These results clearly indicate that the OERSs of RuX2+andOsX2+(RuH•+ andOs•+) are stable and the reduction powers of these OERSs can be systematically tuned by the addition of the substituents into the diimine ligands. The reduction potential of the3MLCT excited state of the complexes [ E1/2(∗PS2+/PS•+)] is calculated using the fol- lowing equation using the first redox potential (Table II) and E00 (Table II) and are summarized in Table II: FIG. 4 . Cyclic voltammograms of RuX2+andOsX2+measured in Ar-saturated N,N-dimethylacetamide containing Et 4NBF 4(0.1M) measured using glassy car- bon working (diameter, 3 mm) and Pt wire reference electrodes at a scan rate of 200 mV s−1.E1/2(∗PS2 +/PS●+)=E1/2(PS2 +/PS●+)+E00. (1) The oxidation potential of BIH could be determined by mea- suring a rapid CV using a micro glassy-carbon working (diameter, 33μm) and Pt wire reference electrodes at a scan rate of 200 V s−1 asE1/2(BIH•+/BIH) = 0.02 V vs Ag/AgNO 3(Fig. S3a). The CV of BI+was irreversible even when measured at a scan rate of 200 V s−1 (Fig. S3b), for which the peak potential was Epred(BI+/BI•) =−2.05 V vs Ag/AgNO 3. The UV–Vis absorption spectra of all OERSs of RuX2+and OsX2+were obtained using a flow electrolysis cell with a photodi- ode array detector and a D 2//I2mixed lamp.23As a typical exam- ple, the absorption spectrum and the relationship between current and absorption under an applied voltage for RuH2+andOsH2+are shown in Figs. 5 and 6, respectively. The first spectral changes with isosbestic points started close to the onset potential of the first reduc- tion wave in both cases ( Ewas∼−1.50 V vs Ag/AgNO 3and∼−1.55 V vs Ag/AgNO 3forRuH2+andOsH2+, respectively) and finalized before the onset potential of the second reduction ( Ewas∼−1.78 V and∼−1.73 V for RuH2+andOsH2+, respectively) in which the number of electrons accepted by one molecule was ∼1 in both cases. From these results, the molar extinction coefficients of the OERSs for RuX2+andOsX2+can be calculated at the absorption maximum in all wavelengths measured in the flow electrolysis experiments. The results of the other complexes are shown in Figs. S4–S8 and show similar tendencies to those of RuH2+andOsH2+. Photochemical formation of the OERSs using a Xe lamp As a typical example of photochemical reduction of the metal complexes, Fig. 7(a) shows the UV–Vis absorption changes for a DMA solution containing RuH2+(0.3 mM) and BIH (0.1M) during irradiation [ λex= 480 nm (3.7 ×10−9E s−1)] under an Ar atmosphere using a 500 W Xe lamp with a band-pass filter, and fitting results using the spectrum of the OERS ( RuH•+) obtained by flow elec- trolysis (Fig. 5). It is clear that RuH•+was quantitatively produced J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-4 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Redox potentials and molar extinction coefficients of the one-electron reduced species (OERS) of RuX2+ andOsX2+. E1/2aE1/2(∗PS2+/PS•+) Complex V vs Ag/AgNO 3 v00b104cm−1E00b(eV) V vs Ag/AgNO 3 RuCO 2Me2+−1.20 −1.32 −1.55 1.58 1.96 0.76 RuH2+−1.64 −1.82 −2.09 1.74 2.16 0.52 RuMe2+−1.74 −1.92 −2.18 1.66 2.06 0.32 RuOMe2+−1.80 −1.96 −2.18 1.68 2.08 0.28 OsCO 2Me2+−1.11 −1.25 −1.53 1.35 1.67 0.56 OsH2+−1.56 −1.75 −2.09 1.42 1.77 0.21 OsMe2+−1.67 −1.85 −2.16 1.40 1.73 0.06 aFrom Fig. 4. bDetermined using the emission spectra measured at 77 K (Fig. 3). during the irradiation and the amount of RuH•+produced can be calculated [Fig. 7(b)]. At the initial stage, in which less than 5% of RuH2+was reduced, a linear relationship between the irradiation time and the concentration of RuH•+can be seen, and the quantum yield for the formation of RuH•+[ΦOERS(total) ] can be determined as 1.10. Under irradiation longer than 15 min, the formation rate slowly decreases owing to the inner filter effect accumulated by the OERS itself. In the case of OsH2+, the results are shown in Fig. 8, and theΦOERS(total) was 0.16. A similar analysis was applied to the other complexes (Fig. S8), and the results are summarized in Table III. The FIG. 5 . UV–Vis absorption spectra of reduced RuH2+(right) obtained by the flow electrolysis and dependencies of current and absorption on the applied voltage (left). FIG. 6 . UV–Vis absorption spectra of reduced OsH2+(right) obtained by the flow electrolysis and dependencies of current and absorption on the applied voltage (left).only exception was OsCO 2Me2+. The UV–Vis absorption changes for a DMA solution containing OsCO 2Me2+and BIH during irradi- ation were different from those during the flow electrolysis, which indicates the decomposition of the complex during the irradiation FIG. 7 . (a) UV–Vis absorption changes of a N,N-dimethylacetamide solution con- taining RuH (0.3 mM) and BIH (0.1M) during irradiation at λex= 480 nm (3.7 ×10−9E s−1) under an Ar atmosphere (dots) and fitting results with the spec- trum of RuH•+(curves) and (b) the relationship between the irradiation time and the concentration of RuH•+. FIG. 8 . (a) UV–Vis absorption changes of a N,N-dimethylacetamide solution con- taining OsH2+(0.3 mM) and BIH (0.1M) during irradiation at λex= 480 nm (2.6 ×10−8E s−1) under an Ar atmosphere (dots) and fitting results with the spectrum of the OsH•+(curves) and (b) the relationship between the irradiation time and the concentration of OsH•+. J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-5 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Photochemical formation of the one-electron reduced species (OERS) of the complexes. Complex ΦOERS(total) KSV(M−1) kq(107M−1s−1) ηq −ΔGpet(eV)a RuCO 2Me2+1.7 6.7 ×1036.4×1021.00 0.75 RuH2+1.1 2.4 ×1032.6×1021.00 0.51 RuMe2+1.0 1.4 ×1031.9×1020.99 0.31 RuOMe2+1.0 1.8 ×1029.4×10 0.95 0.27 OsH2+0.16 2.6 ×10 6.2 ×10 0.72 0.20 OsMe2+∼0.01 1.8 7.5 0.15 0.05 a−∆Gpet=−E1/2(BIH•+/BIH) + Ered(∗PS2+/PS•+)−wp+ w r:E1/2(BIH•+/BIH) = −0.02 V vs Ag/AgNO 3,E1/2(∗PS2+/PS•+) in Table I, wr= 0 eV, wp= 0.03 eV. (Fig. S10). Therefore, OsCO 2Me2+was not used for the following experiments. The first step of the photochemical formation of the OERS is a reductive quenching process of the3MLCT excited state of the com- plex by BIH. This process was investigated by the emission spectral measurements of the excited complexes in the absence and pres- ence of BIH with various concentrations, including a Stern–Volmer analysis. As typical examples, the results of RuH2+and OsH2+ are shown in Figs. 9 and 10. In both cases, linear Stern–Volmer plots were obtained, and their Stern–Volmer constants ( KSV) were FIG. 9 . Emission spectra (a) and Stern–Volmer plots (b) of RuH2+measured in N,N-dimethylacetamide at room temperature in the absence or presence of BIH with various concentrations. FIG. 10 . Emission spectra (a) and Stern–Volmer plots (b) of OsH2+measured in N,N-dimethylacetamide at room temperature in the absence or presence of BIH with various concentrations.2.4×103M−1forRuH2+and 2.6×10 M−1forOsH2+. Linear Stern–Volmer plots for the other complexes were also obtained (Fig. S11). The quenching rates ( kq),KSV, and the free energy change of the photochemical electron transfer ( ΔGpet) are summarized in Table III. The quenching fractions ( ηq) of the excited complexes by BIH (0.1M) were calculated using the following equation and are summarized in Table III: ηq=kqτem[BIH]/(1 +kqτem[BIH]). (2) Time-resolved spectroscopy of the photochemical formation processes of the OERSs using laser light The oxidation processes of BIH in various photosensitized reactions were carefully investigated and reported as illustrated in Scheme 2.22,24The one-electron oxidation product (BIH •+), which is produced by the electron transfer to the3MLCT excited state ofPS, is rapidly deprotonated by a base, such as another BIH molecule, to BI •. As this radical species has a strong reduction power [Epred(BI+/BI•) =−2.05 V vs Ag/AgNO 3], it gives another electron toPS2+in the ground state to produce PS•+. SCHEME 2 . Oxidation process of BIH by the excited photosensitizer (*PS2+). J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-6 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 11 . Time-resolved visible absorption spectra after laser-excitation ( λex = 532 nm) of (a) RuH2+or (b) OsH2+(0.03 mM) in the presence of BIH (0.1M) in N,N-dimethylacetamide under an Ar atmosphere. Therefore, the one-photon excitation of PS2+in the presence of BIH can give two molecules of PS•+. To separately handle these two processes, i.e., the photochemical electron transfer from BIH to PS2+and the reduction of another PS2+in the ground state by BI •, we measured time-resolved visible absorption (TR-AB) spectra after the laser-excitation of RuX2+andOsX2+using the randomly inter- leaved pulse train method in a flow cell. Figure 11 shows the TR-AB spectra in the cases of RuH2+andOsH2+as typical examples. The DMA solution containing the complex (0.3 mM) and BIH (0.1M) was irradiated under an Ar atmosphere by 532 nm pulse light from a Nd:YAG laser (pulse width 350 ps) and the solution was flowed at 4 ml min−1to avoid the accumulation of the OERS. In both cases, just after laser irradiation, the spectrum was fitted with a linear com- bination of spectra for the3MLCT excited state (Fig. S12), which was measured using the same TR-AB spectroscopy in the absence of BIH and the OERS of the complexes (Fig. 5). However, after sev- eral tens of nanoseconds after the laser flash, the spectra were similar to the OERS [Figs. 7(a) and 8(a)] as shown in Fig. 1(a). This is rea- sonable because the lifetimes of the3MLCT excited states of RuH2+ andOsH2+in the presence of 0.1M of BIH were 3.8 ns and 11.7 ns, respectively. If we carefully check these spectra, however, a slightly higher absorbance at λ>600 nm can be observed in both transient spectra (red curves in Fig. 12) in comparison to the OERSs (blue). After several microseconds, this higher absorbance at λ>550 nm disappeared and only the absorption spectrum of the OERS was observed in both RuH2+andOsH2+(Fig. 13). These results clearly FIG. 12 . Transient absorption spectra at 10 ns [(a) RuH2+] or 40 ns [(b) OsH2+] after the laser excitation ( λex= 532 nm) in the presence of BIH (0.1M) in N,N- dimethylacetamide under an Ar atmosphere (red lines), and absorption spectra of the one-electron reduced species (OERS) of these complexes (blue lines). FIG. 13 . Transient absorption spectra at 9 μs after laser excitation ( λex= 532 nm) in the presence of BIH (0.1M) in N,N-dimethylacetamide under an Ar atmosphere (red lines), and absorption spectra of the one-electron reduced species (OERS) of (a)RuH2+and (b) OsH2+(blue lines). indicate that another species, which has an absorption at λ>550 nm, should exist in the solution for several microseconds after the laser irradiation. Candidates for this species are BIH •+and/or BI •, which are produced through the one-electron oxidation by the excited metal complex and the following deprotonation process. The absorption spectra of BIH •+and BI •were estimated using the time-dependent density-functional theory (TDDFT) calculation of the optimized structure by density functional theory (DFT) in DMA (PCM) with UB3LYP exchange correlations functional and a basis set of 6- 311G++(d,p) (Fig. 14). These results clearly indicate that BI •should affect the transient absorption at λ>550 nm, although the effect of BIH •+should be small owing to its very weak absorption in this region. Notably, the contribution of BI •to the absorption spectra is small, especially between 400 nm and 550 nm, owing to its weak absorbance [Fig. 14(a)]. This is supported by the TR-AB spectra, measured at several tens of nanoseconds after the irradiation, which is similar to the TR-AB spectra of the OERSs (Fig. 12). Estimation of the escape rate constant of the OERS and BIH •+from the solvated ion pair The escape rate from the solvent cage ( kesc) can be estimated from the Eigen equation [Eqs. (3) and (4)],25 kesc=3D r2δ(r) 1−e−δ(r), (3) δ(r)=zDzAe2 εrkBT, (4) where zDand zAare the charges of the product species, Dis the sum of the diffusion constants, ris the distance between the ions in the ion pair, εis the dielectric constant of the solvent (38.2 in the case of DMA26),kBis the Boltzmann’s constant, and Tis the absolute temperature (298 K). The diffusion constant of the chem- ical species calculated using the molecular radius ( r) is represented by the Stokes–Einstein equation [Eq. (5)], D=kBT 6πηr, (5) J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-7 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 14 . Oscillator strengths of BI •(a) and BIH+•(b) obtained using the TDDFT calculation of the optimized structure by the DFT in N,N-dimethylacetamide (PCM) with UB3LYP exchange correlations functional and a basis set of 6-311G++(d,p) (red line), and their UV–Vis absorption spectra estimated by the summation of the Gaussian functions based on the oscillation strengths (black line). where ηis the viscosity of the solvent. We can estimate the depen- dence of the kescon the distance between the OERS of the complex (PS•+)and BIH •+in the ion pair using Eq. (3). This clearly indicates that the escape rate strongly depends on the distance between PS•+ and BIH •+in the ion pair. Assuming that the ionic radii of RuH•+and BIH •+are 7.1 Å and 5.3 Å (longest direction), respectively, based on the MM2 calcu- lation results,27the sum of the diffusion constants can be estimated as DRuH●++DBIH●+=kBT 6πη(1 rRuH●++1 rBIH●+)=7.5×10−10m2s−1. (6) If the electron transfer would proceed with the collision of the excited RuH2+and BIH, the minimum distance between them would be [ r= (7.1 + 5.3) Å], and kesccan be estimated as 2.5×109s−1. DISCUSSION The quantum yield for the formation of OERS strongly depends on both the central metal ion and the ligands (Table III). Scheme 3 illustrates the elementary processes in the photochemical reduction ofPS2+with BIH as a reductant, in which kxis the rate constant for each process. There are three primary causes to lower the ΦOERS : (1) competition with the radiative and non-radiative decays ( krandknr), which lowers the reductive quenching efficiency ( ηqin Table III) of the excited state of PS2+(∗PS2+) by BIH, for which the rate constant iskq. (2) Back-electron transfer from the OERS of PS2+(PS•+) to the one-electron oxidation species of BIH (BIH •+) inside the sol- vent cage ( krec1), in which PS•+and BIH •+are a solvated radicalion pair ([ PS•+⋯BIH•+]) formed just after the photochemical elec- tron transfer. (3) Another back-electron transfer through the dif- fusion collisions between the free PS•+and the free BIH •+(krec2), which forms through the escape from the solvation cage ( kesc). This back-electron transfer process competes with the deprotonation process from BIH •+(kdp), which makes BI •another electron donor toPS2+in the ground state ( ket). We separately investigate these competitive processes below. Figure 15 shows the relationship between kqand the free energy change of a photochemical electron transfer from BIH to∗PS2+ (ΔGpet, Table III). In the case of RuCO 2Me2+,kqwas the largest at 6.4×109M−1s−1. Since this rate is similar to the diffusion rate, which was estimated using Eq. S1 (supplementary material) as 7.0×109M−1s−1, the electron transfer rate between∗PS2+and BIH should be diffusion controlled for RuCO 2Me2+. Since the electron transfer proceeded slower if the −ΔGpetvalue was smaller in the other cases, electron transfer itself should be a rate limiting pro- cess of these quenching reactions of∗PS2+. The efficiency of this reductive quenching process [quenching efficiency, ηq, in Table IV) should directly affect ΦOERS (total)]. In other words, its effect can be removed from the formation efficiency of PS•+by dividing ΦOERS (total) with ηq(Table IV). As shown in Scheme 3, there are two formation processes for the OERS; the photochemical formation through the reductive quenching of the excited complex by BIH, and the following reduc- tion process of the ground state of another complex molecule by BI •. As BI•, the reductant of the latter process is produced by the former photochemical process, the amount of OERS produced by the latter process must be equal or be less than that for the former photochem- ical process. The time-conversion curves of the absorbance at λdet= 520 nm and 530 nm for RuH2+andOsH2+, respectively, are shown in Fig. 16. The OERSs of RuH2+andOsH2+have a strong absorp- tion, whereas the other transient species (the3MLCT excites states, SCHEME 3 . Elementary processes in the photochemical reduction of PS2+with BIH. J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-8 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 15 . Relationship between the reductive quenching rate constant ( kq) and the free energy change of photochemical electron transfer from BIH to∗PS2+(ΔGpet). TABLE IV .ΔA1/ΔA2,ΦOERS(pe) /ηq, andΔGbet. Complex ΔA1/ΔA2 ΦOERS(pe) /ηq −ΔGbet(eV)a RuCO 2Me 0.52 0.85–0.88 1.21 RuH 0.51 0.55–0.56 1.65 RuMe 0.52 0.50–0.52 1.75 RuOMe 0.53 0.50–0.53 1.81 OsH 0.51 0.11 1.57 OsMe ... ∼0.05 1.68 a−∆Gpet=−E1/2(PS2+/PS•+) + E1/2(BIH•+/BIH)−wp+ w r:E1/2(BIH•+/BIH) =−0.02 V vs Ag/AgNO 3,E1/2(PS2+/PS•+) is the first reduction potential of the complex (Table II), wr= 0.03 eV, wp= 0 eV. BIH•+, and BI •) have much lower absorption (Figs. 12 and 13). This clearly indicates that the amount of the OERS produced by the second process is similar to the former process, i.e., ΔA1 (absorption changes during irradiation of laser pulse) was about FIG. 16 . Absorbance changes at 520 nm ( RuH2+, left) and at 530 nm ( OsH2+, right) in the time-resolved visible absorption experiments ( λex= 532 nm) to the N,N-dimethylacetamide solution containing the complex (0.3 mM) and BIH (0.1M). ΔA1andΔA2mean the absorption change during the irradiation and the total absorption change, respectively.half of ΔA2(total absorption change); ΔA1/ΔA2= 0.52 ( RuH ) and 0.51 ( OsH ). This means that the back-electron transfer process from the free OERS of the complex to the free BIH •+was much slower than the deprotonation process from BIH •+even at relatively high concen- trations of both the OERS and the free BIH •+at several nanoseconds after the laser excitation. In the steady-irradiation using a Xe lamp, accumulation of the OERS was proportional to the irradiation time in the initial stage (Figs. 6 and 7 and Fig. S6). This also supports that the deprotonation from the free BIH •+was much faster than the back-electron transfer process in this stage, as the increase in the concentration of the OERS did not have a strong effect on the formation speed of the OERS, even though this increase should be faster than the speed of the back-electron transfer. Therefore, con- tribution of the back-electron transfer from the free OERS of the complexes to the free BIH •+should be negligible in the photochem- ical reactions. In other words, the quantum yield for the forma- tion of the OERS by the photochemical electron transfer process from BIH to the excited complex {including the separation pro- cess of the ion pair [ ΦOERS(pe) ]} should be almost half the value of the total ΦOERS (total) obtained in the steady-irradiation experi- ments (Table IV). A similar method was applied to the other com- plexes (except for OsMe2+), and the ratios between ΦOERS(pe) and ΦOERS(total) were similar, i.e., ΔA1/ΔA2was∼0.51–0.53 (Table IV). In the case of OsMe2+,ΦOERS(pe) could not be determined from its ΦOERS(total) from the TR-AB spectra owing to the low ηqof the OERS ofOsMe2+. However, it is reasonable that the lower yields for the OERS and BIH •+in the case of OsMe2+should be lower than the contribution of the back-electron transfer from the OERS to BIH •+, i.e., the ratios between ΦOERS(pe) andΦOERS(total) should be close to 0.5:1. In the case of RuH2+as a typical example, a more detailed investigation was performed on both the results of the TR-AB spectroscopy using the Nd/YAG laser light and the steady irradi- ation using the Xe lamp. At first, the TR-AB spectra were ana- lyzed using a multivariate curve resolution with the alternating least squares method (MCR-ALS), in which the spectrum obtained by DFT [Fig. 14(a)] was used as the initial spectrum of BI •, with the spectra of the3MLCT excited state (∗PS2+) and the OERS ( PS•+) obtained using TR-AB (Fig. S11) and the flow electrolysis methods [Fig. 5(a)]. All of these spectra are illustrated in Fig. 17(a). Changes in the relative concentrations of each active species were obtained by using the absorption changes at wavelengths in which each species had a strong absorption, i.e., 470 nm for∗PS2+, 520 nm for PS•+, and 650 nm for BI •[Figs. 17(b) and 17(c)]. The absorption spec- trum of BIH •+was not added to this fitting because its absorption is weak in the visible region [Fig. 14(b)]. The results are shown in Figs. 17(d)–17(f). The similarities of the spectra of∗RuH2+andRuH•+between the experimental results and the fitting ones [Figs. 17(a) and 17(d)] suggest that the fitting procedure went well. Although the spec- trum of BI •obtained by the fitting is broader compared to the calculated spectrum, the absorption maxima were similar. The time- conversion curves of∗RuH2+andRuH•+clearly show that the reductive quenching finished within 10 ns after the laser flash to pro- duce RuH•+, which is in good agreement with the emission decay measurement in the presence of BIH. After this initial period, BI • formed up to 200 ns and RuH •+increased. After this, BI •decreased, J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-9 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 17 . Initially used spectra of∗RuH2+,RuH•+, and BI •(a), time-conversion curves of absorbances (b), and (c) final results of the normalized concentration change and the absorption spectra optimized by using the multivariate curve resolution with the alternating least squares calculation of the observed transient absorption spectra (d)–(f). while RuH•+continuously increased up to 6 μs. These results indi- cate the time scales of the reductive quenching of the3MLCT excited state of RuH2+by BIH, deprotonation from BIH •+, and the electron transfer from the free BI •to another RuH2+molecule in the ground state, respectively. Based on Scheme 3, the concentration changes of the active species after laser irradiation can be described by the set of the differential equations shown as follows: d[PS∗] dt=−(kr+knr+kq[BIH])[∗PS2+], (7) d[PS●+⋯BIH●+] dt=−(kesc+krec1)[PS●+⋯BIH●+] +kq[BIH][∗PS2+], (8) d[BIH●+] dt=kesc[PS●+⋯BIH●+] −(kdp[BIH]+krec2[PS●+])[BIH●+], (9) d[BI●] dt=kdp[BIH][BIH●+]−ket[PS2+][BI●], (10) d[PS●+] dt=kesc[PS●+⋯BIH●+]+ket[PS2+][BI●] −krec2[BIH●+][PS●+], (11)in which kr+knr=τem−1. In Eqs. (10) and (11), BIH was included as the base species in the deprotonation process. The set of rate equations was numerically solved under the initial conditions and the binding conditions. As the sum of the RuH2+and its derivative species should be preserved within the entire time range, the equa- tions can be solved under the binding conditions of the following: [PS2+]0=[PS2+](t)+[∗PS2+](t)+[PS●+⋯BIH●+](t). (12) The initial concentration of the excited species ([∗PS2+]0) was evalu- ated from the irradiated laser intensity ( Iexc= 0.14 E m−3at 532 nm) (Fig. S10) in the absorbance of the sample solution at the excitation wavelength ( Aexc). The initial concentration of the other transient species was assumed to be zero [Eq. (13)], [∗PS2+](0)=Iexc(1−10−Aexc). (13) The transient absorption spectrum observed after a sufficiently long period (>8μs) after laser excitation coincides with PS•+observed by the flow-electrolysis method (Fig. 13). It was confirmed that only PS•+existed for a longer period after laser excitation and the concentration of PS•+could be evaluated from the transient absorption. Under the initial and binding conditions, the concentration changes that were estimated using the MCR-ALS method were ana- lyzed by the nonlinear least square model fitting, in which the calcu- lation of the set of rate equations was conducted using the deSolve package (ver. 1.27.1) on GNU R (version 3.6.0). The fitting result was shown in Fig. 18. Generally, the fitting results are in good agree- ment with the experimental results except for BI •in the initial stage. J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-10 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 18 . Estimated concentration change by multivariate curve resolution with the alternating least squares method (dots) and the nonlinear least square model fitting results by deSolve on R (solid lines) of the transient species in the time-resolved visible absorption experiment of RuH2+with BIH. This could be caused by the effect of BIH •+as its absorption change was not included into the fitting owing to its low absorption in this wavelength region. In the initial stage, however, the concentration of BIH•+should be relatively high, with a weak absorption at around 700 nm [Fig. 14(b)]. A similar fitting process could be applied to OsH2+(Fig. S13). The concentration of the transient species, i.e.,∗OsH2+, [OsH•+⋯BIH•+],OsH•+, BIH•+, and BI •over time is shown in Fig. 19. The rate constants including the values obtained using the fit- ting results ( kdp,krec1,krec2,ket) are summarized in Table V. The rate constants related to the excited state decay ( kr,knr,kq) were employed from the experimental values obtained by the emission decay measurement and the quenching experiment. Notably, the kdp values were similar between the RuH andOsH systems. This also indicates the reliability of the analyses of the TR-AB results. The escape rate constant of the ions from the solvated radical ion pair [RuH•+–BIH•+] was estimated using the Eigen equation [Eq. (3)]. Notably, the OERS species of all the complexes and BIH •+have one plus charge; therefore, electrostatic repulsion should occur between the OERS and BIH •+, and if the distance between the ions is large, the escape rate drastically becomes slower (Fig. 15). In the case ofOsH2+, estimation of the maximum escape rate constant from FIG. 19 . Estimated concentration change by multivariate curve resolution with the alternating least squares method (dots) and the nonlinear least square model fitting results by deSolve on R (solid lines) of the transient species in the time-resolved visible absorption experiment of OsH2+with BIH.TABLE V . Estimated rate constants by the kinetic analysis. [BIH] = 0.1 M, the solvent isN,N-dimethylacetamide. Rate constant (s−1) RuH2+OsH2+ kr+knr+ k q[BIH] 4.0 ×1086.4×107 kesc ≤2.5×109≤2.5×109 krec1 (0.78–0.82) ×kesc 8.1×kesc kdp[BIH] 6.1 ×1064.0×106 krec2 9.5×1099.5×109 ket 2.7×1093.9×109 [OsH•+–BIH+•] gave a similar value to that of [ RuH•−–BIH•+] (kesc≤2.5×109s−1) owing to the similar sizes of the OERS species. In the case of RuH2+, the ΦOERS(pe) /ηqvalue was 0.55–0.56 (Table IV); thus, the rate constant of the back-electron transfer in the solvated radical ion pair ( krec1) can be estimated as ∼(45/55– 44/56)×kesc, i.e., krec1≤2.1×109s−1. The investigation of OsH2+ gave krec1= 89/11×kesc. Both processes are highly exergonic reac- tions: ΔGbet=−1.68 eV ( RuH2+) and 1.60 eV ( OsH2+), i.e., krec1 ≤2.0×1010s−1. As the deprotonation process from BIH •+ (kdp[BIH•+] = 6.1×106s−1) was much slower compared to the other processes of the solvated ion pairs, contribution of this pro- cess should be negligible for the separation process of the ion pair and back-electron transfer in the solvated ion pair. In other words, deprotonation from BIH •+mostly proceeds through collisions with other BIH molecules as a base after each ionic species was sepa- rately solvated. It was reported that the second formation process of the OERS species of the complexes, i.e., electron transfer from BI•to metal complexes in the ground state, proceeded quantita- tively in the photocatalytic CO 2reduction reactions, in which the complexes worked as a redox photosensitizer or a catalyst and BIH as a reductant. Although the Gibbs free energy changes of the sec- ond electron transfer process ( −∆Get2), which are calculated using the reduction potentials of the complexes in DMA [ E1/2(PS2+/PS•+) =−1.11 V∼−1.80 V vs Ag/AgNO 3, Table II] and the reduction potential of BI+[Ep(BI+/BI•) =−2.05 V vs Ag/AgNO 3] with the Rehm–Weller equation [Eq. (14)], are different ( ∼0.2–∼0.8 eV), all were exergonic; therefore, this second electron transfer should quan- titatively or almost quantitatively proceed with high speed in all the cases, e.g., ket= 2.7×109s−1:RuH2+, 3.9×109s−1:OsH2+(Table V). This is supported by the fact that all ΔA1/ΔA2values were closed to 0.5 (Table IV), −ΔGet=−E1/2(BI+/BI●)+Ered(PS2+/PS●+)−wp+wr, (14) in which wrandwpwere 0 eV and 0.03 eV, respectively. Although another back-electron transfer process through colli- sions of the separately solvated OERS of the complexes and BIH •+ was an almost diffusion controlled reaction owing to the high −ΔGbet values in all the cases (Table IV), its substantial rate should be much slower compared to the deprotonation process from BIH •+because the concentration of BIH as the base ( ∼0.1M) was much higher when compared to the maximum accumulated concentration of the OERS J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-11 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of the complexes in the TR-AB experiments [for example, [ RuH•+] <25×10−6M (Fig. 19)]. This should be the main reason why this back-electron transfer process did not contribute to the formation quantum yield of the OERS of the complexes. It is interesting to see the effects of solvent on the rate con- stant of the deprotonation process from BIH •+. Many reports describe photocatalytic CO 2reduction using BIH as a reduc- tant in mixed solutions of N,N- dimethylformamide (DMF)– triethanolamine (TEOA) or DMA–TEOA (typically 5:1 v/v). We have already reported that, in the DMF–TEOA (5:1 v/v), includ- ing 0.1M of BIH, the rate constant of the deprotonation process (kdp) from BIH •+was 1.1×105M−1s−1,22which is much smaller than that in DMA including 0.1M of BIH ( kdp= 6.1×107M−1s−1). As DMA and DMF have similar properties such as basicity, TEOA should suppress the deprotonation from BIH •+. This is a surpris- ing result because many reports, including ours, describe that TEOA was added as a base for accelerating deprotonation from the oxida- tion products of the reductants (including BIH) and suppressing the back-electron transfer from the reduced PS and/or the reduced cat- alyst to the oxidation product of the reductant. To the best of our knowledge, this is the first evidence that TEOA suppresses depro- tonation from the oxidation product of the reductant. However, the addition of TEOA improved the efficiencies of photocatalytic reactions in many reported systems probably because of a different role(s). We reported that in photocatalytic systems with a Re(I) com- plex or a Mn(I) complex as a catalyst, a deprotonated TEOA works as a ligand of the complexes and captures CO 2into the Re–O bond even at low concentrations of CO 2.28,29 The free energy change of the back-electron transfer from PS•+to BIH •+(ΔGbet) is expected to affect the ΦOERS(pe) /ηqvalue because the back-electron transfer process in the solvated radical ion pair [ PS•+⋯BIH•+] competes with the accumulation of the free PS•+to lower ΦOERS(pe) (Table IV). The relationship between ΔGbet andΦOERS(pe) /ηqfor which the minimum value is ΦOERS(total) /ηq ×1/2 and the maximum is ΦOERS(total) /ηq×ΔA1/ΔA2is plotted in Fig. 20. With the same central metal ion, the system with larger −ΔGbettended to have a lower ΦOERS(pe) /ηq. On the other hand, FIG. 20 . Relationship between ΔGbetandΦOERS(pe) /ηqfor which the minimum value is ΦOERS(total) /ηq×1/2 and the maximum is ΦOERS(total) /ηq×ΔA1/ΔA2.the difference in the central metal ion more drastically affected the ΦOERS(pe) /ηqvalues even in cases with similar ΔGbet, i.e., OsH2+ [ΦOERS(pe) /ηq= 0.11, ΔGbet=−1.60 eV] and OsMe2+[ΦOERS(pe) /ηq ∼0.05 ,ΔGbet=−1.71 eV] vs RuH2+[ΦOERS(pe) /ηq= 0.55–0.56 , ΔGbet=−1.68 eV]. Much higher ΦOERS(pe) /ηqvalues in the case of RuH2+compared to those of OsX2+(X = H, Me) clearly indicate that there should be factors other than ΔGbetthat strongly affect ΦOERS(pe) /ηq. As previously described, the free BIH •+was almost all depro- tonated to BI •, which reduced the complex in the ground state quantitatively (Schemes 2 and 3) in all cases. Therefore, the differ- ence in the ΦOERS(pe) /ηqvalues between RuX2+andOsX2+should be caused by the competitive processes between the back-electron transfer from the OERS of the complex to BIH •+in the radical ion pair and the escape of the solvated radical ion pair forming free OERS and BIH •+, i.e., the back-electron transfer was more favor- able in OsX2+, whereas these two processes are in competition for RuX2+, excluding RuCO 2Me2+in which the escape processes pro- ceed much faster than back-electron transfer. For example, krec1/kesc forRuH2+andOsH2+are 0.8 and 8.1, respectively. Notably, the charges, shape, and ΔGbetforRuH2+andOsH2+are the same or similar. In general, there are two possible mechanisms to induce this difference between RuX2+andOsX2+. One difference is the heavy- atom effect of the central metal ion. In the solvated radical ion pair, the OERS of the metal complex and BIH •+interact with each other; therefore, unpaired electrons mainly located on the diimine ligand of the OERS of the metal complex and the nitrogen atom of BIH •+can- not behave independently. As the photochemical electron transfer proceeds from BIH in the singlet ground state to the3MLCT excited state of the metal complex, the ion pair should have a triplet-state character at least in the initial stages: in this3[PS•+–BIH•+], back- electron transfer is spin-forbidden. As dictated by quantum chem- istry, spin-forbidden reactions can be partially allowed by intro- ducing large spin–orbit coupling, and the larger heavy-atom effect induces larger spin–orbit coupling. The Os atom has a stronger heavy-atom effect compared to the Ru atom (spin–orbit coupling constants are 3531 cm−1and 1081 cm−1, respectively).30Therefore, we can assume that the stronger heavy-atom effect of the central Os(II) ion accelerates the “spin-forbidden” back-electron transfer. It was reported that in the photochemical reduction of metal por- phyrins by benzo-1,4-quinone, the back-electron transfer rate from the reduced porphyrins with a heavy metal, such as Ru ( kbec1= 5 ×1011s−1), was faster than those with a light metal, such as Mg, Zn, and Al (∼8×1010s−1).31 Another mechanism to explain the difference is the difference in the distance between the OERS of the metal complex and BIH •+ in the solvated radical ion pair just after photochemical electron transfer from BIH to the excited metal complex. There are many reports about the rates of electron transfer from various electron donors to the excited Ru(II) diimine complexes. They highlight the difference in electron-transfer distances causing deviation in the experimental results using the Marcus theory for electron transfer. In other words, electron transfer can proceed between a donor and an acceptor at a longer distance, if the −ΔGpetvalue of the electron transfer is larger. In the case of RuH2+,−ΔGpetof the photochemical electron transfer from BIH is much larger than those in the cases J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-12 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ofOsH2+andOsMe2+(Fig. 16, Table III); therefore, the distance between the excited RuH2+and BIH in the formed complex should be larger than those in the cases of the Os complexes. If this dif- ference in the distance reflects the distance between the OERS of the complex and BIH •+in the solvated radical ion pair produced by the photochemical electron transfer, the back-electron transfer in [RuH•+⋯BIH•+] is probably slower than in [ OsX•+⋯BIH•+] (X = H, Me) owing to the larger distance and the similar −ΔGbet. At this stage, we cannot identify how these mechanisms con- tribute to the efficiency difference of the photochemical reduc- tion between RuX2+andOsX2+. However, we can generally point out that efficiencies of various redox photosensitized reactions and redox photocatalytic reactions can be drastically improved by increasing the oxidation power of the excited photosensitizer and/or avoiding using a central metal (ion) with a strong heavy-metal effect. EXPERIMENTAL SECTION General procedures and synthesis 1H nuclear magnetic resonance (NMR) spectra were measured in chloroform- dusing a JEOL ECA400II (400 MHz) system. CVs were measured using Et 4NBF 4(0.1M) as a supporting electrolyte under an Ar atmosphere using a BAS CHI620EX or CHI760Es elec- trochemical analyser with an Ag/AgNO 3(0.01M) as a reference electrode, a Pt counter-electrode, and a glassy-carbon working elec- trode (diameter, 3 mm) at a scan rate of 200 mV s–1or a micro glassy-carbon electrode (diameter, 33 μm) at a scan rate of 2 V s–1. DMA was dried over a 4 Å molecular sieve and distilled at reduced pressure. TEOA was distilled at a reduced pressure. Tetraethylammonium tetrafluoroborate (Et 4NBF 4) was recrystal- lized twice from acetonitrile ethyl acetate and dried in vacuo at 100○C overnight just before use. All other reagents were of reagent- grade quality and were used without further purification. The reduc- tant, BIH,20,32and PF 6−salts of RuH2+,33RuMe2+,33RuCO 2Me2+,34 RuOMe2+,35OsH2+,36andOsMe2+37were synthesized according to the references. Measurements of photophysical properties UV–Vis absorption spectra were measured with a JASCO V-670 spectrophotometer. Emission spectra were measured by HORIBA Fluorolog-3 and Fluorolog-NIR spectrofluorometer and the emission quantum yield by a quantum yield analyzer Quantaurus-QY Plus C13534-01 (HAMAMATSU) with a 150 W Xenon arc lamp light source (400 nm) and a A10080- 01 monochrometer. Emission lifetimes were measured using a HORIBA Jobin-Yvon FluoroCube time-correlated single photon counting system. The sample solutions for the emission decay and quantum yield measurements were degassed for 20 min under Ar prior to the measurements. Emission-quenching experiments were performed in Ar-saturated solutions containing a complex and BIH. Quenching rate constants, kq, were obtained from the slopes of Stern–Volmer plots of the luminescence intensity against the concentration of BIH. The steady-state IR absorption spectra were measured with a JASCO FT/IR-610 IR spectrometer. In situUV–Vis absorption spectral changes under photo-irradiation con- ditions were measured by a photodiode array spectrometer Photal MCPD-9800 (Ohtsuka Electronics Co., Ltd.). Flow-electrolysis UV–Vis and IR absorption spectra under bulk electrolysis con- ditions were measured by the flow electrolysis apparatus as fol- lows:23An ALS/CHI BAS CHI-720D electrochemical analyzer was employed as a potentiostat and the absorption spectra were moni- tored by the photodiode array spectrometer Photal MCPD-9800 for UV–Vis, and the JASCO FT/IR-610 IR spectrometer for IR mea- surement. The sample solutions were prepared as a DMA solution containing 0.1M Et 4NBF 4as a supporting electrolyte, which was bubbled with Ar for over 40 min prior to the measurement. The sample solution was transferred at a flow rate of 0.2 ml min−1–0.5 ml min−1using a JASCO PU-980 HPLC pump into a VF-2 elec- trolysis cell (EC Frontier) and then a quartz flow cell (1.5 mm light path) or an IR flow cell with CaF 2windows (1.5 mm light path). The flow electrolysis cell was constructed with a carbon-felt action electrode, an Ag/AgNO 3(0.01M) reference electrode, and a Pt wire counter electrode. The number of electron(s) ( ne), which were con- sumed in the reduction process of the complex, was evaluated using Eq. (15). The reduction current ( i) was calculated from the plateau of the current vs loaded potential plots of the solution, and the background currents ( i0) were evaluated from the plots of the elec- trolyte solution without the complex. F,c, and vare the Faraday constant, the concentration of the metal complex and the flow rate, respectively, ne=i−i0 cvF. (15) TR-AB spectroscopy TR-AB spectra were measured by a transient absorption spec- troscopy system picoTAS-ns (UNISOKU Co., Ltd.). The excitation light source was a passive Q-sw micro-chip laser (532 nm, pulse duration 350 ps FWHM, 10 Hz operation, pulse energy 22 mJ pulse−1). The excitation laser power was measured by a thermal laser power sensor USP-S401C (ThorLabs, Inc.). The sample solu- tion for the TR-AB measurement was bubbled with Ar for 20 min prior to the measurements and was circulated at a flow rate of 6 ml min−1using a bimorph pump BPF-465P (NITTO KOHKI Co.) path through a quartz optical flow cell (2 ×2 mm2light path). Franck–Condon analysis and determination of the free energy change of the photochemical electron transfer The Gibbs free energy changes (– ΔGpet) of the photoinduced electron transfer process between the excited PSand BIH were eval- uated using the redox potentials, the excited state energies, and the following equations: −ΔGpet=−E0(BIH●+/BIH)+E1/2(∗PS2+/PS●+)−wp+wr, (16) E1/2(∗PS2+/PS●+)=E1/2(PS2+/PS●+)+E00. (17) J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-13 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The Coulombic terms, wrandwp, correspond to the reactant and the product ion pairs in the electron transfer process, respectively, and were calculated using wr=zBIHz∗PSe2 dcεS=0, (18) wp=zBIH+ ●zPS− ●e2 dcεS, (19) where ziis the charge of the donor and acceptor species i, eis the elementary charge, ε0is the permittivity of vacuum, εS= 38.2 (the solvent dielectric constant of DMA), and dcis the distance between the centers of the electron donor and acceptor pair. In the case of the reductive quenching of the excited RuH2+by BIH, wr= 0 because the charge of BIH was zero ( zBIH= 0). The distance between RuH2+ and BIH ( dc) was calculated by the sum of these radii 7.1 Å for RuH2+and 5.3 Å for BIH in the energy-minimized structure of the ground state molecule calculated by the MM2 force field calculation using Chem3D v. 12.0, Cambridge Soft Corporation. Substituting these parameters into Eq. (S4), the value of wpwas evaluated to be 0.03 eV. This wpvalue was used as an approximate for the other sys- tems using RuX2+andOsX2+. The experimental error owing to this approximation should be small owing to the low value of wpand the similar molecular sizes of the complexes. The 0–0 band energy gaps between the3MLCT excited state and the ground state ( E00) were obtained from a simulation of the emission spectrum using the double-mode Franck–Condon line-shape analysis by the following equation:38 I(˜ν)=∑5 νM=0 νL=0(E00−νMhωM−νLhωL E00)(SνM M νM!)(SνL L νL!) ×exp⎡⎢⎢⎢⎢⎣=4 log 2(˜ν−E00−νMhωM−νLhωL ˜ν1/2)2⎤⎥⎥⎥⎥⎦, (20) where I(˜ν) is the relative emission intensity at the wavenumber ˜ν,E00 is the 0–0 band gap energy, μMandμLare the average of medium and low-frequency acceptor modes, respectively, that are coupled to the electronic transition, SMandSLare the electron-vibration cou- pling constant (Huang–Rhys factor) of medium and low-frequency acceptor modes,39respectively, and ˜ν1/2is the half-width of the 0–0 vibrational band. For the precise E00evaluations (Table I), the emission spectra were measured in a DMA solution at 77 K using the liquid nitrogen cryostat (CoolSpeK USP-203S, UNISOKU Co., Ltd.) and simulated using Eq. (S5) (Fig. 3). The evaluated E00values and the excited state reduction potentials [ E1/2(∗PS2+/PS•+)] are summarized in Table II. Measurement of the formation quantum yield of PS •+under steady-state light irradiation The sample solution was prepared as an Ar saturated DMA solution containing the metal complex and 0.1M of BIH. The irra- diation light sources were a Xe arc lamp for 480 nm and a high- pressure mercury lamp for 436 nm, which were fitted with a heatremoving filter, a monochromatic band-pass filter, and/or a proper neutral density filters (ASAHI SPECTRA). The sample temperature was controlled at 25 ±0.1○C using an IWAKI CTS-134A tem- perature bath. The amount of the produced OERS was evaluated using the absorption spectral change of the sample solution dur- ing light irradiation. The irradiation light intensity was measured by a K 3Fe(C 2O4)3chemical actinometer.40The quantum yield for the formation of the OERS [ ΦOERS(pe) ] is calculated as ΦOERS=amount of formed OERS (mol) mumber of absorbed light qanta (E). (21) QUANTUM CHEMICAL CALCULATION The calculations were performed by the Gaussian 16 program41 using the UB3LYP exchange correlations functional. Geometry opti- mizations were performed in DMA using a general basis set of 6-311G++(d,p). The geometries were fully optimized without sym- metry constraints. TDDFT calculation of the excited states was also performed in a DMA solution using the same 6-311G++(d,p) basis set used for geometry optimizations. Multivariate curve resolution by the alternating least squares method (MCR-ALS) of the TR-AB spectra In order to analyze the formation processes of OERS of RuH2+ andOsH2+in detail, transient absorption spectra of DMA solutions containing the metal complex and BIH (Fig. 10) were investigated as follows. The observed transient absorption in the longer wave- length region (550 nm–700 nm) continued to increase from 0 ns to 500 ns after laser excitation and then decayed monotonously. This excitation and decay behavior could not be reproduced only by the absorption contributions of the photoexcited state and the OERS state of the metal complex. An additional contribution of other intermediate species such as BI •is required. The absorption spectrum of the BI •was estimated by the quantum mechanical cal- culation for absorption bands in this region [Fig. 13(a)]. The TR-AB spectra were analyzed as a MCR-ALS.42–46Absorbance A(λ,t) at wavelength λand time twas expressed by the following equation according to Lambert–Beer’s law: A(λ,t)=l∑iεi(λ)Ci(t), (22) in which λ,εi(λ), and Ci(t) are the light path length, the molar extinc- tion coefficient, and the concentration of the component i, respec- tively. The transient absorption was observed at a series of discrete wavelengths λi(i= 1, 2, ...,m) and time ti(i= 1, 2, ...,n), and the spectral data can be tabulated as the n×mmatrix [Eq. (23)], A=⎛ ⎜ ⎝A(λ1,t1)⋯A(λm,t1) ⋮ ⋱ ⋮ A(λ1,tn)⋯A(λm,tn)⎞ ⎟ ⎠ =l⎛ ⎜ ⎝C1(t1)⋯CN(t1) ⋮ ⋱ ⋮ C1(tn)⋯CN(tn)⎞ ⎟ ⎠⎛ ⎜ ⎝ε1(λ1)⋯ε1(λm) ⋮ ⋱ ⋮ εN(λ1)⋯εN(λm)⎞ ⎟ ⎠=lCST, (23) J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-14 © Author(s) 2020The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp in which Ci(tj) andεi(λk) are the concentration at tjand the extinc- tion coefficient at λkof the component I, respectively, and Nis the number of the components. The error between the observed absorp- tion spectra (A l−1) and the calculated values of the concentrations and the extinction coefficients ( CST) were defined by the Frobenius norm [Eq. (24)], f(C,S)=∥Al−1−CST∥2 F. (24) Starting from the initial guess of the C and S, iterative calculation was conducted to minimize the Frobenius norm (ALS method) using the ALS package (version 0.0.6) on GNU R (version 3.6.0). For the fit- ting calculation, observed absorbance changes occurred at the wave- lengths (650 nm, 520 nm, and 470 nm), which were as expected for the absorption maxima of the transient species (∗PS2+,PS•+, BI•, respectively). SUPPLEMENTARY MATERIAL See the supplementary material for emission decays, relation- ships between redox potentials of the complexes and Hammett sub- stituent constants, flow electrolysis, CVs of BIH and BI+, TR–Vis, emission quenching, and absorption spectrum of the excited state. ACKNOWLEDGMENTS This work was supported by JSPS KAKENHI Grant Nos. JP17H06440 in Scientific Research on Innovative Areas “Inno- vations for Light-Energy Conversion (I4LEC),” JP20K20367, and JP20H00396. The authors thank Professor Kazuyuki Ishii (Univer- sity of Tokyo) for useful discussion. 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Sexena, Handbook of Atomic Data (Elsevier Scientific Publishing Company, 1976). 31A. Harriman, G. Porter, and A. Wilowska, J. Chem. Soc., Faraday Trans. 2: 79, 807 (1983). 32E. Hasegawa et al. , J. Org. Chem. 70, 9632 (2005). 33P. A. Mabrouk and M. S. Wrighton, Inorg. Chem. 25, 526 (1986). 34M. Schwalbe et al. , Eur. J. Inorg. Chem. 2008 , 3310. 35Y.-R. Hong and C. B. Gorman, J. Org. Chem. 68, 9019 (2003). 36E. C. Constable, P. R. Raithby, and D. N. Smit, Polyhedron 8, 367 (1989). 37S. R. Johnson et al. , Inorg. Chem. 27, 3195 (1988). 38G. H. Allen et al. , J. Am. Chem. Soc. 106, 2613 (1984). 39K. Huang, A. Rhys, and N. F. Mott, Proc. R. Soc. London, Ser. A 204, 406 (1950). 40C. G. Hatchard and C. A. Parker, Proc. R. Soc. London, Ser. A 235, 518 (1956). 41M. J. Frisch et al. , Gaussian 16, Revision B.01, Gaussian, Inc., Wallingford, CT, 2016. 42H. Mori, M. Kuroda, and K. Adachi, Least Square Method and Alternating Least Square Method (Kyoritsu Shuppan, Co., Ltd., 2017). 43C. Ruckebusch et al. , J. Photochem. Photobiol. C 13, 1 (2012). 44L. R. Terra, M. N. Catrinck, and R. F. Teófilo, Chemometrics Intell. Lab. Syst. 167, 132 (2017). 45I. H. M. van Stokkum, K. M. Mullen, and V. V. Mihaleva, Chemometrics Intell. Lab. Syst. 95, 150 (2009). 46H. Motegi et al. , Sci. Rep. 5, 15710 (2015). J. Chem. Phys. 153, 154302 (2020); doi: 10.1063/5.0023593 153, 154302-15 © Author(s) 2020

5.0016756.pdf

AIP Conference Proceedings 2265 , 030001 (2020); https://doi.org/10.1063/5.0016756 2265 , 030001 © 2020 Author(s).Study of local structure of Sr0.95Pr0.05TiO3 thin film: Absence of antiferrodistortive phase Cite as: AIP Conference Proceedings 2265 , 030001 (2020); https://doi.org/10.1063/5.0016756 Published Online: 05 November 2020 Vivek Dwij , Binoy Krishna De , Mukul Gupta , Niti, and V. G. Sathe ARTICLES YOU MAY BE INTERESTED IN Structural dielectric and impedance spectroscopy of YBaCuFeO 5 AIP Conference Proceedings 2265 , 030002 (2020); https://doi.org/10.1063/5.0016649 Magnetic modulation enhanced magneto-dielectricity in CaMn 7O12 AIP Conference Proceedings 2265 , 030008 (2020); https://doi.org/10.1063/5.0017757 Nanoscopic diffusive dynamics in bio-mimetic membrane systems AIP Conference Proceedings 2265 , 020001 (2020); https://doi.org/10.1063/5.0017330Study of Local Structure of Sr 0.95Pr0.05TiO 3Thin Film: Absence of Antiferrodistortive Phase Vivek Dwija), Binoy Krishna De, Mukul Gupta, Niti, V.G. Sathe 1UGC-DAE Consortium for Scientific Research, Univers ity Campus, Khandwa Road, Indore 452001, India. a) Corresponding author: vivekd@csr.res.in Abstract. 5% Pr doped SrTiO 3 thin film has been synthesized using Pulse Laser deposition. X ANES and Raman spectroscopic measurements were carried out to investigate the local structure of the thin film. The Raman results revealed absence of Antiferrodistortive (tetragonal) phase in t he synthesized thin film till 80K in contradiction to the corresponding bulk compound. This effect has been attributed to the strong polar order (FE) emerging from Ti off- centering which competes and canc els Antiferrodistortive (AFD) instability arising from octahedral tilt (present in the parent SrTiO 3). Dominance of polar order occurring due to larger Ti off-cent ering has been conformed using Raman spectroscopy and XANES studies carried out on the film. INTRODUCTION Relaxor ferroelectrics have been extensively investigated due t o its large scale applications in devices. Disorder is the key to achieve relaxor ferroelectric behavior. These mat erials show broad dielectric maxima and remarkable frequency dispersion which in turn allows temperature stability in a broad range as well as enhanced characteristic properties for practical applica tions. The mechanism behind thi s large maxima is ascribed to dynamics of nano scale polar regions known as PNRs.SrTiO 3 being incipient ferroelectric is extremely susceptible to exte rnal perturbations and thus chemical doping in this compound provides ideal condit ions for investigating role of the disorder. SrTiO 3 undergoes antiferrodistortive (AFD) structural transition at 10 5 K due to tilting of TiO 6 octahedral [1]. Below AFD transition, structure of SrTiO 3 changes from cubic to tetragonal resulting in activations of f irst order E g a n d B 1g Raman modes.Our recent investigation [2] suggests that due to i ncorporation of smaller Pr atom, TiO 6 octahedra can undergo larger tilt which increas es the AFD transition temperat ure to room temperatur e (7.5% Pr doping). Interestingly, SrTiO 3shows giant photoconductivity in tetragonal symmetry [3]. This photoconductivity can be utilized for solar cell applications. Pr doping leads to polari zation in the system making Pr doped SrTiO 3 possible candidate for photo-ferroelectric material. However, for larger Pr content, the polarization values decrease significantly due to commencement of global AFD above room temp erature. Therefore, we choose 5% Pr doped SrTiO 3, which has AFD transition around 260 K and has significant fer roelectric polarization values to utilize the near room temperature polarizati on for solar cell applications. Therefore, we synthesized the thin film of the 5% Pr doped SrTiO 3 (SPT5) material on Si (001) substrate (SPT5/Si(001)) and Raman spectroscopic study was carried out to investigate the polar and AFD order in the film. EXPERIMENTAL PROCEDURE 5% Pr doped SrTiO 3 bulk has been prepared using conventional solid state reaction method which is reported elsewhere [2]. The film has been deposited on commercially proc ured Si (001) single crystal substrates. Thin film of Pr doped SrTiO 3(SPT5/Si(001))was deposited using Pulse Laser Deposition techni que equipped with a 248nm KrF excimer Laser of energy of 220mJ and a repetition rate of 1 0Hz. The chamber was evacuated to abase pressure of ~10-5 mbar before the deposition. The oxygen partial pressure of 0.3 50 mbar and substrate temperature of 750oC DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030001-1–030001-4; https://doi.org/10.1063/5.0016756 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030001-1was maintained during the film deposition. The film was found t o be ~100 nm thick. The film has been characterized using high resolution X-ray diffraction (HRXRD) s ystem with Cu K α radiation. Raman spectrum were obtained using Horiba JY HR-800, micro Raman spectrometer equip ped with a473 nm excitation laser and having over all spectral resolution of ~ 1 cm-1. Temperature dependent Raman measurements were carried out by mounting the film in Linkam, THMS600 stage with a temperature stability of േ 0.1K. X-ray absorption near edge spectrum (XANES) were recorded in total electron yield (TEY) mode at BL1 beamline at Indus-2 [4]. RESULTS AND DISCUSSIONS FIGURE 1. a) X-ray diffraction of 5% Pr doped SrTiO 3 thin film deposited on Si (001). F or better comparison the x-r ay diffraction data of the Si(100) s ubstrate is also presented at the bottom. b) Raman spectrum of s ingle crystal substrate of Sr TiO 3, bulk 5% Pr doped SrTiO 3, and deposited filmSPT5/Si(001) . Inset shows zoomed view aroun d polar TO 2 mode. Region between 480-540cm-1having strong signal from substr ate has been removed for clarit y. Figure 1(a) gives the X-ray diffraction patterns ofthe SPT5/Si( 001) film along with that on Si (001) single crystal. It shows domination of the (100) family of peaks apart from a weaker peak repres enting (110) reflection. It may be noted that in bulk compound (110) reflection is the most prominent peak while the (100) peak is extremely feeble, thus, x-ray diffraction results confirm that the film i s not epitaxial in nature but highly oriented along (100) direction. A hump near 70ois also observed which is attributed to the contribution from t he Si substrate. As such the substrate should show a narrow peak, however, a slight misalign ment of the substrate with respect to incident x-ray beam results in hump like feature. The x-ray diffraction result s are similar to that reported previously for SrTiO 3 thin film on Si substrate [5].Importantly, the peaks were found to b e shifted towards higher 2 when compared to bulk compound indicating decrease inout of plane lattice parameter i ndicating reduction in unit cell volume in low dimensions. In order to examine the local structure of the mate rial Room temperature Raman spectrum was collected on the SrTiO 3 single crystal substrate, 5% Pr doped SrTiO 3 bulk and thin film that is presented in Figure 1 (b). Compared to bulk only two Raman modes are observed at 176 and 7 96 cm-1 in the thin film samples and rest of the Raman modes were not detected possibly due to the domination of R a m a n s i g n a l f r o m t h e s u b s t r a t e a n d f e e b l e intensity of the respective Rama n modes. Mode due to Si substra te were observed at 302 cm-1, 440 cm-1 and 520 cm- 1 (truncated from the spectrum). In the SP T5/Si(001) film, the R aman mode falling at 176 cm-1 has been identified as polar TO 2 mode and 796 cm-1 as LO 4 mode as observed in bulk sample [2]. In thin film, the polar T O2 mode position was found to be shifted towards higher wave number when compare d to bulk sample suggesting increased force constant that is attributed to reduction in unit cell volume as observed in x-ray diffraction studies. Also both TO 2 and LO 4 were found to broaden due to reduced dimensions. Low frequency TO 2-TO 1 mode has also been observed as shown in the inset of figure 1(b) but these modes could be iden tified clearly only in low temperature Raman 030001-2measurements. The observation of polar TO 2 mode is a direct evidence of presence of polar instability in the thin film sample. The TO 2 mode showed asymmetric line profile which is a manifestation o f Fano effect [2]. Low temperature Raman spectrum w as collected for investigating presence of long range tetragonal distortion in the system. Selected low tempera ture Raman spectra are shown in figure 2(a). Surprisingly, the film does not show any signatures of tetragonal symmetry (E g a n d B 1g R a m a n m o d e ) i n t h e R a m a n s p e c t r u m t i l l 8 0 K w h i c h w a s observed I the corresponding bulk compound [2]. At 80 K, along with TO 2 mode we were able to track the TO 2- TO 1 mode at low frequency falling near 90 cm-1. Both these modes are strongly affected by the polar ordering similar to the modifications observed in the phonon or lattice due to spin ordering in magnetic systems [6,7]. In our previous study on bulk Pr doped SrTiO 3, it was shown that the variation in the position as well as in tensity of the polar TO 2 mode is an indicator of the growth and fluctuations of the PNR . With increasing temperature, the mode position was found to harden anomalously along with decrease in asymmetry and intensity of the TO 2 mode indicating decay of strength of the polar order [2]. Here, pola r order is generated from the disorder in Ti oscillator in the system which is reflected in asymmetric profile of the pola r TO 2 mode. This asymmetry is manifestation of coupling of polar mode with disor der in Ti sub-lattice [2] aris ing from the local randomfields. FIGURE 2 . (a) Raman spectrum collected on SPT5/Si(001) at selected temp eratures; shaded region marks TO 2–TO 1(left) and polar TO 2(right) modes,respectively.(b) Co mparison of O K-edge for film and SrTiO 3 single crystal and inse t shows variation in excitations associated with the t 2g and e gbands. XANES measurements at Ti L-edge and O-K edge were carried out i n order to verify the oxidation state of Ti atom in the synthesized thin film and were normalized using Ath ena software [8]. The XANES of the bulk compound and SrTiO 3 single crystal substrate was also recorded for comparison. The Ti and Oxygen edges in all the three samples arefound to be consistent with Ti(+4) oxidation s tate. Figure 2(b) shows spectrum collected at O K- edge on the SPT5/Si(100) film along with SrTiO 3 substrate. The intensity ratio of the peaks related tot 2g and egbandsin SPT5/Si(100) is found to be totally different than that in Sr TiO 3.This observation points out towards a significant change in Ti-O hybridization due to a local structu ral rearrangement that is attributed to the reduction in unit cell volume in film when compared to bulk compound as dete cted by room temperature x-ray diffraction and Raman spectroscopy analysis.The variation in Ti-O hybridization can arise due to dramatic variation in tilt system or a small shift in the Sr(Pr) or Ti atoms from its mean position. In Raman studies we observed hardening of Polar TO 2 mode indicating lower octahedral tilt therefore the increased o verlap in Ti-O orbitals points towards shift in cations from its mean position. The shift in mean position of the catio ns generates local polar lattice instabilities over the AFD. As such in bulk, polar and AFD competes and cooperate at d ifferent length scales [2] . Since the energy gain from stabilization of AFD is larger, so a long range AFD is sta bilized with short range polar order [1,2]. On the 030001-3contrary in the present case, the significant reduction in unit cell volume is realized by reduction in the octahedral tilt angle and enlargement of Ti off-centering as found in Rama n spectrum. This completely suppress the AFD in this film. CONCLUSIONS We have successfully synthesized thin film of 5% Pr doped SrTiO 3 using pulsed laser deposition method. Local structure of 5% Pr doped SrTiO 3 thin film has been investigated using Raman and XANES. Raman i nvestigations showed polar modes and absence of modes related with the AFD (t etragonal symmetry) from 300 K down to 80 K in contrast to the bulk compound and SrTiO 3. Thus, it may be concluded that due to variation in local symm etry in thin film form due to reduced unit cell volume the polar order competes and cancels the AFD. The decrease in octahedral tilt and increased Ti off-centering is attributed to s t r a i n i n t h e t h i n f i l m s a m p l e . T h i s e f f e c t h a s b e e n attribute to strong competition between AFD and polar instabili ties, in the 5 % Pr doped SrTiO 3 film the polar order dominates and cancels the AFD instability. The film showed abse nce of long range tetragonal distortion and hence these films can be a good candida te for understanding photo-fer roelectric properties in p seudo cubic symmetry. REFERENCES 1. S. Tyagi, G. Sharma, V. G Sathe , J. Phys. Condens. Matter. 30 ( 2018). 2. Vivek Dwij et al Raman spectroscopic investigation of Pr doped SrTiO 3 and origin of Fano resonance, arXiv: 1809.09403v4 (2019). 3. Hayato Katsu, Hidekazu Tanaka an d Tomoji Kawai; Jpn. J. Appl. P hys. Vol. 39 (2000) pp. 2657–2658. 4. D. M. Phase, M. Gupta, S. Potdar , L.Behera, R.Sah, A. Gupta, AIP Conf. Proc., 1591, 685-686(2014). 5. F. Sánchez et al Applied Physics Letters 61, 2228 (1992). 6. Binoy Krishna De, Vivek Dwij, Shekhar Tyagi, Gaurav Sharma and V.G. Sathe, AIP Conference Proceedings 1942 , 030008 (2018). 7. G. Sharma et al, Journal of Alloys and Compounds 732 358-362, ( 2018). 8. B. Ravel, M. Newville, J. S ynchrotron Radiat. 12 (2005) 537–541. 030001-4

5.0029099.pdf

J. Chem. Phys. 153, 164702 (2020); https://doi.org/10.1063/5.0029099 153, 164702 © 2020 Author(s).Ag5-induced stabilization of multiple surface polarons on perfect and reduced TiO2 rutile (110) Cite as: J. Chem. Phys. 153, 164702 (2020); https://doi.org/10.1063/5.0029099 Submitted: 09 September 2020 . Accepted: 02 October 2020 . Published Online: 22 October 2020 P. López-Caballero , S. Miret-Artés , A. O. Mitrushchenkov , and M. P. de Lara-Castells The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ag5-induced stabilization of multiple surface polarons on perfect and reduced TiO 2rutile (110) Cite as: J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 Submitted: 9 September 2020 •Accepted: 2 October 2020 • Published Online: 22 October 2020 P. López-Caballero,1 S. Miret-Artés,1 A. O. Mitrushchenkov,2 and M. P. de Lara-Castells1,a) AFFILIATIONS 1Instituto de Física Fundamental (AbinitSim Unit), CSIC, Serrano 123, 28006 Madrid, Spain 2MSME, University Gustave Eiffel, CNRS UMR 8208, University Paris Est Creteil, F-77454 Marne-la-Vallée, France 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: Pilar.deLara.Castells@csic.es ABSTRACT The recent advent of cutting-edge experimental techniques allows for a precise synthesis of subnanometer metal clusters composed of just a few atoms, opening new possibilities for subnanometer science. In this work, via first-principles modeling, we show how the decoration of perfect and reduced TiO 2surfaces with Ag 5atomic clusters enables the stabilization of multiple surface polarons. Moreover, we pre- dict that Ag 5clusters are capable of promoting defect-induced polarons transfer from the subsurface to the surface sites of reduced TiO 2 samples. For both planar and pyramidal Ag 5clusters, and considering four different positions of bridging oxygen vacancies, we model up to 14 polaronic structures, leading to 134 polaronic states. About 71% of these configurations encompass coexisting surface polarons. The most stable states are associated with large inter-polaron distances ( >7.5 Å on average), not only due to the repulsive interaction between trapped Ti3+3d1electrons, but also due to the interference between their corresponding electronic polarization clouds [P. López-Caballero et al. , J. Mater. Chem. A 8, 6842–6853 (2020)]. As a result, the most stable ferromagnetic and anti-ferromagnetic arrangements are ener- getically quasi-degenerate. However, as the average inter-polarons distance decreases, most ( ≥70%) of the polaronic configurations become ferromagnetic. The optical excitation of the midgap polaronic states with photon energy at the end of the visible region causes the enlargement of the polaronic wave function over the surface layer. The ability of Ag 5atomic clusters to stabilize multiple surface polarons and extend the optical response of TiO 2surfaces toward the visible region bears importance in improving their (photo-)catalytic properties and illustrates the potential of this new generation of subnanometer-sized materials. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029099 .,s I. INTRODUCTION The very recent development of highly selective experimental techniques making possible the synthesis of subnanometer metal clusters is pushing our understanding of these, more “molecu- lar” than “metallic,” systems far beyond the present knowledge in materials science. Subnanometer metal clusters with sizes below 10 Å–15 Å (i.e., ∼100–150 atoms) have physical and chemical prop- erties differing significantly from those of nanoparticles and bulk materials due to the quantum confinement effects.1,2When the cluster size is reduced to a very small number of atoms, the d- band of the metal splits into a subnanometer-sized network ofdiscrete molecule-like dorbitals, with the inter-connections having the length of a chemical bond (1 Å–2 Å). The spatial structures of the molecular orbitals of these clusters make all the metal atoms coop- eratively active and accessible, leading to the appearance of their novel properties and air stability (see, e.g., Refs. 3–6 and references therein). Their possible applications range from visible-light photo- activation of semiconductor surfaces7,8to innovative drugs for can- cer therapies.9For instance, new catalytic and optical properties are acquired by titanium dioxide (TiO 2) surfaces when decorated with the Cu 5atomic cluster.8,10The Cu 5cluster shifts the absorption of sunlight toward visible light, where the Sun emits most of its energy.8 The coated titanium dioxide stores the absorbed energy temporarily J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp in the form of charge pairs, i.e., electrons and holes, in the direct vicinity of the surface, which is a perfect prerequisite for follow-up chemistry.11 Looking for further applications, we have recently modified the TiO 2surface with a single atomic silver cluster Ag 5.6In this way, we have found that Ag 5-modified TiO 2surfaces are both visible-light photo-active materials, potential photocatalysts for CO 2reduction, and capable of hosting surface polarons.6As reported by Selloni and collaborators for various noble metal clusters on anatase TiO 2sur- faces, the charge transfer between the metal cluster and TiO 2surface is at the very basis of the underlying mechanisms (see, e.g., Refs. 12 and 13). Moreover, we have also provided new relevant fundamental insights into a general polarization phenomenon accompanying the formation of surface polarons.6This phenomenon has been mani- fested as a depopulation of titanium 3 dorbitals in favor of oxygen 2 p orbitals, as experimentally observed through x-ray absorption spec- troscopy at the Ti K-edge and theoretically predicted by applying state-of-the-art computational modeling.6In our picture, a surface polaron is visualized as a slow electron projecting a “shadow” on its traveling way, causing not only nuclear motion (i.e., a structural distortion of the crystal lattice) but also electronic rearrangements that are manifested in a carrying “polarization cloud.”6Yet, there are a number of important open questions to be addressed, such as: (1) how Ag 5-induced surface polarons interfere with those formed due to surface defects such as bridging oxygen vacancies (referred to O-vacancy-induced) in reduced samples?; (2) could supported Ag5clusters promote O-vacancy-induced polarons transfer from the subsurface regions?; (3) are co-adsorbed Ag 5clusters capable of inducing the formation of multiple (coexisting) surface polarons?; (4) the couplings between Ag 5-induced surface polarons are ferro- magnetic or anti-ferromagnetic?; (5) how these couplings and the stability of surface polarons depend on the inter-polarons distance, considering also the possible interference between the correspond- ing polarization clouds? To address these questions, we conduct an extended ab ini- tioinvestigation considering perfect and reduced TiO 2rutile (110) decorated with Ag 5atomic clusters. For this purpose, we apply den- sity functional theory (DFT) and an approach combining DFT with reduced density matrix (RDM) theory. We have chosen a DFT-D3 ansatz14,15on the basis of its excellent performance in describing the adsorption of subnanometer silver clusters on rutile TiO 2.7For an appropriate determination of the electronic structure associated with polarons formation in TiO 2surfaces,6,8,16we employ the HSE06 hybrid functional of Heyd, Scuseria, and Ernzerhof17,18on top of the structures optimized via the DFT-D3 approach. We have also calcu- lated the UV–Vis absorption response of the multiple polarons states by employing the reduced density matrix (RDM) theory within the Redfield approximation,19combined with DFT calculations using the HSE06 functional. This combination of RDM and DFT, pro- posed by Micha and collaborators,20–22has been successfully applied to silver6,7,23–25and copper8,10clusters on semiconductor TiO 2and silicon surfaces.6–8,10,23–26 This article is organized as follows. In Sec. II, we provide further details of the computational and theoretical approaches developed and applied in this work. Then, the results from our calculations are presented and discussed in Sec. III. This section is split into several subsections. Section III A outlines the polaron concept and proves the performance of our approach by applying it to reducedTiO 2rutile (110), in comparison with reported results. Next, we investigate the Ag 5-induced formation of multiple surface polarons, either by interacting with O-vacancy-induced polarons or by co- adsorption of Ag 5clusters. Finally, Sec. IV summarizes our main findings, presenting a practical perspective of their implications. II. COMPUTATIONAL METHODS A. Periodic calculations Periodic electronic structure calculations are performed with the Vienna Ab initio Simulation Package ( VASP 5.4.4),27,28following a computational approach similar to that reported in previous works on Ag 5– and Cu 5–TiO 2(110) interactions.6,8,10Specifically, struc- tural optimizations and the calculation of interaction energies are carried out with the Perdew–Burke–Ernzerhof (PBE) density func- tional29and the Becke–Johnson (BJ) damping14for the D3 disper- sion correction. This combination is referred to as the PBE-D3(BJ) scheme. These calculations are carried out including spin polariza- tion without imposing restrictions on the overall magnetic moment. The magnetic moments in converged calculations differed by <0.015 μBon average from the “nominal” values (e.g., 2 μBfor two polarons in ferromagnetic configurations). In order to describe the localized 3 d-electrons on specific Ti cations, the corresponding Ti–O distances are initially elongated to mimic polaronic structures.30Next, the resulting structures are opti- mized with the Hubbard DFT+U term31added and including spin- polarization. The value of U (4.2 eV) reported in previous studies on Cu 5–TiO 2(110) interactions is adopted.8,32The optimized geome- tries obtained at the PBE+U/D3 level are used in the final HSE06 calculations of the electronic structures. The HSE06 approach is applied using a HF/GGA mixing ratio of 25:75 with the screening parameter of 0.11 bohr,−1as recommended in Ref. 18. It provides a value of the direct band gap for rutile TiO 2(110) (∼3.3 eV), which is in a good agreement with the experimental observations. The HSE06 ansatz employed in the final calculations of the orbitals (but not the adsorption energies or the optimized structures) did not include D3 corrections. It is clear that the inclusion of such dis- persion contribution would modify the absolute values of the total energies but not the energy differences between the orbital levels neither the orbital shapes, which are the relevant magnitudes in both the electronic density of states (EDOS) and the UV–Vis spectra calculations. Electron–ion interactions are described by the projector augmented-wave method,28,33using PAW-PBE pseudo-potentials as implemented in the program. The electrons of the O(2 s, 2p), C(2s, 2p), Ti(3 s, 4s, 3p, 3d), and Ag(4 d, 5s) orbitals are treated explic- itly as valence electrons. A plane wave basis set with a kinetic energy cutoff of 700 eV is used. A Gaussian smearing of 0.05 eV is employed to account for partial orbital occupations. The Brillouin zone is sampled at the Γpoint.34,67The convergence criterion is fixed to a value of 10−4eV for the self-consistent electronic minimizations. All the atoms from the supercells are relaxed with a force threshold of 0.02 eV/Å. Stoichiometric (perfect) TiO 2rutile (110) is modeled via peri- odic slabs, using a 4 ×2 supercell (four TiO 2trilayers giving ∼13 Å slab width). The reduced surface is modeled by removing J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp a single bridging oxygen anion from the slab. The adsorption of the Ag 5clusters are assumed on one side of the slab, with 38 Å of vacuum above it. This large vacuum region ensures that the super- cell is not interacting with its periodic images in the direction per- pendicular to the surface. Using a larger 8 ×2 supercell model in structural optimizations, we have tested that the increase in the supercell size does not modify the polarons positions. The influ- ence of the number of layers was analyzed in a previous study by Deskins et al.35using also a 4 ×2 supercell model but varying the corresponding slab thickness. It was shown that the decrease in the number of trilayers from nine to five leads to an underestimation of the stabilization energy of configurations bearing surface polarons by∼0.1 eV.35Hence, we can expect that the relative energies of surface polarons are underestimated, at least, by the same value (0.1 eV). The adsorption energy of nAg5clusters ( n= 1, 2) on TiO 2 (perfect and reduced) surfaces is obtained as Eads={n E Ag5/TiO 2(110)}(min)−{n E Ag5/TiO 2(110)}(long-range ), where{n E Ag5/TiO 2(110)}denotes the total energy of the system, with nAg5clusters at the minimum energy position (referred to as “min”) or at a distant position from the surface so that the cluster–surface interaction becomes negligible (referred to as “long-range”). The average inter-polarons distance (referred to as ⟨dpp⟩) is calculated as follows: ⟨dpp⟩=Npol ∑ ijdi−j Npol, (1) where Npolis the number of polarons and di−jstands for the distance between the ith and jth polarons. The periodic images are considered in this definition as well. The average distance between the polarons and either the vacancy or the central atoms of two adsorbed silver clusters is estimated in a similar way. B. Reduced density matrix treatment UV–Vis absorption spectra are calculated using the com- putational approach previously applied to the Cu 5– and Ag 5– TiO 2(110) systems in Refs. 8 and 6, respectively. The relaxation processes involved are described by the reduced density matrix (RDM) approach in the Redfield approximation,19based on the orbitals taken from calculations employing the HSE06 hybrid func- tional. This combined RDM–DFT treatment developed by Kilin and Micha20,21is already a well-established tool in describing the optical spectra of subnanometer-sized metal clusters adsorbed on semiconductor surfaces36(see, e.g., Refs. 7, 23–25, 37). Very briefly, in the presence of a monochromatic electromag- netic field Eof frequency Ω, the evolution equation for the reduced density ρin the Schrödinger picture takes the form ˙ρjk=−i ̵h∑ l(Fjlρlk−ρjlFlk)+∑ l,mRjklmρlm, ˆF=ˆFKS−ˆD⋅E(t), E(t)=E0(eiΩt+e−iΩt),where ˆFKSdenotes the effective Kohn–Sham Hamiltonian (the indices refer to its representation in the Kohn–Sham basis set), with ˆDas the electric dipole moment operator and Rjklm as the Red- field coefficients, i.e., the Kohn–Sham components of the relaxation tensor. The latter are defined in Ref. 19 and are implemented as described in Ref. 20. Within the Redfield approximation, the relaxation tensor embodies not only fast electronic dissipation due to electronic fluc- tuations in the medium, but also the relatively slow relaxation due to vibrations of the atomic lattice. It is convenient to perform a coordinate transformation into a rotating frame accounting for the electromagnetic field oscillation. This is described via the following equations: ˜ρij(t)=ρij(t)exp(iΩt),εi>εj, ˜ρij(t)=ρij(t)exp(−iΩt),εi<εj, ˜ρii(t)=ρii(t), where εiis the energy of the ith Kohn–Sham orbital. Time averaging over the fast terms in the equation of motion for the RDM yields ˜ρSS jj=Γ−1 jHOMO ∑ k=0gjk(Ω),j≥LUMO, ˜ρSS jj=1−Γ−1 j∞ ∑ k=LUMOgjk(Ω),j≤HOMO, as stationary-state solutions for the diagonal elements.20Here, HOMO and LUMO denote the lowest-energy unoccupied and the highest-energy occupied molecular orbital, respectively. Γjis a depopulation rate, and the sum terms gjkare given by gjk(Ω)=γΩjk γ2+Δjk(Ω)2, where γdenotes the decoherence rate and Ω jkare the Rabi frequen- cies given by Ω jk=−Djk⋅E0/̵h, with Δjk(Ω) = Ω −(εj−εk) as detunings. The diagonal elements provide the populations of the KS orbitals. The population relaxation rate̵hΓand the decoherence rate ̵hγare kept fixed to values of 0.15 meV and 150 meV (27 ps and 27 fs), respectively. These values have been chosen according to the known rates for phonon decay and electronic density excitations in semiconductors (see, e.g., Ref. 38). In terms of the stationary populations, the absorbance is given by7,24,25,39,40 α(Ω)=HOMO ∑ j=0∞ ∑ k=LUMOfjk(˜ρSS jj−˜ρSS kk)1 π̵hγ/2 (̵hΔjk)2+(̵hγ/2)2, where fjkis an oscillator strength per active electron.41The solar flux absorption spectrum is then expressed as F(̵hΩ)=α(Ω)Fsolar(̵hΩ)̵hΩ, J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where the solar flux is approximated by the black body flux distribu- tion, normalized to an incident photon flux of 1 kW/m2, Fsolar(̵hΩ)=(̵hΩ)3 π3̵h3c3CT exp(̵hΩ/kBT)−1, with CTthe flux normalization constant and the temperature Tset to 5800 K. III. RESULTS AND DISCUSSION A. Polaron concept: Small polarons formation on reduced TiO 2rutile (110) The polaron concept characterizes an electron moving in a dielectric crystal.42Its formation is a well-known and fundamental phenomenon in transition metal oxides. In particular, Selloni and collaborators have contributed very significantly to the fundamen- tal understanding of polarons formation in TiO 2(both rutile and anatase) surfaces (see, e.g., Refs. 43–45). Typical defects of rutile TiO 2surfaces such as bridging oxygen vacancies46lead to two excess electrons. The latter becomes self-trapped in Ti3+3d1states and couple to lattice distortions forming small polarons,47as illustrated in panel (a) of Fig. 1. The existence of intrinsic small polarons as self-trapped electrons in rutile TiO 2(i.e., forming stable Ti3+ 3d1states) has been both theoretically predicted44,45,48–53and exper- imentally confirmed.54–56Agreeing well with previous studies,35,53,56 we find that the subsurface positions of polarons are favored on the rutile surface. For instance, notice that the energies of the left- and right-hand structures shown in panel (a) of Fig. 1 differ by ∼0.36 eV. Consistently, a previous work35reported that the ener- gies of configurations bearing two surface polarons were 0.28 eV– 0.44 eV higher in energy than the most stable one with two subsur- face polarons. Actually, as predicted from first-principles molecular dynamics calculations at experimental conditions, the polarons stay most of the time ( ∼75%) at the subsurface layer but also hop to the top-most layer.56Accordingly, as shown in panel (a) of Fig. 1 the supplementary material, the energy of the configuration featuringpolarons at both subsurface and surface layers differs very little (0.02 eV) from that of the most stable state. The two structures fea- turing surface polarons were also reported in a previous study by Deskins et al. (labeled 3-1′and 1-3 in Ref. 35), but higher relative energies were estimated (0.02 eV vs 0.33 eV and 0.36 eV vs 0.87 eV). We attribute such differences to the dispersion forces as they are expected to reduce the so-called “structural cost” (as defined in Ref. 53) in surface polarons formation. In order to screen the self-trapped Ti3+3d1electrons, the O2− anions depart from their equilibrium positions, as indicated by red arrows in panel (b) of Fig. 1. Notice that the Ti–O bond lengths are longer (by about 0.1 Å) when the oxygen anions are bonded to the Ti cation hosting the polaron (shown with a yellow sphere). Similar values of Ti–O bond elongations have been reported for both anatase57and rutile.58This lattice distortion (known as the phonon cloud) always accompanies the polaron formation wherever it settles down.6The presence of multiple polarons is also revealed at the electronic density of states (EDOS) (see supplementary material, Fig. 2) through the appearance of a single midgap peak arising from almost degenerate energy levels of the two polarons. In accord with previous determinations (see, e.g., Ref. 59), it is located at about 1 eV below the conduction band. Hence, the overall agree- ment with the previous studies of the polarons formed on rutile TiO 2(110) allows us to further confirm the validity of our theo- retical approach to characterize Ag 5-induced surface polarons in Secs. III B and III C. B. Ag 5-induced stabilization of surface polarons on reduced TiO 2rutile (110) When supporting a single Ag 5atomic cluster onto the reduced TiO 2(110) surface, the planar trapezoidal- and pyramidal-shaped structures shown in Fig. 2 are both predicted to be highly stable (see Table I). As for the case of Ag 5adsorption on stoichiometric TiO 2,6 the pyramidal-shaped isomer, lying flat over the surface, is found to be the most stable structure with an adsorption energy of −4.51 eV, differing by ∼1.3 eV by that of the trapezoidal isomer (see also Sec. 2 of the supplementary material). As can be observed in Table I, the adsorption energies are only slightly affected by the presence of an FIG. 1 . Polarons hosted on reduced TiO 2rutile (110). (a) Polaronic Ti3+3d1states induced by the presence of a bridging oxygen vacancy on TiO 2rutile (110). The left-hand structure is the most stable, holding two subsurface polarons. (b) Picture illustrating the lattice distortions accompanying the polarons formation. The positions of the Ti atoms where the polarons are localized (referred to as Ti3+) are represented by yellow spheres. Red and blue spheres indicate the positions of oxygen anions and titanium cations, respectively. J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Panels (a)–(c) [(d)–(f)] correspond to the trapezoidal-shaped (pyramidal-shaped) Ag 5isomer, respectively. Upper panels [(a) and (d)]: structures for trapezoidal- and pyramidal-shaped Ag 5clusters adsorbed on reduced TiO 2rutile (110), bearing an oxygen vacancy. The different positions considered for the oxygen vacancy are also indicated. Red, blue, and gray spheres stand for oxygen, titanium, and silver atoms, respectively. Middle panels [(b) and (e)]: energy difference between a given polaronic structure and the most stable configuration ( ΔE) vs the average inter-polarons distance (see also Tables 1–8 of the supplementary material). The oxygen vacancy positions of the polaronic configurations are indicated by different colors. Bottom panels [(c) and (f)]: an enlarged view of the graphics shown at the middle panels. The two most stable structures are also shown, with the Ti3+atoms hosting the polarons marked in yellow color. The labels “s0,” “s1,” and “s2” stand for the surface and subsurface (first and second layer) locations of the polarons. The values of the adsorption energies are also indicated. oxygen vacancy. As mentioned above, an oxygen vacancy of reduced TiO 2rutile (110) induces the formation of two small polarons mostly located at a subsurface layer.56Further modification of the sur- face via decoration with the open-shell Ag 5cluster leads to the appearance of a third polaron through the donation of its unpairedelectron to the TiO 2surface. The donated electron becomes local- ized in one specific 3 dorbital lying at the surface, centered at one Ti atom in the nearby of both trapezoidal and pyramidal-shaped Ag5isomers, as illustrated in panels (c) and (f) of Fig. 1. The most relevant frontier orbitals are shown in Fig. 3, with the Ti3+3d1 J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Adsorption energies (in eV) of one Ag 5cluster on reduced and perfect TiO 2rutile (110), with the latter including co-adsorption of a second Ag 5cluster. For comparison purposes, the adsorption energy for a single cluster is −4.53 eV ( −3.03 eV) for the pyramidal-shaped (trapezoidal-shaped) isomer.6 Ag5/reduced TiO 2(pyramidal) −4.51 Ag5/reduced TiO 2(trapezoidal) −3.21 Ag5(co-adsorption)/perfect TiO 2(pyramidal) −4.34 Ag5(co-adsorption)/perfect TiO 2(trapezoidal) −3.00 electrons characterizing the singly occupied (referred to as SOMOs) orbitals. Considering the four different positions of the bridging oxygen vacancies shown in the upper panels of Fig. 2, we have sampled over 70 polaronic configurations. The initial structures were created by enlarging Ti∗–O bond distances, with Ti∗denoting the specific sites where the polarons are assumed to be hosted. These calculations were performed by applying the PBE+U/D3(BJ) approach. Panels (b) and (e) of Fig. 2 show the relative stability of a given configu- ration (with the most stable structure as the reference) as a func- tion of the average inter-polarons distance. For both trapezoidal- and pyramidal-shaped isomers, we find that the stability of a given structure is higher when the polarons are located further away fromeach other. This is due to both the repulsion between the trapped Ti3+3d1electrons and the interference between the associated polar- ization clouds.6As reported in Ref. 6, the polarization phenomenon accompanies the formation of surface polarons, modifying the elec- tronic structure in their neighborhood: self-trapped Ti3+3d1elec- trons repel near oxygen anions and attracts near titanium cations, which, in turn, affects their electronic structure, causing the transfer of electronic charge from Ti4+cations to O2−anions. This surface polaron property favors large inter-polarons distances as the inter- ference between the polarization clouds associated with different polarons is better avoided. Yet, there are a number of exceptions to this “rule of thumb” when the polarons are located on titanium atoms next to the bridging oxygen vacancies (see supplementary material, Sec. 2). In fact, the interaction between the vacancy and the polaron is attractive. As comprehensively reported by Reticcioli et al. ,53the O-vacancy-polaron distance between a polaron is another important factor determining the stability of a given pola- ronic structure over the bare TiO 2(110) surface. In order to bet- ter visualize its influence for the Ag 5-decorated reduced surface, supplementary material Fig. 5 presents the relative energies vs the average polaron-vacancy distance. It clearly illustrates an enhanced stability as the polaron-vacancy distance decreases in a quasi-linear dependence for the lowest-energy states. It is worth stressing that structures with two surface polarons are more stable than those featuring just one surface polaron, despite the larger inter-polarons distance for the former. This finding points FIG. 3 . Picture showing iso-surfaces of the frontier “singly occupied” molecular orbitals (SOMO, SOMO −1, and SOMO −2) and the highest-energy and second “doubly occupied” molecular orbitals (HOMO and HOMO −1) for the most stable structures of Ag 5-modified reduced TiO 2rutile (110). Upper panel: planar trapezoidal-shaped isomer. Bottom panel: pyramidal-shaped isomer. J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp out that Ag 5clusters are capable not only of inducing their “own” surface polarons via transfer of their unpaired electrons but also of promoting polarons transfer from the subsurface to surface sites (see also supplementary material, Sec. 2). As previously reported for CO as the adsorbate,60the favored surface location is caused by the attractive interaction of Ag 5clusters with surface polarons. From our extended sampling of polaronic configurations, the repulsion that each polaron exerts on each other is clearly appar- ent and especially noticeable at short distances. For this rea- son, when examining the ferromagnetism featured by the differ- ent configurations, the associated inter-polarons distances should be accounted for. The quasi-degeneration of polaronic configu- rations is clearly apparent when the average inter-polarons dis- tance is large (more than 3.5 Å): 45% (55%) of the polaronic configurations are anti-ferromagnetic (ferromagnetic). However, at short distances, below 3.5 Å, the ferromagnetic couplings domi- nate the whole scenario so that 70% of the polaronic configura- tions become ferromagnetic. Assuming a Maxwell–Boltzmann dis- tribution, the polaronic states encompassing short inter-polarons distances become accessible upon increasing the temperature above 490 K. 1. Frontier orbitals, electronic density of states, and UV–Vis spectra Having analyzed the Ag 5-modified reduced TiO 2(110) surface in the ground electronic state, we focus now on its optical excita- tions. Figure 4 (bottom panel) shows the photo-absorption spectra of the Ag 5-modified reduced TiO 2surface, with the Ag 5cluster located at the most stable (i.e., pyramidal-shaped) arrangement. For the sake of clarity, the corresponding electronic density of states (EDOS) is shown in the upper panel of Fig. 4 (see also Fig. 3 for a picture of the most relevant frontier orbitals). As can be observed in the EDOS, the localized Ti3+states (characterizing the SOMOs) appear 1 eV– 1.5 eV below the bottom conduction band. Similar values have been reported by Di Valentin et al. for Ti(5f) 3 d1states in hydroxylated and reduced rutile TiO 2(110) surfaces.45A more detailed description of the frontier orbitals and EDOS spectra is presented in supplemen- tary material, Sec. 2, extending the analysis to the trapezoidal-shaped isomer. Moving to the photo-absorption process, the first thing to notice from Fig. 4 is that the excitation of the Ti3+3d1electrons (located at the SOMOs) with a photon energy at the end of the visible region (marked with yellow, orange, and green arrows) leads to their spreading over the top-most layer of the TiO 2surface. This behavior can be explained as follows: the hole generated due to the depopu- lation of the titanium 3 dorbitals when a surface polaron is created becomes filled when the polaron becomes photo-excited. The same phenomenon has been reported for the case when the Ag 5atomic cluster is supported on stoichiometric TiO 2rutile (110).6However, notice that the polaron charge spreads less efficiently on reduced TiO 2rutile and that oxygen vacancies hinder somewhat the hole fill- ing process itself. The transfer of localized 3 d1electrons to the con- duction band via visible light excitation has been also experimentally observed in reduced TiO 2nanoparticles.61 In contrast to the bare TiO 2surface, Fig. 4 (bottom panel) shows that the Ag 5-modified reduced surface absorbs at the visi- ble region with the most intense peak located at about 2.3 eV. This FIG. 4 . Ag 5-lying down-TiO 2+ oxygen vacancy system. Upper panel: Electronic Density Of States (EDOS). The zero of energy corresponds to the Fermi level, defined here as the lowest unoccupied level. The energy positions of the HOMO, HOMO −1, SOMOs, and LUMO are indicated by arrows of different colors. The pro- jected density of states onto O(2 p), Ti(3 d), and orbitals centered at the Ag 5cluster is also shown. Bottom panel: photo-adsorption spectra. The insets present iso- density surfaces of the orbitals involved in the photo-excitation processes marked in the photo-absorption spectrum. feature has been both theoretically predicted6,7and experimentally confirmed.6The Ag 5modification is also responsible for the increase of the absorption in the UV region. The most intense peak (marked with a red arrow) involves a single electron “jumping” from 5 s orbitals centered at the Ag 5cluster (HOMO) to the conduction band, leaving a long-lived (localized) “hole” at the cluster. The den- sity of the acceptor state (i.e., the LUMO+223 orbital) extends all along the subsurface layers of the TiO 2slab, bearing a depleted (empty) region at the cluster–surface interface, and thus hinder- ing the electron–hole recombination. Since the latter process is a key factor limiting the photo-catalytic activity of the unmodified TiO 2surface, the Ag 5-induced ability to separate charge-carriers is expected to improve the visible photocatalytic properties of the modified material (see also Refs. 6 and 8). C. Interaction of Ag 5-induced surface polarons on perfect TiO 2rutile (110) New polaronic structures have been determined when support- ing two Ag 5clusters over stoichiometric TiO 2rutile (110). Each Ag 5 J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp atomic cluster induces the formation of one surface polaron, allow- ing us to analyze the inter-polarons interaction. Considering both planar trapezoidal- and pyramidal-shaped Ag 5structures, the rela- tive positions of the two Ag 5clusters have been varied to create a total of six different structures. For each structure, we modeled up to 13 polaronic configurations, with the follow-up PBE+U/D3(BJ)calculations leading to a total of 59 polaronic states. Panels (b) and (e) of Fig. 5 show the relative stability of a given configuration (with the most stable structure as the reference energy) as a function of the inter-polarons distance. As for a single cluster,6the pyramidal- shaped structure is more stable also when two clusters are sup- ported on TiO 2rutile (110) (see Table I and supplementary material, FIG. 5 . Panels (a)–(c) [(d)–(f)] correspond to the trapezoidal-shaped (pyramidal-shaped) Ag 5isomer, respectively. Upper panels [(a) and (d)]: structures for two trapezoidal- and pyramidal-shaped Ag 5clusters supported on TiO 2rutile (110). Blue, red, and gray spheres stand for oxygen, titanium, and silver atoms, respectively. Middle panels [(b) and (e)]: difference between the adsorption energy of a given polaronic structure and that of the most stable configuration ( ΔE) vs the inter-polarons distance (see also Tables 9–14 of the supplementary material). Energy differences of configurations featuring different relative positions are indicated with different colors (see supplementary material, Sec. 3 for details). Bottom panels [(c) and (f)]: an enlarged view of the graphics presented in the middle panels. The two most stable structures are also shown, with the Ti3+atoms hosting the polarons marked in yellow color. The labels “s0” stand for the surface locations of the polarons. The values of the adsorption energies are also indicated. J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Sec. 3 for details). As can be observed in Table I, co-adsorption effects make the adsorption energies smaller ( ∼0.2 eV for the most stable isomer). Similar to the case of Ag 5supported over reduced TiO 2rutile (110), the analysis of the results presented in Fig. 5 for the trapezoidal isomer shows that the stability of a given configuration is higher as the inter-polarons distance increases. Still, despite having associ- ated smaller inter-polarons distances in some cases, configurations featuring two surface polarons are more stable than those bearing just one surface polaron, making evident the capability of the Ag 5 clusters to promote surface polarons formation. Interestingly, pola- ronic structures having one Ti atom in between two surface polarons are more stable than those with an oxygen atom instead, indicating that Ti atoms better screen the polaronic charges from each other. Similarly, when Ag 5clusters are lying flat on the TiO 2surface, the further the polarons locate, the higher the stability of the corre- sponding structure. Yet, as for the trapezoidal-shaped Ag 5isomer, the exceptions make even more evident the ability of Ag 5clusters to stabilize surface polarons (see supplementary material, Sec. 3 for details). Altogether, our results for two Ag 5clusters supported on TiO 2highlight the role of the repulsion between the polarons as trapped Ti3+3d1electrons in favoring rather large inter-polarons distances for the most stable structures. However, as for a single Ag5cluster adsorbed on reduced TiO 2rutile (110), the polariza- tion phenomenon accompanying polarons formation is also partly responsible. The Ti3+3d1electrons attract Ti4+cations and repelsO2−anions located nearby them. As a result, there is a transfer of electronic charge from Ti4+3dorbitals to O2−2porbitals and an average depopulation of 3 dorbitals has been found. Consequently, as illustrated in supplementary material, Fig. 12, oxygen anions, thus, hold excess of electronic charge, which increases when two polarons are located in adjacent positions and destabilizes the cor- responding polaronic configuration. For the case of free-Ag 5TiO 2 surfaces, it has been reported that the polaronic charge itself is not only located on the “nominal” Ti3+site, but also shared with the surrounding atoms.53,62,63This would explain the increase of excess charge on the oxygen anions when the polarons are settled in adjacent positions over the bare TiO 2(110) surface. Another factor influencing the stability of a given structure is the average distance between the polarons and the central atoms of the silver clusters. Thus, a “central position” allows for a more attractive cluster–polaron interaction than the polaron location at one end side of the clusters (see supplementary material, Sec. 3). In order to illustrate this, supplementary material Fig. 11 presents the average distance between the polarons and the central atoms for the trapezoidal-shaped Ag 5isomer: it can be observed that shorter dis- tances favor the stability of the polaronic structures, which tend to be less stable as the polaron-Ag 5distance increases. Large inter-polarons distances on the average result in the weight of anti-ferromagnetic and ferromagnetic polaronic con- figurations being almost the same. However, considering surface polarons located at distances shorter than 3.5 Å from each other, 75% of the configurations are ferromagnetic (70% for the case of FIG. 6 . Iso-surfaces of the frontier “singly occupied” molecular orbitals (SOMO and SOMO −1) and the highest-energy and second “doubly occupied” molecular orbitals (HOMO and HOMO −1) for the most stable structures of the 2Ag 5–TiO 2system, considering both trapezoidal-shaped (upper panel) and pyramidal-like (bottom panel) structures. J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the reduced sample). Hence, we can conclude that the dominant coupling is ferromagnetic. Accurate estimations of these couplings from the energy splitting between ferromagnetic and antiferromag- netic states will involve future research, considering larger supercell models and both constrained and unconstrained magnetic moment calculations. 1. Frontier orbitals, electronic density of states, and UV–Vis photo-absorption spectra As mentioned above, the modification of TiO 2rutile (110) with two Ag 5clusters leads to the formation of two polarons, holding surface positions. The polaronic Ti3+3d1states characterize the SOMO −1 and SOMO shown in Fig. 6. For the trapezoidal-shaped Ag5isomer, one polaron is located just in front of one Ag 5cluster. The second polaron is also at the surface, but hosted in a different row (see also supplementary material, Fig. 10). For the case of two Ag5clusters lying flat over the surface, both polarons are located FIG. 7 . Electronic Density Of States (EDOS) and UV–Vis photo-absorption spec- trum of the 2Ag 5-lying down-TiO 2system (corresponding to the most stable polaronic state). Upper panel: electronic density of states. The zero of energy corresponds to the Fermi level, defined here as the lowest unoccupied level. The projected density of states onto O(2 p), Ti(3 d), and the orbitals centered at the Ag 5clusters is also shown. The energy positions of the HOMO, HOMO −1, SOMOs, and LUMO are indicated by arrows of different colors. Bottom panel: photo-adsorption UV–Vis spectra. The insets show iso-density surfaces of the orbitals involved in the photo-excitation processes that have been indicated by arrows.below the clusters, but shifted from the position of their central Ag atoms. As can be observed from the electronic density of states (EDOS) of the system composed of two Ag 5clusters lying flat over the TiO 2 surface (upper panel of Fig. 7), the formation of two polarons as two Ti3+3d1states with very similar energies (i.e., the SOMOs) causes the appearance of two close peaks located 1 eV–1.5 eV below the conduction band (see supplementary material, Fig. 13 for the EDOS of the trapezoidal-shaped Ag 5isomer). The peaks associated with the HOMO and HOMO −1, bearing also very close energies, are located in between the SOMOs and the LUMOs at the EDOS spec- tra. Notice from the photo-absorption UV–Vis spectrum (bottom panel of Fig. 7) that the excitation of the polaronic Ti3+3d1states with photon energies at the visible region (about 3 eV–3.15 eV) leads to the enlargement of the polaronic wave function over the top-most layer of the TiO 2surface (see also supplementary material, Sec. 3 for details), as previously discussed for a single Ag 5clus- ter on reduced TiO 2. The improvement of the optical response of the material at the visible region is also very clear: The highest absorption peak is located at ∼2.4 eV, involving transitions from the highest-energy HOMOs, centered at the Ag 5cluster (see Fig. 6), to delocalized states in the conduction band formed by 3 d(Ti) atomic orbitals (the LUMO+99 orbital, see the insets at the bottom panel of Fig. 7). IV. CONCLUSIONS In this work, we have shown that the TiO 2surface modification with subnanometer silver clusters6serves to induce the formation of multiple surface polarons via the donation of the cluster unpaired electrons as well as the promotion of polarons transfer from the sub- surface to surface sites. From a practical perspective, this capability to stabilize multiple surface polarons has an interest in building the so-called polaronic 2D materials ,64favoring a more efficient charge transfer to adsorbed catalytic species, and thus improving the cat- alytic properties of the material. For instance, it has been suggested that surface polarons could be employed to solve one of the major problems of water splitting (i.e., the reduction of overpotentials in the oxygen evolution reaction).65 Owing to both the repulsion between the polarons as trapped Ti3+3d1electrons and the interference between the correspond- ing polarization clouds,6the average inter-polaron distances are larger for the most stable states. As a result, ferromagnetic and anti-ferromagnetic states are energetically almost degenerate. How- ever, ferromagnetic states become dominant in polaronic structures implying relatively short inter-polarons distances ( <3.5 Å), which could be populated, e.g., upon increasing the temperature, com- bined with the application of (strong) magnetic fields. The pos- sibility of controlling the ferromagnetism of these polaronic 2D materials might become relevant in future studies of new quantum technologies, but further research is necessary. As theoretically predicted6,7and experimentally observed,6we confirm that Ag 5clusters either in perfect or in reduced samples improve the optical response of the material by extending its absorp- tion toward the visible region and augmenting it in the UV. Thus, the photo-excitation of Ag 5-induced surface polarons with photon energy at the end of the visible region leads to the enlargement of the J. Chem. Phys. 153, 164702 (2020); doi: 10.1063/5.0029099 153, 164702-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp polaron wave function and the formation of large surface polarons. It is also worth stressing that our results are consistent with the general polarization phenomenon of surface polarons.6In the exper- iment, the intensity of the associated x-ray peak was only significant on Ag 5-modified TiO 2samples.6This can be explained, as shown in this work, due to the fact that Ag 5clusters favor multiple surface polarons formation. Moreover, the coupling with light is expected to be more effective in the silver-modified material. This work along with our very recent studies on subnanometer- sized metal (silver and copper) clusters5–8,10confirm the great poten- tial that lies in this new generation of materials, not only from an application perspective but also from a fundamental view, as demonstrated through the theoretical prediction and experimental demonstration of a surface polaron property6(see also Ref. 66). All our findings have been issued from extended state-of-the-art first- principles calculations and not from theoretical models, pointing out the key role of first-principles tools in shaping the modern field of subnanometer science. SUPPLEMENTARY MATERIAL See the supplementary material for supplementary figures, including additional details, as indicated in Sec. III. Supplemen- tary Tables 1–14 summarize the results found for all polaronic structures. DEDICATION This paper is dedicated to the memory of Professor Carmela Valdemoro (1932–2017) who worked at the Spanish Research Coun- cil (in the Institute of Materials Science and the Institute of Fun- damental Physics) for more than 25 years. Professor Valdemoro was a pioneer of reduced density matrices-based methodologies for electronic structure studies. She was an enthusiastic, creative, gen- erous, and encouraging researcher, having inspired generations of scientists. ACKNOWLEDGMENTS The authors thank Diego R. Alcoba, Pablo Villarreal, Luis Lain, and Alicia Torre for joining us in our dedication. This work was supported by the Spanish Agencia Estatal de Investigación (AEI) and the Fondo Europeo de Desarrollo Regional (FEDER, UE) under Grant No. MAT2016-75354-P. The CESGA super-computer cen- ter (Spain) is acknowledged for having provided the computational resources used in this work. P.L.C. expresses her gratitude for a grad- uate student contract in the “Garantía Juvenil” program from the Comunidad de Madrid. M.P. de L.C. is greatly thankful to David A. Micha and Tijo Vazhappilly for having shared the original code to calculate the absorption coefficients. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1P. Jena and Q. Sun, “Super atomic clusters: Design rules and potential for building blocks of materials,” Chem. Rev. 118, 5755–5870 (2018). 2M. Zhou, C. Zeng, Y. Chen, S. Zhao, M. Y. Sfeir, M. 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5.0027987.pdf

Appl. Phys. Lett. 117, 150502 (2020); https://doi.org/10.1063/5.0027987 117, 150502 © 2020 Author(s).Magnetic order in 3D topological insulators —Wishful thinking or gateway to emergent quantum effects? Cite as: Appl. Phys. Lett. 117, 150502 (2020); https://doi.org/10.1063/5.0027987 Submitted: 02 September 2020 . Accepted: 24 September 2020 . Published Online: 14 October 2020 A. I. Figueroa , T. Hesjedal , and N.-J. Steinke ARTICLES YOU MAY BE INTERESTED IN Quantum neuromorphic computing Applied Physics Letters 117, 150501 (2020); https://doi.org/10.1063/5.0020014 A perspective on hybrid quantum opto- and electromechanical systems Applied Physics Letters 117, 150503 (2020); https://doi.org/10.1063/5.0021088 A tunable plasmonic resonator using kinetic 2D inductance and patch capacitance Applied Physics Letters 117, 151103 (2020); https://doi.org/10.1063/5.0026034Magnetic order in 3D topological insulators—Wishful thinking or gateway to emergent quantum effects? Cite as: Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 Submitted: 2 September 2020 .Accepted: 24 September 2020 . Published Online: 14 October 2020 .Corrected: 21 October 2020 A. I.Figueroa,1,a) T.Hesjedal,2,a) and N.-J. Steinke3,a) AFFILIATIONS 1Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSICand BIST, Campus UAB, Barcelona 08193, Spain 2Clarendon Laboratory, Department of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom 3Institut Laue-Langevin, 71 Avenue des Martyrs, 38000 Grenoble, France a)Authors to whom correspondence should be addressed: adriana.figueroa@icn2.cat ;Thorsten.Hesjedal@physics.ox.ac.uk ; and steinkenj@ill.eu ABSTRACT Three-dimensional topological insulators (TIs) are a perfectly tuned quantum-mechanical machinery in which counterpropagating and oppositely spin-polarized conduction channels balance each other on the surface of the material. This topological surface state crosses the bandgap of the TI and lives at the interface between the topological and a trivial material, such as vacuum. Despite its balanced perfection, itis rather useless for any practical applications. Instead, it takes the breaking of time-reversal symmetry (TRS) and the appearance of anexchange gap to unlock hidden quantum states. The quantum anomalous Hall effect, which has first been observed in Cr-doped (Sb,Bi) 2Te3, is an example of such a state in which two edge channels are formed at zero field, crossing the magnetic exchange gap. The breaking of TRS can be achieved by magnetic doping of the TI with transition metal or rare earth ions, modulation doping to keep the electronically active channel impurity free, or proximity coupling to a magnetically ordered layer or substrate in heterostructures or superlattices. We review thechallenges these approaches are facing in the famous 3D TI (Sb,Bi) 2(Se,Te) 3family and try to answer the question whether these materials can live up to the hype surrounding them. Published under license by AIP Publishing. https://doi.org/10.1063/5.0027987 Prototypical three-dimensional (3D) topological insulators (TIs)1 of the (Bi,Sb) 2(Se,Te) 3family of solid solutions, most notably Bi 2Se3, Bi2Te3,a n dS b 2Te3, had a successful career as efficient thermoelectric materials2,3before the theoretical prediction of their topological sur- face states (TSSs) in 2009.4The TSS results from their large spin–orbit coupling and is made up of spin-momentum locked, counterpropagat-ing streams of oppositely spin-polarized electrons. Elastic backscatter-ing by nonmagnetic impurities is forbidden by time-reversalsymmetry (TRS), in principle resulting in high carrier mobilities. The existence of the gapless TSS in 3D TIs was first experimentally demon- strated using angle-resolved photoemission spectroscopy, 5instead of, as one might have expected, in transport measurements. The reasonlies in the rather poor electronic properties of (Bi,Sb) 2(Se,Te) 3-based materials, which are in fact narrow-gap semiconductors with strong, unintentional charge doping due to chalcogen vacancies. These high levels of bulk carriers can be most efficiently overcome by counter-doping, 6in particular in very thin films in which the relative bulk carrier concentration is naturally suppressed.In order to observe the exciting physical phenomena TIs are syn- onymous for, such as the quantum anomalous Hall effect (QAHE),7 the topological magnetoelectric effect,8the physics related to chiral edge states,9and spintronic effects,10TRS has to be broken and an exchange gap introduced in the TI.11The exchange gap was initially achieved by direct magnetic doping of the TI, by both transition-metals and rare-earth ions, in bulk crystals as well as thin films. 12–20 The QAHE was recently observed in the intrinsic magnetic TIMnBi 2Te4.21Proximity-coupling to magnetically ordered substrates or layers, i.e., ferromagnets, ferrimagnets, or antiferromagnets, is another way to break TRS. One inherent advantage of this approach is thatthere are no dopants interfering with the electrically active part of theTI, which could be compromised by the added impurities. Finally, thecombination of the different approaches in the form of heterostruc- tures and superlattices opens up new ways to achieve efficient TRS breaking without, in principle, compromising the electronic propertiesof the TI too much. Nevertheless, neither approach is yielding anobservable QAHE at decent (liquid He and above) temperatures Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-1 Published under license by AIP PublishingApplied Physics Letters PERSPECTIVE scitation.org/journal/apldespite the fact the TSS is observable at room temperature. On the other hand, the magnetic transition temperature TCitself is not setting the limit either; in fact, it can reach very high values. For instance, Cr0:15ðBi0:1Sb0:9Þ1:85Te3, which has been the most successful QAHE system so far, has a TCof 16 K, whereas very low temperatures (<300 mK) are required to observe the QAHE.7,22,23 There are several outstanding recent reviews of magnetic TIs, their exotic phenomena, and applications, which can be found in Refs. 24–26 . In this review, we focus on the discussion of the various growth and doping approaches for achieving an exchange gap in TI thin films, with a particular emphasis on heterostructures, and we finally try to answer the question whether there is hope for these materials to fulfill t h e i rs t a rp o t e n t i a l . In the context of thermoelectric applications and fundamental studies of the (Bi,Sb) 2(Se,Te) 3family of solid solutions, a number of single crystal,27,28nanostructure,29–34and thin film growth techniques have been employed;35–40however, for brevity and also given the ver- satility of the method, our review focuses solely on results obtained using molecular beam epitaxy (MBE)41,42(for reviews on TI thin film growth, see, e.g., Refs. 43–45 ). High-quality single crystalline TI thin film growth by MBE has been reported on a wide variety of single- crystalline substrates, e.g., Si,46–51Al2O3,52–56GaAs,57–60Ge,61,62 CdS,63SrTiO 3,64graphene (on 6H-SiC),65lattice-matched InP,66,67 and BaF 2,68as well as amorphous SiO 2/Si for back-gated electrical transport measurements69–71and fused silica,72indicating that these materials may grow on virtually anything.73 In fact, the in-plane lattice mismatch spans a remarkable range in the context of conventional thin film growth (see Table I in Ref. 44), reaching from 0.2% for Bi 2Se3on InP all the way up to 43.8% for Bi2Te3on graphene.44The reason for this enormous tolerance of lat- tice mismatches lies in the layered nature of the rhombohedral crystal s t r u c t u r e[ s p a c eg r o u p D5 3d(R/C223m)] of the (Bi,Sb) 2(Se,Te) 3compounds, w h i c hi sc h a r a c t e r i z e db yt h ev a nd e rW a a l s( v d W )g a p . Figure 1(a) illustrates a Bi 2Te3unit cell, in which the Te–Bi–Te–Bi–Te quintuple layer (QL) building blocks are illustrated, which are separated from one another by the vdW gap across which the Te–Te bonding is weak. In consequence, the bonding between the first QL and the substrate is also characterized by weak vdW forces. The vdW gap can be clearly seen in transmission electron microscopy images shown in Fig. 1(b) . The vdW epitaxy growth mechanism74–76does not require lattice matching between the film and the substrate but is nevertheless char- acterized by in-plane (rotational) alignment between the film and the substrate. van der Waals epitaxy has seen a renaissance with the advent of 2D materials, for which TIs provide a suitable substrate for the direct 2D materials growth or for integration with TIs in hetero- structures for future electronic devices.77The flip-side of the lack of strong ionic or covalent bonding across the film-substrate interface is a lack of control over the TI film growth. The growth of (Bi,Sb) 2(Se,Te) 3thin films by MBE is usually car- ried out with a considerable chalcogen overpressure on the order of >5:1 due to the high rate of re-evaporation. The growth rate of typi- cally less than 1 nm/min is controlled by the group-V element flux and varies as a function of substrate temperature. The substrate tem- perature, which typically lies in the range between 200/C14C and 300/C14C, has been shown to be the main parameter determining the quality of the films.44,78It has proven advantageous to first grow a seed layer at a temperature of /C2450/C14C lower than the final growth temperature,followed by an anneal under chalcogen flux.55,79The growth can be monitored in situ by reflection high-energy electron diffraction (RHEED). Figure 1(c) shows streaky RHEED patterns on Bi 2Te3onc- plane sapphire obtained along the ½11/C2220/C138and½10/C2210/C138azimuths, which repeat every 60/C14. The streaky patterns are indicative of flat, crystalline surfaces and they remain intact for doped films as well, up to the criti- cal concentration above which the streaks become diffuse (and the films rough), indicative of their non-substitutional incorporation.14–16 Note that RHEED cannot be continuously used for (Bi,Sb) 2(Se,Te) 3 growth as the electron beam visibly damages the films, most likely dueto local heating. For the evaporation of Bi and Sb, standard effusion cells are commonly used, whereas Se and Te are often (but not neces- sarily) evaporated out of special cracker cells. For the evaporation of dopants, high temperature effusion cells have proven advantageous since they are thermally better shielded, thereby reducing the uninten- tional evaporation of the high vapor-pressure chalcogens from sur- rounding areas. In fact, Se is very hard to contain and someunintentional Se “co-evaporation” is unavoidable, as shown in high- resolution compositional analysis using Rutherford backscattering spectroscopy with 2.3 MeV He ions and particle-induced x-ray emis- sion with 1 MeV protons. 16 The structural properties of the films are usually further investi- gated ex-situ using x-ray diffraction (XRD) with Cu K a1radiation. Figure 1(d) shows the spectrum for a Bi 2Te3film on BaF 2(111), which is characterized by the (1 1 1) substrate peaks and the (allowed) (0 0 3l) film peaks, representative of c-axis oriented, rhombohedral single crystalline material. Upon doping, the analysis of the resulting peak shifts and peak broadening can be used to quantify the undesired deg- radation in crystalline quality, as well as giving clues on the doping sce-nario. 18,80,81Furthermore, the in-plane lattice constants, as well as the rotational symmetry of the system, can be obtained from asymmetric 2D reciprocal space mapping or u-scans, as shown in Fig. 1(e) . Whereas the in-plane orientation of vdW epitaxially grown films, in principle, lock to the symmetry of the underlying substrate, this lock-ing may not be perfect. In particular (Bi,Sb) 2(Se,Te) 3films on c-plane sapphire are known to exhibit rotational twins, as can be seen in the 60/C14repeat of the streaky patterns in RHEED [cf. Fig. 1(c) ]—instead of the threefold symmetry expected from the R/C223mcrystal structure. In contrast, films grown on BaF 2(111) are twin-free,68which alters the electronic properties of the film as well.82Figure 1(e) shows a compari- son of the threefold symmetric f10/C2215gBi2Te3peaks on a smooth BaF 2surface (120/C14apart) and the additional peaks due to twinning (60/C14apart) on a rougher BaF 2surface (which dominates the growth on, e.g., c-plane sapphire). The film morphology, and the distribution of rotational domains, can be conveniently visualized using atomic force microscopy (AFM)owing to the dominating QL-terrace structure with its 1 nm-high steps. Figure 1(f) shows the characteristic, triangular terraced islands, which are rotated with respect to each other by 60 /C14in a Bi 2Te3film on c-plane sapphire. The triangular shape in the case of a hexagonal sys- tem can be understood in the framework of the Burton-Cabrera-Franktheory, when for particular crystallographic orientations, the distance between corners of the structure is larger than the diffusion length of the surface adatoms. 36In general, the growth is of the spiral surface growth type.83The growth spirals are believed to be due to the pinning of 2D growth fronts at irregular substrate steps.84The formation of screw dislocations in Bi 2Te3h a sb e e nl i n k e dt ov a r i a t i o n si nApplied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-2 Published under license by AIP Publishingstoichiometry of the deposited nucleation centers due to Te re- evaporation during the initial stages of film growth.36 This characteristic morphology of (Bi,Sb) 2(Se,Te) 3thin films poses a major challenge for their controlled doping [see illustration inFig. 1(g) ]. While during standard layer-by-layer growth the dopants can be incorporated in a deterministic way, allowing for the so-calledd-doping of individual layers, 85,86the exposed vdW gap of the terraced islands with its relatively weak electrostatic bonding forces and large FIG. 1. Structural properties and growth of Bi 2Te3thin films by MBE. (a) Crystal structure. On the left, the unit cell consisting of three Te–Bi–Te–Bi–Te quintuple layer (QL) blocks is shown. The QL blocks (Bi and Te are shown in blue and red, respectively) are separated by the weakly bonding Te–Te van der Waals gap. (b) Cross-s ectional high- angle annular dark field scanning transmission electron microscopy. The high-resolution image on the left shows the fine structure of the QL blocks and the van der Waals gap (no intensity). The overview scan on the right shows the perfectly ordered QL stack. (c) RHEED images of Bi 2Te3onc-plane sapphire along the ½11/C2220/C138and½10/C2210/C138azimuths, showing streaky patterns, indicative of flat, crystalline surfaces. (d) Out-of-plane h-2hXRD spectrum of Bi 2Te3on BaF 2(111). The film peaks (red) and substrate peaks (blue) are indicated. (e) In-plane XRD uscan of a f10/C2215gBi2Te3peak. On smooth substrates (black line), only one domain is found (120/C14apart), consistent with the threefold crystal symmetry. On rough substrates (blue line), two in-plane domains are found (60/C14apart). (f) The surface morphology of Bi 2Te3onc-plane sapphire, as obtained by AFM, is dom- inated by two domains of triangular islands, measuring /C241lm across. On the right, the close-up reveals details of the typical 1-QL-step terrace structure as well as the spiral- like growth center. (g) Schematic illustration of a Bi 2Te3island. The stepped structure exposes the van der Waals gap, which can act as an entry point for the efficient, but uncontrolled, diffusion of dopants into the material.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-3 Published under license by AIP Publishinglayer spacings provides an entry portal for the uncontrolled incorpora- tion of dopants.87This is particularly concerning since the diffusion constants in these layered hosts are highly anisotropic, with the diffu-sion along the planes (in the vdW gap) being particularly fast. 87–89The diffusing dopants are then incorporated into the (Bi,Sb) 2(Se,Te) 3host, illustrated by the concentration gradient in Fig. 1(g) ,a st h es o l u b i l i t y limit for interstitial metal incorporation is usually low. This is cer- tainly no surprise and a long-standing problem87in the thermoelec- trics community where the electric contacting with the obviouschoices of low electrical resistivity and high thermal conductivity metals has been challenging (with a few exceptions such as Ta 90) mostly resulting in the degeneration (or even complete dissolu-tion 91) of the electrodes in the case of, e.g., Sn,92Ag,93and Au.91 For instance, Cu is notorious for its high diffusivity in Bi 2Te3of 10/C06cm2s/C01parallel to the plane and 3 /C210/C015cm2s/C01along the c-axis at room temperature87and the formation of binary chalcoge- nides. For electrodes, specific diffusion barriers have been devel- oped (e.g., Ni barriers for Sn electrodes); however, some of themrely on the formation of secondary chalcogen compounds at the interface 92and so do not provide a solution for achieving con- trolled, substitutional doping of TIs. The need to break TRS in TIs led to a very active quest for suit- able dopants to introduce magnetic order. Magnetic TIs (MTIs) wereachieved with magnetic dopants, both with transition metals (TMs) and rare earths (REs). With TMs, magnetic order has been observed in Fe-, Mn-, Cr-, and V-doped compounds. For Fe-doping in Sb 2Te3and Bi 2Se3,t h e compounds remained paramagnetic while the electron concentration was increased, despite successful substitutional doping94,95and pd- hybridization predicted by spin density calculations.96In contrast, Kulbachinskii et al. reported ferromagnetic (FM) ordering up to 12 K in (Fe,Bi) 2Se3single crystals.97 In Mn-doped Tis, FM order has been observed in bulk crystals and thin films. In addition, there are reports on the surface magnetismin Mn:Bi 2Se3with observations of both an enhanced moment and TC due to surface segregation of the Mn ions,98as well as a diminished surface moment and soft magnetic behavior with no hysteresis possi-bly caused by partial antiferromagnetic (AF) ordering near the sur-face. 18A bandgap has been observed in Mn-doped TIs, but it is likely not of magnetic origin.99 Of all the single-ion dopants tried so far, Cr and V possess the most robust long-range FM order with out-of-plane anisotropy and a typical (doping concentration-dependent) TCof 59 K (104 K) for Cr (V) concentrations of x¼0.29 (0.25) in Sb 2/C0x(V/Cr) xTe3.23In Table I , we summarize the properties of the most common TI dopants. TABLE I. Summary of the physical properties of transition metal and rare earth element doped TI thin films. The doping concentration is given in % of the (Bi þSb) sites for sub- stitutional (SUB) doping, unless otherwise noted [e.g., for interstitial (INT) doping]. The magnetic moment is given per doping ion (in lB) and the transition temperature (trans. temp. TCorTNin K). The out-of-plane (OOP) magnetocrystalline anisotropy (MCA) is indicated when stated in the reference. Dopant TI hostDoping conc. (at.%) ValenceMagnetic moment (lB)Magnetic orderTrans. temp. (K) MCAOpen loop Comments References Cr Bi 2Se3 /C205.2*<3þ 2†FM 20‡… Yes*Poor cryst. quality for >5% Mn 80 †Max. for 2% Mn;‡max. for 5.2% Mn Cr Bi 2Se3 12 2 þ/C24 2.1 FM … OOP Yes SUB and INT 18 Cr Bi 2Se3 0.6†Mixed 2.49*FM 46*OOP†Effective conc. for modulation doping 104 2þ,3þ 1.3‡31‡*Surface value;‡bulk value Cr Sb 2Te3 7.5–21 2 þ 2.8 FM 28–125 OOP Yes SUB 105 Cr (Bi,Sb) 2Te3 5 <3þ 3.19 FM 20 OOP Yes Zener-type pd-exchange interaction 106 Cr (Bi,Sb) 2Te3 /C2015 3 þ 2.9 FM /C2059*OOP Yes*14–59 K for 5–15% Cr 23 VB i 2Se3 /C2012 … … FM /C2016*OOP Yes*10 K for 1%, 16 K for 6% 107 V (Bi,Sb) 2Te3 /C2013 Mixed 1.5 FM /C20104*OOP Yes*23–104 K for 4–13% V 23 3þ,4þ SUB VS b 2Te3 5 <3þ 1.84 FM 45 OOP Yes Zener-type pd-exchange interaction 106 Mn Bi 2Te3 /C2010 … … FM*/C2017 OOP Yes SUB and INT possible; FM*:/C212% Mn 108 Mn Bi 2Se3 … mostly 1.6*FM 1.5 OOP No XAS:109SUB and INT 19 2þ 5.1†*XMCD;†SQUID Mn Bi 2Te3 /C2013 … … FM*/C2015 OOP Yes vdW gap INT 110 *FM/C213% Mn; max TCfor 9% Gd Bi 2Te3 /C2030 3 þ 7 AFM /C02.5 … No XAS:111SUB 14,112 Dy Bi 2Te3 /C2036 3 þ 4.3–12.6 AFM /C01.2 … No XAS:111SUB 15 Ho Bi 2Te3 /C2021 3 þ 5.15 AFM /C00.837 … No XAS:111SUB 16 Eu Bi 2Te3 /C2042 þ … … … … … SUB, EuTe for 9% Eu 113 Eu Bi 2Se3 /C2021 … … FM 8–64 … Yes Nonuniform Eu SUB; FM /C2110% Eu 114Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-4 Published under license by AIP PublishingIn line with these findings, the QAHE has been observed in both Cr- and V-doped samples but not in Mn or Fe doped ones (except for the intrinsic magnetic TIs based on MnBi 2Te4).21,100However, the fact that even in these materials the QAHE is limited to very low tempera- tures, far below TC, calls the nature and robustness of the magnetic order into question. Early theoretical and experimental studies7,115,116on Cr- and V- doped TIs proposed that long-range magnetic ordering of the Van Vleck type could be established.117This type of magnetic order is sta- b i l i z e db yi n t r a - a t o m i cm i x i n go ft h e d-orbital ground state with higher energy excitations close in energy, which lead to a positive exchange integral. Observations of long-range magnetic order in Cr- and V-doped thin films and crystals were first investigated by super- conducting quantum interference device (SQUID) magnetometry, x-ray magnetic circular dichroism (XMCD), and polarized neutron reflectometry (PNR),18,80,105,106which reported a clear ferromagnetic transition and open hysteresis loops with out-of-plane anisotropy and, crucially, a doping concentration-dependent TC. This is incompatible with the notion of magnetic order dominated by intra-atomic interac- tions and, instead, strongly hints at a magnetic ordering mechanism that is dependent on either carrier or spin concentrations. T h ee x a c tf e a t u r e so ft h em a g n e t i co r d e r i n ga p p e a rt od e p e n do n the details of the modification of the electronic band structure due tothe dopant. For instance, in V-doped samples, impurity bands caused by the V 3 dstates are found near the Fermi energy, E F, and Dirac point of the system.118,119These states appear to stabilize the magnetic ordering. Both V and Cr show significant pdhybridization between the host 5 pstates and the dopant 3 dstates in Sb 2Te3based com- pounds as has been observed by a number of groups using XMCD.20,101,118,120Figure 2(a) shows the XMCD signal due to induced spin polarization in the Sb p-states for a Cr-doped Sb 2Te3 thin film. As the conduction and valence bands are mainly formed by the host lattice p-states, this leads to the question if it is not the carriers that mediate magnetic order in these systems. In the tetradymite dichalcogenides, near the center of the Brillouin zone, the valence band is formed by the anion p-states of Te or Se, whereas the conduc- tion band is formed by the p-states of the Sb and Bi cations, though it has recently been shown that the bonds of magnetically doped com- pounds have a strong covalent character.106 The important role carriers play in stabilizing the magnetic order was shown by the gate-voltage dependence of the magnetic relaxation time103and strongly supported by the dependence of TCon the con- centration of Bi, acting as a counter dopant, in V- and Cr-doped films.Yeet al. showed, by systematically varying the dopant and Bi concen- t r a t i o ni nC r : ðSb;BiÞ 2Te3and V: ðSb;BiÞ2Te3, not only that the mag- netic ordering is stabilized with the increasing doping concentration but also that TCis systematically suppressed when the carrier-hole con- centration is decreased by substituting Sb with Bi.101,120Figure 2(b) shows Arrott plots from which the values of TCare determined for systematically varied Cr and Bi concentrations in Cr x(Sb1/C0yBiy)2/C0xTe3 crystals. An increase in the Cr concentration stabilizes the magnetic order, whereas an increase in Bi strongly suppresses it. First principles calculations show the extent of the magnetic coupling in the c-axis direction across the vdW gap. The picture that emerges is that of a dominant carrier-mediated interaction based on the hybridization of thed-states of the dopants with the carrier bands formed by the p-states of the host lattice, similar to the ordering in dilute magneticsemiconductors.121,122In addition, it seems that the introduction of Bi weakens the pd-hybridization between the 5 pstates and the 3 d states.101Apart from this carrier-mediated exchange, there is evidence of exchange interactions mediated by the host ions.118,119 Notwithstanding, it is clear that the necessary tuning to reduceunwanted bulk carriers will always be detrimental to the long-range magnetic order in these compounds. It seems that the surface magnetism can differ from the bulk behavior, in particular, in Cr-based compounds. For instance, in Ref. 123, it was shown that the surface magnetization lies in-plane, rather than following the bulk out-of-plane anisotropy. However, the study by Ye et al. using surface-sensitive XMCD in total-electron yield mode on Cr-doped ðSb;BiÞ 2Te3did not find any evidence for a variation of the Cr-magnetism at the surface, neither did Duffy et al.20In the work by Liu et al. ,104it was reported that TCfor modulation-doped Bi 2Te3 increased if the Cr dopants were introduced at the sample surface. U s i n gm u o ns p i nr o t a t i o n( lþSR) techniques, the magnetic vol- ume fraction can be directly tracked as a function of temperature in Cr- and V-doped thin films of Sb 2Te3and other TIs.102,124These mea- surements show that ferromagnetic ordering develops rather gradually over a wide temperature range and that for lower doping concentra- tions, a significant fraction of the material remains paramagnetic. The magnetic transition can also be tracked through the slow relaxation rate during the phase transition in lSR measurements [see Fig. 2(c) ]. This relaxation rate is a measure of the static and ls-dynamic mag- netic disorder in the system. This peak is very broad, and, for lower doping concentrations, disorder persists down to at least 4 K, the low- est temperature reachable in the system. Furthermore, a change of the internal magnetic field experi- enced by the muons in the samples provides evidence for a percola- tion transition, above which magnetically ordered clusters gradually appear in a paramagnetic sea, growing and percolating as the temperature is decreased.102For lower doping concentrations, the internal field remains shifted even at the lowest tempera- tures.102,124This behavior is likely due to residual paramagnetic regions, which remain between the ferromagnetic patches. Note that there can also be a large difference in percolation between sur- face and bulk due to different screening behaviors.125Lachman et al.103investigated the magnetic domain pattern in a Cr-doped (Sb,Bi) 2Te3film using scanning SQUID-on-tip microscopy [see Fig. 2(d) ]. In their measurements, the nearly non-interacting, slowly fluctuating, superparamagnetic domains were directly imaged. These measurements show directly that the magnetic order in TM-doped TIs is not robust over long lengthscales. Instead, a picture of superparamagnetic or superferromagnetic domains emerges, which gradually form during a broad transition, and with significant disorder, and paramagnetic regions remaining intact until far below TC[see illustration in Fig. 2(e) ]. With an increase in doping concentration, the distances between the dopant ions shrink and additional carriers are introduced into the system. The net effect is the stabilization of the magnetic order, but at the cost of severely compromising the quantum properties of the material. The phenomenon of nanoscale magnetic phase separation is known from dilute magnetic semiconductors, such as GaMnAs. There, depending on the extent of the localization of the (hole) car- riers, either the magnetic transition follows the percolation dynamics of bound magnetic polarons in the case of strongly localized holes126Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-5 Published under license by AIP Publishingor, in the case of hole states extended over distances longer than the average dopant distance, the Zener model becomes valid.121However, even in the Zener model description, paramagnetic regions may persist below TCin hole-free regions.122 Rare earth dopants are extremely attractive given their large, localized magnetic moments and their similarity to the Bi and Sb ionsin the host lattice. The ionic radius of Dy 3þis 105 pm, much more similar to that of Bi3þ(117 pm) than to Cr2þin the high (low) spinstate of 94 (87) pm. Doping with a high moment ion has the advantage that as the exchange gap is dependent on the size of the magnetization, the doping concentration can be reduced for the same gap size, thereby preserving crystal quality. Of the rare earth lanthanide series,the high moment materials Gd, Dy, and Ho were all used to dopeBi 2Te3.14–16However, none of these materials display long-range mag- netic order by themselves.14–17,127Doping concentrations of up to 35% were achieved in Gd-doped Bi 2Te3thin films; much above the bulk FIG. 2. Magnetic properties of MTI films. (a) X-ray absorption (XAS) and x-ray magnetic circular dichroism (XMCD) spectra at the Sb M4;5edges of an in situ cleaved Cr:Bi 2Te3film. The measurement was carried out in a field of 8 T and at a temperature of 3 K. The XMCD signal (green) is obtained by subtracting the XAS spectrum measur ed with right-circularly polarized x-rays (blue) from the one measured with left-circularly polarized x-rays (red).20(b) XMCD intensities (symbols) and SQUID magnetization data (lines) obtained for different Cr doping concentrations xand different Sb to Bi ratios in counter-doped Cr x(Sb 1/C0yBiy)2/C0xTe3crystals.101From the plot of the inverse intensity (i.e., magnetization) in the panel on the right, the transition temperature is estimated, which shows a clear trend: TCincreases with x, whereas an increase in Bi:Sb ratio strongly reduces it. (c) Temperature-dependent transverse-field lþSR measurement showing the magnetic transition in a Cr:Sb 2Te3film.102The data shown in red represent the behavior of the near-surface region of the film, whereas the (higher muon energy) data shown in black include the information from the middle of the fi lm as well. The relax- ation rate k, which is a measure of the static and dynamic ( ls) magnetic disorder in the system, is very broad across the magnetic transition. (d) SQUID-on-tip microscopy showing the magnetization reversal dynamics of a Cr:(Bi,Sb) 2Te3film. (Left) magnetic images, i.e., the Bzcomponent of the film magnetization, obtained in applied magnetic fields increasing (0.5 mT field steps) at a temperature of 250 mK. (Right) change in the film’s magnetic flux, illustrating the reversal of isolated domain s.103(e) Illustration of the magnetic order in TM-doped TIs, showing magnetically ordered and paramagnetic regions coexisting at temperatures far below TC. Panel (a) adapted from Duffy et al., Phys. Rev. B 95, 224422 (2017). Copyright 2017 Authors, licensed under a Creative Commons Attribution CC BY 4.0 license. Panel (b) reproduced with permission from Y eet al. , Nat. Commun. 6, 8913 (2015). Copyright 2015 Authors, licensed under a Creative Commons Attribution CC BY 4.0. license Panel (d) adapted from Lachman et al. , Sci. Adv. 1, e1500740 (2015). Copyright 2015 Authors, licensed under a Creative Commons Attribution CC BY 4.0 license.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-6 Published under license by AIP Publishingsolubility limit.81The details of the magnetic configurations are given inTable I . Of the investigated RE dopants, Dy shows the most interest- ing properties: it is the only ion that leads to a concentration- dependent moment, its susceptibility is large, and thin films can be magnetically saturated in fields as low as /C242T . P N R m e a s u r e m e n t s have shown that the induced moment can be as large as /C241/3 of the full Dy3þmoment at fields as low as 0.65 T.128lþSR measurements showed that while no long-range order can be established, strong, short-range magnetic correlations establish themselves with decreasing temperature.128This makes Dy-doped TIs a prime candidate for fur- ther exploitation in heterostructures, where order can be introduced via proximity coupling.129,130 Incorporating TIs into magnetic heterostructures has been explored both as an alternative to magnetic doping and as an added materials design opportunity to tune the magnetic properties. Magnetic heterostructures offer the advantage of, e.g., opening an exchange gap in the TI with less disorder than in doped systems since no extra scattering centers need to be introduced. It is proposed that when a non-magnetic TI is combined with another magnetically ordered material, proximity effects at the interface involving the sur- face Dirac fermions will align spin moments of the TI band with the itinerant carriers and give rise to exchange splitting. Similar exchange mechanisms are used in stacks with MTIs to alter their magnetic prop- erties. Engineering heterostructures by incorporating magnetic and non-magnetic TIs has therefore opened up new avenues to access emergent quantum phenomena. In the following, we review some of these aspects of magnetic heterostructures incorporating TIs. Several strategies have been followed in order to exploit magnetic proximity effects at the interface of such heterostructures with the goal of tuning and controlling the magnetic and quantum transport prop- erties of MTIs. Modulation doping in heterostructures with Cr-doped (Bi,Sb) 2Te3films has increased the QAHE temperature from /C2430 mK up to 2 K.131The designed structures are illustrated in Figs. 3(a) and 3(b), and the characteristic magnetotransport data showing the QAHE are plotted in Fig. 3(c) . This approach has also allowed for the observa- tion of the axion insulator state in MTIs132—a magnetoelectric phe- nomenon characterized by a large longitudinal resistance and zeroHall plateau (where both the Hall and longitudinal conductivity become zero), which is illustrated in Fig. 3(d) . In order to raise the temperature at which quantum and magneto- electric effects in MTIs can be observed, it seems natural to explore ways to increase their magnetic ordering temperature by, e.g., interfacial mag- netic interactions. Growth of Cr-doped Bi 2Se3films onto the high TC (/C24550 K) ferrimagnetic insulator Y 3Fe5O12(YIG) was found to enhance theTCof the MTI from /C2420 K up to /C2450 K.133The evaporation of a thin FM layer, such as Fe or Co, onto the pristine surface of in-vacuum cleaved Mn-doped Bi 2Te3crystals or Cr-doped Bi 2Se3,C r - d o p e dS b 2Te3,and Dy- doped Bi 2Te3thin films was also demonstrated to alter their TCnear the surface of the TI through proximity.20,127,134,135Figures 3(e) and3(f)depict the increase in TCfor Cr-doped Bi 2Se3and Sb 2Te3fi l m sw i t haC oo v e r - layer, as obtained from XMCD measurements. Growth of high-quality Dy:Bi 2Te3/Cr:Sb 2Te3heterostructures resulted in successful introduction of long-range magnetic order in Dy:Bi 2Te3as a result of proximity coupling to the higher transition temperature Cr:Sb 2Te3layer.129 Alternative approaches to raise TCof MTIs include their cou- pling to antiferromagnets. An example is superlattices of the AF CrSb and Cr-doped (Bi,Sb) 2Te3,w h e r e TCwas found to increase from/C2438 K up to /C2490 K.136,137The successful design of such structures relies on the fact that these materials are chemically very compatible and well lattice-matched, resulting in abrupt interfaces which support the magnetic exchange across the interface. This concept of using materials with related or at least comparable crystal structure and simi- lar atomic compositions has also been explored, both theoretically and experimentally, in heterostructures of MnBi 2Te4/Bi2Te3,M n B i 2Se4/ Bi2Se3,M n S e / B i 2Se3, and similar compounds.138–141In particular, het- erostructures of, e.g., MnBi 2Te4and Bi 2Te3, which are part of the (MnBi 2Te4)(Bi 2Te3)mhomologous series,142are particularly suited as there are no real interfaces (the MnBi 2Te4layer is effectively termi- nated by Bi 2Te3layers). This makes the magnetic film a natural exten- sion of the TI and allows for efficient TRS breaking. In fact, some of these systems exhibit ferromagnetism up to room temperature and a clear Dirac cone gap opening of /C24100 meV.140 The latter examples have an additional advantage in terms of the electrical characteristic of the magnetic layer. They use magnetic insu- lators (MIs), instead of metallic films, which preserve the band struc- ture of the TI, and thus the TSS, as they avoid hybridization with the bulk states of the AF or FM in contact. Using a MI is also convenient for the fabrication and incorporation of such structures into quantum devices. Intensive investigation of proximity effects has been done in heterostructures comprising TIs and MIs, such as EuS/Bi 2Se3,E u S / (Bi,Sb) 2Te3,( B i , S b ) 2Te3/Y3Fe5O12,( B i , S b ) 2Te3/Tm 3Fe5O12,B i 2Te3/ Fe3O4,( B i , S b ) 2Te3/Cr2Ge2Te6, and (Zn,Cr)Te/(Bi,Sb) 2Te3/ (Zn,Cr)Te.124,143–151However, the QAHE has only been reported in (Zn,Cr)Te/(Bi,Sb) 2Te3/(Zn,Cr)Te,150which results from the appropri- ate design of the heterostructure to guarantee smooth interfaces (as it satisfies the conditions described above), as well as the maximizationof the exchange gap at the top and bottom surface states by using a MI/TI/MI sandwich stack. Proximity effects due to exchange interactions are not the only interesting phenomena that emerge at TI interfaces. The helical locking of the TSS, illustrated in Fig. 3(g) , and the strong spin–orbit coupling are very relevant for their application in topological spintronics. The exis- tence of such a spin texture means that an electron current flowing onthe surface of a TI can generate a non-equilibrium spin density [see Fig. 3(h)] with both in-plane and out-of-plane components. The spin- polarized surface currents are expected to efficiently induce out-of-plane and in-plane torques in an adjacent magnetic layer. Experiments of spin- pumping using ferromagnetic resonance (FMR) techniques in FM/TI and FM/TI/FM stacks, as the one depicted in Fig. 3(i) ,h a v ec o n fi r m e d t h ea b i l i t yo faT It oa b s o r ba n dt r a n s f e rap u r es p i nc u r r e n t . 152–156 Figure 3(j) shows signatures of transfer of angular momentum between the FMs Co 50Fe50and Ni 81Fe19(Py, Permalloy), mediated by Bi 2Se3,a s observed in the amplitude and phase of precession of each FM layer recorded in time-resolved FMR measurements.153,155 The absence of bulk conduction in ideal TIs further increases their spin-charge conversion efficiency. Exceptionally large spin- charge conversion values at TI/FM interfaces have been experimen- tally demonstrated by spin–orbit torque (SOT) measurements.157–161 Figure 3(k) shows a diagram of a Py/Bi 2Se3bilayer, where the non- equilibrium magnetization produced by an in-plane charge current in SOT measurements is represented. The measured resonance line shape plotted in Fig. 3(l) has two components: a symmetric or antidamping- like SOT (in-plane) and an antisymmetric or field-like SOT (perpen- dicular). Analysis of these FMR curves yields spin-charge conversionApplied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-7 Published under license by AIP Publishingparameters that are found to be much larger than those reported for heterostructures of FM with heavy metals.157SOT in such FM/TI het- erostructures can be used to efficiently manipulate the magnetization of the FM—a highly relevant feature for their applications in spin- tronic devices. Effective magnetization switching by SOT at room tem-perature has been demonstrated in, e.g., Py/Bi 2Se3,160CoTb/Bi 2Se3, and CoTb/(Bi,Sb) 2Te3structures.162 Despite the huge effort striving to make practical use of the quan- tum and spin phenomena at the surfaces and interfaces of TIs anddespite the remarkable progress in the growth of thin films and com- plex heterostructures, there are a number of limitations intrinsic tothese materials and many questions are still waiting to be addressed. Most of the experiments reviewed here rely on highly (charge) doped TIs in which the TSS coexist with bulk bands at the Fermi level, andfor their analysis, it is assumed that the FM/TI interface is perfectlyabrupt. However, magnetic phase segregation and interdiffusion arecommon issues in doped TIs and at FM/TI interfaces, and only recently, the systematic analysis of FM/TI interfaces has started, FIG. 3. Emergent phenomena in TI heterostructures. Schematic Cr-modulation-doped (a) quantum anomalous Hall and (b) axion insulators. Panels (c) and (d) s how the mag- netic field ( B) dependence of the Hall conductivity ( rxy) and longitudinal conductivity ( rxx) in the respective heterostructure films illustrated in (a) and (b). Panels (e) and (f) depict linear fits of parameters extracted from Arrott plots obtained from XMCD measurements in Cr-doped (e) Bi 2Se3and (f) Sb 2Te3thin films. The temperature at which the intercept goes through zero yields TC. The inset illustrates the process of in situ evaporation of a thin Co layer on top of the Cr-doped TI film. (g) Schematic illustration of the spin-momentum locked helical spin texture of the TSSs in a TI. (h) Schematic of surface spin density on two opposite surfaces of a TI thin film for a charge current flowing along the /C0xdirection and for a charge current flowing along the þxdirection. (i) Schematic of a FM/Bi 2Se3/FM heterostructure for spin pumping experiments with FMR and time-resolved FMR. (j) Phase (top panel) and amplitude (bottom panel) of precession of magnetization for Py (pink circles) and Co 50Fe50(green triangles) layers for tTI¼ 4 nm. Dashed lines show the positions of the resonance amplitude peak. (k) Schematic diagram of SOT measurements in a Py/Bi 2Se3bilayer. The yellow and red arrows denote spin moment directions. (l) Measured field dependent (top panel) spin-torque FMR at room temperature for a Py (16 nm)/Bi 2Se3(8 nm) bilayer, together with fits of the symmetric and antisymmetric resonance components. Bottom panel: Measured dependence on the magnetic field angle ufor the symmetric and antisymmetric resonance components. Panels (a)–(d) reprinted with permission from Mogi et al. , Nat. Mater. 16, 516 (2017). Copyright 2017 Springer Nature. Panel (e) adapted and reprinted with per- mission from Baker et al. , Phys. Rev. B 92, 094420 (2015).135Copyright 2015 American Physical Society. Panel (f) adapted from Duffy et al. , Phys. Rev. B 95, 224422 (2017). Copyright 2017 Authors, licensed under a Creative Commons Attribution CC BY 4.0 license. Panels (i) and (j) reprinted with permission Figueroa et al. , J. Magn. Magn. Mater. 400, 178 (2016). Copyright 2016 Elsevier. Panels (k)–(l) reprinted with permission from Mellnik et al. , Nature 511, 449 (2014). Copyright 2014 Springer Nature.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 150502 (2020); doi: 10.1063/5.0027987 117, 150502-8 Published under license by AIP Publishingincluding practical strategies for improving their quality.161,163The fact that the QAHE and other quantum, magnetoelectric, and spin phenomena continue to be only observable at very low temperaturescan, in fact, be associated with this disorder at the topological interface.Therefore, there is a clear path to improve the quality of these systems, and there is still real potential for significantly increasing the energy and temperature scales at which these intriguing quantum effects canbe observed, opening the door for useful quantum devices in thefuture. 26So, coming back to the question we raised in the abstract, i.e., w h e t h e rt h e r ei sh o p ef o rm a g n e t i c a l l yd o p e d3 DT I si nt h e (Sb,Bi) 2(Se,Te) 3family, we hope to have been able to provide some evi- dence for answering the question with a cautious “yes.” The authors express their deep gratitude to the medical care personnel during the Covid-19 pandemic. We acknowledge continuoussupport for the TI project at Oxford from the John Fell Fund (Oxford University Press), the synchrotron facilities Diamond Light Source, ALS, BESSYII, ALBA, and the ESRF and the neutron sources ISIS and ILLfor beamtime leading to the reported insights. A.I.F. acknowledgesfunding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 796925, the FET-PROACTIVE project TOCHA underGrant Agreement No. 824140 and the H2020 European ResearchCouncil PoC project SOTMEM under Grant Agreement 899896. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007). 2H. J. Goldsmid and R. W. Douglas, Br. J. Appl. Phys. 5, 386 (1954). 3H. J. Goldsmid, A. R. Sheard, and D. A. Wright, Br. J. Appl. Phys. 9, 365 (1958). 4H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. 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5.0016989.pdf

AIP Conference Proceedings 2265 , 030621 (2020); https://doi.org/10.1063/5.0016989 2265 , 030621 © 2020 Author(s).Up-converting rare earth phosphor yttrium doped TiO2 material for dye sensitized solar cells application Cite as: AIP Conference Proceedings 2265 , 030621 (2020); https://doi.org/10.1063/5.0016989 Published Online: 05 November 2020 Navdeep Kaur , Aman Mahajan , and D. P. Singh ARTICLES YOU MAY BE INTERESTED IN New candidate for red phosphor applications AIP Conference Proceedings 2265 , 030221 (2020); https://doi.org/10.1063/5.0016601 Ambient-air fabrication with inorganic/polymer hole transport layer: Towards low cost perovskite solar cells AIP Conference Proceedings 2265 , 030296 (2020); https://doi.org/10.1063/5.0017180 Electrospun 1D TiO 2 nanorods for enhancing the performance of dye sensitized solar cells AIP Conference Proceedings 2265 , 030620 (2020); https://doi.org/10.1063/5.0016993Up-converting Rare Earth Phosphor Yttrium doped TiO 2 material for Dye Sensitized Solar Cells Application Navdeep Kaur1, Aman Mahajan1, a) and D. P. Singh1 1Department of Physics, Guru Nanak Dev University, Amritsar 143005, India a)Corresponding author: aman.phy@gndu.ac.in Abstract. In this paper, the up-conversion effect of rare earth (RE) met als that luminesces from over near infrared through visible to UV region of solar spectrum; on the optical properties of TiO 2 were investigated for their potential application as photoanode material in dye sensitized solar cell s (DSSCs). A series of yttrium (Y) RE metal doped 3 m thick TiO 2 at varied concentrations of RE p recursor (1-20 mM); were prepa red using hydrothermal technique. Further, the structural and optical properties of Y doped TiO 2 were studied through FESEM, UV-Vi s and photoluminescence (PL) spectroscopic techniques; and clai ms the optimized porosity wit h enhanced optical absorption ability of TiO 2 attributed to the up-conversion from NIR to UV light of Y; as well as reduced recombination as Y acts as scavenger for photo- generated electrons in TiO 2. Hence, the development of such visible light activated Y dope d TiO 2 materials could pave a way to efficient utilization of solar spectrum for the developm ent of high performing DSSCs. 1. INTRODUCTION Over the last few decades, global energy crises have led to the worldwide research in the utilization of predominant, environment-friendly and inextinguishable renewabl e solar energy1. DSSCs have achieved growing focus among different energy photo-voltaic devices; owing to it s inexpensiveness, ease of fabrication and better operation under dim light conditions2. In typical DSSCs, TiO 2 adsorbed dye sensitized photoanodes absorbs incident sunlight, and transport the photo-generated electron throughout the circuit to Pt counter electrodes, via intermediate redox electrolyte. TiO 2 regarded as the key component of DSSC, exhibits wide band gap (~ 3.2 eV), photo-chemical stability, bio-compatibility, non-toxicity and low cost3. However, TiO 2 photoanodes absorb only 5% of the solar energy, and the rest is unused. Besides, TiO 2 rapidly recombines the photo-generated charge carriers due to frequent trapping and de-trapping events; resulting in decreased photovo ltaic performance of DSSCs4. Thus, exploration of variedly modified TiO 2 is a necessity nowadays to overcome the aforementioned issue via incorporation of metals and non-metals, up and down converting phosphors, hierarchicall y nano-structured and scattering top layer depositions etc.; that can efficiently absorb wide spectrum of sunlight as well as reduces recombination5-7. Recently, RE elements such as lanthanum, cerium, erbium, yttriu m , n e o d y m i u m e t c . ; p o s s e s s i n g u n i q u e u p - conversion luminescence from over the near infrared light throu gh the visible to the UV region of solar spectrum; have attracted much attention in the same direction8, 9. The up-conversion of RE i.e. conversion of low energy photons (NIR) into high energy photons (UV); emerges due to own 4f and vacant 5d orbitals; generating intermediate energy states and empowering multiple electron con figuration10. Y have been chosen as a dopant in TiO 2 among RE elements with its recognitions of sharp, near monochr omatic emission lines, exceptional electronic and optical properties; that would facilitate their possible in teraction with TiO 2 surface to further enhance their light absorption range and inhibit back recombination reactions11. Till now, various strategies to incorporate Y in TiO 2 have been explored in literature12, 13. In the present paper, we report the optimized concentration of Y doping in TiO 2 prepared using hydrothermal method. The enhanced optical properties with acute structural a nalysis of optimized Y doped TiO 2, would pave a way for their effective utilization as photoanode material in e fficient DSSCs. DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030621-1–030621-4; https://doi.org/10.1063/5.0016989 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030621-12. EXPERIMENTAL PROCEDURE Fluorine doped tin oxide (FTO) glass substrate (sheet resistivi ty ~7 Ω sq.-1), titanium (IV) isopropoxide (TTIP), yttrium(III) oxide (Y 2O3), nitric acid (HNO 3), and Di-tetrabutylammonium cis-bis(isothiocyanato)bis(2,2'- bipryridyl-4,4'-dicarboxylato)ruthenium(II) (N719) dye of analy tical grade were purchased from Sigma Aldrich. Titanium dioxide (TiO 2) paste was procured from Dyesol, Australia. TiO 2 compact layer was deposited using spin coating method on pre-cleaned FTO substrates subjected to annea ling at 450ºC for 30 min. TiO 2 thin film (3 m thick) was doctor bladed on annealed films followed by annealin g at 450ºC for 30 min. in air. The prepared TiO 2 films were doped with Y using hydrothermal method. TiO 2 films were dipped in different molar solution of Y 2O3 in HNO 3 (1, 5, 10, 15 and 20 mM) poured in teflon autoclave and kept at 100oC for 12 hours. Further, the Y doped TiO 2 films were annealed at 500ºC for 2 h. The prepared films were characterized to study their optical properties through UV-Vis and PL spectroscopy. UV-VIS-NIR 3600 spectromete r was used to record the absorption spectra. Field emission scanning electron microscope (FESEM) images were obtained from FESEM-Carl Zeiss, Supra 55. PL study was performed on PerkinElmer, LS 55 Fluorescence Spect rometer at an excitation wavelength of 620 nm. 3. RESULTS AND DISCUSSION The doping of Y in TiO 2 was conf irm ed thro ugh p relim inary i nvesti gatio ns of X-ray d iff raction; showing the anatase phase formation and as well X-ray photoelectron spectro scopy confirming the doping of Y element in TiO 2. Fig. 1 represents the UV-Vis absorption spectra of different un doped and Y doped TiO 2 films at varied concentrations. Undoped TiO 2 clearly shows a broad and intense absorption band around 390 n m (UV region only); attributed to the characteristic absorption of electron from va lence band (V.B.) to the conduction band (C.B.) of anatase TiO 2. A red shift in the absorption edge towards the visible region is observed with the doping of Y in TiO 2 up to 10 mM; as the stronger charge transfer interactions takes place between TiO 2 and Y via intra 4f electrons. Further, at higher concentration (15 and 20 mM) of Y doping, th e absorption edge shows a blue shift; ascribed to the occupied Ti 3d states of C.B. of TiO 2. FIGURE 1 . UV-absorption spectra of undoped TiO 2 (a) and Y doped TiO 2 at (b) 1 mM, (c) 5 mM, (d) 10 mM, (e) 15 mM and (f) 20 mM concentration. FIGURE 2. UV-absorption spectr a of N719 dye loaded undoped TiO 2 (a) and Y doped TiO 2 at (b) 1 mM, (c) 5 mM, (d) 10 mM, (e) 15 mM and (f) 20 mM concentration. UV-Vis absorption spectra of undoped and Y doped TiO 2 adsorbed N719 dye (Fig. 2) depicts broad absorption in the visible region (400-650 nm); attributed to th e transitions of N719 dye molecules. It is observed that the absorbance increases up to 10 mM concentration of Y and the n suddenly decreases in comparison to undoped TiO 2. The enhanced absorption occurs due to the transition of 4f el ectrons of Y favoring higher adsorption of dye on TiO 2 surface; and further reduced absorbance occurs due to the grad ual movement of C.B. of TiO 2. FESEM images of Y doped TiO 2 at 10 and 15 mM concentration is shown in Fig. 3. Y doped TiO 2 (10 mM) (Fig. 3 (a)) exhibits uniformly dispersed porous TiO 2. With increase in doping concentration (15 mM) (Fig. 3 (b)), the conglomeration of TiO 2 nanoparticles occurs; resulting in its reduced porosity. 030621-2. FIGURE 3. FESEM images of Y doped TiO 2 at (a) 10 mM and (b) 15 mM concentration. Fig. 4 shows the luminescence PL spectra of undoped and Y doped TiO 2; representing broad emission peak of undoped TiO 2 around 450-550 nm associated with the recombination within the band gap as well as surface defect of host TiO 2. Y doped TiO 2 exhibits similar PL spectra with lower PL intensity as that of TiO 2; indicating that new defect levels are not created in band gap of TiO 2 with Y doping. Moreover, the PL intensity of Y doped TiO 2 decreases with increase in doping concentration up to 10 mM; an d further increase in Y doping increases PL emissions. The decreased PL intensity results from the trapping of photo-generated electrons within Y; hence inhibiting maximum back recombination reactions for 10 mM conce ntration of Y in TiO 2. FIGURE 4. PL spectra of undoped TiO 2 (a) and Y doped TiO 2 at (b) 1 mM, (c) 5 mM, (d) 10 mM, (e) 15 mM and (f) 20 mM concentration. CONCLUSIONS In conclusion, optimized 10 mM concentration of Y doped TiO 2 sufficiently enhances its light absorption capability, observed from UV-Vis spectra; via converting unused NIR light into visible region of the solar s pectrum. A s w e l l , Y d o p i n g i n T i O 2 inhibits the recombination of photo-generated electrons shown from the reduced emissions PL intensity. Moreover, the higher doping concentrati on of Y leads to the agglomerated TiO 2 nanoparticles; which reduces the light harvesting. Hence, highe r light absorption and reduced recombination opts out the optimized Y doped TiO 2 (10 mM) as efficient photoanodes in DSSCs. ACKNOWLEDGEMENT One of the authors, Navdeep Kaur, is thankful to UGC New Delhi, India for awarding Senior Research Fellowship (SRF). Authors are also thankful to SERB, New Delhi, for providing financial assistance under project no. EMR/2016/003408. (b) (a) 030621-3REFERENCES 1. J. Gong, K. Sumathy, Q. Qiao and Z. Zhou, Renewable and Sustain able Energy Reviews 68, 234-246 (2017). 2. A. Hagfeldt, G. Boschloo, L. Sun , L. Kloo and H. Pettersson, Ch emical Reviews 110 (11), 6595-6663 (2010). 3. A. Gupta, I. C. Maurya, Neetu, S. Singh, P. Srivastava and L. B ahadur, ENERGY AND ENVIRONMENT FOCUS 6 (2017). 4. M. Adachi, Y. Murata, J. Takao, J. Jiu, M. Sakamoto and F. Wang , Journal of the American Chemical Society 126 (45), 14943-14949 (2004). 5. F. Zhao, R. Ma and Y. Jiang , Applied Surface Science 434, 11-15 (2018). 6. P. Rai, Sustainable Energy & Fuels 3 (2018). 7. J. C. Goldschmidt and S. Fische r, Advanced Optical Materials 3 (4), 510-535 (2015). 8. G. Xie, J. Lin, J. Wu, Z. Lan, Q. Li, Y. Xiao, G. Yue, H. Yue a nd M. Huang, Chinese Science Bulletin 56 (1), 96-101 (2011). 9. J. Nadolna, T. Grzyb, J. Sobczak, W. Lisowski, M. Gazda, B. Oht ani and A. Zaleska-Medynska, Applied Catalysis B: Environmental 163 (2014). 10. J. Yu, Y. Yang, R. Fan, D. Liu, L. Wei, S. Chen, L. Li, B. Yang and W. Cao, Inorganic Chemistry 53 (15), 8045-8053 (2014). 11. F. Zhou, C. Yan, Q. Sun and S. Kom arneni, Microporous and Mesop orous Materials 274 (2018). 12. K. L. Reddy, S. Kumar, A. Kumar an d V. Krishnan, Journal of Haz ardous Materials 367, 694-705 (2019). 13. W. Kallel, S. Chaabene and S. Bouattour, Physicochemical Proble ms of Mineral Processing (2019). 030621-4

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J. Chem. Phys. 153, 164710 (2020); https://doi.org/10.1063/5.0024499 153, 164710 © 2020 Author(s).Assessment of PBE+U and HSE06 methods and determination of optimal parameter U for the structural and energetic properties of rare earth oxides Cite as: J. Chem. Phys. 153, 164710 (2020); https://doi.org/10.1063/5.0024499 Submitted: 07 August 2020 . Accepted: 08 October 2020 . Published Online: 27 October 2020 Shikun Li , Yong Li , Marcus Bäumer , and Lyudmila V. 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Moskaleva1,2,a) AFFILIATIONS 1Institute of Applied and Physical Chemistry, Faculty 02, University of Bremen, 28359 Bremen, Germany 2Department of Chemistry, University of the Free State, P.O. Box 339, Bloemfontein, South Africa a)Author to whom correspondence should be addressed: lyudmila.moskaleva@gmail.com ABSTRACT Rare earth oxides are attracting increasing interest as a relatively unexplored group of materials with potential applications in heteroge- neous catalysis and electrocatalysis; therefore, a credible and universal computational approach is needed for modeling their reactivity. In this work, we systematically assessed the performance of the PBE+U method against the results of the hybrid HSE06 method with respect to the description of structural parameters and energetic properties of the selected hexagonal lanthanide sesquioxides and the cubic fluorite- type cerium dioxide. In addition, we evaluated the performance of PBE+U in describing the electronic structure and adsorption properties of the CeO 2(111) and Nd 2O3(0001) surfaces. The HSE06 method reproduces rather well the lattice parameters and selected energetic prop- erties with respect to the experimental values. The PBE+U method is able to reproduce the results of HSE06 or the experimental values only if the U parameter is selected from an appropriate range of values. The U value around 3 eV gives the best description of the lattice parameters of most bulk oxides. 2 eV–3 eV is also found to be the optimal range of U for the reaction energies of bulk La 2O3, Ce 2O3, Nd 2O3, Er 2O3, and Ho 2O3. U = 1 eV gives the best results for Pr 2O3, Pm 2O3, Eu 2O3, Tm 2O3, and Lu 2O3, whereas Gd 2O3could not be accurately described by the PBE+U method. The U values ( ∼3 eV) found optimal for most bulk oxides also work well in the calculations of adsorption of small molecules on Nd 2O3(0001) and CeO 2(111), although larger U values are required to obtain sufficient localization of 4felectrons. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024499 .,s I. INTRODUCTION Rare earth metal oxides (REOs) have attracted increasing atten- tion over the past few decades due to their extensive application as luminescent materials, ceramics, magnets, conductors, and cata- lysts. REOs have attracted interest as catalysts for many reactions, such as low-temperature carbon monoxide oxidation,1–3carbon dioxide methanation,4and oxidative coupling of methane (OCM).5 The OCM reaction converting methane to value-added chemicals, ethylene and ethane,6–9possesses a potentially enormous economic value. Some of the known REOs-based catalysts were shown to behighly active for the OCM reaction, especially lanthanum oxide doped by Na or alkaline earth metals (Sr, Mg, and Ba), as was indi- cated by Zavyalova et al.10who performed a statistical analysis on the basis of a database consisting of catalytic data of catalysts studied in the past. The C2 (ethane and ethylene) yield of the unpromoted La2O3, Sm 2O3, and Nd 2O3could even reach 10%–12%, whereas Sm 2O3promoted by alkali metal Li achieved the highest C2 yield of 22%.11,12However, further improvement in the catalytic activ- ity of REOs is difficult to achieve because a detailed understanding of the microscopic structures and catalytic properties of REOs is lacking. J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp At present, theoretical modeling, especially density functional theory (DFT), is the state-of-the-art approach to investigate the reaction mechanisms and electronic structure and to assist in the design of highly efficient catalysts. The first member of the REO series La 2O3has been successfully modeled in several theoretical studies focusing on the surface properties, surface active sites, sur- face oxygen vacancy formation, and hydrogen abstraction from methane.13–16Owing to its high activity for a range of reactions, cerium dioxide (CeO 2) has also been well studied, e.g., with respect to Ce 4 felectron localization in the bulk oxide,17the activation of O 2on low-index ceria surfaces,18and oxygen vacancy forma- tion and methane activation on the surfaces doped by Pd19–21or Fe/Co/Ni.22When referring to higher lanthanide oxides, Sm 2O3, Nd 2O3, etc., standard DFT in the form of the local-density approx- imation (LDA) or generalized gradient approximation (GGA) fails to properly describe their electronic structure, magnetic, and ther- modynamic properties owing to the self-interaction error, which becomes large in systems with strongly correlated 4 felectrons.23 A series of advanced DFT-based methods, such as self- interaction correction,24GW,25,26dynamical mean-field theory (DMFT),27,28DFT+U29approach, hybrid functionals, and well- designed pseudopotentials and basis sets,30have been developed to correct the self-interaction error. At this moment, DFT+U is most widely used for reasonably large systems (because of its relatively low computational cost) and the use of hybrid functionals, mainly as a benchmark method, comes second.31Hybrid functionals incor- porating a fraction of HF exact exchange partially eliminate the self-interaction error and further improve the description of the localization of 4 felectrons. B3LYP and wB97XD hybrid functionals have been extensively applied to molecular systems,32while HSE03, HSE06, and PBE0 are more commonly used for periodic solids. Brugnoli et al.33performed an assessment of several functionals for Ce2O3and CeO 2with respect to lattice parameters, vibrational prop- erties, bandgaps, and reaction energies. PBE and M06L could better reproduce the reduction energies of CeO 2to Ce 2O3but performed poorly for most other properties, while the HSE06 hybrid functional showed the best performance among the investigated functionals for vibrational properties, bandgaps, and properties of surfaces with O vacancies (vacancy formation energy and bandgap).33However, the usage of a hybrid functional combined with a plane-wave basis set is very computationally expensive and, hence, currently prohibitive in most practically relevant cases. In this context, the DFT+U method invoking a correction on top of LDA or GGA and based on a simple Hubbard model, in which an additional on-site Coulomb interaction term U and site exchange term J are exerted on the on-site two-electron integrals, has been applied in a majority of published calculations on REO systems. The U correction seldom increases the elapsed time with respect to the standard LDA or GGA functional.29The Hubbard parameter could be treated as a penalty function reducing one-electron potential for particular orbitals in a Hartree–Fock-like manner. Dudarev et al.34 simplified the method by replacing two independent U and J param- eters with their difference, U −J (further simplified as U because J is usually set to zero). The introduction of the effective parameter U facilitates the localization of 4 felectrons35,36and qualitatively cor- rects the description of polarons (a polaron is a quasiparticle, which could be described as a combination of a trapped electron and a resulting defect/polarization of the lattice).37LDA+U and GGA+U approaches adopting various U values have been used to calculate the geometric, electronic, and thermo- chemical properties of stoichiometric or non-stoichiometric CeO 2 and Ce 2O3, which have been then compared to the experimental values.38–43A recent study44analyzed the performance of DFT+U for CeO 2and for the selected catalytically relevant transition metal oxides (including oxides of Ti, Co, Ni, Mo, and Mn). That study argued that the performance of the DFT+U method depends heav- ily on the proper choice of the parameter U, which can be obtained empirically from fitting a property of interest to experiment or to calculations at a higher level of theory. Fitting to different properties (bandgap or reaction energy) or choosing different reference values (experimental or theoretical) may lead to different optimal values of the parameter U. Hence, it is crucial to define the optimal range of U with respect to the desired properties of interest (in this work, the focus lies on structural parameters and thermodynamic properties, such as formation energies). Most recent DFT+U studies of systems including ceria as one of the components, e.g., Refs. 18, 20, 45, and 46, adopted U values close to 4.5 eV following the work of Fabris et al.17who recom- mended the U values of 5.3 for LDA+U and 4.5 eV for GGA+U derived using the linear-response approach of Cococcioni and de Gironcoli.47Da Silva et al.45compared the performance of hybrid functionals (HSE and PBE0), GGA+U, and LDA+U methods in the prediction of the lattice parameters, electronic structure, and thermodynamic energies of CeO 2. The optimal U value giving the best description of the above properties with the experiment was found to lie around U = 2 eV, which is below the value obtained by the linear-response method. The HSE hybrid functional was shown to correctly predict the lattice constant and the localiza- tion of a single 4 felectron in Ce 2O3.45The formation energies of both cerium oxides were predicted very accurately with HSE with respect to the available experimental data, while for certain other energetic parameters, deviation from the experiment was substantial. Although La 2O3and CeO 2have been given sufficient atten- tion in the past, theoretical studies on other REOs are rather scarce. Consequently, it is necessary to evaluate the ability of DFT+U and HSE approaches to adequately describe the material properties of interest of REOs and to recommend a set of optimal values of the parameter U. In this work, we aim at (i) the assessment of PBE+U and HSE06 in the description of several material properties of bulks or surfaces of lanthanide oxides and (ii) the determination of a consis- tent set of optimal values of the U parameter in order to efficiently and accurately apply the PBE+U method to the calculations of lan- thanide oxides. Therefore, we systematically optimized the crystal volumes of a series of lanthanide oxides by HSE06 and PBE+U including various U values ranging from 1 eV to 8 eV, where the electronic structures and reaction energies of model reactions were examined in a further step. A set of optimal U values for each lan- thanide element was separately determined by taking the HSE06 results as reference. The net ionic charges and work function of two selected surfaces, CeO 2(111) and Nd 2O3(0001), and the adsorp- tion of CH 3and NH 3on these surfaces were investigated using the obtained optimal U values to evaluate the applicability of these U val- ues in reproducing surface properties compared to the HSE06 hybrid functional. J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp II. THEORETICAL MODELS AND METHODS While cerium forms a stoichiometric dioxide (CeO 2) with Ce in the oxidation state (IV), for most other elements of the lanthanide series, sesquioxides (Ln 2O3, where Ln stands for any lanthanide element) with the metal oxidation state (III) are more common. Exceptions are Tb and Pr, which usually exist as non-stoichiometric oxides with the oxidation states between (III) and (IV) but can under certain conditions form stoichiometric dioxides. We have selected sesquioxides as the primary target of our investigation for consis- tent comparison of REOs because sesquioxides are known for all lanthanides. The sesquioxides may take one of the three distinct polymorphous forms, hexagonal, monoclinic, and cubic,48labeled A-type, B-type, and C-type, respectively (Fig. 1). We have selected A-type polymorphs of bulk sesquioxides to examine the perfor- mance of PBE+U and HSE methods for the geometric, electronic, and thermodynamic properties aiming at a consistent comparison within the Ln 2O3series. The initial lattice parameters of the A-type Ln2O3were taken from the optimization values obtained by Wu et al. using the PBE method.49Because CeO 2is a more common oxide for Ce, this dioxide has been singled out for a comparison to the hexagonal Ce 2O3. The spin-polarized calculations for all of the considered A-type Ln 2O3(except La 2O3and Lu 2O3) bulk oxides and the Ln 2O3surfaces lead to a ferromagnetic (FM) spin solution, while La 2O3, Lu 2O3, and CeO 2bulk and surface were calculated as non-magnetic spin states. We have chosen the following imaginary reference reaction: 2LnF 3+ 3H 2O→Ln2O3+ 6HF (1) to judge the performance of PBE+U and HSE methods with respect to the accuracy of the calculated thermodynamic properties of Ln2O3and to determine the optimal U values, while another ref- erence reaction 2CeF 3+ 4H 2O→2CeO 2+ H 2+ 6HF (2) was used for CeO 2. The crystal structures of most LnF 3(Ln: from Sm to Lu) are isostructural with YF 3crystals, i.e., orthorhombic trifluorides, although the hexagonal structure, like that of LaF 3, was also found for the trifluorides of Sm, Eu, Ho, and Tm.50,51 Herein, the YF 3-type structure was adopted for all LnF 3to ensure consistency. To check whether the conclusions inferred from calculations of bulk oxides are transferable to different coordination environments, we have also examined the properties of selected REOs surfaces.The Ce 2O3(0001), Nd 2O3(0001), and CeO 2(111) surfaces were con- structed from the bulk Ce 2O3, Nd 2O3and CeO 2optimized at the HSE06 level chosen as representatives of REOs sesquioxides and dioxides, respectively. The p(2×2) Ce 2O3(0001) and Nd 2O3(0001) surface slab models comprised two repeated units of which only the top one was allowed to be relaxed to the equilibrium geometry. The CeO 2(111) surface was modeled as a p(3×3) supercell comprising nine layers of which the bottom three layers were constrained to the optimized bulk crystal structure. The periodically repeated slabs were separated with a vacuum spacing of 15 Å for all of the sur- face models to minimize the spurious interaction between the two adjacent slabs. The work function and the adsorption energies of NH 3and methyl radical CH 3were calculated to evaluate the agree- ment between PBE+U with various U values and the HSE method. The work function was defined as the difference between the Fermi energy and the product of the charge of an electron and the electro- static potential in the vacuum nearby the surface.52The adsorption energy was calculated as Eads=E(adsorption complex) −E(surface) −E(NH 3or CH 3). The energies of the adsorbate molecules, E(NH 3 or CH 3), were obtained by the relaxation of NH 3or CH 3in a 20×20×20 cubic cell. The weak interaction between the NH 3 or CH 3adsorbates and the surfaces was further visualized by IGM methodology,53as implemented in the Multiwfn package.54 All calculations in this study were carried out with the Vienna ab initio simulation package (VASP)55,56using the projector aug- mented wave (PAW)57,58method and a plane-wave basis set. The considered valence electrons of O, F, and lanthanide atoms in the PAW potentials are given in Table S1 of the supplementary mate- rial. The simplified rotationally invariant LSDA+U introduced by Dudarev et al.34was combined with the Perdew, Burke, and Ernzer- hof (PBE)59,60exchange–correlation functional so that only the dif- ference U−J (J = 0) was applied in the calculations. For comparison with the PBE+U results, the benchmark calculations were done with the Heyd–Scuseria–Ernzerhof (HSE06)61,62exchange–correlation hybrid functional. k points were selected using Monkhorst–Pack63 grids for the integrations within the Brillouin zone. (8 ×8×6), (5×5×8), and (7×7×7) meshes were, respectively, used in the unit-cell geometry optimization of the bulk Ln 2O3, LnF 3, and CeO 2, whereas for single-point energy and DOS calculations at the PBE+U level, (12 ×12×8), (7×7×12), and (11 ×11×11) meshes were employed, respectively. The HSE06 calculations were carried out with (6 ×6×4), (4×4×6), or (6×6×6) meshes for Ln 2O3, LnF 3, or CeO 2including the grid reduction factor64(equal to 2) in the cell optimization. The (2 ×2×1) k-point mesh was used to sample the Brillouin zone for the CeO 2(111) surface model, and FIG. 1 . Crystal structures of (a) A- type Ln 2O3, (b) fluorite-type CeO 2, and (c) YF 3-type LnF 3. Oxygen atoms are shown in dark red, cerium atoms in light blue, fluorine atoms in light green, and lanthanide atoms in dark green. J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the (3×3×1) k-point mesh was used for the Ce 2O3(0001) and Nd 2O3(0001) surface models. The cell optimization of bulk crystals was conducted with a larger plane-wave cutoff energy of 600 eV for the sake of the elimination of Pulay stress,65whereas 470 eV cut- off energy was used for the remaining calculations. In most of the bulk calculations, we used the Methfessel–Paxton order 1 smear- ing scheme66with a smearing parameter of 0.05 eV apart from the energy and DOS calculations within the PBE+U adopting tetra- hedron method with Blöchl corrections.67Furthermore, Gaussian smearing66with the smearing width of 0.01 eV was used for all of the calculations of the surfaces and free molecules. The dispersion correction using Grimme’s D3 approach68,69and the dipole cor- rection70was included in the surface calculations. The relaxation of the unit cell and surfaces (with the conjugate gradient method) was performed until the force acting on each atom was less than 0.02 eV/Å, while the energy was converged with a tolerance of 10−5eV. VASPKIT71was applied to post-process the PDOS results of the surfaces. III. RESULTS AND DISCUSSION A. Equilibrium lattice parameter 1. A-type Ln 2O3and YF 3-type LnF 3 The equilibrium lattice parameters computed by PBE, PBE+U, and HSE methods for A type Ln 2O3and fluorite-type CeO 2are sum- marized in Table I and in Fig. 2; the calculated lattice parameters of LnF 3are given in Fig. 2 and in Table S2 of the supplementary material. The equilibrium volumes of A-type Ln 2O3decrease across the lanthanide (Ln) series. The obtained tendency is similar to the previously reported results using the PW9172or PBE49functional combined with PAW pseudopotentials and treating f-electrons as core electrons. Such “large-core” pseudopotentials mimic the so- called “lanthanide contraction.” It is also observed for YF 3-type LnF 3 that the equilibrium volumes calculated by HSE06 method decrease with the atomic number of the Ln element. For Ce 2O3, which is a hexagonal sesquioxide with one formula unit per unit cell, the HSE06 calculation of this work predicts a 0and c0to be 3.86 Å and 6.09 Å, respectively, in line with the results of Hay et al.73and Da Silva et al.45HSE06 predicts a 0.03 Å (0.8%) smaller value of the lattice parameter a 0than the experimental value, a 0 = 3.89 Å, whereas the c 0parameter is 0.03 Å larger than the exper- imental value, c 0= 6.06 Å.74,75Similarly, HSE06 underestimates the a0, c0, and volume of Pr 2O3by 1.0%, 0.2%, and 1.7% compared to the experimental values,763.86 Å, 6.01 Å, and 77.54 Å3, respectively. The experimental lattice parameters of La 2O3(a0= 3.94 Å and c 0 = 6.14 Å)15and Nd 2O3(a0= 3.83 Å and c 0= 5.99 Å)77are also under- estimated by HSE06, but the error is within 0.03 Å. On the basis of this comparison with the available experimental data, it could be concluded that the HSE06 method predicts the structural parameters with reasonable accuracy. The optimized lattice parameters (a 0and c 0) and the equilib- rium volumes obtained under the PBE+U treatment increase with increasing U for La 2O3, Ce 2O3, Pr 2O3, Nd 2O3, Pm 2O3, and Sm 2O3. However, for the later Ln 2O3from Eu 2O3to Lu 2O3, the optimized unit cell volumes decrease with increasing U. There are two main competing effects: one is that the increased on-site Coulomb repul- sion acts to reduce electron delocalization and to increase the latticeparameter, whereas the other effect acting in the opposite direction is the reduced shielding of valence electrons (lanthanide contrac- tion), which becomes more pronounced as 4 fstates become more localized with increasing U and also increases with the increasing atomic number from early to late lanthanides. In addition to these two effects, the degree of covalency of the Ln–O bond also changes slightly with increasing U, as illustrated by the COHP analysis below. The combination of the above effects results in different trends for the lattice volume vs U dependence for REOs of earlier and later Ln. Specifically, for Ce 2O3, the minimum value of the volume com- puted by PBE+U (U = 1 eV) is overestimated by 1.9% in comparison with the HSE result; at the same time, the calculated value of a 0 at U = 1 eV, 3.89 Å, agrees quantitatively with the experimental value. Upon increasing U from 3 eV to 8 eV, the overestimation of the volumes by PBE+U with respect to HSE06 rises from 3.8% to 6.7%, respectively. La 2O3, Pr 2O3, and Nd 2O3show similar behav- ior to Ce 2O3so that PBE+U with U = 1 eV predicts geometries in better agreement with the experiment than HSE06. In other words, PBE+U with U = 1 eV–3 eV results in accuracy similar to that of the HSE06 method for the lattice parameters of La, Ce, Pr, and Nd sesquioxides. More generally, for the first half of the Ln series from La to Eu, and also for Lu, the best agreement between PBE+U and the experimental results in terms of the optimized cell parameters of Ln2O3could be achieved with the U values of 1 eV–3 eV. For Sm 2O3, the HSE calculations unfortunately have not converged so that the direct comparison of the two methods was not possible. For Eu 2O3, Gd 2O3, and Er 2O3the best agreement between PBE+U and HSE06 was achieved using U = 7 eV–8 eV, but with the U value of 3 eV, the overestimation of the lattice volumes was not too severe, under 2.0%. The lattice parameters of Ho 2O3and Tm 2O3can also be accurately calculated using the PBE+U method with the U parameter of 3 eV–4 eV, deviating only by 0.2% and 0.6% for the cell volume, respectively. Therefore, we conclude that with respect to optimization of lattice parameters, the U value around 3 eV should be a safe choice for the PBE+U calculations of lanthanide sesquioxides. The PBE functional without the U correction yielded closer lat- tice parameters to those computed by HSE06 than PBE+U did, but not closer to the experimental values than PBE+U with U = 1 eV– 3 eV for La 2O3, Ce 2O3, Pr 2O3, and Nd 2O3. When comparing to the HSE06 results as reference, PBE performs better than PBE+U (adopting the optimal parameter U) for Pm 2O3, but worse for the second half of the Ln series from Eu to Lu. Referring to lanthanide fluorides, the HSE06 method was able to accurately predict the lattice volumes of EuF 3(205.05 Å3), GdF 3 (203.63 Å3), and HoF 3(194.84 Å3) with respect to the correspond- ing experimental values, 204.3 Å3, 201.6 Å3, and 192.8 Å3.51For ErF 3, TmF 3, and LuF 3, the description of the volumes using HSE06 gets worse, but the overestimation error of HSE06 is still below 5% in comparison to the experimental values.51When applying the PBE+U method, we found that parameter U has little effect on the volumes of LnF 3. In other words, the volumes of LnF 3predicted by pure PBE and PBE+U methods are close to each other, as given in Table S2 of the supplementary material. With respect to the HSE06 functional, the PBE functional in most cases overestimates the lattice volumes of LnF 3with the error of around 3%. J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Equilibrium lattice parameters a 0(in Å), c 0(in Å), and volume (in Å3) of Ln 2O3(Ln = La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Ho, Er, Tm, Lu) and CeO 2calculated by PBE, PBE+U (U = 1 eV–8 eV), and HSE06 methods. PBE+U REOs Lattice parameter PBE U = 1 U = 2 U = 3 U = 4 U = 5 U = 6 U = 7 U = 8 HSE06 La2O3 a0 3.94 3.95 3.95 3.96 3.97 3.97 3.98 3.99 3.99 3.91 c0 6.18 6.18 6.20 6.21 6.22 6.22 6.23 6.24 6.25 6.14 Volume 83.02 83.36 83.86 84.28 84.82 85.16 85.57 85.93 86.30 81.38 Ce2O3 a0 3.83 3.89 3.90 3.91 3.92 3.92 3.93 3.94 3.94 3.86 c0 6.08 6.12 6.15 6.16 6.17 6.18 6.19 6.19 6.23 6.09 Volume 77.17 80.05 80.85 81.46 81.96 82.39 82.81 83.24 83.74 78.49 Pr2O3 a0 3.84 3.86 3.87 3.88 3.89 3.90 3.89 3.90 3.90 3.82 c0 6.02 6.03 6.05 6.06 6.08 6.09 6.11 6.12 6.13 6.00 Volume 77.52 77.64 78.31 78.91 79.45 79.86 80.24 80.58 80.91 76.26 Nd 2O3 a0 3.80 3.83 3.84 3.84 3.85 3.86 3.86 3.87 3.87 3.80 c0 6.03 6.05 6.07 6.08 6.10 6.09 6.11 6.11 6.11 6.02 Volume 75.53 76.72 77.29 77.80 78.25 78.49 78.83 79.16 79.46 75.30 Pm 2O3 a0 3.78 3.82 3.83 3.83 3.84 3.84 3.84 3.85 3.85 3.79 c0 5.96 6.01 6.00 6.01 6.03 6.05 6.05 6.05 6.06 5.97 Volume 73.81 75.94 76.11 76.45 76.81 77.34 77.40 77.56 77.76 74.23 Sm 2O3 a0 3.77 3.78 3.79 3.79 3.80 3.80 3.81 3.81 3.80 . . . c0 5.95 5.96 5.98 5.97 5.99 6.01 6.01 6.02 6.01 . . . Volume 73.29 74.30 74.73 74.85 75.15 75.28 75.46 75.74 75.50 . . . Eu2O3 a0 3.79 3.81 3.81 3.81 3.81 3.80 3.80 3.79 3.78 3.78 c0 5.97 5.93 5.93 5.92 5.95 5.94 5.98 6.03 6.02 5.90 Volume 74.37 74.58 74.69 74.46 74.73 74.51 74.69 75.08 74.18 73.03 Gd 2O3 a0 3.74 3.75 3.75 3.75 3.74 3.74 3.74 3.73 3.72 3.72 c0 5.96 5.96 5.96 5.96 5.96 5.96 5.96 5.94 5.93 5.93 Volume 72.34 72.51 72.51 72.46 72.17 72.17 72.17 71.64 71.13 71.20 Ho 2O3 a0 3.67 3.67 3.67 3.68 3.68 3.64 3.60 3.50 . . . 3.67 c0 5.89 5.88 5.88 5.84 5.84 5.82 5.79 5.77 . . . 5.86 Volume 68.41 68.25 68.32 67.54 67.55 66.23 64.85 61.09 . . . 67.66 Er2O3 a0 3.66 3.67 3.66 3.66 3.65 3.65 3.64 3.64 3.63 3.64 c0 5.84 5.84 5.84 5.83 5.83 5.82 5.82 5.82 5.82 5.81 Volume 67.79 67.89 67.77 67.59 67.32 67.18 66.94 66.77 66.48 66.63 Tm 2O3 a0 3.66 3.67 3.66 3.65 3.64 3.62 3.59 3.57 3.55 3.62 c0 5.78 5.82 5.82 5.82 5.81 5.79 5.79 5.76 5.72 5.82 Volume 66.50 66.79 66.60 66.15 65.71 64.89 64.11 63.20 62.20 65.75 Lu2O3 a0 3.61 3.60 3.58 3.56 3.55 3.53 3.50 3.47 3.43 3.59 c0 5.83 5.82 5.78 5.77 5.76 5.73 5.70 5.67 5.62 5.75 Volume 65.56 65.03 64.24 63.39 62.63 61.58 60.35 59.10 57.22 64.16 CeO 2 a0 5.48 5.48 5.49 5.49 5.50 5.50 5.51 5.51 5.52 5.39 Volume 164.26 164.26 165.30 165.75 166.23 166.50 167.16 167.65 168.05 156.97 2. COHP analysis of covalent bonding in Ln 2O3 To investigate the contribution of 4 fstates in the covalent part of M–O bonding and its variation with the increasing Hubbard parameter U, we selected Ce 2O3and Ho 2O3as representative early and late Ln sesquioxides, respectively. The crystal orbital Hamilton population (COHP)78diagrams and the integrated values (ICOHP) could, respectively, give information on the bonding/anti-bonding contributions to the band-structure energy and on the strength ofcovalent bonding, which is one of the factors that may affect the change in the unit cell volume at different U values. The COHP of one pair of Ln–O nearest neighbor contacts (Ln = Ce/Ho) were cal- culated by using the LOBSTER package79,80for various U values, but for the same lattice geometry optimized at the HSE06 level (see Fig. 3 and Figs. S2 and S3). For the Ln–O bonds, we find that, as expected, the main contribution to covalent bonding comes from O2 p–Ln5 d interactions and a significant contribution originates from O2 s– Ln5dinteractions. There is also small contribution from Ln4 fstates J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Equilibrium volumes of A-type Ln 2O3and YF 3-type LnF 3(Ln = La, Ce, Pr, Nd, Pm, Eu, Gd, Ho, Er, Tm, Lu) calculated by the HSE06 method. (see Figs. S2 and S3). Whereas the qualitative features of the COHP diagrams for Ce 2O3and Ho 2O3show little variation with U, the ICOHP values (the larger the ICOHP value, the stronger the cova- lent bonding) vary slightly with U. COHP analysis for individual orbital contributions allows us to deduce the specific contribution of Ln4 fstates to covalent bonding. For Ce 2O3, the net ICOHP val- ues increase slightly with increasing U (see Fig. 3). However, as seen from Table S3, the Ce5 d–O2s, Ce5 d–O2p, and Ce4 f–O2pcon- tributions to bonding have a different dependence on U. Whereas the participation of 4 fstates in covalent bonding becomes less pro- nounced with increasing U, the Ce5 d–O2sand Ce5 d–O2pbonding slightly increases with U. We relate this effect to the change of the conduction band character from mainly Ce4 fto mainly Ce5 dupon increasing U. For Ho–O bonds in Ho 2O3, the projected COHP diagrams in Fig. S3 and ICOHP values in Table S3 show that the 4 fstates of Ho are even more localized than for Ce and their almost negligi- ble contribution to covalent bonding decreases with U. The degree of Ho5 d–O2sand Ho5 d–O2pcovalent bonding also decreases with U. The trend is opposite to what was found for Ce 2O3. 3. Fluorite-type CeO 2 Among all lanthanide oxides, CeO 2is by far the most inves- tigated. Many recent DFT studies used state-of-the-art approaches to obtain the equilibrium lattice parameter a 0and the volume V 0of CeO 2. Hay et al.73derived a value of a 0of 5.41 Å for CeO 2using HSE by fitting the energy vs volume curve using the Birch–Murnaghan equation of state.81,82Da Silva et al.45obtained a value of a 0= 5.40 Å by HSE03, very close to the above result, using a different optimiza- tion strategy based on minimizing the stress tensor and all internal degrees of freedom. Our predicted lattice parameter a 0= 5.39 Å obtained on the basis of stress tensor optimization and using the HSE06 hybrid functional is very close to the result of Da Silva et al.45 Another study recently reported a value of a 0of 5.408 Å using the HSE06 functional and atom-centered basis sets.33The experimen- tal lattice parameter a 0has been measured by several groups to be5.39 Å,835.406 Å,84and 5.411 Å.85Apparently, the HSE06 func- tional performed quite well in all mentioned calculations despite the use of different types of basis sets and geometry optimization strategies. The lattice parameters predicted at the PBE+U level are overestimated even at low U values and increase with the increasing U. The a 0value of CeO 2at U = 1 eV is 5.48 Å, which is overesti- mated by 1.7% compared to the HSE result. It is almost the same as the value obtained using the PBE functional without the inclusion of U, 5.47 Å. The lattice parameter increases only slightly to 5.49 Å when U is set to 2 eV–3 eV. These results suggest that also for Ce(IV) oxide, the best agreement of the calculated lattice parameter with the experimental values is achieved using small U values (1 eV–3 eV). However, the deviation from experiment and from the HSE06 result is larger for CeO 2than for Ce 2O3. B. Thermodynamic properties 1. A-type Ln 2O3 In many cases, the property of interest is not the lattice geom- etry, but rather energetic parameters such as reaction energies as determined in modeling of catalytic reactions on surfaces. To assess the performance of PBE+U for reaction energies and to evaluate the effect of U on the thermodynamic properties of A-type Ln 2O3, we selected a hypothetical reference reaction, 2LnF 3+ 3H 2O→Ln2O3 + 6HF. We have preferred this reaction over the usually considered enthalpy of formation of an oxide from the elements because this model avoids the calculation of metallic Ln and O 2owing to a known failure of DFT in describing the electronic structure of the O2ground state. Furthermore, the chosen reaction is better bal- anced in a sense that LnF 3and Ln 2O3are both ionic compounds, whereas H 2O and HF are both molecular compounds known to be described well by standard DFT. Furthermore, the electronic ener- gies of LnF 3almost keep unchanged as the U value increases (see Fig. S1 of the supplementary material), which permits us to assume that mainly the electronic structure of the lanthanide oxide but not of the fluoride is affected by the varied U value. Therefore, we can determine which range of U is optimal for an accurate prediction of the energy of the chosen model reaction (with the HSE result serving as a benchmark), and we can expect the transferability of that conclusion to chemical reactions on the surfaces of REO oxides. This transferability will be subsequently tested for several selected examples. The performance of PBE+U and HSE06 methods with respect to the thermodynamic properties of A-type Ln 2O3has been eval- uated using reference reaction (1). The reference reaction energy calculated by using the HSE06 hybrid functional steadily decreases with increasing the Ln atomic number, except for two cusps seen for Eu and Tm (see Fig. 4). This general decrease is in agreement with the experiment (Table II). The cusp positions can be understood when looking at the total energies of bulk Ln 2O3and LnF 3calcu- lated by HSE06. Both energies show a qualitatively similar change pattern, i.e., the energy smoothly decreases from La to Eu with a big drop at Gd (due to the stabilization of the half-filled fshell) and then steadily increases from Gd to Lu. We acquired the experimental reaction energy of reaction (1) based on the available experimental enthalpy of formation of A-type J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . The -COHP curves and -ICOHP values for Ce–O or Ho–O bonds in (a) Ce2O3and (b) Ho 2O3employing vari- ous U values. The calculations were per- formed at a fixed geometry optimized at the HSE06 level. Ln2O386and LnF 387(Ln = La, Pr, Nd, Gd, Ho, Er) and gas molecules (H2O and HF).88As given in Table II, PBE could get close to the experimental values for La 2O3, Nd 2O3, and Er 2O3but performs worse for Pr 2O3, Gd 2O3, and Ho 2O3compared to the HSE06 func- tional. Overall, the calculation error of the reaction energy for HSE06 is less than 1.0 eV relative to the experimental values throughout Ln2O3(Ln = La, Pr, Nd, Gd, Ho, Er). We also note that the calculated HSE06 results are systematically lower than the experimental values by 0.3 eV–1 eV. A recent study89suggested that the performance of HSE06 could be further improved by adjusting the fraction of exact exchange in the functional and found that 15% exact exchange in theHSE06 functional gives a relatively good accuracy for the energy of the conversion reaction from CeO 2to Ce 2O3, 2CeO 2→Ce2O3+ 0.5O 2, (3) and also predicts electronic DOS in good agreement with the experi- mental data. While the reaction energies calculated using the hybrid functional do not always perfectly agree with the experimental val- ues, we are not certain about the accuracy of those scarcely available experimental data on the thermodynamic properties of lanthanide J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . (a) Energies (in eV) of the refer- ence reaction 2LnF 3+ 3H 2O→Ln2O3 + 6HF and total electronic energies of (b) A-type Ln 2O3and (c) YF 3-type LnF 3(Ln = La, Ce, Pr, Nd, Pm, Eu, Gd, Ho, Er, Tm, Lu) calculated by the HSE06 method. compounds. Besides, the lattice structure types of LnF 3(Ln = La, Pr, Nd, Gd, Ho, Er), which could possibly belong to YF 3-type or LaF 3- type, are unfortunately not given in Ref. 87 from which we took the experimental enthalpies of formation. Therefore, in the following analysis of PBE+U results, the parameter U will be determined by taking the HSE results as reference in dealing with REO systems, rather than by fitting to the experimental values obtained by dif- ferent techniques or under different experimental conditions and sometimes not known with sufficient accuracy. At the PBE+U level, the closest agreement with the HSE06 results (within 0.3 eV) was obtained for La 2O3, Ce 2O3, Nd 2O3, and Ho 2O3using the U values of 2 eV–3 eV, whereas the deviation increases for higher values of U (see Figs. 5(a) and 5(c)). The min- imum deviation between the PBE+U and HSE06 results is achieved using U = 6 eV for Er 2O3, while the deviation is higher but still less than 0.4 eV in the range of U (2 eV–6 eV) [see Fig. 5(a)]. The HSE06 energies of reaction (1) for Pr 2O3, Pm 2O3, and Lu 2O3were repro- duced quantitatively with U = 1 eV; the error of PBE+U increases with increasing U and reaches its maximum beyond 2.0 eV for TABLE II . Calculated (by PBE or HSE06 methods) and experimental values (in eV) of the reaction energies for the reference reaction 2LnF 3+ 3H 2O→Ln2O3+ 6HF (Ln = La, Pr, Nd, Gd, Ho, Er). La2O3 Pr2O3 Nd 2O3 Gd 2O3 Ho 2O3 Er2O3 PBE 7.54 7.72 6.74 5.09 8.30 6.15 HSE06 8.02 7.62 7.62 7.25 6.96 6.83 Expt. 7.23 6.82 6.65 6.93 6.24 5.98U = 7 eV [see Figs. 5(b) and 5(c)]. Similarly, U = 1 eV gives the best agreement with the HSE06 result for Eu 2O3and Tm 2O3, whereas for higher U values, the deviation of the reaction energy from the bench- mark does not exceed 2 eV [see Figs. 5(b) and 5(c)]. For Gd 2O3, we find poor agreement between PBE+U and HSE06, namely, PBE+U underestimates the energies by around 2 eV with respect to the HSE06 almost independently of the chosen U value [see Fig. 5(c)]. Therefore, it could be concluded that PBE+U with U = 2 eV–3 eV gives the best agreement with the benchmark HSE06 for the cho- sen reference reaction for La 2O3, Ce 2O3, Nd 2O3, and Ho 2O3, while U = 1 eV is optimal for Pr 2O3, Pm 2O3, Eu 2O3, Tm 2O3, and Lu 2O3. The reaction energies of Er 2O3could be reasonably calculated in a wide range of U (2 eV–6 eV). For Gd 2O3, the reaction energy could not be accurately described by PBE+U in the whole range of U values considered. As seen from Fig. 5(a), for Ce 2O3, the U values of 2 eV–4 eV give a very good agreement between PBE+U and HSE06 reaction energies, within 0.2 eV. The same optimal range of U was also deter- mined in Ref. 41 for the formation energy of Ce 2O3and for the energy of reaction (3) calculated at the PW91+U level. Note that reaction (3) also involves Ce in two different oxidation states, i.e., it refers to a redox process. In Ref. 45, the optimal value of U was determined to be 1 eV–2 eV for reaction (3) using the PBE func- tional. Hence, it is conceivable that the same optimal range of U applies to other reactions involving Ce 2O3and could perhaps also work well for A-type sesquioxides of other Ln elements. 2. Fluorite-type CeO 2 We have chosen an analogous model reaction to that discussed above for A-type Ln 2O3, for the formation of CeO 2from a reaction J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . Relative energies (in eV) of the reference reaction 2LnF 3+ 3H 2O→Ln2O3+ 6HF (Ln = La, Ce, Pr, Nd, Pm, Eu, Gd, Ho, Er, Tm, Lu) calculated by PBE (U = 0 eV) or PBE+U (U = 1 eV–8 eV) with respect to the HSE06 method. For clarity, the results are separately shown in (a)–(c). between CeF 3and H 2O: 2CeF 3+ 4H 2O→2CeO 2+ H 2+ 6HF. The performance of PBE+U with respect to the reaction energy of the above transformation is similar to the case of Ce 2O3despite differ- ent coordination environments and the oxidation state of ceria. The reaction energies calculated at the PBE+U level increase with the parameter U. The reaction energy obtained with U = 3 eV, 22.76 eV, is close to the HSE06 result, 21.96 eV (see Fig. 6). It shows that for the above reaction, the optimal U for Ce is 3 eV. In a previous study by Loschen et al. ,41the PW91 + U method with U = 2 eV was able to reproduce the experimental formation energy of CeO 2 from Ce metal and O 2(without considering the correction to bind- ing energy of O 2, which is poorly predicted by DFT-GGA). We have already mentioned above the conversion reaction of CeO 2to Ce2O3, reaction (3), for which the optimal range of U = 1 eV–2 eV in combination with the PBE functional was previously found to give best agreement with the experiment.45This reaction could also berewritten in a different form, replacing 0.5 O 2by H 2O–H 2, giving 2CeO 2+ H 2→Ce2O3+ H 2O. The latter reaction was used in Ref. 90 to investigate the optimal value of U, and it was reported that U = 0.2 eV in combination with PBE gives the best match to the experimental value of 138 kJ/mol. Moreover, the calculated value using U = 1–2 eV was still close to the experimental value within the error of 0.5 eV,90which is in line with the optimal U range found in Ref. 45 for reaction (3). The HSE06 hybrid functional com- bined with the atom-centered basis set underestimates the energy of reaction (3) by around 1.0 eV in comparison to the experimental value.33When adopting the HSE06 value as a benchmark, the opti- mal values of U in Ref. 45 for reaction (3) were close to 3 eV, in line with the result of our investigated reference reaction (2). Therefore, it appears that the obtained optimal range (2 eV–4 eV) of U values for Ce on the basis of our selected reference reaction (1) also works well for the energies of other reactions of Ce oxides, in particular, J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Calculated energies of a reference reaction for fluorite-type CeO 2by PBE (U = 0 eV) or PBE+U (U = 1 eV–8 eV) as well as the HSE06 method. Reference reaction: 2CeF 3+ 4H 2O→2CeO 2+ H 2+ 6HF. for the redox reactions discussed above. This supports our idea that the optimal U ranges derived above for a series of Ln elements could be reliably used in GGA+U (PBE or PW91) calculations of various reaction energies for the systems involving the oxides of the same Ln element and may even be transferable to oxidation states other than +3. C. Surface properties 1. Degree of electron localization In Secs. III A and III B, we have assessed the performance of PBE+U and HSE06 for A-type Ln 2O3and fluorite-type CeO 2with respect to the lattice parameters and reaction energies. In order to evaluate the transferability of the derived optimal parameter U for calculations of surface properties, CeO 2and Nd 2O3were selected as representative dioxide and sesquioxide, respectively. We con- structed the respective surface slabs and calculated net ionic charges of Ce and Nd ions, the work function, and the adsorption energies of representative adsorbates: the CH 3radical and NH 3molecule. Net ionic charges are calculated by the subtraction of valence charges obtained from Bader projections91of the computed total electron charge density from the number of valence electrons in the PAW potentials. The net ionic charges could be used to esti- mate electron localization degree of Ce4 felectrons. Referring to the defect-free CeO 2(111) surface, the average net ionic charge of the surface Ce ions at the PBE+U level increases with the increas- ing parameter U from +2.26 e(U = 0 eV, i.e., PBE) to +2.44 e(U = 8 eV), which is slightly below the value obtained with the HSE06 method +2.51 e. Our PBE+U results agree very well with the values found by Castleton et al.36for defect-free bulk CeO 2at the LDA+U level. The calculated net ionic charges are quite different from nom- inal +4 e, suggesting that the Ce–O bonding is to some extent cova- lent, as also supported by the COHP analysis above. It should be noted, however, that the theoretically calculated net atomic chargessensitively depend on the scheme used to calculate them, in this case from the density partitioning scheme according to Bader charge analysis.92–95 When one surface lattice oxygen is released from the surface, one oxygen vacancy forms and two excess electrons will be left on the surface. For the CeO 2(111) surface, the two extra electrons will mainly localize on the two surface Ce ions surrounding the oxygen vacancy, which leads to a decrease in their net ionic charges.96We removed one oxygen atom connected to three Ce atoms in the top layer from the surface to form one surface oxygen vacancy. The net ionic charge of Ce near the defect site [see Fig. 7(a)] was taken as the average value of the two surface Ce ions holding the excess electrons near the oxygen vacancy. This charge on the Ce ions near the defect site decreases with increasing U from +2.21 e(U = 0 eV, i.e., PBE) to +2.07 e(U = 8 eV), which confirms that increasing U leads to a stronger localization of Ce4 felectrons on them. At the same time, the net ionic charge of Ce far from the defect is increasing with the U value from +2.24 e(U = 0 eV, i.e., PBE) to +2.43 e(U = 8 eV). The difference between the net ionic charges of the Ce ions far from the O vacancy and near the O vacancy becomes larger with the increasing parameter U, which further indicates that 4 felectrons become more localized for U values of 4 eV–7 eV, consistent with the results of Castleton et al.36 A similar variation trend of the net ionic charge of Nd ions was found on the Nd 2O3(0001) surface [see Fig. 7(b)]. The net ionic charge of Nd ions far from the defect increases from +1.90 e(U = 0 eV, i.e., PBE) to +2.03 e(U = 8 eV), while the charge on Nd atoms near the defect decreases from +1.70 e(U = 0 eV) to +1.63 e(U = 3 eV) and then levels off at U = 4 eV so that the charge difference between the two types of Nd ions also increases, similar to what has been found for the CeO 2(111) surface, suggesting that increasing U leads to a stronger localization of Nd 4 felectrons for the Nd 2O3(0001) surface. As an alternative method for evaluating the change in 4 felec- trons’ localization with respect to the change of the U parameter, we have also examined contributions of 4 forbitals to local magnetic moments of surface Ce and Nd atoms, as shown in Fig. 8. On the defective CeO 2(111) surface, the two extra electrons acquired by the system after the removal of an O atom reduce two Ce ions to a triva- lent state, corresponding to the 4 f1electronic configuration. The average magnetic moment of Ce ions surrounding the O vacancy is 0.09 μBat U = 0 eV and rapidly increases with U reaching values beyond 0.9 μBup from U = 4 eV. Moreover, the magnetic moments indicate that considerable 4 felectron localization is already achieved at U = 3 eV. On the defect-free Nd 2O3(0001) surface, Nd ions show a mag- netic moment close to 3 for all values of U, which is consistent with the anticipated Nd4 f3configuration and an average of three 4 felec- trons on each Nd ion. On the defective Nd 2O3(0001) surface, two extra electrons are added to the system. These electrons partially localize on three Nd ions near the defect. The average magnetic moment on these three atoms first increases from 3.13 μBto 3.16 μB for U between 0 eV and 2 eV and then slowly decreases to 3.02 μB for U = 2 eV–8 eV, while Nd atoms far from the defect show smaller magnetic moments decreasing from 3.09 μBto 3.01 μBas U increases from 0 eV to 8 eV. Therefore, magnetic moments suggest only par- tial localization of 4 felectrons for the whole range of U values. The difference of the magnetic moments between Nd atoms near and far J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . Average net Ce/Nd ionic charges on the (a) CeO 2(111) and (b) Nd 2O3(0001) surfaces as a function of U, including Ce/Nd ions on the defect-free surface (dark blue squares), two Ce with the lowest net ionic charge or three Nd ions near the oxygen vacancy site (light blue circles), and Ce/Nd ions far from the oxygen vacancy site (orange triangles). Top view of (c) CeO 2(111) and (d) Nd 2O3(0001) surfaces with one surface oxygen vacancy highlighted in orange. Ce atoms are shown in blue, Nd atoms in green, and O atoms in dark red. from the defect reaches its maximum at U = 2 eV–5 eV, indicating that this range of U corresponds to the highest degree of localiza- tion. At larger U values, the gap between the blue and orange curves in Fig. 8(b) rapidly decreases and almost vanishes for U = 8 eV, pointing to the loss of 4 flocalization at large U. This result is some- what different from the conclusions made on the basis of net ionic charge analysis, which is, however, a more indirect method because it includes contributions of all orbitals in the Bader volume rather than the atom-centered Wigner–Seitz spheres used in the magne- tization calculations, where projections of local magnetic moments onto 4 forbitals were considered. Therefore, we conclude that for Nd 2O3(0001), the U range of 2 eV–3 eV should be optimal to capture the maximum electron localization of 4 felectrons and also to prop- erly describe the lattice parameters and reaction energies, whereas for CeO 2(111), the optimal range of U values required to properlydescribe 4 flocalization (U = 4 eV–8 eV) is higher than the range of U values that works best for the lattice parameters and reaction energies (U = 1 eV–3 eV). 2. Surface reactivity The work function reflects the ease of releasing one electron from the surfaces, namely, the reactivity of the surfaces. It slowly changes from 6.40 eV (U = 1 eV) to 6.23 eV (U = 8 eV) and is thus almost independent of the parameter U for the CeO 2(111) sur- face, differently from the Nd 2O3(0001) and Ce 2O3(0001) surfaces, for which the work function increases with U (see Fig. 9). This dif- ferent behavior of the two cerium oxide surfaces is rather expected since the Ce4 fstates of the CeO 2(111) surface are mainly unoccu- pied and tetravalent Ce cannot be further oxidized (see Fig. S6 of J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Average contributions of 4 forbitals to local magnetic moments of (a) two Ce atoms with the highest magnetic moment near the vacancy site on the defective CeO 2(111) surface and (b) Nd atoms on the defect-free Nd 2O3(0001) surface (dark blue squares), three Nd atoms near the oxygen vacancy site (light blue circles), and Nd atoms far from the oxygen vacancy site (orange triangles) as a function of U. the supplementary material). The dependence of the work function of the Ce 2O3(0001) and Nd 2O3(0001) surfaces on the parameter U originates from the partially occupied and partially empty Ce4 for Nd4fbands. The deviation of the PBE and PBE+U for the work function from the HSE06 result is up to 1.2 eV for the CeO 2(111) surface, indicating that the description of the process of electron emission from the surface by PBE+U is not satisfactory. For the Nd 2O3(0001) and Ce 2O3(0001) surfaces, pure PBE also significantly underestimates the work function by 2.6 eV and 2.0 eV, respec- tively, but the error can be reduced, when applying a U parameter. FIG. 9 . Work function for the CeO 2(111) surface (dark blue squares), Nd 2O3(0001) surface (orange triangles), and Ce 2O3(0001) surface (light blue circles) as a func- tion of the U parameter. All of the horizontal dashed lines represent the respective HSE06 results indicated by the corresponding colors.PBE+U with the U values of 5 eV–8 eV thus helps to achieve bet- ter agreement with the HSE06 result for the Nd 2O3(0001) surface, while for the Ce 2O3(0001) surface, U = 5 eV gives quantitative agree- ment with the results of the hybrid functional. This range of opti- mal U values is substantially higher than the optimal range deter- mined based on the fitting of structural parameters and reaction energies. 3. Adsorption properties Next, we have studied the adsorption of two probe molecules, CH 3and NH 3, on the above REO surfaces to evaluate the perfor- mance of PBE+U and HSE06 with respect to the adsorption prop- erties of REOs surfaces. Both the CH 3radical and NH 3molecule adsorb on top of a surface cerium atoms owing to the Lewis acidity of the CeO 2(111) surface. Although the PBE+U adsorption ener- gies of CH 3and NH 3on the CeO 2(111) surface slightly increase and then decrease with the U, the variation range is only within 0.1 eV, which is similar to the work function. A similar observation was reported in Ref. 44, where the adsorption energy of formalde- hyde on CeO 2(111) was also found independent of the parameter U. However, the activation and reaction energy for the first C–H bond breaking and the adsorption energy of the produced adsorbed CHO−and H+decreased with the U, where U = 3 eV–4 eV was found to fit best to the HSE06 results.44Therefore, the inclusion of U will have an apparent influence in the redox processes on CeO 2(111) but not on the weak adsorption of molecules or radi- cals. Owing to the insufficient experimental data on the adsorption energy of CH 3and NH 3on ceria, we studied the adsorption of the H 2O molecule to assess the performance of HSE06. The exper- imental desorption energy of H 2O is estimated to be ∼0.9 eV at low coverage, while the calculated result by HSE06 is 0.3 eV or 0.49 eV depending on the adsorption geometry of water on the (2×2) CeO 2(111) surface without the dispersion correction.97Our calculations, which included the dispersion correction, resulted in J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the adsorption energy of water up to −1.45 eV on the (3 ×3) CeO 2(111) surface with the HSE06 functional, while the PBE func- tional predicted the adsorption energy of −0.8 eV in a better agree- ment with the experiment. The large deviation between the result of our HSE06 calculation and that of Ref. 97 mainly originates from the effect of H 2O coverage98on the two different sizes of the unit cell and the dispersion correction, which would contribute an additional 0.18 eV (according to Ref. 97) to the adsorption energy of one water molecule. Figure 10 visualizes the interaction between the adsorbates and surfaces through the non-covalent interaction analysis in the form of the Independent Gradient Model (IGM).53This approach is based on the electronic charge density ρand its derivatives. Figure 10 plots the isosurface of the δginter(ρ) descriptor, defined as δginter(ρ) = ∣∇ρIGM, inter∣− ∣∇ρ∣, i.e., the difference between the vir- tual upper limit of electron density representing the non-interacting system ∣∇ρIGM, inter∣and the true electron density gradient ∣∇ρ∣. To differentiate between stabilizing attractive and destabilizing inter- actions, it is necessary to analyze the Laplacian of the charge density∇2ρ, which could be decomposed into three eigenvalues, ∇2ρ=λ1+λ2+λ3. The values of sign( λ2)ρdenoting the inter- action types are projected on the isosurface via a blue–green–red color scale of which the blue color indicates strong attractive inter- action (hydrogen bond, halogen bond, etc.), while the red color indi- cates strong nonbonding overlap. The CH 3adsorption isosurface totally colored in green indicates a weak interaction of CH 3with the CeO 2(111) surface ( Eads>−0.5 eV). The NH 3adsorption is stronger than that of CH 3(Eads>−0.9 eV), which is reflected in the blue color of the isosurface representing the existence of the attractive interaction between N and Ce atoms, as shown in Fig. 10(b).Different from the adsorption on the CeO 2(111) surface, there is a positive linear relationship between the CH 3or NH 3adsorp- tion energy on the Nd 2O3(0001) surface and the parameter U (see Fig. 11). CH 3adsorption turns out to be stronger than NH 3adsorp- tion on the Nd 2O3(0001) surface by about constant difference of 0.5 eV through the whole range of U values from 1 eV to 8 eV, while the adsorption energies in both cases significantly depend on U and change from −1.25 eV to−0.6 eV and from −0.75 eV to 0 eV for CH 3and NH 3, respectively. Adsorption of both species weakens with increasing U, which might be related to increasing the degree of localization of 4 felectrons and their weakened contribution to bonding. At U = 3 eV, NH 3still binds relatively strongly on the sur- face through the attractive Lewis acid–base interaction between the N atom of the ammonia surface and the Nd3+center, whereas the CH 3adsorption isosurface of the same δginter(ρ) value occupies less space compared to the NH 3adsorption isosurface, showing that the CH 3adsorption on the Nd 2O3(0001) surface falls outside the weak interaction to some extent. While we find that the adsorption energies calculated by PBE(+U) for the CH 3or NH 3adsorption on CeO 2(111) deviate from the HSE06 values by up to 0.5 eV, it is possible that the hybrid func- tional shows systematic overbinding. This could be concluded from comparing the H 2O adsorption energy on the CeO 2(111) surface to the experimental value. In that case, the PBE functional performs better than the HSE06 hybrid functional (see above). Therefore, the PBE functional could be reasonably applied in the surface adsorp- tion energy calculations, but the inclusion of a Hubbard correction U = 3 eV in combination with the PBE functional is suggested as a compromise solution to obtain a reasonable electron localization and adsorption energy of NH 3and CH 3. FIG. 10 . The equilibrium adsorption geometries and IGM isosurfaces of (a) CH 3and (b) NH 3molecules on the CeO 2(111) surface, (c) CH 3and (d) NH 3on the Nd2O3(0001) surface calculated by PBE+U with U = 3 eV. Oxygen atoms are shown in dark red, hydrogen atoms in white, nitrogen atoms in dark blue, carbon atoms in silver, cerium atoms in light blue, and neodymium atoms in dark green. The δginter(ρ) = 0.01 a.u. isosurface is shown for the adsorption complexes. A blue–green–red color scale rep- resents the values of sign( λ2)ρranging from −0.05 a.u. to 0.05 a.u. on the isosurface, where the blue color indicates strong attractive interaction, while the red color indicates strong nonbonding overlap. J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 11 . Adsorption energy of CH 3(dark blue squares) and NH 3(orange triangles) on the (a) CeO 2(111) surface and (b) Nd 2O3(0001) surface as a function of U and the corresponding HSE06 results shown by the horizontal dashed lines. IV. CONCLUSIONS The structural parameters and thermodynamic properties of A-type lanthanide sesquioxides and fluorite-type cerium dioxide as well as the surface properties of CeO 2(111) and Nd 2O3(0001) have been systematically investigated at the DFT level using HSE06 and PBE+U functionals combined with plane-wave basis sets. The HSE06 functional reproduces well the “lanthanide con- traction” but in most cases underestimates the lattice parameter of Ln 2O3by less than 0.03 Å. The equilibrium volumes of La 2O3, Ce2O3, Pr 2O3, Nd 2O3, Pm 2O3, and Sm 2O3increase with the increas- ing value of the parameter U, where the best agreement with the HSE06 geometries is reached using the U values of 1 eV–3 eV. On the contrary, for Ln elements to the right of Eu, the equilibrium vol- umes of their sesquioxides decrease with the increasing U values. The U value of 7 eV–8 eV results in the best lattice geometries for Eu2O3, Gd 2O3, and Er 2O3, but small U values around 3 eV result in a relatively small error, whereas U = 3 eV–4 eV is optimal for Ho 2O3, Tm 2O3, and CeO 2, and U = 1 eV–3 eV works best for Lu 2O3. There- fore, we conclude that the U value around 3 eV could be universally applied to most lanthanide oxides to give an accurate description of their structural parameters. As for the thermodynamic properties of bulk oxides, evalu- ated by using the energy of the reference reaction (1), the HSE06 results were close to the experimental values (the error being less than 1.0 eV). Treating the HSE06 results as a benchmark, the opti- mal U values for PBE+U are found to be 2 eV–3 eV for La 2O3, Ce2O3, Nd 2O3, and Ho 2O3, as well as 1 eV for Pr 2O3, Pm 2O3, Eu2O3, Tm 2O3, and Lu 2O3. For Er 2O3, the U values of 2 eV–6 eV were able to satisfactorily reproduce the HSE06 results. For Gd 2O3, PBE+U was not able to closely reproduce the reaction energies with respect to the HSE06 results. The optimal range (2 eV–4 eV) of the U values of PBE+U obtained by reference reaction (2) for CeO 2is consistent with that of Ce 2O3and could be safely applied in reac- tion energy calculations involving both Ce(IV) and Ce(III), such as reaction (3).PBE(+U) and HSE06 have also been assessed for their descrip- tion of the electronic and adsorption properties of the selected REOs surfaces, CeO 2(111), Ce 2O3(0001), and Nd 2O3(0001). The range of U of 3 eV–5 eV is required to achieve considerable localization of Ce4fand Nd4 felectrons; in this regard, U = 5 eV had been earlier recommended as optimal for Ce4 f.36A similar conclusion holds for the work function of Ce 2O3(0001) and Nd 2O3(0001) surfaces, which is predicted in agreement with the HSE06 values by PBE+U when applying the U values in the range of 4 eV–7 eV. The adsorption energies of NH 3and CH 3on the CeO 2(111) surface are only slightly affected by the parameter U, and the deviation of the adsorption energies calculated with the PBE functional from those calculated with HSE06 is less than 0.5 eV. For H 2O adsorption, PBE performs better than HSE06 with respect to the experimental results. How- ever, the surface properties of other REOs still need to be studied systematically in further research. In conclusion, the U values around 3 eV are a reasonable choice for the PBE+U calculations of geometric parameters and for the reaction energy calculations of most REOs in the bulk form as well as for their surfaces, while larger U values are required for a better description of the electronic structure. The assessment of PBE+U against the HSE06 benchmark along with the determina- tion of the optimal parameter U for REOs prepares us for addressing more complex chemical systems, such as catalytic reactions on REOs surfaces, on the basis of first-principles DFT calculations. SUPPLEMENTARY MATERIAL See the supplementary material for the valence configurations in the PAW potentials employed, the equilibrium volumes, and bulk energies of YF 3-type LnF 3; the results obtained for -COHP of Ce 2O3 and Ho 2O3including the contributions from individual orbitals, as well as pDOS of CeO 2(111), Nd 2O3(0001), and Ce 2O3(0001); and atomic fractional coordinates of the equilibrium structures of A-type Ln2O3, fluorite-type CeO 2, and YF 3-type LnF 3(HSE06 level). J. Chem. Phys. 153, 164710 (2020); doi: 10.1063/5.0024499 153, 164710-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ACKNOWLEDGMENTS The authors gratefully acknowledge Professor Dr. Hong Jiang (Peking University) for reading this manuscript and insightful dis- cussions. We thank the North German Association for High Perfor- mance Computing (HLRN) for providing computational resources (Project No. hbc00029). L.V.M. acknowledges the financial support from the German Research Foundation (DFG) (Project No. MO 1863/3-1). S. K. Li also gratefully acknowledges financial support from the China Scholarship Council (Grant No. 201706060196) for a four year study at the University of Bremen. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1J.-X. Liu, Y. Su, I. A. W. Filot, and E. J. M. Hensen, J. Am. Chem. Soc. 140, 4580 (2018). 2L. Nie, D. Mei, H. 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5.0022463.pdf

J. Appl. Phys. 128, 234104 (2020); https://doi.org/10.1063/5.0022463 128, 234104 © 2020 Author(s).Influence of annealing temperature on the existence of polar domain in uniaxially stretched polyvinylidene-co- hexafluoropropylene for energy harvesting applications Cite as: J. Appl. Phys. 128, 234104 (2020); https://doi.org/10.1063/5.0022463 Submitted: 21 July 2020 . Accepted: 21 November 2020 . Published Online: 16 December 2020 Rolly Verma , and S. K. Rout COLLECTIONS Paper published as part of the special topic on Phase-Change Materials: Syntheses, Fundamentals, and Applications ARTICLES YOU MAY BE INTERESTED IN Ferroelectric domain structure of Bi 2FeCrO 6 multiferroic thin films Journal of Applied Physics 128, 234103 (2020); https://doi.org/10.1063/5.0029812 Microstructural evolution in chemical solution deposited PbZrO 3 thin films of varying thickness Journal of Applied Physics 128, 235302 (2020); https://doi.org/10.1063/5.0028523 Enhanced dielectric permittivity and relaxor behavior in thermally annealed P(VDF-TrFE) copolymer films Applied Physics Letters 117, 232903 (2020); https://doi.org/10.1063/5.0010569Influence of annealing temperature on the existence of polar domain in uniaxially stretched polyvinylidene-co-hexafluoropropylene for energyharvesting applications Cite as: J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 View Online Export Citation CrossMar k Submitted: 21 July 2020 · Accepted: 21 November 2020 · Published Online: 16 December 2020 Rolly Verma and S. K. Routa) AFFILIATIONS Department of Physics, Birla Institute of Technology, Mesra, Ranchi, Jharkhand 835215, India Note: This paper is part of the Special Topic on Phase-Change Materials: Syntheses, Fundamentals, and Applications. a)Author to whom correspondence should be addressed: skrout@bitmesra.ac.in ABSTRACT The structural and electroactive properties of the as-prepared random copolymer polyvinylidene-co-hexafluoropropylene thin film are explored as a function of thermal treatment at various temperature regions. The thermal treatment of the polymer thin film not onlychanges the structural conformations that is very natural but establishes a polar domain in the non-polar α-phase. Here, we discover an anomalous temperature-dependent crossover behavior from the non-polar α-phase to an appreciable enhancement in ferroelectric and piezoelectric responses. The maximum unipolar strain ( S max¼/C05:01%), an ultrahigh value of normalized piezoelectric coefficient (d* 33¼/C0556 pm/V), high electromechanical coupling factor ( Kp¼0:78) factor including the high dielectric constant ( ϵ0¼23 at 100 Hz) at a relatively low electric field of 900 kV/cm may, therefore, be an effect of the established polar domain for the sample annealed at 105 °C.The direct piezoelectric charge coefficient ( d 33), a key factor for the performance of a prepared polymer thin film system as an energy harvester, lies in the range of /C010+2 pC/N .Also, the annealed sample exhibited a persistent polarization after several cumulative cycles of applied stress. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022463 I. INTRODUCTION Organic ferroelectrics comprising the polyvinylidene (PVDF) family with its copolymer are multifunctional candidates having a wide range of applications that include industrial automation, petro-chemical industries (VDF-based seals, gaskets, lining, etc.), as electri-cal and electronics devices, aircraft jacketing, high temperaturewiring, lightweight and flexible sensitive transducers, and laser beamprofile sensors, just to name a few. In modern flexible electronicssystems, for numerous transduction applications such as micro-positioning system, energy harvesting, etc., polyvinylidene (PVDF)family with its copolymer are a ubiquitous component. Their ferro-electric properties originate from the charge distribution in the VDF ( /C0CH 2/C0CF2/C0) repeat unit creating a large switchable dipole moment organized in several crystal phases with zero, moderate, andlarge polarizations. 1Such a polymer matrix having substantialelectroactive properties, mechanical flexibility, chemical resilience, ease of solubility in common organic solvents facilitating thin filmconstruction and most importantly lead-free constituents made it amost promising alternative of ceramic dielectrics. PVDF family with its copolymer is a semi-crystalline material and its crystallinity is limited to 50% –60% only. The latter may be due to the presence of head to head ( /C0CF 2/C0CF2/C0) or tail to tail (/C0CH 2/C0CH 2/C0) configuration defects. Various experimental tech- niques have been used to obtain high crystallinity utilizing numer- ous course of action such as epitaxial growth,2mechanical stretching,3electrical poling,4,5crystallization at high pressures,6 etc. Conventionally, PVDF with its copolymer is known to exhibit four crystalline phases —α,β,γ, and δ-phases depending on the competition between the phase separation and the crystallization process.7Prior reports address that the high ferroelectric and piezo- electric performance could mainly be achieved by preserving theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-1 Published under license by AIP Publishing.polar β-phase in the PVDF polymer system.8The addition of polar solvents and controlling the solvent evaporation rate,9controlling the surface energy of top layers of PVDF thin film,10and blending with poly(methyl methacrylate) (PMMA) are a few of the processesthat can give stabilized β-polymorph. 11The phase-field model qualitatively discussed the effect of PMMA on the crystallization rate and the spherulite shape of PVDF, which outperforms the fer- roelectric behavior. Poled spin-coated PVDF polymer melt exhib-ited a maximum polarization of 8 mC/m 2at an electric field of 40 MV/cm.12The addition of barium titanate nanoparticles induces the nucleation of β-polymorph in the 3D printable PVDF and exhibited a maximum piezoelectric coefficient ( d31¼18 pC/N) comparable to the stretched commercial PVDF film sensors.Barrau et al. 13investigated the development of the γ-phase PVDF by the incorporation of carbon nanotubes and obtained themaximum effective piezoelectric coefficient ( d * 33) of 13 pm/V. However, pure PVDF is not suitable for large piezoelectric response for energy harvesting applications, and its stabilized β-crystals create a long-range ferroelectric order leading to a large polariza-tion hysteresis ( P r). For example, PVDF-templated nanowires exhibited a higher value of remnant polarization ( Pr¼19μC/cm2) than the saturation one ( Ps¼9μC/cm2).14Liet al.15revisited the δ-phase PVDF thin film and accounted the displaced charge density at a zero field to 7 μC/cm2for which the maximum polari- zation was only 10 μC/cm2at an external stress of 250 MV/m. As a consequence, a highly non-linear and large hysteretic ferroelectric response is obtained giving unfavorable high loss that hinders itspractical functioning. In order to combat the concerns for highloss, relaxor ferroelectrics has turned out to be the most appropri-ate alternative material for high performance device applications. 16 The relaxor behavior is induced in the PVDF polymer bycopolymerizing it with large and bulky monomers such as tri-fluoroethylene (TrFE), 9,17chlorotrifluoroethylene (CTFE),18hexa- fluoropropylene (HFP),19etc. The first-ever relaxor behavior was reported in an irradiated P(VDF-TrFE) copolymer.20But, its high coercive field (of the order of 100 MV/cm) and large remnant polarization value ( Pr¼60/C080 mC/m2) made it an unsuitable candidate. Later, other random copolymers P(VDF-HFP) haveattracted attention that could exhibit giant electromechanicalresponse in two times greater effectiveness than TrFE and CTFE. 21,22The addition of large monomer unit hexafluoropropy- lene (HFP) facilitates decoupling of dipoles and breaks the ferro-electric macrodomains to nanodomains thus showing relaxorbehavior. 22–24In 1997, Tajitsu and his colleagues investigated the effect of the HFP content on the PVDF/TrFE polymer and found apparent dependence of Pron HFP content. Prdecreases to almost zero at 15 mol. % HFP content prepared by a solvent castingmethod. 25They concluded that the presence of the bulky CF 3 group disturbs the organization of the crystalline region in the HFP modified PVDF/TrFE polymer. The above copolymer system expe- riences nonlinearity in electric field induced ferroelectric polariza-tion with delay saturation and high breakdown strength, both beingpre-requisites for piezoelectric nanogenerators for energy harvest-ing implementations. 22 The P(VDF-HFP) copolymer system displays large strains on the order of several percentages when stressed with an externalapplied field. They, therefore, induced large electric field-inducednormalized strain under an applied external stress. An ultrahigh electrostrictive response of 1700 pm/V and 1200 pm/V were pro- claimed for 5 mol. % and 15 mol. % P(VDF-HFP) copolymersystems, respectively. 26,27However, the major drawback is their unenviable large remnant polarization of 60 mC/m2and 48 mC/ m2, respectively. Later, Wegener et al.28realized a polarization of 24 mC/m2and a coercive field of 100 MV/m in the P(VDF-HFP) copolymer film. A recent study by Nikruesong with his colleaguesdemonstrated a maximum strain of 4% in the P(VDF-HFP) copoly-mer fiber compressed at 60 °C. 29For the uniaxially stretched 15 mol. % P(VDF-HFP) copolymer, an escalated value of piezoelec- tric coefficient ( d31¼30 pC/N) was reported.26These observations specify the excellent pursuit of the P(VDF-HFP) polymer system inferro- and piezoelectric disciplines. 27The most thermodynamically stable polymorph of the P(VDF-HFP) polymer system is the non-polar α-phase. The dipoles of α-phase are considered to be inactive with regard to favorable piezoelectric response. The polar analog of α-phase is the δ-phase which is known since 1978 but their crystal has recently been refined. 15,30The lattice constant and chain con- formation of the δ-phase are identical to the α-phase and exhibit no significant difference in their x-ray diffraction pattern. The only difference is the rotation of every second chain around 180° of the chain axis which makes it polar30and is evident by its ferroelectric polarization. Furthermore, the current state-of-the-art of ferroelec-tric polymer reads out the requirement of a very high electric field (∼order of several megavolts per cm) to achieve large polarization and strain density for electromechanical (EM) response whichlimits their lifetime and reliability. 31Jayasuriya et al. obtained maximum polarization of 50 mC/m2for melt-quenched 15 mol. % PVDF-HFP copolymer at a very high electric field of 200 MV/cm. Also, large maximum polarization comes with large values of Pr (40 mC/m2). Hence, a comprehensible mechanism to acquire a large ferroelectric and piezoelectric response in polymer ferroelec-trics with a simultaneous decrease in dielectric loss remains a fron-tier to be conquered. In this perspective, we reinvestigated the polarization and strain behavior of the neat random copolymer P(VDF-HFP) thin-film as a function of thermal treatment at various temperatureregions ranging from room temperature (RT) to its melting point.The present work includes an effective fabrication process followed by annealing, plays a crucial role in the construction of the porous free polymer thin film, being pre-requisite for high breakdownstrength. However, the characterization of the PVDF based copoly-mer thin film system under an external factor is a troublesome assignment. A major issue is the accumulated experimental error and the calculation error that arises due to its high mechanical loss,electrical conductance, effect of electrode layer thickness, incom-plete poling, mismatch in the experimental conditions, etc. 33,35 Roebben et al.32demonstrated that the stiffer sample such as ceramics, the error in the measurement is of the order of 10−5, while it is about 10−3in soft, high damping, and light material such as ferroelectric polymers. Owczarek et al.33realized a statisti- cal error ( ≲5%) in the electrical permittivity measurement for flexi- ble ferroelectric polymers. Ikeda et al.34carried out the theoretical analysis and concluded that the space charge created at the elec- trode–polymer interface introduces a significant error in the D –E hysteresis curve of the PVDF/TrFE copolymer system. ConsideringJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-2 Published under license by AIP Publishing.these factors, all the samples of P(VDF-HFP) copolymer thin film are sputtered to a practical minimum thickness layer of the silver electrode to nullify the mechanical effects caused due to the heavi-ness of the electrode on the P(VDF-HFP) copolymer thin-filmsystem. Furthermore, each characterization was repeated for morethan four to five times at similar application conditions to obtain the high precision in the measured value. The results executed are consistent with the previous literature reports. Furthermore, weobserved a crossover between two different regimes of the non-polar α-phase and a significant enhancement in piezoelectric responses. We demonstrated that the thermal treatment of the polymer thin film not only changes the structural morphology which is very natural but induces a favorable orientation of thepolar domain in the non-polar α-phase, which may lead to an appreciable enhancement in piezoelectric responses. Remarkably,the temperature dependence of crossover features appears to be unprecedented for the PVDF-HFP copolymer thin film and to our knowledge has not been reported so far. We expect that ourapproach will open a new range of future possibilities and providea new opportunity for large scale industrial applications in polymerferroelectrics. II. EXPERIMENTAL PROCEDURE Polyvinylidene-co-hexafluoropropylene P(VDF –HFP) (Sigma- Aldrich, purity> 99%) pellets and DMF (dimethylformamide)solvent were used to prepare a 20 wt. % P(VDF-HFP) mixture at room temperature. The mixture was heated at 40 °C for 1 h till the polymer melt dissolves completely and forms a transparent viscoussolution. 0.5 ml of the solution was put on a glass slide and uniax-ially stretched by a gravitational pull for 60 s. The surface of glassslide was cleaned thoroughly using acetone before the solution was casted. The uniaxially stretched sample form a thin layer onto a glass slide and was thermally treated at various temperature regionsranging from room temperature (RT) to 150 °C below its meltingpoint. Furthermore, the samples are allowed to cool slowly at room temperature for 12 –13 h. The slow cooling affects the crystallization process. The 60 s uniaxial stretching process gave a thickness of30μm (approx.). The thickness of the solution cast film varied from 30 μm in the middle to about 40 μm at the edge of the film. However, the thickness varies at the edge only and the rest part of the film is uniform. The phase structure of the polymer film was examined by x-ray diffraction patterns performed on an x-ray diffractometer(SmartLab 9 kW, Rigaku, Japan) using CuK αradiation with a scan rate of 2°/min in the range of 10° –80°. Surface morphology was studied by using a standard field-emission scanning electron micro- scope (FESEM, Zeiss SIGMA, Germany). Fourier transform infra-red (FTIR) spectroscopy was carried out by Shimadzu Corpn.,Japan, IR-Prestige 21 in the mid-IR spectral range from 400 cm −1 to 4000 cm−1. Raman spectroscopy was carried out by in Via, Renishaw, UK using 514 nm laser in the spectral range of 100 to 3000 cm−1. Frequency-dependent dielectric studies at room tem- perature were performed using a standard impedance analyzer(SI 1260, Solartron, UK). Silver electrodes were painted on both sides of the disk for dielectric measurement. The electromechanical coupling factor ( K p), mechanical quality factor ( Qm), and figure ofmerit (M) were evaluated by the resonance –antiresonance method using an impedance analyzer (Alpha-A High Performance Frequency Analyzer, Novocontrol Technologies, Germany). Theferroelectric behavior, piezoelectric response (S- –E) curve, (P, S, and E denote the polarization, strain, and the electric field, respec-tively) and positive-up-negative-down (PUND) measurement at room temperature were carried out by using a ferroelectric tester (Premier II, Radiant Technology, USA) The electric field with anincrement of 10 –1300 kV/cm was applied across the polymer thin- film with a triangular waveform at a frequency of 100 Hz. Thespecimen was placed between electrodes in a customized specimen holding apparatus. Samples were immersed in silicon oil to prevent arching. The normalized piezoelectric coefficient ( d * 33) was calcu- lated after the measurement of strain values. III. RESULTS AND DISCUSSION A. Structural analysis 1. Crystallization and fabrication Figure 1 illustrates the schematic representation of the step by step fabrication process for constructing the P(VDF-HFP) copoly- mer thin film. The polymer melt is uniaxially stretched and ther-mally treated at various temperature regions ranging from roomtemperature (RT) to below its melting point (150 °C). In such away, the CF 2dipoles present in a polymer matrix are supposed to align in one direction and, consequently, a polar domain could be achieved. This process is considered as a simple and most effectivetechnique to boost the ferroelectric and piezoelectric responses.The visual appearance of uniaxially stretched and RT dried P(VDF-HFP) polymer thin film is milky opaque, while the high temperature annealed one is transparent as shown in Fig. 2 . The whitish appearance of the RT dried sample is due to the presenceof pores (discussed in Sec. III B ) at the solid –air interface, which reflects and refracts the visible radiation causing a milky opacity. 35 The annealing of the sample followed by slow cooling induces transparency, crystallinity, flexibility, and excellent mechanical strength, which facilitates increased breakdown strength (BDS) andfatigue resilience. 2. X-ray diffraction analysis The phase and crystallinity of the as-prepared sample are investigated by analyzing the x-ray diffraction pattern. Figure 3 shows the x-ray diffraction peak of all the samples of uniaxiallystretched and subsequently annealed P(VDF-HFP) copolymer thinfilms. The simultaneous appearance of amorphous hump and sharp intense peaks clearly indicates the submersion of polar crys- tallites in an amorphous matrix. Herein, the analysis of x-raydiffraction peaks unveiled the effect of thermal treatment on thepolar domain of the as-prepared samples at different temperatureregions. The uniaxial stretching of polymer melts could lead to the alignment of CF 2dipole moment in the polymer matrix. This is obvious as the high intense peak positioned at 20.5° accountable tothe polar β-phase for the overlapped (110)/(200) plane observed in the RT dried sample, responsible for the electrical properties of the P(VDF-HFP) copolymer thin film. Furthermore, as the sample is annealed at a variant temperature region, the intensity of the x-rayJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-3 Published under license by AIP Publishing.diffraction peak decreases and shift toward lower 2 θvalues con- firming the expansion of cell volume. Figure 4 clearly illustrates the variation of XRD patterns with the change in the annealing tem- perature for all the prepared P(VDF-HFP) copolymer thin filmsamples. From the graph, it is noted that an apparent congruencybetween shift in the diffraction peak and the variation in annealingtemperature is missing. In the mid-temperature region around 105 °C, entire conformational change from the polar β-phase tonon-polar α-phase is clearly evident with the appearance of diffrac- tion peaks positioned at 2 θ¼19:9 /C14corresponding to the (110) plane, strictly referring to the most stable non-polar α-phase.35The other less intense diffraction peaks are observed at 18°, 36,° and39.4° referred to the (020), (310), and overlapped (401)/(132)planes, respectively, all characteristics of the non-polar α-phase. Surprisingly, the remarkable ferroelectric response (discussed in Sec.III D ) for the sample annealed at 105 °C raises the question on the conventional non-polar nature of the α-phase. To answer this question, we carried out a series of experiments and came up withthe assumption that the thermal treatment of the sample at 105 °C enables the parallel crating of the polymer chain and set up its dipole oriented in the same direction, thus establishing a polardomain in the non-polar α-phase. Also, a significant rise in inten- sity and sharpness of diffraction peak has been observed attributingto the increase in the crystalline domain size. 36The schematic rep- resentation of the arrangement of polar dipoles of α-crystals is rep- resented in Fig 5 . The diagram depicts that the dipoles created by carbon –fluo- rine (C –F) bonds are oriented in the same direction in the polar α-phase, thus constitute a polar axis, while the dipoles are neutral- ized in the non-polar α-phase. Furthermore, as the annealing tem- perature increases above 105 °C, the diffraction peak moves towardhigher 2 θvalues clearly indicating the reappearance of the polar β-phase. A similar result was reported by He et al. with 10 mol. % P(VDF-HFP) in which enhanced β-phase content was obtained by annealing the material below the crystal phase transition tempera- ture. 26The tendency of the conformational change of the P (VDF-HFP) copolymer thin film on annealing at different tempera-ture regions can be ascribed to two reasons. First, the high viscosity of the polymer melt at RT provides sufficient stress during uniaxial stretching to form β-crystals. 37Second, the change in the annealing FIG. 1. Schematic diagram of the step by step process of the preparation of the P(VDF-HFP) polymer thin film. (a) Heating the mixture of P(VDF-HFP) pellets and DMF (dimethylformamide) solvent at 40 °C for 1 h. (b) Viscous solution of the polymer melt formed, (c) casting of the polymer melt on the glass slide, (d) uni axial stretching of the polymer cast by gravitational pull. (e) Annealing of the uniaxially stretched polymer at different temperatures (RT –150 °C). (f) Peeling of the annealed polymer thin film, (g) schematic representation of the free-standing prepared P(VDF-HFP) copolymer thin film. FIG. 2. Visual appearance of the uniaxially stretched P(VDF-HFP) polymer thin film (a) dried at room temperature (RT) and (b) subsequently annealed at hightemperatures, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-4 Published under license by AIP Publishing.FIG. 3. The x-ray diffraction peaks of uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) room tempera ture (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, and (f) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-5 Published under license by AIP Publishing.temperature causes an obvious change in the solvent evaporation rate which reconfigures the polymer crystals.38However, the quan- tification of the solvent evaporation rate is a separate subject ofresearch and is not the scope of our present study. The above result is analogous to the previous literature reports. 7,39–41 The polar domain β-phase is known to have all trans- conformation (TTTT) with two formula units per unit cell and anorthorhombic space group Cm2m, while the α-phase shows a trans-gauche (TG +TG−) pattern with four formula units per unit cell and a monoclinic space group P21/c. The crating of the polymer chain is parallel in β-crystals, while it is anti-parallel in α-crystals.42Furthermore, the mechanical deformation caused by the uniaxial stretching broadens the diffraction peaks, thereby cre- ating the lattice micro-strain.43The lattice strain ( K) induced broadening of peaks can be represented by the well-knownWilliamson and Hall 10technique [Eq. (1)] and the crystalline domain size ( L) has been evaluated using the FWHM ( β-value)from the Scherrer12method [Eq. (2)], K¼β 4 tan θ, (1) L¼kλ βcosθ, (2) where βis the full width at half maximum intensity (FWHM) mea- sured on the corrected diffraction profile, kis the shape factor cons- tant which is 0.89, λis the wavelength of the x-ray (1.54 nm) for a Cu-target, and θis the Bragg diffraction angle. The crystalline domain size and the other calculated physical parameters listed in Table I can be used as an engineering parame- ter for scanning the electroactive properties of the material. From the table, it can be concluded that the sample undergoes a continu- ous lattice expansion, where the strain decreases (tensile stress) asthe annealing temperature increases from 60 °C to 105 °C since theatoms experience more driving energy to modify their arrange-ments at high temperatures. 44However, further increment of annealing temperature unexpectedly increases the tensile stress. The latter may be attributed to the Ostwald ripening phenome-non. 44,45In our current study, in order to calculate the fraction of the crystalline phase listed in Table I , we use the relation given as % of crystallinity ¼Ic IcþIa/C2100/C20/C21 , (3) where Icis the intensity of crystalline peaks and Iais the intensity of amorphous humps. 3. FTIR spectral analysis The phase identified by x-ray diffraction is confirmed by FTIR spectroscopic measurement. Figure 6 shows the FTIR spectra for all the six samples of uniaxially stretched and subsequently annealed P (VDF-HFP) polymer thin films. The IR vibrational band clearly identifies the morphological changes that occur during the crystalli-zation process at different temperatures. The FTIR peaks are broadas most of the material part is amorphous. The vibrational bandidentified at 775 and 996 cm −1are associated with the non-polar α-phase. The bands positioned at 442, 510, 945, and 1167 cm−1are the characteristic bands of the β-phase. The vibrational band present at 1453 cm−1is associated with the in-plane bending and scissoring of the CH 2group. The band 510 cm−1is assigned to the CF2bending of the β-phase. The band at 480 cm−1is bending and wagging of CF 2in the α-phase. The modes appearing at 600 cm−1 are assigned to the amorphous component of the material.46The vibrational band appeared at 840 cm−1is associated with the out-of-phase combination of CH 2rocking and CF 2asymmetric bending common to both β- and γ-phases.35,46 Furthermore, at the temperature region around 105 °C, the characteristic bands of the β-phase disappeared. The vibrational bands are positioned at 410, 432, 532, 614, 761, 853, and 976 cm−1 strictly associated with the non-polar α-phase as depicted in Fig. 4 . However, as the annealing temperature increases above 105 °C, the IR spectra recaptures the characteristics bands of β-phase FIG. 4. Shifting of x-ray diffraction peaks for all the uniaxially stretched P (VDF-HFP) random copolymer thin film samples annealed at different tempera-ture regions. FIG. 5. Schematic diagram for the creation of a polar domain in non-polar α-crystals.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-6 Published under license by AIP Publishing.positioned at 442, 510, 840, 945, and 1167 cm−1, respectively. The FTIR spectral bands justified the conformational change from theβ-phase to α-phase for the sample annealed at the temperature region of 105 °C. The above result corroborates the x-ray diffraction analysis. 4. Raman spectroscopic analysis The electroactive properties of the P(VDF-HFP) copolymer thin film are primarily based on its internal phase conformations. The Raman spectroscopic analysis provides the complete under-standing of the conjugated structure and the chained skeleton ofthe prepared polymer thin film. Figure 7 depicts the Raman vibra- tional shifts for the P(VDF-HFP) copolymer thin film annealed at various temperature regions. Due to structural defects and the reduction in molecular symmetry, the sample exhibits manyRaman active bands with broad and sharp peaks. The sharpness ofpeaks indicates the presence of a significant amount of a well crys-talline phase in the polymer matrix. 47,48A major five well-defined sharp peaks explicated as P 1,P2,P3,P4, and P 5(Table II ) are ana- lyzed to describe the change in internal structures of all the pre-pared samples. The frequency region defined as peak P 1comes in the range of 790 /C0825 cm/C01is associated with the combination of CH 2rocking and CF 2stretching of paraelectric α-crystals.49,50The peak P 2positioned at 840 cm/C01is associated with out-of-plane CH 2rocking and CF 2stretching of ferroelectric β-crystals. However, this peak is shifted to the lower frequency region of835 cm /C01for the sample annealed at 105 °C clearly indicating the absence of the β-phase. The above result corroborates the x-ray dif- fraction analysis and FTIR spectral analysis. Also, a significant risein its intensity has been observed which may be attributed to thepresence of large crystalline domains in the sample. Further analy-sis of the graph reveals that the peak P 1experiences the redshift due to the weakening of the covalent bond between carbon and electronegative atoms (fluorine and hydrogen) as the annealingtemperature increases from RT. The band P 3ranging 887 /C0 895 cm/C01corresponds to C –H bond rocking vibrations. The most intense band P 4present at 1350 cm/C01is associated with sp3[C–C] hybridization and cleft at P 5at 1450 cm/C01is indicative of sp2 bonded carbon.47The other vibrational bands situated in the range of 977 /C0986 cm/C01, 1069 /C01075 cm/C01correspond to CH 2vinyli- dene wagging and out of plane C –H bonding, respectively. The band around 500 cm−1is indicative of some amorphous sp3 bonded carbon atoms.47,49Hence, an obvious effect of thermaltreatment can be seen on the interaction of intermolecular covalent bonds and structural morphology. The above result is consistentwith the x-ray diffraction and FTIR spectroscopic analysis. The band P 3ranging 887 –895 cm−1corresponds to C –H bond rocking vibrations. The most intense band P 4p r e s e n ta t1 3 5 0 c m−1 is associated with sp3[C–C] hybridization and cleft at P 5at 1450 cm−1is indicative of sp2bonded carbon.47The other vibrational bands situated in the range of 977 –986 cm−1, 1069 –1075 cm−1 corresponds to CH 2vinylidene wagging and out of plane C –H bonding, respectively. The band around 500 cm−1is indicative of some amorphous sp3bonded carbon atoms.47,49Hence, an obvious effect of thermal treatment can be seen on the interaction ofintermolecular covalent bonds and structural morphology. The a b o v er e s u l ti sc o n s i s t e n tw i t ht h ex - r a yd i f f r a c t i o na n dF T I R spectroscopic analysis. B. Microstructural analysis The surface morphology of polymer thin films is analyzed by FESEM micrograph represented in Fig. 8 . It has been observed that the uniaxially stretched sample crystallized at room temperature(RT) presented a porous microstructure formed by stretched spher-ulites. 35The magnified image of pores formed is shown in the inset of micrograph of the RT dried thin film. This porosity may be useful in membrane applications, but the electroactive properties are encumbered as it reduces the mechanical strength and thebreakdown strength to relatively very low fields. Hence, the forma-tion of the porous-free thin film with a stabilized electroactive phase has been one of the most important targets of our research group. Further analysis of the FESEM micrograph reveals that thethermal treatment of polymer thin film promoted the compactmorphology with reduced porosity. 51The latter may be attributed to the change in the diameter of spherulites on annealing the sample above the RT. C. Dielectric properties To investigate the effect of thermal treatment on the ability of a ferroelectric polymer to polarize in response to the alternating field, its relative permittivity ( ϵ0) is measured. The variation in rela- tive permittivity ( ϵ0) and dielectric loss (tan δ) at room temperature (RT) under a frequency range of 102/C0106Hz are depicted in Fig. 9 . We note an apparent change in relative permittivity ( ϵ0) with the increase in the annealing temperature of the samples but the change is not gradual. The maximum relative permittivityTABLE I. The physical parameters of all the prepared samples of the uniaxially stretched P(VDF-HFP) polymer thin film. Samples ’annealing temperatured-spacing (Å)Average crystalline domain size (Å)Lattice strain (%)Crystallinity (%)β-fraction (%) RT 4.30 35.4 0.32 53 85 65 °C 3.43 100 1.10 51 73 85 °C 3.64 250 0.437 50 70 105 °C 4.93 600 0.186 53 … 125 °C 4.32 280 0.321 51 75150 °C 4.39 320 0.606 53 78Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-7 Published under license by AIP Publishing.FIG. 6. FTIR spectra for all the samples of uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) room temp erature (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, (f ) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-8 Published under license by AIP Publishing.FIG. 7. Raman vibrational shift of uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) room temperatur e (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, (f) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-9 Published under license by AIP Publishing.(ϵ0= 14.5) at a frequency of 102Hz has been noted for the sample annealed at 150 °C, while the sample which is dried at RT lies in its close approximation ( ϵ0= 12). This clearly signifies that the relative permittivity ( ϵ0) of the P(VDF –HFP) copolymer thin film largely depends on the percentage of crystalline domain that the materialacquired at a particular annealing temperature. Factors such as change in mean crystallite size, presence of polar and non-polar phase conformations, structural defects, existence of polar domains,etc. that tend to change the volume of the microregion must alsoinfluence the relative permittivity ( ϵ 0) of the samples. Further anal- ysis of graph reveals that the values of relative permittivity ( ϵ0) for all the prepared samples decrease with the increase in the fre- quency. The observed decrement may be attributed to the straininduced in the covalent bonding as the high frequency range limitsthe contribution of polarization of delocalizing electrons amongconjugated molecular orbitals. 24,52Furthermore, high dielectric constant comes with high dielectric loss and is found to be maximum (tan δ= 0.15) for the sample annealed at 150 °C. Also, an obvious increment in tan δhas been observed with the increase of the frequency, which may be ascribed to the increased electricalconductivity at high frequencies. 29 D. Ferroelectric properties To further explore the influence of annealing temperature on polarization components, the large signal ferroelectric behavior as afunction of the external applied field (P-E hysteresis loop) is evalu-ated and discussed. All the samples are of uniform thickness of 30μm with the electrode dimension of 90 mm 2. It is investigated that all the samples which are thermally treated after uniaxialstretching withstand up to an electric field of 1100 –1300 kV/cm and exhibit a typical ferroelectric hysteresis loop, while the samplewhich is crystallized at RT withstand up to 350 kV –cm only. The latter may be attributed to the presence of pores and inhomoge- neous microstructure as evidenced in FESEM micrograph. As thesamples are annealed above the RT, the porosity is reduced thatenables the polymer thin film to hold out against the compressiveforce exerted by the external electric field resulting in higher break- down strength. 53The formation of microcracks in few samples during the measurement process, the broadening of loop at a highelectric field are some of the factors that stagnate the experimentalspeculation. Hence, in order to keep up the consistencies in experi-mental results of all the prepared samples, the ferroelectrichysteresis loops are characterized under an electric field of 900 kV/cm at a time period of 100 ms illustrated in Fig. 10(a) . It has been observed from the graph that the polarization component is strictly dependent on the annealing temperatures ofthe sample. As the annealing temperature of the sample increasesto 150 °C, the maximum polarization value increases from P max¼ 10 mC/m2(sample annealed at 65 °C) to 20 mC/m2. The enhance- ment of polarization may be attributed to the formation of the sta-bilized polar β-phase. For the sample annealed at the mid-temperature region around 105 °C exhibited a significantlyhigh degree of polarization ( P max¼18 mC/m2) consisting of only non-polar α-phase (discussed in Sec. III A ). Interestingly, the idea of stabilized β-phase for enhanced polarization is not justifiable here. Nevertheless, this high degree of polarization may be attrib-uted to thermal treatment of the sample at a particular temperaturethat not only changes the phase conformation but created a polar domain in the non-polar α-phase. It is assumed that the annealing of polymer thin film at the temperature region around 105 °C mayinduce parallel molecular packing of α-crystals, whose dipoles are not compensated and point in a single direction stems the polarnature. 42Furthermore, the induced ferroelectric state with the change in applied field strength can be well understood by analyz- ing the variation in field-induced relative permittivity ( ϵ0) illus- trated in Fig. 10(b) . From the graph, a gradual increase in permittivity with the applied electric field has been observed for all the prepared sample of polymer thin film. This clearly indicates that the prepared polymer thin film has the major contribution of90° ferroelectric domain alignment as 180° domain switching doesnot affect the increase of permittivity because of symmetry aboutthe polar axis for 180° reversal. 54The 90° domain switching is important in technological exploitation of the materials during poling and the evaluation of their electromechanical properties.This is in accord with the Devonshire ’s free energy function that greater the polarization developed by a material in an applied fieldof a given strength, the greater will be the permittivity. 55 The capacitive response of the P(VDF –HFP) copolymer thin film samples has been investigated by the instantaneous current – time (I –t) graph depicted in Fig. 11(b) . The instantaneous current – time (I –t) curve is for a triangular waveform [illustrated in Fig. 11(a) ] of a maximum amplitude of 2700 V applied to the pre- pared annealed sample (1050 V for the RT dried sample) of a uniform thickness of 30 μm. The area of the electrode is about 90 mm2. In this approach, measuring the instantaneous current is a proxy for the rate at which sample delivers charge at a particular voltage. It consists of two sharp current peaks at a particular time interval. The two sharp current peaks at the leading and trailingedges of the applied voltage pulse determine the duration of thecapacitive response of the sample. 56The high current peak implies the enhance possibility of acquiring maximum current when the capacitor is constructed. The analysis of graph reveals that the sample annealed at 150 °C holds the maximum current of 0.03 mA,while it decreases to 0.02 mA for the sample annealed at 105 °C.The time interval between two sharp current peaks indicates theswitching time of long-range ferroelectric order in the P (VDF-HFP) copolymer thin film system. 57 However, during the typical bipolar hysteresis (polarization – electric field curve) measurement, both intrinsic and extrinsicTABLE II. Raman vibrational shifts of five major peaks P 1,P2,P3,P4, and P 5for all prepared samples of the uniaxially stretched P(VDF-HFP) polymer thin film system. SampleP1 (cm)−1P2 (cm)−1P3 (cm)−1P4 (cm)−1P5 (cm)−1 RT 819 845.95 888.36 1314.82 1438.66 65 °C 822.13 844.66 886.44 1352.77 1437.7385 °C 813.15 844.93 887.77 1349.48 1434.77105 °C 811.39 835.48 886.69 1333.65 1436.74125 °C 802.32 845.05 881.16 1358.98 1432.02 150 °C 809.80 841.32 886.17 1333.81 1436.65Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-10 Published under license by AIP Publishing.FIG. 8. FESEM micrograph for all the six samples of uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) r oom tem- perature (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, (f) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-11 Published under license by AIP Publishing.factors including the non-ferroelectric component of the material contribute the measured value. For the P(VDF-HFP) based ferro-electric copolymer system, their relatively high conductivity (/difference10 /C07s/cm) sometimes, would lead to an overestimated polariza- tion that may lay down the true polarization.58Hence, it is impera- tive to separate the different components of the electrical responseof the sample to quantify the true polarization value. The PUND(positive up negative down) measurement distinguishes the true polarization from non-ferroelectric switching contributed by theextrinsic factor. The process begins with the application of a sequence of five pulses of square waveform to the polymer thinfilm sample and the obtained current transients are measured. These sequences of pulse allowed us to separate the switching and non-switching current response. The applied waveform is distinctfrom the triangular bipolar waveform put in for the typical hystere-sis measurement. An application of voltage with a square waveformprovides a longer polarization time and hence executes more accu- rate polarization values, better reflecting the real condition with FIG. 9. (a) Dielectric constant vs frequency graph at room temperature all the six samples of uniaxially stretched P(VDF-HFP) random copolymer thin film sam ples annealed at different temperature regions. (b) Dielectric loss (tan δ) vs frequency graph for all the prepared samples of the P(VDF-HFP) copolymer thin film. FIG. 10. (a) Hysteresis loop vs electric field graph. (b) Increment in permittivity as a function of applied electric field for the prepared P(VDF-HFP) copoly mer thin film system.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-12 Published under license by AIP Publishing.higher accuracy.58The amplitude of the applied square wave signal is 2700 V for all the thin film samples of a uniform thickness of 30μm. The area of the electrode is 90 mm2. The time period of the applied train pulse is 100 ms with a delay time for each pulse takenas 10 ms. Figure 12 represents the schematic representation of the applied pulses of the square waveform over time. The first pulse is the preset pulse to initiate the sample into the polarization state. No measurement has been made as a resultof this pulse. The second and third pulses measured the switchableand non-switchable polarization in the þV maxdirection, while the fourth and fifth pulses measured switchable and non-switchable polarization in the /C0Vmaxstate. A non-switching pulse is appliedafter every pulse. It establishes eight measured parameters, com- monly found in the ferroelectric literature. The characteristics of each of the applied pulses are explicated in Table III . The graph discerns the four +P*(switchable polarization + non-switchable polarization) and four +^P(non-switchable polarization) terms based on polarization switching and non-switching pulses. Thepolarization produced by the switching and non-switching pulses that does not reinstate to zero as the applied pulse is returned to zero volts are classified as +P * rand+^Pr, respectively, on “Radiant Technologies tester. ” Figure 13 specifies the PUND measurement at 900 kV/cm for all the P(VDF –HFP) copolymer thin film samples annealed at various temperature regions and at 350 kV/cm for the RT dried sample, respectively. The analysis of the graph reveals a substantialdecrease in the polarization value for all the prepared samples ascompared to the typical hysteresis measurement [ Fig. 10(a) ]. The observed decrement may be attributed to the change in the applied waveform (square waveform) during PUND analysis. The true FIG. 11. (a) Voltage –time (V –t) graph for the stimulus triangular waveform of the peak amplitude of 2700 V for all the prepared annealed samples (1050 V for the RT dried sample) applied over time of 100 ms. The inset shows the magni- fied image showing the applied waveform for all the samples of the P(VDF-HFP) copolymer thin film annealed at different temperature regions. (b)Instantaneous current as a function of time (I –t) graph for all the prepared samples of the P(VDF-HFP) copolymer thin film annealed at different tempera- ture regions. The inset shows the I –t curve for the RT dried sample. FIG. 12. Schematic representation of the applied pulses of square waveform over time. TABLE III. Classification of each of the pulses applied during PUND analysis. PulseVoltage applied Measured value 1 −Vmax None 2 Vmax P* (remnant + non- remnant polarization) Switching pulse 0.0 P* r 3 Vmax ^P(non-remnant polarization) Non-switching pulse 0.0 ^Pr 4 −Vmax −P* (remnant + non-remnant polarization) Switching pulse 0.0 /C0P* r 5 −Vmax /C0^P(non-remnant polarization) non-switching Pulse 0.0 /C0^PrJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-13 Published under license by AIP Publishing.FIG. 13. PUND analysis for uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) room temperature (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, (f) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-14 Published under license by AIP Publishing.switchable (remnant polarization) polarization is calculated using Eqs. (4)and(5), dP¼P*/C0^P, (4) dPr¼P* r/C0^Pr, (5) where P*= (switchable polarization + non-switchable polarization) and^P= non-switchable polarization. P* r= (switchable remnant polarization + non-switchable remnant polarization, ^Pr= non- switchable remnant polarization. Theoretically (the experimental values are listed in Table IV ), the values of dPand dPrlie in the same range. Furthermore, the experimental results showed that the pulse delay below 10 ms orabove 30 ms is not suitable to reach the polarization equilibrium.The best result for polarization equilibrium has been observed atthe pulse delay of 10 ms. From the values listed in Table IV , it can be concluded that only 20% –30% of the polarization is a switchable polarization for all the samples of the P(VDF-HFP) copolymer thinfilm with a very low value of remnant one ( P r). The relatively smaller value of Prcompared to the maximum polarization ( Pmax) defy the inclusion of any leakage current in the true polarization measurement.59Furthermore, the sharp response of polarization switching, an outcome of fast rotation of the dipoles (C –F and C –H bonds around the central polymer chain) along with a relativelysmall zero-field ferroelectric microdomain ( dP r) suggests the appear- ance of relaxor behavior.29,60The above results supported the crucial effect of the thermal treatment of polymer thin film to obtain theuniform and ordered polarization component. For an additionallook over on the relaxor behavior of the sample, unipolar andbipolar strain values are evaluated (discussed in Sec. III E). E. Piezoelectric properties The analysis of bipolar strain as a function of the electric field for all P(VDF –HFP) copolymer thin film are illustrated in Fig. 14 . Unlike ferroelectric ceramics, all the samples of polymer thin filmexhibited an inverted butterfly-shaped bipolar strain curve withnegligible positive strain typical for polymer ferroelectrics. As the electric field is applied to the sample, the strain curve first increasesin the positive y axis from zero forming a peak at the coercive field, then reverses its direction into the negative y axis, reaches its maximum strain value at the highest field and reduces almost line-arly to zero as the polarization switching completes, thus formingan inverted butterfly loop. The negative piezoelectric strain may beattributed to the sudden contraction of the lattice along the direc- tion of polarization with the applied field. The instantaneous con- traction of the crystal lattice may be due to the presence ofrelatively strong intra-molecular bonds within the chains of P(VDF-HFP) random copolymer thin film creating an uneven distri-bution of bond energies between inter- and intra-molecular chains causing the Vander Waal gap to shrink more than the expansion of the intramolecular bond promoting the highly anisotropic physicalproperties. 61,62Furthermore, a symmetric switching strain curve is a characteristic of normal ferroelectrics containing both positiveand negative strain in equal proportion, but our present random copolymer P(VDF-HFP) thin-film system experiences a very minimal positive strain. The negligible positive strain is a conse-quence of the fast switching of dipoles as discussed in PUND anal-ysis section, thus suggesting the appearance of relaxor phase in thepolymer system. The sharp switching of the dipoles may be due to the presence of bulkier monomer HFP that enlarges the interchain distance. They facilitate dipole switching by truncating the largepolar crystals and decouples the ferroelectric domains, thusshowing a relaxor-like behavior. 23 Further analyzing the graph, it has been noted that the RT dried sample exhibits the uneven bipolar strain with a maximumstrain of 0.04 for the positive cycle of the applied field and 0.06 forthe negative cycle of the applied field. This irregular behavior maybe attributed to the presence of internal bias field. 57As the sample is annealed, a uniform bipolar strain has been observed. Comparing all the samples, maximum bipolar strain ( Smax¼0:04) has been achieved for the sample annealed at 105 °C at an electricfield of 900 kV/cm. Hence, it can be concluded that thermal treat-ment of polymer thin film in this particular temperature induces a large polar domain that produces a giant electrostrictive strain. Electrostriction is the ability of the piezoelectric materials to expand in the direction of electric field and the reversal of theapplied electric field does not reverses the direction of deformation.In the late 1990s, a systematic investigation on electrostriction was TABLE IV . The measured parameters from PUND analysis for all prepared samples of the uniaxially stretched P(VDF-HFP) copolymer thin film system annealed at different temperature regions. Sample annealing temperature(Remnant polarization + non-remnant polarization) ( P*) (mC/m2)Switchable remnant Polarization (P* r)Non-remnant polarization ( ^P) (mC/m2)Non- remnant switchable polarization (^Pr)Switchable polarization (dP) (dP¼P*/C0^P)Switchable polarization (dPr) (dPr¼P* r/C0^Pr)Polarization switched (%) RT 3.89 1.5 2.71 0.35 1.18 1.15 28.2 65 °C 11.99 3.53 11.09 2.65 0.9 0.88 7.385 °C 10 3.5 8.18 1.84 1.82 1.66 16.5105 °C 16.5 6.53 13.4 3.42 3.10 3.11 18.8125 °C 11.92 5.07 9.34 2.45 2.58 2.62 22.01 150 °C 17.98 8.0 15.27 5.27 2.71 2.73 15.1Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-15 Published under license by AIP Publishing.FIG. 14. Bipolar strain behavior of uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) room temperatur e (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, (f) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-16 Published under license by AIP Publishing.carried out in a ferroelectric polymer and realized the high electro- strictive strain ( ∼4%) in polyvinylidene fluoride (PVDF).63The electrostrictive properties of the prepared sample of the P(VDF-HFP) copolymer thin film is computed by its electrostrictioncoefficient ( Q 11) values using the expression as S¼Q11/C2P2, (6) where PandSare the polarization and strain values obtained from the bipolar strain curve, respectively.31,57The change in the values of field-induced bipolar strain and the corresponding electrostric- tion coefficient ( Q11) with the variation in the annealing tempera- ture region for all the prepared samples of the P(VDF-HFP)copolymer thin film are depicted in Fig. 15 . From the graph, it is noted that Q 11is maximum for the sample annealed at 85 °C ( Q11¼/C00:15). The increased Q11values may be attributed to the induced large bipolar strain in the sampleunder the influence of an electric field. As the annealing tempera-ture of the sample increases, the values of electrostriction coeffi- cient (Q 11) decrease, which may be ascribed to the enhancement in the values of maximum polarization (P max)[Fig. 10(a) ] on the thermal treatment of the sample. Furthermore, the direct piezoelectric effect where the applied stress accounts for the generation of the electrical charge is one of the most decisive factors affecting the performance of the piezoelec- trics as a mechanical energy harvester. The parameter that couplesthe stress and the displacement of charges is called the direct piezo-electric charge coefficient (d 33) measured using a Berlincourt d 33 meter (YE2730A METER APC International Ltd., USA). In contrast to the bulk piezoelectric ceramics where the applied stress leads to the expansion of the material with positive longitudinal piezoelectriccoefficient ( d 33.0), the PVDF-HFP copolymer thin films undergo shrinkage of dipoles, substantiates a negative longitudinal piezoelec- tric coefficient (d 33,0).64The measured d33values for all the pre- pared polymer thin film samples lie in the range of /C010+2 pC/Nand is maximum for the sample acquiring a higher degree of crystal- linity. Ever since the report of the direct piezoelectric coefficient d33 of the P(VDF-HFP) copolymer thin-film is remarkably low com- pared to that of bulk piezoelectric ceramics. This may be attributedto the semi-crystalline (amorphous matrix and the crystalline part are intertwined) nature of the polymer thin film system. When a compressive stress is applied to the sample, a part of the deformationis subsumed by the contraction of the amorphous matrix which doesnot participate in the generation of surface charges, thereby decreasesthe value of d 33. It is suggested that increasing the crystallinity of the polymer thin film samples could be the most effective technique to enhance the direct piezoelectric response of the polymer thin filmsystem. On the basis of observed experimental values of the directpiezoelectric coefficient d 33, we quantified the longitudinal piezoelec- tric voltage coefficient,65g33, g33¼d33 ϵ0ϵo, (7) where ϵ0is the measured relative permittivity and ϵois the permittiv- ity of free space, a vital parameter that directly represents the figure of merit (d /C2g) for the performance of the piezoelectric P (VDF-HFP) copolymer thin film system as an energy harvester. Thecomputed values of the piezoelectric voltage coefficient (g 33) and the corresponding figure of merit are listed in Table V . While the high d 33value quantifies the potency of the piezo- electric materials to scavenge the natural dissipating energy to elec-trical energy, the normalized piezoelectric strain coefficient (d * 33)i s the crucial parameter that matters in the actuator system which is computed employing the relation d* 33¼S(Emax)/Emax, (8) where Sis the field induced strain under the maximum unipolar loading. Figure 16 instantiated the unipolar strain ( S–E), S¼D t/C2100, (9) where Dis the piezoelectric displacement and tis the thickness of the prepared polymer thin films) curve under an applied unipolar electric field of 900 kV/cm. FIG. 15. Electrostriction coefficient (Q 11) as a function of annealing temperature of the uniaxially stretched P(VDF-HFP) thin film system.TABLE V . Comparison of piezoelectric parameters with the change of annealing tem- perature for all the prepared samples of the P(VDF-HFP) copolymer thin film system. Sample annealingtemperature (°C)Piezoelectric charge coefficient ( d 33) (pC/N)Piezoelectric voltage coefficient ( g33) (Vm/N)Figure of merit (d33×g33) pJm N2/C0/C1 65 °C 10 107.6 × 10−31.07 85 °C 8 100.43 × 10−30.80 105 °C 12 104.30 × 10−31.25 125 °C 8 90.3 × 10−30.72 150 °C 12 93.51 × 10−31.12Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-17 Published under license by AIP Publishing.The large unipolar strain ( Smax¼/C05:01 %) and ultrahigh normalized piezoelectric coefficient (d* 33¼/C0556 pm/V) has been obtained for the polymer thin film annealed at 105 °C whenstressed with a unipolar field of 900 kV/cm. The observed high pie-zoelectric response may be attributed to the induced polar domaindue to the thermal treatment of the polymer sample at a particular temperature. Figure 17 represents the computed values of normal- ized piezoelectric coefficient d * 33, the direct piezoelectric coefficient (d33) and the induced unipolar strain as a function of different annealing temperature regions for the P(VDF-HFP) copolymerthin film system.F. Electromechanical coupling factor ( Kp) The exploitation of piezoelectric properties for energy harvest- ing applications is primarily governed by the electromechanicalcoupling coefficient (K p).It is the parameter which defines the rate of transducing electrical energy into mechanical energy and vice versa of the piezoelectric materials. We quantified the electrome-chanical coupling factor by the resonance method [using Eq. (9)]. The resonance method is the most preferable method as it circum-vents the issues developed by the leakage current during the experi- ment. 66The frequency-dependent impedance and phase angle values has been measured using an electric field of 0.01 V/mm.Since the piezoelectric materials are mechanical vibrators, thevibrating frequency is specifically defined by the geometry of thesample. The geometrical shape of the specimen is like a circular disk of thin films with diameter (2R ¼12 mm) and uniform thick- ness (h ¼30μm). The electrode area is about 90 mm 2. To nullify the mechanical effect on the resonance vibrational mode of thesample caused due to the stiffness and heaviness of silver electrodeon a soft, light P(VDF –HFP) copolymer thin film system, all the prepared samples of the P(VDF-HFP) copolymer thin film are sputtered with a practical minimum thickness layer of the silverelectrode. The complete electrode samples provide a stabilizedimpedance –frequency curve as the physical phenomena occurring at the metal –polymer interface creates a self-polar axis, thus execut- ing a strongly excited mode. 67The schematic representation of sample geometry and its vibrational mode direction are representedinFig. 18 . Unlike ceramics, the ferroelectric polymer material executes a negative strain behavior with the application of an external electric field. Therefore, when an electric field is applied in the thicknessdirection (z axis) of the sample, the disk contracts along the direc-tion of polarization (z axis) and expands in the radial directionalong the x- and y(r) axis due to Poisson ’s ratio effect. 68The planar electromechanical coupling factor (Kp)signifies the coupling of mechanical and electrical energy in a thin round disk polarized inthickness direction and vibrates in the radial direction. The radial FIG. 16. Monopolar strain as a function of the applied electric field for all the uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at different temperature regions. FIG. 17. Graph showing the variation in normalized piezoelectric coefficient (d/C3 33), the direct piezoelectric coefficient ( d33), and the induced unipolar strain values as a function of the different annealing temperature region for the P(VDF-HFP) copolymer thin film. FIG. 18. Schematic representation of the sample geometry describing the direc- tion of modes of vibration during the measurement of planar electromechanicalcoupling factor ( K p).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-18 Published under license by AIP Publishing.expansion/contraction accompanied by contraction/expansion along the thickness direction results in resonances in the samplewhich is evaluated by the impedance-frequency curve illustrated inFig. 19 . The graph exhibits two prominent frequency region: (a) the frequency region of minimum impedance and maximum current designated as resonant frequency ( f r), (b) the frequency region of maximum impedance and minimum current is designated as anti-resonant frequency ( f a). These two regions are defined as the area of operation in the applied vibrational mode.66,69It has been observed from the graph that all P(VDF-HFP) copolymer thin film samples execute resonance and antiresonance frequency mode atthe high frequency region ( ∼20–40 MHz). Hence, at the device level, this material could be a suitable candidate for high frequencyenergy harvesters. 7Furthermore, the impedance curve exhibited broad peaks at the resonance and anti-resonance frequency regiondue to the semi-crystalline behavior of the prepared sample .It causes a local composition fluctuation which results in local statisti- cal distribution of crystal microregions. Hence, the resonant fre-quency corresponding to each crystal microregion is distributed ina gaussian fashion around a mean resonant frequency. The electromechanical coupling factor (K p)and the mechani- cal quality factor ( Qm)calculated using Eqs. (10) and(12) are listed FIG. 19. Impedance characteristics and change of phase angle as a function of frequency for uniaxially stretched P(VDF-HFP) random copolymer thin film sampl es annealed at temperature regions (a) room temperature (RT) (b) 65 °C, (c) 85 °C, (d) 105 °C, (e) 125 °C, (f) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-19 Published under license by AIP Publishing.inTable VI , 1 K2 p¼0:395fr fa/C0frþ0:574, (10) K2 eff¼f2 a/C0f2 r f2 a: (11) It has been observed that as the annealing temperature increases from 65 °C to 105 °C, the electromechanical coupling coefficientincreases from ( K p¼0:71) to ( Kp¼0:78), respectively. However, a further increase of annealing temperature decreases the Kpvalue to 0.63 for 125 °C, which again rises to 0.76 for the sample annealed at 150 °C. This apparent effect of annealing temperature on the electromechanical coupling coefficient (Kp)may be attrib- uted to the observed variation in the polar crystalline domain withthe change in the annealing temperature. The sample annealed at105 °C exhibits the maximum electromechanical coupling coeffi- cient ( K p¼0:78), which is much higher than the ferroelectric ceramics system.70The result justifies the presence of a favorable polar domain in the sample annealed at 105 °C. The mechanical loss as a consequence of internal friction during vibration is evaluated by its quality factor Qm. It is the measure of the sharpness of the resonance peak obtained from the impedance-frequency curve and is calculated by Eq. (12), Qm¼fr Δf, (12) where fris the resonance frequency and Δfis the bandwidth at 3 dB corresponding to an amplitude reduction of 1/p2 in relation to the resonance as shown in Fig. 20 . Furthermore, the degree of poling the sample can be estimated by evaluating the phase angle ( f) derived from the impedance- frequency curve.70,71An ideal piezoelectric material exhibits a phase angle change from −90° to 90° during resonance and anti- resonance frequencies, respectively. It has been observed from thegraph that the maximum phase angle for all the prepared samplevaries between 72° and 76° approximately. As the phase angle decreases from 90°, more difficult is to pole the sample. The decre- ment in phase angle has been initiated after 30 MHz frequency forall the samples indicating the reduced movement of ions at a higher frequency. Furthermore, it is noted that the resonance peak is shifted to the higher frequency ( f r¼23:87 MHz) for the sample annealed at 105 °C. The shifting of resonance peaks toward highfrequencies explicate the possibilities of ionic mobility in the mega-hertz frequency region. 71This implies that the sample annealed at 105 °C is having a well-developed polar crystalline domain com- pared to the other prepared polymer thin film system. This hasalready been verified by ferroelectric analysis and giant electrostric-tive strain values. G. Fatigue analysis Despite the remarkable ferroelectric and piezoelectric responses of the prepared P(VDF-HFP) copolymer thin filmsamples, the reliability of these materials under service condition isof vital concern. Typically, ferroelectrics undergoing repeated switching tends to lose available polarization on prolonged cycling, known as electric fatigue. This effect severely limits the practicalapplications of the ferroelectric materials. As mentioned in previ-ous literature reports, the maximum polarization of organic ferro-electric polymer reduced to half in less than 10 6cycles.72 In this pursuit a systematic investigation of polarization fatigue has been carried out. All the samples are silver-pasted bothsides to avoid the electrodic effect during the measurementprocess. 73The samples are cycled for more than 106times at a fre- quency of 100 Hz and 900 kV/cm external loading. The degrada- tion of polarization values as a function of cumulative cycle of applied stress for all the P(VDF-HFP) copolymer thin film samplesannealed at different temperature region is depicted in Fig. 21 . The RT dried sample does not sustain the repeated cycling of electric field due to existence of pores in the microstructure of the material. These pores accumulates the space-charge that concentrate the elec-tric field around the pores, leading to breakdown under repeatedapplied stress. 74On the contrary, the samples which are annealed at various temperature regions exhibited a notable fluctuations inTABLE VI. The electromechanical parameters for all the prepared P(VDF-HFP) copolymer thin film samples required for energy harvesting applications. Sample annealingtemperaturePlanar electromechanical coupling factor ( K p) K2 effMechanical quality factor Qm RT 0.71 0.54 6.8765 °C 0.72 0.55 7.6 85 °C 0.73 0.56 7.74 105 °C 0.78 0.58 8.66125 °C 0.63 0.48 8.06150 °C 0.76 0.57 8.52 FIG. 20. Qmas a measurement of the sharpness of the resonant peak obtained from impedance vs frequency graph.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-20 Published under license by AIP Publishing.maximum polarization after repeated cycle of external stress. After successive application of 106cycles, for the sample annealed at 65 °C, the polarization degrades by 5% while the degradationincreased to 10% for the sample annealed at 150 °C. Further, thesample annealed in the mid-temperature region around 105 °C, exhibited the polarization degradation of 3.5% only. The change may attribute to the percentage of crystalline domain in particulartemperature. Therefore, it can be concluded that the fatigue rate FIG. 21. The fatigue analysis for all the uniaxially stretched P(VDF-HFP) random copolymer thin film samples annealed at temperature regions (a) 65 °C, (b) 85 °C, (c) 105 °C, (d) 125 °C, (e) 150 °C, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 234104 (2020); doi: 10.1063/5.0022463 128, 234104-21 Published under license by AIP Publishing.largely depends on the annealing temperatures of the prepared polymer thin film sample. Further analysis realized an abrupt increase in the polarization values after 103cycles in all the charac- terized samples which is a consequence of the recrystallization ofthe P(VDF-HFP) copolymer thin film on repeated application ofthe electric field. However, the continuous repetition of external loading starts degrading the saturated polarization. The degradation may be due to the ionization of organic compound or chemicaldecomposition by a high voltage applied stress. 75Factors such as surface morphology, inhomogeneous microstructures, electrodingtechniques, porosity, etc. may also be responsible for the degrada- tion of polarization on prolonged cycling. Furthermore, the fatigue result shows that the onset of degradation starts after 10 4cycles for all the polymer thin film samples which indicate that a similar deg-radation mechanism is responsible for the polarization fatigue. Inparticular, an acceptable fatigue resistance of the sample up to 10 6 cycles made it a good candidate for practical applications. IV. CONCLUSION We verified the enhanced electroactive responses of the uniax- ially stretched P(VDF-HFP) copolymer thin film by means of thermal treatment at various temperature regions. The annealing ofthe uniaxially stretched P(VDF-HFP) copolymer thin films fostereda porous free microstructure with sustained polarization afterseveral cumulative cycles of applied stress. The dielectric, ferroelec- tric, and piezoelectric properties were investigated to elucidate the optimum annealing temperature region required for enhanced pie-zoelectric properties of the P(VDF –HFP) copolymer thin film system. The direct piezoelectric charge coefficient (d 33) measured using a Berlincourt d 33meter lies in the range of /C010+2pC/N and their corresponding piezoelectric voltage coefficient (g33) and figure of merit were computed. Moreover, the experimental resultsclearly demonstrated that the annealing of the polymer thin filmnot only changed the phase conformations but induced a polardomain in the non-polar α-phase as verified by ferroelectric and piezoelectric characterizations. An ultrahigh value of normalized piezoelectric coefficient ( d * 33¼/C0556 pm/V), large value of electro- mechanical coupling factor ( Kp¼0:78) including the high dielec- tric constant ( ϵ0¼23 at 100 Hz) at a relatively low electric field of 900 kV/cm and an acceptable value of direct piezoelectric charge coefficient (d 33¼/C012 pC/N) for the sample annealed at 105 °C may, therefore, be contemplated as an effect of established polardomain. These excellent properties could give the as-prepared P(VDF-HFP) copolymer thin film system competitive performance preferable for a variety of applications. ACKNOWLEDGMENTS All authors gratefully acknowledge the financial support from the KIRAN DIVISION, Ministry of Science and Technology, Department of Science and Technology (DST), Government of India through Project No. SR/WOS-A/PM-75/2018 (G) and Science andEngineering Research Board (SERB), Department of Science andTechnology (DST), Government of India through Project No. EMR/ 2016/005281. We thank the Central Instrumentation Facility, BIT Mesra for material characterizations. We also thank Dr. JoydeepDhar (Asst. Professor), Department of Chemistry, BIT Mesra for technical help during material processing. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could beconstructed as a potential conflict of interest. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1R. Dong, V. Ranjan, M. Buongiorno Nardelli, and J. Bernholc, Phys. Rev. B 94, 014210 (2016). 2M. R. 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5.0017171.pdf

AIP Conference Proceedings 2265 , 030533 (2020); https://doi.org/10.1063/5.0017171 2265 , 030533 © 2020 Author(s).Investigation of cation distributions and temperature-dependent magnetic properties of polycrystalline CoFe2O4 Cite as: AIP Conference Proceedings 2265 , 030533 (2020); https://doi.org/10.1063/5.0017171 Published Online: 05 November 2020 A. Barik , M. R. Sahoo , Sweta Tiwary , R. Ghosh , and P. N. Vishwakarma ARTICLES YOU MAY BE INTERESTED IN Multiferroic magnetoelectric composites: Historical perspective, status, and future directions Journal of Applied Physics 103, 031101 (2008); https://doi.org/10.1063/1.2836410 Study of magnetization and magnetoelectricity in CoFe 2O4/BiFeO 3 core-shell composites Journal of Applied Physics 123, 064101 (2018); https://doi.org/10.1063/1.5008542 Enhanced magnetoelectricity in bismuth substituted SrFe 12O19 hexaferrite Journal of Applied Physics 126, 074104 (2019); https://doi.org/10.1063/1.5095979 Investigation of Cation Distributions and Temperature- dependent Magnetic Properties of Polycrystalline CoFe 2O4 A. Barik1, M. R. Sahoo1, Sweta Tiwary1, R. Ghosh1, and P. N. Vishwakarma1, a) 1Department of Physics and Astronomy, National Ins titute of Technology Rourkela, Odisha- 769008, India a)Corresponding author: prakashn@nitrkl.ac.in Abstract. Phase-pure cobalt ferrite [CoFe 2O4 (CFO)] is prepared by sol-gel autocombustion technique in orde r to study its structural and magnetic properties. The presence of mixed spin el cubic structure (space group= Fd-3m ) is confirmed from the Rietveld refinement of X-ra y diffraction (XRD) patterns and Raman spectra study. X-ray photoelectron spectrum (XPS) suggest the existence of Co+2 and Fe+3 ions. Analysis of the magnetic data shows that coercivity, rem anence and saturation magnetization decreases with increase in temperature. INTRODUCTION Magnetic oxide with general fourmula AB 2O4 has attracted much attention b ecause of there impressive struc tural, electric and magnetic properties. These materials are found to have multiple applications in many fields of science (medicine, magnetic recordings, etc.) and continues to attract with their complex and fascinating fundamental properties1-2. Among all, cobalt ferrite CoFe 2O4 (CFO) have earned special interest because of its high coercivity, cu bic magnetocrystalline anisotropy, moderate saturation magnetizatio n. It exhibit ferrimagnetism with high Curie temperature ܶ ~ 790 K (bulk) and crystallizes in the partially inverse spinel structure ( Fd-3m ) with general formula (Co 1-x Fe x)A [Co xFe2-x]B O4, where A and B represents tetrahedral and octahedral sites res pectively. For inverse spinel structure x=1 and if x=0 it is normal spinel otherwise it is in mixed spinel structure3. The poperties of CFO depends strongly on the synthesis process, cation distribution, particl e sizes etc. Therefore in the present paper, we have thoroughly investigated the distribution of cations in CFO, whi ch is rather interesting as it has a huge impact onto its electric and magnetic properties. Simultaneously, a comprehensi ve study has been made on its magnetic properties in a wide temperature range ( above room temperature). EXPERIMENTAL DETAILS Polycrystalline CFO sample is synthesized via sol-gel autocombu stion method. All the reagents [Co(NO 3)2.6H 2O, Fe(NO 3)3.9H 2O, and glycine (C 2H5NO 2) used as fuel] are of analytical grade. Stoichiometric amounts of metal nitrates are dissolved in distilled water along with glycine. The soluti on is then heated under continuous stirring, to undergo combustion, resulting fine black powder. The obtained powder is properly ground and pressed into pellets. The pressed pellets are put for sintering at a temperature 1173 K for 2 h. The phase purity of the sample is ascertained at room temperatu re by Rigaku Ultima IV X-r ay diffractometer with Cu Kα (1.5406 Å) radiation with Ni β-filter. X-ray diffraction (XRD) data are recorded in the 2 ߠ range 20-700 at a slow scan rate of 20/min with a step of ∆2ߠ=0.020 and analyzed with the Rietveld refinement technique using FULLPROF program. Raman spectra is collected on the smooth samp le surface using micro-Raman spectrometer (WITEC ALPHA 300R), in the wave number range 100-1000 cm-1, with an excitation wavelength of 532 nm and a power of 5 mW (optimized by considering the signal to noise rat io and sample degradation). X-ray photoelectron spectroscopy (XPS) studies are carried out on a Photo-emission Electron Spectroscopy (PES ) beamline (BL-14) of Indus-2 synchrotron source. The binding energy of the sample ha s been calibrated by taking the C 1s peak as reference DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030533-1–030533-4; https://doi.org/10.1063/5.0017171 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030533-1(284.6 eV). The magnetic properties under the application of a magnetic field up to 15 kOe are recorded from 300 K to 900 K using a vibrating sample magnetometer (Lakeshore 7404) . RESULTS AND DISCUSSIONS Fig. 1(a) depicts the XRD patter n of CFO. The distinct diffract ion peaks are indexed to the cubic spinel structure of CFO. The Rietveld analysis revealed that the sample is in a single phase and crystallizes in a cubic structure with Fd-3m (No. 227) space group. The conve ntional reliability indices for goodness of fitting and the structural parameters as obtained from Rietveld analysis are given in Fig. 1(a). The average crystallite size of CFO is calculated from the (311) diffraction peak using Scherrer equation and is found to be 65 nm. Using Rietveld r efinement technique the cation distribution between the octahedral ( O) and tetrahedral ( T) sites are estimated and is observed that both O and T sites are occupied by Co+2 and Fe+3 ions which suggest the mixed spinel structure. To further vali date the phase- purity and mixed spinel of CFO, Raman spectrum is measured at r oom temperature and corresponding result is presented in Fig. 1(b). The Raman peaks observed at 179, 309, 4 69, 568, 618 and 690 cm-1 corresponds to optical active Raman modes (A 1g, Eg, 3T 2g) of the cubic mixed (or inverse) spinel CFO with space group Fd-3m and thus confirm the formation of single phase CFO4. (a) (b) FIGURE 1. Room temperature Rietveld refined XRD patterns (a) and deconvo luted Raman spectrum (b) of CFO confirms the single phase mixed spinel structure. In order to gain insight into the oxidation states and distribu tion of Co and Fe at O and T sites XPS measurements are carried out and corresponding results are presented in Fig. 2. The wide-scan XPS spectrum of CFO demonstrated that the sample contains Co, Fe and O elements (see Fig. 2(i)). Fig 2 (ii) depicts the Fe 2p core-level spectrum of CFO. The spin-orbit splitting of Fe 2p peaks located at 711.6 and 72 4.8 eV (energy difference ∆ܧ=13.2 eV) corresponds to Fe 2p 3/2 and Fe 2p 1/2 respectively with two shakeup satellites (marked as sat. 1 and sat. 2) which attributes the existence of Fe+3. The Fe 2p 3/2 and Fe 2p 1/2 are individually deconvoluted into two peaks corresponding to F e+3 at O and T-sites (mention in Fig 2 (ii)). Similarly in high resolution XPS spec tra for Co 2p, the peaks are found at 780.7 eV (Co 2p 3/2) and 795.8 eV (Co 2p 1/2) with ∆ܧ=15.1 eV having two satellite peaks as shown in Fig. 2 (iii). P eaks that are observed at 779.8 and 782.3 eV are for O and T –sites Co 2p 3/2 respectively and at 795.3 and 797.2 eV for O and T –sites Co 2p1/2. The observed values of ∆ܧ obtained for Fe and Co are comparable to the reported values5. Here, the fitting of XPS spectra discarded the presence of Fe+2 and Co+3 respectively. The deconvolution of oxygen (O 1s) spectrum is shown in Fig. 2 (iv), which exhibit a dominant peak at 529.5 eV ascribed to the lattice oxygen. The second peak at 531.7 eV is may be due to defects or contaminations. Thus, the XRD, XPS and Raman spectroscopy studies gives a clear indication of mixed cubic spinel structure of CFO corrobo rates the magnetic data (discussed latter). 030533-2 FIGURE 2. XPS spectra of CFO measured at room temperature: (i) wide-scan (0-1000 eV), (ii) Fe 2p, (ii i) Co 2p and (iv) O 1s. The temperature dependence of zero-field-cooled (ZFC) and field -cooled (FC) magnetization ( ܯிሺܶሻ and ܯிሺܶሻ) measured on warming in a field of 500 Oe is shown in Fig. 3 ( a). The Curie temperature ( ܶ) is extracted from ܯிሺܶሻ, i.e., inflection point which gives a minimum in ௗெ ௗ் vs T (as shown in inset Fig. 3 (a)) and is found to be 840 K, which is consistent with those of CFO nanoparticles6. The abrupt change in ZFC magnetization at around 630 K is ascribed to the magnetic glasiness of the sample, whic h need further investigation. T he isothermal field driven dc magnetization M(H) are performed starting from 300 K to 823 K are presented in Fig. 3 (b). It can be observed that the hysteresis loops close at re latively low fields and at high temperature the magnetizations approaches to saturation at nearly 15 kOe. The values of saturation magnetization ( ܯௌ) (obtained from extrapolation of M vs 1/H to 1/H=0), coercitivity ( ܪ) and remanent magnetization ( ܯ) at room temperature are 83.32 emu/g, 936.16 Oe and 30.13 emu/ g respectively and found to decreas e with rise of temperature (in sets of Fig.3 (b)). Distribution of cations in CFO can be obtained theoretically by using Neel’s two sub-lattice model according to which the total magnetization ( ܯ௧௧) of a spinel ferrite is represented by ܯ௧௧ൌܯെܯ, where ܯ a n d ܯ r e p r e s e n t t h e m a g n e t i c m o m e n t s a t tetrahedral and octahedral sites respectively. On the other han d, experimental magnetic moment per formula unit in Bohr magneton is calculated by the relation as mention in refer ence 7 and at room temperature it is around 3.5 ߤ. The cation distribution is calculated as (Co 0.25Fe0.8)T [ C o 0.75Fe1.2]O O4 and magnetic moment per formula unit is obtained to be 3.57 ߤ which is nearly same as the experimental value. Based on the r esults from the XRD, Raman, XPS and magnetic studies, the CFO is found to have mixed spinel cubic structure. 030533-3 (a) (b) FIGURE 3. (a) Typical ܯி (T) and ܯி (T) curves of CFO measured at 500 Oe. The ܶ value is indicated in the first derivative of magnetization ௗெ ௗ் vs temperature (T). (b) Magnetic field dependence magnetizatio n M(H) measured at various isothermal temperatures. Upper inset shows the dependence of coercive fiel d (ܪ) and remanence ሺܯሻ on temperature. Temperature dependence of satura tion magnetization ሺܯௌሻ are shown in lower inset. The so lid lines are only guide to ey es. CONCLUSION The polycrystalline CFO is succes sfully synthesiz ed by sol-gel auto combustion method. A detailed analysis of Rietveld refinement of XRD patterns along with the deconvoluted Raman spectrum confirms the mixed cubic spinel structure of CFO having space group Fd-3m . XPS results revealed the presence of Co+2 a n d F e+3 ions and are distributed in both tetrahedral and octahedral sites. Based on the temperature dependence of magnetic properties study, the Curie temperature is obtained to be ~840 K and the coercivity, remanence and saturation magnetization are found to decrease with rise of tempera ture and approaches to zero nea r Curie temperature. ACKNOWLEDGMENTS The financial support from DST, New Delhi and UGC DAE CSR, Mumb ai in the form of projects “EMR/2014/000341” and “CRS-M-223/2016/724” are gratefully ackno wledged. The authors are thankful to U. K. Goutam from RRCAT, Indore for XPS measurement. The author Sweta Tiwary would also like to thank CSIR, India, for CSIR-SRF fellowship (09/983(0021)/2k18-EMR-I) and financial assistance. REFERENCES 1. D. Zhang, Z. Liu, S. Han, C. Li, B. Lie, M. P. Stewart, J. M. T our, and C. Zhou, Nano Lett. 4, 2151 (2004). 2. T. Maehara, K. Konishi, T. Kamimori, H. Aono, T. Naohara, H. Ki kkawa, Y. Watanabe, and K. Kawachi, Jpn. J. Appl. Phys., Part 1 41, 1620 (2002). 3. A. Franco junior, and F. C. e Silva, , Appl. Phys. Lett. 96, 172505 (2010). 4. P. N. Anantharamaiah, and P.A. Joy, J. Appl. Phys. 121, 093904 (2017). 5. Z. Zhou, Y. Zhang, Z. Wang, W. Wei, W. Tang, J. Shi and R. Xiong, Appl. Surf. Sci. 254, 6972 (2008) 6. J. A. Paulsen, C. C. H. Lo, J. E. Snyder, A. P. Ring, L. L. Jon es, and D. C. Jiles, IEEE Trans. Magn. 39, 3316 (2003). 7. M. Atif, M.W. Asghar, M. Nadeem , W. Khalid, Z. Ali, and S. Bads hah, J. Phys. Chem. Solids. 123 , 36 (2018). 030533-4

5.0022126.pdf

J. Appl. Phys. 128, 194304 (2020); https://doi.org/10.1063/5.0022126 128, 194304 © 2020 Author(s).Topological insulator nanoribbon Josephson junctions: Evidence for size effects in transport properties Cite as: J. Appl. Phys. 128, 194304 (2020); https://doi.org/10.1063/5.0022126 Submitted: 17 July 2020 . Accepted: 09 November 2020 . Published Online: 20 November 2020 Gunta Kunakova , Ananthu P. Surendran , Domenico Montemurro , Matteo Salvato , Dmitry Golubev , Jana Andzane , Donats Erts , Thilo Bauch , and Floriana Lombardi Topological insulator nanoribbon Josephson junctions: Evidence for size effects in transport properties Cite as: J. Appl. Phys. 128, 194304 (2020); doi: 10.1063/5.0022126 View Online Export Citation CrossMar k Submitted: 17 July 2020 · Accepted: 9 November 2020 · Published Online: 20 November 2020 Gunta Kunakova,1,2 Ananthu P. Surendran,1Domenico Montemurro,1,3 Matteo Salvato,4 Dmitry Golubev,5 Jana Andzane,2Donats Erts,2Thilo Bauch,1and Floriana Lombardi1,a) AFFILIATIONS 1Quantum Device Physics Laboratory, Department of Microtechnology and Nanoscience, Chalmers University of Technology, SE-41296 Göteborg, Sweden 2Institute of Chemical Physics, University of Latvia, Raina Blvd. 19, LV-1586 Riga, Latvia 3Dipartimento di Fisica “Ettore Pancini, ”Università degli Studi di Napoli Federico II, I-80125 Napoli, Italy 4Dipartimento di Fisica, Università di Roma “Tor Vergata, ”00133 Roma, Italy 5QTF Centre of Excellence, Department of Applied Physics, Aalto University, P.O. Box 15100, FI-00076 Aalto, Finland Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Author to whom correspondence should be addressed: floriana.lombardi@chalmers.se ABSTRACT We have used Bi 2Se3nanoribbons, grown by catalyst-free physical vapor deposition to fabricate high quality Josephson junctions with Al superconducting electrodes. In our devices, we observe a pronounced reduction of the Josephson critical current density Jcby reducing the width of the junction, which in our case corresponds to the width of the nanoribbon. Because the topological surface states extend overthe entire circumference of the nanoribbon, the superconducting transport associated with them is carried by modes on both the top andbottom surfaces of the nanoribbon. We show that the J creduction as a function of the nanoribbon width can be accounted for by assuming that only the modes traveling on the top surface contribute to the Josephson transport as we derive by geometrical consideration. This finding is of great relevance for topological quantum circuitry schemes since it indicates that the Josephson current is mainly carried by thetopological surface states. © 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/10.1063/5.0022126 I. INTRODUCTION The study of the proximity effect between a superconductor and a semiconductor or an unconventional metal has latelyreceived a dramatic boost due to the increasing possibilities to man-ufacture a larger variety of interfaces and materials. Novel phenom- enology of the proximity effect is currently coming from the integration of semiconducting nanowires, with strong spin –orbit coupling, as barriers, as well as the edge and surface states of two-dimensional (2D) and three-dimensional (3D) TopologicalInsulators (TIs) 1–5and Dirac semimetals.6,7In these cases, the Josephson transport properties of the hybrid devices will manifest neat fingerprints related to the formation of Majorana boundstates, which is of great interest for topological quantum computation.8–10Lately, superconductor –TI–superconductor Josephson junctions with 2D and 3D TIs have shown a 4 πperiodic Josephson current phase relations,11–14which could be associated with the presence of Majorana modes15and gate-tunable Josephson effects2,16–19when the TI is tuned through the TI ’s Dirac point. Still several aspects of the physics of the Josephson effect related to the topological protected edge/surface states and the con-tribution of the unavoidable bulk remain to be clarified. In thisrespect, the use of 3D TI nanoribbons could be advantageousbecause of the reduced number of transport channels involved in the transport. Here, the transport is ruled by the quantization ofJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 194304 (2020); doi: 10.1063/5.0022126 128, 194304-1 ©A u t h o r ( s )2 0 2 0the nanoribbon ’s propagation modes, which could give new hints about the Josephson phenomenology associated with the Topological Surface States (TSSs). In this work, we have fabricated Josephson junctions by using Bi2Se3Topological Insulator Nanoribbons (TINRs) with widths spanning from 50 nm to almost a micron. Because the TSSs extendover the entire circumference of the TINR, the superconducting transport associated with them is carried by modes on both the top and bottom surface of the nanoribbon. As shown in Fig. 1(a) , in our TINR Josephson junction, the current flows between thesuperconducting contacts fabricated on the top surface of the nanoribbon. For the TINR with a circumference of C¼2Wþ2t (where Wis the width and tthe thickness of the nanoribbon), the transverse momentum k y, perpendicular to the current (see Fig. 1 ), is quantized as ky¼2π(nþ1=2)=C, (1) where nis an integer.21Therefore, the modes with ky/difference0 remain on the top surface, while the modes with ky/C290 are winding around the perimeter of the TINR [see Fig. 1(a) ]. Here, we show that the value of the Josephson current density strongly depends on the junction width, which in our case corresponds to the width of the nanoribbons [ Fig. 1(a) ]. We discuss the possible origin of this phenomenology also in connection with the fact that only small ky value modes are involved in the Josephson transport, which are the ones that travel on the top surface of the nanoribbon. This number reduces by reducing the nanoribbon width [see Eq. (1)]. This finding is of great relevance since it indicates that (a) the Josephsoncurrent is mainly carried by the surface states and (b) because ofthe selectivity in k y, the number of modes involved in the transport scales much faster compared to the width. This is of great relevance for topological quantum circuitry schemes.II. EXPERIMENTAL DETAILS AND RESULTS Nanoribbons of Bi 2Se3with a variation of widths from /difference50 nm to about 1 μm were grown as reported in Ref. 22. For the fabrication of Josephson junctions, nanoribbons were transferred topre-patterned substrates of Si/300 nm SiO 2and SrTiO 3(STO). We used two different TINR growth batches for the two sub- strates, respectively. For the two batches, the carrier densities ofthe topological surface states measured through Shubnikov –de Haas oscillations varied by a factor of 2. 22A standard process of electron beam lithography was used, followed by the evaporation of 3 and 80 nm thick layers of Pt and Al, respectively. Before the evaporation of the metals, the samples were etched for 30 s by Arion milling to remove the native oxide layer of the nanoribbons.SEM images of the fabricated Al =Bi 2Se3=Al Josephson junctions for different nanoribbon widths are shown in Figs. 1(b) and 1(c). The Pt interlayer between the Bi 2Se3and Al has an important role in the formation of a transparent interface, as it was shown in ourprevious works. 3,23,24 Junctions were measured at a base temperature of 19 mK in anrf-filtered dilution refrigerator. The planar Josephson junction can be described as SIS0I0/C0N/C0I0S0IS. Here, S0is the proximized TINR that lies underneath the superconductor (S) —Al, and N rep- resents the not-covered TINR part between the two Al electrodes[see schematics in the inset of Fig. 1(b) ]. I is the interface of the barrier between the Al and the Bi 2Se3nanoribbon, while I0repre- sents the barrier interface formed between the Bi 2Se3under the Al and the Bi 2Se3of the normal metal region N (nanogap). We have previously demonstrated that both I and I0are highly transpar- ent3,23and that we have full control of the strength and phenome- nology of the proximity effect. When the voltage drop across the Al=Bi2Se3interface is negligible, compared to that across the TI in the nanogap, as in our case,25the physics of the planar junction is effectively that of a S0I0/C0N/C0I0S0. A typical current –voltage characteristic (IVC) of one of the TINR based Josephson junctions is shown in Fig. 2(a) . The IVCs of FIG. 1. (a) Schematics of a Bi 2Se3nanoribbon Josephson junction. Arrows indicate transport modes carrying supercurrent by the topological surface states at the nanorib- bon top surface ( ky/difference0) and around the perimeter ( ky/C290). (b) and (c) Partly colored SEM images of the fabricated Josephson junctions of Bi 2Se3nanoribbons with dif- ferent widths W.Lindicates the length of a junction. The scale bar is 500 nm. The inset in panel (b) is a schematic cross section of a junction, and the dotted –dashed green line highlights the location of a trivial 2DEG.20Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 194304 (2020); doi: 10.1063/5.0022126 128, 194304-2 ©A u t h o r ( s )2 0 2 0our junctions have a hysteresis that points to an increased electron temperature as the junction is switched to the resistive state.26A clear gap structure is seen at 2 Δ0(where Δ0is the induced gap of the order of 170 μV); several bumps in the IVC at 2 Δ0=nare also visible: as we have previously demonstrated they are connectedto multiple Andreev reflections that are made possible by the hightransparency of the I and I 0barriers.23The IVCs measured at various temperatures are shown in Fig. 2(b) . At higher tempera- tures, we observe a finite resistance in the supercurrent branch.This could be attributed to premature switching and consecutiveretrapping of the superconducting phase difference. 27–30The so-called phase diffusion regime is characteristic for a moderate to low quality factor Josephson junction. The quality factor can beestimated using Q¼ω PRC, with the plasma frequency ωP¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πIc=Φ0Cp , where Φ0¼h=2eis the superconductive flux quantum, and R≃100Ωis the real part of the shunting admit- tance of a dc biased Josephson junction.31,32Here, the capacitance is dominated by the shunting capacitance through the substrate.For STO with a relative dielectric constant of 25 000 at low temper-atures, we approximate C≃1 pF resulting in a quality factor smaller than one for junctions having a critical current lower than 50 nA. Alternatively, for the Si =SiO 2substrate, it is more difficult to estimate the shunting capacitance due to the conducting substrate.However, we expect a much smaller capacitance value resultingin quality factors smaller than one for critical currents alreadybelow 500 nA. The value of the critical current of the junction I cis obtained from the forward scan, and the critical current density Jcis calcu- lated accordingly by dividing Icby the width of the nanoribbon (W). The normal state resistance RNis determined by the inverse of the slope, calculated from the IVC region at voltages above 2 Δ0 in S0(represented by the area of Bi 2Se3underneath the Al).Figures 3(a) and3(b) show the dependence of the Jcas a function of the Wfor devices with different lengths Lfabricated on Si=SiO2and STO substrates, respectively. In Fig. 3(b) , we have included a 10 μm wide junction where the Al electrodes are pat- terned along the length of TINR; for this device, no windingmodes are expected to contribute to the Josephson transport. One FIG. 2. (a) Current –voltage characteristic of a Bi 2Se3nanoribbon junction ( Ic¼0:36μA,L¼70 nm, t¼16 nm, W¼430 nm) measured at T¼20 mK. The solid black lines are linear fits of the IVC at high bias voltages. The departure from linearity observed at V ¼340μV corresponds to twice the induced gap, with gap Δ0¼170μV (see the dashed line). (b) Low bias current –voltage characteristics at various temperatures T¼100,300,and 390 mK of a Bi 2Se3nanoribbon junction ( Ic¼10 nA, L¼70 nm, t¼13 nm, W¼60 nm). The hysteretic current –voltage characteristics develop a finite resistance in the superconducting branch for increasing temperature. FIG. 3. (a) Critical current density as a function of TINR width for Josephson junction devices realized from the same growth batch on a Si =SiO2substrate. (b) Critical current density as a function of TINR width for Josephson junctionsrealized from a second growth batch [different from those shown in panel (a)] on a SrTiO 3substrate. All measurements were performed at T¼20 mK. The blue and red dots are for junction lengths of 50 –80 nm and 100 –110 nm, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 194304 (2020); doi: 10.1063/5.0022126 128, 194304-3 ©A u t h o r ( s )2 0 2 0clearly sees that if the length is fixed, the Jcsharply decreases as a function of the width Wof the nanoribbon. We note that the value of Jccan change by a factor of 5 –6b yg o i n gf r o mt h en a r - rowest nanori bbons of 60 nm to the widest ones. In cont rast, as shown in Figs. 4(a) and 4(b), the specific resistance, obtained by considering the product RN/C2(W) for the devices on the Si =SiO2 and STO substrate, respectively, is almost independent of W.T h i s fact allows one to exclude that the reduction of the Jcat small widths Whas its origin in strong modifications/deterioration of the junction specific resistance for narrow TINR. III. DISCUSSION AND CONCLUSIONS What is the origin of this peculiar Jc(W) phenomenology? As we have discussed earlier, the explanation of the phenomenon can have its ground in the quantization of the nanoribbon ’s propaga- tion modes. One can derive that the relative number of modesn top=ntottraveling only on the top surface reduces with the junction width for a fixed junction length L. Here, ntotis the total number of modes, ntot¼kFC=2π/C01=2, (2) and ntopis the number of modes traveling only on the top surface of the nanoribbon, ntop¼kFCW =4πL((W=2L)2þ1)/C01=2/C01=2: (3) The relations above can be obtained from geometric considerations; seeFig. 5(a) . Here, kFis the Fermi vector. InFig. 5(b) , we show the width dependence of the relative number of modes ntop=ntotfor three different junction lengths.The curves of Fig. 5(b) have been obtained by considering kF¼0:55 nm/C01, a value that is typical for our nanoribbons.20For L¼100 nm, we obtain a reduction of the relative number of modes traveling only on the top surface by a factor of 5 whenreducing the junction width going from 900 nm to 50 nm. For comparison in Fig. 5 , we also show the full calculation of the supercurrent density following Ref. 7taking into consideration also the angle dependence of the transmission coefficients of eachtransport mode traveling only on the top surface. We see that thefull calculation of J Cand the relative number of transport modes using Eqs. (2)and(3)give the same qualitative behavior. This dependency qualitatively reproduces the measured Jcvs width dependence shown in Figs. 3(a) and 3(b). Indeed, the solid lines represent the relative number of transport modes forL¼100 nm. This suggests that only the modes traveling on the top surface contribute to the Josephson current. We note that the criti- cal current density of the 10 μm wide device is in agreement with FIG. 5. (a) Sketch of a planar TI junction with electrode separation Land width W. The dashed lines indicate quasiparticle trajectories. The maximum transver- sal momentum for which the transport mode still propagates only on the topsurface is indicated by k y. For larger transversal momentum, the quasiparticle trajectory has to wind around the TINR and does not contribute to the critical current (red cross). (b) Relative number of transport modes nrel¼ntop=ntot propagating only on the top surface (dashed lines) and the corresponding nor- malized critical current density (solid lines) as a function of junction width forthree different junction lengths. FIG. 4. (a) Specific resistance as a function of TINR width for Josephson junc- tion devices realized from the same growth batch on a Si =SiO2substrate. (b) Specific resistance as a function of TINR width for Josephson junctions realized from a second growth batch [different from those shown in panel (a)] on a SrTiO 3substrate. All measurements were performed at T¼20 mK. The blue and red dots are for junction lengths of 50 –80 nm and 100 –110 nm, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 194304 (2020); doi: 10.1063/5.0022126 128, 194304-4 ©A u t h o r ( s )2 0 2 0the saturation value of the expected current density, where the contribution of winding modes to the total Josephson current is negligible. A possible explanation for this finding could be related to the lower mobility of the Dirac states at the interface between the TINRand the substrate. Indeed, our magnetotransport measurements have shown the formation of a trivial 2D gas at the interface with the substrate overlapping with the Dirac states at the nanoribbonbottom [see the inset of Fig. 1(b) ]. 20This interaction, which leads to a lower mobility (diffusive transport regime) of the Dirac states,might be responsible for the transport modes winding around the nanoribbon (high k ymodes) contributing less to the Josephson transport. Although we cannot exclude that the transport throughthe trivial 2D gas contributes to the Josephson current, it wouldonly cause a constant offset of the J Cvalues without affecting the overall width dependence. We observe that the overall values of JC for the devices on the STO substrate are higher than those on Si=SiO2. This can be partially attributed to the larger values of the top surface Dirac carrier density of the batch used to realize thedevices on the STO substrate. The larger value of the trivial 2Dcarrier density we typically observe in devices fabricated on STO substrates 33could be further responsible for the difference observed in the critical current densities between devices on STO andSi=SiO 2substrates. To conclude, we have fabricated high transparency 3D TINR Josephson junctions showing a peculiar phenomenology that can be associated with the transport through the topological surfacestates. This is a step forward toward the study of topological super-conductivity in a few mode devices instrumental for topologicalquantum computation. ACKNOWLEDGMENTS This work has been supported by the European Union ’s Horizon 2020 Research and Inn ovation Program (Grant Agreement No. 766714/HiTIMe) and by the European Union ’s project NANOCOHYBRI (Cost Action CA 16218). G.K. acknowledges the European Regional Development Fund project(No. 1.1.1.2/VIAA/1/16/198). This work was supported by theEuropean Union H2020 under the Marie Curie Actions(No. 766025-QuESTech). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1J. Hajer, M. Kessel, C. Brüne, M. P. Stehno, H. Buhmann, and L. W. Molenkamp, “Proximity-induced superconductivity in CdTe –HgTe core – shell nanowires, ”Nano Lett. 19, 4078 –4082 (2019). 2L. A. Jauregui, M. Kayyalha, A. Kazakov, I. Miotkowski, L. P. Rokhinson, and Y. P. 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5.0024431.pdf

Appl. Phys. Lett. 117, 132405 (2020); https://doi.org/10.1063/5.0024431 117, 132405 © 2020 Author(s).Electrostatic-doping-controlled phase separation in electron-doped manganites Cite as: Appl. Phys. Lett. 117, 132405 (2020); https://doi.org/10.1063/5.0024431 Submitted: 07 August 2020 . Accepted: 15 September 2020 . Published Online: 29 September 2020 Dong-Dong Xu , Ru-Ru Ma , You-Shan Zhang , Xing Deng , Yuan-Yuan Zhang , Qiu-Xiang Zhu , Ni Zhong , Xiao-Dong Tang , Ping-Hua Xiang , and Chun-Gang Duan COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Tracking ferroelectric domain formation during epitaxial growth of PbTiO 3 films Applied Physics Letters 117, 132901 (2020); https://doi.org/10.1063/5.0021434 Size-derived reaction mechanism of core-shell aluminum nanoparticle Applied Physics Letters 117, 133902 (2020); https://doi.org/10.1063/5.0015367 Synthetic chiral magnets promoted by the Dzyaloshinskii–Moriya interaction Applied Physics Letters 117, 130503 (2020); https://doi.org/10.1063/5.0021184Electrostatic-doping-controlled phase separation in electron-doped manganites Cite as: Appl. Phys. Lett. 117, 132405 (2020); doi: 10.1063/5.0024431 Submitted: 7 August 2020 .Accepted: 15 September 2020 . Published Online: 29 September 2020 Dong-Dong Xu,1Ru-Ru Ma,1You-Shan Zhang,1Xing Deng,1Yuan-Yuan Zhang,1Qiu-Xiang Zhu,1NiZhong,1,2,a) Xiao-Dong Tang,1Ping-Hua Xiang,1,2,a) and Chun-Gang Duan1,2 AFFILIATIONS 1Key Laboratory of Polar Materials and Devices, Department of Electronics, East China Normal University, Shanghai 200241, China 2Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China a)Authors to whom correspondence should be addressed: nzhong@ee.ecnu.edu.cn andphxiang@ee.ecnu.edu.cn ABSTRACT The coexistence of distinct insulating and metallic phases within the same manganite sample, i.e., phase separation scenario, provides an excellent platform for tailoring the complex electronic and magnetic properties of strongly correlated materials. Here, based on an electric- double-layer transistor configuration, we demonstrate the dynamic control of two entirely different phases—canted G-type antiferromagnetic metal and C-type antiferromagnetic charge/orbital ordered insulator phase—in electron-doped system Ca 1/C0xCexMnO 3(x¼0.05). The reversible metal-to-insulator transition, enhanced colossal magnetoresistance ( /C2427 000% for Vg¼3.0 V), and giant memory effect have been observed, which can be attributed to an electronic phase separation scenario manipulated by a tiny doping-level-variation of less than 0.02 electrons per formula unit. In addition, the controllable multi-resistance states by the combined application of magnetic and electrostatic fields may serve as an indicator to probe the dynamic multiphase competition of strongly correlated oxides. These results offer crucialinformation to understand the physical nature of phase separation phenomena in manganite systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024431 The dynamic reversible regulation of electronic and magnetic states in strongly correlated electron systems has been a hot topic in condensed matter physics. 1–9In perovskite manganites, rich electronic and magnetic states are enabled by the complex interactions among charge, spin, orbital, and lattice degrees of freedom, as exemplified in various spin, charge, and orbital ordering, Mott insulators, and multi-ferroics. 10–15In particular, phase separation, a core issue in the physics of manganites caused by those ordering competitions, has drawn extensive attention and become the most cited mechanism of manyexotic physical phenomena including metal-insulator transition, colos- sal magnetoresistance effect, and magnetorelaxor behavior. 16–21It is well recognized that the phase separation scenario typically involvestwo entirely different phases, i.e., ferromagnetic metal and antifer-romagnetic charge/orbital ordered insulator, showing strong ten- dencies toward intrinsically nanometer- or micrometer-scale inhomogeneity in manganites. 22Nevertheless, the existing mecha- nisms invoked so far remain controversial due to the intrinsic complexity of the manganite models. One of the generally accepted theories is electronic phase separation, a purely electronicresponse. 22Carrier density plays a crucial role in determining electronic and magnetic ground states in hole- or electron-dopedmanganites. However, the lattice disorder induced by the most commonly used carrier density tuning methods (chemical/electro- chemical doping) inevitably intertwines in these systems. Consequently, the lattice disorder can significantly change the elec- tronic and magnetic states because the electrons and their corre-lated interactions in these materials are tightly coupled to the lattice. 23–26Thus, it is a hard task to isolate the carrier density tuning effect from lattice disorder influenced by external factors. The recent development of electric-double-layer transistors (EDLTs) both circumvents these challenges and provides unique opportunities for realizing the purely tuning effect of carrier den-sity. 1,27Electrostatic doping via giant surface charge accumulation is considered a “clean” “impurity-free” way to probe the phase separa- tion phenomenon without disorder. However, in the past few years,the ionic liquid gated manganites usually followed the electrochemical mechanism (H þor O2/C0intercalation/extraction),28–34which makes the phase separation issue more complicated because of the composi- tion alteration or structural changes. On the other hand, much atten- tion has been focused on hole-doped manganites, correspondingto the Mn 3þ-rich composition. However, there have been no reports on the electron-doped manganite models based on the EDLT Appl. Phys. Lett. 117, 132405 (2020); doi: 10.1063/5.0024431 117, 132405-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplconfiguration in spite of the large difference in the electron- and hole- doped regions in phase diagrams. In this paper, we focus on the electrostatic doping-induced elec- tronic phase separation in electron-doped Ca 1/C0xCexMnO 3(CCMO) thin films. In bulk samples, non-doped CaMnO 3is a non-collinear G- type antiferromagnetic insulator ( G-AFI) [ Fig. 1(a) , orange arrows].14,35 Through the substitution of Ce4þfor Ca2/C0(0.01<x<0.07), doped charge carriers ( egelectrons) mediate the double-exchange interaction providing a weak ferromagnetic component. Distinct from Ca-dopedLaMnO 3, the ferromagnetic component originates from the canted G- type antiferromagnetic metal ( cG-AFM) [ Fig. 1(a) , green arrows] instead of a ferromagnetic phase. The higher concentration of Ce4þ (x>0.07) leads to the formation of charge and/or orbital ordered (CO/OO) C-type antiferromagnetic insulators ( C-AFI) [ Fig. 1(a) ,g r a y arrows].36Recently, Xiang et al. have experimentally elucidated the phase diagrams of strained CCMO epitaxial films.13,37,38For the Ca0.95Ce0.05MnO 3(x¼0.05, CCMO5) film, a metallic transport char- acteristic is observed only on a practically lattice-matched NdAlO 3 (NAO) substrate.11When the doping concentration x¼0.06, the perovskite manganite Ca 0.94Ce0.06MnO 3(CCMO6) undergoes a transi- tion from cG-AFM to C-AFI, showing extreme sensibility to electron doping, whereas little is known about the intermediate states especially for a tiny doping-level-variation in both experiment and theory. Therefore, CCMO5 films are an excellent model system for the electrostatic doping study. In this work, we demonstrate the dynamiccontrol of electrostatic-doping-induced electronic phase transition by doping a tiny amount of electrons (less than 0.02 electrons per formula unit) into CCMO5 films using an EDLT configuration. Remarkably,the insulation of metallic CCMO5 films, enhancement of magnetore-sistance, and magnetoresistive memory effect caused by electronic phase separation are observed. The electric field suppresses the metal- lic phase and stabilizes a metastable insulator at low temperature, andthe magnetic field has the opposite effect, leading to controllablemulti-resistance states of CCMO5 thin films. These results providecrucial information to understand the physical nature of phase separa- tion behavior in manganite systems. CCMO epitaxial thin films with nominal Ce concentrations of x¼3, 4, 5 and 6% (CCMO3, CCMO4, CCMO5, and CCMO6) were fabricated by pulsed laser deposition on (001)-oriented NAO sub-strates. Typical x-ray diffraction 2 h–xscans with clear Laue fringes around the (002) peak certify the structural coherence of CCMO films [Fig. 1(c) ]. With the increasing Ce doping concentration, the diffrac- tion peak shifts to a lower angle, gradually showing an enlarged out-of-plane cell parameter and cell volume. This is due to the increasedMn (III) (0.72 A ˚) content with a larger effective ion radius relative to that of Mn (IV) (0.67 A ˚), indicating the activation of Ce ion doping. Because of the huge magnetic moment of the NAO substrate, it isalmost impossible to trace the magnetic properties of the film directly.Thanks to the strong coupling between the electronic and magneticstates, transport properties with rich electronic and magnetic informa- tion are recognized as a powerful tool to probe the complex phase transformation in manganates. Hence, the low-temperature transportproperties were performed to study the ground states of our fabricatedCCMO films (Fig. S1 in the supplementary material ). The results reveal a consecutive phase transitions upon increasing the Ce 4þdop- ing level: from G-AFI to cG-AFM and then to C-AFI. Hall measure- ments (Fig. S2 in the supplementary material ) suggest that Ce doping is fully active and the substitution of Ca2þby Ce4þresults in the release of two electrons into the Mn 3d e gorbitals, which determines the electronic and magnetic states of CCMO films. The phase diagram is summarized in Fig. 1(b) . To further investigate the electrostatic doping effect on CCMO5 thin films, an EDLT device was fabricated utilizing ionic liquid N,N-diethyl-N-methyl-N-(2-methoxyethyl)-ammonium bis-(trifluorome- thanesulfonyl)imide ([DEME][TFSI]) as the dielectric layer [ Fig. 2(a) ]. The CCMO5-channel size is 10 lmw i d e ,1 0 0 lml o n g ,a n d4 0 n m thick (Fig. S3 in the supplementary material ). To suppress the possible electrochemical reaction, we introduce electrons into CCMO5 films at FIG. 1. Chemical doping-dependent electronic, magnetic, and structural properties of Ca 1/C0xCexMnO 3films. (a) Sketch of spin structures. Orange, green, and grey arrows rep- resent the G-type antiferromagnetic, canted G-type antiferromagnetic, and C-type antiferromagnetic configurations, respectively. Canting angles are amplified for clarity. (b) A summarized phase diagram of CCMO films on (001)-oriented NAO substrates. The abbreviations denote the G-type antiferromagnetic insulator ( G-AFI), canted G-type antifer- romagnetic metal ( cG-AFM), charge and/or orbital ordered C-type antiferromagnetic insulator (CO/OO C-AFI), paramagnetic insulator (PI), and paramagnetic metal (PM). (c) X-ray diffraction 2 h–xscans of CCMO thin films.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132405 (2020); doi: 10.1063/5.0024431 117, 132405-2 Published under license by AIP Publishing230 K by low gating voltages 0 V /C20Vg/C203 V (within the electrochemi- cal stability window of [DEME][TFSI]).39The relatively low tempera- ture not only suppresses the electrolysis of residual water but also ensures the accumulation of cations [DEME] at the solid–liquid interface (Fig. S4 in the supplementary material ). The temperature- dependent four-terminal resistivity was measured in the range of230–10 K. As shown in Fig. 2(b) , the pristine CCMO5 film shows a metallic transport characteristic with the positive temperature deriva- tive of q(dq/dT>0). After the application of gating biases, the contin- uous increase in d q/dThas been observed, indicating the gradual formation of insulating states from metallic CCMO5 films. Finally, aresistivity of more than three orders of magnitude was achieved at 10 K, which corresponds to the significant changes of ground states in CCMO5 films. It can be attributed to the electronic phase separation,which has been well studied in chemical-doped CaMnO 3systems. According to the phase diagram of the CCMO films on NAO,11 CCMO5 is located in the vicinity of the phase boundary betweencG-AFM and C-AFI. As in the case of the CCMO bulk, 14doping more electrons to e gorbitals of Mn4þions by electrostatic gating is likely to increase the number of double exchange interactions ofMn 3þ-O-Mn4þ, which would introduce a magnetic frustration to the three-dimensional G-type AFM that originates from superexchangeinteractions of Mn4þ–O–Mn4þtype. Eventually, with the increase in gating voltage, the gated CCMO5 film tends to stabilize the C-typeAFI phase containing weak ferromagnetism chains. The electrostatic doping-controlled phase separation can be attributable to the coexis- tence and competition between double exchange and superexchangeinteractions. In addition, the effect of the Jahn-Teller distortion andorbital reconstruction rising from the Mn 3þegelectrons may also play a role along with the increase in Mn3þions.14In the present work, the electrostatic doped electrons can lead to the consecutive creation and growth of the C-AFI phase from a cG-AFM host matrix, and the C- AFI fraction plays a critical role in blocking the percolative conductionof the whole film. Figure 2(c) shows the coexistence and competition between cG-AFM and C-AFI ground states in CCMO5 films, which can be controlled and stabilized by electrostatic doping. Recent studieson electrolyte-gated manganates have been reported that an electro-chemical process including O 2/C0intercalation/extraction may account for the regulation of electronic and magnetic states.7,40,41To confirm the electrostatic operation, we retested the transport curve after settinggate bias V g¼0 V. A reverse process, i.e., insulator-to-metal transition, was observed, and the transport curve returned to the original state[Fig. 2(b) , dotted line]. That is, the emerging insulating states are metastable depending on the creation and elimination of the FIG. 2. Electrostatic doping-induced reversible electronic phase separation. (a) Schematic views of the CCMO5-channel EDLT device. At Vg>0 V, the electric-field-driven [DEME] cations and [TFSI] anions form electric double layers at the surface of the CCMO5 thin film and the gate, respectively. (b) Resistivity vs temper ature ( q–T) curves showing the insulation of the metallic CCMO5 film in a continuous reversible manner. (c) Schematic diagrams of electric/magnetic field-controlled el ectronic phase separation involving cG-AFM and C-AFI ground states at various gating voltages. The abbreviations denote the electric field (E) and magnetic field (H). (d) Temperature dependence of activation energy Ea. (e) ln( rT1/2) as a function of 1/ T1/4according to Mott’s variable range hopping model. (f) Summarized phase diagram of electrostatic-doped CCMO films. The triangular symbols represent the transition temperatures determined from the transport properties.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132405 (2020); doi: 10.1063/5.0024431 117, 132405-3 Published under license by AIP Publishingelectric-double-layer at the liquid-solid interface. Moreover, after removing the ionic liquid, the CCMO5 channel shows high quality surface morphology after gating, indicating the negligible etching effect caused by electrochemical reactions (Fig. S5 in the supplementary material ). The good cyclability of the electrolyte gating process is dem- onstrated in Fig. S6 in the supplementary material . In order to clarify the links between transport properties and phase transitions, the activation energy Eaas a function of temperature was evaluated from q-Tcurves through a thermal activation model: r¼r0exp(/C0Ea/jBT). We determine the phase-transition tempera- tures, i.e., N /C19eel temperature TNor CO/OO transition temperature Tco, from Ea-Tcurves because of the excellent consistency of phase- transition temperatures derived from transport and magnetic proper- ties, respectively, according to our pervious results.11Figure 2(d) shows the temperature dependence of Eafor CCMO5 films tuned by different gating voltages. At Vg¼0 V, the negative Eavalues suggest a metallic conduction except for a small kink. The kink stemming from the con- ductivity anomaly is a sign of phase transition from paramagnetic metal (PM) to cG-AFM at TN¼100 K. When increasing gate bias, the Ea-Tcurves change significantly: (i) the Eavalues convert to positive, indicating a metal-to-insulator transition; (ii) the peak width shows a clear broad trend toward that of CCMO6 gradually. The broad peaksofE a-Tcurves can be interpreted as indicative of CO/OO C-AFI tran- sition by comparison with the same phenomena of chemical-doped CCMO films in previous studies.11In other words, the electrostatic doping induced a series of intermediate states between chemical- doped CCMO5 and CCMO6. (iii) The phase-transition temperatures TNincrease linearly, disclosing the improved stability of CO/OO C-AFI states via electrostatic doping. We note that Eastrongly depends on temperature, implying that the electrical conduction of the gated CCMO5 films does not follow an Arrhenius behavior. In other electron-doped manganites, an adia-batic small polaron conduction mechanism was proposed to fit the electrical conductivity curves. However, a large deviation was observed below T Nin our films as displayed in Fig. S7(a) in the supplementary material . Consequently, Mott’s variable range hopping (VRH) model has been attempted and indeed completely dominates the transport property in the CCMO5 film below TN[Fig. S7(b) in the supplementary material ]. We then used the VRH model expression: r¼r0exp[/C0(T0/T)1/4] to fit the mixed phase states of gated CCMO5 films by different biases.42,43ln(rT1/2) as a function of 1/ T1/4 (T<60 K) is plotted in Fig. 2(e) . A well-behaved linear relationship suggests that the conducting states of electrostatic-doped CCMO5 films are governed by disorder-induced localized states, as well as chemical-doped CCMO films (Fig. S8 in the supplementary material ). The fitted electronic states exhibit a series of intermediate behaviors approaching to those of CCMO6 films, which coincides with the continuous phase transition scenario. According to the VRH model, the deduced average hopping energy Was a function of the gate voltage and Ce4þdoping level also supports this point (Fig. S9 in the supplementary material ). From this perspective, it can be roughly estimated that the electrostatically doped-electron-concen- tration is less than 2% electrons per formula unit. This doping level is close to the value of our numerical estimation 0.66% per Mn atom (see the supplementary material ) for Vg¼3 V according to Ref. 39. Figure 2(f) summarizes the phase diagram of electrostatic-doped CCMO5 films.Since the electrostatic field favors the CO/OO states in C-AFI domains, while the external magnetic field tends to suppress it as pre- viously reported,37,44the phase competition process can be dynami- cally probed by the combined application of magnetic and electrostaticfields. Figures 3(a)–3(e) exhibit the magnetotransport properties under the magnetic fields of 0 T (solid lines) and 7 T (dotted lines). When V g /C202.0 V, a clear insulator (0 T)-to-metal (7 T) transition was observed due to the charge/orbital ordering collapse [ Figs. 3(b) and3(c)]. The recovery of metallic behavior over the whole temperature range indi-cates that the charge ordered state generated by electrostatic dopingcan be totally extinguished by a magnetic field [ Fig. 2(c) ]. However, at V g¼3.0 V, the volume of the charge/orbital ordered phase becomes the majority giving an insulating state even under a magnetic field of7T[Fig. 3(e) ]. A neck and neck state with d q/dT/C250f o r cG-AFM and CO/OO C-AFI phases seems to have been achieved at V g¼2.5 V, H¼7T[Fig. 3(d) ]. As shown in Fig. 3(f) , an enhanced negative magnetoresistance effect was also observed depending on the gating voltage. The enlargedmagnetoresistance effect is interpreted as the integrated impacts ofelectrostatic and magnetic fields. The stronger electrostatic field con- tributes to a higher resistance state, and the magnetic field favors a low resistance state. Consequently, the magnetoresistance value of [ q(0T) /C0q(7 T)]/ q(7T) reaches more than 27 000% at 10 K for V g¼3.0 V. To examine the magnetoresistive characteristics, the resistivity of gated CCMO5 films was measured with successive scans of magnetic fields (0 T !9T!/C0 9T!0T )a t1 0K( Fig. 4 ). Evidently, the elec- trostatically doped CCMO5 films exhibits a large memory effect ofmagnetic field history, i.e., magnetorelaxor behavior. In fact, the mem-ory effect has been also found in other phase separation systems, such as Nd 0.7Sr0.3MnO 3and La 0.67Ca0.33MnO 3.18,24In those systems, the volume fraction of COO and ferromagnetic domains can be tuned and FIG. 3. Magnetotransport properties of gated CCMO5 films. (a)–(e) Resistivity vs temperature curves under the magnetic fields of 0 T (solid lines) and 7 T (dotted lines). The magnetic field-induced melting of CO/OO phase leads to low resistance states in competition with the electrostatic doping effect. (f) The enhanced magneto-resistance effect depending on the electric-double-layer gating. The magnetoresis-tance (MR) is defined as [ q(7T)/C0q(0 T)]/q(0T).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132405 (2020); doi: 10.1063/5.0024431 117, 132405-4 Published under license by AIP Publishingfixed by the applied magnetic field history. Similarly, the memory effect in gated CCMO5 films can be interpreted in the view of phaseseparation behavior. The suppression of the insulating charge-orderedphase by an external magnetic field is an irreversible process.Removing the magnetic field will lead to a partial recovery of insultingdomains, which is accompanied by the decrease in metallic domains. In summary, we have demonstrated the electrostatic-doping-con- trolled multiphase coexistence and competition in electron-dopedmanganite CCMO5 via the EDLT configuration. The C-type antiferro- magnetic charge/orbital ordered insulator phase can be induced grad-ually from the canted G-type antiferromagnetic metallic host matrix. A series of intermediate states with different volume ratios of insulatorand metal phases have been achieved by doping a tiny amount of elec-trons (less than 0.02 electrons per formula unit) into CCMO5 thinfilms, accompanied by reversible metal-to-insulator transition,enhanced colossal magnetoresistance ( /C2427 000% for V g¼3.0 V), and giant memory effect. The combined application of magnetic andelectrostatic fields provides further insights into the complex phaseseparation of strongly correlated systems. See the supplementary material for experimental methods, trans- port properties of the Ca 1/C0xCexMnO 3and electrolyte-gated films, image of the EDLT device, surface morphology of Ca 0.95Ce0.05MnO 3, fitted electrical conductivity curves, and estimation of the doping level. The authors would like to thank Dr. A. Sawa for useful discussions and thank Dr. W. Yin for his assistance in electricalmeasurements. This work was supported by the National KeyResearch and Development Program of China (Grant No.2017YFA0303403), the National Natural Science Foundation ofChina (Grant Nos. 11874149, 12074119, and 11774092), theShanghai Science and Technology Innovation Action Plan (GrantNo. 19JC1416700), the ECNU (East China Normal University)Multifunctional Platform for Innovation (006), and theFundamental Research Funds for the Central Universities. D.-D. Xuacknowledges the ECNU Academic Innovation PromotionProgram for Excellent Doctoral Students (Grant No. YBNLTS2019-030).DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Z. Bisri, S. Shimizu, M. Nakano, and Y. Iwasa, Adv. Mater. 29, 1607054 (2017). 2Y. Deng, Y. Yu, Y. Song, J. Zhang, N. Z. Wang, Z. Sun, Y. Yi, Y. Z. Wu, S. Wu, J. Zhu, J. Wang, X. H. Chen, and Y. Zhang, Nature 563, 94 (2018). 3A. M. Goldman, Annu. Rev. Mater. Res. 44, 45 (2014). 4Z. Li, S. Shen, Z. Tian, K. Hwangbo, M. Wang, Y. Wang, F. M. Bartram, L. He, Y. Lyu, Y. Dong, G. Wan, H. Li, N. Lu, J. Zang, H. Zhou, E. Arenholz, Q. He,L. Yang, W. Luo, and P. Yu, Nat. Commun. 11, 184 (2020). 5N. Lu, P. Zhang, Q. Zhang, R. Qiao, Q. He, H. B. Li, Y. Wang, J. Guo, D. Zhang, Z. Duan, Z. Li, M. Wang, S. Yang, M. Yan, E. Arenholz, S. Zhou, W. Yang, L. Gu, C. W. Nan, J. Wu, Y. Tokura, and P. Yu, Nature 546, 124 (2017). 6M. Wang, X. 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5.0013852.pdf

J. Chem. Phys. 152, 231102 (2020); https://doi.org/10.1063/5.0013852 152, 231102 © 2020 Author(s).Non-adiabatic transitions in the reaction of fluorine with methane Cite as: J. Chem. Phys. 152, 231102 (2020); https://doi.org/10.1063/5.0013852 Submitted: 14 May 2020 . Accepted: 31 May 2020 . Published Online: 16 June 2020 Bin Zhao , and Uwe Manthe ARTICLES YOU MAY BE INTERESTED IN Using principal component analysis for neural network high-dimensional potential energy surface The Journal of Chemical Physics 152, 234103 (2020); https://doi.org/10.1063/5.0009264 State-pairwise decoherence times for nonadiabatic dynamics on more than two electronic states The Journal of Chemical Physics 152, 234105 (2020); https://doi.org/10.1063/5.0010081 Temperature dependence on bandgap of semiconductor photocatalysts The Journal of Chemical Physics 152, 231101 (2020); https://doi.org/10.1063/5.0012330The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Non-adiabatic transitions in the reaction of fluorine with methane Cite as: J. Chem. Phys. 152, 231102 (2020); doi: 10.1063/5.0013852 Submitted: 14 May 2020 •Accepted: 31 May 2020 • Published Online: 16 June 2020 Bin Zhaoa) and Uwe Manthea) AFFILIATIONS Theoretische Chemie, Fakultät für Chemie, Universität Bielefeld, Universitätsstr. 25, D-33615 Bielefeld, Germany a)Authors to whom correspondence should be addressed: bin.zhao@uni-bielefeld.de and uwe.manthe@uni-bielefeld.de ABSTRACT Reactions of methane with different atoms are benchmark examples of elementary reaction processes intensively studied by theory and experiment. Due to the presence of conical intersections and spin–orbit coupling, non-adiabatic transitions can occur in reactions with F, Cl, or O atoms. Extending detailed quantum theory beyond the Born–Oppenheimer approximation for polyatomic reaction processes, non-adiabatic wave packet dynamics calculations studying the F(2P3/2)/F∗(2P1/2) + CHD 3→HF + CD 3reaction on accurate vibroni- cally and spin–orbit coupled diabatic potential energy surfaces are presented. Non-adiabatic transitions are found to increase the reac- tivity compared to Born–Oppenheimer theory and are more prominent than in triatomic reactions previously studied. Furthermore, the lifetimes of reactive resonances are reduced. The reactivity of F(2P3/2) is found to exceed the one of F∗(2P1/2) even at low collision energies. Published under license by AIP Publishing. https://doi.org/10.1063/5.0013852 .,s Fundamental research on elementary chemical reactions focuses on a detailed quantum state-resolved understanding of increasingly complex processes. Moving beyond triatomic and tetra-atomic systems, reactions of methane with H, F, Cl, or O atoms have become benchmark examples intensively investi- gated by sophisticated experiments1–8and detailed theory.5,7–12 First principle based theory of reactive scattering typically uti- lizes the Born–Oppenheimer approximation, and calculations sim- ulate the quantum dynamics on a single adiabatic potential energy surface (PES). However, atoms such as F or Cl show degener- ate electronic states affected by spin–orbit coupling and the PESs describing their reaction with methane show conical intersections in the entrance channel.8,10,13These conical intersections give rise to non-adiabatic transitions that potentially affect the reaction dynamics.14 The effect of non-adiabatic transitions on elementary chemical reactions has been rigorously studied for the triatomic F + H 2and Cl + H 2reactions.15–20The Born–Oppenheimer approximation was found to work well for the reaction of ground state F(2P3/2) atoms16 and to slightly overestimate the reaction probabilities for the reac- tion of Cl(2P3/2).19An important question that motivates the present work is whether these conclusions can be transferred to reactions of methane and other polyatomic molecules. While the reactions ofhalogens with hydrogen and methane show similar energetics and other properties, the change from the linear H 2molecule to the non-linear CH 4molecule results in fundamental topological differ- ences that affect the definition of the diabatic electronic states.21–23 Furthermore, the significantly increased number of degrees of freedom present in the methane potentially enhances energy dissipation. The F + CHD 3reaction received particular interest due to the counter-intuitive mode selective chemistry observed in experi- ment.4,24,25In an attempt to theoretically understand the underlying dynamics, the reaction was investigated by quasi-classical trajectory calculations26–28and reduced dimensionality wave packet dynam- ics calculations.29These studies employed the Born–Oppenheimer approximation and considered only the lowest adiabatic PES. Tra- jectory surface hopping calculations based on accurate vibronically and spin–orbit coupled PESs indicated a significant amount of non- adiabatic transitions.14However, zero point energy leakage was found to be a major problem in the trajectory calculations limiting their predictive power. This work presents the first detailed quantum dynamics calcu- lations studying a reaction of methane, the F + CHD 3→HF + CD 3 reaction, beyond the Born–Oppenheimer approximation. Specif- ically, the reaction of fluorine atoms in the2P3/2ground state J. Chem. Phys. 152, 231102 (2020); doi: 10.1063/5.0013852 152, 231102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp and the2P1/2spin–orbit excited electronic state with CHD 3reac- tants in the ro-vibrational ground state is investigated. Seven- dimensional wave packet dynamics calculations are performed on vibronically and spin–orbit coupled diabatic PESs, which have been constructed based on accurate high-level ab initio calculations.8,13,22 The six-state diabatic model developed in Refs. 8, 13, and 22 employs a coordinate-dependent potential matrix Vpotto describe the vibronic interaction and a coordinate-independent spin–orbit coupling matrix VSO. Efficient wave packet propagation is facilitated by the operator splitting e−iˆHΔt=e−iˆTΔt 2e−iVSOΔt 2e−iVpotΔte−iVSOΔt 2e−iˆTΔt 2 (1) (ˆHand ˆTdenote the Hamiltonian and the kinetic energy oper- ator, respectively), where the coordinate-independent propagator e−iVSOΔt/2is evaluated using a closed matrix form and the e−iVpotΔt propagation is performed in the diagonal (adiabatic) representation ofVpot. The calculations explicitly consider all nuclear degrees of freedom except the symmetry breaking bends of the CD 3fragment and the C–D-stretches. Due to the very early transition state of the reaction, these vibrational modes do not relevantly affect the reac- tion probabilities and cross sections and are frozen at their reactant equilibrium values. Spin–vibronic angular momentum coupling is irrelevant at the energy resolution considered here,33and reaction probabilities have been computed for vanishing total nuclear angular momentum. Integral cross sections have been calculated using the non-adiabatic reaction probabilities and the transition state based rotational sudden approach.32Further details regarding methods, numerical parameters, and convergence testing are given in the supplementary material. Two different effects resulting from non-adiabatic transitions will be investigated. First, the validity of the Born–Oppenheimer approximation for the reaction of the ground state F(2P3/2) atoms will be tested by comparison with fully vibronically and spin– orbit coupled calculations. Differences between existing results for triatomic reactions and the present findings for a polyatomic reaction will be outlined. Second, the reaction of the spin–orbit excited F∗(2P1/2) atoms, a processes prohibited within the Born– Oppenheimer approximation, will be studied. To provide an overview of the system, adiabatic PESs in the entrance channel and the transition state region are shown in Fig. 1. Cuts along the F–C-distance Rare shown for C 3vsymmetric geome- tries corresponding to the H–CD 3–F (on face ) and F–H–CD 3(on H) arrangements in panels (a) and (b), respectively. The two low- est adiabatic PESs correlate with incoming F(2P3/2) atoms, while the third adiabatic PES correlates with spin–orbit excited F∗atoms in the2P1/2state. Conical intersections of the two lowest adiabatic PESs can be found at on H geometries. Cuts along an angular coordi- nate θdescribing the rotation of the F atom around the methane in a C S-symmetric arrangement are depicted in panel (c) for a F–C- distance of 3.68 Å. Two conical intersections can be seen at the on Hgeometries with θ= 0 and θ= 2π/3. The shape of the adiabatic PES shown in panel (c) provides important information regarding the non-adiabatic dynamics to be expected. Wave packet dynam- ics on conically intersecting PESs generally follows a simple set of rules (see, e.g., Refs. 30 and 31 for a detailed discussion): wave pack- ets on the ground adiabatic PES tend to stay on the same PES, while wave packets on an excited adiabatic PES decay to the lower FIG. 1 . Diabatic PESs for F + CHD 3: Cuts along the F–C-distance Rcorre- sponding to H–CD 3–F and F–H–CD 3arrangements at C 3vsymmetric geometries are displayed in panels (a) and (b), respectively (CHD 3is frozen at its equilib- rium geometry). The angular dependence of the two lower adiabatic PESs at the R-value of the conical intersection is shown in panel (c). Here, a cut along the angle θdescribing the rotation of the F atom around the CHD 3in a C S-symmetric arrangement is depicted. J. Chem. Phys. 152, 231102 (2020); doi: 10.1063/5.0013852 152, 231102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp adiabatic PES if attracted to a funnel resulting from a conical inter- section. Two such funnels are clearly visible in Fig. 1(c). However, a clear downward route from the excited adiabatic PES to the ground adiabatic PES via a funnel is available only for θvalues in specific intervals. F + CHD 3→HF + CD 3reaction probabilities for CHD 3in the ground ro-vibrational state are calculated at different levels of the- ory. The results are shown in Fig. 2. For the reaction of ground state F(2P3/2) atoms, the fully non-adiabatic theory (black line) is found to yield reaction probabilities 15% larger than the ones obtained within the Born–Oppenheimer approximation (thin red line). Moreover, the prominent resonance structures visible at low energies in the Born–Oppenheimer approximation are much less pronounced in the non-adiabatic theory. Thus, the lifetime of the reactive reso- nances that give rise to the resonance structures is reduced by the non-adiabatic coupling. The increase in reactivity resulting from non-adiabatic transi- tions can be rationalized with the help of the PESs displayed in Fig. 1 and the general rules regarding wave packet motion on coupled PESs described above. Two adiabatic PESs correlate with the F(2P3/2) + CHD 3asymptote, but only one of them is reactive.13,14,22The incoming reactants are statistically distributed between these two PESs. Thus, within the Born–Oppenheimer approximation, the F(2P3/2) + CHD 3reaction probability is only half of the reac- tion probability computed on the reactive PES. Considering non- adiabatic transitions, incoming reactants on the non-reactive upper PES can switch to the reactive lower PES and vice versa. As noted above, wave packets moving on the lower adiabatic PES tend to remain on the same PES and non-adiabatic transition almost exclu- sively occur from the upper to the lower PES. Consequently, the contribution to the F(2P3/2) + CHD 3reaction probability from reac- tants starting at the reactive PES is not significantly diminished by FIG. 2 . F + CHD 3→HF + CD 3reaction probabilities: results of non-adiabatic calculations for F(2P3/2) + CHD 3and F∗(2P1/2) + CHD 3are displayed by black solid line and dotted blue line, respectively, and compared to results for F(2P3/2) + CHD 3obtained within the Born–Oppenheimer approximation (thin red line).non-adiabatic transition compared to the Born–Oppenheimer approximation. In addition, reactants that start on the non-reactive upper PES and react after a transition to the lower PES contribute to the reaction probability. The results seen in Fig. 2 imply that about 15% of the reactants starting on the upper PES take this route. This moderate probability for transfer from the upper to the lower adia- batic PES matches with the limited size of the regions on the upper adiabatic PES, where funnels direct the reactants toward a conical intersection. The reaction probability of spin–orbit excited F∗(2P1/2) atoms is found to be comparatively small at all collision energies despite the electronic excess energy available. The reactive ground state PES and the excited PES asymptotically correlating with F∗do not show conical intersections, which could facilitate ultra-fast non-adiabatic transitions. A detailed analysis of the data displayed in Fig. 2 shows that the relative reactivity of F∗atoms compared to F atoms increases with an increase in the collision energy. This result is in agreement with simple Landau–Zener-type arguments that increasing velocities should result in increasingly diabatic behavior and, thus, favor non- adiabatic transitions. To connect to experiment, integral cross sections for F(2P3/2) + CHD 3→HF + CD 3and F∗(2P1/2) + CHD 3→HF + CD 3reac- tive scattering are shown in Fig. 3. The cross sections for reactive scattering of ground state F atoms are more than an order of mag- nitude larger than the ones for spin–orbit excited F∗atoms at all collision energies studied. This finding differs from the experimen- tal and theoretical results obtained for the reactions F + D 2→DF + D17and Cl + H 2→HCl + H.19,20In both triatomic reactions, the reaction probabilities of F∗(2P1/2)/Cl∗(2P1/2) atoms exceed the reac- tion probabilities of F(2P3/2)/Cl(2P3/2) atoms for very low collision energies. In the title reaction, the extremely low barrier (signifi- cantly lower than in F + D 2) and the presence of prominent reactive FIG. 3 . Integral cross sections for the F(2P3/2) + CHD 3→HF + CD 3(solid black line) and F∗(2P1/2) + CHD 3→HF + CD 3(dotted blue line) reactions obtained from the non-adiabatic theory using the transition state rotational sudden approach.32 J. Chem. Phys. 152, 231102 (2020); doi: 10.1063/5.0013852 152, 231102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp resonances supported by the prereactive van der Waals well effi- ciently facilitate reaction of ground state F(2P3/2) atoms even at very low collision energies. Thus, the electronic excess energy available in the F∗(2P1/2) atom is not required to surmount the reaction barrier even at very low collision energies. In conclusion, non-adiabatic transitions have been found to affect the reaction of ground state fluorine atoms with methane. An increased reactivity and decreased lifetimes of reactive reso- nances compared to results obtained within the Born–Oppenheimer approximation are observed. The present findings should be con- trasted with the result obtained for triatomic reactions: “the overall dynamics of the F + H 2reaction will be well described on a single, electronically adiabatic PES.”16The differences in the relevance of non-adiabatic effects in the reaction of F atoms with hydrogen or methane are noteworthy. Future studies considering the reaction of F atoms and other open shell atoms such as Cl or O with polyatomic molecules will have to show whether the present conclusions can be transferred to a larger class of reactions. We hope that the results of first-principle theory regarding the relative reactivity of F(2P3/2) and F∗(2P1/2) atoms in polyatomic reactions will motivate experimental research. See the supplementary material for methods, numerical details, and additional results. The authors want to thank Minghui Yang for making the code used for the calculations in Ref. 29 available to them. 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5.0023002.pdf

Appl. Phys. Lett. 117, 154001 (2020); https://doi.org/10.1063/5.0023002 117, 154001 © 2020 Author(s).Coherent coupling of a trapped electron to a distant superconducting microwave cavity Cite as: Appl. Phys. Lett. 117, 154001 (2020); https://doi.org/10.1063/5.0023002 Submitted: 25 July 2020 . Accepted: 30 September 2020 . Published Online: 13 October 2020 April Cridland Mathad , John H. Lacy , Jonathan Pinder , Alberto Uribe , Ryan Willetts , Raquel Alvarez , and José Verdú COLLECTIONS Paper published as part of the special topic on Hybrid Quantum Devices ARTICLES YOU MAY BE INTERESTED IN Quantum neuromorphic computing Applied Physics Letters 117, 150501 (2020); https://doi.org/10.1063/5.0020014 Photon-mediated entanglement scheme between a ZnO semiconductor defect and a trapped Yb ion Applied Physics Letters 117, 154002 (2020); https://doi.org/10.1063/5.0019892 A perspective on hybrid quantum opto- and electromechanical systems Applied Physics Letters 117, 150503 (2020); https://doi.org/10.1063/5.0021088Coherent coupling of a trapped electron to a distant superconducting microwave cavity Cite as: Appl. Phys. Lett. 117, 154001 (2020); doi: 10.1063/5.0023002 Submitted: 25 July 2020 .Accepted: 30 September 2020 . Published Online: 13 October 2020 April Cridland Mathad,1 John H. Lacy,2 Jonathan Pinder,3 Alberto Uribe,3 Ryan Willetts,3 Raquel Alvarez,3 and Jos /C19eVerd /C19u3,a) AFFILIATIONS 1Swansea University, Singleton Campus, Swansea SA2 8PP, United Kingdom 2Williams College, Williamstown, Massachusetts 01267, USA 3Department of Physics and Astronomy, University of Sussex, Falmer BN1 9QH, United Kingdom Note: This paper is part of the Special Issue on Hybrid Quantum Devices. a)Author to whom correspondence should be addressed: jlv20@sussex.ac.uk ABSTRACT We theoretically investigate the coupling of a single electron in a planar Penning trap with a remote superconducting microwave (MW) cavity. Coupling frequencies around X¼2p/C11 MHz can be reached with resonators with a loaded quality factor of Q¼105, allowing for the strong coupling regime. The electron and the cavity form a system of two coupled quantum harmonic oscillators. This is a hybrid andlinear microwave quantum network. We show that the coherent interaction can be sustained over distances of a few mm up to several cm.Similar to classical linear MW circuits, the coherent quantum exchange of photons is ruled by the impedances of the electron and the cavity. As one concrete application, we discuss the entanglement of the cyclotron motions of two electrons located in two separate traps. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023002 Cryogenic Penning traps allow for a very accurate control of the dynamics of a trapped electron, at the fundamental level of quantumjumps between the Fock states of the harmonic trapping potential. 1 The particles can be captured for very long periods (months); the con-tinuous Stern–Gerlach effect permits the detection and manipulationof the electron’s spin, 2and the Purcell effect enhances the coherence time of its motional quantum state.3The latter can be monitored non- destructively, thereby effectively performing a quantum nondemoli-tion (QND) measurement of microwave (MW) quanta. 1Hence, cryogenic Penning traps are excellent quantum laboratories andtrapped electrons are solid candidates for quantum technology. 4–6We are developing a trapped electron for quantum metrology applications,specifically as a transducer of quantum microwave (MW) radiation. 7 For this, we use the “geonium chip” planar Penning trap.8–10The elec- tron can operate both as a detector and also as an emitter of MW pho-tons. Being a quantum harmonic oscillator, in principle, its state canbe mapped one-to-one onto the quantum state of a single mode MWradiation field. 11This makes the electron a linear and reversible quan- tum microwave transducer, unlike single MW photon counters basedupon three-level systems 12,13and similar technologies. In those cases, only MW fields consisting of one (and only one) photon can beobserved, and these are irreversibly lost after the detection. In contrast,the trapped electron might “witness” a more complex quantummicrowave field and further reuse or redistribute it to other devices within a MW quantum network. In this Letter, we theoretically inves-tigate the basic scheme for such reversible and QND measurements ofMW radiation. The quantum object to be “measured” is assumed tobe a distant superconducting microwave cavity, coupled to the electronthrough a transmission line of some finite length. Figure 1(a) shows a geonium chip with its five basic trapping electrodes. These result from the projection of a five pole cylindricPenning trap onto a flat surface. 8The buried wires shown in Fig. 1(b) provide the required DC trapping voltages and also coupling for radiofrequency (RF) signals. They are connected through vias to the trap’selectrodes. A static magnetic field, ~B¼B 0^uz, forces the electron to follow a closed (cyclotron) orbit around its axis. The magnetic fieldsource is shown in Fig. 1(c) . It is made of NbTi superconducting wire and spans about /C2410/C210 cm 2. Its construction and calibration have been described in Refs. 14and15. It is placed underneath the geonium chip and is constantly powered with DC supplies. An example of ameasured field is shown in Fig. 1(d) . Homogeneous magnetic fields up to 0.5 T ( ¼5000 G) at the electron’s trapping position can be reached with this source. A new magnetic source is under development, 15 which operates in persistent mode and is magnetized with an espe-cially devised flux pumping technique. 16The electrons can be captured at some height y0above the central conducting strip. The value of y0is Appl. Phys. Lett. 117, 154001 (2020); doi: 10.1063/5.0023002 117, 154001-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldetermined by the applied DC voltages.8The particle’s motion consists of three independent oscillations: the mentioned cyclotron motion, with frequency xp, and the axial and magnetron motions. The latter two will not be further considered in this Letter. At B0¼0:5T , w e have xp=2p¼13:99 GHz.8 Superconducting microwave resonators can be fabricated in differ- ent shapes, for instance, as 3D cavities made of aluminium17or copper with a tin coating,18reaching internal quality factors as high as Q/C24109, and also as flat, chip devices, such as coplanar-waveguide (CPW) cavities, achieving quality factors Q2½105;106/C138.19,20While our electron might be coupled to either kind of resonator, for simplicity, we concentrate on the CPW cavities; our analysis will apply also for other types of MW res-onators. Their high Qallows for transferring quantum information between different components in a microwave quantum circuit, such as in circuit-quantum electrodynamics (cQED). 21Due to the long coher- ence time of their spin state, electrons in different forms, such as trapped on the surface of liquid helium, within molecular ions, or as spin ensem-bles in solid state systems, have been proposed and are being tested as quantum memories for cQED. 22–26Ar e v i e w27of other atomic systems proposed as potential quantum memories for cQED has been published. A scheme for interfacing electrons in a Paul trap with superconducting qubits has also been discussed.28Our trapped electron can be coupled to a microwave quantum circuit via the interaction of the photons with its cyclotron motion or its spin. The latter is very weakly coupled, and hence, we focus on the former. The electrodes of the geonium chip define the central conducting strip of a CPW transmission line,8where MW photons can exist in either one of the two possible propagation modes,29t h eo d do rt h ee v e n mode, of a CPW line. Thus, our chip might be designed and operated also as a planar microwave cavity. However, the quality factor of a super-conducting CPW resonator is reduced by three to four orders of magni- tude in the presence of a magnetic field. 19This makes it inadequate for storing MW photons. Furthermore, we want to investigate the QND measurement of the quantum state of a microwave system at anarbitrary distant location. Thus, we assume that our chip is coupled to a remote CPW superconducting cavity, as sketched in Fig. 2(a) .T h et r a p and the cavity are connected through a superconducting transmission line of some finite length LTL. The geonium chip confines the magnetic field into the trapping region, which decays rapidly away from it, as shown in Fig. 1(d) . Hence, ~Bdoes not reduce the quality factor of the superconducting cavity. The interaction between the distant MW quan- tum device (the cavity) and the electron is mediated by photons. For simplicity, we assume that the photons when arrived at the geonium chip exist in one mode of the CPW shaped by the trap’s electrodes, as mentioned above. However, in general, this radiation can be delivered tothe trapped electron by other types of transmission lines coupled to the electrodes or to a small antenna fabricated in the chip, such as the anten- nas used in near field scanning microwave microscopes. 30 The interaction of a trapped electron with a radio frequency (RF) resonator made of lumped elements was first investigated by Dehmelt and Walls.31This interaction is commonly employed for the electronic detection of the trapped particles and can be described as the coupling of two equivalent electric circuits.31In our system of Fig. 2(a) ,t h eR F resonator is substituted by the superconducting CPW microwave cavity. The latter is equivalent to a parallel tank circuit, with inductance L, capacitance C, and losses modeled by the resistance R.M o r e o v e r ,t h e electron’s cyclotron motion acts as a series tank circuit, with equivalent inductance Leand capacitance Ce.32The electron-cavity interaction cor- responds to the coupling in parallel to their equivalent electric circuits,31 as represented in Fig. 2(b) . Hence, the system’s overall impedance is 1 ZLðxÞ¼1 ZcavityðxÞþ1 ZeðxÞ,w h e r e ZcavityðxÞis the cavity’s input impedance seen from the trap and ZeðxÞ¼ixLeþ1 ixCethe electron’s cyclotron impedance. ZLðxÞdelivers a resonance spectrum as plotted in Fig. 2(c) . In the case of resonant coupling, two symmetric peaks appear around the cavity’s resonance frequency, xcavity. The energy exchange rate between the particle and the resonator is given by the width Xof the dip between both maxima. In Penning trap experiments such as in Refs. 33–35 ,t y p i c a lv a l u e so ft h eu s e dR Fr e s o n a t o r s36areQ/C20104.T h u s ,i n FIG. 1. (a) Top view of the geonium chip with the trap’s electrodes. (b) Chip’s rear side showing the buried wires. (c) Magnetic field source made of NbTi wire. (d) Example of the measured magnetic field (dots) and fitted Biot–Savart functions (continuous curve). FIG. 2. (a) Sketch of the geonium chip connected to a remote superconducting CPW cavity. The open ends avoid photon leaks by reflecting the microwaves inside the system. (b) Equivalent electric circuit of the coupled electron and CPW cavity.(c) Impedance of the cavity þelectron’s cyclotron. This example assumes strong, coherent coupling. (d) Transverse dimensions of a CPW transmission line.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 154001 (2020); doi: 10.1063/5.0023002 117, 154001-2 Published under license by AIP Publishingthose cases, once delivered to the resonator, the particle’s energy is irre- versibly dissipated almost instantly37ands¼1=Xis denoted as the “resistive cooling time constant.” In order to get the geonium chip down to 80 mK, we use a minia- ture adiabatic demagnetization refrigerator.38This will also require MW attenuators39,40to avoid residual thermal photons entering into the sys- tem. With these, at 80 mK, the number of blackbody photons at 13.99 GHz is sufficiently low for both constituents of Fig. 2(a) to reach the ground state.1This system becomes a quantum circuit and can be excited with a quantum of current, i.e., one single microwave photon. The photon then oscillates between both components at the coupling frequency X. From the general expression of s,32we have XðdÞ¼jZcavityðdÞj Le,w i t h d¼xcavity/C0xp. The coupling Xis maximum when the electron and the cavity are resonant xcavity¼xp.I nt h a tc a s e , the impedance of the cavity is Zcavityð0Þ¼R¼QxcavityL.41When d6¼ 0;jZcavityðdÞjdecreases, the coupling strength is reduced, and the two maxima become asymmetric, as shown in Fig. 2(c) .F o rt h ee/C0–cavity coupling to be coherent, Xmust be faster than the photon loss rate of the resonator, responsible for the irreversible energy dissipation causing t h e“ r e s i s t i v ec o o l i n g . ”T h ec a v i t y ’ sp h o t o nl o s sr a t ei sg i v e nb yt h ew i d t h of its resonance: C¼xcavity=Q.H e n c e ,t h ec o n d i t i o nf o rc o h e r e n tc o u - pling isX C/C291. For the resonant case, d¼0, we have X C¼Q2L Le/C291: (1) In Eq. (1),Qis the quality factor of the loaded MW cavity, as seen by the electron, that is, including possible photon losses along the con- necting transmission line. It must be observed that, as sketched in Fig. 2(a) , the trap is enclosed within a rectangular metallic box.14This also acts as a microwave resonator far detuned from xp.10Hence, it prevents the electron from emitting cyclotron radiation into free spaceor into any other modes but those of the trap’s CPW line, coupling to the remote MW cavity. The ratio X=Cof Eq. (1)can be calculated with the general expression 7,32Le¼m q2D2 eff. The symbols qandmrepresent the charge and mass of the electron, respectively. The “effective coupling dis- tance,” Deff, is inversely proportional to the strength of the electric field at the position of the electron, ECPWðy0Þ, of a 1 V microwave propagat- ing along the CPW. In general, both quantities are related through32 ECPWðy0Þ¼1V Deff. For the slow axial motion xz, the corresponding RF electric field has been calculated using electrostatic techniques.32That approach must be modified for the electron’s cyclotron motion, which belongs to the MW domain. The electric field components Eodd CPW and Eeven CPW have been computed analytically using special MW techni- ques.29With those results, we obtain the cyclotron effective coupling distances, for each of the two possible CPW modes, Dodd eff/C16/C17/C01 ¼4 pWX1 n¼1sinnpW a/C18/C19 sinnpWþS a/C18/C19 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2þa2 k2 p/C0a2/C232 p c2s8 >>>>< >>>>: /C2exp/C02py0 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n2þa2 k2 p/C0a2/C232 p c2vuut0 B@1 CA9 >= >;; (2)Deven eff/C0/C1/C01¼4 pWX1 n¼0sin2nþ1 2pW a/C18/C19 cos2nþ1 2pWþS a/C18/C19 2nþ1 28 >>< >>: /C2exp/C02py0 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nþ1 2/C18/C192 þa2 k2 p/C0a2/C232 p c2vuut0 B@1 CA9 >= >;: (3) In Eqs. (2)and(3)cis the speed of light in vacuum. The transverse dimensions of the CPW are defined in Fig. 2(d) .Sis the width of the central conducting strip, Wis the gap between the latter and the “ground planes,” and ais the total chip’s width. In the geonium chip, the wavelength k pof the radiation of cyclotron frequency (/C23p¼xp=2p) depends on the substrate’s electric permittivity /C15r,i t s thickness d, the thickness of the conducting layer t, and the dimen- sions S,W.42 In order to illustrate our system, we have calculated in detail one example. It is shown in Fig. 3 . We have assumed a distant supercon- ducting k=4 short circuited CPW microwave cavity,41with dimen- sions: Scavity¼10lm,Wcavity¼4:8lm,tcavity¼200 nm, and a sapphire substrate of dcavity¼0:3 mm. These values are motivated by some real resonators.19,20Our cavity has the characteristic impedance ofZ0¼50X, as obtained from previously compiled CPW design for- mulas.42Further, we assume a loaded quality factor (as seen by the electron) of Q¼100 000. This corresponds to C=2p¼140 kHz, a resonance resistance of R¼6.37 M X, and an inductance of L¼0.72 nH. Moreover, we have assumed a geonium chip of the same conduct-ing layer and an alumina substrate of thickness 0.7 mm, as shown in Fig. 1(a) .W i t ht h e s e ,w ec o m p u t e X=Cfor 4 values of the width: S trap¼0:5;0:25;0:15, and 0.1 mm, with gaps Wtrap¼59;34;22; and 15 lm, respectively. Each Wtrapis adjusted to ensure that the CPW formed in the geonium chip has a characteristic impedance of Z0¼50X. The increasing values of X=Cfor smaller Strapobserved in Figs. 3(b) and3(c)reflect the bigger strength, close to the chip’s surface, of the microwave electric field ECPW when the “mode volume” of the CPW line in the trap is reduced. At very low positions, X=Cis higher for the odd than for the even mode; the latter actually disappears at y0!0. This is due to the vanishing electric field component parallel to the conducting surface: Ex CPW!0a ty0!0. The active mode, odd or even, depends on how the system is connected and grounded.42 FIG. 3. Coupling frequency of one electron vs photon losses as a function of the trapping position, for various values of the CPW width in the trap chip. We assumed¼0 and Q¼105. Calculated with Eqs. (2)and (3), truncating the series at n¼60 000. (a) Odd mode. (b) Even mode.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 154001 (2020); doi: 10.1063/5.0023002 117, 154001-3 Published under license by AIP PublishingThe electron can be positioned at y0/C2150lm, by carefully choosing the trapping voltages.8,10At such low positions, the electrostatic trap- ping potential becomes anharmonic,8a n da t8 0m K ,t h ee x p e c t e dr e l a - tive frequency shifts can be estimated toDxp xp’6/C210/C011. Such fluctuations of xpare negligibly small. Therefore, when placed at y0/C2150lm, the electron’s cyclotron motion still acts as a quantum harmonic oscillator. As shown in Fig. 3 ,a ts u c hl o w y0, the cavity and the cyclotron oscillator interact coherently. After exchanging quantum information, y0can be increased (typically to 0.5–1 mm) where the trapping potential becomes harmonic. There, xpcan be measured accurately8,10and the cyclotron quantum state can be read out and further manipulated. The electron and the cavity are two coupled quantum harmonic oscillators. This is a linear system, where the coupling strength Xis invariant under a number of photons. The coherent dynamics of two coupled quantum harmonic oscillators has been calculated.11An equiva- lent system has been implemented with two ions held in two Paul trapsat a distance of 40 lm. 43In our case, the maximum cavity-electron sepa- ration permitting coherent coupling is bound by the achievable loadedquality factor Q. The dependence Q¼QðL TLÞcan be obtained from the transmission-line impedance equation.41T h u s ,a tt h et r a p ,t h ec a v i t y ’ s unloaded input impedance Z0 cavityðxÞis transformed into ZcavityðxjLTLÞ¼Z0Z0 cavityðxÞcothðcLTLÞþZ0 Z0cothðcLTLÞþZ0 cavityðxÞ: (4) The imaginary part of the propagation constant, c¼aþib,s h i f t st h e cavity’s resonance frequency. We, therefore, assume that the cavity, asseen from the trap, has been designed to be resonant with x p.T h er e a l partaaccounts for the photon losses. The dissipation caused by the walls of the connecting transmission line is linked to the superconduc-tor surface resistivity: R S/C205 4p2/C110/C028x2Ohm (with xgiven in rad/ s).44At 13.99 GHz, it is negligible when compared to the dielectric losses. The latter, in the case of a superconducting coaxial line operat-ing at x p, gives rise to the attenuation constant a¼xpffiffiffiffiffiffiffiffiffiffiffiffiffiffil0/C1/C15TLptand,41with l0being the magnetic permeability, /C15TLthe electric permittivity, and tan dthe loss tangent. With that aand with the same cavity as in Fig. 3 , assuming20Qunloaded ¼106,w ec o m p u t e Q¼QðLTLÞfrom the spectra ZcavityðxjLTLÞ. The results are given in Fig. 4(a) .We have assumed quartz with /C15TL¼4:64 (relative) and tand¼3/C210/C06;Y A Gw i t h /C15TL¼10:40 and tan d¼1/C210/C07,a n d sapphire with /C15TL¼9:4 and tan d¼4/C210/C08, (all values are mea- sured at /C2410 K).45From the figure, we observe that, for quartz, the loaded quality factor drops below 105at a distance of LTL’0:8c m , while both YAG and sapphire sustain Q>105at separations above 10 cm. Within that range of distances, coherent e/C0-cavity coupling might be achieved, as shown in Fig. 3 .I tm u s tb eo b s e r v e dt h a ti nt h i s discussion, we have assumed one or small numbers of MW photons.When the quantum state of the exchanged MW signal consists of alarge number of photons, then the coherence or the “quantum nature” of the state will, in general, degrade more rapidly than for low photon numbers. 46A larger number of photons will make the exchanged MW signal increasingly nonmonochromatic, and in that case, the disper-sion of the coupling transmission line will need to be considered. Through the coherent interaction of the trapped electron with a remote superconducting resonator, the former can be further coupled to other components in a microwave quantum network. The electron’s cyclotron oscillator enables the linear mapping of the quantum state ofany multi photon microwave field onto an atomic degree of freedom.This is possible while keeping the photons stored within the cyclotron quantum state for periods of the order of a minute. 1The linearity of the electron also makes the computation of the coupling with anyother quantum devices in a MW network straightforward: the cou- pling strength Xis simply given by the transformed impedance [through Eq. (4)] of that device as seen by the electron. Furthermore, in contrast to systems where the atomic species are attached to theCPW microwave cavity, 22–26our electron quantum MW transducer and the cavity are two fully independent devices. This makes increas- ingly complex network topologies possible, enabling many applica-tions beyond quantum MW memories. A fundamental one is theentanglement of two electrons stored in different traps. This can be achieved with a configuration as sketched in Fig. 4(b) .T h e r e ,t w ot r a p s are connected to two cavities of the same resonance frequencies andquality factors. Cyclotron–cyclotron entanglement occurs through asimilar mechanism as demonstrated with two laser-cooled ions in dis- tant Paul traps. 47For that, cyclotron oscillators must be prepared in the first excited state, np¼1, with both particles at a high y0, i.e., ini- tially invisible to the cavities, due to the vanishing coupling ( Fig. 3 ). Such state initialization can be achieved by pumping photons from a FIG. 4. (a) Loaded Qvs trap-cavity distance. Calculated for three different dielectric materials in the coax line. (b) Two geonium chips connected to two different dist ant identical cavities.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 154001 (2020); doi: 10.1063/5.0023002 117, 154001-4 Published under license by AIP Publishingvery weak source connected to the traps, while monitoring the quan- tum number npnondestructively.1Thereafter, the electrons are simul- taneously dropped for a short period of time to a very low position,around 50 lm, hence switching on briefly the strong coupling with the MW cavities. Hence, in our system, the unavailable fast spontaneousemission (essential in the Paul trap experiment 47) is substituted by the rapidly switchable strong coupling to the remote resonators. Thisforces both cyclotrons to deliver the microwave photons simulta-neously (in principle with equal probability) to any of the cavities. The irreversible detection of the emitted photons, either in cavity 1 or 2 (for instance as in Ref. 48), erases the “which way” information, thereby entangling the particles. 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5.0014928.pdf

J. Chem. Phys. 153, 104108 (2020); https://doi.org/10.1063/5.0014928 153, 104108 © 2020 Author(s).Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation Cite as: J. Chem. Phys. 153, 104108 (2020); https://doi.org/10.1063/5.0014928 Submitted: 25 May 2020 . Accepted: 21 August 2020 . Published Online: 09 September 2020 Nicholas H. Stair , and Francesco A. Evangelista ARTICLES YOU MAY BE INTERESTED IN Richardson–Gaudin mean-field for strong correlation in quantum chemistry The Journal of Chemical Physics 153, 104110 (2020); https://doi.org/10.1063/5.0022189 Top reviewers for The Journal of Chemical Physics 2018–2019 The Journal of Chemical Physics 153, 100201 (2020); https://doi.org/10.1063/5.0026804 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation Cite as: J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 Submitted: 25 May 2020 •Accepted: 21 August 2020 • Published Online: 9 September 2020 Nicholas H. Staira) and Francesco A. Evangelistab) AFFILIATIONS Department of Chemistry and Cherry Emerson Center for Scientific Computation, Emory University, Atlanta, Georgia 30322, USA a)Electronic mail: nstair@emory.edu b)Author to whom correspondence should be addressed: francesco.evangelista@emory.edu ABSTRACT This work explores the ability of classical electronic structure methods to efficiently represent (compress) the information content of full configuration interaction (FCI) wave functions. We introduce a benchmark set of four hydrogen model systems of different dimensionalities and distinctive electronic structures: a 1D chain, a 1D ring, a 2D triangular lattice, and a 3D close-packed pyramid. To assess the ability of a computational method to produce accurate and compact wave functions, we introduce the accuracy volume, a metric that measures the number of variational parameters necessary to achieve a target energy error. Using this metric and the hydrogen models, we examine the performance of three classical deterministic methods: (i) selected configuration interaction (sCI) realized both via an a posteriori (ap-sCI) and variational selection of the most important determinants, (ii) an a posteriori singular value decomposition (SVD) of the FCI tensor (SVD-FCI), and (iii) the matrix product state representation obtained via the density matrix renormalization group (DMRG). We find that the DMRG generally gives the most efficient wave function representation for all systems, particularly in the 1D chain with a localized basis. For the 2D and 3D systems, all methods (except DMRG) perform best with a delocalized basis, and the efficiency of sCI and SVD-FCI is closer to that of DMRG. For larger analogs of the models, the DMRG consistently requires the fewest parameters but still scales exponentially in 2D and 3D systems, and the performance of SVD-FCI is essentially equivalent to that of ap-sCI. Published under license by AIP Publishing. https://doi.org/10.1063/5.0014928 .,s I. INTRODUCTION An outstanding challenge in modern electronic structure the- ory is solving the many-body Schrödinger equation for strongly cor- related electrons. The availability of accurate computational meth- ods for strongly correlated systems is imperative to study a myriad of important phenomena such as bond breaking and photochemi- cal processes,1,2molecular magnetism,3high-temperature supercon- ductivity,4and many others.5–8In brief, strong correlation arises when the cost of promoting electrons to higher energy orbitals is small in comparison to the electron pairing energy (Coulombic repulsion). Consequently, strongly correlated electrons cannot bequalitatively described by a mean-field picture because the wave function may contain nontrivial contributions from many Slater determinants.9,10In this situation, electronic structure methods that build on a mean-field reference cannot effectively approximate the wave function with a polynomial number of parameters and, there- fore, often yield inaccurate energies and molecular properties. The full configuration interaction (FCI) expansion captures all correlation effects for Nelectrons in Lspatial orbitals. Restrict- ing FCI to a complete active space (CAS), with self-consistent field (CASSCF)11or without orbital optimization, is also a common strat- egy when strong correlation effects are limited to a few orbitals. However, the size of the FCI (or CASSCF) determinant space scales J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp like a binomial coefficient in Nand L, making these methods intractable for most systems of chemical interest12containing more than∼18 electrons in 18 orbitals (18e, 18o)—although massively parallel computations have recently managed to push this figure to (22e, 22o).13 Fortunately, for many ground and low-lying states, the com- plexity of the wave function is reduced by symmetry restrictions, sparsity bread by non-interacting determinants, and regular struc- ture resulting from the local nature of Coulombic correlation. Much work has thus been devoted to the development of methods that can exploit sparsity or use decomposition techniques to compactly approximate the wave function. However, it is not generally known what approaches are the most efficient (i.e., which ones can reach a target accuracy using the fewest parameters), given the physical dimension of the system, the degree of correlation strength, and the choice of molecular orbitals. Understanding the degree to which different wave functions may be compressed is important to guide future development of both classical and quantum computational methods.14In partic- ular, there is a growing need for benchmark sets that may be used to compare classical and new quantum algorithms in vari- ous regimes of electron correlation. Since many classes of emerging quantum algorithms—such as variational quantum eigensolvers15–18 and quantum subspace diagonalization techniques19–22—use param- eterized Ansätze , one way to compare them to classical algorithms is to quantify their efficiency in terms of classical resources needed to achieve a target energy accuracy. Such characterization is also useful in answering whether or not a quantum algorithm has an advantage over a purely classical approach. The goal of this work is to examine how to best compress the FCI wave function of strongly correlated systems using classical methods. To this end, we introduce a benchmark set and a simple metric to analyze the performance of a method. We consider three families of deterministic methods that systematically approach FCI in a near-continuous fashion: (i) selected CI (sCI),23–26(ii) singular value decomposition FCI (SVD-FCI) (related to the methods dis- cussed in Refs. 27–29), and (iii) the density matrix renormalization group (DMRG).30These exemplify different strategies to approx- imate the exact wave function; however, they all converge to the exact energy in the limit of no truncation, and their accuracy can be controlled by a single parameter. Selected CI schemes approximate the FCI solution using a sub- set of the full determinant space. Therefore, they are most efficient when the exact wave function has a sparse structure. In contrast to other forms of truncated CI, selected CI methods identify an optimal determinant basis using an iterative selection procedure, which gradually expands the determinant space. Although selected CI23–26was proposed decades ago, in recent years, it has received renewed attention with new deterministic,31–42stochastic,43–48and semistochastic49–52variants being proposed. The closely related fam- ily of determinant-based Monte Carlo methods53–60has also been explored. The singular value decomposition finds use in several areas of quantum chemistry as a way to achieve low-loss compression of data.61–68The SVD can be applied to compress the FCI state once the coefficient vector is reshaped as a matrix, a representation naturally suggested by string-based CI algorithms.69Taylor pro- posed to reduce the memory requirements of FCI by performinga SVD at each iteration of the Davidson procedure.28Another method that employs a compressed representation of the FCI vector is rank-reduced FCI (RR-FCI), originally proposed by Koch27and recently extended by Fales and co-workers.29RR-FCI approximates the FCI solution with a polar decomposition of the FCI vector (rep- resented as a matrix) combined with variational minimization of the energy. We note that both Taylor’s “gzip” approach and the RR-FCI approach cannot be justified on the basis of a symmetry or a physi- cal principle, although a variant of RR-FCI that exploits locality has been proposed (see Ref. 70). Tensor network states (TNSs) represent a broad family of meth- ods that approximate the FCI coefficients (viewed as a tensor) with a collection of tensors connected by contractions. The simplest type of TNS is a matrix product state (MPS), the underlying Ansatz71–73of the density matrix renormalization group (DMRG).30MPSs are able to maximally exploit local orbital entanglement, that is, for states that satisfy an area law for the entanglement entropy, MPSs can yield near-exact results in 1D and quasi-1D systems.74,75The generaliza- tion of the MPS Ansatz to two- (2D) and three-dimensional (3D) TNSs using high-order tensor factorizations is also an active area of research.75–77In practice, the variational optimization of TNSs suf- fers from very high scaling and is less efficient relative to MPSs. The DMRG (as applied to quantum chemistry)78–82has been tremen- dously successful in describing the ground states of quasi-linear molecular systems. For example, the DMRG has enabled the inves- tigation of long hydrogen chains,78,83–86oligoacenes,87–89and large biochemically relevant transition metal complexes with up to 100 orbitals.90,91 To test the performance of electronic structure theories in the strongly correlated regime, we have introduced a benchmark set of one-, two-, and three-dimensional (3D) hydrogen systems. These systems model strongly correlated electrons in significantly differ- ent regimes and dimensionalities and allow us to explore the physics of Mott insulators and spin frustrated systems in 2D and 3D. 1D hydrogen systems have recently been the subject of comprehensive benchmark studies aimed at treating strong correlation in real mate- rials.92–94Hydrogen lattices with localized spins are also related to the more fundamental Heisenberg and Hubbard models, exhibiting similar spin correlation patterns and band structures. Our bench- mark set contains four H 10models: the well-investigated 1D chain and ring, a 2D triangular lattice (referred to as “sheet” through- out this paper), and a 3D close-packed pyramid. For each model, we consider both the effect of the H–H distance on the strength of correlation and the use of different molecular orbital bases (delocal- ized/localized). We characterize these models by computing various metrics of correlation, including the norm of the two-body cumu- lant, the total quantum information, and spin–spin correlation func- tions. Additionally, to investigate the compression efficiency as a function of system size, we also consider H 12, H 14, and H 16analogs of the four models. Since the methods considered here play an important role as substitutes for FCI in multireference treatments of electron correlation,95–100we are particularly interested in assessing their performance when applied only to valence orbitals. To simulate this scenario, our computations employ a minimal basis set. Note that this treatment may be considered equivalent to diagonalizing a valence effective Hamiltonian101with interactions modified by dynamical correlation effects. It is important to point out that since J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp we only consider zero-temperature quantum chemistry approaches, we focus, in particular, on regimes of electron correlation that range from weak to medium/strong. We intentionally avoid the limit of infinite H–H separation because all the models considered here develop a massively degenerate ground state containing 210states. At large separation, it is ludicrous to characterize a ground state, and one should instead seek to compute thermal averages employing a finite-temperature approach.102–104 To compare the performance of each method, we evaluate the error in the energy and the two-body density cumulant as a func- tion of the number of variational parameters. These errors measure how well electronic correlation effects are preserved in the com- pression and therefore can indicate the quality of an approximate wave function and its properties. From this information, we extract a single metric, the accuracy volume , which measures the number of variational parameters necessary to achieve a target energy error. Although the accuracy volume does not take into account the actual cost of a computation, this metric serves as a proxy for the computa- tional resources required by each method, independently of imple- mentation details. We also compare the energy errors produced with Hartree–Fock theory, second-order Møller–Plesset many-body per- turbation theory (MP2), coupled cluster theory with singles and doubles (CCSD),105CCSD with perturbative triples [CCSD(T)],106 the completely renormalized CC approach with perturbative triples [CR-CC(2,3)],107and the variational two-particle reduced density matrix (V2RDM) method.108–112We have collected the data gener- ated in this study in an online repository113in the hope that it will be useful in future studies. The remainder of this article is organized in the following way. Section II defines the accuracy volume, summarizes the three meth- ods compared in this study, and defines the metrics used to assess correlation strength in the hydrogen model systems. Section III pro- vides the computational details of our study. Numerical results are reported in Sec. IV, in Sec. V we discuss the size consistency and the effective scaling of the accuracy volume, and Sec. VI summarizes our findings and discusses their relevance in the context of classical and quantum algorithms for strongly correlated systems. II. THEORY A. Definition of the accuracy volume For a systematically improvable method X, we indicate the energy computed using Nparparameters as EX(Npar). We then define the accuracy volume, VX(α), to be the smallest number of parame- ters such that the error per electron with respect to the FCI energy (EFCI) is less than or equal to 10−α, VX(α)=Npar:∣EX(Npar)−EFCI∣ N≤10−αEh. (1) For convenience, in the rest of this paper, we always assume that the target energy error is 1 m Ehfor the H 10systems, which corre- sponds to a 0.1 m Eherror per electron ( α= 4), and use the more compact symbol VXinstead. For methods that exploit the sparsity of the FCI wave function (e.g., selected CI), the accuracy volume is a measure of the number of Slater determinants or configuration state functions (equal to the number of parameters). This literal inter- pretation of the accuracy volume does not extend to approximationschemes based on tensor decomposition in which case it only reflects the total number of parameters employed. We intend the accu- racy volume to be used as a performance metric of a method, since it approximately measures the computational resources [memory and central processing unit (CPU)] necessary to achieve a target accuracy. Because the accuracy volume can be equally applied to purely classical and hybrid quantum–classical methods, it provides a straightforward way to compare the two on more equal footing. Our definition of VX[Eq. (1)] considers the energy error per elec- tron to allow the comparison of systems with different numbers of electrons. This approach is consistent with the fact that approximate methods that are size consistent, when applied to noninteracting fragments, give an error that is additive in the error of each fragment. We also choose to define VXas the absolute number of parameters, as opposed to the fraction of the total Hilbert space, since the for- mer is proportional to the computational resources required by a method. In contrast, a comparison based on the fraction of Hilbert space parameters employed by a method would be dependent on the exploitable symmetries for the orbitals that are chosen (e.g., symme- try adapted delocalized vs localized orbitals) making comparisons of different computations less indicative of actual computational resources. B. Overview of the computational methods Given a basis of Kspin orbitals { ψp} with p= 1, . . .,K, we indi- cate a generic N-electron determinant ∣ψi1⋯ψiN⟩using the notation |ΦI⟩where the multi-index I= (i1,. . .,iN) represents an ordered list of indices ( i1<i2<⋯<iN). The set of N-electron determinants (HN) forms a Hilbert space of dimension ∣HN∣=NH. Using this notation, the FCI wave function is written as a linear combination of determinants, each parameterized by a coefficient ( Ci1,...,iN≡CI), ∣ΨFCI⟩=K ∑ i1<i2<⋯<iNCi1⋯iN∣Φi1⋯iN⟩=NH ∑ ICI∣ΦI⟩. (2) An equivalent way to express the FCI wave function employs occu- pation vectors. In this representation, each determinant | ΦI⟩is asso- ciated with a vector of length K, |n⟩= |n1,n2,. . .,nK⟩, where ni ∈{0, 1} is the occupation number of spin orbital ψi. The FCI wave function represented in the occupation vector form is given by ∣ΨFCI⟩=∑ {ni}Cn1,...,nK∣n1,. . .,nK⟩=∑ nCn∣n⟩, (3) where the sum over all occupation vectors ({ ni}≡n) is restricted toN-electron determinants ( ∑jnj=N) of given spin and spatial symmetry. 1. Selected CI Selected CI methods approximate the FCI wave function using a subset M(model space) of the full determinant space, ∣ΨsCI⟩=∑ I∈M˜CI∣ΦI⟩. (4) All flavors of selected CI aim to approximate the FCI vector with the smallest number of elements and differ primarily in the way they J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp determine the set M. For sCI methods, we report Nparas the size of the space M(or equivalently, the size of the vector ˜CI). The first approach we consider consists of an a posteriori selected CI (ap-sCI) compression of the exact FCI wave function. This compressed representation is obtained by sorting the determi- nants according to their weight wI= |CI|2and discarding elements with the smallest weight while satisfying the condition ∑ I∉M∣CI∣2<τsCI. (5) The compressed ap-sCI vector ˜CIis then normalized, and the energy is computed as the expectation value of the Hamiltonian. Even though this compression scheme does not yield a variationally opti- mal solution, the error in the ap-sCI energy is quadratic in the wave function error. Still, this ideal (albeit impractical) version of selected CI is useful in assessing the error introduced by the different selection schemes used in practical sCI approaches. The second approach we consider, the adaptive configuration interaction (ACI),40,42identifies the space Mvia an iterative pro- cedure that seeks to control the energy error. ACI is unique in the regime of selected CI methods as it aims to approximate the FCI energy within a user-specified error tolerance σ, EACI(σ)−EFCI≈σ, (6) where EACI(σ) is the ACI energy. In ACI, the model space is divided into two spaces M=P∪Q, where Pcontains the most impor- tant determinants and Qcontains singly and doubly excited deter- minants spawned from P. New candidate determinants ( ΦA) for the model space are selected from the singly and doubly excited determinants generated from the current Pspace. Each candidate determinant is ranked by its energy contribution, ϵ(ΦA), a quan- tity estimated by diagonalizing the Hamiltonian in the basis of the ACI wave function at the current iteration and ΦA. To determine an improved model space, the candidate determinants are sorted according to | ϵ(ΦA)| and unimportant elements are removed until the sum of their estimated energy is less than or equal to σ, ∑ ΦI∉M∣ϵ(ΦI)∣≤σ. (7) Optionally, additional determinants are included in M at each iteration to ensure spin completeness. After adding these determinants, the Hamiltonian is diagonal- ized and a new Pspace is formed by coarse graining Maccording to determinant weight using a cumulative metric similar to Eq. (7). The course graining step increases the overall efficiency of the proce- dure and reduces the dependency of the final solution on the initial guess (usually the HF determinant or a small multideterminantal wave function). The final ACI energy is computed by diagonalization of the Hamiltonian in the model space basis. However, during the selection process, it is possible to accumulate the estimate of the energy con- tributions from the discarded determinants ( EPT2), and this quan- tity can be added to the ACI energy to obtain an improved energy (EACI+PT2 ). 2. Singular value decomposition FCI In this work, we consider an a posteriori rank reduction of the FCI tensor obtained via a singular value decomposition (SVD-FCI).Our approach is essentially identical to the “gzip” treatment used by Taylor (Ref. 28) with the caveat that we only perform SVD of the final converged wave function rather than at each FCI iteration. The SVD-FCI approach is also inspired by the rank-reduced FCI method (RR-FCI).27,29However, because we do not variationally optimize the SVD-FCI wave function, RR-FCI would yield lower energies than SVD-FCI for a specified rank (particularly, at low ranks). SVD-FCI starts from a string-based representation of the FCI wave function69,114in which each determinant is labeled by separate multi-indices (strings) for alpha and beta electrons ( IαandIβ), and the determinant ∣ΦI⟩=∣ΦIαΦIβ⟩factorizes into products of alpha (ΦIα) and beta ( ΦIβ) spin orbitals. Consequently, the FCI vector CI is represented as a matrix indexed by string configurations ( Iα/Iβ), (C)IαIβ=CIαIβ, and the wave function is written as ∣ΨFCI⟩=Nα ∑ IαNβ ∑ IβCIαIβ∣ΦIαΦIβ⟩, (8) where NαandNβare the number of alpha and beta strings, respec- tively. While the original RR-FCI algorithm is based on variational minimization of the energy, in this work, we consider only an a posteriori compression. To this end, we perform the singular value decomposition of the FCI coefficient matrix, C=USV , where we assume that the entries of Care real. To find the most compact reduced-rank approximations of C, we reconstruct an approximate matrix̃CSVDdefined as ̃CSVD=ŨSV, (9) wherẽSis a truncated version of S. Assuming that the singular values si=Siiare sorted in decreasing order, we keep in ̃Sthe diagonals s1, . . .,sRsuch that the sum of the square of the elements excluded is less than a user-provided threshold ( τSVD), ∑ i=R+1s2 i<τSVD. (10) Therefore, Rrepresents the rank of ̃CSVDand the error in the FCI wave function is given by ∥C−̃CSVD∥F<√τSVD, (11) where∥⋅∥Fis the Frobenius norm. The SVD-FCI energy is computed as ESVD-FCI=(̃CSVD)†H̃CSVD, (12) and although it does not correspond to the optimal energy for a wave function of rank R, this estimate deviates from the variational energy by a quadratic term. For SVD-FCI, we calculate the num- ber of parameters as Npar=R(Nα+Nβ), where we have assumed that the singular values ̃Sare folded into either UorV. Note that with no truncation, the SVD-FCI requires twice the number of parameters as the size of the Hilbert space. We also point out that since the FCI wave function is invariant with respect to unitary rotations of the orbitals, the rank RSVD approximation yields the same approximate wave function in any orbital basis. However, the number of parameters may differ from one orbital basis to another J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp if symmetry is employed and the SVD is applied only to the non-zero blocks of C. 3. Density-matrix renormalization group The matrix product state representation at the basis of the DMRG is a conceptually different form of compression that aims to exploit the local character of entanglement. A MPS decomposi- tion of the FCI tensor in the occupation number representation is given by Cn1,...,nK≈CDMRG n1,...,nK=An1 1An2 2⋯AnK K, (13) where, for a given value of the occupation number nj, a generic term Anj jis aM×Mmatrix, except for the first and last terms, which are a row and a column vector of size M, respectively. Given an occupation number pattern ( n), the corresponding tensor element Cn1...nKis approximated by the product of all the Anj jmatrices. Quan- tum chemistry implementations of DMRG exploit the symmetry group of the Hamiltonian (particle number, spin, and point group) to induce a block-sparse structure in the MPS tensors Anj j, with a consequent reduction in computational and storage costs. We calcu- lateNparfor the DMRG as the sum of the number of parameters in each site tensor Anj jin the converged MPS, taking into account the block structure induced by symmetries (assuming at most abelian point groups). Formally, the MPS representation can be derived by perform- ing a series of successive SVDs on the FCI tensor (appropriately reshaped) at each step retaining only Mterms. Therefore, it is exact in the limit of M→NH. In practice, the DMRG method directly builds the MPS representations via a sweep algorithm using a fixed value of Mspecified by the user. For chemical applications, the qual- ity of the MPS as a function of Mis controlled by two choices: the type (localized vs delocalized) and ordering of the orbitals. These aspects present a challenge for practical calculations since different orbital types and orderings can dramatically affect the final out- come of a calculation. Although there are rules of thumb for spe- cific cases—such as choosing the localized orbitals ordered to be spatially adjacent for elongated molecules115—the choice of these parameters is generally a non-trivial problem beyond 1D. Vari- ous approaches to ordering the delocalized orbitals have also been explored.116–119 C. Metrics of strong electronic correlation 1. Metrics based on mean-field and coupled cluster wave functions In computational quantum chemistry, the prevailing measure of electronic correlation is the correlation energy. This metric dates back to the work of Löwdin120and is defined as the difference between the FCI and mean-field ( EMF) energy, Ecorr=EFCI−EMF. (14) The correlation energy may be further partitioned into dynamical and non-dynamical contributions, as proposed by Sinano ˇglu and others.121,122 One can similarly estimate correlation effects from the mag- nitude of the overlap of the Hartree–Fock determinant with thenormalized FCI wave function, | CHF| = |⟨ΦHF|Ψ⟩|. This metric has been discussed as a diagnostic tool for determining the quality of single-reference electron correlation methods.123However, for infi- nite systems, | CHF|→0, so this metric is probably suited only for comparing systems with the same number of electrons. In the context of coupled cluster theory, several diagnostics have been introduced. The D1diagnostic captures deficiencies in the reference and is defined as the 2-norm of the matrix of singles cluster amplitudes (T)ia=ta i, where the indices iandaspan the occupied and virtual orbitals, respectively. This metric is defined as D1=∥T∥2=√ λmax(TTT), (15) whereλmax(TTT) indicates the largest eigenvalue of the matrix TTT. The D2diagnostic is a measure of correlation, and it is similarly defined using doubles amplitudes ( tab ij) with the above equation modified to make this metric orbital invariant.124 2. Measures based on the two-body density cumulant The norm of the two-body cumulant ( λ2) has become a well- established metric of correlation.125–129This quantity is the portion of the two-body density matrix γ2that is not separable into one-body contributions, and it is defined as λrs pq=γrs pq−γr pγs q+γr pγq s, (16) whereγp qandγrs pqare the one- and two-body reduced density matri- ces, respectively, γp q=⟨Ψ∣a† paq∣Ψ⟩,γpq rs=⟨Ψ∣a† pa† qasar∣Ψ⟩. (17) The information contained in λ2can be distilled down to a single value metric via its Frobenius norm, ∥λ2∥F=√ ∑ pqrs∣λrspq∣2, (18) which captures both spin entanglement and Coulombic correla- tion effects127,129and is null for a single determinant. The two-body density cumulant also has a direct connection to the number of effectively unpaired electrons, which itself has been used as a met- ric of correlation.130–132For two non-interacting fragments A and B with no interfragment spin entanglement, the square Frobenius norm is additive,127,129that is,∥λ2(A)∥2 F+∥λ2(B)∥2 F=∥λ2(A⋯B)∥2 F, where “A⋯B” indicates A and B at infinite separation. Therefore, in our comparison of the models, we report the square Frobenius norm. Moreover, the two body cumulant is directly related to the definition of the intrinsic correlation energy (ICE) proposed by Kutzelnigg.10By expressing the energy in terms of 1- and 2-RDMs and expanding the latter in terms of the two-body cumulants, one may rewrite the two-body contribution to the total energy as a sum of Coulombic, exchange, and correlation contributions, E2=ECoul +Eex+EICE. Here, EICEis a pure two-body potential energy term, J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp which may be expressed using the two-body cumulant represented in coordinate space [ λ2(1, 2; 1, 2)] as EICE=1 4∑ pqrsλpq rs⟨rs∥pq⟩=1 2∫λ2(1, 2; 1, 2) r12dτ1dτ2. (19) This intrinsic correlation energy has the advantage of being defined, irrespective of a reference mean-field wave function. 3. Spin correlation metrics We also characterize electronic states using various metrics based on the spin–spin correlation function as they are helpful in diagnosing spin frustration. The spin–spin correlation function (Aij), defined as Aij=⟨ˆSi⋅ˆSj⟩−⟨ˆSi⟩⋅⟨ˆSj⟩ (20) measures the irreducible correlation of total spin ( ˆSi) for two local- ized spatial orbitals ϕiandϕj. In this work, we employ Pipek–Mezey localized orbitals133to define spin–spin correlation metrics. We also compute the spin–spin correlation density Ai(r), which can be used to graphically represent the spatial correlations of spin with respect to a localized orbital ϕi. For the well localized atomic orbitals, Ai(r) can be approximated as Ai(r)=⟨ˆSi⋅ˆS(r)⟩−⟨ˆSi⟩⋅⟨ˆS(r)⟩≈∑ jAij∣ϕj(r)∣2, (21) where ˆS(r)is the total spin operator in real space, | ϕj(r)|2is the spa- tial density of the jth orbital, and Aijare elements of the spin–spin correlation function. Additionally, we consider three scalar metrics introduced in previous molecular spin frustration studies: (i) the sum of the absolute value of the spin–spin correlations ⟨S2⟩abs,134 ⟨S2⟩abs=∑ ij∣⟨ˆSi⋅ˆSj⟩∣, (22) (ii) the sum of the absolute value of the long range spin–spin correlations ⟨S2⟩abs, lr , ⟨S2⟩abs, lr=⟨S2⟩abs−∑ i∣⟨ˆSi⋅ˆSi⟩∣−2∑ ⟨kl⟩∣⟨ˆSk⋅ˆSl⟩∣, (23) and (iii) the sum of the nearest-neighbor spin–spin interactions ⟨S2⟩nn, ⟨S2⟩nn=∑ ⟨kl⟩⟨ˆSk⋅ˆSl⟩, (24) where iandjindex all orbital sites, and ⟨kl⟩is a double sum over nearest neighbor orbital sites. 4. Metrics based on quantum information theory Metrics inspired by quantum information theory have also been recently used to investigate various phenomena related to strong correlation and entanglement135and find several applications in computational chemistry.118,119,136–138We consider two quantities, the single-orbital entanglement entropy (SOEE) and the total quantum information ( Itot), both of which can be derived from the 1- and 2-RDMs. The SOEE describes the entanglement of a spatial orbital ϕiwith the remaining bath orbitals. For a given spatial orbital ϕi, we can write four occupation patterns for the corresponding αandβspin orbitals ∣p⟩≡∣niαniβ⟩ ∈{∣00⟩,∣01⟩,∣10⟩,∣11⟩}, which we label with the index p= 1, 2, 3, 4. The reduced density matrix ρi pq=Trbath[⟨p∣Ψ⟩⟨Ψ∣q⟩]is computed by projecting the wave function onto single-orbital configurations |q⟩and | p⟩of orbitalϕiand tracing out all other degrees of freedom. For states with the fixed number of electrons, this matrix is diagonal with elements given by ρi 11=1−γiα iα−γiβ iβ+γiαiβ iαiβ, (25) ρi 22=γiα iα−γiαiβ iαiβ, (26) ρi 33=γiβ iβ−γiαiβ iαiβ, (27) ρi 44=γiαiβ iαiβ. (28) The SOEE of orbital ϕiis then computed as the Shannon entropy with respect to the four occupations, Si=−4 ∑ p=1ρi ppln(ρi pp). (29) The total quantum information ( Itot) is given as the sum of the SOEEs for all spatial orbitals, Itot=L ∑ i=1Si. (30) Large values of Itotindicate departure from integer orbital occupa- tions and are associated with strong correlation effects.139We note, however, that the value of Itotis not invariant with respect to unitary rotations of the orbitals and, therefore, will depend on the type of orbital basis employed in a computation. III. COMPUTATIONAL DETAILS The ground-state singlet energies and two-body density cumu- lants of the model systems were calculated using FCI, ACI, and DMRG. The ap-sCI and SVD-FCI wave functions were obtained from FCI wave functions, as described in Secs. II B 1 and II B 2, respectively. All computations employed self-consistent field (SCF) orbitals obtained with the open-source quantum chemistry pack- age P SI4140,141and used a STO-6G basis set.142Canonical (delocal- ized) orbitals were computed using restricted Hartree–Fock (RHF). The localized orbitals were obtained by first performing a restricted open-shell Hartree–Fock (ROHF) computation using maximum multiplicity (e.g., S= 5 for H 10) and then localizing the orbitals with the Pipek–Mezey (PM)133procedure (allowing rotations among all orbitals). Computations based on canonical RHF orbitals were run in D2h symmetry for the H 10chain, ring, and sheet and in C2vsymmetry J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp for the H 10pyramid. The H 12, H 14, and H 16analogs of the four sys- tems were run with the same symmetry as their H 10counterparts with the exception of the H 14pyramid, which used D2hsymmetry. All computations using the localized orbitals were performed in C1 symmetry. The ranges of threshold parameters used for each method are given in the supplementary material. MP2, CCSD, and CCSD(T) computations were performed using the P SI4, while V2RDM calculations employed the open source V2RDM-CASSCF plugin.112CR-CC(2,3) computations were performed using GAMESS.143FCI and ACI computations were performed using our open-source code F ORTE.144All ACI computations included additional determinants to ensure the spin completeness of thePandQspaces. The rank-reduction procedure used for SVD- FCI and the a posteriori determinant screening procedure for ap-sCI were implemented in a development version of F ORTE. Density matrix renormalization group calculations were per- formed with C HEMPS2 .145DMRG calculations associated with a par- ticular final value of Mwere preceded by three preliminary computa- tions with smaller bond dimension and added noise. This procedure has been shown to make the overall DMRG calculation converge more rapidly and produce more accurate results.78,146In the first two preliminary computations, Mis set to 150, 500, 500, and 500 (for H 10, H 12, H 14, and H 16, respectively) to build an initialization for the last two instructions with a larger value of M. In cases where the final value of Mis less than the values specified above, the same value of Mis used for the three preliminary calculations and for the final calculation. As mentioned already, due to the block struc- ture of the DMRG tensors induced by symmetries, the final MPS, in general, does not correspond to a set of dense matrices of dimen- sion M2. For DMRG calculations using a localized basis, orbitals for the 1D chain and ring were ordered to be spatially consecu- tive. The localized orbitals for the 2D sheet and 3D pyramid systems used a Fiedler vector ordering derived from the two electron inte- grals to account for physical proximity and orbital overlap.82Plots of the localized orbitals and the site orderings are reported in the supplementary material. For canonical MOs, orbitals were grouped into blocks by irreducible representation and (within each irre- ducible representation) ordered energetically. For calculations using D2hsymmetry, the irreducible representation blocks were ordered asAg,B1u,B3u,B2g,B2u,B3g,B1g, and Ausuch that blocks corre- sponding to bonding and anti-bonding orbitals were adjacent on the DMRG lattice. This strategy has been shown to be successful for several DMRG studies81,147–149and is rationalized by quantum information principles.119For calculations using symmetries other than D2h, the ordering of the irreducible representations followed Cotton’s ordering. IV. RESULTS In this section, we analyze the results of our study for the H 10 models. Figure 1 shows the structure of the four H 10model sys- tems. The geometry of each model is controlled by a parameter r, which determines the nearest neighbor H–H distance (in Å). The geometries of all models, raw data for the potential energy curves, and energy errors are collected in a GitHub repository.113 The rvalues considered here (0.75 Å–2.0 Å) cover both the weak and strong electron correlation regimes of each model. This FIG. 1 . Structure of the H 10model systems studied in this work. Geometries are parameterized by the nearest-neighbor H–H distance ( r), indicated by green dashed lines. point can be quantified by estimating the U/tratio of the Hubbard Hamiltonian, ˆH=−t∑ i,σ(ˆa† i,σˆai+1,σ+ˆa† i+1,σˆai,σ)+U∑ iˆni↑ˆni↓, (31) where tandUare obtained by fitting the excitation energies (for singlet and triplet states) of the Hubbard dimer to those of the H 2 molecule with bond length r. Using this approach, we find that U/t ranges from about 0.94 at r= 0.75 Å to 8.55 at r= 2.0 Å. A. Ground and low-lying electronic states We have found a variety of interesting characteristics in the ground and low lying excited states of the H 10model systems. Met- rics of correlation for the ground state of the four H 10systems as a function of the rare reported in Table I. As expected, the numbers show an increase in correlation as rincreases across all four systems. However, when comparing different systems, there are interesting discrepancies between the metrics. For example, at r= 1.5 Å, the 1D chain has the second largest absolute value of Ecorr(0.4038 Eh), the largest absolute value of intrinsic correlation energy (1.066 Eh), a high∥λ2∥2 Fvalue (6.11), and the largest D2value (0.70); however, this system unexpectedly displays a relatively large weight of the Hartree–Fock determinant (| CHF| = 0.67). A comparison of the ring with the chain shows that the former is slightly less correlated than the latter. In the case of the 2D sheet at r= 1.5 Å, all metrics of cor- relation indicate that this system has the smallest degree of electron correlation. In contrast, the 3D pyramid displays the strongest corre- lation effects, yielding the largest absolute value of Ecorr(0.4051 Eh), a large intrinsic correlation energy (0.9765 Eh), and the smallest HF J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Properties of the singlet ground states of the four H 10systems at different values of the H–H distance ( r). Summary of correlation metrics: correlation energy ( Ecorr), the squared Frobenius norm of the two-body density cumulant ( ∥λ2∥2 F), coupled-cluster amplitude diagnostics ( D1andD2), magnitude of the Hartree–Fock coefficient in the normalized FCI expansion (| CHF|), and the total quantum information in a RHF canonical basis ( Id tot) and a localized basis ( Il tot). For the H 10pyramid at r= 2.0 Å, the data reported correspond to an excited state adiabatically connected to the ground state at smaller values of r(see the supplementary material for details). System r(Å) EFCI(Eh) Ecorr(Eh) EICE(Eh)∥λ2∥2 F D1 D2 |CHF| Id tot Il tot H101D chain0.75 −5.228 560 −0.1082 −0.2628 0.61 0.018 0.202 0.96 1.24 13.74 1.00 −5.415 393 −0.1678 −0.4351 1.46 0.015 0.302 0.91 2.57 13.52 1.50 −5.036 293 −0.4038 −1.0662 6.11 0.010 0.696 0.67 7.42 11.99 2.00 −4.790 989 −0.7912 −1.6754 13.27 . . . . . . 0.37 11.78 9.22 H101D ring0.75 −5.151 378 −0.1026 −0.2323 0.43 0.000 0.122 0.97 1.01 13.81 1.00 −5.422 958 −0.1475 −0.3650 1.02 0.000 0.189 0.94 2.05 13.67 1.50 −5.048 052 −0.3616 −1.0197 5.96 0.000 0.643 0.67 7.28 12.24 2.00 −4.794 398 −0.7678 −1.6659 13.64 . . . . . . 0.32 11.87 9.35 H102D sheet0.75 −3.917 633 −0.1040 −0.2325 0.35 0.008 0.107 0.98 0.85 13.65 1.00 −4.891 538 −0.1393 −0.3262 0.71 0.014 0.159 0.95 1.58 13.56 1.50 −4.903 192 −0.2868 −0.7820 2.85 0.038 0.337 0.79 5.47 12.92 2.00 −4.739 235 −0.6886 −1.6949 9.22 . . . . . . 0.21 12.36 9.44 H103D pyramid0.75 −2.853 673 −0.1737 −0.4151 1.13 0.015 0.320 0.93 1.77 13.54 1.00 −4.269 379 −0.2397 −0.5811 2.28 0.031 0.486 0.84 3.13 13.40 1.50 −4.733 459 −0.4051 −0.9765 3.54 0.067 0.635 0.62 6.88 12.67 2.00∗−4.694 062 −0.7480 −1.7252 3.59 0.093 0.685 0.25 12.62 9.48 determinant weight (| CHF| = 0.62). However, strong correlation in the 3D system is not reflected in the value of ∥λ2∥2 F(3.54), which is smaller than that of both 1D systems ( ≈6). The quantum information metric (in a delocalized basis) Id totpaints a similar picture: the pyra- mid total information lies in between that of the 1D systems and the less correlated 2D system. However, in a localized basis, the same metric Il totdecreases for all systems as a function of r. This behav- ior is interesting as it suggests that quantum information metrics could potentially be useful for choosing orbitals to use with vari- ous approximate methods. As discussed in more detail in Sec. IV B, the low value of ∥λ2∥2 Fobserved for the 3D pyramid is likely a con- sequence of spin frustration, which results in a rapid decay of spin correlation functions. We also note that after r= 1.5 Å, the ground state of the 3D pyramid crosses several low-lying singlet states and byr= 2.0 Å, it corresponds to the third excited state of Agsymmetry (see the supplementary material for a plot of the low-lying states of the 3D pyramid in the range r= 1.5 Å–2.0 Å). The small discrepancies observed in the various metrics are due to the fact that they measure different aspects of correlation. While Ecorrand | CHF| quantify the deficiency of the mean-field treatment (measured in both energetic and wave function terms), quantities like∥λ2∥2 FandItotcapture only statistical aspects of correlation. The intrinsic correlation energy ( EICE) appears to offer a good compro- mise between the mean-field and statistical measures of correlation; nevertheless, its value is significantly larger than the Ecorrvalues and captures contributions due to Coulomb repulsion (i.e., absent of Coulomb repulsion, EICEis zero even for a correlated state). The D1 andD2metrics measure the importance of orbital rotations ( D1) and correlation effects ( D2) in the CCSD wave function. In particular, since D1is not directly related to electron correlation, its behavioris very different from that of D2, with the latter growing with rin all models. In contrast, D1decreases in the 1D chain, it is identically zero in the 1D ring due to the different symmetry of singly excited determinants, and it grows with rin the 2D and 3D models. Another common approach to diagnose the onset of strong correlation is symmetry breaking of the Hartree–Fock solution. The Coulson–Fischer point (here defined in terms of the restricted →unrestricted symmetry breaking) of the chain and ring models is found at r= 0.85 Å and 1.05 Å, respectively. Consistent with the lower degree of correlation in the 2D sheet, the corresponding UHF solution exhibits spin-contamination at a point farther out in the dissociation curve (1.35 Å). Instead, the 3D pyramid exhibits sym- metry breaking at the smallest distance (0.70 Å) compared to the other three systems. Finally, we characterize the strength of correlations by comput- ing the density of states (DOS) with fixed particle number [ D(E)]. For convenience, we convolute the density of states with a Gaussian function of exponent αand shift the energies by the ground state energy ( E0). This convoluted DOS is expressed in terms of excita- tion energies ΔEi=Ei−E0, where Eiare energies of singlet, triplet, and quintet electronic excited states, and it is given by Dg(E)=∑ iexp(−α(ΔEi−E)2). (32) Note that this quantity is different from the DOS computed for electron attached/detached states. Figure 2 shows the energy spectra in the range of 0–0.5 Eh (0 eV–13.6 eV) relative to the ground state from computations of the lowest 50 singlet, triplet, and quintet states of the H 10systems. At shorter bond lengths ( r= 1.0 Å), the 1D and 2D systems show large J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Density of states with fixed elec- tron number convoluted with a Gaussian function [ Dg(E), defined in Eq. (32)] com- puted from the 50 lowest singlet, triplet, and quintet states of the H 10systems at an H–H distance (a) r= 1 Å and (b) r= 1.5 Å. The density of states was convoluted with a Gaussian function of exponentα= 105E−2 h. gaps between the ground state and the lowest triplet state. How- ever, this gap closes significantly in the 3D pyramid to ∼0.044 Eh. At longer bond lengths ( r= 1.5 Å), the singlet–triplet gap decreases for all systems. Interestingly, the 3D pyramid shows an almost zero gap (∼0.007 Eh), and several singlet near-degenerate states accumulate near the ground state. B. Spin correlation and frustration We have found that there are signs of spin frustration in the 2D sheet and 3D pyramid models. Frustration is indicated by the inability to satisfy antiferromagnetic interactions—a condition which is not mathematically rigorous but that has, nonetheless, been used to define systems as spin frustrated150—and the lack of long range antiferromagnetic ordering beyond nearest-neighbor interactions.The spin–spin correlation densities shown in Figs. 3(a) and 3(b) indicate clear antiferromagnetic ordering beyond nearest neighbors in the 1D chain and ring. Each localized spin is anti-correlated with its nearest neighbor, as depicted by the adjacent red and blue shad- ing. Figure 3(c) shows the spin–spin correlation density for the four symmetry unique sites in the 2D sheet. In contrast to the 1D mod- els, it can be seen that there is no way to simultaneously satisfy all antiferromagnetic interactions for the 2D sheet, and consequently, spin correlations decay more rapidly. This is also the case for the 3D pyramid [see Fig. 3(d)] for which each site is anti-correlated with all other sites, suggesting no antiferromagnetic ordering beyond nearest neighbors. Table II summarizes the spin correlation prop- erties for the H 10, H 12, and H 14systems at r= 1.5 Å. As is the case with the 2-body cumulant norm, the H 101D chain and ring systems have larger absolute spin correlation ⟨S2⟩abs(17.42 and 18.66, respectively) than the 2D and 3D systems (11.55 and 10.86, FIG. 3 . Spin correlation density Ai(r) of the H 10models at an H–H distance r= 1.25 Å plotted for (a) the edge and central localized MO sites of the hydrogen chain, (b) the symmetry unique site of the hydrogen ring, (c) the four symmetry unique sites of the H 10sheet, and (d) the two symmetry unique sites of the H 10pyramid. Positive and negative values of Ai(r) are indicated in red and blue, respectively, and in each plot, the localized orbital ϕiis denoted by an asterisk. J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Ground-state of the four H nsystems at an H–H distance r= 1.50 Å. Sum of the absolute value of the spin–spin correlations ( ⟨S2⟩abs), the sum of the absolute value of the long range spin–spin correlations ( ⟨S2⟩abs, lr ), and the sum of the nearest neighbor spin–spin interactions ( ⟨S2⟩nn) [see Eqs. (22)–(24) for the definition of these metrics]. System n ⟨S2⟩abs ⟨S2⟩abs, lr ⟨S2⟩nn Hnchain10 17.42 5.25 −3.10 12 21.77 7.18 −3.72 14 26.29 9.27 −4.35 Hnring10 18.66 6.51 −3.16 12 24.13 9.39 −3.84 14 28.95 11.91 −4.42 Hnsheet10 11.55 2.46 −1.94 12 14.31 2.59 −2.66 14 17.16 3.63 −3.06 Hnpyramid10 10.86 3.04 −1.19 12 18.06 4.02 −3.63 14 18.30 5.94 −2.40 respectively). The short-range nature of spin correlation of the 2D and 3D H 10systems is also indicated by their smaller value of ⟨S2⟩abs, lr (2.46 and 3.04), compared to the 1D systems (5.25 and 6.51, for the chain and ring, respectively). These results are consis- tent with the spin correlation density analysis in Fig. 3. We note that the scaling of ⟨S2⟩absand⟨S2⟩abs, lr with nfor the sheet and pyra- mid systems is (in most cases) linear or super-linear, which is notexpected for systems absent of long range spin ordering. However, this is likely because the lattice sizes considered are still relatively small, and the addition of two hydrogens at a time does not extend the lattices in a completely uniform manner, thus altering (possibly greatly) the frustrated character. It is also possible to observe a lack of long-range correlation and ordering for the 2D sheet and 3D pyra- mid by considering the radial distribution of spin–spin correlations and absolute spin–spin correlations reported in the supplementary material. It is evident from the various metrics of correlation, the DOS plots, and our analysis of spin correlation that the H 10lattices display a broad range of correlation regimes. Therefore, we believe that it is important to consider the 2D and 3D models in future benchmarks of electronic structure methods because they capture some aspect of the physics of spin frustration that are not displayed by 1D hydrogen models. C. Performance of sCI, SVD-FCI, and DMRG Having characterized the nature of the ground state of the H10models, we now proceed to analyze the efficiency with which sCI, SVD-FCI, and DMRG approximate the wave functions of these systems. In Fig. 4, we plot the energy error [ EX(Npar)−EFCI] as a func- tion of the number of variational parameters for the H 10systems in the regime of strong electron correlation ( r= 1.5 Å). The accuracy volume may be obtained from these plots by finding the number of parameters corresponding to a 1 m Eherror. When using canonical orbitals, we see that the DMRG affords the most compact represen- tation, although Vap-sCI andVACI+PT2 are within a factor of 1.5–2 of VDMRG . ACI without the PT2 correction always requires more vari- ational parameters to match the accuracy of ap-sCI and ACI+PT2, FIG. 4 . Ground-state of the four H 10models at an H–H distance r= 1.5 Å. Energy error with respect to FCI vs number of parameters ( Npar) of approximate methods. The gray shaded region represents chemically accurate energies (error less than 1m Eh). J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp andVACIis 2–3 times VDMRG . For all four H 10systems, SVD-FCI exhibits the worst efficiency, although only by a small margin such thatVSVD-FCI is 2–4 times greater than VDMRG . Note that we include two sets of results for the DMRG: the lowest energy eigenvalue found during all DMRG sweep optimizations (labeled DMRG, see Ref. 145 for details), and the energy obtained from the reduced density matri- ces of the final MPS (indicated with DMRG∗). When a large bond number Mis used, the two energy values are nearly identical, but for smaller values of M, the DMRG∗value may be slightly higher than the DMRG one. When using the localized orbitals, we see that the DMRG again produces the most compact representation and by a much larger margin for all four H 10systems. In particular, for the 1D chain, VDMRG is two orders of magnitude lower than all other methods. Comparing the accuracy volume of DMRG with different orbital bases, one notices that the localized basis is more efficient in the 1D systems, while the delocalized basis leads to smaller VDMRG for the 3D model. For the 2D model, the localized and canonical bases yield comparable VDMRG values. The advantage of using canonical orbitals is inconsistent with previous findings,82and it is likely due to the size of the systems considered here. In this case, the compression afforded by using a localized basis is outweighed by the advantages of using point group symmetry. When comparing results across canonical and localized orbitals for the other methods, we find that the accuracy volume is always smaller in the delocalized basis, so there is no advantage to orbital localization. This is in agreement with past observations100that localization is beneficial for sCI methods only after a certain system size is reached. It is interesting to observe that the accuracy vol- ume for DMRG and sCI mirrors the behavior of the total quantum information for both delocalized and localized bases (see Table. I), suggesting that this metric may be useful for determining the best orbital basis to use at a given geometry.We note that for the sheet and pyramid, there are a few values of σfor which the ACI results do not converge monotonically and lead to small bumps. We have also encountered cases where the iterative ACI algorithm finds the first excited state due to near-degeneracies, an issue that may be resolved using a state-averaged version of the method.42These incorrect energies were not included in Fig. 4. Figure 5 shows the plots of Nparvs the two-body cumulant error ∥Δλ2∥Ffor the four H 10systems at r= 1.5 Å. These plots do not include ACI+PT2 results since second-order corrections to the ACI 1- and 2-RDMs were not available. We find similar trends for the efficiency to represent λ2as we do for the energy, with the caveat that in a canonical basis, ap-sCI generally gives the best compres- sion efficiency. It can be seen that with canonical orbitals, ap-sCI actually preserves the accuracy of the two-body density cumulant after compression better than the DMRG does for the 1D chain, and similarly to the DMRG for the other three systems. There is also a larger disparity in the performance of ap-sCI and ACI for cumulant compression performance, which can be attributed to two reasons. First, ACI adds additional determinants at each iteration to ensure spin completeness (the compressed ap-sCI wave func- tion is not guaranteed to be an eigenfunction of spin). Second, ACI selects determinants according to their energetic contribution and not explicitly their contribution to the wave function. SVD- FCI is the least efficient in compressing the wave function for the 1D systems but does nearly as well as ACI for the 2D sheet and 3D pyramid. However, it possible that if variational optimization is used for SVD-FCI, the cumulant error may increase, similarly to the behavior observed for ACI. When using the localized orbitals, it can be seen that the DMRG likewise shows the best compression effi- ciency with respect to ∥Δλ2∥F, especially for the 1D chain and ring systems. As shown in Table I, the degree of correlation for all mod- els increases as the H–H distance rbecomes larger. In Table III, FIG. 5 . Ground-state of the four H 10models at an H–H distance r= 1.5 Å. Density cumulant error ∥Δλ2∥Fwith respect to FCI vs number of parameters ( Npar) of approximate methods. J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE III . Accuracy volume ( VX) computed for the ground state of the H 10models for various methods. Values are reported for both localized and delocalized molecular orbital bases. The Hilbert space size ( ∣HN∣) for all models in C1symmetry is 63 504. Hilbert space sizes with the largest abelian symmetry exploited in the computations are reported in this table. ACI + PT2 values with a <sign indicate that the energy error with the reported number of parameters is significantly lower than 1.0 m Eh. Finding more precise values of VXfor ACI + PT2 is challenging as the energy error is not monotonic as a function of σ. Delocalized (RHF canonical) Localized (Pipek–Mezey) System ∣HN∣ r(Å) ap-sCI ACI ACI+PT2 SVD-FCI DMRG ap-sCI ACI ACI+PT2 SVD-FCI DMRG 1D chain 31 752 ( D2h)0.75 1491 2066 <335 5292 2600 41 872 46 882 45 052 10 584 468 1.00 5122 6978 <2156 10 584 4896 35 962 42 332 39 510 21 168 388 1.25 11 201 14 231 7347 17 136 9598 29 148 35 306 30 732 34 272 376 1.50 18 176 22 989 16 356 26 964 12 674 20 424 26 008 20 564 53 928 176 1D ring 15 912 ( D2h)0.75 577 873 <181 2784 740 53 358 56 244 53 448 11 088 3359 1.00 2019 2803 663 5328 1522 49 982 53 364 50 084 21 168 3164 1.25 4791 6384 2701 9492 2663 45 452 49 537 43 486 37 800 2688 1.50 8520 11 056 7895 16 296 4034 36 450 41 254 34 134 65 016 1884 2D sheet 15 912 ( D2h)0.75 766 1102 <218 2532 1117 53 252 59 470 58 050 10 080 4649 1.00 1899 2809 <718 4296 1853 51 822 58 256 56 252 17 136 4071 1.25 4139 5283 3478 7848 2626 50 318 57 036 53 852 31 248 4113 1.50 8667 11 122 6468 15 156 4218 47 916 54 466 49 540 60 480 4192 3D pyramid 15 912 ( C2v)0.75 1478 2115 <787 4044 1630 44 062 56 232 55 452 16 128 10 832 1.00 2755 3605 <1607 7056 2250 45 812 55 986 55 078 28 224 12 556 1.25 4997 6530 2869 9864 2927 45 844 56 348 52 728 39 312 10 998 1.50 8097 10 519 6457 13 152 3495 43 932 53 280 48 580 52 416 9489 we can see that in a delocalized basis, the complexity of the wave function, as gauged by VX, also increases as rbecomes larger such that all methods require a larger number of parameters to achieve chemical accuracy. In a localized basis, VDMRG ,Vap-sCI , andVACI decrease with increasing r, suggesting that these methods can exploit the local character of correlation, although in most cases not enough to outweigh the benefits of symmetry-adapted delocalized orbitals. In a delocalized basis, we note that for small values of r, the ACI+PT2 produces very accurate results with very few parameters, outperforming the DMRG using just a few hundred determinants. It is interesting to note that compression efficiency for SVD-FCI decreases dramatically as rincreases, suggesting that the method is not able to take advantage of local correlation. Additionally, it can be seen that at more contracted geometries (smaller values of r), there is less of a disparity between the compression performance of the various approaches. D. Comparison with other electronic structure methods It is interesting to use the H 10models to benchmark the robust- ness and accuracy of conventional methods that employ a fixed number of parameters. Figure 6 compares the energy errors rela- tive to FCI for RHF, MP2, CCSD, CCSD(T), CR-CC(2,3), V2RDM with the two-body positive-semidefinite P, Q, and G conditions (V2RDM-PQG), and V2RDM-PQG with additional three-body pos- itive semidefinite T 2conditions (V2RDM-PQGT2). RHF deviates significantly from FCI for all four systems, even near the H 2equilib- rium geometry ( re= 0.74 Å), where it gives errors of ∼80–100 m Eh.MP2 reduces the energy error near reto about 10 m Eh. While the RHF and MP2 energies do not diverge, they do not capture the dis- sociation of the H 10systems even qualitatively, giving energy errors well over 100–200 m Ehforr≥1.6 Å. The three coupled cluster variants—CCSD, CCSD(T), and CR-CC(2,3)—achieve chemical accuracy for the 1D chain and ring systems for r≤1.0 Å and diverge beginning around r≥1.5 Å, past the Coulson–Fisher point. Performance for CCSD, CCSD(T), and CR- CC(2,3) is slightly worse for the 2D sheet and 3D pyramid, where chemical accuracy is only achieved for r≤0.75 Å, and divergence is seen once again at larger values of r. For all four systems, when r>1.25 Å, the magnitude of the HF coefficient | CHF| in the FCI wave function is less (or significantly less) than 0.9. It is worth mention- ing here, however, that a handful of hydrogen systems have been investigated with variants of CC that provide stable results relative to the examples in Fig. 6, namely, the paired coupled cluster doubles (pCCD)151and the singlet pCCD (CCD0).152 The V2RDM approaches achieve the best descriptions of the potential energy surfaces compared to the other methods used in this section. Enforcing the PQG conditions during the optimization gives a good qualitative description of the dissociations but still pro- duces large quantitative errors in the range of 10–50 m Ehfor the 1D chain and ring systems and 50–200 m Eherrors for the 2D sheet and 3D pyramid. Enforcing the additional T2 condition improves the V2RDM results significantly such that energy errors for the chain and ring systems at r= 1.50 Å are 3.0 m Ehand 7.4 m Eh, respec- tively. It can be seen, however, that for the 2D sheet and 3D pyramid, V2RDM-PQGT2 fails to produce chemically accurate results by a large margin, with errors of the order of 10–50 m Ehat stretched J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . Ground-state potential energy curves of the four H 10models. Energy error (ΔE) with respect to FCI for various electronic structure methods as a function of the H–H distance ( r). The gray shaded region indicates the range of rfor which the restricted Hartree–Fock solution is stable. geometries. Interestingly, the performance of V2RDM is far less sen- sitive to rthan RHF, MP2, or CC as indicated by smaller values of nonparallelism error (the maximum error minus the minimum error over the entire range of r). Additionally, the error for V2RDM has a maximum in the re-coupling region ( r≈1.5 Å), while all other methods generally decrease in accuracy with increasing r. V. SCALING OF THE ACCURACY VOLUME AND SIZE CONSISTENCY In this section, we discuss some of the formal properties of the methods and present numerical results concerning the scaling of theaccuracy volume and size consistency. We begin by comparing the scaling with respect to system size. sCI may be considered a zero- dimensional Ansatz , in the sense that it is particularly efficient in the description of few electrons in many virtual orbitals, especially due to the PT2 correction. If one demands that the sCI energy is size consistent for a set of non-interacting fragments A⋯B⋯C⋯, one concludes that the number of parameters grows as NANBNC. . .. In other words, the sCI accuracy volume (per electron) grows exponen- tially with the number of electrons N(albeit with a smaller prefac- tor than FCI), VsCI∝exp(N). In practice, we find this to be the case for ACI using the localized orbitals. To achieve an accuracy of ∼10−4Ehper electron for a system of five non-interacting H 2 molecules requires 2380 parameters, rather than the 20 parame- ters required by a product state built from solutions for each H 2 molecule. Our analysis also suggests that the SVD-FCI approach suffers from the exponential growth of the accuracy volume (independently from dimensionality), although, to the best of our knowledge, a for- mal analysis has not been reported. Even in the best case scenario, the number of parameters for a rank 1 SVD-FCI approximation scales as NSVD-FCI∝2√NH, which implies VSVD-FCI≥2√NH. We likewise observe that for the same system of dissociated hydrogens, SVD-FCI requires 14 616 parameters to achieve ∼10−4Ehper elec- tron. Like the result for sCI, this indicates that SVD-FCI with a fixed number of parameters is not size consistent. In the case of DMRG, a MPS with bond dimension Mcan describe a system with entanglement entropy Sbound by condi- tion S≤log 2M75or, equivalently, exp( S)≤CM with Ca constant. For gapped systems of dimensionality Dthat satisfy an area law, the entanglement entropy is expected to scale as S∝LD−1(plus loga- rithmic corrections for non-gapped systems), where Lis the length scale of the system.153Therefore, in the DMRG, the bond dimen- sion Mscales at most as exp( γLD−1)∝exp(γN(D−1)/D). Similarly, we estimate that the accuracy volume of DMRG scales as VDMRG ∝NM2=Nexp(2γN(D−1)/D). For one dimensional systems ( D= 1), VDMRG is independent of system size and the ground state can be well approximated by a finite bond dimension. Beyond one dimen- sion, this analysis suggests that the DMRG bond dimension grows exponentially. However, for D= 2, the DMRG is already exponen- tially more efficient than sCI since VDMRG∝Nexp(2γN1/2). This, in practice, implies that the DMRG is still applicable without expo- nential cost to both one-dimensional and “thin” two-dimensional problems.154 The DMRG is formally size consistent, giving additively sep- arable energies for non-interacting fragments A⋯B, so long as the orbitals are localized on either AorB.79,115If the orbitals in the DMRG lattice are ordered by subsystem ([ A⋯B]), then the wave function for noninteracting fragments becomes a product state of the MPS on AandB. Then, a MPS obtained by concatenating the MPS of AandBwith a bond of dimension 1 ( M= 1) is sufficient to represent the product state and satisfy size consistency. This implies that the VA+B DMRG for a system of non-interacting fragments is approx- imately equal to VA DMRG +VB DMRG . In practice, we have found this to be nearly true, a product state for a system of five non-interacting H2molecules has 20 parameters, and the DMRG can reproduce the FCI energy exactly with a bond dimension as small as M= 3 (with 60 parameters). In principle, the DMRG should be able to achieve this J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-13 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp result already with M= 2, but we find that instead it converges to a local minimum rather than the FCI energy. Finally, we present the numerical results for the scaling with system size of the accuracy volume for analogous of our four model systems with up to up to 16 hydrogens. In Fig. 7, we plot the accu- racy volume for n= 10, 12, 14, and 16 at r= 1.5 Å, corresponding to absolute energy errors of 1.0, 1.2, 1.4, and 1.6 m Eh, respectively. For comparison, we have also included the size of the FCI space (in FIG. 7 . Accuracy volume ( VX) for various approximate methods as a function of the number of hydrogen atoms ( n) for the four H nmodels. For comparison, we also report the number of FCI determinants (in C1symmetry) and the curve n4. The 12, 14, and 16 hydrogen chains, rings, sheets, and pyramids are exten- sions of the H 10models in that the additional hydrogens are placed within the same lattice structure. Unless otherwise noted, all results employ canonical RHF orbitals.C1symmetry) and a curve with n4scaling, which is proportional to the number of Hamiltonian matrix elements. We note that the ground states of the H 12ring, H 16ring, and H 16sheet have symme- tries ( B1g), (B1g), and ( B3u), respectively, different from that of all other systems ( Ag). Additionally, we note a small dip in the curve for the 3D systems at 14 hydrogens, which we attribute to the dif- ferent symmetry used for that lattice, D2has opposed to C2v. It can be seen that the DMRG again provides the best compression of the wave function as measured by the accuracy of the energy for differ- ent system sizes. In a localized basis, a polynomial fit of VDMRG as a function of the number of hydrogens ( n) gives a scaling proportional ton2.1for the chain and n3.3for the ring, demonstrating the advan- tage of this methods for one-dimensional systems. It is also worth pointing out that for the larger systems (H 12–H 16), it is advantageous (though still exponentially scaling) to use the localized orbitals with the DMRG even for the 2D systems. This result is consistent with other DMRG studies comparing localized vs canonical orbitals for finite 2D arene systems.82For all the other methods, VXin a delo- calized basis appears to scale exponentially with a prefactor smaller than that of FCI. It may be helpful to the reader to note that for all the systems considered here, we find that in many cases, the FCI computations are still faster than ACI and DMRG even up to 16 electrons. On a single node, FCI computations run in about 1 s up to 3 h for the H10–H 16systems, whereas the implementation of DMRG used in this work can take up to 1–2 days on a single node for the more chal- lenging 2D and 3D systems. We observe the most striking difference in the case of DMRG applied to the 1D systems, where even a very accurate computation in a localized basis can take on the order of 1 s even for H 16. VI. CONCLUSIONS AND FUTURE WORK This work accomplishes two main goals. First, we propose a series of benchmark hydrogen models with a tunable degree of correlation, which cover a wide range of electronic structures. These include 1D hydrogen chains and rings with antiferromag- netic ground states, a 2D triangular lattice (sheet) with spin frus- trated interactions, and a 3D pyramid system that displays both spin frustration and a vanishing energy gap (dense manifold of near- degenerate states). We analyze these systems with various correla- tion metrics and by computing their low-energy spectra and spin– spin correlation functions. The models are found to have drastically different electronic structures depending on the physical dimension. In particular, since 2D and 3D systems exhibit some of the fin- gerprints of spin frustration and are not efficiently approximated with the MPS, they nicely complement benchmark sets based on 1D lattices. Our comparison of different metrics of correlation also highlights the importance of using multiple descriptors to charac- terize electronic states, as our results clearly show that they measure different aspects of correlation. Second, using the hydrogen models, we compare the perfor- mance of selected CI, SVD-FCI, and DMRG in various regimes of strong electron correlation. We focus, in particular, on determin- ing the ability of each method to efficiently compress the informa- tion content of the FCI wave function. To quantify this property, J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-14 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp we introduce a new metric, the accuracy volume ( VX), which cor- responds to the minimum number of variational parameters nec- essary to achieve a target energy error (in our case, defined as 0.1 mEhper electron). As expected, the DMRG affords the most effi- cient representation for the 1D H 10chain and ring using at least an order of magnitude fewer parameters to achieve the same level of energy or two-body cumulant accuracy compared to the other methods. Nevertheless, this efficiency is gradually lost when going from 1D to higher-dimensional systems. In contrast, all flavors of sCI perform best in a delocalized basis but are generally less effi- cient than the DMRG. The SVD-FCI, which we use as a proxy for rank-reduced FCI, is generally found to be the most ineffi- cient approach to approximate the H 10wave functions. However, as mentioned previously in Sec. II B 2, the variational optimiza- tion in RR-FCI would yield lower accuracy volumes than SVD- FCI, likely making RR-FCI more competitive with sCI and DMRG. We have similarly analyzed the ability of each method to accu- rately represent electron distributions, namely, the cumulant of the two-body density matrix. In this case, the trends are similar to those observed for the energy, with the difference that sCI shows better performance for the 2D and 3D systems in a delocalized basis. In analyzing the compressibility of the wave functions for H 12, H14, and H 16analogs of the four H 10models, we have determined that the DMRG consistently shows the smallest accuracy volume and that the performance of SVD-FCI is more on par with that of ap-sCI for the larger systems, suggesting that future developments of RR-FCI methods such as those in Refs. 27 and 29 are certainly worthwhile, especially for systems larger than those considered in this study. Despite the significant reduction in the number of param- eters relative to the FCI wave function afforded by selected CI, SVD- FCI, and DMRG, none of these methods bring a reduction in scal- ing from exponential to polynomial in the general case. Alternative methods, such as higher-dimensional tensor network states, quan- tum Monte Carlo, and quantum computational algorithms, may be required to circumvent the storage cost of an exponentially scaling wave function. We note that while the accuracy volume is a generally applica- ble metric for determining the performance of a method, the bench- mark set considered here uses a minimal basis, is restricted to small systems amenable to FCI computations, and does not include atoms with more complex electronic structures. Therefore, one should be cautious in extrapolating the relative performance of the methods in the case of more complex systems. In future studies, it might also be interesting to investigate the advantages of employing other orbitals bases, such as natural orbitals and split-localized orbitals. In addi- tion, our work has focused only two tensor decomposition methods. It would be interesting to examine the accuracy volume of projected entangled-pair states,155the multi-scale entanglement renormaliza- tion Ansatz ,156tree tensor network states,76,77,139and other more general tensor network states. With appropriate modifications, the accuracy volume is also applicable to stochastic methods,157,158both in real and determinant space, and could provide a way to compare these approaches to deterministic methods. Our work does raise a few important questions as we (poten- tially) approach an era of quantum advantage for molecular com- putations. Although quantum computational algorithms are able to avoid the explicit storage of the wave function, they still sufferfrom non-trivial classical computational overhead. For example, the quantum phase estimation159,160(QPE) algorithm relies on the time evolution of the Hamiltonian, which implies a computational scal- ing and storage costs (ignoring the cost of state preparation) at least proportional to K4in a delocalized basis, although more efficient representations have been recently proposed.161,162For the purpose of comparing the resource cost of classical and quantum algorithms, in Fig. 7, we have also reported an estimate of the resources needed by quantum algorithms computed as n4, where nis the number of hydrogen atoms (equal to the number of spatial orbitals). This plot shows that classical compression approaches use more than n4parameters even with systems as small as 12 electrons. While this prefatory comparison highlights the importance of quantum algorithm development even for modestly sized systems, it also sug- gests a threshold for the maximum number of classical parameters a quantum algorithm should employ. In other words, a successful quantum algorithm should achieve a VXsmaller (and with lower n- scaling) than state-of-the art classical methods such as selected CI and DMRG for a given level of accuracy. The competitiveness of any quantum algorithm could be tested for various regimes of correla- tion by comparing the computational resources (classical variational parameters) required to achieve a 0.1 m Ehenergy error per electron with those reported in Table III. In summary, this study has explored the limits of classical state-of-the-art electronic structure methodologies as applied to strongly correlated electrons. The hydrogen benchmark set and the accuracy volume metric are two new tools that will be useful in guiding the development of the next generation of classical and hybrid quantum–classical methods for strongly correlated systems. An important open problem in electronic structure theory is iden- tifying the practical limits of classical methods and knowing under what circumstances quantum algorithms can overcome these limits. This work approaches this problem from a computational perspec- tive and sheds some light on the first aspect; in future work, we plan to investigate the ability of various quantum algorithms to go beyond the limits of classical methods. SUPPLEMENTARY MATERIAL Ranges of threshold parameters used for each systematically improvable method, examples of the localized orbitals with the cor- responding DMRG ordering, spin–spin radial distribution func- tion for each H 10model, and potential energy curves for the low- lying states of the H 10pyramid are included in the supplementary material. ACKNOWLEDGMENTS The authors would like to thank Sebastian Wouters for help- ful conversations regarding DMRG calculations, Garnet Chan for discussions on the scaling of DMRG, and Mario Motta for his insights into spin frustrated lattices. This work was supported by the U.S. Department of Energy under Award No. DE-SC0019374 and a Camille Dreyfus Teacher-Scholar Award (No. TC-18-045). N.H.S. was supported by a fellowship from The Molecular Sciences Software Institute under NSF Grant No. ACI-1547580. J. Chem. Phys. 153, 104108 (2020); doi: 10.1063/5.0014928 153, 104108-15 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp DATA AVAILABILITY The geometries of the hydrogen models, energy errors at each value of Nparfor ap-sCI, ACI, SVD-FCI, and DMRG, and raw data for the potential energy curves of the H 10models using RHF, MP2, CCSD, CCSD(T), CR-CC(2,3), V2RDM-PQG, V2RDM-PQGT2, and FCI are openly available on GitHub (Ref. 113). 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5.0023242.pdf

Appl. Phys. Lett. 117, 122412 (2020); https://doi.org/10.1063/5.0023242 117, 122412 © 2020 Author(s).Magnon-mediated spin currents in Tm3Fe5O12/Pt with perpendicular magnetic anisotropy Cite as: Appl. Phys. Lett. 117, 122412 (2020); https://doi.org/10.1063/5.0023242 Submitted: 27 July 2020 . Accepted: 10 September 2020 . Published Online: 24 September 2020 G. L. S. Vilela , J. E. Abrao , E. Santos , Y. Yao , J. B. S. Mendes , R. L. Rodríguez-Suárez , S. M. Rezende , W. Han, A. Azevedo , and J. S. Moodera ARTICLES YOU MAY BE INTERESTED IN Robust spin–orbit torques in ferromagnetic multilayers with weak bulk spin Hall effect Applied Physics Letters 117, 122401 (2020); https://doi.org/10.1063/5.0011399 Spin current generation and detection in uniaxial antiferromagnetic insulators Applied Physics Letters 117, 100501 (2020); https://doi.org/10.1063/5.0022391 Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745Magnon-mediated spin currents in Tm 3Fe5O12/Pt with perpendicular magnetic anisotropy Cite as: Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 Submitted: 27 July 2020 .Accepted: 10 September 2020 . Published Online: 24 September 2020 .Corrected: 26 October 2020 G. L. S. Vilela,1,2,a) J. E.Abrao,3 E.Santos,3 Y.Yao,4,5J. B. S. Mendes,6 R. L. Rodr /C19ıguez-Su /C19arez,7 S. M. Rezende,3W.Han,4,5 A.Azevedo,3 and J. S. Moodera1,8 AFFILIATIONS 1Plasma Science and Fusion Center, and Francis Bitter Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2F/C19ısica de Materiais, Escola Polit /C19ecnica de Pernambuco, Universidade de Pernambuco, Recife, Pernambuco 50720-001, Brazil 3Departamento de F /C19ısica, Universidade Federal de Pernambuco, Recife, Pernambuco 50670-901, Brazil 4International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China 5Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 6Departamento de F /C19ısica, Universidade Federal de Vic ¸osa, Vic ¸osa, Minas Gerais 36570-900, Brazil 7Facultad de F /C19ısica, Pontificia Universidad Cat /C19olica de Chile, Casilla 306, Santiago, Chile 8Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA a)Author to whom correspondence should be addressed: gilvania.vilela@upe.br ABSTRACT The control of pure spin currents carried by magnons in magnetic insulator (MI) garnet films with a robust perpendicular magnetic anisotropy (PMA) is of great interest to spintronic technology as they can be used to carry, transport, and process information. Garnet filmswith PMA have labyrinth domain magnetic structures that enrich the magnetization dynamics and could be employed in more efficientwave-based logic and memory computing devices. In MI/non-magnetic (NM) bilayers, where NM is a normal metal providing a strong spin–orbit coupling, the PMA benefits the spin–orbit torque-driven magnetization switching by lowering the needed current and rendering the process faster, crucial for developing magnetic random-access memories. In this work, we investigated the magnetic anisotropies inthulium iron garnet (TIG) films with PMA via ferromagnetic resonance measurements, followed by the excitation and detection of magnon-mediated pure spin currents in TIG/Pt driven by microwaves and heat currents. TIG films presented a Gilbert damping constant of a/C250:01, with resonance fields above 3.5 kOe and half linewidths broader than 60 Oe, at 300 K and 9.5 GHz. The spin-to-charge current conversion through TIG/Pt was observed as a microvoltage generated at the edges of the Pt film. The obtained spin Seebeck coefficient was0.54lV/K, also confirming the high interfacial spin transparency. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023242 Spin-dependent phenomena in systems composed of layers of magnetic insulators (MIs) and non-magnetic heavy metals (NMs)with strong spin–orbit coupling have been extensively explored ininsulator-based spintronics. 1–6Among the MI materials, YIG (Y3Fe5O12) is widely employed in devices for generation and transmis- sion of pure spin currents. The main reason is its very low magneticdamping with the Gilbert parameter on the order of 10 /C05and its large spin decay length, which permits spin waves to travel distances of sev-eral orders of centimeters inside it before they vanish. 7–9When com- bined with heavy metals such as Pt, Pd, Ta, or W, many intriguingspin-current related phenomena emerge, such as the spin pumpingeffect (SPE), 10–14spin Seebeck effect (SSE),7,15–18spin Hall effect (SHE),19–21and spin–orbit torque (SOT).22–25The origin of these effects relies mainly on the spin diffusion length and the quantum-mechanical exchange and spin–orbit interactions at the interface andinside the heavy metal. 26All these effects turn the MI/NM bilayer into a fascinating playground for exploring spin–orbit driven phenomenaat interfaces. 27–30 Well investigated for many years, intrinsic YIG(111) films on GGG(111) (GGG ¼Gd3Ga5O12) exhibit in-plane anisotropy. To obtain YIG single-crystal films with perpendicular magnetic anisot- ropy (PMA), it is necessary to grow them on top of a different Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplsubstrate or partially substitute yttrium ions with rare-earth ions, to cause strain-induced anisotropy.31–33Even so, it is well-known that magnetic films with PMA play an important role in spintronic tech- nology. The PMA enhances the spin-switching efficiency, which reduces the current density for observing the spin–orbit torque (SOT) effect, and it is useful for developing SOT-based magnetoresistive ran- dom access memory (SOT-MRAM).34–36Besides that, PMA increases the information density in hard disk drives and magnetoresistive random access memories,37–39and it is crucial for breaking the time- reversal symmetry in topological insulators (TIs) aiming toward quantized anomalous Hall states in MI/TI.40–42 Recently, thin films of another rare-earth iron garnet, TIG (Tm 3Fe5O12), have caught the attention of researchers due to their large negative magnetostriction constant, which favors an out-of-plane easy axis.4,43,44TIG is a ferrimagnetic insulator with a critical tempera- ture of 549 K, a crystal structure similar to YIG, and a Gilbert dampingparameter on the order of a/C2410 /C02:4,45Investigations of spin trans- port effects have been reported in TIG/Pt45,46and TIG/TI,42,47where the TIG was fabricated by the pulsed laser deposition (PLD) technique. The results showed a strong spin mixing conductance at the interface of these materials that made it possible to observe spin Hall magneto- resistance, spin Seebeck, and spin–orbit torque effects. In this paper, we first present a study of the magnetocrystalline and uniaxial anisotropies, as well as the magnetic damping of sput- tered epitaxial TIG thin films using the ferromagnetic resonance (FMR) technique. For obtaining the cubic and uniaxial anisotropy fields, we analyzed the dependence of the FMR spectra on the film thickness and the orientation of the dc applied magnetic field at room temperature and 9.5 GHz. Then, we swept the microwave frequency for getting their magnetic damping at different temperatures. Subsequently, we focused this investigation on the excitation of magnon-mediated pure spin currents in TIG/Pt via the spin pumping and spin Seebeck mechanisms for different orientations of the dc applied magnetic field at room temperature. Pure spin currents trans- port spin angular momentum without carrying charge currents. They are free of Joule heating and could lead to spin-wave based devicesthat are energetically more efficient. Employing the inverse spin Hall effect (ISHE), 12we observed the spin-to-charge conversion of these currents inside the Pt film, which was detected as a developed microvoltage. TIG films with thicknesses ranging from 15 to 60 nm were depos- ited by rf sputtering from a commercial target with the same nominal composition of Tm 3Fe5O12and a purity of 99.9%. The deposition pro- cess was performed at room temperature, at a pure argon working pressure of 2.8 mTorr and a deposition rate of 1.4 nm/min. To improve the crystallinity and the magnetic ordering, the films were post-growth annealed for 8 h at 800/C14C in a quartz tube in flowing oxy- gen. After the thermal treatment, the films yielded a magnetization sat- uration of 100 emu/cm3, and an RMS roughness below 0.1 nm was confirmed using a superconducting quantum interference device (SQUID) and high-resolution x-ray diffraction measurements, as detailed in our recent article.44Moreover, the out-of-plane hysteresis loops showed curved shapes, which might be related to labyrinth domain structures very common in garnet films with PMA.48The next step of sample preparation consisted of an ex situ deposition of a 4 nm-thick Pt film over the post-annealed TIG films using the dc sput- tering technique. Platinum films were grown under an Ar gas pressureof 3.0 mTorr, at room temperature, and a deposition rate of 10 nm/ min. The Pt films were not patterned. Ferromagnetic resonance (FMR) is a well-established technique for the study of basic magnetic properties such as saturation magneti-zation, anisotropy energies, and magnetic relaxation mechanisms. Furthermore, FMR has been central to the investigation of microwave- driven spin-pumping phenomena in FM/NM bilayers. 11,12,49First, we used a homemade FMR spectrometer running at a fixed frequency of 9.5 GHz, at room temperature, where the samples were placed in themiddle of the back wall of a rectangular microwave cavity operating in the TE 102mode with a Q factor of 2500. Field scan spectra of the deriv- ative of the absorption power ( dP=dHÞwere acquired by modulating t h ed ca p p l i e dfi e l d ~H0with a small sinusoidal field ~hat 100 kHz and using lock-in amplifier detection. The resonance field HRwas obtained as a function of the polar and azimuthal angles ( hH;/HÞof the applied magnetic field ~H, as illustrated in Fig. 1(d) ,w h e r e ~H¼~H0þ~hand h/C28H0. The FMR spectra for TIG(t) films are shown in Figs. 1(a)–1(c) for thicknesses t ¼15, 30, and 60 nm, respectively. The spectra were measured for Happlied along three different polar angles: hH¼0/C14 (blue), hHffi45/C14(green), and hH¼90/C14( r e d ) .T h ec o m p l e t ed e p e n - dence of HR, for each sample, as a function of the polar angle (0/C14/C20hH/C2090/C14)i ss h o w ni n Figs. 1(e)–1(g) . For all samples, HRwas minimum for hH¼0/C14, confirming that the perpendicular anisotropy field was strong enough to overcome the demagnetization field. While the films with t ¼15 nm and 30 nm exhibited the maximum value of HRforhH¼90/C14(in-plane), the sample with t ¼60 nm showed a maximum HRathH/C2460/C14.T oe x p l a i nt h eb e h a v i o ro f HRas a func- tion of the out-of-plane angle hH, it is necessary to normalize the FMR data to compare with the theory described as follows. The most relevant contributions to the free magnetic energy den- sity/C15for GGG(111)/TIG(111) films are /C15¼/C15Zþ/C15CAþ/C15Dþ/C15U; (1) where /C15Zis the Zeeman energy density, /C15CAis the cubic anisotropy energy density for (111)-oriented thin films, /C15Dis the demagnetization energy density, and /C15Uis the uniaxial energy density. Taking into con- sideration the reference frame shown in Fig. 1(d) , each energy density term can be written as:50 /C15Z¼/C0MSHsinhsinhHcos//C0/H ðÞ þcoshcoshH ðÞ ; (2) /C15CA¼K1=12 3/C06cos2hþ7cos4hþ4ffiffi ffi 2p coshsin3/sin3h/C0/C1 ;(3) /C15Dþ/C15U¼2p~M/C1^e3/C0/C12/C0K? 2~M/C1^e3=MS/C0/C12 /C0K? 4~M/C1^e3=MS/C0/C14;(4) where hand/are the polar and azimuthal angles of the magnetization vector ~M,MSis the saturation magnetization, K1is the first order cubic anisotropy constant, and K? 2andK? 4are the first and second order uniaxial anisotropy constants. The uniaxial anisotropy terms come from two sources: growth-induced and stress-induced anisot-ropy. The relation between the resonance field and the excitation fre- quency xcan be obtained from: 51,52 x=cðÞ2¼1 M2sin2h/C15hh/C15///C0/C15h/ðÞ2hi ; (5) where cis the gyromagnetic ratio. The subscripts indicate partial derivatives with respect to the coordinates, /C15hh¼@2/C15=@h2jh0;/0,Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-2 Published under license by AIP Publishing/C15//¼@2/C15=@/2jh0;/0,a n d /C15h/¼@2/C15=@h@/jh0;/0,w h e r e h0and/0 are the equilibrium angles of the magnetization determined by the energy density minimum conditions, @/C15=@hjh0;/0¼0a n d @/C15=@/jh0;/0¼0. The best fits to the data obtained using Eq. (5)are shown in Figs. 1(e)–1(g) by the solid red lines. The main physical parameters extracted from the fits, including the effective magnetiza-tion 4 pM eff, are summarized in Table I . Here, 4 pMeff¼4pM /C02K? 2=MS, where the second term is the out-of-plane uniaxial anisot- ropy field HU2¼2K? 2=M,a l s on a m e d H?. It is important to notice that the large negative values of HU2w e r es u f fi c i e n t l ys t r o n gt os a t u - rate the magnetization along the direction perpendicular to the TIG film’s plane, thus overcoming the shape anisotropy. We used the satu- ration magnetization as the nominal value of MS¼140:0G.A st h ethickness of the TIG film increased, the magnitude of the perpendicu- lar magnetic anisotropy field, HU2, decreased due to the relaxation of the induced growth stresses as expected. To obtain the Gilbert damping parameter ðaÞof the TIG thin films, we used the coplanar waveguide technique in the variable tem-perature insert of a physical property measurement system (PPMS). Avector network analyzer measured the amplitude of the forward com- plex transmission coefficients ( S 21) as a function of the in-plane mag- netic field for different microwave frequencies ðfÞand temperatures (T).Figure 2(a) shows the FMR spectra ðS21vsHÞfor TIG(30 nm) corresponding to frequencies ranging from 2 GHz to 14 GHz at 300 K, with a microwave power of 0 dBm, after normalization by background subtraction. Fitting each FMR spectrum using the Lorentz function,we were able to extract the half linewidth DHfor each frequency, as shown in Fig. 2(b) .T h e n , awas estimated based on the linear approxi- mation DH¼DH 0þ4pa=cðÞ f,w h e r e DH0reflects the contribution of magnetic inhomogeneities, the linear frequency part is caused by the intrinsic Gilbert damping mechanism, and cis the gyromagnetic ratio.53The same analysis was performed for lower temperature data, and it was extended to TIG(60 nm). Due to the weak magnetization ofthe thinnest TIG (15 nm), the coplanar waveguide setup was not able to detect its FMR signals. Figure 2(c) shows the Gilbert damping dependence with T. At 300 K, a¼0:015 for TIG(60 nm), which is in agreement with the values reported in the literature, 4,45and it increases by 130% as Tgoes down to 150 K.54 Next, this work focused on the generation of pure spin currents carried by spin waves in TIG at room T, followed by their propagation FIG. 1. FMR absorption derivative spectra vs field scan H for (a) TIG(15 nm), (b) TIG(30 nm), and (c) TIG(60 nm), at room T and 9.5 GHz. The half linewidths ( DH) for TIG(15 nm) with Happlied along hH¼0/C14;50/C14, and 90/C14are 112 Oe, 74 Oe, and 72 Oe, respectively. For TIG(30 nm), DHvalues are 82 Oe, 72 Oe, and 65 Oe for hH¼0/C14;50/C14, and 90/C14, respectively. For TIG(60 nm), DHvalues are 72 Oe, 75 Oe, and 61 Oe for hH¼0/C14;45/C14, and 90/C14, respectively. These values were extracted from the fits using the Lorentz function. (d) Illustration of the FMR experiment where the magnetization ( M) under an applied magnetic field (H) is driven by a microwave. (e)–(g) show the dependence of the resonance field HRwithhHfor different thicknesses of TIG. The red solid lines are theoretical fits obtained for the FMR condition. Magnetization curves are given in Ref. 44. TABLE I. Physical parameters extracted from the theoretical fits of the FMR response of the TIG thin films with thickness t, performed at room Tand 9.5 GHz. 4pMeffis the effective magnetization, H 1Cis the cubic anisotropy field, and H U2and HU4are the first and second order uniaxial anisotropy fields, respectively. H U2is the out-of-plane uniaxial anisotropy field, also named H?. TIG film’s thickness t 15 nm 30 nm 60 nm 4pMeff(G) /C0979 /C0799 /C0383 H1C¼2K1=MS(Oe) 31 26 /C0111 HU2¼4pMeff/C04pMS(Oe) /C02739 /C02559 /C02143 HU4¼2K? 4=MS(Oe) 311 168 432Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-3 Published under license by AIP Publishingthrough the interface between TIG and Pt and their spin-to-charge conversion inside the Pt film. Initially, we explored the FMR-driven spin-pumping effect in TIG(60 nm)/Pt(4 nm), where the coherentmagnetization precession of the TIG injected a pure spin current J s into the Pt layer, which was converted as a transverse charge current Jcby means of the inverse spin Hall effect, expressed as ~Jc¼hSHr^/C2~Js/C0/C1 ,w h e r e hSHis the spin Hall angle and r^is the spin polarization.55As the FMR was excited using a homemade spectrome- ter at 9.5 GHz, a spin pumping voltage ( VSP) was detected between the two silver painted electrodes placed on the edges of the Pt film, as illus- trated in Fig. 3(a) . It is important to note that when the magnetization vector was perpendicular to the sample’s plane, no V SPwas detected. The sample TIG(60 nm)/Pt(4 nm) had dimensions of 3 /C24m m2and a resistance between the silver electrodes of 48 Xat zero field. VSP showed a peak value of 0 :85lVin the resonance magnetic field for anincident power of 185 mW and an in-plane dc magnetic field (hH¼90/C14)a ss h o w ni n Fig. 3(b) . The signal reversed when the field direction went through a 180/C14rotation. The dependence of VSPon the microwave incident power was linear, as shown in Fig. 3(c) , whereas the spin pumping charge current ( ISP¼VSP=R) had the dependence of VSP/sinhH,a ss h o w ni n Fig. 3(d) , for a fixed micro- wave power of 100 mW. The ratio between the microwave-driven volt- age and the microwave power was 4 lV/W. We also excited pure spin currents via the spin Seebeck effect (SSE) in TIG(60 nm)/Pt(4 nm) at room T. The SSE emerges from the interplay between the spin and heat currents, and it has the potential to harvest and reduce power consumption in spintronic devices.16,18 When a magnetic material is subjected to a temperature gradient, a spin current is thermally driven into the adjacent non-magnetic (NM) layer by means of the spin-exchange interaction. The spin accumula- tion in the NM layer can be detected by measuring a transversal charge current due to the ISHE. To observe the SSE in our samples, the uncovered GGG surface was placed over a copper plate, acting as athermal bath at room T, while the sample’s top was in thermal contact with a 2 /C22mm 2commercial Peltier module through a thermal paste, as illustrated in Fig. 4(a) . The Peltier module was responsible for creating a controllable temperature gradient across the sample. On the other hand, the temperature difference ( DT) between the bottom and top of the sample was measured using a differential thermocouple. The ISHE voltage due to the SSE ( VSSE) was detected between the two silver painted electrodes placed on the edges of the Pt film. The behavior of VSSEby sweeping the dc applied magnetic field (H), while DT,hH,a n d /Hwere kept fixed, was investigated. Fixing /H¼0/C14and varying the magnetic field from out-of-plane ( hH¼0/C14Þ to in-plane along the x-direction ( hH¼90/C14Þ,VSSEwent from zero to its maximum value of 5.5 lVforDT¼20K,a ss h o w ni n Fig. 4(b) . Around zero field, no matter the value of hH, the TIG’s film magneti- zation tended to rely along its out-of-plane easy axis, which zeroes VSSE. For in-plane fields ( hH¼90/C14)w i t h DT¼12K,VSSEwas maxi- mum when /H¼0/C14, and it was zero for /H¼90/C14.T h er e a s o nt h a t VSSEwent to zero for /H¼90/C14may be attributed to the generated charge flow along the x-direction, while the silver electrodes were placed along the y-direction, thus not enabling the current detection [seeFig. 4(c) ]. The analysis of the spin Seebeck amplitude DVSSEvs hH,/H,a n dDTshowed a sine, cosine, and linear dependence, respec- tively, as can be seen in Figs. 4(d) and4(e),w h e r et h er e ds o l i dl i n e s are theoretical fits. The Spin Seebeck coefficient (SSC) extracted from the linear fit of DVSSEvsDTwas 0.54 lV/K. FIG. 2. (a) Ferromagnetic resonance spectra vs in-plane applied field Hfor a 30 nm-thick TIG film at frequencies ranging from 2 GHz to 14 GHz and a temperature of 300 K, after normalization by background subtraction. (b) Half linewidth DHvs frequency for TIG(30 nm) at 300 K. The Gilbert damping parameter awas extracted from the linear fitting of the data. (c) Damping avs temperature Tfor TIG films with thicknesses of 30 nm and 60 nm. FIG. 3. Spin pumping voltage (V SP) excited by a FMR microwave of 9.5 GHz, at room T, in TIG(60 nm)/Pt(4 nm). (a) Illustration of the spin pumping setup. (b) In- plane field scan of V SPfor different microwave powers. (c) Linear dependence of the maximum V SPwith the microwave power. (d) hHscan of the charge current (ISP) generated by means of the inverse spin Hall effect in the Pt film. (e) In-plane field scan of the FMR absorption derivative spectrum for 5 mW.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-4 Published under license by AIP PublishingIn conclusion, we used the FMR technique to probe the magnetic anisotropies and the Gilbert damping parameter of the sputtered TIGthin films with perpendicular magnetic anisotropy. The results showedhigher resonance fields ( >3.5 kOe) and broader linewidths ( >60 Oe) when comparing with YIG films at room T. Thinner TIG films (t¼15 nm and 30 nm) presented a well-defined PMA; on the other hand, the easy axis of the thicker TIG film (60 nm) showed a deviationof 30 /C14from normal to the film plane. By numerically adjusting the F M Rfi e l dd e p e n d e n c ew i t ht h ep o l a ra n g l e ,w ee x t r a c t e dt h ee f f e c t i v e magnetization, the cubic (H 1C), and the out-of-plane uniaxial anisot- ropy (H U2¼H?) fields for the three TIG films. The thinnest film pre- sented the highest intensity for H ?as expected, even so H ?was strong enough to overcome the shape anisotropy and gave place to a perpendicular magnetic anisotropy in all the three thickness of TIGfilms. The Gilbert damping parameters ðaÞfor TIG(30 nm) and TIG(60 nm) films were estimated to be /C2510 /C02, by analyzing a set of FMR spectra using the coplanar waveguide technique at various microwave frequencies and temperatures. As Twent down to 150 K, the damping increased monotonically 130%. Furthermore, spin waves (magnons) were excited in the TIG(60 nm)/Pt(4 nm) heterostructure through the spin pumping and spin Seebeck effects, at room Tand 9.5 GHz. The generated pure spin currents carried by the magnons were converted into charge currentsonce they reached the Pt film by means of the inverse spin Hall effect. The charge currents were detected as a microvoltage measured at the edges of the Pt film, and they showed sine and cosine dependenceswith the polar and azimuthal angles, respectively, of the dc appliedmagnetic field. This voltage was linearly dependent on the microwave power for the SPE and on the temperature gradient for the SSE. Theseresults confirmed a good spin-mixing conductance in the interface TIG/Pt and an efficient conversion of pure spin currents into chargecurrents inside the Pt film, which is crucial for the employment of TIGfilms with a robust PMA in the development of magnon-based spin-tronic devices for computing technologies. This research was supported in the USA by the Army Research Office (Nos. ARO W911NF-19-2-0041 and W911NF-20-2-0061),NSF (No. DMR 1700137), and ONR (No. N00014-16-1-2657), inBrazil by CAPES (No. Gilvania Vilela/POS-DOC-88881.120327/2016-01), FACEPE (Nos. APQ-0565-1.05/14 and APQ-0707-1.05/14), CNPq, UPE (No. PFA/PROGRAD/UPE 04/2017), andFAPEMIG-Rede de Pesquisa em Materiais 2D and Rede deNanomagnetismo, in Chile by Fondo Nacional de DesarrolloCient /C19ıfico y Tecnol /C19ogico (FONDECYT) No. 1170723, and in China by the National Natural Science Foundation of China (No.11974025). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. Pirro, T. Br €acher, A. V. Chumak, B. L €agel, C. Dubs, O. Surzhenko, P. G€ornert, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 104, 012402 (2014). 2A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands, Nat. Phys. 11, 453 (2015). 3L. J. Cornelissen, J. Liu, R. A. Duine, J. B. Youssef, and B. J. van Wees, Nat. 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(Oxford University Press, 2017).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122412 (2020); doi: 10.1063/5.0023242 117, 122412-6 Published under license by AIP Publishing

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AIP Advances 10, 095027 (2020); https://doi.org/10.1063/5.0018639 10, 095027 © 2020 Author(s).Spectromicroscopic measurements of electronic structure variations in atomically thin WSe2 Cite as: AIP Advances 10, 095027 (2020); https://doi.org/10.1063/5.0018639 Submitted: 20 August 2020 . Accepted: 28 August 2020 . Published Online: 28 September 2020 T. Klaproth , C. Habenicht , R. Schuster , B. Büchner , M. Knupfer , and A. Koitzsch 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 Numerical investigations of trajectory characteristics of a high-speed water-entry projectile AIP Advances 10, 095107 (2020); https://doi.org/10.1063/5.0011308 Structural, electronic, and transport properties of Co-, Cr-, and Fe-doped functionalized armchair MoS 2 nanoribbons AIP Advances 10, 095029 (2020); https://doi.org/10.1063/5.0022891 Origin of band inversion in topological Bi 2Se3 AIP Advances 10, 095018 (2020); https://doi.org/10.1063/5.0022525AIP Advances ARTICLE scitation.org/journal/adv Spectromicroscopic measurements of electronic structure variations in atomically thin WSe 2 Cite as: AIP Advances 10, 095027 (2020); doi: 10.1063/5.0018639 Submitted: 20 August 2020 •Accepted: 28 August 2020 • Published Online: 28 September 2020 T. Klaproth,1C. Habenicht,1 R. Schuster,1B. Büchner,1,2M. Knupfer,1and A. Koitzsch1,a) AFFILIATIONS 1IFW Dresden, Helmholtzstraße 20, 01069 Dresden, Germany 2Institute of Solid State and Materials Physics, TU Dresden, 01069 Dresden, Germany a)Author to whom correspondence should be addressed: a.koitzsch@ifw-dresden.de ABSTRACT Atomically thin transition metal dichalcogenides (TMDCs) are promising candidates for implementation in next generation semiconducting devices, for which laterally homogeneous behavior is needed. Here, we study the electronic structure of atomically thin exfoliated WSe 2, a prototypical TMDC with large spin–orbit coupling, by photoemission electron microscopy, electron energy-loss spectroscopy, and density functional theory. We resolve the inhomogeneities of the doping level by the varying energy positions of the valence band. There appear to be different types of inhomogeneities that respond differently to electron doping, introduced by potassium intercalation. In addition, we find that the doping process itself is more complex than previously anticipated and entails a distinct orbital and thickness dependence that needs to be considered for effective band engineering. In particular, the density of selenium vs tungsten states depends on the doping level, which leads to changes in the optical response beyond increased dielectric screening. Our work gives insight into the inhomogeneity of the electron structure of WSe 2and the effects of electron doping, provides microscopic understanding thereof, and improves the basis for property engineering of 2D materials. ©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/10.1063/5.0018639 .,s I. INTRODUCTION Semiconducting transition metal dichalcogenides (TMDCs) such as MoS 2and WSe 2have attracted enormous scientific attention in the last years.1Their layered crystal structure allows downsiz- ing them to single atomic layers without dangling bonds, which can be functionalized in electronic devices with promising performance parameters. This meets the desire to find electronically active materi- als that may substitute silicon at the extreme miniaturization limit.2 Field effect transistors based on atomically thin MoS 2and WSe 2as channel materials with excellent switching characteristics3and high mobility4,5have been demonstrated. Moreover, TMDCs exhibit var- ious other intriguing physical properties holding strong potential for optoelectronic applications, such as thickness dependent bandgaps in the visible energy range6,7and the locking of spin and momentum at the K/K′valleys giving rise to the field of valleytronics.8,9 WSe 2, in particular, possesses several advantages compared to other TMDCs. It is easier to control the character of thecharge carriers ranging from hole dominated, ambipolar to electron dominated.10,11Moreover, the spin–orbit coupling is larger than that in MoS 2, making WSe 2especially suitable for studying spin and valley dependent properties. The great success of silicon as the backbone of semiconducting industry roots partially in the ability to precisely control its prop- erties by alloying, implantation or diffusion of dopant ions, and maintaining high levels of homogeneity. Similar methods of band engineering are needed for TMDCs. Indeed, several attempts have been reported in this regard,12,13e.g., controlled defect creation by sputtering or thermal treatments,14,15substitution with dopants, e.g., Nb,16,17isovalent substitution of the type WSe 1−xSx,18,19and charge transfer doping by donor or acceptor atoms or molecules.20The latter is achieved by solution based methods or by in situ evapora- tion of the pertinent dopant material. It is a well established method for layered materials and has been applied to bulk TMDCs and graphite among others. Alkali metals figure prominently as donors and, among the alkali metals, potassium offers probably the highest AIP Advances 10, 095027 (2020); doi: 10.1063/5.0018639 10, 095027-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv degree of electron transfer. It has been repeatedly used to achieve ntype doping in atomically thin MoS 2and WSe 2up to degener- ate doping levels and the transition to a metallic state.21,22The latter has been directly monitored by angle-resolved photoemission spec- troscopy (ARPES) by the observation of the electron pockets.23–26 Apart from providing electrons, potassium may introduce defects or alter the existing ones.27It may also affect the lateral homogeneity of the doping level. Spatial inhomogeneities play an important role for the prop- erties of TMDCs and much work has been devoted to them.28–35 Inhomogeneities may originate from different sources, such as intrinsic defects and doping variations, strain, and dielectric disor- der.36,37An important consequence is the local fluctuation of the Fermi level with respect to the gap edges, which determines band alignment. Here, we measure the valence band with lateral resolution, extract the energy landscape of its position, and determine how it evolves upon electron doping. For controlled band and defect engi- neering, it is crucial to understand the effects of doping beyond simple band filling taking into account lateral inhomogeneities and chemical modifications. To this end, we utilize photoemission elec- tron microscopy (PEEM), electron energy-loss spectroscopy (EELS), and density functional theory (DFT) and obtain a comprehensive picture of the electronic structure of exfoliated, atomically thin WSe 2. II. METHODS 2H-WSe 2single crystals have been purchased from HQ graphene and exfoliated down to a single layer using poly- dimethylsiloxane (PDMS) sheets. The WSe 2flakes were transferred to a silicon substrate with an oxide thickness of 90 nm. To avoid charging effects during photoemission experiments, the substrate is coated with a 10 nm gold layer. The optical contrast is sufficient to distinguish single layer differences. Atomic force microscopy (AFM) measurements were carried out using a Bruker Dimension Icon microscope in tapping mode. Since photoemission experiments are highly surface sensitive, the characterization via AFM was done after the photoemission mea- surements, including the doping experiments resulting in rather rough surface profiles. The results of the AFM measurements are shown in the supplementary material. The photoemission data were measured using a NanoESCA system (Scienta Omicron) equipped with a helium lamp with an energy of 21.21 eV (He I) and an energy resolution of ≤0.2 eV. The spatial resolution is below 1 μm. After obtaining an initial photoemission dataset of the freshly exfoliated flake, potassium was deposited onto the flake in steps using the SAES dispensers. The deposition rate was controlled by a quartz crystal microbalance and found to be about 0.2 nm/min. The EELS measurements were carried out using a purpose built transmission electron energy-loss spectrometer with a primary elec- tron energy of 172 keV and the energy and momentum resolution of ΔE= 82 meV and Δq= 0.04 Å−1, respectively, at T= 20 K.38,39The films ( thickness≈100 nm) were exfoliated using a scotch tape. Subse- quently, the films were mounted onto standard electron microscopy grids and transferred into the EELS spectrometer. The undopedcrystals were intercalated in situ by potassium vapor from the SAES dispensers. The band structure calculations were performed using the full- potential local-orbital code (FPLO)40employing the Perdew-Wang 92 exchange-correlation-functional.41For 1-layer to 4-layer WSe 2, we constructed super-cells with about 10 nm of vacuum between layers and sampled the Brillouin zone with a (18 ×18×1) k-point mesh. For bulk WSe 2, a (12 ×12×12) k-point mesh was used. Only the lattice constant awas optimized by minimizing the total energy of bulk-WSe 2and was found to be a= 3.302 Å. The parameter c was set to the experimental value of 12.982 Å42since van der Waals interactions are not taken into account. The lattice constants deter- mined in this way were used for the 1-layer to 4-layer calculations as well. All calculations were performed in fully relativistic mode. The calculations have been performed for freestanding flakes, i.e., no interaction with the substrate is taken into account. III. RESULTS AND DISCUSSION An optical microscope image of the prepared flake is given in Fig. 1(a). Figure 1(c) shows the same flake measured by photoe- mission electron microscopy (PEEM) at four different energies. The areas of different thicknesses can be distinguished in the images. From atomic force microscopy and photoemission spectroscopy (PES) measurements, we conclude that area 1 corresponds to a monolayer, area 2 to a trilayer, and area 3 to a four layer. X-ray pho- toemission spectroscopy of the Se 3 dand W 4 fcore levels confirms the absence of oxides and other foreign phases. See the supplemen- tary material for further information and additional AFM, PEEM, and XPS data. A. Doping homogeneity We start by considering the homogeneity of the electronic structure. In order to do so, we apply PEEM to measure the valence band laterally resolved at all positions of the flake. The inset of Fig. 2(a) presents the low energy part of the monolayer valence band integrated around the Γpoint. The energy location of the valence band depends on the position of the Fermi energy inside the gap and represents a signature of the doping level. The position of the lead- ing peak can be evaluated by standard fitting procedures (see the supplementary material for details). Figures 2(a)–2(d) show these positions for the monolayer region of the flake evaluated by pixel- by-pixel analysis before and after potassium intercalation. Figure 4 of the supplementary material presents some explicit examples of the spectra and fitting curves, confirming the high quality and reliability of the fitting procedure. Note that the valence band maximum of the monolayer is situated at Krather than Γ(see Fig. 2 of the supple- mentary material). However, here, we are interested in the relative changes in the energy rather than the absolute values of the valence band maxima. Figures 2(a) and 2(c) show the valence band positions of the undoped and doped flakes on a joint color scale, whereas Figs. 2(b) and 2(d) maximize the local contrast by applying separate scales to the same data. The map of the undoped sample, indeed, shows inho- mogeneities on the μm scale with the energy variations in the order of 50 meV. The overall effect of potassium intercalation is a shift to lower energy [Figs. 2(a) and 2(c)]. This is expected because of AIP Advances 10, 095027 (2020); doi: 10.1063/5.0018639 10, 095027-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . (a) Optical microscope image of the investigated flake on the substrate. (b) Crystal structure of WSe 2. (c) Pho- toemission intensity distributions of the right part of the flake at different bind- ing energies. The areas 1 and 2 can be distinguished. the well known ntype character of alkali metal intercalation.25,26,43 A closer inspection of Figs. 2(b) and 2(d) reveals that the shift is not uniform. In particular, the contrast of some elongated, stripe- like structures increases and they become generally more prominent. This impression is confirmed by examining the valence band posi- tion along the white dashed lines in Figs. 2(b) and 2(d), as shown in Fig. 2(f). The energy difference, i.e., the contrast, across the elon- gated structure becomes larger with doping (purple shaded region). On the other hand, the contrast decreases in the turquoise shaded region, where a more regular patch-like structure is situated for the undoped sample. Hence, the response of the electronic structure for these two regions is qualitatively different. The histograms of the energy position for the undoped and doped cases are depicted in Fig. 2(e). The overall shift amounts to ΔE= 0.22 eV. The error of the individual fits is well below the width of the histograms. Hence, the histograms represent the intrinsic dis- tribution of the valence band positions. A laterally integrated spec- trum may acquire additional broadening due to this inhomogeneity. We interpret the two-fold response of the Fermi level position on electron doping as a consequence of different types of inhomo- geneity, whereas the elongated, stripe like structures are possibly related to the microcracks introduced by the exfoliation transfer pro- cess, and the patches could be due to the variations of dielectric screening imposed by the local flake–substrate interaction. It has been shown previously that edges and microcracks are more eas- ily functionalized by extrinsic manipulations than the smooth inner part of the flake.44–46The edge sites probably are chemically morereactive. This increases the inhomogeneity upon potassium interca- lation. Local dielectric fluctuations, on the other hand, are expected to be smoothened when immersed in an additional charge reservoir, which increases screening. Electron doping by potassium intercalation causes specific effects not only depending on the type of underlying inhomogeneity but also depending on the thickness and with respect to the involved atomic orbital character. B. Suppression of the Se states Figure 3 presents valence band photoemission spectra taken from the monolayer, the trilayer, and the tetralayer before and after doping. The momentum window is again restricted to the vicinity of the Γpoint. The shape of the spectra depends on the number of layers. The leading peaks of the trilayer and four-layer spectra do not correspond to the first peaks of the density of states (DOS) because the photoemission intensity of the uppermost bands is weak and hidden below the tails of the main peak at E≈2 eV (see Fig. S3). When applying potassium intercalation to the sample, several changes of the electronic structure are observed. First of all, the intercalated spectra shift to lower energy for all thicknesses. For a sufficiently large potassium concentration, the conduction band would eventually become occupied and an insulator–metal transi- tion occurs.44,47–49The thickness dependent energy shifts are dis- cussed in detail below. Here, we focus on the changes of the spectral shape. In particular, there are well resolved peaks highlighted by AIP Advances 10, 095027 (2020); doi: 10.1063/5.0018639 10, 095027-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . (a) Valence band position of the undoped flake. Inset: integrated valence band around Γ. (b) Valence band position of the undoped flake with a zoomed color scale. The white line indicated the position of the line scan in (f). (c) Valence band position of the doped flake. (d) Valence band position of the undoped flake with the zoomed color scale. (e) Histograms of the valence band positions of undoped (black) and doped (red) flake. (f) Line scan of the valence band position along the white dashed line in (b) and (d). Black: undoped and red: doped. The purple and turquoise shadings are discussed in the text. arrows for the undoped sample that vanish or become significantly less intense upon doping. We associate the origin of this effect to the orbital character of the involved bands. In the lower part of Fig. 3, we depict the orbital resolved density of states (pDOS) calculated for the undoped material. In order to facilitate meaningful comparison to the experimental data, we restricted k-integration of the DOS to the same window that applies for experiment ( Δk≈0.15 Å). The leading peaks are dominated by the tungsten states in each case. The smaller peaks indicated by arrows in Fig. 3 are mostlyformed by the Se states. We conclude that the potassium interca- lation disturbs the Se lattice more profoundly than the W sublattice. This is naturally explained by the sandwich type monolayer structure where the middle W layer is protected by the outer Se layers. It is also consistent with the previous DFT studies, which show that a chem- ical bond between Se and K is established.21,50The tungsten lattice and the tungsten states appear to be intrinsically protected, which is important because the uppermost valence band is dominated by W 5 d. FIG. 3 . Comparison of doped and undoped photoemission spectra integrated around Γand partial density of states as a function of the number of layers. AIP Advances 10, 095027 (2020); doi: 10.1063/5.0018639 10, 095027-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv If the Se-related pDOS is modified by potassium intercalation, other material properties should be affected. Of particular impor- tance for the TMDC are the optical properties that depend on the electronic structure and must be influenced by the purported sce- nario.6,7Therefore, we investigated the dielectric properties of potas- sium doped WSe 2by EELS in the optical limit, i.e., momentum transfer near zero. Figure 4 shows EELS measurements of undoped (a) and doped (b) bulk samples.51The undoped sample is gapped below∼1.7 eV followed by a sharp intensity onset and a peak at E= 1.8 eV. This peak is of excitonic character.47,52,53The structures at higher energies are due to interband transitions but may also be influenced by excitonic effects. The doped sample is metallic, as can be seen by the presence of the charge carrier plasmon at E≈1 eV.51 No exciton is present anymore, as expected for the metallic samples, and the spectral shape at higher energies has changed with respect to the undoped case. FIG. 4 . Experimental EELS data for (a) undoped and (b) doped WSe 2in its bulk form. The red line represents the joint density of states, which accounts for the higher lying features and their doping dependence. Inset: low energy region of the doped sample covering the charge carrier plasmon.In order to understand these changes and to connect the EELS results with the PES measurements above, we compare the EELS measurements with the joint density of states (JDOS) in Fig. 4. The JDOS is proportional to the imaginary part of the dielectric func- tionϵ2(ω), which is an approximate measure of the so-called loss function, determining EELS intensity. At the same time, ϵ2(ω) is responsible for the optical absorption. The JDOS has been calcu- lated based on the DFT results shown in Fig. 3 and shifted by 0.9 eV to lower energy for better comparison with experiment. The JDOS derived in this way relies on the one-particle band structure and can- not reproduce the intense exciton at 1.8 eV correctly. However, at higher energies, our model matches qualitatively the experimental results and the doping dependence. The JDOS for the doped sam- ple was calculated in the following way: motivated by the reduced Se photoemission intensity after potassium deposition, we weighted the JDOS by the tungsten orbital weight at each k-point, which means that the Se derived states are suppressed. The downturn at E= 2.6 eV–2.8 eV in Fig. 4(b) corresponds to this suppression of the Se bands analogous to the doping dependence of the EDCs in Fig. 3. The effect of potassium doping on the optical properties can be summarized as follows: it reduces excitonic effects by enhancing the screening of many body interactions. From a rigorous treat- ment, which involved the Kramers–Kronig analysis of EELS data of an equivalent sample, we obtained previously an electron density of 1.2e−/u.c.51In addition, the underlying interband transitions them- selves are altered in an energy range that is still accessible by visible light. FIG. 5 . Comparison of undoped and doped photoemission spectra integrated around Γas a function of the number of layers. The leading peaks have been aligned for the undoped samples of different thicknesses in order to highlight the position dependence of the doped case. Inset: energy shift as a function of thickness. AIP Advances 10, 095027 (2020); doi: 10.1063/5.0018639 10, 095027-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv C. Thickness dependence The electron doping generally causes an upward shift of the Fermi level. Now, we consider the thickness dependence of this shift. Figure 5 collects the photoemission spectra of Fig. 3 on a joint energy axis and aligns the leading peak of the undoped spectra for each thickness. The corresponding doped spectra are shifted accord- ingly. From this representation, it becomes clear that the energy shift depends on thickness. The difference of the peak position is maximal for the monolayer and decreases for the three layer and four layer samples. The amount of potassium evaporated per unit area is exactly the same in each case. The effective doping per sample volume, on the other hand, will be lower for the thicker parts of the sample. Hence, the shift of the Fermi level is smaller. This means that the band alignment of adjacent layers of different thicknesses depends on the doping level applied. Such lateral homojunctions have been studied before. A monolayer–bilayer junction features a type I band align- ment accompanied by band bending near the interface.54–58Alkali metal intercalation represents a means of engineering the valence band offset at such junctions. However, it follows from Sec. III A that the interface region itself would be modified even stronger than the inner regions of the films. IV. CONCLUSIONS We measured the electronic structure of atomically thin WSe 2 by PEEM, EELS, and DFT. Lateral inhomogeneity is observed, which spans an energy scale of ∼50 meV and a length scale of μm. Electron doping by potassium intercalation affects the electronic structure in various ways. It increases the inhomogeneity; especially along the stripe-like regions possibly related to the microcracks or domain walls, it suppresses the Se states, which means that the sample is chemically modified, and it alters the thickness dependent valence band offset across the flake. These changes have an effect on the optical properties that goes beyond enhanced dielectric screening. SUPPLEMENTARY MATERIAL See the supplementary material for (i) AFM measurements, (ii) further valence band photoemission data, including angle-resolved measurements, a detailed description of the fitting procedure under- lying Fig. 2 and example spectra, and (iii) XPS data. ACKNOWLEDGMENTS We thank R. Hübel, S. Leger, F. Thunig, and M. Naumann for technical and Ulrike Nitzsche for computational assistance. This work was supported by the German Research Foundation (DFG) under Grant No. SFB 1143. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev, and A. Kis, “2D transition metal dichalcogenides,” Nat. Rev. Mater. 2, 17033 (2017). 2J. A. Robinson, “Perspective: 2D for beyond CMOS,” APL Mater. 6, 058202 (2018). 3S. B. Desai, S. R. Madhvapathy, A. B. Sachid, J. P. Llinas, Q. Wang, G. H. Ahn, G. Pitner, M. J. Kim, J. Bokor, C. Hu, H. S. P. Wong, and A. Javey, “MoS 2 transistors with 1-nanometer gate lengths,” Science 354, 99–102 (2016). 4B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, “Single-layer MoS 2transistors,” Nat. Nanotechnol. 6, 147 EP (2011). 5H.-J. Chuang, X. Tan, N. J. Ghimire, M. M. Perera, B. Chamlagain, M. M.-C. Cheng, J. Yan, D. Mandrus, D. Tománek, and Z. 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AIP Advances 10, 105012 (2020); https://doi.org/10.1063/5.0021775 10, 105012 © 2020 Author(s).Strain and electric field tunable electronic transport in armchair phosphorene nanodevice with normal-metal electrodes Cite as: AIP Advances 10, 105012 (2020); https://doi.org/10.1063/5.0021775 Submitted: 14 July 2020 . Accepted: 21 September 2020 . Published Online: 07 October 2020 Guo-Hong Chen , Yi-Nuo Chen , Yan-Wei Zhou , Yun-Lei Sun , and En-Jia Ye 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 Coupling sensitivity in concentric metal–insulator–semiconductor tunnel diodes by controlling the lateral injection electrons AIP Advances 10, 105002 (2020); https://doi.org/10.1063/5.0022326 Anisotropic behavior of excitons in single-crystal -SnS AIP Advances 10, 105003 (2020); https://doi.org/10.1063/5.0021690 Unusual behavior of coercivity in Hf/GdFeCo bilayer with MgO cap layer by electric current AIP Advances 10, 105202 (2020); https://doi.org/10.1063/5.0023636AIP Advances ARTICLE scitation.org/journal/adv Strain and electric field tunable electronic transport in armchair phosphorene nanodevice with normal-metal electrodes Cite as: AIP Advances 10, 105012 (2020); doi: 10.1063/5.0021775 Submitted: 14 July 2020 •Accepted: 21 September 2020 • Published Online: 7 October 2020 Guo-Hong Chen,1Yi-Nuo Chen,1Yan-Wei Zhou,1Yun-Lei Sun,1,a) and En-Jia Ye2,a) AFFILIATIONS 1School of Information and Electrical Engineering, Zhejiang University City College, Hangzhou 310015, People’s Republic of China 2Jiangsu Provincial Research Center of Light Industrial Optoelectronic Engineering and Technology, School of Science, Jiangnan University, Wuxi 214122, People’s Republic of China a)Authors to whom correspondence should be addressed: sunyl@zucc.edu.cn and yeenjia@jiangnan.edu.cn ABSTRACT Phosphorene, one of the graphene counterparts, is believed to have promising potential to be utilized in nanoelectronics due to its signifi- cant properties. Phosphorene has a nonplanar puckered structure with high anisotropy, which enables the elastic strain or external field to tune its electronic structure. In this work, we propose a nanodevice model based on an armchair phosphorene nanoribbon (APNR) with normal-metal electrodes and study the tuning effect of elastic strain and electric field on the electronic transport properties. We first confirm that the APNR can be driven to be of metallic conduction with linear dispersion around the Fermi level, by applying a critical compressive strain. After applying a perpendicular electric field, the APNR turns out to be a band insulator. Furthermore, we calculate the dc conduc- tance and density of states (DOS) of the nanodevice, where the APNR is connected to normal-metal electrodes. The numerical results show that in the absence of an electric field, the nanodevice possesses peak values of conductance and DOS at the Fermi level. Once the electric field is applied, a gap emerges around the Fermi level in the conductance, which suggests that the nanodevice is turned off by the external electric field. Our investigation on the present system could be useful in the development of a field-effect nanodevice based on monolayer phosphorene. ©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/10.1063/5.0021775 .,s I. INTRODUCTION Phosphorene, a monolayer of phosphorus atoms with the puck- ered structure, has attracted lots of attention due to its signifi- cant electronic properties in recent years.1The successful synthesis of phosphorene triggered more interest in this material.2–7Phos- phorene is believed to be a very promising material for nano- electronics applications, as more and more experimental results are reported. For instance, phosphorene has a higher ON/OFF ratio than graphene as field-effect transistors.8In comparison to other two-dimensional materials such as transition-metal dichalco- genides, phosphorene has a higher carrier mobility.9,10Theoret- ically, first-principles calculations, such as DFT and its many- body extensions, are performed to study the electronic structure ofphosphorene.11–16Unlike graphene, whose electronic structure at a low energy can be determined by a simple tight-binding model with one hopping parameter, phosphorene is more challenging. A four- band tight-binding model containing five neighbor hopping sites was proposed recently.17,18Based on the Hamiltonian model, the tuning effects of the strain or electric field on electronic properties were then studied. Strain has been utilized to modulate the electronic structure of monolayer materials and widely used to boost nanodevice perfor- mance in the electronic industry.19,20Due to the puckered structure, the crystal lattice of phosphorene exhibits remarkable flexibility for elastic planar strain.11A high compressive strain can induce a struc- tural phase transition or a gap transition with Dirac-like cones.21–24 If spin–orbit coupling is considered, a compressive biaxial in-plane AIP Advances 10, 105012 (2020); doi: 10.1063/5.0021775 10, 105012-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv strain and perpendicular tensile strain can even lead to a topologi- cal phase transition.25,26Besides, because of the nonplanar structure, applying an external perpendicular electric field can also modify the electronic structure.27–29For example, the electric field increases the energy gap and affects the edge bands of the zigzag phosphorene nanoribbon (PNR). Similar to graphene nanoribbons, the transport properties of phosphorene nanoribbons (PNRs) are also very sensitive to the geometry and edge shape.30Theoretical efforts on PNRs with var- ious edge shapes, such as zigzag, armchair, and beard, have been made to reveal the electronic transport properties for the appli- cation of future nanoelectronics.31–33However, the ultimate elec- tronic and photoelectronic devices in experiments are connected to metal electrodes.34–36With the layered phosphorene field effect transistors being successfully fabricated,3,8,9the effect of the inter- face of metal–phosphorene contacts plays a pivotal role in electronic transport. People selected low-resistance metal contacts, where the Schottky barrier height is small, to increase electron injection effi- ciency.37,38Most of the research studies are focused on the metal– phosphorene top contact configuration, while the edge contact has not been reported.39,40The edge dangling bonds of phos- phorene can form intimate chemical bonding with the normal- metal electrode for charge transfer. This edge contact configura- tion has a smaller physical separation of the interface and stronger orbital overlaps.36It shows the high capability of electron injection, decreases the contact resistance, and, finally, enhances the device performance. In this paper, numerical calculations are performed to investi- gate the strain and electric field tunable electronic transport in an armchair phosphorene nanodevice with edge contact normal-metal electrodes, based on the tight-binding approach, Green’s function, and Landauer–Büttiker theory. We first study the strain effect on the armchair phosphorene nanoribbon (APNR), and the results confirm that the APNR can be driven to be metallic under a critical compres- sive strain. Then, we propose a nanodevice model, where the com- pressive APNR is connected to normal-metal electrodes, and applied a perpendicular electric field via a top gate. The electric field tuning effect on the electronic transport of the nanodevice is revealed by thedc conductance and density of states (DOS). Our results may be use- ful in the application of field-effect nanodevices based on monolayer phosphorene. II. MODEL AND THEORY A. Hamiltonian model Figures 1(a) and 1(b) show the top and side views of the crys- tal structure of monolayer phosphorene, respectively, with in-plane parameters a= 4.580 Å and b= 3.320 Å, and the vertical distance between sublattices 2 l= 2.150 Å. Along the x-direction (armchair direction), the width (length) of the PNR is indicated by Na(L). The tight-binding Hamiltonian model can be given by H=∑ ⟨i,j⟩tijc† icj+∑ ilEi zc† ici, (1) where c† i(ci) is the creation (annihilation) operator of an electron at site iand lEi zis the on-site electric potential. tijruns over all the nonzero hoppings, as shown in Fig. 1(a). To describe the elec- tronic structure, they are set to be (in units of eV) t1=−1.220, t2 = 3.665, t3=−0.205, t4=−0.105, and t5=−0.055, respectively, in the tight-binding Hamiltonian model. These hopping parame- ters, obtained from first-principles calculation, can well reproduce the band structure around the Fermi level.17,18We can indicate the various atom connections ri, which correspond to those hopping parameters ti. The unit cell of phosphorene contains four atoms, two of which exist in the upper layer and the other two exist in the lower layer, as shown in Fig. 1(b). Therefore, we can obtain the four-band Hamil- tonian in the momentum representation H=∑kc†(k)ˆH(k)c(k). At theΓpoint, the k-space Hamiltonian reads ˆH=⎛ ⎜⎜⎜⎜⎜ ⎝Δ t2+t5 4t4 2t1+ 2t3 t2+t5−Δ 2t1+ 2t3 4t4 4t4 2t1+ 2t3 Δ t2+t5 2t1+ 2t3 4t4 t2+t5−Δ⎞ ⎟⎟⎟⎟⎟ ⎠, (2) FIG. 1 . (a) Top and (b) side views of the lattice structure of monolayer phosphorene. The lattice parameters are a= 4.580 Å and b= 3.320 Å, and the hopping parameters t1, t2,t3,t4, and t5in the Hamiltonian model are indicated. The width of the APNR is indicated by Na. (c) Schematic illustration of the armchair phosphorene nanodevice with edge contact normal-metal electrodes. AIP Advances 10, 105012 (2020); doi: 10.1063/5.0021775 10, 105012-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv where Δ=elEzis the on-site electric potential induced by an external electric field. When Δ= 0, we could diagonalize the Hamiltonian matrix into ˆH′=⎛ ⎜⎜⎜⎜⎜ ⎝ϵc0 0 0 0ϵv 0 0 0 0ϵT 0 0 0 0 ϵB⎞ ⎟⎟⎟⎟⎟ ⎠, (3) withϵc= 4t4+ (2t1+ 2t3) + (t2+t5),ϵv= 4t4−(2t1+ 2t3)−(t2+t5), ϵT=−4t4−(2t1+ 2t3) + (t2+t5), andϵB=−4t4−(2t1+ 2t3)−(t2+ t5). In addition, the energy gap can be given by ϵg=ϵc−ϵv= 2(2 t1 + 2t3+t2+t5) = 1.52 eV. After applying an external electric field Ez, the energy gap is evaluated as Eg=4t1+ 4t3+√ (t2+t5−4t4)2+Δ2 +√ (t2+t5+ 4t4)2+Δ2 ≈4t1+ 4t3+ 2√ (t2+t5)2+Δ2. (4) Obviously, the energy gap is determined by hopping energy tiand the electric potential, which can be tuned by applying a compressive or tensile strain and an electric field. An elastic strain with a strength of ( εx,εy,εz) can be introduced by deforming the initial bond lengths ri= [rix,riy,riz] tor′i= [(1 + εx)rix, (1 +εy)riy, (1 +εz)riz] (i= 1, 2, 3, 4, and 5). Considering the first-order linear expansion, r′ican be given by r′ i≃(1 +αixεx+αiyεy+αizεz)ri, (5) withαiμ=(riμ/ri)2(μ=x,y,z). According to Harrison’s rela- tion,41the hopping energy depends on the bond length as ti∝r−2 i. Therefore, the hopping parameters tiof strained phosphorene can be obtained as t′ i=(ri/r′ i)2ti≃[1−2(αixεx+αiyεy+αizεz)]ti. (6) B. Conductance and local density of states We propose a field-effect nanodevice based on an armchair phosphorene nanoribbon (APNR) with normal-metal electrodes, as shown in Fig. 1(c). The APNR is grown on a substrate, which can offer strain by lattice match. The external electric field can be applied perpendicular to the plane of the APNR. To investigate transport properties, we divide the nanodevice into three parts, the center APNR device (D), left normal-metal electrode (L), and right normal- metal electrode (R). The two electrodes are assumed to be semi- infinite to further connect to the electron reservoirs. The APNR and normal-metal electrodes can be constructed by repeating supercells with widths Na. The Hamiltonian of the nanodevice then can be written as H=⎡⎢⎢⎢⎢⎢⎢⎣HLHLDO HDLHDHDR O H RDHR⎤⎥⎥⎥⎥⎥⎥⎦, (7) where HD,HL(HR), and HDL(HDR) represent the center device, left (right) semi-infinite electrode, and the coupling between them,respectively. A simple Hamiltonian model is used to describe the normal-metal electrodes, HL(R)=tm∑ ⟨i,j⟩c† icj, (8) and tmis the nearest-neighbor hopping energy between normal- metal atoms. Green’s function for the nanodevice is GD=[(E+i0+)I−HD−ΣL−ΣR], (9) where ΣL(R)=VDL(R)Gs L(R)V† DL(R)is the self-energy of the left (right) electrode. Gs L(R)is the surface Green’s function of the left (right) electrode, and VDL(R)is the finite interaction Hamiltonian between the left (right) electrode and the center device. The coupling matrix ΓL(R)(E)=i[ΣL(R)(E)−Σ† L(R)(E)]is defined to obtain transmission and dc conductance,42,43 Gαβ(E)=2e2 hT(E) =2e2 hTr[ΓR(E)GD(E)ΓL(E)G† D(E)], (10) with Plank’s constant h, electron charge e, and transmission param- eterT. In addition, the local density of states (LDOS) is expressed as44,45 dn(r)/dE=−Im(GD)rr/π. (11) III. RESULTS AND DISCUSSION We first investigate the strain effects on the electronic structure of the APNR, by applying tensile and compressive strains with sym- metric and asymmetric strain distribution. Naof the APNR is set to be 9, with a width of 1.324 nm, in our calculations. In Fig. 2, we show the band structure of the APNR under asymmetric strains (a) along the x-axis (εx=±10%,εy=εz= 0) and (b) along the y-axis (εy=±10%,εx=εz= 0) and (c) symmetric in-plane strains ( εx=εy =±10%,εz= 0). As shown, in the absence of strain and the electric field, the APNR is a semiconductor with a direct energy bandgap at theΓpoint. Applying tensile strains ( εμ>0), no matter uniaxial or biaxial, can enlarge the bandgap. Nevertheless, the valance band and conduction band keep parabolic behavior around the Fermi level. On the other hand, compressive strains ( εμ<0), along both the x- andy-axis, can reduce the bandgap, which is in good agreement with previous reports.23Specifically, biaxial compressive strains seem to have more significant effects on the bandgap than uniaxial strains. One can see that when the strength of biaxial compress strains εx =εy=−10%, the bandgap becomes much smaller. In addition, the valance band and conduction band near the Fermi level turn out to be of linear dispersion, which suggests a possible Dirac fermion behavior. Note that further increase in compressive strains leads to a phase transition to a topological insulator, if spin–orbit coupling is considered.25Therefore, the Dirac fermion-like behavior might be the trace of the metal state with high mobility as in graphene. To further confirm the possible compressive strain-induced semiconductor–metal transition, we increase the strength of com- pressive strain and plot the energy dispersion of the APNR under a critical biaxial compressive strain value of εx=εy=−17% in Fig. 3(a). AIP Advances 10, 105012 (2020); doi: 10.1063/5.0021775 10, 105012-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Band structure of the APNR along the Γ−Xdirection in the presence of asymmetric strains (a) along the x-axis (εx=±10%,εy=εz= 0) and (b) along the y-axis (εy=±10%,εx=εz= 0) and (c) symmetric in-plane strains ( εx=εy=±10%, εz= 0). One can see that the energy gap vanishes and a Dirac cone emerges at the Γpoint. In other two-dimensional materials, such as silicene, such a Dirac cone can be gapped by spin–orbit coupling and a non- trivial topological phase can be introduced.46As for phosphorene, strain-induced topological transition has been demonstrated by the- oretical investigation with the inclusion of the spin–orbit coupling.25 In the present case, we confirm the metallic behavior of the APNR with linear dispersion at the Γpoint under a critical compressive strain. Such a metallic conduction dominated by a massless Dirac fermion is desirable in the current transport of nanodevices. Due to the bandgap tuning effect of the electric field and the application of a field-effect nanodevice, we further investigate the electronic transport by applying electric field Ezperpendicular to the FIG. 3 . (a) Band structure of the APNR under biaxial compressive strain εx=εy =−17%, where the energy gap vanishes and the valance and conduction bands show linear behaviors. (b) Band structure of the APNR in the presence of both biaxial compressive strain and external electric field Ezin which the energy gap is reopened by the electric field. APNR plane. Ezis set to be 0.5 V/Å in the numerical calculation so that the energy gap can be comparable to the hopping energy of phosphorene. As shown in Fig. 3(b), once the external electric field is applied, the bandgap is reopened around the Fermi level. In addi- tion, the linear dispersion is destroyed, which makes the system a band insulator. Having the tuning effect of strain and the electric field on the electronic properties of the APNR demonstrated, we propose a nanodevice based on the APNR connected with normal-metal elec- trodes. The APNR is grown on the substrate, which can offer specific strains, and a top gate is used to apply the perpendicular electric field. Figure 4 shows the dc conductance (black solid curves) and density of states (red dashed curves) of the present nanodevice sys- tem, without (a) and with (b) an electric field Ez. As discussed above, the APNR is of metallic behavior in the absence of an electric field. Therefore, one can see that the dc conductance of a uniform APNR (blue dots) is nonzero around the Fermi level in Fig. 4(a). As for FIG. 4 . Dc conductance and DOS of the APNR nanodevice connected by normal- metal electrodes (a) without and (b) with an external electric field Ez. The black solid lines and red dashed ones are the dc conductance and DOS for the present APNR nanodevice. The dashed blue lines are the dc conductance for the ideal uniform APNR. AIP Advances 10, 105012 (2020); doi: 10.1063/5.0021775 10, 105012-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv the nanodevice with normal-metal electrodes, the dc conductance shows oscillations rather than perfect steps due to the interface scat- tering between the APNR and normal-metal electrodes. Note that the dc conductance of the nanodevice keeps nonzero around the Fermi level because of the metallic nature of the APNR. In addition, some peaks emerge in the spectrum of dc conductance, which corre- sponds to the resonance states. These resonance states, which has a maximum value of the DOS, originate from the band mixing of the APNR and normal-metal electrodes. Applying a perpendicular elec- tric field can reopen the bandgap and destroy the linear dispersion of the APNR. As expected, a gap around the Fermi level emerges in the dc conductance of the uniform APNR (blue dots) in Fig. 4(b), as well as the nanodevice (black solid curves). Note that the nonzero DOS (red dashed lines) in the gap mainly comes from the normal-metal electrodes. To further study the field-effect electronic transport of the present nanodevice, we plot the spatial-resolved LDOS at the Fermi level in Fig. 5. In the absence of electric field Ez, the APNR has Dirac fermion-like behavior, which results in a peak value of the DOS, as well as conductance. Therefore, one can see that the LDOS has a larger value of distribution at the APNR and normal-metal electrodes in Fig. 5(a), compared to that in the presence of an elec- tric field in Fig. 5(b). According to the relation of jx∝Ψ†σxΨ, the current density of Dirac equation jxis proportional to the wave function amplitudes ( ∼LDOS) at neighboring atoms.47A nonzero current density jxof the nanodevice can be obtained from the spatial- resolved LDOS, which is in good agreement with the feature of con- ductance and DOS in Fig. 4(a). Once the electric field is applied, the bandgap reopens and the APNR turns out to be a band insulator. Accordingly, the LDOS almost vanishes at the central region of the nanodevice, as shown in Fig. 5(b), and the distribution of the LDOS at the normal-metal electrodes corresponds to the nonzero DOS in the gap in Fig. 4(b). FIG. 5 . Spatial-resolved local density of states for the APNR nanodevice con- nected by normal-metal electrodes around the Fermi level (a) without and (b) with external electric field Ez.To study the universality of the nanodevice model, we calcu- lated the dependence of the conductance around the Fermi level on the size of the APNR. In Fig. 6(a), we show the dependence of the conductance on the width Naof the APNR, with different lengths (L= 6, 9, 12, and 15). In the absence of an electric field, the con- ductance basically obeys a linear law as the width Naincreases and the slope is associated with the length L. It is well known that in the mesoscopic system, the resistance mainly comes from the contact resistance of the interface between electrodes and the central device. Thus, a wider device provides more transport channels for electrons and larger dc conductance for the nanosystem. In Fig. 6(b), we show the dependence of the conductance on the length L, with different widths ( Na= 7, 9, 13, and 17). Normally, the conductance decreases with increasing length, for a certain width. Nevertheless, the conduc- tance exhibits oscillations, whose peaks emerge at the same length for all the different widths. This implies that there are resonant states induced by band mixing between the APNR and normal-metal device, and these resonant states are shifted as the length increases. Figure 6(c) shows the size dependence of conductance, where the width Naand length Lincrease at a certain ratio ( Na/L= 0.5, 1, and 2). One can see that the conductance slowly increases with slight oscillations as the geometric size increases. Such a behavior is the combination of the linearly increasing conductance with the width and the shift of resonant states with the length of the nanodevice. On applying electric field Ez, the conductance vanishes for all the geometrical configurations. Finally, we test the influence of the normal-metal electrodes on the conductance of the nanodevice, for a certain size Na= 9 and L= 9. In Fig. 6(d), the nearest-neighbor hopping energy tm is adjusted from −0.01 eV to −4 eV to simulate various cases of FIG. 6 . The dependence of the conductance around the Fermi level of the APNR nanodevice on its size [(a)–(c)] and the hopping energy of normal-metal atoms tm (d). AIP Advances 10, 105012 (2020); doi: 10.1063/5.0021775 10, 105012-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv normal-metal electrodes. In the absence of an electric field, the con- ductance remains a nonzero constant even for tm→−0.1 eV, which indicates that the electron transport of the nanodevice is quite robust against the change of hopping energy in metal electrodes. Further- more, the electric field can turn off the nanodevice for all these cases. IV. CONCLUSION To summarize, we propose an APNR nanodevice connected by normal-metal electrodes and investigate the electric field tunable transport properties. The APNR is driven to be of metallic con- duction with linear dispersion by biaxial compressive strain with a strength of εxy=−17%. Furthermore, the dc conductance and DOS of the present nanodevice show peaks at the Fermi level, which indi- cates current transport. After applying electric field Ez, the APNR turns out to be a band insulator. Accordingly, the LDOS of the nan- odevice vanishes and conductance shows a gap around the Fermi level. It could be concluded that the electric field can be used to tune the electronic transport of the phosphorene nanodevice, and the results may be useful for the future application. ACKNOWLEDGMENTS The authors would like to acknowledge financial support from the National Natural Science Foundation of China (Grant Nos. 11604293, 11447206, and 11504137). DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1F. Xia, H. Wang, and Y. Jia, Nat. Commun. 5, 4458 (2014). 2S. P. Koenig, R. A. Doganov, H. Schmidt, A. H. Castro Neto, and B. Özyilmaz, Appl. Phys. Lett. 104, 103106 (2014). 3L. Li, Y. Yu, G. J. Ye, Q. Ge, X. Ou, H. Wu, D. Feng, X. H. Chen, and Y. Zhang, Nat. Nanotechnol. 9, 372 (2014). 4H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, ACS Nano 8, 4033 (2014). 5E. S. Reich, Nature 506, 19 (2014). 6H. Liu, Y. Du, Y. Deng, and P. D. Ye, Chem. Soc. Rev. 44, 2732 (2015). 7M. U. Rehman, C. Hua, and Y. Lu, Chin. 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5.0023353.pdf

J. Chem. Phys. 153, 164120 (2020); https://doi.org/10.1063/5.0023353 153, 164120 © 2020 Author(s).Efficient multireference perturbation theory without high-order reduced density matrices Cite as: J. Chem. Phys. 153, 164120 (2020); https://doi.org/10.1063/5.0023353 Submitted: 28 July 2020 . Accepted: 06 October 2020 . Published Online: 29 October 2020 Nick S. Blunt , Ankit Mahajan , and Sandeep Sharma COLLECTIONS Paper published as part of the special topic on Frontiers of Stochastic Electronic Structure CalculationsFROST2020 ARTICLES YOU MAY BE INTERESTED IN Transcorrelated density matrix renormalization group The Journal of Chemical Physics 153, 164115 (2020); https://doi.org/10.1063/5.0028608 A self-consistent field formulation of excited state mean field theory The Journal of Chemical Physics 153, 164108 (2020); https://doi.org/10.1063/5.0019557 Models and corrections: Range separation for electronic interaction—Lessons from density functional theory The Journal of Chemical Physics 153, 160901 (2020); https://doi.org/10.1063/5.0028060The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Efficient multireference perturbation theory without high-order reduced density matrices Cite as: J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 Submitted: 28 July 2020 •Accepted: 6 October 2020 • Published Online: 29 October 2020 Nick S. Blunt,1,a) Ankit Mahajan,2 and Sandeep Sharma2,b) AFFILIATIONS 1Department of Chemistry, Lensfield Road, Cambridge CB2 1EW, United Kingdom 2Department of Chemistry, University of Colorado, Boulder, Colorado 80302, USA Note: This paper is part of the JCP Special Topic on Frontiers of Stochastic Electronic Structure Calculations. a)Author to whom correspondence should be addressed: nicksblunt@gmail.com b)Electronic mail: sanshar@gmail.com ABSTRACT We present a stochastic approach to perform strongly contracted n-electron valence state perturbation theory (SC-NEVPT), which only requires one- and two-body reduced density matrices, without introducing approximations. We use this method to perform SC-NEVPT2 for complete active space self-consistent field wave functions obtained from selected configuration interaction, although the approach is applicable to a larger class of wave functions, including those from orbital-space variational Monte Carlo. The accuracy of this approach is demonstrated for small test systems, and the scaling is investigated with the number of virtual orbitals and the molecule size. We also find the SC-NEVPT2 energy to be relatively insensitive to the quality of the reference wave function. Finally, the method is applied to the Fe(II)-porphyrin system with a (32e, 29o) active space and to the isomerization of [Cu 2O2]2+in a (28e, 32o) active space. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023353 .,s I. INTRODUCTION In studying electronic structure problems, correlation effects are often separated into strong and dynamic correlation. In some systems, a single determinant is sufficient to provide a qualitative description of a system’s electronic structure. However, in strongly correlated systems, this assumption breaks down, and one often has to use a superposition of multiple determinants to describe the ref- erence state. These determinants are obtained by including all (or several) possible occupations within a subset of orbitals known as the active space. One often optimizes the active space orbitals to minimize the energy, which results in a method known as complete active space self-consistent field (CASSCF). The rest of the correla- tion due to excitation into remaining orbitals is known as dynamic correlation and can be included using a variety of methods including multireference configuration interaction (MRCI),1,2multireference perturbation theory (MRPT),3–5and multireference coupled cluster (MRCC).6–11 In recent years, there have been significant improvements in algorithms for performing (near-exact) CASSCF calculations. Meth- ods including the density matrix renormalization group algorithm(DMRG),12,13full configuration interaction quantum Monte Carlo (FCIQMC),14–17and selected configuration interaction (SCI)18–21 can now be used to solve CASSCF problems accurately for active spaces of 40–50 orbitals and possibly beyond.16,17,21–23However, there still remains the important task of including dynamic corre- lation. Traditional implementations of MRCI and MRPT require calculating and storing the three- and sometimes four-body reduced density matrices (RDMs) within the active space, which require O(n6 a)andO(n8 a)storage, respectively. This becomes infeasible for the large active spaces considered above, and separate approaches must be developed. A variety of methods have been proposed and used to avoid the need for higher-order RDMs. These include the use of cumu- lant approximations,24uncontracting terms that require high-order RDMs,25use of matrix product states,26,27approaches based on FCIQMC (where the high-order RDMs are only sampled),28,29a time-dependent formalism,30,31fully uncontracted formulations of MRPT,32external contraction,33,34and others.35–40 Recently, we demonstrated that it is possible to perform strongly contracted MRCI (SC-MRCI) and second-order n-electron valence perturbation theory (SC-NEVPT2) without constructing J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp RDMs but instead using variational Monte Carlo (VMC).41In this approach, rather than constructing RDMs and contracting them with integrals, it is possible to directly sample contributions from determinants in the first-order interacting space (FOIS). Although the number of determinants in the FOIS grows exponentially with the active space size, VMC provides a polynomial scaling method to sample them. These stochastic approaches were referred to as SC-MRCI(s) and SC-NEVPT2(s). In this article, we develop this idea further, focusing specifi- cally on SC-NEVPT2(s). In particular, we present a somewhat dif- ferent algorithm that is more efficient and avoids the need for a trial wave function in the FOIS, as was required in the original SC- NEVPT2(s) approach. We also extend this to include core orbitals, which were not considered in our original implementation. We go on to provide the analysis of this approach, including scaling with the number of virtual orbitals for the N 2molecule and with system size for polyacetylene molecules. We also provide examples demon- strating the performance of SC-NEVPT2 when the reference wave function is in error. We then study two much larger systems than considered previously by this method, namely, Fe(II)-porphyrin in a (32e, 29o) active space and [Cu 2O2]2+in a (28e, 32o) active space, demonstrating that this approach is practical for challenging problems. II. SC-NEVPT2 OVERVIEW We begin by recapping the strongly contracted NEVPT2 method4,5,42and defining the notation to be used throughout. In multireference perturbation theory, one begins by solving the complete active space (CAS) problem, giving a reference wave function ∣ϕ(0) m⟩, ∣ϕ(0) m⟩=∑ ICI,m∣DI⟩, (1) where mis the state label and | DI⟩are determinants in which all core orbitals (denoted i,j,...) are occupied and all virtual orbitals (denoted r,s,...) are unoccupied, while active orbitals (denoted a, b,...) can take any occupation number. Assuming that the set of active orbitals is chosen appropri- ately,∣ϕ(0) m⟩provides a qualitative description of the true wave function. For better accuracy, dynamic correlation must then be included by considering excitations involving core and virtual orbitals. This can be done by second-order perturbation the- ory, after choosing an appropriate reference Hamiltonian ( ˆH0). There is no unique way of defining a reference Hamiltonian, in fact, any Hamiltonian that has ∣ϕ(0) m⟩as its ground state can be selected. Various reference Hamiltonians have been chosen in the literature, and each leads to a different perturbation theory. In this article, we take the reference Hamiltonian to be the one that defines strongly contracted NEVPT (SC-NEVPT) theory [see Eq. (3)]. In the SC scheme, the uncontracted FOIS is partitioned into subspaces S(k) l. Here, kspecifies the change in the number of active space electrons relative to ∣ϕ(0) m⟩(−2≤k≤2), while lspecifies which non-active orbitals are involved in the excitation. For example, a determinant that contains two unoccupied core orbitals iand janda single occupied virtual orbital rbelongs to class S(1) ij,r. A single perturber state is then assigned to each class S(k) l, defined by ∣ψ(k) l⟩=P(k) lH∣ϕ(0) m⟩, (2) where P(k) lis the projector onto the S(k) lsubspace and His the Hamiltonian operator. This definition ensures that perturber states are orthogonal to each other (but not normalized). The above perturber states can be further divided into eight types, depending on the number of core and virtual orbitals involved. We refer to these as v,vv,c,cv,cvv,cc,ccv, and ccvv. For example, we say that a perturber state ∣ψ(1) ij,r⟩is of type ccv. Given the perturber states ∣ψ(k) l⟩, the zeroth-order Hamiltonian for SC-NEVPT is defined as H(0)=∑ mE(0) m∣ϕ(0) m⟩⟨ϕ(0) m∣+∑ l,kE(k) l∣ψ(k) l⟩⟨ψ(k) l∣, (3) which leads to the second-order perturbative energy correction, E(2) m=∑ l,kN(k) l E(0) m−E(k) l. (4) Here, N(k) lare the squared norms of the perturbers, N(k) l=⟨ψ(k) l∣ψ(k) l⟩, (5) E(0) mis the zeroth-order energy for state m, and E(k) lare the per- turber energies. In NEVPT, these perturber energies are defined via the Dyall Hamiltonian, HD, E(k) l=1 N(k) l⟨ψ(k) l∣HD∣ψ(k) l⟩, (6) with HD=core ∑ iϵia† iai+virtual ∑ aϵaa† aaa+Hactive, (7) where Hactive is the core-averaged Hamiltonian in the active space such that HD∣ϕ(0) m⟩=E(0) m∣ϕ(0) m⟩. The primary task is to calculate the second-order energy from Eq. (4). To do so, both the squared norms, N(k) l, and the perturber energies, E(k) l, are required. The exact expressions for E(k) land N(k) l can be obtained in terms of active-space RDMs. However, these include three- and four-body RDMs, whose storage requirements scale as O(n6 a)andO(n8 a)in the number of active-space orbitals, na. Instead, we will take a stochastic approach that avoids the need for higher-order RDMs. III. STOCHASTIC SC-NEVPT2 The estimation of E(2) mcan be performed in two stages. First, we calculate the squared norms, N(k) l, for all perturbers. In the second step, we calculate the summation in Eq. (4) stochastically by sam- pling perturbers ∣ψ(k) l⟩with probabilities proportional to N(k) l. For the selected perturber, we estimate the energy E(k) land accumulate the contribution toward E(2) m(this is fully described in Sec. III D). J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We begin with some important general points. First, we only wish to avoid the use of three- and four-body RDMs; storing one- and two-body RDMs is always straightforward for current active spaces. We, therefore, use the existing SC-NEVPT2 approach to cal- culate all instances of E(k) land N(k) l, which only require the 1-RDM and 2-RDM. This greatly reduces the sampling task to be performed. Using this rule, in the stochastic approach to be described, we can ignore ccvv,cvv, and ccvcontributions entirely. For cc,cv, and vv, we need to sample E(k) lbut not N(k) l. For the remaining two sets (cand v), both E(k) land N(k) lmust be sampled. This is summarized in Table I. For the algorithm to be presented, the only computational requirement on ∣ϕ(0) m⟩is that overlaps such as ⟨n∣ϕ(0) m⟩can be cal- culated and that the 1-RDM and 2-RDM can be constructed. In this article, we solely take ∣ϕ(0) m⟩from SCI. However, this require- ment is met by other wave functions such as those that can be optimized in orbital-space VMC. These include Jastrow antisym- metric geminal power wave functions43,44and other symmetry- projected Jastrow mean-field wave functions,45which can be accu- rate for strong correlation and permit the efficient calculation of⟨n∣ϕ(0) m⟩(though wave functions used in real-space VMC will not). Whenever generating connections by the application of the Hamiltonian, it should be understood that the heat bath criteria are applied. This was described in the initial presentation of our VMC approach, which we refer to for in-depth description.46This ensures that, for a determinant | n⟩, connections | p⟩are not gener- ated if∣⟨p∣ˆH∣n⟩∣<ϵfor some small threshold ϵ. For results in this article, we always take ϵ= 10−8hartree. This typically reduces both the prefactor and scaling of the resulting algorithm, with a negligible effect on the accuracy. A. The continuous time Monte Carlo algorithm In the following, it is necessary to sample from probability distributions ρn, which take the form ρn=∣⟨n∣ψ⟩∣2 ⟨ψ∣ψ⟩, (8) TABLE I . Table showing which E(k) landN(k) linstances are calculated stochastically and by the traditional deterministic approach. The deterministic approach is taken if only 1- and 2-body RDMs are required, otherwise we use the stochastic approach in order to avoid 3- and 4-body RDMs. Perturber type Energies (E(k) l) Norms ( N(k) l) c Stochastic Stochastic v Stochastic Stochastic cc Stochastic Exact cv Stochastic Exact vv Stochastic Exact ccv Exact Exact cvv Exact Exact ccvv Exact Exactwhere |ψ⟩is some wave function. Typically, in VMC, this would be sampled by the Metropolis–Hastings algorithm.47–49However, this can be quite inefficient when working in a discrete basis of Slater determinants. Instead, we use the Continuous Time Monte Carlo (CTMC) algorithm,50,51which was introduced to VMC recently.46 When applied in other areas, this algorithm is sometimes known as Kinetic Monte Carlo (KMC) or the Bortz–Kalos–Lebowitz (BKL) algorithm. We briefly recap it here: 1. From a determinant | n⟩, calculate r(p←n), r(p←n)=∣⟨p∣ψ⟩ ⟨n∣ψ⟩∣, (9) for all determinants | p⟩connected to | n⟩by a single or double excitation (within the relevant space). 2. Calculate the residence time for | n⟩, defined as tn=1 ∑pr(p←n). (10) This will define the weight of contributions from | n⟩in subse- quent estimators. 3. Select a new determinant | p⟩with probability proportional to r(p←n). After a short burn-in period, iterating this procedure will cor- rectly sample ρn, provided that tnare used as weights for contribu- tions to estimators. We denote the total residence time for a random walk by T=∑ntn. It is worth pointing out that, in CTMC, all moves are accepted and there are no rejections, but this comes at the added cost of having to evaluate all the overlap ratios in Eq. (9). However, this additional cost is mitigated in our VMC algorithm because these overall ratios are obtained when evaluating the local energy. B. Previous SC-NEVPT2(s) algorithm For completeness, we briefly recap the stochastic SC-NEVPT2 approach that we presented previously.41This was a trial implemen- tation that excluded excitations involving core orbitals. The energies E(k) lwere estimated by sampling the numerators and denominators, ⟨ψ(k) l∣HD∣ψ(k) l⟩ ⟨ψs∣ψs⟩=∑ n∣⟨n∣ψs⟩∣2 ⟨ψs∣ψs⟩⟨ψ(k) l∣n⟩ ⟨ψs∣n⟩⟨n∣HD∣ψ(k) l⟩ ⟨n∣ψs⟩, (11) ⟨ψ(k) l∣ψ(k) l⟩ ⟨ψs∣ψs⟩=∑ n∣⟨n∣ψs⟩∣2 ⟨ψs∣ψs⟩∣⟨ψ(k) l∣n⟩∣2 ∣⟨ψs∣n⟩∣2, (12) where |ψs⟩is an appropriately chosen sampling wave function, ∣ψs⟩=∑ k,lc(k) l∣ψ(k) l⟩, (13) for some coefficients c(k) l. Here, all quantities are sampled by a sin- gle random walk that takes place in S(0) 0and its FOIS, using the CTMC algorithm to sample∣⟨n∣ψs⟩∣2 ⟨ψs∣ψs⟩. This approach was success- ful for the test systems considered in Ref. 41 but has drawbacks that can be improved upon. First, the approach requires the user to J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp choose the coefficients c(k) lin the sampling wave function. A suit- able choice is far from obvious, and a suitable prescription can vary widely between systems. The final statistical error varies significantly depending on | ψs⟩. In our previous work, we chose c(k) lrandomly with c(0) 0being an order of magnitude larger than other coefficients. This was found to be sufficient for the test systems considered but is far from optimal. Second, using the CTMC algorithm as described in Sec. III A to sample∣⟨n∣ψs⟩∣2 ⟨ψs∣ψs⟩requires calculating overlaps ⟨p|ψs⟩for all | p⟩ connected to the current walker, | n⟩. In contrast, to calculate the expression Eq. (11) for a given | n⟩requires ⟨n∣HD∣ψ(k) l⟩=∑ p∈S(k) l⟨n∣HD∣p⟩⟨p∣ψ(k) l⟩. (14) Because HDonly acts within the active space, the number of over- laps⟨p∣ψ(k) l⟩to calculate is much smaller. Thus, the CTMC approach previously used does notonly use information already available but requires significant extra computation to calculate the additional overlaps. We instead describe an alternative algorithm that avoids both issues. Here, we first estimate the norms N(k) land then use these to sample S(k) l. Each corresponding E(k) lis then estimated by a random walk entirely within S(k) l. C. Sampling N(k) l We take the general case where ∣ϕ(0) m⟩may not be normalized. The squared norms can be sampled using the following approach: N(k) l=⟨ψ(k) l∣ψ(k) l⟩ ⟨ϕ(0) m∣ϕ(0) m⟩(15) =⟨ϕ(0) m∣ˆHˆP(k) lˆH∣ϕ(0) m⟩ ⟨ϕ(0) m∣ϕ(0) m⟩(16) =∑ n∈S(0) 0∣⟨n∣ϕ(0) m⟩∣2 ⟨ϕ(0) m∣ϕ(0) m⟩⟨n∣ˆHˆP(k) lˆH∣ϕ(0) m⟩ ⟨n∣ϕ(0) m⟩(17) =⟨N(k) l[n]⟩ ρn. (18) Here,ρnis the probability distribution to be sampled by a random walk, ρn=∣⟨n∣ϕ(0) m⟩∣2 ⟨ϕ(0) m∣ϕ(0) m⟩. (19) The determinants selected, | n⟩, are referred to as walkers. We emphasize that this random walk takes places entirely within the CASCI space ( S(0) 0). The quantity N(k) l[n]is defined by N(k) l[n]=⟨n∣ˆHˆP(k) lˆH∣ϕ(0) m⟩ ⟨n∣ϕ(0) m⟩(20)=∑p∈S(k) l⟨n∣ˆH∣p⟩∑r∈S(0) 0⟨p∣ˆH∣r⟩⟨r∣ϕ(0) m⟩ ⟨n∣ϕ(0) m⟩. (21) N(k) l[n]is calculated by the following steps: First, generate all deter- minants | p⟩inS(k) lthat are connected to | n⟩(calculating ⟨n∣ˆH∣p⟩for each). Then, for each | p⟩, generate all connected determinants | r⟩ within S(0) 0(calculating ⟨p∣ˆH∣r⟩and⟨r∣ϕ(0) m⟩for each). In practice, instead of calculating N(k) l[n]for each S(k) lsepa- rately, we accumulate all instances simultaneously, that is, for each walker∣n⟩∈S(0) 0, loop over connected determinants | p⟩in all S(k) lis considered, accumulating contributions to N(k) l[n]for each. The norm and energy of the zeroth-order wave function are sampled in an analogous way during the same random walk. In this article, we take ∣ϕ(0) m⟩from SCI, such that the wave function is nor- malized by construction, and its energy known. Nonetheless, this step is important in general. Walker moves within S(0) 0are made using the continuous time Monte Carlo (CTMC) algorithm, described above. Importantly, each r(p←n) is already constructed in order to obtain ⟨ϕ(0) m∣ϕ(0) m⟩ so that the CTMC algorithm can be performed essentially for free. Note that for every S(k) lsampled, the quantity ⟨p∣ψ(k) l⟩ =⟨p∣ˆH∣ϕ(0) m⟩is calculated for at least one determinant | p⟩inS(k) l. We can, therefore, keep a list of determinants that have the largest value of ⟨p∣ψ(k) l⟩for each S(k) lsector (of the determinants reached). These determinants are used to initialize the walkers when sampling the corresponding E(k) l. As noted in Table I, we only need to sample norms for per- turbers of types cand v. However, we also need to generate initial determinants for cc,cv, and vv. Therefore, there are two parameters that specify the sampling in this step, which we denote Nnorm and Ninit. For the first Nnorm iterations, N(k) lis only sampled for cand v-type perturbers. We then perform Ninititerations in which N(k) lis sampled for all five perturber types ( c,v,cc,cv, and vv). The N(k) l estimates for cc,cv, and vvfrom this step are notused, as we have access to the exact values. Instead, we use the generated initial deter- minants when sampling E(k) lin the next step. These final iterations are more expensive. However, we always take Ninit≪Nnorm, and typically, Ninit= 50 is more than sufficient. D. Sampling E(2)and E(k) l We next consider the sampling of E(2) mitself, as defined in Eq. (4). This is done by sampling terms in this summation with a probability proportional to N(k) l, E(2)=∑ k,l≠01 E(0)−E(k) lN(k) l(22) =⎡⎢⎢⎢⎢⎣∑ k′,l′≠0N(k′) l′⎤⎥⎥⎥⎥⎦×∑ k,l≠01 E(0)−E(k) l⋅N(k) l ∑k′,l′≠0N(k′) l′(23) J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp =⎡⎢⎢⎢⎢⎣∑ k,l≠0N(k) l⎤⎥⎥⎥⎥⎦×⟨1 E(0)−E(k) l⟩ ρ(l,k), (24) whereρ(l,k)=N(k) l ∑k′,l′≠0N(k′) l′. It is straightforward to sample from ρ(l,k) since all N(k) lvalues are stored after the initial stage of the algorithm. We also truncate the summation to only include contri- butions with N(k) l≥10−8as an efficiency improvement, which we do not find to affect the accuracy. For each S(k) lselected, the corresponding E(k) lmust then be estimated. This is achieved by a random walk entirely within S(k) l. Specifically, E(k) l=⟨ψ(k) l∣ˆHD∣ψ(k) l⟩ ⟨ψ(k) l∣ψ(k) l⟩(25) =∑ n∈S(k) l∣⟨n∣ψ(k) l⟩∣2 ⟨ψ(k) l∣ψ(k) l⟩⟨n∣ˆHD∣ψ(k) l⟩ ⟨n∣ψ(k) l⟩(26) =⟨ED L[n]⟩ρn, (27) where ρn=∣⟨n∣ψ(k) l⟩∣2 ⟨ψ(k) l∣ψ(k) l⟩(28) and ED L[n]is the local energy at | n⟩with respect to ˆHD, ED L[n]=⟨n∣ˆHD∣ψ(k) l⟩ ⟨n∣ψ(k) l⟩(29) =∑p∈S(k) l⟨n∣ˆHD∣p⟩∑r∈S(0) 0⟨p∣ˆH∣r⟩⟨r∣ϕ(0) m⟩ ∑r∈S(0) 0⟨n∣ˆH∣r⟩⟨r∣ϕ(0) m⟩. (30) The numerator of this expression is calculated by the following steps: First, generate all connections | p⟩within S(k) l(and calculate each ⟨n∣ˆHD∣p⟩). Then, for each | p⟩, generate all connections | r⟩within S(0) 0(and calculate each ⟨p∣ˆH∣r⟩and⟨r∣ϕ(0) m⟩). Similarly, the denom- inator of this expression is obtained by looping over all connected determinants | r⟩inS(0) 0and calculating ⟨n∣ˆH∣r⟩and⟨r∣ϕ(0) m⟩for each. The distribution ρnis again sampled using the CTMC algo- rithm. All required values of r(p←n) are obtained when ED L[n]is calculated such that this can be performed essentially for free. There are two parameters that define the sampling in this step, which we denote Nenergy and NE(k) l. Here, Nenergy is the number of samples taken from ρ(l,k) (i.e., the number of E(k) lselected), while NE(k) lis the number of samples to estimate each E(k) lselected. How- ever, instead of using a fixed iteration count for all E(k) l, it is often more accurate to use a fixed residence time (see the Appendix). In cases where we use a fixed residence time, we will list both the total residence time used, denoted T, and the average iteration count per E(k) lestimate.The final energy is estimated using an average of 1 /(E(0)−E(k) l) over the selected S(k) l, as in Eq. (24). Because E(k) lis obtained as a random variable, this is a biased estimator. For a given E(k) l, the expectation value of 1 /(E(0)−E(k) l)will have an error that will become larger as the statistical error in E(k) lincreases. Because only a small number of samples are used to estimate each E(k) l, this error can become non-negligible for some challenging problems. This is discussed in more detail in the Appendix, where we show that the bias in each 1 /(E(0)−E(k) l)can be largely corrected by including the following term: Ebias corr.=−var[ˆE(k) l] (E(0) m−E(k) l)3, (31) where var [ˆE(k) l]is the variance of the E(k) lestimate. We include this correction term throughout, unless stated otherwise, with an example given in the Appendix. E. Parallelism The above algorithm can be efficiently performed on large-scale computers. In our current implementation, this is done by running the above steps independently on each process. Each process gener- ates its own N(k) land E(k) lestimates and, ultimately, its own E(2)esti- mate at the end of the simulation. These E(2)values are then averaged to produce the final estimate of the SC-NEVPT2 energy, together with an error estimate. This error estimate is simple to obtain since results from different processes are statistically independent. There is no communication between MPI processes at any point during the simulation. This approach has very good parallel efficiency. The only cause of non-ideal parallel performance is that processes will take varying times to complete all iterations. Note that the sampling parameters defined above ( Nnorm,Ninit, Nenergy , and NE(k) l) are the number of iterations performed on each process. F. Scaling In the following, we denote the number of core, active, and virtual orbitals as nc,na, and nv, respectively. In the algorithm presented, a norm estimate N(k) lis obtained for all ( l,k) for which the heat bath criteria are satisfied. However, only a subset of E(k) lare obtained, as sampled according to the dis- tribution in Eq. (24). We, therefore, consider the scaling to calculate N(k) lforall(l,k) values and E(k) lfor a constant number of ( l,k) samples. Consider the cost to calculate all N(k) l[n], for a given ∣n⟩ ∈S(0) 0. The expression to be evaluated is given in Eq. (21). First, all determinants ∣p⟩∉S(0) 0connected to | n⟩are generated. For per- turbers of types c,v,cc,cv, and vv, the number of valid | p⟩scales asO(n3 anc),O(n3 anv),O(n2 an2 c),O(n2 ancnv), andO(n2 an2 v), respec- tively. For each ∣p⟩∈S(k) l, the cost to generate all connected ∣r⟩∈S(0) 0 then scales as O(n3 a)forc-andv-type perturbers and as O(n2 a)for cc-,cv−, andvv-type perturbers. J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . The expected scaling to sample E(k) lorN(k) lestimates. The scaling pre- sented for E(k) lis for a fixed ( l,k), while for N(k) lthe scaling is given for all ( l,k) of the given perturber type. This assumes that all valid excitations are generated, whereas excitations are actually generated by the heat bath criteria, which are expected to reduce scaling. However, the number of samples required to maintain a constant statistical error will usually increase with system size, increasing the overall scaling. Scaling for real examples is investigated in Sec. IV. Perturber type Energies (E(k) l) Norms ( N(k) l) c O(n7 a) O(n6 anc) v O(n7 a) O(n6 anv) cc O(n6 a) O(n4 an2 c) cv O(n6 a) O(n4 ancnv) vv O(n6 a) O(n4 an2 v) E(k) l[n]is calculated by Eq. (30). For a given ∣n⟩∈S(k) l, the cost to generate all connected ∣p⟩∈S(k) lscales as O(n4 a)for all perturber types. Then, for each ∣p⟩∈S(k) l, the cost to generate all connected ∣r⟩∈S(0) 0scales as O(n3 a)forc-andv-type perturbers and as O(n2 a) forcc-,cv−, andvv-type perturbers. The overall scaling for each perturber type, obtained from the above arguments, is given in Table II. The true scaling will be some- what different to this in practice. First, we do not loop over all con- nected determinants but instead use the heat bath criteria, where connections are not generated if they have a Hamiltonian element below some threshold. This is expected to reduce the overall scal- ing (however, in this article, we use CASSCF orbitals; because these are delocalized, the potential benefits are more limited). Second, the above only gives the scaling to calculate E(k) l[n]and N(k) l[n]for a constant number of samples, | n⟩. In general, the number of samples will increase with system size for a constant statistical error. This increases the overall scaling. It is difficult to write down a generalformula to describe this effect. We instead investigate this through examples in Sec. IV. IV. RESULTS In the following, PySCF52,53is used to generate molecular orbitals via CASSCF and molecular integrals for the subsequent SC- NEVPT2(s) calculations. Heat bath CI (HCI) as implemented in the Dice code is used as the CASSCF solver21,54,55and also to generate the zeroth-order wave function ∣ϕ(0) m⟩for the SC-NEVPT2(s) step. To account for burn-in errors, we discard the initial 50 itera- tions for each CTMC random walk, both for N(k) land E(k) lestima- tion, unless stated otherwise. A. Scaling with the number of virtual orbitals: N 2 As a simple first example, we consider N 2in its ground state at R= 2.5 a 0bond length. The active space is (10e, 8o) with two core orbitals. We then consider calculating the SC-NEVPT2 energy for increasing correlation consistent basis sets from cc-pVDZ (18 virtual orbitals) to aug-cc-pV6Z (368 virtual orbitals). For the norm-sampling stage, we use parameters Nnorm = 900 and Ninit= 100. For the energy sampling stage, we use Nenergy = 10 000 and NE(k) l=100. The results are presented in Table III. The final two columns compare the stochastic SC-NEVPT2 energies to those calculated with Molpro,56which agree within 1 or 2 statistical error bars. All timings presented are wall times. The timing and error results from Table III can be used to assess scaling with respect to the number of virtual orbitals. Based on the theoretical scaling in Table II, and for a fixed number of iterations , one would except the sampling of norms (time tnorm) to asymp- totically scale with the number of virtual orbitals as O(nv). The expected asymptotic scaling to generate initial determinants (time tinit.det. ) isO(n2 v). Sampling a constant number of energies (time tenergy ) should be independent of nv. TABLE III . Scaling of SC-NEVPT2(s) timing and error estimates with basis set size, applied to the ground state of N 2atR= 2.5 a 0. The active space is (10e, 8o). tnormis the time to perform 900 iterations to sample N(k) lforc-andv-type perturbers. tinit. det. is the time to perform 100 iterations to generate initial determinants. tenergy is the time to sample 10 000 values of E(k) l. The final two columns compare the subsequent SC-NEVPT2(s) energy estimates to exact results from Molpro.56 Total energy + 109 (hartree) Basis nv tnorm (s) tinit. det. (s) tenergy (s) Statistical error (hartree) SC-NEVPT2(s) Molpro SC-NEVPT2 cc-pVDZ 18 5.103 1.019 234.110 2.0 ×10−4−0.1857(2) −0.185 43 aug-cc-pVDZ 36 10.482 2.454 238.149 2.2 ×10−4−0.2025(2) −0.202 36 cc-pVTZ 50 15.672 3.989 204.581 3.5 ×10−4−0.2844(4) −0.284 98 aug-cc-pVTZ 82 26.841 8.526 225.526 3.6 ×10−4−0.2946(4) −0.294 33 cc-pVQZ 100 30.979 10.756 221.810 5.2 ×10−4−0.3452(5) −0.345 11 aug-cc-pVQZ 150 52.517 21.578 231.923 5.0 ×10−4−0.3494(5) −0.348 80 cc-pV5Z 172 56.710 25.333 224.181 4.1 ×10−4−0.3680(4) −0.367 52 aug-cc-pV5Z 244 94.875 50.289 257.278 3.0 ×10−4−0.3706(3) −0.369 56 cc-pV6Z 270 99.372 55.366 244.260 4.9 ×10−4−0.3828(5) −0.382 85 aug-cc-pV6Z 368 152.935 101.737 271.484 4.6 ×10−4−0.3838(5) −0.384 06 J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Scaling of the norm-sampling time ( tnorm) against the number of virtual orbitals. The system is N 2atR= 2.5 a 0, with a (10e, 8o) active space. A con- stant number of iterations are performed, Nnorm= 900. The scaling is found to be tnorm∼O(n1.1 v). tenergy is seen to be independent of nv, as expected. Meanwhile, the observed scaling of tnorm isO(n1.1 v), while the observed scaling of tinit. det. isO(n1.5 v)in reasonable agreement with the predicted results. The scaling and fit for tnorm are shown in Fig. 1. It is more challenging to reason about how quickly the statistical error should increase. There are two sources of statis- tical error: first from sampling the norms and second from sam- pling the energies. We usually observe that the majority of statis- tical error comes from the energy sampling step, though this will depend on how the simulation parameters are chosen. In the present case, there is a noticeable increase in the statistical error from cc- pVDZ to cc-pVQZ, but interestingly the error becomes somewhat insensitive to nvbeyond this point. There is an error on each of these error estimates, but they are small enough to not affect this conclusion. B. Scaling with molecule length: Polyacetylene To consider scaling with overall molecule size, we consider trans -polyacetylene molecules with two terminal hydrogen atoms. They take the form C 2nH2n+2. We denote the number of carbonatoms as Nand consider cases from N= 4 to N= 28. The cor- responding number of core, active, and virtual orbitals is given in Table IV. The orbital basis set is 6-31g. This is not large enough for accu- rate quantitative results but sufficient for the present scaling study. Similarly, we take a model geometry, where all bond lengths and angles are fixed. Specifically, single C–C bond lengths are 1.45 Å, double C–C bond lengths are 1.34 Å, and C–H bond lengths are 1.08 Å. All angles are set to 120○. For larger values of N, the CASCI problem becomes infeasible to solve by FCI. Instead, we use selected CI (SCI), specifically the heat bath CI (HCI) method. A constant HCI threshold of ϵ= 5×10−5 hartree is used for each value of N. The number of determinants in the HCI wave function is reported as ndetsin Table IV. The same parameters are used for each simulation: Nnorm = 900, Ninit= 100, Nenergy = 1000, and NE(k) l=100. Simulations were run on 32 cores on two Intel E5-2650 nodes. Each of nc,na, and nvscales linearly with the number of car- bon atoms. Therefore, from Table II, the idealized asymptotic scal- ing for a constant number of iterations is O(N7). If we discard the N= 4 data point (to better investigate the asymptotic scaling), then the observed scaling for the total time ( tnorm +tinit. det. +tenergy ) is O(N6.1). This lower scaling is reasonable, given that the theoreti- cal scaling does not account for excitations ignored by the heat bath criteria. There is also an increase in the final statistical error with molecule size. Interestingly, this error decreases from N= 20 to N= 24; we have checked that this is accurate and not the result of error on the error estimate. However, all other data points follow the expected trend of increasing error. The statistical error decreases with the number of samples ( ns) asn−1/2 s and so decreases with simulation time ( t) ast−1/2. Therefore, a sensible measure of overall computational cost is η=t×σ2, (32) where tis the total time and σis the final error estimate. For the polyacetylene data in Table IV, the values of ηare plotted in Fig. 2, which agree well with a linear regression line on this log–log plot. Excluding the first data point ( N= 4), the overall cost scales roughly asO(N8.2). Although this scaling is steep, it is similar to that of traditional SC-NEVPT2 but with the benefit of not requiring higher- order RDMs. In Sec. IV E, we demonstrate that the method is feasible TABLE IV . Simulation time and statistical error for SC-NEVPT2(s) simulations performed on polyacetylene, as the number of carbon atoms ( N) is increased. nc,na, and nvgive the number of core, active, and virtual orbitals, respectively. A constant number of iterations were performed for each simulation (see the main text for simulation parameters). No. of C atoms ( N) nc na nv ndets tnorm (s) tinit. det. (s) tenergy (s) Statistical error (hartree) 4 9 4 31 20 1.556 1.165 3.469 1.1 ×10−4 8 17 8 59 2458 26.469 29.072 90.156 2.2 ×10−4 12 25 12 87 7.9 ×104208.94 251.76 865.784 3.0 ×10−4 16 33 16 115 4.2 ×1051.134 ×1031.279 ×1035.086 ×1034.8×10−4 20 41 20 143 1.4 ×1064.694 ×1034.832 ×1032.006 ×1046.8×10−4 24 49 24 171 2.5 ×1061.981 ×1041.594 ×1046.733 ×1044.7×10−4 28 57 28 199 3.3 ×1065.615 ×1044.222 ×1041.822 ×1051.1×10−3 J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . A measure of computational cost in SC-NEVPT2(s), plotted against the number of carbon atoms ( N) in polyacetylene molecules. The cost is η=σ×t2, whereσis the statistical error and tis the total simulation time. The cost is seen to scale roughly as O(N8.2). for active spaces with 32 orbitals. Given the favorable parallel effi- ciency, we expect active spaces with more than 40 orbitals to be achievable. Nonetheless, we are investigating alternative approaches to reduce this scaling. C. Effect of error in the reference wave function It is interesting to investigate how the accuracy of the reference wave function affects the final SC-NEVPT2 energy. This is impor- tant in our case since for larger active spaces, we use an approximate HCI wave function as the reference, ∣ϕ(0) m⟩. To do this, we have primarily considered the same trans - polyacetylene (TPA) system, as studied in Sec. IV B, for the case with16 carbon atoms, N= 16. We performed the SC-NEVPT2(s) proce- dure using different HCI wave functions, obtained by varying the HCI threshold, ϵ, which controls the accuracy of the wave function. The exact reference is obtained in the small ϵlimit. For TPA (6-31g) results, the following parameters were used. For the norm sampling step of SC-NEVPT2(s), we take Nnorm = 950 and Ninit= 50. For the energy sampling step, Nenergy is set to 700, except forϵ= 5×10−6hartree where Nenergy = 1000. The total resi- dence time is T= 1.0, except for ϵ= 5×10−6hartree where T= 1.5. We use 20 burn-in iterations for norm and energy sampling steps. To address the concern that results may rely on the very small basis set used, we also obtained results for the same TPA system in the cc-pVDZ basis with two ϵvalues. We also performed a simi- lar analysis for the Fe(II)-porphyrin [Fe(P)] system. The system and basis are identical to those fully described in Sec. IV D. The results forϵ= 10−5hartree are identical to those presented in Sec. IV D. We then performed an additional calculation with ϵ= 3×10−5 hartree. The results are given in Table V. For each ϵvalue, we state the HCI variational energy, E(0), which is the reference energy in the subsequent SC-NEVPT2 calculation. We also state the Epstein–Nesbet perturbative correction within the CAS (“HCI PT2”), obtained by the semi-stochastic HCI (SHCI) algorithm.55 This gives a measure of error in the reference but does not include corrections from the FOIS. We then show the SC-NEVPT2(s) energy estimates E(2)and the final energy estimate, obtained asE(0)+E(2). The final column can be used to assess the sensitivity of the total SC-NEVPT2 energy to E(0). Interestingly, this total energy shows lit- tle variation with ϵ. For TPA (6-31g) with ϵ= 5×10−4hartree, the HCI variational energy is in error by ∼38 mhartree, using only 1.2 ×104determinants in a (16e, 16o) active space. However, the final SC-NEVPT2(s) energy is in error by only ∼3 mhartree. For ϵ= 1 ×10−4hartree, where the reference energy is in error by ∼10 TABLE V . Results performed for trans -polyacetylene (TPA) with 16 carbon atoms (C 16H18) and Fe(II)-porphyrin [Fe(P)] in the5Agstate. We vary the accuracy of the reference wave function, obtained using the HCI method. We then perform SC-NEVPT2(s) using each resulting reference wave function. The final column shows the variation in the total SC-NEVPT2 energy, which is seen to have only weak dependence on the quality of the reference. Even when the reference energy ( E(0)) is in error by ∼38 mhartree, the final SC-NEVPT2 energy is in error by ∼3 mhartree for TPA (6-31g). We also include the HCI PT2 correction within the CAS, obtained by the semi-stochastic HCI approach. Energies (hartree) System ϵ(hartree) ndets HCI variational ( E(0)) HCI PT2 SC-NEVPT2(s) ( E(2)) Total ( E(0)+E(2)) TPA (6-31g)5×10−41.2×104−616.2060 −0.0218 −1.2288(4) −617.4348 (4) 3×10−43.7×104−616.2151 −0.0169 −1.2199(5) −617.4351 (5) 1×10−42.4×105−616.2341 −0.0065 −1.2035(5) −617.4376 (5) 7×10−53.3×105−616.2367 −0.0049 −1.2020(6) −617.4386 (6) 3×10−56.3×105−616.2393 −0.0034 −1.1991(5) −617.4384 (5) 5×10−66.4×106−616.2439 −0.0007 −1.1938(5) −617.4377 (5) TPA (cc-pVDZ)1×10−42.2×105−616.4946 −0.0059 −1.9154(6) −618.4100 (6) 1×10−52.3×106−616.5016 −0.0014 −1.9079(7) −618.4096 (7) Fe(P)3×10−52.0×106−2245.0225 −0.0061 −3.1708(10) −2248.1934 (10) 1×10−59.3×106−2245.0269 −0.0033 −3.1653(6) −2248.1922 (6) J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp mhartree, the total SC-NEVPT2 energy is converged to the exact value within statistical error bars. Similarly, TPA (cc-pVDZ) and Fe(P) results show agreement within error bars after varying ϵ. These results show that the SHCI PT2 energy (which cor- rects E(0)itself) should not be included in the final energy estimate. Instead, SC-NEVPT2 energies can be estimated simply as E(0)+E(2). Clearly, including the SHCI PT2 correction would give energies in a significant error for the results presented. The accuracy of E(0)+E(2)can be partially understood because E(2)is formed as a sum of negative quantities, N(k) l/(E(0)−E(k) l). Therefore, as E(0)becomes less negative (larger ϵ), each contribu- tion in the summation becomes more negative. It is not unreason- able to then expect partial cancellation between errors in E(0)and E(2). Nonetheless, the very accurate nature of cancellation here is perhaps surprising. If this result were general, it would be pow- erful and extremely useful. However, a general statement on the accuracy of this cancellation cannot be made without more test- ing, for example, with several different systems and basis sets, which will be a task for future work. However, these are promising initial results, and they justify the HCI wave functions used in Secs. IV D and IV E. D. Fe(II)-porphyrin Next, we perform calculations of the Fe(II)-porphyrin [Fe(P)] system. This has been an important benchmark system for mul- tireference methods in recent years, in part due to the diffi- culty of identifying the spin state ordering.17,21,57,58Experimen- tal results on Fe(P) and related systems have usually found the ground state to be a triplet state, although these results are obtained either from a polar solvent or the crystal phase.59–62Initial the- oretical studies have predicted a quintet5Agground state, while a triplet ground state is observed with larger or more care- ful active space choices.21,57,58Very recently, it has been sug- gested that the true ground state is a quintet, when geometrical effects are properly considered;63we do not consider such effects here. We focus on a (32e, 29o) active space used in early studies by Li Manni et al.17and subsequently by Smith et al.21This active space consists of 20 C 2 pz, 4 N 2 pz, and 5 Fe 3 dorbitals. We then investigate the effect of dynamic correlation through SC-NEVPT2. In particular, we consider the vertical excitation energy, using the same geometry as Smith et al. , which is given in the supplemen- tary material. This geometry was originally described by Groenhof et al. ,64was optimized for the triplet state, and was also used in a DMRG investigation of this system.65At this fixed geometry, pre- vious results suggest that the ground state is a triplet; for example, this was found to be the case with a larger (44e, 44o)21active space. Lee and co-workers also studied this system recently,66giving a use- ful summary of recent results and using auxiliary-field quantum Monte Carlo (AFQMC) to confirm the triplet ground state. How- ever, for this (32e, 29o) active space, and at the CASSCF level of theory, a5Agground state is observed. It is interesting and valu- able to investigate to what extent SC-NEVPT2 can correct this situation. Although Fe(P) has D4hsymmetry, we use D2hinstead. Using D2hsymmetry labels, we calculate the lowest energy states in boththe5Agand3B1gsectors. Note that the irreducible representation B1gofD2hcorresponds to A 2gand B 2ginD4h. The basis set is cc-pVDZ. We use the same CASSCF orbitals optimized by Smith et al. for their CASSCF study of this system, where HCI was used as the solver. The reference wave function in SC-NEVPT2(s) was also obtained with HCI, using a final threshold ofϵ= 10−5hartree, which resulted in a wave function of ∼107deter- minants for both states. For SC-NEVPT2(s) simulations, parameters used by each process were: for the5Agstate, Nnorm = 950, Ninit= 50, Nenergy = 1500, and T= 0.4 (giving NE(k) l≈72 on average), and for the 3B1gstate, Nnorm = 900, Ninit= 100, Nenergy = 1260, and T= 0.4 (giv- ingNE(k) l≈104 on average), performed with 320 MPI processes for both states. The following orbitals were frozen in the SC-NEVPT2(s) calculation: 20 C 1 s, 20 N 1 s, and 1–3 s, 2–3 pon the Fe atom, 33 orbitals in total. The results are presented in Table VI. Using CASSCF only, the 5Agstate is lower in energy than the3B1gstate by∼31 mhartree. Including the SC-NEVPT2 correction, it is seen that the quintet state remains the ground state; however, the energy gap is lowered by approximately 19 mhartree, suggesting an improved result overall. Note that the CASSCF energy is in error by approximately +5 mhartree for the5Agstate and by ∼+9 mhartree for the3B1gstate due to the finite value of ϵused in HCI, although correcting for this does not change our conclusion significantly. It would be simple to improve this by using a smaller value of ϵ, which only has a small effect on the SC-NEVPT2(s) simulation time. This is because coef- ficients in the reference wave function are obtained by a hash table lookup, the time for which has very weak scaling with the number of determinants. Our results show that including dynamic correlation through SC-NEVPT2 does noticeably improve the predicted energy gap in this system, but that the expected ordering only occurs with a larger active space. In particular, including the set of 5 Fe 4 dorbitals together with 10 σbonds between Fe and N atoms (1 Fe 4 px, 1 Fe 4 py, 4 N 2 px, and 4 N 2 py) results in a (44e, 44o) active space,21,65which gives a triplet ground state. Li Manni and Alavi have also studied a separate model of Fe(P), where C βH groups are replaced by hydro- gen atoms. With this, they also predict a triplet ground state with a more compact (32e, 34o) active space, which also includes the Fe 4 d orbitals, and part of the Fe–N σmanifold.57Combined, these results highlight the importance of appropriately choosing the active space in such systems. TABLE VI . Energies for two low-lying states of Fe(II)-porphyrin, obtained with CASSCF and SC-NEVPT2(s), using a common geometry for both states. The (32e, 29o) active space of Li Manni et al.17was used. Irreducible representation labels here refer to the D2hpoint group, which was used for all calculations. Energies (hartree) State CASSCF SC-NEVPT2(s) Total 5Ag −2245.0269 −3.1653(6) −2248.1922 (6) 3B1g −2244.9957 −3.1844(7) −2248.1800 (7) ΔE 0.0312 −0.0190(9) 0.0122(9) J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp E.[Cu 2O2]2+ As a final example, we consider the [Cu 2O2]2+molecule. In particular, we study the isomerization between bis( μ-oxo) andμ-η2: η2-peroxo isomers. The model and process, in particular, when com- bined with appropriate ligands, have an important role as an active site for O 2activation by enzymes such as tyrosinase. Given the pres- ence of transition metals, it is expected that the treatment of static correlation may be important, and it has further been suggested that a balanced treatment of static and dynamic is required for accurate results. Moreover, existing benchmarks are available from previous computational studies,67,68making this a sensible test system. We describe the isomerization process using the same geome- tries of Cramer et al.67In this, the Cu–Cu distance is equal to 2.8 + 0.8 FÅ, while the O–O distance is equal to 2.3 −0.9FÅ. Here, Fis a parameter, which varies from 0 to 1. F= 0 indicates the bis( μ-oxo) geometry, and F= 1 indicates the μ-η2:η2-peroxo geometry. We use the ANO-RCC-VQZP basis set,69,70which corresponds to Cu:[21 s15p10d6f4g2h/7s6p4d3f2g1h] and O:[14 s9p4d3f2g/4s 3p2d2f1g] contractions. This is slightly different to the basis used in other studies such as that by Yanai et al.68 We take the same (28e, 32o) active space of Yanai et al. , consisting of all Cu 3 dand 4 dorbitals and all O 2 pand 3 porbitals. CASSCF orbitals are obtained with HCI using a final thresh- old ofϵ= 10−4hartree. We then use a tighter threshold of ϵ= 2×10−5hartree to generate the reference wave function for SC- NEVPT2(s). This results in HCI wave functions with between 2.0 ×107and 2.6 ×107determinants, depending on F. We then perform SC-NEVPT2(s), using norm parameters Nnorm = 450 and Ninit= 50. The number of energy samples, Nenergy , is between 2000 and 2100, and simulations were run with either 320 or 360 processes, depend- ing on the value of F. The total residence time Twas set to 0.4, which gave NE(k) lbetween 50 and 55 on average (in addition to 50 burn-in iterations). The results are given in Table VII and plotted in Fig. 3. We also include results from previous studies for comparison. In partic- ular, CAS(16,14), CR-CCSD(TQ), and CR-CCSD(TQ) Lresults were TABLE VII . Energies (in kcal mol−1) from various methods, including SC- NEVPT2(s), for the isomerization of [Cu2O2]2+between bis( μ-oxo) andμ-η2:η2- peroxo isomers. Energies are relative to the μ-η2:η2-peroxo isomer ( F= 1.0). Note that we use a different basis set to these two studies. F Method 0 0.2 0.4 0.6 0.8 1 HCI-SCF 22.4 14.3 8.2 3.7 1.0 0 SC-NEVPT2(s) 41.3 (8)33.5(9)26.3(9)19.9(8)9.3(9)0 CAS(16,14)a0.2−7.2−12.7−16.3−14.0 0 CR-CCSD(TQ)a35.1 26.7 18.9 10.7 3.1 0 CR-CCSD(TQ) La38.5 28.8 20.0 11.4 3.6 0 DMRG-CIb−12.8−20.9−21.5−16.7−10.0 0 DMRG-SCFb26.4 17.9 11.0 5.1 1.1 0 DMRG-SC-CTSDb37.4 29.0 22.0 14.4 6.1 0 aResults from Ref. 67. bResults from Ref. 68. FIG. 3 . Energies for the isomerization of [Cu 2O2]2+, relative to the μ-η2:η2- peroxo isomer ( F= 1.0). Data plotted are the same as in Table VII. Results plotted in black are from Ref. 67. Results plotted in green are from Ref. 68. taken from the study of Cramer et al. ,67and DMRG-CI, DMRG- SCF, and DMRG-SC-CTSD results were taken from the study of Yanai et al.68Our CASSCF results, obtained using HCI as a solver, are labeled “HCI-SCF.” It is known to be difficult to obtain the cor- rect isomerization profile for this system. Too small an active space leads to an unphysical minimum. HCI-SCF results show that the more substantial (28e, 32o) active space removes this minimum, as previously found by Yanai using DMRG-SCF. More accurate results are obtained when dynamical correlation is included. Our SC-NEVPT2(s) results are approximately in agreement with exist- ing results. We find slightly larger relative energies than previous results. However, we use a larger basis set, so it is perhaps expected that results will not be identical. Cramer et al. also use a pseudopo- tential for Cu atoms, while we freeze core electrons. Overall, these results show reasonable agreement and demonstrate the usefulness of this approach for a significant active space. V. CONCLUSION In this work, we have developed a stochastic approach to performing strongly contracted NEVPT2. This method reproduces exact SC-NEVPT2 energies within statistical error bars but avoids the prohibitive cost of constructing and storing three- and four-body RDMs. The method has low scaling with the number of virtual orbitals, nv. The cost to sample a fixed number of perturber energies, E(k) l, is independent of nv, while the increase in associated statistical error is low for small basis sets, plateauing off for larger basis sets. The scaling with the number of active space orbitals is more restrictive. In particular, we investigated the scaling of the overall computational cost with molecular size, N, for polyacetylene molecules. In this case, the number of core, active, and virtual orbitals increases linearly with N, and the total cost (after accounting for the increase in statistical error) was found to scale roughly as O(N8.2). We also investigated the sensitivity of the final SC-NEVPT2 energy to a reference wave function of varying accuracy. Interest- ingly, we found final energies to remain accurate, with relatively weak dependence on the quality of the reference energy. If this result J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp were general, then it would be very powerful. We intend to study this for further systems to investigate this possibility. The method was applied to example systems where multi- reference behavior is expected to be important: Fe(II)-porphyrin with a (32e, 29o) active space and [Cu 2O2]2+with a (28e, 32o) active space. The method was successfully applied to these large active spaces, raising the possibility of obtaining SC-NEVPT2 results, with- out approximations, in larger active spaces than previously con- sidered. These calculations were performed with moderate com- puter resources. However, the approach has good parallel efficiency such that it could be used in a straightforward manner on much larger parallel computers, as have been used in many QMC studies previously. There are several areas in which this method could be devel- oped. First, it will be important to develop SC-NEVPT2(s) to work with other types of wave functions, in particular orbital-space VMC wave functions.45Such wave functions can be well suited to strong correlation and with favorable scaling.43,44,71Because only wave function overlaps ( ⟨n∣ϕ(0) m⟩) and 1- and 2-RDMs are needed, this should be a straightforward task. Our code already supports the optimization of such wave functions, including the calculation of the required overlaps and RDMs. Second, we are keen to investi- gate approaches to reduce the scaling, in particular, with respect to the active space size. With these developments, we hope that this may be a robust method to perform NEVPT2 with active spaces of 40–50 orbitals, which we believe would be valuable in the general task of performing strongly correlated electronic structure calculations. SUPPLEMENTARY MATERIAL The supplementary material includes the geometry of the Fe(II)-porphyrin model studied in this article. This geometry was taken from Ref. 64. The geometries for all other systems are stated in the article. ACKNOWLEDGMENTS N.S.B. is grateful to St John’s College, Cambridge for funding and supporting this work through a Research Fellowship. S.S. and A.M. were supported by NSF through Grant No. CHE-1800584. S.S. was also partly supported through the Sloan research fellow- ship. This study made use of the CSD3 Peta4-Skylake CPU cluster at the University of Cambridge and the Summit supercomputer at CU Boulder. APPENDIX: POTENTIAL BIASES Because a large number of energies E(k) lmust be sampled, each with its own independent random walk, only a limited number of samples can be used to estimate each E(k) l. This is different to the typical case in VMC, where a single energy is to be estimated by a long random walk (typically by the Metropolis algorithm). Sta- tistical biases in QMC will become larger as the number of sam- ples becomes smaller. Therefore, for the small number of samples used, there are some potential biases to consider carefully for the algorithm presented.1. Burn-in Each random walk with the CTMC algorithm has a burn-in period. In practice, we have found that results are essentially iden- tical regardless of whether burn-in iterations are discarded or not, suggesting this to be a negligible effect here. Nonetheless, it is sen- sible to account for this possibility where affordable. We, therefore, typically discard the first 50 iterations for each random walk, both inS(0) 0(for N(k) lestimation) and in each S(k) lsampled (for E(k) l estimation). 2. CTMC estimates of E(k) l Some care is required in using the CTMC algorithm. In CTMC, a sample from a given determinant | n⟩is weighted by a correspond- ing residence time, defined as tn=1 ∑pr(p←n). The final point estimate of E(k) lis obtained by ˆE(k) l=∑ntnED L[n] ∑ntn. (A1) Using a constant number of iterations for each E(k) lleads to small statistical bias, which becomes noticeable for very large systems. Instead, each E(k) lshould be estimated with a constant total res- idence time, T=∑ntn. We, therefore, run CTMC random walks until some fixed threshold time is reached, at which point the walk is ended. This is found to resolve all such issues with biases in E(k) l estimates. 3. Bias in (E(0) m−E(k) l)–1estimator Contributions to E(2) meach take the form (E(0) m−E(k) l)−1, where each E(k) lis stochastically sampled. Even if the estimator for E(k) lis unbiased, the final result will be biased because E [1 X]≠1 E[X]. Estima- tors of this type are very common in QMC, and associated biases are typically negligible. In the current case, however, the bias is larger because the number of samples used to estimate each E(k) lis very small (∼50–100) for the calculations presented in this work. To see the issue more clearly, we can consider a Taylor expan- sion of(E(0) m−ˆE(k) l)−1, where ˆE(k) lis a point estimate of E(k) l. We may write ˆE(k) l=E(k) l+δ, whereδdenotes the error. Assuming that ˆE(k) lis unbiased, we have that E[ δ] = 0. One can then write 1 E(0) m−ˆE(k) l=1 E(0) m−E(k) l−δ(A2) =1 (E(0) m−E(k) l)[1−δ E(0) m−E(k) l](A3) =1 E(0) m−E(k) l⎡⎢⎢⎢⎢⎣1 +δ E(0) m−E(k) l+δ2 (E(0) m−E(k) l)2+O(δ3)⎤⎥⎥⎥⎥⎦. (A4) J. Chem. Phys. 153, 164120 (2020); doi: 10.1063/5.0023353 153, 164120-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We can use this to look at the expected value of (E(0) m−ˆE(k) l)−1, E⎡⎢⎢⎢⎢⎣1 E(0) m−ˆE(k) l⎤⎥⎥⎥⎥⎦=1 E(0) m−E(k) l⎡⎢⎢⎢⎢⎣1 +E[δ2] (E(0) m−E(k) l)2+O(δ3)⎤⎥⎥⎥⎥⎦(A5) =1 E(0) m−E(k) l+var[ˆE(k) l] (E(0) m−E(k) l)3+O(δ3). (A6) Therefore, it can be seen that the bias will increase as the energy difference E(0) m−E(k) lbecomes smaller and as the estimate of E(k) l becomes more noisy. The above equation gives an expression to correct much of the bias, Ebias corr.=−var[ˆE(k) l] (E(0) m−E(k) l)3. (A7) Using this expression requires an estimate of the variance of ˆE(k) l. If the Metropolis algorithm were used, the standard estimator for the variance of the mean would be used, ˆσ2 ˆE(k) l=1 Ns(Ns−1)∑ n(ED L[n]−ED L)2, (A8) where Nsis the number of samples and ED Lis the sample mean. Instead, we use the CTMC algorithm, where the estimator for E(k) l is formed as a weighted sum, as in Eq. (A1). An estimator for the variance of a weighted sum is more complicated, and there is no gen- erally accepted formula for all applications. We have tested several estimators and found that the following formula72,73is very accurate for our case, which we, therefore, use: ˆσ2 ˆE(k) l=1 TNs (Ns−1)[∑ n(tnED L[n]−TED L)2(A9) −2ED L∑ n(tn−T)(tnED L[n]−TED L) (A10) +ED L2 ∑ n(tn−T)2]. (A11) Here, T=∑ntnis the total residence time and tnact as weights in the estimator for Ek l, as in Eq. (A1). ED L[n]is the local energy with respect to the Dyall Hamiltonian, as in Eq. (29). In addition, samples ED L[n]are serially correlated, and we account for this by using an automated reblocking procedure.74 4. Example: N 2cc-pVDZ As a simple example to demonstrate these concepts, in particu- lar, the estimation of σ2 ˆE(k) land the bias correction term, we consider N2in a cc-pVDZ basis set. This is the same example considered in Sec. IV A, using a (10e, 8o) active space and two core orbitals. We consider the estimation of a single perturber energy, E(k) l, of type vv, involving the two virtual orbitals that are lowest in energy. For this small example, it is possible to enumerate all determinants inS(k) land calculate the exact E(k) l. By repeating the stochastic esti- mation of E(k) la large number of times, we can investigate the aboveeffects. In particular, we repeat this estimation of E(k) l100 000 times so that we can accurately construct the distribution function and investigate the true variance and bias. For the perturber in question, the exact result is E(k) l =−105.258 96 hartree. Performing the CTMC estimation of E(k) l, exactly as in the SC-NEVPT2(s) algorithm, and then averaging over the 100 000 repeated estimates give E(k) l=−105.258 82 (15)hartree so that the method is unbiased within error bars. An accurate esti- mate of the variance (obtained directly from the constructed prob- ability distribution) is Var [ˆE(k) l]=0.002 29 hartree2, while the estimate from Eq. (A11) is σ2 ˆE(k) l=0.002 33 hartree2. Similarly, the difference between the exact and estimated values of(E(0) m−E(k) l)−1is 3.3(11) ×10−5hartree−1, indicating the possibil- ity of a small bias. Including the above bias correction changes this discrepancy to −1.1(11) ×10−5hartree−1, suggesting an improve- ment. In this case, the correction is extremely small so could be ignored. For non-trivial problems, this correction needs more care- ful consideration. For the [Cu 2O2]2+examples in Sec. IV E, the bias correction in Eq. (A7) is of size ≈0.6 mhartree for each value of F. We, therefore, include this correction term in all results presented in this article. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1H. J. Werner and P. J. Knowles, J. Chem. Phys. 89, 5803 (1988). 2P. J. Knowles and H.-J. Werner, Theor. Chim. Acta 84, 95 (1992). 3K. Andersson, P. A. Malmqvist, B. O. Roos, A. J. Sadlej, and K. Wolinski, J. Phys. Chem. 94, 5483 (1990). 4C. Angeli, R. Cimiraglia, S. Evangelisti, T. Leininger, and J.-P. Malrieu, J. Chem. Phys. 114, 10252 (2001). 5C. Angeli, R. Cimiraglia, and J.-P. Malrieu, J. Chem. Phys. 117, 9138 (2002). 6R. J. Bartlett and M. Musiał, Rev. Mod. Phys. 79, 291 (2007). 7F. A. Evangelista, J. Chem. Phys. 149, 030901 (2018). 8U. S. Mahapatra, B. Datta, and D. Mukherjee, J. Chem. Phys. 110, 6171 (1999). 9J. Meller, J. P. Malrieu, and R. Caballol, J. Chem. Phys. 104, 4068 (1996). 10J. 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5.0023175.pdf

Appl. Phys. Lett. 117, 133103 (2020); https://doi.org/10.1063/5.0023175 117, 133103 © 2020 Author(s).Orbital-fluctuation freezing and magnetic- nonmagnetic phase transition in α-TiBr3 Cite as: Appl. Phys. Lett. 117, 133103 (2020); https://doi.org/10.1063/5.0023175 Submitted: 27 July 2020 . Accepted: 05 September 2020 . Published Online: 28 September 2020 Shenghai Pei , Jiangke Tang , Cai Liu , Jia-Wei Mei , Zenglong Guo , Bingbing Lyu , Naipeng Zhang , Qiaoling Huang , Dapeng Yu , Li Huang , Junhao Lin , Le Wang , and Mingyuan Huang ARTICLES YOU MAY BE INTERESTED IN Spin-reorientation transition induced magnetic skyrmion in Nd 2Fe14B magnet Applied Physics Letters 117, 132402 (2020); https://doi.org/10.1063/5.0022270 Temporal acoustic wave computational metamaterials Applied Physics Letters 117, 131902 (2020); https://doi.org/10.1063/5.0018758 Electrostatic-doping-controlled phase separation in electron-doped manganites Applied Physics Letters 117, 132405 (2020); https://doi.org/10.1063/5.0024431Orbital-fluctuation freezing and magnetic-nonmagnetic phase transition ina-TiBr 3 Cite as: Appl. Phys. Lett. 117, 133103 (2020); doi: 10.1063/5.0023175 Submitted: 27 July 2020 .Accepted: 5 September 2020 . Published Online: 28 September 2020 Shenghai Pei,1,2Jiangke Tang,1,2CaiLiu,3Jia-Wei Mei,2,3Zenglong Guo,2Bingbing Lyu,2Naipeng Zhang,2 Qiaoling Huang,2Dapeng Yu,2,3LiHuang,2Junhao Lin,2 LeWang,3,a)and Mingyuan Huang2,a) AFFILIATIONS 1School of Physics, Harbin Institute of Technology, Harbin 150001, China 2Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China 3Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China a)Authors to whom correspondence should be addressed :wangl36@sustech.edu.cn andhuangmy@sustech.edu.cn ABSTRACT We present a detailed study on the structural phase transition in a-TiBr 3, which is deeply connected with the lattice and orbital degree of freedoms. A chemical vapor transport method is adopted to synthesize the a-TiBr 3single crystal samples, and the structural phase transition at about 180 K is characterized by x-ray diffraction (XRD), magnetic susceptibility, and specific heat capacity. To further the understandingin the physical nature of this phase transition, a systematic Raman spectroscopic study is performed on a-TiBr 3crystals. With temperature decreasing, a large frequency blue shift and peak width narrowing are observed in the vibrational mode associated with Ti in-plane relative movement, which indicates the formation of Ti–Ti bonding and orbital-fluctuation freezing at low temperatures. These results are fully consistent with magnetic–nonmagnetic phase transition resolved by the measurement of magnetic susceptibility and lattice changes by XRD. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023175 Since two-dimensional (2D) crystals with intrinsic magnetism were demonstrated in monolayer CrI 3and Cr 2Ge2Te6for the first time in 2017,1,2searching and exploring magnetic materials in 2D materials have quickly become a research focus and greatly stimulatedresearch interest in studying magnetism in MX 3materials,3–8where M represents 3 dor 4dtransition metals and X denotes halogen elements. For example, few layer CrI 3with a weak interlayer antiferromagnetic interaction, which can be manipulated by the external electric field,magnetic field, and pressure, displays great potential in spintronicdevices. 1,6,9–15Among MX 3magnetic materials, a-TiX 3are 2D mag- netic semiconductors with a honeycomb structure in the ab plane, of which a-TiCl 3is famous for heterogeneous Ziegler–Natta catalysis16–19 and its physical properties have been reported.19–24At about 210 K, a-TiCl 3undergoes a magnetic–nonmagnetic phase transition and the Jahn–Teller effect is considered to be the main reason for the phasetransition. 21Goodenough proposed the formation of the bond between Ti3þions to explain the magnetic collapse at low temperatures.25 Monolayer a-TiBr 3has the same D3dstructure as a-TiCl 3,a s s h o w ni nt h el e f tp a n e lo f Fig. 1(a) ,w h e r eT i3þions form ahoneycomb lattice, and a Ti with the six nearest neighbor Br forms a TiBr 6octahedral frame.26,27At room temperature, the bulk a-TiBr 3 crystal adopts a rhombohedral R/C223 structure, and the x-ray diffraction (XRD) study indicates phase transition from a high-temperature R/C223t o a low-temperature P/C221 structure.27The measurement of magnetic sus- ceptibility resolves a magnetic phase transition at about 185 K.24 However, the physical nature about this phase transition has not beenclearly addressed yet, probably due to the difficult preparation methodand easy-deteriorating feature in the air. With the development of 2Dmaterials, the sample-protection technique by using h-BN encapsula-tion makes it more convenient to study the air-sensitive 2D materials. Experimentally, Raman spectroscopy is a widely used method to study structural phase transitions. In this Letter, we prepare the high-quality a-TiBr 3crystals and study the basic physical properties by XRD, magnetic susceptibility, and specific heat capacity. The structural phase transition at about180 K is clearly observed. In order to understand the physical naturein depth, we perform Raman scattering measurements on a-TiBr 3thin films and analyze the changes of vibrational modes. With the decrease Appl. Phys. Lett. 117, 133103 (2020); doi: 10.1063/5.0023175 117, 133103-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplin temperature, the phase transition is signaled by the dramatic blue shift and narrowing of the full width at half maximum (FWHM) of the peak at about 253 cm/C01, which is associated with the Ti in-plane relative movement. The intensity difference of AgandBgmodes at low temperatures indicates the C3symmetry breaking. By the density func- tional calculations (DFT) and theoretical analysis, we find that theorbital-fluctuation effect leads to the elongation distortion at high temperatures, while below 180 K, the Ti–Ti bonding freezes the orbital-fluctuation effect, which results in the width narrowing of theTi in-plane Raman mode and magnetic collapse and forms the orbitalordering. As a consequence of orbital ordering, the lattice shrinks in the ab plane, and so the Ti Raman mode experiences a significant blue shift. Due to the Poisson effect, the interlayer spacing dis elongated, which is detected by XRD. The preparation methods of a-TiBr 3crystals reported before are very complicated.26–28In this paper, we adopt a chemical vapor trans- port (CVT) method to prepare the high-quality a-TiBr 3crystals (details in the supplementary material ). The polycrystal powder XRD data collected by grounded single crystals are not good due to thequick degeneration and good ductility of sample. Therefore, the temperature-dependent XRD measurement on a tape-protected flat single crystal is carried out and the results are shown in Figs. 1(d) , 1(e), and S1 in the supplementary material . By comparing the temper- ature evolution for (003) and (0012) peaks of a-TiBr 3with the (111) peak for the copper holder with temperature, the discontinuous evolu-tion of (003) and (0012) peaks is identified at about 180 K. The inter- layer spacing d, calculated using Bragg’s equation with the (0012) peakpositions, is plotted as a function of temperature in Fig. 1(f) for both warming and cooling processes. The sharp change of dreveals the structural phase transition at 180 K, which is further confirmed by thetemperature-dependent heat capacity C pwith a k-shape anomaly around 178 K [ Fig. 2(c) ]. The transition temperature is nearly field- independent. To characterize the magnetic property of a-TiBr 3,t h em e a s u r e - ment of magnetic susceptibility is performed, and the result is dis- played in Fig. 1(b) . The sharp change near 180 K is a direct evidence of magnetic phase transition, which is in agreement with the results mea-sured in 1987. 24At low temperatures, the behavior of the Curie tail probably originates from defects or impurities, which is also seen in a-MoCl 3.7We believe that the main defects are the vacancies of halo- gen elements in a-TiBr 3. By fitting the data in the low temperature range with the Curie–Weiss relation, the contributions from defects or i m p u r i t i e sa r er e m o v e d ,a n dt h eb l u el i n ei n Fig. 1(b) denotes the true magnetic susceptibility of a-TiBr 3. Similar to a-TiCl 3, the high temper- ature phase of a-TiBr 3is paramagnetic with localized magnetic moments coupled with each other antiferromagnetically,22,24while the flat curve below 150 K indicates nonmagnetic nature after the structure phase transition. Raman spectroscopy is a powerful tool to study structural phase transitions, and the temperature dependent Raman spectra are mea- sured in a h-BN encapsulated a-TiBr 3sample with a thickness of about 50 nm and displayed in Fig. 2(a) . Here, the sample temperatures are carefully calibrated by the intensity ratio between the anti-Stokes and Stokes Raman modes29,30(see details in the supplementary FIG. 1. (a) Lattice structures of monolayer a-TiBr 3at high temperatures ( R/C223) and low temperatures ( P/C221). (b) The magnetic susceptibility of a-TiBr 3(blue curve) is obtained by subtracting the defect contribution (red curve) from the raw data (black curve). (c) Specific heat of a-TiBr 3; the inset shows the specific heat near the phase transition at 0 and 6 T. [(d) and (e)] Temperature evolution of the (003) peak (d) and the (0012) peak (e) in the XRD pattern. K a1 and K a2 are diffraction peaks excited by characteristic spectral lines of the copper target. (f) The interlayer spacing d calculated using the positions of the (0 0 12) peak.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 133103 (2020); doi: 10.1063/5.0023175 117, 133103-2 Published under license by AIP Publishingmaterial ). A dramatic change of the Raman spectrum can be identified at around 170 K, which is associated with the structure phase transi- tion observed in the XRD, magnetic susceptibility, and heat capacitymeasurements. The minor temperature difference may come from the different sample thicknesses we used, where the thin films are used in Raman, and bulk samples are used in other measurements. At 35 K, seven Raman modes are observed at 64, 92, 109, 133, 181, 245, and 274 cm /C01, and labeled as P1–7 in Fig. 2(a) , respectively. At about 170 K, P3 disappears, P4 shows an abrupt red shift, P5 experiences a significant intensity drop, and P6 and P7 evolve into a broad peak at about 253 cm/C01. In addition, there are several peaks above 300 cm/C01, a ss h o w ni nF i g .S 2i nt h e supplementary material ,w h i c ha r ea s s i g n e d as multi-phonon modes.31–35For example, the mode at 366 cm/C01is from P2 þP7 and the mode at 424 cm/C01is from P5 þP6. To obtain the symmetry of the Raman modes, we also carry out the polarized Raman spectrum measurement, and the Raman spectra from the parallel (XX) and perpendicular (XY) channels at 35 and300 K are displayed in Fig. 2(b) . At room temperature, a-TiBr 3belongs to the R/C223 space group and the irreducible representation of atomic dis- placement at the Cpoint is Copt¼4Agþ4Egþ3Auþ3Eu,36and so the Raman active modes are CR¼4Agþ4Eg.A ss h o w ni n Fig. 2(b) , the peaks at 91 and 175 cm/C01are the Agmode and the peaks at 62, 125 and 253 cm/C01are the Egmode. Other modes are unresolvable due to the small scattering cross section. We also determine the eigenvec- tors of Raman modes with the help of the first-principles calculations,and the calculated frequencies and atomic displacements of all Raman active modes are listed in Fig. S3 in the supplementary material .Among the five observed Raman modes, the broad mode at 253 cm /C01 is related to Ti in-plane relative movement as shown in Fig. 3(a) and other four modes are dominated by vibrations of Br. With the temper-ature decreasing, all doubly degenerated E gmodes evolve into modes (labeled as AgþBg) with different intensities in the XX and XY chan- nels, but the frequencies from both channels are indistinguishable.The lift of degeneracy results from the C 3symmetry breaking, and the space group of a-TiBr 3changes into P/C221 at low temperatures. From the polarized Raman measurements, we notice that the broad mode at 253 cm/C01at room temperature evolves into P7 at low temperatures, while P6 loses its intensity as P3 in the high temperaturephase. Obviously, the Ti in-plane Raman mode displays the most dra-matic change during the structural phase transition among all of theRaman modes: with temperature decreasing, the FWHM narrows a lotand its frequency shows about 20 cm /C01blue shift, which indicates that the structural phase transition is deeply connected with Ti3þ.T h e enlarged Raman spectra of the Ti in-plane Raman mode at differenttemperatures are plotted in Fig. 3(b) , and the broken line denotes its frequency shift. The frequency and FWHM of the Ti in-plane Ramanmode are extracted by Lorentzian fitting and plotted as a function oftemperature in Fig. 3(c) along with the frequency of P4, and the struc- tural phase transition can be clearly resolved. In order to understand the behavior of the Raman spectrum of a- TiBr 3, its electronic structure needs to be considered. As is well known, the spin–orbit coupling (SOC) strength is much smaller than the crys-tal field splitting in Ti-based compounds, and so the SOC effect isneglected. As shown in Fig. 4(b) ,T i 3þhas the electronic configuration of [Ar]3 d1with fivefold degenerated dlevels, which split into t2gandeg levels in the octahedral crystal field, and the single electron occupies the triply degenerated t2glevels. According to the Jahn–Teller theorem, FIG. 2. (a) Evolution of the Raman spectra of a-TiBr 3at different temperatures. (b) Polarized Raman spectra of a-TiBr 3at 35 and 300 K. FIG. 3. (a) The atomic displacement of the Ti in-plane relative movement mode (P7). (b) Evolution of the Raman spectra at different temperatures between 220and 300 cm/C01. (c) Temperature dependence of the frequencies of P4 and P7, and the full width at half maximum (FWHM) of P7.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 133103 (2020); doi: 10.1063/5.0023175 117, 133103-3 Published under license by AIP Publishingthe partially filled t2gshell is unstable (except the half-filled), and the small distortions will occur with the symmetry reduction and degener-acy lift. The orbital degree of freedom can couple with the lattice, and this phenomenon of orbital fluctuation is similar to the dynamic Jahn–Teller effect, which has been found in many Ti-based com-pounds. 37,38As a consequence, the TiBr 6octahedra are elongated along the [111] direction as shown in Fig. 4(a) .27A similar distortion has been found in a-TiCl 3, while the difference is the distortion origi- nated from contraction of TiCl 6octahedra in a-TiCl 3.18 By considering the electronic states of Ti3þas shown in Fig. 4(b) , the 3 d1electron can hop between three t2glevels at high temperatures, which leads to the abnormally large FWHM of the Ti in-plane Raman mode as shown in Fig. 3(b) . With temperature decreasing to below 180 K, the abrupt narrowing of the FWHM of the Ti in-plane Ramanmode indicates that the orbital degree of freedom of t 2gis fully quenched and the Ti–Ti bonding between the two nearest Ti3þwith the shortened distance is formed, as shown in the right panel of Fig. 1(a). A similar phenomenon has been also detected in a-TiCl 3and MoCl 3at low temperatures and a-RuCl 3at high pressures.7,18,39This process of orbital ordering freezes the orbital-fluctuation effect, liftsthe degeneracy of t 2glevels, and breaks the C3symmetry in a-TiBr 3. As a consequence of the orbital-fluctuation freezing, the lattice of a- TiBr 3shrinks in the abplane as shown in Fig. 4(c) , which results in the blue shift of the Ti in-plane mode and P4. By considering thePoisson effect, the lattice expands in the out-of-plane direction asshown in Fig. 4(d) , which is consistent with our XRD results as shown inFig. 1(f) . Figures 4(e) and4(f)are the schematic diagrams of the t 2gorbi- tals. By considering the orbital overlapping, only the hopping between jxy>is relevant, which is the origin of the Ti–Ti bonding. Molecular orbital theory provides a consistent explanation for rearranging of energy levels via metal–metal bonding, which successfully explains the Mo–Mo dimerization in edge-sharing bioctahedral dimolybdenum (III) molecules.40According to the molecular orbital theory, the t2glevels of the two nearest neighbor Ti3þform six levels as r/C3,p/C3d/C3,d,p,a n d r, where we rank them from the highest to the lowest energy, of which the lowest rlevel denotes the overlap of the jxy>levels from two neighbor- ing Ti3þ. As a result, two 3 d1electrons with the opposite spin occupy therlevel at low temperature, which makes the magnetism disappear. In conclusion, Raman spectroscopy is used to study the structural phase transition in a-TiBr 3at around 180 K. The Raman mode associ- ated with the Ti in-plane relative movement displays an abnormally large FWHM in the high temperature phase, which originates from the orbital-fluctuation effect. The structural phase transition is signaled by the sudden FWHM narrowing and significant blue shift of the Ti Raman mode, which results from the disappearance of the orbital degree of freedom and the formation of the Ti–Ti bonding. Thisprocess of orbital ordering freezes the orbital-fluctuation effect, lifts the degeneracy of t 2glevels of Ti3þ, and breaks the C3symmetry in a-TiBr 3. With the orbital-fluctuation freezing, the lattice of a-TiBr 3 contracts in the abplane and expands in the out-of-plane direction, which is fully consistent with our results from Raman and XRD measurements. Furthermore, the Ti–Ti bonding leads to magnetic collapse, which agrees with the results from magnetic susceptibility measurements. Our work demonstrates that Raman spectroscopy is an effective tool to study the orbital-fluctuation effect and orbital ordering. See the supplementary material for the sample-preparation method, details of the measurement, temperature dependence of XRD patterns, Raman spectra and the calculated vibrational modes, andtemperature calibration. AUTHORS’ CONTRIBUTIONS S. Pei and J. Tang contributed equally to this work. This work was supported by the Science, Technology and Innovation Commission of Shenzhen Municipality (Grant No. JCYJ20170412152334605) and Initiative Funds for Shenzhen High Class University (Grant No. G02206301). J.W.M was partially supported by the program for Guangdong Introducing Innovative and Entrepreneurial Teams (No. 2017ZT07C062). DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, D. Zhong, E. 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5.0018577.pdf

J. Chem. Phys. 153, 124117 (2020); https://doi.org/10.1063/5.0018577 153, 124117 © 2020 Author(s).Almost exact energies for the Gaussian-2 set with the semistochastic heat-bath configuration interaction method Cite as: J. Chem. Phys. 153, 124117 (2020); https://doi.org/10.1063/5.0018577 Submitted: 17 June 2020 . Accepted: 08 September 2020 . Published Online: 30 September 2020 Yuan Yao , Emmanuel Giner , Junhao Li , Julien Toulouse , and C. J. Umrigar COLLECTIONS Paper published as part of the special topic on Frontiers of Stochastic Electronic Structure CalculationsFROST2020 ARTICLES YOU MAY BE INTERESTED IN OrbNet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features The Journal of Chemical Physics 153, 124111 (2020); https://doi.org/10.1063/5.0021955 Efficient evaluation of exact exchange for periodic systems via concentric atomic density fitting The Journal of Chemical Physics 153, 124116 (2020); https://doi.org/10.1063/5.0016856 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/10.1063/5.0018930The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Almost exact energies for the Gaussian-2 set with the semistochastic heat-bath configuration interaction method Cite as: J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 Submitted: 17 June 2020 •Accepted: 8 September 2020 • Published Online: 30 September 2020 Yuan Yao,1,a) Emmanuel Giner,2,b) Junhao Li,1,c)Julien Toulouse,2,3,d) and C. J. Umrigar1,a) AFFILIATIONS 1Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853, USA 2Laboratoire de Chimie Théorique, Sorbonne Université and CNRS, F-75005 Paris, France 3Institut Universitaire de France, F-75005 Paris, France Note: This paper is part of the JCP Special Topic on Frontiers of Stochastic Electronic Structure Calculations. a)Authors to whom correspondence should be addressed: yy682@cornell.edu and cyrusumrigar@cornell.edu b)Electronic mail: emmanuel.giner@lct.jussieu.fr c)Electronic mail: jl2922@cornell.edu d)Electronic mail: toulouse@lct.jussieu.fr ABSTRACT The recently developed semistochastic heat-bath configuration interaction (SHCI) method is a systematically improvable selected configu- ration interaction plus perturbation theory method capable of giving essentially exact energies for larger systems than is possible with other such methods. We compute SHCI atomization energies for 55 molecules that have been used as a test set in prior studies because their atom- ization energies are known from experiment. Basis sets from cc-pVDZ to cc-pV5Z are used, totaling up to 500 orbitals and a Hilbert space of 1032Slater determinants for the largest molecules. For each basis, an extrapolated energy well within chemical accuracy (1 kcal/mol or 1.6 mHa/mol) of the exact energy for that basis is computed using only a tiny fraction of the entire Hilbert space. We also use our almost exact energies to benchmark energies from the coupled cluster method with single, double, and perturbative triple excitations. The energies are extrapolated to the complete basis set limit and compared to the experimental atomization energies. The extrapolations are done both without and with a basis-set correction based on density-functional theory. The mean absolute deviations from experiment for these extrap- olations are 0.46 kcal/mol and 0.51 kcal/mol, respectively. Orbital optimization methods used to obtain improved convergence of the SHCI energies are also discussed. Published under license by AIP Publishing. https://doi.org/10.1063/5.0018577 .,s I. INTRODUCTION The recently developed semistochastic heat-bath configuration interaction (SHCI) method1–7is a systematically improvable quan- tum chemistry method capable of providing essentially exact ener- gies for small many-electron systems. It has been successfully applied to a number of challenging problems in quantum chemistry, includ- ing the potential energy curve of the chromium dimer8for which coupled cluster with single, double, and perturbative triple excita- tions [CCSD(T)], the gold standard of single-reference quantum chemistry, does not give even a qualitatively correct description.It has also been used as the reference method for calculations on transition metal atoms, ions, and monoxides9to test the accuracy of a wide variety of other electronic-structure methods. SHCI is an example of the selected configuration interaction (SCI) plus perturbation theory (SCI + PT) methods,10–21which have two stages. In the first stage, a variational wave function is con- structed iteratively, starting from a determinant that is expected to have a significant amplitude in the final wave function, e.g., the Hartree–Fock (HF) determinant. The number of determinants in the variational wave function is controlled by a parameter ϵ1. In the second stage, second-order perturbation theory is used to improve J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp upon the variational energy. The total energy (sum of the variational energy and the perturbative correction) is computed at several values ofϵ1and extrapolated to ϵ1→0 to obtain an estimate for the full con- figuration interaction (FCI) energy. The efficiency of SHCI depends on the choice of the orbitals—natural orbitals lead to the faster con- vergence of the energy relative to HF orbitals and optimized orbitals yield yet faster convergence. In this paper, the SHCI method is reviewed in Sec. II, our orbital optimization schemes are described in Sec. III, the basis-set correc- tion and extrapolation that we use are discussed in Sec. IV, and the details of the calculations are given in Sec. V. In Sec. VI, we apply SHCI to the 55 first- and second-row molecules that served as the training set for the Gaussian-2 (G2) protocol22because accu- rate experimental atomization energies were believed to be known for them. The G2 protocol is one of several quantum chemistry composite methods that combine low-order methods on large basis sets and high-order coupled-cluster methods on smaller basis sets to compute accurate thermochemical properties (see, e.g., Refs. 23–27). These 55 molecules, which we refer to as the G2 set, have previously been used to test the accuracy of coupled-cluster-based methods24 and quantum Monte Carlo (QMC) methods.28–31We employ the correlation consistent basis sets cc-pV nZ for n= 2 (D), 3 (T), 4 (Q), and 5,32keeping the core electrons frozen, to obtain SHCI energies that we believe are well within 1 mHa of the exact (FCI) energies for each of the molecules and basis sets. Hence, these calculations pro- vide a set of reference energies that can be used to test other accurate electronic-structure methods. The molecules in the G2 set are sufficiently weakly correlated that one would expect CCSD(T) to be reasonably accurate but not at the level of 1 mHa. Hence, we also calculate the CCSD(T) energies using the same basis sets in order to use SHCI to evaluate the errors in the CCSD(T) energies, as FCI is not feasible for most of these systems. The SHCI energies are then extrapolated to the complete- basis-set (CBS) limit, both without and with a basis-set correction based on density-functional theory (DFT).33–36Corrections taken from the literature for zero-point energy, relativistic effects, and core-valence (CV) correlation are then applied to obtain our predic- tions for the atomization energies, which are then compared to the best available experimental values. For some systems, the available experimental values differ substantially from each other, and for at least one system, we believe that the theoretical estimates are more accurate than the best experimental value. II. REVIEW OF THE SHCI METHOD In this section, we review the SHCI method, emphasizing the two important ways it differs from other SCI + PT methods. In the following, we use Vfor the set of variational determinants and P for the set of perturbative determinants, that is, the set of deter- minants that are connected to the variational determinants by at least one non-zero Hamiltonian matrix element but are not present inV. A. Variational stage SHCI starts from an initial determinant and generates the vari- ational wave function through an iterative process. At each iteration,the variational wave function, ΨV, is written as a linear combination of the determinants in the space V, ∣ΨV⟩=∑ Di∈Vci∣Di⟩, (1) and new determinants, Da, from the space Pthat satisfy the criterion ∃Di∈Vsuch that ∣Haici∣≥ϵ1 (2) are added to the Vspace, where Haiis the Hamiltonian matrix ele- ment between determinants Daand Di, andϵ1is a user-defined parameter that controls the accuracy of the variational stage.37 (Whenϵ1= 0, the method becomes equivalent to FCI.) After adding the new determinants to V, the Hamiltonian matrix is constructed and diagonalized using the diagonally preconditioned Davidson method38to obtain an improved estimate of the lowest eigenvalue, EV, and eigenvector, ΨV. This process is repeated until the change in the variational energy EVfalls below a certain threshold. Other SCI methods use different criteria, based on either the first-order perturbative coefficient of the wave function, ∣c(1) a∣=∣∑iHaici EV−Ea∣>ϵ1, (3) or the second-order perturbative correction to the energy, −ΔE(2)=−(∑iHaici)2 EV−Ea>ϵ1, (4) where Ea=Haa. The reason we choose instead the selection crite- rion in Eq. (2) is that it can be implemented very efficiently without checking the vast majority of the determinants that do not meet the criterion, by taking advantage of the fact that most of the Hamil- tonian matrix elements correspond to double excitations, and their values do not depend on the determinants themselves but only on the four orbitals whose occupancies change during the dou- ble excitation. Therefore, at the beginning of an SHCI calculation, for each pair of spin–orbitals, the absolute values of the Hamilto- nian matrix elements obtained by doubly exciting from that pair of orbitals are computed and stored in decreasing order by mag- nitude, along with the corresponding pairs of orbitals the electrons would excite to. Then, the double excitations that meet the crite- rion in Eq. (2) can be generated by looping over all pairs of occu- pied orbitals in the reference determinant and traversing the array of sorted double-excitation matrix elements for each pair. As soon as the cutoff is reached, the loop for that pair of occupied orbitals is exited. Although the criterion in Eq. (2) does not include infor- mation from the diagonal elements, this selection criterion is not significantly different from either of the criteria in Eqs. (3) and (4) because the terms in the numerators of Eqs. (3) and (4) span many orders of magnitude, so the sums are highly correlated with the largest-magnitude term in the sums in Eq. (3) or (4), and because the denominator is never small after several determinants have been included in V. It was demonstrated in Ref. 1 that the selected deter- minants give only slightly inferior convergence to those selected using the criterion in Eq. (3). This is greatly outweighed by the improved selection speed. Moreover, one could use the criterion J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp in Eq. (2) with a smaller value of ϵ1as a preselection criterion and then select determinants using the criterion in Eq. (4) or something close to it, thereby having the benefit of both a fast selection method and a close to optimal choice of determinants. We use a similar but somewhat more complicated criterion also for the selection of the determinants connected to those in Vby a single excitation, but this improvement is of lesser importance because the number of deter- minants connected by single excitations is much smaller than the number connected by double excitations. With these improvements, the time required for selecting determinants is negligible, and the most time-consuming step by far in the variational stage is the con- struction of the sparse Hamiltonian matrix. Details for doing this efficiently are given in Ref. 7. B. Perturbative stage In common with most of the other SCI + PT methods, the perturbative correction is computed using Epstein–Nesbet pertur- bation theory.39,40The variational wave function is used to define the zeroth-order Hamiltonian, ˆH(0), and the perturbation, ˆH(1), ˆH(0)=∑ Di,Dj∈VHij∣Di⟩⟨Dj∣+∑ Da∉VHaa∣Da⟩⟨Da∣, ˆH(1)=ˆH−ˆH(0).(5) The first-order energy correction is zero, and the second-order energy correction ΔE(2)is ΔE(2)=⟨ΨV∣ˆH(1)∣Ψ(1)⟩=∑ Da∈P(∑Di∈VHaici)2 EV−Ea, (6) whereΨ(1)is the first-order wave-function correction. The SHCI total energy is ESHCI=EV+ΔE(2)=⟨ΨV∣H∣ΨV⟩+ΔE(2). (7) It is expensive to evaluate the expression in Eq. (6) because the outer summation includes all determinants in the space Pand their number is O(N2 eN2 vNV), where NVis the number of variational determinants, Neis the number of electrons, and Nvis the num- ber of unoccupied orbitals. The straightforward and time-efficient approach to computing the perturbative correction requires storing the partial sum ∑Di∈VHaicifor each unique a, while looping over all the determinants Di∈V. This creates a severe memory bottleneck. An alternative approach, which is widely used, does not require stor- ing the unique abut requires checking whether the determinant was already generated by checking its connection with variational deter- minants whose connections have already been included. This entails some additional computational expense. The SHCI algorithm instead uses two other strategies to reduce both the computational time and the storage requirement. First, SHCI screens the sum1using a second threshold, ϵ2(whereϵ2<ϵ1), as the criterion for selecting perturbative determinants Da∈P, ΔE(2)(ϵ2)=∑ Da∈P(∑(ϵ2) Di∈VHaici)2 EV−Ea, (8)where∑(ϵ2)indicates that only terms in the sum for which ∣Haici∣ ≥ϵ2are included. Similar to the variational stage, we find the con- nected determinants efficiently with precomputed arrays of double excitations sorted by the magnitude of their Hamiltonian matrix ele- ments.1Note that the vast number of terms that do not meet this criterion are never evaluated . Even with this screening, the simultaneous storage of all terms indexed by ain Eq. (8) can exceed computer memory when ϵ2is cho- sen small enough to obtain essentially the exact perturbation energy. The second innovation in the calculation of the SHCI perturbative correction is to overcome this memory bottleneck by evaluating it semistochastically. The most important contributions are evaluated deterministically, and the rest are sampled stochastically. Our orig- inal method used a two-step perturbative algorithm,2but our later three-step perturbative algorithm7is even more efficient. The three steps are as follows: 1. A deterministic step with cutoff ϵdtm 2(<ϵ1), wherein all the variational determinants are used, and all the perturbative batches are summed over. 2. A “pseudo-stochastic” step, with cutoff ϵpsto 2(<ϵdtm 2), wherein all the variational determinants are used, but the perturba- tive determinants are partitioned into batches. Typically, only a small fraction of these batches need to be summed over to achieve an error much smaller than the target error. 3. A stochastic step, with cutoff ϵ2(<ϵpsto 2), wherein a few stochastic samples of variational determinants, each con- sisting of Nddeterminants, are sampled with probability ∣ci∣/∑Di∈V∣ci∣, and only one of the perturbative batches is randomly selected per variational sample. Using this semistochastic algorithm, the statistical error of our calculations for each ϵ1is at most 20 μHa, which is negligible on the scale of the desired accuracy. Having a small statistical error is important for doing a reliable extrapolation to the ϵ1= 0 limit. This is done3by computing ESHCIat five or six values of ϵ1and using a weighted quadratic fit of ESHCIto−ΔE(2)to obtain ESHCIat−ΔE(2) = 0, using weights proportional to (ΔE(2))−2. Figure 1 shows the FIG. 1 . Convergence of SHCI energy of SO 2in the cc-pV5Z basis set. The line is a weighted quadratic fit but is very nearly linear. The statistical error bars are plotted but are invisible on the scale of the plot. J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp convergence of ESHCIfor the system that has the largest extrapolation distance (difference between the energy at the smallest ϵ1used and the estimated energy at ϵ1= 0), namely, SO 2in the cc-pV5Z basis set. We note that, subsequent to our first semistochastic paper,2a completely different, but also efficient, semistochastic approach has been presented in Ref. 18. III. ORBITAL OPTIMIZATION SHCI gives an estimate of the exact FCI energy by extrapolating energies evaluated at several ϵ1>0 toϵ1= 0, the FCI limit. This results in an extrapolation error that disappears in the limit that the extrapolation distance goes to zero. The extrapolation distance can be reduced by decreasing ϵ1, but this is limited by the available computer memory and time. An alter- native approach is to optimize the orbitals to obtain more compact configuration-interaction (CI) expansions with lower variational energies. The first step to orbital optimization is to find the SHCI natural orbitals, i.e., the eigenstates of the one-body reduced density matrix. These orbitals have a definite occupation number for a given varia- tional wave function, and the most occupied ones represent in some sense the most important degrees of freedom. Orbitals can be further optimized by directly minimizing the energy of the variational wave function through the orbital rotation parameters X, E(X)=⟨ΨV∣exp(ˆX)ˆHexp(−ˆX)∣ΨV⟩, (9) where ˆXis a real anti-Hermitian operator such that exp (−ˆX)param- eterizes unitary transformations in orbital space. For a system with Norbreal-valued orbitals, this yields at most Norb(Norb−1)/2 orbital optimization parameters that are the elements of the real antisym- metric matrix X. In reality, the number of parameters will often be less than this due to point-group symmetry. Depending on the particular optimization algorithm used, the gradient and sometimes part of the Hessian of the energy with respect to the orbital param- eters are needed, either of which requires computing both the one- and two-body density matrices of the variational wave function. In addition to the orbital parameters, the CI parameters (which are much more numerous) must be optimized as well. We next discuss some of the optimization methods we have studied. A. Newton’s method Newton’s method is a straightforward method for optimizing the parameters. The parameters xt+1at iteration t+ 1 are given by xt+1=xt−h−1 tgt, (10) where gtandhtare the gradient and the Hessian of the energy with respect to the parameters at iteration t. In practice, it is more efficient to find the parameter changes by solving the set of linear equations, ht(xt+1−xt)=−gt. (11) However, the problem is that the number of parameters is typically much too large for even this to be practical. Typically, even using a rather large value of the threshold parameter ϵ1for the optimization FIG. 2 . Comparison of four orbital optimization schemes for the H 2CO molecule in the cc-pVDZ basis and threshold parameter ϵ1= 2×10−4. All four calculations start with HF orbitals and construct natural orbitals on the first iteration, so they differ only from the second iteration on. The Newton and diagonal Newton curves are nearly coincident for this system. step, there are millions of CI parameters, whereas there are only thousands of orbital parameters. Hence, one resorts to alternating the optimization of the CI parameters using the usual Davidson algorithm and optimizing the orbital parameters in the much smaller space of orbital rotations using Newton’s method. This alternat- ing optimization often converges very slowly because the coupling between the CI parameters and the orbital parameters is strong, as can be seen in Fig. 2. Note that the orbital optimization problem in SHCI is more difficult than that in the usual complete-active- space self-consistent-field (CASSCF) method for two reasons. First, none of the orbital rotations among orbitals of the same symme- try are redundant, so the number of orbital parameters that need to be optimized is much larger. Second, the coupling between the CI parameters and the orbital parameters is stronger. In quantum chemistry problems, the orbital part of the Hessian matrix is often diagonally dominant. In that case, one can save sig- nificant computer time by ignoring the off-diagonal elements. We refer to this as the “diagonal Newton” method, and Fig. 2 shows that for this molecule it converges at the same rate as Newton’s method. The convergence of both methods is limited by the lack of coupling between the CI and orbital parameters. B. AMSGrad AMSGrad is a momentum-based gradient-descent method commonly used in machine learning.41It avoids the expensive Hes- sian calculations since only gradient information is needed. At each iteration, it employs running averages of the gradient components and their squares, determined by the mixing parameters β1,β2∈(0, 1), according to mt=β1mt−1+(1−β1)gt, vt=β2vt−1+(1−β2)g2 t, ˆvt=max(ˆvt−1,vt), xt+1=xt−η√ˆvt+ϵmt.(12) J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The learning parameters η,β1, andβ2together determine the level of aggressiveness of the descent, and ϵis a small constant for numer- ical stability. We have found empirically that with a suitable level of aggressiveness, AMSGrad oscillates for the first few iterations but eventually descends at a much quicker pace per iteration compared to either Newton or diagonal Newton, as can be seen in Fig. 2. In addition, each iteration takes less time since only the gradient is needed. For a variety of systems, we have found that the parameters η= 0.01,β1= 0.5, andβ2= 0.5 give reasonably good convergence, even though they are much different from the values recommended in the literature. C. Accelerated Newton’s method Finally, we have developed a heuristic overshooting method that achieves yet better convergence for most systems. Here, the overshooting tries to account for the coupling between CI and orbital parameters, but it may be more generally useful whenever alternating optimization of subsets of parameters is done. At each iteration, a diagonal Newton step is calculated for the orbital parameters, but, instead of using the proposed step, it is amplified by a factor ftdetermined by the cosine of the angle between the previous step xt−xt−1and the current step xt+1−xt, ft=min(1 2−cos(xt−xt−1,xt+1−xt),1 ϵ), (13) whereϵis initialized to 0.01 and ϵ←ϵ0.8each time cos( xt −xt−1,xt+1−xt)<0. The cosine in the expression is calculated in a “scale-invariant” way to make it invariant under a rescaling of some of the parameters, i.e., in the usual definition cos (v,w) =⟨v,w⟩/√ ⟨v,v⟩⟨w,w⟩, we define the inner product as ⟨v,w⟩ =vThw, where the Hessian hcan again be approximated by its diag- onal. Another scale invariant choice for the inner product is ⟨v,w⟩ =vTggTw, and that works equally well. As shown in Fig. 2, this accelerated scheme optimizes much faster than the previous schemes. For instance, after four iterations, the gain in variational energy is already better than that after 20 iterations using the conventional Newton’s method. Compared to AMSGrad, the higher per iteration cost is more than made up by the greatly reduced number of iterations needed. For this system, not only does the energy drop significantly but also the number of determinants decreases. For the accelerated scheme, the drop is from 145 370 to 93 882 determinants. However, for some systems, the number of determinants increases, thereby partly offsetting the benefit of the energy gain. IV. BASIS-SET CORRECTION AND EXTRAPOLATION We employ the correlation consistent polarized valence (cc- pVnZ) basis sets with n= 2 (D), 3 (T), 4 (Q), and 5. The ener- gies computed for each atom or molecule are extrapolated to the CBS limit using separate extrapolations for the HF energy and the correlation energy,42–44 EHF CBS=EHF n+aexp(−bn), (14) Ecorr CBS=Ecorr n+cn−3, (15)where nis the cardinal number of the basis set. The only exception is Li, for which the lowest HF energy is taken as the CBS energy because the energies for n= 3, 4, and 5 cannot be fit by a decay- ing exponential. Note that the correlation energy extrapolation has two parameters, so it is necessary to use only the n= 4 and 5 basis sets, whereas the HF extrapolation has three parameters, and hence, it is necessary to use the n= 3, 4, and 5 basis sets. Consequently, the extrapolation error is larger for the HF energy than for the corre- lation energy, mostly for molecules containing second-row atoms, as we have verified for some systems by going to the n= 6 basis sets. In order to partially cure this problem, the cc-pV( n+d)Z basis sets, which have one additional set of d basis functions, were intro- duced45for the second-row atoms Al through Ar. For H, He, and first-row atoms, the cc-pV nZ and cc-pV( n+d)Z basis sets are iden- tical. Hence, all the CBS energies presented in this paper use extrap- olated HF energies obtained from Eq. (14) but with EHF nreplaced byEHF n+d, where EHF n+dare the HF energies in the cc-pV( n+d)Z basis sets. We find that although the cc-pV( n+d)Z basis sets of course give lower total energies than the cc-pV nZ basis sets for each n, the estimated CBS energies are higher. Of the systems we study, replac- ing the cc-pV nZ basis sets with the cc-pV( n+d)Z basis sets has the largest effect for SO 2and SO, reducing the atomization energies by 3.68 kcal/mol and 0.82 kcal/mol, respectively. The large change in the estimated CBS energy of SO 2has previously been noted in Refs. 46–48. To estimate the total energies in the CBS limit, we also employ the DFT-based basis-set correction recently developed in Refs. 33– 36. In this scheme, the total SHCI energy in a given basis set is corrected as ESHCI+PBE n=EHF n+d−EHF n+ESHCI n +¯EPBE n[ρ,ζ,μ], (16) where ¯EPBE n[ρ,ζ,μ]is a basis-set-dependent functional of the density ρ(r), the spin polarization ζ(r) = [ρ↑(r)−ρ↓(r)]/ρ(r), and the local range-separation function μ(r), ¯EPBE n[ρ,ζ,μ]=∫ρ(r)¯εsr,PBE c, md(ρ(r),ζ(r),μ(r))dr. (17) In Eq. (17), ¯εsr,PBE c, mdis the complementary short-range correlation energy per particle with a multideterminant reference (md) that was constructed in Ref. 34 based on the Perdew–Burke–Ernzerhof (PBE)49correlation functional and the on-top pair density of the uniform-electron gas. The local range-separation function μ(r) pro- vides a local measure of the incompleteness of the basis set and is defined as μ(r)=√π 2W(r,r), (18) where W(r,r) is the on-top value of the effective two-electron interaction in the basis set, W(r,r)={f(r,r)/n2(r,r), if n2(r,r)≠0 ∞, otherwise,(19) J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp with f(r,r)=∑ pq∈B∑ rstu∈Aϕp(r)ϕq(r)Vrs pqΓtu rsϕt(r)ϕu(r), (20) n2(r,r)=∑ rstu∈Aϕr(r)ϕs(r)Γtu rsϕt(r)ϕu(r), (21) where Vrs pq=⟨pq∣rs⟩are the two-electron integrals and Γtu rsis the opposite-spin two-body density matrix. Since μ(r) is very weakly dependent on Γtu rs, we calculate Γtu rsat the HF level only. Consistently, {ϕp(r)} are the HF orbitals, and ρ(r) andζ(r) are also calculated at the HF level. Since the core electrons are frozen in SHCI, we use the frozen-core variant34,36of this DFT basis-set correction, which means that in Eqs. (20) and (21) the sums over r,s,t,uare restricted to the set of active (i.e., non-core) occupied HF orbitals A. Yet, the local range-separation function μ(r) probes the entire basis set through the sums over p,q, which run over the set of all (occupied + virtual) HF orbitals B. For a fixed basis set, the energy functional ¯EPBE n[ρ,ζ,μ]provides an estimate of the energy missing in FCI to reach the CBS limit. It has the desirable property of vanishing in the CBS limit, i.e., ¯EPBE CBS=0, and thus, the DFT basis-set correction does not alter the CBS limit, i.e.,ESHCI+PBE CBS=ESHCI CBS, but just accelerates the basis convergence. Based on the analysis of basis convergence in range-separated DFT,50we assume an exponential basis convergence of ESHCI+PBE n , which gives us another estimate of the CBS limit of ESHCI n via the extrapolation ESHCI+PBE CBS=ESHCI+PBE n +aexp(−bn), (22) using n= 3, 4, and 5. The only exceptions are Be and Cl, whose cc- pV5Z energy is higher than the cc-pVQZ energy and for which the cc-pV5Z energy is taken as the CBS energy. V. COMPUTATIONAL DETAILS The HF and CCSD(T) calculations are done with PySCF51or MOLPRO.52The starting integrals are computed for HF orbitals. The core orbitals are kept fixed for all the subsequent steps. Then, we construct integrals in the SHCI natural orbital basis bycomputing and diagonalizing the one-body density matrix and rotating the integrals in the HF basis to the natural orbital basis. Next, we use the methods discussed in Sec. III to construct the inte- grals in the optimized orbital basis. We use a fairly large value of ϵ1 (typically 2 ×10−4) to construct the natural orbitals and the opti- mized orbitals. For some systems, the natural orbital basis is rea- sonably close to the optimal one, but for most systems the optimized orbital bases result in considerable gains in efficiency. The final SHCI calculations using the optimized orbitals employ smaller values of ϵ1(typically five values ranging from 2 ×10−4to 2×10−5), which are then used to extrapolate to the ϵ1= 0 limit. The system with the largest extrapolation distance, SO 2in the cc-pV5Z basis, was shown as an example in Fig. 1. The PBE-based basis-set correction described in Sec. IV is cal- culated independently from the SHCI calculations using the soft- ware QUANTUM PACKAGE.53If the HF two-body density matrix is used in Eqs. (20) and (21), the basis-set correction has a compu- tational cost of O(NgN2 eN2 orb), where Ngis the number of real-space grid points used for numerical integration in Eq. (17) and Norbis the total number of orbitals (including core orbitals) in the basis set. The two-electron integrals in the HF orbital basis, involving up to two virtual orbitals, are also needed, and the cost for doing the inte- gral transformation to compute these is O(N2 eN3 orb). However, most of these integrals (aside from those involving the core orbitals) are needed for SHCI anyway. Hence, the DFT-based basis-set correc- tion does not increase the computational time of SHCI calculations appreciably. The geometries are taken from the supplementary material of Ref. 30, which in turn took them from the papers cited therein. They are provided in Ref. 66. The only exceptions are HCO and C 2H4 for which we took the geometry from Ref. 34 because these geome- tries gave lower CBS-extrapolated energies by ∼1.5 mHa. In order to compare to experimental atomization energies, the CBS SHCI ener- gies are corrected for zero-point energies (ZPE), core-valence (CV) correlation, scalar relativity (SR), and spin–orbit (SO) effects. We take the corrections from the literature. Since most of the papers do not have all the 55 molecules we studied, we take the corrections from Refs. 24 and 54 in that order, i.e., we take it from the first of these references, which contains corrections for that molecule. The source of the corrections is indicated in Table I next to the entry for the zero-point energy (ZPE). Similarly, the experimental TABLE I . Deviation of SHCI and SHCI + PBE atomization energies, D0, in the complete-basis-set limit, from the best available experimental energies in units of kcal/mol. The raw SHCI and SHCI + PBE energies are corrected for zero-point energy (ZPE), scalar relativity (SR), spin–orbit energy (SO), and core-valence correlation (CV). For each molecule, the ZPE, SR + SO, and CV corrections are taken from Ref. 54 if available, and otherwise from Ref. 24, as shown next to the ZPE correction. The only exceptions are that the CV corrections for LiH and Li 2were taken from Ref. 24 because Ref. 54 did not freeze the core for these systems. SHCI SHCI + PBE Molecule SHCI De ZPE SR + SO CV Expt. D0 Deviation D0 Deviation LiH 57.71 −1.9954−0.02 0.30 55.705756.00 0.30 56.02 0.32 BeH 50.23 −2.9254−0.02 0.51 47.705847.80 0.10 47.80 0.10 CH 84.11 −4.0454−0.08 0.14 79.975580.13 0.16 80.16 0.19 CH 2(3B1) 190.01 −10.5554−0.23 0.82 179.8355180.05 0.22 179.95 0.12 CH 2(1A1) 181.12 −10.2954−0.17 0.39 170.8355171.05 0.22 171.10 0.27 CH 3 306.93 −18.5554−0.25 1.07 289.1155289.20 0.09 289.18 0.07 J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I .(Continued.) SHCI SHCI + PBE Molecule SHCI De ZPE SR + SO CV Expt. D0 Deviation D0 Deviation CH 4 419.25 −27.7454−0.27 1.26 392.4755392.50 0.03 392.56 0.09 NH 83.09 −4.6454−0.07 0.11 78.365578.49 0.13 78.55 0.19 NH 2 182.50 −11.84540.08 0.32 170.5955171.06 0.47 171.10 0.51 NH 3 297.91 −21.3354−0.25 0.65 276.5955276.98 0.39 276.97 0.38 OH 107.26 −5.2954−0.24 0.14 101.7355101.87 0.14 101.81 0.08 H2O 233.01 −13.2654−0.49 0.38 219.3755219.64 0.27 219.51 0.14 HF 141.76 −5.8654−0.58 0.17 135.2755135.49 0.22 135.37 0.10 SiH 2(1A1) 153.90 −7.3024−0.60 0.00 144.1057146.00 1.90 146.05 1.95 SiH 2(3B1) 133.31 −7.5024−0.80 −0.50 123.4024124.51 1.11 124.42 1.02 SiH 3 228.22 −13.2024−0.80 −0.20 212.2057214.02 1.82 214.02 1.82 SiH 4 324.80 −19.4024−1.00 −0.20 302.6057304.20 1.60 304.27 1.67 PH 2 154.24 −8.4024−0.20 0.30 144.7024145.94 1.24 145.96 1.26 PH 3 241.91 −14.4454−0.44 0.33 227.1057227.36 0.26 227.36 0.26 H2S 183.63 −9.4054−0.93 0.24 173.2057173.54 0.34 173.41 0.21 HCl 107.41 −4.2424−1.00 0.30 102.2155102.47 0.26 102.30 0.09 Li2 24.14 −0.50540.00 0.20 23.905723.84 −0.06 23.84 −0.06 LiF 138.15 −1.3024−0.60 0.90 137.6057137.15 −0.45 137.34 −0.26 C2H2 403.16 −16.5054−0.46 2.47 388.6455388.67 0.03 388.84 0.20 C2H4 561.72 −31.6654−0.50 2.36 532.0455531.92 −0.12 532.09 0.05 C2H6 711.36 −46.2354−0.56 2.42 666.1955666.99 0.80 666.97 0.78 CN 180.24 −2.9554−0.24 1.10 178.1255178.15 0.03 178.58 0.46 HCN 311.91 −9.9554−0.31 1.67 303.1455303.32 0.18 303.76 0.62 CO 258.61 −3.0954−0.46 0.95 256.2355256.01 −0.22 256.47 0.24 HCO 278.10 −8.0954−0.59 1.16 270.7655270.58 −0.18 270.92 0.16 H2CO 373.42 −16.5254−0.65 1.30 357.4855357.55 0.07 357.88 0.40 H3COH 512.44 −31.7224−0.80 1.50 480.9755481.42 0.45 481.52 0.55 N2 227.66 −3.3654−0.14 0.80 224.9455224.96 0.02 225.62 0.68 N2H4 438.61 −32.6854−0.51 1.14 404.8155406.56 1.75 406.60 1.79 NO 152.33 −2.7154−0.23 0.42 149.8155149.81 0.00 150.23 0.42 O2 120.50 −2.2554−0.62 0.24 117.9955117.87 −0.12 117.95 −0.04 H2O2 269.21 −16.4454−0.82 0.36 252.2155252.31 0.10 252.33 0.12 F2 39.09 −1.3054−0.79 −0.11 36.935536.89 −0.04 36.93 0.00 CO 2 388.19 −7.2454−1.01 1.77 381.9855381.71 −0.27 382.46 0.48 Na2 16.74 −0.20240.00 0.30 17.005716.84 −0.16 16.85 −0.15 Si2 76.66 −0.7354−1.01 0.13 74.405775.05 0.65 75.03 0.63 P2 116.66 −1.1154−0.25 0.77 116.0057116.07 0.07 116.29 0.29 S2 103.95 −1.0454−1.40 0.34 100.8057101.85 1.05 101.51 0.71 Cl2 59.92 −0.8054−1.82 −0.13 57.185557.17 −0.01 56.75 −0.43 NaCl 100.03 −0.5024−1.10 −1.20 97.405797.23 −0.17 96.85 −0.55 SiO 192.01 −1.7854−0.90 0.95 189.8057190.28 0.48 190.53 0.73 CS 171.55 −1.8354−0.80 0.75 170.4057169.67 −0.73 169.67 −0.73 SO 126.15 −1.6354−1.09 0.41 123.5057123.84 0.34 123.67 0.17 ClO 65.58 −1.2254−0.81 0.06 63.425563.61 0.19 63.07 −0.35 ClF 62.95 −1.1254−1.39 −0.10 60.355560.34 −0.01 59.99 −0.36 Si2H6 535.40 −30.5024−2.00 0.00 500.1024502.90 2.80 503.34 3.24 CH 3Cl 395.06 −23.1924−1.40 1.20 371.3555371.67 0.32 371.53 0.18 H3CSH 474.48 −28.6024−1.20 1.50 445.1057446.18 1.08 445.91 0.81 HOCl 166.62 −8.1824−1.50 0.40 156.8855157.34 0.46 156.93 0.05 SO 2 260.36 −4.3854−1.79 0.92 254.4656255.11 0.65 255.00 0.54 J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp values quoted in Table I are taken from Refs. 24, 55–57 in that order. VI. RESULTS A. Accuracy of CCSD(T) We have computed the total energies for each of the 55 molecules and their 12 constituent atoms in the four basis sets mentioned in Sec. IV. The accuracy of these energies should be considerably better than 1 mHa, as discussed later in this section. These energies are provided in Ref. 66 and can serve as a refer- ence for other approximate methods. In particular, we have used it to test the accuracy of CCSD(T). None of the 67 systems studied is strongly correlated, so one would expect the CCSD(T) energies to be reasonably accurate. This is in fact the case, as can be seen from Fig. 3, which shows the deviation of the CCSD(T) total ener- gies from the SHCI total energies. CCSD(T) deviates from SHCI by 1 mHa–2 mHa for the lighter systems and 3 mHa–4 mHa for the heavier ones. For systems with two or fewer valence electrons, the two methods agree exactly as they must, and for all the systems with more electrons, CCSD(T) underestimates the correlation energy. The mean absolute deviation (MAD) is roughly independent of the basis size, being 0.99 mHa, 1.06 mHa, 1.09 mHa, and 1.05 mHa, respectively, for the four basis sets. The pattern of the errors is very similar for the four basis sets. Although the absolute value of the cor- relation energy grows with the size of the basis set by a few tens ofpercent going from cc-pVDZ to cc-pV5Z basis sets, the error that CCSD(T) makes does not grow in proportion. The same set of molecules have also recently been computed by another SCI + PT method.59In their calculation, they correlate all the electrons, so the energies they obtain are not directly com- parable to ours. They employ only the cc-pVDZ and cc-pVTZ basis sets, so they cannot extrapolate to the CBS limit. Furthermore, they employ at the most only 106determinants, whereas we employ a few times 108determinants for the larger molecules and basis sets. Consequently, when they compare to CCSD(T) energies, they find two systems for the cc-pVDZ basis set and several systems for the cc-pVTZ basis set where their energies are higher than those from CCSD(T). In contrast, as shown in Fig. 3, we find that our SHCI energies are always lower than CCSD(T) energies and further that the pattern of the energy differences is very similar for the various basis sets. B. Atomization energies Table I shows the difference between the SHCI total energies for the molecules and their constituent atoms, extrapolated to the CBS limit according to Eqs. (14) and (15). It also shows the ZPE, SR + SO, and CV corrections taken from the literature and the final prediction for the SHCI atomization energy, D0, and how much it differs from the best available experimental values. The difference between the SHCI D0and experiment is also plotted in Fig. 4, both before and after the corrections are applied. FIG. 3 . The error in the CCSD(T) total energies obtained by comparison to the SHCI total energies. The CCSD(T) errors are of course zero for systems with one or two valence electrons, and they are positive in all other cases. The errors for each system are very similar for the various basis sets, especially for the larger basis sets. J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . The comparison of SHCI atomization energies in the extrapolated complete-basis-set limit with experiment, with (red dots) and without (green crosses) scalar relativistic and spin–orbit (SR + SO) corrections and core-valence (CV) corrections. Both sets of points include zero-point energy (ZPE) corrections. Systems for which red dots fall in the shaded region are considered to have reached chemical accuracy (1 kcal/mol). There are three possible sources of discrepancy between the cal- culated and the experimental atomization energies: (1) The extrapo- lation to the CBS limit may not be accurate; (2) the literature values of the ZPE, SR + SO, and CV corrections may not be accurate; (3) the experimental values have errors. It seems likely, as discussed below, that all three of these play a role for some of the systems. We show in Fig. 5 the convergence of the atomization ener- gies with basis size. The SHCI atomization energies in fact have two extrapolation errors. The first and more benign error comes from extrapolating SHCI total energies for each basis set to the FCI limit, i.e.,ϵ1→0. This error can be reduced by employing smaller ϵ1and/or using better optimized orbitals. For the four basis sets n= 2 (D), 3 (T), 4 (Q), and 5, the largest extrapolation distances in the total energy of these 55 molecules and 12 atoms are 0.97 mHa, 2.36 mHa, 3.34 mHa, and 2.90 mHa, respectively.60Assuming that the extrapolated energies are in error by no more than a fifth of the extrapolation distance, all these energies should be accurate to con- siderably better than 1 mHa. Furthermore, the typical extrapolation distances are much smaller, especially for the lighter systems: the median distances for the four basis sets are 2.92 mHa, 14.4 mHa, 56.4 mHa, and 77.0 μHa, respectively. The second source of error comes from extrapolation to the CBS limit, using Eqs. (14) and (15), and is less under control. For these 67 systems, the maximum and median CBS extrapolation distances are 21.8 mHa and 6.47 mHa, respectively. This CBS extrapolation error is likely to be importantfor those systems where the extrapolation distance (the energy difference between the black dots and red crosses in Fig. 5) is large. To further study the magnitude of the CBS extrapolation error, we add the PBE-based basis-set correction discussed in Sec. IV to the SHCI energies for each basis set [see Eqs. (16) and (17)] and then extrapolate the corrected energies to the CBS limit according to Eq. (22), which gives us an alternative way to estimate the CBS limit of the SHCI energies. The PBE-based corrections can also be found in Ref. 66. It is apparent from Table I that the deviations of the SHCI and the SHCI + PBE energies from experiment are strongly corre- lated, thereby giving us a reasonable measure of confidence in our two extrapolations as well as an estimate of the extrapolation errors. Figure 6 shows the same information as Fig. 5 after the PBE-based basis-set correction has been included. As summarized in Table II, for each basis set, the MAD from experiment decreases by about a factor of three compared to that without the basis-set correction.61 In particular, SHCI + PBE gives a MAD of only 0.55 kcal/mol already with the cc-pVQZ basis set. The cc-pV5Z basis set has a MAD of only 0.49 kcal/mol. Applying the CBS extrapolation to SHCI + PBE gives a somewhat larger MAD from experiment of 0.51 kcal/mol, as the computed atomization energies are too small for the smaller basis sets but increase with increasing basis size, and for the majority of the molecules, the computed CBS atomization energies are larger than experiment. J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The comparison of SHCI atomization energies with experiment in the individual basis sets and in the extrapolated complete-basis-set limit. The top panel is a blowup of the top portion of the bottom panel. The shaded region indicates chemical accuracy (1 kcal/mol). FIG. 6 . Same as Fig. 5 but with the PBE-based basis set correction applied. The extrapolation distances are much reduced compared to Fig. 5. J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Summary statistics of deviations from experimental atomization energies for the 55 molecules. For each of the basis sets (but not for the CBS limit), the inclusion of the PBE-based basis-set correction reduces the MAD by about a fac- tor of 3. MAD: mean absolute deviation; MAX: maximum absolute deviation; units: kcal/mol. Method MAD MAX SHCI cc-pVDZ 20.77 54.32 SHCI cc-pVTZ 6.83 17.43 SHCI cc-pVQZ 2.47 7.38 SHCI cc-pV5Z 1.20 3.15 SHCI CBS 0.46 2.80 SHCI + PBE cc-pVDZ 6.52 27.72 SHCI + PBE cc-pVTZ 1.47 6.02 SHCI + PBE cc-pVQZ 0.55 3.55 SHCI + PBE cc-pV5Z 0.49 3.36 SHCI + PBE CBS 0.51 3.24 As seen from Figs. 5 and 6, the predicted CBS atomization energy of Si 2H6is more than 3 kcal/mol larger than experiment. However, even the n= 5 value is larger than experiment, so the dis- crepancy cannot be attributed to an inaccurate CBS extrapolation but instead to either inaccurate ZPE, SR + SO, and CV corrections or to errors in the experimental value. The ZPE correction for Si 2H6 is quite large, −30.50 mHa, so even a small fractional error in its esti- mate could account for the discrepancy in the atomization energy. In fact, these statements hold for all the seven molecules in Fig. 6 that have cc-pV5Z atomization energies that are larger than exper- iment by more than 1 kcal/mol. Note that there are several systems for which the atomization energies are overestimated in Figs. 5 and 6 by more than 1 kcal/mol but none for which they are underestimated by more than 1 kcal/mol. The majority of the deviations fall below 1 kcal/mol, reach- ing chemical accuracy as can be seen in Table I and Figs. 5 and 6. As regards those where the deviations are larger than 1 kcal/mol, it should also be kept in mind that in addition to the uncertainties in the corrections, especially the ZPE correction, the experimen- tal values may also be inaccurate, particularly for those atomiza- tion energies that are not available from the ATcT database.55For example, for PH 2, the two available experimental values differ by 4.5 kcal/mol and our computed value differs by +1.5 kcal/mol from Ref. 24 and −3.0 kcal/mol from Ref. 57. For the molecules in the ATcT database, the MAD is only 0.24 kcal/mol before the PBE- based basis set correction is applied and 0.32 kcal/mol after it is applied. Compared to other methods, our MAD of 0.46 kcal/mol is sig- nificantly less than the MAD of 1.2 kcal/mol–3.2 kcal/mol obtained in various QMC studies.28–30Diffusion Monte Carlo works directly in the CBS limit, but the fixed-node approximation is the dom- inant error. Using trial wave functions with Slater determinants chosen from an SCI method, it should be easily possible to reduce considerably the fixed-node error, as demonstrated in Refs. 15, 62, and 63. Our MAD is comparable to results reported from com- posite coupled-cluster-based methods.24,64,65The HEAT studies per- formed all-electron calculations using the coupled-cluster methodwith up to quadruple excitations on a somewhat different set of molecules consisting solely of first-row elements.25Unfortunately, none of the molecules for which we have discrepancies of more than 1 kcal/mol were included. For the 19 molecules also present in the G2 set, the MAD of HEAT, SHCI, and SHCI + PBE is 0.07 kcal/mol, 0.16 kcal/mol, and 0.27 kcal/mol, respectively. It should be noted that HEAT is a composite quantum chemistry method, and for the lower levels of theory, it employs larger basis sets than those we used, thereby significantly reducing the CBS extrapolation error. VII. CONCLUSION AND OUTLOOK The SHCI method enables the calculation of essentially exact energies within basis sets up to cc-pV5Z of all the molecules in the G2 set. After extrapolation to the CBS limit and addition of ZPE, SR + SO, and CV corrections, the MAD from the experimental atomization energies is only about 0.5 kcal/mol. However, depend- ing on whether we use the PBE-based basis-set corrections or not, there are seven or nine molecules where the computed atomiza- tion energy is more than 1 kcal/mol larger than experiment (and none for which it is more than 1 kcal/mol smaller than experi- ment). These differences are mostly due to a combination of errors in the various corrections applied and in the experiments rather than lack of convergence of the SHCI energies to the FCI energies. With additional computational effort, it would be possible to reduce the uncertainties in the computed energies. First, instead of adding on a CV energy correction, one could use the cc-pwCVnZ basis sets to include the correlation contribution from the core electrons. This could also make the basis-set extrapolation more reliable. Although this entails a large increase in the Hilbert space, the increase in the computational cost of the SHCI is not prohibitive because rel- atively few of the core excitations have a large amplitude. Second, relativistic effects could also be included within the SHCI method, as has already been demonstrated.65Third, the computation of the ZPE correction would require calculating derivatives with respect to the nuclear coordinates. This could also be done but would be the most computationally expensive part of the calculation. Fourth, the CBS extrapolation could be improved either by employing bet- ter basis sets or using a better DFT-based basis-set correction that employs the SHCI rather than the HF density matrix. With these improvements, the computed energies could be sufficiently accu- rate to reliably pinpoint errors in experimental values of atomization energies. ACKNOWLEDGMENTS This work was supported in part by the AFOSR under Grant No. FA9550-18-1-0095. Y.Y. acknowledges support from the Molec- ular Sciences Software Institute, funded by U.S. National Science Foundation Grant No. ACI-1547580. Some of the computations were performed at the Bridges cluster at the Pittsburgh Supercom- puting Center supported by NSF Grant No. ACI-1445606. We thank Pierre-François Loos for valuable comments on the manuscript and helping us to converge the HF calculation of Si 2to the correct3Σ− g ground state, and one of the referees for suggesting that we use the cc-pV( n+d)Z basis sets to improve the basis-set convergence. J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp DATA AVAILABILITY The data that support the findings of this study are avail- able within the article and the supplementary material of the arXiv version of this paper.66 REFERENCES 1A. A. Holmes, N. M. Tubman, and C. J. Umrigar, J. Chem. Theory Comput. 12, 3674 (2016). 2S. Sharma, A. A. Holmes, G. Jeanmairet, A. Alavi, and C. J. Umrigar, J. Chem. Theory Comput. 13, 1595 (2017). 3A. A. Holmes, C. J. Umrigar, and S. Sharma, J. Chem. Phys. 147, 164111 (2017). 4J. E. T. Smith, B. Mussard, A. A. Holmes, and S. Sharma, J. Chem. 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S. Levine, D. Hait, M. Head-Gordon, and K. B. Whaley, J. Chem. Theory Comput. 16, 2139 (2020). 60The extrapolation distance depends on the value of ϵ1in Eq. (2) and on how well the orbitals are optimized to improve the convergence of the energy. 61To avoid confusion, we note that in Ref. 34 it was found that CCSD(T)+PBE had a MAD of only 1.96, 0.85, and 0.31 kcal/mol with respect to the CCSD(T) CBS limit for the cc-pVDZ, cc-pVTZ, and cc-pVQZ basis sets, respectively. These con- siderably smaller values compared to those in Table II are the result of a one-body basis-set correction that was always included by adding the cc-pV5Z HF energy to the CCSD(T) correlation energies for the different basis sets. Of course, one could do the same for the SHCI energies in the current paper. 62E. Giner, R. Assaraf, and J. Toulouse, Mol. Phys. 114, 910 (2016). J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 63M. Dash, S. Moroni, A. Scemama, and C. Filippi, J. Chem. Theory Comput. 14, 4176 (2018). 64J. M. L. Martin and G. de Oliveira, J. Chem. Phys. 111, 1843 (1999). 65R. Haunschild and W. Klopper, J. Chem. Phys. 136, 164102 (2012).66Y. Yao, E. Giner, J. Li, J. Toulouse, and C. J. Umrigar, arXiv:2004.10059. See also the supplementary material which is available at arxiv.org/src/2004.10059/anc. This contains files with geometries, HF, CCSD, CCSD(T) and SHCI energies, and PBE-based basis set corrections. J. Chem. Phys. 153, 124117 (2020); doi: 10.1063/5.0018577 153, 124117-13 Published under license by AIP Publishing

5.0009045.pdf

Rev. Sci. Instrum. 91, 114705 (2020); https://doi.org/10.1063/5.0009045 91, 114705 © 2020 Author(s).Microstrip resonator for nonlinearity investigation of thin magnetic films and magnetic frequency doubler Cite as: Rev. Sci. Instrum. 91, 114705 (2020); https://doi.org/10.1063/5.0009045 Submitted: 30 March 2020 . Accepted: 31 October 2020 . Published Online: 13 November 2020 B. A. Belyaev , A. O. Afonin , A. V. Ugrymov , I. V. Govorun , P. N. Solovev , and A. A. Leksikov Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi Microstrip resonator for nonlinearity investigation of thin magnetic films and magnetic frequency doubler Cite as: Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 Submitted: 30 March 2020 •Accepted: 31 October 2020 • Published Online: 13 November 2020 B. A. Belyaev,1,2 A. O. Afonin,1 A. V. Ugrymov,1 I. V. Govorun,1 P. N. Solovev,1 and A. A. Leksikov1,a) AFFILIATIONS 1Kirensky Institute of Physics SB RAS, Krasnoyarsk 660036, Russia 2Siberian Federal University, Krasnoyarsk 660041, Russia a)Author to whom correspondence should be addressed: a.a.leksikov@gmail.com ABSTRACT A structure that consists of a λ/4 stepped-impedance microstrip resonator is proposed as an instrument for the investigation of nonlinear effects in thin magnetic films and also can be used as a microwave frequency doubler. A conversion efficiency of 0.65% is observed at a one- layer 100 nm Ni 80Fe20thin film at an input signal level of 4.6 W for a 1 GHz probe signal. The maximum measured conversion efficiency (1% at 1 GHz) was achieved for the 9-layer Ni 80Fe20film where 150 nm magnetic layers were separated by SiO 2layers. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009045 .,s I. INTRODUCTION Despite that recently most commercial microwave multipli- ers employ semiconductor elements as nonlinear media for gen- eration of high-order harmonics, a number of applications when semiconductor elements are not suitable still exist. For this rea- son, great attention is devoted to investigations of new methods and materials, which can be used for multiplier designing,1–3for exam- ple, graphene4–6or ferroelectric films.7Magnetic materials are well known as nonlinear media;8–13they are well suitable for nonlinear investigations and applications,14–16and it is obvious that attempts were made to design magnetic frequency converters.17,18At the same time, most of the present investigations are performed at frequencies of spin wave mode generation,8which will be promising in the future but are rather high for modern electronics. Recently, mainly two methods are exploited for excitation in magnetic media of nonlinear harmonics using microwave fields. The most common method, a method of broadband stripline spec- troscopy,19–25is the same as for ferromagnetic resonance (FMR) investigations, when a microstrip or a coplanar line is used for RF- field generation. The main advantage of the method is that the line is not a frequency selective structure, so it can be easily simulated and used in a very wide frequency range, allowing one to use it as agenerator and detector of nonlinear harmonics simultaneously and making it the most common method for the investigation of spin waves traveling in magnetic films. In the other method,26,27independent excitation and detec- tion circuits are used. For example, in Ref. 27, for excitation, an X-band waveguide was employed to excite the second harmon- ics in a YIG sample, while detection was performed by a K-band waveguide. The main advantage of the method is that the frequency of the excitation signal is lower than the cut-off frequency of the receiver waveguide which means that most of the energy trans- mitted to the ferrite element will be transferred to the nonlinear oscillation modes. However, two remarks should be made about the above-mentioned method. Any regular waveguide is transparent for high-order harmonics of its principal eigenmode, meaning that all high-order harmonics excited in the ferrite will also be transferred to the receiver waveguide. In other words, the energy transferred to the ferrite can be involved in the excitation of all nonlinear harmon- ics, reducing the efficiency of the device with the second harmonic generation. In addition, a frequency doubler in waveguide imple- mentation will be tremendous in size for L- and S-bands, which is the scope of the current investigation. This problem can be solved, if one would apply planar electrodynamic resonators, for example, microstrip or stripline resonators. These structures are common in Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 91, 114705-1 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi VHF and UHF ranges in communication systems as a part of system components because of their diminutiveness and suitable quality factor. For this reason, much attention was paid to the resonator structure in the presence of the magnetic material, particularly, fre- quency tuning vs a magnetic state of a material28–31as the resonators are prospective in terms of application in active tunable microwave devices.32–34 In the resonator method, one resonator is used to excite the media and to measure its nonlinearity by transferring high-order harmonics to the output of the device. The main advantage of this method is that it operates in the standing wave mode, meaning that the energy stored in the resonator will be proportional to its loaded quality factor. Hence, much more energy will be available to excite nonlinearity in the media. It is well known that a regu- lar microstrip resonator has an equidistant spectrum of eigenfre- quencies, so it has the same disadvantage as the waveguide method. However, the incorporation of steps in the width of the resonator allows one to displace the eigenfrequency of resonant oscillation modes. In addition, in the best case, presented in the current inves- tigation, the frequency of the second oscillation mode will be twice the frequency of the first mode, while frequencies of the 3rd and 4th modes will be shifted to much higher frequencies. Hence, most of the energy injected in the resonator will be used to derive a signal of the second harmonics. A brief explanation of the nonlinear effect is presented in Sec. II. The proposed resonator construction for the investigation of the nonlinearity of magnetic media will be presented in Sec. III. The experimental setup will be discussed in Sec. IV, while the experimental results for Ni 80Fe20thin magnetic films will be pre- sented in Sec. V. The results of the investigation are summarized in Sec. VI. II. FREQUENCY DOUBLING IN A MAGNETIC MEDIUM The dynamics of the magnetization Mare described by the Landau–Lifshitz equation, ∂M ∂t=−γM×Heff+α MM×∂M ∂t, (1) whereγis the gyromagnetic ratio, αis the damping parameter, M is the saturation magnetization, and Heffis an effective field, usu- ally composed of three different terms: the applied constant and time-varying magnetic fields, the shape demagnetizing field, and the magnetic anisotropy field. In an equilibrium state, the magnetiza- tion vector is oriented along the effective field ( M×Heff= 0). In the case of an isotropic magnet under the influence of a circularly polarized microwave field hwith components hxandhyapplied in thex–yplane perpendicular to the Heffvector at frequency ω, the magnetization vector will precess about Heffat the frequency of the exiting field in a circular orbit in the x–yplane. In this case, the amplitudes of the dynamic time dependent part m(t) of the mag- netization vector Mare |mx| = |my| and | mz| = 0 [Fig. 1(a)]. However, if the magnet is anisotropic or the driving microwave field is lin- early polarized, the magnetization oscillation in the x–yplane may become elliptical (| mx|≠|my|). However, since the magnetization vector length must be constant, the non-zero mzcomponent will emerge [Fig. 1(b)]. From this simplified picture readily follows that mzvaries at a doubled frequency 2 ωof the excitation field.26 FIG. 1 . Sketch of magnetization dynamics: (a) an isotropic magnet excited by the circularly polarized microwave field (|m x| = |m y|) and (b) an anisotropic magnet excited by the linearly polarized field (|m y|>|mx|). In the second case, a double frequency component of magnetization m zarises. Using the second-order approximation, the amplitude of the double frequency component mzcan be expressed as mz=1 4M∣m2 x−m2 y∣, (2) where mx=χxx(ω)hx+iχxy(ω)hyand my=iχyx(ω)hx+χyy(ω)hy are the components of the dynamic magnetization determined from the solution of the linearized Landau–Lifshitz equation with χxx, χxy,χyx, andχyybeing the components of the magnetic suscep- tibility tensor.35In general, the magnitude of the magnetic sus- ceptibility reaches its maximum when the frequency of the exci- tation field is close to the ferromagnetic resonance frequency of the magnet. The expression (2) shows why a thin magnetic film is a preferred medium for the excitation of nonlinear magnetiza- tion dynamics: Because of the film’s shape magnetic anisotropy, it demonstrates a very high | my|/|mx| ratio ( x-axis being normal to the film plane). From the expression (2) also follows that the frequency doubling in a magnetic medium is a quadratic effect—the output power at frequency 2 ωincreases quadratically with the input power at frequency ω. Concluding this section, we should note that the general solution of the inherently nonlinear Landau–Lifshitz equa- tion reveals rather complex dynamics of magnetization, with the excitation of high-order harmonics with exponentially decreasing amplitudes.36,37 III. RESONATOR STRUCTURE The microstrip resonator is a well-known instrument for mate- rial investigation.38λ/2 microstrip resonators are easier in fabrica- tion and usually have a higher measured unloaded quality factor thanλ/4 microstrip resonators, which are twice smaller than the Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 91, 114705-2 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 2 . Resonator construction and its photo in the middle. first mentioned. This state is very important in terms of the fabri- cation of a frequency doubler for a commercial microwave system. However, a regular λ/4 microstrip resonator has a frequency of the second oscillation mode equal to 3 f0(f0—frequency of the funda- mental oscillation mode), so one will need to make irregularities in the resonator strip to shift the frequency of the second oscillation mode to a frequency of 2 f0. In Fig. 2, the structure of the proposed microstrip resonator for the investigation of nonlinear dynamics of thin magnetic films is presented, which can be used as a base for a magnetic frequency dou- bler. The resonator is divided into two parts: one ( S1) that is closer to the grounded end of the strip has an air-gapped structure, where a substrate with a film under investigation is placed below the strip, while the second part is a stepped impedance microstrip resonator designed on a dielectric substrate. The thickness of the air gap and the length of the segment of the resonator are strongly dependent on parameters of a thin magnetic film, particularly on the size of the dielectric substrate used in film synthesis. The width of the resonator segment at the air-gapped part is taken as an optimum between the current density in the strip and the amount of magnetic material involved in the harmonic generation, and in our case, when the size of a film is 8 ×11 mm2, it is equal to 1.0 mm. It means that frequen- cies of resonator’s modes can be tuned only by changing parameters of the microstrip part of the structure, particularly, the width and the length of segments S2andS3(w2,w3,l2, and l3). To investigate the generation of the second harmonics in a magnetic film, one will need to shift the frequency of the second oscillation mode to a double frequency of the principal oscillation mode (2 f0). It can be obtained by making a step of the width of the segments S2andS3(w2>w3). It should be noted that, in the best case, the middle of the segment S2should be situated in the antinode of the electric field as the output of the resonator is also connected to the middle of the segment through a SMD capacitor. This solution will allow one to isolate the output of the resonator from the excita- tion signal. It is obvious that a change in the width of the segments will lead to a change in the frequency of the first mode of the res- onator, which can be tuned by lengths l2andl3, whereas parameters w1andl1are fixed. Another feature of the resonator is that the step of the width that shifts even modes to a lower frequency also shifts odd modes to a higher frequency, thereby breaking the frequency multiplicity and the equidistance of the oscillation modes in the res- onator. In terms of the nonlinearity investigations, this means thatthe system will be sensitive only for the second harmonics generated in a film, and all other modes, allowed for generation in the film, are prohibited for the resonator. The resonator is connected to the input port at the open end of the segment S 3, ensuring that it has a proper matching with feedlines at a frequency of the principal oscillation mode. The overall degree of matching is determined by the value of capacitance of the SMD capacitor used to connect with the excitation port. For the current investigation, a resonator with the frequency of the principal oscillation mode equal to 1 GHz was designed and fabricated. The parameters of the resonator were as follows: w1 = 1.0 mm, w2= 9 mm, w3= 0.3 mm, l1= 12 mm, l2= 2.7 mm, and l3= 24.1 mm. A 0.508 mm Rogers RO4003B was used as a dielec- tric substrate for stepped impedance segments of the resonator. The width of the feedlines connecting the external ports with the capac- itors is 1.1 mm, which correspond to the 50 Ω impedance of the microstrip lines fabricated on the substrate. A value of the capaci- tance of SMD components was obtained during the measurements, and it was found that in terms of the second harmonics excited in the film and transferred to the output of the device, the feedlines should be connected to the resonator using 1 pF capacitance. An increase and a decrease in the capacitance lead to a decrease in the level of the second harmonics emerging the device, which is explained by the existence of an optimum degree of matching with feedlines. An increase in the matching degree decreases the loaded Q-factor of the resonator, meaning that less energy can be transferred to a film from the resonator for the harmonic generation in one period of time. On the other hand, a decrease in the matching degree, which increases the value of the Q-factor, reduces the overall level of energy trans- mitted through the resonator due to enlarged reflected power from the resonator. FIG. 3 . Resonator frequency response in the absence of a magnetic film (black line) and in the case of the harmonic generation (red line). Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 91, 114705-3 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi The frequency dependence of the fabricated resonator is pre- sented in Fig. 3, in the absence of a magnetic film and in the state of the second harmonic generation. It can be seen that in the state of the harmonic generation, the frequency of the second oscillation mode of the resonator exactly doubles (2 GHz) the frequency of the princi- pal mode (1 GHz) at which the film is excited. At the same time, the frequency of the third oscillation mode is 5.5 GHz, which suggests that the one nonlinear harmonic was excited in the structure. IV. MEASUREMENT SETUP To purify the probe signal and in order not to overload the input circuit of the spectrum analyzer used to measure the signal, two bandpass filters were designed and fabricated. The first one, con- nected to the input of the resonator, has a passband at the frequency of the first oscillation mode of the resonator, while the frequencies of the second, the third, and the fourth modes are located inside the stopband of the filter. The filter’s task is to purify a probe signal, by filtering harmonic and non-harmonic components of the signal at the output of a microwave generator, which are further additionally amplified before the resonator. The second filter has a passband at the frequency of the nonlinear harmonic under investigation, while the frequency of the probe signal and frequencies of another har- monic signal are located in the stopband of the filter. The filter locks the probe signal in the resonator forcing it to transform into a non- linear harmonic. In addition, the filter is protecting the input circuit of a spectrum analyzer from a probe signal and allows one to make a proper measurement of a harmonic level. First of all, let us consider how a high-level probe signal influ- ences the input schemes of spectrum analyzers and distorts the level of the signal under investigation. A 20 dBm probe signal was used in two schemes that differ from each other only by the sec- ond bandpass filter installed at the output of the resonator filled with a clear quartz substrate. One can see from Fig. 4 that the spec- trum of the probe signal (black line) consists of harmonics generated and amplified before the resonator, as it has been previously men- tioned. Next, the measured signal is strongly distorted in terms of increased noise level for frequencies less than 3 GHz. The inclu- sion of the second filter reduces the level of the signal at 1 GHz on 76 dB and at 3 GHz on 100 dB, and, what is more important, a FIG. 5 . Measurement setup. HC—Helmholtz coil, VNA—vector network ana- lyzer R&S ZVA50, Gen—microwave generator R&S SMA100B, WBPA—wideband power amplifier R&S BBA150, BPF1—bandpass filter 1, BPF2—bandpass filter 2, and SA—spectrum analyzer R&S FSW. 19 dB decrease of the signal level of the second harmonics is observed, which is much closer to a specified level for the generator (>60 dBc) [Fig. 4(a)]. The connection of the first bandpass filter before the resonator reduces the harmonic level in the probe signal on 85 dB, as it can be seen from Fig. 4(b), so one can be sure about the purity of the probe signal. The presented method was previously39used to investigate the nonlinear behavior of a plasma antenna and, for the first time, allowed to confirm that plasma antennas are strictly nonlinear devices, previously presented results left questions and doubts about the nature of the measured harmonics. The whole scheme of the investigation is presented in Fig. 5; it consists of two parts, one with a vector network analyzer as the main sensitive instrument, which is used to measure the resonator fre- quency response and make a slight frequency tuning, in the presence of magnetic materials, and the tuning of the filters in the presence of the resonator. The second part of the scheme is the standard scheme for nonlinearity investigations, which consists of a microwave gen- erator, a broadband microwave power amplifier, and a spectrum analyzer. An external magnetic field used for biasing a thin film under investigation is produced by two orthogonal Helmholtz coils that allow obtaining the angular dependence of harmonic generation for the fixed resonator and a magnetic film inside it. A comparison FIG. 4 . Probe signal spectrum measured by the spectrum analyzer (black lines): (a) in the presence of the bandpass fil- ter 2 (red line) and (b) in the presence of the bandpass filters 1 and 2 (red line). Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 91, 114705-4 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 6 . Comparison of probe (1 GHz) and measured (2 GHz) signals in frequency (left panel) and time (right panel) domains. of the probe and measured signal is presented in Fig. 6 for frequency (left panel) and time (right panel) domains. V. EXPERIMENTAL RESULTS For the current investigation, a 100 nm thin film was deposited from the Ni 80Fe20target in the presence of a bias magnetic field in the chamber during a dc-magnetron deposition on a 0.5 mm quartz substrate. An external magnetic field, applied during depo- sition, induced in the film an in-plane magnetic uniaxial anisotropy. Two samples with lateral size 8 ×11 mm2were further cut off from the film so that sample 1 has the easy axis of magnetization (EA) along the short side of the sample, while, in sample 2, the EA was directed along the long side. As the first step of the investigation, both samples were char- acterized by means of a local FMR spectrometer40to obtain the following magnetic parameters of the film: For sample 1, H FMR = 8.1 Oe,ΔHFMR = 3.7 Oe, M s= 914 emu/cm3, H k= 2.73 Oe, and T k= 89.9○. For sample 2, H FMR = 14 Oe,ΔHFMR = 3.7 Oe, M s= 917.2 emu/cm3, H k= 2.96 Oe, and T k= 1.7○. Here, H FMR is the field of ferromagnetic resonance, ΔHFMR is the resonance linewidth, M sis the saturation magnetization, H kis the anisotropy field, and T kis the angle of the easy axis. In Fig. 7, the results for the level of the second harmonics vs the angle of the external field are presented at the level of the probe signal limited to 100 mW. The measurements were performed at the field being higher than the anisotropy field obtained during FMR spectrum investigation. Figure 7 consists of the four curves: 1. yellow curve: sample 1 short side oriented along the strip; 2. green curve: sample 1 long side oriented along the strip; 3. black curve: sample 2 short side oriented along the strip; 4. red curve: sample 2 long side oriented along the strip. One can see from Fig. 7 that the maximum level of the sec- ond harmonics is observed, when induced magnetic anisotropy and shape anisotropy give the same contribution to the harmonic gen- eration (red curve), while the smallest one is observed when the anisotropies are in opposition (black curve). At the same time, theshape anisotropy gives a higher contribution than the induced one, according to the difference between the maximum levels in the yellow and black curves. The maximum of harmonic generation is observed at an angle of 25○from the hard axis. This behavior of the angular dependencies can be explained as follows: when the film is placed in the resonator so that the direction of the induced anisotropy (EA) is oriented along the resonator strip and the external field is applied perpendicular to it, the coupling between the resonator and magnetic layer is rather small, allowing magnetic dynamics in the film to develop indepen- dently with the minimum influence of the resonator, which brings the system to the highest quality factor, which is proved by a narrow linewidth. In addition, in such a configuration, a bigger amount of magnetic material is incorporated in the harmonic generation, as the shape anisotropy is directed along the resonator strip. On the opposite, when the EA of the induced anisotropy is directed along the microwave field from the resonator, the resonator FIG. 7 . Angular dependences of the harmonic generation for both the samples. Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 91, 114705-5 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 8 . Field dependences of the harmonic generation for both the samples. has a maximum coupling with the film, reducing the quality factor of generation and decreasing the overall level of harmonics. Next, the behavior of black and green curves displays that the volume of the magnetic material, which can be incorporated in the harmonic generation, contributes more than the magnetic state in a film. For the optimal angles, obtained from Fig. 7 (47○—black curve, 25○—red curve, 40○—green curve, 30○—yellow curve), the field dependences of the level of the second harmonics were measured, which are presented in Fig. 8. One can see that the results obtained for the angle dependence are proved for the field dependence: The maximum of the harmonic level is observed for the case when the FIG. 9 . Level of the second harmonics vs input power for both samples and their quadratic fit. FIG. 10 . Conversion efficiency of the second harmonics vs input power for the 9-layer Ni 80Fe20film. shape and induced anisotropies have the same contribution; at the same time, the magnetic field, which corresponds to the maximum harmonic level, is 1.5 times higher (9 Oe for red curve) than for the case where anisotropies have different contributions, as the direction of the applied external field is closer to the hard axis of the film. The power dependencies of the harmonic generation for the two best cases were obtained for the optimal amplitude and the angle of the applied field (9 Oe/25○—sample 1, 6 Oe/40○—sample 2), which are presented in Fig. 9. The classical quadratic behavior of the power dependence is observed, where, for low powers, the level of the second harmonics does not change much, and at powers above 1 W, there is a significant increase in the harmonic level. The maximum measured harmonic level for the presented magnetic films is 30 mW for the probe signal of 4.6 W, which cor- responds to a conversion efficiency of 0.65% at 1 GHz. For the moment, the maximum observed conversion efficiency of the sec- ond harmonics was measured for the 9-layer Ni 80Fe20film where 150 nm magnetic layers were separated by SiO 2layers and were found to be 1% (43 mW for a probe signal of 4.2 W), whose conversion efficiency vs input power is presented in Fig. 10. VI. CONCLUSIONS A construction of a λ/4 stepped-impedance microstrip res- onator based on a hybrid substrate, which has an air gap in the area of the grounded end, designed to install a dielectric substrate containing a thin magnetic film on its surface is presented. Near the field of ferromagnetic resonance in the film, the high-frequency field of the strip resonator gives rise to a nonlinear regime of mag- netization oscillations in the film, which causes the generation of a high-frequency current in the resonator at a double frequency. The proposed structure can be used either for the investigation of nonlin- ear effects in magnetic films or magnetic samples or as a microwave Rev. Sci. Instrum. 91, 114705 (2020); doi: 10.1063/5.0009045 91, 114705-6 Published under license by AIP PublishingReview of Scientific InstrumentsARTICLE scitation.org/journal/rsi frequency doubler for the case, when semiconductor devices cannot be applied. A method for nonlinear investigation is proposed based on the application of bandpass filters and the resonator, which allows for the measuring of pure nonlinear effects of a film, but not a measurement setup. For two 100 nm Ni 80F20thin films with different magnetic configurations, but deposited during the same dc-magnetron depo- sition, the way to obtain the maximum of the second harmonic generation (a conversion efficiency of 0.65% at 1 GHz) is presented. The maximum conversion efficiency (1%) measured at the 1 GHz probe signal was achieved with the 9-layer Ni 80Fe20film where 150 nm magnetic layers were separated by SiO 2layers. ACKNOWLEDGMENTS This work was supported by the Russian Science Foundation under Grant No. 19-72-10047. The equipment of the Krasnoyarsk Regional Center of Research Equipment of Federal Research Cen- ter⟨ ⟨Krasnoyarsk Science Center SB RAS ⟩ ⟩was used during the measurement. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon request. REFERENCES 1J. Zhao, V. K. Chillara, B. Ren, H. Cho, J. Qiu, and C. J. Lissenden, J. Appl. Phys. 119, 064902 (2016). 2Y. Shen, J. Gao, Y. Wang, J. Li, and D. 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AIP Conference Proceedings 2265 , 030590 (2020); https://doi.org/10.1063/5.0024817 2265 , 030590 © 2020 Author(s).Influence of annealing on boron diffusion from obliquely sputtered Co60Fe20B20 thin films Cite as: AIP Conference Proceedings 2265 , 030590 (2020); https://doi.org/10.1063/5.0024817 Published Online: 05 November 2020 Nanhe Kumar Gupta , Vineet Barwal , Sajid Husain , Lalit Pandey , Soumyarup Hait , Vireshwar Mishra , and Sujeet Chaudhary ARTICLES YOU MAY BE INTERESTED IN Anisotropic gilbert damping in B2 ordered full Heusler alloy Co 2MnAl thin films AIP Conference Proceedings 2265 , 030574 (2020); https://doi.org/10.1063/5.0017134 Tunnel magnetoresistance of 604% at by suppression of Ta diffusion in pseudo-spin-valves annealed at high temperature Applied Physics Letters 93, 082508 (2008); https://doi.org/10.1063/1.2976435 Thick CoFeB with perpendicular magnetic anisotropy in CoFeB-MgO based magnetic tunnel junction AIP Advances 2, 042182 (2012); https://doi.org/10.1063/1.4771996Influence of Annealing on Boron Diffusion from Obliquely Sputtered Co 60Fe20B20 Thin Films Nanhe Kumar Gupta1, Vineet Barwal1, Sajid Husain2, Lalit Pandey1, Soumyarup Hait1, Vireshwar Mishra1 and Sujeet Chaudhary1, a) 1Thin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi-110016, India 2Department of Engineering Sciences, Uppsala University, Box 534, SE-751 21 Uppsala, Sweden a) Corresponding author: sujeetc@physics.iitd.ac.in Abstract We report controlled effect of Boron diffusion on annealing of Co 60Fe20B20 (CoFeB) thin films grown on SiO 2/Si (100) substrates using pulsed dc m agnetron sputtering. X-ray di ffraction studies indicated that the crystallization of CoFeB is achieved above 100°C via the formation of bcc CoFe with (110) preferred orien tation. Saturation magnetizatio n ሺߤMs) of the as-deposited film is found to be 1000 kA/m, which enhanc es upon annealing such that a value of ߤMs of 1375 kA/m is observed in the sample anne aled at 400°C while the coercivit y decreases from 12 mT to 2 mT. A lowest value of 0.0050.002 of the damping constant is evidenced for the sample annea led at 400°C. The tunability of the damping constant via the Boron out-diffusion from CFB achieved by controlling th e annealing temperature is certainly important for spintronics device applications . INTRODUCTION The (Co xFe1−x)80B20 alloys have been extensively studied due to the high tunneling magnetoresistance (TMR) and perpendicular magnetic anisotropy (PMA) in thin films in combin ation with ultrathin MgO layer.1-2 A controlled transition of CoFeB from amorphous to crystalline phase is a ne cessary condition to obtain the giant TMR.3 Post deposition thermal treatment in vacuum results in the TMR enhan cement of CoFeB based magnetic tunnel junctions (MTJs).3 Although CoFeB/MgO system is widely used for in-plane anisotro pic MTJs, recent results4 have shown that it can also meet the requirements of perpendicular MTJs having high thermal stability and low current density for magnetization switching. Moreover, CoFeB has lower coercivity ( Hc),5 high spin-dependent scattering,6 stronger spin- tunneling effect1 resulting in a higher output si gnal for giant TMR ratios in Co FeB electrode-based MTJs. From the literature it can be observed that there exists a shar p contradiction with regards to the changes in the TMR ratios subsequent to Boron diffusion at the interface betwe en the ferromagnetic electrode and tunnel barrier. Some studies7–8 have reported that the TMR enha ncement results from Boron diff usion at the interface of CoFeB and tunnel barrier during annealing because it forms an energetical ly favorable poly-crystallin e Mg-B-O layer in case of CoFeB/MgO system. On the other hand, other groups9–10 have claimed that Boron enrichment in the barrier is detrimental to TMR since it significantly suppresses the majori ty-channel conductance. Furthermore, the crystallization of amorphous CoFeB during thermal annealing at the interface has been repor ted to act as a stimulus to TMR effect due to enhanced coherent tunneling.2, 11 However, the crystallization of CoFeB electrode and changes in the barrier properties due to Boron diffusion at the interfa ce with annealing have not been fully characterized until now. In this work, we have studied annealing effect on controll ed Boron diffusion from CoFeB thin films using structural, static and dynamic magnetization characterization t echniques. DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030590-1–030590-4; https://doi.org/10.1063/5.0024817 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030590-1EXPERIMENTAL DETAILS The Co 60Fe20B20 (CoFeB) thin films were grown on natively oxidized Si (100) su bstrates using pulsed dc- magnetron sputtering. Prior to deposition, Si (100) wafers were cleaned with acetone and propanol using ultrasonic cleaning bath and then loaded into the vacuum chamber of the Excel Instruments make sputtering system. Turbomolecular pump along with the rotary pump were used to obt ain a base pressure of ~2.0 10-7 Torr. Working pressure of 1.1 10-3 Torr was maintained by flowing Ar gas at 12 sccm (details of s ample preparation parameters are given in Table-1). As-deposited Co 60Fe20B20 (~15 nm) thin films were annealed for 1hour (h) in the chamber at 100, 200, 300 and 400°C under high vacuum of ~5.5×10−7 Torr. The films were then capped with the thin Al (~5 nm) layer to protect sample surface from the oxidation. The deposit ion rates of the CFB and Al targets at their respective powers have been calibrated us ing X-ray reflectivity (XRR) meas urements, and the crystallinity of the films was studied by X-ray diffraction (XRD) investigations. The static m agnetization was measured at room temperature using a vibrating sample magnetometer (VSM) option in physical proper ties measurement system {PPMS} ( Quantum Design make Evercool- II model system). The dynamic magnetization properties were cha racterized in the frequency range of 5-10 GHz with the lock-in based ferromagnetic resonan ce (FMR) technique using a vector network analyzer (VNA) and co-planer waveguide transmission line method.12 Table 1 Growth and post anneal ing parameters of Co FeB/Al thin films Power applied to sputter the Co 60Fe20B20 target (W) 140 Power applied to sputter the Al target (W) 60 Base Pressure (Torr) ~2×10–7 Working Pressure (Torr) 1.1×10–3 Deposition rate of CoFeB (Å/s) and Al (Å/s) 1.45 and 0.5 Annealing temperature (oC) RT, 100, 200, 300 and 400 Annealing Time (h) 1 RESULTS AND DISCUSSION Figure 1 (a) shows the XRD patterns of the as-deposited and ann ealed CoFeB thin films. The diffraction peaks were absent in RT grown sample indicating the amorphous state o f the CoFeB thin film so obatined. After annealing above 100°C for 1 hour, the appearance of diffraction peaks poi nts towards the transformation of CoFeB from amorphous to crystalline state. For all the annealed samples, a single diffraction peak was observed at 2θ ~ 44.2°, which corresponds to bcc-CoFe phase with (110) preferred orient ation. At low annealing temperature, the presence of hump indicates that the CoFe nanoc rystallites states are embedd ed in the amorphous CoFeB phase. The prevailing large width of the XRD peak is attributed to the disorderness i n the amorphous phase created by the compositional inhomogeneity due to the expulsion of Boron from the CoFe nanoc rystallites as the solubility of B in bcc-CoFe is limited. Figure.1 . (a) GIXRD patterns, (b) ߤMs vs. field and (c) Saturation ma gnetization and Coercivity as a function of annealing temperature (Lines are guided to the eye) plots for CoFeB thin films annealed at va rious temperatures Figure 1 (b) shows the saturation magnetization curves of CoFeB films. Effective saturation magnetization ( ߤMs) increases with increasing anneali ng temperature. This behavior is contrary to that of most of the ferromagnetic thin 030590-2films where the ߤMs decreases since the interdiffusion occurs during annealing. Th e surprising increase in ߤMs of CoFeB film is attributed to the Boron diffusion. During anneali ng above 100°C, the Boron atoms diffuse away from CoFeB, i.e., towards the adjoining interf ace(s) resulting in the reduction of magnetic impurities and thereby increasing the magnetic moment. Based on the rigid band model, the moment of the transition metals Co and Fe increases by decreasing the metalloid (Boron) concentration. Since the size of the Boron atom is much smaller than the other atoms Co, Fe, and Al, its diffusion is easier and faster than other a toms’ resulting in the magnetization variation during annealing. In addition, the coercivity ߤHc also decreases with the increasing annealing temperature. This is attributed to progressive reduction of the grain boundaries, which are kno wn to hinder the domain wall motion as field is cycled, after annealing. Higher coercivity in films desposited at RT an d/or annealed at the lowest substrate temperature of 100°C could be the result of bond angle anisotropy usually obse rved in amorphous or nanocrystalline CoFeB. The FMR spectra were recorded on CoFeB/Al films in the frequenc y (f) range of 5–10 GHz in field sweep mode. Fig. 2(a). shows the normalized FMR spectra recorded at 8 GHz for CoFeB/ Al thin films annealed at different temperatures. The recorded FMR spectra were fitted with the der ivative of Lorentzian function12 to obtain the values of resonance field ሺߤܪሻ and linewidth ሺߤ∆ܪሻ. To extract the anisotropy field ሺߤܪሻ and effective magnetization ሺߤܯሻ, the f versus Hr plots [see Fig. 2(b)] were fitted using the Kittel equation, Figure. 2. (a) Field-swept normalized in-plane FMR spectra of the CoFeB (15 n m) /Al (5 nm) thin films recorded at 8 GHz., (b) Variation of f vs ߤܪ fitted with Eq. (1), (c)Variation of ߤ∆ܪ vs f for CFB/Al thin films fitted with Eq. (2), (d) Dependence of ߤܪ on the annealing temperat ure (e) Variation of ߤ∆ܪ as a function of annealing temperature. (f) Gilbert damping constant ( ) as a function of ann ealing temperature fo r CFB/Al) heterostru ctures. ݂ൌఓబఊ ଶగ ൣሺܪܪሻ൫ܪܪܯ൯൧భ మ. (1) where, ߛ and ߤ are the gyromagnetic ratio and magnetic permeability. Figure 2 (b) shows the f vs ߤܪ plot fitted with Eq. 1. The values of ߤܯ is found to vary from 1.4334 0.0003 T to 1.5069 0.0003 T for the samples deposited /annealed at RT/400°C. The observed small variation i n ߤܪ (3.5mT to 1mT) possibly arises from changes in interfacial microstructure and orbital moment in the films o wing to the presence of a crystalline phase of CFB annealed at different temperature. 030590-3Figure 2(c) shows the variation of ∆H with the resonance frequency. This linear increase might indic ate that the damping of the precession in the system is governed by the intr insic Gilbert’s phenomena, i.e., magnon-electron (ME) scattering. The observed frequency dependence of ∆H is fitted with the equation, ߤ ∆ܪ ൌ ߤ ∆ܪସగఈ ఊ . ( 2 ) Where, ߤ∆ܪ accounts for line broadening owing to the extrinsic contributi ons ( e.g., scattering due to magnetic inhomogeneities, etc.) to the overall damping. Using equation ( 2), the damping constant values were calculated and results are shown in Fig. 2(f). I t can be seen that the lowest damping constant value of 0.005 0.002 is found to be for the sample annealed at 400°C. The magnetic inhomogeneity ߤ∆ܪ is somewhat similar in all the samples except for the film annealed at 400°C. CONCLUSION In summary, we have grown CoFeB/Al thin films using pulsed dc m agnetron sputtering using alloy target. XRD together with VSM measurements r eveal that the annealing from 1 00 to 400◦C temperatures results in crystallization of CoFe with bcc structure having (110) orientation. Improved c rystallinity and increase in ߤMs value indicate towards Boron segregation at the interface of Si/CoFeB/Al. The least value of damping constant is found to be 0.0050.002 for the film annealed at 400°C. The study showed that the annealing temperature is one of the growth parameter which has the potential to obtain CoFeB thin films wi th desired values of static and dynamic magnetization response, which are indispensable for numerous spintronic appli cations. ACKNOWLEDGMENTS One of the authors (NKG) acknowledges the ministry of HRD, Gove rnment of India, for providing the scholarship. REFERENCES 1. S. Ikeda, J. Hayakawa, Y. Ashiza w a , Y . M . L e e , K . M i u r a , H . H a s egawa, M. Tsunoda, F. Matsukura, and H.Ohno, Appl.Phy. Lett. 93, 082508 (2008). 2. W. G. Wang, C. Ni, G. X. Miao, C. Weiland, L. R. Shah, X. Fan, P. Parson, J. Jordan-sweet, X. M. Kou, Y. P. Zhang, R.Stearrett, E. R. Nowak , R. Opila, J. S. Moodera, and J . Q. Xiao, Phys. Rev. B. 81, 144406 (2010). 3. W. G. Wang, J. Jordan-sweet, G. X. Miao, C. Ni, A. K. Rumaiz, L. R. Shah, X. Fan, P. Parsons, R. Stearrett, E. R. Nowak, J. S. Moodera, and J. Q. Xiao, App. Phy. Lett. 95, 242501 (2009). 4. S. Ikeda, K. Miura, H. Yamamoto, K.Mizunuma, H. D. Gan, M. Endo , S. Kanai, J. Hayakawa, F.Matsukura, and H. Ohno, Nature Matter. 9, 721 (2010). 5. S. Husain, N. K. Gupta, V. Barwal, and S. Chaudhary, AIP Conf. Proc. 1953, 120048 (2018). 6. K. Nagasaka, Y. Seyama, L. Varga, Y. Shimizu, and A. Tanaka, J. Appl. Phys. 89, 6943 (2001). 7. J. J. Cha J. C. Read, W. F. Egelhoff, P. Y. Huang, H. W. Tseng, Y. Li, R. A. Buhrman, and D. A. Muller, Appl. Phys. Lett. 95, 032506 (2009). 8. J. C. Read, J. J. Cha, W. F. Egelhoff, Jr., H. W. Tseng, P. Y. Huang, Y. Li, D. A. Muller, and R. A. Buhrman, Appl. Phys. Lett. 94, 112504 (2009). 9. J. D. Burton, S. S. Jaswal, E. Y. Tsymbal, O. N. Mryasov, and O . G. Heinonen, Appl. Phys. Lett. 89, 142507 (2006). 10. 10.T. Miyajima, T. Ibusuki, S. Umehara, M. Sato, S. Eguchi, M. Tsukada, and Y. Kataoka, Appl. Phys. Lett. 94, 122501 (2009). 11. S. Yuasa and D. D. Djayaprawira, J. Phys. D: Appl. Phys. 40, R3 37 (2007). 12. S. Husain, A. Kumar, V. Barwal, N. Behera, S. Akansel, P. Svedl indh and S. Chaudhary, Phys. Rev. B 97, 064420 (2018). 030590-4

5.0015936.pdf

J. Appl. Phys. 128, 113901 (2020); https://doi.org/10.1063/5.0015936 128, 113901 © 2020 Author(s).The electric and magnetic properties of novel two-dimensional MnBr2 and MnI2 from first-principles calculations Cite as: J. Appl. Phys. 128, 113901 (2020); https://doi.org/10.1063/5.0015936 Submitted: 31 May 2020 . Accepted: 26 August 2020 . Published Online: 16 September 2020 Jia Luo , Gang Xiang , Yongliang Tang , Kai Ou , and Xianmei Chen COLLECTIONS Paper published as part of the special topic on 2D Quantum Materials: Magnetism and Superconductivity ARTICLES YOU MAY BE INTERESTED IN Cleavable magnetic materials from van der Waals layered transition metal halides and chalcogenides Journal of Applied Physics 128, 110901 (2020); https://doi.org/10.1063/5.0023729 Exploring the potential of MnX (S, Sb) monolayers for antiferromagnetic spintronics: A theoretical investigation Journal of Applied Physics 128, 113903 (2020); https://doi.org/10.1063/5.0009558 Theoretical modeling of triboelectric nanogenerators (TENGs) Journal of Applied Physics 128, 111101 (2020); https://doi.org/10.1063/5.0020961The electric and magnetic properties of novel two-dimensional MnBr 2and MnI 2from first-principles calculations Cite as: J. Appl. Phys. 128, 113901 (2020); doi: 10.1063/5.0015936 View Online Export Citation CrossMar k Submitted: 31 May 2020 · Accepted: 26 August 2020 · Published Online: 16 September 2020 Jia Luo,1 Gang Xiang,2,3 Yongliang Tang,1Kai Ou,1 and Xianmei Chen1,a) AFFILIATIONS 1School of Physical Science and Technology, Key Laboratory of Advanced Technology of Materials Ministry of Education of China, Southwest Jiaotong University, Chengdu 610031, China 2College of Physical Science and Technology, Sichuan University, Chengdu 610064, China 3Key Laboratory of Radiation Physics and Technology of Ministry of Education, Sichuan University, Chengdu 610064, China Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Author to whom correspondence should be addressed: chenxianmei@swjtu.edu.cn ABSTRACT The structural, electronic, and magnetic properties of two-dimensional (2D) manganese dibromide (MnBr 2) and manganese diiodide (MnI 2) are investigated using first principles calculations. The dynamical and thermal stabilities of 2D MnBr 2and MnI 2have been illustrated from the phonon dispersion and molecular dynamic calculations. From the phonon dispersion, three Raman-active and threeinfrared-active vibration modes are found. The calculated formation energies and cleavage energies indicate that 2D MnBr 2and MnI 2are energetically stable and could be potentially obtained by exfoliation. The hybrid functional theory is employed to discover that 2D MnBr 2 and MnI 2are wide gap semiconductors. The magnetic frustration is revealed by the calculation of magnetic exchange interaction and magnetocrystalline anisotropy interaction. By analyzing different magnetic orders, the relatively weak magnetic exchange is attributed to thecompetition of the direct exchange and the superexchange interaction. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015936 INTRODUCTION Since the advent of graphene, tremendous research attention has been focused on two-dimensional (2D) nanomaterials due to their rich physical and chemical properties. 1–3Novel electronic properties related to 2D materials have been reported, such asquantum control of valley pseudospin, 4quantum phase transitions in 2D superconductors,5charge density wave regulated Mott insu- lators,6and topological field effect transistors.7On the other hand, magnetism in 2D materials has the potential to achieve a large number of exciting applications in computing, sensing, and datastorage. But, long-range magnetic order is not possible in most ofthe available pristine 2D nanomaterials because of the enhanced thermal fluctuations revealed by the Mermin –Wagner theorem. 8 Fortunately, magnetocrystalline anisotropy makes some materials not limited by the Mermin –Wagner theorem. 2D materials with intrinsic ferromagnetism have been reported in Cr 2Ge2Te6,9CrI3,10and their isostructural compounds,11whiletheir Curie temperature was far below room temperature, which limited their applications. Recently, some other novel 2D materials, such as single layer MnSe 2,12α-RuCl 3,13α-Fe2O3,14and vanadium doped VI 3,15have been added to the library of 2D magnetic materi- als. Compared with ferromagnetism, antiferromagnetic order is more common in magnetic materials. Besides the traditional antiparallel arrangement of spins, the emergence of some new antiferromagneticsequence such as Bloch-type and Néel-type skyrmions, 16triangular magnetic order17has provided a new space for antiferromagnetic applications. In particular, memory devices based on antiferromag-nets have already begun to be explored. 18 Inspired by the similarities to CrI 3, transition metal dihalides are currently of much interest.19The optical, magnetic, and electric properties of Mn-dihalide thin flakes have been experimentally studied,20,21and it was found that Mn-dihalides show diverse mag- netic properties and were predicted to be exfoliated from their bulklayered forms. 22,23But, their exchange interaction andJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 113901 (2020); doi: 10.1063/5.0015936 128, 113901-1 Published under license by AIP Publishing.magnetocrystalline anisotropy have not been fully studied. In this paper, we focused on two manganese dihalides: MnBr 2and MnI 2. Using density functional theory (DFT), we first assessed theirdynamical stability and thermal stability, then we investigated theirelectronic structures, and finally, the magnetic exchange interactionand magnetocrystalline anisotropy were studied. Our work is potentially useful for the design and application of devices based on antiferromagnetic two-dimensional nanomaterials. METHODS Spin-polarized first-principles calculations were performed within the density functional theory (DFT) framework, as imple-mented in the Vienna ab initio simulation package (VASP). 24The core electrons were treated within the projector augmented wave (PAW) pseudopotentials,25and the exchange and correlation inter- action between electrons were described through the generalizedgradient approximation (GGA) of Perdew, Burke, and Ernzerhof(PBE). 26In particular, we used the HSE06 hybrid functional including a 25% non-local Hartree –Fock exchange to correct the underestimated bandgaps and got a more accurate electronic struc-ture prediction. 27The kinetic-energy cut-off was set to 400 eV. Electronic energy minimization was performed with a tolerance of10 −7eV, while the force on each atom was converged within 10−3eV/Å. The Brillouin zone (BZ) sampling was performed using a 9 × 9 × 1 grid for unit cell relaxation calculations and 15 × 15 × 1grid for static calculations. A vacuum larger than 18 Å was appliedto avoid the interaction between the monolayers caused by theperiodic boundary condition. The phonon dispersions were calcu- lated using the frozen-phonon method as implemented in the PHONOPY package 28to check the dynamical stability of 2D MnBr 2and MnI 2. The 6 × 6 × 1 super cell and 5 × 5 × 1 Γcentered k-point sampling were used for phonon calculations. The thermalstability of the structures was tested by ab initio molecular dynam- ics simulations at 500 K for 1000 steps with a time step of 2 fs. RESULTS AND DISCUSSION The top and side views of the optimized geometric structure under consideration are presented in Figs. 1(a) and 1(b), respec- tively. The atomic geometry of 2D MnBr 2and MnI 2is similar to the well-known transition-metal dichalcogenides (TMDCs). As shown in Fig. 1 , each manganese dihalide is composed of three atomic planes: the layer of manganese atoms sandwiched betweentwo layers of Br/I atoms and forms geometrically frustrated trian-gular nets. It is found that both MnBr 2and MnI 2prefer the 1-T crystal structure at the monolayer level. Obviously, each Mn atom surrounded by six neighboring halogen atoms forms six Mn –Br/I bonds and an octahedral environment, and each halogen atomforms three Mn –Br/I bonds with neighboring Mn atoms. We deter- mined the stable structures by optimizing all lattice constants andatomic positions. The optimized lattice parameters of 2D MnBr 2 (MnI 2) is 3.885 Å (4.159 Å), and the bond length is 2.703 Å (2.914 Å), which is close to the experimental values of bulkstructures. 22,29 Before investigating the electric and magnetic properties of 2D MnBr 2and MnI 2, we interrogated their dynamical stability and thermal stability. Figures 2 (a) and 2(b) are the phonon dispersioncurves of 2D MnBr 2and MnI 2, respectively. It is observed that no appreciable imaginary frequency is found in the whole Brillouin zone confirming that MnBr 2and MnI 2monolayers are dynamically stable. Figure 2(c) shows vibration modes at the gamma point. The symbols E and A stand for the double-degenerate and nondegener-ate states, respectively. The R in parentheses stands for the Raman-active mode, and the IR in parentheses stands for the infrared-active mode. The first three vibration modes correspond tothe three acoustic branches. The six optical branches are eitherRaman-active or infrared-active. No Raman and infrared mode orsilent mode is shown because of the inversion symmetry of the 2D MnBr 2and MnI 2structures. Both the phonon dispersion of 2D MnBr 2and MnI 2show a bandgap in optical branches, which are different from that of the 2D MoS 2and WS 2with phonon gaps between acoustic and optical branches. As the phonon gap forbidsthe lattice vibration transporting, 2D MnBr 2and MnI 2could be useful in making low dimensional thermal resistance materials. According to the phonon dispersions, we found that the structureof 2D MnCl 2is unstable so we did not include MnCl 2in our dis- cussion. Then, the molecular dynamics simulations are performedat 500 K for 1000 steps with a time step of 2 fs. After the molecular dynamics simulations, the structure is little rippled without being destroyed, and the results suggest that 2D MnBr 2and MnI 2are thermally stable ( supplementary material , Fig. S1). The thermodynamic stability of 2D MnBr 2and MnI 2can be evaluated from their formation energy using the formula30 Eform¼(Etotal/C0nMμM/C0nXμX)/N, where Etotalis the total energy of the structure, nMandnXrepresent the number of Mn and Br/I atoms, respectively, μMandμXcorre- spond to the energy of each atom in the elemental phase of Mn FIG. 1. Top (a) and side (b) views of the atomic structure of 2D MnBr 2and MnI 2. Purple and brown balls represent Mn and Br/I atoms, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 113901 (2020); doi: 10.1063/5.0015936 128, 113901-2 Published under license by AIP Publishing.and Br/I, respectively, and Nis the total number of atoms. The calculated formation energies of MnBr 2and MnI 2are −2.055 eV/atom and −1.662 eV/atom, respectively. Although the formation energies (absolute value) of MnBr 2and MnI 2are smaller than that of graphene, the calculated formation energies arenegative indicating that the MnBr 2and MnI 2monolayers are ener- getically stable, and their values are very close to those of MoS 2and WSe 2, and consistent with previously reported values.22We also calculated the interlayer interactions of the MnBr 2and MnI 2 bilayer structure using the DFT-D3 method for van der Waalscorrection.31,32The calculated cleavage energies of MnBr 2and MnI 2are 14.8 meV/Å2and 15.3 meV/Å2, respectively, which are smaller than that of graphite. The interlayer interactions are weak enough means that both MnBr 2and MnI 2monolayers have a large probability to be obtained through exfoliation. In order to under-stand the bonding character of 2D MnBr 2and MnI 2, we analyzed their electron localization function (ELF), as shown in Fig. 3 . The value of the ELF near Mn is close to 0.5, which indicates that the valence electron localization degree of Mn is low, while the value ofthe ELF near Br is close to 0.8, indicating that the probability of the FIG. 2. The phonon dispersion of 2D (a) MnBr 2and (b) MnI 2. (c) Vibration modes from left to right correspond to the eigenvalues from bottom to top at the gamma point, respectively. The red, green, and blue arrows stand for the vibration direction along x, y, and z, respectively. FIG. 3. (a) The ELF contour map of MnBr 2, (b) The ELF curve of MnBr 2along the bond axes.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 113901 (2020); doi: 10.1063/5.0015936 128, 113901-3 Published under license by AIP Publishing.electronic localization near Br is high. Therefore, we tend to believe that ionic bonds are formed between Mn and Br atoms. Figures 4(a) and 4(b) show the majority and minority spin band structures of 2D MnBr 2and MnI 2. As can be seen obviously, both 2D MnBr 2and MnI 2are indirect bandgap semiconductors with bandgaps of 4.21 eV and 3.34 eV, respectively, so they can be used as wide band-gap semiconductors. In both 2D MnBr 2and MnI 2, the conduction band minimum (CBM) is located at the K point, while the valence band maximum (VBM) is located at the Γpoint. Our spin-polarized calculations demonstrate that, due to the spin nondegeneracy, 2D MnBr 2and MnI 2possess a large magnetic moment, and the total magnetic moments are 5 μBper unit cell, in good agreement with the magnetic moments of free manganeseatoms, described by Hund ’s rules. Most of the calculated magnetic moments in monolayer Mn-dihalides residing on the Mn site (∼4.69μB), and a small part comes from the halogen atoms. From the spin-polarized total and partial density of states (DOS) ofmonolayer MnBr 2and MnI 2(supplementary material , Fig. S2), it is found that there are several pairs of equal-energy Mn dpeaks and Br/I ppeaks located below the Fermi level, indicating that the hybridization of Mn dorbit and Br/I porbit lead to the formation of chemical bonds between Mn and Br/I atoms, which is consistentwith the electron localization function (ELF) analysis. By comparing the ferromagnetic (FM) and antiferromagnetic (AFM) orders, it is found that the 2D MnBr 2and MnI 2are all geo- metrically frustrated. In order to further analyze this frustration behavior, four kinds of magnetic orders are calculated to obtain theexchange interaction intensity in the system, as shown in Fig. 5 .F o r each type of magnetic order, we have the following relationships: E FM¼2E0/C06J1S2/C06J2S2/C06J3S2, EAFM-1¼2E0þ2J1S2þ2J2S2/C06J3S2, EAFM-2¼3E0/C0J1S2/C03J2S2/C03J3S2, EAFM-3¼6E0/C06J1S2/C06J2S2/C012J3S2, where E 0represents the energy without considering spin, J 1,J2, and J 3represent the nearest-neighbor, next nearest-neighbor, and the third nearest-neighbor exchange interaction, respectively, FIG. 4. The spin-polarized band structure of 2D MnBr 2(a) and MnI 2(b) calcu- lated with the high-accuracy HSE06 functional. The blue and red lines corre-spond to the spin-up and spin-down bands, respectively, and the green horizontal dotted line indicates the Fermi energy level. FIG. 5. Spin density isosurfaces of four different magnetic order structures: (a) FM, (b) AFM-1, (c) AFM-2, and (d) AFM-3. The yellow and blue isosurfaces ind icate the spin-up and spin-down densities, respectively. (e) Schematic diagram of nearest-neighbor J 1, next nearest-neighbor J 2, and third nearest-neighbor J 3.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 113901 (2020); doi: 10.1063/5.0015936 128, 113901-4 Published under license by AIP Publishing.and divalent Mn has a 3d5electronic con figuration, so its spin moments S = 5/2. The calculated exchange parameters J 1,J2, and J 3 are−0.1210 meV ( −0.0414 meV), 0.0904 meV (0.0335 meV), and 0.0152 meV (0.0339 meV) for 2D MnBr 2(MnI 2), respectively. The calculated J 1,J2, and J 3are reasonably close to the experiment results in the bulk counterpart, which shows that the two- dimensional structure could have a similar magnetic continuous phase transition property.33In 2D MnBr 2and MnI 2, the Mn atom is in an octahedral crystal field so the d orbitals could split intolower threefold level T 2gand upper two-fold level E g. The five d electrons of Mn2+ion fully occupy the spin-up channel of T 2gand Eg, while the spin-down channel is totally empty. This will make the direct exchange between Mn to be AFM stable, based onthe band coupling model. 34–36In addition, the Mn –Br–Mn and Mn –I–Mn bond angle is 88.122° and 88.929°, respectively, both angles are close to 90°. So, the super-exchange interactions medi- ated by Br and I favor the FM state, according to the Goodenough – Kanamori –Anderson (GKA) rules.37–39As a result, the competition of the direct exchange and the super-exchange interaction leads tothe relatively weak magnetic exchange. The existence of Br and I could lead to significant spin –orbit coupling interactions. Thus, we take into consideration of the mag- netocrystalline anisotropy energy with a fourth order, as describedin the formulas below, E k¼E0/C0λ1S2cos2π 2/C16/C17 /C0λ2S4sin4π 2/C16/C17 , Etilt¼E0/C0λ1S2cos2π 4/C16/C17 /C0λ2S4sin4π 4/C16/C17 , E?¼E0/C0λ1S2cos2(0)/C0λ2S4sin4(0) : Here, Ek,Etilt,and E ?stand for the energy with magnetic moment parallel to the plane, 45° inclined to the plane, and per- pendicular to the plane, respectively. The calculated anisotropy constant λ1andλ2are 6.0800 e −06 meV, −1.0240e −07 meV, and 9.2800e −06 meV, −2.0480e −07 meV for 2D MnBr 2and MnI 2, respectively, which are very small because Mn2+ions are in spheri- cally symmetric states with the orbital and the spin moments of L = 0 and S = 5/2. CONCLUSION In summary, we have verified the dynamical, thermal, and energetical stabilities of 2D MnBr 2and MnI 2and studied their elec- tronic and magnetic properties. With the hybrid functional theory, the calculated magnetic exchange interaction and magnetocrystal-line anisotropy interactions are reasonably close to experiments.The competition of the direct exchange and super-exchange leadsto the relatively weak magnetic coupling comparing with other 2D materials, such as CrI 3. In monolayer CrI 3, the anisotropy intro- duced by spin –orbit coupling provides considerable magnetism. However, our results show that the anisotropy of pristine 2DMnBr 2and MnI 2is very low. We think the introduction of defor- mation, strain, dopants, and vacancies to pristine 2D MnBr 2and MnI 2may be an effective means to generate anisotropy andenhance magnetism. As a result, 2D MnBr 2and MnI 2offer a testbed for the quantum phase transition because of the magnetic frustrated geometry. Considering that the low symmetry magneticorder brings the ferroelectricity, 40,41the 2D MnBr 2and MnI 2could act as low dimensional multiferroic materials in the future. SUPPLEMENTARY MATERIAL See the supplementary material for the atomic structure of MnBr 2after 1000 steps of molecular dynamics calculation at 500 K and the spin-polarized total and partial DOS for monolayer andbulk MnBr 2/MnI 2. ACKNOWLEDGMENTS This work was supported by the Fundamental Research Funds for the Central Universities (Grant Nos. A0920502052001-245, A0920502051904-65, and A0920502052001-140). 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5.0020430.pdf

J. Chem. Phys. 153, 134103 (2020); https://doi.org/10.1063/5.0020430 153, 134103 © 2020 Author(s).Decomposition and embedding in the stochastic GW self-energy Cite as: J. Chem. Phys. 153, 134103 (2020); https://doi.org/10.1063/5.0020430 Submitted: 01 July 2020 . Accepted: 14 September 2020 . Published Online: 01 October 2020 Mariya Romanova , and Vojtěch Vlček COLLECTIONS Paper published as part of the special topic on Frontiers of Stochastic Electronic Structure Calculations ARTICLES YOU MAY BE INTERESTED IN Simple eigenvalue-self-consistent The Journal of Chemical Physics 149, 174107 (2018); https://doi.org/10.1063/1.5042785 Stochastic time-dependent DFT with optimally tuned range-separated hybrids: Application to excitonic effects in large phosphorene sheets The Journal of Chemical Physics 150, 184118 (2019); https://doi.org/10.1063/1.5093707 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/10.1063/5.0018930The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Decomposition and embedding in the stochastic GW self-energy Cite as: J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 Submitted: 1 July 2020 •Accepted: 14 September 2020 • Published Online: 1 October 2020 Mariya Romanova and Vojt ˇech Vl ˇceka) AFFILIATIONS Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106-9510, USA Note: This paper is part of the JCP Special Topic on Frontiers of Stochastic Electronic Structure Calculations. a)Author to whom correspondence should be addressed: vlcek@ucsb.edu ABSTRACT We present two new developments for computing excited state energies within the GW approximation. First, calculations of the Green’s func- tion and the screened Coulomb interaction are decomposed into two parts: one is deterministic, while the other relies on stochastic sampling. Second, this separation allows constructing a subspace self-energy, which contains dynamic correlation from only a particular (spatial or energetic) region of interest. The methodology is exemplified on large-scale simulations of nitrogen-vacancy states in a periodic hBN mono- layer and hBN-graphene heterostructure. We demonstrate that the deterministic embedding of strongly localized states significantly reduces statistical errors, and the computational cost decreases by more than an order of magnitude. The computed subspace self-energy unveils how interfacial couplings affect electronic correlations and identifies contributions to excited-state lifetimes. While the embedding is neces- sary for the proper treatment of impurity states, the decomposition yields new physical insight into quantum phenomena in heterogeneous systems. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020430 .,s I. INTRODUCTION First-principles treatment of electron excitation energies is cru- cial for guiding the development of new materials with tailored opto- electronic properties. Localized quantum states became the focal point for condensed systems1–7since the lifetime of localized exci- tations can be controllably modified by interfacial couplings.8–11 Besides predicting experimental observables, theoretical investiga- tions thus help elucidate the interplay of the electron–electron interactions and the role of the environment. The excited electrons and holes near the Fermi level are con- veniently described by the quasiparticle (QP) picture: the charge carriers are characterized by renormalized interactions and a finite lifetime limited by energy dissipation, which governs the deexci- tation mechanism. Quantitative predictions of QPs necessitate the inclusion of non-local many-body interactions. The prevalent route to describe QPs in condensed systems employs the Green’s function formalism.12,13The excitation energy and its lifetime are inferred from the QP dynamics. The many-body effects are represented by the non-local and dynamical self-energy,Σ. In practice, Σis approximated by selected classes of interactions, forming a hierarchy of systematically improvable methods.12The formalism also allows constructing the self-energy for distinct states with a different form of Σ. Here, we neglect the vibrational effects, and the induced density fluctuations dominate the response of the system to the excitation. The perturbation expansion is conveniently based on the screened Coulomb interaction, W, which explicitly incorporates the system’s polarizability. The neglect of the higher-order (vertex) terms,12can- celing the “self-screening” error and capturing two-particles inter- actions,14,15leads to the popular GW method.12,16–18This approach predicts the QP gap, ionization potentials, and electron affinities, in good agreement with experiments.16,18Within the GW approxima- tion,Σis a product of Wand the Green’s function, G, in the time domain. Conventional implementations of the GW self-energy scale as O(N4) with the system size, and it is usually limited to few-electron systems. Recent developments have significantly reduced the pref- actor of the conventional GW implementations.19–22Furthermore, a hybrid stochastic–deterministic approach attempted to reduce the J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp cost by constructing the stochastic “pseudobands” when summing over quasi-degenerate states.23In contrast, our stochastic treatment recasts the full self-energy expression as a statistical estimator. The method employs sampling of electronic wavefunctions combined with the decomposition of operators.24–26In practice, this formu- lation decreases the computational time significantly and leads to a linear scaling algorithm.24–26 The stochastic approach employs real-space random vectors that sample the Hilbert space of the electronic Hamiltonian. The statistical error in Σis governed by the number of vectors, which are used for the decomposition of Gand the evaluation of W. Besides the calculation parameters, i.e., the number of stochas- tic vectors, the sampling critically depends on how uniformly the Hilbert space is represented. The statistical fluctuations are often large for localized orbitals in condensed systems (e.g., defects or moiré states) since it is problematic to sample both localized and delocalized states evenly in the real-space representation.27This translates to an increased computational cost for defects, impuri- ties, and molecules.26,27Furthermore, too high fluctuations poten- tially hinder proper convergence and may result in sampling bias. Here, we overcome this difficulty by constructing a hybrid deterministic–stochastic approach. We show how to efficiently decompose Gand Win real-time and embed the strongly local- ized states. In this formalism, the problematic orbitals are treated explicitly without relying on their sampling by random vec- tors. A similar embedding scheme was so far employed only in static ground-state calculations.28The decomposition is gen- eral and can be used to sample arbitrary excitations. Naturally, it is especially well suited for spatially or energetically isolated electronic states. Furthermore, we employ the decomposition tech- nique to compute Σstochastically from an arbitrarily large sub- space of interest. We show that the self-energy separation disentan- gles correlation contributions from different spatial regions of the system. After deriving the formalism of the self-energy embedding and decomposition, we illustrate our methods numerically for nitrogen vacancy in large periodic cells of hBN. This system represents a real- istic simulation of a prototypical single-photon emitter.2,29–32First, we demonstrate the reduction of the stochastic error for the defect states in the hBN monolayer. Next, we study the interaction of the defect in the hBN-graphene heterostructure. II. COMPUTING QUASIPARTICLE ENERGIES A. Self-energy and common approximations The Green’s function ( G), a time-ordered correlator of cre- ation and annihilation field operators, describes the dynam- ics of an individual quasiparticle. The poles of Gfully deter- mine the single-particle excitations (as well as many other properties). Solving directly for G is often technically challenging (or outright impossible). Alternatively, the Green’s function is often sought via a perturbative expansion of the electron–electron inter- actions on top of a propagator of non-interacting particles (i.e., the non-interacting Green’s function, G0). The two quantities arerelated via the Dyson equation G−1=G−1 0−Σ, where Σis a self- energy accounting, in principle, for all the many-body effects absent inG0. Calculations usually employ only a truncated expansion of the self-energy. Despite ongoing developments,15,33,34the most com- mon approach is limited to the popular GW approximation to the self-energy, which is composed of the non-local exchange ( ΣX) and polarization ( ΣP) terms. In the time-domain, the latter operator is expressed as ΣP(r,r′,t)=iG(r,r′,t)WP(r,r′,t+), (1) where WPis the time-ordered polarization potential due to the time- dependent induced charge density.25The potential is conveniently expressed using the reducible polarizability χand the Coulomb kernelνas WP(r,r′,t)=∬ν(r,r′′)χ(r′′r′′′,t)ν(r′′′,r′)dr′′dr′′′. (2) Evaluating the action of WPon individual states is the practical bottleneck of the GW approach. Hence, despite that Eq. (1) together with the Dyson equation should be self-consistent, the QP ener- gies are commonly computed by a “one-shot” correction, denoted asG0W0, where W0is computed from the non-interacting starting point.12,17In practice, Eq. (1) thus contains quantities constructed from the mean-field Hamiltonian, H0, comprising one-body terms and local Hartree, ionic, and exchange-correlation potentials. For WP, it is common to employ the random phase approximation (RPA). Beyond RPA, approaches are more expensive and, in gen- eral, do not improve the QP energies unless higher-order (vertex) terms are included in Σ.15,35 Within this one-shot framework, the QP energies become12 εQP=ε0+⟨ϕ∣ΣX+ΣP(ω=εQP)−vxc∣ϕ⟩, (3) whereε0are eigenvalues of H0and vxcis the (approximate) exchange-correlation potential. In Eq. (3), ΣPis in the frequency domain. B. The stochastic approach to the self-energy The stochastic G0W0method seeks the QP energy via ran- dom sampling of wavefunctions and decomposition of operators in the real-time domain.24–26The expectation value of Σis expressed as a statistical estimator. The result is subject to fluctuations that decrease with the number of samples as 1 /√ N. In this formalism, the polarization self-energy expression is separable.24–26Specifically, for a particular H0eigenstate ϕ, the perturbative correction becomes ⟨ϕ∣ΣP(t)∣ϕ⟩=⟨ϕ∣iG0(t)WP(t)∣ϕ⟩ ≃1 N¯ζ∑ ¯ζ∫ϕ(r)ζ(r,t)uζ(r,t)d3r, (4) where uζ(r,t) is an induced charge density potential and ζis a ran- dom vector used for sampling of G0(discussed in detail in Sec. III and the Appendix). The state ζat time tis J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ∣ζ(t)⟩≡U0,tPμ(t)∣ζ⟩, (5) where the U0,tis time evolution operator, detailed in the Appendix, U0,t≡e−iH0t. (6) The projector Pμ(t) selects the states above or below the chemical potential,μ, depending on the sign of t. In practice, Pμ(t) is directly related to the Fermi–Dirac operator.25,36–38Since the Green’s func- tion is a time-ordered quantity, the vectors in the occupied and unoccupied subspace are propagated backward or forward in time and contribute selectively to the hole and particle non-interacting Green’s functions. The induced potential uζ(r,t) represents the time-ordered potential of the response to the charge addition or removal, uζ(r,t)=∫WP(r,r′,t)¯ζ(r′)ϕ(r′)d3r′, (7) where¯ζspans the entire Hilbert space.24–26 In practice, we compute uζfrom the retarded response poten- tial, which is ˜uζ=∫˜WP(r,r′,t)¯ζ(r′)ϕ(r′)d3r′; the time-ordering step affects only the imaginary components of the Fourier trans- forms of uζand ˜uζ(see the Appendix).13,25,26 The retarded response, ˜uζ, is directly related to the time-evolved charge density δn(⃗r,t)≡n(r,t)−n(⃗r,t=0)induced by a perturbing potentialδv, ˜uζ(r,t)=∭ν(r,r′′)χ(r′′,r′′′,t)δv(r′′′,r′)dr′dr′′dr′′′ ≡∫ν(r,r′)δn(r′,t)dr′, (8) where we define a perturbing potential δv=ν(r,r′)¯ζ(r′)ϕ(r′). (9) Note thatδvis explicitly dependent on the state ϕand the¯ζvector; the latter is part of the stochastically decomposed G0operator. The stochastic formalism further reduces the cost of eval- uating ˜uζ. Instead of computing δn(⃗r,t)by a sum over single- particle states, we use another set of random vectors {η}con- fined to the occupied subspace. Time-dependent density n(r,t) thus becomes24–26,37–39 n(r,t)=lim Nη→∞1 NηNη ∑ l∣ηl(r,t)∣2, (10) whereηlis propagated in time using U0,tand H0in Eq. (6) is implic- itly time-dependent. As common,12,17we resort to the density func- tional theory (DFT) starting point. Since H0is therefore a functional ofn(⃗r,t), the same holds for the time evolution operator, ∣η(t)⟩=U0,t[n(t)]∣η⟩. (11) Furthermore, we employ RPA when computing ˜uζ; this corre- sponds to evolution within the time-dependent Hartree approxima- tion.36,40,41Practical calculations use only a limited number of random states. Consequently, the time evolved density exhibits random fluc- tuations at each space–time point. To resolve the response to δv, we use a two-step propagation whose difference is the δnthat typically converges fast with Nη.24–26 III. EMBEDDED DETERMINISTIC SUBSPACE The stochastic vectors {ζ}and{η}, introduced in Sec. II B, are constructed on a real-space grid and sample the occupied (or unoc- cupied) states. The number of these vectors ( NζandNη) is increased so that the statistical errors are below a predefined threshold. Fur- thermore, the underlying assumption is that {ζ}and{η}sample the Hilbert space uniformly. Here, we present a stochastic approach restricted only to a sub- set of states, while selected orbitals, {ϕ}, are treated explicitly and constitute an embedded subspace. We denote this set as the { ϕ}- subspace. In the context of the G0W0approximation, we use the hybrid approach for (i) the Green’s function, (ii) the induced poten- tial, or (iii) both G0and usimultaneously. In the following, we present each case separately. A. First The non-interacting Green’s function is decomposed into two parts (omitting for brevity the space–time coordinates), G0≡˜G0+Gϕ 0, (12) where Gϕ 0is the Green’s function of the constructed explicitly from {ϕ}as Gϕ 0(r,r′,t)=∑ j∈{ϕ}(−1)θ(t)iϕj(r)ϕ∗ j(r′)e−iεjt, (13) whereθis the Heaviside step function responsible for the time- ordering (corresponding to particle and hole contributions to G0). The complementary part, ˜G0, is sampled with random states as in Eq. (4). Unlike in the fully stochastic approach (where no Gϕ 0term is present), the sampling vectors are constructed as orthogonal to the {ϕ}-subspace, i.e., ∣¯ζ⟩=(1−Pϕ)∣¯ζ0⟩. (14) Here,¯ζ0spans, in principle, the entire Hilbert space and Pϕis Pϕ=∑ j∈{ϕ}∣ϕj⟩⟨ϕj∣. (15) The construction of ˜G0remains the same as in the fully stochastic case: ˜G0is decomposed by a pair of vectors¯ζandζ(t), cf. Eqs. (5), (6), and (14). Note that it is possible to generalize the projector Pϕ, Eq. (15), to an arbitrary {ϕ}-subspace. The particular choice of ϕdoes not affect the decomposition of the Green’s function [Eq. (12)] or the time evo- lution ofζ(t). However, the dynamics of Gϕ 0would require explicit action of U0,tonϕthat is, in principle, not an eigenstate of H0. Furthermore, instead of an intrinsically localized {ϕ}-subspace, one J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp can also employ localized orbitals constructed by a unitary transfor- mation of orbitals.42–44We do not pursue this route here and select {ϕ}-subspace composed from the starting point eigenstates. B. Second The retarded induced potential ˜uζis decomposed through par- titioning of the time-dependent charge density, cf., Eq. (8) (omitting the space–time coordinates), n≡˜n+nϕ, (16) where the nϕis the density constructed from occupied states ϕ (which we assume to be mutually orthogonal), nϕ(r,t)=∑ j∈{ϕ}fj∣ϕj(r,t)∣2. (17) Here, fjis the occupation of the jth state. Note that choosing ϕwithin the unoccupied subspace is meaningless in this context since only occupied states contribute to the charge density nϕ. The complemen- tary part, ˜n, is given by stochastic sampling [Eq. (10)] that employs random vectors ˜η, ∣̃η⟩=(1−˜Pϕ)∣η⟩, (18) whereηspans the entire occupied subspace and ˜Pϕ=∑ j∈{ϕ}fj∣ϕj⟩⟨ϕj∣. (19) The time propagation of the charge density is similar to the fully stochastic case. The deterministic and stochastic vectors evolve as ∣ϕj(t)⟩=U0,t[nϕ(t),˜n(t)]∣ϕj⟩, (20) ∣˜η(t)⟩=U0,t[nϕ(t),˜n(t)]∣˜η⟩, (21) where U0,tis explicitly expressed as a functional of the two den- sity contributions from Eq. (16). Note that these expressions are analogous to Eq. (11). C. Third Both partitionings are used in conjunction. While G0con- tains contributions from both occupied and unoccupied states, only the former are included in ˜uζ. The combined partitioning may use entirely different subspaces for the Green’s function and the induced potential. In Sec. V C, we employ the decomposition in both terms because it yields the best results and significantly reduces statistical fluctuations. IV. DECOMPOSITION OF THE STOCHASTIC POLARIZATION SELF-ENERGY In Sec. III we partition retarded induced potential and the Green’s function, intending to decrease stochastic fluctuations in the self-energy. Here, we use the partitioning to achieve the second goal of this paper—to determine the contribution to ΣP(ω). Conceptually, we want to address quasiparticle scattering by correlations from a particular subspace. In the expression for ΣP(ω)[Eq. (4)], this corresponds to accounting for selected charge density fluctuations in ˜uζ. In practice, we construct the subspace polarization self-energy as ⟨ϕ∣Σs P(t)∣ϕ⟩=1 N¯ζ∑ ¯ζ∫ϕ(r)ζ(r,t)us ζ(r,t)d3r, (22) where we introduced the subspace induced potential us ζ, which is obtained from its retarded form [in analogy to Eq. (8)], ˜us ζ(r,t)=∫ν(r,r′)δns(r′,t)dr′. (23) This potential stems from the induced charge density that includes contributions only from selected orbitals { ϕ}. Note that ns(r′,t)is obtained either from individual single-particle states or from the stochastic sampling of the { ϕ}-subspace under consideration. If the set of ϕstates is large, it is natural to employ the stochastic approach; the density is sampled according to Eq. (10) with vectors ηsprepared as ∣ηs⟩=˜Pϕ∣η⟩, (24) where the projector is in Eq. (19) and vectors ηspan the entire occupied subspace. The time evolution of ηsfollows Eq. (11), i.e., it is governed by the operator U0,tthat depends on the total time-dependent density, ∣ηs(t)⟩=U0,t[n(t)]∣ηs⟩. (25) Hence, despite Σs P(t)contains only fluctuations from a particular subspace, the calculation requires knowledge of the time evolution of both nsand n. In practice, we employ a set of two independent stochastic sam- plings: (i) vectors ∣η⟩describing the entire occupied space and (ii) vectors∣ηs⟩confined only to the chosen { ϕ}-subspace. The first set characterizes the total change density fluctuation and enters U0,t in Eq. (25). The stochastic–deterministic GW algorithm is in the Appendix. V. NUMERICAL RESULTS AND DISCUSSION A. Computational details In this section, we will demonstrate the capabilities of the method introduced above. The starting-point calculations are per- formed with a real-space DFT implementation, employing regu- lar grids, Troullier–Martins pseudopotentials,45and the Generalized gradient approximation (GGA)46functional for exchange and corre- lation. We investigate finite and 2D infinite systems using modified periodic boundary conditions with Coulomb interaction cutoffs.47 The numerical verification for the SiH 4molecule is in Sec. V B. To converge the occupied H0eigenvalues to <5 meV, we use a kinetic energy cutoff of 26 Ehand 64×64×64 real-space grid with the step of 0.3 a0. Large calculations for the VNdefect in hBN monolayer and in hBN heterostructure with graphene are in Secs. V C and V D. The heterostructure is built with an interlayer distance of 3.35 Å. In both cases, we consider relaxed rectangular 12 ×6 supercells containing 287 and 575 atoms. We performed the structural optimization in the QuantumESPRESSO code48together with Tkatchenko–Scheffler’s J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp total energy corrections49and effective screening medium method.50 Our band structure calculations in low dimensional systems have been validated against the QuantumESPRESSO calculations.27 The GW calculations were performed using a development ver- sion of the StochasticGW code.24–26The calculations employ an additional set of 20 000 random vectors used in the sparse stochas- tic compression used for time-ordering of ˜uζ.25The time propaga- tion of the induced charge density is performed for a maximum propagation time of 50 a.u., with the time step of 0.05 a.u. B. Verification using molecular states We verify the implementation of the methods presented in Secs. III and IV on the SiH 4molecule. In particular, we test sepa- rately (i) the construction of the embedding schemes, (ii) decom- position of the time propagation, and (iii) the evaluation of the subspace self-energy. The reference calculation employs only one level of stochastic sampling (for the Green’s function, while the rest of the calculation is deterministic). For small systems, this approach converges fast as the stochastic fluctuations are small.25,26Yet, we need Nζ= 1500 vectors to decrease the QP energies errors below 0.08 eV. An illustration of the self-energy for the HOMO state of SiH 4is in Fig. 1; everywhere in this figure, the stochastic error is below 0.13 eV for all frequencies. We first inspect the results for an embedded deterministic sub- space. Figure 1 shows that the different schemes for the explicit treatment of the HOMO state (Sec. III) produce the same self-energy curve. The inclusion of HOMO in Wis combined with the stochastic sampling of the three remaining orbitals by Nη= 16 random vec- tors. Note that it is not economical to sample the action of WPfor FIG. 1 . Verification for the SiH 4molecule. (a) The comparison of the total self- energies of the highest occupied state (HOMO) obtained with reference and embedding schemes. The figure demonstrates that all five approaches yield iden- ticalΣ(ω). In the plot, G denotes the decomposition scheme where HOMO is embedded in the Green’s function, W denotes the calculation with HOMO state embedded in the screened Coulomb potential, and G+W corresponds to simulta- neous decomposition of G and W. The red dashed line represents the sum over subspace contributions shown in the panel below. (b) Subspace self-energy of the HOMO state that contains only a contribution of a specific state i, denoted as Σiin the legend.small systems,25,26and this calculation serves only as a test case. This treatment yields a statistical error of 0.08 eV, i.e., the same as the reference calculation despite the additional fluctuations due to the η vectors. When the HOMO orbital is explicitly included in G, the result- ing statistical error is below the error of the reference calculation (0.05 eV). Such a result is expected since the reference relies on the stochastic sampling of the Green’s function. The same happens when the frontier orbital is in both Gand W(Nη= 16 random vectors sample the induced charge density). Tests for other states are not presented here but lead to identical conclusions. Next, we verify that the density entering the time-evolution operator U0,t[n(t)] can be constructed from different states than it is acting on. Specifically, the induced charge and the time-dependent densities may be sampled and built by different means. To demon- strate this, we propagate each of the H0eigenstates with U0,t[n(t)], where n(t) isstochastically sampled byNη= 16 random vectors. Only the induced charge density, entering Eq. (8), is computed from the {ϕ} eigenvectors. The agreement with the reference self-energy curve is excellent with differences smaller than the standard deviations at each frequency point [see Fig. 1(a)]. Finally, we inspect the subspace self-energy in which U0,t[n(t)] employs the total charge density sampled by Nη= 16 random vec- tors. Figure 1(b) shows four different Σs(ω) curves corresponding to the contributions of individual H0eigenstates. Since the eigenstates are orthogonal, the total self-energy is simply the sum of individ- ualΣs(ω) components. The additivity of the subspace self-energies is demonstrated numerically in Fig. 1(a). The subspace results illus- trate that HOMO and the bottom valence orbital exhibit the largest amplitudes of Σs P; hence, these two states dominate the correlation near the ionization edge. C. Deterministic treatment of localized states The deterministic subspace embedding should numerically sta- bilize the stochastic sampling and decreases the computational cost. To test the methodology on a realistic system, we consider the elec- tronic structure of an infinitely periodic hBN monolayer containing a single nitrogen vacancy ( VN) per a unit cell with dimensions of 3.0 ×2.6 nm2. The system comprises 1147 electrons with the defect state being singly occupied and hence positioned at the Fermi level. In the current calculations, we enforce spin degeneracy. The reason is twofold: (i) half populated states are strongly polarizable and they exhibit stronger stochastic fluctuations in the time evolu- tion. They are thus a more stringent test of the embedding. (ii) In Sec. V D, we compare the monolayer with a heterostructure to deter- mine the role of substrate material on the self-energy. In the het- erostructure, the magnetic splitting of spin-up/down components disappears.51 The relaxed monolayer geometry shows only mild restructur- ing. The vacancy introduces three localized states with the D3h point group symmetry.51,52The singly occupied in-gap state is labeled by its irreducible representation a′′ 2.52The second state ( e′) is doubly degenerate and pushed high in the conduction region. Due to the enforced spin-degeneracy, the electron–electron inter- actions are increased and e′appears higher than in the previous calculations.29,53–55The a′′ 2and e′single-particle wavefunctions are illustrated in Fig. 2. J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp We first focus on the delocalized top valence and bottom conduction states. The fully stochastic calculations converge fast for both of them. The charge density fluctuations are sampled by Nη= 8 vectors, and the Green’s function requires Nζ= 1500 to yield QP energies with statistical errors below 0.03 eV for the valence band maximum (VBM). The error for the conduction band minimum (CBM) is<0.01 eV. The computed resulting quasiparticle bandgap is 6.49±0.04 eV, in excellent agreement with previous calculations and experiments (providing a range of values between 6.1 eV and 6.6 eV).56–58 To investigate the electronic structure in greater detail, we employ the projector-based energy–momentum analysis based on supercell band unfolding.27,59–61In practice, the individual wave- functions within our simulation cell are projected onto the Brillouin zone of a single hBN unit cell. The resulting band structure is shown in Fig. 2(a). Since our calculations employ a rectangular supercells, the critical point K of the hexagonal Brillouin zone appears between theΓand X points (marked on the horizontal axis by ⋆). The figure shows that the fundamental bandgap is indirect; the direct transi- tion is 7.39 ±0.04 eV, extremely close to the results for the pure hBN monolayer (reported to be in a range between 7.26 eV and 7.37 eV).58,62,63 The defect states appear as flat bands, labeled in Fig. 2 by their irreducible representations. As expected from the outset, the stochastic calculations for a′′ 2exhibit large fluctuations. While each sample is numerically stable, random vectors produce a self-energy curve with a significant statistical error at each frequency point, i.e., the time evolution is strongly dependent on the initial choice of { ζ} and {η}. In this example, only the a′′ 2state exhibits such behavior. Figure 2 illustrates the self-energies for the band edge and the defect states. For all the cases, the plots show the spread of the ΣP(ω)curves: these correspond to the outer envelope for 15 distinct cal- culations, each employing 100 ζsampling vectors (combined with Nη= 8 each). For the a′′ 2defect state, the stochastic sampling is possibly biased. The variation is three times as big as the spread of the VBM and almost seven times larger compared to CBM. The large spread is associated with increased statistical errors (50 meV), notably larger than those of delocalized VBM (30 meV) and CBM (9 meV) states. Away from the QP energy, the fluctuations increase even further; the spread of the samples becomes two times larger near the maximum of ΣPat−20 eV. The convergence of the QP energy is poor, and the low sample standard deviation (roughly twice as big as for VBM) suggests incorrect statistics. In practice, each sampling yields a self-energy curve that lies outside of the standard deviation of the previous simulation. The deterministic embedding remedies insufficient sampling without significantly impacting the computational cost. Naturally, we select the a′′ 2defect state and treat it explicitly (while randomly sampling the rest of the orbitals). Hence, according to the notation of Sec. III, the { ϕ}-subspace contains only a single orbital. The decomposition of the non-interacting Green’s function fol- lows Eq. (12) and stabilizes the sampling. The spread of the self- energy curves decreases approximately three times for a wide range of frequencies. With the embedded a′′ 2state, the statistical error of the QP energy decreases smoothly and uniformly with the number of samples. Each new sampling falls within the error of the calcula- tions. Yet, the final statistical error (0.03 eV) is less than 10% larger than for the delocalized states. The decomposition of the induced charge density alone is less promising. Fundamentally, δn(r,t) contains contributions from the entire system, and a small { ϕ}-subspace will unlikely lead to drastic improvement. Indeed, the statistical errors and FIG. 2 . Deterministic embedding for the nitrogen vacancy in hBN monolayer. (a) Black points represent individual states of the unfolded band structure; blue lines are a guide for the eyes. Panels (b)–(d) depict the self-energy curves (solid line) fore′,a′′ 2, and VBM state, respectively; the states are marked in panel (a) by red dashed rectangles. The statistical errors are smaller than the thickness of the line. Gray shaded areas (on each plot) cor- respond to the “spread” of self-energy curves for 15 distinct fully stochastic cal- culations; blue solid lines are the statisti- cal averages. The light red area in panel (c) represents the spread after deter- ministic embedding of the a′′ 2state; the red solid line is the self-energy curve. The third column contains the electron density of the corresponding states pre- sented at the same isovalue. J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp convergence behavior remain the same as for the fully stochastic treatment. Embedding of the localized state in both G0andδn(r,t) is, how- ever, the best strategy. If both decompositions use the same (or over- lapping) {ϕ}-subspace, the induced charge density and the potential δv[Eq. (9)] share (at least some) ϕstates. Consequently, the sam- pling of ΣP(ω) becomes less dependent on the particular choice of ζvectors, which sample only states orthogonal to { ϕ}. Indeed, the embedding of a single localized state in Gand Wresults in a nearly fourfold reduction of the statistical fluctuation that translates to more than an order of magnitude savings in the computational time. The error of the QP energy of the a′′ 2state is approximately half of the error for VBM (16 meV for Nζ= 1500). For completeness, we also applied the three types of embedding on VBM, which does not suffer from bias or large statistical errors. Unlike for the localized state, we observe only a negligible reduc- tion in the fluctuations. Specifically, the stochastic error for VBM decreases from 30 meV to 29 meV regardless of whether GorG +Wembedding is used. For delocalized states, the fully stochastic sampling is thus sufficient as expected from our previous work.25–27 Note that the total cost of the calculation scales quadratically25 with the number of states used to evaluate the retarded poten- tial˜uζ[Eq. (8)]. An optimal computational cost is achieved when strongly localized states are treated via embedding, but the subspace of delocalized states is sampled stochastically.D. Stochastic subspace self-energy Having an improved description of the localized states in hand, we now turn to the decomposition of the self-energy. The real- time approach described in Sec. IV allows inspecting the many-body interactions from a selected portion of the system. In contrast to Subsection V C, the subspace of interest contains a large number of states (irrespective of their degree of localization), and it is randomly sampled. The goal of this decomposition is to understand the role of correlation, especially at interfaces. To test our method, we investigate a periodic hBN monolayer containing a single VNdefect placed on graphene. Such a het- erostructure has also been realized experimentally.64–67The struc- ture contains 2299 valence electrons, and it is illustrated in Fig. 3 together with selected orbital isosurfaces. The a′′ 2state is ener- getically lower than e′(as in the pristine hBN monolayer). Both defects only weakly hybridize with graphene and remain localized within the hBN sublayer. However, the presence of graphene leads to an increased delocalization of the defect states within the mono- layer68and decrease in the hBN fundamental bandgap, attributed to image-charge effects.69 By unfolding the wave functions of the bilayer onto the hBN conventional rectangular unit cell, we obtain the band structure shown in Fig. 3(a). The graphene and hBN bands are distinguished by a state projection on thedensities above and below the center FIG. 3 . Excitations in an hBN-graphene heterostructure containing a nitrogen vacancy. (a) Calculated band structure—the states localized on graphene are in green, while blue points represent hBN.70Lines serve as a guide for the eyes. The individual isosurfaces are shown on the right for VBM, a′′ 2defect state, and e′defect states. Second row contains self-energy curves with vertical lines indicating position of the QP energies: (b) VBM with QP energy −6.9 eV, (c) a′′ 2defect state with QP energy −2.6 eV, (d) e′defect state with QP energy −1.4 eV, and (e) CBM with QP energy −0.6 eV. The frequency axis in panels (b)–(e) is relative to the vacuum level. Black solid lines are the full heterostructure self-energies; green and blue dashed lines are the Σs P(ω)graphene monolayer and hBN monolayer subspace self-energies, respectively. The blue line represents the self-energy of an isolated hBN monolayer with VN(discussed in Sec. V C). The third row of panels contains the ImΣs P(ω). J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp of the interlayer region. Some states, however, extend apprecia- bly across this boundary and lead to the appearance of spurious low-intensity peaks in the unfolded band structure.70Yet, individ- ual bands are easy to distinguish, and one can clearly disentangle individual hBN and graphene states. The graphene portion of the band structure reproduces the well-known semimetallic features with a Dirac point located at the K boundary of the hexagonal Brillouin zone. As discussed above, the K appears in between Γand X of the rectangular cell, and it is labeled by⋆in Fig. 3. The typical Dirac cone dispersion of graphene is only a little affected by the hBN presence, and the Dirac point remains close to the Fermi level despite the charge transfer from thea′′ 2defect51This is not surprising: sparse charge defects lead to only weak doping of graphene. Furthermore, the previous DFT results employed small supercells and may suffer from significant electronic “overdelocalization”71,72that spuriously enhances charge transfer. The hBN part of the band structure (Fig. 3) is only weakly affected by the heterostructure formation. The delocalized states are qualitatively identical to those in the monolayer (Fig. 2). The fun- damental bandgap remains indirect and reduced to 6.32 ±0.04 eV. The screening introduced by graphene thus leads to a small change inEg(<0.2 eV compared to the monolayer). The positions of the defect states, however, change notably. Here, both appear above the Fermi level due to the charge transfer from a′′ 2to the graphene layer. Due to the charge transfer and Fermi level shift, it is clear that graphene is responsible for altering the charge fluc- tuations, i.e., the polarization part of the self-energy. In prac- tice, graphene acts as a dielectric background inducing a signif- icant screening of the Coulomb interaction in hBN. It stands to reason that the localized defect states would be strongly affected by such a polarizable layer and that the corresponding self- energy should be dominated by the spectral features originating in graphene. To investigate the degree of coupling and the contribution to the self-energy from each monolayer, we compute Σs P(ω)using Eq. (22). Here, the { ϕ}-subspace is constructed from stochastic samples of the 576 occupied graphene states (distinguished by the green color in Fig. 3). For each sampling of the Green’s func- tion, the subspace is described by eight random vectors in the {ϕ}-subspace. Additional eight vectors sample the complementary subspace. Figure 3 shows the decomposition of the self-energy, indicat- ing the contribution of graphene. For the delocalized states, the ΣP(ω) curves of the heterostructure (black solid lines) appear sim- ilar to those of an hBN monolayer (blue solid lines). Specifically, we see that ΣP(ω) for VBM [panel (b)] has practically identical fre- quency dependence between −15 eV and 0 eV. While in-plane con- tributions from hBN (blue dashed line) dominate the entire curve, screening from the graphene (green dashed line) substrate is sub- stantial. A naïve comparison between the self-energy curves of the hBN monolayer and the heterostructure suggests that the enhanced maximum in Re ΣPat−25 eV (marked by an asterisk) is caused extrinsically by graphene. However, this is not the case, and the effect is only indirect: the presence of graphene leads to shifting of the hBN spectral features. At the QP energy (marked by the dashedvertical line), graphene contributes to total ΣPof the heterostructure by 36%. The situation is different for CBM [panel (e)]. Toward the static limit ( ω→0), the self-energy curve becomes more nega- tive. This shift is caused directly by the induced density fluctua- tions in the graphene substrate. The substrate screening renormal- izes the quasiparticle bandgap, as has been shown in the literature of low-dimensional materials.9,27,69,73–75 The self-energy curves of the a′′ 2panel (c) and e′[panel (d)] defect states are significantly different from those computed for the monolayer. In both cases, we observe a negative shift related to the QP stabilization by non-local correlations. The ΣPcontri- butions to the defects’ QP energies are roughly three times larger in the heterostructure. Here, the intrinsic hBN interactions are the major driving force. The defect states are mostly affected by the in-plane induced density fluctuations (constituting 64% and 76% of the total polarization self-energy for a′′ 2and e′). Like the VBM, graphene indirectly acts on the defect QP states by enhanc- ing the in-plane fluctuations, rather than direct coupling to the defects. In the same vein, Im Σ(ω=εQP) of the graphene subspace is small, showing that there is only a small lifetime broadening due to the substrate. In detail, the graphene monolayer contributes to the broadening of the VBM state, a′′ 2state, and e′around twice less than the hBN subspace. The only exception is the CBM state for which the substrate contribution dominates (by a factor of ∼3). As shown in this example, the stochastic subspace self-energy efficiently probes dynamical electron–electron interactions at interfaces. The random sampling allows selecting an arbitrary subspace to iden- tify its contribution to the correlation energy and the excited state lifetimes, obtained from real and imaginary parts of Σ(ω=εQP), respectively. VI. CONCLUSIONS Stochastic Green’s function approaches represent a class of low- scaling many-body methods based on stochastic decomposition and random sampling of the Hilbert space. The overall computational cost is impacted by the presence of localized states that increase the sampling errors. Here, we have introduced a practical solution for the G0W0method. The localized states are treated explicitly (deterministically), while the rest is subject to stochastic sampling. We have further shown that the subspace-separation can be applied to decompose the dynamical self-energy into contributions from distinct states (or parts of the systems). Using nitrogen vacancy in hBN monolayer, we demonstrate that the deterministic embedding dramatically reduces the statis- tical fluctuations. Consequently, the computational time decreases by more than an order of magnitude for a given target level of stochastic error. The embedding may be made within the Green’s function and the screened Coulomb interactions independently or simultaneously; the latter provides the best results. For delocal- ized states, embedding is not necessary and performs similar to the standard uniform sampling with real-space random vectors. The mechanism of embedding presented here is general, and it is the first step toward hybrid techniques that treat selected subspace at the distinct (higher) level of theory. The development of hybrid J. Chem. Phys. 153, 134103 (2020); doi: 10.1063/5.0020430 153, 134103-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp techniques employing various orbital localization procedures is cur- rently underway. We further demonstrate that the subspace self-energy con- tains (in principle, additive) contributions from the induced charge density. The self-energy contribution from a particular por- tion of the system is computed by confining sampling vectors into different (orthogonal) subspaces. We exemplify the capabil- ities of such calculations on defects state in the hBN-graphene heterostructure. Here, the charged excitation in one layer directly couples to density oscillations in the substrate monolayer. The elec- tronic correlations for the defects are governed by the interac- tions in the host layer; the substrate only indirectly affects their strength. This example serves as a stimulus for additional study of the defect states in heterostructures. Here, the coupling strength between individual subsystems is tunable by particular stacking order and induced strains. The subspace self-energy represents a direct route to explore such quantum many-body interfacial phe- nomena. ACKNOWLEDGMENTS This work was supported by the NSF through the Materials Research Science and Engineering Centers (MRSEC) Program of the NSF through Grant No. DMR-1720256 (Seed Program). We grate- fully acknowledge support via the UC Santa Barbara NSF Quan- tum Foundry funded via the Q-AMASE-i program under Award No. DMR-1906325. The calculations were performed as part of the XSEDE76computational Project No. TG-CHE180051. Use was made of computational facilities purchased with funds from the National Science Foundation (Grant No. CNS-1725797) and administered by the Center for Scientific Computing (CSC). The CSC is sup- ported by the California NanoSystems Institute and the Materials Research Science and Engineering Center (MRSEC; Grant No. NSF DMR-1720256) at UC Santa Barbara. APPENDIX: THE STOCHASTIC–DETERMINISTIC GW ALGORITHM The new implementation is derived from the fully stochastic algorithm presented in Ref. 25. The steps (1), (2), and (a) are new, i.e., a result of the embedding introduced in this article. Steps (b)–(d) and (3) are similar to the fully stochastic scheme, and they employ hybrid quantities [i.e., decomposed time-dependent density Eq. (16) and/or the decomposed Green’s function Eq. (12)]. Step 4 is identical to the original algorithm. The steps of the algorithm are as follows: (1) Select a set of deterministic states ϕfor embedding or sub- space decomposition. Construct the projector Pϕ[Eq. (15)] for the Green’s function and/or ˜Pϕ[Eq. (19)] for the retarded induced potential. (2) Create N¯ζ0stochastic vectors (typically in the order of 102–103) for sampling the full Hilbert space. If including deterministic states in the Green’s function, construct¯ζvia projection with Pϕ[Eq. (14)]. For each sampling of the Green’s function, perform the next steps:(a) Create a set of N˜η(typically<10) random vectors in the occupied subspace. Use ˜Pϕ[Eq. (19)] and either ●construct ˜ηvia Eq. (18) for the embedding scheme ●or construct ˜ηsvia Eq. (24) for the subspace self- energy decomposition. (b) Calculate δv(r) by Eq. (9) and perturb ˜ηand/orϕas ˜ηλ(r)=e−iλδv(r)˜η(r), (A1) ϕλ(r)=e−iλδv(r)ϕ(r), (A2) withλ≈10−3−10−5E−1 h. (c) Perform time-propagation in discrete time steps dt(typi- cally 0.05 a.u.) for states ˜ηλ(r)andϕλ(r), ∣˜ηλ(t+dt)⟩=e−iHλ(t)dt∣˜ηλ(t)⟩, (A3) ∣ϕλ(t+dt)⟩=e−iHλ(t)dt∣ϕλ(t)⟩. (A4) Two propagations are needed: with λ= 0 and with a finite value ofλ. We employ the random phase approximation, corresponding to the time-dependent Hartree approach. The time evolution thus depends only on the total charge density constructed using Eq. (16). (d) Compute ˜uζ(t)as the difference of perturbed and unper- turbed Hartree potentials at each time step for λ= 0 and for a finite value of λ, ˜uζ(r,t)=vλ H(r,t)−vλ=0 H(r,t) λ, (A5) and then apply a time-ordering operation13in frequency domain, uζ(r,ω)=⎧⎪⎪⎨⎪⎪⎩˜uζ(r,ω),ω≥0, (˜uζ(r,ω))∗,ω<0.(A6) (3) Compute ζ(r,t) for negative and positive times [for hole and particle components of the Green’s function—Eq. (5)]. If embedding in the Green’s function is applied, construct Gϕ components using Eq. (13). Accumulate the matrix element of the self-energy via Eq. (4). (4) When steps (1)–(3) are performed, average the resulting ⟨ϕ|Σ(t)|ϕ⟩from each sampling of the Hilbert space (which employed¯ζandϕorbitals). Perform Fourier transformation of the self-energy, and solve Eq. (3) in the frequency domain. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. J. Chem. 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5.0020253.pdf

J. Appl. Phys. 128, 155301 (2020); https://doi.org/10.1063/5.0020253 128, 155301 © 2020 Author(s).Impact of UV-induced ozone and low-energy Ar+-ion cleaning on the chemical structure of Cu(In,Ga)(S,Se)2 absorber surfaces Cite as: J. Appl. Phys. 128, 155301 (2020); https://doi.org/10.1063/5.0020253 Submitted: 29 June 2020 . Accepted: 26 September 2020 . Published Online: 15 October 2020 Victor R. van Maris , Dirk Hauschild , Thomas P. Niesen , Patrick Eraerds , Thomas Dalibor , Jörg Palm , Monika Blum , Wanli Yang , Clemens Heske , and Lothar Weinhardt ARTICLES YOU MAY BE INTERESTED IN A density-functional theory study of the Al/AlO x/Al tunnel junction Journal of Applied Physics 128, 155102 (2020); https://doi.org/10.1063/5.0020292 The effect of vacancy-impurity complexes in silicon on the current–voltage characteristics of p–n junctions Journal of Applied Physics 128, 155702 (2020); https://doi.org/10.1063/5.0023411 Comparative ab initio study of the structural, electronic, dynamical, and optical properties of group-I based CuMO 2 (M = H, Li, Na, K, Rb) Journal of Applied Physics 128, 155701 (2020); https://doi.org/10.1063/5.0019961Impact of UV-induced ozone and low-energy Ar+-ion cleaning on the chemical structure of Cu(In,Ga)(S,Se) 2absorber surfaces Cite as: J. Appl. Phys. 128, 155301 (2020); doi: 10.1063/5.0020253 View Online Export Citation CrossMar k Submitted: 29 June 2020 · Accepted: 26 September 2020 · Published Online: 15 October 2020 Victor R. van Maris,1,2 Dirk Hauschild,1,2,3 ,a) Thomas P. Niesen,4 Patrick Eraerds,4Thomas Dalibor,4 Jörg Palm,4Monika Blum,5,6 Wanli Yang,5 Clemens Heske,1,2,3 and Lothar Weinhardt1,2,3 ,a) AFFILIATIONS 1Institute for Photon Science and Synchrotron Radiation (IPS), Karlsruhe Institute of Technology (KIT), Hermann-v.-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany 2Institute for Chemical Technology and Polymer Chemistry (ITCP), Karlsruhe Institute of Technology (KIT), Engesserstraße 18/20, 76131 Karlsruhe, Germany 3Department of Chemistry and Biochemistry, University of Nevada, Las Vegas (UNLV), 4505 Maryland Parkway, Las Vegas, Nevada 89154-4003, USA 4AVANCIS GmbH, Otto-Hahn-Ring 6, 81739 München, Germany 5Advanced Light Source, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA 6Chemical Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA a)Authors to whom correspondence should be addressed: dirk.hauschild@kit.edu andlothar.weinhardt@kit.edu ABSTRACT Dry buffer layer deposition techniques for chalcopyrite (CIGSSe)-based thin-film solar cells lack the surface-cleaning characteristics of the commonly used CdS or Zn(O,S) wet-chemical bath deposition. A UV-induced ozone and/or a low-energy Ar+-ion treatment could provide dry CIGSSe surface cleaning steps. To study the impact of these treatments, the chemical surface structure of a CIGSSe absorber is investigated. For this purpose, a set of surface-sensitive spectroscopic methods, i.e., laboratory-based x-ray photoelectron spectroscopy and x-ray-excited Auger electron spectroscopy, is combined with synchrotron-based soft x-ray emission spectroscopy. After treatment times asshort as 15 s, the UV-induced ozone treatment decreases the amount of carbon adsorbates at the CIGSSe surface significantly, while theoxygen content increases. This is accompanied by the oxidation of all absorber surface elements, i.e., indium, selenium, sulfur, and copper.Short (60 s) low-energy Ar +-ion treatments, in contrast, primarily remove oxygen from the surface. Longer treatment times also lead to a removal of carbon, while extremely long treatment times can also lead to additional (likely metallic) Cu phases at the absorber surface as well. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020253 I. INTRODUCTION Chalcopyrite-based thin-film solar cells and modules can be processed with different buffer layers, such as CdS or Zn(O,S) [by chemical bath deposition (CBD)] or In 2S3[by physical vapor deposition (PVD)]. Research and development of such buffer layers[e.g., Zn(O,S), 1–4(Zn,Mg)O,5,6and In 2S37–10] is motivated by the replacement of Cd, the increase of efficiency by increasing the optical transmission, and/or the replacement of wet processes to improve the environmental footprint by avoiding waste water.Among the alternative approaches, atomic layer deposition, 11ionlayer gas reaction,12and PVD13have been prominently reported. Such “dry”processes, however, lack the surface-cleaning properties of the CBD, which may have a negative impact on the performance of the complete solar cell device. In fact, several studies have shown degradation effects of air-exposed co-evaporated CuInSe 2or Cu(In,Ga)Se 2absorbers on the minute and long-term time scale, leading to a reduction of the open circuit voltage (V OC) in the full solar cell .14,15Surface studies have revealed the adsorption of, e.g., carbon hydroxides and water ( “surface adsorbates ”), the formation of oxides, and segregation as a potential reason for thisJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155301 (2020); doi: 10.1063/5.0020253 128, 155301-1 Published under license by AIP Publishing.degradation.16–20In this work, the physical and chemical impact of two different dry treatments on the surface of Cu(In,Ga)(S,Se) 2 (CIGSSe) absorbers is investigated. Possible candidates for dry cleaning treatments include a UV-induced ozone treatment and a low-energy (50 eV) Ar+-ion surface treatment, which are both candidate treatments for inclusion in a dry in-line process. In the UV treatment, photoexcitation by a low- pressure mercury lamp generates ozone (O 3) from ambient oxygen, which is subsequently decomposed into (highly reactive) atomicoxygen. Hydrocarbons at the surface are excited and dissociated by acharacteristic mercury line (253.7 nm), which makes it more likely for them to react with the atomic oxygen, creating volatile molecules (e.g., carbon oxides and hydroxides) that then desorb from the surface. 21–24 UV-induced ozone treatments are already used in production processes for other thin-film disciplines.25,26For indium-tin-oxide (ITO) thin films, it is reported that a UV treatment reduces the relative concentra- tion of carbon atoms and forms a Sn-deficient and O-rich surface,27 but other studies indicated “not much change ”in the chemical compo- sition.28In the soft x-ray synchrotron community, a UV treatment is commonly used to remove carbon adsorbates from beamline optics.29 In the low-energy (50 eV) Ar+-ion treatment, a commercial ion gun is used at very low energies to stimulate adsorbate desorp- tion with minimal (or no) sputter damage to the surface. Thisapproach was first established by Weinhardt et al. 30to remove adsorbates from CIGSSe surfaces without the previously observed surface metallization by sputter-cleaning surfaces at 500 eV.31,32 Most dominantly, sputtering at 500 eV (or above, as commonly used in destructive depth-profile approaches) led to preferentialenrichment of CIGSSe surfaces with Cu, coupled with the creationof a metallic surface layer (as evidenced by the presence of a Fermi edge in UV and inverse photoemission spectra). In this paper, a UV-induced ozone treatment and a low- energy Ar +-ion treatment (including an ultra-long treatment exper- iment of 2 h) is applied to CIGSSe-based absorber surfaces to studytheir impact on the chemical surface structure. For this purpose, x-ray photoelectron spectroscopy (XPS) and x-ray exited Auger electron spectroscopy (XAES) are used to examine CIGSSe surfacesafter each treatment step. This measurement set is combined withsoft x-ray emission spectroscopy (XES) to also give element-specificchemical structure information at and near the surface, but in a complementary fashion to XPS and XAES. 33 II. EXPERIMENTAL SECTION All samples originate from one 10 × 10 cm2absorber sample processed by AVANCIS GmbH using the R&D baseline process.34 The (air-exposed) sample was shipped to KIT, where the sample wastransferred into an argon-filled glovebox directly connected to theultra-high vacuum surface characterization system (Materials forEnergy —MFE lab at KIT). Here, the 10 × 10 cm 2sample was cut into several smaller pieces. Each piece was unloaded separately from the glovebox, put into a UV-induced ozone cleaner (UVO-Cleaner Model 18, Jelight Company Inc.), treated for a given duration (15, 45, 60, 75,90, and 1200 s), and subsequently loaded back into the glovebox tominimize air exposure after the treatment. Note that, for industrial applications in an in-line process, the use of a glovebox would not be required. The ambient air exposure time before and after thetreatment was minimized to less than 10 s each. For the low-energy Ar +-ion treatment, a FOCUS FDG 150 ion source was utilized (Eion=5 0e V , j sample∼300 nA/cm2, treatment times of 60, 120, 180, 3600, and 7200 s) at an angle of 45° with respect to the samplenormal. XPS and XAES were measured with a non-monochromatizedDAR 450 twin anode x-ray source (Mg and Al K α)a n da nA r g u sC U electron analyzer (Scienta Omicron) , calibrated according to Moulder et al. using Au, Ag, and Cu sputter-cleaned metal references.35The base pressure was ∼1×1 0−10mbar in the analysis chamber. After the XPS and XAES measurements, small pieces were cut off from the measured samples, sealed under inert atmosphere in the glovebox, and shipped to Beamline 8.0.1 of the Advanced Light Source (ALS) in Berkeley for XES measurements (with brief airexposure during loading into the XES chamber). A CdS referencesample was used to calibrate the emission energy axis for the here-presented S L 2,3XES spectra.36 III. RESULTS AND DISCUSSION In the XPS survey spectra ( Fig. 1 ), all peaks associated with the absorber elements (copper, indium, sulfur, and selenium) are visible.Only trace amounts of gallium are detected at the absorber surface FIG. 1. Mg K αXPS survey spectra of CIGSSe absorbers that underwent (from bottom to top): no treatment, 90 s UV treatment, 1200 s UV treatment, 180 s low-energy Ar+-ion treatment, and 7200 s Ar+-ion treatment. Prominent XPS and XAES signals are labeled.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155301 (2020); doi: 10.1063/5.0020253 128, 155301-2 Published under license by AIP Publishing.(accordingly, the surface is sometimes called “CISSe ”).34,37,38In addi- tion, sodium- (e.g., Na 1s), oxygen- (e.g., O 1s), and carbon-related (e.g., C 1s) signals can be identified. For the 7200 s Ar+-ion treated sample, a small Ar 2p signal is found (at ∼240 eV, not visible in Fig. 1 ). For the as-received sample (in the following “untreated ”), the carbon and oxygen peaks are clearly visible, indicating that the sample exhibits a significant amount of surface adsorbates (as we will show in the following). As already visible in the survey spectra of Fig. 1 , and more easily seen in the detail spectra of Fig. 2 , the sample that underwent a 90 s UV-induced ozone treatment shows an intensity increase of the O 1s signal (×2.6), while that of C 1s is decreased (/3.0). The increase in the oxygen signal can be attributed to the formation ofmetal oxides and hydroxides, which causes a shift of the peakmaximum to lower binding energies (indicated by the gray dashedline in Fig. 2 ). The reduction of the carbon signal is primarilyassociated with the main C 1s contribution (amorphous carbon and hydrocarbons), while the (much smaller) carbonates/carboxyl component is not significantly altered. In addition, all absorber-related lines (including Na 1s) become more intense due to the lower attenuation at the absorbersurface. For longer UV-treatment times (1200 s), the O 1s peak is further increased, and only a very small C 1s intensity remains. In parallel, the Cu Auger feature is broadened and the spectralshape of the Cu 2p signal is significantly changed, indicating adifferent chemical environment (which will be discussed later),and the Na 1s peak is reduced in intensity. Most likely, volatile sodium-containing components are formed under the UV light, e.g., with participation of H 2O molecules, similar to a rinsing step.17,39We also observe that the intensity decrease of the less surface-sensitive Na KLL Auger line is even more pro-nounced than that of Na 1s, which rules out a possible attenua- tion of the signals by adsorbate layers, and indicates that Na is still localized at the topmost layer of the film after the prolongedUV treatment. For the Ar +-ion treated samples, the O 1s signal is strongly reduced, even for short treatment times (180 s), while the C 1s intensity is reduced slightly. Only significantly longer treatment times (3600 and 7200 s) are able to reduce the concentration of allcarbon -containing species from the surface (by a factor of ∼4). Furthermore, sodium is almost completely removed, suggesting that sodium is only present at the outermost surface. 39,40The removal of C, O, and Na leads to higher intensities of all absorberphotoemission lines (due to reduced signal attenuation). The evolution of the oxygen 1s line during the Ar +-ion treat- ment reveals multiple components. The analysis of absorber ele- ments, discussed later, will show In –O and Se –O bonds, and some surface adsorbates (water, hydroxides) are likely present (andremoved) as well. The largest contribution is removed during thefirst treatment step. We note that the concentration of physisorbedspecies could also be reduced by an annealing step; however, this runs the risk of also inducing annealing-related changes of the absorber (e.g., the Cu profile). To study the impact of the treatments on the CIGSSe absorber elements, we next analyze the In M 4,5N4,5N4,5and Cu L 3M4,5M4,5 Auger features ( Fig. 3 ). The indium Auger feature for the untreated sample already shows some indium-oxygen bonds, recognizable by the additional intensity in the “valley ”at∼405 eV. The UV treat- ment further increases the amount of oxidized indium. For the 90 sUV-treated sample, the spectrum can be fitted by two In MNN components. The first (non-oxidized) spectral component is described by the spectrum after 7200 s of Ar +-ion treatment, while the second component is represented by the same spectrum, butshifted by 2.1 eV to lower kinetic energy to emulate indium oxide.We note that the In MNN Auger transition only involves 3d and “shallow ”4d core levels, resulting in very similar In MNN line shapes for different compounds. The indium oxide component inthe 90 s UV-treated sample accounts for 50% (±2%) of the overallIn signal. From the energetic position of the M 4N4,5N4,5peak, and the modified In Auger parameter (851.5 ± 0.1 eV), it can be con- cluded that this second component indeed represents In –O bonds. In contrast, the Ar+-ion treatment completely removes the indium – oxygen bonds, best seen in the deepening of the valley at ∼405 eV. FIG. 2. Mg K αXPS spectra of the O 1s (left) and C 1s (right) regions for the untreated (black), UV-treated (15, 45, 60, 75, 90 s; from black to red), and Ar+-ion treated (60, 120, 180, 3600, 7200 s; from black to blue) surfaces. The ordinate is given as an “intensity-true ”representation, allowing for a direct comparison of the spectral area under the curves. Black bars indicate chemical species commonly found for C and O adsorbates.35,41–43Arrows serve as a guide-to-the-eye for the spectral evolution, and a dashed line for the O 1s peaksafter UV treatment is used to illustrate a shift to lower binding energy.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155301 (2020); doi: 10.1063/5.0020253 128, 155301-3 Published under license by AIP Publishing.For the 7200 s Ar+-ion treated sample, only one (non-metallic) In species is visible in the In MNN spectrum. In the case of the Cu LMM Auger transitions, UV treatments up to 90 s only lead to a small shift to higher kinetic energy.Likewise, the corresponding Cu 2p 1/2spectra in Fig. 4 do not show a change in the spectral shape [but an increase in the overall inten- sity, due to the high surface-sensitivity (low kinetic energy) of the electrons contributing to the Cu 2p signal and the reduction ofsurface adsorbates]. For longer UV treatment times (1200 s; topspectrum in Fig 4 ), in contrast, the Cu 2p 1/2spectral shape changes significantly and now includes a strong satellite feature (at ∼962.5 eV) that can be ascribed to the formation of CuO.44 Upon Ar+-ion treatment, the Cu LMM spectra in Fig. 3 shift to lower kinetic energies, and, for longer treatment times, the valleyat 919 eV gets less pronounced. In the spectrum of the 7200 s Ar +-ion treated sample, a clear contribution of a second compo- nent can be seen. Likewise, shifts to higher binding energy and aline broadening are observed in the Cu 2p 1/2spectra. We speculate that this is due to Cu atoms in a metallic Cu environment.45While the creation of metallic surface species (in particular Cu, due to preferential sputtering of the other elements in CIGSSe) is a knowneffect when using higher energies and/or longer sputter times, 31,46 it is here observed for the first time when using 50 eV Ar+ions. However, we point out that this is related to the exceedingly long ion-treatment time and rather high current densities. Typically, short treatments (such as the 90 s employed here) are fully suffi-cient to remove the majority of “removable ”surface contaminants, without any evidence of metallic copper (or other metallic compo-nents) in XPS, XAES, UV photoelectron spectroscopy (UPS), and inverse photoemission spectroscopy (IPES). In addition to the copper and indium signals, the absorber constituents sulfur and selenium were analyzed, in particular the Se3p/S 2p and Se 3d regions ( Fig. 5 ). The untreated Se 3d spectrum consists of the chalcopyrite main feature at ∼54 eV and a second feature at ∼59 eV, which is increasing for increasing UV-treatment FIG. 3. Mg K αXAES spectra of the In M 4,5N4,5N4,5(left) and Cu L 3M4,5M4,5 (right) Auger features for the untreated (black), UV-treated (15, 45, 60, 75, 90 s; from black to red), and Ar+-ion treated (60, 120, 180, 3600, 7200 s; from black to blue) surfaces. The In spectra were normalized to the M 4N4,5N4,5peak at ∼408 eV , while the Cu spectra were normalized to the L 3M4,5M4,5peak at ∼917 eV . A fit of the In Auger feature for the 90 s UV-treated sample is also shown (top left). FIG. 4. Mg K αXPS spectra of the Cu 2p 1/2region for the untreated (black), UV-treated (15, 45, 60, 75, 90, 1200 s, from black to red), and Ar+-ion treated (60, 120, 180, 3600, 7200 s; from black to blue) surfaces. The ordinate is givenas an “intensity-true ”representation, allowing for a direct comparison of the spectral area under the curves. Black bars indicate chemical species commonly found for Cu 2p 1/2.43Arrows serve as guide-to-the-eye for the spectral evolution.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155301 (2020); doi: 10.1063/5.0020253 128, 155301-4 Published under license by AIP Publishing.times. For longer UV-treatment times (75 s), additional intensity is found at ∼55 eV, suggesting a third Se component. In order to sep- arate the individual components, the 90 s UV-treated sample wasfitted with a minimal number of spin –orbit–split Voigt doublets (three). The Gaussian and Lorentzian contributions were fixed foreach individual component, the area ratio was kept constant at 3:2 according to the 2j + 1 multiplicity, and the spin –orbit splitting was set to 0.86 eV. The first component at ∼54 eV can be assigned to Se in a selenide environment (e.g., CISSe); the second one at ∼55 eV could indicate a second selenide (e.g., Cu –Se bonds) or elemental Se; also, an inhomogeneous distribution of slightly varying local environments appears possible. The third component (at ∼59 eV) exhibits a shift of ∼4.7 eV with respect to the main component, suggesting Se –O bonds (as already observed for the untreated sample). The Ar +-ion treatment fully removes the Se –O component already after the first 60 s treatment. In parallel, the intensity of the main component increases due to reduced amount of surface adsorbates.Due to the spectral overlap, the Se 3p/S 2p core level region shows a complex behavior upon the UV treatment. Additional intensity is found in the valley between the S 2p and Se 3p 1/2peak at∼165 eV, and the spectral intensity between 168 and 173 eV increases. To analyze these signals, the 90 s UV-treated spectrumwas fitted with a Se 3p doublet (blue; for the Se component in a CISSe environment), a second Se doublet (light blue; for the Se –O component as identified from the Se 3d signal), and three S 2pdoublets. Note that the Se 3d analysis shows the presence of a thirdSe species, but due to the overlap with the S 2p signals and thelarger widths of the Se 3 p lines, the Se 3p/S 2p region can be suffi- ciently well described with only two spectral Se components. For each doublet, two Voigt profiles (Gaussian:Lorentzian ratio fixed,area ratio 2:1, spin –orbit splitting set to 1.2 and 5.7 eV for S 2p and Se 3p, respectively) were used. This approach results in one sulfide-(at 161.6 eV, green) and two S-O-related signals (at 167.4 and 168.6 eV, orange and pink, resp.). The two additional S –O signals can be assigned to sulfites and sulfates, for which more evidencewill be presented in the XES results below. In contrast, the Ar +-ion treatment does not modify the Se 3p/S 2p region significantly. The main effect is an overall inten- sity increase, which can, again, be related to the reduced attenuation of the XPS signal due to the removal of surface adsorbates. To derive chemical information with an increased information depth, x-ray emission spectroscopy (XES) measurements at the FIG. 5. Mg K αXPS spectra of the Se 3p/S 2p (left) and the Se 3d (right) region for the untreated (black), UV-treated (15, 45, 60, 75, 90 s; from black tored), and Ar+-ion treated (60, 120, 180, 3600, 7200 s; from black to blue) surfa- ces. The ordinate is given as an “intensity-true ”representation, allowing for a direct comparison of the spectral area under the curves. At the top, fits of the Se 3p/S 2p (left) and Se 3d (right) region of the 90 s UV-treated sample areshown. FIG. 6. SL 2.3XES spectra (h νexcitation = 180 eV) of the untreated CIGSSe absorber surface (black) and after different UV-treatment times (60, 120, 180, 300, 600, and 1200 s; from black to red), difference spectra (180 s –0 s) and (1200 s –0 s), and Na 2SO 3and In 2(SO 4)3reference spectra.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155301 (2020); doi: 10.1063/5.0020253 128, 155301-5 Published under license by AIP Publishing.SL 2,3edge are presented in Fig. 6 . Using the element-specific and local nature of XES, the S L 2,3emission gives detailed information of the local chemical environment of sulfur in a complementaryfashion to XPS. All spectra are dominated by the S 3s →S 2p emis- sion at 149 eV, typical for a sulfide environment. 47For the untreated sample (black), the spectral structure between 154 and 158 eV can be assigned to In 5s-derived bands, indicating S –In bonds. The broad signal at ∼160.5 eV originates from Cu 3d-derived bands, indicating S –Cu bonds. For increasing UV-treatment time, several new features appear, e.g., at 153.5 and164.0 eV. A difference spectrum of the 180 s UV-treated sample and the 0 s (untreated) sample highlights additional spectral weight between 160 and 164 eV. A comparison with Na 2SO3and In2(SO 4)3reference spectra suggests the formation of sulfur-oxygen bonds, best described as a sulfite (SO 32−). For longer UV-treatment times, additional features at 155.3 and 156.5 eV and a broad maximum at ∼162 eV become more pronounced. The difference spectrum (1200 s –0 s) strongly resembles the reference spectrum of a sulfate, with an admixture of the sulfite spectra features. Notethat the Na 2SO3and In 2(SO 4)3reference spectra are only used to demonstrate the most pertinent sulfite and sulfate features —the formation of other sulfates and sulfites is also possible. Nevertheless, the finding of both sulfite as well as sulfate spectralfeatures supports the XPS findings of two distinct S –O bond signals in Fig. 5 . The UV-induced ozone treatment hence clearly produces S –O bonds with varying degrees of oxidation. IV. SUMMARY AND CONCLUSION The impact of two dry surface-cleaning approaches for CIGSSe solar cell absorbers, namely, a UV-induced ozone treat-ment and a low-energy (50 eV) Ar +-ion treatment was investigated for different treatment times. Even for UV treatments as short as90 s, we find a two-thirds reduction of carbon and hydrocarbons but also an increase of oxygen at the CIGSSe absorber surface. This is accompanied by an oxide formation of indium, selenium, andsulfur after several tens of seconds at the surface. Copper remainsunaffected for the first 3 min of the UV treatment, but also shows oxidized components when treated for 20 min. Furthermore, sulfite and sulfate signatures are found. In contrast, the low-energyAr +-ion treatment readily removes surface oxygen species, while longer treatment times also lead to a removal of carbon. For verylong treatment times, metallic surface phases are also induced. As both treatments appear very effective, already for short treatment times (i.e., carbon removal for UV treatment and oxygen removalfor low-energy ion treatment), we speculate that a sequential execu-tion might represent a promising pathway to minimize carbon andoxygen surface contaminants for optimized heterojunction engi- neering in high-efficiency thin-film solar cells. ACKNOWLEDGMENTS This research received financial support by the Deutsche Forschungsgemeinschaft (DFG) in Project No. GZ:INST 121384/64-1 FUGG and the BMWi project “EFFCIS ”(Nos. 0324076E and 0324076G). 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5.0022870.pdf

J. Appl. Phys. 128, 160901 (2020); https://doi.org/10.1063/5.0022870 128, 160901 © 2020 Author(s).Computer aided design of stable and efficient OLEDs Cite as: J. Appl. Phys. 128, 160901 (2020); https://doi.org/10.1063/5.0022870 Submitted: 24 July 2020 . Accepted: 08 October 2020 . Published Online: 22 October 2020 Leanne Paterson , Falk May , and Denis Andrienko ARTICLES YOU MAY BE INTERESTED IN Introduction to spin wave computing Journal of Applied Physics 128, 161101 (2020); https://doi.org/10.1063/5.0019328 Frontiers in atomistic simulations of high entropy alloys Journal of Applied Physics 128, 150901 (2020); https://doi.org/10.1063/5.0025310 Design strategy for p-type transparent conducting oxides Journal of Applied Physics 128, 140902 (2020); https://doi.org/10.1063/5.0023656Computer aided design of stable and efficient OLEDs Cite as: J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 View Online Export Citation CrossMar k Submitted: 24 July 2020 · Accepted: 8 October 2020 · Published Online: 22 October 2020 Leanne Paterson,1Falk May,2and Denis Andrienko1,a) AFFILIATIONS 1Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany 2Merck KGaA, 64293 Darmstadt, Germany a)Author to whom correspondence should be addressed: denis.andrienko@mpip-mainz.mpg.de ABSTRACT Organic light emitting diodes (OLEDs) offer a unique alternative to traditional display technologies. Tailored device architecture can offer properties such as flexibility and transparency, presenting unparalleled application possibilities. Commercial advancement of OLEDs is highly anticipated, and continued research is vital for improving device efficiency and lifetime. The performance of an OLED relies on an intricate balance between stability, efficiency, operational driving voltage, and color coordinates, with the aim of optimizing these parametersby employing an appropriate material design. Multiscale simulation techniques can aid with the rational design of these materials, in orderto overcome existing shortcomings. For example, extensive research has focused on the emissive layer and the obstacles surrounding blue OLEDs, in particular, the trade-off between stability and efficiency, while preserving blue emission. More generally, due to the vast number of contending organic materials and with experimental pre-screening being notoriously time-consuming, a complementary in silico approach can be considerably beneficial. The ultimate goal of simulations is the prediction of device properties from chemical composition,prior to synthesis. However, various challenges must be overcome to bring this to a realization, some of which are discussed in thisPerspective. Computer aided design is becoming an essential component for future OLED developments, and with the field shifting toward machine learning based approaches, in silico pre-screening is the future of material design. © 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/10.1063/5.0022870 INTRODUCTION Organic light emitting diodes (OLEDs) have attracted much interest in recent years, from the pivotal discovery of electrolumi- nescent properties in certain organic materials 1–3and the fabrica- tion of the first device4to applications in present day technologies. With display applications being an essential component of manymodern electronic devices, the unique properties of OLEDs have propelled them into the industry. Every day, functionality, reliabil- ity, and efficiency drive electronic devices forward, and with thesedevices advancing at an unprecedented rate, it is imperative thatthe technology behind them follows the same trend. The mechani- cally flexible, transparent, and lightweight properties of OLEDs create a whole host of new technologies and applications in thisrapidly expanding market. Therefore, the advancement of thesedevices is highly anticipated. In comparison to rigid inorganicLEDs, OLEDs offer a unique flexible substitute, with possibilities including curved, foldable, and wearable displays. Additionally, asthey do not require a backlight, a better contrast ratio and an overall improved image quality are achieved, compared with the LCD technology. OLEDs can already be found in a host of com- mercial applications from the automotive industry to wearable and mobile devices, such as smartphones and watches. While OLEDsoffer an enticing substitute, they remain in their infancy with respect to their inorganic counterparts, so continued research is essential in improving device efficiency and lifetime. So, what is an OLED? As a sub-category of organic semicon- ductors, OLEDs are carbon-based compounds with structures tail- ored for photo- or electro-luminescence. They utilize a thin layer of a polymeric or small molecule based organic materials to achieve adesirable wavelength of emission. A typical (small molecule) multi- layer OLED structure sandwiched between two electrodes is illus- trated in Fig. 1 , showing each layer with the corresponding function. Upon the application of an external potential, electrons and holes are injected from the cathode and anode, respectively. Electron/holeJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-1 ©A u t h o r ( s )2 0 2 0injection and transport layers then facilitate the movement of the charge carriers to the emissive layer (EML), where they recombine toform excitons, and consequently, a photon of the desirable wave-length is emitted. A blocking layer, in combination with, or additionalto the transport layer for each carrier can also be utilized to assist in the accumulation of carriers in the emissive layer by preventing elec- tron transport to the anode and hole transport to the cathode. Various factors affect the performance of an OLED, and typi- cally, there is never an “ideal ”solution to obtain an optimized device in all relevant parameters. A compromise between stability, efficiency, operational driving voltage, and color coordinate is inev-itable. However, with an appropriate material design, it is possibleto achieve devices that balance these individual limitations. Thishas been demonstrated for an emissive layer, with the novel concept of unicolored phosphor-sensitized fluorescence (UPSF), 5,6 where a phosphorescent donor and fluorescent acceptor achieve a trade-off between stability and efficiency, while preserving thesky-blue emission color. Maximizing the device performance previ-ously focused solely on improving the stability of the organic mate- rials used. However, due to the advent of the new 5G standard for mobile applications, the focus has shifted toward increasing theefficiency and reducing the driving voltage of each layer. The func-tionality of the layer, which is determined by molecular architec-ture, electronic properties, and charge carrier mobilities, relies on the choice of organic materials. The individual choice of each layer and each emitter is crucial in enhancing the OLED in its entirety.Charge transporting and emissive layers have to be designed insuch a way as to maximize their function and stability. By utilizing an electron and hole injection and transport layer, the device per- formance has been dramatically improved already. 2,7–9Therefore,tremendous research efforts have involved tailoring of electron injection and transport layers, hole injection and transport layers,and the various emitters for the emissive layer, as well as potentialdegradation mechanisms 10–13and reactions within the OLED device,13to target stability improvements, all of which are necessary in the path toward high performing and long-lasting OLEDs. Considering the individual emitters, at present, there exist stable and efficient triplet emitters for red,14–23green,24–33and yellow (for use in white OLED-TV stacks), with blue falling short. Blue emitters continue to be problematic and many ongoing research efforts target this specifically.34The focus is most often on the emissive paths, be that fluorescence,35phosphorescence,36–38 thermally activated delayed fluorescence (TADF),39–44or sensitizing approaches.5,6This ranges from understanding degradation mecha- nisms in phosphorescent emitters45–48to utilizing combinations of emitters to overcome individual limitations.5,49,50Although blue emitters are most notably challenging, overall increased stabilityand efficiency of all emitters and the device as a whole remain chal-lenges for commercial OLED development. With many potential candidates for each emitter and many potential emissive paths, it is crucial that simulations provide informa-tion and a deeper insight, which otherwise would not be accessible.Simulations can be used to study each of the OLED layers and thecontending organic materials, as w ell as possible degradation mecha- nisms, all toward a better understanding of the internal processes and highlighting key areas of improvement. There are well establishedmethods for helping to optimize the stacked OLED architecture, bymodeling light outcoupling and the balancing of hole and electron transport. 51,52The next step is to incorporate molecular details and achieve a mechanism of evaluating an OLED chemical design.53,54 FIG. 1. (a) Basic OLED structure: electrons are injected at the cathode to the electron injection layer (EIL) and transported via the electron transport laye r (ETL); holes are injected from the anode to the hole injection layer (HIL) and transported through the hole transport layer (HTL). Both then combine in the emissive lay er (EML) to form an exciton and emit a photon of specific wavelength. (b) Schematic representation of the flow of electrons and holes from electrodes to the EML.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-2 ©A u t h o r ( s )2 0 2 0The goal of simulations is to predict physical properties from chemical composition, also known as the forward problem; this would allow for pre-screening and an overall better insight.Simulations can not only be advantageous but also are an essentialtool in the design of future OLEDs. This extraction of propertiesfrom the molecular structure alone is, of course, non-trivial, and computational pre-screening is not yet accurate enough for this task exclusively. 55However, a collective experimental and in silico approach can be beneficial due to the fact that there are a vastnumber of potential candidates and experimental pre-screening isnotoriously time-consuming. This Perspective will discuss the current shortcomings of OLEDs, in the search for increased device lifetime and efficiency,and how multiscale simulations can assist OLED design, includingpredictions of electronic properties, such as ionization energy, elec-tron affinity, and electron and hole mobility. Additionally, it gives further insight into internal processes and highlights potential areas for improvement in existing devices. The prediction of micro-scopic device properties will be discussed, including the steps thatwould be required for the extraction of valuable properties fromthe molecular structure alone. We will then look at the possible direction the field may take, considering atomistic scale machine learning, with machine learning based approaches gaining signifi-cant momentum in most computational areas. IMPROVING DEVICE PERFORMANCE The path of exciton formation and decay resulting in photon emission, as well as the emission color, depends on the organic mol-ecules and molecular packing. It is crucial to utilize the individualstrengths of organic materials for their designated task. As a result, extensive research efforts have focused on each of the individual layers and the organic constituents. The “one by one ”layer approach is more practical, as chemical properties of the molecules can betuned for specific OLED characteristics. Listed below is a short intro-duction to each of the OLED layers, with an example of how the spe- cific challenges are addressed, paying particular attention to the emissive layer and the challenges surrounding blue emitters. Cathode and electron injection layer (EIL) The injection of electrons into the organic layers has a signifi- cant impact on the efficiency of the OLED. Lowering the energeticbarrier between the cathode and the lowest unoccupied molecularorbital (LUMO) of the adjacent organic layer facilitates the injec-tion of electrons. Low work function metal alloy cathodes, such as Mg:Al, 4are susceptible to atmospheric conditions.56Therefore, for increased stability, cathode bilayer structures, such as MgAg/Ag14,57 and LiF/Al,56,58,59have been frequently used. Numerous other studies have investigated n-type metal oxide semiconductors60and alkali metal containing interlayers.61,62 Electron transport layer (ETL) The electron transport layer facilitates the movement of elec- trons toward the emissive layer and, therefore, is a vital component of the device, confirmed by the extensive research on ETL materi- als.8,63Since balancing electron and hole injection and transport tothe emissive layer is crucial for OLED performance, the challenge revolves around finding suitable and stable materials with high charge carrier mobilities. Within electron or hole transportingmaterials, the presence of energy traps can have a detrimentalimpact on the charge carrier mobilities. For large energy gap mate-rials, this is due to the fact that either the electron affinity or ioni- zation energy lies in a trap region, such that unipolarity prevails. Specifically, when considering the ETL, a shallow electron affinityresults in trap limited electron transport and low electron mobility.It has been shown that there is in fact an energy window, withinwhich there lie materials for trap-free ambipolar transport, result- ing in higher mobilities. 64This is an ionization energy below 6 eV for hole transport and an electron affinity above 3.6 eV, such thatfor an appropriate material design, ideally, this energy windowshould be targeted. Additionally, for optimal emission, the ETLshould block holes and excitons from escaping the emissive layer. Therefore, the ETL layer should have a small injection barrier for electrons from the EIL or cathode, a highest occupied molecularorbital (HOMO) level low enough to effectively block holes fromthe emissive layer, and a high triplet energy level for the case oftriplet excitons (with high diffusion lengths). Considering one material among many various options, Alq 3has been extensively studied for electron transport (as well as hole transport and emis-sive layers), spanning from the first OLED fabrication. 4However, as it has a low triplet energy level,65it cannot be used with an EML creating triplet excitons, unless coupled with an interlayer. For example, the use of a hole and exciton blocking layer, such as BCPin combination with Alq 3, has been demonstrated to increase effi- ciency.14,15,24,57A further option is improving the thermal stability of materials with high electron mobility, such as BPhen, demon- strated with the use of alkali metal n-dopants.66Another alternative is the use of triazine based electron transporters, where substituentscan be used to tune the electron mobility and LUMO energy foradjusted injection. Hole injection layer (HIL) Inserted between the typically used transparent indium-tin-oxide (ITO) anode and the hole transport layer (HTL), the hole injection layer eases hole migration at each of the interfaces. Materials for theHIL should have an ionization energy level situated between that ofthe preceding and succeeding layers. This is slightly easier for holeinjection, in comparison to electron injection, due to the typically lower injection barrier. Nevertheless, it has been shown that the stepwise injection from anode to HIL to HTL can improve perfor-mance; such an instance has been shown with the use of an organicinterlayer, 67such as MTDATA or 2-TNATA.68 Hole transport layer (HTL) For a HTL to be effective, the material should fulfill certain properties. It should have good hole mobility and appropriate HOMO level to ensure a low energetic barrier from the HIL, for hole transport, and a suitable LUMO level to act as an electronblocking layer. Similar to the ETL, HTL materials also need sufficienttriplet energies to confine the exciton within the EML. The most widely used HTL materials are arylamines, such as NPB, due to their high hole mobility and suitable HOMO levels. However, due to itsJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-3 ©A u t h o r ( s )2 0 2 0low glass transition temperature, NPB itself has low thermal stability, which can lead to device degradation. The vast research in recent years69has focused on finding more stable NPB derivatives or alter- natives with similar or improved performance.70–72 Emissive layer (EML) Red, green, and blue emitters are essential for full color dis- plays. The goal is to achieve an emission layer of each color withhigh luminous properties, high efficiency, and high stability. Independent of the emission path, in order to maximize the out- coupling efficiency, the transition dipole moment of the emittershould be aligned horizontally with respect to the substrate plane. 73 Also here, computer simulations of the evaporation process canhelp in predicting how emitter –host interactions can be employed in a rational compound design. 74 Localized energy states, resulting in well-defined singlet and triplet spin states within the OLED, aid with the luminescent prop-erties. 75The excitation of the molecule involves the promotion of an electron from the singlet ground state (S 0) to the first excited singlet state (S 1) or the first triplet excited state (T 1), as shown in Fig. 2 . Excitation occurs with a 25% probability to the S 1state, as there is only one possible configuration (maintaining antiparallelspin), or a 75% probability to the T 1state, with three potential par- allel spin combinations. First generation fluorescent emitting OLEDs typically have high stability but low efficiency. This is due to an unfavorable spintransition between the singlet and triplet states, with rapid radiative(∼ns) decay only readily occurring from the S 1state, resulting in around a 25% efficiency of the emitter. Increasing the overall effi- ciency in fluorescent emitters is possible using triplet –triplet anni- hilation, which is often facilitated by anthracene based hosts. Second generation OLEDs or phosphorescent emitters, com- prised of organometallic complexes, have high efficiency but lower stability. Here, radiative decay is possible from the T 1state, result- ing in slow ( ∼μs) phosphorescent emission. Additionally, the pres- ence of a heavy metal atom, such as iridium or platinum, with anappropriate ligand design, results in spin –orbit coupling. Thisallows for the transition between S 1and T 1states, known as inter- system crossing (ISC), which significantly increases the efficiency. The downfall being that these heavy metal complexes are particu-larly susceptible to environmental factors, which can lead to devicedegradation. The phosphorescent long-lived excited triplet state,typically in the order of microseconds (compared to nanoseconds of the fluorescent S 1state), also leads to degradation, and as such, these emitters have low stability. Third generation OLEDs that use TADF emitters harvest the triplet excitons lost in conventional fluorescent emitters by makinguse of a thermally activated process called reverse inter-system crossing (RISC). Choosing a sufficiently low energy gap between the T 1and S 1levels allows for RISC and can result in highly effi- cient and stable emitters. Finally, a sensitizing approach is possible, combining a donor and acceptor molecule, with the aim of overcoming the individual limitations of both emitters. One such example is the use of a phos- phorescent donor and a fluorescent acceptor. We will discuss theapplication of this mechanism, with a unicolored phosphor-sensitized fluorescence (UPSF) approach, for the realization of asky-blue OLED. The various emissive paths are shown in Fig. 3 , outlining the energy levels and radiative or non-radiative (NR) decay for each type of emitters, or combinations. In its entirety, theemissive layer consists of the chosen type(s) of emitter(s), typicallydispersed within a charge transporting host material to increase efficiency. Usually, this combats adverse factors, such as triplet – triplet annihilation in phosphorescent emitters, for example. Currently, there are stable and efficient red and green emitters available, with extensive research focusing on both. Due to highefficiency, phosphorescent emitters have dominated the field, in the search for red 14–20and green emitters.24–28Various charge trans- porting host materials have been the subject of investigation tounderstand the relationship between host and device performance,targeting hosts to combat potential efficiency losses. For a similarpurpose, double emissive layers have also been studied. 76,77 Additionally, red21–23and green29–33TADF emitters have been shown to be efficient, low cost alternatives to the expensive preciousheavy metals found in phosphorescent emitters. As previously stated, blue emitters are particularly problematic due to higher triplet energy and long triplet lifetimes, leading to the lowest stability. This remains one of the largest hurdles for commer- cial OLED applications to date. Immense efforts have centeredaround finding solutions to limit device degradation in phosphores-cent emitters or utilizing hybrid emissive technology in an attempt to overcome it. The short operational lifetime of blue phosphorescent emitters has resulted in studies of degradation mechanisms. 45–48 Once understood, these limitations can be addressed, potentially increasing stability. Specifically, this has included targeting adversefactors such as (1) chemical degradation, 45(2) triplet –polaron quenching,38,46,47,78,79and (3) triplet –triplet annihilation.80–82 Additionally, simulations have been used to investigate host materials for efficient charge transport within blue phosphorescent emitters,83 providing a link between the electronic structure and molecularpacking, to the rational design of effective host materials with high charge carrier mobilities. Blue TADF emitters 39–44and combinations of TADF with conventional fluorescent84,85or phosphorescent emit- ters84,86have also been investigated. This includes the impact of FIG. 2. Electronic configuration for the ground state (S 0), the first singlet excited state (S 1), and the first triplet excited state (T 1). The arrows represent the elec- tron spin.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-4 ©A u t h o r ( s )2 0 2 0emitter –host interaction42,43and methods of lowering the singlet –triplet energy gap,44potentially enhancing the perfor- mance. However, at present, with triplet lifetimes similar to that of phosphorescent emitters, the low stability problem remains forblue TADF based systems. A phosphor-sensitized fluorescence approach 49,87–90offers an alternative to conventional phosphorescent and TADF emitters for various emitting wavelengths, in an effort to overcome their indi- vidual shortcomings, by coupling both fluorescent and phosphores-cent emitters. By utilizing a phosphorescent donor and afluorescent acceptor, distributed within a host, it is possible to obtain a dual emitting system that is both stable and efficient. A unicolored phosphor-sensitized fluorescence (UPSF) approach,with matching donor and acceptor emission color, was recentlyproposed for blue OLEDs. 5,6The energy transfer processes and radiative decay paths of the phosphorescent donor and fluorescent acceptor, for the UPSF system, are illustrated in Fig. 3 . The phos- phorescent donor can (1) emit from the T 1state, or (2) transfer energy to the S 1state of fluorescent acceptor or (3) the acceptor T 1 state. If the donor and acceptor molecules are adequately spaced(>1 nm), there will be a Förster resonance energy transfer (FRET) from the donor T 1state to the acceptor S 1state, resulting in fluo- rescence emission from the acceptor. On the other hand, if themolecules are closer in distance, a Dexter energy transfer willprevail, between the donor T 1state the acceptor T 1state, resulting in a loss of efficiency as this is a non-radiative state and no RISC is present. Experimentally, it was shown that by increasing the con- centration of fluorescent acceptor molecules, the stability wasincreased and made apparent with a threefold increase in devicelifetime. However, as a result of the probability of the Dexter energy transfer also increasing, the efficiency of the device decreases with increasing acceptor concentration, shown with areduction in the photoluminescence quantum yield (PLQY). Therefore, it was shown that the UPSF OLED has to be designed to optimize the acceptor concentration for increased stability and operation lifetime, and target donor –acceptor combinations with slow Dexter and fast FRET rates for increased efficiency. Our recently published complementary simulation study for this blue UPSF OLED approach 91demonstrates an essential com- putational component of such OLED designs. By providing a deeper insight for better understanding and expanding on thescope of experiment to look at the limitations of the concept, aswell as key areas of potential improvement, computational input can be vital. We developed a multiscale model, as shown in Fig. 4 , to investigate the UPSF OLED and highlight any inherent limita-tions of the concept. Additionally, as the simulations are based onthe experimentally achieved results, it is possible to do so withoutexplicit consideration of chemical design, starting with atomistic morphologies of the UPSF systems and parametrizing the rates (from experimental data) of all of the essential processes involvedto then solving the respective master equation with the use of akinetic Monte Carlo (KMC) algorithm. First, it became apparent that an additional mode of energy transfer was missing from the original description and that transfer between donor molecules was also essential. This was taking placebetween the triplets of the respective donor molecules, as a donorto donor Dexter energy transfer, which was not obvious fromexperimental results. But, it was a mechanism which facilitated increased energy transfer between the donor and acceptor by allow- ing for a step-wise energy transfer process. Second, we couldexpand on the experimental results by studying increased acceptorconcentrations in order to understand any limitation of the UPSF concept. A level of saturation was observed in terms of the number of donor to acceptor energy transfer processes, such that higher FIG. 3. Various emissive paths: fluorescence (fl) with 25% efficiency only emits from the first singlet state. Phosphorescence (Ph) with an intersystem cro ssing (ISC) allows for complete emission from the first triplet state. Thermally activated delayed fluorescence (TADF) makes use of the reverse intersystem cro ssing (RISC) to emit from the first singlet state more efficiently. Unicolored phosphor-sensitized fluorescence (UPSF) with a phosphorescent donor emitting from the f irst triplet state and energy transfer occurring via FRET or Dexter to the singlet or triplet of a fluorescent acceptor. Radiative decay can occur from the singlet of the acceptor bu t the acceptor triplet is a non-radiative (NR) decay pathway.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-5 ©A u t h o r ( s )2 0 2 0concentration of acceptor did not result in a higher number of FRET/Dexter events. Although the saturation of Dexter events mayseem positive, it ultimately means that the UPSF OLED has anupper limit to the lifetime. When considering acceptor concentra- tion to increase, less energy transfer from the donor to acceptor results in increased (long-lived) donor phosphorescent emission.Finally, we were able to demonstrate the potential of an ideal UPSFOLED design with Dexter energy transfer suppression and/or an increase of the radius in which FRET could occur. It was evident that the FRET radius had a significant impact on the OLED perfor-mance, exceeding the improvement from Dexter suppression alone.However, the chemical design of these materials, in particular,tuning of the FRET radius, is experimentally challenging, as it can easily lead to a compromise between efficiency, stability, and emis- sion color. Therefore, we show that by combining experimentaland computational results, it is possible to gain a better under-standing of such a novel concept while highlighting key areas forfurther device improvements. By addressing the role of exciton decay time in the UPSF system, we present one example of how simulations can addressquestions surrounding device stability. However, for a long time,the aspect of OLED stability has been underrepresented in aca-demic research, partially because of a lack of access to highly puri- fied materials or insufficient control over device fabrication methods. This problem can be overcome by collaboration betweenacademia and industry such that nowadays stability limitationsranging from chemical 92to morphological changes93in each of the OLED layers can be addressed. Computationally, it is possible to study chemical stability in terms of single-molecule properties, likebond-dissociation energies94or more complex molecular degrada- tion scenarios involving polarons and excitons.46Simulations on the device level have also shown that the role of exciton –exciton or exciton –polaron interaction can play a substantial part in OLED stability.51Therefore, being able to correctly describe the photophy- sics is a crucial part in understanding OLED stability limitations.This can be described as a multi-scale problem, including thesingle-molecule level, e.g., understanding fast intersystem crossing in TADF emitters, 95or on larger scales, e.g., triplet diffusion in the emissive layer96or exciton-dynamics coupled to charge dynamics on the full device level.97 OLED MATERIAL LIBRARY As shown for the UPSF OLED, an initial computational approach is to construct a model from the available experimentaldata, thus providing deeper insight. However, the ultimate goal ofOLED simulations is gaining extensive understanding between molecular structure and physical properties, ideally before synthe- sis, due to the vast number of possibilities. Therefore, it is vital toachieve reliable and informative simulations that could potentiallyreduce experimental focus to a more manageable subset of candi-dates. That being said, it remains a challenge to predict properties, such as solid-state ionization energies, electron affinities, and charge carrier mobilities. The morphology of the system is crucial in this task, as mor- phological disorder can lead to energetic disorder and energy traps, affecting charge transport. As a result, predictive structure –property simulations are difficult, as realistic morphologies require accurate FIG. 4. Simulation workflow: initial molecular structures are used to construct a morphology for molecular dynamics simulations followed by rate paramete rization from experimental data and then KMC to randomly propagate the system through time, providing OLED properties such as photoluminescence quantum yield (PLQY) and radia- tive decay times and plots such as time resolved photoluminescence (TRPL).Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-6 ©A u t h o r ( s )2 0 2 0modeling capable of predicting local packing arrangements, molecu- lar ordering, and trap concentration.98In order to construct mor- phologies representative of experimental systems, simulations requirewell controlled generation of homogenous amorphous solids. Thiscan be achieved by thermal annealing above the glass transition tem-perature of the material, followed by a fast cooling process, to form a glass structure, in an attempt to imitate the typical deposition process of an OLED layer structure. By doing so, the molecules arelocked in local energy minima. Unfortunately, this often varies sig-nificantly between experiment and simulation, such that the resultingmorphologies regularly disagree. The main challenge is that realistic molecular packing is difficult to achieve and requires long simulation times to study self-assembly. By employing more complex methodol-ogy with increased computational cost, ultra-stable glass structurescan be achieved by depositing particles one-by-one to obtainuniform packing, more closely mimicking that of experimental phys- ical vapor deposition structures. 99 Accurate morphologies rely on efficient classical force-fields, while accurate descriptions of morphological and electronic degreesof freedom require polarizable force-fields. Parametrization of suchforce fields is a tedious task and would be impossible for the vast number of organic compounds required. To overcome this, it is pos- sible to construct building blocks made up of the essential compo-nents of compounds most likely to be experimentally investigated,thereby introducing the concept of an OLED material library. OLED simulation workflow First, in order to generate realistic morphologies, appropriate force fields including accurate descriptions of bonded and non-bonded interactions must be used. Additionally, polarizable forcefields are required, which take into account the charge distribution rearrangement caused by changes in the environmental charge dis- tribution. An amorphous morphology can then be simulated withmolecular dynamics (MD), typically annealing above the glass tran-sition temperature, followed by fast quenching to room temperature using the NPT ensemble. Density functional theory (DFT) based electronic structure methods can be utilized to compute gas-phaseionization energy (IE 0) and electron affinity (EA 0). The choice of functional has to be carefully considered with the importance ofconsidering long-range corrected hybrid functionals for IE 0and EA0values recently shown.100In disordered organic materials, such as those found in OLEDs, charge carriers are localized and propa-gate through the system by successive hops from one molecule toanother. Rates can be computed with the Miller –Abraham expres- sion, 101typically utilized within Gaussian disorder models (GDM), with a lattice arrangement of hopping sites and Gaussian distributed site energies. Alternatively, rates can be described by a thermallyactivated type of transport in terms of the Marcus theory 102 –104or by using Weiss –Dorsey rates105 –109for a wider range of temperature regimes, specifically using the low temperature approximations where Marcus rates are not applicable. Within the high temperature limit of the classical charge transfer theory,103,110the Marcus rate equation is derived from the importance of environmental coupling, using linear response theory to describe a heat bath coupled to electronic tunneling.105 This quantum mechanical tunneling moves the electron from onemolecule to the other, at sufficiently high temperatures, when the nuclear vibrations (also described as bath fluctuations) bring the corresponding energy levels into resonance. The rate for a chargeto hop from site i to site j ( ω ij) is given by ωij¼2π /C22hJ2 ijffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πλijkBTp exp/C0(ΔEij/C0λij)2 4λijkBT/C20/C21 , (1) Here, λijis the reorganization energy, the response to a change of charge state. J ijis the electronic coupling matrix elements, describing the strength of coupling between two localized states. ΔEij¼Ei/C0Ejis the driving force, or site energy difference between two neighboring sites, where E iis the site energy of mole- cule i.111Molecules within a small cutoff range are considered neighbors, between which carrier hopping can occur. The quanti-ties within the Marcus equation are then computed for each neigh- boring pair. A brief outline is included below, and further details and an in-depth explanation of these calculations can be foundelsewhere. 54,112Computation of site energies includes (1) the ioni- zation energy and electron affinity of a single molecule in chargedand neutral geometries and (2) the interaction with the environ- ment, including the electrostatic and induction contributions, with adequate cutoffs for long-range interactions to be considered. Forreorganization energies, considering a charging and dischargingmolecule, the internal contribution due to geometric rearrangementand external contribution due to the environment must both be included. Finally, the electronic coupling elements require approxi- mation of diabatic states, usually with the ionization energy andelectron affinity, for hole or electron transport, respectively. Withthe charge transfer rates computed, it is possible to then model charge dynamics. Each carrier hop or event takes the system from state “a”to“b”with the corresponding rate for this transition and a probability for it to occur, which can be represented by the masterequation. KMC is one method of solving the master equation thateffectively provides charge carrier mobilities in the given system for holes or electrons. The complete multiscale simulation workflow is outlined in Fig. 5 . In order to facilitate a simpler workflow, it would be advanta- geous to have a library containing the vital components of thesecalculations. This would include the classical and polarizable force fields, as well as the structures and input parameters required for computation of electronic properties. In turn, this would establishthe building blocks for further structures and systems to be investi-gated. In our current work, by creating these building blocks forvarious small molecule systems, it is our aim to provide an outlook for a fully functional OLED material library, which is of the upmost importance for future material design. In order for thisconcept to reach its full potential, i.e., the extraction of materialproperties from molecular structure alone, certain aspects of the forward problem should be addressed, which brings us to the adap- tation of the material design strategy for future progress. THE FUTURE OF COMPUTATIONAL OLED DESIGN Ideally, parameter-free computer-based OLED design would be employed, utilizing pre-existing building blocks and tools, begin- ning with accurate prediction of material morphology for a newJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-7 ©A u t h o r ( s )2 0 2 0system followed by the calculation of the energetic landscape. This would lead to rate evaluation of the various processes within thesystem, such as charge or exciton transfer, proceeding to solvingthe time-dependent master equation, all of which provide essentialquantities for material evaluation, such as electron affinity, ioniza- tion potential, and hole and electron mobilities. However, this mul- tiscale procedure, with length and time scales spanning severalorders of magnitude is non-trivial, and as such, it is difficult topredict device properties from the structure alone. A necessity for future OLED development (and for all organic electronics) is the complete underst anding of all fundamental processeswithin the device and constituent materials. Charge and energy transfer simulations play a pivotal role in this pursuit, thereby pro-moting the advancement of simulation techniques and methods toachieve an increasingly comprehensive description. Exciton forma-tion and transfer are particularly important for OLED functionality. The theoretical tools used to determine exciton transport parame- ters include DFT for electronic excitation properties, 113many-body Green ’s function theory, and GW approximation with the Bethe – Salpeter equation (GW-BSE)114for excited states113,115,116and an adaption of Marcus theory to describe exciton diffusion.117Further to this, an essential link between micro and macroscale would be FIG. 5. OLED multiscale modeling simulation workflow: starting from first principles of an isolated molecule combined with atomistic and polarizable forc e fields. The amor- phous morphology is generated with the use of molecular dynamics (MD). After computing the charge transfer rates, kinetic Monte Carlo (KMC) can then b e used to solve the master equation to study charge dynamics (e.g., carrier mobilities), giving macroscopic device characteristics. Additionally, future develo pment can allow for explicit coulomb interaction to be included within KMC; implementation would involve rate updates at each KMC step.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 160901 (2020); doi: 10.1063/5.0022870 128, 160901-8 ©A u t h o r ( s )2 0 2 0employing more realistic charge transport dynamics, with an impor- tant addition, but computationally demanding approach, being the inclusion of explicit coulomb interactions and reevaluation ofcharge transfer rates at each KMC step. The embedding of electrostatics within KMC simulations, as shown in Fig. 5 , would provide valuable insight into charge dynam- ics, specifically, within slab geometries and interface interactions, vital for device characteristics. However, when investigating asystem that is stochastically propagated in time, with each chargecarrier hop leading to a new charge distribution and consequentlyaltering all interactions with other charge carriers, the complexity of the problem is elevated and requires accurate and efficient methodology. The accurate evaluation of electrostatic interactions is crucial but still remains a challenge as the methods to compute electrostat-ics require large system sizes and inherently long-range (complex) interactions, leading to high computational cost. Interaction dis- tance cutoff methods can be applied to reduce the electrostatic con-tributions. But due to the long-range nature of electrostatics, as aresult of the 1/r decay of the Coulomb potential, a cutoff radius isoften insufficient and more exact methods have to be employed. In fact, neglecting long-range contributions has been shown to cause inaccuracies with variation of simulated carrier densities and deviceperformance predictions, when compared to exact methods. 118On the other hand, using exact methods for computing electrostatics typically involves summation techniques, such as Ewald summa- tion119and the more efficient particle mesh Ewald (PME).120,121 However, in large periodic systems, an exact description is often unfeasible or highly computationally demanding. Therefore,further methods have to be considered, specifically for non- periodic systems (requiring more sophisticated summation). Very recently, efficient electrostatic evaluation for the use in KMC applications have been demonstrated by exploiting the factthat each KMC step corresponds to one charge carrier hop, result-ing in a modest charge redistribution. 122,123This has been achieved with a new variant of the Fast Multipole Method122and in a second approach utilizing local charge contributions to thehopping rate before and after a hop, allowing for a newly adaptedcutoff scheme. 123The implementation of electrostatics within KMC simulations is clearly challenging, but this recent progress shows it is achievable. Furthermore, computational material design tools look toward other future developments. A key area and one which has gainedsignificant interest in recent years is machine learning based techni- ques. Application of machine learning for OLED materials would be a significant step forward for their computational design. Forthis to be achievable, accurate molecular/chemical descriptors arecrucial. This is a difficult task, as it requires identification of corre-lations between similar structures with similar properties, linking this to a simple and systematic feature which can be extracted. 124 For OLED materials, these vital descriptors are missing, and only when they can be accurately obtained can the field move towardmachine learning approaches. Nevertheless, recent progress inmachine learning demonstrates the significant potential of these methods. First, local properties such as electrostatic multipoles can already be predicted using kernel-based techniques. 125Secondly, they can present practical strategies to parametrize force fields byproviding coarse-grained (CG) potentials, which are more effi- cient.126Third, properties such as the glass transition temperature can be correlated with the chemical structure, using a quantitativestructure –property relationship approach, 127with predictive model- ing capable of pre-screening thermally stable candidates from onlytopological indicies, 128altogether highlighting the possibilities of machine learning in the context of OLED design. CONCLUSION AND OUTLOOK In summary, OLED applications do offer a unique alternative to traditional display technologies, with their flexible, lightweight, and transparent possibilities. However, due to their commercial infancy, they have obstacles to overcome, particularly regardingdevice lifetime and efficiency. Within this Perspective, we havehighlighted the improvements made to each of the OLED layers, with respect to their constituent materials, most notably the emis- sive layer and the extensive efforts to achieve an efficient and stableblue OLED, including the novel UPSF approach, bringing a blueOLED to light by reaching a balance of stability, efficiency, andemissive color, accompanied by our multiscale computational studies, highlighting the possibilities of the concept with an appro- priate material design. The shift to an OLED material library thensignifies the importance to achieve pre-screening, reducing analmost infinite set of potential candidates for OLED applications,to a more manageable number prior to synthesis. This multiscale procedure is, of course, a non-trivial task, and certain aspects of the forward problem need to be addressed before moving toparameter-free methods, allowing for extraction of device proper-ties from the molecular structure alone. Computational methoddevelopment is therefore essential, only then can the gap between structure and device properties be closed, thereby reducing both experimental and computational cost. ACKNOWLEDGMENTS D.A. acknowledges the BMBF Grant InterPhase (No. FKZ 13N13661) and the European Union Horizon 2020 Research andInnovation Programme ‘Widening Materials Models ’under Grant Agreement No. 646259 (MOSTOPHOS). This research has beensupported by the King Abdullah University of Science and Technology (KAUST), via the Competitive Research Grants (CRG) Program. D.A. acknowledges KAUST for hosting his sabbatical. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1A. Bernanose, M. Comte, and P. Vouaux, “A new method of emission of light by certain organic compounds, ”J. Chim. Phys. 50,6 4 –68 (1953). 2A. Bernanose and P. Vouaux, “Electroluminescence of organic compounds, ” J. Chim. Phys. 50, 261 –263 (1953). 3M. 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5.0029808.pdf

Appl. Phys. Lett. 117, 171603 (2020); https://doi.org/10.1063/5.0029808 117, 171603 © 2020 Author(s).Time-dependent resistance of quasi-two- dimensional electron gas on KTaO3 Cite as: Appl. Phys. Lett. 117, 171603 (2020); https://doi.org/10.1063/5.0029808 Submitted: 16 September 2020 . Accepted: 16 October 2020 . Published Online: 28 October 2020 Gensheng Huang , Pengfei Zhou , Lingyu Yin , Ze Zhou , Shuainan Gong , Run Zhao , Guozhen Liu , Jinlei Zhang , Yang Li , Yucheng Jiang , and Ju Gao ARTICLES YOU MAY BE INTERESTED IN Assessing the limits of determinism and precision in ultrafast laser ablation Applied Physics Letters 117, 171604 (2020); https://doi.org/10.1063/5.0023294 Magnetoresistance effects in cadmium arsenide thin films Applied Physics Letters 117, 170601 (2020); https://doi.org/10.1063/5.0031781 Delta-doped β-Ga2O3 films with narrow FWHM grown by metalorganic vapor-phase epitaxy Applied Physics Letters 117, 172105 (2020); https://doi.org/10.1063/5.0027827Time-dependent resistance of quasi-two-dimensional electron gas on KTaO 3 Cite as: Appl. Phys. Lett. 117, 171603 (2020); doi: 10.1063/5.0029808 Submitted: 16 September 2020 .Accepted: 16 October 2020 . Published Online: 28 October 2020 Gensheng Huang,1Pengfei Zhou,1Lingyu Yin,1ZeZhou,1Shuainan Gong,1Run Zhao,1Guozhen Liu,1 Jinlei Zhang,1 Yang Li,1Yucheng Jiang,1,a) and Ju Gao2,a) AFFILIATIONS 1Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Physical Science and Technology, Suzhou University of Science and Technology, Suzhou 215009, People’s Republic of China 2School for Optoelectronic Engineering, Zaozhuang University, Shandong 277160, People’s Republic of China a)Authors to whom correspondence should be addressed: jyc@usts.edu.cn andjugao@hku.hk ABSTRACT For most conductive materials, resistance remains constant over time in the absence of external physical stimulation. Here, we report the time-dependent resistance of a quasi-two-dimensional electron gas (Q2DEG) on a KTaO 3substrate. Arþ-ion bombardment is used to achieve a Q2DEG of high density. Such a Q2DEG shows a linear increase in resistance with time without further physical stimulus. Both theresistance and its increase rate can be determined by the beam voltage of Ar þbombardment. Furthermore, we find that light illumination strongly influences the increase rate of resistance, with the effect depending primarily on the wavelength of incident light. The Hall effect reveals that this phenomenon can be attributed to the spontaneous decrease in carrier density over time due to the migration of oxygenvacancies. Our work offers a pathway toward a self-excited resistance timer in a Q2DEG system. Published under license by AIP Publishing. https://doi.org/10.1063/5.0029808 Two-dimensional electron gas (2DEG), identified at oxide inter- faces, has drawn considerable attention for its remarkable physicalproperties, 1,2such as superconductivity,3magnetoresistance,4and ferromagnetism,5among others. Specifically, the 2DEG at the interface of LaAlO 3/SrTiO 3(LAO/STO) shows high mobility and persistent photoconductivity, making it a candidate for future electronic andmemory devices. 6,7The electric field and light are often applied to con- trol the carrier density of 2DEG.8–11With the advancements in research, a simpler method has been developed to induce a 2DEG layer on the STO surface by creating oxygen vacancies.12–14Arþbom- bardment is a useful technique to separate oxygen ions from the oxide,thus causing oxygen vacancies. 15The charge carrier density is deter- mined by beam voltage and time.12,13Despite the different preparation methods, these 2DEG systems display similar photoelectric transportproperties. 16Their intrinsic resistance is stable and time-independent. After stimulus by an electric field or by light, a 2DEG system may become metastable, showing a temporary time-dependent resis- tance.17,18Given enough time, it will recover to its intrinsic resistance. Recent research has revealed novel 2DEG systems based on KTaO 3(KTO), such as LaTiO 3/KTO,19LAO/KTO,20EuO/KTO,21 and LaVO 3/KTO.22As a perovskite oxide, the KTO surface can form a high-mobility 2DEG by depositing an oxide layer on it.23Comparedto another perovskite oxide of STO, a great advantage of KTO is the existence of large spin–orbit coupling, making it a candidate for aRashba system. 24This offers the possibility of achieving a spin- polarized current due to the Rashba effect, allowing for spin-electronic applications.25Despite some works on 2DEG based on KTO, few stud- ies have focused on the 2D conductive layer caused only by oxygenvacancies. It is, in fact, very important to clarify the transport propertyof oxygen vacancy-induced 2DEG, which avoids the influence of other oxides and, thus, directly reflects the nature of KTO. In this Letter, we report the achievement of a quasi-2DEG (Q2DEG) of high carrier density on a KTO substrate through Ar þ-ion bombardment. It shows a linear dependence of resistance on the time,referred to as the timer resistance (TR) effect, suggesting a potentialapplication as a timer. Both the resistance and its increase rate (r in)a r e determined by the beam voltage of Arþbombardment. Compared to a traditional timer based on a crystal oscillator,26the Q2DEG on KTO can perform a long-term recording of time without being drivenelectrically. In a stable environment, the device never deviates from a linear increase in resistance, even over a very long period of time. Moreover, the increase rate of resistance is strongly influenced by lightillumination. The wavelength, rather than optical power density, deter-mines the value of r in. The Hall effect demonstrates that a spontaneous Appl. Phys. Lett. 117, 171603 (2020); doi: 10.1063/5.0029808 117, 171603-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apldecrease in carrier density should be responsible for the TR effect, resulting from the migration of oxygen vacancies. KTO single crystals with (001) orientation were used for Arþ bombardment.13,27T h ec o n d u c t i v el a y e r sw e r ef o r m e do nK T Os u b - strates through an ion source with the beam voltages of 250, 300, 350, 400, and 500 V.28The Ar gas pressure and flow rate are 3 /C210/C04 mbar and 6 sccm, respectively. The detail conditions of Arþbombard- ment are shown in Table I . All samples are exposed to the Arþion beam for 10 min. According to the previous work,12,29the penetration depth of Arþions is about 3–10 nm for the KTO samples, suggesting the quasi-2D conduction. This indicates that the Q2DEG can be formed on the surface of KTO.27,30The resulting samples are spin- coated by polymethyl methacrylate (PMMA) for protecting them from oxygen. The transparent PMMA layer cannot influence the elec- tric transport of Q2DEG on KTO, due to the insulation. The lattice orientation was measured by x-ray diffraction (XRD) with the wave- length of 1.5406 A ˚. An atomic force microscope (AFM) was used to measure the surface morphology of KTO. The circuit was connected by a wire bonder, and the resistance-time characteristic was measured using a 6517B Source Meter. All electrical measurements were per- formed using a physical property measurement system with a He atmosphere. Monochromatic light illumination was supplied by power-tunable semiconductor lasers with the wavelengths of 405, 447, 532, 655, and 808 nm. The light transmitted into the sample chamber through an optical fiber bundle. The crystal structure of KTO is illustrated in Fig. 1(a) , exhibiting a cubic perovskite structure. Figure 1(b) shows an XRD h-2hscan of the KTO single crystal. We observed the (100), (200), and (300) XRD diffraction peaks, confirming the single-crystal lattice structure. Due to its large bandgap of 3.61 eV, a pristine KTO single crystal is a transpar- ent insulator.31According to previous studies on the oxide STO, the Arþbombardment can induce oxygen vacancies on the surface and form a Q2DEG layer.12Inspired by this, we used the same method to induce oxygen vacancies on the KTO surface as shown in Fig. 1(c) . Normally, the surface oxygen vacancies after Arþbombardment tend to recombine in the air. In order to stabilize the physical properties of the device, the KTO substrate was spin-coated by PMMA and placed in a chamber of He atmosphere to prevent oxidation. Figure 1(d) shows the surface morphology of KTO revealed through AFM before and after the Arþbombardment. Obviously, the surface is relatively rough before the bombardment, and it becomes smoother after that. Figure 1(e) shows the Raman spectra of KTO before and after the Arþ bombardment. All longitudinal and transverse Raman modes are detected, such as LO 1–LO 3and TO 1–TO 4, in good agreement with the previous reports.32It is found that Arþbombardment cannoteffectively affect the positions of Raman peaks, indicating that the lat- tice symmetry of KTO is highly stable. Here, we focus on the electrictransport characteristics of Q2DEG on KTO at room temperature. In Fig. 1(f) , the resistance is plotted as a function of time for the KTO substrate bombarded at 250 V. At 50 min, with the ambient lightturned off, the resistance of the device starts to increase, as shown inthe inset of Fig. 1(f) . Due to the loss of the influence of ambient light, the device recovers to the intrinsic resistance preference. For ordinarymaterials, the resistance enables one to return to a stable resistancestate, as long as the time is long enough. However, for the Q2DEG on KTO, the resistance shows a linear dependence on the time without leveling off, even after the four-day measurement. The resistanceseems to increase permanently at the same r in. The resistance can be expressed as R¼R0þrint,w h e r eR 0is the initial resistance before the linear increase and t is the time. The device was even placed in dark-ness for over one month. When measuring it again, we observed that its resistance was still increasing linearly with the same r in. This anom- alous metastable phenomenon cannot come from the long-lastingresistance change with light turned off due to persistent photoconduc-tivity, but from the intrinsic electric transport property of Q2DEG onKTO. Such a special TR behavior is completely spontaneous withoutbeing driven electrically, which inspires us to use this feature to imple- ment timer applications. The beam voltage plays an important role in the oxygen vacancy density of KTO. As shown in Fig. 2(a) , the beam voltage strongly influ- ences the resistance and r inof the device. With the beam voltage less than 350 V, the resistance shows a steady linear increase with the timeover a 10 h period. But there is a slight deviation from the linear rela-tionship when the beam voltage is larger than 400 V. Figure 2(b) shows that the resistivity of the Q2DEG on KTO first increases and then decreases as the beam voltage increases. It is obvious that thedevice shows the largest resistivity at the beam voltage of 350 V. At alarger beam voltage, all the K, Ta, and O atoms will be ablated away,thus allowing for no more oxygen vacancies. Figure 2(c) shows the dependence of r inon the beam voltage. The device bombarded at 300 V exhibits the maximum of r in. Large r insignifies a high resolutionTABLE I. Arþbombardment conditions. Beam voltage (V)Accelerator voltage (V)Beam current (mA)Emission current (mA) 250 50 8 9 300 60 10 11350 70 15 17400 80 20 22500 100 30 33 FIG. 1. (a) Lattice structure of KTO, showing a perovskite unit cell. (b) XRD h-2h scan of a KTO single crystal. (c) Schematic of the KTO substrate bombarded by the Arþion beam for Q2DEG. (d) AFM image for KTO surface morphology before and after the Arþbombardment. (e) Raman spectra of KTO before and after Arþ bombardment. (f) Resistance-time characteristic of the Q2DEG on KTO after Arþ bombardment at 250 V with ambient light off. Inset: amplification of the curve in theselected area.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 171603 (2020); doi: 10.1063/5.0029808 117, 171603-2 Published under license by AIP Publishingfor the time recording, but a relatively short work time. For the device bombarded at 250 V, it exhibits the minimum of r inand a long work time. For studying the effect of oxygen on the Q2DEG on KTO, we investigate the electric transport with it exposed to air by removing the PMMA coating in Fig. 2(d) , showing the exponential increase in resis- tance with time. After five days, it becomes almost an insulator. The c o m b i n a t i o no fo x y g e ni na i ra n do x y g e nv a c a n c i e so nK T Oi n d u c e sa drastic decrease in carrier density of the Q2DEG, thus leading back tothe intrinsic insulation state. The recombination of oxygen vacancies should be responsible for the exponential increase in resistance. By contrast, the TR effect displays a linear increase in resistance without oxygen exposure, which cannot be attributed to the oxygen-induced recombination of oxygen vacancies. To use the TR device as a timer, the stable resistance increase is very important. In Fig. 3(a) , we inspect the effect of light on the resis- tance. With the 405 nm laser turned on, the device shows the sharp decrease in resistance and deviates from the intrinsic increase rate of resistance. But, as long as the light is turned off, it will return to the original state. This indicates that the effect of light on the resistance is temporary. Such a stable spontaneous resistance increase offers the possibility of the device as a timer. Moreover, it is observed that the resistance still increases linearly with time under the continuous light illumination. Compared to the case in darkness, r inis significantly decreased by the light. Another intriguing observation is that r inis independent of the optical power density. With it increasing from 3 to 130 mW /C1cm/C02, this device shows nearly the same increase rate of resistance. Note that the optical power density can determine the speed of the transition from the light-induced resistance drop to the stable linear increase. The transition time significantly decreases with the increasing light intensity. Actually, the value of r independs strongly on the photon energy of incident light, as shown in Fig. 3(b) . For the photon energy of 1.53 eV, corresponding to the wavelength of8 0 8 n m ,t h ei n c r e a s er a t eo fr e s i s t a n c eb e c o m e sv e r yc l o s et ot h a ti n darkness. The inset exhibits that it pronouncedly decreases with the increasing photon energy. This enables us to achieve a desired rate byselecting an appropriate wavelength. Resistance of a material is determined by the carrier density and carrier mobility. For the linear dependence of resistance in theQ2DEG on KTO, it is necessary to investigate the time dependence of them. To clarify this point, we conducted Hall effect studies on such a TR device. In Fig. 4(a) , the Hall resistance depends linearly on the magnetic field, and the absolute value of its slope gradually decreases with time. For the Hall effect in a 2D system, we can write down theexpression as R H¼B ns/C1q, where R His the Hall resistance, B is the magnetic field, n sis the sheet carrier density, and q is the unit charge. Figure 4(b) shows the dependence of carrier density and mobility on time. It is observed that the carrier density decreases pronouncedly with time, while the mobility exhibits only a slight increase. The Halleffect results suggest that the reduction of carrier density is the main reason for the linear increase in resistance in the Q2DEG on KTO. A possible explanation for the TR effect is that the oxygen ions in the KTO bulk tend to diffuse to the surface and combine with the FIG. 2. (a) R–R 0as a function of time in darkness at room temperature, after Arþ bombardment at the beam voltages of 250, 300, 350, 400, and 500 V. (b) Resistivity vs beam voltage at room temperature. (c) Dependence of r inon beam voltage. (d) Resistance-time characteristic of Q2DEG on KTO without PMMA coat-ing after the Arþbombardment at 400 V. FIG. 3. (a) Resistance-time characteristic of the Q2DEG on KTO prepared at the beam voltage of 300 V with the 405 nm light on and off at room temperature, wherethe optical power densities are 3, 19, and 130 mW /C1cm/C02. (b) R–R 0of the Q2DEG plotted as a function of time, under the light illuminations of 405, 447, 532, 655, and 808 nm wavelengths. Inset: dependence of r inon the photon energy. FIG. 4. (a) Hall resistance as a function of magnetic field for the Q2DEG on KTO prepared at the beam voltage of 250 V, where the measurement is performed every five hours. (b) Time dependence of sheet carrier density and mobility at room tem-perature. (c) Schematic showing the migration of oxygen vacancies. (d) R–R 0vs t at various temperatures.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 171603 (2020); doi: 10.1063/5.0029808 117, 171603-3 Published under license by AIP Publishingoxygen vacancies in Fig. 4(c) , due to the oxygen gradient from the inside to the surface. This process can also be understood as the migra-tion of oxygen vacancies from the surface to the inside. When oxygenvacancies are concentrated on the surface, there are strong connections between them with conductive channels formed. By contrast, the oxy- gen vacancies that migrate into the bulk are scattered in a low density.The distances between oxygen vacancies are too long to form conduc-tive channels, causing the whole bulk to be still insulating. The migra-tion of oxygen vacancies induces the slow decrease in carrier densityon the surface, which just makes the resistance increase linearly. It isworth pointing out that the Q2DEG on KTO is essentially in a meta-stable state, where the migration of oxygen vacancies is a slow butlong-term spontaneous process. Such a process will not stop until theoxygen vacancies are uniformly distributed in the whole KTO bulk.Figure 4(d) shows the time dependence of R–R 0at various tempera- tures. It is observed that the resistance remains constant with the timeat the temperature of 200 K. With the temperature rising to 320 K, thelinear increase in resistance occurs and r inis significantly enhanced. As we have known, the diffusion is a key factor to drive the migration of oxygen vacancies. An increase in temperature must speed up thediffusion process, thereby inducing a faster increase in resistance.When the temperature reaches 350 K, the resistance may deviate fromthe linear dependence on the time and display almost an exponentialgrowth. Thus, the temperature plays an important role in acceleratingthe recombination of oxygen vacancies. In summary, we achieve the Q2DEG of high carrier density on the KTO substrate by Ar þ-ion bombardment. The linear dependence of its resistance on time is observed as the TR effect, suggesting apotential application as a timer. In a stable dark environment, thedevice shows the constant value of r inover a very long period of time. Continuous illumination is found to control the rate, which is corre- lated with the wavelength of light, but independent of optical power density. The Hall effect demonstrates that the spontaneous decrease incarrier density is the main reason for the TR effect. A possible mecha-nism is proposed to explain the lowering of oxygen vacancy density.For the Q2DEG in a metastable state, the migration of oxygen vacan-cies drives the linear increase in resistance with the time. AUTHORS’ CONTRIBUTIONS G.H., P.Z., and L.Y. contributed equally to this work. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11504254, 11704272, 12074282,62004136, and 11974304). This work was also supported by the Jiangsu Key Disciplines of the Thirteenth Five-Year Plan (No. 20168765), the Jiangsu Undergraduate Training Program forInnovation and Entrepreneurship (No. 201910332021Z), thePostgraduate Research & Practice Innovation Program of JiangsuProvince (No. KYCX20_2753), the Natural Science Foundation ofJiangsu Province (Grant No. BK20190939), and the Natural ScienceFoundation of the Jiangsu Higher Education Institutions of China(Grant No. 19KJB150018). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1A. Ohtomo and H. Y. Hwang, Nature 427, 423 (2004). 2S. Gariglio, M. Gabay, and J. M. Triscone, APL Mater. 4(6), 060701 (2016). 3L. Li, C. Richter, J. Mannhart, and R. C. Ashoori, Nat. Phys. 7(10), 762 (2011). 4S. Seri and L. Klein, Phys. Rev. B 80(18), 180410 (2009). 5F. Bi, M. Huang, S. Ryu, H. Lee, C.-W. Bark, C.-B. Eom, P. Irvin, and J. Levy, Nat. 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5.0019584.pdf

J. Appl. Phys. 128, 184503 (2020); https://doi.org/10.1063/5.0019584 128, 184503 © 2020 Author(s).High-acoustic-index-contrast phononic circuits: Numerical modeling Cite as: J. Appl. Phys. 128, 184503 (2020); https://doi.org/10.1063/5.0019584 Submitted: 25 June 2020 . Accepted: 24 October 2020 . Published Online: 11 November 2020 Wance Wang , Mohan Shen , Chang-Ling Zou , Wei Fu , Zhen Shen , and Hong X. Tang COLLECTIONS This paper was selected as an Editor’s Pick High-acoustic-index-contrast phononic circuits: Numerical modeling Cite as: J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 View Online Export Citation CrossMar k Submitted: 25 June 2020 · Accepted: 24 October 2020 · Published Online: 11 November 2020 Wance Wang,1,2,a) Mohan Shen,1,2Chang-Ling Zou,1,2 Wei Fu,1Zhen Shen,1,2and Hong X. Tang1,b) AFFILIATIONS 1Department of Electrical Engineering, Yale University, New Haven, Connecticut 06511, USA 2Depratment of Optics, University of Science and Technology of China, CAS, Hefei, Anhui 230026, China a)Present address: Department of Physics, University of Maryland, College Park, Maryland 20742, USA. b)Author to whom correspondence should be addressed: hong.tang@yale.edu ABSTRACT We numerically model key building blocks of a phononic integrated circuit that enable phonon routing in high-acoustic-index waveguides. Our particular focus is on the gallium nitride-on-sapphire phononic platform which has recently demonstrated high acoustic confinementin its top layer without the use of suspended structures. We start with the systematic simulation of various transverse phonon modes sup- ported in strip waveguides and ring resonators with sub-wavelength cross section. Mode confinement and quality factors of phonon modes are numerically investigated with respect to geometric parameters. A quality factor of up to 10 8is predicted in optimized ring resonators. Next, we study the design of the phononic directional couplers and present key design parameters for achieving strong evanescent couplingsbetween modes propagating in parallel waveguides. Last, interdigitated transducer electrodes are included in the simulation for direct excita-tion of a ring resonator and critical coupling between microwave input and phononic dissipation. Our work provides a comprehensive numerical characterization of phonon modes and functional phononic components in high-acoustic-index phononic circuits, which supple- ments previous theories and contributes to the emerging field of phononic integrated circuits. Published under license by AIP Publishing. https://doi.org/10.1063/5.0019584 I. INTRODUCTION Surface acoustic wave (SAW) devices are widely used in elec- tronic circuits, finding applications such as filters and oscillators in communication devices 1and a range of sensing applications.2–4 Recently, SAWs were also exploited for coherent control of various quantum systems, including the electron quantum dots,5–7electron spins in diamond,8superconducting qubits,9–11and integrated pho- tonic devices.12–15Thus, SAW provides a promising platform for hybrid quantum systems,16–19where the traveling SAW phonon can serve as a quantum bus to facilitate quantum state transfer. The coupling strength between SAW and matter/photons is enhanced with reduced mode volume of the SAW. The enhanced interactions such as the strong coupling regime of cavity quantum acoustodynamics20,21not only increase coherent coupling rates between quantum bits but also improve the sensitivity of SAW-based measurements.22Also, the full potential of phononic systems can be only unraveled when maneuverability of phonon becomes comparable to its electrical and optical counterparts.Therefore, a strong confinement of the SAW to the diffraction limit and the long lifetime phononic resonators are in high demand. To that end, phonon waveguides23–25and ring resonators26,27based on SAW were proposed and experimentally studied in the 1970s.However, in the following several decades, the experiments anddetailed theoretical studies of the confinement of itinerant phononsin microstructures were less active, especially for the phonon reso- nators of radiating waves in nature. Only in recent years, due to the advances of nano-fabrication technologies, ultra-low loss phononicwaveguides and resonators are achieved with bulk acousticwave 28,29and with SAW,30–41pushing the study of phononic devices, including SAW-related waveguides and resonators back to the frontier of researches as a strong contender for advanced pho- nonic circuits. Interdigital transducers (IDTs) convert the electrical RF signal into SAW or vice versa by employing the piezoelectric materials. Inmost applications, IDTs are fabricated on a uniform film, and the lateral size of IDT is much larger than the wavelength of SAW to excite and collect the quasi-collimated SAW. As a primary sourceJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-1 Published under license by AIP Publishing.or receiver of SAWs, the scaling of IDTs and their integration with other SAW devices is also in critical demand in the development of phononic circuits. A variety of piezoelectric material platforms have been explored for SAW devices, including ST-X quartz,20,22zinc oxide,42–44lithium niobate,10,45,46lithium tantalate,47aluminum nitride,48–51gallium arsenide,9,21,52,53and gallium nitride.41,54–56 Together with substrate materials such as sapphire, diamond, and silicon, various combinations of high-acoustic-index-contrast SAWdevices can provide better confinement and thus low loss, and thedispersion can be engineered. Table I summarizes common piezo- electric materials and substrates. Among these piezo-materials, LiNbO 3and LiTaO 3possess much larger electromechanical cou- pling coefficients ( K2) than other materials, but they have poor temperature stabilities, whereas quartz is in a reverse case withweak coupling but good temperature stability at room tempera- ture. 57ZnO has wide bandgap58and good film stoichiometry59in addition to relatively large K2.60ZnO =diamond is also a quantum interface between SAW and nitrogen-vacancy center in diamond.42 AlN has high refractive index and high phase velocity,56,61which is in favor of high-frequency devices, but in turn, the contrast of acoustic index will be low. It also finds applications in high- temperature environments.49Semiconductor materials GaN and GaAs are compatible with standard wafer-scale fabrication, andGaAs is also naturally suited for InGaAs quantum dots, 53which benefit from the telecom range. The availability of GaN high- performance amplifiers enables monolithic integration withdevices. For the layered system, to our knowledge, ZnO =diamond, ZnO =sapphire, ZnO =SiO 2,42,43,62AlN=sapphire, AlN =SiO 2,49,50 and GaN =sapphire56have been explored. Diamond has the highest known SAW phase velocity but its fabrication requires chemical vapor deposition,62and it suffers from larger propagation loss. The contrast of SiO 2-based platform is not high. Concluding all factors, the GaN =sapphire platform stands out as a high-performance and ready-access choice. In this paper, we numerically investigate the properties of inte- grated phononic waveguides and resonators based on theGaN-on-sapphire platform. Different from ΔV=Vconfinement effect induced by the metal electrodes, 30–33,70we consider the pure geometric confinement of phonon on the surface of the chip, and, thus, the metal-induced loss is eliminated. We prove that the pho- nonic ring resonator can have exceptionally high quality factors ( Q)with the radius of tens of micrometers, in which the whispering gallery modes (WGMs) can be excited either by the evanescent acoustic field through the phononic waveguide coupling or bydirect IDT excitation. These waveguide and resonator structureswill be basic elements in future phononic integrated circuits ofboth fundamental and practical interests. 56,71 II. WAVEGUIDE Efficient confinement and guiding of phonons is essential for scalable operation of phononic devices. In the past few decades,SAW has been mostly employed in applications that utilize its verti-cal confinement at the surface of substrates. However, the lateral confinement to the SAW is seldom studied. Alternatively, three- dimensional confinement of phonon has been mostly achieved insuspended phononic microstructures, which imposes practical con-straints in fabrication yield and structural robustness. Here, we numerically study a phononic architecture based on unsuspended phononic waveguides and resonators. Shown in Fig. 1(a) is a typical strip waveguide. The basic requirement for phonon confine-ment is that the speeds of both transverse and longitudinal wavesin strip material are slower than those in the substrate. 24Thus, we choose the material platform of GaN-on-Sapphire satisfying these requirements, with parameters shown in Table II . An equivalent condition is also drawn in Ref. 72, where the waveguide layer mate- rial needs to be heavier and less stiff. More detailed analyses aboutthe requirements for the existence of Love wave can be found in Refs. 72–74. Conventional Rayleigh waves have been widely studied and applied in SAW devices, while half-space single medium also sup-ports shear-horizontal (SH) waves. In layered systems, in additionto layered Rayleigh waves, there are also Love waves, 57which was first discovered by Love.75Love waves have a shear component with displacements in the surface, which can be regarded as modifiedforms of the SH wave. SH waves and Love waves are now attractinginterest in spin –orbit interactions of phonons 56and sensing applications.47,58 The basic properties of confined guiding modes are studied numerically by the three-dimensional finite-element method(COMSOL Multiphysics v5.2). The waveguide geometrical parame-ters, namely, its width w 0and height h0marked in Fig. 1(a) , deter- mine the properties of the waveguide modes such as frequency, confinement, loss, and energy distributions. In this paper, all thesimulations are carried out using periodic boundary conditionsaccording to the translational symmetry of the structure, and per-fectly matched layers (PMLs) are employed for studying the radia- tive loss, with the detailed numerical model illustrated in Fig. 1(b) . To reveal the basic behaviors of the phononic waveguides, theanisotropy of GaN and sapphire are neglected. Shown in Figs. 2(a) –2(h) are fundamental to higher-order transverse phonon modes supported in a strip waveguide, with its width set at w 0¼5μm (for mode h,w0¼6μm because it does not appear at w0¼5μm) and h0¼0:5μm. The wavelength along propagating direction is set at λ¼2μm. The color mappings show the strength of normalized displacement j~xj=max (j~xj)i ne a c h mode profile, which applies to all other figures of the displacement field in this paper. Note that we label the modes by theTABLE I. SAW waveguide and substrate materials. Vland Vtare longitudinal and transverse wave speed, respectively. Material Vl(m s−1) Vt(m s−1) Piezoelectricity Quartz635 700 3 158 Weak ZnO645 790 2 700 Strong LiNbO 3657 316 4 795 Strong AlN6610 169 6 369 Good GaN667 350 4 578 Good Diamond6718 000 12 000 No Sapphire6810 658 5 796 No SiO 2698 433 5 843 NoJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-2 Published under license by AIP Publishing.corresponding characters a,...,hin the following for analyzing the evolution of mode profiles against geometry parameters. Modea[Fig. 2(a) ] is dominated by out-of-plane (z-direction) displace- ment, which we refer as quasi-Rayleigh mode owing to its reminis- cence of Rayleigh wave. 78Mode bhas dominant in-plane (y-direction) motion and is a quasi-Love mode because of its simi-larity to Love wave. 75The modes aand bcan be regarded as the fundamental phononic modes in the strip with different polariza- tions (z- and y-polarized motion). Due to the lateral confinement and the edges, we found that the phononic modes are distinct fromSAW modes. On the one hand, the modes do not show pure flex-ural wave or in-plane shear waves. In particular, at the edge, it isdifficult to identify the mode orientation because of the strong hybridization of deformation in all three orthogonal directions. On the other hand, higher-order transverse modes appear due to thelateral confinement [ Figs. 2(c) –2(h)]. For example, compared to fundamental mode aandb, in which vibrations on the strip edges are in-phase, the motion of modes canddon the opposite edges is out-of-phase. In the rest of this paper, we call the symmetric (in-phase) mode aas S-Rayleigh mode and call the anti-symmetric(out-of-phase) mode cas A-Rayleigh mode, as well as A-Love mode band S-Love mode d. We also observe that higher-order modes [ Figs. 2(e) –2(h)] exhibit more complex field distributions. For most applications, the concerned properties of guided phonon modes are 24dispersion, confinement, radiation loss, and energy distributions. Therefore, we numerically evaluate the follow-ing properties with varying waveguide geometrical parameters. (1) Mode frequency ffor a given wavelength λor wavenumber k x¼2π=λalong guided direction. (2) Phase velocity vp¼2πf=kxand group velocity vg¼2πdf=dkx. (3) Mode area Aeff¼1 LW(x,y,z)dx dy dz max ( W(x,y,z)), (1) where Lis the waveguide length along the propagation direc- tion and W(x,y,z) is the elastic strain energy density. The integral and maximum are calculated in the full simulated region. The mode area is a concept borrowed from photonics (see Refs. 79and 80) as a measure of lateral confinement, where a smaller Aeffindicates stronger confinement of phonon. (4) Quality factor Q¼2πf/C2Energycolonstored Powercolonloss: (2) In this paper, we only calculate the radiation loss into sub- strate. A discussion of other loss mechanisms can be found in the conclusion Sec. IV. FIG. 1. (a) Schematic illustration of an unsuspended strip phononic waveguide. w0and h0represent the width and the height of the GaN strip, respectively. (b) The geom- etry setup in COMSOL simulation. The phononic waveguide modes propagating in the uniform structure along x!with wavevector k!are calculated by applying a periodic boundary condition, and PMLs are employed to absorb radiative acoustic waves in the substrate. TABLE II. Elastic properties of GaN and Sapphire used in the simulation. Material GaN66,76Sapphire77 Density (g cm−3) 6.15 3.98 Young ’s modulus (GPa) 305 345 Poisson ratio 0.183 0.29Longitudinal wave speed (m/s) 7350.0 10 658.0 Transverse wave speed (m/s) 4578.3 5 796.4Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-3 Published under license by AIP Publishing.FIG. 2. Simulated displacement field profile for low-order phonon modes supported in an unsuspended strip waveguide. The color mappings show the strength o f displace- ments j~xjnormalized by their own maximum max ( j~xj), which applies to all other figures of the displacement field in this paper. Displacement amplitude is internally excited by the simulation software, so only relative strength j~xj=max (j~xj) has significance. max ( j~xj) is extracted among the entire field of each mode, and every following figure is normalized to their own maximum. However, since the color map setups are the same, we let them share one common color bar for simplicity. (a) Sy mmetric quasi-Rayleigh mode a. (b) Anti-symmetric quasi-Love mode b. (c) Anti-symmetric quasi-Rayleigh mode c. (d) Symmetric quasi-Love mode d. (e)–(h) Higher-order phonon modes e–h. The italic characters label different modes and will be used throughout this paper. Wavelengths λalong the guided direction xare all set at 2 μm. The wave- guide height is fixed at h0¼0:5μm while its width is set at w0¼5μm except that the w0of mode his chosen to be 6 μm because that specific mode does not appear forw0¼5μm.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-4 Published under license by AIP Publishing.(5) Energy confinement ratio η¼stripW(x,y,z)dx dy dz allW(x,y,z)dx dy dz(3) indicates the ratio between elastic strain energy stored in the phononic waveguide and the total elastic strain energy. Figures 3(a) –3(c) show the modal frequency ( f), area ( Aeff), and energy storage ratio ( η) of eight phonon modes vs the width w0of strip waveguide, with a fixed height h0¼0:5μm and wavelength along propagation direction λ¼2μm. To clarify complicated mode hybridization, we now explain the notations and colors of plots that are adopted in this paper. By varying the waveguidegeometry by small increments, the mode properties vary smoothlyand thus we can track the frequencies or other parameters as con-tinuous curves, which is called “branch ”and labeled by different colors. As shown in the frequency plot [ Fig. 3(a) ], each branch is assigned with one specific color consistent with all other propertiesplots. As explained above, due to the strong mode hybridization in the strip waveguides, it is very difficult to identify mode orders orpolarizations by branches. Therefore, we only label the modes inthe regime that can be clearly distinguished according to the modesshown in Fig. 2 . Sections labeled with two characters indicate the modes are experiencing hybridization. The model frequency features vs width w 0, as shown in Fig. 3(a) , indicates a transition between single-mode and multi- mode waveguide. At large w0values, all phonon mode branches asymptotically approach slab phonon modes in GaN-on-sapphire thin films. When w0becomes comparable to λ, the curves branch off and the behaviors of each branch can be explained as follows.Since the lateral confinement imposes an approximate quantizationcondition k y/differencenπ w0, an increasing w0results in a reduction of total phonon wavenumber as k¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi k2 xþk2 yq for a fixed kx, and hence a monotonic reduction of frequency approaching a constant. This intuitive interpretation is valid for most modes, however, fails for the A-Love modes b. According to the mode area and energy FIG. 3. Properties of phononic modes in a strip waveguide as the waveguide width w0is varied. (a) –(c) Frequency f, mode area Aeff, and energy confinement ratio η,v s w0, respectively. (d) Evolution of mode hybridizations when w0is varied with an increment of 1 μm. The color mappings show the strength of normalized displacement j~xj=max (j~xj) in each mode profile. Note the max ( j~xj) is selected among all displacement field images for each mode, or equivalently, each column, which plots one common mode, has the same normalization. Among different columns, they use a common color mapping but their normalizations are distinct. (e) The defo rmation of anti- symmetrical flexural Lamb wave (left) and the simulation result (right) for a waveguide with an aspect ratio w0=h0¼0:2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-5 Published under license by AIP Publishing.confinement ratio given in Figs. 3(b) and3(c), the confinement of A-Love mode bis excellent at small w0/C28λ(rapid reducing of mode area), and thereby almost all wave energy is confined in thestrip. It could be interpreted as the effective boundary becomesair-GaN for the A-Love mode bas opposed to sapphire-GaN for the other modes, and consequently, it can be treated as an anti- symmetrical flexural Lamb wave of a plate normal to ydirection [Fig. 3(e) ]. 81For a plate mode, the vibration frequency f/ffiffiffiffiffiffi D ρw0q , in which bending stiffness D¼w3 0E 12(1/C0ν3)82(ρ,E,νare density, Young ’s modulus and Poisson ratio, respectively.). Thus, the fre- quency of A-Love mode bscales with w0and decreases for narrow waveguides. The interactions between the phonon modes lead to crossings and avoided crossings in Figs. 3(a) –3(c). For the avoided crossings, the corresponding mode area and the energy confinement ratioshow crossing behavior, indicating the coupling between modesand the mode hybridization at these device parameters. For thecrossing in frequency, the coupling between modes is negligible, sothe changes in mode area and energy confinement ratio is notnoticeable. Outside of the hybridization regimes, the confinementof quasi-Rayleigh modes ( aandc) weakens (larger mode area) with the increase of width while that of quasi-Love modes ( band d) remains relatively confined. At very small w 0, however, the wave- guide becomes too narrow to support a fully confined mode a, and its mode frequency rapidly approaches its cutoff frequency, whichis the frequency of SAW in the substrate for λ¼2μm. The mode area of A-love mode b, in contrast to mode a, drops because this mode resembles anti-symmetrical flexural Lamb wave at smallwidth, and its confinement is enhanced as width narrows 81(like a vibrating membrane whose energy is tightly concentrated in thewaveguide region thus the influence from the substrate is negligi-ble). The quasi-Love modes ( b,d) exhibit stronger confinement than quasi-Rayleigh modes ’(a,c) due to the much larger ηof quasi-Love modes (except the hybridization regime), as shown in Fig. 3(c) . For a strip waveguide of fixed cross section h 0¼0:5μm and w0¼5μm, the dispersion characteristics shown in Fig. 4 are numerically calculated as a function of the mode wavenumber kx. InFig. 4(b) , the quasi-Love modes banddhave faster phase veloc- ities than their corresponding quasi-Rayleigh modes and theirbehaviors approach the infinite surface according to analytical pre-dictions in Refs. 73,81, and 83. The cutoffs result in sharp rising of group velocities in Fig. 4(c) at small wavevectors, and meanwhile, inFig. 4(d) , mode areas rise sharply at low frequencies, indicating a remarkable divergence of the confinement when the size of stripbecomes much smaller than the wavelength. This is confirmed inFig. 4(e) that energy confinement ratio decreases, i.e., phonon energy spreads into the substrate, at small k x.I nFig. 4(d) , we also found an interesting minimum point of mode area that occurssimultaneously for branches of mode band daround 2.2 GHz. A close comparison with results in Figs. 4(a) and 4(b) reveals that these minima occur at the avoided crossing point of modes bandd when they are maximally hybridized. Therefore, the mode hybridi-zation presents a unique approach to engineer the phononic modes and might find interesting phononic sensing applications.III. RING RESONATOR In this section, we study the properties of ring resonators formed by bending and closing strip waveguide which supportsphononic WGMs. The pioneering works of the acoustic WGMs date back to the discovery by Lord Rayleigh, 84,85who reported the sound wave travels around a concave boundary in St. Paul ’s Cathedral due to the continuous total internal reflection. Theconcept is generalized to electromagnetic waves, 86–88which holds unique characteristics of high Q-factors, small mode volumes, and the ease of fabrication.89These remarkable merits are also pos- sessed by phononic WGMs according to our investigations. Due to the cylindrical symmetry of the ring resonator, the eigenmode profiles of the elastic differential characteristic equationare in the form as u(r,z, f)¼ψ(r,z)eimf, where r,z,fðÞ are the cylindrical coordinators, ψ(r,z) is the field distribution at the cross section, mis the angular momentum number. Utilizing the symme- try, we can numerically solve the mode profiles of a sector unit in fdirection in COMSOL, with periodic boundary condition: u(r,z,f)¼u(r,z,0 )e/C0imf, where m¼2πR=λfor an azimuthal wavelength λ¼2μm along the tangent direction. Figures 5(b) and 5(c) display the typical quasi-Rayleigh and quasi-Love WGMs in a ring resonator. Compared to those in wave-guide [ Figs. 2(a) and2(b)], the displacement fields of WGMs in the ring are more concentrated at the outer rim. Figure 5(d) shows the mode profiles of the first four branches in Fig. 6(a) . In the rest of this section, except for dispersion behaviors, we study the proper-ties of WGMs around fixed geometry parameters R 0,w0,h0 fg ¼50μm, 5 μm, 0 :5μm fg . Similar to the cases of strip waveguides, the colors of different branches are fixed in Figs. 6 –8. Instead of the mode area, the confinement in the resona- tor is characterized by the mode volume,90 Veff¼W(r,θ,z)rdrdθdz max [ W(r,θ,z)]: (4) As shown in Fig. 6(a) , the dependence of mode frequencies with respect to width w0follows the same trend of those of strip waveguides [ Fig. 3(a) ]. However, compared with the crossing in the waveguide, there is avoided crossing between mode aandb, arising from the broken symmetry (hence mode hybridization) in the bending waveguide. The mode volume and energy confinement ratio ηare presented in Figs. 6(b) and6(d), showing similar cross- ing and avoid-crossing behavior as in the waveguide case. For ring resonators, in contrast to the strip waveguides, there is always radiation loss due to the waveguide bending, and thus the Qfactors of WGMs are finite. As shown in Fig. 6(c) , S-Rayleigh a, A-Rayleigh c, and A-Love bare three modes having the highest Q factors. For modes aandb,Qs are sensitive to the width at small w0value but almost constant when w0exceeds 3 μm, implying a saturation of loss approaching the case of a disk resonator. We observe an abrupt increment of Qwhen bandaare hybridized at around w0¼1:7μm, suggesting that modes band acouple to common leaky modes in the substrate and their destructive inter-ference suppresses the phonon loss. Thus, the special width leads to a parameter-tuning type of the bound state in the continuum. 91,92Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-6 Published under license by AIP Publishing.FIG. 4. Dispersion characteristics for phononic modes on strip waveguide, with geometry parameters w0¼5μm and h0¼0:5μm (aspect ratio w0=h0¼10). (a) Frequency fvs guide wavenumber kx. (b) and (c) Phase velocity and group velocity vs kx, respectively. (d) and (e) Mode area Aeffand energy confinement ratio ηvsf, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-7 Published under license by AIP Publishing.Figure 7(a) shows the properties of phononic WGMs with varying radius. For a fixed λ, the frequencies of the ring converge to the case of a straight waveguide at large R0values with their cal- culated modal volumes shown in Fig. 7(b) . An optimal radius exists for each mode because the lateral confinement (i.e., effective mode area Aeff) becomes better for larger R0, while the mode volume is approximately proportional to R0(Veff/C252πR0Aeff). The Qfactors inFig. 7(c) increase exponentially with increasing R0, indicating radiation loss to the substrate caused by bending decreases expo- nentially with the curvature, similar to the radiation loss of optical WGM in dielectric spheres.86,90The increment of Qwhen bandc are hybridized at around R0¼54μm can be explained with the same reason as that of bandainFig. 6(c) . The dispersion characteristics of ring resonator are summar- ized in Fig. 8 .kf¼m=R0in the figure is effective propagatingwavenumber along tangential direction. Most branches resemble their counterparts in the strip waveguide ( Fig. 4 ), including f/C0kf relation, and the decrease of phase velocity and mode volume, and increase of ratio ηwith increasing kforf. We also compute Q factors that grow exponentially at short wavelengths. IV. DIRECTIONAL COUPLER In an integrated phononic circuit, multiport devices trans- porting and routing phonons between components such as the waveguide-waveguide and waveguide-resonator coupler areindispensable circuit elements . A directional coupler consists of two closely placed parallel waveguides. Here, we study the phononic coupling through the coupled-mode theory by analogy to its optical counterpa rt widely used in the photonic FIG. 5. (a) Schematic illustration of a strip ring resonator. (b) S-Rayleigh (mode a) and (c) A-Love (mode b) in ring resonator. The color mappings show the strength of normalized displacement j~xj=max (j~xj) in each mode profile. (d) The evolution of mode hybridization profiles with varying w0. The normalization convention is the same as we used in Fig. 3(d) : the max ( j~xj) is selected among all displacement field images for each mode, or equivalently, each column which plots one common mode has the same normalization. (b) –(d) share a common color bar.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-8 Published under license by AIP Publishing.community.93–95The coupling between the modes in different waveguides arises from the tunneling of the elastic wave between them. The confined waveguide mode has a non-zeroevanescent field that overlaps with the other waveguide, leadingenergy transfer in between. The spatial mode amplitude evolution along the propagation direction in coupled waveguides for a given input signal frequencyis described by the following equations: d dzα1(z)¼/C0ik1α1(z)/C0ig12α2(z), (5) d dzα2(z)¼/C0ik2α2(z)/C0ig21α1(z): (6) Here, the subscripts “1”and “2”represent waveguide 1 (width w1) and 2 (width w2), respectively, k1(2)is the wavenumber of the modein waveguide 1(2), and g12,g21are the coupling coefficients between two modes. For lossless waveguides, coefficient gmust satisfy g12¼g* 21due to the power conservation.95Setting g12g21¼ jgj2and solving Eqs. (5)and(6), we obtain the wavenumber eigen- values of k+¼k1þk2 2+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1/C0k2 2/C18/C192 þjgj2s (7) for hybrid modes in two waveguides. When k1¼k2¼k, the differ- ence between k+reaches the minimum 2 jgj, and the eigenmodes are the a1+a2, as an equal superposition of two waveguide modes. If the phonon input at the first waveguide with a1(0)¼1 FIG. 6. Phononic mode properties in a ring resonator of varying width w0. The ring radius and height are fixed, respectively, at R0¼50μma n d h0¼0:5μm. (a)–(d) show frequency, mode volume Veff,Qfactor, and energy confinement ratio η, respectively. The insets of (a) are the mode profiles of hybridized modes of (a,b)atw0¼1:7μm.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-9 Published under license by AIP Publishing.anda2(0)¼0, we arrive at α1zðÞ¼ cosjgjzðÞ e/C0ikz, (8) α2zðÞ ¼ sinjgjzðÞ e/C0i(kzþπ 2): (9) Thus, the phonons in the first waveguide α1could be fully trans- ported into the second waveguide α2after a coupling distance of π=2jgj, or half of the phonons can be transmitted into mode α2 after a coupling distance π=4jgj. The larger the jgj, the quicker the energy exchanges. A. Coupling between two identical waveguides For two identical waveguides with w1¼w2, the wavenumbers of the modes are the same k1¼k2 ðÞ ; thus, all supported modescan couple between the two waveguides without phase mismatch- ing. Here, we set the width w0¼5μm and height h0¼0:5μm for both waveguides, as shown in Fig. 9(a) . From the analysis above, we can estimate the coupling strength gbetween two waveguides by the splitting of k+. Figure 9(b) shows the coupling strength of various waveguide modes with different gaps, where the label a-gcorresponds to the modes introduced in Fig. 3 . We set the λ¼2μm. Eigenmodes a,b, andcin the coupled waveguides are shown in Fig. 9(c) , featuring in-phase and out-of-phase hybridized modes in the coupled wave- guides as predicted by the coupled-mode theory. Since the evanes- cent field exponentially decays with the distance to the waveguide,a reduction of the gwith increasing gap is expected. As shown by the numerical simulations, there is a clear exponential relation between the coupling strength and the gap, yet the trends of high order modes eandfslightly deviate from the exponential curve at FIG. 7. Phononic mode properties in a ring resonator as functions of ring radius R0with width w0¼5μm and height h0¼0:5μm. (a)–(d) show frequency, mode volume Veff,Qfactor, and energy confinement ratio η, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-10 Published under license by AIP Publishing.FIG. 8. Dispersion characteristics for modes in a ring resonator of radius R0¼50μm, width w0¼5μm, and height h0¼0:5μm. (a) Modal frequencies fvskf—the effective propagating wavenumber defined as m=R0, with mis angular momentum number. (b) and (c) Phase velocity and group velocity vs kf. (d)–(f) Mode volume Veff, Qfactor, and energy confinement ratio ηvsf, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-11 Published under license by AIP Publishing.small/large gaps. This can be attributed to the weaker confinement of the high order modes, for which the perturbative approximation in coupled-mode theory is no longer accurate. B. Coupling between dissimilar waveguides The coupled-mode theory is also applicable to dissimilar modes in dissimilar waveguides and, therefore, opens the possibilityfor mode conversions with different polarization or mode orders aslong as there is a finite mode overlap between them. From Eq. (6), the maximum ratio of energy transmittance between the two modes is sinc 2[1þk1/C0k2 ðÞ2=4g2]1=2. Thus, for efficient energy transfer, we should adjust the waveguide width to match the wave-number of the two dissimilar modes, i.e., fulfill the phase-matchingcondition. In Fig. 10(a) , the width of one waveguide ( w 1)i s changed while the width of the other is fixed ( w2¼4:5μm) in order to meet with the phase-matching condition, k1(w1)¼k2(w2): (10) The modal frequencies of coupled waveguides are shown in Fig. 10(a) as the w1is varied from 1 μmt o2 :5μm. From the plot, we observe four avoid crossings, corresponding to four regions of modal coupling: I: ( b1,a2), II: ( a1,b1), III: ( b1,c2), and IV: ( a1,c2).Figure 10(c) displays the displacement fields of the four coupling regions at the minimum frequency differences. Similar to the iden- tical waveguides, gvalues of different modes also exhibit an expo- nential dependency on gap [Fig. 10(b) ]. Among four cases, coupling II is particularly interesting because it represents a specialmechanism of “self-coupling ”between modes aandbwithin wave- guide 1, corresponding to a double tunneling process that modes a andbin waveguide 1 both couple to waveguide 2, which mediates the coupling of modes aandb. The double tunneling mechanism could also explain the much faster decaying of the couplingstrength for case II. With such a mechanism, we would expect therealization of a single-waveguide mode converter 96in the future with the assistance of an ancillary waveguide. V. COUPLING TO IDT Aside from the vibrational properties of phononic waveguides and ring resonators, the excitation and detection of phonons are also of practical importance and interests, for example, in phononic implementation of microwave delay lines and filters. Based on themodel of ring resonator presented in Sec. III, we add IDT elec- trodes on the top surface of the ring resonator [ Fig. 11(a) ] and numerically investigate the coupling between IDT and phononic ring resonators and its efficiency to excite phonons. In this model, FIG. 9. (a) Schematic of the directional coupler. (b) The estimated coupling strengths jgjas a function of gap. Set the λ¼2μm along the guided direction. (c) The dis- placement field profiles for the in-phase and out-of-phase modes in the directional coupler, which is made by two identical waveguides with gap¼5μm. The color map- pings show the strength of normalized displacement j~xj=max (j~xj) in each profile. The in-phase and out-of-phase figures share the same normalization, and all figures share the common color bar.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-12 Published under license by AIP Publishing.only the electrical effect of the IDT electrodes to the resonator is considered, while the mechanical and other effects due to loadingof the electrodes are ignored (such as mass loading). We first evaluate the quality factor of the mechanical ring res- onator under the presence of IDT electrodes. The width of eachelectrode is 1 =4 of the acoustic wavelength and the electrodes cover half of the ring resonator area. For the S-Rayleigh mode awith a wavelength of 2 μmi na5 μm-wide ring resonator without IDT, the quality factor increases exponentially with the radius of the resona- tor as the quality factor is limited by the radiation loss; with theIDT electrodes, the quality factor saturates when the radius is largerthan 40 μm, as shown in Fig. 11(b) . The presence of the electrodes modifies the local phonon velocity, due to the δv=veffect. 70Hence, the IDTs break the cylindrical symmetry of the perfect ring,leading to extra coupling of the phonon mode in ring to the leaky phonon modes in the bulk substrate. The same effect also happensto the Love mode. When the radius is small and radiation loss is dominant, the quality factors with and without IDT electrodes are almost the same. At a larger radius in our simulation, the IDT elec-trode induced loss becomes comparable with radiation loss and thequality factor with IDT electrodes are slightly lower. To evaluate the external coupling to the phonon resonator, we also simulate the equivalent impedance of the IDT coupled mechanical ring resonator in order to achieve impedance matchingtoZ 0¼50Ωtransmission lines for maximal acoustic launching efficiency. In the COMSOL simulation, by assigning voltage Von the electrodes and measuring the current flow Ion the electrodes at the mechanical resonance frequency, the equivalent impedance, FIG. 10. (a) Modal frequencies in the directional coupler of dissimilar waveguides, plotted against the waveguide width w1, with the other waveguide width and coupling gap fixed ( w2¼4:5μm,gap¼1μm). Due to the modal coupling, four avoided-crossing regions I ,II,III,IV fg are observed when the frequencies of modes in separate waveguides approach each other. The subscripts of waveguide mode labels 1 ,2fg denote the waveguide 1 and 2, respectively. (b) The modal coupling strength jgjas a function of gap. (c) The mode profiles of the directional coupler in the four avoided crossing regions. The color mappings show the strength of normalized displaceme nt j~xj=max (j~xj) in each profile. The in-phase and out-of-phase figures share the same normalization, and all figures share the common color bar.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-13 Published under license by AIP Publishing.Z¼V=I, can be extracted. The microwave reflectivity can be expressed by r¼Z/C0Z0 ZþZ0(11) and the corresponding microwave to mechanical energy conversion efficiency T¼1/C0r2jj. The simulated results are plotted in Figs. 11(c) and 11(d) for quasi-Rayleigh and quasi-Love mode, respectively. The results indicate that the effective impedance of the IDT coupled mechanical resonator decreases exponentially with theradius and the impedance matching condition Z¼Z 0can be satis- fied with a certain radius, which gives 100% microwave-to-phonon conversion efficiency. To explain the dependence of the impedance on the radius, we consider the Butterworth –Van Dyke circuitmodel of the piezomechanical resonator, which consists of a static capacitance C0in parallel with a motional series RLC circuit. The admittance of the circuit is97 Y(ω)¼/C0iωC0þ1 1=(/C0iωCm)/C0iωLmþRm ¼/C0iωC0þiω2 sCmω ω2þiωsω=Qi/C0ω2 s ¼/C0iωC0þiCmω2 sω(ω2/C0ω2 s) (ω2/C0ω2 s)2þω2 sω2=Q2 i þω3 sω2Cm=Qi (ω2/C0ω2 s)2þω2 sω2=Q2 i,(12) FIG. 11. (a) Schematic illustration of the IDT integrated ring resonator, with the electrodes of IDTs connected to external microwave cable (not shown) for si gnal input. (b) The influence of the IDT on the mode Qfactors. (c) and (d) The equivalent impedance and the energy transmission (conversion efficiency) from microwave to phonon as functions of the ring radius for S-Rayleigh and A-Love modes, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 184503 (2020); doi: 10.1063/5.0019584 128, 184503-14 Published under license by AIP Publishing.where Cm,Lm, and Rmare the motional capacitance, inductance, and resistance. The mechanical resonance ωs¼1=ffiffiffiffiffiffiffiffiffiffiffiffiLmCmpand its intrinsic quality factor Qi¼1=ωsRmCm. Note that ωsis the reso- nant frequency without the presence of the IDT electrodes, andsuppose that with IDT on it, the resonant frequency is ω f. This res- onant frequency shift originates from the δv=veffect and the value (ωf/C0ωs)=ωsequals1 2δv=v(as the electrodes cover only half of the ring resonator). In the regime where the quality factor is limited byintrinsic radiation loss instead of the δv=vscattering, we have (ω f/C0ωs)=ωs¼δv=v/C281=Qi. In this case, the admittance can be simplified to Y(ωf)¼/C0iωfC0þωfCmQi, (13) with the real part being dominant, we get the relation between the equivalent impedance and its quality factor Z(ωf)¼1 ωfCmQi: (14) We see that the equivalent impedance is inversely proportional to its quality factor. The motional capacitance Cmis proportional to the electrode area like C0,98so it is linearly proportional to the radius as the surface area increases while the quality factor increases exponentially with radius. So, in Figs. 11(c) and 11(d) , the impedance ’s dependence with radius is mostly exponential, consistent with the quality factor simulated in Fig. 11(b) . The microwave-to-phonon conversion efficiency can also be understood from a coupling condition point of view where the external cou- pling between the electrodes and the resonator is weak, but bychanging the resonator ’s intrinsic energy decay rate, the critical coupling condition can be satisfied and all the microwave energycan be converted into mechanical energy. Therefore, the IDT integrated mechanical resonator can enhance microwave-phonon conversion efficiency, boost the per-formances of applications that require high excitation and collec-tion efficiency, and provide the coherent interface betweenphononic and superconducting circuits. VI. CONCLUSION In conclusion, we have quantitatively studied phononic mode properties in unsuspended strip waveguides and ring resonators.Our numerical results demonstrate the efficient confinement ofphonon in the practical GaN-on-sapphire microstructures, whichprovide a scalable platform of phononic integrated circuits for acoustic signal processing and enhanced phonon –matter or phonon –light interactions. Basic components for a phononic circuit are thoroughly investigated, including the directionalcoupler made by two waveguides, quasi-Rayleigh mode toquasi-Love mode conversion in coupled non-identical waveguides, ring resonator, as well as microwave photon-to-phonon conversion for efficient input and output coupling. It is worth noting that thestudies in this paper can be generalized to other frequencies byrescaling the geometry parameters. For example, 10 GHz phononic waveguide modes could be realized with a thickness of 125 nm and a width of 500 nm, whose geometry is compatible with thephotonic integrated circuits and promises applications in optical Brillouin scattering. Although not addressed in this paper, it is worth noting that there are various other loss mechanisms. 99Some losses do not exist or can be neglected in our strip waveguide SAW devices, forexample, our system does not use any suspended structures so it is not subject to clamping or support losses. For the circuit damping caused by the drive-and-detection circuit, we evaluate the equiva-lent impedance and Qfactor in Sec. V. Scattering losses from crystal defects or thermal phonons are ignored, though thermoelas-tic damping can be one primary limit of the Qfactors at room tem- perature. Recently, studies were done on phononic band structure engineering to reduce the scattering of SAW into bulk modes. 41,46 However, scattering losses from surface roughness due to fabrica- tion disorder can be critical for miniaturizing phononic systemsand thus for high-frequency application, 100since the scattering loss coefficient has a quadratic dependency with respect to the guided wave frequency,101–103it could be the main limitation to efficient photo-phonon interaction.100,104In addition, scaling to smaller geometry increases surface-to-volume ratios, which may give rise tomore surface loss, and this loss mechanism has been found in other micromechanical systems. 105Besides, we do not consider material losses due to any intrinsic microscopic processes. Thedamping from the surrounding medium contributes to additionalloss if the devices are operating under an atmosphere or a viscous medium. 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5.0017514.pdf

AIP Conference Proceedings 2265 , 030223 (2020); https://doi.org/10.1063/5.0017514 2265 , 030223 © 2020 Author(s).Spectroscopic investigations of AgNO3 doped bismuth silicate glasses Cite as: AIP Conference Proceedings 2265 , 030223 (2020); https://doi.org/10.1063/5.0017514 Published Online: 05 November 2020 M. Laya Krishnan , and V. V. Ravi Kanth Kumar ARTICLES YOU MAY BE INTERESTED IN Photoluminescence properties of Sm3+ -doped LiPbB 5O9 phosphor for reddish-orange emitting light applications AIP Conference Proceedings 2265 , 030219 (2020); https://doi.org/10.1063/5.0017735 Green emission features of erbium doped lithium zinc borate glasses AIP Conference Proceedings 2265 , 030218 (2020); https://doi.org/10.1063/5.0016896 New candidate for red phosphor applications AIP Conference Proceedings 2265 , 030221 (2020); https://doi.org/10.1063/5.0016601Spectroscopic Investigations of AgNO 3 Doped Bismuth Silicate Glasses M Laya Krishnan*and V V Ravi Kanth Kumar Department of physics, Pondicherry University V Nagar, Kalapet, Pondicherry -605014, India *Corresponding auther:mlayakrishnan@gmail.com Abstract: AgNO 3 doped bismuth silicate glasses were prepared by traditional me lt quenching techniques. UV-Vis absorption spectra used to study the optical properties of prep ared samples. Absorption spectrum shows 479nm peak due to lower valance of bismuth ion that is Bi2+ or Bi0 Surface Plasmon Resonance (SPR). Optical band gap from Tauc pl ot gives the idea about localized states formation near to valance band and conduction band by the addition of AgNO 3. The 570nm absorption peak in 5LBSAg0.5 is due to silver nano partic le. Most of the silver formed as silver oxide that can be seen in Raman spectra. The Bi 2O3 c a n e x i s t b o t h B iO 3 and BiO 6 unit. The photoluminescence spectra give two main peaks at 435nm and 630nm due to Bi3+ and Bi2+ respectively. The CIE color coordinates (x, y), color purity ( CP) and correlated color temperature (CCT) are calculated for all the s amples. INTRODUCTION Materials contain metal nano particles (NP) exhibit high linear and nonlinear properties and can be used in the several applications [1, 2]. The linear optical phenomenon like localized surface plasmon resonance can be used for sensing and bio sensing applications. The metal NP in glass mat rix shows high third order nonlinearly and this property can be used in photonics applications [3,]. These line ar and nonlinear properties of nanoparticle completely depend on type of the metal, size and shape [4]. There are seve r a l m e t a l l i k e A u , A g , C u , P t e t c . a v a i l a b l e f o r producing metal NP. Among this Ag is the most commonly used met al, because it have high quality factor about 10, and it is cheaper than Au [5]. There are so many techniques to introduce metal NP in to the glass host, such as, ion implantation, melt quenching, thermal evaporation, ion exchange etc. A traditional melt quenching technique is used in the present studies to incorporate the metal NP in to the heavy metal glass . The bismuth can exist in low valance state (Bi+ and Bi0) while melt at high temperature. This bismuth nano particle al so gives the SPR peak [6]. So the present work disusing about the structural and optical changes in the glass while adding different concentration of AgNO 3. EXPERIMENTAL PROCEDURE Glass preparations The conventional melt quenching method is used to prepare a ser ies of four samples of AgNO 3 doped bismuth silicate glass. The chemical formula of this series is (65 Bi 2O3-5LiF-(30-x) SiO 2-xAgNO 3) x=0, 0.1, 0.2, 0.5 and 1. The proper amount of chemical of A R grade are grounded in an a gate mortar each batches 10g of samples are taken in to silica crucible and melted at 11000C for 3h. The melt were stirred to get a homogeneous liquid for every 30min. Then the melt were poured into a preheated steel mould to get d esired shape. The samples are then annealed at 3600C for 2h and polished with SiC and diamond paste. The prepared samples were labeled as 5LBS (AgNO 3=0), 5LBSAg0.1 (AgNO 3=0.1), 5LBSAg0.2 (AgNO 3=0.2), 5LBSAg0.5 (AgNO 3=0.5) and 5LBSAg1 (AgNO 3=1). DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030223-1–030223-4; https://doi.org/10.1063/5.0017514 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030223-1Glass characterizations Uv-Vis absorption spectra were taken in the visible region 350- 700nm using shimadzu UV-VIS-NIR spectrophotometer. Using optical absorption measurements Optic al band gap (E g) and refractive index (n) are calculated. Photoluminescence spectrum is measured by fluorolog 3 spectrofluorometer (HORIBA Jobin Yvon) in the visible region. From measured PL spectra the CIE color coor dinates (x, y), color purity (CP) and correlated color temperature (CCT) were calculated. Raman spectroscopy carried o ut for structural information usig Renishaw Raman microscope (20-2000cm-1). All the experiments are done in the room temperature. RESULTS AND DISCUSSIONS UV-Vis Absorption Spectroscopy The prepared glass samples are transparent and light brown to d ark brown and became opaque (5LBSAg1) with increasing AgNO 3 concentrations. So that for further study 5LBS, 5LBSAg0.1, 5LB SAg0.2 and 5LBSAg0.5 was taken. Fig.1 shows the absorption spectrum of prepared glass s amples. The data for 5LBS taken from, our previously reported journal [7]. All samples possess a broad ab sorption in the visible region centered at 472 nm to 479nm. This is mainly due to the formation of lower valance of Bi ion in mainly Bi2+ or Bi0 [6]. If it is Bi2+ ion the transition may occur from 2P1/2 to 2P3/2 in other words there is any bismuth NPs (Bi0 may the transition from 4S3/2 to2P1/2) present in the glass or Ag NPs it will give surface plasmon r esonance (SPR) peak in this region. From Fig.1 inset graph shows a small kink at 573 nm for 5LBSAg.05 glass du e to the Ag nanoparticle [8]. From the figure we can see that the absorption edge shifted to lower wavelength in dicating the concentration of Bi3+ ion in the prepared glass system is less [7]. Using the absorption coefficien t the optical band gap (Eg) o f the glasses was calculated using Tauc plot [7]. The samples have indirect allowed transitions. The band gap increas es with increasing AgNO 3. Refractive index of the material measured from obtained Eg values [7].All the values ar e tabulated in Table.1. Addition of AgNO 3 w i l l makes the changes in glass structure that can be seen from the optical band gap values. AgNO 3 increases the localized states near to valance and conduction band. This lead s to more disorder in the glass matrix. The structural studies give the explanation about disordered glass matrix. FIGURE 1. Absorption spectra of prepared glass samples TABLE 1. Optical Band gap (Eg) and Refractive Index (n) of is (65 Bi 2O3-5LiF- (30-x) SiO 2-xAgNO 3) x=0, 0.1, 0.2, 0.5 Sample Name Optical band gap, E g (eV) Refractive index, n 5LBS 2.91 2.66 5LBSAg0.1 2.92 2.42 5LBSAg0.2 2.92 2.42 5LBSAg0.5 2.94 2.41 030223-2Raman Spectroscopy The detailed structural studies are done by Raman spectroscopy. It will give the exact idea about non-bridging oxygen (NBOs), bridging oxygen and other building blocks. Fig.2 (a) shows the Raman spectra of prepared glass samples. All samples have broad and same characteristics peak, for detailed study the spectrum under goes deconvolution with Gaussian function and shown in Fig.2 (b) and (c). The peak at 60cm-1 is boson peak normally seen in the glass due to the vibration of short range order. P eak at 135cm-1 is Bi3+ ion vibration in BiO 6 octahedra or BiO 3 pyramidal units. The peak ranging from 150cm -1 to 700cm-1 asc ribed to the bridging vibration of Bi-O-Bi and Si-O-Si linkage[7]. The extra peak at 224cm-1 is due to the AgO in the glass matrix [5]. There is a chance t o form AgO and Ag NPs in the glass and AgO vibration can be seen in the 248, 386 cm-1also. In this region Bi-O-Bi and / or Bi-O- i n B i O 6 octahedra and Si-O-Si are present. The peak above 800cm-1 are non-bridging oxygen vibration is present such as SiO 4 with one, two and three NBO [9]. From the Raman spectrum can s ee that AgNO 3 alter the newtwork and formation of NBOs, AgO and Ag NPs. FIGURE 2. (a) Raman spectra of all glass samples (b) Deconvoluted Raman spectra of 5LBSAg and (c) 5LBSAg0.5 Photoluminesence Spectrocopy Formation of NBOs definetly make some changes the Photoluminesc nece spectra (PL). the PL spectra of all sample are illustratedin Fig.3(a) which are ecxited at 350nm. The PL spctra exhibit three peaks centeded at 435nm, 580nm and 630 nm. All the PL peaks are resposible for Bi ion. Lower v alance of Bi ion can emit in the visible region the first peak is due to the Bi3+ and other twos are Bi2+. The area under the cureve of Bi3+ in PL spectra is calculated and it follow the trend which seen in the absorption spectra. That is shifting of absoprtion edge in the lower wavelength. the ground state energy level of Bi3+ is 1S0 and the excitatin levels are 3P0,3P1,3P2 and 1P1 increasing order of energy. 1S0 t o 3P0 a n d 3P2 are forbiddent transition where as 1S0 t o 3P1 a n d 1P1 are allowed transition via unsymmetrical lattice vibration and spin orbit coupling between 1P1 and 3P1 respectevely. Consider the luminescence due to Bi2+ ion, it will give two peak in the visible region one at 580nm a nd other one at 630nm. The ground state is 2P1/2 and the excitation levels are 2P3/2(I), 2P3/2(II) and 2S1/2 increasing order of energy. The absorption happens between 2P1/2 to 2P3/2(I), 2P3/2(II) and 2S1/2 but the emission from2P3/2(II) to 2P1/2 and 2P3/2(I) to 2P1/2 [7]. FIGURE 3. (a) Photoluminescence spetra of is (65 Bi 2O3-5LiF-(30-x) SiO 2-xAgNO 3) x=0, 0.1, 0.2, 0.5 (b) CIE color coordinates The CIE color cordinates[10] are calculate from obtained PL spe ctra. The spectral power distributions P(λ) multiplied with color matching function such as ݔ̅, ݕത and ݖ̅ gives the tristimulus value X,Yand Z, then CIE color 030223-3cordinates are calculated from this and shown in Fig.3 (b).Ther e is a shift in CIE cordinates from 5LBS to 5LBSAg0.5 along with the correlated color temperature (CCT) and color purity (CP) of emitted light are calculated. All vallues are tabulated in Table2 . The addition of AgNO 3 to the glass can tune the white light to blue color. CONCLUSIONS A set of AgNO 3 doped bismuth silicate glasses with chemical formula (65 Bi 2O3-5LiF-(30-x) SiO 2-xAgNO 3) x=0, 0.1, 0.2, 0.5 and 1 were prepared by melt quenching techniques. Except 5LBSAg1 all samples are transparent brown in color. The absorption spectra shows main peak centered at 47 2 to 479 nm ascribed to lower valance of bismuth ion. Tauc plot gives the band gap of prepared samples, shows in creasing the band gap due to the addition of AgNO 3 in the glass. Higher concentration of AgNO 3 can makes more disorder in the glass matrix whereas refractive indexes of the sample decrease with increasing AgNO 3. From the Raman spectra can be seen there is a extra peak arou nd 234 cm-1 responsible for the AgO bond, also the building blocks present in the samples are BiO 3,BiO 6 ,SiO 4, non- bridging oxygen and bridging oxygens. These could see after dec onvoluting the spectra with Gaussian function. From the PL spectra we can see that there is a main three peak in the visible region due to lower valance of bismuth. One is at 435nm due to Bi3+ and other two peaks at 580nm and 630 nm due to Bi2+. There is a slight change in the intensity of Bi3+ peak with the addition of AgNO 3 but Bi2+ peak have visible changes in their intensity. This variation in PL spectra definitely changes CIE color coordinates. So AgNO 3 can be used for color tuning. And this material can be used in lighting techniques filed. ACKNOWLEDGMENT MLK acknowledges the UGC, New Delhi for providing Rajiv Gandhi National Fellowship (RGNF).We thank Central Instrumentation Facility, Pondicherry University for pr oviding characterization facilities and also DST- SERB no.0320 for financial support. REFERENCES 1. E. C. Romani,D. Vitoreti, P.M . P. Gouvêa, P. G. Caldas, R. P rioli, S. Paciornik, M.Fokine,A. M. B. Braga, A. S. L. Gomes and I.C. S. Carvalho , Opt. Express 20(2012), 5430-543 9 2. R. D. Averitt, S. L. Westcott , and N. J. Halas, J. Opt. Soc. Am. B 16(10), 1824–1832 (1999) 3. F. Z. Henari and A. A. Dakhel, J. Appl. Phys. 104(3), 033110 (2008). 4. K. L. Kelly, E. Coronado, L. L . Zhao, and G. C. Schatz, J. P hys. Chem. B 107(3), 668–677 (2003) 5. A.K Pal, D.B Mohan, Opt,.Mater. 48 (2015) 121–132 6.S.P. Singh and B.Karmakar, SPS Book, Kolkata. 7. M. L.Krishnan, M.M. Neethish , V.V. R.K Kumar, J.Lumin, 201(2 018)442-450 8. S.P. Singh and B. Karmakar, Plasmonics (2011) 6:457–467 9.R.S. Kundu, M. Dult, R. Punia , R. Parmar, N. Kishore, J. Mol. Struct. 1063 (2014) 77–82 10. E. Fred Schubert, Light Emitt ing Diodes, second ed., Cambri dge University press,Cambridge, UK, 2006, p. 292 (Chapter 17) TABLE 2. CIE color coordinates (x, y), Correlated Color Temperature (CC T) and Color Purity of (65 Bi 2O3-5LiF- (30-x) SiO 2-xAgNO 3) x=0, 0.1, 0.2, 0.5 Sample name CIE Color Coordinates (x, y) Correlated Color Tempe rature (CCT), K Color Purity (CP) 5LBSAg (0.29, 0.24) 13064 27 5LBSAg0.1 (0.26, 0.18) - 53 5LBSAg0.2 (0.26, 0.19) - 46 5LBSAg0.5 (0.27, 0.21) 54411 38 030223-4

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J. Chem. Phys. 153, 224107 (2020); https://doi.org/10.1063/5.0029223 153, 224107Analytic expressions for the steady-state current with finite extended reservoirs Cite as: J. Chem. Phys. 153, 224107 (2020); https://doi.org/10.1063/5.0029223 Submitted: 24 September 2020 . Accepted: 15 November 2020 . Published Online: 08 December 2020 Michael Zwolak ARTICLES YOU MAY BE INTERESTED IN On the quantum origin of few response properties The Journal of Chemical Physics 153, 221101 (2020); https://doi.org/10.1063/5.0027545 Reflections on electron transfer theory The Journal of Chemical Physics 153, 210401 (2020); https://doi.org/10.1063/5.0035434 Construction of explicitly correlated one-electron reduced density matrices The Journal of Chemical Physics 153, 224109 (2020); https://doi.org/10.1063/5.0031195The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Analytic expressions for the steady-state current with finite extended reservoirs Cite as: J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 Submitted: 24 September 2020 •Accepted: 15 November 2020 • Published Online: 8 December 2020 Michael Zwolaka) AFFILIATIONS Biophysical and Biomedical Measurement Group, Microsystems and Nanotechnology Division, Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA a)Author to whom correspondence should be addressed: mpz@nist.gov ABSTRACT Open-system simulations of quantum transport provide a platform for the study of true steady states, Floquet states, and the role of temper- ature, time dynamics, and fluctuations, among other physical processes. They are rapidly gaining traction, especially techniques that revolve around “extended reservoirs,” a collection of a finite number of degrees of freedom with relaxation that maintains a bias or temperature gradi- ent, and have appeared under various guises (e.g., the extended or mesoscopic reservoir, auxiliary master equation, and driven Liouville–von Neumann approaches). Yet, there are still a number of open questions regarding the behavior and convergence of these techniques. Here, we derive general analytical solutions, and associated asymptotic analyses, for the steady-state current driven by finite reservoirs with pro- portional coupling to the system/junction. In doing so, we present a simplified and unified derivation of the non-interacting and many-body steady-state currents through arbitrary junctions, including outside of proportional coupling. We conjecture that the analytic solution for proportional coupling is the most general of its form for isomodal relaxation (i.e., relaxing proportional coupling will remove the ability to find compact, general analytical expressions for finite reservoirs). These results should be of broad utility in diagnosing the behavior and implementation of extended reservoir and related approaches, including the convergence to the Landauer limit (for non-interacting systems) and the Meir–Wingreen formula (for many-body systems). https://doi.org/10.1063/5.0029223 .,s I. INTRODUCTION Nanoscale and molecular electronics encompasses a broad range of fundamental studies1–4and applications, such as sensing in aqueous solution.5–12Sensing in inhomogeneous, dynamical envi- ronments, in particular, requires both statistical averaging13–18and the incorporation of dephasing19and other quantum effects due to rapid atomic fluctuations, and potentially quite large energetic changes, e.g., due to ionic motion. The incorporation of fluctua- tions and quantum effects into simulation is challenging as it often requires the inclusion of additional degrees of freedom, such as explicit electronic reservoirs or phonon/vibrational bath modes. Along these lines, many transport processes can be treated within the so-called “microcanonical approach,” where a closed, finite system (including “reservoirs”) is directly simulated.20–27 There have been various implementations of finite, closed reser- voirs within Matrix Product States (MPSs).24,26,28–38These are com- putationally limited due to the rapid generation of entanglementduring particle flow and scattering, but they can incorporate corre- lated impurity centers. Recent many-body implementations include a combined numerical-renormalization group and MPS approach,39 as well as a technique that transforms the canonical basis—a basis that respects the scattering nature of transport and “breaks” the entanglement barrier—and exponentially improves the simulation efficiency.40 Steady states can rapidly form, with a rise time inversely pro- portional to the electronic bandwidth in certain cases.27The micro- canonical approach can thus still be quite powerful. However, sta- tistical averaging over noise and fluctuations is computationally demanding since closed systems inevitably have recurrences, and thus, many separate simulations are required. Scanning over a parameter, such as coupling or temperature (e.g., included within a “microcanonical” picture via purification), can suffer from the same issue, requiring many independent simulations of a real-time evolution into a quasi-steady state. Quantum noise, dephasing, and periodic driving (Floquet states) present further issues, demanding J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-1The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp very large reservoirs or baths to act as effective sinks.41Opening up the reservoirs, via relaxation, can extend the scope of numerical simulations of nanoscale devices and sensors, especially the treat- ment of electronics in noisy environments or in the presence of quantum processes. Ultimately, the efficiency for any given task has to be directly assessed and compared between methods. For instance, within tensor networks, stochastic wave function tech- niques42or averaging over pure states43may be more efficient at solving the open-system equations or handling finite temperature. More broadly, different computational machinery and approaches introduce a range of considerations that factor into the compu- tational cost and accuracy, which will not be our objective here to assess. Rather, we will examine theoretical aspects of a class of open-system simulations. There have been a number of similar proposals for open- system simulations with explicit degrees of freedom to represent the reservoirs, such as “extended reservoirs,”44–46the auxiliary quan- tum master equation approach (AMEA) that permits non-isomodal relaxation in the extended reservoirs,47–52and the driven Liouville– von Neumann (DLvN) that treats non-interacting systems,53–58as well as other approaches59–61(some of which do not obey proper quantum evolution, see the overview in Ref. 45). This particular class of open-system approaches to transport is in addition to a variety of other open-system techniques for transport and dynam- ics,62–86which employ different strategies and, in some cases, apply to particular parameter ranges (e.g., weak coupling). Explicit or extended reservoir approaches (ERAs) are all part of a general idea: a finite reservoir can effectively be transformed into a continuum by broadening the modes via relaxation,87–90with recent rigorous results for bosonic systems as thermal quantum environments.91–93A related open-system approach was developed for classical thermal transport through DNA94,95and topological sys- tems.96,97The theoretical treatment and behavior of this approach follows Kramers turnover for friction-controlled rates,98but applied at the simulation level rather than for condensed-phase chemical reaction rates.99,100The quantum case is analogous and also follows a Kramers turnover,44–46with only the intermediate relaxation regime giving currents commensurate with physical expectations (i.e., the relaxation-free Landauer result for non-interacting systems and the Meir–Wingreen formula for interacting systems). Very recently, these open-system ideas have been translated to novel tensor network approaches. In Ref. 101, an open-system, time- dependent variational principle approach was taken to solve for the dynamics and steady states within the “mixed basis” (the basis that breaks the entanglement barrier in closed system transport by prop- erly accounting for the scattering nature of current-carrying parti- cles40). A similar approach was developed in Ref. 102, which uses just the energy basis and the time-evolving block decimation algorithm with swaps and applies the technique to quantum thermal machines. In Ref. 103, building on earlier work,49the authors implemented the AMEA approach within MPS with swaps of sites and a partic- ular organization of fermionic modes and applied the technique to quantum transport in the Anderson model with a power-law spectral function. Similarly, Ref. 104 employs numerical-renormalization group like ideas (e.g., logarithmic discretization outside the bias window) and a reorganization of modes within an MPS approach; see also Ref. 105 for related work applied to non-interacting systems.ERAs are thus already mature enough to be applied to quantum transport. However, it is clear that there are many issues that remain, some of which may hinder the application of these approaches or inadvertently lead to spurious results. Here, we will further develop the analytic and mathematical basis for using extended reservoirs for transport. Specifically, we will derive analytical expressions, within the “proportional coupling” scenario, for transport through non-interacting and many-body quantum impurities for arbitrary relaxation strength. We conjecture that these are the most gen- eral analytic results of this form when relaxation is isomodal. We also examine the asymptotic limits. All of these expressions should be helpful in assessing and validating numerical implementations. Furthermore, we give a unified and simplified derivation of the non-interacting and many-body solutions to the extended reservoir quantum master equation. This includes an alternative approach of treating the Markovian relaxation directly within the Keldysh for- malism (where previously Green’s functions were computed directly from the Markovian equation of motion). We will discuss how these approaches with Markovian relaxation limit, in both non-interacting and many-body cases, to the relaxation-free Landauer and Meir– Wingreen formulas, respectively (note that non-Markovian relax- ation already obeys exactly a Landauer or Meir–Wingreen formula at finite relaxation). The outline of this article is as follows. In Sec. II, we provide a brief summary of the main results, specifically the expressions for the steady-state current for many-body and non-interacting systems, the analytical expressions for proportional coupling, and the asymptotic expressions in the same scenario. We will also note which equations appear twice in this paper so the correspondence is clear. In Sec. III, we provide the connection between the Lindblad master equation with relaxation for many-body systems and the driven Liouville–von Neumann equation (DLvN) for non-interacting systems. In Sec. IV, we provide a unified, general solution for Markovian and non- Markovian relaxation, and many-body and non-interacting systems, for the steady-state current. In Sec. V, we examine the solution in proportional coupling and derive a fully analytic result for finite reservoirs. In Sec. VI, we present the asymptotic analyses. These results and calculations give a comprehensive and, we hope, acces- sible treatment of the steady-state current within extended reservoir approaches to quantum transport. II. SUMMARY We first summarize the main results. Some of the equations, thus, appear twice in this manuscript, for which we give the corre- spondence here. Details of all quantities are in Secs. III–VI. We examine transport driven through a junction/impurity/ system Sby a chemical potential or a temperature drop between the left ( L) and right ( R) reservoirs, both of which are assumed to be non-interacting with a quadratic coupling to the system. Follow- ing the ERA solution developed in Ref. 44 (including averaging the current from the left and right reservoirs to create a symmetrized version of the current), the steady-state current, I, can always be written as I=ıe 2∫dω 2πtr[{ΓL−ΓR}G<+{˜ΓL−˜ΓR}{Gr−Ga}]. (1) J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-2The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp This equation has a similar structure to the well-known Meir– Wingreen formula,106,107being in terms of S’s Green’s functions [G<,Gr(a)] and reservoir spectral densities [ ΓL(R),˜ΓL(R)]. Crucially, however, it is distinct: ˜ΓL(R)have occupation factors that can- not be disentangled from the actual spectral density when Marko- vian relaxation is present, which has important consequences dis- cussed throughout this work. We emphasize that this equation is valid for both non-interacting and many-body impurities, as well as Markovian or non-Markovian relaxation, and whether there is proportional coupling or not. Equation (1) here is the same as Eq. (36) in Sec. IV. The quantities appearing in Eq. (1) are the electron charge e, the weighted and unweighted spectral densities (Γαand ˜Γα, respectively) for reservoir α=L,R, the system’s full lesser Green’s function G<, and the system’s full advanced and retarded Green’s functions ( GrandGa, respectively). These are all NS×NSmatrices representing correlations between the NSsystem modes. We briefly note here that the many-body current, Eq. (1) lim- its to the normal Meir–Wingreen formula as the reservoir size goes to infinity and then the relaxation strength goes to zero. That is, whether many-body or non-interacting, the exact result for the current in the standard, relaxation-free, Meir–Wingreen setup will result. This also entails that the normal Landauer formula will result in this limit for non-interacting electrons.44This will be discussed in Sec. IV B (see also Ref. 108). For non-interacting systems, Eq. (1) becomes I=e∫dω 2πtr[˜ΓLGaΓRGr−ΓLGr˜ΓRGa]. (2) The Markovian nature of this equation is reflected in the appear- ance of two similar terms rather than a single term multiplied by a difference in Fermi–Dirac distributions. Equation (2) is the same as Eq. (39) in Sec. IV A. For “proportional coupling,” the two reservoirs have identical mode distributions and couplings (up to a proportionality factor) to the system modes (i.e., ΓR=λΓL). Since the reservoirs are finite, this entails that the reservoir modes have the same set of energies and relaxation strengths. Under this condition, one obtains I=ıeλ 1 +λ∫dω 2πtr[Δ˜Γ{Gr−Ga}]. (3) Equation (3) here is the same as Eq. (45) in Sec. V. The difference in the weighted spectral density is ˜ΓL−˜ΓR=ı∑ k∈L(˜fL k−˜fR k)[gr k(ω)−ga k(ω)]∣vk⟩⟨vk∣, (4) where ˜fL(R) kare the Fermi–Dirac occupations evaluated at frequency ωk(ω) for Markovian (non-Markovian) relaxation and bias μL(R), ⟨i∣vk⟩=vikis the coupling between the mode i∈Sand the mode k∈L(ifk∈R,vik→√ λvik), and gr(a) k=1/(ω−ωk±ıγk/2)are the retarded (advanced) Green’s functions for k∈L(orR) with relaxationγk>0 but without contact to the system. The sum is over only the left reservoir since we take proportional coupling and the modes and relaxation are the same in the right reservoir.While we might not be able to solve for Gr(a)due to many-body interactions or involved self-energies, we can still do the integration for Markovian relaxation, yielding I=−2eλ 1 +λ∑ k∈L(˜fL k−˜fR k)⟨vk∣ImGr(ωk+ıγk/2)∣vk⟩, (5) which is analytic ( Gris analytic in the upper half-plane where it is evaluated here). Thus, the important assumption is to consider proportional coupling with Markovian relaxation. Non-Markovian relaxation, for instance, has the integrand dependent on the Fermi– Dirac distribution and, thus, is not readily integrated. Equation (5) here is the same as Eq. (48) in Sec. V. We conjecture that Eq. (5) is the most general analytic result of this form for two reservoir transport with isomodal relaxation, i.e., that relaxing the assumption of proportional coupling will, at best, result in only specific cases of finite reservoirs to be analytically solv- able, since this derivation makes clear what is required. In order to perform the ωintegration, we need that the integrand has functions that are only analytic in the upper or the lower half-plane. If this is not satisfied, for instance, due to the appearance of G<[see Eq. (1)], the integral then relies on specific knowledge about the form of the system’s Green’s functions (opposed to just regions of analyticity). We will be surprised, but delighted, if this conjecture is not true. We can also analyze the asymptotic forms for Markovian relax- ation, starting either from Eq. (3) or Eq. (5). For a weak Markovian relaxation, we find I≈2eλ (1 +λ)2∑ k∈Lγk(˜fL k−˜fR k). (6) This equation is only derived for non-interacting systems, but allows for inhomogeneous γk(where the inhomogeneity is across k, but not between LandR). For a strong relaxation, the current is I≈4eλ 1 +λ∑ k∈L∣vk∣2 γk(˜fL k−˜fR k). (7) This expression holds for both non-interacting and many-body sys- tems and homogeneous or inhomogeneous relaxation. Equations (6) and (7) are the same as Eqs. (54) and (56), respectively, in Sec. VI. The set of these equations [Eqs. (5)–(7)] give compact analytic expressions to assess the behavior, convergence, performance, and numerical implementation of the extended and auxiliary reservoir approaches for many-body or non-interacting systems, as well as the DLvN for non-interacting systems. III. BACKGROUND We first define the equations we solve, since there are alterna- tive forms in the literature. We start with a Lindblad master equation for the full (potentially many-body) density matrix ˙ρ=−ı ̵h[H,ρ]+∑ k∈LRγk+(c† kρck−1 2{ckc† k,ρ}) +∑ k∈LRγk−(ckρc† k−1 2{c† kck,ρ}). (8) The first term in the master equation gives the Hamiltonian evo- lution ofρunder H. The next two terms, which both contain the J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-3The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp anticommutator {A,B}, give the particle injection (depletion) into the reservoir states kat a rateγk+(γk−). The total Hamiltonian is H=∑ k∈LR̵hωkc† kck+∑ k∈LR∑ i∈S̵h(vkic† kci+vikc† ick)+HS, (9) whereωkis the frequency of reservoir mode k∈LR,vik=v⋆ kiis the coupling of that mode to the system Sat site i, and̵his the reduced Planck’s constant. The Hamiltonian HSis forS, which can have many-body interactions that the DLvN does not, and cannot, include (beyond mean field). The operators c† m(cm) are the creation (annihilation) operators for the state m∈LSR . The index mcar- ries a numerical index, as well as all labels (electronic state, spin, reservoir, or system mode). Equation (9), when the continuum limit is taken, is the stan- dard starting point for transport through impurities, where non- interacting left ( L) and right ( R) reservoirs drive a current through a system Svia an applied potential or temperature drop. The reser- voirs are connected to the system only via quadratic (hopping) terms and are not connected to each other. As usual, one wants the relaxation terms to give an equilibrium state of the reservoirs in the absence of S. This requires that γk+ ≡γkfα(ωk) andγk−≡γk[1−fα(ωk)], where fα(ωk) is the Fermi–Dirac distribution in the α∈{L,R}reservoir at different chemical poten- tials or temperatures. One important point is that this equilibrium is actually a pseudo-equilibrium for the reservoirs that does not prop- erly incorporate the broadening of the reservoirs’ states due to the relaxation, which can result in, e.g., zero-bias currents. This clearly unphysical situation has been investigated in Refs. 44 and 45. When there is symmetry between the left and right reservoir modes, the zero-bias anomaly goes to zero.44However, there is still a related anomaly due to smearing of full states above the Fermi level and vice versa for empty.111 Throughout this work, when we refer to Markovian relaxation, we are referring to Eq. (8). When we refer to non-Markovian relax- ation, we are referring to a similar relaxation that gives identical retarded and advanced Green’s functions, but a proper Fermi level. We do not write its equation of motion (as its name implies, it has memory and is not a time-local equation when the external environments are integrated out). The AMEA approach has the same equation as shown in Eq. (8). However, it allows for transitions between kand k′in the relaxation. We will limit ourselves here to energetically local (iso- modal) relaxation ( k=k′),109but many results still hold for this more general situation. Moreover, in Ref. 45, it was shown that there is an exact correspondence between the DLvN equation and the Lind- blad master equation when electrons are taken to be non-interacting. Equation (8), however, is more general since ρis the full, many- body density matrix on the exponentially large Hilbert space of the system Sand the left (right) reservoirs L(R), and thus, this can include many-body interactions [for many-body implementations of Eq. (8), see Refs. 101–104]. We emphasize that the state, ρ, is not the single-particle density matrix appearing in the DLvN, for which we reserve the N×Nsingle-particle correlation matrix C(this is related to a single-particle density matrix by normalizing the trace) where Nis the total number of electronic levels in LSR (treating spin, if present, as a separate level).To make the correspondence of Eq. (8) with the DLvN, one has to take all regions to be non-interacting rather than just the reservoirs (and the reservoir-system coupling). When HSis also a non-interacting Hamiltonian, it can be written as HS=∑ i,j∈S¯Hijc† icj, (10) where ¯Hijis the single-particle Hamiltonian. Note that we restrict i,jto the system sites. When summing over all sites (the system and reservoirs), this gives the global single-particle Hamiltonian H=∑m,n¯Hmnc† mcnwith m,n∈LSR , which we will also use with the same symbol ¯Hsince the system single-particle Hamiltonian is just a submatrix of this larger matrix. These matrices are operators, but on the single-particle space rather than the full Hilbert space. With this introduction, it is now a simple matter to connect Eq. (8) to the DLvN.45Assuming all regions to be non-interacting, one writes the equation of motion for the single-particle correlation matrix as Cnm=tr[c† mcnρ] (11) with m,n∈LSR , which gives ˙C=−ı[¯H,C]/̵h+R[C], (12) where R[C]is the relaxation. In block form, as the DLvN uses, this equation has terms C=⎛ ⎜⎜⎜ ⎝CL,LCL,SCL,R CS,LCS,SCS,R CR,LCR,SCR,R⎞ ⎟⎟⎟ ⎠(13) and R[C]=−γ⎛ ⎜⎜⎜⎜ ⎝(CL,L−CL 0)1 2CL,SCL,R 1 2CS,L 01 2CS,R CR,L1 2CR,S(CR,R−CR 0)⎞ ⎟⎟⎟⎟ ⎠. (14) Cα,α′are for a subset of states, i.e., in the regions α,α′∈{L,S,R}, where we have also taken a uniform γas otherwise one would have to include an inhomogeneous γinto the blocks. This is a trivial matter but would hinder the direct comparison with the normal DLvN. However, for completeness, when inhomogeneous relaxation is present, the elements are (R[C])mn=γm+δmn−Cmn 2(γm++γm−+γn++γn−), (15) where m,n∈LSR andγm±=0 ifm∈S. The “relaxed” distribu- tions are Cα 0=diag[fα(ωk)]. The connection between the Lindblad master equation [Eq. (8)] and Eq. (12) also shows that the DLvN is always a proper quantum evolution, that is, correlation matrix evolution is from a completely positive trace-preserving map on the full density matrix. This proof works for finite or infinite reservoirs, unlike the result in Ref. 56, which holds only in the limit of infi- nite reservoirs. Moreover, since the Lindblad operators are in second quantized form, the evolution will always respect Pauli exclusion (observed empirically in Ref. 53). Further details are contained in Ref. 45. J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-4The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp This connection entails that any solution of Eq. (8) when elec- trons are non-interacting is also a solution of the DLvN [Eq. (12)]. In addition, the solution will also be to the AMEA with only isomodal relaxation. We will start by recalling the computation for the current from Ref. 44. Our presentation will differ from Ref. 44 only in two significant respects: we will give a simultane- ous, unified approach for both the non-Markovian and Markovian extended reservoirs, including for many-body and non-interacting impurities. We will also assume the wide-band limit from the begin- ning. Since external environments, whether Markovian or non- Markovian, relax the reservoir modes, a true steady state exists even for finite reservoirs. We will need the lesser Green’s function G< nm(t,t′)=ı⟨c† m(t′)cn(t)⟩, (16) where the time-dependence signifies Heisenberg picture operators on the global space ( LSR plus external, implicit environments). This Green’s function is connected to the correlation matrix ele- ments through Cnm=−ıG< nm(t,t). (17) We will also need the retarded and advanced Green’s functions Gr nm(t,t′)=−ıΘ(t−t′)⟨{c† m(t′),cn(t)}⟩ (18) and Ga nm(t,t′)=ıΘ(t′−t)⟨{c† m(t′),cn(t)}⟩, (19) respectively. Again, the {A,B}gives the anticommutator. Equa- tions (18) and (19) are related by [Ga nm(t,t′)]⋆=Gr mn(t′,t). Within a stationary state, these Green’s functions depend only on the time difference and, for the retarded and advanced Green’s functions, we haveGr(ω)=[Ga(ω)]†for their Fourier transforms. IV. CURRENT By taking the derivative of the total particle number in the left or right reservoir, the time-dependent current is given by the sum over system–reservoir correlations. For the current from the left into the system, this reads IL=ıe∑ k∈L,j∈S(vkj⟨c† kcj⟩−vjk⟨c† jck⟩) (20) evaluated at time t. The current is related to the lesser Green’s functions through I(t)=e∑ k∈L∑ j∈S[vkjG< jk(t,t)−vjkG< kj(t,t)]. (21) Within the steady state, we take the Fourier transform I=e∑ k∈L∑ j∈S∫dω 2π[vkjG< jk(ω)−vjkG< kj(ω)]. (22) Given the non-interacting nature of the reservoirs, the Dyson equa- tions for these Green’s functions areG< jk(ω)=∑ i∈Svik[Gr ji(ω)g< k(ω)+G< ji(ω)ga k(ω)] (23) and G< kj(ω)=∑ i∈Svki[gr k(ω)G< ij(ω)+g< k(ω)Ga ij(ω)]. (24) These stationary state expressions are found by the equation-of- motion method, which we provide in the Appendix for the readers’ convenience. The current [Eq. (22)] is the inverse Fourier transform at equal time t−t′= 0 and can then be rewritten as I=e∑ k∈L∑ i,j∈S∫dω 2πvjkvki{g< k(ω)[Gr ij(ω)−Ga ij(ω)] −[gr k(ω)−ga k(ω)]G< ij(ω)}. (25) This expression applies to both the Markovian and non-Markovian reservoirs. Already, one can see here the appearance of self-energies Σji=∑kvjkvkigk, with gk=g< k,gr(a) k. The lesser components will depend on non-Markovian or Markovian nature of relaxation. To simplify the expression further while capturing both the Markovian and non-Markovian extended reservoirs, we define a new function ˜fk=⎧⎪⎪⎨⎪⎪⎩fα(ω)non–Markovian fα(ωk)Markovian ,(26) with fα(ω) the normal Fermi–Dirac function with bias μαand α=L(R)is the reservoir component of index k. This sets the occu- pation either according to the physical, non-Markovian equilibrium or the unphysical, Markovian equilibrium. In both situations, the Fermi–Dirac distribution fappears, but either with the frequency appearing in Green’s function, ω, or with the reservoir’s isolated mode frequency, ωk. As we discuss extensively elsewhere,44,45the former correctly occupies the broadened mode, whereas the lat- ter occupies and then broadens. As will be clear below, for a weak external relaxation, these are approximately equal to each other, but diverge for a strong relaxation resulting in unphysical, sometimes quite large and potentially zero-bias, currents. Consolidating the terms by using the Keldysh equation g< k(ω)=gr k(ω)Σ< k(ω)ga k(ω), (27) where, for this non-interacting “isolated” mode, non–Markovian Σ< k=ıγkfα(ω)(28) for the wide-band, non-Markovian extended reservoirs44(herein called just “non-Markovian”) and Markovian Σ< k=ıγk+⋅1 +ıγk−⋅0=ıγkfα(ωk)(29) for the Markovian extended reservoirs. Note that the non- Markovian version connects each reservoir mode to a single external environment with a well-defined chemical potential. Each of these J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-5The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp external environments for Lhave the same well-defined chemical potentialμLand the same for R,μR(this follows the calculations in Ref. 44 and see this same reference for the expressions outside of the wide-band limit). It is non-Markovian due to a surface, the Fermi level, even though it has no other structure. The Fermi level is sufficient to introduce memory (for non-Markovian relaxation with an additional structure in these environments; see the expressions in Ref. 44). The Markovian case represents the reservoir mode in contact with separate full and empty wide-band reservoirs, but with differ- ent couplings, which results in a finite mode occupancy. This case has to be wide band, as otherwise it would not be Markovian. Note that, in Ref. 44, we derived Green’s functions directly from the Lind- blad equation of motion [Eq. (8)]. Here, however, we simply take a Keldysh approach and use the fact that the Markovian master equa- tion [Eq. (8)] is exactly the equation of motion that one gets when each reservoir mode is connected to two reservoirs, one full and one empty giving the factors 1 [i.e., f(ω) = 1] and 0 [i.e., f(ω) = 0]. The lack of a well-defined chemical potential (i.e., they are at ±∞) points to a serious issue that has repercussions in calculations that must be avoided by a careful examination of behavior vs parameters. Equation (27) with Eq. (28) or Eq. (29) yields the identity (for either the non-Markovian or the Markovian reservoirs) g< k(ω)=−˜fk[gr k(ω)−ga k(ω)]. (30) Defining the weighted and unweighted spectral densities [i.e., weighted and unweighted versions of the normal ΓL(R)=ı(Σr L(R) −Σa L(R))=−2IΣr L(R)] as ˜ΓL(R)(ω)=ı∑ k∈L(R)˜fk[gr k(ω)−ga k(ω)]∣vk⟩⟨vk∣ (31) and ΓL(R)(ω)=ı∑ k∈L(R)[gr k(ω)−ga k(ω)]∣vk⟩⟨vk∣, (32) respectively, the current [Eq. (25)] becomes I=ıe∫dω 2πtr[ΓLG<+˜ΓL{Gr−Ga}]. (33) This is structurally similar but distinct from the Meir–Wingreen result [cf., Eq. (18) in Ref. 107]. For Markovian relaxation, one can- not remove the occupation factor from ˜Γ. Thus, while analogous, it is different than their formula, providing the exact solution to transport through a many-body impurity according to Eq. (8). Moreover, we will apply it to finite reservoirs where each mode has a finite lifetime. In the wide-band limit (see Ref. 44 for the more general case), Green’s functions of the reservoir modes are as fol- lows. For both non-Markovian and Markovian extended reservoirs, the retarded (advanced) Green’s functions are gr(a) k(ω)=1 ω−ωk±ıγk/2. (34) The lesser Green’s functions can be found by direct computation or using the Keldysh equation of the reservoir mode in contact with theexternal environment. We use the latter, g< k=gr kΣ< kga k=gr kΣ< k[gr k]∗. This gives g< k(ω)=ıγk˜fk (ω−ωk)2+γ2 k/4. (35) The issue of occupancy becomes clear with this equation. The non- Markovian equilibrium occupies every part of the single-mode den- sity of states according to the Fermi–Dirac distribution, ˜fk=fα(ω). The Markovian equilibrium, though, occupies the whole spectrum according to its completely isolated frequency ωk,˜fk=fα(ωk), vio- lating the fluctuation-dissipation theorem. As long as the relaxation strength,γk, is weak, there is little difference. This directly leads to the condition, γk≪kBT/̵h, that the relaxation has to be much less than the thermal relaxation for the Markovian approximation to be valid.44The condition, in some situations, is not so strict, as many (but not all) models will give the correct Landauer result (for non- interacting systems) even when this is not satisfied. This comes down to the fact that there is often a temperature scale in the problem, T⋆, below which the current becomes independent of temperature (see Sec. IV B). When this is the case, a “too large” relaxation strength can still give the correct result. This is a heuristic, a rule-of-thumb for practical calculation, that should be employed with care. We can give symmetric expressions by averaging the left and right currents I=ıe 2∫dω 2πtr[{ΓL−ΓR}G<+{˜ΓL−˜ΓR}{Gr−Ga}]. (36) This equation is valid for both non-interacting and many-body system Hamiltonians HS, as well as for both non-Markovian and Markovian relaxation. Indeed, it is our starting point for all subse- quent calculations and also appears as Eq. (1) in Sec. II. Again, the difference with the standard approach of Meir– Wingreen,106,107besides being for finite reservoirs, is the lack of ability to pull out the occupancies from the ˜Γ’s. For the non- Markovian reservoirs, the current vanishes when in equilibrium, fL(ω)=fR(ω)≡feq(ω)andG<=−feq(Gr−Ga). For the Markovian case, the current vanishes in equilibrium for symmetric reservoirs in a proportional coupling scenario. For non-proportional coupling, one can often have zero-bias currents, pointing to one of the partic- ular pathologies of the Markovian setup.110These currents, though, are related to̵hγk/kBTand vanish as this goes to zero. We caution that even in the proportional coupling scenario, the Markovian case still gives anomalous currents. We discuss these elsewhere,111but they have the same mathematical origin. The Markovian equation occupies and then broadens, which results in unphysical occupied (and unoccupied) states at high and low energy. Essentially, one will have Lorentzian broadened “Fermi level” rather than a Fermi–Dirac exponential cutoff. From here, there are two special cases that are useful. One is to maintain generality and allow for many-body interactions in S, but assume proportional coupling in the reservoirs, and the other is to assume a globally non-interacting lattice (in the single par- ticle sense, which is quite strict but widely applied for molecular and nanoscale electronic junctions, including in density functional theory approaches). We will first treat non-interacting systems and J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-6The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp then outline the general (non-interacting and many-body) corre- spondence to Landauer and Meir–Wingreen formulas. In Sec. V, we will examine the proportional coupling. A. Non-interacting systems For non-interacting electrons, one can also simplify the current in the standard way.106,107,112Starting from Eq. (36), one uses the Keldysh equation G<=GrΣ<Ga→ıGr(˜ΓL+˜ΓR)Ga(37) and the identities Gr−Ga=−ıGr(ΓL+ΓR)Ga=−ıGa(ΓL+ΓR)Gr, (38) which both employ non-interacting electrons (after the arrow for the first equation, as one can include the many-body lesser self- energy).113Again, the Markovian extended reservoirs require that we retain the occupancy within the spectral density. Employing the rightmost expressions in Eqs. (37) and (38), Eq. (36) becomes I=e∫dω 2πtr[˜ΓLGaΓRGr−ΓLGr˜ΓRGa]. (39) Equation (39) is the exact solution to the Markovian equation of motion [Eq. (8)], when the impurity region is non-interacting, and thus, it is the exact solution to the driven Liouville–von Neumann approach [Eq. (12)] for arbitrary reservoirs and parameters. When the relaxation is non-Markovian, the solution reduces to the more traditional expression I=e∫dω 2π(fL(ω)−fR(ω))tr[ΓLGrΓRGa], (40) since in this case, there is a well-defined Fermi level.114Both of these expressions just require computing the retarded (advanced) Green’s functions Gr(a)=1/(ω−¯HS−Σr(a)), self-energies Σr(a) =∑k∈LRgr(a) k∣vk⟩⟨vk∣, and the weighted spectral density. We note here, of course, that in the non-interacting limit, one can compute transport properties in the normal way. The primary benefit of ERAs is that they provide a numerical framework for many-body systems, as well as time dynamics. Equation (39), though, can be used as a standard tool to assess behavior (such as the presence of anomalous currents111), evaluate performance (e.g., in using different reservoir discretizations115,116), and validate the numerical implementation of ERAs.101,111,116 B. Landauer and Meir–Wingreen correspondence Whether one works with a many-body impurity, Eq. (36) gen- erally or Eq. (45) for proportional coupling, or the non-interacting result, Eq. (39) or Eq. (45) for proportional coupling, the Marko- vian expressions converge to non-Markovian expressions, as this has to do with the convergence of the weighted spectral density— alternatively, the lesser Green’s function—of the finite reservoir. This was examined in detail in Ref. 44. There, it was shown that the one-norm difference of the Markovian and non-Markovian lesser Green’s function is bounded by̵hγk 4kBTlnkBT ̵hγk. (41) That is, it is controlled by the ratio of the relaxation strength and the thermal energy. This also helps put bounds on the necessary reservoir size.44,45Clearly, as the temperature goes to zero, conver- gence requires a very small relaxation strength, as the derivative of the occupation at the Fermi level is diverging. However, while this can influence the accuracy of the calculation if there are sharp fea- tures at the Fermi level, e.g., due to interference effects,58,117–119in many other cases, one may not see its effect. For instance, as seen in Ref. 101, below a certain temperature, T⋆, which potentially could depend on the voltage for sharp features, the current is not chang- ing.120Thus, as long as γk≪kBT⋆/̵h, the current will still be accurate even though the lesser Green’s function has not formally converged. As emphasized above, this has to be applied with care. When the weighted spectral density and lesser Green’s func- tions have approximately converged to their non-Markovian coun- terparts, the current for Markovian relaxation approximately obeys a Landauer or Meir–Wingreen formula (for non-interacting or many- body interactions, respectively). It seems unlikely that an exact Lan- dauer expression with relaxation will be obeyed unless the relaxation is both non-Markovian and given by a single-particle mechanism. The finite extended reservoir, whether Markovian or non- Markovian, will then converge to the relaxation-free infinite reser- voir as N→∞and thenγk→0. As done in Ref. 44, the limit to the Landauer formula can now be taken directly with Eq. (39). First one takes the infinite reservoir limit, then when γkis sufficiently narrow in the sense already discussed, one can replace ˜fkwith f(ω) (for the Markovian case, since the non-Markovian already has this) in Eq. (31). This shows that the current in the steady-state of the Markovian master equation [Eq. (8)] limits to the Landauer result. It requires both that the reservoir goes to infinity (first) and then the relaxation strength to zero.44,108 For many-body systems, one has to work instead with Eq. (36). The same replacement applies, noting that it occurs both in the weighted spectral density and within G<. One can think of this pro- cess in an alternative manner: The lesser Green’s function of the truly isolated reservoir is a delta function in frequency space and one is instead using the Lorentzian representation of the delta func- tion (the retarded and advanced Green’s functions are no differ- ent than normal, except one has the broadening tied to γk/2 rather than “η→0,” withηthe control parameter appearing in Green’s functions whose sole purpose is to ensure causality—or, for what is important here, analyticity in the upper or the lower half-plane). In mathematical terms, it is expressing the following replacement: lim γk→0ı[gr k(ω)−ga k(ω)]˜fα=2πδ(ω−ωk)˜fα =2πδ(ω−ωk)fα(ω), (42) which is exact as the relaxation strength goes to zero. As already noted, the reservoir size has to be infinite already before taking this limit.121 V. PROPORTIONAL COUPLING When the reservoirs are identical in density of states and dis- tribution of couplings to system (allowing for the total coupling J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-7The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp scale to be different), one obtains the proportional coupling case,122 ΓR=λΓL, which allows one to eliminate the need to compute G<. To exactly hold, e.g., in widely employed tight-binding models, this would require all atoms bound to one of the electrodes to also be bound, up to the same proportionality constant, to the other elec- trode. This condition is quite limiting. It can be, however, approxi- mately valid in some cases. There are, for example, situations where one or a few junction states dominate transport, such as a sym- metrically bound HOMO (highest occupied molecular orbital) level. Projecting on the subspace of relevant modes would, in these partic- ular cases, give identical (or proportional) coupling. We investigate proportional coupling here as a test case for numerical implementa- tions, as well as to understand the behavior and convergence to the relaxation-free, continuum (i.e., the exact) results. Following Refs. 106 and 107, we take the steady-state current as a weighted average of left and right currents, I=xIL+(1−x)IRwith x∈[0, 1]. Employing x=λ/(1 +λ) eliminates G<from the expres- sion for the current. However, unlike Refs. 106 and 107, we have to separately address ˜Γ. Proportional coupling requires that the discrete set of modes in LandRare the same, as well as their relaxation (it can be inhomogeneous in kbut identical between modes ωkinLand R). The couplings to the system need to satisfy vk′i=√ λvkiδkk′with k′∈R,k∈L, andδkk′signifying that the numerical part of the index is identical. This entails x˜ΓL−(1−x)˜ΓR=ıλ 1 +λ∑ k∈L(˜fL k−˜fR k)∣vk⟩⟨vk∣ ×[gr k(ω)−ga k(ω)], (43) for which we define x˜ΓL−(1−x)˜ΓR≡λ 1 +λΔ˜Γ (44) atx=λ/(1 +λ). The current can then be written as I=ıeλ 1 +λ∫dω 2πtr[Δ˜Γ{Gr−Ga}]. (45) Equation (45) is the same as Eq. (3) and is the one employed to derive the general result for finite reservoirs for Markovian relaxation (see below). This equation applies to many-body and non-interacting systems with proportional coupling, as well as to non-Markovian and Markovian relaxation. Note that since Δ˜Γgoes to zero as the bias goes to zero for proportionally coupled reservoirs with Marko- vian relaxation, the current is zero when no bias is present. For non-Markovian relaxation, the current is always zero when no bias is present, for both proportional and non-proportional coupling. Anomalies, however, can still be present for both cases, as we will discuss elsewhere.111 For non-Markovian relaxation, Eq. (45) is equivalent to the Meir–Wingreen expression [cf., Eq (21) in Ref. 107], as one can pull out the Fermi–Dirac distributions I=ıeλ 1 +λ∫dω 2π(fL(ω)−fR(ω))tr[Γ{Gr−Ga}]. (46) In this non-Markovian case, this is a “pseudo-Landauer” form (note Γ=ΓL=ΓR/λ) and it has a nice interpretation. It is the spec- tral overlap of the reservoir and broadened system density of stateswithin the bias window that determines the current. For Markovian extended reservoirs, though, one cannot pull out the Fermi–Dirac distributions and this interpretation breaks down. There is still a notion of spectral overlap, but it is the spectral overlap weighted by the difference in the Fermi–Dirac distributions of the isolated reser- voir states. This is not equivalent to the former and gives anomalous behavior that has to be accounted for in practical calculations.111 Unlike the Markovian case within Eq. (45), Eq. (46) cannot be analyzed directly since the Fermi–Dirac distributions depend on ω. When the relaxation is Markovian, one has a sum over k∈L with each term having a product [gr k(ω)−ga k(ω)]⟨vk∣{Gr−Ga}∣vk⟩, (47) where we used that the coupling matrix for each kis an outer prod- uct in order to take the trace. This contains all the ω-dependent factors. We can thus do integrations in either the upper or the lower half-plane depending on the functions in the integrand. For instance, gr kGrcan be integrated in the upper half-plane giving zero (the retarded functions are analytic there) and gr kGacan be integrated in the lower half-plane since we do not know necessarily where the poles of Gaare (but they are all in the upper half-plane). The ω integral in Eq. (45), thus, gives I=−2eλ 1 +λ∑ k∈L(˜fL k−˜fR k)⟨vk∣ImGr(ωk+ıγk/2)∣vk⟩. (48) This derivation demonstrates that this is a very general equation. It assumes proportional coupling and Markovian relaxation but other- wise applies to many-body or non-interacting systems and homoge- neous or inhomogeneous relaxation. For non-Markovian relaxation, one has to work with Eq. (46) instead since the Fermi–Dirac dis- tribution depends on the integration variable. Equation (48) is the same as Eq. (5) in Sec. II. The derivation, though, also demonstrates why a successful analytic calculation results. To obtain the current, one need only to deal with various retarded and advanced Green’s functions. These have analytic properties that, e.g., the lesser Green’s function does not. Hence, this is why we conjecture that Eq. (48) is the most general result of its form for isomodal relaxation. VI. ASYMPTOTIC ANALYSES Taking Eq. (48) for proportional coupling and further assuming that the system is non-interacting, one can directly take the small and large relaxation limits. When the system is non-interacting, the retarded Green’s function is Gr=1 ω−¯HS−Σr(49) with Σr=∑ k∈LR∣vk⟩⟨vk∣ ω−ωk+ıγk/2(50) =(1 +λ)∑ k∈L∣vk⟩⟨vk∣ ω−ωk+ıγk/2, (51) J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-8The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where the second line is for proportional coupling and the factor of (1 +λ) is to account for both reservoirs when the sum goes over onlyL. A. Weak relaxation When the γkare all much smaller than the level spacing, γk≪ωk−ωk′∀k,k′, the self-energy evaluated at ωk+ıγk/2 becomes Σr(ωk+ıγk/2)≈−ı(1 +λ)∣vk⟩⟨vk∣ γk. (52) That is, for very small γk, the kth term is picked out. Moreover, Eq. (52) diverges and we will have that ∥Σr(ωk+ıγk/2)∥≫∥ωk+ıγk/2−¯HS∥, (53) assuming all elements of ¯HSare bounded and we have taken the operator norm. The self-energy from mode kis thus dominant in Green’s function. Either we can then retain all elements of Green’s function and diagonalize it perturbatively or we can note that due to the⟨vk∣○∣vk⟩in the numerator of Eq. (48), we only need to invert Green’s function on a subspace of rank 1. That is, the coupling matrix for one mode k,∣vk⟩⟨vk∣, is of rank 1 (the operator acts on the whole single-particle space of the system and, thus, is an NS×NS matrix with NSlevels in the system) and can be inverted on that subspace. When put into Eq. (48), this gives ⟨vk∣(∣vk⟩⟨vk∣)−1∣vk⟩=1 (where 1 is the identity matrix). For smallγk, Eq. (48) thus yields I≈2eλ (1 +λ)2∑ k∈Lγk(˜fL k−˜fR k). (54) While we provided an identical analytic expression in Ref. 44 for λ= 1, this now gives an alternative demonstration that it applies to all non-interacting junctions in the proportional coupling scenario. It further generalizes the result to inhomogeneous γ. Many-body sys- tems, though, do not seem to be easily treated with this approach for smallγ. As the number of reservoir modes increases towards the continuum limit, we note that the region of validity of Eq. (54) is pushed towards smaller values of γk(e.g., generically as the inverse of the number of modes). Equation (54) is Eq. (6) in Sec. II. B. Strong relaxation Whenγkare all much larger than any frequency scale in the problem,γk≫∥¯H∥∀k∈L(ork∈R), the self-energy is small (it is bounded by a function of 1/ γk). In other words, the system itself (i.e., just the junction) is effectively almost closed, as excita- tions take a long time to decay into the extended reservoirs (and out to the external environments). The denominator of Green’s func- tion is dominated by the extra factor of γkpassed to it through its argument. Thus, Gr(ωk+ıγk/2)≈−2ı γk. (55) Putting this into Eq. (48) gives I≈4eλ 1 +λ∑ k∈L∣vk∣2 γk(˜fL k−˜fR k). (56) This is exactly the expression already derived in Ref. 44, albeit for homogeneous γk, which, however, was not essential in the deriva- tion, andλ= 1. Equation (56) is Eq. (7) in Sec. II.Equation (48), though, holds for many-body systems as well. We can thus go further. The key expression123transforms back to real-time ∫∞ −∞dωγ k (ω−ωk)2+γ2 k/4Gr ij(ω) =2π∫∞ −∞dt e−ıωkt−γk∣t∣/2Gr ij(−t)≈−ı4π γkδij, (57) where we can take the imaginary part after integration since the Lorentzian is real. The exponential cutoff due to the large γklim- its the contribution to the integral to only a short time, 1/ γk. Using Eq. (18), the short time approximation is simply the identity times −ı, but only for positive arguments. Also note that Green’s function is for the system, so that the self-energy from the reservoirs is small and smeared at large γ(that is, large γdoes not introduce a fast pro- cess into Greven though Eq. (18) has the full time evolution from LSR and the external environments). Employing Eq. (57) in Eq. (45) [instead of using Eq. (48)] gives exactly Eq. (56). We stress that this applies to many-body or non-interacting systems within proportional coupling. We note also that the expression [Eq. (56)] is well-behaved in the continuum limit, since the coupling constants squared are related to canoni- cal transforms and they decay with the inverse reservoir size. Also, we expect more general cases, beyond proportional coupling, of the largeγregime can be analyzed, but the form will change since the single sum over k∈Lreflects the proportional coupling. Interest- ingly, a straightforward result, valid for arbitrary many-body or non- interacting systems, with or without proportional coupling, and with homogeneous or inhomogeneous relaxation, results directly from an analysis of Eq. (36).124 It is not a surprise that the large relaxation regime has only a minor dependence on the system, only through the coupling to the reservoir modes but not on the levels and interactions within the system itself. The relaxation overdamps the coherence that is form- ing between the reservoir modes and the system, giving an effective coupling that can be thought of perturbatively and hence the form ∣vk∣2/γk. Strong local dephasing has a similar effect and yields a dif- fusion equation.125Coherence is needed for the current to flow, and thus, the relaxation is limiting the flow from the reservoir states into the system. That being said, some alternative asymptotic regimes can appear for non-Markovian relaxation (see, e.g., Ref. 46). This has to do with the band structure of the reservoirs (e.g., band gaps), the level energies of the system, and symmetries. VII. CONCLUSION We are now in the era of ERAs: Extended reservoir approaches to transport are becoming widespread and employed in various applications, including many-body transport.101–104We show that for local relaxation (i.e., acting independently of each of the modes in the reservoir), a simple analytic expression results for both many- body and non-interacting systems with finite reservoirs proportion- ally coupled to the system. This will allow quite large finite reser- voirs to be examined, especially for convergence to the Landauer (for non-interacting systems) or Meir–Wingreen (for many-body J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-9The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp systems) limits,116as well as understanding the underlying mech- anistic behavior of finite representations of the reservoirs.111The asymptotic forms we derive also will be quite useful in benchmarking and validating numerical implementations, as well the full solutions [Eqs. (36) and (39)]. We expect, therefore, that the results here will help support the use of extended, mesoscopic, or auxiliary reservoir simulations broadly, including its non-interacting incarnation as the DLvN, and bring it to bear on new applications. ACKNOWLEDGMENTS We thank Marek Rams, Gabriela Wójtowicz, Justin Elenewski, Yonatan Dubi, and Vladimir Aksyuk for helpful comments. APPENDIX: EQUATION-OF-MOTION METHOD Due to the non-interacting nature of the reservoirs, the Dyson equations [Eqs. (23) and (24)] are easily obtained by the equation- of-motion method starting from the time-ordered Green’s function GT nm(t−t′)=−ı⟨T{cn(t)c† m(t′)}⟩ =−ıΘ(t−t′)⟨cn(t)c† m(t′)⟩ +ıΘ(t′−t)⟨c† m(t′)cn(t)⟩, (A1) with T the time-ordering operator and n,m∈LSR . The equation of motion is −ı∂t′GT nm(t−t′)=δ(t−t′)δnm−ıΘ(t−t′)⟨cn(t)[H,c† m] t′⟩ +ıΘ(t′−t)⟨[H,c† m] t′cn(t)⟩, (A2) where we used ⟨cn(t)c† m(t′)+c† m(t′)cn(t)⟩t→t′=δnm. In Eq. (A2), the operators should be evolved according to the full Hamiltonian (including also the implicit reservoirs). However, we can use that any single-particle extension of the reservoirs (e.g., their connection to the implicit reservoirs) requires just replacing the reservoir Green’s functions with their dressed versions,113which is the approach we will here. For the current-carrying correlations, Eq. (A2) gives −ı∂t′GT jk(t−t′)=−ıΘ(t−t′)⟨cj(t)[H,c† k] t′⟩ +ıΘ(t′−t)⟨[H,c† k] t′cj(t)⟩ =ωkGT jk+∑ i∈SvikGT ji, (A3) with j∈Sand k∈LR. 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For equal LandRmode frequencies, the setup has already been sufficiently restricted and is of measure zero in parameter space. Proportional cou- pling, of course, is also measure zero in parameter space. We conjecture that the lack of zero-bias anomalous behavior only happens within scenarios of measure zero. Moreover, as we will show elsewhere, there is still anomalous behavior, but it happens in the presence of a bias.111 111G. Wójtowicz, J. E. Elenewski, M. M. Rams, and M. Zwolak (unpublished). 112C. Caroli, R. Combescot, P. Nozieres, and D. Saint-James, J. Phys. C: Solid State Phys. 4, 916 (1971). 113H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics of Semi- conductors , Springer Series in Solid-State Sciences, 2nd ed. (Springer-Verlag, Berlin Heidelberg, 2008). 114Note that to get Eq. (40), we also used that tr [˜ΓLGaΓRGr]is equal to tr[˜ΓLGrΓRGa]. When the relaxation is non-Markovian, this follows by taking ΓR=ΓR+ΓL−ΓL, applying Eq. (38), and canceling ΓLterms by pulling out J. Chem. Phys. 153, 224107 (2020); doi: 10.1063/5.0029223 153, 224107-11The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp fL(which is allowed since this does not depend on k) and using the cyclic prop- erty of the trace. When the relaxation is Markovian, the Fermi–Dirac distribution cannot be pulled out of ˜ΓL. 115M. Zwolak, J. Chem. Phys. 129, 101101 (2008). 116J. E. Elenewski, G. Wójtowicz, M. M. Rams, and M. Zwolak (unpublished). 117G. C. Solomon, D. Q. Andrews, T. Hansen, R. H. Goldsmith, M. R. Wasielewski, R. P. Van Duyne, and M. A. Ratner, J. Chem. Phys. 129, 054701 (2008). 118L.-Y. Hsu and H. Rabitz, Phys. Rev. Lett. 109, 186801 (2012). 119C. J. Lambert, Chem. Soc. Rev. 44, 875 (2015). 120This temperature is dependent on voltage and on features in the density of states. A smooth density of states (in both the reservoirs and the system) over the bias window increases T⋆, as can many-body relaxation. 121If the reservoir size is finite, γk→0 will result in zero current. Nevertheless, it will still limit to a Landauer or Meir–Wingreen formula, as these formulas for non-Markovian relaxation are exact for all relaxation strengths and reservoir sizes.Hence, all that is required is that the Markovian relaxation expressions limit to the non-Markovian ones. 122Note that we take ΓR=λΓLrather than ΓL=λΓR, as is done in Refs. 106 and 107, to maintain consistency with writing expressions using the left reservoir modes in Ref. 44. 123This is Eq. (B17) in Ref. 44 with the minor typo (a missing t in the exponent) corrected. 124This can be found by considering Eq. (57) and a similar equation for the lesser Green’s function. Replacing the latter in Eq. (57) yields ı8πniδij/γk. Using both expressions in Eq. 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5.0026351.pdf

J. Chem. Phys. 153, 164706 (2020); https://doi.org/10.1063/5.0026351 153, 164706 © 2020 Author(s).Stochastic scattering theory for excitation- induced dephasing: Time-dependent nonlinear coherent exciton lineshapes Cite as: J. Chem. Phys. 153, 164706 (2020); https://doi.org/10.1063/5.0026351 Submitted: 23 August 2020 . Accepted: 27 September 2020 . Published Online: 23 October 2020 Ajay Ram Srimath Kandada , Hao Li , Félix Thouin , Eric R. Bittner , and Carlos Silva COLLECTIONS Paper published as part of the special topic on Excitons: Energetics and Spatio-temporal Dynamics ARTICLES YOU MAY BE INTERESTED IN Stochastic scattering theory for excitation-induced dephasing: Comparison to the Anderson–Kubo lineshape The Journal of Chemical Physics 153, 154115 (2020); https://doi.org/10.1063/5.0026467 Exciton–exciton annihilation in a molecular trimer: Wave packet dynamics and 2D spectroscopy The Journal of Chemical Physics 153, 164310 (2020); https://doi.org/10.1063/5.0027837 Efficient multireference perturbation theory without high-order reduced density matrices The Journal of Chemical Physics 153, 164120 (2020); https://doi.org/10.1063/5.0023353The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Stochastic scattering theory for excitation-induced dephasing: Time-dependent nonlinear coherent exciton lineshapes Cite as: J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 Submitted: 23 August 2020 •Accepted: 27 September 2020 • Published Online: 23 October 2020 Ajay Ram Srimath Kandada,1 Hao Li,2 Félix Thouin,3 Eric R. Bittner,2,a) and Carlos Silva3,4,5,b) AFFILIATIONS 1Department of Physics and Center for Functional Materials, Wake Forest University, 1834 Wake Forest Road, Winston-Salem, North Carolina 27109, USA 2Department of Chemistry, University of Houston, Houston, Texas 77204, USA 3School of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, Georgia 30332, USA 4School of Chemistry and Biochemistry, Georgia Institute of Technology, 901 Atlantic Drive, Atlanta, Georgia 30332, USA 5School of Materials Science and Engineering, Georgia Institute of Technology, North Avenue, Atlanta, Georgia 30332, USA Note: This paper is part of the JCP Special Topic on Excitons: Energetics and Spatio-Temporal Dynamics. a)Electronic mail: ebittner@central.uh.edu b)Author to whom correspondence should be addressed: carlos.silva@gatech.edu ABSTRACT We develop a stochastic theory that treats time-dependent exciton–exciton s-wave scattering and that accounts for dynamic Coulomb screening, which we describe within a mean-field limit. With this theory, we model excitation-induced dephasing effects on time-resolved two-dimensional coherent optical lineshapes and we identify a number of features that can be attributed to the many-body dynamics occurring in the background of the exciton, including dynamic line narrowing, mixing of real and imaginary spectral components, and multi-quantum states. We test the model by means of multidimensional coherent spectroscopy on a two-dimensional metal-halide semi- conductor that hosts tightly bound excitons and biexcitons that feature strong polaronic character. We find that the exciton nonlinear coherent lineshape reflects many-body correlations that give rise to excitation-induced dephasing. Furthermore, we observe that the exci- ton lineshape evolves with the population time over time windows in which the population itself is static in a manner that reveals the evolution of the multi-exciton many-body couplings. Specifically, the dephasing dynamics slow down with time, at a rate that is governed by the strength of exciton many-body interactions and on the dynamic Coulomb screening potential. The real part of the coherent opti- cal lineshape displays strong dispersive character at zero time, which transforms to an absorptive lineshape on the dissipation timescale of excitation-induced dephasing effects, while the imaginary part displays converse behavior. Our microscopic theoretical approach is suf- ficiently flexible to allow for a wide exploration of how system-bath dynamics contribute to linear and non-linear time-resolved spectral behavior. Published under license by AIP Publishing. https://doi.org/10.1063/5.0026351 .,s I. INTRODUCTION It is well recognized that many-body phenomena have a profound effect on the linear and non-linear optical lineshapes of semiconductors with reduced dimensionality, in which Coulombcorrelations can be particularly strong due to the decreased screen- ing and quantum confinement effects. One such effect is biex- citon formation in which Coulomb binding of two electron– hole pairs results in new two-electron, two-hole quasiparticles.1–11 Another important process that is highly relevant in exciton J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp quantum dynamics is excitation-induced dephasing (EID),12–27 primarily investigated in two-dimensional (2D) systems such as III–V quantum wells,13,18–20,22–24single-layer transition-metal dichalchogenides,25,26and two-dimensional metal-halide perovskite derivatives.27This can be described as the incoherent Coulomb elas- tic scattering between multiple excitons or between excitons and an electron–hole plasma generated with the excitation optical field. The scattering process gives rise to faster dephasing dynamics com- pared to the low-density pure-dephasing limit and may be the dom- inant dephasing pathway at sufficiently high densities. In many sys- tems, especially those with strong exciton–phonon coupling, the background excitations are transient and co-evolve with optical modes of the system, and consequently, a strictly incoherent kinetic description such as this mesoscopic approach or a kinetic Marko- vian Boltzmann-like scattering theory15cannot describe coherence dynamics. EID can be effectively rationalized from a mesoscopic perspective by means of the optical Bloch equations, which cap- ture the effect of many-body exciton scattering on both popula- tion and coherence dynamics derived from coherent spectroscopy of semiconductors.20,21Recent advances toward a more microscopic perspective have been presented by Katsch et al. in which exci- tonic Heisenberg equations of motion are used to describe lin- ear excitation line broadening in two-dimensional transition-metal dichalchogenides.28Their results indicate exciton–exciton scattering from a dark background as a dominant mechanism in the power- dependent broadening EID and sideband formation. Similar theo- retical modeling on this class of materials and their van der Waals bilayers has yielded insight into the role of effective mass asymme- try on EID processes.29These modeling works highlight the need for microscopic approaches to understand the nonlinear quantum dynamics of complex 2D semiconductors, but the computational expense could become considerable if other many-body details such as polaronic effects are to be included.30As an alternative general approach, an analytical theory of dephasing in the same vein as the Anderson–Kubo lineshape theory,31,32but that includes transient EID and Coulomb screening effects, would be valuable to extract microscopic details on screened exciton–exciton scattering fromtime-dependent nonlinear coherent ultrafast spectroscopy via a direct and unambiguous measurement of the homogeneous excita- tion linewidth.33,34 Here, we employ a quantum stochastic approach, derived from a first-principles many-body theory of interacting excitons, to develop an analytical model that describes linear and nonlinear spectral lineshapes that result from exciton–exciton scattering pro- cesses and, importantly, their dependence on the population time due to the evolution of a non-stationary/non-equilibrium excitation background [see Fig. 1(a)]. Our approach is similar in spirit to the celebrated Anderson– Kubo theory31,32and reduces to that in the limit of a stationary back- ground population at sufficiently long times.35The model captures a microscopic picture of EID by integrating over the interactions of excitons produced via a well-defined coherent pathway (Fig. 2), and background excitons that do not have a well-defined phase relation- ship, and by treating them as a non-stationary source of quantum noise. In doing so, we can directly insert the spectral density of the bath into non-linear spectral response functions and obtain fully analytical expressions for the coherent exciton lineshapes. We implement the model to investigate the evolution of the two-dimensional coherent excitation lineshape in a polycrystalline thin film of a prototypical two-dimensional single-layer metal- halide perovskite derivative, phenylethylammonium lead iodide [(PEA) 2PbI 4] [see Fig. 1(c) for the crystal structure]. We have selected this material as a model system because of its well-defined exciton lineshape that we have modeled quantitatively within a Wannier–Mott framework36and because it displays strong many- body phenomena—strongly bound biexcitons at room tempera- ture37and robust EID effects.27Furthermore, we have concluded that the primary excitations are exciton polarons30,38—quasiparticles with Coulomb correlations that are renormalized by lattice dynam- ics via polaronic effects; both electron–hole and photocarrier– lattice correlations are ingredients of the system Hamiltonian such that the lattice dressing constitutes an integral component of its eigenstates and eigenvalues. This renders the system highly dynam- ically disordered such that the lattice screening effects play an FIG. 1 . (a) Schematic representation of optical absorption of excitons and exciton–exciton scattering with a background population, where the dispersion relation is in the exciton representation and⃗k=⃗ke+⃗khis the exciton wavevector. (b) Time evolution of population N(t)/Neqfrom an initial nonstationary state produced by exciton injection. Individual trajectories are represented by blue dots. Asymptotically, the function reaches a stationary state that yields the Anderson–Kubo limit. (c) Crystal structure of (PEA) 2PbI 4with a schematic representation of Coulomb-correlated exciton–exciton elastic scattering. J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Double-sided Feynman dia- grams for coherent response functions [Eq. (16)] with rephasing phase match- ing (top): (a) R2a, (b) R3a, and (c) R∗ 1band with non-rephasing phase matching (bottom): (d) R1a, (e)R4a, and (f) R∗ 2b. important role in shaping the linewidth27and in dictating nona- diabatic dynamics.39We measure the dephasing dynamics via the homogeneous linewidth extracted by means of two-dimensional coherent excitation spectroscopy.33,34In our measurements, exci- tons generated coherently by a sequence of time-ordered and phase- matched femtosecond pulses scatter from incoherent background excitons and thereby undergo EID, which is perceived via changes in the homogeneous linewidth. We find that EID affects the com- plex lineshape by mixing absorptive and dispersive features in the real and imaginary spectral components; the real component of the two-dimensional coherent spectrum initially displays a disper- sive lineshape that evolves into an absorptive over the timescale in which EID couplings persist, and the imaginary component evolves in the converse fashion. Furthermore, we find that the homogeneous contribution to the spectral linewidth narrows with the population time, indicating a dynamic slowing down of the dephasing rate as the EID correlations active at early time dissi- pate. We find that the dynamic line narrowing phenomenon is reproduced by our stochastic scattering theory, which allows us to explore the effect of dynamic Coulomb screening on EID quantum dynamics. II. THEORETICAL MODEL A complete derivation of our stochastic scattering theory is presented in Ref. 35; here, we outline the elements that allow us to calculate the nonlinear coherent spectral lineshapes. Our model is initiated by assuming that at t= 0, a non-stationary populationof background excitations is created by a broadband laser exci- tation. This physical picture is sketched in Fig. 1. In the current work, excitation occurs with a sequence of phase-matched and time- ordered femtosecond pulses used to measure a coherent nonlinear excitation spectrum, and the excitons produced and measured via a well-defined coherent pathway (see Fig. 2 for the relevant ones in this work) are assumed to scatter elastically with their incoher- ent counterparts—excitons that are produced by the pulse sequence but have no phase relationship to those that produce signals in our experiments. The initial background population can be character- ized by an average population N0and variance σN0, both of which depend upon the excitation pulse as well as the density of states of the material. Optical excitations at k= 0 evolve in concert with a non-stationary ( k≠0) background of excitations in which the inter- action determined by a screened Coulomb potential gives rise to a noisy driving term that effectively modulates the exciton energy gap. These background excitations undergo diffusion as the population relaxes to some stationary distribution. In Ref. 35, we show how one can arrive at a reduced model described by a Hamiltonian of the form (we hereby set̵h=1) H0(t)≈ω0ˆa† 0ˆa0+γ1 4ˆa† 0ˆa† 0ˆa0ˆa0+γ1ˆa† 0ˆa0N(t), (1) where ˆa0and ˆa† 0are exciton operators, N(t) is a stochastic variable representing the number of background excitations [see Fig. 1(b) for a depiction of its non-stationary nature], and γ1is the exciton– exciton interaction, which we obtain from the s-wave scattering length aand reduced mass μwithin the Born approximation,40 J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp γ1=8π̵h2a μ. (2) This assumption does not rely upon the specific form of the exciton– exciton interaction, only that it be of finite range. In the cur- rent context, this interaction will be due to the Coulomb-mediated exciton–exciton scattering that gives rise to EID.27However, it is possible that each distinct exciton within the family of the 2D per- ovskite system considered here30has a distinct and unique value ofγ1, as we reported in Ref. 27, where we demonstrated distinct Coulomb screening of different exciton polarons. For purposes of our theoretical model, we assume that the system has a single exci- ton species that is susceptible to many-body scattering and there- fore EID mediated via γ1. The exciton operators themselves then evolve as ˆa0(t)=exp(−i(ω0+γ1 2ˆn0)t−iγ1∫t 0N(τ)dτ)ˆa0, (3) where ˆn0=ˆa† 0ˆa0is the number operator of k= 0 excitons, which is a Schrödigner operator and is therefore time independent. This devel- opment is within the framework of the interaction representation. These operators are then used to construct the expressions shown schematically in Fig. 2. Finally, it should be noted that the angle brackets ⟨⋯⟩below each diagram denote both the thermal average over the initial conditions and averaging over the stochastic variable, N(t). A central part of our model is that we assume that the background population N(t)=⟨∑k≠0ˆa† kˆak⟩follows from an Ornstein–Uhlenbeck process described by the stochastic differential equation dN(t)=−γN(t)dt+σdW(t), (4) where W(t) represents a Wiener process (continuous-time stochas- tic process), γgives the background relaxation rate, and σgives the variance. This is a reasonable assumption lacking an explicit descrip- tion of the background population and its influence on the system as a source of random collisions. The Ornstein–Uhlenbeck process is a prototypical noisy relaxation process and describes the dynamics of an overdamped oscillator driven by thermal fluctuations. For a stationary background population, i.e., ⟨N(t)⟩= 0, the covariance evolves according to ⟨N(t)N(s)⟩=⟨N(t−s)N(0)⟩ =σ2exp[−γ|t−s|]/2γ. In this limit, our model reduces to the Anderson–Kubo model in which the frequency fluctuates about a stationary average according to an Ornstein–Uhlenbeck process. In this case, the population relaxation time in our model is equiva- lent to the correlation time in the Anderson–Kubo model, and this gives the rate at which the environment relaxes back to its station- ary average, given a small push. Moreover, the fluctuation ampli- tude, Δ2, in the Anderson–Kubo model is equivalent to σ2/γin our model. As we shall show, what appears at first to be a sim- ple modification to the dynamics of a system has rather significant implications in terms of the non-linear spectral response of the system. At time t= 0, we push the background population signifi- cantly away from the steady-state distribution to an initial value of ⟨N(0)⟩=N0, and the population evolves asN(t)=N(0)e−γt+σ∫t 0e−γ(t−s)dW(s) (5) and ⟨N(t)⟩=e−γtN0, (6) where N0is the mean number of background excitations present at time t= 0. In principle, there will be a distribution about this mean characterized by a variance σ2 N0. As a result, we break reversibility and the time symmetry of the correlation functions. Mathemati- cally, this means that ⟨N(t)N(s)⟩≠⟨N(t−s)N(0)⟩since the choice of initial time is no longer arbitrary. In Ref. 35, we discuss the use of Itô calculus41–43to evalu- ate these correlation functions. From a practical point of view, the Itô calculus is a tool for manipulating stochastic processes that are closely related to Brownian motion and Itô’s lemma allows us to eas- ily perform noise-averaged interactions. For the model at hand, the covariance of N(s) and N(t) is given by Cov(N(s),N(t))=⟨(N(s)−⟨N(s)⟩)(N(t)−⟨N(t)⟩)⟩ =σ2 2γ(e−γ∣t−s∣−e−γ(t+s))+σ2 N0e−γ(s+t), (7) withσ2 N0being the variance of N(0) while the third term vanishes. Similarly, the variance Var[N(t)]=(σ2 N0−σ2 2γ)e−2γt+σ2 2γ(8) also depends upon the initial variance in the background population. Mathematically, the Fourier transform of the kernel of the integral in Eq. (5) provides the spectral density of the noisy process. In fact, a trivial modification of the approach would be to replace the kernel in Eq. (5) with another kernel reflecting a more complex spectral density. The resulting expressions for the responses will be more complex indeed. However, Itô’s lemma provides a tractable route for computing the necessary response functions. A. Optical response functions and spectral lineshapes Such expressions are useful since they enter directly into the calculation of response functions for linear and non-linear spec- troscopy. For example, the linear response for optical excitation is given by S(1)(t)=i⟨ˆμ(t)[ˆμ(0),ρ(−∞)]⟩, (9) where ˆμ(t)=μ(ˆa† 0(t)+ˆa0(t))is the excitonic transition dipole operator and ρ(−∞) is the initial density matrix. The absorption spectrum is obtained by Fourier transformation. Averaging over the fluctuations generates terms involving cumulants of the background noise, which result in terms such as ⟨exp[−iγ1∫t 0N(τ)dτ]⟩≈eiγ1g1(t)−γ2 1 2g2(t), (10) J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where⟨⋯⟩denotes averaging over noise. Here, the first cumulant g1(t) gives rise to a characteristic frequency shift as the background population decays, g1(t)=∫t 0⟨N(τ)⟩dτ=No γ(1−e−γt) (11) and g2(t,t′)=∫t 0∫t′ 0Cov[N(τ),N(τ′)]dτ′dτ =σ2 2γ3[2γmin(t,t′)+2e−γt+2e−γt′ −e−γ∣t′−t∣−e−γ(t′+t)−2] +σ2 No γ2[e−γ(t+t′)−e−γt−e−γt′ + 1]. (12) When the two time limits are the same, this reduces to g2(t)=∫t 0∫t 0Cov[N(τ),N(τ′)]dτdτ′ =σ2 2γ3(2γt+ 4e−γt−e−2γt−3)+σ2 N0 γ2(1−e−γt)2. (13) In Ref. 35, we discussed the linear response of our model and its relation to the Anderson–Kubo model. Here, we shall focus solely on the higher-order responses that reveal the dynamic evolution of the two-dimensional coherent excitation lineshape. The third-order response involves the phase-matched interactions of the system with a sequence of three laser pulses, S(3)(τ3,τ2,τ1)=i3⟨μ(τ3)[μ(τ2),[μ(τ1),[μ(0),ρ(−∞)]]]]⟩ . (14) The times 0 <τ1<τ2<τ3define the sequence of the time-ordered interactions in Fig. 2. The expressions for these can be evaluated using the standard rules for the double-sided Feynman diagrams (Fig. 2, cf. Ref. 44) representing various optical paths. For example, the pathways involving only single-excitation take the form Rα(τ1,τ2,τ3)=i3μ4(n0+ 1)2exp⎡⎢⎢⎢⎢⎣−i(ω0+n0γ1/2)3 ∑ j=1(±)jτj⎤⎥⎥⎥⎥⎦ ×⟨exp⎡⎢⎢⎢⎢⎣iγ13 ∑ j=1(±)j∫τj 0N(s)ds⎤⎥⎥⎥⎥⎦⟩ (15) =i3μ4(n0+ 1)2exp⎡⎢⎢⎢⎢⎣−i(ω0+n0γ1/2)3 ∑ j=1(±)jτj⎤⎥⎥⎥⎥⎦ ×exp⎡⎢⎢⎢⎢⎣−iγ13 ∑ j=1(±)jg1(τj)⎤⎥⎥⎥⎥⎦ ×exp⎡⎢⎢⎢⎢⎣−γ2 1 23 ∑ i,j=1(±)i(±)jg2(τi,τj)⎤⎥⎥⎥⎥⎦(16) The sign function ( ±)jtakes “+” and “ −” depending upon the sign of the photon wavevector entering or leaving the system. Note thatg1(τj) = 0 when the system is initially prepared in the ground state since N0= 0. Figure 2 shows the most relevant diagrams for the rephasing (−) and non-rephasing/(+) optical response. The detailed expressions are included in the supplementary material. It is important to note that the exciton–exciton interaction termγ1, and hence the screening due to the exciton–lattice interac- tions, appears in three distinct places in the third-order responses: first, as a frequency shift due to self-interactions between the bright excitons; second, as a frequency shift due to the interac- tions of bright excitons with the evolving background population density; and third, as the leading contribution to the lineshape. In addition, the third term involving g2(t) carries the influence of the initial conditions (via σN0). The effect of many-body exciton– exciton scattering thus leads to time-evolving EID processes. Given these observations, we expect that the homogeneous linewidth will evolve with the population time, dictated by the evolution ofg2(t). III. TWO-DIMENSIONAL COHERENT SPECTROSCOPY A. Predictions from the stochastic model Having established the mathematical model, let us briefly reca- pitulate some of its features. First, we started by assuming that the background population dynamics give rise to a stochastic process N(t) that enters into the Heisenberg equations of motion for the system operators [Eq. (3)]. In particular, we assumed that N(t) cor- responds to an overdamped Brownian oscillator and that, at time t= 0, there is a non-stationary population of background excita- tions. These two mathematical assumptions can be relaxed to some extent if one has a more detailed description of the spectral den- sity of the background process and the initial background pop- ulation. Second, we assume that averages over exponential terms can be evaluated using the cumulant expansion. What then fol- lows are the mathematical consequences as expressed in terms of the spectral responses of the model. In Ref. 35, we explored the linear response, especially as compared to the Anderson– Kubo model.31,32The key features of our model include35the following: 1.Blocking : Increasing the initial background exciton density suppresses the peak absorption intensity. 2.Energy shift : The peak position shifts to the blue with the increasing background population due to the increased Coulombic interactions. 3.Broadening: The spectrum acquires a long tail extending to the blue due to the dynamical evolution of the background. This feature appears in the 2D coherent spectroscopy as an asym- metry along the absorption axis and as phase scrambling in the rephasing and non-rephasing signals (see Figs. S4 and S5 of the supplementary material of Ref. 27 for the exciton-density- dependent lineshapes at τp= 0). 4.Biexciton formation: The peak is split by γ1/2 corresponding to the biexciton interaction.37 These effects are consistent with the experimental observations and theoretical models of 2D semiconductors and transition-metal dicalcogenides.28 J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Theoretical real and imaginary spectra, respectively, of rephasing [(a) and (b)] and nonrephasing [(c) and (d)] phase matching and at population waiting time τp= 0 fs. The bar to the right of the figure displays the vertical false color scale in arbitrary units. Figures 3–5 correspond to the rephasing and non-rephasing behavior of theoretical models as parameterized to approximate the excitons in the 2D metal-halide perovskite system studied in the experimental investigations, which we shall describe later in this FIG. 5 . Exciton 2D coherent lineshape contour at half-maximum intensity as a function of population waiting time derived from the theoretical rephasing absolute spectral evolution in Fig. 4. The center of mass and one of the principle axes are shown for each contour. section. The parameters used to produce these spectra are given in Table I. The two pairs of gray dashed lines correspond to the bare exciton energy at̵hω0= 2.35 eV and the dressed exciton energy at ̵hω0+γ1/2 = 2.36 eV. Figure 3 gives the rephasing [(a) and (b)] FIG. 4 . [(a)–(d)] Real parts of theoretical rephasing spectra at population times τp indicated at the top of each panel. [(e)– (h)] Corresponding imaginary parts of the spectrum. [(i)–(l)] The norm (absolute value) of the optical response. J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Parameters used in the theoretical model to obtain Figs. 3–5. Description Symbol Value Bare exciton energy̵hω0 2.35 eV Noise variance σ20.0025 fs−1 Relaxation rate γ 0.01 fs−1 Exciton/exciton interaction γ1 20 meV Average initial background density N0 2 per unit volume Initial background variance σN0 0.35 per unit volume and non-rephasing [(c) and (d)] spectra computed at τp= 0. Two features highlighted above are immediately striking in the modeled 2D spectra. Both the asymmetry of the signals and the lineshape inversion of the real and imaginary spectral components can be traced specifically to terms within the response functions in Eq. (16) that depend upon the transient background relaxation and exciton self-interactions. Both the phasing and asymmetry evolve with the increas- ing population time, as shown in Figs. 4(a)–4(l). Importantly, the rephasing signal evolves being dispersive at τp= 0 to absorptive at longer times. The non-rephasing signal [Figs. 4(e)–4(h)] has a complementary behavior, evolving from absorptive to dispersive. Figures 4(i)–4(l) give the absolute value of the total response as it evolves over τp. The peak is displaced from the diagonal and its position as well as the linewidth evolves over τp. In Fig. 5, we extract the contour corresponding to the half- maximum intensity at various indicated τppopulation times. Super- imposed over each contour is one of the principal axes of the con- tour scaled according to its magnitude. The central points are the geometric centers of contours. This analysis clearly shows that the peak systematically narrows, rotates, and distorts as the exciton co- evolves with the background population. Moreover, the center peak shifts by about 10 meV toward the red in both absorption and emission spectral dimensions as Coulombic interactions with the evolving background are diminished—this phenomenon is known as excitation-induced shift.10The lineshape evolution predicted by the stochastic model is due to g1(t)≠0. The early-time blue shift as well as more rapid dephasing arise from the many-body effects contained within g1; as this function decays, these effects dissi- pate as shown in Fig. 5. We note that in Eq. (16), if we set g1 = 0, the coherent response functions reduce to a stationary back- ground and the lineshape evolution in Fig. 5 would not arise (see also Ref. 35). It should be noted, however, that we assume here that the initial background excitation is broad compared to its fluctuations about a stationary state. Starting from the opposite regime, one can obtain dynamic broadening (rather than narrowing) as the system relaxes to the stationary state. We specifically choose these condi- tions to best represent the experimental conditions of an ultrafast experiment with ∼20 fs pulses. In order to test the predictions of the non-stationary model described above, we have carried out two-dimensional coherent measurements on (PEA) 2PbI 4(PEA = phenylethylammonium)— a multiple-quantum-well-like single-layer metal-halide perovskitederivative [see Fig. 1(c)]. We choose this material to test the the- oretical framework developed above because of its susceptibility to strong many-body effects27,37,45and dynamic exciton–lattice cou- pling that drives their dynamics.30,36,38,39To further examine EID in this material, we dissect the population-time-resolved nonlinear coherent optical lineshape of the family of exciton polarons30by means of two-dimensional (2D) coherent spectroscopy.46Impor- tantly for this work, the 2D coherent optical lineshape permits sepa- ration of the homogeneous and inhomogeneous contributions to the linewidth33,34,47and is therefore an appropriate technique to spec- trally and temporally resolve dephasing rates, which we exploit here to quantify EID dynamics. We have discussed the linear spectral lineshape of (PEA) 2PbI 4in Refs. 36 and 37, and here, we sum- marize it in Sec. III B. We will then discuss the 2D spectral line- shapes of (PEA) 2PbI 4in Sec. III C. Specifically, we will present the experimentally observed dispersive lineshape as a signature of EID, which we had suggested previously in Ref. 27. We will show that such a lineshape evolves into an absorptive form with the pop- ulation time, as a consequence of g2(t) in the response function that is quenched due to the increased screening of exciton–exciton interactions. B. Linear absorption lineshape We display the linear absorption spectrum of (PEA) 2PbI 4mea- sured at 5 K in the region of exciton absorption [Fig. 6(a)]. It has been reported extensively that this spectrum displays a struc- ture consisting of multiple resonances.30,36,37,48–52We have argued previously that the primary photoexcitations in two-dimensional hybrid metal-halide perovskite derivatives are a family of exciton polarons,30with exciton binding energies differing by ∼40 meV,36 and each with distinct phonon dressing.38Here, we focus on the two primary transitions labeled XAandXB, but we also highlight an additional shoulder of XA, labeled XA′. We had initially hypoth- esized that XA′is the envelope of replicas in a vibronic progression with origin XA,36but we then subsequently found that its elastic scat- tering rate is distinct from XAandXB,27indicating it to be another distinct state within the spectral bandwidth of the excitonic transi- tions. We highlight that this spectral structure is general to other derivatives with different organic cations, including the ones that induce lattice distortions53that modulate the central exciton bind- ing energy,36,54and the relative intensities of the transitions, but not the energy spacing in the spectral structure.30,36 C. Time-resolved 2D coherent excitation lineshape We next consider the complex 2D coherent excitation spec- trum to quantify the consequences of EID in the nonlinear lineshape. We have previously reported that the multiple excitons identified in Fig. 6(a) display strong many-body effects, manifested via the presence of stable biexcitons37and the dominance of EID signa- tures on the homogeneous linewidth.27We have observed that XA and XBdisplay the different dependence of EID on exciton den- sity and on temperature27and have interpreted these phenomena as indicative of specific dynamic Coulomb screening of XAandXB by different polaronic dressing phonons.38Shown in Figs. 6(b) and J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . (a) Linear absorption spectrum of (PEA) 2PbI 4at 5 K. The real part of the corresponding rephasing (b) and non-rephasing (c) spectra at a population time of τp= 0 fs. The bar to the right of the figure displays the vertical false color scale in arbitrary units. 6(c) are the real parts of two different coherent excitation path- ways; the time-ordering of the three optical pulses in the experi- ment and phase-matching conditions defines the specific excitation pathways, based on which rephasing [Fig. 6(b)] and non-rephasing [Fig. 6(c)] spectra are obtained.46In the rephasing experiment, the pulse sequence is such that the phase evolution of polarization after the first pulse and that after the third pulse are of opposite sign, while in the non-rephasing experiment, they are of the same sign [see Eq. (16) and Fig. 2]. Both measurements shown in Fig. 6 are taken at a population waiting time τp= 0 fs and an excitation fluence of 40 nJ/cm−2, which corresponds to an exciton density in which we have identified the effects of elastic exciton–exciton scattering.27 The corresponding diagonal spectral features at the energies of XA, XA′, and XB(indicated by the magenta vertical dotted-lines in Fig. 6) are observed, both in rephasing and non-rephasing spectra. Apart from these diagonal peaks, we observe an off-diagonal excited-state absorption feature (opposite phase with respect to the diagonal fea- tures) corresponding to a correlation between the absorption energy ofXAand emission energy ∼2.3 eV, which has no correspondingdiagonal signal. We have assigned this cross-peak to a biexciton resonance.37 From the norm of the rephasing spectrum at zero time [not shown in Fig. 6 but shown in Fig. 7(i)], one can extract the homogeneous and inhomogeneous linewidths via a global analy- sis of the diagonal and the anti-diagonal lineshape.33,34In Ref. 27, we reported the homogeneous linewidth ( γ0) in the absence of excitation-induced dephasing (zero-density limit) is ∼2 meV for all the observed excitonic resonances, which implies a dephasing time T∗ 2∼500 fs.55This linewidth is comparable to the inhomoge- neous width of ∼6.5 meV, placing this system firmly in a dynamic disorder regime. Upon increasing exciton density, the homoge- neous linewidth derived from the τp= 0 fs spectrum increases due to EID arising from many-body elastic scattering. As men- tioned earlier in this section, by quantifying the contribution of EID to the homogeneous linewidth, we have have concluded that XAand XBdemonstrate distinct exciton–exciton scattering rates, which we attributed to their peculiar phonon dressing38lead- ing to a specific dynamic Coulomb screening of their nonlinear coupling. In Ref. 27, we considered the norm of the rephasing spectrum atτp= 0 fs; however, upon close inspection of Fig. 6, we note that the real part of the lineshape in Fig. 6 displays dispersive shape, i.e., derivative shape about the peak energy, both for diagonal and off- diagonal resonances, in both the rephasing and non-rephasing spec- tra. Note the sign-flip for the off-diagonal feature that is consistent with its assignment to the excited state absorption to the biexcitonic state. Similarly, the imaginary part of the spectra (not shown in Fig. 6 but shown in Fig. 7) displays an absorptive lineshape. The theoretical spectra shown in Fig. 3 suggest that such dispersive lineshapes are a consequence of many-body correlations. In the absence of the lat- ter, the lineshape should be purely absorptive. The spectra in Fig. 6, therefore, reveal phase mixing due to the many-body Coulomb cor- relations responsible for EID, as has been reported in semicon- ductor quantum wells.22In fact, we have demonstrated in Ref. 27 that the EID dominates the non-linear response in the employed pump fluence range. These phenomena are reproduced by the 2D coherent spectra predicted by our stochastic theory, as shown in Fig. 3. The evolution of the rephasing lineshape shown in Fig. 6(b) with the population waiting time τpis displayed in Fig. 7. The top row displays the real part of the spectrum at different val- ues ofτp, the middle row the imaginary component, and the bot- tom row the norm (absolute value) of the complex spectrum. We observe that the phase scrambling phenomenon displayed in the τp= 0 fs spectrum [Fig. 6(b)] dissipates within τp≤240 fs: the real component of the spectrum evolves from an initially disper- sive [Fig. 7(a)] to absorptive [Fig. 7(d)] lineshape, while that of the imaginary part evolves from absorptive [Fig. 7(e)] to disper- sive [Fig. 7(h)] character. We note that although the evolution of the real and imaginary components of the complex lineshape is sub- stantial over this ultrafast time window, the population decay of the diagonal features for XAand XBis weak, observed via the mod- est evolution of the total intensity in Figs. 7(i)–7(l). The decay of theXA′diagonal peak and the biexciton cross peak appears more substantial. The marked evolution of the complex lineshape is also pre- dicted by the stochastic theory as evidenced by the theoretical spectra J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . [(a)–(d)] Real parts of the experi- mentally measured rephasing spectra at population times τpindicated at the top of each panel, measured at 5 K. [(e)– (h)] The corresponding imaginary parts of the spectrum. [(i)–(l)] The norm (abso- lute value) of the optical response. All the spectra are plotted in the same relative vertical color scale to facilitate the com- parison of the time-dependent signal. presented in Fig. 4. Recalling the arguments presented in Sec. III A, such a dynamic is fundamentally driven by the exciton–exciton interactions that are time dependent due to the evolving background population. While the theory also predicts asymmetry in the line- shapes, non-negligible inhomogeneous effects, which are not con- sidered in the theory yet present in the experimental spectra, prevent the observation of clear asymmetry. We also highlight the reduction in the total linewidth of the each diagonal exciton resonance in the absolute value of the response shown in Figs. 7(i)–7(l) with the population time. Inspection of these spectra reveals dynamic narrowing of XAand XB, primar- ily along the anti-diagonal spectral axis. It is more difficult to visually ascertain the linewidth evolution of XA′and the biex- citon cross peak, given the non-negligible decay over this time period. The dynamic line narrowing in Fig. 7 has also been pre- dicted by the theory, again due to the loss of the dephasing path- way in the form of EID. This phenomenon reflects the spectral evolution predicted in Fig. 5, which highlights the dynamic line narrowing. To quantify the measured dynamic line narrowing, we display in Fig. 8 the homogeneous linewidth as extracted from Ref. 27 as a function of population time τp. This is extracted from a global analysis of the diagonal and the antidiagonal lineshape as developed in Refs. 33 and 34: in the limit of similar homogeneous and inho- mogeneous widths as is the case in this material,27,36,37the diagonal lineshape follows a Voigt profile, while the antidiagonal spectrum isthe product of a Gaussian and complementary error function, but both diagonal and antidiagonal widths depend on the dephasing parameter. By this analysis, Fig. 8 shows that the linewidth of XA reduces most drastically, but that of XBalso reduces over a typical time window, while XA′displays no line narrowing. We note that in FIG. 8 . Homogenous linewidths obtained from the lineshape analysis of the abso- lute value of the rephasing spectra (see Ref. 27) plotted as a function of the population time for (a) XA, (b)XA′, and (c) XBexciton lines shown in Fig. 6(a). J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ref. 27, we reported that XAhas a stronger density dependence of EID than XB, which is consistent with the observation derived from Figs. 8(a) and 8(c). We have found XBto be more strongly displaced along phonon coordinates involving octahedral twist in the plane of the inorganic layer and out of plane scissoring of the Pb–I–Pb apex.38 Interestingly, the homogeneous linewidth of XAdisplayed thermal broadening by a dominant phonon mode on the inorganic plane, while the thermal broadening mechanism for XBinvolved a phonon with motion involving the organic cation.37The stronger exciton– phonon coupling implies that XBis more susceptible to dynamic screening than XA, which is consistent with the data in Fig. 8 and Ref. 27. Furthermore, the linewidth of XA′displayed a weaker, non- Boltzmann temperature dependence.27Finally, we point out that the asymptotic value of the homogeneous linewidth for XA,XA′, and XB tends toward the low-exciton-density linewidths that we reported in Ref. 27. The linewidth of XA′remains relatively constant over the probed population time. While this might initially suggest that this resonance is immune to the EID effects, we note that the real part of the rephasing spectrum associated with this particular transition exhibits a dispersive lineshape at all population times, consistent with the inital lineshapes of XAand XB. This indicates the clear presence of EID effects, as also confirmed by the density depen- dent linewidth previously published in Ref. 27. The trend shown in Fig. 8(b), on the other hand, suggests that the inter-exciton scat- tering does not evolve with the population time, at least within the probed time range. Inspection of the lineshape, however, sug- gests that the dispersive lineshape of the real part is preserved at all population times, suggesting that XA′is subjected to EID over a much longer period of time than the other two resonances. Fol- lowing the arguments developed by the theory in this paper, this implies the presence of a background exciton population that con- tributes to the scattering of XA′and whose stochastic evolution is that of the background of the other two resonances. This reiterates our assignment of the multiple resonances within the spectral struc- ture to the excitonic states of distinct character and possibly specific origin.30 D. Consequences for the exciton spectral structure The origin and nature of the spectral fine structure of the excitonic transition—the presence of distinct resonances such as XA,XA′, and XB—has been under discussion.52Early works on 2D hybrid metal-halide perovskites suggested that the spectral struc- ture is the outcome of degeneracy-lifting processes driven by spin- exchange interactions ubiquitous to lead-based semiconductors. Even though the spin-exchange energy was estimated to be of the same order magnitude as the energy spacing within the fine struc- ture,56Kataoka et al. ,57and more recently Urban et al. ,58noted an indiscernible difference in the diamagnetic shift of each of the resonances with the applied magnetic field. Alternatively, a more chemical perspective was also suggested in which the structure was assigned to a vibronic progression within a single exciton state.51,58 This was particularly highlighted in (PEA) 2PbI 4, where Urban et al.58and Straus et al.59identified a vibrational mode at about 40 meV in the off-resonance Raman spectrum associated with the motion of the phenylethylammonium cation. We underline, however, that in order to unambiguously establish the vibronic nature, it is essentialto measure the resonant Raman spectrum, which, in fact, has a dom- inant contribution from the phonon modes of the inorganic lattice. We have reported that this type of measurement over a lower fre- quency range reveals distinct displacements along different phonon modes for XAand XB.38Moreover, we highlight that a similar if not the same spectral structure is observed even in other 2D hybrid metal-halide perovskite derivatives with other organic cations, fur- ther suggesting that the spectral structure is unlikely to be vibronic in nature. We have presented arguments in Ref. 30 as to why we consider that such an interpretation of a vibronic progression for the spec- tral structure in Fig. 6(a) does not explain a series of experimental observations, including distinct polaronic dressing of XAandXB,38 distinct screening of EID for these excitons,27and distinct biexciton binding.37We consider that the clearly peculiar behavior of XAand XBin Fig. 8 adds to the body of work that establishes these excitons as a family of distinct but correlated excitons with strong polaronic character. E. Summary of phenomenology and relationship to predictions from stochastic theory We summarize the phenomena presented in Ref. 27 and in Sec. III C pertaining to the EID effects in the 2D coherent lineshape evolution in (PEA) 2PbI 4. ●Theτp= 0 rephasing absolute spectrum displays a homo- geneous linewidth that depends on exciton density, with XA,XA′, and XBdisplaying distinct density dependence.27 Furthermore, exciton–exciton scattering is activated by phonons on the lead iodide plane for XAbut by motion of the organic cation for XB, while XA′displays a relatively weaker thermal broadening.27We interpreted these distinct behaviors as indicative of peculiar screening by the lattice for each exciton polaron.38 ●The model predicts a lineshape asymmetry, manifested both in the linear (Ref. 35) and nonlinear (Fig. 3) spectra by a tail to higher energy that depends on the background-exciton density N0. This asymmetry is evident in the exciton- density dependence of the τp= 0 rephasing lineshape mea- sured experimentally (see the supplementary material of Ref. 27). ●Both theτp= 0 rephasing and nonrephasing spectra dis- play real and imaginary spectra with the inverted line- shape: the real component displays a dispersive lineshape, while the imaginary one is absorptive (Figs. 6 and 7). This phenomenon is predicted by the stochastic model and arises from the background evolution from the g1term (Fig. 3). ●This lineshape inversion dissipates over an ultrafast timescale in which the homogeneous linewidth decreases (Figs. 7 and 8). The dynamics of the lineshape evolution are once again distinct for the different excitons. The linewidth evolution is predicted by the theory and is ascribed to the time dependence of terms that arise due to the EID effects (Figs. 4 and 5). In the situation of broadband excitation in which the initial distribution of exciton energies is broad, the model distinctly predicts dynamic line narrowing, but J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp it also predicts dynamic line broadening in situations of narrow-band excitation (not shown in this manuscript). The microscopic model developed here captures the essen- tial EID physics and opens new opportunities for novel, detailed understanding of many-body exciton physics and of how system- bath dynamics contribute to the non-linear spectral behavior. We note that the theoretical model considered in the first part of this manuscript lacks some of the necessary ingredients to reproduce rigorously the experimental lineshapes, such as spin- exchange and polaronic effects. Despite this limitation, we have qualitatively identified the physical origin of the observed non- linear lineshape through the stochastic model presented in this article. IV. DISCUSSION We present here a joint theoretical and experimental study of excitation induced dephasing that connects the dynamics of an oth- erwise dark background density of states to the evolution of the 2D coherent spectral lineshape. Such dynamics are input in our analytical model in the form of the spectral density and corre- sponding stochastic equations of motion of the background popu- lation, which dress the quantum operators for the “bright” degrees of freedom. An important feature of our model and its connection to the experimental observation is that the complex and poten- tially intractable dynamics of the dark variables can be reduced to a single stochastic variable and a few physical parameters that can be directly related: exciton–exciton interactions and density of states. Coupled with the Itô calculus, this provides a powerful analytical tool for interpreting the dynamical features in the 2D coherent spectra. Furthermore, the approach can be extended to include additional interactions such as polaronic binding and spin– orbit coupling. We reserve the inclusion of these physics for future investigation. Our model predicts that the homogeneous linewidth evolves with the population time purely due to the dynamics of many- body correlations. Dynamic line broadening effects in 2D coher- ent spectra are often interpreted in the context of spectral dif- fusion.46,60This work demonstrates that in condensed matter systems, competing line narrowing processes due to the many- body interactions can complicate such lineshape evolution. We consider that the fact that the stochastic model developed here and in Ref. 35 predicts that the linewidth changes with time (whether it increases or decreases) is in itself a very important result with profound implications in condensed-matter chemical physics. This stochastic theory as developed here is strictly valid for semiconductors, but it can be readily developed to include further physics appropriate for the description of Moiré excitons in the 2D transition-metal dichalchogenide heterostructures61and signatures of ground-state spin–orbit entanglement in the optical conductiv- ity spectrum of the quasi-one-dimensional Mott insulators,62,63for example. These are the two optically accessible systems in which many-body interactions are dominant. Appropriate details of these physics can be, in principle, included in the spectral density in Eq. (5), and the consequences of these on the nonstationary spectral behavior can be unraveled via our model.V. PERSPECTIVE The spectral density of the environment plays a central role in many complex systems and governs the relaxation and decoher- ence of a quantum subsystem. Typically, we treat the environment as being quasi-stationary. The theoretical model presented in this article presents a means to represent the environment as nonsta- tionary, here in the context of exciton–exciton scattering, leading to a rich evolution of the nonlinear coherent exciton lineshape. Our perspective is that there is ample scope to include richer physics in the spectral density, such as an explicit treatment of polaronic effects in materials such as the 2D metal-halide perovskites consid- ered here.30We consider that this theoretical development presents opportunities to include microscopic understanding of many-body interactions that are dominant in condensed-matter systems on their quantum dynamics. The theoretical development presented here allows the explo- ration of the following open questions in the chemical physics com- munity: in multi-chromophoric systems such as light harvesting complexes, does sculpting of the spectral density determine sensi- tively (compared to the experimental observables) the evolution of the optical exciton lineshape in 2D coherent excitation spectra? Does the additional microscopic detail contained in the spectral density of the environment matter to capture the observed lineshape evo- lution? We believe that our theoretical framework can contribute toward this fundamental understanding. SUPPLEMENTARY MATERIAL The explicit expressions for the response functions described in Fig. 2 are available as supplementary material. AUTHORS’ CONTRIBUTIONS H.L. and A.R.S.K. contributed equally to this work. The experi- mental data were acquired by F.T. under the supervision of A.R.S.K and C.S. The experimental data were analyzed by A.R.S.K and F.T. The theoretical work was carried out by H.L. under the supervision of E.R.B. The project was conceived by A.R.S.K., E.R.B., and C.S. ACKNOWLEDGMENTS We are deeply thankful to Daniele Cortecchia and Annamaria Petrozza for providing the high quality samples for this work. The work at the University of Houston was funded, in part, by the National Science Foundation (Grant Nos. CHE-1664971 and DMR- 1903785) and the Robert A. Welch Foundation (Grant No. E-1337). The work at Georgia Tech was funded by the National Science Foun- dation (Grant No. DMR-1904293). C.S. acknowledges support from the School of Chemistry and Biochemistry and the College of Science at Georgia Tech. APPENDIX: EXPERIMENTAL METHODS 1. Sample preparation The samples were provided by Dr. Daniele Cortecchia for the work presented in Ref. 27, which reports measurements taken concurrently with those reported in this article. Thin films of J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (PEA) 2PbI 4(thickness of 40 nm) were prepared on sapphire sub- strates (optical windows 25 ×0.5 mm2, Crystran) by spin coat- ing a 0.05M solution of the perovskite in N,N-Dimethylformamide (DMF). A quantity of 12.5 mg of (PEA)I (Dyesol) was mixed with 11.5 mg PbI 2(TCI) and dissolved in 500 μl of DMF (Sigma Aldrich, anhydrous, 99.8%). The solution was left to dissolve on a hotplate at 100○C for 1 h. After exposing the substrate to an oxygen plasma, the solution (kept at 100○C) was spin-coated on the sapphire window at 6000 rpm for 30 s, and the film was annealed on a hotplate at 100○C for 15 min. The solution and the film were prepared in a glove-box under N 2atmosphere. 2. Two-dimensional coherent excitation spectroscopy The pulse train (25 fs, attenuated to a fluence ∼40 nJ/cm2, cen- tered at 530 nm) was generated by a home-built single-pass non- collinear optical parametric amplifier pumped by the third harmonic of a Yb:KGW ultrafast laser system (Pharos Model PH1-20-0200- 02-10, Light Conversion) with an output pulse train at 1030 nm, a repetition rate of 100 kHz, an output power of 20 W, and a pulse duration of 220 fs. Two-dimensional spectroscopic measurements were performed using a home-built, pulse-shaper-based multidi- mensional spectrometer that passively stabilizes the relative phase of each pulses.64Our implementation is described in detail in Ref. 37, albeit with a different ultrafast laser source with a much lower rep- etition rate. Each beam was independently compressed using the chirp-scan65to a pulse duration of 25 fs FWHM and was character- ized using the cross-correlated second harmonic frequency resolved optical gating66(SH-XFROG) in a 10 μm-thick BBO crystal placed at the sample position. For details of pulse characterization, includ- ing a typical SH-XFROG trace, we refer the reader to Ref. 27 since the data presented in this manuscript were taken simultaneously as the data presented in that article. The sample was kept at 5 K using a vibration-free cold-finger closed-cycle cryostat (Montana Instruments). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Mysyrowicz, J. B. Grun, R. Levy, A. Bivas, and S. Nikitine, “Excitonic molecule in CuC1,” Phys. Lett. A 26, 615–616 (1968). 2D. Magde and H. Mahr, “Exciton-exciton interaction in CdS, CdSe, and ZnO,” Phys. Rev. Lett. 24, 890 (1970). 3J. B. Grun, S. 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Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Springer Science & Business Media, 2012). J. Chem. Phys. 153, 164706 (2020); doi: 10.1063/5.0026351 153, 164706-13 Published under license by AIP Publishing

5.0014938.pdf

J. Chem. Phys. 153, 144114 (2020); https://doi.org/10.1063/5.0014938 153, 144114 © 2020 Author(s).Self-interaction correction, electrostatic, and structural influences on time- dependent density functional theory excitations of bacteriochlorophylls from the light-harvesting complex 2 Cite as: J. Chem. Phys. 153, 144114 (2020); https://doi.org/10.1063/5.0014938 Submitted: 22 May 2020 . Accepted: 25 September 2020 . Published Online: 13 October 2020 Juliana Kehrer , Rian Richter , Johannes M. Foerster , Ingo Schelter , and Stephan Kümmel ARTICLES YOU MAY BE INTERESTED IN Comparison of intermolecular energy transfer from vibrationally excited benzene in mixed nitrogen–benzene baths at 140 K and 300 K The Journal of Chemical Physics 153, 144116 (2020); https://doi.org/10.1063/5.0021293 Key role of retardation and non-locality in sound propagation in amorphous solids as evidenced by a projection formalism The Journal of Chemical Physics 153, 144502 (2020); https://doi.org/10.1063/5.0019964 Effects of perturbation order and basis set on alchemical predictions The Journal of Chemical Physics 153, 144118 (2020); https://doi.org/10.1063/5.0023590The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Self-interaction correction, electrostatic, and structural influences on time-dependent density functional theory excitations of bacteriochlorophylls from the light-harvesting complex 2 Cite as: J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 Submitted: 22 May 2020 •Accepted: 25 September 2020 • Published Online: 13 October 2020 Juliana Kehrer, Rian Richter, Johannes M. Foerster, Ingo Schelter, and Stephan Kümmela) AFFILIATIONS Theoretical Physics IV, University of Bayreuth, D-95440 Bayreuth, Germany a)Author to whom correspondence should be addressed: stephan.kuemmel@uni-bayreuth.de ABSTRACT First-principles calculations offer the chance to obtain a microscopic understanding of light-harvesting processes. Time-dependent den- sity functional theory can have the computational efficiency to allow for such calculations. However, the (semi-)local exchange-correlation approximations that are computationally most efficient fail to describe charge-transfer excitations reliably. We here investigate whether the inexpensive average density self-interaction correction (ADSIC) remedies the problem. For the systems that we study, ADSIC is even more prone to the charge-transfer problem than the local density approximation. We further explore the recently reported find- ing that the electrostatic potential associated with the chromophores’ protein environment in the light-harvesting complex 2 beneficially shifts spurious excitations. We find a great sensitivity on the chromophores’ atomistic structure in this problem. Geometries obtained from classical molecular dynamics are more strongly affected by the spurious charge-transfer problem than the ones obtained from crystallography or density functional theory. For crystal structure geometries and density-functional theory optimized ones, our cal- culations confirm that the electrostatic potential shifts the spurious excitations out of the energetic range that is most relevant for electronic coupling. Published under license by AIP Publishing. https://doi.org/10.1063/5.0014938 .,s I. INTRODUCTION Natural photosynthesis provides an outstanding example for the efficient conversion of light into other forms of energy. In pho- tosynthesis, light is absorbed by antenna complexes built from chro- mophores that are embedded in a protein structure.1,2Many differ- ent realizations of such structures have been found.3,4The absorbed energy is transferred with remarkable efficiency5,6to other antenna structures and ultimately to a reaction center, where a charge sep- aration step takes place that triggers the conversion into chemical energy.7Because photosynthesis first developed in bacteria, the bac- terial photosynthetic apparatus is relatively transparent. For this rea- son, the bacterial light-harvesting structures have been the focus ofefforts to gain a microscopic understanding of the elemental energy- and charge-transfer (CT) steps8that are decisive in photosynthetic energy conversion. First-principles calculations offer the chance to obtain insight into such systems and processes, reducing the need for model assumptions.9–13Due to its favorable ratio of accuracy to compu- tational cost, time-dependent density functional theory14(TDDFT) is the method of choice for such studies.15–24However, its use for studying light-harvesting questions is so far restricted by the charge- transfer (CT) problem of TDDFT: The local and semilocal exchange- correlation functionals that lead to the highest efficiency in TDDFT calculations systematically and severely underestimate the energies of CT excitations.25–27The CT problem can be solved, e.g., with J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp tuned range-separated hybrid functionals,28–31but at a substantial increase in the computational cost. This effectively limits the size of the systems that can be studied. In this article, we explore real-time propagation32–50as a tool for studying electronic excitations in bacteriochlorophylls. TDDFT in real-time has not been used much for that purpose so far but is attractive because it parallelizes well and can thus allow for the study of large, complex biological structures. Most of our calculations in addition use real-space grids, which allow for another very efficient layer of parallelization. The specific focus of this work is to explore options for calculating the electronic excitations of bacteriochlorphylls from a light-harvesting complex with TDDFT using exchange-correlation approximations of low computational cost. In a first step, we check whether a computationally efficient self-interaction correction (SIC), the average density (AD) SIC,51 remedies the CT problem. We confirm that average density self- interaction correction (ADSIC) leads to more negative eigenvalues and thus improves on the eigenvalue deficiencies of semilocal func- tionals. However, we find that it does not improve the description of CT excitations. In a second step, we thoroughly check the recently reported finding23that for the frequently studied light-harvesting complex 2 (LH2), including the protein electrostatic potential cir- cumvents the problem of spurious CT excitations in the time- dependent local density approximation (TD-LDA). We find a great sensitivity to the chromophore geometry in this problem: Geome- tries obtained from large-scale molecular dynamics (MD) simula- tions with the CHARMM27 force field52,53show the CT problem more pronouncedly than crystal structure geometries and geome- tries obtained from DFT structure optimization. For the latter two, including the protein electrostatic potential shifts spurious CT exci- tations out of the energetic range where excitation energy transfer coupling is expected. II. EXCITATIONS FROM TDDFT IN THE REAL-TIME APPROACH In the real-time Kohn–Sham scheme that we use here, the time- dependent density is obtained by solving the non-linearized Kohn– Sham equations for the sets of Nσoccupied spin ( σ= {↑,↓}) orbitals i̵h∂ ∂tφj,σ(r,t)=hKS,σ(r,t)φj,σ(r,t), (1) and the total number of electrons is N=N↑+N↓. The Kohn–Sham Hamiltonian is hKS,σ(r,t)=−̵h2 2m∇2+vKS,σ(r,t), (2) where the Kohn–Sham potential is defined by vKS,σ(r,t)=vxc,σ(r,t)+vH(r,t)+vext(r,t). (3) Here, vxc,σ(r,t) denotes the spin-dependent exchange-correlation potential, vH(r,t) denotes the Hartree potential, and vext(r,t) denotes the external potential. The time-dependent spin density nσ(r,t)=Nσ ∑ j=1∣φj,σ(r,t)∣2(4)directly follows from the time-dependent spin orbitals, and the total density is n=n↑+n↓. The exact form of vxc,σis unknown, and it must be approx- imated. In this study, the focus is on testing the reliability of simple, computationally efficient functionals. Therefore, we use either the TD-LDA or the TD-ADSIC approximation, as detailed below. We solved Eq. (1) and calculated the spectra as described in Ref. 54. In short, we recorded the TD dipole moment after a boost exci- tation of strength 0.0001 Rydberg and calculated the dipole strength function (DSF), Sθ γ(̵hω)=−2mω 3πe̵h2kIm[δμθ γ(ω)]. (5) The index γand superscript θdenote the Cartesian components of the dipole moment and the boost direction, respectively (i.e., γ andθ∈{x,y,z}).δμ(ω) is the Fourier transform of the induced time-dependent dipole moment, which we obtain from the induced density fluctuation δn(r,t) =n(r,t)−n0(r). Here, n0(r) is the ground-state density, and Imin Eq. (5) denotes taking the imagi- nary part. Summing over all three Cartesian components obtained in three separate calculations, each with a boost in one of the Cartesian directions, provides the overall DSF S(̵hω)=Sx x+Sy y+Sz z (6) that has peaks at the absorption energies of the molecules. As described in Ref. 54, one can avoid doing three separate calcula- tions for the three boost directions and thus reduce the compu- tational effort by applying the boost in one direction that is non- orthogonal to any of the transition dipoles and evaluate the response signal accordingly. The spectra shown below were calculated in this way. For a finite propagation time, the spectral lines have the shape of sine cardinals. As described in Ref. 54, excitation energies and oscillator strengths can therefore be obtained accurately by fitting the numerically obtained S(̵hω) with a sine cardinal superposition. In our experience, good starting values for the fit can be obtained with the Padé approach of Bruner et al. ,47yet the final excita- tion energies and oscillator strengths are obtained more accurately with the sine cardinal fit. The following spectra show, unless noted otherwise, the data obtained from the fit. To guide the eye, we broadened the peaks with Gaussian functions e[−̵h(ω−ω0)/η]2with η= 0.025 eV. For visualizing excitations, we calculated their tran- sition densities, which we obtained from the imaginary part of the Fourier transformed induced density fluctuation.54 III. CHROMOPHORES IN THE LH2 COMPLEX In our study, we focus on bacteriochlorophyll (BChl) chro- mophores from the B850 ring7of the LH2 complex of purple bac- teria Rhodoblastus (Rbl.)acidophilus [formerly Rhodopseudomonas (Rps.)acidophila ], strain 10050, a frequently studied paradigm antenna complex. It is schematically depicted in Fig. 1. This complex is often described by the crystal structure of Papiz et al.55(1nkz), but when we use crystal structure coordinates in this work, we rely on the more recently determined one with higher resolution.23,56The B850 BChls have a phytyl chain that is attached to the molecule via an ester group. The influence of this lipid on the photo-absorption J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . LH2 complex highlighting the position of the B850 BChl molecule that is studied in Sec. V in yellow and schematically showing the protein environment that contributes to the electrostatic potential. spectrum of the pigment is negligible. In order to keep the compu- tational effort low, we truncated it and replaced it by a hydrogen atom. The histidine amino acid that coordinates the Mg atom of the BChl molecule was explicitly considered in the TDDFT calcu- lations in Secs. V and VI as earlier work has shown that it influences excitation energies noticeably.23We separated it from the protein backbone between C αand Cβand saturated it with another hydrogen atom. Figure 2 depicts the resulting structure. FIG. 2 . The examined BChl (spheres) and histidine (rods). FIG. 3 . (a) Atomistic structure of the LH2 complex and the analyzed BChl molecule (yellow) (b) The electrostatic potential generated by the LH2 complex. The iso- surfaces are at −4kbT e(red) and 4kbT e(blue) for a temperature of T= 298 K. The B850 chromophores are embedded in an environment consisting of water, ions, amino acid, lipid chains, and other chro- mophores that are present in the LH2 complex. In Sec. V, we take this environment into account in the form of the electrostatic poten- tial that it generates. We calculated the electrostatic potential as described in Ref. 23. Figure 3(a) displays the atomistic structure of the LH2 complex with one of the studied B850 BChl molecules highlighted in yellow, whereas Fig. 3(b) depicts the corresponding electrostatic potential. IV. TESTING THE TIME-DEPENDENT ADSIC FUNCTIONAL TDDFT using adiabatic semilocal functionals is well known to seriously overestimate static CT57–59and underestimate CT excitation energies.25–27Exchange-correlation approximations that explicitly depend on the orbitals can resolve this problem. This has been explicitly demonstrated, e.g., for range-separated hybrid functionals28,60–65that incorporate Fock exchange and orbital- specific self-interaction correction schemes that employ the opti- mized effective potential.66–70SIC schemes offer the advantage that only NCoulomb integrals need to be evaluated, which is a substantial reduction of numerical effort compared to the ∝N2integrals that Fock exchange requires. The technical chal- lenges71,72of using the optimized effective potential scheme in TDDFT, however, so far prevent its wide spread use in practical applications. ADSIC is a substantial simplification of the full optimized effective potential procedure that also leads to a global multiplica- tive potential and reduces the computational burden to the evalu- ation of just one additional Coulomb integral.51It replaces the full Perdew–Zunger73energy expression ESIC xc=ELDA xc[n↑,n↓]−∑ σ=↑,↓Nσ ∑ j=1(UH[nj,σ]+ELDA xc[nj,σ, 0]), (7) where nj,σ(r)=∣φj,σ(r)∣2, by the averaged expression J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp EADSIC xc=ELDA xc[n]−(N U H[n N]+N↑ELDA xc[n↑ N↑, 0] +N↓ELDA xc[0,n↓ N↓]). (8) For this expression, the functional derivative δEADSIC xc[n]/δnσ(r)can be calculated straight forwardly, vADISC xc,σ=vLDA xc[n]−(vH[n N]+vLDA xc[nσ Nσ, 0]), (9) and the potential is then used in the adiabatic approximation in the TDDFT equations.74 This is the simplest SIC scheme that we know of. Its compu- tational demands are much lower than those of full SIC schemes or of functionals that employ Fock exchange. For these reasons, it has frequently been employed with success, e.g., in studies of ionization processes,74–77but there has also been a report point- ing out caveats.78Recently, ADSIC has been used in the investi- gation of light-harvesting systems where computational efficiency is mandatory due to the size of the studied system.21This moti- vates us to investigate whether the ADSIC scheme can be generally recommended for calculating excitations of natural light-harvesting chromophores. To this end, we calculated the electronic structure and the exci- tations for the BChl molecule B302, using the geometry of Ref. 23. Note that for the sake of transparency, we did not include the envi- ronment or histidine in this calculation, i.e., here, the bare BChl molecule was in vacuum. Numerical details of the calculations are given in the Appendix. Table I shows the frontier orbital eigenval- ues that we find with the LDA and the ADSIC functionals and their energetic differences. As a reference, we also list the correspond- ing energies for the optimally tuned ωPBE functional, which has been shown to describe the electronic structure of BChl reliably and which solves the CT problem.23Two important observations can be made in the table. First, ADSIC lowers the eigenvalues and, thus, brings the HOMO in rather good agreement with the first ioniza- tion potential that one would obtain from the much more advanced optimally tuned ωPBE functional. Second, however, one notices that the energy differences between the eigenvalues in ADISC are very similar to the ones in LDA. Thus, the considerably larger HOMO– LUMO gap of ωPBE is not found with ADSIC. We also found that the spatial density distribution for the HOMO and the LUMO are very similar in LDA and ADSIC. TABLE I . Frontier orbital energies ϵiand their energetic separation for the ADSIC-, LDA-, andωPBE functionals for BChl 302 (in eV). ϵi ADSIC ΔϵADSIC i−1,i LDA ΔϵLDA i−1,iωPBE ΔϵωPBE i−1,i ϵHOMO−1−6.402 ...−5.296 ...−6.549 ... ϵHOMO −5.902 0.500 −4.790 0.506 −5.850 0.700 ϵLUMO −4.716 1.187 −3.618 1.172 −1.593 4.257 ϵLUMO+1 −3.494 1.221 −2.353 1.264 0.073 1.666As the more relevant and stringent test in our present con- text, we then calculated the excitation spectrum with the ADSIC functional. Figure 4 shows the excitation spectrum obtained with TD-LDA as a reference in the upper panel (a). One can clearly identify the Q y- and Q x-excitations with relatively high oscillator strength and between them a spurious CT excitation with a lower oscillator strength, as already discussed in Ref. 23. The ωPBE spec- trum (not plotted) shows just two lines, the Q yone at 1.83 eV and the Q xone at 2.29 eV. The result from the TD-ADSIC cal- culation, shown in the lower panel (b), is sobering. Very clearly, ADSIC does not solve the CT problem. In fact, there are even two spurious CT excitations (labeled CT s) between the Q y- and Qx-peaks. Given that other SIC schemes have been reported to substan- tially improve on the CT problem,66one may wonder why ADSIC fails to do so. A first observation is that due to the explicit depen- dence on the particle number, cf. Eq. (8), the ADSIC scheme is not size consistent. A second observation is that the problems of ADSIC to describe charge transfer correctly can be understood from the perspective of the theory of the optimized effective potential.79 Exact exchange and optimized effective potential SIC develop a term in the Kohn–Sham exchange-correlation response that coun- teracts the external potential that is driving the charge transfer. This field-counteracting term is decisive for the correct descrip- tion of charge transfer.57,59However, the field-counteracting term arises from orbital-dependent terms in the full optimized effective FIG. 4 . TD-LDA (a) and TD-ADSIC (b) spectrum of BChl 302 obtained as described in Sec. II. In addition to the Q y- and Q x-excitations that are qualitatively correctly represented by both functionals, TD-LDA leads to one spurious CT excitation (labeled CT s), and TD-ADSIC even leads to two spurious CT excitations. J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp potential, which are deliberately neglected in the density-averaging procedure51that gives rise to the simple form of the ADSIC functional. Thus, the construction principle of the ADSIC potential inevitably leads to the loss of the field-counteracting term. In order to demonstrate this explicitly, we have evaluated the ADSIC response potential for the model system of a hydrogen chain. This model has been suggested in Ref. 57 for revealing field- counteracting properties. The idea of the model is to not look at the frequency dependent polarizability (photoabsorption) but at the static polarizability to make the situation more transparent. An exchange-correlation approximation that underestimates CT exci- tation energies effectively makes charge transfer too easy. Corre- spondingly, charge transfer will also be too large in the static limit, i.e., the static electric polarization will be overestimated. The hydro- gen chain model allows us to directly visualize the reasons for this overestimation. The model consists of a linear chain of hydrogen atoms (eight in our case) with alternating distances of 2 bohrs and 3 bohrs. In Fig. 5, the atomic positions are indicated by black circles along thez-axis. When a linear external electric potential (indicated by the dashed line in Fig. 5), corresponding to a homogeneous elec- trical field, is applied along the chain, the chain gets polarized, i.e., the electrons move in the direction of the applied field. The exact exchange potential counteracts this polarization. This is seen from the green line in Fig. 5, which shows the exchange response, i.e., the difference between the exact Kohn–Sham exchange poten- tial for the system with and without the polarizing external field.57 The green line shows oscillations but on average has a slope oppo- site to the polarizing field. Thus, it reduces charge transfer. This is one way to understand why functionals that incorporate exact exchange describe charge transfer more realistically, and hydrogen chains have become a well established test system.59,80–86The violet curve in Fig. 5 shows the exchange-correlation response from the ADSIC functional. Clearly, it does not show a field-counteracting behavior but has an average slope in the direction of the polar- izing field. Thus, ADSIC in this respect differs disadvantageously from more elaborate SIC schemes69and does not improve on the CT problem. FIG. 5 . Static response potential from the exact exchange functional (green line) and the ADSIC functional (violet line) for a chain of eight hydrogen atoms in a polarizing electric field (dashed line). See the main text for discussion.V. COMBINING THE PROTEIN ELECTROSTATIC POTENTIAL WITH CHROMOPHORE GEOMETRIES FROM MOLECULAR DYNAMICS Although this first test showed that ADSIC is not an alternative to full SIC schemes or range-separated hybrid functionals, there is still hope that computationally inexpensive functionals can be useful for computing excitations of BChl chromophores under certain cir- cumstances. This hope is motivated by the recent finding that while TD-LDA shows spurious CT excitations for the bare BChl chro- mophores of the LH2 complex, cf. Fig. 4(a), the situation changes when the electrostatic potential generated by the protein environ- ment is taken into account: For the molecular geometry that cor- responds to the experimental crystal structure,56the spurious CT transitions that lie close to the Q y- and Q x-excitations are shifted to substantially higher energies. This is potentially a very relevant find- ing because it can indicate that the energy transfer in the LH2 com- plex can be studied with TD-LDA. TD-LDA in combination with real-time and real-space techniques20,35,37,38,54,87–89reaches a compu- tational efficiency that will allow one to study the whole ensemble of B850 chromophores in the LH2 complex with its many thousands of electrons. Before one can proceed in this direction, one must, however, conduct an important further check. The results of Ref. 23 were obtained for the crystal structure geometry of Ref. 56. Yet, for a realistic description of phenomena like excitation energy transfer, it will ultimately be important to include nuclear geometries other than the crystal structure ones into the calculations, e.g., in order to take into account nuclear dynamics in a TDDFT-based Ehrenfest- like scheme. Therefore, it is important to check whether the protein electrostatic potential shifts the spurious CT excitations of TD-LDA not only for the crystal structure geometries but also for other atomic configurations. As a first test, we picked several coordinates from the MD simulations of Ref. 53. In their study, Mallus et al. analyzed sim- ilarities and differences between the excitonic systems of the LH2 complex of Rbl. acidophilus and other light-harvesting complexes, with a special focus on the energy gaps. Due to the size of the stud- ied system, they calculated the excitations with the semi-empirical Zerner’s intermediate neglect of differential overlap with parame- ters for the spectroscopic property technique in combination with configuration interaction singles (ZINDO/S–CIS). For the purposes of their work, this is a well-established approach, and they obtained valuable insights. We here use BChl coordinates from their work for a different purpose: By calculating the excitations with TD-LDA for BChls in the MD geometries, we can check whether the effect that the protein electrostatic potential shifts the spurious CT excitations is just a lucky coincidence for the crystal structure geometry or a more general finding. To this end, we picked several geometries for the analogs of BChl B302 from the MD data. We selected geometries that differ substantially in their mean square displacement (MSD) for a subunit of six BChls with their coordinating histidines. Within the avail- able set of MD structures, we aim to investigate the ones that differ most from the geometries of the BChls that were used in the pre- vious study.23Thus, we chose the structures with the smallest, a medium, and the highest MSD, respectively. The structural differ- ences are most pronounced in the side chains of the molecules and J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the histidines. For details about the MD simulations, see Ref. 53. Figure 6 shows the TD-LDA spectra, again calculated and plotted as described above, for three of the MD geometries in vacuum. These calculations show typical issues that one can observe in the TD-LDA calculations. The following discussion focuses on the spectral region from 1.5 eV to 2.2 eV. For the first geometry [Fig. 6(a)], the TD-LDA spectrum is similar to what we discussed above for the TD-ADSIC functional. Two spurious CT excitations fall in the energy range between the Qy- and the Q x-excitations. We assigned the labels CT s, Q y, and Qxbased on an analysis of the transition densities, but we do not show the latter in this article for the sake of brevity. A care- ful analysis of the dipole signal indicates that there is another FIG. 6 . TD-LDA excitations in the energy range between 1.5 eV and 2.2 eV of a BChl molecule (with coordinating histidine) in vacuum in three different geometries (a), (b), and (c) as discussed in the main text. The molecular geometries were taken from the MD simulations of Ref. 53 for the LH2 complex of Rbl. acidophilus . See the main text for discussion.weak excitation slightly below the Q ypeak. We did not analyze the character of this excitation in detail, but the peak is visi- ble in Fig. 6 (a), and this spurious excitation is marked with the label S. For the other geometries whose spectra are discussed in panel (b) and (c), we find even more involved situations. Panels (b) and (c) are presented in the same manner as panel (a), i.e., excita- tions whose character we could clearly assign based on the tran- sition density are labeled with, e.g., Q x, while other excitations are just designated by S. Panel (b) shows that spurious excita- tions can also occur at noticeably lower energy than the Q yexci- tation and can be close in energy to it with appreciable oscilla- tor strength. In such a situation, one cannot distinguish between what are spurious CT excitations and what is the Q yexcitation because all excitations couple and have a mixed character. There- fore, all excitations must be considered “spurious” in the sense that the resulting transition densities do not reflect density distri- butions that would be found in reality. If a real-time LDA sim- ulation, e.g., in the spirit of Ref. 90, would be done for the LH2 complex in this type of situation, one would have to expect that the results can be qualitatively wrong, because the electronic cou- pling between different excitations would not be correct. Figure 6(c) finally shows that a similar situation can also occur when the spu- rious CT excitations mix with the Q xexcitation. In (c), the Q y excitation is described qualitatively correct, but the spurious exci- tations mix with the Q xtransition so that three spurious excita- tions are found in the plotted energy range above the Q ytransi- tion. Furthermore, we found a weak spurious excitation below the Qypeak. These findings are in line with previous observations that chromophore excitations can depend sensitively on the molecular geometries.91 As the next step, we calculated the electrostatic poten- tial that corresponds to the protein environment. In doing so, we followed the procedure described in Ref. 23. In order to generate the potential, the atoms of the MD simulations were assigned with standard partial charges of the CHARMM force field.92The partial charges of the ligand molecules were taken from the former study.23The electrostatic potential was calcu- lated with the program APBS93by solving the linearized Poisson– Boltzmann equation. The potential that we finally used consisted of two parts, the BChl environment and the reaction field arising from the polarization of the in the (TD)DFT calculation treated molecules. The electrostatic potential was then added as an external potential in the Kohn–Sham calculation and the excitation ener- gies calculated again with TD-LDA in real time. Figure 7 shows the resulting spectra for the same three geometries that were used in Fig. 6. By comparing, e.g., Fig. 6(a) to Fig. 7(a), one can directly see how the environment’s electrostatic potential shifts the excitation energies. One sees that the Q xand Q yexcitations are hardly affected and that one of the spurious CT excitations is shifted out of the relevant energy range. However, others remain between Q xand Q y. Corresponding observations are made when comparing panels (b) and (c): The environment potential shifts some of the spurious excitations, but at least one is remain- ing in the energetically relevant range in each case. This find- ing is, thus, distinctly different from the one in Ref. 23 for the crystal structure. J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 7 . TD-LDA excitations in the energy range between 1.5 eV and 2.2 eV of a BChl molecule (with coordinating histidine) in the three different geometries (a), (b), and (c) used in Fig. 6, now including the electrostatic potential generated by the protein environment as part of the Kohn–Sham potential. See the main text for discussion. VI. COMBINING THE PROTEIN ELECTROSTATIC POTENTIAL WITH OPTIMIZED CHROMOPHORE GEOMETRIES: TD-LDA VS TD-ADSIC The observations made in Sec. V leave one with the follow- ing question: Is it a special feature of the crystal structure that all spurious excitations are shifted by the electrostatic potential, or is there something special about the MD geometries that pre- vents the spurious excitations from being shifted? In order to answer these questions, we studied a third set of geometries. Start- ing from the BChl 302 geometry that was used in Ref. 23, we optimized the BChl geometry in a Quantum Mechanics/Molecular Mechanics (QM/MM) approach. The BChl and the coordinating histidine were designated as the QM region, and DFT was usedfor this part. Since geometries of organic systems are typically well described by hybrid functionals, the B3LYP functional was used here. In the pre-optimization, a def2-SVP basis set was used,94while in the production run, a def2-TZVP basis94was used. The pro- tein environment and all ligands not located in the QM region were designated as the classical region, which was described by the CHARMM force field.92During the QM/MM optimization, the BChls that are part of the classical region were kept fixed. All QM/MM calculations were done with pDynamo95in combination with ORCA.96 We then calculated the electronic excitations for this geometry with the TD-LDA and the TD-ADSIC functionals, once in vacuum and once with the environment potential. As a reference, we also calculated the spectrum that one obtains with the optimally tuned ωPBE functional for the same geometry (see Ref. 23 for details on the tuning). The optimally tuned ωPBE functional serves as a refer- ence here as this functional is known to resolve the CT problem. The energy range in the following plots was chosen from 1.6 eV to 2.3 eV as in the previous TD-ADSIC calculations of Fig. 4 to ease the com- parison. Figure 8(a) shows the TD-LDA spectrum. The comparison to Fig. 9 reveals that the Q yexcitation energy found with the TD- LDA is close to the one found with ωPBE, the Q xexcitation energy with the TD-LDA is somewhat red-shifted, and there are two spu- rious TD-LDA excitations above Q x. Thus, the overall structure of the spectrum with Q xand Q yas the dominant features is correct when the optimized geometry is used, but the spurious excitations close to the Q xare unphysical. Looking at Fig. 8(b) shows that the TD-ADSIC calculation is much stronger affected by the spurious FIG. 8 . TD-LDA spectrum (a) and TD-ADSIC spectrum (b) (from 1.6 eV to 2.3 eV) for a BChl with the coordinating histidine in the QM/MM optimized geometry. The spectra were computed in vacuum. J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 .ωPBE spectrum (from 1.6 eV to 2.3 eV) for a BChl with the coordinat- ing histidine in the QM/MM optimized geometry. The spectrum was computed in vacuum. excitations. It does not show the characteristic features of the Q x and Q yexcitations due to strong mixing with spurious excitations, i.e., the spectrum is qualitatively wrong. When taking into account the environment potential, cf. Fig. 10, the spurious excitations are shifted further up in the TD-LDA spectrum [panel (a)]. Qualitatively, the main spectral features in the stud- ied energy range are thus described correctly (we note in passing that it is well known that taking into account environmental effects can shift excitation energies46). In the TD-ADSIC spectrum [panel (b)], on the other hand, the spurious excitations are still very pro- nounced and close to the main peaks. Based on this result for a single FIG. 10 . TD-LDA spectrum (a) and TD-ADSIC spectrum (b) (from 1.6 eV to 2.3 eV) for a BChl with the coordinating histidine in the QM/MM optimized geometry. The spectra were computed in the electrostatic environment potential.BChl, there is little hope that a TD-ADSIC calculation will be able to describe the coupling between several BChls correctly. In order to explicitly check the inter-chromophore coupling and to test how well inter-chromophore CT physics is captured by the ADSIC approximation, we further investigated a system of two bacteriochlorophylls (the analogs of B302-B303) from the B850 ring of the LH2 complex. We optimized the geometry follow- ing the QM/MM protocol described at the beginning of this sec- tion. Figure 11 shows the spectrum that we obtain for this system with TD-LDA [panel (a)] and TD-ADSIC [panel (b)]. The refer- enceωPBE calculation (not shown for brevity) shows four excita- tions in the relevant energy range. They are grouped as one expects based on a simple coupling argument, i.e., the two Q yexcitations couple [excitation energies and oscillator strength (1.78 eV, 0.726) and (1.84 eV, 0.051), respectively] and the two Q xexcitations cou- ple [excitation energies and oscillator strength (2.12 eV, 0.016) and (2.15 eV, 0.144), respectively]. Both the TD-LDA spectrum and the TD-ADSIC spectrum seen in Fig. 11 deviate qualitatively from this result as both show more than four excitations in this energy range. From previous analysis,23we know that the additional spurious excitations can typically be described as combinations of spurious CT excitations and Q-band excitations of a single BChl, e.g., the Qyat B302 coupled to a CT excitation at B303, as well as inter- chromophore charge transfer. Again, we observe that the prob- lem is more severe with the TD-ADSIC approximation than with the TD-LDA. FIG. 11 . TD-LDA spectrum (a) and TD-ADSIC spectrum (b) (from 1.6 eV to 2.3 eV) for a pair of BChl molecules corresponding to the B302-B303 chro- mophore arrangement (with their coordinating histidines) in the B850-ring of the LH2 complex. The spectra were computed in vacuum. J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp When we calculate the spectrum for the coupled chromophores in the electrostatic potential generated by the LH2 complex envi- ronment, the situation improves considerably, as seen in Fig. 12. The TD-LDA spectrum in Fig. 12(a) qualitatively shows the correct physics with a total of four excitations that correspond to the cou- pled Q yand Q xtransitions. In the TD-ADSIC spectrum (b), one can also see this coupling, and on first sight, the spectrum looks better than what one may have expected in view of Fig. 10(b). How- ever, the reason for the improvement to some extent is just due to the fact that the geometry of the BChls in the B302-B303 sys- tem is different from the single B302 geometry as now the com- bined system was optimized. Furthermore, closer inspection reveals that there is still an issue with the TD-ADSIC spectrum as there are two spurious excitations at 2.05 eV and 2.09 eV, respectively. They carry only little oscillator strength and thus are visible in the plot only as a small broad peak. However, in a calculation with more than two BChls, e.g., if one were to build up the complete B850 ring in a simulation, these spurious states could couple to other transitions and thus can lead to qualitatively wrong predic- tions. Therefore, we conclude that in the context of BChl exci- tations and coupling, TD-LDA yields a more reliable description than TD-ADSIC. Finally, we note that all the results of this section show that the DFT optimized geometries lead to results that are similar to the results that were obtained earlier23for the crystal structure, whereas FIG. 12 . TD-LDA spectrum (a) and TD-ADSIC spectrum (b) (from 1.6 eV to 2.3 eV) for a pair of BChl molecules corresponding to the B302-B303 chromophore arrangement (with their coordinating histidines) in the B850-ring of the LH2 complex. The spectra were computed in the electrostatic environment potential.the results of Sec. V that were based on the MD geometries dif- fer. In order to understand these differences,97we analyzed the MD geometries of the BChls with the Cemer–Pople puckering ampli- tude method.98This method shows up differences in the planarity and the distortion of ring systems for molecules such as the BChls. This analysis revealed that all of the MD geometries show a much more pronounced curvature than the crystal structure and also than the QM/MM optimized geometries. We therefore conclude that this is most likely an unrealistic feature of the classical MD geometries generated by the classical MD force field. VII. CONCLUSION We have investigated two different approaches for studying electron excitation dynamics in bacteriochlorophyll chromophores from natural light-harvesting systems. Our focus was on explor- ing methods that have the potential to be so computationally effi- cient that they might be used to study supramolecular structures as large as the LH2 complex in photosynthetic bacteria, i.e., with many thousands of electrons, in real-time DFT. We checked whether the TD-ADSIC approximation is promising for this purpose. Our results showed that TD-ADSIC does not resolve the problem of low- lying spurious CT excitations that occurs with local and semilocal exchange-correlation approximations, and we explained why that is. Furthermore, TD-ADSIC in the cases that we studied here leads to a less realistic description of chromophore coupling than TD- LDA. We further showed that for the chromophores of the LH2 complex that we studied, the electrostatic potential of the (protein) environment shifts the spurious CT excitations out of the stud- ied energetic range when DFT (QM/MM) optimized chromophore geometries are used. The same effect was previously observed for geometries obtained from x-ray crystallography. These findings raise hope that excitation dynamics in chromophore assemblies such as the B850 and B800 rings of the LH2 structure can be simulated meaningfully in real-time, real-space TD-LDA calculations when the environment potential is explicitly included in the calculations. The advantages of real-time and real-space techniques, such as their good intrinsic parallelizability and the unbiased representation of func- tions on the grid, may thus be brought to bear on light-harvesting questions. AUTHORS’ CONTRIBUTIONS J.K. and R.R. contributed equally to this work. ACKNOWLEDGMENTS The authors are grateful to Maria I. Mallus and Ulrich Kleinekathöfer for making available to us data from their MD sim- ulations for the LH2 complex of Rbl. acidophilus . R.R., J.M.F., I.S., and S.K. acknowledge financial support by the program “Biologi- cal Physics” of the Elite Network of Bavaria. The authors further acknowledge support through the computational resources provided by the Bavarian Polymer Institute and by the Bavarian State Min- istry of Science, Research, and the Arts in the Collaborative Research Network “Solar Technologies go Hybrid.” J. Chem. Phys. 153, 144114 (2020); doi: 10.1063/5.0014938 153, 144114-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp APPENDIX: NUMERICAL DETAILS We performed the calculations with the program suite BTDFT54on a real-space grid with a grid spacing of Δx= 0.18 Å ≈0.34 a 0. The grid was delimited by an ellipsoid with half-axes of Rx=Ry=Rz= 22 a 0(single BChl) or Rx=Ry= 34 a 0andRz= 27 a 0 (two-BCL supermolecule), respectively. The ion potentials and core electrons were represented by norm-conserving Troullier–Martins pseudo potentials99with the following cut-off radii: C 1.09 a 0(sand p), H 1.39 a 0(s), O 1.10 a 0(sandp), N 0.99 a 0(sandp), Mg 2.18 a0(s) and 2.56 a 0(pandd), and Na 3.09 a 0(s,p, and d). We chose the s-component as the local component in the Kleinman–Bylander transformation for all atoms but oxygen, for which we chose the p- component. The environment potentials were constructed following the protocol described in detail in Ref. 23. For the real-time propagation, we used a time step of Δt = 0.02 fs and propagated for 100 fs. At the beginning of the real- time propagation, we excited the system using a boost excitation54 with a strength of Eboost = 0.0001 Ry ≈0.001 36 eV in the (1, 0, 1)- direction. For the ADSIC calculations in Sec. 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J. Appl. Phys. 128, 140902 (2020); https://doi.org/10.1063/5.0023656 128, 140902 © 2020 Author(s).Design strategy for p-type transparent conducting oxides Cite as: J. Appl. Phys. 128, 140902 (2020); https://doi.org/10.1063/5.0023656 Submitted: 31 July 2020 . Accepted: 26 September 2020 . Published Online: 12 October 2020 L. Hu , R. H. Wei , X. W. Tang , W. J. Lu , X. B. Zhu , and Y. P. Sun ARTICLES YOU MAY BE INTERESTED IN Oxygen vacancies: The (in)visible friend of oxide electronics Applied Physics Letters 116, 120505 (2020); https://doi.org/10.1063/1.5143309 Defects in Semiconductors Journal of Applied Physics 127, 190401 (2020); https://doi.org/10.1063/5.0012677 Beyond Graphene: Low-Symmetry and Anisotropic 2D Materials Journal of Applied Physics 128, 140401 (2020); https://doi.org/10.1063/5.0030751Design strategy for p-type transparent conducting oxides Cite as: J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 View Online Export Citation CrossMar k Submitted: 31 July 2020 · Accepted: 26 September 2020 · Published Online: 12 October 2020 L. Hu,1 R. H. Wei,1 X. W. Tang,1W. J. Lu,1X. B. Zhu,1,a) and Y. P. Sun1,2,3 ,a) AFFILIATIONS 1Key Laboratory of Materials Physics, Institute of Solid State Physics, HFIPS, Chinese Academy of Sciences, Hefei 230031, China 2High Magnetic Field Laboratory, HFIPS, Chinese Academy of Sciences, Hefei 230031, China 3Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China a)Authors to whom correspondence should be addressed: xbzhu@issp.ac.cn and ypsun@issp.ac.cn ABSTRACT Transparent conducting oxides (TCOs), combining the mutually exclusive functionalities of high electrical conductivity and high optical transparency, lie at the center of a wide range of technological applications. The current design strategy for n-type TCOs, making widebandgap oxides conducting through degenerately doping, obtains successful achievements. However, the performances of p-type TCOs lag far behind the n-type counterparts, primarily owing to the localized nature of the O 2 p-derived valence band (VB). Modulation of the VB to reduce the localization is a key issue to explore p-type TCOs. This Perspective provides a brief overview of recent progress in the field ofdesign strategy for p-type TCOs. First, the introduction to principle physics of TCOs is presented. Second, the design strategy for n-typeTCOs is introduced. Then, the design strategy based on the concept of chemical modulation of the valence band for p-type TCOs isdescribed. Finally, through the introduction of electron correlation in strongly correlated oxides for exploring p-type TCOs, the performance of p-type TCOs can be remarkably improved. The design strategy of electron correlation for p-type TCOs could be regarded as a promising material design approach toward the comparable performance of n-type TCOs. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023656 I. INTRODUCTION Transparent conducting oxides (TCOs) denote a class of oxide materials that simultaneously exhibit high electrical conductivityand optical transparency to visible light. TCOs are critical tonumerous state-of-the-art technological applications, ranging from touchscreen screens, flat panel displays, and photovoltaic cells to light emitting diodes and transparent electronics. 1–4However, the properties of electrical conductivity and optical transparency aremutually exclusive and there is a challenge to combine them. The visible light spectrum is located within a wavelength range of 400 nm to 700 nm. The corresponding photon energy (/C22hω) of visible light is between 3.1 eV and 1.75 eV, indicating that the low-energy ( E L) and high-energy ( EH) levels of the optical transparency window for TCOs should satisfy EL,1:75 eV and EH.3:1 eV. Physically, ELis determined by screened plasma fre- quency ( ωp) due to a fundamental excitation of the free carrier, ωp¼(effiffiffiffiffiffiε0εrp )ffiffiffiffin m*p, with ebeing the elemental charge, ε0being the permittivity of vacuum, εrbeing the relative permittivity of thematerial, nbeing the density of free carriers, and m*being the carrier effective mass.5The material is highly reflective to incident light when /C22hωis smaller than /C22hωp. In contrast, the light can propa- gate through the material, and consequently, the material becomes transparent when /C22hω./C22hωp. The strong optical absorption due to direct interband transition across the optical bandgap (E g) usually determines EHfor optical transparency.5Accordingly, the electrical conductivity ( σ) of a material is defined as σ¼neμ, where μis the carrier mobility and given by μ¼eτ=m*, with τbeing the carrier relaxation time.6It is obvious that a high nto realize high σwould also result in a high /C22hωpto reduce the optical transparency in the visible region. For instance, the values of nin conventional metals (such as Au, Ag, Cu, and Al, etc.) are in the range of 1022–1023cm/C03, which lead to /C22hωpin the ultraviolet region.6As a result, these metals are highly reflective and opaque in the visible region. In contrast, the compounds such as Al 2O3, SiO 2, and ZrO 2 have large bandgaps (E g.3:1 eV) to guarantee high optical trans- parencies in the visible region. However, the large bandgaps stronglyJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-1 Published under license by AIP Publishing.suppress the thermal excitation of electrons from the valence band (VB) to the conduction band (CB) resulting in low electrical con- ductivities as well. The main design strategy to comprise the two incompatible properties of high electrical conductivity and optical transparencyin TCOs is to use wide bandgap oxides as the host materials favor- ing transparency and make them conducting by doping with either electrons (n-type) or holes (p-type). A series of high-performancen-type TCOs has been developed and already used in many opto-electronic devices. 7–10In contrast, the performances of p-type TCOs lag far behind their n-type counterparts, which has strongly impeded many technological applications. The fundamental obsta- cle is to achieve p-type TCOs roots in the intrinsic electronic struc-ture of most metal oxides with a strongly localized oxygen2p-derived VB. In this regard, engineering the VB to reduce the localization behavior lies at the heart of the development of p-type TCOs, and at the same time fulfilling the promise of optical trans- parency and p-type dopability. Following these rules, differentdesign strategies have been proposed toward p-type TCOs. In this Perspective, we present an overview of the design strat- egy for p-type TCOs. This section contains a brief introduction to TCOs and the related principle physics of optical transparency and electrical conductivity. Section IIof this Perspective mainly covers the design strategies for n-type TCOs. We start with Sec. II A that presents the basic ideas of degenerately doping wide bandgap semi- conductors. We then proceed to Sec. II B, which shows that the correlated metals can be developed as n-type TCOs rely on strongelectron –electron interactions. Section IIIis devoted to the design strategy of “chemical modulation of the valence band ”for p-type TCOs. In Sec. IV, special attention is paid to the design strategy of electron correlation engineering for exploring high-performance p-type TCOs. In Sec. V, some promising topics that have not yet fully explored for high performance p-type TCOs are proposed. II. DESIGN STRATEGIES FOR n-TYPE TCOs A. Degenerately doping wide bandgap oxides Figure 1 shows the schematic diagram of the design strategy for n-type TCOs. The wide bandgap oxides ( E g.3:1e V ) s u c h a s In2O3,Z n O ,S n O 2,a n dB a S n O 3have been chosen as the host oxides to guarantee high optical transparency in the visible region. The optical bandgap in In 2O3(/difference3:75 eV) is not equal to the fundamental bandgap of /difference2:9 eV because the optical transi- tion across the fundamental bandgap is forbidden.11Free electron carriers are introduced by degenerately donor doping to achievehigh electrical conductivity. This route successfully develops a series of high performance n-type TCOs such as Sn-doped In 2O3 (ITO), Al-doped ZnO (AZO), F-doped SnO 2(FTO), and La-doped BaSnO 3(LBSO).7–9From the viewpoint of electronic structures, these host oxides typically possess an oxygen2p-derived VB and the metal sorbital-derived CB. The spherical spread vacant sorbitals exhibit large spatial distributions, result- ing in highly dispersive CB with large bandwidths and conse-quently small electron m *.10Furthermore, when these host oxides are doped with appropriate donors, the dopant energy levels locate slightly below the minimum of CB (CBM).7As a result, the electrons can be easily excited into the CBM to act as free carriers,resulting in the increase of σ. Hitherto, ITO represents the most widely used n-type TCOs due to the best balance of electrical conductivity and optical transparency. The high n(σ)o f 1021cm/C03(104Sc m/C01) can be achieved in industry ITO thin films. Meanwhile, the corresponding /C22hωpof/difference1:61 eV safely lies in the near-infrared spectral region, which ensures the realization of a high optical transparency in the visible region.4 It is evident that the performances of TCOs strongly depend on both electrical and optical properties. To quantitatively evaluatethe performance of TCOs, it is convenient to compare the values offigure of merit (FOM) that correlates the sheet resistance R S(where RS¼1/σd,dis the film thickness) and the transmittance in the visible s region ( T). Two types of FOM are often calculated by FOMH¼T10=RSand FOMG¼/C01/(RSlnT), which are proposed by Haacke and Gordon, respectively.12,13The larger values of FOM indicate higher performances of the TCOs. The typical ITO thin films can exhibit a high (low) T(sheet resistance RS) of 85% (10Ω). The values of FOMHand FOMGfor the ITO thin films are /difference1:67/C2104and/difference4/C2106MΩ/C01, respectively. Despite the success of degenerately doping wide bandgap oxides for n-type TCOs, there are still several drawbacks. First, this design strategy critically relies on a delicate balance in a material between chemical stability, bandgap for transparency, and dopability for con-ductivity. The carrier concentrations are governed by the limitationof the dopants solubility and/or the self-compensation effect. As a result, ndoes not exceed 3 /C210 21cm/C03. Second, the donor dopants would also behave as scattering centers of the itinerant electrons and FIG. 1. Schematic diagram of the design strategy for n-type TCOs.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-2 Published under license by AIP Publishing.consequently enhance ionized and neutral impurity scattering, which strongly suppress the carrier mobility.7,14Third, the commercially available n-type TCOs also suffer from the cost issue (scarcity ofindium) and toxicity issue (containing toxic element F). B. Correlated metals as n-type TCOs m *is usually regarded as an intrinsic and passive material parameter that is determined by the periodic potential of the crystaland characterizes the dynamic properties of electrons or holes under driving fields. Therefore, it is often expected the smaller the better facilitating for high carrier mobility. Recently, Zhang et al. have pro- vided an alternative design strategy toward combining high opticaltransparency and high electrical conductivity by virtue of theenhancement of m *due to strong electron –electron interactions to simultaneously optimize /C22hωpandσ.15It is experimentally demon- strated that the correlated metals of SrVO 3and CaVO 3thin films exhibit a carrier concentration of .3/C21022cm/C03,/C22hωpof/difference1:33 eV, and excellent performances comparable to ITO thin films.15 In solid state physics, the electron –electron interaction described by the band theory can often be neglected or representedby mean periodic potential and simplified to one electron approxi-mation. 6However, the band theory fails to incorporate many-body effects emerging from the strong electron –electron interaction ( U).16 The strength of electron correlation due to electron –electron interac- tion can be quantified by Zk¼m* band/m*,w h e r e m* bandandm*are free noninteracting electron effective mass extracted from the bandtheory and renormalized electron effective mass incorporating theelectron correlation, respectively. 17As a rule, Zkis equal to 1 for the noninteracting electronic systems. However, the electron correlation in correlated systems would impede the motion of electrons and sup-press the itinerancy to favor localization, leading to the Z kvalue below unity.17For correlated metals of SrVO 3and CaVO 3,t h e values of Zkare determined to be /difference0:33 and /difference0:31, respectively.15,18 Furthermore, the electron correlation can be tuned by the U=W ratio with Wt h ee l e c t r o n i cb a n d w i d t h .16Decreasing the electron cor- relation, either by increasing Wor decreasing U,w o u l di n c r e a s e Zk and decrease m*. Boileau et al. h a v er e p o r t e dt h a tt h ee l e c t r o n i cc o r r e - lation in the SrVO 3thin films can be used as a supplementary level for tuning the optical properties of plasma frequency through strainengineering. 19Stoner et al. have used Zkas a key material design parameter to optimize the optoelectronic performance of SrMoO 3 (Zk¼0:48) thin films as n-type TCOs.20Furthermore, Paul and Birol have used the first-principle density-functional theory and the dynamical mean-field theory to demonstrate that not strain butdimensionality in vanadate, niobate, and molybdate perovskite oxideswould strongly affect the trends in both Z kand/C22hωp.21 III. DESIGN STRATEGY OF CHEMICAL MODULATION OF THE VALENCE BAND FOR p-TYPE TCOs Besides n-type TCOs, p-type TCOs are also the key elements for numerous technological applications. Optoelectronic applica- tions such as organic and thin film solar cells and transparent elec-tronics would usher in an era of transparent gadgets to alter ourdaily lives if p-type TCOs can exhibit comparable performances with n-type TCOs. 22Otherwise, only unipolar devices based on n-type TCOs can be realized and available as transparent passiveelectronic applications. However, it has been proven to be very dif- ficult to directly transform the n-type TCOs to p-type TCOs via acceptor doping. This discrepancy originates from that the VB ofmost metal oxides is composed of strongly localized O 2 p-derived orbitals (as shown in Fig. 1 ). Specifically, the localized nature of O 2porbitals and a high electronegativity of oxygen lead to a large hole m *and difficulty in introducing shallow acceptors. In this respect, the design strategy for p-type TCOs would critically rely on reducing the localization behavior of VB. In1997, Kawazoe et al. proposed the concept of “chemical modula- tion of the valence band ”(CMVB) by modifying the VB through the hybridization of oxygen 2 porbitals with metal dorsorbital toward p-type TCOs. 23Figure 2 shows the schematic diagram of CMVB. The energy levels of cations with closed shells arerequired to be close to the oxygen 2 plevels. The cations of Cu þ, Agþ,C d2þ,I n3þ,S n4þ,a n dS b5þwith closed shells of d10s0and Inþ,S n2þ,B i3þ,a n dS b3þwith closed shells of d10s2can be used as candidate building blocks for p-type TCOs, avoiding colorationdue to d–dtransitions in an open shell. Under this condition, the metal dorsstates would hybrid with the O 2 pstates to promote the modification of VB. As a result, the VB maximum (VBM) become more dispersive to have a lower effective mass, which is advantageous for hole mobility. Another important requirement related to optical transpar- ency is that E H(EL) should be larger(smaller) than 3.1(1.75) eV. The strong interband absorption (screened plasma frequency) should reside in the ultraviolet (infrared) region in order to notreduce transparency in the visible region. Last but not least, thep-type TCOs should be easily doped p-type. Physically, the p-typeTCOs should have low formation energies of acceptor-like defects either as intrinsic acceptors or as extrinsic dopants soluble in the host oxides, which behaves as shallow acceptors and cannot becompensated by “hole killers ”such as oxygen vacancies. 24–26 FIG. 2. Schematic diagram of CMVB.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-3 Published under license by AIP Publishing.Therefore, exploring p-type TCOs must simultaneously satisfy design criteria regarding electronic structure, optical absorption, and bulk and defect phase thermochemistry. Following the design strategy of CMVB, several p-type TCOs such as Cu (Ag)-based delafossite oxides, oxychalcogenides, and SnOhave been developed. 23,27–36However, the performances of the p-type are significantly lower than the n-type TCOs due to the imbalance between electrical conductivity and optical transparency.37–40For example, p-type TCOs of CuAlO 2thin films exhibit a high T (/difference70%) in the visible region; however, the hole concentrations and resultant electrical conductivities ( /difference1017cm/C03and/difference1Sc m/C01)a r e several orders of magnitude lower than that of n-type TCOs (/difference1021cm/C03and/difference104Sc m/C01) due to the deep level of acceptor defects.23,41,42In contrast, Mg-doped CuCrO 2thin films exhibit a high σof 220 S cm/C01with a low Tof/difference30%.28High-throughput material screening has also identified several promising p-type TCOs.43–46These p-type TCOs candidates either require further experimental confirmation or exhibit very low σ.45So far, there exists a huge performance gap between the p-type and n-type TCOs. IV. DESIGN STRATEGY OF ELECTRON CORRELATION FOR p-TYPE TCOs Another design strategy for p-type TCOs is to utilize the electron correlation for exploring p-type TCOs. As stated above, the presence of electron correlation would result in the enhancement of m*. Furthermore, it may also lead to significant changes in the electronicstructure of materials. As schematically shown in Fig. 3 ,m a n y transition-metal oxides (TMOs) with a partially filled d-electron band were not conductors as predicted by the band theory but indeed insu- lators. 16,47It is well established that the Coulomb interaction between the electrons ( U)i so fg r e a ti m p o r t a n c ef o ru n d e r s t a n d i n gt h ea p p e a r - ance of insulating phases and consequent metal –insulator transitions. As illustrated by the Hubbard model, the electron correlation wouldsplit the dband into an upper Hubbard band (UHB) and a lower Hubbard band (LHB) with separation of U. 16Consequently, these TMOs possess a d-derived CB (UHB) and the VB is composed of the hybridization of oxygen 2 porbitals with the dorbitals (LHB). In this sense, the electron correlation could promote VB modification toform the basis for p-type conduction, which acts in a similar role as the aforementioned CMVB. Hitherto, several TMOs such as NiO, Cr 2O3,L a C r O 3,L a V O 3, Ca3Co4O9,N a xCoO 2,B i 2Sr2Co2Oy,B i 2Sr2CaCu 2Oy,a n dV 2O3havebeen reported as p-type TCOs.48–57Based on the relative magnitudes ofU,W, charge-transfer energy between O 2 panddorbitals ( Δ), and doccupation, these TMOs can be either Mott insulators or p-type correlated metals.16We will discuss these p-type TCOs in Secs. IV A andIV B, emphasizing the importance of electron correlation. A. Mott insulators as starting points for p-type TCOs 1. NiO Sato et al. reported the first p-type TCOs of NiO thin films with ρof/difference0:14Ωcm,nof/difference1:3/C21019cm/C03,a n d Tof/difference40%.48The p-type conductivity arises from the Ni vacancies formed in oxygen-rich atmosphere. NiO is a prime example of the Mott insulator withan optical bandgap of 3.4 –4.0 eV. 58,59The conventional band theory predicts NiO to be metallic but electron correlation drives it to be insulating. In NiO, the Ni eglevel is split into an UHB and a LHB due to electron correlation. The energy level of LHB is close to that ofO2plevel. As a result, it is expected that the LHB composed of e g orbital can hybridize with the O 2 porbital to form the VB. The detailed electronic structure of NiO is a long-standing yet not completely resolved topic in condensed matter physics.58,60–64 Usually, NiO is classified as the charge-transfer insulator ( U.Δ) with the optical gap interpreted as p!dtransition.58,60However, experimental and theoretical studies also reveal that the character of the insulating state in NiO could be described as a mixture of charge- transfer and Mott –Hubbard character, i.e., NiO lies in the intermedi- ate Mott –Hubbard (MH)/charge-transfer (CT) insulator regime.63,64 Hole carriers in NiO can also be introduced by doping NiO with monovalent cations such as Liþand Kþto enhance p-type conductivity.65–70Kuiper et al. used oxygen K-edge x-ray-absorption spectroscopy (XAS) to reveal that the holes com-pensating the Li þimpurity charge in Li xNi1/C0xO are of primarily the O 2 pcharacter.65Pickering et al. used nickel K-edge x-ray absorption fine structure spectroscopy (XAFS) to support a model in which the holes introduced by Liþare of predominantly Ni d character.66Chen and Harding performed theoretical calculations on Li 0:125Ni0:875O using both the HSE06 hybrid functional and the density-functional theory (DFT)+U method to show that the holes localize on the nickel ion (which is thus formally Ni3þ) rather than in the O 2 pband.67Zhang et al. have investigated the effect of Li doping on the electronic, optical, and transport properties of NiOepitaxial thin films. 68It is found that the Li doping significantly increases the p-type conductivity but with low hole mobilities (,0:05 cm2V/C01s/C01). As shown in Fig. 4 , a combination of x-ray photoemission and O K-edge XAS investigations reveals that theFermi level gradually shifts toward VBM and a new hole statedevelops with Li doping. 68Wrobel et al. have utilized XAFS to show that Kþcan be fully incorporated in the NiO matrix and accompanied by a significant distortion around the dopants.69NiO has been used as p-type TCOs in electronic devices such as trans-parent pndiodes, thin film transistors (TFTs), and solar cells. 71–74 2. Cr 2O3 Cr2O3is an antiferromagnetic insulator with an Egof /difference3:4e V .75The strong electron correlation in Cr 2O3plays an essen- tial role in determining the electronic structure. The presence of U FIG. 3. Schematic diagram of VB modification due to electron correlation.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-4 Published under license by AIP Publishing.parameter effectively imposes an energetic penalty on partially occu- pied Cr 3 dstates to increase the energy gap between occupied and unoccupied states. As a result, the Cr 3 dstates are predominant in the CB and the VB is composed of a significant amount of mixingbetween the O 2 pstates and Cr 3 dstates. Figure 5 shows the Cr 3 d and O 2 ppartial electronic density of states (PEDOS) of Cr 2O3.76 The direct bandgap of 2.99 eV extracted from the PBE+U results is underestimated as compared to the experimental bandgap, which istypical for PBE+U calculations. 76,77The Cr 3 dstates at VBM indi- cate the d!dtransitions giving rise to the bandgap of Cr 2O3, which is the typical character of the MH insulator.76,77It is alsonoted that the O 2 pstates mainly contribute to the VB at the same energy level of Cr 3 dstates, leading to hybridization between them as the typical character of the CT insulator. Cr 2O3is usually regarded as a borderline between MH and CT insulators.78,79 In this respect, Cr 2O3can been chosen as the host matrix for p-type TCOs. Arca et al. demonstrated the possibility of Cr 2O3as p-type TCOs by magnesium (Mg) and nitrogen (N) co-doping toimprove its electrical conductivity and retain the transparencyproperties. 80The co-doping in Cr 2O3thin film results in electrical resistivity of 3 Ωcm and transmission up to 65% for a 150 nm thick film. Theoretical analysis indicates that the Mg ion on a Crsite has a formation of 0.67 eV and an ionization level of 0.42 eVunder O-rich conditions. 77The conduction mechanism in the Mg-doped Cr 2O3thin films is small-polaron hopping of holes, which results in low hole mobility of /difference10/C04cm2V/C01s/C01and sig- nificantly suppresses the electrical conductivity.81Arca et al. found that Ni is a very effective p-type dopant in Cr 2O3, capable of reach- ing conductivities of 28 S cm/C01.82The modification of the VB in Cr2O3by Ni-doping and concomitant higher solubility account for the improvement in the electrical properties. Dabaghmanesh et al. utilized the concept of CMVB to explore a new Cr 2O3-based p-type TCO by anion alloying with sulfur.83The increase of VB dispersion can be realized by forming hybridized orbitals betweenO2pstates and chalcogen porbitals (S, Se, and Te), which are more delocalized than O 2 porbitals. Cr 4S2O4has been theoretically predicted as a new p-type TCO host candidate, having an opticalbandgap of 3.08 eV and hole m *of 1.8 m0. As concerned with prac- tical application, Cr 2O3is of great interest for optoelectronic devices such as the hole transporting layer in organic solar cells and the building block for transparent p –n junctions.84–86 FIG. 4. (a) O 1s, (b) Ni 2 p3=2, (c) valence band spectra for Li xNi1/C0xO with dif- ferent x, (d) a detailed comparison for Ni 2p3/2 from NiO and 0.09, (e) OK-edge XAS spectra for Li xNi1/C0xO films, and (f) schematic energy diagram for Li doped NiO. Reprinted with permission from Zhang et al. , J. Mater. Chem. C 5, 2275 (2018). Copyright 2018 Royal Society of Chemistry. FIG. 5. Calculated spin polarized PEDOS for bulk α-Cr2O3. The green and red lines are the Cr 3 dand O 2 pstates, respectively. The top of the valence band is aligned to 0 eV , and the dotted line indicates the calculated position of theFermi Level. Reprinted with permission from Carey et al. , J. Phys. Chem. C 120, 19160 (2016). Copyright 2016 American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-5 Published under license by AIP Publishing.3. Perovskite LaCrO 3and LaVO 3 The low σin the p-type TCOs primarily results from the doping bottleneck in increasing n. The hole carriers are usually introduced by intrinsic defects (cation vacancies or interstitial oxygen) and often suffer from the solubility limit of the acceptors. Perovskite oxides with ABO 3stoichiometry possess a robust struc- ture allowing the easy chemical substitution for carrier doping.87 Recently, Zhang et al. have reported new p-type TCOs based on perovskite LaCrO 3(LCO) with high hole concentrations of 1020–1021cm/C03.88Figure 6(a) shows the XPS VB and O K-edge XAS of the La 1/C0xSrxCrO 3film.89The VB spectra of LCO exhibit three prominent peak features labeled A at /difference6e V , B a t /difference3:2e V , and C at /difference1:5 eV. As shown in Fig. 6(b) , theoretical calculations (PBE+U) indicate that feature A is mostly composed of the O 2 p orbital and feature B is O 2 pnonbonding derived, and feature C is dominated by the occupied Cr 3 dt2gorbital. The broad feature E derived from O K-edge XAS can be denoted as unoccupied Cre g"and t2g#hybridized with O 2 p. As a result, the Cr 3 dt2g" orbitals are occupied and form the VBM, while other Cr eg"and t2g#orbitals are unoccupied and form the CBM. Substitution of Sr for La introduces hole carriers that shift the Fermi level towardthe VB and creates a split-off empty Cr t 2g/O 2 phybridized state just above the Fermi level.88,89First-principles calculations have revealed that the formation energies of Sr, Ca, and Ba divalent substitution for La in LCO behave as shallow acceptors, leading to p-type conductivity.90 The Coulomb repulsion energy Uof the 3 dtransition-metal ions exhibits a systematic decrease from the right to the left side of the 3 dr o wo w i n gt ot h es y s t e m a t i ce x p a n s i o ni nt h es i z eo f wavefunction.16This suggests that there is a straightforward route to optimizing the performance of p-type TCOs based onelectron correlation through the material design. Uin LaVO 3(LVO) is relatively smaller than that in LCO, which might be beneficial for carrier mobility. The LVO is a prototypical MH insulator due to electron correlation in which the CB and VB are,respectively, composed of UHB and LHB (as shown in Fig. 7 ). 91 Huet al. have proposed that the LVO can be the starting point for exploring p-type TCOs.52Substitution of La3þby Sr2þintro- duce a high hole carrier concentration of /difference1021cm/C03at VBM. It is demonstrated that La 2=3Sr1=3VO 3(LSVO) thin films can serve as new p-type TCOs with very high FOM.52The highest FOMH and FOMGare 25.52 and 6776 M Ω/C01, respectively. Although there exists a huge gap between p-type and n-type TCOs, the LSVO thin films have remarkably reduced the performance gap (as shown in Fig. 8 ). As stated above, the electron correlation can be quantified by a renormalization factor Zk, which can be determined by consider- ing the unscreened plasma frequencies from DFT calculation /C22hωDFTand experiment /C22hωEXPbyZk¼m* band/m*=(/C22hωEXP//C22hωDFT)2.15 The values of /C22hωEXPcan be extracted from reflectivity spectra by the observed minimum or optical conductivity.17,19Figure 9 shows the reflection spectra of the LSVO thin film in thevisible-infrared range. The spectra at high and low wavelengthcan be identified as the free carrier reflection and interbandtransition, respectively. The minimum of the reflection spectra for the LSVO thin film locates at λ¼573 nm and /C22hω EXPcan be determined to be /difference2:164 eV. /C22hωDFTis determined to be /difference4:360 eV by first-principles calculations performed within DFT (calculation details in the supplementary material ). A FIG. 6. (a) Valence band XPS and O K-edge XAS spectra for the La1/C0xSrxCrO 3film series; (b) analogous theoretical densities of states based on PBEsol+U (U ¼3 eV). Reprinted with permission from Zhang et al. , Phys. Rev. B91, 155129 (2015). Copyright 2015 American Physical Society. FIG. 7. Combined valence and conduction bands along with a schematic band structure of the strained epitaxial LaVO 3thin film. (Inset) Zoomed view of the band edge portion of combined spectra. Reprinted with permission from Jana et al. , Phys. Rev. B 98, 075124 (2018). Copyright 2018 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-6 Published under license by AIP Publishing.renormalization constant of Zk¼0:2 4 6i sf o u n df o rL S V O ,a much reduced value as compared to conventional metals (Zk¼1), which directly reveal pronounced correlation effects without complete carrier localization.B. p-type metals as high performance p-type TCOs It is well known that most of p-type TCOs suffer from the doping trouble, which accounts for the poor electrical conductivity.Several p-type metals with intrinsic high carrier concentrations such as Ca 3Co4O9,B i 2Sr2Co2Oy,N a xCoO 2,B i 2Sr2CaCu 2Oy, and V2O3have been reported as p-type TCOs, which can easily evade the hole doping conundrum for most p-type TCOs.53–57 Among them, Ca 3Co4O9,B i 2Sr2Co2Oy, and Na xCoO 2belong to p-type layered cobalt oxides that characterize by two- dimensional CoO 2layer with edge-shared CoO 6octahedrons. The corresponding crystal structures are featured by an alternativestacking of the CoO 2layer and rock salt layer (CaO –CoO –CaO, Na, or SrO –BiO–BiO–SrO) along the caxis.92These layer cobalt oxides have attracted much attention because of their unusual elec- trical properties due to strong electron correlation, such as super- conductivity in water intercalated Na 0:35CoO 2and large thermoelectric powers.93–95The CoO 2layer is considered to be the conduction layer, while the rock salt layer is regarded as just acharge reservoir. Therefore, the electronic structure of the layered CoO 2triangular lattice lies at the heart of understanding the layer cobalt oxides. As shown in Fig. 10 , the common feature of the elec- tronic structure for these layer cobalt oxides is that the Fermi levellocates at nearly VBM consisted of the Co t 2gorbitals to support p-type conductivity with high hole concentrations.95,96 NaxCoO 2exhibits a large value of the electronic specific-heat coefficient γof/difference48 mJ/mol K2, which is an order of magnitude larger than that of simple metals indicating a strongly correlatedoxide. 97The angle-resolved photoemission study of Na 0:7CoO 2 indicates the existence of a larger Hubbard Usupporting the strongly correlated nature and reveals a hole-type Fermi surfacewith an extremely narrow quasiparticle band. 98The value of Zkis determined to be 0 :15+0:05 for layer cobalt oxide of [Bi 2Ba2O4] [CoO 2]2corresponding to strong electron correlation.99From the theoretical aspects, the electron correlation is found to be very large (U=t/difference26) and essential to achieve consistence with experimental results. As concerned with the optical properties, ELis much lower than 1.75 eV extracted from the reflectivity spectrum, which results FIG. 8. Graphical representation of electrical resistance and optical transmission for the LSVO thin films (blue squares) and other p-type TCOs (black triangles). The typical n-type TCOs of the ITO thin film are added for comparison. The figure of merit (in units of M/C01) with specific values proposed by Hackle (FOMH, dashed line) and Gordon (FOMG, solid line) is also shown for evaluat- ing the performance of p-type TCOs. Reprinted with permission from Hu et al. , Adv. Electron. Matter. 4, 1700476 (2018). Copyright 2018 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. FIG. 9. Reflection spectra of the LSVO thin film in the visible-infrared range. FIG. 10. Schematic illustration of (a) cluster levels in the CoO 6octahedron, (b) the density of states in the network of the CoO 6octahedrons, and (c) the XPS spectrum expected to be observed for the network of the CoO 6octahedrons. Reprinted with permission from Takeuchi et al. , Phys. Rev. B 69, 125410 (2004). Copyright 2004 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-7 Published under license by AIP Publishing.from the enhancement of m*due to electron correlation through tuning the n=m*ratio.100–103The optical absorptions of the layer cobalt oxides in the visible region can be ascribed to moderateinterband transitions, which cannot strongly reduce the opticaltransparency in thin films. Aksit et al. have reported that the Ca 3Co4O9thin films synthesized through the solution process are shown to be high performance p-type TCOs with RS(T)o f 5.7 kΩ/sq (67%) and FOMGof 151 M Ω/C01.53Wei et al. have reported the Bi 2Sr2Co2Oyt h i nf i l m sa san o v e lp - t y p eT C Ow i t h ah i g hF O MGof 800 M Ω/C01, possessing potential applications for photoelectric devices because of their low electrical resistivity and moderate optical transmittance (as shown in Fig. 11 ).54 Compared with Ca 3Co4O9and Bi 2Sr2Co2Oy,N a xCoO 2exhibits much better electrical conducti on, which is highly desirable for TCOs. Yuan et al. have shown that the Na xCoO 2thin films exhibit a low electrical resistivity of /difference1:2mΩcm and an average transmittance in the visible range of /difference37%, which result in a very high FOMGof 4191 M Ω/C01.55 Bi2Sr2CaCu 2Oy(BSCCO) is p-type high- TCsuperconductor exhibiting a strong electron correlation.104,105The electrical con- ductivity of BSCCO at room temperature is higher than that of the layered cobalt oxides. Furthermore, ELof BSCCO is determined to be/difference0:8 eV by setting ϵ1¼0(,1:75 eV) and the optical absorp- tion in the visible spectrum range is weak.106Wei et al. have reported the BSCCO thin films as the first p-type transparent superconductor with high performance ( σof 200 –1064 S cm/C01,n of.1021cm/C03, and Tof.50%) (as shown in Fig. 12 ).56 V2O3is a prototypical MH system, exhibiting a correlated metal phase at room temperature.107Huet al. have proposed the strategy of electron correlation engineering for exploring high perfor- mance p-type TCOs in strongly correlated oxides.57By decreasingelectron correlation from Cr 2O3(Zk¼0) to V 2O3(Zk¼0:12), the performance of V 2O3thin films as p-type TCOs can be remarkably enhanced (as shown in Fig. 13 ). The V 2O3thin films exhibit high hole concentrations ( .1022cm/C03)a n dal o w ELof/difference0:97 eV, which leads to a good balance between the electrical conductivity and optical transparency.57This result indicates that electron FIG. 11. (a) Photograph of the transparent BSC222 thin films on a paper with background of letters [from left to right are the thin films on STO(100), (110), and (111) substrates, respectively]; (b) –(d) transmittance measurements of all STO substrates and BSC222 films on STO substrates. Reprinted with permis-sion from Wei et al. , Chem. Commun. 50, 9697 (2014). Copyright 2014 Royal Society of Chemistry. FIG. 12. (a) Optical transmittance spectra for the bare STO (001) and BSSCO thin films with different thickness (with the unit of nm as shown in the brackets).(b) Photograph of the transparent BSSCO thin films with single coating fabri- cated by different solution molar concentration. (c) The fitting results of Taucs relation for the BSSCO thin films on LaAlO 3(001) substrates with 1 and 3 coat- ings to give the direct allowed bandgap. (d) T emperature dependent resistivity inthe range of 350 –5 K for the BSSCO thin films. (e) The Seebeck coefficient vs measured temperature behavior for the thin film of 0.1M/1C. (f) Evolution of the room temperature Hall resistance with the magnetic field (B) for the thin film of0.1M/1C. (g) Thickness dependence of the room temperature hole concentrationand hole mobility for the BSSCO thin films. Reprinted with permission from Wei et al. , Appl. Phys. Lett. 112, 251109 (2018). Copyright 2018 AIP Publishing LLC. FIG. 13. Graphical representation of Tandσfor the V 2O3thin films (red semi- solid stars) and other typical p-types TCOs. The numbers in parentheses showthe corresponding values of FOM Hand FOMGin the units of M Ω/C01. The n-type TCO of the ITO thin film is also added for comparison. The blue arrow indicates the enhancement of performance through electron correlation engi- neering. Reprinted with permission from Hu et al. , Phys. Rev. Appl. 12, 044035 (2019). Copyright 2019 American Physical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 140902 (2020); doi: 10.1063/5.0023656 128, 140902-8 Published under license by AIP Publishing.correlation engineering paves an effective way for exploring high performance p-type TCOs. V 2O3h a sb e e nu s e di no p t o e l e c t r o n i c devices such as silicon hybrid solar cells to promote photovoltaicperformances. 108 V. SUMMARY AND OUTLOOK In this Perspective, we provide a brief review on the principle physics of TCOs and the design strategies based on CMVB and electron correlation for p-type TCOs. In general, the essential issuefor obtaining p-type TCOs is to mitigate the localization behaviorof VBM composed of the O 2 porbital. The concept of CMVB, engineering the VBM by alloying the oxygen 2 porbitals with the d orsorbital of metal cations, has spurred the development of a series of p-type TCOs. 100,101 The electron correlation provides an alternative route for VBM modification. Meanwhile, the electron correlation woulddirectly determine the performance of p-type TCOs. On one hand, the increase of m *caused by electron correlation will reduce the electrical conductivity of the material. On the other hand, theincrease of m *is beneficial to the opening of the transparent window by decreasing ELin the visible spectral region. It is intui- tive to infer that the Mott insulator ( Zk¼0) would exhibit poor performance of p-type TCOs because the itinerant carriers are localized owing to the electron correlation. In contrast, the conven-tional conducting oxides ( Z k¼1) fail to provide p-type conductiv- ity and usually are unable to open the transparent window in the visible spectrum region ( EL.1:75 eV). In this respect, the performance of p-type TCOs in strongly correlated oxides can be optimized through electron correlationengineering at appropriate Z k. Therefore, we propose the design strategy of electron correlation for p-type TCOs as a promising materials design approach toward the comparable performance of n-type TCOs. So far, the reported p-type TCOs based on electroncorrelation is very limited. There are plenty of correlated oxidespossibly suitable for developing as p-type TCOs, which remainslargely unexplored. Last but most important, the electron correla- tion ( Z k) could be regarded as a key material design parameter to optimize the optoelectronic performance of p-type TCOs instrongly correlated oxides. It is well known that the electron corre-lation can be tuned by the U=Wratio. In this way, Z kcan be varied by altering the material (changing U), strain, and dimen- sionality (changing W), which provides multiple tuning parame- ters. Further investigation in this field is urgently needed. SUPPLEMENTARY MATERIAL See the supplementary material for the theoretical calculation details of /C22hωDFTfor LSVO. ACKNOWLEDGMENTS This work was supported by Joint Funds of the National Natural Science Foundation of China (NNSFC) and the ChineseAcademy of Sciences Large-Scale Scientific Facility (Grant No. U1532149) and the National Key R&D Program of China (Grant No. 2017YFA0403600).DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1T. Minami, Semicond. Sci. Technol. 20, S35 (2005). 2E. Fortunato, D. Ginley, H. Hosono, and D. C. Paine, MRS Bull. 32, 242 (2007). 3C. G. Granqvist, Sol. Energy Mater. Sol. Cells 91, 1529 (2007). 4K. Ellmer, Nat. Photon. 6, 809 (2012). 5M. Fox, Optical Properties of Solids (Oxford University Press, 2001). 6C. Kittel, Introduction to Solid State Physics , 8th ed. (Wiley, Hoboken, NJ, 2005). 7P. P. 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5.0022270.pdf

Appl. Phys. Lett. 117, 132402 (2020); https://doi.org/10.1063/5.0022270 117, 132402 © 2020 Author(s).Spin-reorientation transition induced magnetic skyrmion in Nd2Fe14B magnet Cite as: Appl. Phys. Lett. 117, 132402 (2020); https://doi.org/10.1063/5.0022270 Submitted: 21 July 2020 . Accepted: 13 August 2020 . Published Online: 28 September 2020 Y. Xiao , F. J. Morvan , A. N. He , M. K. Wang , H. B. Luo , R. B. Jiao , W. X. Xia , G. P. Zhao , and J. P. Liu COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Synthetic chiral magnets promoted by the Dzyaloshinskii–Moriya interaction Applied Physics Letters 117, 130503 (2020); https://doi.org/10.1063/5.0021184 Nutation wave as a platform for ultrafast spin dynamics in ferromagnets Applied Physics Letters 117, 132403 (2020); https://doi.org/10.1063/5.0013062 Bending strain tailored exchange bias in epitaxial NiMn/ γ′-Fe4N bilayers Applied Physics Letters 117, 132401 (2020); https://doi.org/10.1063/5.0018261Spin-reorientation transition induced magnetic skyrmion in Nd 2Fe14B magnet Cite as: Appl. Phys. Lett. 117, 132402 (2020); doi: 10.1063/5.0022270 Submitted: 21 July 2020 .Accepted: 13 August 2020 . Published Online: 28 September 2020 Y.Xiao,1,2F. J.Morvan,2A. N. He,2M. K. Wang,2H. B. Luo,2R. B. Jiao,2W. X. Xia,2,a) G. P. Zhao,1,a) and J. P. Liu3 AFFILIATIONS 1College of Physics and Electronic Engineering and Institute of Solid State Physics, Sichuan Normal University, Chengdu 610066, China 2Key Laboratory of Magnetic Materials and Devices, Ningbo Institute of Material Technology and Engineering, CAS, Ningbo 315201, China 3Department of Physics, University of Texas at Arlington, Arlington, Texas 76019, USA a)Authors to whom correspondence should be addressed: xiawxing@nimte.ac.cn andzhaogp@uestc.edu.cn ABSTRACT The easy axis of Nd 2Fe14B is known to deviate from the c-axis when the temperature decreases to under the spin reorientation point TSR, 135 K. In this work, magnetic domain evolution in Nd 2Fe14B was in situ observed by using Lorentz transmission electron microscopy at variable temperatures and magnetic fields. It appears that most inverse domains shrink to stripes and disappear suddenly to achieve thesaturation state under a magnetic field, and the saturation field increases with the decreasing temperature due to the increased anisotropy. Magnetic bubbles with zero topological number are formed at temperatures higher than T SR, whereas magnetic skyrmions are found at temperatures around TSRdue to the spin reorientation. The tunable anisotropy and saturation magnetization at TSRare the main causes of forming magnetic skyrmions. This finding exhibits the feasibility of generating skyrmions in the ordinary rare-earth permanent magneticmaterials. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022270 A magnetic skyrmion as a topologically protected magnetic struc- ture has been intensively investigated in recent years due to its richphysics in spintronics and potential applications in the fields of infor- mation recording and logical devices. 1–5So far, Bloch-type magnetic skyrmions have been found to exist in non-centrosymmetric magnetswith the bulk Dzyaloshinskii–Moriya Interaction (DMI), such as B 20- type structures MnSi, FeGe, Cu 2OSeO 3;2,3,6–8N/C19eel-type skyrmions have been found in metal/ferromagnetic bi- and multilayers due to a large interfacial DMI resulting from strong interfacial spin–orbit cou- pling, such as Pt/Co/MgO and Pt/Co.9,10In addition to the above two kinds of skyrmions with definite chirality, magnetic skyrmions or sky- rmionic bubbles were also found in perpendicular magnetic films without the DMI, in which the energy competition among perpendic-ular anisotropy, magnetic dipole–dipole interaction, and Zeeman energy stabilizes the skyrmion. 11–17In these films, a rich variety of topological magnetization textures such as biskyrmions in La2–2xSr1þ2xMn 2O711and MnNiGa,12skyrmions with the multiple helicity reversals in BaFe 12-x-0.05 ScxMg0.05O19,13and frustrated kagome Fe3Sn214were found though the mechanism has not been clarified. Thereafter, searching skyrmions in normal perpendicular anisotropymaterials without the DMI and clarifying the mechanism of new emergent topological features are challenging works in the research ofskyrmions, which is the main motivation of this work. T h ei n t e r m e t a l l i cc o m p o u n dN d 2Fe14B is a well-known rare-earth p e r m a n e n tm a g n e tw i t hb r o a da p p l i c a t i o n s .T h ee x c e l l e n tm a g n e t i c property results from the large saturation magnetization and high uni- axial magnetic anisotropy. It has been pointed out that in the tetragonalcrystal structure, the Fe moments at six different sublattice locations are coupled together by a large Fe–Fe exchange interaction, providing the most magnetization of the crystalline unit. The interplay of the rare- earth crystalline electric field (CEF) and the Nd–Fe exchange interac- tions are the main cause of the crystalline anisotropy. 18When the tem- perature is lower than 135 K, designated as the spin-reorientation point, TSR, the Nd and Fe moments are aligned noncollinear and the easy magnetization direction (EMD) that originally orients toward the c-axis becomes an easy cone, with the highest inclination angle being 28/C14at 4.2 K18–21Furthermore, the saturation magnetization increases monoto- nously with the decreasing temperature. As discussed in the formation of skyrmions in films with perpendicular anisotropy and without the DMI, the anisotropy and magnetic dipole–dipole interaction are key Appl. Phys. Lett. 117, 132402 (2020); doi: 10.1063/5.0022270 117, 132402-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplfactors for forming skyrmions. Therefore, elemental doping and varia- tion of temperatures have been used to adjust the intensity and directionof anisotropy to stabilize skyrmions. 14–16The spin reorientation of Nd2Fe14B happens to offer a way to tune anisotropy and magnetization, which helps to obtain magnetic skyrmions in this material. In this paper, the domain structure of the Nd 2Fe14Bm a g n e tw a s in situ observed by Lorentz transmission electron microscopy (LTEM) at variable temperatures and magnetic fields, where the domain evolu-tion caused by the spin-reorientation was shown. Trivial magneticbubbles without topological protection and magnetic skyrmions (orskyrmionic bubbles, i.e., type I bubbles in the early bubble literature 26) were observed at temperatures higher than TSRand those around TSR, respectively, which deepens the understanding of the spin reorienta- tion in Nd 2Fe14B and broadens the material range for the formation of magnetic skyrmions. The Nd 2F14B alloy was prepared by induction melting with high purity raw materials. The ac susceptibility was measured by using aPhysical Property Measurement System (PPMS, Model-9) at tempera-tures from 80 K to 200 K, where the amplitude and frequency of theapplied ac fields were 80 A/m and 1000 Hz, respectively. The grains inthe TEM specimen with c-axes parallel to the electron beam were observed. The magnetic domain structure was observed by using Talos F200x in the Lorentz TEM mode, where the fields were applied by the objective lens. The temperature is the displayed value in the tempera-ture controller of a Gatan cryogenic holder (Model 900). In Nd 2Fe14B, the 4f electron distribution of Nd3þis aspherical and the spin–orbit coupling is very strong, by analyzing the crystal-field splitting of Nd 3þtogether with a mean-field description of Nd–Fe exchange;18it is shown that the CEF dominates the directional preference of the overall magnetic moment. As the temperaturedecreases, the contribution of higher-order CEF terms to the anisot- ropy increases, resulting in the negative anisotropy constant K 1at 135 K, which is the starting point of the spin-reorientation. Thechanges of anisotropy constants with temperature are shown inthesupplementary material , Fig. S1. The temperature dependence of the real component ( v 0) and the imaginary component ( v00)o fa cs u s - ceptibility were measured for the bulk Nd 2Fe14B ingot, and the results are shown in Fig. 1(a) . In the temperature range from 80 K to 200 K, a peak in v0a n da ni n fl e c t i o np o i n ti n v00at 133 K are observed, where the derivative of v0shows a minimum. Therefore, the spin reorienta- tion temperature of our sample, TSR, is 133 K. The slight deviationfrom 135 K is probably due to the polycrystalline ingot. It was reported that with the further decreased temperature, the EMD tilts away from thec-axis in the four symmetry-equivalent {110} planes,18–20as sche- matically shown in Fig. 1(b) . The variation of the property of Nd 2Fe14B with the temperature will cause different domain structures under the same magnetic field H.Figures 2 and S3 show the process observed by LTEM, where under-focused images are shown and the defocused values are all 181 lm. The specimen thickness was estimated to be 52.5 nm based on the SEM method. Figures 2(a)–2(d) a r et h ec a s e sa t1 5 0K ,w h i c hi sh i g h e r than TSR. The sample was first saturated by the magnetic field applied downward and, then, the field was reduced to 0 T, with the remanent magnetic domain pattern presented in Fig. 2(a) .T h i sk i n do fm a z e domain is a typical domain structure of films with perpendicular anisotropy subjected to the demagnetization energy,22where the mag- netization directions of four representative domains, ‹,›,fi,a n dfl, are indicated. When Hwas increased to 0.5 T as shown in Fig. 2(b) (hereafter, the fields were all applied upward), the domain area in the downward direction shrank, as shown in ‹›, while the upward domains expanded, as shown in fifl. The process from Figs. 2(a) and 2(b) was reversible, i.e., when the magnetic field was reduced to 0 T, the domain returned to the state of Fig. 2(a) .A s Hincreased from 0.5 T to 0.75 T, the domain walls merged and the domain shape changed significantly. This corresponds to the Barkhausen jump and is an irreversible process. In Fig. 2(c) , most of the downward domains disappeared and some of them shrank to form stripes. A further increase in the field led to the shrinking of the stripe domain to the magnetic bubble as shown in Fig. 2(d) forH¼1.0 T; then the bubble suddenly disappeared when the field reached 1.22 T and the samplewas saturated. The change of the magnetic domain at 130 K, which is roughly theT SR,i ss h o w ni n Figs. 2(e)–2(h) . Although the domain evolution is basically similar to the case of 150 K, the downward stripe domains shrank to magnetic bubbles as shown in Fig. 2(g) (the topological fea- tures are described in detail below); then, the bubble decreased in sizes [Fig. 2(h) ] and finally disappeared quickly, leading to the completely saturate state at H¼1.17 T. The magnetization reversal at TSRis easier than that above TSRdue to the deviation of the EMD from the c-axis, which is in agreement with the peak of the real component ( v0)i n Fig. 1(a) . When the temperature decreased to 120 K, as shown in Figs. 2(i)–2(j) , the contrasts gradually became blurred compared to those in Figs. 2(a)–2(d) and S3(a)–S3(d). A similar phenomenon was also observed at the temperatures of 110 K and 95 K, as shown in Figs. S3(e) and S3(i), respectively, which was caused by the confused config- uration of magnetic moments due to the spin reorientation, i.e., the EMD became a cone below TSR. With further decreased temperature, magnetic bubbles were observed before saturation in the case of 120 K (saturation field, Hs¼1.23 T), while they were not observed in the cases of 110 K ( Hs¼1.29 T) and 95 K ( Hs¼1.55 T). It is noteworthy that before the disappearance of magnetic domain walls at low temper- ature well below TSR, the wall contrasts such as ·and/C181in Fig. S3(l) presented different characteristics than those at other temperatures. Figures 3(a) and3(b) show the profiles of two kinds of domain walls for comparison. Figure 3(a) is the line intensity profile along the yellow arrow in inset /C176[taken from Fig. 2(a) , at 150 K], which is the general wall contrast in the maze domain. It is shown that at positions FIG. 1. (a) The temperature dependence of the ac susceptibility ( v0andv00) mea- sured for a bulk Nd 2Fe14B ingot. The spin reorientation temperature TSRwas deter- mined by a minimum in the curve of d v0/dTvsT. (b) Schematics of the Nd 2Fe14B EMD at the low temperature.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132402 (2020); doi: 10.1063/5.0022270 117, 132402-2 Published under license by AIP Publishing1–2–3–4 in the inset of Fig. 3(a) , the brightness varies in subsequence of gray-bright-dark-gray, which is caused by the spin configurationillustrated in Fig. 3(a) . The directions of the magnetic moments at positions “1” and “4” are parallel to the electron beam; thus, the elec- tron beams are not affected by the Lorentz force, where the image con- trast is gray and the intensity is nearly the mean value. Between “1”and “4,” the domain wall is a Bloch type, where the center moment isperpendicular to the electron beam. The electron beam is deflected bythe Lorentz force of the in-plane moment so that the contrast of the domain wall becomes half-bright and half-dark in the Lorentz image.Figure 3(b) shows the profile of brightness intensity along the yellow arrow in inset ·[taken from Fig. S3(l), at 95 K], where only one peak exists, in contrast to the two peaks of Fig. 3(a) . The single-bright con- trast in Fig. 3(b) is due to the spin reorientation, i.e., after all the down- ward moments have rotated upward, there is still an angle betweenmagnetic moments as shown in the inset of Fig. 3(b) . The inverse in- FIG. 2. The under-focused LTEM Fresnel images of Nd 2Fe14B with the out-of-plane external magnetic fields applied at different temperatures. The upper-, middle-, and lower- rows were taken at 150 K, 130 K, and 120 K, respectively. The four columns from left to right were taken at fields of 0 T, 0.50 T, 0.75 T, and 1.0 T, respectivel y.‹–‡represent domains or domain walls at different positions, the white dotted frames are topologically trivial magnetic bubbles, the red dotted frames are skyrmi ons, and the yellow dotted frame in (a) is a Bloch-type domain wall shown in Fig. 3(a) . The black arrows in (d) indicate the diffraction contrasts, and the corresponding positions in other figures are the same. FIG. 3. (a) and (b) Intensity profiles along arrows in the yellow boxes /C176[from Fig. 2(a) , at 150 K] and the yellow arrow ·[from Fig. S3(l), at 95 K], respectively. The insets show the corresponding spin configurations. (c)–(e) are enlarged images of –inFig. 2(g) and†and‡inFig. 2(h) , respectively. (f)–(h) are the spin textures obtained from the TIE analysis of (c)–(e), respectively. The inset of panel (g) shows the color wheel. Both bubbles and skyrmions show spin configurations with multi farious topological fea- tures, which are multiple concentric circles or arcs.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 132402 (2020); doi: 10.1063/5.0022270 117, 132402-3 Published under license by AIP Publishingplane components form the single-bright contrast at the domain wall position, similar to the normal 180/C14wall. Similarly, the single-dark contrasts, such as /C181in Fig. S3(l), demonstrate inverse relative in-plane directions. From the data in Fig. 2 and those at 110 K and 95 K (supplemen- tary Fig. S3), one can see that the saturation magnetic field rises gradu- ally with the decreased temperature (T <TSR), in agreement with the reported increase in the anisotropy field.23We have carried out micro- magnetic simulation of this process (supplementary Fig. S5), which agrees with the experiment. In particular, some magnetic bubbles are observed as shown in Figs. 2(d) ,2(g),2(h),2(k),2(l), S3(c), and S3(d). The topological fea- tures of the spin textures of some representative bubbles are analyzed in detail by using the transport-of-intensity equation (TIE), which is the popular method of reconstructing in-plane magnetization compo- nents using under-focus, just-focus, and over-focus Lorentz images.16 Three bubbles, –inFig. 2(g) and†‡inFig. 2(h) , are selected and the TIE analysis results are shown in Figs. 3(f)–3(h) , respectively. The color wheel in Fig. 3(g) represents the direction of the magnetic moment, where the color brightness represents the magnitude of the in-plane component and the black indicates no in-plane component. From Fig. 3(c), it is shown that the changes of wall contrasts in the left part and right part are asymmetrical, resulting in the spin texture in Fig. 3(f) .I n the left part, the winding directions of in-plane magnetic moments in three arcs from outside to inside are counterclockwise, clockwise, and counterclockwise, respectively, while the moments in the right part have symmetrical winding directions. Subsequently, two Bloch lines exist as indicated by dotted white lines in Figs. 3(c) and3(f),24,25where the central magnetic moments run through the domain wall perpen- dicularly. In other words, the magnetic moments at the wall center do not form a closed circle and the magnetic bubbles like –are topologi- cally trivial. For †inFig. 3(d) , the variation of domain wall contrasts in the radical direction is axial symmetric; the TIE results [ Fig. 3(g) ] show that the in-plane magnetic moment has three concentric rings, exhibiting that the spin texture is a skyrmionic bubble with topological protection. The winding directions of the magnetic moments from out-side to inside are counterclockwise, clockwise, and counterclockwise, respectively, and two adjacent rings are opposite, implying helicity reversals inside the skyrmion. The color of the middle ring is brighter and, thus, the lateral magnetization is larger than the rings on both sides. This kind of skyrmion pattern is similar to the skyrmions reported in BaFe 12-x-0.05 ScxMg0.05O1913and frustrated kagome Fe3Sn2.14While‡inFig. 3(e) has opposite contrast variation in com- parison with †, the winding directions of in-plane magnetization or the chirality of ‡are opposite to that of †as exhibited in Fig. 3(h) . According to the above analysis and the contrasts of magnetic bubbles, the skyrmionic bubbles are marked by dotted red boxes and trivial bubbles by white boxes in Figs. 2 and S3. It is shown that with the increased magnetic field, at 300 K and 150 K, the stripe domains shrink to trivial bubbles before disappearing; at 130 K, most stripe domains shrink to skyrmionic bubbles except one trivial bubble; then, the sizes of skyrmionic bubbles reduce and trivial bubble disappears [Figs. 2(g) and2(h)]; at temperature 120 K, one skyrmionic bubble is still observed; at temperatures of 110 K and 95 K, neither trivial bub- bles nor skyrmionic bubbles are observed. Therefore, in Nd 2Fe14B material, the skyrmionic bubble mostly appears at temperature around spin reorientation, TSR. At temperature above TSR, trivial bubbles areeasily observed. Both the skyrmionic bubble and trivial bubble have a multiple-ring structure or multifarious topological features, which is analogous to those observed in centrosymmetric magnets without the DMI, such as BaFe 12-x-0.05 ScxMg0.05O19and Fe 3Sn2.12–14The multifari- ous topological features are difficult to understand by only the energycompetition between the anisotropy energy and the magnetic dipole–- dipole interaction. Intuitively, the topological characteristic in Nd 2Fe14Bi sc a u s e db yt h ec o m p l i c a t e dc r y s t a ls t r u c t u r ea n d ,a c c o r d - ingly, the magnetic anisotropy determined by the interplay of the rare-earth CEF and the Nd–Fe exchange interactions. Furthermore, the skyrmion density in this work is lower than those in Co, 26BaFeO,27 and recent centrosymmetric materials,11–17which is considered to be due to the large anisotropic energy in Nd 2Fe14Bc o m p a r e dw i t h demagnetization energy. The large anisotropy at low temperature should be the reason why the skyrmion is rarely observed at tempera- tures well below TSR. Doping other kinds of rare-earth elements can help to reduce the anisotropy, which is probably an effective way toincrease the skyrmion density in Nd 2Fe14B-based materials. In conclusion, the spin-reorientation of Nd 2Fe14B is investigated byin situ LTEM observation. With the decreasing temperature, the sample becomes hard to saturate due to the increased anisotropy. At atemperature above T SR, only topological trivial magnetic bubbles are observed. At a temperature around TSR, magnetic skyrmions are observed due to the modulated anisotropy and magnetization origi- nated from the spin orientation. The results in this work suggest apractical way to obtain skyrmions in rare-earth permanent magneticmaterials. See the supplementary material for magnetization processes at 300 K, 110 K, and 95 K, the selected-area electron diffraction pattern of the sample, and micromagnetic simulation results. This work was supported by the National Natural Science Foundation of China (Nos. 51971005, 51771127, 51571126, and 51772004) and the project of Sichuan Provincial EducationDepartment (Grant Nos. 18TD0010 and 16CZ0006). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962). 2S. M €uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, and P. B €oni,Science 323, 915 (2009). 3X. Z. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010). 4L. C. Shen, J. Xia, G. P. Zhao, X. C. Zhang, M. Ezawa, O. A. Tretiakov, X. X. Liu, and Y. Zhou, Phys. Rev. B 98, 134448 (2018). 5L. C. Shen, J. Xia, G. P. 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5.0020706.pdf

J. Chem. Phys. 153, 104107 (2020); https://doi.org/10.1063/5.0020706 153, 104107 © 2020 Author(s).Fast, accurate enthalpy differences in spin crossover crystals from DFT+U Cite as: J. Chem. Phys. 153, 104107 (2020); https://doi.org/10.1063/5.0020706 Submitted: 03 July 2020 . Accepted: 19 August 2020 . Published Online: 08 September 2020 Miriam Ohlrich , and Ben J. Powell ARTICLES YOU MAY BE INTERESTED IN Exploring Hilbert space on a budget: Novel benchmark set and performance metric for testing electronic structure methods in the regime of strong correlation The Journal of Chemical Physics 153, 104108 (2020); https://doi.org/10.1063/5.0014928 Hybridizing pseudo-Hamiltonians and non-local pseudopotentials in diffusion Monte Carlo The Journal of Chemical Physics 153, 104111 (2020); https://doi.org/10.1063/5.0016778 Richardson–Gaudin mean-field for strong correlation in quantum chemistry The Journal of Chemical Physics 153, 104110 (2020); https://doi.org/10.1063/5.0022189The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Fast, accurate enthalpy differences in spin crossover crystals from DFT+U Cite as: J. Chem. Phys. 153, 104107 (2020); doi: 10.1063/5.0020706 Submitted: 3 July 2020 •Accepted: 19 August 2020 • Published Online: 8 September 2020 Miriam Ohlricha) and Ben J. Powellb) AFFILIATIONS School of Mathematics and Physics, University of Queensland, Saint Lucia, Queensland 4072, Australia a)Electronic mail: miriam.ohlrich@uq.net.au b)Author to whom correspondence should be addressed: powell@physics.uq.edu.au ABSTRACT Spin crossover materials are bi-stable systems with potential applications as molecular scale electronic switches, actuators, thermometers, barometers, and displays. However, calculating the enthalpy difference, ΔH, between the high spin and low spin states has been plagued with difficulties. For example, many common density functional theory (DFT) methods fail to even predict the correct sign of ΔH, which determines the low temperature state. Here, we study a collection of Fe(II) and Fe(III) materials, where ΔHhas been measured, which has pre- viously been used to benchmark density functionals. The best performing hybrid functional, TPSSh, achieves a mean absolute error compared to experiment of 11 kJ mol−1for this set of materials. However, hybrid functionals scale badly in the solid state; therefore, local functionals are preferable for studying crystalline materials, where the most interesting spin crossover phenomena occur. We show that both the Liechten- stein and Dudarev DFT+U methods are a little more accurate than TPSSh. The Dudarev method yields a mean absolute error of 8 kJ mol−1for Ueff= 1.6 eV. However, the mean absolute error for both TPSSh and DFT+U is dominated by a single material, for which the two theoretical methods predict similar enthalpy differences—if this is excluded from the set, then DFT+U achieves chemical accuracy. Thus, DFT+U is an attractive option for calculating the properties of spin crossover crystals, as its accuracy is comparable to that of meta-hybrid functionals, but at a much lower computational cost. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020706 .,s I. INTRODUCTION Spin crossover (SCO) is a phenomenon where the equilibrium state of a material can transition between a high spin (HS) and low spin (LS) state with changes in temperature, pressure, applied mag- netic fields, or light irradiation.1Many SCO materials are pseudo- octahedral coordination complexes of the first row transition metal ions with open d-orbitals ( d4−d7).2The HS state occurs when the crystal field splitting, Δcf, is smaller than the d-orbital electron pair- ing energy, and the LS state occurs in the opposite case.3In SCO materials, the enthalpy difference between the HS and LS states, ΔH=HHS−HLS, is typically less than 50 kJ mol−1.4As SCO often occurs in large coordination complexes and coordination polymers, this is challenging for first principles approaches to accurately and reliably capture. Density functional theories (DFT) utilizing hybrid functionals that include some exact exchange can predict ΔHaccurately enough to allow the prediction of many material properties. Notably, Jensenand Cirera5reported that the meta-hybrid TPSSh, with 10% exact exchange, gave a mean absolute error relative to experiment (MAE) of 11 kJ mol−1forΔHfor a range of Fe based SCO complexes. How- ever, many of the most interesting properties of SCO materials, from both the fundamental6–16and applications15,17–25perspectives, result from the interplay between the changes in enthalpy and entropy when a single molecule changes spin-state, and frustrated elastic interactions between complexes in the solid state. The elastic inter- actions are fundamentally a property of the solid state, rather than a single complex. However, in the solid state, the exact exchange component of hybrid functionals becomes prohibitively time con- suming. This has motivated several groups to investigate whether similar or higher accuracy results can be achieved without the use of hybrid density functionals.26–32 Pure density functionals, such as the local density approxima- tion (LDA) and generalized gradient approximations (GGA), tend to over-delocalize the valence electrons. This is because these approxi- mations for the exchange correlation functional do not fully cancel J. Chem. Phys. 153, 104107 (2020); doi: 10.1063/5.0020706 153, 104107-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp out the self-interaction term in the Hartree potential. Thus, there is some residual repulsion between each electron and itself in the model, which forces the electrons further away from the nucleus.33 This is why DFT calculations based on pure functionals fail to accu- rately predict the properties of materials involving transition metals with open dorforbitals, where the valence electrons are strongly interacting and localized.34,35 One way to counter this over-delocalization is to use an LDA or GGA functional with an added Hubbard model-like potential term (the DFT+U method).33,36–38This potential term includes electro- static interactions between two electrons in the same orbital and the energy associated with the exchange of electrons between orbitals on the same atom. The inclusion of this on-site Coulomb interaction (U) and the electron exchange interaction ( J) increases the elec- tron localization, approximately correcting the over-delocalization of valence electrons in DFT.33,36 Since DFT+U uses a pure density functional for all but the strongly correlated dorbital electrons and the added potential terms only act locally, the computational time for DFT+U is compara- ble to pure functionals such as LDA and GGA, even in the solid state. In contrast, the computational times for hybrid functionals dramatically increase with the size of the crystal.33 Most previous papers that have discussed the use of DFT+U for spin crossover materials have tuned the Uand Jparameters to reproduce the properties of a single material.26–31The work of Vela et al.32is the only systematic study of the DFT+U parame- terization we are aware of. They included an empirical treatment of the vibronic contributions, and to implement this, they needed to adjust the calculated single molecule frequencies to match the measured vibrational entropy. If such measurements are not avail- able for molecular crystals, then this approach is not possible. It is also not clear how to extend this method to frameworks and coordination polymers, where many important SCO phenomena are observed.7–10,25We are, therefore, motivated to ask the fol- lowing: Can a DFT+U method with no experimental input and a common value of Uand Jachieve results of comparable or bet- ter accuracy than those of hybrid functionals for the spin crossover enthalpy difference for a set of spin crossover materials? To answer this question, we investigated two DFT+U methods: the Liecht- enstein method,38which treats Uand Jas separate parameters, and the Dudarev method,34which uses only the difference between them, Ueff=U−J. II. METHODS A. Training set To benchmark our DFT+U calculations, we need a collection of spin crossover materials for which the spin transition enthalpy differences have been experimentally determined. For ease of com- parison, we selected the same set that Jensen and Cirera5used to benchmark a range of functionals (Table I and Fig. 1). However, only the iron complexes were investigated, as different UandJvalues are needed to accurately describe complexes with different central ions. B. Computational details Where absent, hydrogen atoms were added to structures using the “HADD” function in OLEX2.55All DFT calculations wereTABLE I : Spin crossover materials investigated. We present their delectron con- figuration ( dn); the measured enthalpy differences ( ΔH), the range including both the experimental error (where reported) and differences between experiments, with MAEs calculated relative to the midpoint of this range; and the reference codes for the compounds in the Cambridge Structural Database (CSD). Ligands are abbreviated as follows: acac = acetylacetonate, trien = triethylenetetramine, papth = 2-(2-pyridylamino)-4-(2-pyridyl)thiazole, tacn = 1,4,7-triazacyclononane, 2-amp = 2-aminomethylpyridine, HB(pz) 3= hydrotris(pyrazol-1-yl)borate, py(bzimH) = 2-(2′- pyridyl)benzimidazole, and tppn = tetrakis(2-pyridylmethyl)-1,2-propanediamine. No. Material dnΔH(kJ/mol) CSD 1 [Fe(acac) 2trien](PF 6) d57–1739,40actrfe41 2 [Fe(papth) 2](BF 4) d61642colijao43 3 [Fe(tacn) 2](Cl 2) d621–2444dettol45 4 [Fe(2-amp) 3](Cl 2) d618–2546fepicc47 5 [Fe(HB(pz) 3)2] d616–2248,49hpzbfe50 6 [Fe(py(bzimH)) 3](2ClO 4)d620–2151kokfof52 7 [Fe(tppn)](2ClO 4) d625–3053iqiceq54 carried out in the Vienna Ab initio Simulation Package (VASP).56–59 First, structural relaxations were carried out for each crystal using the PBE (Perdew-Burke-Ernzerhof) functional to provide both the HS and LS structures. The unit cell was constrained to the measured dimensions as we do not expect PBE+U to capture the weak inter- actions that are important in determining some of the crystal struc- tures considered here. Then, DFT+U calculations were performed, also using the PBE functional, while gradually varying UandJ. This approach allows us to focus on the changes in the electronic struc- ture due to the local interactions and has a negligible impact on the calculated enthalpy differences (see Sec. III A). The HS and LS states FIG. 1 . Structures of complexes investigated. Hydrogen atoms and counter ions are excluded for clarity. J. Chem. Phys. 153, 104107 (2020); doi: 10.1063/5.0020706 153, 104107-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp were prepared by specifying the initial magnetic moments of the Fe atoms. Explicit inspection of a representative sample of calculations and the obvious adiabatic continuity of the data, below, demonstrate that the correct local minimum was achieved in all cases. All cal- culations used a plane wave basis set with a plane wave cutoff of 500 eV. The stopping condition for the minimization of the density functional was that two successive steps differed in energy by 10−6 eV or less, and the Brillouin zone was sampled only at the Γpoint, as benchmarking indicated that ΔHonly depends very weakly on the number of k-points (see Fig. S1of the supplementary material). Input files and selected output for these calculations are available for download.60 III. RESULTS AND DISCUSSION A. Dudarev approach Ueff=U−Jwas increased in increments of 0.1 eV from the relaxed HS and LS structures. The difference between the calculated ΔHand that measured experimentally for each material is plotted in Fig. 2. Increasing Ueffleads to a linear decrease in the spin crossover enthalpy difference for all materials. For materials 2–7, all of the gra- dients are very similar, whereas the gradient for material 1(black) is clearly different. A straightforward analysis of the Hubbard (or more strictly, Kanamori) model61shows that different gradients should be expected for different (formal) numbers of the d electrons, in line with this finding. Orbital relaxation and hybridization mean that such a simple calculation does not correctly predict the magnitude of the gradient. Nevertheless, it explains the linear variation of ΔHwith Ueff, the very similar gradients of materials 2–7, and the different gradient of 1. To determine the optimal value of Ueff, we calculated the mean absolute error over the entire set of materials (Fig. 3). The DFT+U method, carried out using the Dudarev approach, marginally out- performs the TPSSh functional: the lowest MAE for the DFT+U method is 8 kJ mol−1forUeff= 1.63 eV compared to an MAE of 11 kJ mol−1for TPSSh. Note also that the minimum is extremely soft—therefore, the accuracy is not strongly affected by different choices of Ueff. The largest absolute error for any single mate- rial with this value of Ueffis 30 kJ mol−1, which also marginally FIG. 2 . Error in the calculated enthalpy difference between the HS and LS states, ΔH, relative to the experimentally measured values (Table I) using the Dudarev approach with a PBE functional. The error bars represent the range of experimental values in Table I. FIG. 3 . Mean absolute error in the spin crossover enthalpy difference for the Dudarev approach, using the PBE functional (blue squares), calculated over mate- rials 1–7. For comparison, the MAE for the TPSSh functional for the same set of materials (11 kJ mol−1; red line) is also shown.5ForUeffbetween 1.5 eV and 1.8 eV, the MAE from the DFT+U calculation is lower than that for the TPSSh functional. The lowest MAE (8 kJ mol−1) achieved for Ueff= 1.63 eV. outperforms the largest absolute error for TPSSh, which was 32 kJ mol−1. The largest absolute error for both the TPSSh func- tional and the Dudarev approach occurs for material 5(which is a clear outlier in Fig. 2). Both the Dudarev approach and TPSSh functional give similar energies for this material—both of which are higher than the experimental energy. Thus, one may begin to sus- pect that the reported experimental value for material 5may not be accurate. Excluding this material, the optimal value of Ueffis 1.6 eV, with an MAE of 4.7 kJ mol−1—chemical accuracy—and the MAE for the TPSSh functional is 7.5 kJ mol−1. Hence, the optimized value of Ueffis only changed marginally, and the Dudarev method still yields more accurate results than the TPSSh functional. Finally, we investigated the impact of using structures opti- mized with the pure PBE functional in the above calculations, rather than with the DFT+U, as one would typically do. We therefore reop- timized the HS and LS structures for all seven materials in the train- ing set using DFT+U with Ueff= 1.6 eV. While this caused small changes in the calculated ΔHfor most materials, it had a negligible effect on the MAE for the entire set (which was 9 kJ mol−1for the pure PBE structures vs 10 kJ mol−1for DFT+U). FIG. 4 . Mean absolute error in the spin crossover enthalpy difference for the Liecht- enstein approach using the PBE functional. These values are calculated over materials 1–7at increments of 0.05 eV in Uand J. The white line represents the contour with the same MAE as reported previously for the TPSSh functional5 (11 kJ mol−1for compounds 1–7). All values of UandJwithin this loop give lower MAEs than the TPSSh functional, the minimum MAE being 8.7 kJ mol−1, which occurs at U= 1.7 eV and J= 0. This outperforms the TPSSh functional and is very similar to the MAE found using the Dudarev method. J. Chem. Phys. 153, 104107 (2020); doi: 10.1063/5.0020706 153, 104107-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp B. Liechtenstein approach In the Liechtenstein approach, Uand Jare varied indepen- dently. The MAE over all seven materials is reported in Fig. 4. ForU= 1.7 eV and J= 0, the Liechtenstein method also gives a marginally lower MAE (8.8 kJ mol−1) than the TPSSh func- tional. The largest absolute error is 29 kJ mol−1, which again occurs for material 5. Excluding this material yields a minimum MAE of 5.4 kJ mol−1when U= 1.7 eV and J= 0. Hence, the values of UandJ do not shift in this case, and the results are still better than the TPSSh functional (where the MAE is 7.5 kJ mol−1excluding material 5). IV. CONCLUSION Both the Dudarev and Liechtenstein DFT+U methods give lower values for the MAE than those found by Jensen and Cirera5 using the TPSSh functional. However, the Dudarev method gives a slightly lower MAE than the Liechtenstein method and has a lower computational time, so based on the results for these seven materi- als, it is the recommended method. As we suspect that the reported experimental enthalpy difference for material 5may not be accurate, we recommended using Ueff= 1.60 eV for future calculations of the spin transition enthalpy difference in SCO materials. The optimized values of Ueff=U−Jdetermined for the DFT+U approaches are in good agreement with previous DFT+U calcu- lations where Ueffwas optimized for only a single material. Val- ues obtained for individual materials range from Ueff= 1.55 eV to 2.5 eV.26–29,31Vela et al.32found a larger value of Ueffthan ours (2.65 eV). This is reasonable, as they are subtracting an estimate of the vibronic contribution to ΔHmade by combining experiment and theory. As ΔHmonotonically decreases with Ueff, this implies that one should expect the effective Uof Vela et al. to be larger than ours. We also note that our approach gives a very similar accuracy to the more complicated method of Vela et al. : we find an MAE of 4.7 kJ mol−1(excluding material 5), whereas they reported an MAE of 4.3 kJ mol−1(for a different, but overlapping, set of materials). Overall, using the Dudarev method with Ueff= 1.6 eV is an attractive prospect. It provides a computationally inexpensive way to predict the enthalpy difference between the HS and LS states for crystal structures where no experimental input is available. SUPPLEMENTARY MATERIAL See the supplementary material for calculations showing the independence of the HS-LS entropy difference on the k-mesh employed. ACKNOWLEDGMENTS We thank Thom Aldershof, Ross McKenzie, Tanglaw Roman, Amy Thompson, and Jacob Whittaker for helpful conversations. This work was supported by the Australian Research Council through Grant No. DP200100305. DATA AVAILABILITY The data that support the findings of this study are openly available in UQ eSpace at https://doi.org/10.14264/e619afa.REFERENCES 1G. G. Levchenko, A. V. Khristov, and V. N. Varyukhin, “Spin crossover in iron(II)-containing complex compounds under a pressure,” Low Temp. Phys. 40, 571–585 (2014). 2J. P. Sauvage, Transition Metals in Supramolecular Chemistry , Perspectives in Supramolecular Chemistry (Wiley, Chichester, England, 1999). 3F. A. Cotton, G. Wilkinson, and P. L. Gaus, Basic Inorganic Chemistry , 3rd ed. (John Wiley, New York, 1995). 4J. Turner and F. A. Schultz, “Coupled electron-transfer and spin-exchange reactions,” Coord. Chem. Rev. 219-221 , 81–97 (2001). 5K. P. Jensen and J. Cirera, “Accurate computed enthalpies of spin crossover in iron and cobalt complexes,” J. Phys. Chem. A 113, 10033–10039 (2009). 6E. Trzop, D. Zhang, L. Piñeiro-Lopez, F. J. Valverde-Muñoz, M. 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10.0001703.pdf

Landau levels, edge states, and gauge choice in 2D quantum dots Asadullah Bhuiyan , and Frank Marsiglio Citation: American Journal of Physics 88, 986 (2020); doi: 10.1119/10.0001703 View online: https://doi.org/10.1119/10.0001703 View Table of Contents: https://aapt.scitation.org/toc/ajp/88/11 Published by the American Association of Physics TeachersLandau levels, edge states, and gauge choice in 2D quantum dots Asadullah Bhuiyan and Frank Marsiglio Department of Physics, University of Alberta, Edmonton, Alberta T6G 2E1, Canada (Received 17 May 2020; accepted 22 July 2020) We examine the behavior of a charged particle in a two dimensional quantum dot in the presence of a magnetic field. Emphasis is placed on the high magnetic field regime. Compared to free spacegeometry, confinement in a dot geometry provides a more realistic system where edge effects arise naturally. It also serves to remove the otherwise infinite degeneracy due to the magnetic field; nonetheless, as described in this paper, additional ingredients are required to produce sensible results. We treat both circular and square geometries, and in the latter, we explicitly demonstrate the gauge invariance of the energy levels and wave function amplitudes. The characteristics of bulkstates closely resemble those of free space states. For edge states, with sufficiently high quantum numbers, we achieve significant differences in the square and circular geometries. Both circular and square geometries are shown to exhibit level crossing phenomena, similar to parabolic dots,where the confining potential is a parabolic trap. Confinement effects on the probability current are also analyzed; it is the edge states that contribute non-zero current to the system. The results are achieved using straightforward matrix mechanics, in a manner that is accessible to novices inthe field. On a more pedagogical note, we also provide a thorough review of the theory of single electron Landau levels in free space and illustrate how the introduction of surfaces naturally leads to a more physically transparent description of a charged particle in a magnetic field. VC2020 American Association of Physics Teachers . https://doi.org/10.1119/10.0001703 I. INTRODUCTION The motion of a charged particle in the presence of a mag- netic field is a subject of considerable interest in many areasof physics. In particular, in condensed matter, the presenceof the magnetic field alters the particle’s behavior in a pro-found way. Ignoring the spin degree of freedom, it is knownfrom classical physics that a vector potential is required toinclude the effect of a magnetic field, and more than onechoice of vector potential is possible. Each of these choices constitutes a particular gauge. For a uniform magnetic field, ~B¼B^z, common choices in Cartesian coordinates are the Landau gauge, ~A 1¼xB^y(or~A2¼/C0yB^x), or the symmetric gauge, ~AS¼/C0yB=2^xþxB=2^y. Naturally, all physical quantities obtained in a calculation should be independent of the gauge choice. Typically, agauge choice is made to take advantage of some symmetryin the problem so that one can proceed analytically. Just asoften, the physical system is altered to take advantage of thesymmetry afforded by the gauge choice. A good example isLaughlin’s seminal paper, 1which addressed the quantization of the Hall conductivity in a two-dimensional metal, using a ribbon bent into a loop (i.e., open boundary conditions in one direction and periodic boundary conditions in theother). The magnetic field is perpendicular to the plane ofthe metal, which implies the physically questionable notionof a field either emanating from or converging to an axis atthe center of the loop. That this configuration did not coin-cide precisely with the experiment was immaterial—theconcept of quantization was properly established, and what mattered was that the Landau gauge allowed the argument to proceed analytically. Similarly, Halperin 2followed up with a thin film with annular geometry and adopted thegauge choice ~A¼½B 0r=2þU=ð2prÞ/C138^hto produce a uni- form field B0in the z-direction along with a central flux of magnitude U.Other physical geometries, suitable to confined geometries called “quantum dots,” were adopted thereafter—see, forexample, Refs. 3and4. In this paper, we will review and expand upon some of these calculations, representing charged particles in a confined geometry subjected to a con-stant magnetic field. A very good reference for the quantumdot work is Chakraborty’s book, Quantum Dots , 5which also contains a number of very useful reprinted articles. Aside from providing a better-defined geometry for the subject ofthis work, quantum dots also feature in many useful applica-tions, as discussed, for example, in Ref. 6. As mentioned above, we will focus on the orbital motion of the charged particle (hereafter, these will be electrons) and neglect the spin degree of freedom. Spin will cause a breaking of degen-eracy in the presence of a magnetic field owing to the directcoupling with the field through the Zeeman term. We begin in Sec. IIwith a textbook review of the particle in free space; this is where the idea of Landau levels first arose. Most of the material in this section is relegated to an Appendix , where we highlight both the Landau and symmet- ric gauges and seemingly derive very different results for thewave functions. We further note that confinement within aparabolic trap can also be treated analytically, following for example, Rontani. 7Next, we will review and expand upon Lent’s treatment of the circular quantum dot4and illustrate how this problem can be tackled with undergraduate tools.Finally, we will show numerical results for the properties ofan electron in the square quantum dot with an applied mag- netic field in both aforementioned gauges. This will hope- fully make clear the role of degeneracy, and we explicitlyillustrate the equivalence of the results in the two gauge for-mulations by removing the degeneracy. We will present the results of calculations for simple properties only, such as probability density and current density. Having obtained theexact energy spectrum along with the eigenstates of courseallows one to compute any desired single particle property. 986 Am. J. Phys. 88(11), November 2020 http://aapt.org/ajp VC2020 American Association of Physics Teachers 986The actual fabrication of two-dimensional quantum dots is described, for example, in Ref. 8. Note that all these confined geometries represent “infinitely” confined systems. That is, whether through a par- abolic trap or one of the “wells,” the particle cannot escape. A more realistic quantum dot would have a limited confine- ment potential and allow for the possibility that the electroncould be “ionized” from the sample. Then, the problem will change somewhat, as the energy spectrum will be limited to a finite range; we do not consider that possibility here. II. REVIEW OF LANDAU LEVELS IN FREE SPACE In the presence of a magnetic field B, the canonical momentum pis shifted by the magnetic vector potential Ato give us a Hamiltonian of the form H¼p2 2m¼ðp/C0qAÞ2 2m; (1) where qis the charge and mis the mass of the particle of interest, and p/C17p/C0qAis the kinetic momentum operator. Hereafter, we adopt q¼/C0efor the electron, where e>0i s the magnitude of the charge of the electron. In the presence of a magnetic field, prepresents the true momentum of the particle rather than p. In this problem, we consider an elec- tron free to move in a two-dimensional system perpendicular to a uniform magnetic field, B¼B^z. For such a magnetic field, the three common gauge choices were stated in the introduction. The two Landau gauges are handled very simi-larly mathematically, and so, we will focus only on A 1. However, A1andASare handled very differently mathemati- cally, and we now discuss each in turn. A number of resour- ces are available that provide considerable details for this problem. These include textbooks such as Refs. 9–12 and online lecture notes by Tong13and Murayama.14Systematic derivations of the eigenvalues and eigenstates for each gauge are provided in the Appendix . A succinct summary of the Appendix is as follows: when the electrons are free to move throughout the x-yplane, the eigenvalues are infinitely degenerate, and we obtain thesame standard expression in either gauge, E nL¼/C22hxcnLþ1 2/C18/C19 ;nL¼0;1;2;…; (2) where xc/C17eB=mis the classical cyclotron frequency and nLis the Landau level quantum number. This infinite degen- eracy is often referred to as “Landau degeneracy.” On the other hand, the wave functions in the two gauges appear very differently: Eq. (A31) for the symmetric gauge and Eq. (A45) for the Landau gauge. In both cases, there are “hidden” quantum numbers, ‘for the symmetric gauge and kyfor the Landau gauge. The derivations in the Appendix highlight some trouble- some issues. The infinite Landau degeneracy is difficult, but even more so is the dependency of the wave function on gauge choice. The energies are gauge invariant because achange in gauge choice A!A 0is equivalent to performing a unitary operation on the Hamiltonian, A!A0¼Aþrk)Hr;p/C0qA0/C2/C3 ¼eiqk=/C22hHr;p/C0qA ½/C138 e/C0iqk=/C22h; (3)where kis an arbitrary real function of the position and time andqis the charge of the particle. The wave function, on the other hand, is not gauge invariant, and for a change in gauge,A!A 0, there is a corresponding expected change in the wave function,15 A0¼Aþrk)W0¼e/C0iqk=/C22hW: (4) However, this means that the probability density, jWj2,i s expected to be gauge invariant. Yet, even within the symmet- ric gauge, Eq. (A39) , different choices of origin ( x0,y0) seemingly result in wave functions that differ by far morethan a phase (and therefore probability densities that do dif-fer from one another). The reason for this problem is that a degeneracy exists in the solution. An explicit calculation of the wave function transformation corresponding to that inEq.(4)between the Landau gauge and the symmetric gauge has been given recently in Ref. 16, and this problem was first highlighted in Ref. 17, where the author presented a more general gauge condition to replace the simple wave function relation in Eq. (4)when degeneracy is present. We will return to this issue later. The infinite Landau degeneracy inboth cases allows the wave functions in the two gauges to differ by more than the simple “textbook” phase, as dis- cussed by Swenson 17and Wakamatsu et al.16 To summarize, the two gauges discussed in the Appendix represent the same uniform magnetic field B¼B^z. They result in the same energy spectrum, and yet, they produce two very different-looking eigenstates (see Fig. 1). There are gauge-invariant similarities as well; both gauges produceinfinitely degenerate Landau levels in free space, and both gauges have degeneracies that are expected to be approxi- mately G/C17BA=U 0in a finite space with area A. Equation (4)suggests that the two eigenstates should differ by a sim- ple phase factor. That does not happen here because of the degeneracy associated with the eigenstates. The dilemma issolved by taking an appropriate linear combination of the degenerate eigenstates (which is still an eigenstate), and these can be chosen to result in a state that differs from onein the other gauge by a simple phase factor, as illustrated inRefs. 16and17. We will encounter this dilemma once more in a situation where the degeneracy has (nominally) been lifted and outline a resolution by means different from thatused in Refs. 16and17. III. LANDAU LEVELS IN A CIRCULAR QUANTUM DOT We will revisit the issue of degeneracy in different gauges in Sec. IVwhen we discuss an electron confined to a square dot. First, however, partly to review results due to Lent 4and partly to introduce some technical aspects and more general results to be used later, we present a discussion of results for the circular quantum dot. We will use dimensionless unitswherever possible. However, it is important to convert our results to physical values. As we have already seen, the flux quantum plays an important role in these problems; its valuein SI units is h=e/C254140 T (nm) 2. We will use a dimension- less measure of flux, G¼BA=ðh=eÞ, where the scalar Ais the area of the sample to which the electron is confined andthrough which a (constant) magnetic field penetrates. For cir- cular geometry, we use c/C17Bpa 2=ðh=eÞ, where ais the radius. So, for example, applying a 10 T magnetic fieldthrough a sample with radius 100 nm gives c/C2576. 987 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 987A. Matrix element calculation To study the effects of hard-wall confinement on Landau levels in a circular quantum dot, we use the rotationallyinvariant symmetric gauge, a natural choice for such a con-finement geometry. We outline our matrix mechanicsscheme before moving on to numerical results. Matrixmechanics techniques have been utilized for problems likethis before, albeit for a study of disordered systems, with adifferent gauge and confining potential. 18Otherwise, they have been used more recently in more pedagogical contextsin a variety of contexts, e.g., in Refs. 19–21 . The Hamiltonian for this system is given by H¼p2 2mþxc 2Lzþ1 8mx2 cr2þVðrÞ; (5) where our confining potential V(r) is defined as VðrÞ¼0i f r/C20a 1 else;/C26 (6) where ais the radius of the circular dot. To solve the Schr €odinger equation with this Hamiltonian, we follow the usual procedure of expanding in a basis (see, for example,Refs. 19and20). Our basis states of choice are the eigen- states of the infinite circular well, a natural basis for our con- fining geometry. These eigenstates are well known, hr;/jn;‘i¼ei‘/ ffiffiffiffiffiffi 2ppffiffiffi 2p aJ‘bn;‘r=a/C0/C1 J‘þ1ðbn;‘Þ; (7) where n¼1;2;3; :::; ‘ ¼0;61;62; :::, and bn;‘are the nth zeros, in ascending order, of the ‘th order Bessel function. The corresponding eigenenergies are given by En;‘¼/C22h2b2 n;‘ 2ma2: (8) Carrying out the standard procedure for numerical matrix mechanics, we expand the eigenstate jWiof the Hamiltonian (Eq.(5)) in the circular well basis,jWi¼X1 n0¼1X1 ‘0¼/C01cn0;‘0jn0;‘0i; (9) where jn0;‘0iis the ket corresponding to the circular well basis state, Eq. (7). Acting on this state with the Hamiltonian, Eq. (5), and then taking the inner product of both sides with some arbitrary bra basis state hn;‘jresults in the following matrix equation: X1 n0¼1X1 ‘0¼/C01Hn‘;n0‘0cn0;‘0¼En;‘cn;‘; (10) where Hn‘;n0;‘0¼hn;‘jHjn0;‘0i ¼dn;n0d‘;‘0Eð0Þ n;lþ1 2xchn;‘jLzjn0;‘0i þ1 8mx2 chn;‘jr2jn0;‘0i (11) are the matrix elements of the Hamiltonian given by Eq. (5). Here, Eð0Þ n;lis simply the infinite circular well eigenenergy, Eq.(8), and dijis the standard Kronecker delta function. It should now be apparent that the entire matrix is diagonal in ‘, i.e., Hn‘;n0‘0¼d‘;‘0Hn;n0, and the eigenvalue problem for a given ‘only needs to be solved. While a diagonalization is now required for each ‘separately, this simplification greatly reduces the computational cost. Evaluating the last two inner products of Eq. (11), we obtain all the matrix elements for the Hamiltonian, Hn;n0¼dn;n0/C22h2b2 n;‘ 2ma2þ/C22hxc‘ 2"# þ1 8mx2 ca2q2n;n0: (12) Note that we have suppressed the dependency on ‘in the matrix labels, but it is carried as a parameter in all of these matrices. Using energy units of /C22h2=ð2ma2Þ, we can write the Hamiltonian in dimensionless form Hn;n0 /C22h2=ð2ma2Þ/C17hn;n0¼dn;n0b2 n;‘þ2c‘hi þc2q2 n;n0; (13) Fig. 1. Contour plots of the free-space probability density vs position for (a) the symmetric gauge centered about the origin, with ‘¼5, and (b) the Landau gauge, with kylB¼5. The probability densities in each gauge have strikingly different spatial structures. It is clear that these eigenstates are not related by just a simple phase factor. 988 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 988with q2 n;n0¼hnjr2jn0i a2¼2ð1 0J‘ðbn0;‘qÞq2J‘ðbn;‘qÞqdq J‘þ1ðbn0;‘ÞJ‘þ1ðbn;‘Þ:(14) The integral in Eq. (14) is non-elementary but can be evalu- ated analytically, and we get q2 n;n0¼dn;n0b2 n;‘þ2‘2/C02 3b2 n;‘"# þð1/C0dn;n0Þ /C28bn0;‘bn;‘ ðb2 n0;‘/C0b2 n;‘Þ2J‘/C01ðbn0;‘ÞJ‘/C01ðbn;‘Þ J‘þ1ðbn0;‘ÞJ‘þ1ðbn;‘Þ"# ; (15) where cwas earlier defined as c¼Bpa2 h=e¼U U0: (16) We expect cto give us a rough measure of the degeneracy in the confined system as it is defined as the ratio of magnetic flux to flux quanta. Recall that a constant G¼U=U0(Eqs. (A38) and(A48) ) was argued to be a measure of degeneracy in the confined system. Here, cplays this role. We now have all the matrix elements analytically to insert into Eq. (10) for diagonalization. The big advantage of hav- ing adopted the symmetric gauge with this geometry is thatthe resulting Hamiltonian is diagonal in one of the quantum numbers, ‘. This represents a tremendous computational gain. We mention it here because we will not have this possi-bility when we study the square geometry in the Sec. IV. There we will have to diagonalize N 2/C2N2matrices, where Nis a large number (here typically something like 50 or 200). Also, for circular geometry, we can monitor the quan-tum number ‘precisely, which means that we can track the radial quantum number n ras well. In the symmetric gauge, the Landau level quantum number nLis given simply by nL¼nrþð‘þj‘jÞ=2. These quantum numbers are more precisely explained in the context of the free-space problem in the Appendix . We will not have this possibility with the square geometry. To be clear, in practice, in this section, we diagonalize N/C2Nmatrices and only show results that have converged as Nincreases (recall that Nshould be infinite, but it is chosen to be finite and represents the number of radial basis states (i.e., number of n) for a given ‘in Eq. (7)to be used in the matrix diagonalization). We will do this for a number of dif-ferent ‘’s (typically Nof these), again so that all presented properties are converged as a function of basis size. B. Eigenenergy spectra Having carried out this program, we can now order the eigenvalues in increasing value. We do this to mimic theresult we would have achieved had we not recognized that the matrix is diagonal in ‘(as will be the case with the square geometry). The eigenvalue results for a few values of care shown in Fig. 2and show in all cases plateaus (i.e., degener- ate levels) separated by regions in which the energies increase to the next plateau. The number of points from thestart of one plateau to the start of the next plateau is essen- tially c. The so-called “plateaus” are actually not degenerate to infinite precision, but they are degenerate to the precisionof our computer (16-digit accuracy). These plateaus appear to be the Landau levels, originally infinitely degenerate, but now with a degeneracy of order c, and, in practice, lower. The increase in eigenvalues towards the end of the plateau regions is indicative that these eigenfunctions can feel the edge of the confining potential, and for this reason, they areusually referred to as “edge states” because, as we shall see, their probability density is concentrated there. One might ask, where have the rest of the (originally) infinite states in a given Landau level gone? Fig. 2gives the impression that there were csuch states, and about three quarters of them have remained degenerate, while the remaining one quarter have had their energies elevated due to the edge. A clue that this interpretation is incorrect arises from something barely discernible on this plot but clearly visiblein the inset, which is that the increase in eigenvalues at the ends of the plateaus becomes increasingly less smooth for successive plateaus. The reason for this roughness is made clear in Fig. 3(a), where now the energy levels are plotted as a function of ‘. Again, we emphasize that we are fortunate that this simplification is possible here because different matrices were diagonalized for individual values of ‘.I nF i g . 3(b), we show the same energy levels vs r ‘=a/C17ffiffiffiffiffiffiffi ‘=cp to show how the maximum in the wave function moves towards the edge as ‘increases (with the cautionary proviso that r‘ represents the maximum only for cases well away from the wall—which is why r‘=aends up exceeding unity in this plot—but the qualitative trend is correctly portrayed). Figure 3illustrates that the eigenenergies smoothly increase when plotted vs ‘. The raggedness that exists in Fig. 2is due to the toggling back and forth between edge states of similarFig. 2. Dimensionless eigenvalues ( En=ð/C22h2=ð2ma2ÞÞ) as functions of eigen- value number nfor various values of c. Here, nis used as a label for the hori- zontal axis to indicate that the energies are plotted in ascending order anddoes not refer to any previously defined quantum numbers also labeled with n. The truncation size of the matrices produced is N¼200 in all cases. Note that these plots were produced as a result of generating and diagonalizing matrices for ‘2½ /C0 200;200/C138followed by aggregating the resulting eigene- nergies acquired from all matrices and organizing them in ascending order. This procedure results in over 80 000 eigenvalues, and the lowest 200 of these for each value of care shown here. Note that the degeneracy number increases with the value of caccording to our expectations, and in fact, there are approximately (precisely for the LLL energies) ceigenvalues occurring before the next plateau begins. These plateaus occur at dimensionless ener- gies given by those expected from the free space Landau levels: E n=ð/C22h2=ð2ma2ÞÞ ¼ 2cð2nLþ1Þ, for nL¼0;1;2:::. Shown in the inset are energies occurring after the first, second, and fourth plateau for c¼50;100, and 150, respectively. Notice the disorderly behavior of the energies forc¼50;100, whereas energies between the first and second plateaus for these curves are ordered quite smoothly. 989 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 989energy from the various Landau levels, uniquely identified in Fig.3(and not in Fig. 2). The Landau levels are distinct and clear, except now they carry on indefinitely, to the left inFig.3(a)and to the right in Fig. 3(b). As already stated and further corroborated below, the increase in energy is due tothe confinement of the edge. The “toggling” is evident if oneexamines the states at a dimensionless energy value like/C251400 in Fig. 3(a). The next energy level that would be placed in Fig. 2would come from one of the two branches either emanating from the first or the second Landau level,and these would go back and forth. At a higher value, say/C252000, toggling would occur between three branches and so on, leading to increasing “raggedness” in Fig. 2as the energy goes up, which is precisely what we observed. The standard way with which to summarize the impact of a magnetic field on the energy spectrum is to plot the eigen-values as a function of magnetic field. This is called aFock–Darwin spectrum—see the Appendix for further expla- nation. Following Lent, 4we show this spectrum in Fig. 4, where a condensation of levels occurs.3Ascincreases, more and more levels coalesce as the free-space Landau levelsbecome more applicable since the wave functions becomemore compact and away from the edge. The thick dashed redlines (color online) indicate the expected Landau levels indicative of free space. To the far left, with the exception of ‘¼0, the eigenvalues come in pairs, corresponding to posi- tive and negative ‘, which are degenerate in the absence of a magnetic field. Such condensation (seen in the far right onthis plot) is absent with parabolic confinement, which is pre-sumably why these are called “Landau” levels and not“Fock” levels.Figure 4shows also illustration of the toggling phenome- non we have just discussed. Consider c>30. For fixed c,i f we move up the plot in energy (vertically), we go from acondensed band of (quasi) degenerate states to a region ofnon-degenerate edge states that all belong to the lowest (con-fined) Landau level (the lowest band in Fig. 3). Still increas- ing in energy, we reach another degenerate set beforereaching non-degenerate edge energies once more. However, Fig. 3. Lowest 1000 eigenenergies plotted vs (a) ‘and (b) r‘=a¼ffiffiffiffiffiffiffiffiffiffi j‘j=cp , forc¼150 and matrix truncation size N¼200. In both plots, each smooth energy band corresponds to a distinct Landau quantum number nL¼nrþð j‘jþ‘Þ=2, as labeled. ‘is the angular momentum quantum number, and r‘is the center of the probability density of the corresponding eigenstate as long as the eigenstate is not too close to the walls. This means that r‘gives us a good idea about where the eigenstates corresponding to the eigenenergies are located in real space, as long as r‘=a/H113510:9. Energies with r‘=a>1 should be interpreted as corre- sponding to eigenstates piling up near the boundary. This plot (as opposed to Fig. 2) makes clear that the original Landau level with infinite degeneracy (all negative values of ‘) has very large j‘jstates piled up near the boundary, with increasingly higher energies, thus breaking the original free Landau (infinite) degeneracy. As is clear from the figure, though, a quasi-degeneracy of order c(proportional to the applied magnetic field) remains. Notice that in (a), there is an additional positive ‘state as we jump from one Landau level to the next. This observation is in agreement with the expression derived for the symmetric gauge energies in free space. Fig. 4. The first 100 levels of the Fock–Darwin spectrum for the circularwell. Also shown (with red dashed lines (color online)) are the free Landau levels. In this case, in contrast to the spectrum shown in Fig. (19)with para- bolic confinement, the levels condense to these free Landau levels as the magnetic field increases, as illustrated by Lent (his Fig. 1) (Ref. 4). These condensed bands are the (quasi) degenerate plateaus seen for fixed cin Fig.2. The discrete levels that fill the gaps between condensed bands are the edge states that come after plateaus, also seen in Fig. 2. 990 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 990now, edge energies from both nL¼0 and nL¼1 fill the energy gap between degenerate bands. This pattern repeats as we increase in energy, with the nth gap containing edge states from all confined Landau levels with nL¼n/C01 and lower. This phenomenon has been referred to as “level cross-ing” 7in the context of Fock–Darwin states. The ordering of these crossed levels becomes less smooth as we go to higher level gaps. Level crossing is conceptually identical to theedge-state toggling that we discussed earlier. C. Probability densities We have been discussing edge states and using the param- eter r ‘/C17lBffiffiffiffiffiffiffi 2j‘jp (recall lB/C17ffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C22h=ðeBÞp is the magnetic length) to indicate the location of the wave function. Ofcourse, r ‘is the radius of the maximum of the wave function (from some specified origin) in free space only, so here we examine the actual probability densities. Recall that theprobability density is gauge invariant, except in cases of degenerate wave functions. Here, the degeneracy is removed because of the confining cylinder, but as already discussed, a“practical” degeneracy remains, so the statement aboutgauge invariance is no longer true, 17as we shall see below in Sec.IV. In Fig. 5, we show contours of jWnr;‘ðr;/Þj2forc¼150 for various values of nrand‘. These indicate that the wave function amplitude moves out from the center as j‘j increases, as noted analytically for the free space result. Moreover, it is clear that the number of nodes increases as nr increases, and in fact, this quantum number tells us thenumber of nodes. We should note that Fig. 5looks very “orderly” when it comes to the progression of the number of nodes (as one moves to the right, increasing nr) and the radius of maximum amplitude (as one moves down, increas- ingj‘j). This is because we have complete control of the quantum numbers, as stressed with respect to Fig. 3. This scenario is different when we simply order all the states according to their energies (this procedure was followed to produce Fig. 2). This will make the progression of states somewhat disordered, for a reason entirely different from the disorder already noted in connection with Fig. 2. That disor- der was visible to the eye. Now, we are referring to the quasi-degenerate states in the plateau regions, where the dis-order is not visible but nonetheless exists at a level lower than 10 /C016(so, it is invisible even to the computer, using double precision accuracy). For this reason, even in the pla- teau regions in Fig. 2, the states may be “out of order,” and a state with a higher j‘jmay be ranked lower (according to energy) compared with a lower j‘jstate. To avoid this, we artificially include a very shallow parabolic trap, whose sole purpose is to break the remaining quasi-degeneracy so that the progression of states is orderly. The resulting contour plots are shown in Fig. 6. We will discuss in more detail the introduction of this degeneracy-breaking potential in Sec. IV C. Including it as we do here does not change any of the results concerning energy, which are visible to the eye. Instead, this potential Fig. 6. Contour plots of jWnðr;/Þj2as a function of eigenvalue number nfor various values of c, as indicated at the top of each column. Trends similar to those shown in Fig. 5are evident, but in addition, one can note changes in ‘ and nrquantum numbers. For example, jW50j2forc¼50 has nr¼0 and ‘¼þ1 and is located at the start of the second plateau visible in Fig. 2.I ti s identical to jW2ðr;/Þj2forc¼50 displayed here in the second row. The wave function W2ðr;/Þhas quantum numbers nr¼0 and ‘¼/C01 (recall that jWj2was insensitive to the sign of ‘). Note that this plot is orderly and sym- metric because we have artificially broken the remaining quasi-degeneracy apparent in Figs. 2and3with a very shallow parabolic trap, centered at the middle of the cylinder. This shallow trap changes nothing that is visible to the eye, but it does remove the remaining quasi-degeneracy so that a proper ordering is established. See the text and Sec. IVfor further explanation of this additional potential, which has the sole purpose of removing the degeneracy. Fig. 5. Contour plots of jWnr;‘ðr;/Þj2for various values of nrand‘, for c¼150. The number of nodes increases with increasing nr, more clearly seen for low values of j‘j, and the radius of the maximal amplitude increases with increasing j‘j. The notion that large j‘jstates are edge states is clear from this plot, as the entire probability density resides on the circumference for these states. 991 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 991separates quasi-degenerate energy levels at the 10/C010level, so the computer “knows” which states are supposed to come first and does not resort to a somewhat random linear combi- nation, as happens when there is a degenerate subspace ofsolutions. D. Probability current in a circular dot In the Integer Quantum Hall Effect, edge states (eigen- states localized along a boundary) are often cited as a key ingredient to explain the quantization of Hall conductance. 13 Classically speaking, the idea is that if an electron encoun- ters a boundary while undergoing a cyclotron orbit, it will reflect off the wall and continue its orbit. These reflectionscause the electron to undergo “skipping” orbits along theboundary, resulting in chiral edge currents along the bound- ary of the sample. One would expect that edge states would contribute non-zero current to the system, while states in thebulk would not contribute any current. Lent 4illustrated such currents, and since the matrix mechanics technique allows for easy numerically exact calculations of the probabilitycurrent, we will illustrate them here as well. Emphasis will be placed on current densities corresponding to eigenstates of the lowest Landau level (LLL). The probability current density Jof an electron in an eigenstate Wimmersed in a magnetic field represented by a gauge choice Ais given by 12,22 J¼1 2mW/C3pW/C0WpW/C3þ2ejWj2A/C2/C3 : (17) With this equation and Eq. (4), it can easily be shown that Jis a gauge-independent quantity for non-degenerate eigenstates. However, the free-space infinite degeneracypresent in this problem relates the eigenstates of the two gauges in a not so trivial manner, as mentioned earlier. Consequently, the expressions for the current density ineach gauge radically differ. We will return to the issue of numerical degeneracy when we investigate probability cur- rents in a square quantum dot. Here, we present an analysisof the behavior of the current density in the symmetric gauge. Since we are working in two dimensions, Jis a surface current density with dimensions IL /C01. Using the symmetric gauge ~As, the probability current density associated with an eigenstate Wnr;‘ðr;/Þ¼ei‘/=ffiffiffiffiffiffi 2pp wnr;‘ðrÞis, in polar coordinates, Jnr;‘ðrÞ¼jwnr;‘ðrÞj2 2p/C22h‘ mrþ1 2xcr/C20/C21 ^/ ¼jwnr;‘ðrÞj2 2pKzðrÞ mr^/; (18) where KzðrÞ¼ /C22h‘þ1 2mxcr2is the kinetic angular momen- tum in the ^zdirection. For further study, we focus on the cur- rent density in the LLL with nL¼0. The LLL consists of states with nr¼0 and ‘/C200, as is further explained in the Appendix . To predict the behavior of the LLL current density in the bulk of our dot, we construct an expression for Jin free space. It readily follows from Eq. (18) that the current den- sity in the LLL is given byJðFSÞ 0;‘ðrÞ¼1 2prffiffiffi 2p lB/C18/C192j‘je/C0r2=2l2 B l2 Bj‘j!/C22h‘ mrþ1 2xcr/C20/C21 ^/;‘/C200: (19) We have added a superscript “(FS)” to emphasize that this expression holds in free space only. Note that this expressionis still useful to understand the behavior of bulk states in aquantum dot only because we have located the origin for thesymmetric gauge precisely at the center of the quantum dot. This expression for the probability current will only apply if the wave functions remain essentially zero near the edge ofthe quantum dot, which will be true as long as j‘jis small, or more precisely, r ‘/C28a. Examining the wave function in this manner is a way of defining what a “bulk” state is forthis system. One can verify that JðFSÞ 0;‘is localized near r/C25r‘¼lBffiffiffiffiffiffiffi 2j‘jp but vanishes precisely at r¼r‘. With the exception of ‘¼0, it can also be shown that r¼r‘is also a point of inflection for the current density. We expect thisbehavior for lower ‘bulk states in our confined system, which we plot along with edge states in Fig. 7. Through the inset in Fig. 7, we see that the total area bounded by the current density of the bulk states (aside from‘¼0) appears to vanish (equal areas of positive and nega- tive current densities cancel one another) due to the inflec-tion point present at r¼r ‘. This cancellation does not occur once r‘approaches and exceeds the radius aof the dot, and so, edge current densities bound the non-zero area. Thebounded area in Fig. 7is the net current flow in the ^/direc- tion, and as mentioned earlier, we would expect that this cur-rent is non-zero only near the edge. The notion of non-zeroedge currents and the results of Fig. 7appear to be in qualita- tive agreement, but this agreement can easily be quantified.To obtain the current from a surface current density, we inte-grate it with respect to a scalar differential strip orthogonalto the direction of flow. Thus, the state current I nr;‘of a given eigenstate in our confined system is given by Fig. 7. Dimensionless probability current density magnitude in the confined LLL plotted vs r/afor various ‘/C200 and fixed c¼150. Convergence was achieved with N¼200. Note how the ‘¼0 state has current density concen- trated near the origin, while other (negative) ‘states have current density located (naturally) where their probability density dominates but with an inflection point (as predicted from Eq. (18)right at r=a¼r‘=a¼ffiffiffiffiffiffiffiffiffiffi j‘j=cp ,a s long as the probability density is well away from the edge). These states are representative of bulk currents and behave according to Eq. (19). The larger j‘jstates are representative of edge currents, for which the current density is concentrated near the edge and flows exclusively in the negative ^/ direction. 992 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 992Inr;‘¼ða 0Jnr;‘ðrÞdr: (20) Since the current densities are quite localized, we can accu- rately estimate what to expect for a state far from the edgeusing the free space expression for current density, Eq. (19). The free space current for an LLL state is calculated via IðFSÞ 0;‘¼ð1 0drJðFSÞ 0;‘ðrÞ ¼ð1 0dr1 2prffiffiffi 2p lB/C18/C192j‘je/C0r2=2l2 B l2 Bj‘j!/C22h‘ mrþ1 2xcr/C20/C21 ^/; ‘/C200: (21) Defining IðFSÞ 0;‘/C17IðFSÞ 0;‘/C1^/and using current units /C22h=ð2pma2) for convenience, we evaluate the integral in Eq. (21) as follows: IFSðÞ 0;‘/C30/C22h 2pma2/C17iFSðÞ 0;‘¼ð1 02cj‘jþ1x2j‘j j‘j!e/C0cx2‘ xþcx/C20/C21 dx ¼2cj‘jþ1‘ j‘j!ð1 0x2j‘j/C01e/C0cx2dx þcð1 02cj‘jþ1x2j‘j j‘j!e/C0cx2xdx |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼1 ¼c‘ j‘j!ð1 0xj‘j/C01e/C0xdx/C18/C19 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼Cj‘jðÞ ¼ j‘j/C01ðÞ !þc ¼csign ‘ðÞþ1/C2/C3 )iFSðÞ 0;‘¼2cif‘>0 c if‘¼0 0i f ‘<0:8 >< >:(22) Therefore, the state current magnitude in free space is given by IðFSÞ 0;‘¼fc if‘>0 fc=2i f ‘¼0 0i f ‘<0;8 >>< >>:(23) where fc/C17xc=ð2pÞ. Thus, in our confined system, we expect the bulk states in the LLL to contribute zero current,with the exception of the ‘¼0 state. The results are pre- sented for positive ‘(but n r¼0) states as well. These states exist outside the LLL, but since we did not have to restrict the sign of ‘to proceed with our calculation, we show results for both signs for completeness. Note that these positive ‘ states contribute current that flows in the positive ^/direc- tion. In contrast to the negative ‘states, the positive ‘states all circulate in the direction expected classically (through theLorentz force) but are energetically unfavourable. Numericalresults for probability current in our confined system areshown in Fig. 8. Finally, we show a vector plot of the current for a LLL bulk state in (a) ( ‘¼/C025) and a LLL edge state in (b) (‘¼/C0300) in Fig. 9, with the plots on the right showing thesame result for the region in the red square box (color online) expanded in more detail. Our numerical results are in qualitative agreement with those shown by Lent, 4although he utilized smaller values of c(hisb/C172c), and he showed pictorial vector plots for fairly small magnitudes of ‘(which he calls m). He also provides a nice description in terms of classical orbits, which we willnot repeat here; the reader is referred to Ref. 4for this description.Fig. 8. Dimensionless probability current (Eq. (20)) vs angular momentum quantum number ‘forc¼150. With the exception of ‘¼0, which contrib- utes precisely ccurrent, bulk states with negative values of ‘contribute zero current to the system, while those with positive values of ‘contribute a value of 2cto the current. This result is in accordance with what we calculated for the free space current in Eq. (23). For large values of j‘j, we see that edge states contribute current that flows in the negative ^/direction for negative values of ‘and in the positive direction in excess of 2 cfor positive values of‘.A sj‘jincreases further, the edge states become more localized at the boundary and contribute increasingly more current (in either direction) to the system. Note that only states with ‘/C200 belong to the lowest Landau level. Fig. 9. A vector field plot of the probability current for a bulk state in (a) (‘¼/C025,nr¼0) and an edge state in (b) ( ‘¼/C0300, nr¼0). In both cases, we used c¼100, and convergence was attained with a matrix size of N¼200. The expanded portions show what should already be clear from Fig.7. Although the edge state in (b) contributes non-zero current along the boundary, it does not exhibit the classically expected skipping orbit trajectory. 993 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 993IV. LANDAU LEVELS IN A SQUARE QUANTUM DOT Relatively, little work has been done to date for a square geometry. In this section, we present results in both the symmet-ric and Landau gauges and illustrate the difficulties encounteredfor large fields (or samples) due to the practical degeneracy thatremains in these cases. Since the square geometry is rather diffi-cult in either gauge, we thought it worthwhile to provide results for the same geometry in both gauges. That this works provides a concrete illustration of the remarkable machinery of gaugeinvariance, as now the gauge choice results in a very non-sym-metric-looking potential. In fact, as we will shortly see, it doesnot work for large values of G, for the reasons just described above associated with the numerical degeneracy, i.e., non-degenerate levels that the computer is unable to recognize. Aswas the case with circular confinement, our very slight perturb-ing confining potential will fix the problem (i.e., lift the degener-acy) so that identical results are achieved in both gauges. Wehave included a sample code in the supplementary material(Ref. 33) that shows how the calculation to be described can be implemented in MatLab. A. Matrix element calculation 1. Symmetric gauge The confining potential is defined as Vðx;yÞ¼0i f 0 <x<aand 0 <y<a 1 otherwise :/C26 (24) Here, ais the length of a side of the two-dimensional infi- nite square well representing the square quantum dot.Given this potential, a convenient set of normalized basisstates is / nx;nyðx;yÞ¼h x;yjnx;nyi¼2 asinnxpx a/C18/C19 sinnypy a/C18/C19 ; (25) where nxandnyare positive integers. For convenience, we modify the symmetric gauge ASsuch that it is centered in the potential well, A0 S¼B 2x/C0a 2/C18/C19 ^y/C0y/C0a 2/C18/C19 ^x/C20/C21 : (26) With this gauge A0 S, our Hamiltonian takes the form H¼p2 2me/C0i/C22hxc 2x/C0a 2/C18/C19 @y/C0y/C0a 2/C18/C19 @x/C20/C21 þ1 8mex2 cx/C0a 2/C18/C192 þy/C0a 2/C18/C192"# þVðx;yÞ; (27) where we use meas the mass of the electron to avoid confu- sion with matrix indices. Following the steps leading to Eq.(10), we then expand the eigenstates of (27),jWi,a s jWi¼X nx;nycnx;nyjnx;nyi: (28)Thus, in energy units of E0/C17/C22h2p2=ð2mea2Þand using n/C17ðnx;nyÞandm/C17ðmx;myÞfor short, our dimensionless matrix elements, hn;m/C17Hn;m=E0, for the Hamiltonian for this problem are given by hn;m¼dnx;mxdny;myn2 xþn2 yþG2 21 3/C01 ðpnxÞ2/C01 ðpnyÞ2 !"# þ2G p/C18/C192 ðð1/C0dnx;mxÞdny;mygeðnx;mxÞ þð1/C0dny;myÞdnx;mxgeðny;myÞÞ /C0i16G p3ð1/C0dny;myÞð1/C0dnx;mxÞ /C2goðny;myÞfðnx;mxÞ/C0goðnx;mxÞfðny;myÞ/C0/C1/C2/C3; (29) with geðn;mÞ0i f nþm¼odd 1 ðn/C0mÞ2/C01 ðnþmÞ2ifnþm¼even;8 >< >: (30) goðn;mÞ¼1 ðn/C0mÞ2/C01 ðnþmÞ2ifnþm¼odd 0i f nþm¼even;8 >< >: (31) foðn;mÞ¼nm n2/C0m2ifnþm¼odd 0i f nþm¼even;8 < :(32) where G¼Ba2 h=e/C17U U0; (33) as defined earlier, and the subscripts “e” and “o” in the defi- nitions, Eqs. (30),(31), and (32), serve to remind us that these are non-zero for even or odd sums of the indices only,respectively. 2. Landau gauge We now use the (square-well-centered) Landau gauge given by A 0 L¼Bx/C0a 2/C18/C19 ^y: (34) With this gauge and the confining potential defined in Eq. (24), the Hamiltonian is H¼p2 2me/C0i/C22hxcx/C0a 2/C18/C19 @yþ1 2mex2 cx/C0a 2/C18/C192 þVðx;yÞ: (35) The Hamiltonian, again given in units of /C22h2p2=ð2mea2Þ, becomes 994 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 994hn;m¼dnx;mxdny;myn2 xþn2 yþG2 31/C06 ðpnxÞ2 !"# þ8G p/C18/C192 ð1/C0dnx;mxÞdny;mygeðnx;mxÞ þi32G p3ð1/C0dnx;mxÞð1/C0dny;myÞgoðnx;mxÞfoðny;myÞ; (36) where the functions ge,go,a n d fowere all defined earlier. This is a very different matrix from that generated for the symmetricgauge, and moreover, it is very asymmetric in xandy.Y e t ,w e expect to obtain the same results as in that gauge. B. Eigenenergy spectra The eigenenergies calculated in both gauges were found to be identical to numerical precision, so the results below arerepresentative of both gauges. This outcome is expected of course, as energy is an observable quantity and so must be gauge-invariant, regardless of how different the matrix repre-sentations look in the two gauges. This is because theHamiltonians in each gauge choice are unitary transforma-tions of one another, as shown in Eq. (3). Note that a lower truncation value is necessarily used here since matrices with block matrices as elements have to be constructed and diago- nalized, which is much more computationally intensive thanin the problem with circular confinement. Thus, a truncationofN¼100 for one of the indices, say n x, implies a total trun- cation size of N2for both nxandny. We will refer only to the block matrix truncation size ( N) in the following results (so, N¼100 values of nxrequires diagonalization of a 10 000 /C210 000 matrix). Some of the lowest computed eigenvalues are shown in Fig. 10for various values of Gas indicated. This figure shares many things in common with its circular counterpart,Fig. 2, and so it should since for bulk-like states, we have argued that they resemble the free space results and henceare not affected by the edge, i.e., the electrons do not even know if they are confined in a circular or square geometry. However, the concept of a good quantum number ‘does not exist here, at least not explicitly. So, we do not have thebenefit of a figure like Fig. 3for the square geometry. One key similarity is the role of the ratio of the total flux to theflux quantum, which gives precisely the number of statesbetween plateaus, regardless of geometry. In Fig. 11,w e show the Fock–Darwin spectrum for the case of square con- finement. As with circular confinement, the energy levels eventually condense and become coalesced into the degener-ate Landau levels, as indicated. Both the circular and squareconfinement show this condensation phenomenon, whereasthe parabolic confinement does not. Like its circular (Fig. 4) and parabolic (Fig. 19in the Appendix ) counterparts, we see that this spectrum exhibits level crossing between the con- densed bands at the free space Landau energies. However,the energies vary in a more complicated manner withincreasing flux G, in contrast to results seen in both the para- bolic and circular dots. C. Probability densities While the Hamiltonian in both gauges may be related by a unitary operation, in free space, the wave functions ofeach gauge are not related so trivially. This problem arises because of the infinite degeneracy present in the free spaceformulation of the problem. In confined space, however,this infinite degeneracy is broken. Without this degener-acy, one would expect that Eq. (4)holds true and that the probability densities should be identical for both gaugechoices. However, as we have seen previously with thecircular dot, a finite quasi-degeneracy persists to numericalprecision. Here, we detail how to get around this issue andfind agreement between the probability densities in eachgauge.Fig. 10. Eigenvalues shown in ascending order vs quantum number for G¼50, 100, and 150. This figure is the square version of its circular counter- part, Fig. 2, and we do not have the counterpart to Fig. 3,w h i c hw o u l da l l o w us to sort out here the two quantum numbers that are present in that case. Just as in the circular well, the energies plateau at free space Landau level ener- gies, which are given by EnL/C14 /C22h2p2=2ma2/C0/C1 ¼4G=pnLþ1 2/C0/C1 . Two noticeable features are the plateaus, as in the circular case, especially for large values of G, and the number of states present between plateaus, precisely Ghere, just as it was cfor the circular case. Also noticeable is a similar raggedness for energy values beyond the higher plateaus, as seen clearly for energies occur-ring past the fourth plateau for G¼50 in the inset. This raggedness is presum- ably due to the same toggling/level crossing phenomenon that we could explicitly identify in the circular case. We used N¼150 here. Fig. 11. The Fock–Darwin spectrum for the two-dimensional infinite square well. Here, N¼100. This plot shares with its counterpart for the circular well, Fig. 4, the idea that for a large enough field, the levels condense to a set of degenerate Landau levels (indicated by thicker dashed red lines (color online) for the first three levels). For weaker fields, the levels are fairly dis- ordered, reflecting the more complicated geometry of the square. Level crossing occurs between the condensed energy bands. Note that the zero field energies no longer consistently come in pairs, as the degeneracy pattern for an infinite square well is more complicated than that for an infinite circu- lar well. 995 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 9951. Symmetric gauge As mentioned previously, for the bulk-like states, we expect the results for the square well to be very similar to those of the circular well, simply because the magnetic field keeps the electron sufficiently contained in the central regionof the well so that the electron is not sensitive to the geome- try of the confining potential. In Fig. 12, we show contours of the probability density for a variety of values of G. This plot is to be compared with Fig. 6, which showed probability densities for the circular well. In that plot, however, we could specify both n rand‘, whereas here we can only order the plots according to the value of the energy. Some featuresof this plot are immediately apparent. First, results with small quantum number and small Gare consistent with what we found for circular geometry. This is expected. More excited states become fairly diffuse throughout the square and retain the symmetry of the square; this is also expected. However, as Gincreases, the results become somewhat irregular. The prob- lem here was already alluded to in the case of the circular well: a high degree of practical degeneracy remains as Gincreases. We fixed the problem with little explanation in that case, sonow we discuss this issue in more detail. Because of the remaining practical degeneracy, the diago- nalization routine ends up picking some linear combination of these numerically degenerate states and ordering them in some fashion. The arbitrary linear combination easily leads to a wave function that does not have the symmetry of theconfining potential, as is evident in Fig. 12. We will see the same problem emerge in the Landau gauge, which does not have the symmetry of the confinement potential, so it is pru- dent to emphasize that this aspect (having or not having a gauge with the symmetry of the confining potential) is not so important. Here, where the gauge choice is the more sym- metric one, we still encounter this problem, and it is because of the practical degeneracy (i.e., energies that are within <10 /C016of one another) that remains in the solutions. The practical degeneracy remains because, for large values of G, there is enough space in the confined potential for the center of the wave function to be arbitrarily located. We have resolved this problem by adding an additional perturbative confining potential with the symmetry of the square. Our choice is gauge invariant and is given by H0¼1 2mex2 0x/C0a 2/C18/C192 þy/C0a 2/C18/C192"# ; (37) where x0is the characteristic frequency to characterize the perturbation potential. It is important to emphasize that this additional potential is minute: typically, ðx0=xcÞ2/C2510/C06 forG¼50 so that the energies are not really affected at any level the eye can detect, but the degeneracy is lifted suffi- ciently to allow the diagonalization algorithm to properly and unambiguously order the otherwise quasi-degenerate eigenstates. Most importantly, the centers of all wave func- tions are now correctly placed at the center of the square. When this very weak potential is included, we obtain the results for the probability densities shown in Fig. 13, where now the results look correct and more sensible (the same per-turbative potential was actually used in Fig. 6). For the sake of completeness, we include the required matrix elements for our current problem, ðh 0Þn;m¼H0 n;m E0¼p2/C152 4" 1 12dnx;mxdny;my1/C06 ðpnxÞ2 þ1/C06 ðpnyÞ2! þ2 p2ðdny;my1/C0dnx;mx ½/C138 /C2geðnx;mxÞþdnx;mx1/C0dny;my/C2/C3 /C2geðny;myÞÞ# ; (38) where geðn;mÞis as defined in Eq. (30) and /C15/C17/C22hx0=E0 ¼ð4G=pÞðx0=xcÞ; with this definition, /C152provides the rela- tive energy scale of the perturbing potential. These Hamiltonian matrix elements should be added to the previ- ous ones given in Eq. (29). 2. Landau gauge Without using the perturbing potential in the Landau gauge, the probability density does not agree with that obtained in the symmetric gauge. The results for the Landau gauge (without the perturbing potential) are shown in Fig. 14, where significant discrepancies with Fig. 12are apparent, particularly for larger values of G. The results of Fig. 14show somewhat random character, particularly for larger values of G,i na manner similar to that seen earlier (Fig. 12) in the symmetric gauge, and for the same reason—persistent numerical degen- eracy. Nonetheless, the addition of the perturbing potential, Fig. 12. Probability density, jWnðx;yÞj2, contour plots for various values of eigenvalue number n, as ordered by the diagonalization subroutine. Here, the confining region is a square, outlined in black. The results only look sen-sible in the first two columns; the low quantum number results resemble those obtained from the circular geometry, as expected, and then, the proba- bility densities become more square-like, reflecting the geometry of the con- finement, also as expected. As one increases G, however, the results become less understandable and appear to be wrong. The difficulty, as discussed in the text, is the high degree of degeneracy that remains when large values of Gare used. We used N¼150 for all the square well results. 996 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 996Eq.(37) with /C15¼10/C03, solves the problem as expected, and the results identical (i.e., numerically, to about five digitaccuracy) to those of Fig. 13are attained. These are shown in Fig. 15. D. Probability current in a square dot Although probability current density is a gauge-invariant quantity, the infinite degeneracy in free space results in twovery different looking expressions for the Landau gauge and symmetric gauge. We have already resolved this issue for the probability densities in confined space, and so, here weconfirm that the current densities in each gauge are identical as well. The expression for the probability current density is given by Eq. (17). Adopting the expansion of the wave function in the square well (28) with basis states (25) and arbitrary gauge choice A, the probability current density of the nth total quantum state J ncan be expressed as Jnðx;yÞ¼/C22h meImðW/C3 nHnÞþeA /C22hjWnj2;/C20/C21 where (39) Fig. 13. Probability density, jWnðx;yÞj2, contour plots for the same parame- ters as in Fig. 12but now with an additional confining potential given by Eq. (37) with x0=xc¼1:3/C210/C06. Now, there is a more systematic progres- sion; for example, for the ground state as one increases G, the wave function becomes more localized at the center of the square as Gincreases. In fact, the result would be essentially the same for a circular geometry. As we move down, i.e., to more excited states for a given G, the wave function becomes more delocalized and takes the shape of the edges in the square; these are very clearly edge states. One notable exception is the result for jw50j2with G¼50, where the wave function becomes more concentrated; the difference here is that we have suddenly moved on to the next Landau level (recall how we could track this phenomenon explicitly in the case of the circular well). Note that in general the results compare quite well with those of the circular well (see Fig. 6), except the obvious squareness of the contours of the higher excited states. Fig. 14. Contour plots of the probability density, jWnðx;yÞj2, for various val- ues of n, calculated with the Landau gauge, and as ordered by the diagonali- zation subroutine. The results agree with those in the symmetric gauge, Fig. 12, only in the first two columns. Once again, the reason for the discrepancy in the remaining columns is the degeneracy that is still present, and this defect is remedied in the same manner as with the symmetric gauge results. Fig. 15. Contour plots of the probability density in the Landau gauge withthe perturbative trap included. While these results are identical to those obtained in the symmetric gauge in Fig. 13, we have repeated them here to emphasize that identical results have been achieved with different gauges. 997 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 997Wnðx;yÞ¼2 aX1 mx;my¼1cðnÞ mx;mysinmxpx a/C18/C19 sinmypy a/C18/C19 ;and (40) Hnðx;yÞ/C17rWnðx;yÞ¼2 aX1 mx;my¼1cðnÞ mx;mypmx a /C2cosmxpx a/C18/C19 sinmypy a/C18/C19 ^x þ2 aX1 mx;my¼1cðnÞ mx;mypmy asinmxpx a/C18/C19 /C2cosmypy a/C18/C19 ^y: (41) Note that the eigenvector coefficient corresponding to the nth eigenvector, cðnÞ mx;my, is, in general, a complex number. For the symmetric [Eq. (26)] and Landau [Eq. (34)] gauge choices, the probability current density is given by JðSÞ nðx;yÞ¼/C22h mea/C20 aImðW/C3 nHnÞþpGjWnj2 /C2x a/C01 2/C18/C19 ^y/C0y a/C01 2/C18/C19 ^xÞ/C21 ; (42) for the shifted-symmetric gauge and JðLÞ nðx;yÞ¼/C22h mea/C20 aImðW/C3 nHnÞþ2pGjWnj2x a/C01 2/C18/C19 ^y/C21 ; (43) for the shifted-Landau gauge. While the structure of the equations for the two gauge choices [Eqs. (42) and(43)]i s quite different, the probability current density is a gauge- invariant quantity, and we have confirmed that both results are identical. Equipped with Eqs. (39)–(43) , it is straightforward to eval- uate the probability current density for a given eigenstate. Two vector field plots of typical results are shown in Fig. 16 [in units of h=ðmea3Þ]. We see that (a) is a bulk state, while (b) is an edge-state. In fact, the former result is qualitatively indistinguishable from that attained for a circular geometry in Fig. 9, as is apparent from the result. In all cases, we have also used a perturbing potential as described earlier, with /C15¼10/C03. In (b), there is a clear difference from the circular case, and the current tends to follow the geometry of the square boundary. While one may suspect from this observa- tion that edge state currents simply follow the geometry ofthe boundary, a closer look at the expanded portion in (b) shows a very interesting structure. Rather than smoothly fol- lowing the geometry of the boundary at the corners of thesquare well, the probability current density appears to form stationary vortices rotating in a direction opposite to that of the nearby principal current density flow. This result for thesquare geometry is a very different scenario than that encountered with the circular geometry and is reminiscent of the corner modes noted recently for topological insulators. 23 Here, these are not zero energy modes, but appear as a sim- ple consequence of the square geometry, and occur at a large quantum number. Thus, they may in fact be the precursor ofsemi-classical behavior (e.g., the skipping orbits described in Ref.4). Further investigation is clearly required.One can also characterize the probability current density at slices through the sample. Defining J x;n/C17Jn/C1^xand Jy;n/C17Jn/C1^y, we show Jx;nas a function of y(a vertical slice atx¼a=2) in Fig. 17through the center of the square. For low quantum numbers, the result will resemble that of thecircle, shown in Fig. 7. Here, we cannot classify the states according to their ‘quantum number, as we did in that case. Also, note that Fig. 7displays results across a radius, i.e., half the sample, whereas for the square, Fig. 17shows results across the entire sample and therefore has an inherent Fig. 16. A vector field plot of the probability current density with G¼100 for a bulk state in (a) ( n¼10) and an edge state in (b) ( n¼95). Further details of the region enclosed by the red square (color online) are provided in each case to the right. Convergence was attained with a matrix size of N¼150, and /C15¼10/C03was used for the perturbing parabolic trap. Although this result was generated with the symmetric gauge, we confirmed that an identical result is found for the Landau gauge. In the first case, an inner shell of clockwise-circulating current is followed by a concentric shell of counter-clockwise-circulating current; this result is very similar to what was found with circular geometry (see Fig. 9(a)). However, the edge state current density is very different. As in the circular case, it is primarily a clockwise- circulating current, but the corners cause a vortex-anti-vortex pair to be created, as is clear from the blow-up on the right. Fig. 17. The x-component of the current density Jx;nas a function of ytaken through x¼a=2. This line represents a vertical slice through the center of the sample. We show results for various eigenvalue numbers n, as labeled. The results obtained in the Landau (symmetric) gauge are shown as curves (points) and obviously agree with one another. Here, we used G¼100 and N¼150. Also, note that Jy;n/C240 over this slice. 998 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 998asymmetry. Of course, this asymmetry is present because, in spite of the square geometry, the current is circulating. Forthe same reason, J y;nalong this vertical slice was found to be zero (within numerical noise). Slices related through symme-try operations yield entirely equivalent results. For example,J y;nplotted across a horizontal slice at y¼a=2 looks identi- cal to Jx;natx¼a=2. As seen before in the circular well, the current contribution for a particular state becomes non-zeroas we move from the bulk region to the edge region in Fig.17. Moving from y¼a=2t oy¼a, we see that the total cur- rent over this region vanishes for the bulk states (with theexception of the ground state) due to a sign change across a node ( J x;10, for example, has equal amounts of positive and negative current densities between y¼a=2 and y¼a). States near the edge ( Jx;95, for example) do not experience this nodal sign change, resulting in a non-zero current contri-bution, as indicated by a positive-only current density neary¼a. Also, note that the edge-state current has opposite sign on opposite edges of the confinement, confirming its inherentchirality. In Fig. 18, we show results across another slice, a diagonal across the square from the upper left to the lower right. Weshow only J y;n, but in this case, Jx;nis simply the mirror of Jy;nabout the vertical axis. This plot is not so different from the previous one but can give us a glimpse of the vorticesseen only at the corners of the square. The occurrence ofthese vortices clearly merits a further study. E. Other gauge possibilities Finally, we should note that we explored other gauge choices, namely, A 00 S¼B 2x/C0bx 2/C18/C19 ^y/C0y/C0by 2/C18/C19 ^x"# ; (44) for the symmetric gauge, and A00 L¼Bx/C0bx 2/C18/C19 ^y; (45) for the Landau gauge, where bxandbyare arbitrary. In par- ticular, we formulated the problem for bx¼by¼0, knowingthat this choice was putting ourselves at a disadvantage. Nonetheless, even this choice works but at a cost that much larger matrices are required. This requirement is expected since now the gauge potential is very asymmetric throughoutthe square—in either case, the parabolic confinement arising from the applied magnetic field is centered at one corner of the square. While we refrain from showing results here, wefound identical results as earlier for weak magnetic fields(where we could pursue convergence as a function of matrix size). It is noteworthy that in this regime, there is no practical degeneracy, and so, the perturbation potential, Eq. (37),i s not required. V. SUMMARY We have presented an elementary discussion of the issues concerning gauge invariance, first in free space and, morepertinently, in confined geometries. Practical degeneracies still exist in confined geometries, or quantum dots, in partic- ular where a level floor in the potential well exists; this prob-lem is not present with a parabolic trap, for example. We used a quantum dot with circular symmetry to illustrate some of these degeneracies. However, in this case, it is easyto overlook some of the subtleties, as we conveniently havea good quantum number ( ‘) that both can simplify the calcu- lation and can be used to organize the results in an unambig- uous manner. In this way, even states with energydifferences that cannot be distinguished to numerical preci- sion will, nonetheless, be ordered properly through our knowledge of these quantum numbers. The circular potential also begs for the symmetric gauge to be utilized. Doing so simplifies the problem immensely, so only a one-dimensional equation requires solution. We could have used the Landau gauge, but then the problemwould have been significantly more difficult. For the case ofa two-dimensional square, however, the problem is more dif- ficult right from the start, regardless of which gauge is cho- sen. Partly for this reason, it became a good testing groundfor comparing gauge choices, especially given our method of solution, matrix mechanics. To our knowledge, this problem has not been previously solved in either gauge. Choosing a geometry also highlights that a gauge choice also involves a choice of origin for the gauge. Whether we use the symmetric or Landau gauge, the natural origin is the center of the sample, but we showed that this particularchoice is not required. Non-optimal choices (i.e., not the cen-ter of the sample) generally require more basis states with our method, so there is an additional numerical cost for a non-optimal choice. On the flip side, our method still allowscalculation in any gauge, so students can readily check vari- ous gauge choices. One can compute any property desired (we showed probability and current densities in this study)since we have the eigenvalues and eigenvectors. A further study will explore the susceptibility and other properties that can readily be measured. Finally, we also suggested a simple way of removing the degeneracy at the 10 /C010level (so the computer could tell the difference). As plotted, the eigenvalues will still appear to be degenerate, and this depiction is the correct physics. However, for purposes of organization, it is necessary tohave a method to distinguish these from one another, and the (very) shallow gauge-invariant parabolic potential that we proposed does the trick.Fig. 18. Jy;nas a function of x, taken along the diagonal from the upper left of the square down to the lower right. As xvaries from 0 to a,y¼a/C0x varies also, from ato 0. As in Fig. 17, the results are shown for various quantum numbers and in both gauges. We used G¼100 and N¼150. Jx;n mirrors this result about the vertical axis. 999 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 999The results for a square geometry are new; we expected that, for the properties studied in this work, the square geometrywould not produce anything qualitatively new beyond theresults for the circular quantum dot. Instead, as we saw, the cur-rent density for the edge states contains vortex-anti-vortex pairsat the corners, which are completely absent for the circular dotfor the same range of quantum numbers. We plan to carry out amore in-depth investigation of the conditions under which such modes appear. ACKNOWLEDGMENTS This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).These studies were initiated through a number of formerstudents; in particular, Sophie Taylor carried out some initialstudies that were very helpful to get us started. The authorsalso thank Ania Michalik for helpful calculations anddiscussions. Mason Protter and Joel Hutchinson wereinstrumental in the initial stages with assistance withnumerically technical matters. The authors also thank JosephMaciejko and Jorge Hirsch for very helpful discussions on various aspects of this problem. APPENDIX: REVIEW OF LANDAU LEVELS IN FREE SPACE 1. The symmetric gauge a. Eigenstates Substituting A Sinto the Hamiltonian (1)leads to H¼p2 2m/C0i/C22hxc 2ðx@y/C0y@xÞþ1 8mx2 cðx2þy2Þ; (A1) where @x/C17@=@xand similarly for yand later for radial coordinates as well. We have introduced the classical cyclo-tron frequency x c/C17eB=m. Converting to polar coordinates, we get H¼/C0/C22h2 2m@2 rþ1 r@r/C18/C19 þL2 z 2mr2þxc 2Lzþ1 8mx2 cr2; (A2) where Lz¼/C0i/C22hðx@y/C0y@xÞ¼/C0 i/C22h@/is the standard opera- tor for the zcomponent of the angular momentum. Note that this operator has arisen because of our gauge choice, andtherefore, its eigenvalues should not be considered to repre-sent the real, physical angular momentum of the electrons inthis system. This will be clear when we review the Landaugauge choice below. Therefore, the results for the L zoperator are gauge-dependent. To consider the true angular momen-tum of the system, we must take into account the effect ofthe magnetic field. Just as the canonical momentum isreplaced by the kinetic momentum in the presence of a mag-netic field, the canonical angular momentum ~Lis replaced by the kinetic angular momentum K, L¼r/C2p)K¼r/C2p¼r/C2ðpþeAÞ ¼Lþr/C2A: (A3) Since we are working in two dimensions, the kinetic angular momentum is perpendicular to the plane and is given byKðrÞ¼ L zþ1 2mxcr2/C18/C19 ^z¼KzðrÞ^z; (A4) where KzðrÞis the magnitude of the kinetic angular momen- tum in the ^zdirection. We can also rewrite Eq. (A2) in terms ofKzðrÞ, H¼/C0/C22h2 2m@2 rþ1 r@r/C18/C19 þK2 zðrÞ 2mr2: (A5) This new term can be interpreted as a kinetic centrifugal potential VlðrÞ/C17K2 zðrÞ=2mr2. Returning to our initial problem, to calculate the eigen- states and eigenvalues of Eq. (A2), we first note that since ½H;Lz/C138¼0, we can write the eigenstates Wðr;/Þas Wðr;/Þ¼ei‘/wðrÞ; (A6) where /C22h‘is the eigenvalue of the Lzoperator. Applying peri- odic boundary conditions in the variable /;Wðr;/þ2pÞ ¼Wðr;/Þrequires ‘to be an integer, and the eigenvalues of Lzare, thus, quantized, with ‘the azimuthal or angular momentum quantum number. Inserting Eq. (A6) into Eq. (A2) results in the radial ordinary differential equation as follows: /C0/C22h2 2md2 dr2þ1 rd dr/C18/C19 þ/C22h2‘2 2mr2þ1 2mxc 2/C18/C192 r2þ/C22hxc‘ 2"# /C2wðrÞ¼EwðrÞ: (A7) Equation (A7) is precisely of the form of a 2D isotropic radial quantum harmonic oscillator Hamiltonian with fre-quency x c=2, with an energy shift of /C22hxc‘=2. Although the solution to the above is known well in the literature, oftenonly operator-algebra-based derivations are shown (see Refs.9,13, and 14, for example). While those derivations avoid brute-force series methods, the behavior of the eigenstate is not so clear from the resulting operator-based expressions. For this reason, we will carry out an explicit derivation ofthe eigenstate here. We first introduce a characteristic magnetic length scale, ‘ B/C17ffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C22h=ðeBÞp , and make a change of variables, x¼ffiffiffiffiffiffiffiffiffimxc 2/C22hr ! r/C17r=ffiffiffi 2p ‘B/C0/C1 : (A8) The length ‘Brepresents a characteristic length scale for the wave function of a charged particle in a magnetic field. Forexample, for B/C251T ,l B/C2526 nm. This change-of-variables simplifies Eq. (A7) to the following: d2 dx2þ1 xd dx/C0‘2 x2/C0x2þK‘/C20/C21 wðxÞ¼0; (A9) where K‘/C172ð2E=ð/C22hxcÞ/C0‘Þ. We simplify further with the following functional substitution: wðxÞ¼uðxÞ=ffiffiffixp; (A10) converting our original ordinary differential equation (ODE) in Eq. (A9) to a 1D-like radial Schr €odinger equation as follows: 1000 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 1000d2 dx2/C0‘2/C01=4 x2/C0x2þK‘/C20/C21 uðxÞ¼0: (A11) In the above equation, one can identify (convergent) asymp- totic solutions for large and small x, Forx!1 ;u00ðxÞ/C24x2uðxÞ) uðxÞ/C25e/C0x2=2; (A12) Forx!0;u00ðxÞ/C24‘2/C01=4 x2uðxÞ) uðxÞ/C25xj‘jþ1=2: (A13) The above equations motivate the following ansatz for u(x): uðxÞ¼e/C0ðx2=2Þxj‘jþ1=2vðxÞ: (A14) Substituting the above into Eq. (A11) leads to v00ðxÞþ2jljþ1=2 x/C0x/C20/C21 v0ðxÞþK‘/C02ðjljþ1Þ ½/C138 vðxÞ¼0: (A15) With another change of variables x¼ffiffiffiyp, the above equa- tion reduces to a not-so-familiar ODE, yv00ðyÞþ j ‘jþ1/C0y ½/C138 v0ðyÞþð 1=4ÞK‘/C02ðj‘jþ1Þ ½/C138 vðyÞ ¼0: (A16) This ODE has the form of a confluent hypergeometric equa- tion, which can be written as (see Ref. 24, Sec. 13.2) zw00ðzÞþð b/C0zÞw0ðzÞ/C0awðzÞ¼0: (A17) There are two independent solutions to this equation; one is discarded because of the required behavior at the origin (seeRef.25), and the other is the Kummer function, wðzÞ¼Mða;b;zÞ¼X 1 j¼0ðaÞjzj ðbÞjj!; (A18) where ðaÞjdenotes the Pochhammer symbol, which can be computed as follows: aðÞj¼aaþ1ðÞ aþ2ðÞ /C1/C1/C1aþj/C01 ðÞ ;j¼1;2;3; :::; aðÞ0¼1: (A19) Comparing Eqs. (A16) and(A17) , it is clear that we may write the solution to Eq. (A16) as vðyÞ¼M/C0K‘/C02ðj‘jþ1Þ 4;j‘jþ1;y/C18/C19 : (A20) As seen in Eq. (A18) , our solution for v(y) is an infinite series. This infinite series leads to a troublesome issue: forlarge argument y,v(y) will have asymptotic behavior like vðyÞ¼X 1 j¼0½/C0ðK‘/C02ðj‘jþ1Þ=4/C138ðjÞ j‘jþ1 ½/C138ðjÞyj j! /C24X1 j¼0yj j!¼ey!1 ;fory!1 : (A21)Thus, we see that vðxÞ/C24ex2after returning to the xvariable viay¼x2. This behavior implies that our solution for u(x) will have divergent asymptotic behavior for large argument, uðxÞ¼e/C0x2=2xj‘jþ1=2vðxÞ/C24e/C0x2=2xj‘jþ1=2ex2 ¼xj‘jþ1=2ex2=2!1 ;forx!1 ; (A22) and in particular, the squared magnitude of this function is not integrable. This problem can be remedied in the usualway by requiring that the first parameter of Kummer’s func-tion ( ain Eq. (A18) ) is a non-positive integer /C0n r, where nr¼0;1;2;…. This condition truncates the infinite series since, by Eq. (A19) , /C0nrðÞj¼0 for j>nr;nr¼0;1;2; :::: (A23) Thus, we demand that the following condition must hold for u(x) to be normalizeable: 1=4ðÞ ½ K‘/C02j‘jþ1 ðÞ /C138 ¼nr;nr¼0;1;2;…:(A24) AsK‘is defined in terms of E, this condition quantizes our allowed energies. We see that after enforcing Eq. (A24) ,E q . (A16) reduces to the generalized Laguerre equation as follows:26 yv00ðyÞþ j ‘jþ1/C0y ½/C138 v0ðyÞþnrvðyÞ¼0: (A25) The solution to this equation is given by vðyÞ¼Anr;‘Lj‘j nry½/C138; (A26) where La n½z/C138is the generalized Laguerre polynomial and Anr;‘ is a constant to be determined by normalization. The general- ized Laguerre polynomials27obey the following orthogonal- ity condition over z2½0;1Þ,26 ð1 0e/C0xxaLa n0z½/C138La nz½/C138dz¼ðnþaÞ! n!dn0n;a>0: (A27) Changing variables back to xfrom y, we acquire the solution to Eq. (A11) through Eq. (A14) , uðxÞ¼Anr;‘e/C0ðx2=2Þxj‘jþ1=2Lj‘j nrx2½/C138: (A28) With the above, we obtain the solution to our original radial ODE (Eq. (A7)) by undoing our functional substitution wðxÞ¼uðxÞ=ffiffiffixpand changing variables back to rfrom x using Eq. (A8), wnr;‘ðrÞ¼Anr;‘rffiffiffi 2p lB/C18/C19j‘j e/C0r2=4l2 BLj‘j nrr2 2l2 B"# : (A29) Using Eq. (A27) , the normalization constant Anr;‘can be found, 1¼A2 nr;‘ð1 0jwnr;‘rðÞj2rdr ¼A2 nr;‘ð1 0r2 2l2 B !‘jj e/C0r2=2l2 BL‘jj nrr2 2l2 B"# !2 rdr ¼A2 nr;‘l2Bð1 0xj‘je/C0xLj‘j nrx½/C138/C16/C172 dx¼A2 nr;‘l2Bnrþj‘j ðÞ ! nr! )Anr;‘¼1 lBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nr! nrþj‘j ðÞ !s : (A30) 1001 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 1001Reattaching the angular component ei‘/and normalizing over/2½0;2p/C138, we finally obtain the normalized eigenstate for the symmetric gauge Hamiltonian (Eq. (A2)), Wnr;‘ðr;/Þ¼1 lBffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nr! nrþj‘j ðÞ !s rffiffiffi 2p lB/C18/C19j‘j /C2e/C0r2=4l2 BLj‘j nrr2 2l2 B"# ei‘/ ffiffiffiffiffiffi 2pp : (A31) The quantum number nris the radial quantum number and simply counts the number of nodes that the radial part of thewave function has ( n ris a non-negative integer). The corre- sponding probability density takes the appearance of concen- tric rings that are radially localized increasingly outwardsfrom r¼0 with increasing j‘j. The wave function has n r nodes, and the probability density jWj2depends on j‘jand not the sign of ‘. Examining the wave function alone, its dependence on the value of ‘may lead one to think that there is a double degeneracy for each energy. This is not the case here, as explained below. b. Eigenenergies The expression for the eigenenergies corresponding to the eigenstates given by Eq. (A31) is obtained by rearranging Eq.(A24) forE, Enr;‘¼/C22hxc 2ð2nrþj‘jþ1Þþ/C22hxc‘ 2; (A32)where the appropriate quantum numbers are now sub- scripted. In Eq. (A32) , the first term gives the standard eige- nenergy of a 2D radial quantum harmonic oscillator. The second term comes from the coupled momentum-gauge termthat manifests in the Hamiltonian after expanding the kinetic momentum operator. This term represents the interaction between the canonical momenta and magnetic field and iswhat distinguishes this problem from a standard 2D isotropic quantum harmonic oscillator. If we combine these two terms, we get E nr;‘¼/C22hxcnrþj‘jþ‘ 2þ1 2/C18/C19 ; (A33) so for all ‘/C200;Enr;‘¼/C22hxcnrþ1 2/C0/C1 , and the energy spec- trum is in fact infinitely degenerate. This infinite degeneracy is another distinct feature of the problem of a free electron ina magnetic field. A new quantum number n L/C17nrþð jljþlÞ=2 can be defined as well, and then, the energies are simply EnL¼/C22hxcnLþ1 2/C18/C19 ; (A34) where nL¼0;1;2; :::is the Landau level quantum number. Clearly, further degeneracies are possible through the use of‘>0 for ‘/C20n L. This fact is shown in the table below, where all energies have an infinite degeneracy due to the negative ‘states that carry on indefinitely to the right for each energy. Additionally, however, as nLincreases, positive ‘states are also degenerate in the growing number as indicated, Energy ð/C22hxc=2ÞnL ðnr;‘Þ .........… 52 ð0;2Þð1;1Þð2;0Þð2;/C01Þð2;/C02Þ… 31 ð0;1Þð1;0Þð1;/C01Þð1;/C02Þ … 10 ð0;0Þð0;/C01Þð0;/C02Þ …:(A35) This infinite degeneracy exhibits a peculiar pattern that is asym- metric with the sign of ‘. The electron, in response to the mag- netic field, appears to prefer a direction for Lz, requiring more energy to exist in the positive ‘states than the negative ‘states. Physically, circulating with positive angular momentum in the presence of a magnetic field directed in the positive zdirection costs energy for an electron, whereas circulating with negativeangular momentum does not. This result is consistent with theclassical idea that electrons speed up or slow down depending on whether they circulate with positive or negative angular momentum, respectively. 28 c. The lowest Landau level To better understand the wave functions and issues associ- ated with them, we focus on the lowest Landau level (LLL). For the LLL, nr¼0 and ‘/C200. Then, E0¼/C22hxc=2, and theassociated Laguerre polynomial is identically unity. So, an infinite set of degenerate LLL wave functions is given sim-ply by W LLLðr;/Þ¼ei‘/ ffiffiffiffiffiffi 2pprffiffiffi 2p lB/C18/C19j‘je/C0r2=4l2 B lBffiffiffiffiffiffi j‘j!p ;‘/C200:(A36) The corresponding probability density jWLLLj2is a radial Gaussian function centered on r¼0 but with a maximum at r‘¼lBffiffiffiffiffiffiffi 2j‘jp with a spread about this maximum of /C24Oð lBÞ. Asj‘jincreases, the location of the maximum of jWLLLj2, i.e., r‘, moves outwards from zero . On the other hand, increasing the magnetic field B(thus decreasing lB) leads to a decrease in the radius of the maximum in jWLLLj2. Also, note that as j‘jincreases, the difference between centers of two successive ‘eigenstates Wnr;‘andWnr;‘þ1will be much 1002 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 1002less than the spread lB[see Eq. (A36) ] and will decrease like 1=ffiffiffiffiffi j‘jp for large j‘j, implying that the corresponding eigen- states will increasingly overlap with one another. The values ofr‘also reveal that the amount of magnetic flux UBassoci- ated with each wave function in the LLL is quantized in unitsof the magnetic flux quantum U 0¼h=e. This statement is easily shown: consider the magnetic flux UBbetween a circle of radius r‘and that of (smaller) radius r‘þ1(recall that ‘/C200, so that r‘>r‘þ1), UB¼Bpðr2 ‘/C0r2 ‘þ1Þ¼2pBl2 Bðj‘j/C0j‘þ1jÞ ¼2pB/C22h eB¼U0: (A37) Thus, there is exactly a single quantum of magnetic flux between successive (negative) ‘states in the LLL. For the quantum dot geometries considered in Secs. IIIandIVin this paper, the system is finite, and the electron is confined tosome extent, so there must be some finite number of ‘’s that constitute the degeneracy number for a given Landau level (since the wave functions grow outward with increasing j‘j). The standard argument (already provided above) is that theexpected degeneracy Gof the LLL, given a sample of area A, is the ratio of the total magnetic flux Uto the flux quantum, G¼ U U0¼BA h=e: (A38) As demonstrated later, this result is not quite generally true, as sample edges confine the electron and result in higher energies for states near the edges. d. Gauge invariance and choice of origin It is important to note that these results are all gauge- dependent, even given that we use the symmetric gauge. Inparticular, the choice of symmetric gauge with A sdefined in the first paragraph of the introduction represents just one possible choice (out of infinitely many) where the vector field is referred to the origin. Since for this problem there isno preferred origin, the wave functions will reflect thischoice and be gauge-dependent. For this reason, we highlighted from zero in italics above. An alternative would have been to use A S¼B 2ðx/C0x0Þ^y/C0ðy/C0y0Þ^x ½/C138 ; (A39) and then, the wave functions would have been different, even though this choice of vector potential represents thesame problem, i.e., a charged particle in free space in thepresence of a uniform magnetic field B(see, for example, Ref.10for a discussion of this point). e. Electron in a parabolic quantum dot—Fock–Darwin states In the symmetric gauge, the addition of an isotropic para- bolic confining potential can be studied analytically. Thischoice results in the so-called Fock–Darwin states. 7Fock first addressed the problem of the eigenstates of a charged particle in a uniform magnetic field in 1928.29He discovered a gener- ally non-degenerate spectrum, particularly because the chargedparticle was confined to a parabolic potential. Somewhat later,Landau30examined a similar (simpler) problem in free space and found a degenerate spectrum given in Eq. (A34) ,w h i c h now bears his name. A year later, Darwin31independently obtained results similar to those of Fock, and figures displaying energy levels as a function of the applied field are now called Fock–Darwin spectra. The presence of a confining potential inthe form of a parabolic trap requires very little additional work, so we include a description of this case in this Appendix ,a l s o featured, for example, in Ref. 7. In the symmetric gauge, with an additional confining potential of the form 1 2mx2 0r2=2, the Hamiltonian given in Eq.(A1) becomes H¼p2 2mþxc 2Lzþ1 2mX2r2; (A40) where X/C17ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 0þx2 c=4p . With this parabolic confinement, the eigenenergies are Enr;l¼2/C22hXnrþ1 2j‘jþxc 2X‘/C20/C21 þ1 2/C18/C19 : (A41) Figure 19(a) shows a typical Fock–Darwin plot. A variation on this plot is shown in Fig. 19(b) .A si nF i g . 19(a) , Eq. (A41) is used for the energies, but now they are normalized to the energy /C22hxcand are plotted as a function of the (nor- malized) confining potential characteristic frequency. 2. The Landau gauge a. Eigenstates and eigenenergies The derivation of the eigenstates/eigenenergies in the Landau gauge is much more straightforward than their sym-metric gauge counterparts. For this reason, details of the der- ivation will be kept brief. As examined in the main text for the square geometry, we illustrate explicitly for confined geometries that correct gauge-invariant properties are obtained for various gauge choices. Here, we will not dem-onstrate this fact explicitly but instead point out that differ- ences are apparent in the free space case, and these differences are clearly rooted in the infinite degeneracy thatoccurs in this case. We, therefore, proceed in this section with the Landau gauge discussed in the Sec. I,~A 1¼xB^y. Now, our Hamiltonian is H¼1 2mp2 xþp2 yhi ¼1 2mp2 xþðpyþeBxÞ2hi ; (A42) where pyis the kinetic momentum operator in the ^ydirec- tion. Immediately, we see that ½H;py/C138¼0, so we can sepa- rate our eigenstate Was Wðx;yÞ/eikyywðxÞ; (A43) where the operator pyis replaced by its eigenvalue /C22hky. Substituting this expression into the Schr €odinger equation leads to the one-dimensional differential equation as follows: /C0/C22h2 2md2 dx2þ1 2mx2 cðxþkyl2 BÞ2/C20/C21 wðxÞ¼EwðxÞ; (A44) which is the usual 1D harmonic oscillator (albeit not cen- tered at zero) equation. The derivation of the solution of the1D quantum harmonic oscillator is very well known (see, for example, Ref. 15) and will not be repeated here, which is 1003 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 1003Wn;kyðx;yÞ¼eikyy 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2nn!ffiffiffipplBp e/C0ðxþkyl2 BÞ2=2l2 BHnxþkyl2 B lB/C20/C21 ; (A45) where HnðzÞare the Hermite polynomials.32This method of solution has introduced two quantum numbers, ky, a real number, and n, a non-negative integer. The latter quantum number counts the number of nodes of the probability den- sity. The corresponding energies are given by33 En;ky¼/C22hxcnþ1 2/C18/C19 : (A46) Note the lack of dependence on the quantum number ky. Consequently, the energies are infinitely degenerate forevery non-negative integer n. Here, ncorresponds to the Landau level quantum number n Lintroduced for the symmet- ric gauge energies in Eq. (A34) . Note that the wave functions are one-dimensional-like and certainly do not resemble those obtained previously. The reason that this wrong-looking result is possible is because of the degeneracy present(energy does not depend on k y), which is further discussed in the main body of the text. b. Lowest Landau level In the Landau gauge, the LLL wavefunction must have n¼0 for any ky2R, giving us WLLLðx;yÞ¼eikyye/C0ðxþkyl2 BÞ2=2l2 BffiffiffiffiffiffiffiffiffiffiffiffiffiffipplBp : (A47) Usually, periodic boundary conditions are imposed in the y- direction, resulting in the requirement that ky¼2pmy=a, where ais the length of the sample (i.e., “ribbon”) in the y- direction and now myis an integer. We can estimate the degeneracy here as well. The magnetic flux UBbetween the centers of two successive myeigenstates Wn;myþ1andWn;myisUB¼Baðxkþ1/C0xkÞ¼Ba2pðmyþ1Þ al2 B/C02pmy al2 B/C18/C19 ¼B2p/C22h eB¼h e¼U0; (A48) so, as in the symmetric gauge, each Landau gauge eigenstate contains roughly a single quantum of flux, U0. 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(a) Fock–Darwin spectra obtained by using the symmetric gauge for a charged particle in a parabolic trap, i.e., Eq. (A41) . A defining feature of this spectrum is the level-crossing phenomenon that the confined energies exhibit between free space Landau level energies, as discussed in Ref. 7. The dashed (red) lines (color online) correspond to the free Landau levels. Note, however, that there is no condensation of levels as the field increases, no matte r how strong the field. This lack of condensation is in contrast to what happens with more confining traps, to be discussed later. (b) The same Fock–Darwin spec tra obtained as in (a), but now Enr;‘=ð/C22hxcÞis plotted vs x0=xcfor some selected levels. The numbers in brackets label the ðnr;‘Þquantum numbers. The curves of a given color (color online) all emerge from the horizontal dashed line drawn for that same color; these denote the infinitely degenerate free Landau le vel ener- gies ( x0¼0), while the labeled curves with points indicated with the same symbol (and color, online) indicate how the degeneracy is broken with increasing confinement (increasing x0). It is clear from those levels drawn here that the confinement plays a more important role for negative values of ‘and increasing values of j‘j. This plot also exhibits level crossing, as indicated by the intersection of curves with different symbols (and color, online). Note that we could include all negative ‘states, and these would rise even more than those shown as the confinement potential becomes stiffer. However, at the far left of (b) (x0!0), we approach the free space limit, where the levels all become degenerate (this limit is indicated by the dashed horizontal lines). 1004 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 100417R. J. Swenson, “The correct relation between wavefunctions in two gauges,” Am. J. Phys. 57, 381–382 (1989). 18T. Ohtsuki and Y. Ono, “Numerical study of electronic states in confined two dimensional disordered systems under high magnetic fields,” Solid State Commun. 65, 403–407 (1988). 19F. Marsiglio, “The harmonic oscillator in quantum mechanics: A third way,” Am. J. Phys. 77, 253–258 (2009). 20B. A. Jugdutt and F. Marsiglio, “Solving for three-dimensional central poten- tials using numerical matrix methods,” A m .J .P h y s . 81, 343–350 (2013). 21R. L. Pavelich and F. Marsiglio, “Calculation of 2D electronic band struc- ture using matrix mechanics,” Am. J. Phys. 84, 924–935 (2016). 22G. M. Wysin, See lecture notes, “Probability current and current operators in quantum mechanics, pp. 1–17 (2011), https://www.phys.ksu.edu/per- sonal/wysin/notes/qmcurrent.pdf . 23Zhongbo Yan, Fei Song, and Zhong Wang, “Majorana corner modes in a high-temperature platform,” Phys. Rev. Lett. 121, 096803 (2018). 24NIST Digital Library of Mathematical Functions , edited by F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W.Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain (DLMF, Gaithersburg Maryland, 2020). 25R. Shankar, Principles of Quantum Mechanics (Springer, New York, 2004). 26E. Weisstein, “Associated Laguerre polynomial,” from MathWorld–A wolfram web resource (1996), http://mathworld.wolfram.com/ AssociatedLaguerrePolynomial.html .27Note that there are multiple definitions used for Laguerre polynomials in literature. The Laguerre polynomials are usually defined in physics texts in a manner that often differs from alternate definitions by a factorial factor. For example,ArfL2‘þ1 n/C0‘/C01¼1=ðnþ‘Þ!ArfL2‘þ1 n/C0‘/C01;so, in our case,ArfL1 n/C01 ¼1=n!GriffL1 n/C01whereGriffL1 n/C01denotes the notation used by Griffiths and Schroeter (Ref. 15)a n dArfL1 n/C01denotes the notation used by A r f k e n ,i nG .A r f k e n , Mathematical Methods for Physicists ,3 r de d . (Academic Press, Toronto, 1985). In Eq. ( A27), we use the definition from Arfken. 28David J. Griffiths, Introduction to Electrodynamics , 3rd ed. (Prentice Hall, Upper Saddle River, New Jersey, 1999). 29V. Fock, “Bemerkung zur quantelung des harmonischen oszillators im magnetfeld,” Z. Phys. 47, 446–448 (1928). 30L. Landau, “Diamagnetismus der metalle,” Z. Phys. 64, 629–637 (1930). 31C. G. Darwin, “The diamagnetism of the free electron,” Proc. Cambridge Philos. Soc. 27, 86–90 (1931). 32Like the Laguerre polynomials, different definitions exist for the Hermite polynomials that affect normalization. In Eq. ( A45), we use the physicists’ convention, where HnðzÞhas a coefficient of znequal to 2n;i.e. H0zðÞ¼1;H1zðÞ¼2z;H2ðzÞ¼4z2/C02;etc. This definition can be found in Griffith’s Introduction to Quantum Mechanics (Ref. 15). 33See supplementary material at https://doi.org/10.1119/10.0001703 for sample code. 1005 Am. J. Phys., Vol. 88, No. 11, November 2020 A. Bhuiyan and F. Marsiglio 1005

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Rev. Sci. Instrum. 91, 113201 (2020); https://doi.org/10.1063/5.0020661 91, 113201 © 2020 Author(s).The Panopticon device: An integrated Paul-trap–hemispherical mirror system for quantum optics Cite as: Rev. Sci. Instrum. 91, 113201 (2020); https://doi.org/10.1063/5.0020661 Submitted: 02 July 2020 . Accepted: 15 October 2020 . Published Online: 11 November 2020 G. Araneda , G. Cerchiari , D. B. Higginbottom , P. C. Holz , K. Lakhmanskiy , P. Obšil , Y. Colombe , and R. Blatt COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Panopticon: not a Transformer, but transforming ion trapping technology Scilight 2020 , 461104 (2020); https://doi.org/10.1063/10.0002713 A large field-of-view high-resolution hard x-ray microscope using polymer optics Review of Scientific Instruments 91, 113703 (2020); https://doi.org/10.1063/5.0011961 An ultra-high-vacuum rotating sample manipulator with cryogenic cooling Review of Scientific Instruments 91, 116104 (2020); https://doi.org/10.1063/5.0021595Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi The Panopticon device: An integrated Paul-trap–hemispherical mirror system for quantum optics Cite as: Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 Submitted: 2 July 2020 •Accepted: 15 October 2020 • Published Online: 11 November 2020 G. Araneda,1,a) G. Cerchiari,1 D. B. Higginbottom,2P. C. Holz,1 K. Lakhmanskiy,1P. Obšil,3 Y. Colombe,1 and R. Blatt1,4 AFFILIATIONS 1Institut für Experimentalphysik, Universität Innsbruck, Technikerstrasse 25, 6020 Innsbruck, Austria 2Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada 3Department of Optics, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic 4Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Technikerstrasse 21a, 6020 Innsbruck, Austria a)Author to whom correspondence should be addressed: gabriel.aranedamachuca@physics.ox.ac.uk. Current address: Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU, United Kingdom. ABSTRACT We present the design and construction of a new experimental apparatus for the trapping of single Ba+ions in the center of curvature of an optical-quality hemispherical mirror. We describe the layout, fabrication, and integration of the full setup, consisting of a high-optical access monolithic “3D-printed” Paul trap, the hemispherical mirror, a diffraction-limited in-vacuum lens (NA = 0.7) for collection of atomic fluorescence, and a state-of-the art ultra-high vacuum vessel. This new apparatus enables the study of quantum electrodynamics effects such as strong inhibition and enhancement of spontaneous emission and achieves a collection efficiency of the emitted light in a single optical mode of 31%. ©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/10.1063/5.0020661 .,s I. INTRODUCTION AND MOTIVATION Atomic ions confined and laser-cooled in Paul traps provide an experimental platform to study quantum objects with a unique level of control granted by the localization of the atomic wavefunction over a few nanometers, the long trapping time, and the high fidelity manipulation of internal and external degrees of freedom.1These features have been used extensively to build quantum information processors (see, for example, Ref. 2) and for the study of funda- mental phenomena in quantum optics and atomic physics (see, for example, Ref. 3). One of the most fundamental and intriguing phenomena in quantum optics is spontaneous emission and the possibility to modify it.4–7Spontaneous emission of atoms can be enhancedor inhibited in several ways. One way to modify spontaneous emission rate is through dipole–dipole interactions between dif- ferent atoms, namely, sub- and superradiance-type effects.8–11This has been extensively studied in neutral atom systems and used for diverse applications (see, for example, Refs. 12–20). In general, it is difficult to observe large modifications of spontaneous emission with trapped ions using this approach, since the Coulomb interaction restricts the minimum achievable inter-ion distance. Nevertheless, some degree of modification has been shown for the case of two ions. In Ref. 21, enhancement of ∼1.5% and inhibition of ∼1.2% were observed by locating two ions 1470 nm away from each other. Another way to enhance or reduce spontaneous emission rate of an atom is to modify the electromagnetic vacuum mode structure interacting with the atom.22,23The electromagnetic mode Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-1 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi structure can be altered by placing the atom close to dielec- tric interfaces,24between two mirrors,25or inside photonic struc- tures producing a bandgap.26In particular, large enhancement of emission from a single atom has been achieved using high- finesse cavities.27–30Recently, a fivefold enhancement in sponta- neous emission rate has been demonstrated by placing a single atom in a fiber cavity.31Furthermore, in the realm of solid state emitters, enhancement by a factor of more than 100 has been observed using microcavities.32In ion trap systems, strong cou- pling between a single trapped ion and a fiber cavity has recently been observed.33While a resonant cavity can increase spontaneous emission rate into a particular mode, inhibiting spontaneous emis- sion requires the suppression of the coupling to all modes. Only a few experiments have demonstrated large inhibitions (see, for example, Ref. 34 where the spontaneous emission rate of a single Rydberg atom was reduced by a factor of ∼20 or Ref. 35 where the emission rate of a solid state emitter was reduced by a factor of∼10). It is a common assumption in the field of cavity electrodynam- ics that full inhibition of spontaneous emission can only be achieved by placing the atom between mirrors that cover the full solid angle around it, restricting all the vacuum modes resonant with the atom (see, for example, Ref. 6). However, given the point symmetry of atomic emission, it is possible to fully control the emission rate by covering only half of the solid angle. This can be realized by plac- ing the atom in the center of a concave hemispherical mirror, as theoretically predicted in Refs. 36 and 37. Taking this approach, both inhibition and enhancement of spontaneous emission can be achieved depending on the distance Rbetween the atom and the surface of the mirror, provided that the mirror is close enough to the atom to allow the temporal interference of the field emitted in opposite directions. This condition can be written as 2 R/c≪1/Γ, where cis the speed of light and Γis the free-space decay rate of the observed atomic transition. If R=nλ/2, where nis an integer and λis the wavelength of the transition, an atom located in the center of curvature of the mirror is at a node of the vacuum mode den- sity and inhibition is observed. Conversely, if the radius is R=nλ/2 +λ/4, an atom located in the center of curvature is at an anti-node and enhancement is observed. If the mirror is not a hemisphere (NA = 1), but a spherical mirror with NA <1, only a partial modification is expected. Figure 1 shows the expected modification of the decay rate of a dipole transition depending on the numerical aperture of a perfectly reflective spherical mirror. Furthermore, in addition to the modification of spontaneous emission, a hemispherical mirror is predicted to lead to the ground and excited state level shifts.36,37 In this article, we present the new “Panopticon”38apparatus for the integration of a high-quality hemispherical mirror (NA ≈1) and a Paul trap in order to realize the situation described above. In addition to the fundamental components of such a setup, i.e., the hemispherical mirror and a Paul trap, a high numerical aperture lens (NA = 0.7) is used to collimate light emission and direct it to a detector, as shown in Fig. 2. The design and construction of such an apparatus presents multiple technical challenges that are discussed in the following: ●In Sec. II A, we discuss the construction and characteri- zation of the hemispherical mirror. The effects discussed above vanish rapidly when the mirror deviates from a perfect FIG. 1 . Modification of spontaneous emission rate of an atom in the center of a hemispherical mirror. Spontaneous emission of an atom at the center of curva- ture of a spherical mirror is inhibited or enhanced depending on the radius and numerical aperture of the mirror. If the radius of the mirror is R=nλ/2, the cen- ter of curvature is a node of the vacuum mode density and inhibition is expected (blue curve). If the radius of the mirror is R=nλ/2 +λ/4, the center of curvature is an anti-node of the vacuum mode density and enhancement is expected (orange curve). Full inhibition (respectively, a doubling of the decay rate) is obtained when half of the space is covered by the mirror (NA = 1). In this plot, a mirror with perfect reflectivity is assumed. hemisphere. The target rms surface error for the fabricated mirror isλ/10 over the whole mirror surface. Furthermore, we describe how thermal expansion can be used to adjust the radius of the mirror in order to access both inhibition and enhancement of spontaneous emission. ●In Sec. II B, we discuss the design and characterization of the high-NA lens used for light collection. The lens design aims at diffraction-limited performance, enabling interfer- ence, and imaging experiments such as the ones presented in Refs. 39–41. FIG. 2 . Main components of the Panopticon setup. The green arrows show the propagation of the light emitted by the trapped atomic ion. Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-2 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi ●In Sec. II C, we discuss the design of the Paul trap. The trap must provide full optical clearance between the trapped ion and the hemispherical mirror. Additionally, it must provide optical clearance for the high-NA lens and for the required laser beams. Ideally, the trap design should allow us to trap several ions simultaneously in a stable fashion and provide enough flexibility for multi-ion quantum optics experiments such as the one presented in Ref. 40. ●In Sec. II D, we discuss the integration of the mirror, trap, and lens. The position of the optical elements should be adjustable in situ with nanometer precision. ●In Sec. III, we present the design of the vacuum vessel. All the above-mentioned elements have to be placed in an ultra- high vacuum environment, which contains an atomic source and provides the required electrical connections and optical access. Besides the fundamental interest in measuring effects predicted by quantum electrodynamics,36,37being able to control spontaneous emission of a dipole transition has several applications. For example, reducing the decay rate by two orders of magnitude on a dipole tran- sition would increase the T1time of the transition up to microsec- onds, allowing the implementation of a qubit on such a transition. Using a dipole transition would be beneficial because it permits fast qubit control without the use of strong laser beams. Another applica- tion can be found in laser cooling, as tuning the decay rate of a dipole transition would allow to directly modify, e.g., the Doppler cooling rate.42,43Furthermore, Doppler cooling typically needs repumper laser beams to depopulate additional states connected to the cooling transition.43By suppressing the spontaneous decay through these additional channels, it would be possible to Doppler cool without the need of repumping beams. The system that we present here also provides high light col- lection efficiency. For an optimally oriented linear dipole, i.e., with its dipole moment oriented orthogonal to the optical axis used for collection, the expected single mode photon collection efficiency is 31.7%.44Remarkably, given the high quality of the overall optical system, with wavefront errors below λ/10 for all the components, the collected light can be captured in a single mode and coupled effi- ciently to a single mode fiber. The high collection efficiency and the cleanness of the optical mode are expected to improve the signal-to- noise ratio of any optical measurement of atomic properties. Addi- tionally, the improved collection and absorption rates enabled by this setup could be used to implement a quantum network without cavities.45,46 We note that some of the features exhibited by a setup employ- ing a hemispherical mirror, such as high collection efficiency and improved single photon absorption, can be achieved using an atom in the focus of a parabolic mirror.47,48Indeed, in Ref. 48, collection efficiencies as high as 54.8% have been reported using a parabolic mirror. However, in this approach, the enhancement and inhibition of spontaneous emission due to QED effects vanish for macroscopic parabolic mirrors.47Other approaches consist in using a combina- tion of a high-NA lens collecting the light emitted by an atom and a mirror that retro-reflects the collected light. Although changes in the decay rates have been observed using this method,49the modifi- cations are limited by numerical apertures achievable in trapped ion systems for the collection lens. Furthermore, previous attempts toposition an ion in the center of a spherical mirror have demonstrated stable trapping and improved light collection,50but they have not achieved the mirror surface quality, radius tunability, and numerical aperture required to show inhibition or suppression of spontaneous emission. II. THE PANOPTICON SETUP The main feature of the Panopticon setup is the ability to con- fine a single trapped atomic ion in the center of a hemispherical mirror. The ion trap needs to provide full optical access between the trapped ion and the mirror. Additionally, in order to capture a large portion of the emitted field, a high-NA aspheric lens (NA = 0.7) is positioned opposite to the mirror such that its focal point lies at the position of the ion. Therefore, the trap also needs to provide optical access for the solid angle captured by the asphere. Figure 2 shows a schematic with the main components of the Panopticon setup. In Secs. II A–II D, we describe the design, construction, and characteri- zation of each of these main components, as well as their integration and positioning. A. The hemispherical mirror The main component of the Panopticon setup is the concave hemispherical mirror. The results about the fabrication and charac- terization of this mirror have been previously reported in Ref. 51. To observe a large modification of spontaneous emission of an atom located in its center, the surface of the mirror can deviate only minimally from that of a sphere over its full numerical aperture. To our knowledge, the most precise macroscopic spherical objects ever fabricated are spheres with a surface deviation of only 17 nm peak-to-valley.52This surface quality is achieved by randomly rotat- ing the sphere between two polishing tools for periods of several days. Unfortunately, this technique cannot be applied to concave spherical surfaces, and in general, until now, there was no tech- nique able to produce a surface precision similar to that of a convex sphere. In collaboration with the Australian National University (ANU), we have produced hemispherical mirrors with rms form errors consistently below 25 nm, a maximum peak-to-valley error of 88 nm, and a radius of curvature ≈12.5 mm. The mirrors were fabricated by diamond turning from a cylindrical aluminum 6061 substrate using a CNC (computer numerical control) nano-lathe.53 To achieve low surface errors, an in situ white-light interferometer was implemented on the lathe, permitting the calibration of the tool with sub-nanometer precision prior to the machining. A detailed description of the fabrication and characterization of the mirrors can be found in Refs. 51 and 54. Figure 3 shows one of the fabricated mirrors, with a sur- face rms error of 18.1 nm ( λ/27 forλ= 493 nm) and a peak- to-valley error of 116.5 nm estimated using stitch interferometry (see Ref. 51 for details). Two drilled holes (3 mm diameter) pro- vide laser access to the center of curvature where the ion will be located (see Sec. II D). One of the holes is located along the opti- cal axis of the mirror and the other at 62○with respect to the optical axis. The NA of the mirror is 0.996 (half aperture 85○), which is not a limitation of the fabrication process and is chosen such as to provide 1.1 mm optical clearance to the trapped ion. This gives laser access orthogonal to the mirror’s optical axis. The Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-3 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 3 . Fabricated hemispherical mirror. (a) Photograph of a fabricated hemispher- ical mirror. A euro coin is used for comparison. (b) Schematic showing a cross section of the mirror with its relevant dimensions. Note that the 3 mm diameter drilled holes provide laser access to the center of curvature (CoC) of the mir- ror where the ion will be located. Reproduced with permission from Higginbottom et al. , “Fabrication of ultrahigh-precision hemispherical mirrors for quantum-optics applications,” Sci Rep 8, 221 (2018). Copyright 2018 licensed under a Creative Commons Attribution (CC BY) license. reflectivity of the mirror is set by the reflectivity of the substrate (0.92 for aluminum 6061 at 493 nm); it could be improved by applying a highly reflective thin film coating. However, the concave shape of the mirror makes it difficult to realize this in a uniform fashion, and special techniques would be required.55Figure 4 shows the expected modification of spontaneous emission considering the measured surface form, the reflectivity, and the effect of the drilled holes. Con- sidering form and reflectivity, the maximum expected enhancement and inhibition of spontaneous emission rate in the 493 nm transition in138Ba+correspond to 88% of their ideal values. The main devia- tion from the ideal case comes from the limited reflectivity of the material. The mirrors have been shown to be resilient to a cycle of vacuum bake-out under typical conditions (maximum temperature of 200○C). 1. Temperature stabilization and tuning of the radius of curvature To tune the radius of curvature of the mirror, we rely on the uniform thermal expansion of the substrate. The coefficient of thermal expansion of aluminum 6061 at room temperature is 23.5 ×10−6K−1.56A change in radius of curvature of λ/4≈123.3 nm (withλ= 493 nm) requires a temperature modification of 0.42 K. The mirror was heated when measured with a single-shot white-light ZYGO interferometer; this showed radius tunability over the desired FIG. 4 . Expected performance of the fabricated mirror. Modification of spontaneous emission rate considering the achieved form, and the achieved form and reflectivity of the substrate. The numerical aperture NA = 0.996 is not distinguishable from NA = 1.0 in the plot. Considering form and reflectivity, the maximum expected enhancement and inhibition of spontaneous emission rate correspond to 88% of the ideal value. range with no measured surface distortion due to thermal expan- sion. Nevertheless, small variations in environmental temperature in the experiment may lead to the drifts of the radius of curvature; therefore, active stabilization of the temperature of the mirror is necessary. To control and stabilize the temperature of the mirror, we designed a resistively heated holder, as shown in Fig. 5. The holder is heated by applying current to a self-wrapped heating wire.57The total resistance of the heating wire is 65 Ω, and the current applied varies between 0.1 A and 0.2 A depending on the set temperature. The temperature of the holder is measured in the back of the holder using two temperature sensors,58which are glued to the holder.59 The mirror is not in direct contact with the holder, it lies on ceramic rods60and glass spheres61(see Fig. 5), providing a distance of 1.1 mm between the outer surface of the mirror and the inner surface of the holder. This reduces the contact thermal conductivity between the mirror and the holder while maintaining the radiative heat trans- fer and provides a homogeneous temperature over the volume of the mirror, albeit with a slow temperature tuning rate. The tem- perature is stabilized with a feedback loop using a high-precision proportional integral differential controller (PID).62In-vacuum tests have shown mirror temperature fluctuations below 1 mK over 10 h for different set points between 293 K and 303 K. The heating time constant of the mirror has been measured to be 5.39 h, while the cooling time constant (cooling is achieved passively) is 3.89 h. These times may be reduced by increasing the thermal contact between the mirror and the holder, which would, however, lower the tempera- ture stability. The PID can be operated in the sample and holder or constant current regime, which is useful for sensitive atomic mea- surements where a constant magnetic field could be required. The achieved temperature stability corresponds to fluctuations in the mirror’s radius of curvature below 20 nm, which are lower than the measured surface error of the mirror. As mentioned above, the mir- ror is grounded. This is done to avoid any charging that could affect the trapping potential. The grounding is done with a thin (25 μm Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-4 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 5 . Mirror holder and heater. Components of the mirror’s aluminum holder and heater used to achieve a mirror temperature stability below 1 mK. The right-most panel shows the setup used during the temperature stability tests performed in vacuum, with a thermistor attached directly to one of the mirrors. diameter) gold wire in the front side of the mirror, fixed with con- ductive epoxy. The grounding wire has a negligible effect in the temperature stability and heating and cooling constants. B. The aspheric lens To capture and collimate the light emitted by the ion, an aspheric lens is located opposite to the hemispherical mirror, as shown in Fig. 2. The main advantages of using an aspheric lens instead of a multi-lens objective are the simple, compact design, and the reduced spherical and other optical aberrations. A low- aberration optical system is a crucial requirement for future experi- ments related to the spatial properties of emitted and absorbed pho- tons for interference experiments with photons emitted by atoms in different traps coupled with optical fibers. The asphere, designed and fabricated by Asphericon GmbH, has an NA = 0.7, a working distance of 9.60 mm, and an effec- tive focal length of 16.05 mm. The design was optimized using Zemax OpticStudio to achieve a diffraction-limited performance with ultra-low wavefront aberrations at 493 nm. The dimensions of the designed asphere are shown in Fig. 6. The surface facing the ion has a designed constant radius of curvature RB= 202.353 mm, whereas the opposite surface is a rotation symmetric asphere defined by the following equation: z(r)=r2 RF(1 +√ 1−(1 +k)r2 R2 F)+8 ∑ i=2A2ir2i. (1) The values of parameters RF,k, and A2iare listed in Table I. Three pieces of the designed asphere were fabricated using a S-TIH53 glass substrate63with the refractive index ng≈1.87 (λ= 493 nm) and the Abbe number vg= 23.59 (λ= 546 nm). The manufacturing process includes CNC grinding and polishing as a first step. Thereafter, ion beam figuring is used to reduce sur- face irregularities.64In this process, ion beams are directed at spe- cific regions of the surface where surface errors have been detected.Finally, an anti-reflective coating for 493 nm is applied using plasma- assisted physical vapor deposition.65All these steps were performed by Asphericon GmbH. Figure 7(a) shows a photograph of one of the aspheres, and Fig. 7(b) shows the measured wavefront distortion.66 rms wavefront distortions below 37 nm ( λ/14) and peak-to-valley distortions below 700 nm over the full numerical aperture were FIG. 6 . Aspheric lens dimensions. The surface facing the ion is spherical with a radius of curvature RB, whereas the aspheric surface is defined by Eq. (1). The ion is positioned 9.6 mm away from the front surface. The aspheric surface design parameters are listed in Table I. Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-5 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi TABLE I . Aspheric lens design parameters. Parameter Value RF 14.56 mm k −0.776 A4 3.202 280 6 ×10−6mm−3 A6 −2.900 266 1 ×10−8mm−5 A8 −9.624 991 0 ×10−11mm−7 A10 −1.023 645 6 ×10−13mm−9 A12 4.551 145 9 ×10−16mm−11 A14 3.320 125 2 ×10−18mm−13 A16 −8.764 529 8 ×10−21mm−15 achieved for the three aspheres. From Fig. 7(b), it is clear that the largest wavefront distortions occur close to the lens edge. In fact, by reducing the aperture of the lens to NA = 0.65, the rms wavefront error is reduced to λ/24. The reflection from each coated surface was measured to be smaller than 0.4% for incidence angles up to 40○, and the on-axis transmission of the lens was ≈95%, in agreement with the expected values. Both measurements were done with a laser beam with a 493 nm wavelength. FIG. 7 . Fabricated aspheric lens. (a) Photograph of one of the three fabricated aspheric lenses together with a hundred Chilean pesos coin, giving the scale (the coin has a diameter of 23.25 mm). (b) Measured wavefront distortion of one of the aspheres. Numbers in the color scale are in units of λ= 493 nm. Provided by Asphericon GmbH.The lens is mounted in an aluminum holder, which allows for laser access to the focal point of the lens from several directions (see Sec. II D), and with negligible reduction in the numerical aperture. C. The ion trap The main challenge in the design of an ion trap compatible with the Panopticon setup is that the trap should provide full opti- cal access not only between the trapped ion and the hemispherical mirror but also between the ion and the aperture defined by the aspheric lens. This could be achieved using, for example, a nee- dle trap.67However, this approach allows the trapping of only one ion without excess of micromotion and displacement of the trapped ions. It is desirable to be able to trap several ions without micro- motion, since the setup could then be used for a broader range of quantum optics experiments. Some comply partially with the crite- ria for optical access and stable trapping of several ions, including, for example, the miniaturized segmented “High Optical Access Trap 2.0” developed by the Sandia National Laboratories.68However, this trap only possesses a clear aperture equivalent to NA = 0.25. The approach that we take here is the fabrication of a monolithic slot- ted pseudo-planar macroscopic trap. A simplified scheme of the trap geometry, with only the minimal electrode configuration needed to trap an ion, is shown in Fig. 8. Later in this article, we discuss the actual fabricated trap with additional features. In the design, an ion is trapped above the front surface, providing full clearance on the side of the hemispherical mirror and, through the slot, clearance corresponding to the numerical aperture NA = 0.7 of the aspheric lens. The geometry shown in Fig. 8 generates a confining RF pseudo- potential in the xand ydirections, whereas trapping along the z direction is provided by the DC potential generated primarily by the DC electrodes DC1 and DC2. The DC voltages applied to the electrodes DC3, DC4, DC5, and DC6 contribute weakly to the trap- ping potential in the zdirection and are meant for micromotion compensation. The ring shape of the RF electrode, although counter- intuitive, is the result of systematic optimization, starting from two π-shaped separated RF electrodes producing a cross-shaped slot (see Fig. 9), varying the dimensions of the slot, and keeping a fixed slot vertical (along x) aperture. The obtained shape with a rectangular slot exhibits a reduced residual axial ( zdirection) pseudo-potential, allowing chains of ∼20 ions to be trapped with negligible axial micromotion. The fabrication is done with subtractive 3D laser micro- machining technology. The basic idea is to laser-machine a mono- lithic dielectric substrate to the required shape, including “trenches” separating different regions. In the following, we will refer to this technique as “3D-printing.” The surfaces are then coated with gold, creating an independent electrode in each region surrounded by trenches. Figure 10 depicts how 3D-printing and coating are com- bined to create electrodes without the need for shadow masking and reduce the number of angled metal evaporation steps needed. The bottom of the carved trenches can be grounded or connected to one of the contiguous electrodes by shaping “ramps” between the bottom and the surface in the 3D-printing design. The required precision of the dielectric substrate fabrication can be achieved by “laser carving” techniques provided by compa- nies such as FEMTOprint S.A. in Switzerland or Translume Inc. in USA. The 3D-printing technology used by these companies employs Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-6 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 8 . Ion trap design. (a) Simplified design of the trap. The RF, DC, and ground (G) electrodes lie on the facets of a 3D-printed substrate. All dimensions in mm. Pink text indicates electrode names. (b) Transverse cuts of the trap showing the optical clearance. The trap design provides full clearance toward the hemi- spherical mirror ( y>0). On the other side (y<0), the optical clearance is slightly larger than required by the aspheric lens with NA = 0.7, corresponding to aθ= arcsin(NA) ≈44.4○opening angle (blue area). The RF pseudo-potential minimum, and therefore the position of the ion, is located at a distance y = 0.157 mm from the front plane of the trap. strongly focused high-energy laser pulses to locally change the phys- ical structure of the substrate. A chemical process is then used to remove the treated material. Using this technique on, e.g., fused silica, the carved features can reach an average surface roughness below 100 nm, whereas the untouched surfaces can reach an aver- age roughness below 5 nm. Additional polishing can reduce the FIG. 9 . Slot shape optimization. The ring shaped RF is the result of optimization starting from two separated π-shaped RF electrodes and varying the lengths d1 andd2while keeping the other dimensions of the electrode fixed.roughness below 50 nm, though with detrimental effects in the sur- rounding areas. The scale of the surface roughness achieved has been demonstrated to be adequate for use in ion traps with a mini- mum ion-electrode distance of ∼100μm (see, for example, Ref. 69), whereas in our trap that distance, as described below, is 453 μm. A more detailed description of this fabrication technology applied to the fabrication of ion traps can be found in Ref. 69. The conductive coating is applied in our clean room by means of electron beam evaporation of a thin layer of titanium (∼2 nm), used as an adhesion layer, followed by a thick layer of gold (∼200 nm). Gold evaporation is applied from three different angles on each side of the substrate. Gold electroplating could also be used afterward to increase the thickness of the electrodes to ∼5μm. 1. Simulations of the trapping potential The trap geometry presented in Fig. 8 is the result of a sys- tematic optimization. The different dimensions and positions were varied to maximize the trapping frequencies for a138Ba+ion, as well as the trap depth, and to reduce the trap capacitance and the residual Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-7 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi FIG. 10 . Electrode fabrication through 3D-printing and coating. First, a dielectric substrate is “3D-printed” to create the trenches needed to separate electrodes. Then, the surfaces are coated with a conductor (gold) in an evaporation step. The profile of the trenches combined with the directionality of the evaporation step provides electric insulation between different regions, defining trap electrodes. The recess in the profile allows us to apply evaporation from different angles without shorting the electrodes. The dimensions of the trench profile chosen for the fabricated trap are shown in Fig. 12(d). FIG. 11 . Simulated trapping potential. The parameters are those of “config. 1” in Table II. (a) Cross section through the x–yplane. The white cross shows the posi- tion of the minimum. (b) Cross section through the y–zplane. (c) Trapping potential along the x,y, and zaxis. Close to the center, in a range of ±50μm, the potentials are well approximated by harmonic potentials.axial RF field. The ion-electrode distance, which is constrained by the required optical clearance and laser access to the ion, was maxi- mized. The electric potentials produced by the trap electrodes were simulated via finite element analysis using the software COMSOL Multiphysics 4.4. In these simulations, the grounded hemispherical mirror was included. Figure 11 shows the total trapping potential. The RF pseudo- potential minimum is located 157 μm away from the front plane of the trap, and the distance of a trapped ion to the closest electrode is 453μm. For comparison, in our current “Innsbruck-style” blade trap,70this distance is 707 μm. Small ion-electrode distances tend to increase the ion heating rate due to, e.g., surface noise.71Keeping this distance relatively large is therefore important. Figure 11(c) shows the trapping potentials along each coordi- nate axis. The potential is symmetric about the trap center along directions xand z, but not along y, as typically observed in sur- face traps.67The trapping frequencies can be varied with the volt- ages applied to the different electrodes. Table II shows the simu- lated trapping frequencies and trap depths obtained for a138Ba+ ion. The trap depths are in the eV regime, comparable to 3D rod or blade traps,72which makes the trap suitable for operation at room temperature. Simulations show that independent control of the voltages applied to the electrodes DC3, DC4, DC5, and DC6 is sufficient TABLE II . Trap driving parameters and resulting trap frequencies and depths, obtained by finite-element simulations for138Ba+. Three different configurations are shown. DC1 and DC2 refer to the “endcap” electrodes shown in Fig. 8, while DC3, DC4, DC5, and DC6 are the electrodes in the back plane. In these three configurations, the electrodes G1, G2, G3, G4, and the mirror are grounded. Config. 1 Config. 2 Config. 3 RF freq. Ω RF/2π(MHz) 16.0 16.0 16.0 RF amplitude URF(V) 1000 1500 2000 DC1,2 (V) 200 300 400 DC3,4,5,6 (V) 82 123 164 ωx/2π(MHz) 1.33 2.16 2.96 ωy/2π(MHz) 1.57 2.34 3.07 ωz/2π(MHz) 0.51 0.62 0.72 Trap depth (eV) 2.4 4.9 8.2 Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-8 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi for compensation of micromotion in all directions. Further simu- lations with different trench geometries and dimensions were car- ried out. The most relevant trench parameter, i.e., the electrode– electrode separation provided by the trench, was varied between 50μm and 150 μm. Variations in this range have a negligi- ble effect on the trapping potential so that the final dimensions were selected based on the breakdown test results described in Sec. II C 2. 2. The fabricated trap While the fabricated trap has the same core geometry as pre- sented in Fig. 8, it includes additional features. Figure 12 shows the schematics and renders of the actual design. The ground electrodesG1 and G2 in the front plane are divided into smaller electrodes GR1, GR2, GR3, and GR4 [Fig. 12(a)] to add control of the RF potential minimum position while keeping micromotion compen- sated, allowing the possibility of moving the ion in any direction several micrometers. These electrodes, together with all other elec- trodes in the front plane (EC1, EC2, RF, and GRB), are extended to reach the back plane. There, together with the back plane elec- trodes CU1, CU2, CU3, CD1, CD2, and CD3, they are prolonged into conductive traces extending beyond the radii of the hemispher- ical mirror and of the aspheric lens [Fig. 12(b)]. This flat and elon- gated design provides enough space for wire-bonding, which con- nects the electrodes to the voltage supplies through a printed circuit board (PCB, see Sec. II C 3). In this way, the optical clearance to the FIG. 12 . Full design of the ion trap. (a) Front side of the final trap design. Cylindrical and conical grooves are carved in the substrate in order to minimize the scatter of laser beams propagating close to the surface. All front electrodes extend to the back plane via the edges of the trap. (b) In the back plane, all the electrodes are extended to one extremity of the substrate, where wire-bonding is used for connection to voltage supplies. (c) Transverse cut at the trap center showing the optical clearance provided by the trap. (d) Dimensions of the trenches separating the electrodes. Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-9 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi trapping region is not affected by the bonding wires. The extended area is also used to mount the trap to an aluminum holder using through-holes. Additionally, grooves in the front face are carved in the sub- strate (during the 3D-printing process). These grooves improve the clearance of the laser beams that propagate close to the sur- face, reducing the light scattered by the trap. This is of particular importance when addressing single ions using a strongly focused 1.7μm laser. The conical vertical grooves shown in Fig. 12(a) match the divergence of such a beam with additional 100 μm clearance from the point where the beam intensity has decayed by a fac- tor of 1/ e2. The horizontal cylindrical grooves give enough opti- cal clearance for lightly focused axial cooling and optical pump- ing beams. Figure 12(c) shows transverse cuts at the trap center, showing the position of the ion and the optical clearance provided by the design. The dimension of the trenches separating adjacent electrodes, shown in Fig. 12(d), was chosen after performing in- vacuum DC and RF breakdown tests on a simplified trap. The separation between electrodes varied between 50 μm and 150 μm. These tests did not shown any sign of electric breakdown between electrodes with DC voltages as high as 700 V and RF voltages as high as 1200 V at 20 MHz. The separation was finally chosen to be 100μm. The bottom of the trenches is connected to the com- mon ground GR B. A complete 3D drawing of the designed trap can be found in the supplementary material. Figure 13 shows a pho- tograph of one of the traps fabricated by FEMTOprint S.A. before metalization. We have performed simulations of the generated trapping potential for the fabricated trap design, including all the aforemen- tioned features. The differences with the results of the simulations of the simplified version presented in Sec. II C 1 are negligible. 3. Trap holder, in-vacuum low-pass filter, and voltage driving The ion trap is mounted on an aluminum holder plate (2 mm thickness) using three screws, as shown in Fig. 14(a). The holder plate also acts as a heat sink for the trap and is contacted to the body of the vacuum chamber. This plate also holds a filter PCB. The trap is wire-bonded to the PCB using gold wires. The PCB, made of alu- mina73with a thickness of 1 mm, has 16 silk-printed gold traces, wire-bond pads, paths, and soldering pads74for DC and RF routing. There are eight vias to route some of the paths to the opposite side of the PCB. All the DC paths are capacitively coupled to a common ground using 1 nF high-voltage surface mounted capacitors,75sup- pressing RF pick up. Additionally, 1 kΩ ex-vacuum resistors are used in order to filter high frequency noise induced in the DC electrodes (cutoff frequency 1 kHz). Figures 14(b) and 14(c) show a schematic FIG. 13 . Fabricated ion trap. Fabricated ion trap before metalization. FIG. 14 . Trap holder and filter PCB. (a) Both the trap and the PCB filter are mounted on an aluminum holder. The trap is wire-bonded to the PCB, which has conducting paths, capacitors, and pins for connection to the DC and RF sources. (b) and (c) Details of the PCB paths, position of the capacitors, and pins. view of the PCB. At the edge opposite to the trap, the PCB has 16 gold pins76soldered,77allowing easy connection and disconnection of each line to copper wires attached to the electrical feedthroughs on the base vacuum flange (Fig. 16). None of the PCB traces or the wires are exposed to the atomic flux from the atom source (see Sec. III A). Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-10 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi The DC voltages are generated outside the vacuum chamber using a high-precision high-voltage source.78The RF voltage is gen- erated using a high-precision signal generator79followed by a 28 dB amplifier80and a helical resonator.81 D. System integration The optical setup, consisting of the hemispherical mirror, the aspheric lens, and the ion trap, has to be set in place in a robust and stable manner while still providing enough degrees of freedom for correct alignment. The holders and positioners have to be compat- ible with the ultra-high vacuum environment needed to trap single atomic ions in a stable way. To do so, we designed the mounting system shown in Fig. 15. In this setup, all the elements are carefully designed in order to provide the optical clearance required by the mirror and the lens, and laser access. The positions of the mirror and the lens can be set independently using xyz-nanopositioners82 FIG. 15 . Panopticon optical setup. The mirror and lens are mounted on indepen- dent xyz-nanopositioners, whereas the position of the ion trap is fixed. The blue lines show the four laser axes.shown in this figure. The positioners have a step size of 1 nm in each direction, a position readout resolution of 1 nm, and a maximum displacement of 12 mm. III. THE VACUUM VESSEL The Panopticon optical setup is placed in an ultra-high vacuum environment. The vacuum vessel, shown in Fig. 16, is built around an 8 in.-CF spherical octagon chamber.83 A customized 6-way cross is attached to the main chamber, providing enough flanges to connect a vacuum valve,84a vacuum gauge,85the electric feedthroughs for the wiring of the nanoposi- tioners, a non-evaporable getter (NEG) pump,86and a viewport. The main pumping is done with a combined ion and a NEG pump87 attached to the main chamber. During the NEG activation, the pump can reach temperatures close to 450○C. This temperature is not compatible with the maximum temperature to which the nanopositioners can be exposed (150○C). In order to avoid dam- aging the nanopositioners, the pump is retracted from the main chamber using a spacer, and a two-layer aluminum heat shield is placed between the pump and the positioners (see Fig. 16). Accord- ing to finite-element simulations performed by the pump manufac- turer, this is sufficient to prevent damage of the nanopositioners. We plan to perform additional test to confirm this predictions, and if necessary, the shielding and the pump position will be modified. FIG. 16 . Main components of the vacuum vessel (see details in the main text). Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-11 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi The vacuum vessel has all the necessary viewports to provide laser access to the center of the trap using the planned beam direc- tions (see Fig. 15), including a CF-160 viewport on top of the cham- ber. All the viewports88are anti-reflection coated for all the wave- lengths needed to load, cool, and control138Ba+ions, i.e., 413 nm, 493 nm, 614 nm, 650 nm, and 1762 nm. For these wavelengths, the reflectivity is below 1%. The viewport used for transmitting the light emitted by the ion and collimated by the aspheric lens has an optical quality surface, with wavefront aberrations below λ/10 over all the surface (with λ= 493 nm).89An additional Germanium viewport90 is attached to the main chamber, to allow monitoring of the temper- ature of the atom oven using a thermal camera outside the chamber (see Sec. III A). The electrical connections needed to drive the ion trap, to heat an atom dispenser oven, and to measure and control the temperature of the mirror are routed through feedthroughs in the customized CF-160 bottom flange, assembled by Vacom GmbH (Fig. 16). This flange has a SUB-15 feedthrough for all the mirror temperature sta- bilization connections, a two-wire (12 kV, 13 A) feedthrough for the oven connections, two four-wire (12 kV, 13 A) for the trap DC connections, and a 2-wire (5 kV, 25 A) for the RF trap connec- tions. This customized flange has also a CF-16 viewport intended for 1.7μm laser addressing of individual trapped ions. The con- nections for the nano-positioners are in one of the arms of the cross. A. Atom source To provide a source of atoms inside the vacuum chamber, we have designed a loading stage that simultaneously contains a resistively heated Ba dispenser oven and a laser ablation target. The resistively heated oven is a reliable way to produce a flux of neutral atoms in the center of the trap, which can then be ionized using a 413 nm laser through a two-step excitation pro- cess.70This process is, however, slow and produces an excess of heat inside the vacuum chamber that can have detrimental effects on the operation of ion traps, such as undesired thermal expan- sion of the electrodes and thermal expansion of the hemispherical mirror. Laser ablation from a target is an alternative method, which has been proved to be more efficient and less detrimental to ion trap operation (see, for example, Ref. 91). In this approach, short and strong laser pulses are applied on a Ba target to produce neutral and ionized atoms, which eventually reach the center of the trap. It has been experimentally shown that instead of using a pure Ba tar- get, targets containing BaTiO 3or BaO produce higher Ba+yields,92 making loading more efficient while lifting the requirement of an additional photo-ionization laser. At the same time, using BaTiO 3 instead of pure Ba, prevents fast oxidation of the sample during preparation of the vacuum setup. To guarantee isotope-selective loading, we still plan to use photo-ionization of ablated neutral atoms. Figure 17 shows the designed loading stage, where both a resis- tive dispenser oven93and a BaTiO 3ablation target94are located. The holder is made of macor ®ceramic, which exhibits a low thermal and electric conductance. Both the target and the oven are enclosed in a copper shield, which simultaneously acts as a thermal radiation shield and prevents the ablated Barium from coating surfaces in the FIG. 17 . Loading stage. Design of a loading stage combining a resistive Barium dispenser oven and a target for laser ablation (see details in the main text). chamber and in the trap. Two circular apertures in the front with 1 mm diameter, one for each atom flux production process, are used to collimate the atomic flux toward the center of the ion trap. As the ablation process spreads atoms in every direction, a mirror and a hole in the shield are used (Fig. 17). This configuration prevents a high atomic flux from exiting the shield and directly coating the top CF-160 viewport used for the ablation laser beam. Although the mirror used for the ablation laser is directly exposed to the atomic flux produced by the ablation producing a systematic degradation on the mirror’s reflectivity, this can be compensated directly by increas- ing the power of ablation of the laser. Additionally, there is a small square aperture in the side of the stage, which allows for monitoring of the oven temperature with a thermal camera through an infra-red Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-12 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi transmissive Germanium viewport. This can be used to character- ize the heating dynamics of the oven in ultra-high vacuum and to optimize the loading process avoiding excess heating.95 A broad range of pulsed laser sources can produce the pulses needed for ablation. An example of such a source is a pulsed diode pumped solid state laser (DPSS), with a wavelength of 515 nm and an energy per pulse of 170 μJ.96This wavelength is compatible with our coated viewports. B. Vacuum preparation and ex-vacuum elements The cleaning of all the vacuum components is done following the procedure described in Ref. 97. The main components of the vacuum chamber are pre-baked at 450○C, whereas when adding the in-vacuum optical system and the nanopositioners, the system must be baked at 150○C. Three pairs of magnetic field coils are attached to the outside of the main chamber flanges in order to provide a homogeneous mag- netic field at the center of the trap. These coils may be replaced by rings of permanent magnets in order to achieve a better suppression of RF magnetic field noise. Four legs are attached to the exterior of the vacuum chamber to support it on an optical table. IV. SUMMARY AND OUTLOOK In this article, we have presented the design and construction of a new setup that will allow us to study quantum electrodynamics effects. Furthermore, the setup features an unprecedented single- mode collection efficiency. We have presented the design and con- struction of the main optical components of the setup, namely, a hemispherical mirror and an aspheric lens, and shown how the strict requirements on their optical performance can be fulfilled. Both the mirror and the lens achieve a wavefront distortion below λ/10. The fabrication of macroscopic concave hemispherical mirrors with unprecedented precision will give us access to the enhancement and inhibition of spontaneous emission of a single atom of more than 96% of its free-space value. The fabrication technique could be extended to obtain other concave surfaces, allowing for more exotic quantum electrodynamics situations. One example would be a hemi- spherical mirror with a λ/4 radius step, as shown in Fig. 18 (details can be found in Ref. 98). Such a mirror could be used to enhance spontaneous emission in the modes collected by a lens while inhibit- ing the rest, resulting in collection efficiencies close to 100% for a perfectly shaped mirror. For a mirror with a λ/4 step, fabricated with the surface accuracy demonstrated in Sec. II A, combined with the NA = 0.7 lens presented in Sec. II B, an unprecedented 68.5% single- mode collection efficiency is expected for a linear dipole oriented perpendicular to the optical axis, in comparison with the 31.7% expected with a non-stepped mirror.54 We have also presented the design of a monolithic high-optical access ion trap. Such a trap provides the optical access required for the observation of the quantum electrodynamics effects described above. The trapping parameters are similar to those in state-of-the- art macroscopic traps, such as our “Innsbruck-style” blade trap.70 The design of this new trap employs 3D-printing technologies. The basic concept of using trenches to separate electrodes can be extended to more complex trap geometries, including segmented traps or traps designed for trapping 2D or 3D ion crystals, with FIG. 18 . Step mirror concept. A mirror with the radius of curvature R1=nλ/2 around the center and R2=nλ/2 +λ/4 in the periphery would enhance emission of photons in the solid angle collected by the lens while inhibiting emission of uncollected light. high-optical access to the trapped particles. The trap fabrication pro- cess does not require the alignment of different parts or shadow masking, which makes it reliable and repeatable. The exceptionally high-optical access achieved by the trap could also be used in ion setups that integrate an optical cavity.99,100 We have also presented the design of the complete setup, which includes a state-of-the-art vacuum vessel and an assembly for fast loading of ions in the trap via laser ablation. With several of these setups, one could perform experiments related to remote entanglement distribution and quantum networking without optical cavities.45,46 The physical phenomena that we aim to study with this setup, such as strong inhibition and enhancement of spontaneous emis- sion, are not restricted to atomic ions but could be observed from any quantum emitter. The optical setup presented here could be used with other systems with promising prospects in quantum network- ing and communications, such as quantum dots or diamond spin qubits.101,102 SUPPLEMENTARY MATERIAL The supplementary material includes a STEP file with the trap design. This file can be opened with any standard CAD software such as SolidWorks, AutoCAD, or SALOME. Additionally, we include a 3D PDF with a 3D view of the trap. For a proper view of this file, we recommend using Adobe Acrobat Reader. ACKNOWLEDGMENTS We thank Christoph Wegscheider and the rest of our mechan- ical workshop team for their advice and help with the fabrication of the mechanical components. We gratefully acknowledge the con- tribution of W. Shihua and colleagues at the National Metrology Centre, A∗STAR in Singapore. Without their expertise and the use of their high-NA optical interferometer and contact-probe mea- surements, the first stages of this work would not have been pos- sible. We also thank the FEMTOprint, Elceram, and Asphericon Rev. Sci. Instrum. 91, 113201 (2020); doi: 10.1063/5.0020661 91, 113201-13 © Author(s) 2020Review of Scientific InstrumentsARTICLE scitation.org/journal/rsi teams for their advice and collaboration. We thank Lukas Slod- icka, Martin Van Mourik, Pavel Hrmo, Ezra Kassa, Simon Ragg, and Chiara Decaroli for fruitful discussions. This work was sup- ported by the European Commission through Project No. PIED- MONS 801285 and by the Institut für Quanteninformation GmbH. This work was also supported by the European Commission through the Marie Skłodowska-Curie Action under Grant No. 801110 (Erwin Schrödinger Quantum Fellowship Programme). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, “Quantum dynamics of single trapped ions,” Rev. Mod. Phys. 75, 281 (2003). 2P. Schindler, D. Nigg, T. Monz, J. T. Barreiro, E. Martinez, S. X. Wang, S. Quint, M. F. Brandl, V. Nebendahl, C. F. Roos et al. , “A quantum information processor with trapped ions,” New J. Phys. 15, 123012 (2013). 3B. Appasamy, J. Eschner, Y. Stalgies, I. Siemers, and P. E. 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5.0016071.pdf

J. Appl. Phys. 128, 155101 (2020); https://doi.org/10.1063/5.0016071 128, 155101 © 2020 Author(s).High thermopower and power factors in EuFeO3 for high temperature thermoelectric applications: A first-principles approach Cite as: J. Appl. Phys. 128, 155101 (2020); https://doi.org/10.1063/5.0016071 Submitted: 01 June 2020 . Accepted: 01 October 2020 . Published Online: 15 October 2020 P. Iyyappa Rajan , Carlos Baldo , Enamullah , S. Mahalakshmi , R. Navamathavan , and T. Adinaveen High thermopower and power factors in EuFeO 3 for high temperature thermoelectric applications: A first-principles approach Cite as: J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 View Online Export Citation CrossMar k Submitted: 1 June 2020 · Accepted: 1 October 2020 · Published Online: 15 October 2020 P. Iyyappa Rajan,1,a) Carlos Baldo, III,1,2Enamullah,3S. Mahalakshmi,4,a) R. Navamathavan,5,a)and T. Adinaveen6 AFFILIATIONS 1Asia Pacific Center for Theoretical Physics, POSTECH Campus, Pohang, 37673, South Korea 2Department of Physics, Mapua University, Intramuros, Manila 1002, Philippines 3Department of Physics, School of Applied Sciences, University of Science and Technology, Meghalaya, Ri Bhoi 793101, India 4Chemistry Division, School of Advanced Sciences, Vellore Institute of Technology (VIT), Chennai Campus, Chennai 600127, India 5Division of Physics, School of Advanced Sciences, Vellore Institute of Technology (VIT), Chennai Campus, Chennai 600127, India 6Department of Chemistry, Loyola College (Autonomous), Nungambakkam, Chennai 600034, India a)Authors to whom correspondence should be addressed: rajanselvam29@gmail.com ;mahalakshmi.sn@vit.ac.in ; andnavamathavan.r@vit.ac.in ABSTRACT Thermoelectric materials that can work at operating temperatures of T≥900 K are highly desirable since the key thermoelectric factors of most thermoelectric materials degrade at high temperatures. In this work, we investigate the high temperature thermoelectric performanceof EuFeO 3using a combination of first-principles methods and semi-classical Boltzmann transport theory. High temperature thermoelectric performance is achieved owing to the presence of corrugated flatbands in the valence band region and extremely flatbands in the conduction band region. The lowest energetic structure of EuFeO 3lies within a G-type antiferromagnetic configuration, and the effect of compressive and tensile strains ( −7% to +7%) along the ( a, b) axes on thermoelectric performance is systematically analyzed. An extremely high value of the Seebeck coefficient (more than 1000 μV/K) is consistently recorded in the high temperature region between 900 K and 1400 K in this material. Furthermore, electrical conductivities and power factors are high and electronic thermal conductivities are low in the considered range of temperatures. The calculated theoretical minimum lattice conductivity is small, estimated at around 1.47 –1.54 W m−1K−1. A com- pressive strain of −3% is revealed to be the optimum level of strain for enhancing the key thermoelectric factors. Overall, p-type doping shows better thermoelectric performance than n-type doping in EuFeO 3. Published under license by AIP Publishing. https://doi.org/10.1063/5.0016071 I. INTRODUCTION Growing interest in the study of thermoelectrics (TEs) is driven by the need for alternative sources of electrical power due tothe increasing demand for energy and the depletion of non- renewable resources. 1–4TE materials have become viable candidates due to their capability of converting waste heat into useful electricalenergy. 2,4,5New, high performance TE materials have been reported and can generally be classified into three categories basedon their working temperature: low temperature ( T< 500 K) materi- als such as Bi 2Te3, medium temperature (500 K ≤T≤900 K) mate- rials such as PbTe, and high temperature ( T≥900 K) materialssuch as SiGe alloys.6The availability of TE materials operating at low and mid-temperatures is larger than that for TE materials oper- ating at high temperatures. Nevertheless, TE materials running at T≥900 K are used in key technologies such as in the operation of deep-spacecraft missions, nuclear reactors, and high temperature industrial reactors.1A well-known example is the use of SiGe ther- moelectrics in the spacecraft designs of NASA in the Voyager 1,Voyager 2, and Galileo space probes. 1 Thus, research into new high temperature TE materials with superior properties is receiving a considerable amount of attention. In this work, we investigate exceptional TE performance in aJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-1 Published under license by AIP Publishing.distorted perovskite-based material through first-principles density functional theory (DFT) calculations. Perovskite oxides, with a general formula ABO 3, display a myriad of interesting properties such as magnetoresistance, superconductivity, electrical conductivity,thermal stability, low dielectric loss, and ferroelectricity. 7Additionally, rare earth orthoferrites, usually expressed as RFeO 3,a r ek n o w nt ob e distorted perovskites with extensive applications in solid oxide fuel cells, gas sensors, and photocatalysis.8–11EuFeO 3is one such com- pound from the RFeO 3family that belongs to the orthorhombic crystal system. The Eu ions occupy a dodecahedral A site and the Feions occupy an octahedral B site. The unit cell contains a corner sharing FeO 6octahedron, resulting in a 3D perovskite structure.7The rotation patterns of a strained EuFeO 3and other perovskite films have been studied and reported by Choquette et al.12In an earlier paper, Choquette et al.13observed the development of EuFeO 3thin films by molecular beam epitaxy, and the EuFeO 3bandgap displayed a blue shift in comparison to LuFeO 3. Structural elucidation of rare earth orthoferrites was carried out by Marezio et al., who found that in moving from LuFeO 3to SmFeO 3, the distortion of FeO 6octahe- dron was negligible and nearly constant.14Other than the above- mentioned reports, a few studies have been undertaken for EuFeO 3. Experimental studies of EuFeO 3as a TE material have not been per- formed so far, and we have not found any DFT calculation reportson EuFeO 3that take into account the correct electronic structure. In this work, we present the effectiveness of EuFeO 3as a potential high temperature TE material with the help of first-principles DFT calcula- tions combined with semi-classical Boltzmann transport theory. II. CALCULATION DETAILS A conventional unit cell of EuFeO 3, as illustrated in Fig. 1(a) , follows an orthorhombic structure with space group number 62 (Pnma) containing 4 Eu atoms, 4 Fe atoms, and 12 O atoms. First-principles DFT calculations were implemented using theVienna ab initio simulation package (VASP), 15,16with inputs derived from experimentally resolved x-ray diffraction data14for structural optimization. We employed the projector augmented wave (PAW) method,16,17and a plane wave kinetic energy cutoff of 600.00 eV was used. For the exchange correlation effects, we utilized the Perdew – Burke –Ernzerhof (PBE) potentials, based on the generalized gradient approximation (GGA).18The DFT calculations were carried out in such a way that the exact electronic structure and the ground state magnetic configuration were appropriately realized. This was doneby including the 4f electron states as valence electrons, in addition tothe 5s, 5p, and 6s states for Eu, 3d, and 4s states for Fe, and 2s and2p states for O atoms. To get the appropriate bandgap, we followed the PBE + U approach proposed by Dudarev et al., 19with an effective Hubbard parameter ( Ueff) of 5.0 for 3d states and 11.0 for 4f states. The high Ueffvalue of 11.0 eV for Eu 4f states makes the relaxation convergence easier. This value for Ueffhas been similarly used in Ref. 20to treat Gd 4f states in gadolinium-doped BiFeO 3, which has a similar number of 4f electrons as europium. Brillouin zone integra- tion was chosen to be the Γ-point with 6 × 6 × 4 k-points for relaxa- tion and 10 × 10 × 10 k-points for transport calculations. The EuFeO 3orthorhombic structure was relaxed using the conjugate gradient algorithm method,21wherein only the structural atomic positions underwent the relaxation, not the experimental latticeconstants. The total energy convergence criterion was fixed at 10−6eV and the Hellmann –Feynman forces were relaxed below 0.01 eV/Å. The total energy of the different magnetic configurationsof EuFeO 3(FM, A-type AFM, C-type AFM, and G-type AFM) was examined using spin-polarized calculations, while the non-magneticconfiguration was examined through non-spin-polarized calcula- tions. The results revealed that the lowest energy lies within the G-type AFM configuration of EuFeO 3. The stability order was found to be as follows: non-magnetic > FM > A-type AFM > C-typeAFM > G-type AFM. Therefore, the G-type AFM structure of ortho-rhombic EuFeO 3was considered for all the calculations. After relax- ing the structure of EuFeO 3to the ground state, tensile and compressive bi-axial strains of +7%, +5%, +3%, +1%, −1%,−3%, −5%, and −7% were applied along the in-plane aand baxes and the relaxation was further carried out. The basis of applying bi-axialtensile and compressive strains in EuFeO 3is discussed in Sec. III. The final output structural parameters obtained from the DFT calcu- lations are listed in Table S1 in the supplementary material .T h e s e structural parameters were then fed into the BoltzTrap2 program.22 The thermoelectric performance of a thermoelectric material is determined by analyzing its transport properties, e.g., the Seebeck coefficient ( S), electrical conductivity ( σ), and electronic thermal conductivity ( κe). We calculated these coefficients using a semi- classical Boltzmann transport equation under a constant relaxationtime ( τ) approximation (cRTA) and rigid band approximation as implemented in BoltzTrap2 code. 22The code relies on a Fourier expansion of the electronic band structure energies along with thespace group symmetry information. These initial data were obtainedfrom first-principles simulations, which were required as input datafor the code. Thereafter, the electrical conductivity tensor ( σ γδ) could be obtained by performing the Fourier expansion as σγδ(i,k)¼e2τi,kvγ(i,k)vδ(i,k), (1) where eis the electronic charge, τi,krepresents the electron relaxation time (from the electron –phonon coupling), and vγ(i,k)i st h e γ-component of the group velocity for an electron in band index I, vγ(i,k)¼1 /C22h@[i,k @kγ: (2) From the previous electrical conductivity matrix, it is possible to cal- culate the relevant transport tensor, which depends upon the tem-perature ( T) and the chemical potential ( μ). Hence, the final expressions for σ,v,S,a n d κ etensors are σγδ(T,μ)¼1 (2π)3X ið σγδ(i,k)/C0@f(T,μ) @[/C18/C19 dk, (3) vγδ(T,μ)¼1 (2π)3TX ið σγδ(i,k)[[(k)/C0μ]/C0@f(T,μ) @[/C18/C19 dk, (4) κe,γδ(T,μ)¼1 (2π)3TX ið σγδ(i,k)[[(k)/C0μ]2/C0@f(T,μ) @[/C18/C19 dk, (5)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-2 Published under license by AIP Publishing.and Sij(T,μ)¼X γ(σ/C01)γivγj, (6) where frepresents the Fermi –Dirac distribution function. III. RESULTS AND DISCUSSION A. Band structure of EuFeO 3 The electronic band structures are presented in Fig. 1(b) . The total density of states (DOS) of EuFeO 3is presented in Fig. S1 in the supplementary material . The band structures ofEuFeO 3were calculated within the band path using the SeeK-path software tool.23 Figure 1(c) shows the effect of strain on the relative total energy and bandgap of EuFeO 3. The encircled points in Fig. 1(c) are direct bandgaps, and the other points are indirect bandgaps.There is an increase in the bandgap as the applied strain movesfrom tensile to compressive strain. However, there is a threshold value at which a greater compressive strain has no effect in the bandgap. The reason for the increasing bandgap due to the com-pressive strain can be seen in the band structures of EuFeO 3shown inFigs. 2(a) –2(c). Here, we find that the valence band maximum (VBM) moves downward away from the Fermi level particularly at theΓand A points, whereas the conduction band minimum FIG. 1. (a) Crystal structure of an orthorhombic EuFeO 3(Eu, rose; Fe, brown; and O, red). (b) Electronic band structure of EuFeO 3under zero strain. (c) Effect of strain on the total energy and bandgap of EuFeO 3, where the encircled points are direct bandgaps and the rest of the points are indirect bandgaps. The negative percentage values correspond to compressive strain and positive values to tensile strain.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-3 Published under license by AIP Publishing.(CBM) moves upward as we change from 3% tensile to 3% com- pressive strain. This, in effect, widens the bandgap of the material. At zero strain, a large bandgap of 2.78 eV was calculated, whichwas close to the experimental value of 2.5 eV. 13Though, it appears that our calculations may have overestimated the bandgap ofEuFeO 3, there have been a few cases, such as in PbSe and Bi 2Se3,24 where the PBE approach has overestimated the bandgap. Heyd – Scuseria –Ernzerhof (HSE) calculations could have been used to cal- culate the accurate bandgap, but this was too computationallyexpensive for our system. Nevertheless, we have shown that the cal-culated value for the bandgap using PBE potentials is close to the recently reported bandgap of epitaxial EuFeO 3thin films in Ref. 13. In addition, it is possible that this overestimation might not behugely associated with the PBE potentials that deal with a perfectcrystal structure. In many cases, the synthesized samples may havesome structural/crystal defects (vacancies) that shift the bandgap toward the visible region. From the band structure diagrams in Fig. 1(b) , we find that the valence band of the material displays flat- ness with some local peaks and troughs. Such a band structure withmulti-valley features is referred to as corrugated flat bands as dis-cussed in Ref. 25. In that study, Mori et al . reported that the topmost valence band in PtSb 2, derived from first-principles calcu- lations, is almost entirely from Sb 5p orbitals, it lacks d orbitalcharacter, and is therefore non-bonding in nature. Similar cases havebeen reported, for instance, in Refs. 26and27. In our case, the cor- rugations found in the VBM are brought about by the significant hybridization of Fe and Eu d states with O 2p states. The absence ofsuch a hybridization process is the reason why the CBM is nearlyflat, which is primarily attributed to the Eu 4f and Fe 3d states. Theexistence of such corrugated flatbands in the VBM results in multiple Fermi surface pockets scattered throughout the Brillouin zone. 25 These multiple Fermi pockets arise due to the flat nature of the con- duction band and the corrugated flatness of the valence band. Thepresence of multiple Fermi surfaces with high group velocity givesrise to an enhanced Seebeck coefficient and a high electric conduc- tivity, 25which leads to a high power factor. Aside from the extremely flat and corrugated flatbands, a large bandgap further ensures a highSeebeck coefficient. 26This was similarly observed in Ref. 27,w h e r e nitrogenated holey graphene displayed flatbands near the Fermi leveland showed a higher bandgap, thereby giving rise to a large Seebeck coefficient. The presence of these features in the band structure, combined with its large bandgap, strongly supports our idea thatEuFeO 3is a promising TE material, with a high Seebeck coefficient and a large electrical conductivity, as shown in the following sec- tions. An enhanced power factor is required for a good thermoelec- tric material as it can produce more power. B. Thermoelectric properties The effectivity of a TE material is commonly measured using the figure of merit ZT, ZT¼S2σT κ: (7) Here, we define the following thermoelectric properties: Sis the Seebeck coefficient, σis the electrical conductivity, and κis the FIG. 2. Magnified view of the electronic band structure of EuFeO 3under (a) 3% tensile, (b) zero, and (c) 3% compressive strain.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-4 Published under license by AIP Publishing.total thermal conductivity of the material. κis further expressed as the sum of the electron ( κe) and lattice ( κl) contributions to the thermal conductivity. Recently, TE performance has been measuredusing another TE quantity, the power factor PF, which can be computed directly as PF¼S 2σ: (8) Thus, we can say that an efficient TE candidate is a material that has a large Seebeck coefficient, a high electrical conductivity, ahigh power factor, and a sufficiently low thermal conductivity. Inthis work, we demonstrate the potential TE capability of EuFeO 3at high temperatures by highlighting its remarkable TE properties. We begin with its high Seebeck coefficient obtained for T≥900 K, as shown in Fig. 3(a) . As seen in this graph, the calculated values of Sare found to be at least 800 μV/K across all given temperatures and strains. Additionally, the calculation is employed using differ- ent values of strain, which varies from tensile (+7%, +5%, +3%, +1%) to compressive ( −7%,−5%,−3%,−1%). From Fig. 3(a) ,i t can be seen that Sis greatest across all temperatures when a com- pressive strain −3% is applied, as indicated by the blue arrows. However, application of tensile strain lowers the Seebeck coefficient. This effect of the strain on the Seebeck coefficient can be described in two modes, which are both traced to the band structure. To illus-trate this, we show in Figs. 2(a) –2(c)the band structures of EuFeO 3 under 3% tensile, zero, and 3% compressive strains. First, the pres- ence of compressive strain enhances the corrugations in the VBM and the flatness in the CBM. As can be seen in the band structures, there are more corrugations (local peaks and troughs) in thevalence band as we move from tensile toward compressive strain.These enhanced corrugations near the Fermi level, as discussedearlier, induce a greater velocity difference in the carriers, which consequently leads to an improvement in the Seebeck coefficient. Additionally, the increase in Sdue to the 3% compressive strain is brought about by the overlapping of multiple bands in the VBMand the CBM. As discussed earlier, the bandgap of the material increases as the applied strain is altered from tensile to 3% com- pressive strain, where it reaches its maximum value. This increasein the bandgap can also improve the Seebeck coefficient. In ourwork, we limit the discussion to the two cases of zero strain and3% compressive strain. As shown in Fig. 1(c) , the relative total energies of 5% and 7% compressive strains are significantly high compared to that of zero strain. Therefore, we did not considerthese strains for comparison. In Figs. 3(b) and3(c), we show the effect of doping on SforT≥900 K under zero and 3% compressive strain, respectively. For a given curve of Svsμ−E F, the kind of doping in the material can be deduced from the sign of S:n e g a - tive values indicate n-type dopi ng and positive values indicate p-type doping. Through careful inspection, we find that for bothstrain cases, p-type doping shows slightly higher values of Sthan n-type doping for temperatures above 1000 K. As observed earlier, a larger Sis achieved for the −3% compressive strain than for the unstrained case. There is no doubt that high Sis attain- able from EuFeO 3at high temperatures. Nevertheless, we have discovered that a similar magnitude of Scan also be obtained, even at relatively low temperatures. We show this in Fig. S2 in thesupplementary material . Here, we briefly examine how S FIG. 3. (a) Seebeck coefficients in EuFeO 3under different values of strains for 900 K≤T≤1400 K. (b) and (c) Seebeck coefficients with respect to μ−EFin EuFeO 3under (b) zero and (c) 3% compressive strains for 900 K ≤T≤1400 K.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-5 Published under license by AIP Publishing.changes for 300 K ≤T≤1400 K under zero strain. Interestingly, we find that Sexceeds 1000 μV/K, even for T< 900 K. However, a plateau around μ−EFis clearly present where S=0 f o r t h e c a s e s where T< 900 K. This distinctive feature implies the presence of a wide bandgap in EuFeO 3that prevents current flow through the material at temperatures lower than 900 K. Apparently, this plateau vanishes at T≥900, which indicates that electronic exci- tation becomes easier due to an increase in thermal energy.Therefore, this poses a requirement of thermal activation for theflow of current. Consequently, it is now clear why a larger Sis obtained in EuFeO 3for temperatures beyond 900 K and which makes it an excellent high temperature TE material.Another equally important factor in achieving high ZTand power factors is the electrical conductivity σ. In the case of semicon- ducting materials, σvaries due to the density and dynamics of elec- trons and holes. Figures 4(a) and 4(b) show the behavior of the electrical conductivity per relaxation time ( σ/τ0) in EuFeO 3under zero and 3% compressive strains, respectively. It is demonstrated that the temperature has no significant effect on electrical conduc- tivity, although σslightly decreases with increasing temperature. Under 3% compressive strain in EuFeO 3,σis more pronounced than the unstrained EuFeO 3at the lower μ−EFregion around 1.5 eV. However, σincreases from −1.0 eV to −6.0 eV in both cases. Doping, on the other hand, has a clear effect on σ. As shown in FIG. 4. (a) and (b) Electrical conductivity per relaxation time ( σ/τ0) in EuFeO 3under (a) zero and (b) 3% compressive strains for 900 K ≤T≤1400 K. (c) and (d) Electronic thermal conductivity per relaxation time ( κe/τ0) in EuFeO 3under (c) zero and (d) 3% compressive strains for 900 K ≤T≤1400 K.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-6 Published under license by AIP Publishing.Figs. 4(a) and4(b), p-type doping shows higher electrical conductiv- ities than n-type doping, particularly at energies near the bandgap. The third factor essential in achieving large ZTis the thermal conductivity ( κ) of a material. Related to the flow of heat in the materials, κis broken down into electronic κeand lattice thermal conductivities κl. The former is primarily due to conduction elec- trons, while the latter has its origin from lattice phonon vibrations in the material. For a good TE material, it is expected that the thermal conductivity should be sufficiently low as it is inverselyproportional to ZT.Figures 4(c) and 4(d) show the variation in electronic thermal conductivity ( κ e/τ0) of EuFeO 3within the tem- perature range of 900 K to 1400 K under zero and 3% compressive strains, respectively. It is evident that for both cases, κeis lower at 900 K. Additionally, we find that as the temperature increases, κealso slightly increases. Yet, overall, κedoes not significantly alter with changes in temperature. We were not able to calculate the lattice contribution to the thermal conductivity, due to the limita-tion of the Boltztrap2 code that can calculate κ eonly. Nonetheless, we are confident that EuFeO 3should have a rela- tively high thermopower, due to the presence of extremely flat and corrugated flatbands near the Fermi level and the existence of a wide bandgap. A similar case has also been reported in the TEperformance of crystalline cinnabar ( α-HgS). 28If this is so, the increase in the thermopower ultimately contributes to the increasein the high PFs. In a higher temperature region, TE materials with high PFs would be the right choice rather than looking into the figure of merit ( ZT). With regard to the lattice thermal conductiv- ity of EuFeO 3, previous experimental and lattice-statics simulation studies on the κlof RFeO 3materials have shown that it decreases at higher temperatures.29–31In our work, we provide the values of κlby calculating the theoretical minimum value based on Cahill ’s method.32To estimate this parameter, we first calculated the elastic constants ( Cij) and then checked the orthorhombic stability conditions followed by calculating the mechanical and sound/vibrational properties. The MechElastic code 33was used for this purpose, and the orthorhombic stability conditions34were satis- fied for EuFeO 3under zero and strained conditions. The calcu- lated elastic constants ( Cij) and mechanical properties of EuFeO 3 under zero and compressive strains up to 5% are listed in Tables I and II. The omission of calculating these parameters for tensile strains is discussed later in this section. It is clearly perceived thatthe elastic constants ( C ij), mechanical properties, sound/vibra- tional velocities, and the Debye temperature steadily increasedwith increasing compressive strain. Once these parameters were obtained, the theoretical minimum lattice thermal conductivityTABLE I. Calculated elastic constants ( Cij) required to satisfy the orthorhombic stability conditions of EuFeO 3under zero and compressive strains. (Cij) (GPa)Strain in EuFeO 3 0 −1% −3% −5% C11 286.532 93 296.076 21 309.484 03 325.202 62 C12 128.984 11 143.701 72 174.023 15 204.019 74 C13 119.705 60 129.789 59 150.938 64 176.351 83 C22 268.828 63 282.561 97 305.026 91 327.886 79 C23 125.451 61 132.401 52 147.400 07 167.197 35 C33 272.766 89 285.218 59 311.357 46 341.586 65 C44 109.718 55 112.992 99 118.638 98 125.919 93 C55 88.194 57 92.955 27 101.430 15 110.964 05 C66 76.844 91 84.222 46 100.908 71 119.040 31 TABLE II. Calculated mechanical properties, vibrational/sound velocities, Debye temperatures, and theoretical minimum lattice thermal conductivities o f EuFeO 3under zero and compressive strains. Strain in EuFeO 3 0 −1% −3% −5% Bulk modulus (GPa) Voigt 175.157 186.182 207.844 232.202 Reuss 175.048 186.013 207.628 232.053 average 175.102 186.098 207.736 232.127 Shear modulus (GPa) Voigt 85.217 88.565 94.429 100.992 Reuss 83.338 86.528 90.827 94.117 average 84.278 87.547 92.628 97.554 Young ’s modulus (GPa) Voigt 219.978 229.332 246.029 264.613 Reuss 215.772 224.738 237.805 248.724 average 217.875 227.035 241.917 256.668 Poisson ratio Voigt 0.291 0.295 0.303 0.310 Reuss 0.295 0.299 0.309 0.321 average 0.293 0.297 0.306 0.316 Transverse velocity (m/s) 3388.133 00 3418.653 65 3445.447 95 3462.949 81Longitudinal velocity (m/s) 6257.521 30 6358.171 45 6515.463 85 6672.631 66 Average velocity (m/s) 3780.951 01 3816.952 33 3851.379 37 3875.819 36 Debye temperature (K) 497.670 15 505.789 63 517.341 06 527.907 94Minimum lattice thermal conductivity (W/m K) 1.4762 1.4946 1.5184 1.5402Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-7 Published under license by AIP Publishing.(κl,min) was estimated, as given in Table II . The calculated κl,min was very small (1.4762 W m−1K−1for zero strained EuFeO 3)a n d increased with increasing compressive strain. Given this circum-stance, rather than thermal conductivity, we then assert that thepower factor ( PF) is more crucial in generating high thermoelec- tric power for a given ZT. Recently, it has been discovered that enhancing the PFis a clever way of increasing the efficiency and the output power gener-ation. 35In fact, this approach offers an ingenious technique other than lowering the thermal conductivity. Plots of PFper relaxation time ( S2σ/τ0) of EuFeO 3with respect to μ−EFunder zero and 3% compressive strains are shown in Figs. 5(a) and5(b), respectively.The PFreaches a maximum at the negative chemical potential μ−EF=∼(−1.4 eV), and so we considered this point of chemical potential as a critical point for high TE performance of EuFeO 3. For positive chemical potentials such as μ−EF= +2.04 eV and μ−EF= +2.72 eV, although the calculated PFs were considerably high, these values were comparably lower than those obtained for negative chemical potentials. These results confirm that high PFs were obtained for p- and n-type doped material, but those obtainedfor p-type doping were greater. We have therefore chosen topresent the thermal and strain dependence of the PFs for the case ofμ−E F=∼(−1.4 eV), where the maximum PFwas obtained. Figures 5(c) and 5(d) show how the maxima of PF(S2σ/τ0)a t FIG. 5. (a) and (b) Power factor per relaxation time (S2σ/τ0) in EuFeO 3under (a) zero and (b) 3% compressive strains for 900 K ≤T≤1400 K. (c) and (d) Power factor per relaxation time ( S2σ/τ0)a tμ−EF=∼(−1.4 eV) as a function of temperature for different values of strain in EuFeO 3along the components (c) xxand (d) yy.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 155101 (2020); doi: 10.1063/5.0016071 128, 155101-8 Published under license by AIP Publishing.μ−EF=∼(−1.4 eV) behaves with respect to temperature for differ- ent values of strain along the components ( xxand yy). In these figures, we can immediately infer that the PFs are anisotropic in behavior along the xxand yycomponents. Here, the PFcomponent along yyshows higher values than the component along xx. On the other hand, the component along zzshows similar values of PFto the component along xx; hence, we did not further investigate the application of strain along the c-axis. Note that the PFs are larger along the component yythan along the component xxin all cases of strain. Remarkably, the effect of the bandgap is also reflected inthe PFvariation brought about by the strain. The bandgap increases as we move from tensile strain (positive) to compressive strain (negative), and the PFs also significantly increase. EuFeO 3 with a wide bandgap with moderate strain ( −3% compressive) turns out to have a better PF, which slightly increases with respect to the temperature. The tensile strain shows lower PFs than the compressive strain and so we omitted the calculation of elastic con- stants, mechanical properties, and the theoretical minimum latticethermal conductivities for the tensile strained EuFeO 3configura- tions. Also, the lattice thermal conductivity difference betweentensile and compressive strain is expected to be only a small differ- ence, which could be realized from the difference between the zero and compressive strained EuFeO 3. Although the power factors for 5% and 7% compressive strains appear to be higher than that ofthe 3% compressive strain due to larger electrical conductivities with increasing compressive strain, the relative total energies of the former strains are comparatively higher than in the ground state.While considering the overall variations with respect to strain, thecompressive strain of −3% in EuFeO 3approaching a limiting wide bandgap value shows the highest Seebeck coefficient values, optimal electrical conductivities, and PFs in comparison to zero and other strained conditions in EuFeO 3. IV. CONCLUSION We have found that the presence of corrugated flatbands in the valence band region and the extremely flatbands in the conduc- tion band region in the band structure of EuFeO 3reveal its poten- tial as a thermoelectric material in the high temperature region.This is confirmed from the very high Seebeck coefficients and elec-trical conductivities, low electronic thermal conductivities, and sig-nificantly enhanced power factor values calculated in EuFeO 3 under compressive strain. The theoretical minimum lattice conduc-tivity was estimated at around 1.47 –1.54 W m −1K−1. EuFeO 3is an experimentally relevant system, and we have predicted that theexceptionally high thermopower and power factors in this widebandgap semiconducting band structure, with corrugated flat and extremely flatbands arising due to large numbers of unpaired 4f and 3d electrons, could be a breakthrough in the field of hightemperature thermoelectric materials. SUPPLEMENTARY MATERIAL See the supplementary material for some of the relevant ther- moelectric properties, structural parameters, and density of states discussed in this work.ACKNOWLEDGMENTS This research work and the authors Dr. P. Iyyappa Rajan and Dr. Carlos Baldo III were supported by appointments to the YoungScientist Training (YST) Program at the Asia Pacific Center for Theoretical Physics (APCTP) through the Science and Technology Promotion Fund and Lottery Fund of the Korean Government.This work also received additional support from the Korean LocalGovernments of Gyeongsangbuk-do Province and Pohang City. DATA AVAILABILITY The data that support the findings of this work are available within the article and the supplementary material . REFERENCES 1D. M. Rowe, CRC Handbook of Thermoelectrics (CRC Press, 1995). 2S. Twaha, J. Zhu, Y. Yan, and B. Li, “A comprehensive review of thermoelectric technology: Materials, applications, modelling and performance improvement, ” Renew. Sust. 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5.0017299.pdf

AIP Advances 10, 095135 (2020); https://doi.org/10.1063/5.0017299 10, 095135 © 2020 Author(s).Thermal stability of the crystallographic structure of nanocrystalline Nd0.7Sr0.3MnO3 manganite with enhanced magnetic properties Cite as: AIP Advances 10, 095135 (2020); https://doi.org/10.1063/5.0017299 Submitted: 07 June 2020 . Accepted: 09 July 2020 . Published Online: 28 September 2020 Subrata Das , Bashir Ahmmad , and M. A. Basith 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 Catalog of magnetic topological semimetals AIP Advances 10, 095222 (2020); https://doi.org/10.1063/5.0020096 Magnetic characterization of rare-earth oxide nanoparticles Applied Physics Letters 117, 122410 (2020); https://doi.org/10.1063/5.0023466 A monolithic artificial iconic memory based on highly stable perovskite-metal multilayers Applied Physics Reviews 7, 031401 (2020); https://doi.org/10.1063/5.0009713AIP Advances ARTICLE scitation.org/journal/adv Thermal stability of the crystallographic structure of nanocrystalline Nd 0.7Sr0.3MnO 3manganite with enhanced magnetic properties Cite as: AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 Submitted: 7 June 2020 •Accepted: 9 July 2020 • Published Online: 28 September 2020 Subrata Das,1,a) Bashir Ahmmad,2 and M. A. Basith1,b) AFFILIATIONS 1Nanotechnology Research Laboratory, Department of Physics, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh 2Graduate School of Science and Engineering, Yamagata University, 4-3-16 Jonan, Yonezawa 992-8510, Japan a)Author to whom correspondence should be addressed: subratadas.buet@gmail.com b)Electronic mail: mabasith@phy.buet.ac.bd ABSTRACT We report the effect of temperature on the crystallographic structure and magnetic properties of ultrasonically prepared nanostructured Nd 0.7Sr0.3MnO 3perovskite manganite. The crystal structure of as-synthesized nanoparticles remains unaltered over a wide scanning tem- perature range. Temperature dependent magnetization measurements demonstrate that the Curie temperature (T c) of Nd 0.7Sr0.3MnO 3 nanoparticles is in the range of 211 K–220 K under largely varying applied magnetic fields. Below T c, the soft ferromagnetic nature of these nanoparticles is confirmed by the field-dependent magnetization measurements. The absence of the charge-ordered state is also revealed in this nanomanganite down to 20 K, which is strikingly different from analogous Nd–Sr based nanocrystals. The experimentally observed effec- tive paramagnetic moment and saturation magnetic moment have matched quite well with the values calculated theoretically. The T cvalues up to a temperature of 220 K, nearly perfect ferromagnetically ordered Mn ions below T c, high saturation magnetization, and magnetic soft- ness of synthesized nanostructured Nd 0.7Sr0.3MnO 3manganite can be associated with their good crystallinity as well as the nominal internal and surface disorder effect owing to intermediate particle size ( ∼75 nm to 150 nm). Our investigation elucidates the promising potential of nanocrystalline Nd 0.7Sr0.3MnO 3particles for numerous technological applications. ©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/10.1063/5.0017299 .,s I. INTRODUCTION The mixed valence perovskite manganite, R 1−xDxMnO 3 (R = rare earth cations, e.g., La, Gd, Sm, and Nd; D = alkali cations, e.g., Sr, Ca, and Ba), is a strongly correlated electron system exhibit- ing numerous intriguing multifunctional properties such as the magnetocaloric effect, multiferroicity, colossal magnetoresistance, etc.1,2In recent years, various prototype spintronic devices have been developed exploiting the rich physics involved in manganites.3,4 Notably, a number of recent investigations have demonstrated that nanostructured perovskite manganites manifest novel magnetic and electronic properties, e.g., superparamagnetism, super-spin glass state, etc., which are significantly different from their bulk coun- terparts.4–6However, most of the manganite systems at nanoscalehaving a ferromagnetic ground state exhibit lower ferromagnetic to paramagnetic transition temperature (T c) and weaker ferromag- netism as compared to analogous bulk samples.7–9One of the rea- sons for such a variation in T cis the formation of a magnetically dead layer by the crystallographic defects, the effect of which enhances for reduced particle size.5,6It is worth noting that the complex magnetic and electronic phase diagrams of rare earth based perovskite man- ganites evolve mainly from double exchange and super exchange interactions.10,11Along with the doping concentration of alkaline earth cations, the strength of these interactions is also highly influ- enced by different chemical factors such as the mean radii of rare and alkaline earth cations and cationic size mismatch.12,13 According to previous investigations, Nd3+(ionic radius: 1.163 Å) based manganites demonstrate various intriguing and AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv complex magnetic and magnetotransport properties as compared to other perovskite manganite systems with larger rare earth cations, e.g., La3+(ionic radius: 1.216 Å).14–18For instance, colossal magne- toresistance has been observed in the whole ferromagnetic metal- lic domain of bulk polycrystalline Nd 1−xSrxMnO 3perovskite man- ganites (for x = 0.3–0.48).12,19A giant magnetocaloric effect and complex ferromagnetic state have been reported in Nd 1−xSrxMnO 3 manganites for x = 0.3, which reveals their promising potential for practical applications and intrigues considerable research interest to further explore their functional properties at nanoscale.20–22 However, until now, perovskite Nd–Sr manganites have been extensively investigated mainly in the bulk form as thin films and mono- or poly-crystalline samples. Recently, Arun et al. have presented the magnetic characteristics of nano-grained Nd 0.67Sr0.33MnO 3prepared via the wet chemical based sol–gel tech- nique having an average particle size of 22 nm.23The reported T c of Nd 0.67Sr0.33MnO 3manganite is 66 K, which is much lower than room temperature. Moreover, the magnetization of these mangan- ite nanoparticles (size ∼22 nm) was also conspicuously low. Hence, it might be technologically worthwhile to conduct further exten- sive research to synthesize Nd–Sr manganite nanostructures with T c close to room temperature and improved magnetic properties. Notably, in another previous investigation,24the magneto- transport properties of a related nanocrystalline manganite Nd 0.5Sr0.5 MnO 3have been studied albeit their temperature-dependent crys- tallographic structure was not explored extensively. Since a lat- tice strongly couples with electron, spin, and orbital freedom,4it is worthwhile to comprehensively investigate the crystallographic structure of these nanosized perovskite manganites over a broad temperature range to get better insight into their physical prop- erties. Therefore, in the present investigation, nanostructured per- ovskite Nd 0.7Sr0.3MnO 3manganite with an intermediate particle size (∼75 nm to 150 nm) was synthesized directly from their bulk pow- der material via ultrasonication25to systematically investigate the effect of temperature variation on their crystallographic structure as well as magnetic characteristics under different external fields. We observed consistency in the crystallographic structure and lat- tice parameters of as-synthesized nanoparticles over a wide range of scanning temperatures. Furthermore, magnetization measurements of nanostructured Nd 0.7Sr0.3MnO 3manganite confirmed ordering of Mn ions below T calong with their high saturation magnetization and magnetic softness. II. EXPERIMENTAL DETAILS Nanocrystalline Nd 0.7Sr0.3MnO 3manganite was synthesized directly from their bulk powder materials via the ultrasonica- tion method, as was developed in previous investigations for the synthesis of nanostructured Gd 0.7Sr0.3MnO 3perovskite manganite and multiferroic Bi 0.9Gd 0.1Fe1−xTixO3nanoparticles.25–27Initially, bulk polycrystalline powders of Nd 0.7Sr0.3MnO 3were produced by adopting the solid state reaction technique, as was described else- where.28,29A homogeneous mixture of stoichiometrically weighed analytical grade Nd 2O3, SrCO 3, and MnCO 3(Sigma-Aldrich) pow- ders was prepared and calcined at 900○C for 12 h in a programmable furnace.30Then, the calcined powders were ground for 2 h to enhance the homogeneity of the mixture and again calcined at 1100○C for 6 h. Afterward, pellets (10 mm diameter and 1 mmthickness) of the powder materials were prepared and sintered at 1300○C for 5 h in air for obtaining optimum density and improving the crystal quality by minimizing defects.30Thereafter, the pellets were broken to powder by grinding and mixed with isopropanol. The solution was then ultrasonicated for 60 min in an ultrasonic bath (power: 80 W), and decantation was carried out after 4 h rest, as was described in our previous investigations.26,27Finally, the col- lected supernatant was dried at 80○C for 5 h before performing the required characterizations. The crystallographic structure of as-prepared nanoparticles was characterized by obtaining their powder X-ray diffraction (XRD) patterns over the temperature range of 100 K–300 K. The Rietveld refinement of the XRD spectra was carried out using the Full- Prof Suite software.31Field emission scanning electron microscope (FESEM, JEOL, JSM 5800) imaging along with the energy disper- sive X-ray (EDX) analysis was performed to investigate the surface morphology and elemental composition of as-synthesized nanopar- ticles. For analyzing the surface chemical states of the nanoparticles, X-ray photoelectron spectroscopy was conducted at room temper- ature. A superconducting quantum interference device (SQUID) magnetometer (Quantum Design MPMS-XL7, USA) was used for performing the temperature dependent magnetization measure- ments of Nd 0.7Sr0.3MnO 3nanoparticles under different external magnetic fields ranging from 0.2 kOe to 10 kOe via both zero- field cooled (ZFC) and field cooled (FC) processes.27Further- more, the field dependent magnetization measurements were car- ried out at 300 K, 200 K, 100 K, and 20 K using the same SQUID magnetometer. III. RESULTS AND DISCUSSIONS A. Crystal structure analysis The effect of scanning temperature on the crystallographic structure and phase of as-synthesized Nd 0.7Sr0.3MnO 3nanoparti- cles has been inspected by analyzing their powder XRD patterns by the Rietveld refinement method.31Figures 1(a)–1(f) show both the observed and Rietveld refined powder XRD spectra of the nanoparticles as obtained at the temperatures of 100 K, 150 K, 200 K, 225 K, 250 K, and 300 K, respectively. The Bragg reflec- tion peaks manifested by Nd 0.7Sr0.3MnO 3nanomanganite at all temperatures conform to the single phase orthorhombically dis- torted perovskite structure (space group Pnma).32No undesired secondary phases were detected in the observed XRD spectra of as- synthesized nanoparticles suggesting their high phase purity. The high intensity of the sharp diffraction peaks is a clear indication of the excellent crystallinity of synthesized Nd 0.7Sr0.3MnO 3nanoparti- cles at all temperatures. Notably, the powder XRD pattern remains almost unaltered throughout the whole temperature range. As evi- dent from Fig. 1, splitting of reflection peaks or appearance of any extra reflections did not occur due to the change in scanning tem- perature, which clearly indicates that the structural phase transition of Nd 0.7Sr0.3MnO 3nanoparticles did not occur due to the decrease in the measurement temperature from room temperature to far below their Curie temperature (T c),∼220 K. Such a behavior of sub- 150 nm Nd 0.7Sr0.3MnO 3manganite is in good agreement with the result of a previous investigation which had demonstrated that the room temperature crystal structure of analogous La 0.5Ca0.5MnO 3 AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 1 . Rietveld refined powder XRD patterns of Nd 0.7Sr0.3MnO 3nanoparticles obtained at different temperatures. manganite nanocrystals does not evolve on cooling unlike their bulk counterpart.33 The lattice parameters of the Nd 0.7Sr0.3MnO 3nanoparticles obtained at different temperatures via Rietveld refinement along with the reliability (R) factors are presented in Table I. The small values of R factors indicate an excellent fit with the orthorhom- bic crystal phase throughout the whole scanning temperature range. The lattice parameters of the nanoparticles are in a sequenceb/√ 2<a<c. Such a trend has also been reported for a number of bulk as well as nanostructured Nd–Sr based manganites hav- ing different doping concentrations.32,34Clearly, the change in lat- tice constants is quite nominal over the whole temperature range. Figure 2(a) shows the variation of the unit cell volume of prepared Nd 0.7Sr0.3MnO 3nanoparticles with temperature, which demon- strates that the cell volume shrinks only by 0.4% after cooling down the temperature to 100 K from 300 K. Notably, the orthorhom- bic deformation percentage of perovskite manganites indicates the degree of Jahn–Teller distortion around the Mn3+ions in their unit cells.35To investigate the temperature-effect on the distor- tion of as-synthesized Nd 0.7Sr0.3MnO 3nanoparticles, we have deter- mined their orthorhombic deformations (D%) at different scanning temperatures and inserted in Table I. D% is defined as35 D=1 33 ∑ i=1∣ai−¯a ai∣×100, (1) where a1= a,a2=b/√ 2,a3= c, and ¯a=(abc/√ 2)1/3. Notably, the percentage of orthorhombic deformation only nominally changed despite large variation in the measurement temperature, which reflects the structural stability of the synthesized manganite. Furthermore, the temperature evolution of orthorhombic strains ( Os/⊙◇⊞and Os∥) of Nd 0.7Sr0.3MnO 3nanoparticles is shown in Fig. 2(b). The orthorhombic strain, Os∥= 2( c−a)/(c+a), is defined as the strain in the ac plane, whereas Os/⊙◇⊞=2(a +c−b√ 2)/(a+c+b√ 2)gives the strain along the b-axis with respect to the ac plane.33,36Interestingly, a sudden decline in the curve of Os∥can be observed near its T c, albeit no systematic change is discernible in the values of Os/⊙◇⊞over the temperature range. Such a temperature variation of Os∥is quite unique and has not been observed in similar bulk and nanocrystalline per- ovskite manganites to the best of our knowledge. It may be antic- ipated that the instantaneous change in the spin alignment of Nd 0.7Sr0.3MnO 3manganite due to temperature might play a role in lowering the orthorhombic strain, Os∥, during magnetic phase transition.33 Notably, both the unaltered crystallographic phase and a nom- inal change in lattice parameters of Nd 0.7Sr0.3MnO 3nanoparti- cles due to temperature variation are conspicuously different from those of the bulk materials of the Nd 1−xSrxMnO 3perovskite man- ganite family.4,34For instance, Eremenko et al. had demonstrated TABLE I . Lattice parameters (a, b, and c) and orthorhombic deformation (D%) of Nd 0.7Sr0.3MnO 3nanoparticles determined at different temperatures from Rietveld refined powder XRD spectra and the weighted profile factor (R wp) and Goodness-of-Fit (GoF) parameter of Rietveld refinement. Scanning a b c Orthorhombic temperature (K) (Å) (Å) (Å) deformation D (%) R wp GoF 100 5.4459 (1) 7.6934(1) 5.4559(1) 0.11 3.64 2.31 150 5.4455 (1) 7.6927(1) 5.4567(1) 0.12 3.61 2.41 200 5.4479 (1) 7.6950(1) 5.4589(1) 0.12 3.70 2.45 225 5.4525 (1) 7.6950(1) 5.4611(1) 0.13 3.02 1.54 250 5.4535 (1) 7.7014(1) 5.4627(1) 0.11 3.37 1.94 300 5.4541 (1) 7.7043(1) 5.4648(1) 0.11 3.40 2.20 AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Variation of (a) cell volume and (b) orthorhombic strains of Nd 0.7Sr0.3 MnO 3nanoparticles with temperature. that below 158 K, the crystal symmetry of bulk single crystalline Nd 0.5Sr0.5MnO 3perovskite manganite reduces significantly and phase transition occurs from orthorhombic to monoclinic.34Such a structural transformation of bulk perovskite manganites due to cooling has been associated with the charge and orbital ordering arising from the Jahn–Teller effect.33,34Therefore, we may infer that the temperature-independent crystal structure of nanostruc- tured Nd 0.7Sr0.3MnO 3manganite might prevent the evolution of the charge ordered phase at a low-temperature in these nanoparticles. B. Morphological and elemental studies The FESEM image of as-synthesized Nd 0.7Sr0.3MnO 3per- ovskite nanomanganite is shown in Fig. 3(a), which demonstrates that the surface of the nanoparticles is satisfactorily homogeneous and non-porous. From the particle size distribution histogram, as shown in Fig. 3(b), we may estimate that the particle size of as-prepared Nd 0.7Sr0.3MnO 3manganite lies mostly in the range of∼75 nm to 150 nm with an average of ∼120 nm. Further- more, to determine the elemental composition of the as-prepared Nd 0.7Sr0.3MnO 3particles, the EDX analysis was carried out at room temperature at different points of the surface over the range of ∼1 μm. It is worth noting that the composition remained the same over the scanning range. The experimentally obtained atom (%) yielded a Nd:Sr:Mn ratio of 0.70:0.31:1, which is well consistentwith our desired stoichiometry. However, the elemental composi- tion suggested that our synthesized Nd 0.7Sr0.3MnO 3nanoparticles have some oxygen deficiency (Nd 0.7Sr0.3MnO 3−ε) with ε=∼1.86. Notably, oxygen deficit is quite common in the formation of nano- dimensional mixed-valence perovskite manganites, and it plays a role in determining their magnetic and transport behavior by chang- ing the mixed Mn3+/Mn4+valence states.37,38 C. Chemical state analysis The surface chemical composition of as-synthesized Nd 0.7Sr0.3 MnO 3nanoparticles was studied by XPS, and the full survey XPS spectrum is shown in Fig. 4(a). As shown in this figure, strong peaks for Nd, Sr, Mn, and O core levels were detected in the XPS spec- trum, which conform to the desired surface chemical states of our as-prepared sample and ensure purity.22,39The discerned C 1 ssignal is due to the presence of adventitious carbon layers on the surface of the sample, which is quite common in XPS studies.40,41Figures 4(b)– 4(e) show the high resolution XPS core spectra of Nd 4 d, Sr 3 d, Mn 2p, and O 1 sobtained for the Nd 0.7Sr0.3MnO 3nanoparticles, respec- tively. The core level spectrum of Nd has been identified by the Gaussian peak at the binding energy of 121.5 eV corresponding to the state of Nd 4 d3/2[Fig. 4(b)].42The Sr 3 dXPS spectrum as shown in Fig. 4(c) contains a doublet at the binding energies of 131.8 eV and 133.5 eV, which can be assigned as Sr 3 d5/2and Sr 3 d3/2lines with the energy for spin–orbit splitting being 1.7 eV for Sr 3 d.39,43 FIG. 3 . (a) FESEM image and (b) the cor- responding particle size distribution his- togram of Nd 0.7Sr0.3MnO 3nanoparticles. AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 4 . Room temperature XPS spectra of Nd 0.7Sr0.3MnO 3nanoparticles: (a) an XPS survey spectrum and [(b)–(e)] high-resolution core spectra of Nd 4 d, Sr 3 d, Mn 2 p, and O 1s, respectively. The inset of (d) shows the doublet of the Mn 2 pstate. Figure 4(d) shows the high resolution XPS spectrum of the Mn content in as-synthesized Nd 0.7Sr0.3MnO 3nanoparticles. As can be determined from the inset of Fig. 4(d), the energy separa- tion of the Mn 2 pdoublet peak ( ΔEB.E.=B.E.2p1/2−B.E.2p3/2) in Nd 0.7Sr0.3MnO 3is 11.6 eV, which is consistent with the valuesreported in the literature.43To further investigate the presence of the Mn3+/Mn4+mixed valence state in Nd 0.7Sr0.3MnO 3nanomanganite, we have de-convoluted the asymmetric Mn 2 p3/2peak into two sym- metric Gaussian peaks. Notably, the Mn 2 p3/2peak consists of two peaks at the binding energies of 641.9 eV and 643.4 eV, which can AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv be attributed to the oxides of Mn3+and Mn4+, respectively.43,44The XPS spectrum of the O 1 score level of nanosized Nd 0.7Sr0.3MnO 3 manganite also shows a slightly asymmetric peak, which has been Gaussian fitted by two symmetrical peaks at 529.1 eV and 531.1 eV [Fig. 4(e)]. The peak at the lower binding energy, i.e., 529.1 eV, can be ascribed to the O 1 score spectrum, while the higher bind- ing energy peak indicates oxygen deficiency, i.e., oxygen vacancy related defects45,46in the sample, which was also evident from our EDX analysis. D. Magnetic properties The temperature dependent direct current (dc) magneti- zation measurements of as-synthesized Nd 0.7Sr0.3MnO 3particles were performed under different external magnetic fields rang- ing from 0.2 kOe to 10 kOe via both ZFC and FC methods. Figures 5(a)–5(e) show the variation in magnetization (M) as a func- tion of temperature (T) recorded under 0.2 kOe, 0.5 kOe, 1 kOe, 5 kOe, and 10 kOe magnetic fields, respectively. To carry out the measurement by the ZFC method, the nanoparticles were ini- tially cooled down to 10 K from 300 K without applying an external field, and then, the data were collected while heating under the applied magnetic field. In the FC mode, the magne- tization measurement was performed under the same field while cooling.27 To determine the Curie temperature (T c) of Nd 0.7Sr0.3MnO 3 nanoparticles, we have plotted the derivative of their FC magneti- zation, i.e., dM FC(T)/dT, as a function of temperature in the insets of Figs. 5(a)–5(e). The T cvalues calculated from the minima of dM FC(T)/dTvs T curves vary in the range of ∼211 K to 220 K under different external fields. It is intriguing to note that the reported Tcfor 22 nm Nd 0.67Sr0.33MnO 3nanoparticles was 66 K,23which is much lower than room temperature if compared with the T c observed in our as-synthesized sub-150 nm Nd 0.7Sr0.3MnO 3parti- cles. The increased particle size was found to enhance T cfor anal- ogous Nd 0.5Sr0.5MnO 3nanocrystals.24It is well known that the ferromagnetic transition of manganite nanoparticles is governed by their double exchange Mn3+–O–Mn4+interaction.24Therefore, high T cof as-prepared Nd 0.7Sr0.3MnO 3nanoparticles is an indi- cation of strong double exchange interaction between their core spins. We also anticipate that the effect of uncompensated spins on the surface of as-synthesized nanoparticles was notably reduced due to their intermediate particle size, which eventually weak- ened their magnetic frustration and resulted in reasonably high Tc.4,47In addition to the surface effect, the magnetic behavior of nanomanganites is also highly influenced by the internal disorder existing at the core of their nanostructures. The reasonably nar- row peaks of the dM FC(T)/dTcurves [insets of Figs. 5(a)–5(e)] indicate sharp ferromagnetic transition, which reveals a nominal degree of disorder inside the sub-150 nm Nd 0.7Sr0.3MnO 3parti- cles.24For a further insight, we have shown the inverse suscep- tibility χ−1vs temperature curve obtained at 10 kOe via the FC process in Fig. 5(f). Notably, for perfectly ordered ferromagnets, the relation between magnetic susceptibility and temperature rea- sonably follows the Curie–Weiss law in the paramagnetic state, which isχ=C T−Θ, (2) where C=Nμ2 eff/3kBdenotes the Curie constant, Θis the param- agnetic Curie–Weiss temperature, kBis the Boltzmann constant, μeff is the effective paramagnetic moment, and N is the number of mag- netic ions per unit cell.47It can be noted from Fig. 5(f) that above Tc, the χ−1(T) curve of Nd 0.7Sr0.3MnO 3showed an excellent fit with the Curie–Weiss law, confirming their low internal magnetic disor- der. As shown in this figure, the x-axis intercept of the fitted straight line in the paramagnetic region provided the value of Θ(212 K) of as-prepared nanoparticles, which agrees well with our observed T c of∼220 K. We have determined the μeffof as-synthesized Nd 0.7Sr0.3MnO 3 particles from the slope of the fitted line of Fig. 5(f), and it was found to be∼5.44 μB. Theoretically, the effective magnetic moments, i.e., μth effof Mn3+and Mn4+ions, can be expressed as μth eff=g√ s(s+ 1)μB assuming their orbital angular components to be quenched by inter- atomic forces.48Here, gdenotes their gyromagnetic factor ( g= 2 for both Mn3+and Mn4+ions) and sis their spin ( s= 2 and 3/2 for Mn3+and Mn4+, respectively).49Using this expression, μth eff(Mn3+) and μth eff(Mn4+)are calculated as ∼4.90 μBand∼3.87 μB, respec- tively. Furthermore, the effective magnetic moment of Nd3+ions is given by μth eff(Nd3+)=g√ J(J+ 1)μB, where its gyromagnetic factor g = 8/11 and total angular momentum J = 9/2.24From this expression, μth eff(Nd3+)is found to be ∼3.62 μB. Thus, in a mean field approximation, considering the rigid coupling between Mn3+, Mn4+, and Nd3+ions, the theoretical effective paramagnetic moment of Nd 0.7Sr0.3MnO 3can be calculated as μth eff(Nd 0.7Sr0.3MnO 3) =√ 0.7[μth eff(Mn3+)]2+ 0.3[μth eff(Mn4+)]2+ 0.7[μth eff(Nd3+)]2 =5.52 μB. (3) Clearly, our experimentally obtained value of μeff(∼5.44 μB) agrees quite well with the theoretical one ( ∼5.52 μB), which justi- fies our speculation that the effect of surface disorder in the sub- 150 nm Nd 0.7Sr0.3MnO 3particles was reasonably small. Another notable outcome of the temperature dependent susceptibility study [Fig. 5(f)] is that the as-synthesized manganite nanoparticles do not undergo ferromagnetic to charge-ordered antiferromagnetic tran- sition at low-temperatures, which can be attributed to their high structural stability.33 We have further conducted field dependent magnetization measurements of nanosized Nd 0.7Sr0.3MnO 3perovskite mangan- ite at different temperatures, i.e., 300 K, 200 K, 100 K, and 20 K, with applied magnetic fields of up to ±50 kOe. The magnetization vs magnetic field (M–H) curves are shown in Fig. 6. Clearly, the M–H curve obtained at 300 K is unsaturated and linear without any detectable hysteresis loop, revealing the paramagnetic behav- ior of as-prepared nanoparticles at room temperature.25Below T c, the shape of the M–H hysteresis loops indicates their soft ferro- magnetic nature. Hence, the spins of these nanoparticles might get easily aligned by the applied magnetic field. Consequently, the M–H loops tend to saturate promptly with increasing field albeit full sat- uration cannot be achieved at 200 K up to ±50 kOe. Interestingly, AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-6 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 5 . [(a)–(e)] ZFC and FC magnetization of Nd 0.7Sr0.3MnO 3nanoparticles as a function of temperature measured in the field range of 0.2 kOe–10 kOe. Insets: the derivative dMFC/dT vs temperature. (f) χ−1vs temperature curves for Nd 0.7Sr0.3MnO 3nanoparticles measured at 10 kOe. The straight line is a Curie–Weiss fit providing the value of Θ andμeff. upon lowering the temperature to 100 K and 20 K, the loops com- pletely saturate with high saturation magnetization, M s, under quite nominal magnetic fields of ∼10 kOe, which can be imputed to the reduction of thermal fluctuation at lower temperatures. Notably,the M svalue of Nd 0.7Sr0.3MnO 3nanoparticles at 20 K is found as ∼82 emu/g from which the saturation magnetic moment per for- mula unit, μS, of this nanomanganite can be calculated as ∼3.6 μB/f.u. following the expression μS=Ms/NA, where N Adenotes AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-7 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 6 . The M–H hysteresis loops of Nd 0.7Sr0.3MnO 3nanoparticles measured at 20 K, 100 K, 200 K, and 300 K with an applied magnetic field of up to ±50 kOe. Avogadro’s number.24Generally, for ferromagnetically ordered mixed-valence perovskite R 1−xDxMnO 3manganites, μSapproxi- mates to (4 −x)μB/f.u.47Thus, the theoretical value of μSof syn- thesized Nd 0.7Sr0.3MnO 3nanomanganite can be considered as ∼3.7 μB/f.u., which is well consistent with our experimentally obtained one,∼3.6μB/f.u. This consistency implies that the Mn3+and Mn4+ ions in as-prepared nanoparticles are highly ordered at low tem- peratures, which is strikingly different from the previously inves- tigated Nd 0.5Sr0.5MnO 3nanocrystals.24It has been reported that 50% Sr2+doping in Nd 1−xSrxMnO 3(x = 0.5) nanomanganite intro- duces a fraction of the charge-ordered state at low temperatures, which gives rise to canting of its Mn moment, resulting in high magnetic disorder.24However, in our investigation, doping of 30% Sr2+in Nd 1−xSrxMnO 3(x = 0.3) nanoparticles manifested perfect ferromagnetic ordering of Mn ions without introducing any charge- ordered state, which can be associated with their lattice structure,specifically orthorhombic strains, O s/⊙◇⊞and O s∥. Notably, for charge- ordering to set in, O s/⊙◇⊞needs to be significantly larger than Os∥.33 As was evident from our structural analysis, the difference between Os/⊙◇⊞and O s∥of the as-synthesized nanoparticles is quite negligi- ble below T c, which may justify the absence of the charge-ordered state. Finally, we have carried out a comparative analysis between the magnetic properties of as-synthesized Nd 0.7Sr0.3MnO 3nanopar- ticles and previously reported similar nanomanganite systems and presented the outcome in Table II. Notably, both the T cand Ms values of the nanomaterials synthesized in the present investiga- tion are considerably higher as compared to other rare earth cations, e.g., La3+and Sm3+based nanoparticles. For instance, the reported Tcand Msof 100 nm Sm 0.8Ca0.2MnO 3nanoparticles are 64 K and∼38 emu/g, respectively, which are significantly smaller than those of our synthesized Nd 0.7Sr0.3MnO 3nanomanganite although the average particle size of these two systems is comparable.47 Moreover, our as-prepared sub-150 nm particles have also demon- strated higher T cand better ferromagnetic properties as compared to related Nd–Sr manganites having comparatively smaller parti- cle size. As an example, the Msvalue of 22 nm Nd 0.67Sr0.33MnO 3 nanoparticles is reported to be ∼45 emu/g at 25 K under a suffi- ciently high external field of 90 kOe,23whereas as was mentioned earlier, at 20 K, the Msof our relatively larger Nd 0.7Sr0.3MnO 3 nanoparticles was observed to be as high as ∼82 emu/g under an applied magnetic field of only ∼10 kOe. The rationale behind such high magnetization can be associated with the good crystallinity of our synthesized sample as well as their intermediate particle size, which might have reduced the effect of crystallographic defect induced uncompensated surface spins and prevented magnetic frus- tration.5,6The high magnetization and magnetic softness of synthe- sized Nd 0.7Sr0.3MnO 3nanoparticles can be of great technological importance for a wide range of electronic and power distribution applications such as power supply and audio transformers, electric motors, recording heads, magnetic modulators, etc. They might also have potential applicability in the fields of data storage and pro- cessing, spintronics, magnetic sensors, magnetic resonance imaging (MRI), etc. TABLE II . A comparative analysis of the magnetic properties of as-synthesized Nd 0.7Sr0.3MnO 3nanoparticles and some other reported nanomanganites. Magnetic Average particle Tc Magnetization at Measurement system size (nm) (K) 10 kOe (emu/g) temperature (K) References Sm 0.8Ca0.2MnO 3 23 55 ∼18 20 47 Sm 0.8Ca0.2MnO 3 100 64 ∼38 20 47 La0.7Ca0.3MnO 3 10 120 ∼30 10 50 La0.8Ca0.2MnO 3 18 231 ∼48 10 51 Sm 0.1Ca0.9MnO 3 25 95 ∼4 10 52 Sm 0.1Ca0.9MnO 3 60 108 ∼12 10 52 Nd 0.67Sr0.33MnO 3 22 66 ∼22 25 23 Nd 0.5Sr0.5MnO 3 30 230 ∼23 75 24 Nd 0.5Sr0.5MnO 3 55 235 ∼46 75 24 Nd 0.7Sr0.3MnO 3 120 220 ∼82 20 Present work AIP Advances 10, 095135 (2020); doi: 10.1063/5.0017299 10, 095135-8 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv IV. CONCLUSIONS The Rietveld refined powder X-ray diffraction patterns, energy dispersive X-ray analysis, and X-ray photoelectron spectroscopy have confirmed the successful synthesis of Nd 0.7Sr0.3MnO 3nanopar- ticles having an intermediate particle size ( ∼75 nm to 150 nm) with the desired crystallographic phase, elemental composition, and sur- face chemical states. Notably, these ultrasonically prepared nanopar- ticles did not undergo any crystallographic phase transition upon lowering the scanning temperature from room temperature to far below their Curie temperature. The effective paramagnetic moment as-obtained from the temperature-dependent magnetic study was in good agreement with the theoretically calculated value, which revealed the nominal effect of internal and surface disorder in the as-synthesized nanoparticles. Furthermore, the Nd 0.7Sr0.3MnO 3 nanoparticles demonstrated high ferromagnetically ordered Mn3+ and Mn4+ions below the Curie temperature, large saturation mag- netization, and excellent magnetic softness over a wide range of temperatures. Both the temperature and field-dependent magneti- zation measurements confirmed the absence of the charge-ordered antiferromagnetic state at low temperatures as was also suggested by the crystallographic analysis. It is worth noting that the mag- netic properties of 30% Sr2+doped Nd 0.7Sr0.3MnO 3nanomangan- ite are more promising compared to those of 50% Sr2+doped Nd 0.5Sr0.5MnO 3nanocrystals.24Therefore, considering the observed intriguing properties, we anticipate that synthesizing 30% Sr2+ doped Nd 0.7Sr0.3MnO 3nanomanganite with an intermediate par- ticle size might be worthwhile for a broad range of electronic and power distribution applications as well as for magnetic memory and spintronic devices. ACKNOWLEDGMENTS This work was financially supported by the Ministry of Educa- tion, Government of the People’s Republic of Bangladesh, Order No. 37.20.0000.004.033.020.2016./PS20191017. The Institute for Molec- ular Science (IMS), supported by MEXT, Japan is sincerely acknowl- edged for providing facilities of SQUID magnetometer. The authors would like to thank Dr. Tapas Paramanik, S. N. Bose National Centre for Basic Sciences, for supporting XRD measurements. 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AIP Conference Proceedings 2292 , 030002 (2020); https://doi.org/10.1063/5.0030932 2292 , 030002 © 2020 Author(s).Nuclear structure of some even and odd nuclei using shell model calculations Cite as: AIP Conference Proceedings 2292 , 030002 (2020); https://doi.org/10.1063/5.0030932 Published Online: 27 October 2020 Bhopendra Singh , S. Suman Rajest , K. Praghash , Uppalapati , and Srilakshmi R. Regin ARTICLES YOU MAY BE INTERESTED IN Shadowing and non-shadowing in dynamical systems AIP Conference Proceedings 2292 , 020012 (2020); https://doi.org/10.1063/5.0030805 Inverse shadowing property of transitivity with G-and topological G AIP Conference Proceedings 2292 , 020013 (2020); https://doi.org/10.1063/5.0031013 Two-phase relative permeability prediction from capillary pressure data for one Iraqi oil field: A comparative study AIP Conference Proceedings 2292 , 030001 (2020); https://doi.org/10.1063/5.0030501 Nuclear Structure of Some Even and Odd Nuclei Using Shell Model Calculations Bhopendra Singh1,a) S. SumanRajest 2.b) K. Praghash 3,c) Uppalapati4,d) and Srilakshmi R.Regin5,e) 1Associate Professor, Amity University, Dubai. 2 Reacher, Vels Institute of Science, Technology & Advanced Studies (VISTAS), Chennai, Tamil Nadu, India. 3Assistant Professor, Koneru La kshmaiah Education Foundation, Vaddeswa ram, AP, India. 4Assistant Professor, CSE Department, VFSTR Deemed to be University, Vadl amudi, Guntur, India. 5Assistant professor, Adhiyamaan College of Engineering, Hosur, Tamil Nadu, India. Corresponding Author: a) bsingh@amityuniversity.ae b) sumanrajest414@gmail.com c)prakashcospra@gmail.com d)druppalapati2019@gmail.com e)regin12006@yahoo.co.in Abstract: In this work, we determined the electron dispersing structure factors, just as the vitality levels of certain cores. The computation of electron dispersing structure factors needs numerous issues to be remembered for request to make these figures attainable and quick in time in light of enormous measure of terms speak to arithmeti c, quantum mechanical speculations, atomic shell model hypotheses and equations. In the current work, we exa mined the impacts of the higher setup outside the shell model space and the inactive center which included Tassie Model (TM) to discuss the Longitudinal C2 el ectron scattering form factors for the nuclei: 116Sn, 92Mo,90Zr ,39K and 32S, which calculated for nuclei under consideration, are compared with those of experimental data. The HO and SKX possibilities have been utilize d to compute the wave elements of outspread single-molecule framework components. Some hypot hetical vitality levels of the 52Cr, 32S and 181Ta nuclei are calculated compared with their experimental data. The shell model for windows code NuShellX@MSU has been used in this study. INTRODUCTION The atomic Shell model gives the significant hypothetical apparatus to understanding the atomic properties. It very well may be utilized in the least complex types of individual particles to give a subjective origination, yet it is likewise utilized as a reason for substantially more intri cate and complete estimations. There has all the earmarks of being constrained inside the not so distant future to the extension of its application [1]. The scattering of electrons from the nuclei provides important information about the electromagnetic currents inside the nuclei. Electronic scattering can provide a good tool for this calculation because it is sensitive to the locative dependence of the current and charge density [2, 3]. Important information about the nuclear structure can be obta ined by the scattering of electrons at high energy. Information obtained at high-energy electron scattering depends on the wavelength of the de Broglie wavelength associated with th e electron compared to the range of nu clear forces. If the incident electron energy is 100 MeV and more, the de Broglie wavelength will be in the spatial extent of the target nucleus [23-27]. Thus , the electron with these energies is the be st probe for studying the nuclear structure [4]. Electron scattering is the most important tool to study the nuclear structure for many reasons, the electron and nucleus interaction is well known as the electron interacts electroma gnetically with the local charge and the current and magnetic density of Proceedings of the 2020 2nd International Conference on Sustainable Manufacturing, Materials and Technologies AIP Conf. Proc. 2292, 030002-1–030002-7; https://doi.org/10.1063/5.0030932 Published by AIP Publishing. 978-0-7354-4024-1/$30.00030002-1the nucleus. measurements can be obtained without significantly impairing the structure of target nuclei. While in the case of scattering of a nucleon from nuclei, neither the interaction nor the structure of the target is well known therefore it is very compli cated to distinguish between them by analyzing the experimental data [28-36]. With electron scattering One can instantly conn ect the cross section with the transitio n matrix elements of the operators of local charge and current density and consequently directly related to the nuclear struct ure of the target itself [5, 6]. In electron scattering, one can distinguish two types of scattering: first the nucleus is left on its ground state; this process is called “Elastic Electron Scattering”. In the second type, the nucleus is left in its different excited states, this process is called “ Inelastic Electron Scattering” [7]. Inelastic Longitudinal Form Factors Inelastic form factors involving angular momentum Jand momentum transfer q can be written in terms of the elements of the reduced matrix in both angular momentum and isospin [8]. 2 1,022 )(ˆ 0)1( )1 2(4)( iqTfTT TT T J ZqFL TJ T Zi Zf TT iL J i ffz f¦  ¸¸ ¹· ¨¨ ©§   S 2 2)( )( qF q Ffs cm (1) The diminished grid components of the longitudinal administrator in the turn and isospin space are given between conditions of the last and starting numerous particles of the framework remembering the setup blend for terms of OBDM components increased by the single molecule lattice components of the longitudinal operator [9], i.e.a TbbaJfi OBDM i TfL TJ baTJ L TJˆ ),,,,( ˆ ,¦ (2) The OBDM elements are given in terms of the isospin reduced matrix elements [10], i.e. 2)0 (2001)()( ' ¸¸ ¹· ¨¨ ©§   T OBDM TT TTOBDM Zi Zf T T Zz fW (3) 2)1 (6011)(z f ' ¸¸ ¹· ¨¨ ©§  T OBDM TT TT Zi Zf T T ZW where ZW are the isospin operators of single particle. The OBDM( T') is defined [10] as : >@ 1 2 1 2~ ),,,,(, 'u 'c' c T Ji a af Tjjfi OBDMTJ j j (5) The operator jacreates a nucleon in the single nucleon state j and the operatorjca~ annihilates a nucleon in the state jc. Tassie Model (TM) has been utilized to describe the progress of gamma-and the excitation of cores by electron dissipating. As indicated by the aggregate modes, the center polarization change thickness is given by the Tassie shape [11]. r)r,,(r) 1(21)r,,(1 dfi dfio J zcore JtzUW U1 (4) Where 1 is proportionality constant and oU is the ground state two – body charge density distribution. The Coulomb form factor for this model becomes ¿¾½ ¯®1¸¸ ¹· ¨¨ ©§  ³ ³f f 011 022/1 )r( ),,( rr r )r( r1 1 24)( qjrfi dq d qjZ JqFJ oJ Jtms J iL JzU US)()( qFq Ffs cmu (5) 030002-2The proportionality constant 1 can be determined from the form factor evaluated at q=k, we obtain ³³ f f 1 01102 )r( )r,,( rr41 2)( )r,,( )r( rr k j fi dkJZk F fi kj d J oJi L Jms Jt J Z USU (6) RESULTS, DISCUSSION AND CONCLUSIONS Energy levels Some theoretical energy levels of the 52Cr nucleus compared with the experimental data [12] are shown in figure1. The levels are calculated with FPPN model space and gxlpn as two- body interaction. The active orbitals for FPPN model space are P: 1f7/2, 2p3/2,1f5/2, 2p1 /2 and N: 1f7/2, 2p3/2, 1f5/2, 2p1/2. Gr eat understanding was gotten for the utilized cooperation. The understanding is generally excellent for most states as contrasted and the trial information. The supreme contrasts among hypothetical and exploratory qualities are nearly between 0.040MeV and 0.4 MeV. Figure 2 Show same comparison for the excited energy levels of 32S nucleus [37-46]. Th e hasp interaction [13] has been used. The HASP model space defined by the orbitals 1d3/2,2s1/2,2p3/2,1f7/2. This model includes configuration mixing between 1d3/2,2s1/2 of SD model space and 2p3/2,1f7/2 of FP model space. There are reasonable agreement between theoretical and experimental levels [14] for most states. The energy levels of 92Mo nucleus (fig.3) give good agreement compared with the experimental levels [15]. The calculations performed with the N50J Model Space (2P3/2,1F5/2,2p1/2,1g9/2). The calculations of the 39K energy levels (figs. 1 to 4) using hasp interaction give very poor agreement with the experimental data [16]. FIGURE 1. Excitation energies for the 52Cr with their corresponding experimental values [12] FIGURE 2. Excitation energies for the 32S nuclei compared with experimental values [14] FIGURE 3. Excitation energies for the 92Mo nuclei compared with experimental values [15] 030002-3 FIGURE 4. Excitation energies for the 39K nuclei with experimental values[16] Table 1. shows some of the energy levels of 181Ta. The levels are calculated with PBPOP model space, with restrict protons to contribute in 2d5/2, 2d3/2,3s1/2 and 1h9/2 shells, and the neutrons contributed in 2f5/2,3p3/2 and 3p1/2 shells. The interaction failed to expect the ground state, the theoretical ground state Jʌ is 9/2–, while it is 7/2+ in the experimental data. Many of other states give poor agreement with the experimental data [17]. There is a computational difficult to calculate the levels with another model space. Table 1. Excitation energies for the 181Ta nuclei with their corres ponding experimental values [17] 181Ta Ex(MeV) Jʌorder EXP pbpop 7/2+1 0 9/2–1 0.006 0 7/2–1 0.7729 0.176 5/2 –1 0.542 0.298 13/2–1 0.337 0.364 11/2–1 0.158 0.397 3/2–1 ʊ 0.504 15/2–1 0.542 0.603 17/2–1 0.772 0.748 1/2–1 ʊ 0.776 21/2–1 1.307 0.983 19/2–1 1.027 1.198 25/2–1 1.932 1.466 23/2–1 1.608 1.469 LONGITUDINAL ELECTRON SCATTERING FORM FACTORS We have been determined C2 segments of th e electron dispersing st ructure factors for the 116Sn, 92Mo,90Zr ,39K and 32S cores. The HO and SKX (X=20) possibilities have been utilized to figure the wave elements of outspread single- molecule framework components. Fig. 5, shows the figuring of the longitudinal C2 ( 2ଵା 2) inelastic electron dispersing structure elements of 116Sn core. The powerful charges that utilized is 0.5 for every one of protons and neutrons. The computations are lower than trial result by a factor of around 3 at the primary greatest and around 4 at the subsequent most extreme. The counts are performed by GLEKPN [18] Model Space (P: 1F7/2,1F5/2, 2P3/2,2P1/2,1G9/2 and N:1G9/2,1G7/2,2D5/2, 2D3/2,3S1/2), and glekpn as two body cooperation. The vitality levels of 116Sn (not appeared) give poor concurrence with the trial information, as model the energies of 2ଵାand4ଵା states in hypothetical figures are 1.167MeV and 1.555MeV, separately, while it is 1.294MeV and 2.391MeV in test information. The inelastic longitudi nal structure factors for the C2 ( 2ଵା 4) state in the 92Mo core is introduced in Fig. 6. In this figure, Calculations of the Tassie model with HO (strong bend) and SKX (ran bend) possibilities give a decent concurrence with the trial info rmation at the main most extreme when we utilize compelling charges (0.5). The two possibilities give poor understanding at the subsequent most extreme. The figures are performed by n50j 030002-4model space [19]. Figure 7, shows the estimations of the longitudinal C2 ( 2ଵା 5) inelastic electron dispersing structure elements of the 90Zr. The estimations are additionally performed by n50j collaboration. The counts of Tassie Model with HO and SKX give a decent concurrence with the exploratory information at the main most extreme, while the computation with SKX is nearer to the test information at the subsequent greatest. The determined vitality levels (not appeared) utilizing this association give excellent concurrence with test information. FIGURE 5. The C2 ( 2ଵା 2) form factors for the 116Sn nucleus FIGURE 6. The C2 ( 2ଵା 4) form factor for the 92Mo nucleus compared with Experimental values [20] compared with Experimental values [20] FIGURE 7. The C2 ( 2ଵା 5) in 90Zr nucleus compared with Experimental values [20]. The longitudinal C2 ( 1/2ଵା 1/2) inelastic electron disper sing structure factors in the 39K core is appeared in Fig.8. utilizing the HASP model space. The computations of Tassie model give a decent concurrence with the exploratory information. As appeared in the figure , the computations with SKX are nearer to the exploratory information at the subsequent most extreme. The longitud inal C2 structure factor with cent er polarization impact (TM) for the FIGURE 8. The same fig.5 for the ( 1/2ଵା 1/2 ) in 39K nucleus. Experimental values [21] are indicated by the filled circles FIGURE 9. The C2 ( 2ଵା 0) in 32S nucleus with the Experimental values [22] 030002-5transition to the ( 2ଵା0) in the 32S is appeared in figure 9 contrasted and the trial information. The figuring gauges the test information in the first and the second most extreme ar ea; these estimations with cent er polarization impact are awesome particularly at the second greatest district and it is commonly worthy. 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Anandakumar, “Handover based spectrum allocation in cognitive radio networks,” 2013 International Conference on Green Computing, Communicatio n and Conservation of Energy (ICGCE), Dec. 2013.doi:10.1109/icgce.2013.6823431. doi:10.4018/978-1-5225-5246-8.ch012 [28]. S, D., & H, A. (2019). AODV Route Discov ery and Route Maintenance in MANETs. 2019 5th International Conference on Advanced Computing & Communication Systems (ICACCS). doi: 10.1109/icaccs.2019.8728456 [29]. AdewunmiTaiwo, A. and Ato Bart-Plange, A. Introduction of Smok eless Stove to Gari Producers at Koryire in the Yilokrobo Municipality of Ghana. International Journal of Advanced Engineering, Manageme nt and Science, vol. 2, no. 4, pp.163- 169, 2016. [30]. Ahmed, E. R., Islam, A., Zuqibeh, A., & Alabdullah, T. T. Y. Risks management in Islamic financ ial instruments. Advances in Environmental Biology, Vol. 8, no. 9, pp. 402-406, 2014. [31]. Ahmed, E. R., Islam, M. A., Alabdullah, T. T. Y & bin Amran, A. Proposed the pricing model as an alternative Islamic benchmark. Benchmarking: An International Journal, Vol. 25, no. 8, pp. 2892-2912, 2018. [32]. Alabdullah, T. T. Y., Ahmed, E. R., &Thottoli, M. M. Ef fect of Board Size and Duality on Corporate Social Responsibility: What has Improved in Corporate Governance in Asia?. Journal of Accounting Science, Vol. 3, no.2, pp. 121–135, 2019. [33]. D. Mukherjee and B. V. R. Reddy, “Des ign and development of a novel MOSFET structure for reduction of reverse bias pn junction leakage current,” International Journal of Intelligence and Sustainable Computing, vol. 1, no. 1, p. 32, 2020. [34]. E. Malar and M. Gauthaam, “Wavelet analysis of EEG for the identification of alcoholics using probabilistic classifiers and neural networks,” International Journal of Intelligence and Sustainable Comput ing, vol. 1, no. 1, p. 3, 2020. [35]. Haldorai, A. Ramu, and S. Murugan, “Social Aware Cognitive Radio Networks,” Social Network Analytics for Contemporary Business Organizations , pp. 188–202. doi:10.4018/978-1-5225-5097-6.ch010 [36]. J. Selvaraj and A. S. Mohammed, “M utation-based PSO techniques for optimal location and parameter settings of STATCOM under generator contingency,” Int ernational Journal of Intelligence and Su stainable Computing, vol. 1, no. 1, p. 53, 2020. [37]. K. Jayasudha and M. G. Kabadi, “Soft tissues deformation and removal s imulation modelling for virtual surgery,” International Journal of Intelligence and Sustaina ble Computing, vol. 1, no. 1, p. 83, 2020. [38]. M. Kasiselvanathan, V. Sangee tha, and A. Kalaiselvi, “Palm pattern recognition using scale invariant feature transform,” International Journal of Intelligence and Sustaina ble Computing, vol. 1, no. 1, p. 44, 2020. [39]. Mohmed Raffi, S., & P Lodha, D. A Review Paper on Hydrological Modelling and Clim ate Change in India. 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Suman Rajest, Mustag aliyeva Gulnara, Khairzhanova Akhmar al “A Short View on the Backdrop of American’s Literature”. Journa l of Advanced Research in Dynamical and Control Systems, Vol. 11, No. 12, pp. 182-192, 2019. [45]. Rao, A. N., Vijayapriya, P., Kowsalya, M., & Rajest, S. S. (2020). Computer Tools for En ergy Systems. In International Conference on Communication, Computi ng and Electronics Systems .pp. 475-484), 2020 , Springer, Singapore. [46]. Gupta J., Singla M.K., Nijhawan P., Gangu li S., Rajest S.S. An Io T-Based Controller Real ization for PV System Monitoring and Control. In: Haldorai A., Ramu A., Khan S. (eds) Business Intelligence for Enterprise Internet of Th ings. EAI/Springer Innovations in Communication and Computing. Springer, Cham 2020. 030002-7

5.0016958.pdf

AIP Conference Proceedings 2265 , 030725 (2020); https://doi.org/10.1063/5.0016958 2265 , 030725 © 2020 Author(s).Defect-state mediated excitonic transitions and associated electrical nature in exfoliated MoS2 nanostructures Cite as: AIP Conference Proceedings 2265 , 030725 (2020); https://doi.org/10.1063/5.0016958 Published Online: 05 November 2020 S. Reshmi , Manu Mohan , and K. Bhattacharjee ARTICLES YOU MAY BE INTERESTED IN A- and B-exciton photoluminescence intensity ratio as a measure of sample quality for transition metal dichalcogenide monolayers APL Materials 6, 111106 (2018); https://doi.org/10.1063/1.5053699 Evidence for two distinct defect related luminescence features in monolayer MoS 2 Applied Physics Letters 109, 122105 (2016); https://doi.org/10.1063/1.4963133 Passivating the sulfur vacancy in monolayer MoS 2 APL Materials 6, 066104 (2018); https://doi.org/10.1063/1.5030737 Defect-state mediated excitonic transitions and associated electrical nature in exfoliated MoS 2 nanostructures S Reshmi1, Manu Mohan1, K. Bhattacharjee1, a) 1Department of Physics, Indian Institute of Spac e Science and Technology, Valiyamala, Trivandrum, 695547,India a)Corresponding author: kuntala.b@iist.ac.in; kuntala.iopb@gmail. com Abstract. Layered materials like graphene, transition metal dichalcogenid es(TMDs) etc. are of great importance in nanoelectronics and optoelectronics. Various exfoliation techni ques used for the isolation into thin layers of these materials can impart different type of atomic defects in the sy stem. The presence of defects can highly modify the properties displayed by them. Si gnificant features related to t he defect states can also show up in the optical spectroscopy results. Here, we d iscuss about the defect related modifications of photoluminescence peaks obtained f r o m o u r a s - s y n t h e s i z e d M o S 2 sample comprised of nanostructures and nanosheets and the asso ciated electrical behavior. INTRODUCTION Since the successful isolation of graphene from graphite, vario us other classes of layered materials like transition metal dichalcogenide s (TMDs), hexagonal Boron Nitrid e (h-BN), black phosphorous etc. were successfully synthesized. Unlike graphene, TMDs can show metall ic as well as semiconducting behaviour depending on the structural polytype. 1T phase shows metallic n ature, whereas, 2H phase shows semiconducting nature whose ener gy bandgap can be tuned accordi n g t o t h e t h i c k n e s s . A m o n g t h e c l a s s o f TMDs, MoS 2 is one of the most studied materials. Various exfoliation techniques can generate atomic defects on t hese materials. It can be observed that these intrinsic defects can largely influence the properties of mono/ few layers compared to the bulk materials. The defects in TMDs have thus gained much attention [1], [2], [3]. Based on prior studies, it is known that intrinsic defects in MoS 2 dominates the metal/MoS 2 c o n t a c t r e s i s t a n c e a n d p r o v i d e a l o w S c h o t t k y b a r r i e r , independent of the work function of the metal employed [4]. MoS 2 devices fabricated with various metal contacts have been reported to show low Schottky barriers [5]. [ref 12 from McDonnel paper]. It has also been reported that MoS 2, based on the S/Mo ratio, can display n type to p t ype conductivity [4]. In this work, we report defect mediated modifications in the photoluminescenc e (PL) spectroscopy of ourliquid phase exfoliated (LPE) nanostructured MoS 2 samples and the associated el ectrical nature depending on sulp hur (S) vacancy states. EXPERIMENTAL TECHNIQUES The samples were prepared by liquid phase exfoliation using cosolvency technique. The solvents used were 2 propanol and water in the ratio 2:3 with initial concentratio n 1mg ml-1 of MoS 2 bulk powder (<2 μm; 99% purity, 2H, Sigma Aldrich).MoS 2 dispersion was made by ultra-sonication for 60 min with conseq uent centrifugation at 7500 rpm for 10 min to separate bigger partic les. The supernatant solution was then further stirred at 600 rpm for 30 min followed by microwave (MW) treatm ent in an ‘Anton Paar Mono- wave 300 Monowave EXTRA’ microwave oven at 175 °C for a duration of 10 m in or 30 min. Exfoliated samples were then characterized by photoluminescence (PL) spectroscopy using Horiba Scientific Fluromax 4 Spectroflurometer. Electrical characterisations on the samples were carried outusinga two-probe probe station connected to a power supply from Agilent Technologies (B2912A p recision source/measure unit). Samples for electrical characterisations were prepared by drop castingthe e xfoliated MoS 2 solution containingheterodimensional MoS 2 nanostructures [6]onto a Si wafer containing 270 nmthick SiO 2layeron the surface. 140 nm thick Au electrodes were thermally deposited us ing Fullinger thermal evaporation set up to DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030725-1–030725-4; https://doi.org/10.1063/5.0016958 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030725-1 carry out the electrical measurements on the MoS 2 nanostructures comprising mostly of 2H (S1) [6] or 1T (S2) phase [7] depending on the MW irradiation time of 10 min or 30 min respectively. RESULTS AND DISCUSSIONS FIGURE 1 . (a) SEM image of MoS 2 flakes, (b, c) TEM micrographs of MoS 2 spheres and rod/tube str uctures respectively. Figure 1 shows the evidence of formation of MoS 2 flakes, spheres and rod/tube like structures in our MW assisted LPE samples. As can be seen from Figure 1, the diamete r of the spheres as well as rod/tube like structures is in the range of 150-200 nm [6], [7]. The synthesi s process itself can induce defects of various types in the system, which would be associated with discrete energy l evels appearing within the bandgap of TMDs [8], [9]. Most often these defect states are associated with di stinct optical signatures and therefore, can modify the optical emission spectrum of the materials accordingly prov iding a detailed insight to the defect states. The PL spectra obtained for both the exfoliated samples, S1 [(a)] a nd S2 [(b)], are shown in Figure 2. The excitation wavelength used was 280 nm and the emission spectra were captu red in the range of 600 to 800 nm. FIGURE 2. PL spectrum obtained for 10 minut e (S1) and 30 minute (S2) MW t reated samples are s hown in (a) and (b) respectively. The sample was excited at 280 nm and the de excit ation spectrum obtained is p lotted in the range 600 to 800 nm The valance band maxima in the case of TMDs split into two beca use of spin orbit splitting and interlayer coupling. The excitonic transitions occurring from these states to the conduction band minima gives a signature feature in the emission spectrum. Micro PL studies done previou sly [10] described the occurrence of more than one defect related PL peaks from MoS2 layers. In our sample S1 [Figure 2(a)] which is MW treated for 10 min [ 6], MoS 2 nanostructures exhibit the A exciton peak at 2 eV. About 100 meV below the A exciton peak, a peak at 1.91eV is observed and is assigned to the defect bound exciton transitions. [10], [11], [12], [13]. A nother distinct peak feature around 200 meV below the A excitonic peak was observe d at 1.81 eV. The peaks at 1.91 and 1.81 eV are designated as D1 and D2 respectively in Figure 2[(a), (b)]. Due to the similarity of th e decay and de polarisati onbehaviour of these two peaks, Saigal et al. associated the origin of both the peaks wi th radiative recombination of excitons bound to defect states of S for monolayer MoS 2 [10]. A similar trend was observed for our sample S2 as well which is MW exposed for a duration of 30 min [7]. However, we observe a shift in the PL peak positions associated with change in intensities of the peaks for the sample S2 compared to the optical emission spectrum obtained fr om sample S1. The emission peaks at 1.96, 1.87, 1.78 eV from S2 are assigned to the A exciton transition and defect mediated transitions, D1 and D2 030725-2 respectively. S vacancy is the most common defect observed in M oS2 [14] , [15]. Both D1 and D2 peaks were examined while annealing with and without S vapor. It is report ed that after annealing in vacuum the PL intensities of both D1 and D2 f eatures increase relative to A, in contrast the PL intensities of both decrease when annealed under Svapor [10]. The variation in intensities of bot h the peaks gave a conclusive idea about their o r i g i n t o b e a s s o c i a t e d w i t h t h e S d e f e c t s t a t e s [ 1 0 ] . I n a d d i t ion, theoretical studies have reported a lower formation energy for S vacancies in comparison to metal vacancy [14], [15]. All these factors together are assigned to the origin of the peaks D1 and D2 associated with t he defect states related to S vacancy [10]. D1 peak is attributed due to single S vacancy (V S) while D2 is associated to double S vacancy V 2S [10]. FIGURE 3. Electrical characterisation of as prepared MoS 2 samples. Depending on the defect states, t he electrical character of the samples also changes. The electronic properties show a variation from sample to sample [16]. A detailed underst anding about the interfaces formed between metal and 2D materials is crucial while incorporating these mat erials into device structures. The variation of contact resistances with the metal to the 2D material chosen is very critical in determining the device performance as well. For MoS 2, a majority of studies report an n type behaviour with Au cont act. However, a localised deviation from this behaviour is also observed on a s ingle sample. [4],[17]. Defect states as well as local stoichiometry can drastically influence the metal/MoS 2 contact barrier and simultaneously n or p type nature of the sample. McDonnel et al., [4] has shown that the p otential barrier between MoS 2 and metals cannot be simply calculated by comparing the metal work functions to M oS2 electron affinity. They investigated the sample surface with scanning tunneling spectroscopy (STS) techn ique by utilizing a tungsten probe. It was observed that naturally occurri ng local defects dominate the ob served electronic properties. The observed I-V characteristics by McDonnel et al [4]., showed a similar revers e and forward bias currents contrary to what would be expected from the theo retically calculated Schottky ba rrier. They even observed both n and p type behaviour on the same sample at different positions with Au con tact which they have related with different concentrations of S on the MoS 2 surface. An overall electrical behaviour of our exfoliated MoS 2 nanostructures is obtained using a two probe measurement system with Au as electrodes. For both S1 and S2, w e observe an n type behaviour with slight change in reverse current values. However, sample S1 exhibits a slightly greater n type nature than sample S2 [Figure 3]. From Figure 2, it can be observed that the intensit y of D1 peak is more in comparison to D2 peak for S1 and vice versa for S2. Since we have assigned V S defect to D1 and V 2S defect to D2, stoichiometrically, S1 will have more S/Mo ratio compared to S2, however, both the sam ples will have a stoichiometric S/Mo ratio which is less than 2:1. Previous studies have shown that the st oichiometry of n type regions of MoS 2 is close to 1.81:1 (S/Mo) and 2.3:1 (S/Mo) for p type regions. Consistent w ith this result, we could observe a lower value of reverse current in S2 compared to that of S1, indicating a r educed n type nature of S2. S1 on the other hand, shows a distinct n type nature owing to the presence of lower S deficiency sites w.r.t S2. Considering the difference in S/Mo ratio, we ascribe the dissimilarities in the electrical characteristics of the samples relating a more n type nature of S1 with a higher S/Mo ratio compared to S 2 where S/Mo ratio seems to be less than S1. Therefore, a general n type nature depicted by both S1 and S2 h ere could be a direct consequence of the S vacancy states. However, these interesting results require more experimental introspections in terms of direct probing of the V S and V 2S defect states which we plan to c arry out in future course of a ctions. CONCLUSION The defect mediated optical emissions occurring in MW assisted LPE MoS 2 s a m p l e s a n d t h e a s s o c i a t e d change in the electrical nature are investigated using optical spectroscopy and two probe measurement systems. The defect states arising from single and double S vacancy stat es display distinct peaks in the PL spectra. In general an n type nature is shown by the samples with slight va riation in the reverse current values depending on 030725-3 the overall variations in the S defect states of the exfoliated M o S 2 nanostructures. The 10 min MW treated sample shows a more n type behavior compared to the 30 min samp le which is related to the concentrations of the S vacancy states [4], [10]. Depending on the MW exposure ti me, 30 min sample is associated with more S defect states than the 10 min one as can be seen from the inten sities of PL transitions of D1 and D2, inducing an overall change in the carrier concentrations. We ascribe the el ectrical bahaviour of the samples as a direct consequence of the defect states related to S vacancies where a more or a less n type nature is associated with the variation in the overall stoichiometry of the MoS 2 samples. A comparison between samples S1 and S2 is made based on their defect mediated optical transitions and the influence of the defect state concentrations on the electrical behaviour. ACKNOWLEDGMENT M s . R . S . a n d D r . K . B . s i n c e r e l y t h a n k D r . P a l a s h K u m a r B a s u for providing with the facility of Microwave treatment. REFERENCES 1. Kunstmann J, Wendumu T B and Seifert G 2017 Localized defect st ates in MoS2monolayers: Electronic and optical properties Phys. Status Solidi Basic Res. 254 2. Wu Z and Ni Z 2017 Spectroscopic investigation of defects in tw o-dimensional materials Nanophotonics 6 1219–37 3. Wu Z, Jiang J, Zheng T, Lu J, Ni Z, Zhao W and You Y 2017 Defec t Activated Photoluminescence in WSe2 Monolayer J. Phys. Chem. C 121 12294–9 4. M c D o n n e l l S , A d d o u R , B u i e C , W a l l a c e R M a n d H i n k l e C L 2 0 1 4 D efect-Dominated Doping and Contact Resistance in MoS 2 ACS Nano 8 2880–8 5. Das S, Chen H Y, Penumatcha A V and Appenzeller J 2013 High per formance multilayer MoS2 transistors with scandium contacts Nano Lett. 13 100–5 6. Reshm i S, Akshaya M V, Satpati B, Roy A, Kum ar Basu P and Bhatt acharjee K 2017 Tailored MoS 2 nanorods: a simple microwave assisted synthesis Mater. Res. Express 4 115012 7. Reshmi S, Akshaya M V., Satpati B, Basu P K and Bhattacharjee K 2018 Structural stability of coplanar 1T-2H superlatticeMoS 2 under high energy electron beam Nanotechnology 29 205604 8. Haldar S, Vovusha H, Yadav M K, Eriksson O and Sanyal B 2015 Sy stematic study of structural , electronic , and optical properties of atomic-scale defects in the two-dimensional transition metal dichalcogenides MX 2 ( M = Mo , W; X = S , Se , Te ) Phys. Rev. B - Condens. Matter Mater. Phys. 92 1– 12 9. Yuan S, Rold R and Katsnelson M I 2014 Effect of point defects on the optical and transport properties of MoS 2 and WS 2 Phys. Rev. B - Condens. Matter Mater. Phys. 90 1–5 10. Saigal N and Ghosh S 2016 Evidence for two distinct defect rela ted luminescence features in monolayer MoS2 Appl. Phys. Lett. 109 11. Plechinger G, Schrettenbrunner F-X, Eroms J, Weiss D, Schüller C and Korn T 2012 Low-temperature photoluminescence of oxide-covered single-layer MoS2 Phys. status solidi - Rapid Res. Lett .6 126–8 12. Tongay S, Suh J, Ataca C, Fan W, Luce A, Kang J S, Liu J, Ko C, Raghunathanan R, Zhou J, Ogletree F, Li J, Grossman J C and Wu J 2013 Defects activated photolumines cence in two-dimensional semiconductors: Interplay between bound, charged, and free exci tonsSci. Rep. 3 2657 13. C how P K, Jacobs-Gedrim R B, Gao J, Lu T M, Yu B, Terrones H an d Koratkar N 2015 Defect-Induced Photoluminescence in Mono layer Semiconducting Transition Metal Dichalcogenides ACS Nano 9 1520–7 14. Zhou W, Zou X, Najmaei S, Liu Z, Shi Y, Kong J and Lou J 2013 I ntrinsic Structural Defects in Monolayer Molybdenum Disulfide Nano Lett. 13 2615–22 15. Hong J, Hu Z, Probert M, Li K, L v D, Yang X, Gu L, Mao N, Feng Q, Xie L, Zhang J, Wu D, Zhang Z, Jin C, Ji W, Zhang X, Yuan J and Zhang Z 2015 Exploring atomic defe cts in molybdenum disulphide monolayers Nat. Commun. 6 6293 16. Bao W, Cai X, Kim D, Sridhara K and Fuhrer M S 2013 High mobili ty ambipolar MoS2 field-effect transistors: Substrate and dielectric effects Appl. Phys. Lett. 102 042104 17. Bampoulis P, Bremen R Van, Yao Q, Poelsema B, Zandvliet H J W a nd Sotthewes K 2017 Defect Dominated Charge Transport and Fermi Level Pinning in MoS 2 / M etal Contacts ACS Appl. Mater. Interfaces 9 19278–86 030725-4

5.0015279.pdf

J. Chem. Phys. 153, 090903 (2020); https://doi.org/10.1063/5.0015279 153, 090903 © 2020 Author(s).Perspective on Kramers symmetry breaking and restoration in relativistic electronic structure methods for open-shell systems Cite as: J. Chem. Phys. 153, 090903 (2020); https://doi.org/10.1063/5.0015279 Submitted: 26 May 2020 . Accepted: 12 August 2020 . Published Online: 03 September 2020 Joseph M. Kasper , Andrew J. Jenkins , Shichao Sun , and Xiaosong Li ARTICLES YOU MAY BE INTERESTED IN Ab initio valence bond theory: A brief history, recent developments, and near future The Journal of Chemical Physics 153, 090902 (2020); https://doi.org/10.1063/5.0019480 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 Explicitly correlated coupled cluster method for accurate treatment of open-shell molecules with hundreds of atoms The Journal of Chemical Physics 153, 094105 (2020); https://doi.org/10.1063/5.0012753The Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp Perspective on Kramers symmetry breaking and restoration in relativistic electronic structure methods for open-shell systems Cite as: J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 Submitted: 26 May 2020 •Accepted: 12 August 2020 • Published Online: 3 September 2020 Joseph M. Kasper, Andrew J. Jenkins, Shichao Sun, and Xiaosong Lia) AFFILIATIONS Department of Chemistry, University of Washington, Seattle, Washington 98195, USA a)Author to whom correspondence should be addressed: xsli@uw.edu ABSTRACT Without rigorous symmetry constraints, solutions to approximate electronic structure methods may artificially break symmetry. In the case of the relativistic electronic structure, if time-reversal symmetry is not enforced in calculations of molecules not subject to a magnetic field, it is possible to artificially break Kramers degeneracy in open shell systems. This leads to a description of excited states that may be qualitatively incorrect. Despite this, different electronic structure methods to incorporate correlation and excited states can partially restore Kramers degeneracy from a broken symmetry solution. For single-reference techniques, the inclusion of double and possibly triple excitations in the ground state provides much of the needed correction. Formally, however, this imbalanced treatment of the Kramers-paired spaces is a multi- reference problem, and so methods such as complete-active-space methods perform much better at recovering much of the correct symmetry by state averaging. Using multi-reference configuration interaction, any additional corrections can be obtained as the solution approaches the full configuration interaction limit. A recently proposed “Kramers contamination” value is also used to assess the magnitude of symmetry breaking. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015279 .,s I. INTRODUCTION A proper treatment and understanding of symmetry is an important consideration for electronic structure theory. Symme- try forms a powerful tool to characterize the electronic states of chemical species and understand the selection rules that govern transitions between them. Unfortunately, for approximate varia- tional wavefunctions, the imposition of symmetry constraints can only raise the energy of the obtained solution. This means that we can get a “better” lower-energy wavefunction by ignoring these constraints and is known as Löwdin’s “symmetry dilemma.”1 In 1930, Kramers published a theorem that for systems with an odd number of electrons, the eigenvalues of a Hamiltonian that only includes electric fields (but not magnetic) are at least doubly degenerate.2Much later, Klein proved that for all non- degenerate states of any n-electron system, the expectation value of the magnetic moment is zero as a corollary.3However, it was Wigner who showed that the degeneracies required by Kramerstheorem are due to invariance under time-reversal symmetry (TRS).4 By relating the degeneracy of open-shell systems to symme- try, the problematic breaking of Kramers (or time-reversal) sym- metry has frequently been likened to that of spin-contamination.5 As with spin-symmetry, time-reversal symmetry can be rigorously enforced by additional constraints on the form of the wavefunction or allowed to vary freely, leading to both Kramers-restricted and Kramers-unrestricted formalisms. The arguments surrounding the use or non-use of Kramers-unrestricted methods are thus very simi- lar to those with the unrestricted Hartree–Fock (UHF) and restricted open-shell Hartree–Fock (ROHF) theories. While there are merits to both approaches, it has been found to often be of little consequence on the wavefunctions obtained at correlated levels of theory, such as coupled cluster.6Indeed, methods based on both unrestricted- and restricted-open shell wavefunctions still enjoy considerable popularity today. However, within relativistic quantum chemistry, much of the prior work has focused on using a Kramers-restricted J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp formalism. While it is possible to base ab initio correlated relativis- tic methods on a Kramers-restricted reference, it is often convenient to formulate electronic structure methods assuming an unrestricted, variational framework. This allows certain methods that require a variational ground state (such as response theory) to be applicable and a simpler development for computing molecular properties. It is this case that we now wish to explore and examine the extent to which it is also possible to recover the correct Kramers symmetry approximately in correlated wavefunction methods. In this perspec- tive, we discuss the use of Kramers-unrestricted formalisms, includ- ing time-dependent density functional theory (TDDFT), coupled– cluster (CC) theory, and multireference configuration interaction (MRCI). II. TIME-REVERSAL SYMMETRY AND KRAMERS THEOREM The time-reversal symmetry operator ˆΘis an anti-unitary oper- ator, which has the effect of reversing linear momentum p→−pas well as angular momentum j→−j, in essence playing time backward. The action of the time-reversal symmetry operator ˆΘon a fermionic one-electron wavefunction ψcan be written as follows: ˆΘ∣ψ⟩=∣¯ψ⟩, (1) ˆΘ2∣ψ⟩=−∣ψ⟩. (2) Note thatψmay be either a spinor or bispinor wavefunction. | ψ⟩and ∣¯ψ⟩represent a Kramers pair with the same energy. ˆΘalso transforms t→−tfor any operator O, ˆΘO(t)ˆΘ−1=¯O(−t). (3) To properly address the time-reversal symmetry, the under- lying wavefunction needs to be in the (bi-)spinor formalism. For electrons, a two-component spinor formalism can be used to rep- resent the wavefunction. That is, the molecular orbitals (MOs) are allowed to be a linear combination of both the αandβspin degrees of freedom, ψk(r,t)=⎛ ⎝ϕα k(r,t) ϕβ k(r,t)⎞ ⎠, (4) where the spatial functions {ϕα k(r,t)}and{ϕβ k(r,t)}are expanded in terms of a common set of basis functions { χμ(r)}, ϕα k(r,t)=∑ μcα μk(t)χμ(r), (5) ϕβ k(r,t)=∑ μcβ μk(t)χμ(r). (6) The spinor formalism gives rise to the two-component electronic structure methods, where the Hamiltonian matrix has a spin- blocked form, (FααFαβ FβαFββ). (7)The anti-unitary property of ˆΘmeans that we can represent ˆΘas a product of a unitary operator Ûand the complex conjugation operator ˆK, ˆΘ=ˆUˆK. (8) In the case of time-reversal symmetry for electrons, this is repre- sented as iσyˆK, whereσyis the Pauli spin matrix, given by σy=(0−i i0). (9) In the case of the two-component ansatz, it can be seen that the effect of ˆΘon a given electron is to flip the spin and take complex conjugates, ˆΘ(ϕα ϕβ)=(−ϕ∗ β ϕ∗ α). (10) When the Hamiltonian obeys time reversal symmetry, so [ˆH,ˆΘ]= 0, this implies that ˆH=ˆΘˆHˆΘ−1, and we can determine necessary constraints, ˆΘ(FααFαβ FβαFββ)ˆΘ−1=(Fββ−Fβα −Fαβ Fαα). (11) Equation (11) shows that the requirement of time-reversal symme- try can be imposed by two constraints such that we only have two independent blocks in the Fock matrix: a same-spin block and a mixed-spin block (i.e., Fαα=FββandFβα=−F∗ αβ). Note that since ˆH is Hermitian, the off-diagonal mixed-spin block must also be related by complex conjugation. This structure leads to paired molecular orbitals, or “paired generalized Hartree–Fock (GHF),” which is anal- ogous to the ROHF theory. In the case of relativistic wavefunctions, where spin is no longer a good quantum number, this is also known as a Kramers-restricted formalism.7–16 Before discussing symmetry breaking and restoration, we note that Kramers theorem only applies when the molecule is not sub- ject to magnetic fields. If a molecule is in the presence of magnetic fields, then the true solution need not obey time reversal symme- try as[ˆH,ˆΘ]≠0. On the other hand, if a molecular system is only subject to electric fields, the Hamiltonian can be shown to be time-reversal invariant since the vector potential can be chosen to be zero. We emphasize that this holds true even when relativistic effects such as spin–orbit coupling are included, such as for the Dirac–Coulomb and Dirac–Coulomb–Breit Hamiltonians. To see this, it is a straightforward exercise to check how each of the dif- ferent operators in the Hamiltonian are transformed under ˆΘ. For example, because the spin–orbit operator is of the form L⋅s= (r ×p)⋅s, under time reversal, r→r,p→−p, and s→−s, and there is no overall change. For the remainder of our discussion, we assume that the system is not subject to magnetic fields and thus that the Hamiltonian under consideration is time-reversal invariant so that [ˆH,ˆΘ]=0. III. SYMMETRY BREAKING AND RESTORATION For approximate electronic structure methods, there might be artificial symmetry breaking due to an unbalanced treatment of the J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp different configurations for open-shell systems. This problem is par- ticularly acute for single-reference methods where the reference is typically chosen to be the lowest energy single Slater determinant. When variational constraints are removed, as is the case in four- and two-component Dirac–Hartree–Fock compared to more common real-valued restricted Hartree–Fock (RHF) or unrestricted Hartree– Fock (UHF), symmetry breaking is likely to occur and is nearly guaranteed for open-shell systems. One possible remedy is to apply a projection operator to form a solution that is a good eigenfunction of the operator and remove the offending components. This is frequently done to form good eigenfunctions of spin operators such as ˜S2and ˜Sz.17–19Analogous projection methods to form eigenfunctions of theˆJ2operator would be a very interesting development, though evaluation of ⟨J2⟩remains a challenging problem. Alternatively, one can use a multi-configurational approach such as “average-of- configuration,”7–9,12–16,20,21as implemented in the DIRAC code.22 This method uses a minimal active space and fractional occupancy to build in the effect of optimizing the orbitals for multiple config- urations. In this way, the Kramers-unrestricted problem becomes a Kramers-paired, multi-configurational solution. For open shell systems, molecular orbitals (MOs), ψp′, in a vari- ational solution of Hartree–Fock in spinor basis, such as the general- ized Hartree–Fock (GHF) or two-component Dirac–Hartree–Fock methods, generally do not satisfy time-reversal symmetry. That is, for a given ∣ψp′⟩, there is no single MO ∣¯ψp′⟩that is the Kramers partner. Instead, the time-reversed spinor ∣¯ψp′⟩=ˆΘ∣ψp′⟩is a linear combination of several other occupied and virtual orbitals. These expansion coefficients can be easily determined from the overlaps of the time-reversed spinor ∣¯ψp′⟩with each of the broken symmetry spinors∣ψq′⟩, ∣¯ψp′⟩=∑ q′∣ψq′⟩⟨ψq′∣¯ψp′⟩. (12) More generally, for a given basis function space, the set of orbitals satisfying the time-reversal symmetry, ψpand ¯ψp, can be expressed as linear combinations of the broken symmetry spinor MOs ( ψp′), ∣ψp⟩=∑ p′(∣ψp′⟩⟨ψp′∣ψp⟩+∣¯ψp′⟩⟨¯ψp′∣ψp⟩), ∣¯ψp⟩=∑ p′(∣ψp′⟩⟨ψp′∣¯ψp⟩+∣¯ψp′⟩⟨¯ψp′∣¯ψp⟩).(13) This transformation from broken-symmetry spinor MOs to Kramers-restricted orbitals is unitary and can be equivalently expressed as orbital rotations, exp(iλ)=1 +iλ+1 2!(iλ)2+⋯=∞ ∑ n=0(iλ)n n!, where λ=∑ p′,q′λp′q′a† p′aq′, (14) whereλp′q′is an element of a Hermitian matrix.For the configuration interaction (CI) method, we can conve- niently write the electronic wavefunction ∣M′⟩as a linear combina- tion of configurations (single Slater determinants), Φj′, constructed with broken-symmetry spinor MOs, ∣M′⟩=∑ j′Cj′M′∣Φj′⟩. (15) Any excited configurations constructed from spinor MOs with bro- ken symmetry will not have Kramers symmetry either. Similar to orbital rotation, the variation of expansion coefficients Cj′M′can be expressed as a unitary transformation, exp(iS)=1 +iS+1 2!(iS)2+⋯=∞ ∑ n=0(iS)n n!, where S=∑ L′,M′SL′M′∣L′⟩⟨M′∣. (16) Equations (14) and (16) suggest that the two types of rotations are redundant in the full CI limit, and indeed, the full CI result does not depend on the particular choice of basis functions, only the span. In other words, Kramers symmetry can be recovered either in the orbital rotations of Hartree–Fock or in the CI framework. In the full CI limit, [ˆH,ˆΘ]=0, meaning that the eigenfunction of the Hamiltonian is also an eigenfunction of the time-reversal symmetry operator. Therefore, minimization of the energy function in the CI framework also recovers the time-reversal symmetry even if the ref- erence spinor orbitals do not have Kramers symmetry. For truncated CI, the configurational space generated from excitation operators may not be large enough to create a complete redundancy of orbital rotation. This means that when any truncated configuration space is used, TRS may not be completely recovered if the reference is symmetry-broken. The complete active space self-consistent field (CASSCF) pro- cedure is rather unique in that it includes both types of rotations, obtaining both orbital rotations in the SCF portion and CI-type rota- tions in configuration interaction of the active space orbitals. As a result, the CASSCF procedure should also be able to recover TRS, despite limiting the CI expansion to the active space. Additionally, we note that the state averaged (SA) CASSCF procedure can force a solution where the average energy of a set of degenerate states is equal and thus treat all Kramers pairs equally. The flexibility to recover TRS in two different ways can be even further exploited in the multi-reference configuration interaction (MRCI) method built on top of a CASSCF wavefunction. We note that the results from the SA-CASSCF procedure and that of the average of configuration (AOC) SCF approach are similar, though not identical. While AOC enforces symmetry via constraints on the orbitals , SA-CASSCF recovers symmetry by averaging over states . In AOC-SCF, a set of open shell orbitals ( S) is defined, and the average energy of a set of CI states in this orbital space is recast in terms of orbitals with fractional occupation. The average energy can be written as Eav=∑ fs⎡⎢⎢⎢⎢⎣∑ phpp+1 2∑ S′QS′ pp+1 2(aS−1)QS pp⎤⎥⎥⎥⎥⎦, (17) J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp fS=NS MS, (18) aS=⎧⎪⎪⎨⎪⎪⎩1 if fS=0 MS(NS−1) NS(MS−1)iffS≠0,(19) where fSis the fractional occupation in shell S,MSandNSare the number of orbitals and electrons in shell S, respectively, and the two- electron contribution is QS=fS∑r∈S⟨pr||qr⟩. This energy is minimized in an SCF procedure and the results in orbitals that have TRS. In the non-relativistic case, this can be used to enforce degeneracy of, for example, open shell porbitals. In the rela- tivistic case, TRS symmetry of the orbitals can be recovered yielding the correct symmetry of p1/2andp3/2orbitals. Complete open shell CI (COSCI)21using the AOC-SCF orbitals as a reference gives a set of CI states that also obey TRS. In contrast to AOC-SCF enforcing TRS of the orbitals, SA- CASSCF may use broken-symmetry orbitals to recover TRS of the orbitals and states. Additionally, the AOC-SCF/COSCI approach is applicable to the open-shell orbitals only, while in SA-CASSCF, extension of the active space can build in correlation during the orbital optimization procedure. That is, SA-CASSCF allows for a flexible choice of which orbitals are included and which states are averaged. While averaging over orbitals or states is equiv- alent in the one-electron case, for multi-electron systems, these choices can lead to different results. In both cases, it is neces- sary to use explicit multi-reference techniques to describe excited states. To quickly assess spin-contamination, it is commonplace to calculate ⟨S2⟩and see how it deviates from the ideal values. In analogy with spin-contamination, it has also been recently pro- posed to calculate “Kramers contamination” by deviations in the expected value of an operator that is the generator of the many- electron time-reversal operator.5,23–25That is, one can evaluate the quantity ⟨Ψ∣Θ2 +∣Ψ⟩, (20) where the many-electron time-reversal generator Θ+is given by a sum of the one-electron time-reversal operators, Θ+=N ∑ iΘi. (21) For true multi-configurational wavefunctions, it has been found that ⟨Θ2 +⟩=−k2, where kis the number of open-shell electrons. However, for a single determinant wavefunction, ⟨Θ2 +⟩=−k. IV. COMPUTATIONAL RESULTS For this perspective, we use relativistic two-component time- dependent Hartree-Fock (TDHF), equation-of-motion coupled- cluster theory (EOM-CCSD), CASSCF, and MRCI to examine how these different approaches recover time-reversal symmetry. Specif- ically, we use the one-electron exact-two-component (X2C)26–39 transformation in all implementations of relativistic methods. Our goal is to see how well the different Kramers-unrestricted meth- ods obtain the correct degeneracies of both ground and excited states for several small open-shell systems. The systems usedin our consideration here were chosen primarily to be compu- tationally feasible to do larger configuration interaction spaces such as triples and quadruples but also to have reliable exper- imental data with which to compare. However, we do admit that these results may not be representative of larger molecu- lar systems where covalency and reduced point group symme- try effects play a larger role. Calculations with X2C-EOM-CCSD40 were run in Chronus Quantum,41while the X2C-TDHF,36X2C- CASSCF,42and X2C-MRCI43calculations were run in a locally modified development version of the Gaussian software pack- age.44Two-component molecular orbitals are plotted, as described in Ref. 45. A. Kramers symmetry in p-shell—sodium doublet The first system we discuss is the neutral sodium atom, which has a doubly degenerate ground state and a well-known doublet transition from 3 s→3p. Spin–orbit coupling splits the p-manifold of the lowest excited state into two sets of sublevels based on their total angular momentum J: 3p1/2and 3 p3/2with degeneracies of two and four, respectively, if the Kramers symmetry is preserved. The orbitals obtained from the SCF procedure using X2C-HF with the 6-31G basis set show symmetry breaking. As shown in the spinor MO diagram in Fig. 1, the two 3 sspinor orbitals, which should form a Kramers pair, are separated by around 5 eV. We note that this large value is not necessarily indicative of extreme symmetry breaking as the energies of spin-orbitals are greatly influ- enced by occupation in the Hartree–Fock approximation. As a result of the spin polarization of the unpaired electron in the 3sorbital, the 3 porbitals, which would ideally have degenera- cies of two and four, are found in a degeneracy pattern of three and three. Spin–orbit coupling further splits these orbitals on the order of meV; however, the spin polarization effect dominates. When calculating the Kramers contamination using Eq. (20), a value of −1.000 286 is obtained, rather than the ideal value of −1. Although the deviation from the exact Kramers symmetry seems small, the manifestation on the energies and degeneracies is significant. As the 3 selectron should be formally unpaired, the increased contamination value is due to the lack of ideal pairing of spinors in the ten electrons in the nominal 1 s22s22p6closed-shell configuration. It is evident that spinor orbitals from variational X2C-HF cal- culations for an open-shell system do not satisfy the Kramers sym- metry. Discussions in Sec. III suggest that procedures that allow either orbital rotations or expansion in a configuration interaction or coupled-cluster framework could recover the time-reversal symme- try. We start off with calculations in the 6-31G basis using single ref- erence techniques that include orbital transitions between occupied and virtual spinors. Table I lists eigenvalues from solving X2C-CIS, X2C-TDHF, X2C-CISD, and X2C-EOM-CCSD, respectively. The first eigenroots from X2C-TDHF and X2C-EOM-CCSD are zeros, suggesting that the ground state time-reversal symmetry is properly recovered. By constrast, neither X2C-CIS or X2C-CISD fully recov- ers the degeneracy of the ground state. In fact, in the X2C-CISD calculation, the degeneracy is worse due to the additional correlation included in the reference determinant that is not in the correspond- ing Kramers-paired determinant. In X2C-TDHF, this “excitation” involves transitions from the occupied 3 sspinor to the two virtual J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 1 . Spinor MO diagram of the 3 sand 3porbitals in the X2C-HF ground state of a sodium atom. Note that each level is a spinor orbital and can only be singly occupied. MOs that compose the time-reversed spinor. The relative contri- butions of 91.2% and 8.8% are the linear combination coefficients of the broken symmetry MOs and match exactly those determined using Eq. (12). Compared to X2C-HF spinor orbital energies, the time-reversal symmetry in the J= 1/2 and J= 3/2 manifolds are largely recovered, showing the expected two- and four-fold degen- eracy. Although the linear response X2C-TDHF method allows only single excitations/de-excitations, the energy of a Kramers pair is recovered to within a few meV error. Including double excitations in X2C-EOM-CCSD, all Kramers pairs are obtained within a 20 μeV error. These same trends mirror the two-component GHF non- relativistic analogs in which there is spin contamination rather than Kramers contamination. The values are given in Table II. How- ever, instead of two nearly degenerate sets of two and four states each, there is a single set of six nearly degenerate states. Indeed, these calculations show that just as unrestricted calculations are still commonplace in the non-relativistic regime, the use of Kramers- unrestricted calculations perform similarly to their non-relativistic counterparts.As discussed previously, another way to recover the appro- priate Kramers degeneracy is through the use of multi-reference methods such as CASSCF where the orbitals are reoptimized and the wavefunction takes on the multi-determinantal character of the true ground state. In order to recover the Kramers symmetry, state- averaged CASSCF treatment must be used as optimization for only a single state will favor the symmetry-broken solution.42This allows for a balanced treatment of all Kramers pairs such that each orbital manifold feels the same mean-field potential. For the following cal- culations, we include all the relevant states to the terms. That is, for a minimal active space of SA-CASSCF(1,2), we state average over both microstates in the2S1/2term, and in the larger SA-CASSCF(1,8) calculation, we average over all eight states in the relevant2S1/2, 2P1/2, and2P3/2terms. The Kramers contamination for the state- averaged CASSCF wavefunction is reduced by an order of magni- tude from the HF solution to −1.000 03 for both SA-CASSCF(1,2) and SA-CASSCF(1,8). As noticed from X2C-TDHF and X2C-EOM-CCSD results, single- and double-excitations contribute the most to the recovery of Kramers symmetry. One could use additional excitations from TABLE I . Lowest energy excited states of Na atom (in eV). Term Experiment46X2C-CIS X2C-TDHF X2C-CISD X2C-EOM-CCSD 2S1/2 0.0000 0.0000 0.0000 0.0000 0.0018 0.0000 0.0314 0.0000 2P1/2 2.1023 1.9949 1.9956 2.0282 1.9971 1.9954 1.9959 2.0282 1.9971 2P3/2 2.1044 1.9964 1.9972 2.0299 1.9989 1.9974 1.9976 2.0299 1.9989 1.9978 1.9979 2.0300 1.9989 1.9981 1.9981 2.0300 1.9989 J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp TABLE II . Lowest energy excited state of Na atom using GHF reference (in eV). Term CIS TDHF CISD EOM-CCSD 2S 0.0000 0.0000 0.0000 0.0000 0.0018 0.0000 0.0318 0.0000 2P 1.9919 1.9911 2.0250 1.9926 1.9919 1.9911 2.0250 1.9926 1.9919 1.9911 2.0250 1.9926 1.9928 1.9928 2.0252 1.9926 1.9928 1.9928 2.0252 1.9926 1.9928 1.9928 2.0252 1.9926 the CAS reference to recover the Kramers symmetry, giving rise to the multi-reference configuration interaction (MRCI) method.43In MRCI, additional electron correlation is included naturally accom- panied with the additional recovery of correct degeneracy. As seen in Fig. 2, the balanced treatment of the smaller SA-CASSCF(1,2) performs very similarly to the larger SA-CASSCF(1,8), indicating FIG. 2 . Energy range of nearly degenerate states in the three lowest terms of Na:2S,2P1/2, and2P3/2. The use of state-averaged (SA) CASSCF reference determinants can also recover the degeneracy.that only the minimal active space for the ground state is neces- sary to recover symmetry. The microstates in the2S,2P1/2, and 2P3/2terms computed using MRCI on both the state-averaged X2C- CASSCF(1,2) and X2C-CASSCF(1,8) references are degenerate to within the convergence tolerance (10−6hartree). In Fig. 2, the states are grouped together according to the assigned terms as in Table I, 2S,2P1/2, and2P3/2. Since formally the states within these terms should be degenerate, the difference between the highest and low- est energies in the group is used as a measure of the symmetry breaking. Comparing the CI results using HF and SA-CASSF ref- erences, it is clear that using the less contaminated reference (SA- CASSCF) leads to better degeneracy recovery, with the energy range of states in the same term decreased by several orders of magnitude. In particular, it can be seen that at the quadruple level, there is little effective difference between the choice of references as the error is primarily due to the numerical precision in solving for the eigenvec- tors and eigenvalues when iteratively diagonalizing the CI matrix. This is because the correlation space now includes the Kramers- paired space, so the resulting wavefunctions will have the expected symmetry. As discussed in Sec. III, both orbital rotation and CI expansion can recover the Kramers symmetry. An interesting aspect to investi- gate is which approach is more effective. Since the SA-CASSCF can be converged to a given accuracy, a loosely converged solution can be thought of as only partially optimizing the orbitals to recover the correct symmetry. A natural question then is if this has any impact on how quickly symmetry is recovered in the MRCI framework. That is, does the amount of contamination in the reference affect how well the degeneracy is recovered? Surprisingly, for Na, the rate at which degeneracy is recovered for the SA-CASSCF(1,2) and SA- CASSCF(1,8) systems is insensitive to how well the reference was converged. As seen in Fig. 3, this difference in initial contamina- tion remains even at the triples level. Only in the calculations with quadruple excitations where there is almost no remaining symme- try breaking do the calculations provide indistinguishable results. This suggests that in the MRCI framework, it is more efficient to recover symmetry through orbital rotation in the reference than through the CI excitations, which require high order that is often not affordable. B. Kramers symmetry in d-shell—Sc2+ To examine how the recovery of Kramers degeneracy might be affected by more pronounced relativistic effects, we continue down the periodic table to Sc2+ion, which has a ground state elec- tronic configuration of 3 p63d. The unpaired electron now has the entire 3 dmanifold as well as the low-lying 4 s, creating a much more complicated multireference problem. The MO diagram for the X2C-HF reference is given in Fig. 4 and shows the 4 sorbitals in-between the occupied 3 dorbital and the higher lying unoccu- pied 3 dorbitals. When calculating the Kramers contamination using Eq. (20), a value of −1.001 21 is obtained, rather than the ideal value of −1. The difference is an order of magnitude worse than for Na. The results for the single-reference X2C-TDHF and multiref- erence X2C-SA-CASSCF are given in Table III. The active space for the X2C-SA-CASSCF(1,12) system included both the 4 sand J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp FIG. 3 . Energy range of nearly degenerate states in the three lowest terms of Na:2S,2P1/2, and2P3/2. A lower convergence threshold in the state-averaged (SA) CASSCF reference results in lower degeneracy. 3dorbitals, and state-averaging was performed over the 12 rele- vant states (2D3/2,2D5/2, and2S1/2). While the effect of symmetry breaking is relatively small for the 3 d→4stransition for the2S1/2 term, the fine structure within the d-manifold is unable to be mean- ingfully captured using the single reference X2C-TDHF. This is because the spin–orbit splitting and Kramers symmetry breakingare of similar magnitude. Only one “excitation” has a zero eigen- value using X2C-TDHF corresponding to the Kramers paired orbital of the occupied 3 d. The other two excitations in the2D3/2man- ifold involve transitions to 3 dorbitals from a different Kramers pair, and so these orbitals are not able to fully relax and recover the time-reversal symmetry. As expected, the X2C-SA-CASSCF is FIG. 4 . Spinor MO diagram of the 4 sand 3dorbitals in the X2C-HF ground state of a Sc2+ion. Note that each level is a spinor orbital and can only be singly occupied. Several representative MOs are shown. J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp TABLE III . Sc2+ion with 6-31G basis. All energies are in eV. Term Degeneracy X2C-TDHF X2C-SA-CASSCF Experiment47,48 2D3/2 4 0.0000 0.0000 0.0 0.0000 0.0000 0.0315 0.0000 0.0365 0.0000 2D5/2 6 0.0401 0.0283 0.024 504 0.0422 0.0283 0.0468 0.0283 0.0481 0.0283 0.0612 0.0283 0.0758 0.0283 2S1/2 2 3.2629 3.1842 3.166 472 3.2689 3.1842 able to recover the correct degeneracies to within the convergence tolerance. When using X2C-MRCI, it is again evident that the remain- ing amount of symmetry breaking in the reference is retained for FIG. 5 . Energy range of nearly degenerate states in the three lowest terms of Sc:2D3/2,2D5/2, and2S1/2. A lower convergence threshold in the SA-CASSCF reference results in lower degeneracy.the first several excitation levels. In Fig. 5, the X2C-MRCI results are shown with different SA-CASSCF convergence tolerances. All calculations used a frozen core of the 1 s, 2s, and 2 plevels. In con- trast to the previous results with Na, however, it is also apparent that the restoration of symmetry is not uniform, particularly, for the states in the2D3/2and2D5/2manifolds. These converge to simi- lar solutions when triples are included. The seeming inability of the calculations to recover Kramers degeneracy to a greater degree is an artifact of using a frozen core, so not all possible excited deter- minants are included in the correlation space. That is, the neglect of determinants that are in the Kramers-paired space of the frozen core will lead to some symmetry breaking that cannot be recov- ered. This problem appears to be less severe for the2S1/2state, which has the nominal configuration of 3 p64s. Since neither of the 4sorbitals are included to a significant degree in the ground state reference, there is little energetic benefit to cause symmetry breaking in these orbitals. The results shown here demonstrate that compar- atively small correlation spaces are necessary to recover most sym- metry. This intuitively makes sense because Kramers pairs that are either always both occupied or both unoccupied do not cause sig- nificant symmetry breaking as there is little energetic benefit to be gained. C. Kramers symmetry in f-shell—uranium (V) ion The uranium (V) ion is another interesting system since the valence shell is dominated by strong spin–orbit coupling and contains only a single outermost electron in the 5 fshell. Unlike the previous examples where spin is still approximately a good quantum number, for U(V), it is imperative to consider the degeneracy in the Jmanifold. The Kramers contamination value is calculated to be −1.009 12 rather than the ideal value of−1. While X2C-TDHF was shown in the previous examples to be able to partially recover broken Kramers symmetry of state with low angular momentum, its ability in the high angular momentum case is more limited despite having a similar Kramers contamina- tion value. As shown in Table IV, the energy range of states that belong to the same term is ∼0.3 eV for both2F5/2and2F7/2terms. However, the deviation is less than the spin–orbit coupling, which splits the2F5/2and2F7/2terms by nearly an eV. Therefore, even though the symmetry breaking in U5+might be more severe than in Sc2+due to the increased degeneracy of the ground state, the overall energetic groupings of the microstates are more clear since spin–orbit coupling plays a dominant role. Interestingly, the sym- metry breaking is decreased when using X2C-TDDFT compared to X2C-TDHF. While a full investigation of how Kramers symme- try breaking is manifested in DFT calculations has not been con- ducted, our initial results suggest that this could be due to the exchange–correlation potential acting to smear out the density over the multiple configurations within the 5 fmanifold. Table IV shows that the number of microstates in each term is correctly predicted by X2C-TDHF and X2C-TDDFT, as expected, since all are gener- ated by single excitations from the reference. In contrast, X2C-SA- CASSCF(1,14) is able to completely recover the broken Kramers symmetry through orbital rotation and state-averaging over all 14 microstates. J. Chem. Phys. 153, 090903 (2020); doi: 10.1063/5.0015279 153, 090903-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsPERSPECTIVE scitation.org/journal/jcp TABLE IV . Free U5+ion with SARC-DKH249basis. DFT calculations used the SVWN5 functional,50and the SA-CASSSCF used the CAS(1,14) active space. All energies are in eV. Term Degeneracy X2C-TDHF X2C-TDDFT X2C-SA-CASSCF Experiment51 2F5/2 6 0.000 00 0.000 00 0.000 00 0.0 0.000 00 0.000 08 0.000 00 0.168 61 0.000 18 0.000 00 0.232 76 0.000 82 0.000 00 0.285 33 0.153 07 0.000 00 0.326 59 0.395 50 0.000 00 2F7/2 8 0.942 00 0.942 34 0.971 94 0.943 35 1.015 22 0.955 43 0.971 94 1.066 19 1.003 24 0.971 94 1.183 74 1.005 19 0.971 94 1.188 78 1.012 23 0.971 94 1.261 54 1.029 83 0.971 94 1.271 05 1.072 09 0.971 94 1.292 11 1.097 73 0.971 94 V. CONCLUSION The restoration of correct symmetry is important for spec- troscopic or other physical interpretation of electronic states. In relativistic calculations, time-reversal, or Kramers symmetry, gives rise to well-known degeneracy patterns. For calculations of open- shell molecules, Hartree–Fock or other reference wavefunctions that break Kramers symmetry can have this symmetry partially restored through the inclusion of excitation operators, or through reoptimization of orbitals as in a state-averaged CASSCF. X2C-TDHF has been shown to be able to partially recover the broken Kramers symmetry. While it works well for low-angular momentum orbitals such as the s- and p-shells, its effectiveness is limited for the d- and f-shells, where a higher degree of degener- acy limits the description within a single-determinant framework. Nevertheless, the use of DFT seems to have a mediating effect on symmetry breaking, although much further study in this area would be required to see if this is only true for some functionals. Includ- ing higher excitation operators in coupled-cluster enables a better symmetry recovery. In contrast, multi-reference techniques, such as X2C-CASSCF and X2C-MRCI, are much better able to recover the Kramers symmetry through orbital rotation and state-averaging. This is the “average-of-configuration” approach. The MRCI frame- work allows one to shift the cost of restoring symmetry between the orbital rotation and including additional excitations in an aid to performance and can allow for the use of smaller active spaces in a Kramers-unrestricted framework. Formalisms that can utilize broken symmetry solutions effec- tively allows for flexibility in method development for molecu- lar response properties and extension to regimes where Kramers symmetry need not hold. While considerable work has been done comparing the unrestricted- and restricted-open shell frameworks in non-relativistic calculations, much less work has been done comparing Kramers-unrestricted and Kramers-restricted formalisms in relativistic calculations. Most calculations utilize either a Kramers-restricted formalism or a pseudo-restricted formalism suchas average-of-configuration. The recently proposed “Kramers con- tamination” value does indeed capture the contamination in the broken symmetry reference, though it remains unclear how much deviation in the number of unpaired electrons would be consid- ered problematic, since the size of the open-shell is also crucial. We believe that future studies exploring how methods using a Kramers- symmetry broken reference such as relativistic coupled-cluster the- ory will be important to new advances in relativistic calculations and magnetic properties. Understanding the methodological perfor- mance in recovering Kramers symmetry will aid in differentiating artificial from physical symmetry breaking. AUTHORS’ CONTRIBUTIONS J.M.K. and A.J.J. contributed equally to this work. ACKNOWLEDGMENTS The development of relativistic electronic density functional theory is funded by the U.S. Department of Energy (Grant No. DE-SC0006863). The development of relativistic multi-reference method is supported by the U.S. Department of Energy (Grant No. DE-SC0021100). The development of the computational spectro- scopic method is supported by the National Science Foundation (Grant No. CHE-1856210). The development of the Chronus Quan- tum open source software package is supported by the National Science Foundation (Grant No. OAC-1663636). Computations were facilitated through the use of advanced computational, storage, and networking infrastructure provided by the Hyak supercomputer system at the University of Washington, funded by the Student Technology Fee. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. J. Chem. 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5.0015291.pdf

J. Appl. Phys. 128, 165703 (2020); https://doi.org/10.1063/5.0015291 128, 165703 © 2020 Author(s).Investigation of electron–phonon interaction in bulk and nanoflakes of MoS2 using anomalous “ b” mode in the resonant Raman spectra Cite as: J. Appl. Phys. 128, 165703 (2020); https://doi.org/10.1063/5.0015291 Submitted: 27 May 2020 . Accepted: 12 October 2020 . Published Online: 28 October 2020 Rekha Rao , Ram Ashish Yadav , N. Padma , Jagannath , and A. Arvind ARTICLES YOU MAY BE INTERESTED IN Understanding the breakdown asymmetry of 4H-SiC power diodes with extended defects at locations along step-flow direction Journal of Applied Physics 128, 164501 (2020); https://doi.org/10.1063/5.0020066 Structure evolution, bandgap, and dielectric function in La-doped hafnium oxide thin layer subjected to swift Xe ion irradiation Journal of Applied Physics 128, 164103 (2020); https://doi.org/10.1063/5.0025536 Interface-induced localization of phonons in BeSe/ZnSe superlattices Applied Physics Letters 117, 183104 (2020); https://doi.org/10.1063/5.0026067Investigation of electron –phonon interaction in bulk and nanoflakes of MoS 2using anomalous “b” mode in the resonant Raman spectra Cite as: J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 View Online Export Citation CrossMar k Submitted: 27 May 2020 · Accepted: 12 October 2020 · Published Online: 28 October 2020 Rekha Rao,1,2,a) Ram Ashish Yadav,3N. Padma,2,4Jagannath,4and A. Arvind5 AFFILIATIONS 1Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 2Homi Bhabha National Institute, Mumbai 400094, India 3Government Girls College, Vidisha, Madhya Pradesh 464001, India 4Technical Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, India 5Process Development Division, Bhabha Atomic Research Centre, Mumbai 400085, India a)Author to whom correspondence should be addressed: rekhar@barc.gov.in ABSTRACT Electron –phonon interaction in bulk and nanoflakes of MoS 2is investigated using Raman spectroscopy. Resonant Raman spectroscopic studies carried out on bulk and liquid exfoliated nanoflakes of MoS 2revealed a second order Raman mode (called the “b”mode), whose fre- quency in the case of nanoflakes was found to be largely different from that in bulk MoS 2. Temperature dependent Raman spectra show larger variation in the frequency of the “b”mode in bulk MoS 2as compared to that in nanoflakes of MoS 2. This anomalous behavior of the “b”mode could be attributed to the stronger electron –phonon coupling occurring in bulk MoS 2, due to higher electron concentration in the same, as compared to that in nanoflakes of MoS 2. A larger sulfur vacancy in bulk MoS 2as compared to that of nanoflakes was found to be responsible for higher electron concentrations. These findings are supported by energy dispersive x-ray analysis and x-ray photoelectronspectroscopic studies carried out on bulk and nanoflakes of MoS 2. The present study suggests a more sensitive probe for the estimation of electron concentrations in the low limit range by following the “b”mode in resonance Raman spectra. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015291 INTRODUCTION Electron –phonon interaction (EPI) is an important phenome- non in materials, which governs many properties ranging from electrical conductivity to superconductivity. As EPI changes the electronic as well as phonon energies and lifetimes, it affects theelectronic and optical properties, and hence the applicability ofmaterials in electronic and optoelectronic devices. EPI, in general,can be varied by doping, 1–3applying gate voltage,4irradiation,5,6 and also changing a thermodynamic parameter like pressure.7 There are extensive efforts toward understanding electron –phonon coupling (EPC), both from fundamental interest as well as from theapplication point of view. In graphite, EPC is responsible for thelarge dispersion of the Raman active modes, and the value of EPC can be directly obtained from experimental phonon dispersions. 8 In materials of technological interest, such as topologicalinsulators,9–12two-dimensional (2D) materials like graphene,13,14 monolayer of MoS 2,15,16phosphorene,17and WS 2,18there are reports of tuning EPI by varying the carrier concentration by eitherchanging the material composition or electrical/optical means. EPIis an essential concept to be understood since it affects electronmobility and heat dissipation in devices. The compound molybdenum disulfide (MoS 2) belongs to the family of transition metal dichalcogenides (TMDs) possessing a bandgap and hence is useful for switching applications, unlike gra-phene. It is a layered material in which a layer of Mo atoms is sand-wiched between the two layers of S atoms. This trilayer S –Mo–S unit is called the single layer of MoS 2. While bulk MoS 2is an indi- rect bandgap semiconductor (1.23 eV), it becomes a direct bandgapsemiconductor (1.8 eV) in a monolayer form. 19While within a layer, Mo and S atoms are bonded by strong covalent bonds, twoJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-1 Published under license by AIP Publishing.layers of MoS 2are bonded by the weak van der Waals interaction so that they can be easily exfoliated into a 2D form by different exfoliation methods. Easy exfoliation, bandgap tunability, highcarrier mobility, high absorption coefficient, etc .,make it a highly successful, potential, and exotic candidate for electronic and opto-electronic device applications. EPI in bulk, monolayers/bilayers of MoS 2and WS 2have been investigated using various techniques. While there are severalmethods like angle resolved photoelectron spectroscopy (ARPES),electron energy loss spectroscopy (EELS), etc., to explore electronicstructures, vibrational spectroscopic techniques like Raman spectro- scopy, IR spectroscopy, neutron/x-ray inelastic scattering, and inelastic helium atom scattering (HAS) are used for exploring EPCthrough phonons. There are several EPI studies on bulk and mono-layer MoS 2, of which some prominent investigations are discussed here. Ultrafast visible/far IR spectroscopic studies on bulk MoS 2 have shown the frequency of the IR active E1umode at 382 cm−1to be increasing with electron –phonon coupling.20HAS studies on bulk MoS 2have shown an increase in EPC with the carrier concen- tration.21Recently, complete phonon dispersion of bulk MoS 2has been determined using x-ray inelastic scattering and first-principles calculations, which show a strong 2D character of the vibrational modes.22EPI is expected to be different in the 2D form as com- pared to that in bulk, as the electronic structure and the phononspectra are different in the former. 23While there is little tunability of EPI in bulk materials, reduced dimensionality in the 2D form allows for the tunability of EPI by precise control of carrier concen-trations on applying a gate voltage. In monolayer MoS 2, EPC parameter has been experimentally obtained using ARPES and alsoverified by density functional theory (DFT) calculations by Mahatha et al. 24There are DFT calculations for estimating electron mobility and electron –phonon coupling for monolayer MoS 2, sili- cene, and graphene, which showed anisotropic nature of the cou-pling of charge carriers to phonon modes. 25 Raman spectroscopy has been an important technique for studying phonon interaction with charge carriers and for estimat- ing electron concentration in 2D materials. The response of eachphonon to the charge carriers is different, which is reflected in fre-quency variation and in full width at half maximum (FWHM) ofRaman modes. In general, using FWHM contribution to EPI, it is possible to estimate the EPC as has been done in systems exclu- sively using Raman spectroscopy in graphene and monolayer ofMoS 2. EPC is estimated in graphene13,14and tuned using an elec- tric field. Earlier Raman spectroscopic investigations on monolayer MoS 2by Chakraborty et al.26have reported systematic variation of Raman mode frequencies with electron doping, brought about byapplying the gate voltage in a field effect transistor configuration.Electron –phonon coupling constants were measured for the Raman active modes and confirmed by first principles calculations. 26 The resonant and non-resonant Raman study of MoS 2has been well established and used as a guide for the Raman study ofother 2D materials. Among the higher order modes, there is ahighly dispersive weak mode around 420 cm −1, whose assignment has been controversial. While Sekine et al.27have assigned it to a combination of a dispersive quasi-acoustic (QA) phonon and a dispersion-less transverse optical (TO) phonon along the caxis, a recent report28has attributed it to combinations of the longitudinalacoustic (LA) and transverse acoustic (TA) phonons at K points of the Brillouin zone. In this paper, we identify anomalous behavior of this second order “b”mode in bulk MoS 2as compared to that in nanoflakes of MoS 2synthesized by the liquid exfoliation method. This behavior of the “b”mode is attributed to electron –phonon interaction. The resent study suggests that the “b”mode observed in resonant Raman spectra is more sensitive to electron concentra- tions in MoS 2than the two first order Raman modes and is useable in estimating electron concentrations in the low concentrationregime. EXPERIMENTAL METHODS Nanoflakes of MoS 2were synthesized by dispersing MoS 2 powder (Sigma Aldrich) in iso-propyl alcohol (IPA) and sonicating them for a few hours.29A few microliters of the supernatant was further diluted and dispersed in IPA from which a few microliterswere drop cast on the freshly cleaned SiO 2(270 nm) substrate. They were subsequently allowed to dry naturally till the IPA evapo- rated fully. The presence of flakes was detected initially using an optical microscope (Olympus). The morphology and thickness ofthe flakes were estimated using an atomic force microscope (AFM)(Solver pro SPM). Raman spectroscopic studies were carried outusing a Jobin-Yvon, HR-800 Evolution Raman spectrometer with 633 nm laser and 1800 rulings/mm grating which gives a coverage of 0.35 cm −1per pixel. Spectra were recorded using an objective of 50× (long working distance) with a laser power on the sampleabout 0.4 mW to avoid the heating effect. Temperature dependent measurements from 77 K to 295 K were carried out using a Linkam THMS 600 temperature stage with temperature stability of ±0.5 Kon the sample. Observed Raman modes were fitted to theLorentzian line shape function to obtain the peak positions. Stoichiometry of nanoflakes was estimated by energy disper- sive x-ray (EDX) analysis using Philips XL30ESEM and Oxford X-Max N. X-ray photoelectron spectroscopy (XPS) measurements of the drop cast films of bulk and nanoflakes on glass substrates wereperformed at 4403.7 eV photon energy (synchrotron source) on thePES-BL14 beamline at INDUS II (RRCAT, India). 30Lorentzian fitting of core level spectra was carried out after calibrating with a C1 s peak at 284.6 eV. RESULTS AND DISCUSSION Resonance Raman spectra: Anomalous “b”mode in MOS 2 TMDs can exist in several different structural phases due to different stacking orders of the layers, depending on which theycrystallize in polymorphs 1T, 2H, or 3R structures, which are tet-ragonal, hexagonal, and orthorhombic, respectively. 31While 2H and 3R phases are semiconducting in nature, 1T-MoS 2is metallic. The structure of MoS 2used in the present study was confirmed to be 2H from x-ray diffraction data provided in the supplementary material . The most common structure of MoS 2is hexagonal P63/mmc (point group D4 6h) and it has two S –Mo–S units per unit cell. The factor group predicts 18 normal vibrational modes at ΓJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-2 Published under license by AIP Publishing.point given by the following irreducible representation: Γ¼A1gþ2A2uþ2B2gþB1uþE1gþ2E1uþ2E2gþE2u:(1) Assignments of first order Raman and infrared modes at ambient conditions have been carried out by Wieting andVerble. 32Out of these vibrational modes, only A1g,E1g,a n d E2g modes are Raman active. A1gand E2gare due to the out of plane vibration of S atoms and due to the in plane vibration of both Mo and S atoms, respectively. In the backscattering geometry, the E1g mode is not allowed in the Raman spectra. Thickness dependent Raman spectroscopic studies have established that, while the A1g mode frequency decreases, the E2gmode increases in frequency as the thickness reduces from bulk to monolayer, with most of the changes taking place for a thickness less than five/six layers.33 When the incident photon energy matches the electronic energy gap or any excitonic states, there is a coupling of phonons to elec-tronic transitions, resulting in resonant Raman spectra. Due to the modification of the electronic structure with the layer thick- ness, resonance conditions for both first and second order Ramanscattering are modified with thickness. Due to layer tuned reso-nance enhancement, the intensity of second order Raman modesas well as defect induced first order Raman modes is also reported to be sensitive to the layer thickness below about six layers in MoS 2as well as WSe 2, and it is suggested that it can be utilized for estimating the layer thickness.34The earlier study on the com- parison of resonance Raman and XPS spectra of bulk and exfoli-ated nanosheets of MoS 2reported the nanosheets to be exhibiting mainly metallic 1T character.35They also suggested the bulk and nanosheets of MoS 2to be behaving differently under high pressures. In the present study, the optical image and the corresponding AFM image along with the height profiles of the liquid exfoliated nanoflakes of MoS 2are shown in Fig. 1 . The thickness of thenanoflakes on which Raman spectroscopic investigations are carried out is estimated to be about 20 nm, which is more than about 30 layers of MoS 2. Therefore, the positions as well as the intensity of first order Raman modes are expected to be same asthat of bulk MoS 2.Figures 2(a) –2(c) show the resonant Raman spectra of bulk MoS 2and nanoflakes with first order and higher order Raman peaks. The modes are labeled as per available assign- ments in the literature.28,36The first order modes E2gand A1gin bulk MoS 2are observed at 382 cm−1and 407.3 cm−1, respectively, while they are observed at 385.5 cm−1and 409.5 cm−1, respectively, in nanoflakes. The difference in peak position of E2gand A1gin nanoflakes is around 24.2 cm−1confirming multilayered or bulk like structure of MoS 2flakes.33Other than the symmetry allowed first order Raman modes, Mignuzzi et al. have identified several defect-induced Raman modes arising from zone-edge phonons inthe off resonance Raman spectra. 37Among the defect induced peaks, the relative intensity of the LA(M) mode at around 227 cm−1was reported to be highly sensitive to defect density and was used to quantify defects in monolayer MoS 2. However, in the present data recorded at resonance, the relative intensity of the LA(M) mode at around 227 cm −1with respect to the allowed first order Raman band does not show significant changes from bulk to nanoflakes of MoS 2. Most of the second order bands are assigned as per Chen and Wang.38Remaining higher order modes in Fig. 2 are labeled as per assignments provided in the comprehensive mul- tiphonon analysis by Livneh and Spanier.28In addition to the first and higher order vibrational modes of MoS 2, the low frequency asymmetry of the first order Raman modes is observed whichcould be also defect induced peaks. 39It can be seen that the peak positions of most of the modes in nanoflakes, including that of the first order E2gand A1gmodes, show slight shift to higher frequen- cies compared to that in bulk. The relative intensity of the E2g mode to that of A1gis much smaller in bulk as compared to that in nanoflakes of MoS 2. But, the most prominent difference between Raman spectra of bulk and nanoflakes of MoS 2is the shift in peak FIG. 1. (a) Optical image of MoS 2 nanoflakes and (b) the corresponding AFM image and its height profileacross the yellow line.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-3 Published under license by AIP Publishing.positions of the second order “b”mode, which is assigned to be a combination of LA and TA modes at the K point.28 The “b”mode is observed at around 423.5 cm−1in nanoflakes, while it is observed at 414.8 cm−1in bulk. Sekine et al. have reported dispersive nature of the “b”mode with excitation energy and have shown a decrease in its frequency with increasing excita- tion laser energy.27A recent work by Carvalho et al.40has shown that the frequency of some of the second order features shift with achange in the excitation energy in both bulk and monolayer MoS 2, which is a characteristic of double resonance processes. Theirexperimental work is supported by DFT calculations of double res- onance Raman scattering in monolayer and bulk MoS 2. Among the second order features, the range of 400 –470 cm−1consists of the “b”mode, 2LA(K) around 455 cm−1and 2LA(M) around 465 cm−1 in bulk MoS 2. Both position and line shapes of the modes in this region are wavelength dependent. Among the second order modes, the “b”mode and the 2LA(K) mode have been reported to have large dispersion in both monolayer and bulk MoS 2.40We also find significant changes in the line shape of the modes in this regionbetween the bulk and nanoflakes of MoS 2. There is an increase in the frequency of all the modes in this region, and the relative inten- sity of the 2LA(K) mode with respect to 2LA(M) decreases in nanoflakes compared to bulk MoS 2. However, we focus our atten- tion only on the “b”mode as the 2LA(K) mode appears as a part of the broad cluster of three second order modes. Though the Raman spectra in bulk and the nanoflakes could change because of differ- ent resonant conditions, we attribute the changes in Raman spectraobserved in our bulk and nanoflakes of MoS 2to electron –phonon coupling. This is because our nanoflakes are 20 nm thick havingabout 30 layers of MoS 2. An earlier absorption study has reported measured peak positions of absorption bands of A exciton and B exciton showing layer thickness dependence only below threelayers. 41Thicker flakes were found to have bulk like peak positions of A and B exciton bands. DFT calculations have shown that theoptical properties particularly the bandgap, changes with layer thickness only below eight layers. 42It is reported that when the layer number reaches eight times monolayers, the bandgap isalmost equal to that of bulk 2H-MoS 2. In view of this, we expect our nanoflakes to have bandgap and excitonic resonance same asbulk MoS 2. So, the changes in line shape and position of “b”mode cannot be attributed to different resonant conditions. Furthermore, Kutrowska-Girzycka et al.43found that the “b”mode has unusually large temperature dependence in monolayer MoS 2. Highly disper- sive nature of the “b”mode and its temperature dependence was speculated to depend on its electron concentration.43The harden- ing behavior of this mode observed in nanoflakes in the presentstudy, compared to that in bulk MoS 2, could be due to changes in the charge carrier density in nanoflakes, which could modify theelectron –phonon coupling in the same. Electron –phonon interaction in MoS 2 Electron –phonon interactions in graphene12and monolayer MoS 226have been investigated in earlier studies by following the systematic variation of both Raman mode frequencies and FWHM, for different electron doping brought about by applying varying gate voltage in a field effect transistor configuration. In monolayer FIG. 2. Resonant Raman Spectra of bulk and nanoflakes of MoS 2in the regions (a)100 –300 cm−1(b) 320 –490 cm−1, and (c) 540 –700 cm−1. Note the vibrational mode around 420 cm−1(labeled “b”mode) which appears at different positions in bulk and nanoflakes of MoS 2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-4 Published under license by AIP Publishing.MoS 2, while A1gmode was observed to soften (by ∼4c m−1) and broaden with electron doping (for electron concentration ranging from 0 to 2.0 × 1013/cm2), the E2gmode was not found to be very sensitive to electron doping. Only the A1gmode was found to be sensitive to electron concentration due to its stronger EPC. In thepresent case, the peak positions and FWHM observed for bulk andnanoflakes at ambient conditions are tabulated in Table I . We note an increase in the E 2gmode frequency by about 3.5 cm−1on thin- ning from bulk to nanoflakes, while the A1gmode increases by about 2.2 cm−1(which corresponds to a change in the electron con- centration of 0 –0.5×1013/cm2in monolayer MoS 2).26The FWHM of both the first order modes in bulk MoS 2are also lower than that of nanoflakes which is expected as the bulk MoS 2is expected to have less defects. It is also found that the frequency and FWHMfor the “b”mode in bulk and nanoflakes of MoS 2are distinctly dif- ferent. The “b”mode is broad in bulk, with FWHM 25.5 cm−1as compared to 13.7 cm−1in nanoflakes and lower in frequency in the bulk MoS 2by about 9 cm−1than that in nanoflakes. Contrary to our expectation, we find the “b”mode in bulk MoS 2to be broader compared to that of nanoflakes which could be having moredefects. While the shift in the A 1gmode could be due to electron concentration being different, the E2gmode shows a larger shiftwhich is not explainable as due to electron concentration. The behavior of A1gand E2gmodes show that the effect is not directly due to electron –phonon interaction as it is expected to affect the A1gmode more. In order to investigate the possible reasons for the differ- ences in Raman spectra, temperature dependent Raman spectro- scopic investigations were carried out on both bulk and nanoflakes of MoS 2. An earlier temperature dependent study on monolayer and multilayer MoS 2also has reported differences in temperature dependences of Raman mode frequencies in mono-layer and multilayer MoS 2.44While bulk MoS 2had similar tem- perature dependence for both the first order modes, monolayer MoS 2showed higher temperature dependence for the A1gmode which was attributed to stronger electron –phonon coupling. However, as the experiments were done off resonance, the “b” mode was not observed in the spectra. In the present study, in order to confirm if the different peak position of the “b”mode is due to electron –phonon coupling, we followed the resonance Raman spectra of bulk and nanoflakes of MoS 2in the tempera- ture range of 77 –295 K. Figures 3(a) and 3(b) show Raman spectra at various temperatures from ambient temperature to 77 K for bulk and nanoflakes of MoS 2, respectively. It is seen that all the modes harden with decreasing temperature andbecome narrower for both the cases as expected. Figures 4(a) and 4(b) show temperature dependence of frequencies of both the first order Raman modes as well as the anomalous “b” mode for bulk and the nanoflakes of MoS 2respectively. Similarly, Figs. 4(c) and 4(d) show the corresponding variation of FWHM for bulk and nanoflakes of MoS 2, respectively. It may be noted that fitting errors in FWHM of “b”in bulk MoS 2 are large because of poor intensity. It can be seen that while A1gand E2gmodes exhibit nearly similar temperatureTABLE I. Parameters of Raman spectra of bulk and nanoflakes of MoS 2at ambient conditions. ModesBulk MoS 2 Nanoflake of MoS 2 E2g A1g bE 2g A1g b Frequency (cm−1) 382 407.3 414.8 385.5 409.5 423.5 FWHM (cm−1) 6.0 3.0 25.5 9.5 10.5 13.7 FIG. 3. T emperature dependence of Raman spectra of (a) bulk and (b)nanoflakes of MoS 2in the range of 77 K –295 K.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-5 Published under license by AIP Publishing.dependences for both bulk and nanoflakes, a clear difference is observed in the case of the anomalous “b”mode, which shows much larger temperature dependence for bulk MoS 2.I n addition, the measurement on temperature dependence of FWHM of the two main modes and the anomalous “b”mode[Fig. 4(c) ], revealed the FWHM of the “b”mode in bulk to be increasing more rapidly as compared to that in nanoflakes[Fig. 4(d) ]. The parameters related to the temperature depen- dence of mode frequencies and FWHM for bulk and nanoflakes of MoS 2are tabulated in Table II .F W H Mo faR a m a nm o d ei n FIG. 4. T emperature dependence of Raman mode frequencies in (a) bulk MoS 2(b) nanaoflakes of MoS 2. Variation of FWHM of Raman modes of (c) bulk MoS 2and (d) nanoflakes of MoS 2in the temperature range of 77 –295 K.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-6 Published under license by AIP Publishing.general has contributions from various factors as given below45, FWHM Γ¼Γ0þΓanhþΓEPC, (2) where Γ0is the intrinsic FWHM, Γanhdenotes the increase in FWHM due to phonon –phonon interaction, and ΓEPCis due to electron –phonon interaction. The FWHM in nanoflakes can have additional contributions from defects but bulk MoS 2is expected to be more crystalline. This is the reason for higher FWHM of Agand E2gmodes in nanoflakes which is contained inΓ0as intrinsic FWHM and the “b”mode also is expected to be broader. Here, we have used the approach similar to that ofRef. 46, where temperature dependence of the G mode of graphite is used to extract anharmonic contribution to the FWHM in the G mode of graphene. The difference in tempera-ture dependence of FWHM of Raman modes of graphite andgraphene is used to extract electron –phonon coupling constant in graphene. In the present case, Γ anhis expected to be same in bulk and nanoflakes of MoS 2, as the nanoflakes are 20 nm thick, ΓEPCis the only term expected to be different in the two cases. This is justified because earlier thickness dependentRaman spectroscopic studies on MoS 247have revealed that the effect of layer thickness is seen in the Raman spectra only below six layers. Above that thickness, Raman spectra is similar to that of bulk MoS 2with both FWHM and the peak positions matching with bulk MoS 2. In view of this, we have considered Γanhto be same for bulk and nanoflakes of MoS 2. Therefore, among all the terms, only ΓEPCis expected to be more in bulk MoS 2which could be contributing to additional FWHM of the “b”mode and hence implying stronger EPC in bulk than that in nanoflakes. It may be noted from Figs. 4(b) and 4(d) that temperature dependence of both frequency and FWHM “b” mode of MoS 2nanoflakes show linear variation48in agreement of its assignment as a combination of two acoustic modes dis-cussed earlier. It is interesting to note that the same “b”mode has a larger variation in frequency as well as FWHM in bulkMoS 2as shown in Figs. 4(a) and 4(c). Frequency variation in bulk MoS 2is much larger and could be fitted to a quadratic dependence which is indicative of additional contribution.While linear temperature dependence of phonon frequency innanoflakes of MoS 2indicates three phonon decay process as per its assignment, additional vari ation in bulk must have originated from other than phonons. Figure 4(c) shows additional FWHM that cannot be accounted by phonons, indicating contributionof EPI in bulk MOS 2. Further analyses of the variation of fre- quency and FWHM of the “b”mode of bulk MoS 2to evaluate EPC are not carried out here as the mode is weak, broad, and overlapping with stronger modes.An earlier study on the dependence of Raman mode frequen- cies on sulfur vacancies49demonstrated a drastic decrease in fre- quency, relative intensity, and broadening of the E2gmode with sulfur vacancy generation brought about by electron irradiation and insignificant changes in A1gmode. Some other studies have reported excess electron concentration with an increase in thesulfur vacancy. 50,51In this context, it can be pointed out that the observed changes in Raman modes from nanoflakes to bulk MoS 2 in the present study could be due to the presence of larger sulfurvacancy and hence excess electrons in bulk MoS 2as compared to that in nanoflakes. This could possibly explain the differencebetween first order Raman spectra of bulk and nanoflakes of MoS 2 at room temperature. The low temperature measurements offer a clue that the behavior of the “b”mode could also be influenced by this higher electron concentration in bulk MoS 2. In order to confirm this, as electron concentration measurements in nanoflakesare difficult, we carried out EDX and XPS measurements on bulkand nanoflakes of MoS 2. EDX analysis carried out on bulk and nanoflakes of MoS 2 revealed the Mo:S ratio to be 1:1.93 in bulk and that to be 1:2.2 in nanoflakes,29as depicted in EDX spectra in Fig. 5 . This clearly shows the sulfur content to be lesser in the bulk as compared tothat in the nanoflakes. A higher S:Mo ratio in nanoflakes is also reported in earlier studies. 52,53In order to further confirm this, we also carried out XPS studies on bulk and nanoflakes of MoS 2. Figure 6 shows the Mo and S XPS core level spectra of bulk and nanoflakes of MoS 2. The Mo 3d core level spectrum of bulk MoS 2 powder is deconvoluted into two major peaks at 228.6 and231.9 eV, corresponding to Mo 3d 5/2and 3d 3/2orbitals of the Mo4+ oxidation state of 2H-MoS 2.54,55The peak at 226.0 eV is assigned to the 2 s orbital of sulfur. The broad and weak peak at 234.7 eVcould be attributed to the Mo 6+oxidation state related to MoO 3. The XPS spectrum of S 2p core level is deconvoluted into two peaks at 161.7 and 163.2 eV corresponding to the spin –orbit dou- blets of S 2p 3/2and S 2p 1/2orbitals. In the case of MoS 2nanoflakes, the Mo 3d 5/2and 3d 3/2core level peaks are shifted to 229.6 and 232.8 eV, respectively.29Similarly, the S core level peaks of 2p 3/2 and 2p 1/2orbitals are shifted to 162.7 and 163.8 eV, respectively. It should be noted here that binding energy values of Mo and S ofbulk MoS 2are slightly lower than those reported earlier.51,54This could be attributed to the lower sulfur content (or higher Mo:Sratio) as shown by EDX analysis. 56The S core level peaks of nano- flakes also reveal two small peaks at 164.7 and 167 eV. These can be attributed to higher oxidation states of sulfur,57,58where the higher S:Mo ratio and the oxygen environment provided by thesolvent during synthesis by ultrasonication could be allowing sulfur to form bonds with oxygen atoms. As mentioned before, a lower sulfur content can result in an increased electron concentration inTABLE II. A comparison of temperature dependences of bulk and nanoflakes of MoS 2. Bulk MoS 2 Nanoflake of MoS 2 Modes A1g E2g b Modes A1g E2g b dω/dT (cm−1/K) −0.008 −0.012 0.06 −0.0003 (quadratic fit) d ω/dT (cm−1/K) −0.01 −0.013 −0.026 (linear fit)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-7 Published under license by AIP Publishing.FIG. 5. EDX spectra of (a) bulk and (b) nanoflakes of MoS 2. FIG. 6. XPS core level spectra of (a) Mo 3d for bulk MoS 2, (b) S 2p for bulk MoS 2, (c) Mo 3d for nanoflakes of MoS 2, and (d) S 2p for nanoflakes of MoS 2.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 165703 (2020); doi: 10.1063/5.0015291 128, 165703-8 Published under license by AIP Publishing.MoS 2,50,51and this increased electron concentration could be causing the down shift of Mo core level spectra in bulk MoS 2in the present study.56,59The lower Mo:S ratio of nanoflakes, as shown in EDX spectrum, could be resulting in the shift to higher bindingenergies of the Mo and S core level spectra of the nanoflakes ascompared to those of bulk MoS 2. Higher electron concentration in bulk can be suggested to be responsible for renormalization of the “b”mode frequency. Therefore, the present investigations point out that introducingMo/S vacancy or changing the stoichiometry brings about changesin electron concentration resulting in different EPI. While the first order modes of MoS 2show small changes when the change in the electron concentration is small, these are better manifested in “b” mode characteristics which we propose as a better fingerprint ofestimating electron concentration. An earlier report on calculations of phonon dispersion for bulk and monolayer of MoS 2suggested a reduction in the A1g mode frequency with the addition of electrons.26In the present study on bulk and nanoflakes of liquid exfoliated MoS 2, our results suggest that the “b”mode frequency, which is actually a combina- tion mode at the K point, decreases with an increase in the electron concentration, showing better sensitivity to the same. The present investigations facilitate fine tuning of calculations of phonondispersion relations incorporating electron –phonon interactions. CONCLUSION We have investigated the electron –phonon interaction in bulk and nanoflakes of MoS 2using resonant Raman scattering, which is identified by large shifting of the “b”mode. The electron –phonon interaction is identified to be more in bulk MoS 2as compared to that in nanoflakes synthesized by liquid exfoliation. A change in electron concentration is attributed to different concentrations ofsulfur vacancy in both which is confirmed by EDX measurements.The “b”mode is found to be more sensitive to electron –phonon coupling than the stronger A 1gand E2gmodes and hence can be used to estimate small changes in electron concentration. While the A1gmode softens at the Гpoint in the Brillouin zone, the “b”mode that is assigned to be a combination of LA and TA modes at theK point softens with electron accumulation. SUPPLEMENTARY MATERIAL See the supplementary material for the XRD characterization of the studied bulk and nanoflakes of MoS 2. ACKNOWLEDGMENTS The authors thank Dr. K. G. Girija, Chemistry Division, for AFM data. The authors thank Head, Solid State Physics Divisionfor support and encouragement. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1Y. He, M. Hashimoto, D. Song, S.-D. Chen, J. He, I. M. Vishik, B. Moritz, D.-H. Lee, N. Nagaosa, J. Zaanen, T. P. Devereaux, Y. Yoshida, H. Eisaki, D. H. Lu, and Z.-X. Shen, Science 362, 62 (2018). 2T. Sohier, E. Ponomarev, M. Gibertini, H. Berger, N. Marzari, N. Ubrig, and A. F. Morpurgo, Phys. Rev. X 9, 031019 (2019). 3A. V. Lugovskoi, M. I. Katsnelson, and A. N. Rudenko, Phys. Rev. B 99, 064513 (2019). 4H. Wang, E. Lhuillier, Q. Yu, A. Mottaghizadeh, C. Ulysse, A. Zimmers, A. Descamps-Mandine, B. Dubertret, and H. Aubin, Phys. Rev. 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5.0017351.pdf

AIP Conference Proceedings 2265 , 030312 (2020); https://doi.org/10.1063/5.0017351 2265 , 030312 © 2020 Author(s).Study of scandium nitride thin films deposited using ion beam sputtering Cite as: AIP Conference Proceedings 2265 , 030312 (2020); https://doi.org/10.1063/5.0017351 Published Online: 05 November 2020 Susmita Chowdhury , Rachana Gupta , Shashi Prakash , Layanta Behera , D. M. Phase , and Mukul Gupta ARTICLES YOU MAY BE INTERESTED IN Synthesis of Nb 2N by rapid thermal annealing of interstitial Nb(N) thin film AIP Conference Proceedings 2265 , 030257 (2020); https://doi.org/10.1063/5.0016619 Electrical and optical properties of scandium nitride nanolayers on MgO (100) substrate AIP Advances 9, 015317 (2019); https://doi.org/10.1063/1.5056245 Interfacial chemistry and electronic structure of epitaxial lattice-matched TiN/Al 0.72 Sc0.28 N metal/semiconductor superlattices determined with soft x-ray scattering Journal of Vacuum Science & Technology A 38, 053201 (2020); https:// doi.org/10.1116/6.0000180Study of Scandium Nitride Thin Films Deposited using Ion Beam Sputtering Susmita Chowdhury1, Rachana Gupta1, Shashi Prakash1, Layanta Behera2, D. M. Phase2 and Mukul Gupta2* 1Institute of Engineering & Technology, DAVV, Khandwa Road, Indore, 452017, India 2UGC-DAE Consortium for Scientific Research, Khandwa Road, Indore, 452001, India *Corresponding author: mgupta@csr.res.in Abstract. In this work, ion beam sputtering technique has been utilized t o deposit Sc and ScN x thin film samples. For growth of ScN x samples, N 2 partial pressure (PN 2) was varied in small intervals between 5 ൈ10-6 to 1.0ൈ10-4 Torr. X-ray reflectivity, x-ray diffraction, x-ray near edge absorption spectroscopy (at N K an d Sc L 3,2 edges) and resistivity measurements were performed to characterize the samples. We found that samples deposited below PN 2 = 2.5ൈ10-5 Torr were under stoichiometric but when PN 2 increases to 5 ൈ10-5 Torr or beyond, they become over stoichiometric. This reveals that the formation of ScN phase (at 300 K) requires a very precise control of PN 2 to achieve stoichiometric ScN. INTRODUCTION Scandium nitride (ScN) is a transition metal nitride (TMN) that exhibits refractory properties that are common to early TMNs e.g. ScN has high electron mobility, high hardness ( 23 GPa), high melting temperature (> 2873 K) and corrosion resistance [1,2]. Generally, the resistivity ( ) of early TMNs is low – almost metal like (e.g. of ScN 0.1 to 16 m cm [1]), at the same time ScN exhibits a band gap similar to s emiconductors [1-3]. This makes ScN a metallic semiconductor. The rocksalt type structure of ScN is b oon as it provides epitaxy with some III-V semiconductor (e.g. GaN) realizing the possibility of semicondu ctor heterostructures [3]. Thin films of ScN have been grown using various techniques such as molecular-beam epit axy [4], plasma assisted physical vapor deposition [5], hybrid vapor phase epitaxy [6], reactive dc magnetron sput tering [7]. Generally, stoichiometric ScN films are formed at high substrate temperatures (Ts) 1000 K [1]. Such high Ts is not desirable for application of S cN in semiconductor heterostructures due to interdiffusion at the int erfaces. In order to suppress such interdiffusion, it is desirable to grow ScN films at low Ts. In this scenario, a depo sition technique with high adatom energy is preferred so as to enhance the adatom mobility. Ion beam sputtering (IBS) is one such technique where ion energy can be varied [8]. In this work, we attempted room temperature growth of ScN thin films using IBS and obtained results are presented and discussed. EXPERIMENTAL SECTION Pure scandium (Sc) and a series of ScN x t h i n f i l m s w e r e d e p o s i t e d o n s i n g l e c r y s t a l s i l i c o n ( 1 0 0 ) w a f e rs and amorphous quartz using a home-made radio frequency (rf) ion bea m sputtering (rf-IBS) at room temperature (300K, no intentional heating). A Veeco 3 cm rf (13.56 MHz) ion source was used to ionize the argon (Ar) gas (99.9995% pure) and the resultant beam was incident at an angle of 45° on the Sc (purity 99.9%) target (10 ൈ10 cm2). The argon (Ar) ion energy used was 1 keV at 40 mA. Firstly, a pure Sc fil m was deposited and subsequently, N 2 f l o w (99.999% pure) was introduced at different pressures (using a U HV leak valve) for the deposition of ScN x thin films. Starting with a base pressure of about 1 ൈ10-7 Torr, N 2 partial pressure (PN 2) was varied from 5 ൈ10-6 to 1ൈ10-4 Torr as shown in Table 1. The N 2 gas flow was exposed in the deposition chamber in such a way t hat it floods the substrate and get flushed out immediately with the turbo molecu lar pump (2000 l min-1) placed beneath the UHV leak valve. The phase formation of samples was confirmed by x-r ay diffraction (XRD) measurements (using Cu K DAE Solid State Physics Symposium 2019 AIP Conf. Proc. 2265, 030312-1–030312-4; https://doi.org/10.1063/5.0017351 Published by AIP Publishing. 978-0-7354-2025-0/$30.00030312-1x-rays) and the dispersive part of scattering length density (S LD), thickness and roughness were obtained from x-ray reflectivity (XRR). The local electronic structure at N K-edge and Sc L 3,2 edges were probed using soft x-ray absorption near edge spectroscopy (XANES) at BL-01 beamline at Indus-2, RRCAT, Indore, India [9]. XANES measurements were performed in total electron yield mode and th e pressure during measurements was about 1ൈ10-8 Torr. Resistivity measurements were performed using four probe method at room temperature. RESULTS AND DISCUSSIONS Figure 1(a) shows the XRR patterns of the as-deposited Sc and S cNx samples at various PN 2. They were fitted using Parratt 32 software based on Parrat’s formalism [10] and the fitted parameters are listed in Table 1. As can be seen from Table 1, the presence of a surface oxide layer was ob served in all samples but its contribution reduces with a gradual increase in PN 2. Sc is prone to get oxidized easily as the enthalpy of formati on (∆H) of scandium oxide (Sc 2O3) is extremely small at about -1909 kJ/mol [11], whereas that o f ScN is relatively high at -318 kJ/mol [12,13]. Therefore, not only surface oxidation of Sc takes plac e on exposure to atmosphere, but also during the deposition. This is evident from the fact that the dispersive p art of SLD of Sc comes out to be 2.79 ൈ10-5 Å-2 which is about 15% higher than its bulk value of 2.41 ൈ10-5 Å-2 [11]. It may be noted that the SLD of Sc 2O3 is 3.18ൈ10-5Å-2 [11], and since our obtained value is higher than that for bulk Sc, it signifies that some oxidation of Sc has already taken place during the growth itself. 0.04 0.06 0.08 0.10 0.12 0.14 Sc(a) ScN-1 ScN-2 ScN-3 ScN-4 ScN-5 qz (Å-1)X-ray reflectivity (arb. units) Open diamond = Experimental Solid line = Fitted 30 35 40 45 5 0ScOx/Sc2O3-x ScNSc (111)(002) ScN-1 Sc(b)(100) ScN-2 (200) ScN-4ScN-3 ScN-5 (Degree)Intensity (arb. units) FIGURE 1. X-ray reflectivity (a) and x-ray diffraction (b) patterns of Sc and ScN x thin films measured using Cu K x-rays, deposited at various PN 2 = 0 (Sc), 5 ൈ10-6 (ScN-1), 1 ൈ10-5 (ScN-2), 2.5 ൈ10-5 (ScN-3), 5 ൈ10-5 (ScN-4), and 1 ൈ10-4 Torr (ScN-5). Here, it is worth mentioning that the O 2 base pressure in our chamber (measured using a residual gas an alyzer) was about 7.5 ൈ10−10 Torr [14], and it seems that it is high enough to cause partial oxidation of Sc. However, as N 2 is introduced, it seems to prevent such oxidation process to a lar ge extent. For the ScN x films, when the substrates are TABLE 1. XRR and resistivity parameters of ScN x thin films deposited at various N 2 flow. Sample PNଶ (Torr) Thickness (2 Å) Oxide layer thickness ( 2 Å) Roughness (1 Å) SLD (േ0.02ൈ10-5 Å-2) Resistivity (േ0.1 mΩ cm) Sc 0 806 43 12 2.79ൈ10-5 0.4 ScN-1 5ൈ10-6 1195 42 14 2.86ൈ10-5 0.3 ScN-2 1ൈ10-5 822 41 13 2.88ൈ10-5 0.5 ScN-3 2.5ൈ10-5 805 25 12 3.50ൈ10-5 0.4 ScN-4 5ൈ10-5 742 20 14 3.78ൈ10-5 0.5 ScN-5 1ൈ10-4 628 15 16 3.81ൈ10-5 1.0 030312-2exposed with N 2 gas, it can be seen (from Fig. 1(a)) that the critical q z shifts to higher values due to an increase in SLD. As can be seen from Table 1, the increase in SLD is margin al upto ScN-2 sample, but it abruptly changes from the ScN-3 samples approaching to bulk SLD of stoichiometric ScN (3.49ൈ10-5Å-2) [15]. For ScN-4 and ScN-5 sample, the SLD further increase even beyond bulk ScN value. It may be noted that determination of SLD from XRR measurements can have errors of about 5% (much larger than those obtained by fitting procedure as shown in Table 1) due to alignment of samples [16]. Hence, within error bars, it can be said that samples deposited above PN 2= 2.5ൈ10-5 Torr are suitable for the formation of ScN x films. It is to be mentioned that the surface roughness of all the samples was between 12 to 16 Å which is marginally high er than the substrate roughness of about 12 Å. Figure 1(b) shows the XRD pattern of Sc and ScN x samples deposited at various N 2 flows. XRD pattern of Sc shows prominent peaks appearing at 2 = 30.87° and 33.64° and can be assigned to (100) and (002) pla ne of hexagonal close packed (hcp) Sc. A faint feature at 2 = 35.2° may arise due to surface oxidation of Sc as evident fr om XRR results described above. It may be noted that we noticed an ove rall shift of about 0.3° in our XRD pattern (as compared to bulk Sc, JCPDF-655711), which may be due to due to partial oxidation of whole Sc film or some stress produced during the growth of film. 395 400 405 410 415fc b (arb. units) Sc Energy (eV)ea d(a) ScN-1 ScN-3ScN-2 ScN-4 ScN-5 395 400 405 410 415ScN-3ScN-2ScN-1edc b a Sc(b) f h g ScN-4 ScN-5 Energy (eV) ddE FIGURE 2. Normalised XANES spectra (a) and their first order derivatives (b) of Sc L-edge and N K-edge of as deposited Sc and ScN x thin films at different N 2 flows. After introducing N 2 a t a v e r y s m a l l a m o u n t ( 5 ൈ10-6 T o r r ) , t h e X R D p a t t e r n o f S c N - 1 s a m p l e s s h o w a broadening of peak width and reduction in their intensity. This may happen as N occupies interstitial sites within hcp Sc leading to nanocrystallization or amorphization which genera lly happens in metal nitrides [17]. This becomes even more pronounced as the amount of N 2 is raised to 1 ൈ10-5 (ScN-2) or 2.5 ൈ10-5 Torr (ScN-3). Although between them, the structure seems to transform from amorphous Sc like t o amorphous ScN like. On further increasing PN 2 to 5ൈ10-5 Torr (ScN-4), prominent reflections of ScN phase can be seen a t 2 = 34.12° and 39.32°corresponding to (111) and (200) planes of ScN with a rock-salt (RS) type struct ure [18]. The average lattice parameter (LP) for ScN-4 sample comes out to be 4.56 Å, which is somewhat larger t han the value reported for stoichiometric ScN at 4.52 Å [3]. We note that for the ScN-5 sample (PN 2 = 1ൈ10-4 Torr), the average LP increases further to 4.60 Å. Therefore, it appears that formation of a stoichiometric ScN mi ght take place between PN 2 = 2.5 to 5.0×10-5 Torr and both ScN-4 and ScN-5 films are somewhat over stoichiometric with N concentration. In order to get an insight of about the electronic structure ou r samples, we did N K and Sc L 3,2 edge XANES measurements as shown in Fig. 2(a). It may be noted that these edge features appear very close to each other and therefore they may overlap and create a confusion (e.g. edge ju mp energy is at 401.6 eV for N K-edge; 402.2 eV for Sc L 3 edges) [19]. Nevertheless, the energy difference is much large r than the resolution ( 0 . 3 e V ) a n d b y systematically comparing Sc and ScN x samples, information about element specific hybridized electro nic states can be obtained. Figure 2(b) shows the derivative of XANES spectra of Sc and ScN x thin film samples. In case of Sc samples, prominent features are assigned as ‘a’, ‘b’, ‘c’ and ‘ d’ and they correspond to Sc L 3(t2g), L 3(eg), L 2(t2g) and L2(eg) edges, respectively. The splitting in the L 3 and L 2 edges can be understood due to surface oxidation of Sc 030312-3(also evident from XRR and XRD results), leading to hybridizati on between O 2p and Sc 3d states. The spin-orbit splitting between the L 3(t2g) and L 2(t2g) edges was found to be about 4.4 eV which matches well with th e reported value of 4.5eV [7,19]. The crystal field splitting (10Dq) also matches well with the reported value of 1.8 eV observed in Sc 2O3 [20]. Such well-resolved crystal field splitting can also be s een in ScN-1, ScN-2 and ScN-3 samples, signifying surface oxidation of these samples. However , in ScN-4 and ScN-5 samples splitting of Sc L 3,2 becomes weak. In addition, features ‘e’ and ‘f’ can also be see n in these samples. They evolved due to the N 2p and Sc 3d hybridization leading to 10Dq = 2.8 eV which is higher th an the reported value of 2.1 eV for stoichiometric ScN [7]. The additional features ‘g’ and ‘h’ prominent in Fig. 2(b) may be due to the N-K edge features common in TMNs [20] but it needs further analysis. Our XANES results are i n a g r e e m e n t w i t h X R R a n d X R D r e s u l t s . I t appears that in samples ScN-1 to ScN-3, under-stoichiometric ScN x (x < 1, i.e. Sc rich) phas e is formed but in ScN-4 and ScN-5, it becomes somewhat over-stoichiometric ScN x (x > 1, i.e. N rich). As mentioned already, we did resistivity measurements using fou r probe method on samples deposited on (insulating) quartz substrates. The resistivity data of all sam ples is included in Table 1. As can be seen there, resistivity of Sc in our case is quite high at 0.4 m cm than its bulk value of 0.05 m cm [21]. This may happen due to partial oxidation of Sc. In case ScN-1 to ScN-4 samples, the resistivity is 0.4 m cm but in ScN-5 sam ple, it increases to about 1 m cm. A large variation can be seen in the resistivity of ScN typically between 0.1 to 2 m cm [1] and obviously it is not sensitive to the stoichiometry o f ScN. The values in our samples match with typical values for ScN, exhibiting the semi-conducting natures of our s amples. CONCLUSIONS In conclusions, we prepared and studied Sc and ScN x thin films deposited using ion beam sputtering at different N 2 partial pressures. It was found that Sc and N poorer ScN x phases (x < 1) not only show prominent surface oxidation but also get partially oxidized during the growth itself. Howev er, on increasing the N 2 flow, such oxidation becomes significantly weak. Our results obtained from XRR, XRD and XANES measurements confirm this behavior. All our ScN x samples exhibited low roughness and are semiconducting. ACKNOWLEDGEMENTS SC and RG are grateful to UGC-DAE CSR, Indore for providing fin ancial support through CSR-IC-BL-62/CSR- 179-2016-17/843 project. Thanks are due to R. Rawat, V. R. Reddy, D. 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5.0018581.pdf

J. Chem. Phys. 153, 124903 (2020); https://doi.org/10.1063/5.0018581 153, 124903 © 2020 Author(s).Photophysics of graphene quantum dot assemblies with axially coordinated cobaloxime catalysts Cite as: J. Chem. Phys. 153, 124903 (2020); https://doi.org/10.1063/5.0018581 Submitted: 15 June 2020 . Accepted: 04 September 2020 . Published Online: 28 September 2020 Varun Singh , Nikita Gupta , George N. Hargenrader , Erik J. Askins , Andrew J. S. Valentine , Gaurav Kumar , Michael W. Mara , Neeraj Agarwal , Xiaosong Li , Lin X. Chen , Amy A. Cordones , and Ksenija D. Glusac COLLECTIONS Paper published as part of the 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 Tuning the charge transfer character of the multiexciton state in singlet fission The Journal of Chemical Physics 153, 094302 (2020); https://doi.org/10.1063/5.0017919 Relativistic two-component projection-based quantum embedding for open-shell systems The Journal of Chemical Physics 153, 094113 (2020); https://doi.org/10.1063/5.0012433The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Photophysics of graphene quantum dot assemblies with axially coordinated cobaloxime catalysts Cite as: J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 Submitted: 15 June 2020 •Accepted: 4 September 2020 • Published Online: 28 September 2020 Varun Singh,1,2 Nikita Gupta,1,2 George N. Hargenrader,1,2Erik J. Askins,1,2 Andrew J. S. Valentine,3 Gaurav Kumar,4 Michael W. Mara,2,5 Neeraj Agarwal,6 Xiaosong Li,3 Lin X. Chen,2,5 Amy A. Cordones,4 and Ksenija D. Glusac1,2,a) AFFILIATIONS 1Department of Chemistry, University of Illinois at Chicago, 845 W Taylor Street, Chicago, Illinois 60607, USA 2Chemical Sciences and Engineering Division, Argonne National Laboratory, 9700 Cass Ave., Lemont, Illinois 60439, USA 3Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, USA 4Stanford PULSE Institute, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA 5Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA 6School of Chemical Sciences, UM DAE Centre for Excellence in Basic Sciences, University of Mumbai, Kalina, Santacruz (E), Mumbai 400098, India Note: This paper is part of the JCP Special Topic on 65 Years of Electron Transfer. a)Author to whom correspondence should be addressed: glusac@uic.edu ABSTRACT We report a study of chromophore-catalyst assemblies composed of light harvesting hexabenzocoronene (HBC) chromophores axially coor- dinated to two cobaloxime complexes. The chromophore-catalyst assemblies were prepared using bottom-up synthetic methodology and characterized using solid-state NMR, IR, and x-ray absorption spectroscopy. Detailed steady-state and time-resolved laser spectroscopy was utilized to identify the photophysical properties of the assemblies, coupled with time-dependent DFT calculations to characterize the relevant excited states. The HBC chromophores tend to assemble into aggregates that exhibit high exciton diffusion length ( D= 18.5 molecule2/ps), indicating that over 50 chromophores can be sampled within their excited state lifetime. We find that the axial coordination of cobaloximes leads to a significant reduction in the excited state lifetime of the HBC moiety, and this finding was discussed in terms of possible electron and energy transfer pathways. By comparing the experimental quenching rate constant (1.0 ×109s−1) with the rate constant estimates for Marcus electron transfer (5.7 ×108s−1) and Förster/Dexter energy transfers (8.1 ×106s−1and 1.0×1010s−1), we conclude that both Dexter energy and Marcus electron transfer process are possible deactivation pathways in CoQD-A. No charge transfer or energy transfer intermediate was detected in transient absorption spectroscopy, indicating fast, subpicosecond return to the ground state. These results provide important insights into the factors that control the photophysical properties of photocatalytic chromophore-catalyst assemblies. Published under license by AIP Publishing. https://doi.org/10.1063/5.0018581 .,s I. INTRODUCTION Due to their chemical tunability and modular design, molec- ular photocatalysts that combine light-absorbing chromophores with metal-based electrocatalysts are extensively utilized in solar energy conversion.1–5Time-resolved studies of energy and electronmigration in these chromophore-catalyst dyads provide fundamen- tal mechanistic insights into the factors that control overall quantum efficiencies of molecular photocatalysts. For example, the mecha- nism of electrocatalysis by cobalt-based electrocatalysts has been extensively studied using time-resolved optical and x-ray spectro- scopies.6–13Seminal work by Gray and Dempsey has resulted in J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the identification of the key Co(I) and Co(III)-hydride intermedi- ates formed upon reduction or protonation of cobaloxime-based complexes and in the evaluation of the kinetic barriers associated with the competing heterolytic and homolytic catalytic pathways.6,7 Subsequent experiments probing the Co K-edge transitions in the x-ray region have resulted in the enhanced structural informa- tion regarding the reduced Co(II) and Co(I) species in cobaloxime, cobalt polypyridyl, and Co macrocyclic complexes.10,12These stud- ies have shown that the Co(I) intermediate loses axial ligands and adopts a square planar geometry. The dissociated axial ligand, when held covalently to the complex, was proposed to serve as a proton relay for the hydrogen evolution reaction.12Time-resolved studies are also exceptionally useful in identifying the kinetics of photoinduced charge separation and undesired charge recombi- nation in chromophore-catalyst assemblies.14–18In some of these reports, short lifetimes for charge-separated species were observed and assigned to the fast charge recombination. Mulfort found that the axial/equatorial chromophore/catalyst geometry does not sig- nificantly change the rate of charge recombination.14,19Wasielewski showed that this charge recombination occurs in competition with the cobaloxime dissociation from the chromophore assembly due to the loss of the axial pyridine ligand.18A detailed understanding of structural and electronic factors that control recombination kinetics is still lacking. Graphene quantum dots (GQDs), defined as two-dimensional polyaromatic hydrocarbon flakes of varying size, have emerged aspromising chromophores for light harvesting applications. The opti- cal bandgap of GQDs can be readily tuned either by changing the size of the aromatic flake, from 6 eV in benzene to 0 eV in graphene,20 by introduction of functional groups to GQD edges,21or by intro- duction of twist to the aromatic plane.22Recent developments in the bottom-up synthesis of well-defined GQDs23and their self-assembly into one-dimensional wires with large aspect ratios24–26have enabled the development of materials with excellent photoconductivity. Fur- thermore, the successful implementation of GQDs into the walls of metal–organic frameworks27and branched macromolecules28has provided a pathway toward three-dimensional GQD materials for light harvesting applications. The coupling of GQDs with other light harvesting elements, porphyrins, has shown that the directional- ity of energy transfer can be changed by varying the size of the aromatic GQD core.29Preliminary studies involving the coupling of GQDs with the electrocatalytic metal center, Re-based catalyst for reduction of CO 2to CO, have shown that the photoinduced charge transfer to the metal can take place, leading to photocatalytic performance.30,31 Previously, we investigated the electrochemical and photo- physical properties of GQDs.32,33Light harvesting characteristics of GQD assemblies, such as alkyl-substituted hexabenzocoronene (HBC), were evaluated by determining the exciton size and dynam- ics using ultrafast pump–probe spectroscopy.32The exciton coher- ence length was found to be 1–2 monomeric units, indicating a high degree of static and dynamic disorder in these assemblies. However, SCHEME 1 . Structures of cobalt quantum dots and model compounds. J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the exciton diffusion length measurements, associated with an inco- herent hopping process, showed that up to 30 GQD molecules in a one-dimensional π-stacked assembly can be sampled within the exciton lifetime. In this work, we build on these studies by investi- gating the kinetics of photoinduced charge separation and recom- bination in chromophore-catalyst assemblies shown in Scheme 1. Being well-studied catalytic motifs, cobaloxime-based hydrogen- evolving catalysts Co-A and Co-B were selected for this study. Light-harvesting HBC chromophores were functionalized with pyri- dine moieties (HBCPy) to enable axial coordination of cobaloximes CoQD-A and CoQD-B. Using time-resolved optical and mid-IR spectroscopy, we show that the photogenerated excitons, initially centered on the HBC moiety, undergo a fast decay. However, the absence of spectroscopic signatures for charge or energy intermedi- ates in our data prevented us from identifying whether it is the elec- tron or energy transfer that resulted in the excited-state quenching. The results are discussed using Förster/Dexter models for energy transfer and the Marcus model for electron transfer. These results point to the challenges associated with obtaining long-lived charge- separated states in chromophore-catalyst assemblies. II. EXPERIMENT AND THEORY Model compounds were synthesized as described in the supplementary material. Chemicals and solvents were purchased from Sigma-Aldrich and were used without further purifications. Solution state1H and13C NMR were recorded on a Bruker DPX400 spectrometer operating at 400 MHz for1H and 100 MHz for13C. Solid-state NMR spectra were recorded on a Bruker DRX700 spec- trometer operating at 176.0 MHz for13C and 224.0 MHz for11B with a spinning speed of 30 kHz. MALDI was recorded on a Bruker ultrafleXtreme MALDI-Tof-Tof. UV–vis spectra were recorded on either a Cary 300 Bio spectrometer or an Ocean FX spectrophotome- ter. Fluorescence spectra, fluorescence lifetimes, and quantum yields were recorded on a Horiba PTI QuantaMaster 8000 spectrometer. IR measurements were performed on a Thermo Scientific Nicolet iS5 FTIR spectrometer. XPS measurements were done on a Kratos AXIS-165. Co L-edge and N K-edge x-ray absorption spectroscopy (XAS) measurements were done at beamline 8-2 at the Stanford Syn- chrotron Radiation Lightsource (SSRL). The powder samples were mounted on conductive carbon tape, and the XAS spectra were mea- sured by scanning the incident x-ray energy and detecting the total electron yield, at room temperature in vacuum. The spectra are nor- malized to the incident x-ray flux measured before the sample and represent an average of 4–9 measurements per sample. The spheri- cal grating monochromator resolution was ∼0.2 eV–0.3 eV, and the energy was calibrated to the L 3-edge peak of NiF 2at 852.7 eV and its second harmonic at 426.35 eV. A second order polynomial baseline was subtracted from all spectra, which were subsequently normal- ized to the average intensity in the region >810 eV (up to 878 eV) at the Co L-edge or to the peak intensity in the range of 405 eV–420 eV at the N K-edge. The cobalt K-edge x-ray absorption near edge structure (XANES) and extended x-ray absorption fine structure (EXAFS) were collected at beamline 12-BM of the Advanced Photon Source of Argonne National Laboratory. Si(111) crystals were used as themonochromator. Powder samples were mixed with boron nitride to fulfill the transmission detection requirements with the x-ray atten- uation of 0.1–1 at the Co K-edge. FK102 Co(III) TFSI salt {tris (2-(1H-pyrazol-1-yl)pyridine)cobalt(III) tri[bis(trifluoromethane) sulfonimide]} was used as a reference sample for the oxidation state of cobalt. Energy calibration was performed using a Co foil behind the sample to collect transmitted x-ray signals after the sam- ple. XANES/EXAFS data analysis was performed with the Athena and Artemis packages based on IFEFFIT and FEFF programs. The- oretical models were constructed using Gaussian; the theoretical EXAFS spectra were constructed using FEFF and were fit to the experimental data using Artemis.34,35 All electrochemical measurements were carried out under an argon atmosphere at room temperature in a three-electrode cell consisting of glassy carbon as a working electrode, an auxiliary platinum wire as a counter electrode, and Ag/AgCl as a refer- ence electrode. Cyclic voltammograms were recorded on a GAMRY Instrument Interface 1010E. Solute concentrations were 1.0 mM for the cobaloxime (Co-A and Co-B) and 0.1M for the supporting electrolyte, tetrabutylammonium perchlorate (TBAP). For HBCPy, CoQD-A, and CoQD-B, thin films of the compounds were prepared on the glassy carbon electrode using 1 μl of 1 mM solution in chlo- roform. The thin films were coated with 2 μl of 10% Nafion solution in ethanol and were dried under argon before starting the electro- chemical measurements. The scan rate for all measurements was 100 mV/s. The setup used to perform UV–vis transient absorption mea- surements was described previously.36In brief, 800 nm laser pulses of 100 fs duration were produced at 1 kHz repetition rate by a mode- locked Ti:sapphire laser and regenerative amplifier (Astrella, Coher- ent Inc.). The output from the Astrella was split into pump and probe beams. The pump beam was sent into a BBO crystal to dou- ble the excitation pulse photon energy to 400 nm. The probe beam was focused into a 4 mm CaF 2crystal that was continuously trans- lated with a linear stage (Newport MFA-CC) to generate the white light continuum between 350 nm and 750 nm, which was focused into the sample. Care was taken to ensure that the CaF 2crystal axis matched the polarization of the incident light to avoid any wave- length polarization dependence. Thin film samples were excited by 400 nm light at pump intensities between 200 nJ and 1200 nJ per pulse in a nitrogen-purged 1 cm quartz cuvette, and a high angle of incidence between pump and probe (15○) was used to reduce pump scattering. Thin film measurements were performed at magic angle polarization between pump and probe. After passing through the sample, the probe continuum was coupled into an optical fiber and input into a CCD spectrograph (Ocean Optics, Flame-S–UV–VIS– ES). The data acquisition was achieved using in-house LabVIEW (National Instruments) software routines. The group velocity dis- persion of the probing pulse was determined using nonresonant optical Kerr effect measurements. Time-resolved mid-infrared absorption spectroscopy (TRIR) was performed using 800 nm laser pulses of 100 fs duration, pro- duced at 1 kHz repetition rate by a mode-locked Ti:sapphire laser and regenerative amplifier (Astrella, Coherent Inc.). The output from the Astrella was split into pump and probe beams. The pump beam was sent into a BBO crystal to double the excitation pulse pho- ton energy to 400 nm, and every other pulse was blocked using a mechanical chopper wheel synchronized to the pulses. The probe J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp beam was sent to an optical parametric amplifier (TOPAS-C, Light Conversion) to obtain the mid-IR probe pulse, which was split into reference and probe beams. The probe was focused into the sample. The films were excited at 600 nJ per pulse in a nitrogen purged custom sample holder with 4 mm CaF 2glass windows. Thin film measurements were performed at magic angle polariza- tion between pump and probe. After passing through the sample, the probe along with the reference was refocused into a Czerny– Turner spectrograph (Chromex Inc.), which separated the mid-IR colors in space and refocused the light onto two mercury cad- mium telluride detector arrays of 32 pixels each (MTC-14-2 ×32, Infrared Systems Development Corporation) for the probe and ref- erence beams. The data acquisition was achieved using in-house LabVIEW (National Instruments) software routines. Compressed air was filtered (PCRSBX1A64-FM, Puregas) to be water and CO 2 free, and flowed into the instrument area that was completely enclosed. Calculations for nitrogen K-edge XAS were performed using GAUSSIAN 16.37The 6-31g(d) basis set was employed for all light atoms, while the LanL2DZ effective core potential and associated basis set38were used for Co atoms. Density functional theory (DFT) calculations were performed with the CAM-B3LYP functional.39 This functional was chosen because it is range-separated, to bet- ter describe charge-transfer excited states.40The IEFPCM41solva- tion model for dichloromethane was used for all calculations. The dodecyl side chains of the GQDs were truncated to propyl groups. Energetic minima found by ground-state geometry optimizations were confirmed by normal mode calculations at the same level of theory. Excited states were evaluated using time-dependent density functional theory (TDDFT). Calculations related to theoretical IR and UV–vis absorption were performed using the Gaussian 09 software42package with the resources of the Extreme cluster at the University of Illinois at Chicago. Structure optimization and frequency calculations were performed using the B3LYP43,44hybrid function and 6-31g(d,p) basis set for carbon, hydrogen, nitrogen, oxygen, boron, fluorine, and chlorine atoms, and the LanL2DZ basis set for cobalt. The sol- vation method used was the IEFPCM41model for dichloromethane (DCM). Methyl groups were used instead of dodecyl chains to save computational time. Vibrational frequency analysis was done to show the absence of imaginary frequencies. A scaling factor of 0.966 was used in frequency calculations for the 6-31g(d,p) basis set.45 III. RESULTS AND DISCUSSION A. Synthesis and characterization The synthesis of HBCPy was performed starting with dibro- mohexaphenylbenzene (compound 1, Scheme S1, supplementary material), which was prepared via a Diels–Alder addition/ decarbonylation sequence developed by Ito and co-workers.46Sub- sequent Suzuki coupling47between compound 1 and a pyridine- based boronic ester yielded a soluble polyphenyl derivative, which was further subjected to oxidative dehydrogenation48to yield HBCPy (Scheme S1, supplementary material). Two types of cobaloximes were coordinated axially to HBCPy using the approach developed previously for pyridine based complexesCo-A49and Co-B.50These reactions yielded disubstituted CoQD-A and CoQD-B derivatives. Due to the low solubility of GQDs shown in Scheme 1, we were not able to characterize these compounds using standard solution- based techniques. The solid-state13C NMR of HBCPy reveals two broad peaks associated with the aliphatic (34 ppm) and aromatic (120 ppm) carbons (Fig. S1, supplementary material). Coordination with cobaloximes is supported by the appearance of an additional peak at 155 ppm, associated with the C = =N group of the oxime ligands. Similarly, solid-state11B NMR spectra identified the pres- ence of oxime-based boron atoms, consistent with the CoQD-B structure (Figs. S2 and S3, supplementary material). Mass spectrom- etry was used successfully to characterize HBCPy, but molecular masses of CoQD-A and CoQD-B were not observed, likely due to the labile bond between HBCPy and cobalt.51Infrared (IR) spec- troscopy was combined with density functional theory calculations to identify key functional groups in model compounds (Sec. S2.2, supplementary material). The C = =C stretching region of HBCPy consists of modes arising from the aromatic HBC core (1377 cm−1) and the pyridine moiety (1595 cm−1) in HBCPy. Upon coordina- tion with cobaloximes, the pyridine-based C = =C stretch shifts from 1595 cm−1to 1616 cm−1and 1613 cm−1for CoQD-A and CoQD-B, respectively (Figs. S4 and S5). In addition, CoQD-A and CoQD-B exhibit C= =N stretching modes arising from the equatorial oxime ligands (at 1571 cm−1and 1624 cm−1for CoQD-A and CoQD-B, respectively). Additional characterization of GQDs was performed using core spectroscopies, namely, x-ray photoelectron spectroscopy (XPS) and x-ray absorption spectroscopy (XAS). Due to the small concentra- tion of N elements in the sample, the XPS N1s region of HBCPy contains a very weak pyridinic peak at 398.7 eV (Fig. S9). Once coordinated with Co, the N1s region exhibits more intense peaks with binding energies of 398.5 eV and 399.1 eV (in CoQD-A and CoQD-B, respectively). These peaks are associated with N-atoms of equatorial oxime ligands and confirm that the coordination of Co to GQDs took place. Similarly, the Co2p XPS region of CoQD-A and CoQD-B contains two peaks associated with p 1/2and p 3/2peaks of the cobaloxime moiety. The N1s and Co2p peaks of CoQD-A and CoQD-B are shifted to slightly lower binding energies rela- tive to the model compounds Co-A and Co-B (Figs. S9 and S10), indicating lower electron density on N and Co atoms in GQD samples. The attachment of the Co(DMG) 2complex to HBCPy was con- firmed using N K-edge XAS, combined with electronic structure calculations, as shown in Fig. 1. The pyridyl N-atoms of HBCPy have an absorption peak at 397.7 eV. Upon Co binding, the pyridyl N transitions shift to higher energy and overlap with the stronger dimethylglyoximate N peak, consistent with the depletion of the electron density in pyridinic N upon coordination. Cobaloxime N does not change significantly, as can be observed by comparing Co(DMG) 2Cl2and CoQD-A. The lack of a peak at 397.7 eV in the CoQD-A spectrum, therefore, indicates that there is no significant amount of free HBCPy present in the sample. The oxidation and spin states of CoQD-A and Co-A were characterized using Co K- and L-edge XAS, as shown in Fig. 2. The L-edge spectrum consists of L 3and L 2edges formed due to the spin–orbit coupling of the 2p hole and some fine structure associated with the multiplet nature of the generated 2p53d7final J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . N K-edge spectra of Co(DMG) 2Cl2(purple), HBCPy (black), Co-A (blue), and CoQD-A (red), where solid lines show experimental data, while vertical lines show the calculated transitions associated with the N-atoms localized on the pyridine (black) and cobaloxime (green) moieties. state [Fig. 2(d)]. The L-edge XAS spectra are highly sensitive to the Co oxidation and spin state, and the spectra of CoQD-A and Co-A are similar to those of previously reported low-spin Co(III) compounds.52,53The low-spin character of CoQD-A and Co-A isexpected, as previously shown using magnetic susceptibility mea- surements for cobaloximes.54The Co K-edge XANES of Co-A and CoQD-A, shown in Figs. 2(a)–2(c), is also consistent with Co(III) metal centers. Figure 2(b) shows the pre-edges of the K-edge spec- tra, which consist mostly of quadrupole-allowed 1s →3d electronic transitions. For the reference Co(III) complex (FK102 Co(III) salt), one pre-edge peak is observed at ∼7710 eV; this is expected for a low- spin, d6metal, which in O hgeometry would have a t 2g6eg0electronic configuration, which will yield a single-peak due to the excitation of the 1s core electron into the empty e gmanifold. The Co-A and CoQD-A samples also show a peak at a comparable energy, along with an additional peak of ∼7712.5 eV. This peak is likely not the standard 1s →3d transition but instead transition into a ligand π∗ orbital with mixed metal 4p character, which has been observed previously for low-spin, d6Fe complexes.55Based on the edge ener- gies and pre-edge peaks, the Co-A and CoQD-A samples can be confirmed to have low-spin, Co(III) metal centers. The extended x-ray absorption fine structure (EXAFS) was measured to determine the local structure about the Co cen- ter in each of the Co and QD samples (Fig. 3). Fourier trans- form of this interference pattern yields a pseudo-radial distribu- tion function that is centered about the absorbing Co atom with peaks that reflect Co-atom scattering distances. Structural mod- els were constructed in Gaussian and used to generate theoret- ical EXAFS spectra using FEFF according to the EXAFS equa- tion. The EXAFS fits (fixing the optimized structural parame- ters and fitting for the mean squared variation in pathlength, σ2) are shown as dashed lines in Fig. 3, and the corre- sponding fit parameters are shown in Table S2 (supplementary material). The EXAFS spectra of Co-A and CoQD-A are adequately reproduced using the DFT structures, demonstrating that the DFT methods produce excellent structural models for these systems. B. Electrochemistry Cyclic voltammograms (CVs) of model Co-complexes and the corresponding GQDs are shown in Fig. 4. CoQD-A exhibits an FIG. 2 . (a) XANES spectra (normalized to an edge-jump of 1) at the Co K-edge. (b) Inset at the pre-edge features. (c) Differential XANES spectra of Co-A (blue), CoQD-A (red), and the reference tris(2-(1 H-pyrazol-1-yl)pyridine)cobalt(III) tri[bis(trifluoromethane)sulfonimide [FK 102 Co(III) TFSI salt] (green). (d) Co L-edge spectra for Co-A (blue) and CoQD-A (red). J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . EXAF spectra of Co-A (blue) and CoQD-A (red); solid lines show the experimental data and dashed lines show the fit to the model calculated using DFT over a range of R (1–3) and k (3–11.5). (a) R space, (b) K space, and (c) Im[χ(R)]. electrochemically irreversible process at −0.25 V vs Ag/AgCl, which is assigned to the Co(III/II) reduction coupled with the loss of an axial chloride ligand, based on the comparison with the model Co-A and the previous literature report.49The second reduction peak appears at −1.01 V vs Ag/AgCl, and it is electrochemically reversible and is assigned to the Co(II/I) reduction. While the Co(II/I) reduction is chemically reversible in the case of the Co-Amodel, the same process is chemically quasi-reversible in CoQD- A, as evidenced by the ratio of anodic/cathodic peak currents (I a/Ic = 0.28). The loss of anodic current may occur due to the partial detachment of the cobaloxime moiety from HBCPy upon reduc- tion. This assignment is supported by a relatively small binding constant (K = 200M−1in DMF)50reported previously for axial pyridine coordination to cobaloximes. In the case of B-series, the FIG. 4 . (a) CV of Co-A (1 mM solution) and CoQD-A (thin film deposited onto the working electrode) using 0.1M TBAP in dimethylformamide. (b) CV of Co-B (1 mM solution) and CoQD-B (thin film deposited onto the working electrode) using 0.1M TBAP in acetonitrile as an electrolyte. (c) CV of the HBCPy thin film using 0.1M TBAP in acetonitrile as an electrolyte. J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Electron transfer, energy transfer, and experimental rate constants for CoQD-A. ETλRE(eV) VDA(cm−1)Eo QDa(V) Eo Cob(V) E00c(eV) w(eV)ΔGET(eV) kET(s−1)τCo(ns) Experimental 1.1 1.7 1.35 −0.25 2.6 0.05 −0.95 5.7 ×1080.9 EnT R0(nm) JF(nm4M−1cm−1)τ(ns) kF EnT(s−1) JD(cm) kD EnT(s−1) kq(s−1) 0.8 1.9 ×101310.8 8.1 ×1063.0×10−31.0×10101.0×109 aAnodic peak potential (due to chemical irreversibility) referenced to Ag/Ag+. bThe standard reduction potential referenced to Ag/Ag+. cE00was obtained from the fluorescence maximum of HBCPy ( λ= 479 nm). While E 00is usually obtained as an intercept between absorption and emission maxima, this approach was not used because this would cause an overestimate of E 00(absorption maximum of HBCPy corresponds to transitions to S 3and S 4states, not the S 1state). electrochemical behavior of CoQD-B was not consistent with the model compound Co-B: the Co(II/I) reduction that appears at −0.84 V vs Ag/AgCl for Co-B is barely visible in the case of CoQD- B. Again, we assign this behavior to the detachment of the Co(II)cobaloxime from HBCPy prior to the electrochemical experiment. HBCPy exhibits a chemically irreversible oxidation at +1.35 V vs Ag/AgCl, which is consistent with the oxidation of the aromatic core.33The relevant standard reduction potentials are tabulated FIG. 5 . UV–vis absorption of (a) HBCPy, (b) CoQD-A, and (c) CoQD-B in chloroform. The vertical lines represent the calculated electronic transition obtained using the computational method with B3LYP functional and 6-31g(d,p)/LanL2DZ basis set with IEFPCM = dichloromethane as the solvation method. Orbitals are represented to show charge distribution in electronic states. J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp in Table I and are used in the subsequent text to evaluate the kinetics of photoinduced electron transfer from GQD to cobaloxime in CoQD-A. C. Steady-state electronic spectroscopy The absorption profile of GQDs obtained in chloroform at low concentration ( ∼2μM) is shown in Fig. 5, along with electronic tran- sitions and relevant orbitals calculated using time dependent DFT calculations. The experimental spectrum of HBCPy exhibits a peak with a maximum at 373 nm and the absorption tail that extends up to 500 nm. This spectrum is consistent with calculations, which predict intense transitions at 380 nm and 384 nm, as well as the low oscil- lator strength S 1transition at 445 nm. The orbitals involved in the S0→S1transition (HOMO-1 →LUMO and HOMO →LUMO+1) indicate that the S 1state exhibits a charge transfer character, where the electronic density moves from the hexabenzocoronene (HBC) core to the pyridine ring. Upon coordination with cobaloximes, the experimental absorption spectra of CoQD-A and CoQD-B undergo a small red shift of the intense bands and an increase in the oscillator strength in the tail region. Again, the experimental findings corre- late well with calculations, which predict the red shift of the intense bands and an increase in the transition dipole moment for the HBC→Py charge transfer bands at 456 nm and 440 nm for CoQD-A and CoQD-B, respectively. In addition to these bright states, CoQD- A and CoQD-B also exhibit a number of dark states (Fig. S14, supplementary material). For example, CoQD-A exhibits six dark states below the lowest bright HBC →Py charge transfer band at 456 nm, most of which are localized on the cobaloxime moiety. The lowest S 1state involves a number of different orbitals, as illustratedin Table S3 (supplementary material), which complicates the charac- terization of this dark state. Overall, this transition exhibits a ligand- to-metal charge transfer (LMCT) character, where the electronic density is shifted from oxime, pyridine, and chloride ligands to the cobalt d-orbitals. This lowest energy dark state may be involved in the fast fluorescence quenching of HBC fluorescence observed in CoQD-A and CoQD-B, as described below. To further examine the influence of coordination of cobaloxime, we performed fluorescence measurements of HBCPy, CoQD-A, and CoQD-B [Figs. 6(c) and 6(d), all samples exhibited the same opti- cal density at the excitation wavelength of 377 nm]. HBCPy exhibits an emission spectrum with a maximum at 479 nm that we assign to the emission from the HBC →Py charge transfer state. Almost complete quenching of this fluorescence was observed for CoQD- A and CoQD-B, respectively [Figs. 6(c) and 6(d)]. In the case of CoQD-A, the experimental rate constant for fluorescence quenching (kq= 1.0×109s−1, Table I) was calculated from fluorescence life- times of the HBCPy ( τ) and CoQD-A ( τCo) (Fig. S12) using the following equation: kq=1 τCo−1 τ. (1) As will be discussed below, this quenching is consistent with the photoinduced electron transfer from HBC to the cobalt cen- ters in CoQD-A and CoQD-B. However, the observed fluorescence quenching may also be associated with the energy transfer from HBC to the cobaloxime-based excited states. The calculations discussed in the previous paragraph indicate the presence of cobaloxime-based LMCT states that could serve as energy acceptors. This possibil- ity is supported by the fact that a broad emission band centered around 600 nm was observed for both CoQD-A and CoQD-B, which FIG. 6 . Absorption profile of (a) HBCPy, CoQD-A, and Co-A, and (b) HBCPy, CoQD-B, and Co-B in chloroform. Emis- sion of (c) HBCPy, CoQD-A, and Co- A, and (d) HBCPy, CoQD-B, and Co-B in chloroform. The insets in (c) and (d) show the zoomed region. J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp resembles the emission from the model compounds Co-A and Co-B, respectively. D. Energy vs electron transfer The fluorescence quenching observed in Fig. 6 could be associ- ated with either an energy or an electron transfer process. To evalu- ate the possible mechanism, we compared the experimental quench- ing rate for CoQD-A with the rate constants estimated for Marcus electron transfer (ET), as well as Dexter and Förster energy trans- fer (EnT) processes (additional calculation details are presented in Sec. S3, supplementary material). The ET is expected to take place from the photoexcited GQD moiety to the cobalt center as follows: CoIIIQD∗- A→CoIIQD⋅+- A kET. (2) The rate constant for ET ( kET) in CoQD-A was evaluated using the Marcus ET theory,56expressed as follows: kET=√ 4π3 h2λREkBT∣VDA∣2exp(−(ΔGET+λRE)2 4λREkBT). (3) Here,λREis the solvent reorganization energy, VDAis the elec- tronic coupling matrix between QD and Co moieties, and ΔGETis the thermodynamic driving force for ET. The solvent reorganization energy,λRE, is obtained from the following equation: λRE=e2 8πε0(1 εo−1 εs)(1 rd+1 ra−2 r)+λi. (4) Here, e is the charge of the electron; ε0is the permittivity of free space;εoandεsare the optical and static dielectric constants of the solvent, respectively; rdandraare radii of QD and Co, respectively; andris the center-to-center distance between QD and Co obtained from the optimized structure of CoQD-A. The value for inner- sphere reorganization energy λiwas approximated to be 0.1 eV.57 The electronic coupling factor ( VDA) between QD and Co in Eq. (3) is given by58 VDA=μtrΔE√ (Δμ)2+ 4(μtr)2. (5) Here,μtris the transition dipole moment associated with the QD-to-Co charge transfer excited state. The value for μtrwas obtained from TDDFT calculations for CoQD-A (S 3excited state, more information available in the supplementary material). ΔEis the energy of the QD-to-Co charge transfer excited state S 3, while Δμis the difference between the dipole moments for S 3and S 0states of CoQD-A, evaluated as er (where e is the electron charge and r is the center-to-center distance between QD and Co). The thermody- namic driving force for ET ( ΔGET) was determined using the Gibbs energy of photoinduced electron transfer,59 ΔGET=Eo QD−Eo Co−E00+w, (6) where Eo QDand Eo Coare the standard reduction potentials for QD/QD.+and Co(III/II) processes, estimated from the cyclic voltammograms in Fig. 4. E00is the energy of the QD excited state derived from the peak fluorescence of HBCPy (Fig. 6), while wisthe work term associated with the distance of charge separation. The calculated free energies (Table I) indicate thermodynamic feasibility of PET from QD to Co in CoQD-A. Upon calculating the terms in Eqs. (3)–(6) (additional details in Sec. S3, supplementary material), akETvalue of 5.7 ×108s−1was obtained for CoQD-A using the Marcus theory. The energy transfer (EnT) can take place from π,π∗states local- ized on the QD moiety to either ligand-centered (LC) or ligand-to- metal charge transfer (LMCT) states of Co, CoQD∗- A→Co∗QD - A kEnT. (7) We evaluate here the rates for both Förster60and Dexter61 energy transfer mechanisms in CoQD-A. The rate of Förster EnT was determined using the following equation: kF EnT=8.5×10−5 τ(κ2ϕDJF ε4 0r6), (8) whereτis the fluorescence lifetime of the QD, ϕDis the fluorescence quantum yield of the QD, and JFis the Förster overlap integral. The orientation factor κ2is described by the following equation: κ2=(cosθT−3cosθDcosθA)2. (9) Here, θDandθAare the angles made by the transition dipoles of donor and acceptor, respectively, along the line joining the centers andθTis the angle between the transition dipoles. This value was obtained from transition dipole moments obtained using TDDFT calculations (Sec. S3, supplementary material). The Förster overlap integral JFis defined by the following equation: JF=∫∞ 0FD(λ)εA(λ)λ4dλ. (10) Here, FDis the area normalized emission spectrum of the QD, εAis the molar extinction coefficient of Co-A, and λis wavelength in nm. The JFvalue was calculated to be in the order of ∼1013nm4 M−1cm−1, a relatively low value due to a poor spectral overlap between emission of HBCPy and absorption of Co-A (Fig. S11A, supplementary material). Furthermore, the orthogonal arrangement of transition dipoles of QD and Co moiety in CoQD-A (Fig. S11B, supplementary material) yields a κ2value of 0.046, which is also very small. Due to these poor conditions for Förster EnT, the rate constant kF EnTwas found to be 8.1 ×106s−1, a value that is several order of magnitudes lower than the experimental rate constant for fluorescence quenching in CoQD-A (Table I). The Dexter EnT requires close proximity between donor and acceptor, which is the case in CoQD-A as the QD is axially coor- dinated to Co via a pyridine linkage. The rate of Dexter EnT was calculated using the following equation:58 kD EnT=4π2V2 DA hJD. (11) Here, JDis the Dexter overlap integral calculated from the flu- orescence spectrum of QD [ F(¯ν)] and absorption of Co-A [ ε(¯ν)], JD=∫F(¯ν)ε(¯ν)d¯ν ∫F(¯ν)d¯ν∫ε(¯ν)d¯ν. (12) J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp From Eq. (11), a rate constant of 1.0 ×1010s−1was obtained for Dexter EnT. The experimental EnT rate constant is significantly higher than the calculated values for kF EnT, indicating that the Förster EnT can be ruled out as a possible quenching mechanism. On the other hand, the calculated rate constants for both Marcus ET and Dexter EnT are of the same order of magnitude as the experimental value for fluo- rescence quenching, making it difficult to distinguish which process takes place. The negative ΔGET(−0.95 eV) value and close prox- imity of HBCPy and cobaloxime ( ∼2 Å distance between pyridinic nitrogen and cobalt in CoQD-A) makes Marcus ET and Dexter EnT possible in CoQD-A. E. Time-resolved spectroscopy To investigate the photophysical properties of CoQD-A and CoQD-B, time-resolved transient absorption (TA) experiments were performed by probing in UV–vis and mid-IR spectral regions. Due to the low solubility of CoQD-A and CoQD-B in common organic solvents, TA measurements were performed on drop-casted thin films onto transparent substrates. Preliminary experiments were performed on HBCPy to evaluate the excited-state dynamics of the GQD core in the absence of the metal moiety (Fig. 7). TA spectra of HBCPy consist of the ground-state bleach feature centered at 388 nm and a broad excited-state absorption feature that spans the entire visible range [Fig. 7(a)]. The transient signal decays with the same kinetics at all wavelength, which is also evident by the isosbestic point with zero ΔA at 475 nm. Most of the transient signal disap- pears within the first nanosecond after the excitation pulse, which is quite different from the fluorescence dynamics observed for HBCPy in solution (10.8 ns lifetime, Fig. S12). We assign the differences in photophysics to the presence of inter-chromophore interactions in thin films via π-stacking interaction between aromatic cores. A similar increase in the excited-state relaxation was observed in our previous studies of other GQD assemblies.32 The excited-state dynamics of HBCPy assemblies showed a pump-fluence dependence [Fig. 7(b)], consistent with the presence of annihilation of excitons that migrate along the chromophore stack. This pump intensity dependent dynamics was used to deter- mine the exciton mobility in HBCPy aggregates, using the exciton– exciton annihilation approach we recently applied to a similar sys- tem of molecular aggregates.32The experimental data in Fig. 7(b) were fit to a kinetic model involving two types of excitons: initially formed mobile excitons (with density N1, lifetimeτ1, and diffusion coefficient D) and trapped excitons (with density N2and lifetime τ2) that are formed from mobile excitons with lifetime τtr. The exciton–exciton annihilation of the mobile excitons was modeled using one-dimensional exciton diffusion, N(t)=N1(t)+N2(t), (13) dN 1(t) dt=−N1(t) τ1−N1(t) τtr−√ 2D πt⋅N1(t)2, (14) dN 2(t) dt=N1(t) τtr−N2(t) τ2. (15) FIG. 7 . (a) Transient absorption spectra for a HBCPy thin film in a poly(methyl methacrylate) matrix. The sample was excited at λexc= 400 nm, and the pump intensity was 200 nJ with magic angle orientation of pump and probe beams. (b) Exciton–exciton annihilation fitting for a HBCPy thin film sample with pump intensities varying from 200 nJ to 1200 nJ and probed at 388 nm. The circles rep- resent transient absorption data at the magic angle geometry of pump and probe beam intensities, and the solid lines are the fits obtained from Eqs. (13)–(15). Traces are offset by 0.01 from each other with markers for zero to the right of each trace. Differential equations (13)–(15) were solved numerically using the initial exciton density as described previously.32The fit of the experimental data was performed simultaneously for all pump flu- ences by varying only four parameters: τ1,τ2,τtr, and D. Figure 7(b) shows a satisfactory agreement between the experiment and the- ory. The fitted parameters are shown in Table II along with pre- viously measured values for a similar thin film without pyridine groups (HBC) using the same model.32The addition of the pyridine groups to HBC does not affect significantly the values for τ1andτtr. The exciton trapping (associated with τtr) in similar organic chro- mophore assemblies is often assigned to interchromophore motions that ultimately lead to the formation of “relaxed” excitonic species that cannot undergo further hoping due to the poor energy mis- match between donor and acceptor states.62,63Based on the simi- larity inτ1andτtrvalues for HBC and HBCPy, it appears that the pyridine groups in HBCPy do not alter these relaxation pathways J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE II . Comparison of exciton diffusion parameters for GQDs. The values for HBC are obtained from Ref. 32. τ1(ps)τ2(ps)τtr(ps) D(mol2/ps)aLD(mol)aτMET (ps)τ3(ps) HBC 75 750 70 10 39 HBCPy 67 160 76 18.5 50 CoQD-A 67b160b76b18.5b51 <0.5 CoQD-B 67b160b76b18.5b101 <0.5 aMol = molecule. bValues were set to those found for HBCPy and kept constant during the fit. significantly and that the aggregate structure and dynamics are sim- ilar in two chromophore aggregates. On the other hand, the intro- duction of pyridine moiety causes Dto double in HBCPy compared to HBC. The larger Din HBCPy indicates faster exciton hopping along the aggregate and is likely due to the improved value for theFörster energy transfer overlap integral JFrelative to HBC. Larger overlap integral is consistent with the higher value calculated for the transition dipole moments for the S 1state in HBCPy (0.46 D) com- pared with HBC (0.20 D). This improvement in the exciton diffusion coefficients results in a larger exciton diffusion length in HBCPy FIG. 8 . Transient absorption spectra for (a) CoQD-A and (b) CoQD-B thin films in the poly(methyl methacrylate) matrix. The samples were excited at λexc= 400 nm, and the pump intensity was at 200 nJ with magic angle orientation of pump and probe beams. Exciton–exciton annihilation fitting for (c) CoQD-A and (d) CoQD-B thin film samples with pump intensities varying from 200 nJ to 1200 nJ and probed at 388 nm. The circles represent transient absorption data at the magic angle geometry of pump and probe beam intensities, and the solid lines are the fits obtained from Eqs. (16)–(19). Traces are offset by 0.01 from each other with markers for zero to the right of each trace. J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-11 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (where 50 molecules are sampled within the exciton lifetime) relative to HBC (39 molecules sampled). Qualitatively, the TA spectra of CoQD-A and CoQD-B thin films are similar to HBCPy, with the presence of bleach signal at ∼388 nm and a broad excited-state absorption throughout the vis- ible range (Fig. 8). The two thin films differ from HBCPy in that the signal decays faster (Fig. S13), consistent with the shorter flu- orescence lifetimes obtained for CoQD-A and CoQD-B samples in solution (Fig. S12). The faster decay of CoQDs indicates that exci- tons are finding additional decay pathways that are not present in HBCPy, such as PET and EnT discussed earlier. The kinetic profiles at different pump fluences [Figs. 8(c) and 8(d)] were modeled using a kinetic scheme analogous to the exciton–exciton annihilation model used for HBCPy, with the addition of another decay pathway (via electron or energy transfer from mobile excitons N1with lifetime τMET) to generate a new state (with density N3and lifetime τ3), N(t)=N1(t)+N2(t)+N3(t), (16) dN 1(t) dt=−N1(t) τ1−N1(t) τtr−√ 2D πt⋅N1(t)2−N1(t) τMET, (17) dN 2(t) dt=N1(t) τtr−N2(t) τ2, (18) dN 3(t) dt=N1(t) τMET−N3(t) τ3. (19)It was assumed that the lifetimes τ1,τ2and the diffusion coef- ficient Din CoQD-A and CoQD-B are the same as obtained for HBCPy, and the experimental data in Fig. 8 were fit using two parameters: τMET andτ3. The results of the fit are shown in Table II and indicate that the N3state is very short-lived, which agrees with the lack of spectroscopic signature of this transient in our TA data. In specific, transient species is formed at a slower rate ( τMET = 51 ps for CoQD-A) than the rate of its loss via the recombination pathway (τ3<0.5 ps), making it difficult to build up a detectable concentra- tion of the transient species needed to surpass the detection limit of our instrument. Thus, while spectroscopic signatures for ET64and EnT16transients have been described previously, we were unable to detect them due to the short-lived nature of the N3state. The excited states of cobaloximes, if formed via the Dexter mechanism, are expected to be short-lived, on the order of several tens of picosec- onds, as reported previously by Guldi.16The recombination lifetimes obtained in our fits ( <0.5 ps) are much shorter, suggesting that the ET pathway is more likely. No matter what the deactivation pathway is, it is clear that the long-lived charge separated states desired for photocatalysis applications do not form in CoQDs. It is interesting to note that our result correlates very well with numerous previous reports involving cobaloxime complexes coordinated with organic and inor- ganic photosensitizers. For example, Mulfort and Tiede found that Co-A and Co-B covalently attached to ruthenium tris bipyridine both showed faster decay kinetics than the chromophore alone, and only saw short-lived spectroscopic signature for CoIin one Co-A type compound.14,19It is plausible that, at least in some reports on fast charge-separation in chromophore-cobaloxime systems, FIG. 9 . TRIR [(a), (c), and (e)] and FTIR [(b), (d), and (f)] spectra of GQD thin films drop cast on CaF 2glass. HBCPy (left), CoQD-A (middle), and CoQD-B (right). The samples were excited at λexc= 400 nm with a pump intensity of 600 nJ at magic angle orientation of pump and probe beams. J. Chem. Phys. 153, 124903 (2020); doi: 10.1063/5.0018581 153, 124903-12 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp the quenching mechanism involves a Dexter energy transfer from photoexcited chromophore to a manifold of dark cobaloxime-based states with LMCT and d π-dπ∗character. These states are expected to undergo efficient non-radiative decay to ground state and are likely to outcompete the productive charge-separation pathways. Time-resolved infrared (TRIR) spectroscopy was also utilized in the attempt to identify the spectroscopic signature of ET or EnT intermediates formed in CoQDs (Fig. 9). The ground state IR spectra of HBCPy and corresponding CoQDs (panels B, D, and F) were assigned using DFT-based vibrational mode analysis (Sec. S2.2, supplementary material). All three spectra consist of aromatic ring C− −C modes of HBC (at 1300 cm−1–1400 cm−1) and pyridine (at∼1600 cm−1) moieties. An additional strong vibrational mode at∼1450 cm−1is associated with the C − −H bending modes of the alkyl group and are not expected to be relevant for the TRIR spec- tra. Importantly, the CoQD spectra contain an additional absorption band of∼1550 cm−1(for CoQD-A), which is assigned to the C = =N stretching modes of the cobaloxime moiety. These modes serve as an excellent marker for the involvement of cobaloxime moieties in the photophysics of the model compounds: if the transient bleach of these modes were observed in TRIR, it would indicate that the energy or the electron transfer to the cobaloxime took place. TRIR spectra of HBCPy consist of a broad feature that covers the entire probe window and peaks at 1545 cm−1. Similar broad tran- sients have been reported previously by Asbury in perylene-imide chromophore assemblies,65,66and the observed vibrational broad- ening was explained by the fast dephasing caused by the fact that a large number of molecules is sampled throughout the material, either through large exciton size or fast exciton hopping. Our previ- ous study of exciton size and dynamics in HBC-based chromophore assemblies32showed strong evidence that the exciton delocalization length involves 1–4 chromophore units. Furthermore, experimental and computational studies indicate that the dynamic exciton deco- herence in organic chromophore assemblies take place at timescales shorter than 100 fs.67Based on these studies, we do not expect that the mode broadening arises due to large exciton delocalization. On the other hand, the fast exciton hopping may be associated with the observed broadening, particularly since our kinetic data mod- eling implies a large exciton diffusion length (L D) of 50 molecules (Table II). In the presence of cobaloximes, TRIR spectral features do not change significantly [Figs. 9(c) and 9(e)], but the transient signal decays faster, consistent with the excited state quenching observed in the steady-state and time-resolved measurements in the UV–vis range. Unfortunately, the decay of the broad vibrational features associated with the HBCPy unit was not accompanied with the con- comitant growth of the cobaloxime-based modes at 1571 cm−1. This result is again explained by the unfavorably low population of ET or EnT products controlled by the τMET andτ3kinetic parameters. IV. CONCLUSIONS In summary, our study of photophysical properties of dyads composed of cobaloxime catalysts and GQD chromophores shows that the fast nonradiative decay pathway exists in these systems. By comparing the experimental quenching rate with the rates estimated using models for ET and Ent, we conclude that the Marcus ET andDexter EnT are both plausible mechanisms for GQD excited-state quenching. The EnT mechanism is also supported by TDDFT calcu- lations, which predict the presence of several low-lying cobaloxime- centered dark excited states that could serve as Dexter EnT accep- tors and mediate subsequent nonradiative decay to the ground state. The current study points to additional challenges associated with the achievement of long-lived charge-separation in chromophore- catalyst systems: the plausible energy transfer mechanisms need to be avoided. However, avoiding energy transfer can be a challenge for many metal-based catalysts, considering that they often exhibit low-energy metal-centered ligand states that could serve as energy acceptors.68 SUPPLEMENTARY MATERIAL See the supplementary material for additional details on syn- thesis, characterization, and computational calculations. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences, through Argonne National Laboratory under Contract No. DE-AC02-06CH11357. G.K. and A.A.C. were supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences and Biosciences, through SLAC National Accelerator Laboratory under Contract No. DE-AC02-76SF00515. This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Con- tract No. DE-AC02-06CH11357. Use of the Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator Laboratory, is supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Contract No. DE-AC02- 76SF00515. K.D.G. acknowledges the National Science Foundation (Grant No. 1806388). We thank the beamline scientist Dr. Sungsik Lee for his help with cobalt K-edge XAS measurements. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon request. REFERENCES 1T. M. McCormick, B. D. Calitree, A. Orchard, N. D. Kraut, F. V. Bright, M. R. Detty, and R. Eisenberg, J. Am. Chem. Soc. 132, 15480 (2010). 2A. Fihri, V. Artero, M. Razavet, C. Baffert, W. Leibl, and M. Fontecave, Angew. 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5.0020852.pdf

Appl. Phys. Lett. 117, 122403 (2020); https://doi.org/10.1063/5.0020852 117, 122403 © 2020 Author(s).All-optical probe of magnetization precession modulated by spin–orbit torque Cite as: Appl. Phys. Lett. 117, 122403 (2020); https://doi.org/10.1063/5.0020852 Submitted: 04 July 2020 . Accepted: 08 August 2020 . Published Online: 21 September 2020 Kazuaki Ishibashi , Satoshi Iihama , Yutaro Takeuchi , Kaito Furuya , Shun Kanai , Shunsuke Fukami , and Shigemi Mizukami COLLECTIONS This paper was selected as Featured ARTICLES YOU MAY BE INTERESTED IN Robust spin–orbit torques in ferromagnetic multilayers with weak bulk spin Hall effect Applied Physics Letters 117, 122401 (2020); https://doi.org/10.1063/5.0011399 Nano-second exciton-polariton lasing in organic microcavities Applied Physics Letters 117, 123302 (2020); https://doi.org/10.1063/5.0019195 Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745All-optical probe of magnetization precession modulated by spin–orbit torque Cite as: Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 Submitted: 4 July 2020 .Accepted: 8 August 2020 . Published Online: 21 September 2020 Kazuaki Ishibashi,1,2Satoshi Iihama,3,4,a) Yutaro Takeuchi,5Kaito Furuya,5Shun Kanai,4,5,6,7 Shunsuke Fukami,2,4,5,6 and Shigemi Mizukami2,4,6 AFFILIATIONS 1Department of Applied Physics, Graduate School of Engineering, Tohoku University, 6-6-05, Aoba-yama, Sendai 980-8579, Japan 2WPI Advanced Institute for Materials Research (AIMR), Tohoku University, 2-1-1, Katahira, Sendai 980-8577, Japan 3Frontier Research Institute for Interdisciplinary Sciences (FRIS), Tohoku University, Sendai 980-8578, Japan 4Center for Spintronics Research Network (CSRN), Tohoku University, Sendai 980-8577, Japan 5Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication (RIEC), Tohoku University, Sendai 980-8577, Japan 6Center for Science and Innovation in Spintronics (CSIS), Core Research Cluster (CRC), Tohoku University, Sendai 980-8577, Japan 7Frontier Research in Duo (FRiD), Tohoku University, Sendai 980-8577, Japan a)Author to whom correspondence should be addressed: satoshi.iihama.d6@tohoku.ac.jp ABSTRACT Laser-induced magnetization precession modulated by an in-plane direct current was investigated in a W/CoFeB/MgO micron-sized strip using an all-optical time-resolved magneto-optical Kerr effect microscope. We observed a relatively large change in the precession frequency, owing to a current-induced spin–orbit torque. The generation efficiency of the spin–orbit torque was evaluated as /C00.3560.03, which was in accordance with that evaluated from the modulation of damping. This technique may become an alternate method for the evaluation ofspin–orbit torque. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020852 Spin–orbit torque (SOT) has attracted significant attention as it allows simple, reliable, and fast manipulation of magnetization in thin films. 1–3Conventionally, SOT has been investigated using electrical means such as spin-torque ferromagnetic resonance4–7and second harmonic Hall effect measurement.8–11In the spin-torque ferromag- netic resonance, magnetization precession is excited by the SOT gener- ated from an injected in-plane RF current. Subsequently, thegeneration efficiency of the SOT, i.e., the effective spin-Hall angle in nonmagnetic heavy metals, can be evaluated by analyzing its spectrum amplitude and shape. In the second harmonic Hall effect measure- ment, the magnetization angle is adiabatically changed by the SOT induced by a low-frequency in-plane alternating current. This change in the magnetization angle is detected through the planar Hall effect or an anomalous Hall effect voltage. Even though these techniques are widely utilized, parasitic electrical voltages are induced by spin-charge conversion as well as the thermoelectric effect. 5,10 A direct observation of magnetization precession modulated by the SOT is free from such parasitic effects, and thus, it is a promisingapproach. The time-resolved measurement of magnetization preces- sion modulated by the SOT has been previously reported.12–14These studies mainly focused on the change in the relaxation time of magne- tization precession by the SOT.12,13However, no study has focused on the change in precession frequency due to SOT. In general, the evalua- tion of frequency is more precise than that of the precession relaxation time. Therefore, it is intriguing to observe and understand the effect ofSOT on frequency. In this Letter, we report, for the first time, an obser- vation of the modulation of magnetization precession frequency owing to the SOT, from which the generation efficiency of the SOT was obtained. Thin-film stacks of W(5)/CoFeB(2.4)/MgO(1.3)/Ta(1) (thickness in nm) were fabricated via DC/RF magnetron sputtering on Si/SiO 2 substrates. Here, we used a W underlayer, which is reported to showlarge SOT efficiencies. 15–17The fabrication condition was similar to that for samples exhibiting a high effective spin-Hall angle.18The thickness of the W layer was determined from our previous findings on the W/CoFeB structure in which a 5-nm-thick W layer showed Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apllarge SOT efficiency.18The samples were patterned into rectangular strips with a width wof 10lm and a length Lof 40lm by photoli- thography and Ar ion milling. The magnetization precession dynamicswere investigated using an all-optical time-resolved magneto-opticalKerr effect (TRMOKE) microscope. The setup was similar to thatreported previously. 19–21The wavelength, pulse duration, and pulse repetition rate of the emission laser were approximately 800 nm, 120 fs, and 80 MHz, respectively. The wavelength of the pump laser waschanged to 400 nm using a BaB 2O4(BBO) crystal, and its intensity was modulated using a mechanical chopper at a frequency of 370 Hz.The pump and probe beams were focused on the sample surface usingan objective lens, as shown in Fig. 1(a) . The diameter of the probe beam spot was approximately 1 lm. The pump fluence was less than 2.0 mJ/cm 2, and the probe fluence was /C241 mJ/cm2.T h ep u m p - induced change in the Kerr rotation angle of the reflected probe beam was detected using a balanced photodiode detector, as a function ofpump–probe delay time. Figures 1(b) and1(c)show the schematic dia- grams of the coordinate system and the experimental geometries. Anexternal magnetic field H extwas applied with an out-of-plane angle hH and an in-plane-angle /¼90/C14. The angle of the magnetization direc- tion hin an equilibrium state was determined through the external magnetic field and magnetic anisotropy. We tested two geometries, as shown in Figs. 1(b) and1(c), to clarify these differences. One geometry is that the sample strip was parallel to the x-axis [ Fig. 1(b) ](Jck^x), where Jcdenotes the charge current density. The other geometry is that the sample strip was parallel to the y-axis [ Fig. 1(c) ](Jck^y). Note that only the former case was reported previously.12,13All measure- ments were performed at room temperature. Figure 2(a) shows the typical time-domain measurement for the sample with various electrical currents Ifor the Jck^ygeometry. Here, DhKis the pump laser-induced change in the Kerr rotation angle, which is proportional to the z-component of the magnetization. At a delay time of a few hundred femtoseconds, ultrafast demagnetization was induced through pump laser illumination. Subsequently, the mag- netization recovery and the damped precession of magnetization wereobserved. The magnetization precession triggered by ultrafast demag-netization is well documented in a previous study. 22The relaxation time sand frequency fof the precession were evaluated by the least-squares fitting of the damped sinusoidal function23to the TRMOKE signal DhK DhK¼AþBexp/C0/C23tðÞ þCexp/C0t s/C18/C19 sin 2 pftþ/0 ðÞ : (1) Here, the first two terms represent the change due to the recovery from the demagnetization and are characterized by the amplitudes ofAand Band the recovery rate /C23. The last term in Eq. (1)represents the change due to the damped magnetization precession. Cand/ 0 denote the precession amplitude and initial phase, respectively. The dashed black curves in Fig. 2(a) denote the fitting curve, calculated using Eq. (1).Figure 2(b) shows the typical normalized signals with I¼0;65 mA applied to the x-axis at hH¼25/C14andl0Hext ¼352 mT ( Jck^x). In this figure, the remagnetization background sig- nals, i.e., the first and second terms in Eq. (1), were subtracted. The modulation of the relaxation time of the precession was clearlyobserved. This trend was similar to that observed in previous stud- ies. 12,13Figure 2(c) shows the typical normalized signals with I¼0; 65 mA applied to the y-axis at hH¼4/C14andl0Hext¼270 mT (Jck^y). In contrast to Fig. 2(b) , a distinct change was observed in the precession frequency. Note that the magnetization is not saturatedalong the magnetic field direction, namely, h6¼h HinFig. 1 ,b e c a u s e the applied magnetic field is smaller than the out-of-plane demagnetiz- ing field of the sample. The reason for the choice of the field angle fortwo different experimental geometries is discussed later. Figures 3(a) and3(b)show the modulation of the inverse relaxa- tion time 1 =swith Iunder J ck^xatl0Hext¼6352 mT. The reversal ofl0Hextchanges the sign of the slope of the inverse relaxation time vs FIG. 1. (a) Schematic of the sample stacking structure and the optical setup. Schematic of the coordinate system and the experimental geometries with a directcurrent applied along the x-axis (b) and y-axis (c). FIG. 2. (a) Typical time-domain data with various direct currents Ialong the y-axis at a fixed field angle hH¼4/C14and the external magnetic field l0Hext¼270 mT for the W/CoFeB/MgO/Ta films for the geometry of Jck^y. The dashed curves in (a) represent data calculated using Eq. (1)and were fitted to the experimental data. (b) Normalized time-domain data ^DhKwith I¼0;65 mA applied to the x-axis at hH¼25/C14andl0Hext¼352 mT. (c) ^DhKwith I¼0;65 mA applied parallel to they-axis at hH¼4/C14andl0Hext¼270 mT.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-2 Published under license by AIP Publishingcurrent, indicating that the observed change in the relaxation time is caused by the SOT. Figures 3(c) and3(d) show the modulation of fre- quency fwith Iunder Jck^yatl0Hext¼6270 mT. The frequency shift in both Figs. 3(c) and3(d) exhibited negative slopes, in contrast to the data of 1 =svsI[Figs. 3(a) and3(b)]. To understand the frequency and relaxation time modulated by the SOT, we describe our analysis using the Landau–Lifshitz–Gilbert(LLG) equation, which includes the SOT, dm dt¼/C0cl0m/C2Heffþam/C2dm dt/C0cl0Hsm/C2ðm/C2rÞ; (2) where mis the unit magnetization vector, Heffis the effective magnetic field, ais the Gilbert damping constant, ris the spin polarization vec- tor, and cis the gyromagnetic ratio. Here, we consider only the damping-like SOT in Eq. (2). Assuming that the SOT is caused by the spin-Hall spin current generated from the W layer, the SOT effective field Hsis expressed as Hs¼/C22hngI 2el0MstCFBtWw; (3) where /C22h,e,Ms;tCFB,a n d tWare the Dirac constant, electron charge, saturation magnetization, CoFeB layer thickness, and W layer thick-ness, respectively. grepresents the fraction of the electrical current flowing into the W layer. nis the generation efficiency of the SOT. It should be noted that nis determined not only by the spin-Hall angle, but also by the spin transparency at the interface as well as the damping-like SOT generated at the interface; therefore, nis termed as the effective spin-Hall angle. In the absence of the electrical current,the precession frequency f, the inverse relaxation time 1 =s,a n dt h e i r field components H 1,H2can be expressed by the following equations:24 1 s¼1 2l0caðH1þH2Þ; (4) f¼l0c 2pffiffiffiffiffiffiffiffiffiffiffiH1H2p; (5) H1¼Hextcosðh/C0hHÞ/C0Meffcos2h; (6)and H2¼Hextcosðh/C0hHÞ/C0Meffcos 2 h; (7) where Meffis the effective demagnetizing field. The magnetization angle hwas determined based on the balance between the Zeeman energy and the effective demagnetizing energy, 2Hextsinðh/C0hHÞ/C0Meffsin 2h¼0: (8) In the presence of an electrical current, Hsaffected the precession frequency and relaxation time, depending on the geometries, as dis-cussed below. First, we present our analysis on the geometry of J ck^x(rk^y). The SOT behaves as a damping-like torque, and the additional termproportional to Iwas added to Eq. (4), in which the geometry [ Fig. 1(b)] is similar to that of the anti-damping SOT switching in so-called “Type-y” devices. 3Consequently, the theoretical slope ds/C01=dIwas obtained using the LLG equation [Eq. (2)], ds/C01 dI¼/C0c/C22hng 2eMstCFBtWwsinh: (9) Equation (9)indicates that the linear modulation of the inverse relaxa- tion time was caused by the SOT. The generation efficiency of theSOT nwas evaluated as /C00.3560.07 using Eqs. (8)and(9)with the experimental ds /C01=dIvalue obtained from the data shown in Fig. 3(a) . We performed a least-squares fitting to the data of 1 =svsI using a quadratic polynomial /cþbIþaI2with adjustable parame- tersa,b,a n d c[the curves shown in Figs. 3(a) and3(b)] to extract the slope b/C17ds/C01=dIfrom the experimental data. It is to be noted that the parabolic term aI2was negligibly small and originated from Joule heating. The other parameters used were c¼185 Grad/s/T, Meff¼512 kA/m, and Ms¼1051 kA/m. Additionally, gof 0.57 was used, which was evaluated from the measured resistivity of the W film,205lXcm, and the CoFeB film, 128 lXcm. Next, we present the analysis of the geometry of J ck^y(rk^x). The direction of the SOT term in Eq. (2)is in the y–z plane in the Jck^xgeometry [ Fig. 1(b) ], while the direction of the SOT is parallel to^xin the Jck^ygeometry [ Fig. 1(c) ]. Those SOT terms induce addi- tional effective fields and change the equilibrium magnetization angle. In the Jck^y(Jck^x) geometry, the SOT term induces the effective field in the y–z(x–y) plane and changes the h(/). The his determined by the balance of the torque stemming from the Zeeman and effective demagnetizing energies. Hence, the linear change in frequency as a function of Iwas induced via a change in honly for the Jck^ygeome- try, because the change in frequency caused by the change in /is inde- pendent of the polarity Ifor the Jck^xgeometry. The relationship between the magnetization angle hand electrical current Iwas derived from Eq. (2)under an equilibrium condition as follows: 2Hextsinðh/C0hHÞ/C0Meffsin 2h/C02Hs¼0: (10) This equation is identical to Eq. (8)when no electrical currents are applied. We differentiated Eq. (10) with respect to IatI/C250, and the relationship between handIis obtained as follows: @h @I¼/C22hng 2el0MstCFBtWwH 2: (11) The theoretical slope df/dIwas derived from Eqs. (5)–(7) and(11), FIG. 3. Modulation of the inverse relaxation time 1 =swith direct currents Iapplied along the x-axis under hH¼25/C14; an external magnetic field l0Hextof (a) 352 mT and (b) /C0352 mT. Modulation of the frequency with direct currents Iapplied along they-axis under hH¼4/C14; an external magnetic field l0Hextof (c) 270 mT and (d) /C0270 mT. Curves denote quadratic polynomials cþbIþaI2fitted to experimental data.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-3 Published under license by AIP Publishingdf dI¼l0c 4pffiffiffiffiffiffi H2 H1r @H1 @hþffiffiffiffiffiffi H1 H2r @H2 @h ! /C2/C22hng 2el0MstCFBtWwH 2:(12) We did not take into account the effects of the Oersted field, Joule heating, and the field-like SOT in Eq. (2); however, these effects were negligible at around I/C250o n df/dIin this geometry. This is because the possible changes in fcaused by the three above-mentioned factors should be independent of the polarity of I.M e a n w h i l e ,t h eO e r s t e d field and the field-like SOT are small in this study,18and the parabolic change in fvsImostly originated from Joule heating, as mentioned earlier. Therefore, we interpreted a linear change in the frequency asthe effect of the damping-like SOT, as described in Eq. (2).T h i si sc o n - trary to the case of the J ck^xgeometry, in which the Oersted field and the field-like SOT induce frequency modulation. Here, it should be noted that the frequency modulation df/dIcalculated by using Eq. (12) is increased when the magnetic field direction is close to the film nor-mal, whereas the damping modulation ds /C01=dIexhibits a maximum when the magnetization direction is parallel to the film plane [Eq. (9)]. We evaluated the value of df/dIfrom the data of fvsIusing the qua- dratic polynomial fit, as similarly performed for the data of 1 =svsI [the curves in Figs. 3(c) and3(d)]. Subsequently, the generation effi- ciency of the SOT nwas evaluated as /C00.3560.03 using Eqs. (6)–(8) as well as (12)and the experimental df/dIvalue obtained from the data is shown in Fig. 3(c) .T h e nevaluated by the frequency modulation is in accordance with the nevaluated by the modulation of damping and in agreement with a previous study.17 Figure 4(a) shows the values of df/dIevaluated from the experi- mental data measured at various external magnetic fields. The curves denote the values calculated using Eq. (12).Figure 4(b) shows the gen- eration efficiency of the SOT nevaluated using Eq. (12) at various external magnetic fields. The nvalues were independent of the external magnetic fields. Moreover, the nvalue obtained from the frequency modulation was agree well with that from the inverse relaxation timemodulation within the experimental errors. Therefore, the generationefficiency of the SOT was precisely determined from the frequencymodulation. In summary, we performed time-resolved measurements of mag- netization precession modulated by the SOT in two different geome-tries, i.e., J ck^xandJck^y, in the W/CoFeB/MgO structure. In the first case, Jck^x, the modulation of the relaxation time for magnetiza- tion precession was observed, which was consistent with the previous studies. In the second case, Jck^y, the modulation of the frequency for magnetization precession induced by the SOT was clearly observed.The generation efficiency of the SOT was estimated from the analysisof the change in precession frequency, which was almost independentof the external magnetic field. This study suggests that all-optical TRMOKE measurements of the precession frequency shift are effective tools for evaluating the generation efficiency of the SOT. This study was partially supported by KAKENHI (Nos. 19K15430 and 19H05622), the ImPACT Program of CSTI, Advanced Technology Institute Research Grants, and the Centerfor Spintronics Research Network (CSRN). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 2L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 3S. Fukami, T. Anekawa, C. Zhang, and H. Ohno, Nat. Nanotechnol. 11, 621 (2016). 4L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011). 5K. Kondou, H. Sukegawa, S. Kasai, S. Mitani, Y. Niimi, and Y. Otani, Appl. Phys. Express 9, 023002 (2016). 6C.-F. Pai, Y. Ou, L. H. Vilela-Le ~ao, D. C. Ralph, and R. A. Buhrman, Phys. Rev. B92, 064426 (2015). 7Y. Wang, P. Deorani, X. Qiu, J. H. Kwon, and H. Yang, Appl. Phys. Lett. 105, 152412 (2014). 8J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani,and H. Ohno, Nat. Mater. 12, 240 (2013). 9K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Bl €ugel, S. Auffret, O. Boulle, G. Gaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013). 10C. O. Avci, K. Garello, M. Gabureac, A. Ghosh, A. Fuhrer, S. F. Alvarado, and P. Gambardella, Phys. Rev. B 90, 224427 (2014). 11M. Hayashi, J. Kim, M. Yamanouchi, and H. Ohno, Phys. Rev. B 89, 144425 (2014). 12A. Ganguly, R. M. Rowan-Robinson, A. Haldar, S. Jaiswal, J. Sinha, A. T. Hindmarch, D. A. Atkinson, and A. Barman, Appl. Phys. Lett. 105, 112409 (2014). 13S. Mondal, S. Choudhury, N. Jha, A. Ganguly, J. Sinha, and A. Barman, Phys. Rev. B 96, 054414 (2017). 14T. M. Spicer, C. J. Durrant, P. S. Keatley, V. V. Kruglyak, W. Chen, G. Xiao, and R. J. Hicken, J. Phys. D 52, 355003 (2019). 15C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Appl. Phys. Lett. 101, 122404 (2012). 16K.-U. Demasius, T. Phung, W. Zhang, B. P. Hughes, S.-H. Yang, A. Kellock, W. Han, A. Pushp, and S. S. P. Parkin, Nat. Commun. 7, 10644 (2016). 17C. Zhang, S. Fukami, K. Watanabe, A. Ohkawara, S. DuttaGupta, H. Sato, F. Matsukura, and H. Ohno, Appl. Phys. Lett. 109, 192405 (2016). 18Y. Takeuchi, C. Zhang, A. Okada, H. Sato, S. Fukami, and H. Ohno, Appl. Phys. Lett. 112, 192408 (2018). 19S. Iihama, Y. Sasaki, A. Sugihara, A. Kamimaki, Y. Ando, and S. Mizukami, Phys. Rev. B 94, 020401(R) (2016). 20Y. Sasaki, K. Suzuki, A. Sugihara, A. Kamimaki, S. Iihama, Y. Ando, and S. Mizukami, Appl. Phys. Express 10, 023002 (2017). 21A. Kamimaki, S. Iihama, Y. Sasaki, Y. Ando, and S. Mizukami, Phys. Rev. B 96, 014438 (2017). 22M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201 (2002). 23S. Iihama, S. Mizukami, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. B 89, 174416 (2014). 24S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106, 117201 (2011). FIG. 4. External magnetic field Hextdependence of (a) df/dIand (b) the generation efficiency of the SOT ninJck^ygeometry under hH¼4/C14. Curves in (a) denote the calculated values of df/dIusing Eq. (12).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122403 (2020); doi: 10.1063/5.0020852 117, 122403-4 Published under license by AIP Publishing

5.0028002.pdf

AIP Advances 10, 115014 (2020); https://doi.org/10.1063/5.0028002 10, 115014 © 2020 Author(s).Accuracy of XAS theory for unraveling structural changes of adsorbates: CO on Ni(100) Cite as: AIP Advances 10, 115014 (2020); https://doi.org/10.1063/5.0028002 Submitted: 01 September 2020 . Accepted: 26 October 2020 . Published Online: 11 November 2020 Elias Diesen , Gabriel L. S. Rodrigues , Alan C. Luntz , Frank Abild-Pedersen , Lars G. M. Pettersson , and Johannes Voss COLLECTIONS Paper published as part of the special topic on Chemical Physics , Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics ARTICLES YOU MAY BE INTERESTED IN Solvation at metal/water interfaces: An ab initio molecular dynamics benchmark of common computational approaches The Journal of Chemical Physics 152, 144703 (2020); https://doi.org/10.1063/1.5144912 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 The concept of spin ice graphs and a field theory for their charges AIP Advances 10, 115102 (2020); https://doi.org/10.1063/5.0010079AIP Advances ARTICLE scitation.org/journal/adv Accuracy of XAS theory for unraveling structural changes of adsorbates: CO on Ni(100) Cite as: AIP Advances 10, 115014 (2020); doi: 10.1063/5.0028002 Submitted: 1 September 2020 •Accepted: 26 October 2020 • Published Online: 11 November 2020 Elias Diesen,1,a) Gabriel L. S. Rodrigues,2 Alan C. Luntz,1 Frank Abild-Pedersen,1 Lars G. M. Pettersson,2 and Johannes Voss1,b) AFFILIATIONS 1SUNCAT Center for Interface Science and Catalysis, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA 2Department of Physics, AlbaNova University Center, Stockholm University, SE-10691 Stockholm, Sweden a)Author to whom correspondence should be addressed: diesen@slac.stanford.edu b)E-mail: vossj@slac.stanford.edu ABSTRACT Studying surface reactions using ultrafast optical pump and x-ray probe experiments relies on accurate calculations of x-ray spectra of adsor- bates for the correct identification of the spectral signatures and their dynamical evolution. We show that experimental x-ray absorption can be well reproduced for different binding sites in a static prototype system CO/Ni(100) at a standard density functional theory generalized- gradient-approximation level of theory using a plane-wave basis and pseudopotentials. This validates its utility in analyzing ultrafast x-ray probe experiments. The accuracy of computed relative core level binding energies is about 0.2 eV, representing a lower limit for which spectral features can be resolved with this method. We also show that the commonly used Z +1 approximation gives very good core binding energy shifts overall. However, we find a discrepancy for CO adsorbed in the hollow site, which we assign to the significantly stronger hybridization in hollow bonding than in on-top. ©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/10.1063/5.0028002 INTRODUCTION Surface-sensitive x-ray techniques represent an invaluable tool in surface science. The tunability and intensity of modern syn- chrotron sources result in high-resolution x-ray spectra, allowing the identification of adsorbed atomic and molecular species, as well as their adsorption sites. The chemical environment of a core-ionized atom can be determined using x-ray photoelectron spectroscopy (XPS), while additional information about the local electronic struc- ture can be obtained from x-ray absorption (XAS) and emission spectra (XES). Detailed knowledge about the formation and popu- lation of bonding and antibonding states has been gained for a num- ber of atomic and molecular adsorbates (e.g., C, O, N, CO, and N 2 on transition metal surfaces) relevant for common heterogeneous catalytic processes.1,2 The advent of free-electron lasers delivering ultrashort x- ray laser pulses offers new opportunities for gaining knowledge about chemical reactions on the sub-picosecond time scale where bonds typically form and break. Reactions on the surface, and/ordesorption from it, can be initiated using an ultrashort optical laser pulse, while a probing x-ray pulse with a duration of ∼10 fs gives time-resolved spectra during the evolution of the system, with sub- ps resolution.3–7This has led to a new understanding of surface reaction pathways.4,8–10However, the interpretation of the x-ray probe spectrum relies on accurate theoretical modeling of spectra from whatever species and their structure that may be present and dynamically evolving on the surface. When moving toward systems relevant for industrial catalysis, the number of intermediates and non-equilibrium structures in the given reaction can be high and reliable simulations of the spectra are crucial. Due to the complicated electronic response to core-hole cre- ation, calculating XAS/XES of condensed phase species formally requires a many-body formalism, and this is often accomplished by solving the Bethe–Salpeter equation.11,12However, because many- body effects are generally negligible for the lower levels of adsor- bates on metal surfaces, much simpler density functional theory (DFT) calculations are often used to simulate XAS and XES. While this does not allow the treatment of metastable structures where AIP Advances 10, 115014 (2020); doi: 10.1063/5.0028002 10, 115014-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv electronic excitations are important, the short lifetime13,14(∼fs) of such excitations of adsorbates on metal surfaces, compared to the time resolution ( ∼100 fs) in pump-probe experiments, means they have a negligible effect on the spectrum compared to adsorbate vibrational excitation, diffusion, and desorption. DFT is known to capture trends in bonding of adsorbate systems remarkably well, and calculated trends of XAS/XES of adsorbates on a surface have shown a good overall agreement with experiment.15,16However, the interpretation of the ultrafast dynamics occurring on surfaces that is observed by x-ray spectroscopy requires a quantitative theoreti- cal understanding of the relationship between the structural changes occurring on the surface and the x-ray spectra. In order to assess how quantitative this can be, we investigate the accuracy of a standard DFT analysis of XPS and XAS, employing a Generalized-Gradient- Approximation (GGA) functional, a plane-wave basis set, and core- hole pseudopotentials. This approach is, therefore, scalable to fairly large, periodic systems including itinerant magnetism within the framework commonly used to study periodic solids. We show that this gives reliable and accurate results for the XAS structure depen- dence on a system where static experimental data are available for the same adsorbate/metal system at a number of different binding sites that have all been well characterized by a number of different surface science techniques. We choose the CO/Ni(100) system since the CO molecule binds with very similar adsorption energies in top, hollow, and bridge sites. By changing the CO coverage, the molecule moves to different adsorption sites, and the co-adsorption of hydrogen can be used to give a few different stable, high-coverage phases.17Detailed experimental XAS spectra are, therefore, available for these well- characterized systems for comparison with our calculations.18,19 Without hydrogen, CO at a coverage of 0.5 ML binds on-top and forms a c(2 ×2) structure, while at θ=0.67 ML coverage, a p(3√ 2 ×√ 2)R45 phase is formed with all CO molecules at bridge sites; the FIG. 1. Structures used for spectrum calculations.coverage depends on the adsorption temperature and background CO pressure.20Initially, adsorbing 1 ML of hydrogen at 80 K and then dosing with CO lead again to on-top CO in a c(2 ×2) structure with 0.5 ML saturation coverage.21Annealing this structure to 170 K leads to a c(2√ 2×√ 2)R45 phase, with the half CO occupying top sites and the other half hollow sites. Atomic hydrogen is found in all remaining hollow sites.17 Our structures are shown in Fig. 1. Since XAS has been recorded for all these surface phases using an identical experimental setup,18,19it is an ideal system for comparing to theoretical spectra. We, thus, calculate O and C K-edge XAS spectra for all the men- tioned surface structures using the DFT methods described below. For comparison, the adsorption onset was also calculated for a pure 1/4 ML CO coverage at different binding sites. METHODS We use the DFT code Quantum ESPRESSO,22,23which uses a plane-wave basis for the valence electrons and pseudopotentials to represent the core electrons. Plane waves and pseudopotentials are the standard approach for solid state systems due to the computa- tional efficiency; since we need fairly large supercells for calculating x-ray spectra, this is an obvious advantage. For treating the core-excited system, we use a frozen-core-hole pseudopotential for the core-ionized atom. The XAS absorption onset equals the binding energy of the core electron relative to the Fermi level,2which for a metallic system is equal to the experimental XPS level. Calculating it requires two single-point DFT calculations: one for the neutral ground state and the other for the core-excited state. Since different pseudopotentials (with and without core hole) are being employed in these two DFT calculations, absolute binding energies cannot be predicted, but the relative shifts due to structural changes can be obtained. To preserve the charge neutrality of our supercell in the DFT calculations, the core-excited electron is added as an extra valence electron at the Fermi level. Physically, this corresponds to immediate screening by the metal electrons; the electron screening response to the creation of a core hole occurs on a time scale much faster than the core-hole lifetime (6–7 fs for C24,25). Therefore, using a charged cell to describe adsorbates on metals incorrectly neglects the availabil- ity of electrons from the metallic Fermi level, and it has previously been seen that neutral cells indeed do yield better agreement with experiments.15 Instead of introducing an explicit core hole in electronic structure calculations, core-electron binding energy shifts may be obtained using the Z +1 approximation,1,26where the core-excited atom is represented by the next element in the periodic table. This has been found to work well both for gas phase molecules and adsor- bates on metal surfaces.27In our calculations, this amounts to using the pseudopotentials of N and F for C and O, respectively, instead of the explicit core hole. Calculating XAS requires scattering matrix elements involv- ing unoccupied states far above the Fermi level in addition to the above described computation of the core-electron binding energy. Instead of explicitly calculating the unoccupied band struc- ture, which requires a much larger computational effort than sim- ply converging occupied electronic bands, we calculate the spec- trum using a Lanczos recursion Green’s function technique, as AIP Advances 10, 115014 (2020); doi: 10.1063/5.0028002 10, 115014-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv TABLE I. Core-level binding energy shifts in eV of C 1s and O 1s. From experimental XPS measurements,17calculated with a full-core-hole (FCH) pseudopotential and calculated in the Z +1 approximation. Site C Expt. C FCH C Z +1 O Expt. O FCH O Z +1 c(2×2) 0 0 0 0 0 0 p(3√ 2×√ 2) −0.4 −0.47 −0.36 −0.9 −1.10 −1.09 c(2×2)+H 0.4 0.22 0.18 0.7 0.49 0.48 c(2√ 2×√ 2)+H (∗top) 0.3 0.19 0.13 0.6 0.43 0.41 c(2√ 2×√ 2)+H (∗hollow) −0.8 −0.60 −0.28 −1.9 −1.86 −1.83 1/4 ML bridge −0.43 −0.40 −1.01 −1.08 1/4 ML hollow −0.64 −0.46 −1.90 −1.96 implemented in the xspectra code.28–30For spectrum calculations, a half-core-hole pseudopotential was used on the excited atom (transition-potential approach16), which has been shown to give reliable results.16,31The spectra were then shifted according to our calculated absorption onsets ( Δ-Kohn–Sham method). This gives a better common reference scale for comparing different adsorp- tion sites.31All spectra were finally shifted by a common refer- ence energy, chosen by matching the calculated and experimental c(2×2) spectra. By comparison with the experimental c(2 ×2) spectra, a broadening of 0.4 eV FWHM for C and 0.8 eV for O was applied to account for core-hole lifetime, vibrational,1and instru- mental broadening. The x rays are polarized parallel to the surface, probing the LUMO (2 π∗) of the CO molecule, as in the experiments. The revised Perdew-Burke-Ernzerhof (RPBE) functional was used throughout this study due to its accuracy in determining sur- face adsorption energies.32All calculations are spin-polarized (unre- stricted Kohn–Sham). For c(2 ×2) phases, we use a four-layer 2 ×2 slab model for initial geometry optimization with 20 Å of vac- uum separating the slab from its periodic image. The experimental lattice constant (3.52 Å) is used, and a 4 ×4×1 Monkhorst–Pack grid for Brillouin zone integration. For excited-state calculations, we double the cell in the x- and y-directions to 4 ×4×4 (correspond- ingly reducing the k-point grid to 2 ×2×1) to reduce the interaction of the core hole with its periodic images. For p(3√ 2×√ 2)R45, a 32- atom cell and a 2 ×4×1 k-point grid were used for optimization; for c(2√ 2×√ 2)R45 23 atoms and 3 ×6×1 k-points were used for optimization. For excited states, the supercells of Fig. 1 were used with a k-point grid of 2 ×2×1 points. XAS spectra and densities of states were computed using a refined 11 ×11×1 k-point grid while keeping the density fixed (non-SCF or Harris calculation). Ultrasoft pseudopotentials33were used for atoms without core excitations. For sites where core excitations were considered, ultrasoft half- and full- core-hole pseudopotentials for O, and due to better tested perfor- mance, norm-conserving half- and full core-hole pseudopotentials for C were generated using the “atomic” code included with the Quantum Espresso distribution.22Full- and half-core pseudopoten- tials are generated by occupying the 1 sshell in the single-atom DFT solution by only, respectively, 1 and 1.5 electrons. The constructed 2s-pseudo wavefunctions are nodeless, enabling representation in a plane-wave basis with a reasonable kinetic energy cutoff on the order of 500 eV. The plane-wave and density cutoffs were correspondingly chosen for surface calculations to be 500 eV and 5000 eV, respec- tively. The geometries were optimized until all forces were less than 0.03 eV/Å with the bottom two layers kept fixed.The core-level binding energy for 1/4 ML coverage was cal- culated using the GPAW34,35code, using PAW:s with an explicit core-hole. The finite-difference mode with a real-space grid spac- ing of 0.2 Å was used; all other parameters were the same as in the Quantum Espresso calculations described above. FIG. 2. pDOS at the core-excited atom in the c(2√ 2×√ 2)R45 structure for the case of excited C (upper panel) and O (lower), respectively, using a core-hole pseudopotential vs the Z +1 approximation. The arrows indicate the relative pDOS peak shifts between the top and hollow binding sites for C and O, respectively. These shifts are of similar magnitude comparing Z +1 and core-hole pseudopo- tential approaches in the case of O. There is an ∼0.5 eV shift in the lowest (4 σ) C peak for the core-hole pseudopotential approach, but there is only a negligible shift for the Z +1 approximation. AIP Advances 10, 115014 (2020); doi: 10.1063/5.0028002 10, 115014-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv RESULTS Table I shows the calculated relative binding energies, com- pared with experimental XPS energy shifts taken from Ref. 18. We use the c(2 ×2) phase as a reference for all other energies. The calcu- lated shifts are consistently within 0.2 eV of the experimental value, which is within the accuracy of typical DFT calculations for this type of surface bond.32There is no consistent error in the simula- tions between the different phases, or between the C and O shifts, indicating that the theoretical methods are free of systematic errors regarding the excitation of a certain atom or certain kind of bond- ing. Co-adsorbing hydrogen represents a significant change in the surface composition and the electronic structure, but the accuracy remains similar. Note that this uncertainty is still fairly large com- pared to the energy resolution of modern experiments. The good agreement with experimental values indicates that the uncertainty of GGA-level DFT for adsorption energies of CO on transition met- als36,37does not severely affect our results since the XPS binding energy shifts are significantly larger than the adsorption energy dif- ferences [which we find to be at most 70 meV for CO/Ni(100)]. Furthermore, some of the error is expected to be similar in the initial and final states and, thus, cancels out. The Z +1 approximation yields remarkably good agreement in the case of core-excited O, where the bond is more ionic and the core-hole effect is largely electrostatic. For C, it also gives accurate results in most cases, but the high degree of hybridization for C∗ in the hollow site turns out to be incompletely described in the Z +1 approximation. This can be clearly seen in the atomic orbital- projected density of states (pDOS), as shown in Fig. 2. While the oxygen levels are shifted in a similar way between the top and hol- low sites when using a core-hole pseudopotential and the Z +1 approximation, the pDOS at the core-ionized carbon atom shows no such simple relation. In particular, the lowest shown (4 σ-derived) level has the same energy in the two sites using the Z +1 approx- imation, while the more realistic core-hole pseudopotential givesdifferent hybridization in different chemical environments so that the valence electronic structure in the final state is not the same in the Z +1 vs core-hole case. The electrostatic contribution to the Kohn–Sham effective potential is very similar for the core-hole and Z+1 approximations and nearly identical at sufficient distance from the nucleus. The exchange-correlation contribution to the potential, however, is more negative near the core in the Z +1 case due to more core-electronic charge (e.g., we calculate the 2 s-state of an isolated N+-ion to be ∼2.5 eV lower in energy than for core-hole C). The 2s-states have a non-zero amplitude at the core and are, therefore, affected more strongly than the 2 p-states by the lower Kohn–Sham effective potential, which may affect the hybridization of strongly interacting CO in the hollow site.38This shows that caution may be needed when using the Z +1 approximation for calculating quanti- tative shifts in the core-level binding energy in situations where the hybridization of low-lying energy levels takes place. While both the 1/4 ML bridge and hollow sites give similar core-level binding energies as p(3√ 2×√ 2) and c(2√ 2×√ 2), respectively (except for hollow C Z +1, for the reasons discussed above), for the top site, the presence of hydrogen gives a significant energy shift compared to the pure c(2 ×2) reference. Disregarding the hydrogen would, thus, underestimate the energy shift between the top and hollow sites, which occurs in the actual experimental c(2√ 2×√ 2) co-adsorbed structure. This illustrates the sensitivity of core binding energies to the surface structure and co-adsorbates, and the power of the core-hole pseudopotential DFT approach in discriminating such different structures and identifying the likely observed experimental structure. Figure 3 shows the calculated XAS at the C and O K-edges, respectively, using the calculated XPS shifts, and compared to the experimental data of Ref. 19. As also seen experimentally, the main resonance narrows when co-adsorbing H, consistent with a weak- ened chemisorption bond. The double-peak structure of the c(2√ 2 ×√ 2)R45 phase, where the CO molecules occupy two different FIG. 3. Calculated XAS around the C K-edge (left) and the O K-edge (right) compared to experiment.19A common shift has been applied to align the c(2 ×2) spectra with the experimental ones. The dotted lines show the contributions of the top and hollow sites in the c(2√ 2×√ 2)R45 unit cell. AIP Advances 10, 115014 (2020); doi: 10.1063/5.0028002 10, 115014-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv sites, is also reproduced. However, when using the calculated bind- ing energies to shift the spectrum, the double peak is not as clearly seen in the calculated C K-edge spectrum as in the experiment. This stems from the fact that, as seen in Table I, the experimental C∗shift between top and hollow is 1.1 eV, while the calculated one is only 0.79. In the Z +1 approximation, the shift is just 0.41 eV, and the double-peak structure would not be resolved at all. CONCLUSIONS Ultrafast optical pump and x-ray probe experiments offer unique capabilities for gaining insight into reactions on surfaces; however, correct identification of surface species often requires accurate calculated spectra. We have shown that DFT using core- hole pseudopotentials in a plane-wave code gives quite exact core- level binding energy shifts (within 0.2 eV) and x-ray absorption spectra for the experimentally well-studied CO/Ni(100) system. Thus, calculated XPS shifts and x-ray spectrum simulations are pow- erful tools to identify binding sites from experimental spectra as we have benchmarked here for static experiments on CO/Ni(100). Even if DFT fails to predict small chemisorption energy differences between sites, the XPS shifts are typically larger, which we expect to be useful in guiding the analysis of ultrafast experiments with a dynamic evolution between different binding sites .Resorting to much more expensive approaches to account for explicit many-body effects can likely be avoided as a first level of approximation. Our shifts are accurate enough to distinguish between different binding sites and reproduce the shape and location of the XAS K- edge. Compared to experiments, the main uncertainty comes from the calculated absorption onset, while the shape of the spectral peaks is very well reproduced. The limitations that turn up in our c(2√ 2 ×√ 2)R45 phase must be kept in mind when comparing different XAS peaks coming from the same atomic species at different sites. Since the uncertainty of about 0.2 eV is independent for the two sites, this method can reliably identify peaks more than ≈0.4 eV apart. This also illustrates the value of any additional experimen- tal information—the O K-edge spectra show a very clear distinction between the peaks, leading to an unambiguous identification of the correct structure. We further see that the Z +1 approximation, while intuitively appealing and in most cases quantitatively accurate, must be used with caution in situations with the strong covalent bonding and hybridization of low-lying energy levels, where the core-excited state needs to be accurately represented. ACKNOWLEDGMENTS This research was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division, Catalysis Science Program to the Ultrafast Catalysis FWP 100435, at SLAC National Accelerator Laboratory under Contract Grant No. DE-AC02-76SF00515. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract Grant No. DE-AC02-05CH11231. G.L.S.R. and L.G.M.P. acknowledge support from the Knut and Alice Wallenberg Founda- tion through Grant No. KAW-2016.0042. The GPAW calculationsused resources provided by the Swedish National Infrastructure for Computing (SNIC) at the HPC2N and PDC centers. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1A. Nilsson, J. Electron Spectrosc. Relat. Phenom. 126, 3 (2002). 2A. Nilsson and L. G. M. Pettersson, Surf. Sci. Rep. 55, 49 (2004). 3M. Beye, H. Öberg, H. Xin, G. L. Dakovski, M. Dell’Angela, A. Föhlisch, J. Gladh, M. Hantschmann, F. Hieke, S. Kaya, D. Kühn, J. LaRue, G. Mercurio, M. P. Minitti, A. Mitra, S. P. Moeller, M. L. Ng, A. Nilsson, D. 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5.0033284.pdf

Appl. Phys. Lett. 117, 182104 (2020); https://doi.org/10.1063/5.0033284 117, 182104 © 2020 Author(s).Electron and hole mobility of rutile GeO2 from first principles: An ultrawide-bandgap semiconductor for power electronics Cite as: Appl. Phys. Lett. 117, 182104 (2020); https://doi.org/10.1063/5.0033284 Submitted: 16 October 2020 . Accepted: 20 October 2020 . Published Online: 06 November 2020 K. Bushick , K. A. Mengle , S. Chae , and E. Kioupakis COLLECTIONS Paper published as part of the special topic on Ultrawide Bandgap Semiconductors UBS2021 ARTICLES YOU MAY BE INTERESTED IN Band offset determination for amorphous Al 2O3 deposited on bulk AlN and atomic-layer epitaxial AlN on sapphire Applied Physics Letters 117, 182103 (2020); https://doi.org/10.1063/5.0025835 Graphene-induced crystal-healing of AlN film by thermal annealing for deep ultraviolet light- emitting diodes Applied Physics Letters 117, 181103 (2020); https://doi.org/10.1063/5.0028094 Full-composition-graded In xGa1−xN films grown by molecular beam epitaxy Applied Physics Letters 117, 182101 (2020); https://doi.org/10.1063/5.0021811Electron and hole mobility of rutile GeO 2from first principles: An ultrawide-bandgap semiconductor for power electronics Cite as: Appl. Phys. Lett. 117, 182104 (2020); doi: 10.1063/5.0033284 Submitted: 16 October 2020 .Accepted: 20 October 2020 . Published Online: 6 November 2020 K.Bushick, K. A. Mengle, S.Chae, and E. Kioupakisa) AFFILIATIONS Department of Materials Science and Engineering, University of Michigan, Ann Arbor 48109, USA Note: This paper is part of the Special Topic on Ultrawide Bandgap Semiconductors. a)Author to whom correspondence should be addressed: kioup@umich.edu ABSTRACT Rutile germanium dioxide (r-GeO 2) is a recently predicted ultrawide-bandgap semiconductor with potential applications in high-power elec- tronic devices, for which the carrier mobility is an important material parameter that controls the device efficiency. We apply first-principles calculations based on density functional and density functional perturbation theory to investigate carrier-phonon coupling in r-GeO 2and predict its phonon-limited electron and hole mobilities as a function of temperature and crystallographic orientation. The calculated carriermobilities at 300 K are l elec;?~c¼244 cm2V/C01s/C01,lelec;k~c¼377 cm2V/C01s/C01,lhole ;?~c¼27 cm2V/C01s/C01, and lhole ;k~c¼29 cm2V/C01s/C01.A t room temperature, carrier scattering is dominated by the low-frequency polar-optical phonon modes. The predicted Baliga figure of merit ofn-type r-GeO 2surpasses several incumbent semiconductors such as Si, SiC, GaN, and b-Ga 2O3, demonstrating its superior performance in high-power electronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0033284 Power electronics are important for the control and conversion of electricity, but inefficiencies cause energy loss during each step in the conversion process, resulting in a combined efficiency of /C2480% or less.1In the United States, the existing electricity grid is outdated for modern electricity usage and must be replaced with power-conversion electronics that are able to control the power flow more efficiently.Addressing inefficiencies to improve energy sustainability motivates the ongoing search for new materials for power-electronics devices. Ultrawide-bandgap semiconductors with gaps wider than GaN(3.4 eV) have been the focus of power-electronics materials research. 2 Important material parameters to consider for power-electronicsapplications include the possibility of doping (usually n-type, but ambipolar dopability is also desirable for CMOS devices), high carrier mobility lfor fast switching and low Ohmic losses, high thermal con- ductivity for efficient heat extraction, and a high critical dielectric breakdown field E Cand dielectric constant e0to enable high-voltage operation. The Baliga figure of merit, BFOM ¼1=4e0lE3 C,q u a n t i fi e s the performance of materials in power-electronic devices.3,4The BFOM depends most sensitively on the breakdown field, which increases superlinearly with the increasing bandgap and motivates thesearch for dopable ultrawide-gap semiconductors.Thebpolymorph of gallium oxide ( b-Ga 2O3) has been the recent focus of attention thanks to the availability of native substrates and the n-type dopability with Si or Ge.2While its electron mobility is lower than that of Si, SiC, or GaN, its ultrawide-bandgap of /C244.5 eV produ- ces a high breakdown field and a BFOM superior to these incumbenttechnologies. 5–9However, its low thermal conductivity that prevents heat extraction and the unfeasibility of p-type doping (due to the for- mation of self-trapped hole polarons10) limit its applicability.10–12To overcome these challenges and advance the frontier of power electron-ics, new ultrawide-bandgap semiconducting materials must be identi-fied and characterized. Recently, Chae et al. found that rutile germanium dioxide (r- GeO 2) is a promising ambipolarly dopable semiconductor13with an ultrawide bandgap (4.68 eV). Donors such as Sb Geand F Oare shallow (activation energy /C2425 meV), while Al Geand Ga Geacceptors are deeper with modest ionization energies of /C240.4–0.5 eV.13However, the co-incorporation of acceptors with hydrogen and subsequent annealing can produce high acceptor concentrations that exceed the Mott-transition limit and enable p-type conduction. Moreover, its measured thermal conductivity of 51 W/m K at 300 K surpasses thatofb-Ga 2O3and facilitates more efficient thermal management.14 Appl. Phys. Lett. 117, 182104 (2020); doi: 10.1063/5.0033284 117, 182104-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplRutile GeO 2displays similar chemical and structural properties to rutile SnO 2, an established n-type transparent conductor.15,16However, the wider bandgap of r-GeO 2is promising for deep-ultraviolet (UV) lumi- nescence and efficient power-electronics applications.13,17,18Yet the elec- trical conductivity of r-GeO 2and, thus, its viability and efficiency for power-electronics applications remain unexplored. In this work, we apply predictive atomistic calculations to deter- mine the phonon-limited electron and hole mobilities of r-GeO 2as a function of temperature and crystallographic orientation. We quantify the intrinsic phonon and carrier-phonon-coupling properties that impact carrier transport. We find that the room-temperature carrier mobility is dominated by the low-frequency polar-optical phonon modes, and the resulting mobility values at 300 K are lelec;?~c¼244 cm2V/C01s/C01,lelec;k~c¼377 cm2V/C01s/C01,lhole;?~c¼27 cm2V/C01s/C01, andlhole;k~c¼29 cm2V/C01s/C01. The electron mobility is higher than that of b-Ga 2O3, while the hole mobility at high temperature surpasses the values for p-type GaN. Our results demonstrate that r-GeO 2exhib- its a superior BFOM than current semiconductor technologies such as S i ,S i C ,G a N ,a n d b-Ga 2O3and is, therefore, of great interest for high- power and high-temperature electronics applications. To accurately predict the carrier and phonon properties of r- GeO 2, we use first-principles calculations based on density functional (DFT) and density functional perturbation theories (DFPT) in thelocal density approximation within Quantum ESPRESSO 19–21and the iterative Boltzmann transport equation (IBTE) with the EPW code.22–24We do not consider spin–orbit coupling. In previous work,18we calculated the quasiparticle band structure of r-GeO 2for the experimental lattice parameters25using the G 0W0method. We find that the effective masses obtained from G 0W0are similar to those obtained with a hybrid DFT functional (Table SI in the supplementary material ), verifying the reliability of our calculated band parameters. For phonon calculations, the lattice parameters and atomic positions were relaxed to prevent imaginary phonon frequencies, resulting in lattice parameters of a¼4:516 ˚Aa n d c¼2:978 ˚A that differ from experiment25byþ2.5% and þ4.1%, respectively. The phonon disper- sion, phonon frequencies at C, and sound velocities are included in Fig. S1 and Tables SII and SIII in the supplementary material The qua- siparticle energies, phonon frequencies, and carrier-phonon coupling matrix elements were calculated on 4 /C24/C26 Brillouin zone (BZ) sam- pling grids (using the charge density generated on an 8 /C28/C212 BZ sampling grid for higher accuracy) and interpolated to fine BZ sam- pling grids with the EPW code. Carrier velocities were evaluated with the velocity operator,26,27and the Fr €ohlich correction was applied to the carrier-phonon coupling matrix elements g.28The phonon-limited carrier mobilities were calculated over the 100–1000 K temperature range for a carrier concentration of 1017cm/C03.24The electron and hole mobilities were converged for carrier and phonon BZ sampling grids of 92/C292/C2138 and 48 /C248/C272, respectively. We sampled states within energy windows of 530 meV around the carrier Fermi energies, which accounts for energy differences during scattering of up to /C22hxmaxþ5kBTat 1000 K, where /C22hxmaxis the highest optical phonon energy and kBis the Boltzmann constant. With these calculation parameters, the mobility values at 300 K were converged within 1% for electrons and 2% for holes. We first discuss the mobility obtained from the iterative solution of the IBTE. The converged carrier mobilities at 300 K are lelec;?~c ¼244 cm2V/C01s/C01,lelec;k~c¼377 cm2V/C01s/C01,lhole;?~c¼27 cm2V/C01s/C01,a n d lhole;k~c¼29 cm2V/C01s/C01. Qualitatively, we expect a lower mobility for transport directions in which carriers have a higher effective mass. Quasiparticle band structure calculations find that the electron mass is anisotropic ( m/C3 e;?~c¼0:36m0andm/C3 e;k~c¼0:21m0) and approximately a factor of 2 lighter for the k~cdirection, which is consistent with the approximately 2 times higher mobility along k~c. We also find that the scattering rates for electrons are relatively isotro- pic (Fig. S2), indicating that the anisotropy of the calculated electron mobility is primarily driven by the anisotropy of the effective mass. On the other hand, the hole mobility is approximately isotropic, which is consistent with the relatively small directional dependence of theeffective mass ( m /C3 h;?~c¼1:29m0andm/C3 h;k~c¼1:58m0) and scattering rates (Fig. S2). Figure 1 shows the temperature dependence of the mobility. We can accurately fit the temperature dependence with a combination of exponential equations characteristic of electron-phonon scattering by two optical modes, 1/C14 lTðÞ¼1 l1e/C0T1 Tþ1 l2e/C0T2 T; (1) where l1,l2,T1,a n d T2are fitting parameters to describe the temper- ature dependence. l1andl2are in units of mobility, while T1andT2 are in units of temperature. Table I lists the fitted values for each car- rier type and direction. The two sets of li;TiðÞ correspond to the low- and high-energy polar-optical modes that dominate carrier scattering,as we discuss in greater detail below. Our fitted parameters demon- strate that the low-energy polar-optical modes characterized by the T 1 parameter dominate near room temperature. The high-energy polar- optical modes only become important at high temperatures (higher than 500 K) and to a larger extent for electrons than for holes. Next, we analyze the carrier-phonon coupling matrix elements and scattering lifetimes to understand how different phonon modes affect carrier scattering in r-GeO 2. We first determine the carrier- phonon coupling matrix elements for the bottom conduction and top valence bands for wave vectors along the C—X (?~c)a n dC—Z (k~c) directions [ Figs. 2(a) and2(d)]. Our results show that the polar-optical modes exhibit the strongest carrier-phonon coupling, as expected in polar materials such as r-GeO 2. However, the higher-frequency modes are not as effective at scattering low-energy carriers at room tempera- ture; they require either high temperatures to enable appreciable pho- non occupation numbers and scatter carriers by phonon absorption or high carrier energies to scatter electrons to lower-energy states by pho- non emission. Taking the thermal occupation of phonon modes ( nq) at room temperature ðkBT¼26 meV Þinto account, we find the dom- inant modes for carrier scattering by phonon absorption [ g2nq,Figs. 2(b)and2(e)] and phonon emission [ g2nqþ1 ðÞ ,Figs. 2(c) and2(f)]. Electrons and holes most strongly absorb the polar-optical modes with frequencies of 41, 49, 94, and 100 meV. As with phonon absorption, carriers scatter most strongly by phonon emission by the polar-optical phonons, as well as the Raman-active modes at 87 meV and 102 meV. We thus predict that carriers primarily scatter by absorption (and to a lesser extent by emission) of the A 2uand three E upolar-optical modes. The eigenmodes of these four dominant scattering phonons are shown inFigs. 2(g)–2(j) . Typically, the polar-optical modes limit the room-temperature mobility in oxide materials, and our electron-phonon coupling resultscombined with the mobility calculations indicate that this is the caseApplied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 182104 (2020); doi: 10.1063/5.0033284 117, 182104-2 Published under license by AIP Publishingin r-GeO 2as well.29Further evidence for the role of the polar-optical modes is provided by the mode-resolved carrier scattering rates as a function of carrier energy ( Fig. 3 ). Here, we refer to modes by their numerical index (in order of increasing frequency) since the direc-tional dependence of the LO-TO splitting reorders the mode frequen- cies. Mode 7 has the strongest effect at low energies and corresponds to the 41 meV E umode. In both the valence band Fig. 3(a) and con- duction band Fig. 3(b) , there is a noticeable step in the magnitude above 40 meV, where phonon emission becomes possible. The same is true for Mode 17 around 100 meV, which corresponds to both the94 meV A 2umode and the 100 meV E umode, depending on the direc- tion. Modes 9 and 10, which show the next largest contribution to thetotal electron linewidth, are both (depending on the direction) the 49 meV E uphonon mode. Finally, Mode 13 is associated with the Raman active 67 meV E gmode, though its contribution is approxi- mately one order of magnitude smaller than the more dominant polar optical modes. The carrier mobility is a necessary parameter to understand the performance of a semiconductor in electronic devices. The ultrawidebandgap of r-GeO 2(4.68 eV)17makes it especially suitable for high- FIG. 1. Electron and hole mobility lof r-GeO 2along the ?~candk~cdirections as a function of temperature for a carrier concentration of n¼1017cm/C03. The solid curves are fits to the data according to Eq. (1)andTable I . The dashed curves cor- respond to the low-energy and the dotted curves to the high-energy polar-opticalmodes. FIG. 2. (a)–(c) Square of the intraband electron-phonon coupling matrix element g and scattering of electrons via phonon absorption g2 el/C0phnq/C16/C17 and phonon emis- sion g2 el/C0phnqþ1/C0/C1/C16/C17 for the bottom conduction band from Ctoqas a function of the phonon wave vector qalong the C—X (?~c) andC—Z (k~c) directions, showing the phonon modes with the largest coupling strengths. Phonon occupa- tions are calculated using room temperature ðkBT¼26 meV Þ. Panels (d)–(f) con- tain the corresponding values for the hole-phonon interactions (i.e., the top valenceband). All four IR-active (polar-optical) modes show strong coupling to carriers for wave vectors near C, while the strongest-coupled Raman-active modes show a weak dependence with respect to q(optical deformation potential coupling). (g)–(j) Atomic displacements corresponding to the dominant polar phonon modes. Thelarger purple atoms are germanium, and the smaller red atoms are oxygen.TABLE I. Fitted parameters for the resistivity model given by1/C14 lTðÞ¼1 l1e/C0T1 T þ1 l2e/C0T2 T, where lðTÞ;l1;andl2are in units of cm2V/C01s/C01andT;T1;andT2in K to describe the mobility vs temperature for electrons and holes along the two main crystallographic directions. Parameters Electron, ?~cElectron, k~cHole,?~cHole,k~c l1(cm2V-1s/C01) 43.8 85.9 2.08 2.27 T1(K) 576 525 765 761 l2(cm2V/C01s/C01) 21.1 22.2 0.669 0.603 T2(K) 1221 1268 2691 2395Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 182104 (2020); doi: 10.1063/5.0033284 117, 182104-3 Published under license by AIP Publishingpower and high-temperature applications.13,18Table II lists the mate- rial parameters of r-GeO 2relevant for n-type and p-type power elec- tronics and compares them to incumbent technologies. Thebreakdown fields of b-Ga 2O3(with a gap of 4.5 eV) and r-GeO 2are evaluated using the breakdown field vs bandgap relation by Higashiwaki et al.9The electron mobility of r-GeO 2is lower than thato fS i ,S i C ,a n dG a Nb yo v e r7 0 % ,30–32but higher than that of b-Ga 2O3. Although the experimental electron mobilities of b-Ga 2O3 have typically been obtained with Hall measurements, it is the drift mobility that should be applied to evaluate the BFOM. A drift mobility of 80 cm/C02V/C01s/C01was measured33inb-Ga 2O3at 300 K, while the highest Hall mobility at 300 K is 184 cm/C02V/C01s/C01.30,34However, if the Hall factor at 300 K ( rH¼1:68)35is applied to convert the Hall to drift mobility ( lHall¼ldriftrH), the highest measured room- temperature drift mobility of Ga 2O3is 109 cm/C02V/C01s/C01.M o r e o v e r , recent calculations of the phonon-limited carrier mobility of b-Ga 2O3 from first principles using a similar approach to the present work36 report intrinsic electron and hole drift mobilities at 300 K of 258 cm/C02 V/C01s/C01and 1.2 cm/C02V/C01s/C01, respectively. Overall, r-GeO 2displays the largest BFOM out of the materials considered here as it exhibits the largest ECvalue (since it has the widest bandgap), the highest e0, and a higher electron mobility than b-Ga 2O3. Though ultrawide-bandgap materials such as nitrides and b- Ga2O3are promising for numerous technological applications, one consistent shortcoming is the relative difficulty of p-type doping and low hole mobilities. This is evident in the differences between electron and hole mobility data in Table II . Our findings of a sizable hole mobility in r-GeO 2, in combination with the results of Chae et al. on the feasibility of ambipolar doping,13demonstrate that it may be a can- didate material for high-power CMOS devices, in contrast to b-Ga 2O3, which is limited by self-trapped hole polarons.10Moreover, the hole mobility of r-GeO 2is comparable to that of GaN at temperatures below /C24500 K, matching or exceeding experimental values and within a factor of two of theoretical values (Fig. S3). At higher temperatures, r-GeO 2shows higher hole mobility than experimentally reported GaN values despite having a wider bandgap. The combination of a higher BFOM with the prediction of p-type doping and the possibility of efficient hole conduction demonstrate the promise of r-GeO 2as a superior semiconductor compared to incumbent technologies such as b-Ga 2O3and GaN for high-power and high-temperature electronic applications. In summary, we calculate the phonon-limited electron and hole mobilities of r-GeO 2as a function of temperature and crystallographic orientation from first principles and provide atomistic insights into the dominant carrier-scattering mechanisms. The combination of its ultrawide bandgap of 4.68 eV with its predicted electron mobility thatFIG. 3. Phonon-mode-resolved carrier imaginary self-energies (inverse scattering life- times) for (a) holes and (b) electrons. The total self-energy (black) is decomposedinto the contributions by the dominant phonon modes. Mode 7 corresponds to the41 meV E umode, Modes 9 and 10 correspond to the 49 meV E uphonon mode, Mode 13 corresponds to the 67 meV E gmode, and Mode 17 corresponds to the 94 meV A 2umode and the 100 meV E umode. The 41 meV E umode dominates scat- tering at low carrier energies (less than approximately 100 meV), while at higher ener-gies, scattering by the 94 meV A 2umode and the 100 meV E umode dominates. TABLE II. Baliga figure of merit for r-GeO 2in comparison to other common ultra-wide bandgap semiconductors. Electron mobilities and dielectric breakdown fields for all materials are at room temperature and for carrier densities of 1016cm/C03except those of b-Ga 2O3(1016cm/C03and 1012cm/C03forlHalland 1017cm/C03forldrift), GaN (1017cm/C03forlh), and r-GeO 2(1017cm/C03). MaterialStatic dielectric constant, e0Electron mobility, l(cm2V/C01s/C01)Hole mobility, l(cm2V/C01s/C01)Dielectric breakdown field, EC(MV cm/C01)n-BFOM (106V2X/C01cm/C02)p-BFOM (106V2X/C01cm/C02) Si 11.9371240 (drift)310.398.8 4H-SiC 9.731980 (drift)312.593300 GaN 10.4 ( k~c)381000 (drift)3231 (Hall)393.398300 260 b-Ga 2O3 10.040184; 180 (Hall)30,34… 6.4911 000; 10 000 6300; 4600 109; 80 (drift)33,34 258 (drift)361.2 (drift)365.83611 000 52 r-GeO 2 14.5 (?~c)41244 (drift, ?~c) 27 (drift, ?~c) 7.0 27 000 3000 12.2 (k~c)41377 (drift, k~c) 29 (drift, k~c) 7.0 35 000 2700Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 182104 (2020); doi: 10.1063/5.0033284 117, 182104-4 Published under license by AIP Publishingis higher than that of b-Ga 2O3enables a BFOM that surpasses estab- lished power-electronics materials such as Si, SiC, GaN, and b-Ga 2O3. In addition, our results for its sizable hole mobility in combination with the theoretical prediction of its ambipolar dopability13indicate potential applications for p-type and CMOS devices. Our results high-light the advantages of r-GeO 2compared to incumbent material tech- nologies for power-electronics applications. See the supplementary material for a comparison of the effective masses calculated with G 0W0and the hybrid DFT functional, an ana- lytical discussion of the calculated phonon frequencies and soundvelocities, the data for the directionally resolved electron self-energydue to electron-phonon interaction, and a comparison of the hole mobility of r-GeO 2to available data for GaN. AUTHORS’ CONTRIBUTIONS K.B. and K.A.M. contributed equally to this work. This work was supported as part of the Computational Materials Sciences Program funded by the U.S. Department ofEnergy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0020129. It used resources of the National Energy Research Scientific Computing (NERSC) Center, a DOE Office of ScienceUser Facility supported under Contract No. DE-AC02–05CH11231.K.A.M. acknowledges the support from the National Science Foundation Graduate Research Fellowship Program through Grant No. DGE 1256260. K.B. acknowledges the support of the DOEComputational Science Graduate Fellowship Program throughGrant No. DE-SC0020347. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1C. Murray, S. Guha, D. Reed, G. Herrera, K. Kleese van Dam, S. Salahuddin, J. Ang, T. Conte, D. Jena, R. Kaplar, H. Atwater, R. Stevens, D. Boroyevich, W.Chappell, T.-J. K. Liu, J. Rattner, M. Witherell, K. Afridi, S. Ang, J. Bock, S. Chowdhury, S. Datta, K. Evans, J. Flicker, M. Hollis, N. Johnson, K. Jones, P. Kogge, S. Krishnamoorthy, M. Marinella, T. Monson, S. Narumanchi, P.Ohodnicki, R. Ramesh, M. Schuette, J. 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5.0021650.pdf

J. Appl. Phys. 128, 135705 (2020); https://doi.org/10.1063/5.0021650 128, 135705 © 2020 Author(s).Vacancy defects induced changes in the electronic and optical properties of NiO studied by spectroscopic ellipsometry and first-principles calculations Cite as: J. Appl. Phys. 128, 135705 (2020); https://doi.org/10.1063/5.0021650 Submitted: 13 July 2020 . Accepted: 21 September 2020 . Published Online: 07 October 2020 Kingsley O. Egbo , Chao Ping Liu , Chinedu E. Ekuma , and Kin Man Yu ARTICLES YOU MAY BE INTERESTED IN Design strategy for p-type transparent conducting oxides Journal of Applied Physics 128, 140902 (2020); https://doi.org/10.1063/5.0023656 Capillary-fed, thin film evaporation devices Journal of Applied Physics 128, 130901 (2020); https://doi.org/10.1063/5.0021674 Design and fabrication of highly selective and permeable polymer membranes Journal of Applied Physics 128, 131102 (2020); https://doi.org/10.1063/5.0015975Vacancy defects induced changes in the electronic and optical properties of NiO studied by spectroscopic ellipsometry and first-principlescalculations Cite as: J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 View Online Export Citation CrossMar k Submitted: 13 July 2020 · Accepted: 21 September 2020 · Published Online: 7 October 2020 Kingsley O. Egbo,1 Chao Ping Liu,1,2 Chinedu E. Ekuma,3,4 and Kin Man Yu1,a) AFFILIATIONS 1Department of Physics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon 999077, Hong Kong 2Research Center for Advanced Optics and Photoelectronics, Department of Physics, College of Science, Shantou University, Shantou, Guangdong 515063, People ’s Republic of China 3Department of Physics, Lehigh University, Bethlehem, Pennsylvania 18015, USA 4Institute for Functional Materials and Devices, Lehigh University, Bethlehem, Pennsylvania 18015, USA a)Author to whom correspondence should be addressed: kinmanyu@cityu.edu.hk ABSTRACT Native defects in semiconductors play an important role in their optoelectronic properties. Nickel oxide (NiO) is one of the few wide-gap p-type oxide semiconductors and its conductivity is believed to be controlled primarily by Ni-vacancy acceptors. Herein, we present a systematic study comparing the optoelectronic properties of stoichiometric NiO, oxygen-rich NiO with Ni vacancies (NiO: VNi), and Ni-rich NiO with O vacancies (NiO: VO). The optical properties were obtained by spectroscopic ellipsometry, while valence band spectra were probed by high-resolution x-ray photoelectron spectroscopy. The experimental results are directly compared to first-principles densityfunctional theory + Ucalculations. Computational results confirm that gap states are present in both NiO systems with vacancies. Gap states in NiO: Vo are predominantly Ni 3 dstates, while those in NiO: V Niare composed of both Ni 3 dand O 2 pstates. The absorption spectra of the NiO: VNisample show significant defect-induced features below 3.0 eV compared to NiO and NiO: VOsamples. The increase in sub-gap absorptions in NiO: VNican be attributed to gap states observed in the electronic density of states. The relation between native vacancy defects and electronic and optical properties of NiO are demonstrated, showing that at similar vacancy concentration, the optical constantsof NiO: V Nideviate significantly from those of NiO: VO. Our experimental and computational results reveal that although VNiare effective acceptors in NiO, they also degrade the visible transparency of the material. Hence, for transparent optoelectronic device applications, anoptimization of native V Nidefects with extrinsic doping is required to simultaneously enhance p-type conductivity and transparency. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021650 INTRODUCTION Nickel oxide (NiO) is a material with vast technological potential and has been widely studied as a prototypical p-type transition metal (TM) oxide.1,2It is an antiferromagnetic oxide with its Néel tempera- ture, T Nat about 523 K. Previously described as a Mott –Hubbard insu- lator, NiO is now best characterized as a strongly correlatedcharge-transfer insulator, with its O 2 pband lying between the Ni 3 d upper and lower Hubbard bands. 3T h ee x c i t i n gn a t u r eo ft h ee l e c t r o n i c structure of NiO makes it an extensively studied system.4–8Morerecently, NiO has also found many applications in optoelectronic devices. For example, p-type NiO has been broadly applied as the hole transport layer in solar cells,9–12ultraviolet photodetectors,13–15 transparent junction diodes,16thin-film transistors,17and visible light transparent solar cells.18,19On the other hand, semi-insulating stoichio- metric NiO and Ni-rich NiO have been explored as major componentmaterials for resistive switching and capacitance modulationapplications. 20–22These technological potentials have continued to attract research interests in NiO, with particular efforts to better under- stand its doping23and transport properties.24Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-1 Published under license by AIP Publishing.Native defects play a central role in controlling the electrical and optical properties of oxide semiconductors,25,26for example, Shook et al. recently showed the effects of vacancies on the optoelec- tronic properties of Cu and Ag-based p-type transparent conductingoxides. 27In NiO, optoelectronic and transport properties are strongly affected by its native defects.28It has been demonstrated that Ni vacancy ( VNi) defects lead to p-type conductivity in NiO. Lany et al.29and Dawson et al.20showed that under an O-rich growth condition, the defect formation energy of VNiis very low, while cor- responding hole killers such as oxygen vacancies VOhave high for- mation energy. These findings are consistent with the observed p-type conductivity observed in O-rich NiO materials.30,31Moreover, native vacancy defects have also been shown to strongly affect theoptical character and bandgap nature of NiO. Ni-vacancy ( V Ni) states were believed to cause degradation in its visible light transpar-ency. Newman and Chrenko 32noted that NiO with excess oxygen appeared darker, and they attributed this to background absorption in the spectral range of 0.1 to 3.0 eV due to defects with increasingO/Ni ratio. Ono et al. 33studied the relationship of the transmittance and electrical properties of NiO and suggested that VNi-induced p-d charge transfer transitions may have led to the darker color of the sputter-deposited thin films. However, a critical study on optical properties, including the complex dielectric functions and opticalconstants, has not been explored for these NiO systems with Nivacancies showing p-type character. Hence, detailed knowledge of the effects of native defects such as V Niand VOon the optical properties of NiO are still lacking. In this study, a comprehensive investigation of the correla- tion between vacancy defects and the optical behavior and elec-tronic properties of NiO was performed using Spectroscopic Ellipsometry (SE) and X-ray Photoemission Spectroscopy on NiO thin films sputter deposited in stoichiometric, Ni-rich, and O-richenvironments. We note that when NiO is grown in an O-rich orNi-rich environment, aside from the favorable formation of Nivacancies or O vacancies, respectively, it is also possible that other native defects such as interstitials and complexes would be formed. However, in NiO grown in an O-rich environment, it hasbeen established that the formation energy of Ni vacancies ismuch lower than that of other defects. 29Hence, it is believed that they are responsible for its observed p-type behavior. Thus, following previous reports that vacancy type defects have lower formation energies,20,29,34interstitial type defects are not consid- ered in this work. In the following, Ni-rich and O-rich NiO thinfilms are referred to as NiO: V O(NiO with O vacancies) and NiO: VNi(NiO with Ni vacancies), respectively. The term “stoichiomet- ric”here refers to films sputtered from a NiO ceramic target using Ar gas only. Standard SE was used to obtain the complex dielec-tric functions, optical constants, and absorption coefficient ofNiO samples with different vacancy defects. 35SE is a well-known optical technique for determining the dielectric function and, hence, the optical constants of thin films with high precision. Ithas been used to study the optical properties of several transpar-ent metal oxides. 36,37In addition, first-principles density func- tional theory (DFT) + Ucalculations of the optical properties and electronic structures of NiO with Ni and O vacancies are also carried out, and these computed results are directly compared tothe experimental data.EXPERIMENTAL AND COMPUTATION PROCEDURE Materials synthesis and characterization Stoichiometric NiO, Ni-rich NiO (NiO: V O), and O-rich NiO (NiO: VNi) thin-film samples were synthesized using a radio- frequency magnetron sputtering system at a substrate temperatureof 280 °C. Ni metal target and NiO ceramic targets were usedduring the sputtering process to deposit samples on soda-lime glass. The oxygen flow ratio, r(O 2)=f(O2)/[f(Ar)+ f(O2)], in the sputtering gas was controlled by separately regulating the flow rateof pure Ar and O 2during deposition. Stoichiometric NiO samples were deposited in a pure Ar environment [ r(O2) = 0] from a stoi- chiometric NiO (99.99%) ceramic target, while O-rich NiO: VNi samples were deposited with r(O2) = 4%. Ni-rich NiO: VOsamples were deposited by co-sputtering a Ni metallic target and the NiOtarget. Ni content is controlled by the DC sputtering power for themetal Ni target. Electrical properties of deposited films were investigated by Hall-effect measurements in the van der Pauw geometry with a magnetic field of 0.6 T using a commercial (Ecopia HMS-5500)system. NiO: V Nisamples showed p-type conductivity (with resistivity, ρ∼3.6Ωcm), while the ρof NiO: VOand stoichiomet- ric NiO was too high to measure with our Hall system (ρ>1 04Ωcm). Because of the low carrier mobility in NiO: VNi (<1 cm2/Vs), Hall effect measurement cannot reliably determine the conductivity type of the sample at room temperature. Theconductivity type was verified by thermopower measurements using a commercial MMR SB1000 Seebeck system. We confirmed that NiO: V Nifilms are p-type with a positive Seebeck coefficient of +84 μV/K. Film stoichiometry and thickness were characterized by Rutherford backscattering (RBS) using a 3.04 MeV He++beam, and the spectra were analyzed using the SIMNRA software.38We note that while RBS can be sensitive to high Z impurities to the part permillion levels, the determination of O stoichiometry in NiO towithin 1% precision is challenging. Hence, only a qualitativemeasure of the O stoichiometry in NiO samples is given here by RBS. Film thicknesses in the range of 90 –150 nm were obtained by RBS and were confirmed by spectroscopic ellipsometry thicknessfitting. Grazing incidence x-ray diffraction (GIXRD) with an inci-dence angle of 1° was used to characterize the film structure. Thesurface morphology of all films was studied by Atomic Force Microscopy (AFM). The optical properties of the films were studied by standard (isotropic) Spectroscopic Ellipsometry (SE) in the spectral range of0.73 eV to 6.5 eV using a rotating-compensator instrument(J. A. Woollam, M-2000). The angle of incidence was varied from 55° to 75° with an increment of 10°. To eliminate back-side reflec- tion, the glass substrate backside was taped with a translucentplastic tape. 39The electronic structure of deposited films was obtained by high-resolution x-ray photoelectron spectroscopy (XPS) using a monochromatic Al K αx-ray source (1.487 keV), and emitted photoelectrons were collected and analyzed using a con-centric hemispheric analyzer system. In XPS measurements, thecore level electron binding energies were referenced to the adventi-tious C 1 sat 284.8 eV to correct for electrostatic charging effects for insulating films.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-2 Published under license by AIP Publishing.First principle calculation First-principles calculations were performed using density functional theory (DFT)40,41as implemented in Vienna Ab Initio Simulation Package (VASP ).42We used the Perdew –Burke – Ernzerhoff (PBE) functional43for the exchange and correlation, an effective Coulomb interaction U(DFT + U) to account for electron correlations, and exchanges of the valence dshell elec- trons. We select for the Ni dshell the Coulomb Uand exchange energy parameter Jof 5.10 and 0.95 eV, respectively, using the rotationally invariant method of Liechtenstein et al.44The choice of the effective interaction value is motivated by the experimentas discussed below. The unit cell of NiO as shown in Fig. 1(a) is rock salt with a rhombohedral symmetry due to the type-II anti-ferromagnetic order along the [111] direction. We used a 108-atom rhombohedral supercell to model the defects by ran- domly removing a pair of Ni or O atom, i.e., one vacant site ineach of the magnetic sublattices using the special quasirandomstructures 45with the mcsqs utility in the Alloy Theoretic Automated Toolkit.46The choice of a vacant site on each of the magnetic sublattice at a time is to maintain the antiferromagnetic symmetry of the parent material.47The resulting crystal structures for both Ni and O vacancies contain isolated defect sites that aremore than 15 Å apart. Herein, our simulation is for the lowest possible defect concentration for the 108 atoms supercell, which corresponds to 3.70% of the total Ni or O sublattice sites. To opti- mize the crystal structures, we employed a Γ-point sampling of the reciprocal space to relax the structures until the energy(charge) was converged to within ∼10 −3(10−6) eV and the forces dropped to 10−3eV/Å. The relaxed structure shows that nearest- neighbor oxygen atoms around the Ni defect sites moved outwardby∼0.19 Å. On the other hand, the Ni atoms in the immediate surroundings of the O atom defect sites relaxed inward by ∼0.11 Å. These two opposing relaxation patterns are due to the electrostatic interactions between the defect sites and proximityatoms, which induced spatial distribution of the defect levels. Forexample, the effective charge of V Oi sp o s i t i v e ,t h es a m es i g na s the neighboring Ni atoms and will repel each other leading to the relaxation pattern described above. To obtain the electronic struc-tures, we employed a 3 × 3 × 3 k-point grid to represent the recip-rocal space and performed in the antiferromagnetic phase using a collinear spin-polarized approach with a kinetic energy cut-off of 550 eV for the planewave basis set. The optical spectra wereobtained using independent pa rticle approximation as imple- mented in the VASP code. FIG. 1. (a) The unit cell (denoted with a solid blue line) of the antiferromag-netic NiO crystal structure. (b) X-raydiffraction patterns from NiO, NiO: V Ni, and NiO: VOthin films. (c) Atomic force microscopy (AFM) images of thesamples; the root-mean-square rough-ness (d rms) obtained from each AFM image is specified.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-3 Published under license by AIP Publishing.SE analysis The dielectric functions ( ε=ε1+iε2) of NiO, NiO: VNi, and NiO: VOfilms were evaluated by spectroscopic ellipsometry (SE). In the SE analysis, we assumed an optical model comprising a surface roughness/thin films/glass substrate structure. The refractive index(n) and extinction coefficient ( k) of the glass substrate were obtained as described previously. 36The Bruggeman effective medium approximation (EMA) with a 50:50 vol. % mixture of the bulk layer, and void was used to model the optical properties of the surface roughness layer.35The Tauc –Lorentz oscillator model was used to fit the amplitude ratio Ψand the phase difference Δspectra for stoichiometric NiO and NiO: VOfilms, which have no signifi- cant absorption for E≤Eg. In this model, the imaginary part of the dielectric function, ε2(E) is expressed as a product of the Tauc optical gap and the Lorentz model, ε2(E)¼AE0Γ(E/C0Eg)2 [(E2/C0E2 0)2þΓ2E2]1 E(E.Eg), 0( E/C20Eg),8 >< >:(1) where A,E0,Γ, and Egrepresent the Lorentz amplitude parame- ter, center transition energy, broadening parameter, and Taucoptical gap, respectively. 48The Tauc –Lorentz ε1(E) spectra are then obtained using the Kramers –Kronig relation. At high energies, ε1 describes a constant, energy independent contribution to ε1(E). Hence, in the Tauc –Lorentz model, the dielectric function isdescribed by five parameters ( A,E0,Γ,Eg, and ε1). Two Tauc – Lorentz dielectric functions ( Table I ) were used to accurately fit Ψ andΔof NiO and NiO: VOsamples. The fitted and experimental Ψ andΔspectra of NiO are shown in Fig. 2 . The surface roughness, ds, of 4.2 ± 0.1 nm obtained from the SE analysis of the NiO film is in close agreement with the value of roughness drms= 6.0 nm observed in AFM [ Fig. 1(c) ], confirming the validity of the SE anal- ysis. Also, the mean-square-error (MSE) that is a measure of good-ness of fit has a low value of 2.80. The MSE sums the differences between the measured data and the model generated data over all the measurement wavelengths of the ellipsometric ΨandΔspectra; the MSE is minimized by adjusting the fit parameters using theLevenberg –Marquardt non-linear regression algorithm .The Tauc – Lorentz fitting parameters for NiO and NiO: V Oare summarized in Table I . For O-rich NiO samples (NiO: VNi), the dielectric function was determined from fittings of the SE data using the Cody – Lorentz (C –L) model, and the results are shown in Fig. S1 in the supplementary material . Gaussian and Lorentz oscillators were also applied in modeling the system to improve the fitting of the dielec- tric function, especially at higher energies above the bandgap.Ferlauto et al. 49developed the Cody –Lorentz model to describe amorphous materials, showing a significant sub-gap absorptionand absorption on the band tail regions. Though the Tauc –Lorentz (T–L) model has been used widely to model dielectric functions of transparent metal oxide materials, it assumes ε 2¼0 for energies E/C20Egas shown in Eq. (1)and, hence, may not fully account for absorption behavior below the bandgap. On the other hand, theTABLE I. The peak fitting parameters extracted from the dielectric function of thin films. Material Peak A E0(eV) Γ(eV) Eg(eV) ε1(∞) Eu(eV) Et(eV) Ep(eV) NiO T –L1 39.778 4.048 1.354 2.679 2.251 ……… T–L2 22.940 6.530 5.164 1.316 0 ……… NiO:V O T–L1 48.606 4.138 1.480 2.803 2.283 ……… T–L2 18.970 7.935 5.426 0.675 0 ……… NiO:V Ni C–L1 36.623 3.074 7.000 3.074 2.122 0.549 0.840 0.835 L1 1.504 2.386 4.987 … … ……… G1 2.186 8.399 4.120 … … ……… T–L, Tauc –Lorentz; C –L, Cody –Lorentz; L, Lorentz; G, Gaussian. FIG. 2. The measured (black square) and fitted (red solid line) SE spectra of (a) amplitude ratio ( Ψ) and (b) phase difference ( Δ) for NiO thin film measured at an incident angle of 65°.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-4 Published under license by AIP Publishing.Cody–Lorentz dielectric function model can provide an improved description of the absorption coefficient in the region below the bandgap by including parameters to model absorptions at energiesbelow E g. In contrast to other samples, NiO: VNishowed higher sub-gap absorptions, which necessitated the use of the Cody – Lorentz model to enable a more accurate fitting of absorptions below the bandgap. Like the Tauc –Lorentz, the Cody –Lorentz oscil- lator defines an energy bandgap and Lorentzian absorption peakparameters. However, for the region just above E g, the Cody–Lorentz absorption formula is given by ε2(E)/(E−Eg)2, while the Tauc –Lorentz absorption uses the dependence ε2(E)/[(E/C0Eg)2/E2]. The details of the Cody –Lorentz model fitting parameters are described in the supplementary material . The extracted fitting parameters for the Cody –Lorentz models are also shown in Table I .I nTable I , for the NiO and NiO:V Ofitting using the Tauc –Lorentz oscillator, the first Tauc –Lorent oscillator, T-L1, is used to model interband absorption close to the absorption edge of the material. The second Tauc –Lorentz oscillator T-L2 models other absorption at higher energies above the absorption edge asso-ciated with electronic interband transitions. RESULTS AND DISCUSSION Structural characterization XRD patterns of NiO, NiO: V O, and NiO: VNithin films shown inFig. 1(b) reveal that all films are polycrystalline, showingdiffraction peaks from the rock salt (111), (200), and (220) planes. A decrease in crystallinity is evident in films with defects. The sig- nificant decrease in the XRD peak intensity per unit film thicknessand increase in the FWHM for the NiO: V Niand NiO: VOthin films suggest smaller grain size in the films and are likely due to struc-tural disorders. In particular, the NiO: V Nifilm shows much broader diffraction peaks, indicating a much smaller grain size of ∼9 nm compared to the ∼18 nm gain size for the “stoichiometric ” NiO film. The atomic force microscopy (AFM) results presentedinFig. 1(c) show that the “stoichiometric ”film has a root-mean-square (rms) roughness of ∼6.0 nm compared to the rms roughness of 1.2 and 0.7 nm for NiO: V Oand NiO: VNi, respec- tively. The smoother film surface of the films with vacancy defectsis consistent with their smaller grain size revealed by XRD. Optical properties The dielectric functions of the deposited thin films obtained by model fittings of the SE results are compared with results com-puted using the calculated electronic wave functions and energies.The measured [(a) and (b)] and calculated [(c) and (d)] real ε 1and imaginary ε2parts of the complex dielectric functions as a function of photon energy for three different NiO samples are shown inFig. 3 . We note that the computed spectra for NiO show similar features as the experimental spectra up to a photon energy of∼4.30 eV, but they show a significant deviation especially in ε 1for FIG. 3. Photon energy dependence of the real ε1(a) and imaginary ε2(b) parts of the complex dielectric functionfor NiO, NiO: Vo, and NiO: V Niextracted from the SE analysis. The correspond- ing calculated spectra are presented in(c) and (d), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-5 Published under license by AIP Publishing.higher photon energies. This deviation could be due to several sources in both computation and experimental techniques. For example, the computed spectra in NiO show more structures dueto the small broadening parameter of ∼10 −3used in our calcula- tions. The small grain size of the films ( ∼< 20 nm) may play a role in broadening the experimental spectra, rendering these sharp fea- tures not observable. The deviation could also be attributed to the inability of DFT-based optical calculations to properly describehigh photon energy states. 50The corresponding optical constants, nand k, obtained from ellipsometry fitting are shown in Fig. S2 in thesupplementary material .Table II compares the refractive index, n,at 2 eV for NiO obtained in this work and values reported previ- ously in the literature. Focusing first on the stoichiometric NiO sample, the disper- sive part of the experimental dielectric function is presented inFig. 3(a) . The photon-energy dependence of ε 1is typical of most transparent metal oxides with a peak structure at ∼3.6 eV. In the computed spectrum [ Fig. 3(c) ], this peak appears at ∼3.2 eV. This is in agreement with previously reported calculations for NiO.56 The experimental ε2spectrum for stoichiometric NiO shows negli- gible features between 1 eV and 3 eV with a characteristic peak at ∼4.0 eV, which occurs at around 3.90 eV in the calculated spec- trum. This peak is related to the absorption edge in NiO. Thereported experimental NiO ε 2spectrum is in good agreement with previous reports.51–53 To explore the impact of defects on the optical properties, we compare in Fig. 3 the experimental and computed NiO ε1andε2 spectra with those from NiO: VNiand NiO: VO.I n Fig. 3(a) , the experimental ε1spectra of the NiO: VOsample closely follow that of the stoichiometric NiO sample. However, a slight increase in the NiO: VOspectrum at lower photon energy is observed compared to NiO. On the other hand, the ε1spectrum of NiO: VNishows a stronger deviation from that of NiO. A significant increase in ε1is observed for NiO: VNiat photon energies below 2.3 eV. The com- puted results in Fig. 3(c) also show similar ε1features for the NiO: VOand stoichiometric NiO samples which are very different from that for the NiO: VNisample with the same defect concentration. Due to the presence of O-vacancies in NiO: VO, slight differences in the computed ε1spectrum for NiO: VOare observed compared to NiO spectrum. A comparison of the computed ε1spectrum for NiO: VOwith previously reported calculation shows that the peak position of our calculated data shifts slightly to ∼3.5 eV compared to ∼4e V shown in the previous calculation. This slight discrepancy may bedue to different U-value employed in the previous calculation. 57A similar feature at ∼1.9 eV observed in the calculation by Petersonet.al. can be observed in our data, though, with smaller intensity likely due to differences in defect concentration in the calculations.57 While the computed and experimental ε1for NiO and NiO: VOsamples show similar features, we observe significant deviations between the computed and experimental ε1in NiO: VNi. In partic- ular, we note that the peak around ∼3.0 eV is not observable in the calculated data. A possible reason may be that the vacancy concen- tration used in the calculation is larger than that in the grownsamples. It is also possible that independent particle approximationused in the calculation did not properly capture the low-photonspectra for NiO: V Ni. The ε2spectra also reveal the different behavior of the studied native defects. The experimental and calculated ε2spectra are shown in Figs. 3(b) and3(d), respectively. A slight blue-shift of the fundamental absorption peak at ∼4.0 eV and noticeable non-zero absorptions below ∼3.0 eV are observed in the experi- mental NiO: VOcompared to stoichiometric NiO. The sub-bandgap absorption can be attributed to VOdefect states in the bandgap. In contrast, the experimental ε2spectrum for NiO: VNiexhibits a sig- nificant red-shift of the band-edge, a larger non-zero absorptionbelow ∼3.0 eV and a reduced amplitude of the characteristic absorption peak at ∼4.0 eV. In the computed data in Fig. 3(d) ,t h e ε 2spectrum for NiO: VOclosely follows that of the stoichiometric NiO but deviates from that of NiO: VNi.We notice the emergence of a small local maximum around ∼2 eV (1 eV) in the computed ε2spectrum for NiO: VO(NiO: VNi) that is not resolved in our experimental data. Petersen et al. reported a similar feature at ∼2 eV albeit with higher intensity in their NiO with O vacancies calculation.57We believe that because of the small amplitude of this structure, it might have been masked by the broadening parameter used in obtaining experimental data, which instead gives rise to a broadshoulder around 1.3 eV in the experimental ε 2spectra of NiO: VNi [Fig. 3(b) ]. Figures 4(a) and4(b) show absorption coefficient αobtained from the experimental and calculated dielectric functions, respec- tively, for various samples. The experimental αpresented in Fig. 4(a) show an onset of absorption around ∼3.5 eV for NiO and a slightly lower value of ∼3.4 eV for NiO: VNi. Below the absorp- tion edge in the experimental absorption spectra, stoichiometric NiO and NiO with native vacancies show very different behaviors. Sub-bandgap absorption due to defect states are observed in NiOwith vacancy defects. Figure 4(a) shows that NiO: V Nihas a larger subgap absorption compared to the NiO: VOsample. For example, at a photon energy of 2.5 eV, αis∼1×1 05and 3 × 104cm−1for NiO: VNiand NiO: VO, respectively. The absorption edge of NiO obtained from the calculated absorption coefficient in Fig. 4(b) is slightly lower at ∼3.1 eV com- pared to the experimental value due to the choice of our U-value in the calculation. Similar changes in the experimental absorption spectra for the samples with vacancy defects are seen in the calcu-lated data. While stoichiometric NiO has no significant absorptionbelow the onset for both experimental and calculated spectra,NiO: V Oshows some absorption features between 2 and 3 eV before the onset of absorption at ∼3.5 eV for the experimental and 3.1 eV for the calculated spectra. NiO: VNi, on the other hand, shows a large sub-gap absorption below the onset at 3.4 eV in theTABLE II. Comparison of reported refractive index of NiO at 2 eV . Refractive index at 2 eV Method/Model Reference 2.37 SE (Tauc –Lorentz model) 51 2.40 SE (Tauc –Lorentz model) 52 2.33 Reflectance 53 2.38 SE 54 2.40 SE 55 2.40 SE (Tauc –Lorentz model) This workJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-6 Published under license by AIP Publishing.experimental data while in the calculated data a large sub-gap absorption is also seen which increases with photon energy.Subgap absorption similar to our experimental data has beenreported previously for p-type NiO grown in O-rich environ- ment. 31,58These changes in the absorption spectra for NiO: VNiis attributed to the action of acceptor-type VNiproducing Ni3+and a free hole by trapping an electron from Ni2+leading to an increase in charge transfer transitions in the visible range.33 The high sub-bandgap absorption coefficient observed in the NiO: VNithin film leads to large reduction in visible light transpar- ency of the NiO: VNithin films. Figure S3 in the supplementary material shows the corresponding measured transmittance, T spectra for the NiO, NiO: VO, and NiO: VNifilms. The transmit- tance of the stoichiometric NiO is ∼75% in the visible range. For the samples with vacancy defects, Tdecreases to ∼60% for NiO: VO and∼40% for NiO: VNi. This decrease in transmission intensity is less profound compared to those presented by Ono et al.33We note that in Ono ’s work, NiO films were sputtered in pure O 2plasma while the O-rich NiO films we synthesized were sputtered in an Ar + O 2mixed gas with only 4% O 2. This suggests that NiO films in Ono ’s work were much more O-rich with a much high VNiconcentration. Similar reduced transmittance in the visible regime for p-type O-rich NiO and NiO based alloys was also reported in previous studies.30,59,60To further investigate the nature of high sub-gap absorption of the films with vacancydefects, we characterize the electronic structures with first-principles calculations and x-ray photoelectron spectroscopy measurements. Electronic structure: Density of states and XPS To investigate the electronic properties, we obtain the valence band spectra from XPS measurements while the electronic structures were obtained from first-principles calculations. Figure 5 shows the spin-polarized density of states ( DOS)f o rs t o i - chiometric and disordered NiO calculated using the GGA + U FIG. 4. Absorption coefficient spectra of NiO, NiO: Vo, and NiO: VNisamples obtained from (a) spectroscopic ellips-ometry and (b) ab initio calculations. FIG. 5. Calculated electronic structures of (a) stoichiometric NiO and NiO:x with 3.7% vacancy defects for (b) x = VOand (c) x = VNi. The vertical dashed line is the Fermi level, which is set at zero energy.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-7 Published under license by AIP Publishing.functional. The value of the bandgap for stoichiometric NiO with our calculations using Uof 5.1 eV is ∼2.80 eV. The calculated NiO electronic properties and the predicted Egare in good agreement with previous calculations with similar U-value.61,62As the U-value is increased, the calculated Egis increased, e.g., Eg/difference 3:0+0:2e V f o r U∼6.0–6.5 eV has previously been reported.63–65A larger value of U∼8 eV in our calculation can lead to a larger bandgap, Eg∼4.2 eV, which is closer to the gener- ally accepted experimental value in the range of 3.6 –4.3 eV.53,66,67 However, the agreement of the absorption spectra obtained with such a large Uvalue with experimental data is rather poor. Overall, we do not expect the application of the GGA + Uto the Nidshell to obtain the accurate bandgap value for NiO; as it is well known that bandgaps are significantly underestimated in semi-local functionals even for materials without transition metaldstates such as Si and Ge. Nevertheless, our calculated total density of states for NiO shown in Fig. 5(a) agree with the results reported in the literature. 29,61,68 Modeling defects in an antiferromagnetic system require some care, especially at the dilute limit. Antiferromagnetic systems likeNiO have symmetric spin-up and spin-down sublattices; as such,there is no sublattice preference for defect levels, i.e., the majority spin on a given sublattice is the minority spin on the other sublat- tice, and vice versa. Hence, a single Ni or O vacancy, especially atthe dilute limit, will break the symmetry between the two magneticsublattices in NiO. Our calculations show that single-atom vacancyleads to half-metallicity, while current experiments show a robust antiferromagnetic insulating state. The observation of half- metallicity by a one-atom vacancy in computations has been shownin previous studies. 28,47We can ascertain the stability of half- metallicity through random defect modeling and spin-polarizedphotoemission measurement, which will be the focus of a future study. At the dilute limit, one would expect Ni/O vacancy levels to be randomly distributed without sublattice preference; this couldpreserve the antiferromagnetic configuration of pristine NiO. Inour defect modeling, we have adopted one vacant site in each ofthe magnetic sublattices to preserve the antiferromagnetic symme- try of NiO. We present in Figs. 5(b) and 5(c) the calculated electronic structures of NiO with native O and Ni vacancy defects, respec-tively. One can see that the DOS for both spin channels are identi- cal, confirming the stability of antiferromagnetic states. Distinct features, especially in-gap levels appear in both NiO: Vo and NiO: V Ni. Overall, the defect levels have more impact on the conduction band states. From the projected DOS [Fig. S4 in the supplementary material ], the defect-induced in-gap levels above the Fermi level in NiO: Vo is dominated by Ni-3 dstates. In contrast, the defect-induced in-gap states in NiO: VNiare dominated by a strong hybridization between Ni-3 dand O-2 pstates. Interestingly, more pronounced in-gap states are visible in the NiO: VNicompared to the NiO: Vo with the same defect concentration (3.7%). Specifically, the NiO: VNisystem leads to a half-metallic solution, in basic agree- ment with the works of Ködderitzsch et al.,47and Park et al. ,28 which is due to spin polarization of the atoms in the proximity of the vacant Ni sites. Figure 6 shows the XPS spectra close to the Fermi level for the NiO, NiO: Vo, and NiO: VNithin films. The calculated DOS of NiOconvoluted with a Gaussian broadening, γ= 0.8 using the Galore code,69are also shown for comparison. Two peak energy regions can be distinguished in the calculated DOS between −1.0 eV to −3.0 eV and −5.0 eV to −8.0 eV. From the PDOS [in Fig. S4 in the supplementary material ], pronounced Ni-3 dorbital states are con- centrated in the energy of −1.0 eV to −3.0 eV range with a peak at ∼−2.2 eV, while in the energy range between −5.0 eV to −8.0 eV, O2pstates also have significant contribution to the spectrum. The experimentally obtained XPS data agree closely with the broadened calculated NiO DOS. All three valence band spectra of NiO, NiO:V Ni, and NiO: VOshow a peak at ∼−2 eV, corresponding to pre- dominant contributions from the Ni 3 dstates and consistent with the peak position of the calculated DOS. However, features due to O2pstates around −5.0 eV to −8.0 eV are less pronounced. This may be attributed to the Al K α(1486.7 eV) source used in the mea- surement which has a more favorable photoionization cross sectionfor the delectrons. A similar observation has been made for Cu based oxide systems. 70 The observed in-gap states in both NiO: VNiand NiO: Vo systems confirm the sub-gap absorption in our optical spectra.Also, the half-metallic solution for NiO: V Nisupports the rather high p-type conduction observed in NiO: VNithin films as com- pared to stoichiometric NiO and NiO: Vo thin films. Hence, achiev- ingp-type conductivity in NiO by increasing VNiwill necessarily FIG. 6. Experimental XPS valence-band spectra of NiO, NiO: Vo, and NiO: VNi thin films near the Fermi level. The calculated NiO total density of state convo- luted with a Gaussian broadening of 0.8 using the Galore code is also shownfor comparison (top spectrum, labeled NiO DOS).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135705 (2020); doi: 10.1063/5.0021650 128, 135705-8 Published under license by AIP Publishing.degrade its transparency in the visible range due to this formation of half metallicity in the electronic structure. Therefore, optoelec- tronic applications requiring high transparency and good p-type conduction in NiO would require a systematic combination ofsmall amounts of V Ninative defects with extrinsic doping to reduce the gap states introduced in the material while maximizing conduc- tivity. This alternative route to simultaneously optimize p-type con- ductivity and transparency of NiO would be a subject of furtherstudy. Elements such as Li, Cu, or Ag have been studied as acceptordopants in NiO. 31,71,72A systematic investigation of the optical properties of NiO combining VNiand these acceptor elements may pave the way to realizing transparent and conducting p-type NiO for transparent electronic applications. CONCLUSION We have experimentally investigated changes in the complex dielectric function, absorption coefficient, and electronic structure of NiO due to vacancy defects, and results are supported by com-putational studies. The complex dielectric functions and opticalconstants of stoichiometric NiO and NiO with vacancy defects were obtained by fitting the amplitude ratio and the phase differ- ence of spectroscopic ellipsometry measurements using oscillatormodels. We show that the optical constants of p-type O-rich NiO (NiO: V Ni) deviate substantially from the stoichiometric NiO. By comparing the effects of native vacancies in NiO, we establish that VNipresent in NiO grown under an O-rich environment leads to a significant sub-bandgap absorption as compared to VOin NiO grown under Ni-rich condition. The calculated density of statesconfirm a high concentration of gap states in NiO: V Ni, resembling a half-metallic system as compared to a smaller subgap state in NiO: VO. Hence, this significant density of gap states in NiO: VNi arising from the introduction of VNiacceptors in NiO can explain the high absorption in the visible range for this material.Interestingly, while Ni 3 dstates are responsible for the gap states in NiO: V O, both Ni 3 dand O 2 pcontribute to those in NiO: VNi. Our results establish the relation between vacancy native defects and the optical and electronic properties of NiO. Though native VNiis an effective acceptor leading to p-type conductivity in NiO, they also give rise to a high density of gap states in the electronic structure and hence degrade the visible transparency of the material. Therefore, for applications as a p-type transparent conductor in solar cells and other optoelectronic devices, it is essential to alsoexplore the combination of extrinsic doping with native vacancydefect control in NiO. SUPPLEMENTARY MATERIAL See the supplementary material for details of spectroscopic ellipsometry data analysis using the Cody –Lorentz model for NiO: V Nithin film. Plots of the optical constants of the thin films, trans- mission intensity, and the calculated partial density of states of the the systems are also shown in the supplementary material . ACKNOWLEDGMENTS This work was supported by the General Research Fund of the Research Grants Council of Hong Kong SAR, China, under ProjectNos. CityU 11267516 and CityU-SRG 7005106. C.P.L. acknowl- edges the support of the start-up fund from the Shantou University under Project No. NTF18027, Guangdong Basic and AppliedBasic Research Foundation (Project No. 2020A 1515010180), theMajor Research Plan of the National Natural Science Foundationof China (NNSFC; Project No. 91950101), and the Optics and Photoelectronics Project (No. 2018KCXTDO11). Supercomputer and computational resources were provided by the LehighUniversity (LU) High-Performance Computing Center. C.E.E.acknowledges LU Start-up and Summer Research Fellowship.K.O.E was supported by the Hong Kong Ph.D. Fellowship (No. PF16-02083) Research Grants Council, University Grants Committee, Hong Kong. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material . REFERENCES 1J. Feinleib and D. Adler, “Band structure and electrical conductivity of NiO, ” Phys. Rev. Lett. 21, 1010 –1013 (1968). 2S. Lany, “Band-structure calculations for the 3d transition metal oxides in GW, ” Phys. Rev. B Condens. Matter Mater. Phys. 87,1–9 (2013). 3G. A. Sawatzky and J. W. Allen, “Magnitude and origin of the band gap in NiO,”Phys. Rev. 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5.0024791.pdf

J. Chem. Phys. 153, 154107 (2020); https://doi.org/10.1063/5.0024791 153, 154107 © 2020 Author(s).Ground and excited state first-order properties in many-body expanded full configuration interaction theory Cite as: J. Chem. Phys. 153, 154107 (2020); https://doi.org/10.1063/5.0024791 Submitted: 11 August 2020 . Accepted: 25 September 2020 . Published Online: 16 October 2020 Janus J. Eriksen , and Jürgen Gauss ARTICLES YOU MAY BE INTERESTED IN Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 Core excitations with excited state mean field and perturbation theory The Journal of Chemical Physics 153, 154102 (2020); https://doi.org/10.1063/5.0020595The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Ground and excited state first-order properties in many-body expanded full configuration interaction theory Cite as: J. Chem. Phys. 153, 154107 (2020); doi: 10.1063/5.0024791 Submitted: 11 August 2020 •Accepted: 25 September 2020 • Published Online: 16 October 2020 Janus J. Eriksen1,a) and Jürgen Gauss2,b) AFFILIATIONS 1School of Chemistry, University of Bristol, Cantock’s Close, Bristol BS8 1TS, United Kingdom 2Department Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, 55128 Mainz, Germany a)Author to whom correspondence should be addressed: janus.eriksen@bristol.ac.uk b)E-mail: gauss@uni-mainz.de ABSTRACT The recently proposed many-body expanded full configuration interaction (MBE-FCI) method is extended to excited states and static first- order properties different from total, ground state correlation energies. Results are presented for excitation energies and (transition) dipole moments of two prototypical, heteronuclear diatomics—LiH and MgO—in augmented correlation consistent basis sets of up to quadruple- ζ quality. Given that MBE-FCI properties are evaluated without recourse to a sampled wave function and the storage of corresponding reduced density matrices, the memory overhead associated with the calculation of general first-order properties only scales with the dimension of the desired property. In combination with the demonstrated performance, the present developments are bound to admit a wide range of future applications by means of many-body expanded treatments of electron correlation. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024791 .,s I. INTRODUCTION The ultimate benchmark of electronic structure theory,1–10full configuration interaction (FCI) theory has attracted renewed inter- est over the past decade due to the availability of modern, scalable hardware11,12as well as the emergence of a myriad of new ways in which the exact solution to the electronic Schrödinger equation within a given one-electron basis set may be recast.13–51The major- ity of these efforts have focussed on the computation of ground state energies, although studies of the ordering and spacing of excited states have also started to receive attention.52–58The study of general first-order properties at the near-exact level, on the other hand, has been left somewhat ignored following its pinnacle in the 1990s.59–65 Given that excitation energies and molecular multipole moments are physical observables, whereas the ground state energy is not, a strong incentive from the chemical sciences arguably exists in favor of the development of new theoretical tools that may universally allow for the calculation of a wide range of properties in order to complement and guide experimental endeavors. To that end, wenote how the simulation of electronic properties at various levels of theory has continued to gather momentum over the years, as evi- denced by a number of recent benchmarks on the topic, e.g., assess- ments of Kohn–Sham density functional theory (KS-DFT) dipole moments.66–68In the present work, however, our focus will differ pronouncedly from such endeavors as we will be strictly concerned with the prediction of first-order properties of near-FCI rather than approximate (truncated) quality. As an example of an exception to the general trend discussed above, quantum Monte Carlo approaches have long retained an interest in the near-exact estimation of general expectation val- ues.69–71Expressed on its usual form as a linear combination of Slater determinants in a discrete basis set, the exact N-electron wave function may be described in a systematically improvable fashion by means of initiator FCI quantum Monte Carlo72,73(i-FCIQMC). In the complete absence of the initiator bias and in the limit of long samplings, a stochastic sampling of the FCI wave function, in turn to construct the two-electron reduced density matrix (2-RDM), Γpq,rs, becomes exact. From the 2-RDM, the one-electron analogue J. Chem. Phys. 153, 154107 (2020); doi: 10.1063/5.0024791 153, 154107-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp (1-RDM),γpq, required for the calculation of, e.g., electronic dipole moments,74may be trivially obtained through a partial trace opera- tion. In practice, the so-called replica trick75is employed whereby two independent, randomly initialized FCIQMC calculations are performed for a given state to arrive at significantly less biased 2- RDMs,76using coefficients from both replica runs. The calculation of transition RDMs, from which transition dipole moments are com- puted, is more involved but may these days also be realized by FCIQMC.77 Recently, many-body expanded FCI78–81(MBE-FCI) theory has been proposed by us as an alternative to selected CI,82–84operat- ing instead by performing an orbital-based decomposition of the FCI correlation problem. By solving for correlation energies without recourse to the full N-electron electronic wave function, the MBE- FCI method has emerged as an accurate approximation to exact theory, applicable in extended basis sets80and for moderate-sized molecular systems.85Specifically, a strict partitioning of the com- plete set of molecular orbitals (MOs) of a system into a reference and an expansion space is enforced. A complete active space CI (CASCI) calculation is then performed in the former of these two spaces, while the residual correlation in the latter space is recovered by an orbital-based MBE. In the present work, we will report on an exten- sion of MBE-FCI theory to excited states and first-order properties for arbitrary states with the aim of computing near-exact properties in a unified framework on par with alternatives from the literature, e.g.,i-FCIQMC theory. II. THEORY In the MBE-FCI method, the FCI correlation energy is decom- posed as E0 FCI=E0 ref+∑ pϵ0 p+∑ p<qΔϵ0 pq+∑ p<q<rΔϵ0 pqr+⋯. (1) Here, the MOs of the expansion space (of size Mexp) of unspecified occupancy are labeled by generic indices { p,q,r,s,. . .} andϵ0 pdesig- nates the correlation energy of a CASCI calculation for the ground state (0) in the composite space of the orbital pand all of the refer- ence space MOs. For a general tuple of mMOs, [Ω] m, itsmth-order increment, Δϵ[Ω]m, is defined through recursion. As outlined in Ref. 85, MOs are screened away from the full expansion space at each order according to their relative (absolute) magnitude, which in turn results in a reduced number of increment calculations at the orders to follow. Ultimately, these successive screenings lead to the convergence of an MBE-FCI calculation. In analogy with Eq. (1), excitation energies may now be computed by an expansion of the energetic gap between the ground and the excited state, E0n, rather than the correlation energy, E0n FCI=E0n ref+∑ pϵ0n p+∑ p<qΔϵ0n pq+∑ p<q<rΔϵ0n pqr+⋯. (2) As a CASCI calculation in an active space absent of electron correla- tion yields no correlation energy [comprising only the Hartree–Fock (HF) solution] and hence no excited states, ϵ0n pin Eq. (2) will bedefined on par with ϵ0 pin Eq. (1). For the calculation of static proper- ties, we will here exemplify how this may be achieved in the context of MBE-FCI theory by focussing on electronic dipole and transition dipole moments. From the wave function coefficients of an individ- ual CASCI calculation, the corresponding 1-RDM, γn, for the state n may be readily computed, from which an electronic dipole moment is given as μn p=−∑ rTr[μrγn] (3) in terms of dipole integrals, μr, in the MO basis for each of the three Cartesian components. Letting these quantities take up the role of correlation or excitation energies in Eqs. (1) and (2), respectively, results in the following decomposition of the FCI electronic dipole moment: μn FCI=μn ref+∑ pμn p+∑ p<qΔμn pq+∑ p<q<rΔμn pqr+⋯. (4) Adding the nuclear component, μnuc=∑KZKrK, returns the molec- ular dipole moment. Being a vector rather than a scalar quantity, the screening procedure proceeds along all three Cartesian components (x,y,z) in the case of dipole moments and must be simultane- ously fulfilled for all if a given MO is to be screened away from the expansion space. For the calculation of ground state dipole moments, increments are evaluated against the HF dipole moment—in line with the corre- lation energies entering Eq. (1)—while excited state electronic dipole moments are evaluated in the absence of a zero point. Finally, tran- sition dipole moments, t0n, are evaluated on par with Eq. (4), except for the fact that the individual increments are computed on the basis of transition 1-RDMs, γ0n, which may be arrived at using the wave functions of both states involved in a given CASCI calculation. III. COMPUTATIONAL DETAILS In comparison with MBE-FCI for the computation of ground state correlation energies, the overhead in terms of added compute time and memory demands is minimal, also when contrasted with i-FCIQMC and related methods that sample the full 2-RDM and retrieve the 1-RDM, before contracting this with the appropriate property integrals. By virtue of the small active spaces involved in all individual CASCI increment calculations of an MBE-FCI expan- sion, the additional cost associated with the computation of γnor γ0n, depending on the property in question, is negligible, as we will show below. With regard to storage requirements, these are the same for correlation and excitation energies, while for (transition) dipole moments, the only difference is that the increment quantities to store are now tensorial, rather than scalar—specifically for μnand t0n, the memory requirements increase by a factor 3 over an MBE- FCI calculation of the correlation energy. All results to follow have been obtained in a massively parallel manner using the open-source PyMBE code,86which in turn employs the PySCF code87–89for all electronic structure kernels. All calculations of the present work, for which timings are reported, were run on Intel Xeon E5-2697v4 (Broadwell) nodes (36 cores @ 2.3 GHz, 128 GB). J. Chem. Phys. 153, 154107 (2020); doi: 10.1063/5.0024791 153, 154107-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. RESULTS Figure 1 presents results for the first and second excited states (1Σ+symmetry) of LiH in the aug-cc-pV XZ basis sets,90,91 with MBE-FCI results for excitation energies, dipole moments, and transition dipole moments compared to corresponding i-FCIQMC results from Ref. 77. Calculations for a given state and a given prop- erty were run independently. The MBE-FCI calculations have all been run in C2vsymmetry using the MOs of a state-averaged com- plete active space self-consistent field92(CASSCF) calculation in a (4e,7o) active space coinciding with the employed reference space, and the screening thresholds are discussed in the supplementary material. The results in Fig. 1 and Fig. S1 of the supplementary material collectively show how the performance of MBE-FCI for ground state correlation energies is reflected not only in a corre- sponding accuracy for ground state dipole moments but also in excited state properties as well. In general, the convergence profiles in the six individual plots of Fig. 1 are all different, as are the MO manifolds being screened away in the calculations, attesting to the facts that the properties in question are inherently unrelated and that the MBE-FCI method is flexible enough to cope with this in an orbital expansion framework. All of the results of the present work have been obtained using theπ-pruning of Ref. 80 (generalized to C∞v/C2vpoint groups), which is a prescreening filter that prunes away all increment calcu- lations that fail to simultaneously include the x- and y-componentof a given pair of degenerate π-orbitals. The use of this π-pruning filter results in much shorter (faster) expansions for linear molecules belonging to non-Abelian point groups while at the same time war- ranting convergence onto states spanned by the correct irreducible representation (1Σ+/A1in our case). Table I presents timings in units of core hours that clearly show not only the minimal overhead asso- ciated with computing excited and non-energetic properties but also the reduction in compute time that results from the use of π-pruning [indicated by ( π) in the table]. However, in order to obtain accu- rate results across all of the tested properties (e.g., for the second root), a tighter screening procedure is needed in combination with π-pruning, cf. Figs. S2 and S3 of the supplementary material. Note that the timings in Table I correspond to executions on a single core; in practice, MBE-FCI calculations are performed on all cores across a number of nodes. For instance, using two of the Broadwell nodes described in Sec. III, the timings in Table I are reduced by roughly a factor of 72. We next turn to the problem of MgO in an aug-cc-pVDZ basis set, for which the frozen-core FCI correlation problem is described by the distribution of 16 electrons in 48 orbitals. MBE- FCI results—obtained using reference spaces spanned by (state- averaged) CASSCF(8 e,8o) calculations—are presented in Fig. 2, again in comparison with i-FCIQMC results from Ref. 77. In con- trast to the earlier LiH example, the i-FCIQMC results for MgO are noted in Ref. 77 to be somewhat less converged and hence associated with an increased degree of uncertainty (particularly for FIG. 1 . Excitation energies ( E0n, upper panels), dipole moments ( μn, center pan- els), and transition dipole moments ( t0n, lower panels) for the first two excited states (1Σ+symmetry) of LiH in the aug- cc-pV XZ basis sets. Solid and dashed lines denote MBE-FCI and i-FCIQMC77 results, respectively. J. Chem. Phys. 153, 154107 (2020); doi: 10.1063/5.0024791 153, 154107-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Total timings (core hours) for the LiH calculations in Fig. 1.a,b Property Basis set E0nμnt0n Ground state (1Σ+) aug-cc-pVDZ 3.9 4.9 . . . aug-cc-pVDZ ( π) 0.1 0.1 . . . aug-cc-pVTZ ( π) 10.2 11.6 . . . aug-cc-pVQZ ( π) 443.4 465.1 . . . First excited state (1Σ+) aug-cc-pVDZ 10.6 12.9 12.9 aug-cc-pVDZ ( π) 0.3 0.3 0.3 aug-cc-pVTZ ( π) 16.6 19.4 19.4 aug-cc-pVQZ ( π) 866.2 985.6 980.5 Second excited state (1Σ+) aug-cc-pVDZ 10.1 12.8 12.9 aug-cc-pVDZ ( π) 0.2 0.3 0.3 aug-cc-pVTZ ( π) 14.5 16.4 17.2 aug-cc-pVQZ ( π) 813.4 904.9 897.3 aIntel Xeon E5-2697v4 (Broadwell) nodes (36 cores @ 2.3 GHz, 128 GB). b(π) indicatesπ-pruning. μ1). Be that as it may, i-FCIQMC was still deemed more accurate than, e.g., high level coupled cluster (CC) results at the CC sin- gles, doubles, and triples (CCSDT)93,94, in particular in the case of excited state dipole moments, which from the stochastic wave func- tions in Ref. 77 were shown to depend crucially on highly excited FIG. 2 . Correlation energies ( ΔEn, upper panel) and dipole moments ( μn, lower panel) for the ground and first two excited states (1Σ+symmetry) of MgO in the aug-cc-pVDZ basis set. Solid and dashed lines denote MBE-FCI and i-FCIQMC77 results, respectively.TABLE II . Total timings (core hours) for the MgO calculations in Fig. 2.a Property State Enμn 0 (1Σ+) 4 412 6 566 1 (1Σ+) 11 320 16 720 2 (1Σ+) 24 026 36 497 aIntel Xeon E5-2697v4 (Broadwell) nodes (36 cores @ 2.3 GHz, 128 GB). determinants that are less successfully described by means of CCSDT. For instance, CCSDT differs from i-FCIQMC by +0.04, −0.71, and −0.78 debye in the prediction of the three dipole moments in question.77 From the MBE-FCI results in Fig. 2 (all obtained using π- pruning and the same screening thresholds as for LiH), we observe how the method correctly converges onto the individual states of interest. Despite the fact that the three states lie in close proximity of each other, MBE-FCI succeeds in distinguishing between them. In relation to the troublesome dipole moment for the first excited state, this is also the result in Fig. 2 for which the largest discrep- ancy with respect to i-FCIQMC is observed. In order to determine a more accurate value of μ1, more walkers would need to be employed ini-FCIQMC and a more conservative screening protocol would need to be used in the context of MBE-FCI (as for LiH in Fig. S3 of the supplementary material). Finally, consumed core hours are presented in Table II. As for LiH, these results once again convinc- ingly illustrate the efficacy of the MBE-FCI method as well as the low penalty associated with computing dipole moments over correlation and excitation energies. V. SUMMARY AND CONCLUSIONS In summary, we have reported on a new extension of MBE- FCI theory to molecular first-order properties, valid for both ground and excited states. The performance of the resulting implementation has been verified through calculations on the first three roots of the LiH and MgO diatomics by comparing our results to state-of-the- arti-FCIQMC. On the basis of the proven accuracy and efficacy, we foresee that MBE-FCI has the potential to act as a new near-exact benchmark method for first-order properties of small- to modest- sized molecular systems in the years to come. Although the theory depends on a choice of reference space that necessarily encompasses the main determinant(s) of the target state, we are currently working on automatic selection procedures that will allow for the black-box division of a system’s total set of MOs into optimal reference and expansion spaces. SUPPLEMENTARY MATERIAL All results have been tabulated in the supplementary material, cf. Tables S1 and S2. Additional results are presented for the ground state of LiH (Fig. S1), for its excited states in the aug-cc-pVDZ basis set in the absence of π-pruning (Fig. S2), and for its second excited state using tighter screening thresholds (Fig. S3). J. Chem. Phys. 153, 154107 (2020); doi: 10.1063/5.0024791 153, 154107-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp ACKNOWLEDGMENTS J.J.E. is grateful to the Alexander von Humboldt Foundation as well as the Independent Research Fund Denmark for financial sup- port. The authors furthermore acknowledge access awarded to the Galileo supercomputer at CINECA (Italy) through the 18th PRACE Project Access Call and Johannes Gutenberg-Universität Mainz for computing time granted on the Mogon II supercomputer. DATA AVAILABILITY The data that support the findings of this study are available within this article and its supplementary material. REFERENCES 1P. J. Knowles and N. C. Handy, “A new determinant-based full configuration interaction method,” Chem. Phys. Lett. 111, 315 (1984). 2P. J. Knowles and N. C. Handy, “Unlimited full configuration interaction calcu- lations,” J. Chem. Phys. 91, 2396 (1989). 3J. Olsen, B. O. Roos, P. 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5.0009566.pdf

J. Appl. Phys. 127, 243901 (2020); https://doi.org/10.1063/5.0009566 127, 243901 © 2020 Author(s).High quality epitaxial Mn2Au (001) thin films grown by molecular beam epitaxy Cite as: J. Appl. Phys. 127, 243901 (2020); https://doi.org/10.1063/5.0009566 Submitted: 01 April 2020 . Accepted: 04 June 2020 . Published Online: 22 June 2020 S. P. Bommanaboyena , T. Bergfeldt , R. Heller , M. Kläui , and M. Jourdan 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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Jourdan1 AFFILIATIONS 1Institute of Physics, Johannes Gutenberg-University, 55099 Mainz, Germany 2Institute for Applied Materials-Applied Materials Physics, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany 3Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany Note: This paper is part of the special topic on Antiferromagnetic Spintronics. a)Author to whom correspondence should be addressed: sprakash@uni-mainz.de ABSTRACT The recently discovered phenomenon of Néel spin –orbit torque in antiferromagnetic Mn 2Au [Bodnar et al., Nat. Commun. 9, 348 (2018); Meinert et al., Phys. Rev. Appl. 9, 064040 (2018); Bodnar et al., Phys. Rev. B 99, 140409(R) (2019)] has generated huge interest in this mate- rial for spintronics applications. In this paper, we report the preparation and characterization of high quality Mn 2Au thin films by molecular beam epitaxy and compare them with magnetron sputtered samples. The films were characterized for their structural and morphologicalproperties using reflective high-energy electron diffraction, x-ray diffraction, x-ray reflectometry, atomic force microscopy, and temperatur e dependent resistance measurements. The thin film composition was determined using both inductively coupled plasma optical emission spectroscopy and Rutherford backscattering spectrometry techniques. The MBE-grown films were found to show a superior smooth morphology and a low defect concentration, resulting in reduced scattering of the charge carriers. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009566 INTRODUCTION Though ferromagnetic materials (FMs) have dominated the field of spintronics since its inception, the last few years witnessed a rather sudden interest in antiferromagnetic materials (AFMs) forspintronics applications. Attractive features such as insensitivity tostrong external magnetic fields, lack of stray fields, and ultrafast magnetization dynamics have gradually paved the way to an emerg- ing realm of “antiferromagnetic spintronics ”and provide strong motivation for the investigation of AFMs on par with FMs for next generation electronics. The recent prediction 1and experimental realization of Néel order switching in antiferromagnets with spe-cific crystallographic symmetry using current induced staggeredspin–orbit fields 2–5is a true testimony to the hitherto untapped potential held by AFMs. This aptly named Néel spin –orbit torque (NSOT) is capable of switching the Néel vector between two stableconfigurations in centrosymmetric AFMs where the two magneticsublattices form inversion partners while having a locally broken structure inversion symmetry ( Fig. 1 ). Presently, CuMnAs andMn 2Au are the only two AFMs known to possess the requisite sym- metry. Subsequently, the reoriented antiferromagnetic order parame-ter can be monitored by measuring the corresponding change inresistance (anisotropic magnetoresistance or AMR effect) of the struc- ture. Reversible switching between the two stable states using extremely short current pulses enables these AFMs to be used forinformation storage as demonstrated recently in a fully integratedmemory device. 6 Mn 2Au is considered to be a favorable option due to its large spin-flop transition field, high Néel temperature,7and lack of toxic elements. Until now, all available Mn 2Au thin films were grown by MBE in the (101)-orientation8or by sputtering in the (001)-orientation,3,9,10as well as in the (110)-,10(103)-, (101)-, and (204)-orientation.11However, (001)-axis grown films are favorable to observe NSOT switching as, in this case, the quasi-easy plane isparallel to the substrate plane. (001)-oriented Mn 2Au thin films were first grown by Jourdan et al. via RF magnetron sputtering from a stoichiometric target onto an epitaxial Ta (001) buffer layer deposited on r-plane Al 2O3substrate.9The films were found to beJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 243901 (2020); doi: 10.1063/5.0009566 127, 243901-1 Published under license by AIP Publishing.quasi-epitaxial with a relatively rough morphology. Meinert et al. reported epitaxial Mn 2Au thin films grown by DC magnetron co-sputtering of Au and Mn onto epitaxial ZrN (001) buffer layerdeposited on the MgO (001) substrate. 3They found a trade-off between morphology and crystallinity. Here, we show that it is pos- sible to overcome these problems by improving upon our previous growth technique.9Smooth surface features are important to observe NSOT induced antiferromagnetic domain wall motion andgood crystal quality is required to study the band structure of mate-rials using photoemission. We deposited Mn 2Au thin films using molecular beam epitaxy and magnetron sputtering and incorpo- rated a post-deposition annealing step into the fabrication proce-dure in both cases. The films were subjected to variouscharacterization procedures to understand their crystallographicphase, topography, and composition. The preparation and charac- terization details are presented in the following sections. THIN FILM GROWTH PARAMETERS 10 × 10 × 0.53 mm, single side polished r-plane Al 2O3sub- strates from CrysTec GmbH were used for these experiments. Thesubstrates were annealed in UHV at 550 °C for 30 min prior to dep-osition to get rid of adsorbed water and organic contamination. Mn 2Au has a tetragonal structure with a = 3.32 Å and c = 8.53 Å. Body centered cubic Ta has a lattice constant of 3.30 Å and, there-fore, serves as an ideal buffer layer for Mn 2Au (001) due to the small lattice mismatch of 0.6%. First, a 13 nm thick epitaxial Ta (001) buffer layer was deposited on Al 2O3at 450 °C via RF magne- tron sputtering. The sample was then transferred to the MBEchamber under UHV conditions. Subsequently, a 45 nm Mn 2Aufilm was grown over the Ta buffer by co-evaporation of high purity Mn and Au from separate sources in high vacuum. The base pres- sure of our MBE chamber is 7 × 10−10mbar while the working pressure is 1 × 10−7mbar. Mn was evaporated from a high temper- ature effusion cell with a stable evaporation rate, which was prede- termined by x-ray reflectometry (XRR) analysis of a pure Mn film. An electron beam evaporator was used to evaporate Au and its ratewas continuously monitored during deposition using an oscillatingquartz crystal monitor. Mn and Au were simultaneously depositedon the Ta buffer at a rate of 0.23 Å/s and 0.15 Å/s, respectively cor- responding to a ratio of 1.5:1, which results in stoichiometric Mn 2Au as we will explain below. The substrate temperature was maintained at 270 °C during deposition. Subsequently, the filmswere annealed at 450 °C for 60 min. Both temperature and durationof this post-deposition annealing step were found to be crucial for the formation of a smooth and well-ordered crystal. Additionally, a sample with the same thickness was prepared by RF magnetronsputtering from a stoichiometric Mn 2Au target for comparison. The base pressure of our sputtering chamber is 1.5 × 10−8mbar. 45 nm Mn 2Au was sputtered onto r-plane Al 2O3/13 nm Ta (001) at 300 °C and 0.1 mbar Ar pressure with a deposition rate of 0.75 Å/s. The Ta buffer layer was grown in the same way as explained above.The films were subjected to the same post-deposition annealingstep as the MBE samples. CRYSTALLOGRAPHIC AND MORPHOLOGICAL CHARACTERIZATION In situ reflective high-energy electron diffraction (RHEED) was performed after cooling the samples to room temperature FIG. 1. (a) Unit cell of Mn 2Au. It is seen to be centrosym- metric with the two magnetic sublattices A and B formingspatial inversion partners with respect to the central Au atom, which serves as an inversion center. The inversion symmetry is locally broken on individual magnetic sublatti-ces along [001]. (b) The two orthogonal easy axes ofMn 2Au (represented by dotted lines). The magnetic moments can be reversibly switched between these two states via NSOT . FIG. 2. (a) RHEED pattern of Al 2O3(1–102)/13 nm Ta (001)/45 nm Mn 2Au (001). (a) MBE sample. (b) Sputtered sample. The arrangement of the spots on semicircles andthe Kikuchi lines indicate a smooth morphology with a lowdefect concentration at the surface.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 243901 (2020); doi: 10.1063/5.0009566 127, 243901-2 Published under license by AIP Publishing.FIG. 3. ϴ/2ϴXRD specular scans of Al 2O3(1–102)/13 nm T a (001)/45 nm Mn 2Au (001) demonstrating the (001)-orientation of the sputtered (upper panel) as well as of the MBE-grown (lower panel) samples. A very small fraction of (101)-orientation is visible for sputtered samples, which does not appear in thecase of MBE-grown thin films. The rocking curves of the (002) Mn 2Au peak shown in the insets reveal a mosaicity of 0.5° for both types of samples. The (002) peak of Ta occurs at 55° but has a very low intensity due to the small thickness of this layer. Therefore, it is not visible on this scale. FIG. 4. XRDw-scans of off-specular Mn 2Au (101) and T a (220) peaks of Al 2O3 (1–102)/13 nm Ta (001)/45 nm Mn 2Au (001) MBE sample. The scans confirm the fourfold symmetry of the Mn 2Au crystal and shows a very high degree of in-plane order with Mn 2Au [100] ∥Ta [100] alignment. FIG. 5. Resistivity vs temperature measurements of 45 nm Mn 2Au (001) films. The buffer layer contribution was eliminated by measuring resistance vs temper- ature of an Al 2O3(1–102)/13 nm T a (001) sample and subtracting it from the resistance of Al 2O3(1–102)/13 nm Ta (001)/45 nm Mn 2Au (001) using a parallel resistance model. Compared with the sputtered sample, reduced residual resis- tivity of the MBE thin film indicates a lower degree of disorder achieved by MBE deposition. FIG. 6. Comparison of the XRR scans of Al 2O3(1–102)/13 nm T a (001)/45 nm Mn 2Au (001) films grown via MBE and sputtering techniques. The sputtered sample shows very weak oscillations corresponding to the thickness of the Mn 2Au layer at lower angles, whereas prominent oscillations are observed in the case of the MBE sample even at higher angles. This is due to the smoothermorphology of the MBE sample compared to the sputtered sample.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 243901 (2020); doi: 10.1063/5.0009566 127, 243901-3 Published under license by AIP Publishing.to check both crystal quality and surface features. Sharp diffraction spots on semi-circles are observed in both cases (Fig. 2 ), indicating a highly crystalline and smooth surface. Both samples also reveal the presence of Kikuchi lines,implying a low concentration of crystallographic defects at thesurface compared to our previously reported Mn 2Au films.9 This was further verified using XRR measurements as showninFig. 6 . The x-ray diffraction (XRD) specular ϴ/2ϴscans shown in Fig. 3 reveal the presence of (001)-oriented Mn 2Au in both samples. The sputtered sample additionally shows a very small frac- tion of (101)-orientation. Compared to our previous samples,9thisundesired orientation has been largely eliminated by optimizing the deposition temperature and by adding the post-deposition anneal- ing procedure. Both samples have a mosaicity of 0.5° as can beinferred from the width of the rocking curve of the (002) peak(ω-scan) shown in the inset. Figure 4 shows the in-plane order of Mn 2Au with the help of off-specular XRD w-scans of Mn 2Au (101) and Ta (220) peaks. Mn 2Au [100] is oriented parallel to Ta [100] as expected based on the lattice match. The same type of investigation of sputteredsamples results in very similar data, leading to the conclusion that the degree of in-plane crystallographic ordering of both MBE-grown and sputtered samples is very high. FIG. 7. (a) Atomic force microscopy scan of MBE-grown Al 2O3(1–102)/13 nm Ta (001)/45 nm Mn 2Au (001) film. The regular ripple structure slightly tilted with respect to the horizontal is an artifact of the measurement. The plot on the right side depicts a height profile along the horizontal marked in the topography imag e. (b) Atomic force microscopy scan of a sputtered Al 2O3(1–102)/13 nm Ta (001)/45 nm Mn 2Au (001) film. The plot on the right side depicts height profiles along the lines marked in the topography image.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 243901 (2020); doi: 10.1063/5.0009566 127, 243901-4 Published under license by AIP Publishing.The MBE co-deposition of Mn and Au from separate sources with independent rate control offers the opportunity of controlled variation of the sample composition. However, at thesame time, it is challenging to obtain a stoichiometric composi-tion of the Mn 2Au thin films. Furthermore, the analysis of the composition of thin films requires special methods, as many of the standard techniques for bulk samples such as x-ray fluores- cence (XRF) and energy dispersive x-ray spectroscopy (EDX), ingeneral, deliver unreliable results for thin films. However, after acareful adjustment of the deposition rates, we were able to obtainMBE-grown Mn 2Au thin films with a stoichiometric composition (66.6 ± 1.2 at. % Mn and 33.4 ± 1.7 at. % Au) as determined by inductively coupled plasma optical emission spectroscopy(ICP-OES). The sputtered samples were found to have a compo-sition of 66.1 ± 1.2 at. % Mn and 33.9 ± 1.7 at. % Au. This wasfurther verified using Rutherfo rd backscattering spectrometry (RBS), which found the stoichiometry values to be within the error margin mentioned above. Interestingly, both analysistechniques revealed that for annealing temperatures higher than450 °C, Mn begins to desorb from the film changing both its stoi-chiometry and crystallographic phase. For instance, a sample annealed at 500 °C was found to be polycrystalline and compris- ing of 56.6% Mn and 43.3% Au. Additionally, RBS revealed thatthis sample contained an 8 nm thick Mn –Au–Ta alloy layer between the buffer and Mn 2Au layers due to thermally activated diffusion of Ta atoms into the overlying film. Although x-ray diffraction probes the crystallographic order of the samples, its sensitivity for low degrees of disorder, which never-theless influences the transport properties, is limited. A figure ofmerit for the concentration of impurities, which results in scatter- ing of the charge carriers in metals, is the residual resistance ratio (RRR) = Resistance 300 K/Resistance 4K. Thus, we performed temper- ature dependent sheet resistivity measurements of both MBEdeposited and sputtered Mn 2Au films grown under optimized con- ditions ( Fig. 5 ) using the van der Pauw method. Though the resis- tivity of both films is the same at 300 K, the MBE film has a lower resistivity at 4 K (a temperature where phonon scattering issuppressed), implying a lower level of intrinsic disorder. The MBEfilm shows RRR ≃6.8, while the value for the sputtered film is RRR≃4.6. These values represent a significant improvement in quality over that of the previously reported films, which showed a residual resistivity of 7 μΩcm and RRR ≃3. 9 Furthermore, superior smoothness of our MBE-grown samples is evident from the presence of prominent Kiessig fringes corresponding to the 45 nm Mn 2Au layer in its XRR scan in Fig. 6 . The sputtered sample mainly shows oscillations corresponding tothe 13 nm Ta buffer layer. Though faint fringes corresponding tothe Mn 2Au layer are also present at lower angles, they fade away rather quickly with increasing angle due to its relatively poor morphology. More direct information about the morphology of our films was also obtained by atomic force microscopy and the results areshown in Fig. 7 . The MBE film has a more homogeneous topogra- phy compared to the sputtered sample, which is consistent with the XRR results discussed above. Whereas the typical peak to valley difference in the MBE-grown samples amounts to approximately 1 –2 nm, it is aboutfour times larger in sputtered samples. This is expected to result i nl a r g e rA F Md o m a i n si nt h eM B Es a m p l e sa sw e l la si n reduced domain wall pinning. However, this will require furtherinvestigations similar to our previous work on the sputteredsamples. 4,12 CONCLUSION We demonstrated a substantially improved crystallinity and morphology of Mn 2Au (001) thin films for antiferromagnetic spintronics due to optimized growth conditions and using MBEas an alternative deposition technique. By adding an annealingprocedure after the sample deposition, both morphology and epi-taxial quality were improved. Furthermore, by using MBE instead of sputtering, the degree of diso rder in the thin films, which is relevant for electrical transport, was clearly reduced. Most impor-t a n t l y ,t h eM B E - g r o w nt h i nf i l m ss h o was i g n i f i c a n t l ys m o o t h e rm o r p h o l o g y .T h i si sp r o m i s i n gf o rf u t u r es p i n t r o n i c sd e v i c e s ,which potentially require reduced domain wall pinning, larger AFM domains, and often rely on well-defined smooth interfaces of different functional layers. ACKNOWLEDGMENTS This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) —TRR 173-268565370 (Project Nos. A05 and B12). Financial support by the European Commission FET Open RIA ASPIN (Project No. 766566) is acknowledged. S.P.B. acknowledges financial support from the Max PlanckGraduate Center. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1J.Že l e z n ý ,H .G a o ,K .V ý b o r n ý ,J .Z e m e n ,J .M a šek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth, Phys. Rev. Lett. 113, 157201 (2014). 2S. Yu. Bodnar, L. Šmejkal, I. Turek, T. Jungwirth, O. Gomonay, J. Sinova, A. A. Sapozhnik, H.-J. Elmers, M. Kläui, and M. Jourdan, Nat. Commun. 9, 348 (2018). 3M. Meinert, D. Graulich, and T. Matalla-Wagner, P h y s .R e v .A p p l . 9, 064040 (2018). 4S. Yu. Bodnar, M. Filianina, S. P. Bommanaboyena, T. Forrest, F. Maccherozzi, A. A. Sapozhnik, Y. Skourski, M. Kläui, and M. Jourdan, Phys. Rev. B 99, 140409(R) (2019). 5P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills, R. P. Campion, V. Novák, K. Olejník, F. Maccherozzi, S. S. Dhesi, S. Y. Martin, T. Wagner,J. Wunderlich, F. Freimuth, Y. Mokrousov, J. Kune š, J. S. Chauhan, M. J. Grzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and T. Jungwirth, Science 351, 587 (2016). 6K. Olejník, V. Schuler, X. Marti, V. Novák, Z. Ka špar, P. Wadley, R. P. Campion, K. W. Edmonds, B. L. Gallagher, J. Garces, M. Baumgartner, P. Gambardella, and T. Jungwirth, Nat. Commun. 8, 15434 (2017). 7V. M. T. S. Barthem, C. V. Colin, H. Mayaffre, M. H. Julien, and D. Givord, Nat. Commun. 4, 2892 (2013). 8H.-C. Wu, Z.-M. Liao, R. G. S. Sofin, G. Feng, X.-M. Ma, A. B. Shick, O. N. Mryasov, and I. V. Shvets, Adv. Mat. 24, 6374 (2012).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 243901 (2020); doi: 10.1063/5.0009566 127, 243901-5 Published under license by AIP Publishing.9M. Jourdan, H. Bräuning, A. Sapozhnik, H.-J. Elmers, H. Zabel, and M. Kläui, J. Phys. D Appl. Phys. 48, 385001 (2015). 10M. Arana, F. Estrada, D. S. Maior, J. B. S. Mendes, L. E. Fernandez-Outon, W. A. A. Macedo, V. M. T. S. Barthem, D. Givord, A. Azevedo, and S. M. Rezende, Appl. Phys. Lett. 111, 192409 (2017).11X. F. Zhou, J. Zhang, F. Li, X. Z. Chen, G. Y. Shi, Y. Z. Tan, Y. D. Gu, M. S. Saleem, H. Q. Wu, F. Pan, and C. Song, Phys. Rev. Appl. 9, 054028 (2018). 12A. A. Sapozhnik, M. Filianina, S. Yu. Bodnar, A. Lamirand, M.-A. Mawass, Y. Skourski, H.-J. Elmers, H. Zabel, M. Kläui, and M. Jourdan, Phys. Rev. B 97, 134429 (2018).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 127, 243901 (2020); doi: 10.1063/5.0009566 127, 243901-6 Published under license by AIP Publishing.

5.0011754.pdf

J. Chem. Phys. 152, 244305 (2020); https://doi.org/10.1063/5.0011754 152, 244305 © 2020 Author(s).Bond dissociation energies of the diatomic late transition metal sulfides: RuS, OsS, CoS, RhS, IrS, and PtS Cite as: J. Chem. Phys. 152, 244305 (2020); https://doi.org/10.1063/5.0011754 Submitted: 27 April 2020 . Accepted: 02 June 2020 . Published Online: 22 June 2020 Jason J. Sorensen , Erick Tieu , and Michael D. Morse ARTICLES YOU MAY BE INTERESTED IN Bond dissociation energies of diatomic transition metal selenides: ScSe, YSe, RuSe, OsSe, CoSe, RhSe, IrSe, and PtSe The Journal of Chemical Physics 152, 124305 (2020); https://doi.org/10.1063/5.0003136 ReSpect: Relativistic spectroscopy DFT program package The Journal of Chemical Physics 152, 184101 (2020); https://doi.org/10.1063/5.0005094 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 Bond dissociation energies of the diatomic late transition metal sulfides: RuS, OsS, CoS, RhS, IrS, and PtS Cite as: J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 Submitted: 27 April 2020 •Accepted: 2 June 2020 • Published Online: 22 June 2020 Jason J. Sorensen, Erick Tieu, and Michael D. Morsea) AFFILIATIONS Department of Chemistry, University of Utah, Salt Lake City, Utah 84112, USA a)Author to whom correspondence should be addressed: morse@chem.utah.edu ABSTRACT The spectra of RuS, OsS, CoS, RhS, IrS, and PtS have been recorded near their respective bond dissociation energies using resonant two- photon ionization spectroscopy. The spectra display an abrupt drop to baseline when the bond dissociation energy (BDE) is exceeded. It is argued that spin–orbit and nonadiabatic interactions among the myriad of states that result from the ground and low-lying separated atom limits cause the molecules to predissociate rapidly as soon as the ground separated atom limit is exceeded in energy. Thus, the observed sharp predissociation thresholds are assigned as the 0 K BDEs of the molecules. With this assumption, the BDEs are assigned as follows: 4.071(8) eV (RuS), 4.277(3) eV (OsS), 3.467(5) eV (CoS), 3.611(3) eV (RhS), 4.110(3) eV (IrS), and 4.144(8) eV (PtS). Using thermochem- ical cycles, the gas-phase enthalpies of formation at 0 K, ΔfH0 K○, were calculated to be 531.8(4.3) kJ mol−1(RuS), 651.2(6.3) kJ mol−1 (OsS), 365.3(2.2) kJ mol−1(CoS), 481.5(2.1) kJ mol−1(RhS), 546.7(6.3) kJ mol−1(IrS), and 438.9(1.5) kJ mol−1(PtS). The ionization energies of RuS, CoS, and RhS were also calculated using data on the BDEs of the associated cations and were found to be 8.39(10) eV (RuS), 8.40(9) eV (CoS), and 8.46(12) eV (RhS). Combining these data with predissociation measurements of other transition metal sulfide BDEs, the periodic trends in the transition metal sulfide BDEs are discussed and the BDEs of the transition metal sulfides are com- pared to those of the corresponding selenides. The BDEs of the sulfides are found to be 15.4% greater than those of the corresponding sulfides. Published under license by AIP Publishing. https://doi.org/10.1063/5.0011754 .,s I. INTRODUCTION One of the central pillars of the field of chemistry is the study of chemical bonding and how these bonds are formed and bro- ken. Diatomic molecules provide the simplest system in which the role that electrons play in forming chemical bonds may be fully characterized. Diatomic molecules are therefore of fundamental importance for understanding the nature of the chemical bond. The chemical bond may be characterized in many ways, including the vibrational frequency, bond length, bond order, bond covalency, and bond energy. Of all the descriptors that can be used to describe and classify chemical bonds, the bond dissociation energy (BDE) remains the most quantitatively useful. In the case of transition metal atoms bonded to main group elements, the BDE has also been par- ticularly difficult to measure with precision. In this work, we present highly accurate and precise measurements of the bond dissociationenergies of RuS, OsS, CoS, RhS, IrS, and PtS. The corresponding solid-phase and nanoscopic materials are of particular importance in catalysis,1–3electrochemistry,4–7and supercapacitors.8–10 Developing a detailed chemical understanding of these bulk and nanoscopic transition metal sulfides relies on computational models to predict and describe their chemical and material proper- ties.11–14Usually, these calculations employ density functional the- ory (DFT) techniques, as DFT methods provide one of the few techniques capable of treating electronic contributions in larger systems. This approach greatly simplifies the treatment of elec- tron correlation compared to wavefunction-based methods but is highly dependent on the exchange and correlation functionals used. Although some work has been done to deduce the best functionals for these computational approaches,15–18additional data are needed to determine which functionals work best with transition metal systems. J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Progress in developing more accurate DFT methods is cur- rently limited by the lack of accurate and precise empirical bench- marks that can be used to test these methods. In his Nobel address, John Pople remarked that an acceptable standard for chemical accu- racy in a given computational method is that it produces ther- mochemical results within 1 kcal/mol (0.04 eV) of the accepted experimental benchmark.19This standard is readily achievable for molecules comprised of lighter main group elements, such as organic species. When applied to transition metal species, how- ever, most measurements themselves lack the desired 1 kcal/mol accuracy. Prior to our recent work,20–31most bond energy measure- ments have employed Knudsen effusion mass spectrometry, where the molecules effusing from a high-temperature oven are monitored by mass spectrometry, and the intensities of the various species are used to determine reaction equilibrium constants, enthalpies, and entropies. These are then used to obtain bond dissociation ener- gies. In nearly all of these measurements, the uncertainty limits are in excess of the 1 kcal/mol standard themselves and therefore pro- vide poor benchmarks for computational chemists. This leaves an experimental void of highly accurate and precise thermochemical benchmarks that are essential to furthering the development of DFT methods. Recently, we have employed the technique of resonant two- photon ionization (R2PI) spectroscopy to measure bond disso- ciation energies of various transition metal–main group element diatomics to far greater precision and accuracy than previous meth- ods.20–31The method utilizes laser ablation to produce the molecules of interest and does not require any underlying thermochemical knowledge to measure the BDE. The only requirement that must be met is that the molecule has a sufficient density of electronic states arising from the ground separated atom limit, or low-lying excited atom limits, as to yield a quasi-continuous spectrum. The method operates by exciting a molecule with a tunable excitation photon and a few tens of nanoseconds later, ionizing the excited molecule with a fixed-energy second photon. The ions are detected in a mass spec- trometer, and the ion signal is plotted against the wavenumber of the first laser pulse to provide an R2PI spectrum. As the first laser pulse is scanned in the vicinity of the BDE, a quasi-continuous spec- trum is observed until the dissociation threshold is reached, at which point the signal drops to baseline as the molecule dissociates before it can be ionized. The dissociation process relies on the multitude of potential energy curves that are present at these high energies and the nonadiabatic and spin–orbit interactions between them that permit the molecule to hop onto potential energy curves that disso- ciate to the ground separated atom limit. This sharp predissociation threshold corresponds to the BDE. From the measured BDE, several other thermodynamic val- ues can be calculated. The heat of formation of the molecule, ΔHf,0 K(MS(g)), at 0 K can be calculated from ΔHf,0 K(MS(g))=ΔHf,0 K(M(g))+ΔHf,0 K(S(g))−D0(MS), (1.1) where ΔHf,0 K(M(g)) and ΔHf,0 K(S(g)) are the heats of formation of the individual gas-phase elements and D0(MS) is the 0 K BDE of the MS molecule. Another thermochemical cycle shows that one can go from the gas-phase MS molecule to the separated M++ S + e– species by first dissociating the molecule and then ionizing the metal atom, or by ionizing the molecule and then dissociating the cation.The energy required is the same, either way, D0(MS)+IE(M)=D0(M+−S)+IE(MS). (1.2) Because the atomic ionization energies are all very well known, our measured bond dissociation energies, D0(MS), may be combined with either the ionization energy of the diatomic molecule, IE(MS), or the bond energy of the cation, D0(M+−S), to allow the remaining quantity to be calculated. If both IE(MS) and D0(M+−S) are known, then Eq. (1.2) allows the four measured quantities to be checked for consistency. It also allows the uncertainty limit of the least precise value to be improved. This thermochemical cycle has been used to confirm the values obtained through this technique in the case of V 2 to an accuracy of ±0.002 eV.32–35 In addition to presenting measurements of the bond dissocia- tion energies of RuS, OsS, CoS, RhS, IrS, and PtS, the above ther- mochemical cycles are utilized to obtain enthalpies of formation for each molecule, and the ionization energies of RuS, CoS, and RhS. The dissociation energies along with overall bonding trends of transition metal sulfides are also discussed. II. EXPERIMENTAL The experimental procedure employed to obtain these bond dissociation energies is the same as reported in our previous pub- lications and will only be briefly described here.31The molecules are first synthesized in a vacuum chamber using a laser ablation source. In this source, a pulse of helium gas seeded with 0.05%–0.7% of H 2S gas is pulsed into a stainless steel block through a reaction channel ∼2 mm in diameter, just under the surface of the block. A rotat- ing and translating metal sample disk, composed of the metal of interest, is pressed against the block, and 8 mJ–15 mJ of Nd:YAG third harmonic radiation (355 nm) is focused through a vaporiza- tion channel at right angles to the reaction channel onto the metal disk. The ablation laser pulse is timed such that the ablated material is picked up by the gas pulse. As it travels down the 1 cm reaction channel, the ablated material reacts with the H 2S gas to form var- ious metal compounds; we optimize production for the diatomic metal sulfide of interest by varying the H 2S concentration, the car- rier gas pressure, and the ablation laser power. The gas pulse is then supersonically expanded into vacuum ( ∼10−5torr), where it is rota- tionally cooled to roughly 10 K–20 K. The rapid expansion into vac- uum halts any reactions and allows for the study of reactive species, such as the diatomic MS molecules, in their ground states. The expanding gas cone is then roughly collimated by a 1 cm diameter skimmer to form a molecular beam, which passes into an analytical chamber where the molecules can be spectroscopically probed and detected. When the molecules enter the Wiley–McLaren ion source,36 housed in the analytical chamber, they undergo R2PI analysis. Two lasers are fired at the molecular beam. First, an excitation laser pulse from a tunable optical parametric oscillator (OPO) laser is coun- terpropagated down the axis of the beam, and then, 20 ns–90 ns later, an ionization laser pulse from a KrF (5.00 eV) or F 2(7.90 eV) excimer laser is fired perpendicularly to the path of the molecular beam. The energy of the ionization laser is selected such that a sin- gle photon does not carry sufficient energy to ionize the molecule on its own, but the combination of one photon from each laser J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp source does provide sufficient energy for ionization. In the overlap- ping region of the two lasers, only the molecules that are excited by the first photon can be ionized by the second. This occurs within the Wiley–McLaren ion source, which directs the newborn ions up the flight tube of a reflectron time-of-flight mass spectrometer oriented perpendicular to the path of the molecular beam. Ions are then sep- arated by mass, as lighter ions reach the detector earlier than heavier ions. The ion signals of selected masses are recorded as a function of the wavelength of the OPO laser. To measure the BDE, the OPO laser is scanned over the energy range of the expected BDE, as estimated from previous mea- surements, calculated values, or trends of similar molecules. Scan- ning from lower to higher energy, a quasi-continuous spectrum is observed arising from the high density of states. When the ground separated atom limit is reached, spin–orbit and nonadiabatic inter- actions between states allow the molecule to dissociate faster than it can be ionized by the second laser pulse. This causes the ion sig- nal to abruptly drop to baseline. The sharp predissociation threshold that is observed is then assigned as the BDE. To calibrate the mea- sured BDE, the spectra of one or two atomic species are collected either in conjunction with the molecule or directly afterward. The calibrant metal atoms are usually the main transition metal species, impurities in the metal sample, a metal component in an alloy, or a metal sample with known transitions in the scanned region. By fit- ting the observed electronic transitions to the known atomic energy levels,37the spectra are readily calibrated using either a constant or linear shift. Calibration errors are less than 5 cm−1for all molecules reported here. In the case of PtS, prominent discrete features in the spectrum make the assignment of a precise dissociation threshold difficult. In the previously studied examples of Ni 2,38Pt2,39and CrW,40we have observed short-lived states that lie above the BDE. These can be identified through their lifetimes, with states above the dissoci- ation limit having lifetimes that are an order of magnitude shorter than those below the limit. The sharp decrease in lifetime can then be used to pin down the BDE. To measure the lifetime, we vary the delay between the OPO laser and the ionization laser and plot the ion signal as a function of the delay. The data give an exponential decay curve of signal intensity vs delay between the two lasers. This is fitted to the exponential form using the Levenberg–Marquardt nonlinear least-squares algorithm,41allowing the exponential lifetime, τ, to be extracted. III. RESULTS The spectra of RuS, OsS, CoS, RhS, IrS, and PtS in the energy range near their respective BDEs are shown in Figs. 1–6. In the lower energy portion of each spectrum, a highly congested or quasi- continuous spectrum is observed. The sharp drop in molecular ion signal to baseline, marked with an arrow, gives the predissociation threshold that is assigned as the BDE. The assigned uncertainty is displayed as the horizontal bar at the top of the arrow. Contribu- tions to the uncertainty include factors such as laser linewidth, laser calibration errors, and rotational broadening of the predissociation threshold. Although a small fraction of the molecules probed may be in vibrationally excited states, this does not contribute to the error. These molecules will predissociate at a lower photon energy because FIG. 1 . R2PI spectrum of RuS (blue trace) showing the predissociation threshold and corresponding bond dissociation energy D 0at 32 835(65) cm−1. The atomic spectrum of Ru (red trace) displays the atomic transitions used to calibrate the x axis. In Figs. 1–6, the horizontal bar atop the arrow designates the assigned error limits. of the vibrational energy already stored in the molecule. The final drop to baseline therefore reflects the BDE of the molecules in their ground vibronic levels. A detailed discussion of the assignment of error limits is provided in a previous publication.30The measured BDEs are compiled in Table I along with the enthalpies of formation FIG. 2 . R2PI spectrum of OsS (blue trace) showing the predissociation threshold and corresponding bond dissociation energy D 0at 34 500(25) cm−1. The atomic spectra of Os (red trace) and V (black trace) display the atomic transitions used to calibrate the x axis. J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . R2PI spectrum of CoS (blue trace) showing the predissociation threshold and corresponding bond dissociation energy D 0at 27 965(40) cm−1. The atomic spectrum of Co (red trace) displays the atomic transitions used to calibrate the x axis. of the gaseous MS molecules. These were calculated using Eq. (1.1) in combination with the enthalpies of formation of the gaseous atoms. In addition to the enthalpies of formation, the thermochem- ical cycle of Eq. (1.2) may be employed to extract the ionization FIG. 4 . R2PI spectrum of RhS (blue trace) showing the predissociation threshold and corresponding bond dissociation energy D 0at 29 123(25) cm−1. The atomic spectrum of Rh (red trace) displays the atomic transitions used to calibrate the x axis. FIG. 5 . R2PI spectrum of IrS (blue trace) showing the predissociation threshold and corresponding bond dissociation energy D 0at 33 149(25) cm−1. The atomic spectra of Ir (red trace) and Rh (black trace) display the atomic transitions used to calibrate the x axis. energies of RuS, CoS, and RhS using the well-known atomic ioniza- tion energies,37the BDEs measured here for RuS, CoS, and RhS, and the BDEs of Ru+–S,42Co+–S,43and Rh+–S,44which have been mea- sured by guided ion beam mass spectrometry. These are presented in Table II. FIG. 6 . R2PI spectrum of PtS (blue trace) showing the predissociation threshold and corresponding bond dissociation energy D 0at 33 425(65) cm−1. The atomic spectra of Pt (red trace) and Rh (black trace) display the atomic transitions used to calibrate the x axis. Measured lifetimes are given for the various vibronic levels as indicated. J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Bond dissociation energies and enthalpies of formation of gaseous metal sulfides. ΔfH0K0(M(g)) ΔfH0K0(MS) Molecule D 0(cm−1) D 0(eV) (kJ mol−1) (kJ mol−1) RuS 32 835(65) 4.071 (8) 649.8 (4.2)a531.8 (4.3) OsS 34 500(25) 4.277 (3) 789.1 (6.3)b651.2 (6.3) CoS 27 965(40) 3.467 (5) 425.1 (2.1)c365.3 (2.2) RhS 29 123(25) 3.611 (3) 555.2 (2.1)b481.5 (2.1) IrS 33 149(25) 4.110 (3) 668.5 (6.3)a546.7 (6.3) PtS 33 425(65) 4.144 (8) 564.0 (1.3)a438.9 (1.5) aValues of ΔfH0K0(M(g)) for Ru, Ir, and Pt taken from Ref. 63. bValues of ΔfH0K0(M(g)) for Rh and Os taken from Ref. 64. cValues of ΔfH0K0(Co(g)) and ΔfH0K0(S(g)) taken from Ref. 65. TABLE II . Calculated ionization energies of select metal sulfides, IE(MS).a D0(MS) eV D 0(M+–S) Molecule (this work) (eV) IE (M) (eV)bIE (MS) (eV) RuS 4.071 (8) 3.04(10)c7.360 50 (5) 8.39(10) CoS 3.467 (5) 2.95(9)d7.881 01 (12) 8.40(9) RhS 3.611 (3) 2.61(12)e7.458 90 (5) 8.46(12) aIonization energies were obtained using the thermochemical cycle, Eq. (1.2), using the values of D 0(MS) reported here in conjunction with values of D 0(M+–S) obtained by guided ion beam mass spectrometry and the precisely known atomic ionization energies, IE(M). bFrom Ref. 37. cReference 42. dReference 43. eReference 44. As we have discussed in our previous papers,26,28restrictions based on the conservation of Ω can sometimes exist that could pre- vent the molecule from dissociating promptly at the thermochemi- cal threshold. These have been clearly observed in the examples ofV2and Zr 2.34,45We have confirmed that no such restrictions are present in the molecules investigated here; thus, we are confident that the measured predissociation thresholds correspond to the ther- mochemical bond energies of these diatomic sulfides. In this evalu- ation, we have considered the Ω values of the possible ground states of the molecule and have determined the possible Ω′values that can be reached in electric dipole allowed transitions. We then compared these values to the Ω values that derive from the ground separated atom levels and find that in all cases the Ω′values match values that are obtained from the separated atoms in their ground levels. Thus, Ω can be conserved in a dissociation process that produces sepa- rated atoms in their ground spin–orbit levels. For this evaluation, the experimentally known ground levels of CoS (X4Δ7/2),46,47RhS (X4Σ– 1/2),48,49and PtS (X3Σ– 0+)50,51were used. For OsS (5Σ+ 0+, 3Φ4, or3Σ– 0+),52the calculated ground state of OsO was employed. For RuS (X5Δ4)53and IrS (X2Δ5/2),54,55the experimentally known ground state of the corresponding oxide was used. IV. DISCUSSION A. Lifetime study of PtS The discrete nature of the PtS spectrum may be observed in Fig. 6, which also provides the excited-state lifetimes of selected vibronic bands. All of the bands investigated, except for the last fea- ture, have lifetimes in the range of 300 ns to 1.3 μs. The last observed band, however, has a dramatically shorter lifetime of 70 ns. The short lifetime of this state suggests that it may lie above the ground sepa- rated atom limit. With this in mind, we have chosen a value halfway between the last two vibronic bands as the BDE of PtS and have assigned an error limit that is sufficiently large to encompass both of these bands. B. Comparison to previous studies Previous studies of the BDEs of RuS, CoS, and RhS are com- piled in Table III. In the case of RuS, the only previous report is a DFT study by Sun, Wang, and Wu.56Their result, D 0(RuS) = 3.83 eV, underestimates the BDE of RuS by 0.24 eV compared to our measurement of 4.071(8) eV. TABLE III . Previous studies of the bond dissociation energies of RuS, CoS, and RhS. Bold values indicate experimental results, and plain texts indicate computational results. Molecule D 0(eV) Method Authors Reference RuS 4.071(8) Predissociation threshold This work 3.83 B3LYP/LANL2DZ Sun, Wang, and Wu 56 CoS 3.467(5) Predissociation threshold This work 3.39(0.15) Knudsen effusion Drowart et al. 57 3.25 DFT Wu, Wang, and Su 59 4.52 DFT Bridgeman and Rothery 58 2.94 CASSCF Bauschlicher and Maitre 1 RhS 3.611(3) Predissociation threshold This work 3.23 B3LYP/LANL2DZ Sun, Wang, and Wu 56 J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The BDE of CoS has been measured using Knudsen effusion mass spectroscopy, giving a value of 3.39(15) eV,57which com- pares well to our value of 3.467(5) eV. There have been three com- putational studies of this molecule. In the first, a complete active space self-consistent field (CASSCF) calculation by Bauschlicher and Maitre obtained a BDE of 2.94 eV, 0.53 eV lower than our result.1 Subsequently, Bridgeman and Rothery employed a local spin den- sity DFT method to obtain a BDE of 4.519 eV,58overestimating our result by more than 1 eV. Most recently, Wu, Wang, and Su report a B3LYP DFT value of 3.25 eV, which is the most accurate compu- tational value to date.59Regardless, all three of these computational values remain outside Pople’s criterion for chemical accuracy,19with the closest lying 0.22 eV below our measured value. For RhS, Sun, Wang, and Wu report the only previous value for the BDE, 3.23 eV.56This significantly underestimates the strength of the bond compared to our value of 3.611(3) eV. No previous results for the BDEs of OsS, IrS, or PtS have been reported. C. General observations Combining our previous measurements of transition metal sul- fide bond energies with the present results allows a fairly com- plete picture of the bonding trends among the transition metal sul- fides to be assembled.20,28,29,31This is illustrated in Figs. 7–9 for the 3d, 4d, and 5d MS molecules, respectively. These figures dis- play the BDEs of the MS molecules as one moves across the tran- sition metal periods. For all three periods, the vertical axis spans the same range. It is immediately obvious that for all three periods, the early transition metal sulfides (groups 3, 4, and 5) are much more strongly bound than the subsequent metal sulfides. This is read- ily understood from the molecular orbital configurations of these molecules. FIG. 7 . Trends in the BDEs of the 3d transition metal sulfides, MS. Red circles are measured via the observation of a predissociation threshold in this work (CoS), Ref. 31 (VS), Ref. 29 (FeS, NiS), and Ref. 20 (ScS, TiS). Blue squares are Knud- sen effusion measurements from Ref. 83 (ScS), Ref. 84 (TiS), and Ref. 57 (VS, CrS, MnS, FeS, CoS, NiS, CuS). FIG. 8 . Trends in the BDEs of the 4d transition metal sulfides, MS. Red circles are measured via the observation of a predissociation threshold in this work (RuS, RhS), and Ref. 20 (YS, ZrS, NbS). Blue squares are Knudsen effusion measure- ments from Ref. 85 (YS) and Ref. 86 (ZrS). Open circles are from Sun, Wang, and Wu’s DFT calculation, Ref. 56. If we just consider the valence orbitals of the constituent atoms [(n+1)s, nd on the metal; 3s, 3p on sulfur], the resulting molecular orbitals consist of four σorbitals, two pairs of πorbitals, and one pair of δorbitals. The 1 σorbital is composed mostly of 3s character on sulfur and behaves much like a core orbital. It is unimportant in FIG. 9 . Trends in the BDEs of the 5d transition metal sulfides, MS. Red circles are measured via the observation of a predissociation threshold in this work (OsS, IrS, PtS), Ref. 28 (WS), and Ref. 20 (HfS, TaS). Blue squares are experimental results from Ref. 85 (LaS, Knudsen effusion) and Ref. 82 (AuS, Morse extrapolation of several electronic states). ReS has not been investigated either experimentally or computationally. J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Ground configurations and terms of the diatomic transition metal sulfides, MS.a 3d 4d 5d Group Molecule and term Reference Molecule and term References Molecule and term References 3 ScS 3 σ1,2Σ+66 YS 3 σ1,2Σ+67 LaS 3 σ1,2Σ+68 4 TiS 3 σ11δ1,3Δ 69 ZrS 3 σ2,1Σ+70 HfS 3 σ2,1Σ+71 5 VS 3 σ11δ2,4Σ–72 NbS 3 σ11δ2,4Σ–73 TaS 3 σ21δ1,2Δ 74 6 CrS 3 σ11δ22π1,5Π 75 MoS [3 σ11δ22π1,5Π] 56 and 76 WS 3 σ21δ2,3Σ–60 and 61 7 MnS 3 σ11δ22π2,6Σ+77 TcS [3 σ11δ22π2,6Σ+] 56 ReS [3 σ11δ22π2,6Σ+] ReO 52 8 FeS 3 σ11δ32π2,5Δ 78 RuS [3 σ21δ22π2,5Σ+] 56 OsS [3 σ21δ22π2,5Σ+] OsO 52 9 CoS 3 σ21δ32π2,4Δ 46 RhS 3 σ11δ42π2,4Σ–48 IrS [3 σ11δ42π2,4Σ–] IrO 52 10 NiS 3 σ21δ42π2,3Σ–79 PdS [3 σ21δ42π2,3Σ–] 56 PtS 3 σ21δ42π2,3Σ–50 11 CuS 3 σ21δ42π3,2Π 80 AgS 3 σ21δ42π3,2Π 81 AuS 3 σ21δ42π3,2Π 82 aMost configurations and terms are from spectroscopic determinations. Configurations and terms in square brackets are computational results. For ReS, OsS, and IrS, neither exper- imental nor computational results exist. For these molecules, computational results on the corresponding metal oxides are listed. For ReO, spectroscopic results identify the ground state as 3 σ21δ3,2Δ5/2(Ref. 62). For IrO, spectroscopic results show that the ground level has Ω = 5/2 and derives from a spin–orbit induced mixture of the2Δ5/2and4Δ5/2terms of the 3σ21δ32π2configuration (Ref. 55). an analysis of the chemical bonding in these species. Considerably above the 1 σlie the 2 σand 1 πorbitals; these are combinations of the nd metal orbitals and the 3p orbitals of sulfur and are strongly bonding in character. Above these lie the 3 σand 1 δorbitals, which are primarily metal (n+1)s and nearly purely metallic nd δin char- acter, respectively. These are primarily nonbonding orbitals. Above these are the antibonding 2 πand 4 σorbitals, which are the anti- bonding counterparts of the strongly bonding 1 πand 2 σorbitals. The bonding orbitals are more localized on the electronegative sulfur atom, while the antibonding orbitals are more localized on the metal atom. Table IV provides the ground electronic configurations andterms for the MS molecules, with spectroscopically derived results provided for most molecules. Computational results are given when experimental studies are absent, in square brackets. For ReS, OsS, and IrS, no data exist. Computational results are given for the cor- responding oxides instead. Predissociation-based measurements of the BDEs of the transition metal sulfides and selenides are provided in Table V. In all of the transition metal sulfides, the 1 σ, 2σ, and 1 πorbitals are filled. The group three metal sulfides (ScS, YS, and LaS) have one additional electron, which is placed in the 3 σnonbonding orbital. Moving to the right in the series, the group 4 sulfides (TiS, ZrS, TABLE V . Bond dissociation energies of transition metal sulfides and selenides (eV).a 3d period 4d period 5d period Group Metal D 0(MS) D 0(MSe) Metal D 0(MS) D 0(MSe) Metal D 0(MS) D 0(MSe) 3 Sc 4.852 (10)b4.152 (3)cY 5.391 (3)b4.723 (3)cLa 4 Ti 4.690 (4)b3.998 (6)dZr 5.660 (4)b4.902 (3)dHf 5.780 (20)b5.154 (4)d 5 V 4.535 (3)e3.884 (3)dNb 5.572 (3)b4.834 (3)dTa 5.542 (3)b4.705 (3)d 6 Cr Mo W 4.935 (3)f4.333 (6)f 7 Mn Tc Re 8 Fe 3.240 (3)g2.739 (6)gRu 4.071 (8)h3.482 (3)cOs 4.277 (3)h3.613 (3)c 9 Co 3.467 (5)h2.971 (6)cRh 3.611 (3)h3.039 (9)cIr 4.110 (3)h3.591 (3)c 10 Ni 3.651 (3)g3.218 (3)gPd Pt 4.144 (8)h3.790 (30)c aOnly predissociation-based values are listed. bData for ScS, YS, TiS, ZrS, HfS, NbS, and TaS taken from Ref. 20. cData for ScSe, YSe, RuSe, OsSe, CoSe, RhSe, IrSe, and PtSe taken from Ref. 21. dData for TiSe, ZrSe, HfSe, VSe, NbSe, and TaSe taken from Ref. 30. eData for VS taken from Ref. 31. fData for WS and WSe taken from Ref. 28. gData for FeS, FeSe, NiS, and NiSe taken from Ref. 29. hData for RuS, OsS, CoS, RhS, IrS, and PtS from this work. J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and HfS) add a second electron to the nonbonding (3 σ, 1δ) set and the group 5 sulfides (VS, NbS, and TaS) add a third electron to the nonbonding orbital set. These molecules have a triple bond, formed from a single σ-bond (from the doubly occupied 2 σ2orbital) and twoπ-bonds (from the quadruply occupied 1 π4orbital). As a result, these species are the most strongly bound metal sulfides from their period. Beginning with the group 6 sulfides, CrS and MoS, electrons begin filling the antibonding 2 πorbital. In part, this is because the placement of the last electron in the nonbonding set of orbitals would result in either a 3 σ21δ2,3Σ–or a 3 σ11δ3,3Δground term. These terms cannot correlate with the ground separated atom limit, nd5(n+1)s1,7S + 3p4,3P, which generates only quintets, septets, and nonets. The 3 σ21δ2,3Σ–term correlates instead to the nd5(n+1)s1, 5S + 3p4,3P excited separated atom limit, 7593 cm−1(Cr) and 10 762 cm−1(Mo) above the ground limit. The 3 σ11δ3,3Δterm cor- relates with the nearby but slightly higher nd4(n+1)s2,5D + 3p4,3P excited separated atom limit, 7750 cm−1(Cr) and 10 965 cm−1(Mo) above the ground limit. Either possibility requires a high promo- tion energy that is simply too large for3Σ–or3Δto emerge as the ground state. The ground states of CrS and MoS instead place the last electron in the antibonding 2 πorbital. This allows the ground5Π state to correlate with ground-state atoms but results in a precipitous decrease in BDE. Following our initial study of the WS BDE,28spectroscopic work has shown that the tungsten sulfide molecule instead places its last electron in the 3 σorbital, giving a 3 σ21δ2,3Σ–ground term that avoids population of the antibonding 2 πorbital.60,61Unlike CrS and MoS, this ground molecular term does correlate with the ground separated atom limit of W 5d46s2,5D + S 3p4,3P, which generates two3Σ–terms. As a result, the steep decrease in BDE seen in CrS and MoS is not present in WS. Like the group 3, 4, and 5 transition metal sulfides, WS also has a triple bond. For groups 7–10, all of the MS molecules are either known or calculated to place two electrons in the antibonding 2 πorbital, giving them a double bond. As a result, for all three periods, these species have smaller BDEs than their triply bonded relatives. In this connection, it should be noted that no computational or spectro- scopic results exist for ReS, OsS, or IrS, but calculations indicate that the ground states of the corresponding oxides also place two elec- trons in the antibonding 2 πorbital. Experimental results are at odds with the computational results, however, suggesting that ReO has a 3σ21δ3,2Δ5/2ground level62and that IrO has an Ω = 5/2 ground level deriving from a spin–orbit induced mixture of the2Δ5/2and4Δ5/2 levels that arise from the 3 σ21δ32π2configuration.55These species, and the corresponding sulfides, deserve further computational and spectroscopic study. It is interesting that the late 3d sulfides exhibit a trend of increasing BDE as one moves to the right in the series, while the 4d sulfides display a decreasing trend. The 5d sulfides display lit- tle variation in BDE over this range. We believe that the strongly decreasing trend in BDE for the 4d series, in going from RuS to RhS to PdS, reflects the significant decrease in 4d orbital size and energy as one traverses this series of atoms. By the time one reaches Pd, the 4d orbitals have decreased sufficiently in energy that all of the atomic valence electrons occupy the 4d orbital, giving Pd a ground configuration of 4d105s0,1S.37As a result, palladium is the only tran- sition metal that lacks valence s electrons in its ground state.37Theclosed-shell nature of atomic Pd, along with the low energy of the 4d orbitals in this atom, tends to make Pd-X bonds weak, regardless of the nature of atom X. Within the late members of the 4d series, and continuing on with Ag, the 4d orbitals are becoming more core-like and are less involved with the chemical bonding than in the 3d or 5d series. The coinage metal sulfides, CuS, AgS, and AuS, add one more electron to the 2 πantibonding orbital to form ground-state config- uration and terms of 3 σ21δ42π3,2Π. These have a bond order of 1.5 and display much lower BDEs than the doubly bonded metal sulfides of groups 8–10. The weakening of the MS bond continues with the group 12 metals, Zn, Cd, and Hg, as the 2 πorbital is filled to give 3σ21δ42π4,1Σ+ground states.56,59These species are omitted from Figs. 7–9. The bond energy trends of the transition metal sulfides are remarkably similar to the trends in the BDEs of the correspond- ing selenides. This is illustrated in Fig. 10, where the predissociation measurements of BDEs of the 3d MS and MSe molecules are plot- ted. Similar plots may be constructed for the 4d and 5d series. For each period, the trends for the metal sulfides and selenides are eerily similar. To examine this relationship further, we have constructed a plot correlating the BDEs of the transition metal sulfides and selenides, which is displayed in Fig. 11. For simplicity, this includes only the BDEs that have been measured in this laboratory using the predis- sociation method that is employed here. The figure shows a very consistent trend that the M–S bond is ∼15% stronger than the M–Se counterpart. This is shown by the least-squares linear fit through the data points, which is constrained to pass through the origin and has a fitted slope of 1.154 ±0.006. This plot demonstrates that the underlying bonding mechanisms are extremely similar between the transition metal sulfides and selenides. FIG. 10 . Comparison of the bond dissociation energies of the 3d transition metal sulfides (blue circles) and the corresponding transition metal selenides (red squares). For simplicity, only predissociation-based measurements are included. J. Chem. Phys. 152, 244305 (2020); doi: 10.1063/5.0011754 152, 244305-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 11 . BDEs of the transition metal sulfides plotted against the BDEs of the corresponding selenides. A linear fit constrained to pass through the origin shows that the BDE of the transition metal sulfide is on average 15.4(6)% larger than the BDE of the corresponding selenide. V. CONCLUSION The bond dissociation energies of RuS, OsS, CoS, RhS, IrS, and PtS have been spectroscopically determined using resonant two- photon ionization spectroscopy through the observation of a sharp predissociation threshold. We argue that this predissociation thresh- old provides a precise measure of the bond dissociation energy because the dense manifold of states arising from the ground and low-lying separated atom limits provide a multitude of pathways for the molecule to dissociate as soon as its bond dissociation energy is reached and exceeded. The first measurements of the bond dissociation energy of RuS, OsS, RhS, IrS, and PtS are provided in this report. The bond dissocia- tion energy of CoS measured here agrees with previous experimental values, albeit with a 30-fold improvement in precision. The ioniza- tion energies of RuS, CoS, and RhS are reported, along with the heats of formations of all the molecules studied, using thermochemical cycles and other measurements in conjunction with the BDEs mea- sured in this study. 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6.0000409.pdf

J. Vac. Sci. Technol. A 38, 063403 (2020); https://doi.org/10.1116/6.0000409 38, 063403 © 2020 Author(s).Effect of duty cycle on the thermochromic properties of VO2 thin-film fabricated by high power impulse magnetron sputtering Cite as: J. Vac. Sci. Technol. A 38, 063403 (2020); https://doi.org/10.1116/6.0000409 Submitted: 21 June 2020 . Accepted: 18 September 2020 . Published Online: 05 October 2020 Pi-Chun Juan , Kuei-Chih Lin , Cheng-Li Lin , and Wei-Fan Lin ARTICLES YOU MAY BE INTERESTED IN Plasmonic nitriding of graphene on a graphite substrate via gold nanoparticles and NH 3/Ar plasma Journal of Vacuum Science & Technology A 38, 063001 (2020); https:// doi.org/10.1116/6.0000405 Acoustoelectric drag current in vanadium oxide films Journal of Applied Physics 128, 155104 (2020); https://doi.org/10.1063/5.0015215 Coupled oscillations of VO 2-based layered structures: Experiment and simulation approach Journal of Applied Physics 127, 195103 (2020); https://doi.org/10.1063/5.0001382Effect of duty cycle on the thermochromic properties of VO 2thin-film fabricated by high power impulse magnetron sputtering Cite as: J. Vac. Sci. Technol. A 38, 063403 (2020); doi: 10.1116/6.0000409 View Online Export Citation CrossMar k Submitted: 21 June 2020 · Accepted: 18 September 2020 · Published Online: 5 October 2020 Pi-Chun Juan,1,2,a)Kuei-Chih Lin,3Cheng-Li Lin,4and Wei-Fan Lin1 AFFILIATIONS 1Department of Materials Engineering and Center for Plasma and Thin Film Technologies, Ming Chi University of Technology, New Taipei 243, Taiwan 2Department of Electronic Engineering and Institute of Electro-Optical Engineering, Chang Gung University, Taoyuan 333, Taiwan 3Department of Electronic Engineering, Ming Chuan University, Taoyuan 333, Taiwan 4Department of Electronic Engineering, Feng Chia University, Taichung 407, Taiwan a)Electronic mail: pcjuan@mail.mcut.edu.tw ABSTRACT Thermochromic VO 2thin films are fabricated by using high power impulse magnetron sputtering. The effect of the duty cycle with different on/off time ratios on the thermochromic properties is investigated. Though the transmittance increases with decreasing duty cycle, a moder- ate duty cycle is suggested. It is found that V 2O3crystallinities appear at a low duty cycle, while the inter-diffusion between TiO 2and VO 2 layers becomes worse at a high duty cycle. In this study, the VO 2/TiO 2/glass stacked structures reach a solar regulation efficiency (ΔTsol= 9.5%) and an applicable luminous transmittance (T lum= 43.1%) in a low-temperature state under the duty cycle of 2.5%. The crys- talline behavior of the monoclinic phase shown in the x-ray diffraction pattern is further examined by a high resolution transmission elec- tron microscope. The changes in binding energies of V 2p and O 1s orbits are compared throughout the films. The thermochromic properties as a function of TiO 2thickness are also discussed. Published under license by AVS. https://doi.org/10.1116/6.0000409 I. INTRODUCTION Nowadays, energy saving materials are paid much attention to global environmental concern of carbon assumption. A family ofvanadium oxides such as VO 2and lower oxidation phases of V 2O3, V3O5,V4O7, etc., show a reversible phase transformation, either an insulator-to-metal (IMT) or a semiconductor-to-metal transition (SMT) at a transition temperature (T C).1Among these vanadium oxides, only VO 2shows its T Cin the temperature range of 300 –400 K, which is 341 K for bulk crystals.2Beyond the tempera- ture of T C, the electrical conductivity of thermochromic materials changes abruptly from a low to high value, which blocks the infra-red transmission (700 < λ< 3000 nm) with an acceptable transpar- ence in the visible wavelength range of 400 –800 nm. This specific behavior is recognized as the phase change of the crystalline struc- ture from a monoclinic (M, P2 1/c) to rutile (R, P4 2/mnm) phase.3A high power impulse magnetron sputtering (HIPIMS) system equipped with a high density plasma source is attractive due to an obvious improvement in the degree of crystallization for deposited films.4In the glow discharge, the high power can be adjusted by pulses with a low repetition frequency and duty cycle, which allows a high target current and thus a production of highly energetic ions.5Such a power supply mode results in a high density plasma above the cathode target. The energetic ion bombardment is known to increase the surface diffusion and hence increase the nucleation density.6The high ionic energies during deposition can enhance the crystallization of deposited films and reduce the required amount of thermal budget. Therefore, films with superior properties with less atomic interdiffusion can be expected.7In this study, VO 2films are fabricated by two successive process steps: first, HIPIMS at room temperature without any a substrate heatingARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-1 Published under license by A VS.and then a post-annealing process by rapid thermal annealing (RTA). Several advantages of low deposition temperature (T dep) can be achieved such as facilitation of large-scale production by reduc-ing the energy consumption, improvement of temperature uniform-ity on substrate, and limitation of element ’s out-diffusion from soda-lime glass into VO 2films.8Additionally, in order to further reduce the inter-diffusion at the glass/VO 2interface and to improve the low transmittance in the visible region of VO 2thin-films, a bilayer structure of VO 2/TiO 2is advised.9The buffer layer of TiO 2 can effectively restrain the mobile ions of glass from diffusing intothe VO 2(M) layer during both the deposition and annealing pro- cesses.10Other schemes of the bilayer proposed are placing the thermal stable oxide on top of the VO 2film11and using the VO 2/ TiO 2period structure for metamaterials.12The former is to protect from environmental aggressions without degrading the optical con-trasts due to the thermochromic transition. 13The latter aims to tune the iso-frequency surface and control light-matter interaction. For the low-cost industrial production of VO 2-based devices, the insertion of an antidiffusion barrier layer and the reduction in thethermal budget required during the crystallization process are ourmain considerations. II. EXPERIMENT Thermochromic VO 2thin films are deposited by the HIPIMS technique. The duty cycle is defined as the ratio of on time to thetotal time in one frequency pulse. Three duty cycles of 50/1000, 50/ 2000, and 50/3000 ( μs/μs), which is equal to 5%, 2.5%, and 1.6%, respectively, are used to study the effect on thermochromic proper-ties. Before VO 2deposition, a TiO 2thin-film is first deposited by DC sputtering under 300 °C with thicknesses of 100, 200, and300 nm. The purpose of a TiO 2buffer layer is to improve the crys- tallinity of successive films. For the deposition of VO 2films, the flow rates of Ar and O 2gases are set at 50 and 2 SCCM, respec- tively. The time averaged power of the HIPIMS system is kept at400 W, while the calculated peak power density increases from 1.7 to 5.1 kW/cm 2with decreasing the duty cycle from 5% to 1.6% for a 3-in. target, which meets the requirement of at least 1 kW/cm2 for high density reactive ions in the plasma.14This implies that the plasma density near the target is sufficiently high that it not onlysputters the target but also ionizes the sputtered material to a high degree. After the deposition of VO 2thin-films, a rapid thermal annealing of 500 °C for 3 min is performed for the formation ofcrystals of thermochromic VO 2thin-films. The low thermal budget ensures feasible applications in soda-lime plain glass. The identification of the crystal phase is characterized by x-ray diffraction (XRD; PANalytical-X ’Pert PRO MPD) in a thin-film mode. The x-ray source is Cu K α1and the wavelength is 1.541 87 Å. The operation voltage and current are set to 45 kV and 20 mA,respectively. The XRD patterns show that the crystal phase graduallychanges from V 2O3to VO 2and finally to V 2O5as the O 2/Ar ratio increases. The surface morphology of V xOyis examined by a field-emission scanning electron microscope (FE-SEM; JEOL JSM6701F) with the operation voltage of 10 kV. To improve the crystal-linity of the monoclinic phase at 2 θ= 27.8° for VO 2thin-films (JCPDS-ICDD card No. 43-1051), a TiO 2buffer layer is inserted between a glass substrate and a VO 2thin-film. The crystalline phaseof TiO 2is identified to be the anatase phase (JCPDS-ICDD card No. 21-1272). The laminated structure of VO 2/TiO 2/glass shows superior thermochromic properties near the IR region. The optical transmit-tance characteristics are monitored using an UV-visible-near-IRspectrophotometer at a normal incidence from 350 to 2700 nm onan incorporated heating stage. The spectra of the laminated struc- ture are taken from two temperatures: 25 and 85 °C. Transmission electron microscopy (TEM; JEOL JEM-2010F) with the operationvoltage of 200 kV is used to confirm the microstructures of thestack films. The cross-sectional images of TEM specimens are pre-pared by mechanical pretreatments followed by ion milling. The localized diffraction patterns are evaluated by two-dimensional Fourier transform, i.e., Fast Fourier Transform (FFT). The combina-tion states of V 2p and O 1s electrons are examined by x-ray photo-electron spectroscopy (XPS; Perkin-Elmer ESCA System) using anMg K αline at 1253.6 eV, a pass energy of 46.9 eV, and a take-off angle of 90°. The C 1s peak at 285.3 eV is used as the reference binding energy. The depth profile is performed by the ionic sput-tering of Ar gas. The milling rate is set as low as 0.2 nm/s, whichcan prevent the preferential etching of elements from surfacedamage. From the XPS results, the 3/2 and 1/2 spin –orbit doublet components of V 2p are positioned in the range of 516 –517 eV and 523 –524 eV, respectively, which are associated with the V 4+ valence state.15The detailed data are discussed in Sec. III . III. RESULT AND DISCUSSION Figure 1(a) shows the deposition rates of vanadium oxide films at different O 2/Ar ratios (in %) of 2% –10% using a metallic sputtering target of vanadium. Different duty cycles with increasingthe total time of one pulse are applied, i.e., 1000, 2000, 3000 μs. Since there is no abrupt change in the deposition rate, the poison effect can be neglected. Usually, the reduction in the target poison- ing effect is attributed to the mechanism of ion bombardmentupon the target surface. 16One should notice that the deposition rate using a low duty cycle of 1.6% leads to gradual saturation with increasing O 2/Ar ratio. This indicates that a sufficiently large ionic energy resputters the deposition film; Fig. 1(b) shows the calculated peak power density with decreasing the duty cycle from 10%(50μs/500 μs) to 1.6% (50 μs/3000 μs) with a 500 μs interval at one pulse width. Except the duty cycle of 10% and output powers lower than 200 W, all peak power densities with these duty cycles are greater than 1 kW/cm 2. The high peak power densities greater than 1 kW/cm2are sufficient to produce high density reactive ions, leading to high sputter rates, advantageous bombardment, andhigh reactivity. 15The peak power densities are 1.62, 3.23, and 4.92 kW/cm2under 400 W, which corresponds to the duty cycles of 5%, 2.5%, and 1.6%, respectively, that are used in this study. Figures 2(a) and 2(b) are the SEM images of ×50 K and ×100 K for samples with the duty cycle of 1.6%, (c) and (d) aresamples with 2.5% and (e) and (f) are samples with 5%, respec- tively. The grain size of samples increases with decreasing duty cycle. Irregular shapes on the film surface are shown in sampleswith the duty cycle of 5%, whereas the rounded shapes of grainsare observed at samples with lower duty cycles. In order to figure out the surface topography, atomic force microscopy (AFM) has been performed. Figure 3 shows the AFM morphologies of VO 2ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-2 Published under license by A VS.(75 nm)/TiO 2(300 nm)/glass structures with duty cycles of (a) 5%, (b) 2.5%, and (c) 1.6%. The surface of VO 2is pretty rough and the Rmaxvalues for (a), (b), and (c) are 174 nm, 186 nm, and 191 nm, respectively. Though the heights perpendicular to the structuresurface with duty cycles of 5% and 2.5% are both about 80 nm, thetopographies of these structures look very different. For (a) with a duty cycle of 5%, the distribution of grains seems uniform, but large grains are observed at some localized regions in (b) with a dutycycle of 2.5%. When the duty cycle is lowered to 1.6%, the heightgreatly rises to about 120 nm, which results from some flake-like shapes of broken films that are randomly distributed. The flake-like microstructure can be explained by nonuniform strain relaxation, asevidenced by local crack formation. 17Thus, the more duty cycle decreases, the less uniformity in surface topography, suggesting thatthis is due to serious ion bombardment on the surface. Figure 4 shows XRD patterns of VO 2/TiO 2/glass structures as functions of duty cycle and TiO 2thickness. Samples with different duty cycles are measured in Fig. 4(a) . At a high duty cycle of 5%, two additional reflections are detected at diffraction angles of2θ= 33.1° and 53.9° which corresponds to the V 2O3phase (JCPDS-ICDD card No. 34-0187). At duty cycles of 2.5%, the reflec- tions are detected at diffraction angles of 2 θ= 27.8°, 37.0°, and 55.3°, which correspond to the monoclinic VO 2phase (JCPDS-ICDD card No. 43-1051). At the lowest duty cycle of 1.6%, two additional reflec-tions at diffraction angles of 2 θ= 29.0° and 30.1° are suggested to be V 7O13(JCPDS-ICDD card No. 18-1449), which is close to the stoi- chiometry O/V ratio of 2. In the region of the monoclinic phase near a major diffraction angle of 2 θ= 27.8°, the diffraction peaks behave differently for these three duty cycles as shown in the inset.At a high duty cycle of 5%, the location of peaks is slightly shifted toa diffraction angle of 2 θ= 27.3°, which belongs to the other phase of the monoclinic structure (M, P2 1/c) (JCPDS-ICDD card No. 33-1440). The density of this monoclinic structure is 4.57 (g/cc),which is lower than 4.66 (g/cc) for the structure with a duty cycle of2.5%. Since the peak is broadened, a nano-crystalline structure is expected. At a low duty cycle of 1.6%, the peak near the diffraction angle of 2 θ= 27.8° comprises three possible crystalline phases, i.e., M FIG. 1. (a) Deposition rates of vanadium dioxides at different O 2/Ar ratios from 2% to 10% with a 2% interval during thin-film deposition under three different duty cycles. (b) is the peak power density as a function of applied power and duty cycle. FIG. 2. SEM images of samples with the duty cycle of 1.6% by magnification of (a) ×50 K and (b) ×100 K. (c) and (d) are samples with 2.5% and (e) and (f )are samples with 5%, respectively.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-3 Published under license by A VS.(P2 1/c), R (P4 2/mnm) (JCPDS-ICDD card No. 44-0253), and M (P2/m). Moreover, the values of transition temperature (T C)a r e5 2 and 59 °C for VO 2/TiO 2/glass structures with duty cycles of 1.6% and 2.5%, respectively. Because the SMT behavior of VO 2thin films depends on the lattice parameter owing to the dimer V-V atomchain along this direction,18,19the lower critical temperature of SMT is believed to be caused by the compressive strain due to V-V atomicchains stretching along the c axis. Contrarily, the V-V atomic chains are located in-plane and experience a tensile stress. 20Thus, the out-of-plane tensile strain can shorten the c axis length of VO 2and trigger phase transition at lower temperatures.21,22Based on the theory of cubic crystal structures, if the out-of-plane b axis has thecompressive strain, then the base a-c plane has the tensile strain with a- and c-axes extensions. 23,24As a result, the structure formed with a low duty cycle may correspond to the tensile stain along the c axis,which causes crack propagation and the creation of flake-like micro-structure under in-plane stress. In order to study the effect of TiO 2 thickness on the crystallinity of VO 2thin films, a set of samples with three TiO 2thicknesses of 100, 200, and 300 nm has been compared FIG. 3. AFM morphologies of VO 2(75 nm)/TiO 2(300 nm)/glass structures with duty cycles of (a) 5%, (b) 2.5%, and (c) 1.6%. FIG. 4. XRD patterns of VO 2/TiO 2/glass structures as functions of (a) duty cycle and (b) TiO 2thickness. The inset is the enlarged portion in a 2 θrange of 26°–29°.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-4 Published under license by A VS.inFig. 4(b) . It is clear that the peak intensity of monoclinic (011) and (211) planes increases with increasing the TiO 2thickness. An additional monoclinic structure (M, C2/m) (JCPDS-ICDD card No.65-7960) having the diffraction angles of 2 θ= 29.0°, 30.2° 38.0°, 44.1°, and 49.5° is observed in samples with TiO 2thickness of 100 nm. In this monoclinic structure, the density is as low as 4.03 (g/cc), which is much smaller than that of the space group of P2 1/c. Figure 5 shows TEM images of VO 2(75 nm)/TiO 2(300 nm)/ glass structures for a duty cycle of 2.5%. The nanocrystalline struc-ture is shown for VO 2films. Two selected area electron diffraction (SAED) patterns are denoted by A and B at different locations inside the films. The localized diffraction patterns of these two regions are analyzed by FFT. Several reflections are detected andd-values in Figs. 5(b) and 5(c) are measured to be 0.242, 0.245, and 0.316 nm for location A and 0.242, 0.247, and 0.314 nmfor location B, respectively. The calculation of d spacing is obtained from the inversion operation of FFT. These d-values can be attributed to the (011) and ( /C22211) planes for locations A and B. Figures 6(a) and 6(b) show TEM images of VO 2(75 nm)/TiO 2 (300 nm)/glass structures for duty cycles of 5% and 1.6%, respec- tively. Both also show the nanocrystalline structures. An irregular shape right on the film surface in Fig. 6(a) is consistent with the SEM images as discussed in Fig. 2 . Only some small regions are crystallized and their d-values are measured to be 0.329, 0.326,and 0.328 nm, which come from the refraction of the ( /C22101) plane and meet with the monoclinic structure (space group: C2/m) as discussed in Fig. 4(a) . For the samples for a duty cycle of 1.6% in Fig. 6(b) , a severe film lamination deviates from the VO 2film itself and the lattice is more distorted than that using other duty cycles.A wide range of d-values are detected, of 0.318 –0.324 nm, which might come from the reflections of ( /C22101), (110), and (011) planes with the combination of three different crystal structures as dis-cussed in Fig. 4(a) . Figures 7(a) and7(b) show the thermal switching properties of VO 2/TiO 2/glass structures as functions of duty cycle and TiO 2thick- ness under 25 °C/85 °C switching operation, respectively. For the practical use of thermochromic materials, the optical properties ofVO 2are suggested to be Tlum> 40% and ΔTsol>1 0 % .25The lumi- nous transmittance ( Tlum) and solar modulating ability ( ΔTsol)a r e obtained from the measured spectra using the following equations:26 Tρ¼Ðψρ(λ)T(λ)λÐψρ(λ)dλ, (1) ΔTsol¼ΔTsol,25/C14C/C0ΔTsol,85/C14C, (2) where T(λ) represents the transmittance at wavelength λ.ρdenotes lum orsolfor calculations. ψlumis the standard efficiency function for photopic vision and ψsolis the solar irradiance spectrum for an air mass of 1.5 with a global tilt which is the sun standing 37° above the horizon. The luminous and solar transmittance are integrated in the wavelength ranges of 380 –780 and 340 –2700 nm, respectively. The optical features of samples are summarized in Table I . For the samples with a TiO 2thickness of 300 nm, the luminous transmit- tance increases from 43% to 66% in a low-temperature state as the duty cycle decreases from 2.5% to 1.6%, while corresponding solarmodulating abilities of 9.5% and 6.0% are achieved. The transition temperature with a duty cycle of 1.6% is lower than that of 2.5% as shown in the inset of Fig. 7(a) , and the stress mechanism is illus- trated in Fig. 4 . Besides, the lower diffraction intensity of the FIG. 5. TEM images of VO 2(75 nm)/TiO 2(300 nm)/glass structures for the duty cycle of 2.5%. (a) Two selected area electron diffraction (SAED) patterns aredenoted by A and B. (b) and (c) are the d-spacing values calculated by theinverse-FFT operation from A and B locations, respectively.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-5 Published under license by A VS.monoclinic phase at 2 θ= 27.8° and larger laminated crystallinities for 1.6% cause low solar modulating ability and high luminous trans-mittance, respectively. If the duty cycle is kept at 2.5%, the luminoustransmittance is around 40% –50% no matter what the TiO 2 thickness is. However, it is explicit that the solar modulating ability increases with increasing TiO 2thickness. The solar modulating ability drops to about 2.4% when the TiO 2thickness is only 100 nm. The larger optical window of ΔTsolin samples with a TiO 2thickness of 300 nm is due to the peak intensities of the diffraction angle (2θ= 27.8°) being relatively larger than those with TiO 2thicknesses as shown in Fig. 4(a) . FIG. 6. TEM images of VO 2(75 nm)/TiO 2(300 nm)/glass structures for duty cycles of (a) 5% and (b) 1.6%. The d-spacing values are calculated by theinverse-FFT operation. FIG. 7. Transmittance properties of switching for the VO 2(75 nm)/TiO 2 (300 nm)/glass structure as a function of (a) duty cycle and (b) TiO 2thickness. The visible range of wavelength is set to 400 –800 nm, shown between the dashed lines. The inset shows the values of transition temperature for dutycycles of 1.6% and 2.5%. TABLE I. Visible transmittance and solar modulation ability of different samples. Conditions Duty cycle (s/s)Tlum(%) Tsol(%) ΔTsol VO 2/TiO 2(nm/nm) 25 °C 85 °C 25 °C 85 °C VO 2/TiO 2= 75/300 Duty cycle = 50/2000 43.1 41.8 44.9 35.5 9.5Duty cycle = 50/3000 65.7 58.6 62.6 56.6 6.0 On/Off = 50/2000 VO 2/TiO 2= 75/100 47.5 48.0 38.6 36.2 2.4 VO 2/TiO 2= 75/200 39.4 40.2 39.2 34.4 4.8ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-6 Published under license by A VS.Figure 8 shows the depth profiling of atomic percentages with four O, V, Ti, and Si elements by the integration of peak intensitymeasured from XPS. The oxygen percentage remains almost cons-tant with increasing milling depth, except for a slight increase at the surface. The increase in oxygen at the surface is due to the surface contacting with the atmosphere with an oxygen-ambient environ-ment. The percentage versus depth amount is similar for all samples,which implies that a sufficiently large amount of stoichiometric VO 2 is achieved due to highly energetic oxygen ions reacting with vana- dium. When the duty cycle decreases, the outdiffusion of titanium becomes more serious, which contributes to a small amount of vana-dium in the reduction state. Because there is no ternary phase ofTi-incorporated VO 2detected by x-ray diffraction in Fig. 4(a) , oxygen which is lost in V 7O13crystals by the oxygen scavenging effect is believed to form a stable TiO 2compound inside films. It suggested that the diffusion of Ti ions into the VO 2films can stabi- lize the monoclinic phase.27 Figure 9 shows the depth profiles of binding energies of the O 1s orbit VO 2(75 nm)/TiO 2(300 nm)/glass structure with duty cycles of (a) 5%, (b) 2.5%, and (c) 1.6%. For the duty cycles of 2.5%and 1.6% in (b) and (c), the peak is located at a binding energy of529.9 –530.0 eV, which is associated with the O 2−ionic state.28The binding energies of oxygen for the higher duty cycle of 5% first located at 529.0 eV and then gradually shifting to an energy state of 530.6 eV ( ΔE∼1.6 eV) means that a large portion of in-lattice oxygen with a 2−state decreases inside films. The loss of oxygen content in the VO 2thin-film with a 2−valence state results in the formation of nonstoichiometric vanadium oxides (VO x,x < 2 )s u c ha st h eo x y g e n - poor phase of V 2O3as expressed in the XRD pattern of Fig. 4(a) . Because the peak intensity of V 7O13is comparatively smaller than that of the VO 2phase in Fig. 4(a) , the right-hand shift to an energy state around 529.5 eV from 530.0 eV is minor, as shown in Fig. 9(c) . Figure 10 shows depth profiles of binding energies of the V 2p3/2orbit VO 2(75 nm)/TiO 2(300 nm)/glass structure with dutycycles of (a) 5%, (b) 2.5%, and (c) 1.6%. According to the Handbook of X-ray Photoelectron Spectroscopy , the ranges of two valence states of vanadium compounds, i.e., +3 and +4 positionedat 514.5 –515.0 eV and 515.5 –516.8 eV are reported from some clas- sical samples. 29In order to investigate the valence charges inside films, the depth profiling has been performed. For the high duty cycle in Fig. 10(a) , the peak is promptly shifted from the binding energy of 516.0 –515.0 eV at the upper portion of films, which is again confirmed to be the V 2O3crystals. When the milling time increases, the position of the peak intensity switches back to the lower bond of the binding energy with a +4 valence state. The area FIG. 8. Atomic percentages vs milling depth. Four percentages of elements, O, V , Ti, and Si are obtained from the integration of peak intensities by XPS. FIG. 9. Depth profiles of the binding energy of O 1s orbit for the VO 2(75 nm)/ TiO 2(300 nm)/glass structure with duty cycles of (a) 5%, (b) 2.5%, and (c) 1.6% in the milling time range of 0 –15 min with a 0.5 min interval.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-7 Published under license by A VS.under the peak intensity is gradually broadened with increasing the milling depth, implying that a multi-valence occurs. For theduty cycles of 2.5 and 1.6%, the peak is also broadened and shiftedto lower binding energies below 514.5 eV. Since the outdiffusionsof Ti from the TiO 2buffer layer and V atoms from the VO 2layer are serious at low duty cycles, as mentioned in Fig. 8 , the valence state of atoms during postannealing should be addressed. Fromthe formation energy point of view, the Gibbs free energy of TiO 2 formation is ΔG0=−745.8 kJ/mol under 800 °C, while the Gibbs free energy of VO 2is−521.7 kJ/mol. That means that TiO 2is a more stable compound than VO 2.30Thus, when the interdiffusionbetween layers occurs at the high temperature treatment, the Ti atom is likely to form a metal oxide and leave V atoms in low valence states. IV. CONCLUSION Thermochromic VO 2/TiO 2stacks as functions of the duty cycle and the TiO 2thickness are characterized. The high density plasma of the HIPIMS technique aims to approach the scheme of VO 2crystallization with a low thermal budget. A moderate duty cycle is advised with proper thermochromic properties of solar reg-ulation efficiency and an applicable luminous transmittance.Though the crystallization is dependent on the duty cycle, the ener- getic ion bombardment causing damages to film integrity should be carefully considered. By careful handling with the duty cycle,VO 2thin films fabricated by the HIPIMS technique is a feasible option, with potential for further thermochromic applications. ACKNOWLEDGMENTS The authors wish to acknowledge the Ministry of Science and Technology of Taiwan, Republic of China for supporting this workunder Grant No. MOST 108-2221-E-131-018. REFERENCES 1C. Z. Wu, F. Feng, and Y. Xie, Chem. Soc. Rev. 42, 5157 (2013). 2M. Nakano et al. ,Adv. Electron. Mater. 1, 1500093 (2015). 3F. Morin, Phys. Rev. Lett. 3, 34 (1959). 4Mark J. Miller and Junlan Wang, J. Appl. Phys. 117, 034307 (2015). 5Y. Choi, Y. Jung, and H. Kim, Thin Solid Films 615, 437 (2016). 6J. T. Gudmundsson, N. Brenning, D. Lundin, and U. Helmersson, J. Vac. Sci. Technol. A 30, 030801 (2012). 7A. Gupta, J. Narayan, and T. Dutta, Appl. Phys. Lett. 97, 151912 (2010). 8V. Elofsson, D. Magnfält, M. Samuelsson, and K. Sarakinos, J. Appl. Phys. 113, 174906 (2013). 9M. Samuelsson, D. Lundin, J. Jensen, M. A. Raadu, J. T. Gudmundsson, and U. Helmersson, Surf. Coat. Technol. 205, 591 (2010). 10J. Zheng, S. Bao, and P. Jin, Nano Energy 11, 136 (2015). 11S. Saitzek, G. Guirleo, F. Guinneton, L. Sauques, S. Villain, K. Aguir, C. Leroux, and J.-R. Gavarria, Thin Solid Films 449, 166 (2004). 12H. N. S. Krishnamoorthy, Y. Zhou, S. Ramanathan, E. Narimanov, and V. M. Menon, Appl. Phys. Lett. 104, 121101 (2014). 13S. Saitzek, F. Guinneton, L. Sauques, K. Aguir, and J.-R. Gavarria, Opt. Mater. 30, 407 (2007). 14J. E. Greene, J. Vac. Sci. Technol. A 35, 05C204 (2017). 15K. P. Kelley, E. Sachet, C. T. Shelton, and J. Maria, APL Mater. 5, 076105 (2017). 16P. C. Juan, K. C. Lin, C. L. Lin, C. A. Tsai, and Y. C. Chen, Thin Solid Films 687, 137443 (2019). 17A. Sohn, T. Kanki, H. Tanaka, and D. Kim, Appl. Phys. Lett. 107, 171603 (2015). 18N. B. Aetukuri et al. ,Nat. Phys. 9, 661 (2013). 19L. L. Fan et al. ,Nano Lett. 14, 4036 (2014). 20B. Hong, Y. Yang, K. Hu, Y. Dong, J. Zhou, Y. Zhang, W. Zhao, Z. Luo, and C. Gao, Appl. Phys. Lett. 115, 251605 (2019). 21T. Yajima, Y. Ninomiya, T. Nishimura, and A. Toriumi, Phys. Rev. B 91, 205102 (2015). 22A. Sohn, T. Kanki, K. Sakai, H. Tanaka, and D.-W. Kim, Sci. Rep. 5, 10417 (2015). 23T. Slusar, J. Cho, B. J. Kim, S. J. Yun, and H. T. Kim, APL Mater. 4, 026101 (2016). FIG. 10. Depth profiles of binding energy of V 2 p3/2orbit for the VO 2(75 nm)/ TiO 2(300 nm)/glass structure with duty cycles of (a) 5%, (b) 2.5%, and (c) 1.6% in the milling time range of 0 –15 min with a 0.5 min interval.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-8 Published under license by A VS.24J. M. Longo and P. Kierkegaard, Acta Chem. Scand. 24, 420 (1970). 25S. Y. Li, G. A. Niklasson, and C. G. Granqvist, Thin Solid Films 520, 3823 (2012). 26G. Sun, X. Cao, X. Gao, S. Long, M. Liang, and P. Jin, Appl. Phys. Lett. 109, 143903 (2016). 27H. Paik et al. ,Appl. Phys. Lett. 107, 163101 (2015).28P. Juan, F. Mong, and J. Huang, J. Appl. Phys. 114, 084110 (2013). 29J. F. Moulder, W. F. Stickle, P. E. Sobol, and K. D. Bomben, Handbook of X-Ray Photoelectron Spectroscopy , edited by J. Chastain (Perkin-Elmer, MN, 1992), pp. 74 –75. 30J. Zheng, X. Hou, X. Wang, Y. Meng, X. Zheng, and L. Zheng, Int. J. Refrac. Met. Hard Mater. 54, 322 (2016).ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000409 38,063403-9 Published under license by A VS.

5.0015200.pdf

Appl. Phys. Lett. 117, 092406 (2020); https://doi.org/10.1063/5.0015200 117, 092406 © 2020 Author(s).Modulation of spin conversion in a 1.5nm- thick Pd film by ionic gating Cite as: Appl. Phys. Lett. 117, 092406 (2020); https://doi.org/10.1063/5.0015200 Submitted: 25 May 2020 . Accepted: 24 August 2020 . Published Online: 02 September 2020 Shin-Ichiro Yoshitake , Ryo Ohshima , Teruya Shinjo , Yuichiro Ando , and Masashi Shiraishi ARTICLES YOU MAY BE INTERESTED IN Current-induced out-of-plane effective magnetic field in antiferromagnet/heavy metal/ ferromagnet/heavy metal multilayer Applied Physics Letters 117, 092404 (2020); https://doi.org/10.1063/5.0016040 Observation of carrier concentration dependent spintronic terahertz emission from n-GaN/ NiFe heterostructures Applied Physics Letters 117, 093502 (2020); https://doi.org/10.1063/5.0011009 Current-in-plane spin-valve magnetoresistance in ferromagnetic semiconductor (Ga,Fe)Sb heterostructures with high Curie temperature Applied Physics Letters 117, 092402 (2020); https://doi.org/10.1063/5.0015358Modulation of spin conversion in a 1.5 nm-thick Pd film by ionic gating Cite as: Appl. Phys. Lett. 117, 092406 (2020); doi: 10.1063/5.0015200 Submitted: 25 May 2020 .Accepted: 24 August 2020 . Published Online: 2 September 2020 Shin-Ichiro Yoshitake, Ryo Ohshima, Teruya Shinjo, Yuichiro Ando, and Masashi Shiraishia) AFFILIATIONS Department of Electronic Science and Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan a)Author to whom correspondence should be addressed: shiraishi.masashi.4w@kyoto-u.ac.jp ABSTRACT Gate-induced modulation of the spin–orbit interaction (SOI) in a 1.5 nm-thick Pd thin film grown on a ferrimagnetic insulator was investigated. Efficient charge accumulation by ionic gating enables a substantial upshift in the Fermi level of the Pd film, which wascorroborated by the suppression of the resistivity in the Pd. Electromotive forces arising from the inverse spin Hall effect in Pd under spinpumping were substantially modulated by the gating, in consequence of the modulation of the spin Hall conductivity of Pd as in an ultrathin Pt film. The same experiment using a thin Cu film, for which the band structure is largely different from Pd and Pt and its SOI is quite small, provides further results supporting our claim. The results obtained help in developing a holistic understanding of the gate-tunable SOI insolids and confirm a previous explanation of the significant modulation of the spin Hall conductivity in an ultrathin Pt film by gating. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015200 Electric gating using a solid gate insulator is the most pivotal technical tool in modern electronics. An efficient modulation of theconductivity of a channel layer of a field-effect transistor (FET) by solid gating is the key operating principle of FETs. Gating by ionic liquids or gels, which is a modern advanced technique for electric gat-ing, has been garnering attention in a broad range of condensed- matter physics fields. Compared with solid gating, ionic gating enables more efficient charge accumulation in adjacent solids, which, in turn,enables pioneering explorations that have led to discoveries such asthe appearance of a superconducting state in an oxide insulator 1and in transition-metal dichalcogenides,2the paramagnetic–ferromagnetic transition in InMsAs,3strong modulation of the Curie temperature of nanometer-thick Co (ultrathin Co),4and the vanishing of the inverse spin Hall effect (ISHE) through the large modulation of the spin–orbit interaction (SOI) and the spin Hall conductivity in ultrathin Pt.5The formation of a thin electric double layer6facilitated these notable achievements. Among these achievements, a combination of ultrathin Pt films and ionic gating opened a new frontier in spin conversion science, which had been gathering tremendous attention because spin conver-sion 7allowed the development of, for example, spin-torque diodes8 and spin–orbit torque switching devices.9,10Tunable spin conversion expands this research field. Both resistance and the spin Hall conduc- tivity in a 2 nm-thick Pt film, in which the intrinsic spin Hall effectgoverns the spin conversion, were simultaneously modulated by ionicgating. 5This achievement was attributed to a sufficient charge accu- mulation in the Pt, resulting in a large upshift of the Fermi level by theionic gating. The ISHE is governed by the density of states (DOS) ofthed-orbitals, 11,12and the DOS of the d-orbitals of Pt vanishes above the Fermi level.13T h i sp h y s i c si sr e s p o n s i b l ef o rt h es u p p r e s s i o no ft h e spin Hall conductivity, i.e., the vanishing of the ISHE by positive gat- ing. The formation of an ultrathin film with a low charge carrier den-sity and the use of ionic gating for efficient charge accumulation arekey to the tunable ISHE. Given this brief survey of studies on spin conversion and its con- trol by gating, acquiring additional insights and a more detailed under-standing of the physics in spin conversion science using gating effects by combining ultrathin Pd films and ionic gating can represent a substan- tial contribution to the spintronics field because the band structure ofPd is quite similar to Pt. 13Indeed, a similar gate-tunable SOI can appear if the current understanding of the physics of gate-tunable SOI is cor-rect. In the present study, thin Pd films were prepared and subsequentlyequipped with ionic gates. A similar modulation of the resistance and an electromotive force (EMF) due to spin conversion was detected in the Pd using the ionic gating technique, which underscores the validityof the understanding of the phenomenon constructed previously. 5To support our assertion, the same experiments using thin Cu films werealso implemented with no suppression of the EMF for the gating. Figure 1(a) shows a schematic of the fabricated sample and the measurement setup. Yttrium–iron–garnet (YIG) grown on 0.7 lm-thick, Appl. Phys. Lett. 117, 092406 (2020); doi: 10.1063/5.0015200 117, 092406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/apl3 mm-long, and 1 mm-wide gadolinium–gallium–garnet (GGG) (Granopt, Japan) was used as a spin source. The YIG was polished with an agglomerate-free alumina polishing suspension (50 nm par- ticle size) for 40 min and then annealed at 1273 K in air for 90 min.The Pd was deposited onto the YIG at a rate of 0.02 nm/s via electron-beam evaporation. The thickness of the Pd film was changed from 1.5 to 30 nm. Afterwards, Ti(5 nm)/Au(50 nm) elec- tric pads were formed on the sides of the sample via electron-beam evaporation. An ionic gel was prepared by mixing PS-PMMA-PSpolymer (Polymer Source, USA), DEME–TFSI ionic liquid (Kanto Chemical, Japan), and ethyl propionate (CH 3CH 2COOC 2H5, Nacalai Tesque, Japan) in a weight ratio of 9.3:0.7:20, respectively. Insulating double-sided adhesive tape was placed on the sides of the Pd channel (inside the area covered by Ti/Au electric pads) to pro-vide additional mechanical support for the gate electrode film ontop of which it was placed. The gate electrode film was mounted directly above the Pd channel after the ionic gel was applied. For the measurement, a sample was mounted in a TE 011cavity of an electron-spin resonance system (JEOL JES-FA200). The applied microwave power was set to 10 mW, and the microwave frequencywas 9.12 GHz. The gate voltage was applied at room temperature; after the ionic gel developed an electric double layer, the sample was cooled to 250 K and resistivity, ferromagnetic resonance (FMR), and EMF measurements were carried out after the gate leakage current was con- firmed to be suppressed; i.e., ions in the ionic gel were not mobile. The sample temperature was then increased to room temperature, the gate voltage was changed, and the sample was again cooled to 250 K. A flashing flow of N 2gas was supplied to the cavity containing the sam- ple. EMFs due to spin conversion in the Pd were monitored under an external dc magnetic field along 0/C14and 180/C14, and the EMF at 180/C14 was subtracted from that at 0/C14to eliminate unwanted thermal contri- butions and determine the precise amplitude of the EMF. Figure 1(b) shows the dependence of the resistivity of the Pd films; the resistivity monotonically increases with decreasing filmthickness. Thickness dependence of the resistivity in thin metallic films can be reproduced by a theoretical fitting function, 14 q¼qbulk1/C01 2þ3 4k t/C18/C19 1/C0pe/C0nt=k/C16/C17 e/C0t=k/C20/C21/C01 ; (1) where q,qbulk,k,t,n,a n d pare the resistivity of the film, the bulk resis- tivity of Pd (22 lXcm in the fitting result), the electron mean free path, the sample thickness, the grain-boundary penetration parameter, and the fraction of carriers specularly scattered at the surface, respec- tively. As shown in Fig. 1(b) , the experimental result is well reproduced using this fitting function. From the best fit using Eq. (1),k,n,a n d p were estimated to be 44 nm, 0.035, and 0.97, respectively. From the fit- ting result, the increase in the resistivity of the thinner Pd films is ratio- nalized by the enhancement of surface scattering, as reported for Pt5 and other metals.14More importantly, the 1.5 nm-thick Pd film is con- tinuous because the theoretical fitting holds in the thinnest case, its resistivity being 1250 lXcm, which indicates that the 1.5 nm-thick Pd film operates in the intrinsic spin Hall regime at room temperature15 as designed. From here on, we focus on the 1.5 nm-thick Pd becausegating effects are more salient in thinner films, as explained above. Figure 2(a) shows the gate dependence of the resistance of the 1.5 nm-thick Pd film. Although a negative gate voltage can be applied, we shed light mainly on the results under an applied positive gate volt- age because the Pd film degrades under negative gate voltage applica- tions (see the supplementary material ) due to unwanted surface reactions with water molecules present in the ionic liquid. 6The resis- tance decreases from roughly 3.3 k Xto 2.0 k X, the fraction decrease being 40%. The decrease in resistivity is attributed to efficient charge accumulation as in ultrathin Pt films.5Figure 3 shows the FMR spectra and EMF from the 1.5 nm-thick Pd at external magnetic fields of 0/C14 and 180/C14as a function of gate voltages (see also supplementary mate- rialfor line shape analyses for the FMR spectra and the EMFs). Prominent EMFs are measured at 0/C14and 180/C14under FMR of the YIG; more importantly, their polarities are opposite to each other, which is a compelling result that the measured EMFs are ascribed to the ISHE f o rP d .T h ea m p l i t u d eo ft h eE M Fa t Vg¼0 V is estimated to be 3 lV, which is comparable to that observed for ultrathin Pt (2 lVf o rt h e 2 nm-thick Pt5). Furthermore, the amplitude of the EMFs FIG. 1. (a) Schematic of the ultrathin Pd sample for gate-tunable spin conversion measurements. (b) Thickness dependence of the resistivity of Pd thin films. Thered solid line is the result of the theoretical fitting using Eq. (1)in the main text.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092406 (2020); doi: 10.1063/5.0015200 117, 092406-2 Published under license by AIP Publishingmonotonically decreases for positive gate voltages [see also Fig. 2(b) ]. T h es u d d e nd e c r e a s ei nt h eE M Fa t Vg¼1.25 V is ascribable to the waiting time of the measurements, which were long because we needed to refill the liquid N 2. The dependences of the electric current in the Pd film and the normalized electric current on gate voltage are shown in Figs. 2(c) and2(d), where normalization was applied using the amplitude at Vg ¼0 V. Because the ISHE is an effect where electric current is generated by the conversion from a spin current induced by the SOI, and not aneffect for which voltage is generated, the EMFs were converted to anelectric current by considering the resistivity modulation as a function of the gate voltage. As is evident, the electric current monotonically decreases for positive gate voltages and the suppression ratio of the electric current is about 80%. Given that the band structure of Pd is quite similar with that of Pt, the results in the study are understood asfollows: the intrinsic mechanism governs the spin conversion in the 1.5 nm-thick Pd within the spin-Hall-based spin conversion regime. 16 The intrinsic spin Hall effect (and its reciprocal effect, ISHE) originatesfrom the inter- d-band excitation, as previously described. Because the ISHE observed in an ultrathin Pt film is also intrinsic in nature, theshift of the Fermi level to a position where the DOS of the d-orbitals is largely suppressed is a key factor in the suppression of the ISHE due to that of the spin Hall conductivity. 5The same mechanism holds in the 1.5 nm-thick Pd; i.e., these results provide additional compelling evi- dence for a gate-tunable SOI realized in an ultrathin Pd film as for an ultrathin Pt film. For further supporting evidence, we performed the same experi- ment using a thin Cu film. Unlike Pt and Pd, Cu possesses a weak and constant d-orbital contribution at and above the Fermi level,16–18 resulting in a small and non-tunable spin Hall conductivity. Therefore, an ISHE-induced electric current from a thin Cu film is thought to be quite weak and almost non-tunable. In the experiment, an 8.1 nm-thick Cu film was used because surface oxidation takes place quickly (see also the supplementary material for details of the experiments). Results concerning the gate voltage dependence of the resistance and normalized electric current generated by the ISHE are shown in Figs. 4(a)and4(b). The resistance of the Cu film monotonically decreases albeit weakly for the positive gate voltages, as observed in the Pd (this study) and the Pt. 5A weak gate voltage dependence of the resistance of the Cu occurs because the Cu film is considerably thicker than those of the Pd and the Pt.5With the modulation of the resistance being realized as planned, an upshift in the Fermi level was expected given an efficient carrier accumulation in the Cu film. However, the ISHE- induced electric current does not exhibit a salient gate-voltage depen-dence because of its band structure. It has been well known that the surface oxidation of Cu and the formation of CuO/Cu give rise to siz- able spin-to-charge conversion even in Cu, 19and an application of a strong gate voltage enables O2/C0migration to the top surface of Pt,20 resulting in the enhancement of the spin-to-charge conversion. Given that (i) ionic gating allows more efficient generation of an electric field, probably enhancing O2/C0migration to the top surface of the Cu in our study, (ii) the surface of the Cu in this study is most likely naturally oxidized, and (iii) thicknesses of the Cu and the Pt were comparable to that of the Cu in our study, it is intriguing that such enhancement was not observed and the gate dependence of the ISHE was as expected for Cu. Further study is still awaited. Anyhow, the difference in the ISHE- induced electric current for gate voltages applied to the Pd and the Cu thin films validates our holistic understandings of the gate-tunableSOI for ultrathin single metal. Finally, we briefly discuss plausible applications of the gate- tunable ISHE observed in Pd and Pt based on the holistic under-standings established in this study. An example is an application for spin–orbit torque magnetic random-access memories (SOT-MRAMs). SOT-MRAMs possess superiority in endurance compared to conventional MRAMs because spin current gives much less damages to tunneling barriers in magnetic tunnel junc- tions (MTJs) and the spincurrent is generated by using the SOI of FIG. 2. Gate voltage dependences of (a) resistances, (b) EMFs, (c) magnitudes of the electric current generated by the ISHE, and (d) the normalized electric current,of the 1.5 nm-thick Pd.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092406 (2020); doi: 10.1063/5.0015200 117, 092406-3 Published under license by AIP Publishingheavy materials. In conventional SOT-MRAMs, an MTJ is aligned at the crossing point of a word line and a bit line to control spincurrent injection. Since the SOI is gate-tunable as we have beenclarifying, we need only one line to store and rewrite informationby equipping a gate electrode to control the SOI by the gating. Furthermore, since the SOI modulation allows the modulation of spin diffusion length, the gate-tunable SOI may allow the creationof a spin transistor using metallic materials. In addition, furtherinvestigation for a quest of appropriate material selections andcombinations is awaited because energy dependence SOI is theo-retically predicted in Fe- and Pt-doped Au 21and they can be the other potential candidate for generating the gate-tunable ISHE. In summary, the gate-induced modulation of the SOI in a 1.5 nm-thick Pd thin film grown on a ferrimagnetic insulator wasinvestigated. Efficient charge accumulation due to ionic gating enableda substantial upshift in the Fermi level of the Pd films, allowing sub-stantial suppression of the EMFs arising from the ISHE in Pd becauseof the modulation of the spin Hall conductivity by the gating. Thesame experiment using a thin Cu film revealed no sizable modulation of the EMFs in Cu by gating. Considering the band structure and the small SOI of Cu, the weak gating effect of the EMFs from the Cu isrationalized by the physics of a gate-tunable SOI observed in Pd andPt. Hence, the whole result described in the paper supports a holisticunderstanding of the gate-tunable SOI in solids and corroborates pre-vious explanations of the significant modulation of the spin Hall con-ductivity in ultrathin Pt films by gating. 5 FIG. 3. Observed FMR spectra and EMFs of the 1.5 nm-thick Pd under the FMR of the YIG, when the gate voltage is changed from 0 to þ2 V in increments of þ1 V. The upper panels show the FMR spectra of the YIG, the middle panels show the raw data when the external magnetic field was applied at 0/C14(the black solid line) and 180/C14(the red solid lie), and the lower panels show the net EMFs for each gate voltage. See also the main text on how to estimate the net EMFs. FIG. 4. Gate voltage dependences of (a) resistances and (b) the normalized elec- tric current of the 8.1 nm-thick Cu.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092406 (2020); doi: 10.1063/5.0015200 117, 092406-4 Published under license by AIP PublishingSee the supplementary material that describes the surface reac- tion of the thin Pd under an application of negative gate voltage, details of the experiments using thin Cu films, and line shape analyses of the FMR spectra from the YIG and EMFs from the Pd and the Cu. The authors acknowledge the support by a Grant-in-Aid for Scientific Research (S) No. 16H06330, “Semiconductor spincurrentronics,” and MEXT (Innovative Area “Nano Spin Conversion Science,” KAKENHI No. 26103003). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1K. Ueno, S. Nakamura, H. Shimotani, H. T. Yuan, N. Kimura, T. Nojima, H. Aoki, Y. Iwasa, and M. Kawasaki, Nat. Nanotechnol. 6, 408 (2011). 2Y. Saito, Y. Nakamura, M. S. Bahramy, Y. Kohama, J. Ye, U. Kasahara, Y. Nakagawa, M. Onga, M. Tokunaga, T. Nojima, Y. Yanase, and Y. Iwasa, Nat. Phys. 12, 144 (2016). 3H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature 408, 944 (2000). 4K. Shimamura, D. Chiba, S. Ono, S. Fukami, N. Ishiwata, M. Kawaguchi, K. Kobayashi, and T. Ono, Appl. Phys. Lett. 100, 122402 (2012). 5S. Dushenko, M. Hokazono, K. Nakamura, Y. Ando, T. Shinjo, and M. Shiraishi, Nat. Commun. 9, 3118 (2018).6S. Z. Bisri, S. Shimizu, M. Nakano, and Y. Iwasa, Adv. Mater. 29, 1607054 (2017). 7Y. Otani, M. Shiraishi, A. Oiwa, E. Saitoh, and S. Murakami, Nat. Phys. 13, 829 (2017). 8A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005). 9I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V. Costache, S.Auffret, S. Bandiera, B. 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Adv. 4, eaar2250 (2018). 21G . - Y .G u o ,S .M a e k a w a ,a n dN .N a g a o s a , Phys. Rev. Lett. 102, 036401 (2009).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 092406 (2020); doi: 10.1063/5.0015200 117, 092406-5 Published under license by AIP Publishing

5.0024071.pdf

Appl. Phys. Lett. 117, 142401 (2020); https://doi.org/10.1063/5.0024071 117, 142401 © 2020 Author(s).Room-temperature magnetic skyrmion in epitaxial thin films of Fe2−xPdxMo3N with the filled β-Mn-type chiral structure Cite as: Appl. Phys. Lett. 117, 142401 (2020); https://doi.org/10.1063/5.0024071 Submitted: 05 August 2020 . Accepted: 07 September 2020 . Published Online: 05 October 2020 B. W. Qiang , N. Togashi , S. Momose , T. Wada , T. Hajiri , M. Kuwahara , and H. Asano COLLECTIONS This paper was selected as Featured Room-temperature magnetic skyrmion in epitaxial thin films of Fe 2/C0xPdxMo3N with the filled b-Mn-type chiral structure Cite as: Appl. Phys. Lett. 117, 142401 (2020); doi: 10.1063/5.0024071 Submitted: 5 August 2020 .Accepted: 7 September 2020 . Published Online: 5 October 2020 B. W. Qiang,1 N.Togashi,2S.Momose,1T.Wada,1 T.Hajiri,1 M.Kuwahara,2,3 and H. Asano1,a) AFFILIATIONS 1Department of Materials Physics, Nagoya University, Nagoya 464-8603, Japan 2Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan 3Institute of Materials and Systems for Sustainability, Nagoya University, Nagoya 464-8603, Japan a)Author to whom correspondence should be addressed: asano@numse.nagoya-u.ac.jp ABSTRACT We report experimental observations of chiral magnetic skyrmion phases in thin films of molybdenum nitride with a filled b-Mn-type structure. A series of Fe 2/C0xPdxMo 3N(x¼0.15, 0.32, and 0.54) thin films are grown epitaxially with the (110) orientation on c-plane sapphire substrates by reactive magnetron sputtering, and their structural, magnetic, and transport properties are investigated. Studies using the Topological Hall effect and Lorenz transmission electron microscopy imaging for films with x¼0.32 identified the existence of two types of skyrmion phases with a size as small as 60 nm; one is a dense skyrmion phase at temperatures below 100 K, and the other is an isolatedskyrmion phase in a higher temperature range to well beyond room temperature. These epitaxial thin films in the family of molybdenumnitrides open the way for the study of skyrmions, manipulation of their properties, and the exploration and optimization for skyrmion-based applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024071 Skyrmions with topologically non-trivial spin textures 1–3have been observed or proposed to exist in various magnets due to different mechanisms, such as magnetic dipolar interaction,4,5antisymmetric spin exchange or the so-called Dzyaloshinskii–Moriya interaction,6–9 and frustrated exchange interaction.10In metallic compounds, sky- rmions induce emergent magnetic fluxes that act on the conductionelectrons, thereby inducing the so-called topological Hall effect, 11–14 and the skyrmion spin texture can be confirmed by Lorentz transmis-sion electron microscopy. 1,3,15Skyrmions have great potential for spin- tronic applications, owing to topologically protected rigidness, emergent electromagnetic phenomena, and ultra-low current driven motion. Among the various skyrmions, those induced by Dzyaloshinskii–Moriya (DM) interaction are particularly interesting due to their relatively small size (the same as the helical wavelength k) and fixed unique helicity (spin swirling direction). A number of helical magnets mediated by the DM interaction and with non-centrosymmetric crystallographic symmetries are theoretically pre- dicted and experimentally examined to host stable skyrmions. 6–10In B20-type bulk alloys,6,7,11–13all of which have the same cubic chiralspace group of P213, skyrmions are formed only in a very small region in the temperatures and magnetic field phase diagram just below the ferromagnetic transition temperatures ( TC). Among the B20-type bulk alloys, FeGe hosts near-room-temperature skyrmions ( TC¼278 K andk¼70 nm).13CoxZnyMn z(xþyþz¼20) alloys are cubic chiral magnets of the b-Mn type alloys16–19with the space group P4132 or P4332 and are found to show a stable skyrmion lattice at high tempera- ture (150 K–475 K, depending on the composition) with larger sky- rmion sizes between 115 nm and 190 nm. Tetragonal Heusler alloys Mn–Pt–Sn20with D2dcrystal symmetry are shown to be antiskyr- mions with TC¼400 K and k¼150 nm. It is of particular importance to develop new materials that can create small-sized skyrmions in a wide temperature range including room temperature for both funda-mental studies and practical applications. The emergence of skyrmions from rich parent phases in molyb- denum nitride has been reported very recently. The parent compound Fe 2Mo 3N has been reported to be ferromagnetic21and to have the filled b-Mn-type structure with the cubic chiral space group P4132 or P4332, where Fe atoms lie at the 8 cpositions of a cubic unit cell and form a single network, as shown in Fig. 1(a) . The space within this Appl. Phys. Lett. 117, 142401 (2020); doi: 10.1063/5.0024071 117, 142401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplnetwork is filled by Mo 6N octahedra with Mo atoms occupying the 12dsite and interstitial nitrogen atoms on the 4 asite. Partial substitu- tion of the 8 cpositions by metals, such as Ni, Co, Pt, Pd, and Rh, has also been reported21–23to modify the magnetic properties, such as magnetization, coercivity, and Curie temperature that result from thecompetition between magnetic exchange and DM interactions. Thehighest Curie temperature for the bulk ferromagnetic material (Fe 1.25Pt0.75Mo 3N) was reported to be 220 K.22However, the prepara- tion and characterization of thin films of these magnetic skyrmionmaterials have not been reported. It is, thus, imperative to investigate thin films of the chiral magnets and explore modification of the skyrmion phases. In this study, we focus on the Fe 2/C0xPdxMo 3N system because the parent compound Fe 2Mo 3N has been reported to be ferromagnetic and partial substitution of the 8 csite by Pd is expected to enhance or tune the DM spin–orbit interactions, thereby possibly stabilizing the skyrmion phases.24As e r i e so fF e 1/C0xPdxMo 3N( F P M N )t h i nfi l m s were investigated using Hall transport and Lorentz transmission elec-tron microscopy (LTEM) measurements. As a result, the formation of chiral skyrmions is unambiguously demonstrated in the epitaxial thin films at and above room temperature. Thin films of the FPMN system (50 nm thick) were prepared on c-plane sapphire substrates by reactive magnetron sputtering using a composite target under an Ar/N 2atmosphere. The composite target was composed of a FeMo 1.75or FeMo 2alloy disk with Pd chips on it. The chemical metal compositions of the films were analyzed and determined using both inductively coupled plasma spectroscopy (ICP)and energy dispersive x-ray spectroscopy (EDX). The crystal structure was analyzed using out-of-plane and in-plane x-ray diffraction (XRD) measurements with Cu K aradiation. The magnetic properties of the thin films were characterized using superconducting quantuminterference device (SQUID) magnetometry. Real-space observations were performed using LTEM combined with magnetic transport-of- intensity equation analyses. The h=2hXRD pattern for a film with x¼0.54 shown in Fig. 1(b) is indicative of the single-phase b-Mn-type structure with the (110) orientation. The inset of Fig. 1(b) shows the xdependence of the out-of-plane lattice parameter c 0, for a 50 nm thick film on Pd doping, c0shows a linear dependence on x, which is consistent with the data for the bulk.22The in-plane uscan of a film with x¼0.54 shown in Fig. 1(c) reveals the epitaxial relationship of FPMN[1 10]//Al 2O3 [1100] with a 30/C14rotation and twinning over the area of 2.5 mm2. From the in-plane 2 hv=uscan shown in Fig. 1(d) , the in-plane lattice parameter a0w a s0 . 6 7 5 2 n m ,s i m i l a rt ot h e c0value of 0.6755 nm, which indicates the small epitaxial strain between the film and the substrate. The skyrmion spin texture in b-Mn-type chiral magnets has a skyrmion number of /C012, where each skyrmion is quantized with an emergent gauge flux of 2 flux quanta (2 /0¼h=e).26This quantization produces an emergent magnetic field of Beff¼2/0=A,w h e r e Ais the area of the skyrmion that deflects electrons and leads to the topological Hall effect (THE).25,26The total Hall resistivity qHis then qxy¼qH¼R0HþRsMþqTH; (1) where R0is the ordinary Hall coefficient, Rsis the anomalous Hall effect coefficient, qTHis the topological Hall resistivity due to the sky- rmion spin texture, and a magnetic field His applied perpendicular to the film plane. Here, we considered R0Has a linear contribution to qH andRsMas a linear function of M.I nFig. 2(a) , the Hall resistivities, after subtraction of the ordinary Hall resistivities of FPMN, are plotted as a function of the external field. The magnetization was observed as afunction of the field to obtain the anomalous Hall resistivity, and theanomalous Hall resistivity was calculated using q AH¼RsMand is shown in Fig. 2(b) . The topological Hall resistivity qTHcan be obtained by the subtraction of R0HþRsMfrom qH,a ss h o w ni n Fig. 2(c) .qTH is observed for the x¼0.32 and 0.54 films, and qmax THexists at near zero FIG. 1. (a) Schematic representation of the crystal structure of filled b-Mn-type nitride. (b) Out-of-plane 2 hscan for a FPMN ( x¼0.54) thin film. Inset: dependence of out-of-plane lattice constant c0for the FPMN thin film on the Pd atomic concentra- tion. (c) In-plane /scan for the FPMN ( x¼0.54) (110) thin film and a-Al2O3(0001) structure. (d) In-plane 2 hv//scan for the FPMN ( x¼0.54) thin film. Inset: in-plane (110) direction of the FPMN thin film on the a-Al2O3(001) substrate. FIG. 2. (a) Hall resistivity with normal Hall resistivity subtracted of the FPMN thin film with various xvalues at 25 K. (b) Anomalous Hall resistivity calculated from hysteresis loops of magnetizations. (c) Topological Hall resistivity obtained by sub-traction of the anomalous Hall resistivity from the total Hall resistivity after subtrac-tion of the normal Hall resistivity.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 142401 (2020); doi: 10.1063/5.0024071 117, 142401-2 Published under license by AIP Publishingfield, which indicates the existence of zero-field skyrmions in these films. At large H,qTHvanishes when all the spins are aligned. qTHis almost zero at x¼0.15 and is prominent at x¼0.32 ( qmax TH ¼0:192lX/C1cm), but then decreases at x¼0.54 ( qmax TH ¼0:025lX/C1cm), which shows a clear dependence on x.qTHis deter- mined by qTH¼nSkXPR0Beff; (2) where nSkXis the relative skyrmion density (for skyrmion crystals, nSkX¼1; for isolated skyrmions, nSkX<1) and Pis the local spin polarization. Beffis then related to the size of the skyrmion, which is the same as the helicity wavelength k, in chiral magnets. A smaller size skyrmion can be acquired by an increase in the concentration of heavymetals, 24i.e., the Pd content in this case, to enhance the DM interac- tion, which leads to a larger Beffto show a prominent THE; however, the small skyrmion size can be largely attributed to a small ferromag-netic exchange interaction to lower the magnetic Curie temperature,T C.24,27,28qTHdisappears in the film at x¼0.15 and is prominently observed at x¼0.32, which reveals an emergence of skyrmion state induced by Pd doping. Among the three types of films with different x values, a strong THE shows a well-controlled point at x¼0.32, where qmax TH¼0:192lX/C1cm, which is of comparable magnitude to those for other skyrmion materials.11–14 Figures 3(a)–3(d) show the magnetic field dependence of qTHfor the 50 nm thick x¼0.32 film at several temperatures. The THE is clearly observed at all temperatures. Below 100 K, a large zero-fieldtopological Hall resistivity is observed, which almost disappears above100 K. The maximum value of the topological Hall resistivity, q max TH,a t 300 K is reduced to 1 =20 of that at 4 K ( qmax TH¼0:192lX/C1cmat 4 K and qmax TH¼0:014lX/C1cm at 300 K), which indicates that the density of skyrmions is decreased by an increase in the temperature. Figure 3(e) shows qmax THandR0as a function of the temperature. The sign of qmax THis reversed at ca. 100 K with R0maintaining its sign. The sign reversal of qmax THobserved here is presumably ascribed to B eff, which resulted from the change in the electronic structure and/or magnetic structure such as spin reorientation.20The inset in Fig. 3(e) shows the temperature dependence of the magnetization for the50 nm thick film with x¼0.32 with the 100 Oe magnetic field perpen- dicular to the film plane, indicating that T Cis around 600 K. The observed TCv a l u eo f6 0 0 K ,w h i c hi sm u c hh i g h e rt h a nt h o s e (160–240 K) for bulk materials with x>0.5,22is considered to be pri- marily due to strong ferromagnetic exchange interaction on the Fe- rich 8 csublattice in the b-Mn-type structure, and/or the epitaxial strain effect,29and slight nitrogen nonstoichiomerty.30Based on the THE data, a schematic phase diagram of FPMN ( x¼0.32 and d¼50 nm) thin films is proposed, in addition to a plot of the criti- cal field HT C, beyond which all the spins are aligned making the THE vanish, and HT M, the magnetic field showing qmax TH, as shown inFig. 3(f) . The orange area below 100 K represents the dense sky- rmion phase referred to as the Sk-I phase, and the yellow area from 100 K to 350 K shows the skyrmion phase referred to as theSk-II phase. Zero-field-stabilized skyrmions only exist in the low-temperature Sk-I phase. To verify the formation of the skyrmions and validation of the phase diagram shown in Fig. 3(f) , real-space observations were con- ducted using Lorentz TEM for the FPMN ( x¼0.32 and d¼50 nm) film, the results of which are summarized in Figs. 3(g)–3(j) .Figure 3(g) shows an LTEM image at 90 K with a magnetic field of 205 mT perpendicular to the film plane. A high density-skyrmion phase, of FIG. 3. (a)–(d) Temperature dependence of the topological Hall resistivity of the FPMN ( x¼0.32) thin film. (e) The temperature dependence of qmax THandR0is shown by the square and circle symbols, respectively. The inset of (e) shows the temperature dependence of the magnetization for the 50 nm thick film with x¼0.32 with the 100 Oe mag- netic field perpendicular to the film plane, indicating that the TCvalue is around 600 K. (f) The phase diagram in the H–Tplane. The square and triangle symbols denote the absolute values of HT CandHT M, respectively. The phase diagram represents different phases at different magnetic fields and temperatures: dense skyrmion, isolated skyrmion, and a ferromagnetic phase (FM). (g) and (h) LTEM images at 90 K with a magnetic field of 205 mT perpendicular to the film plane. (h) Magnified view of (g). (i) a nd (j) LTEM images at room temperature (300 K) with a magnetic field of 20 mT perpendicular to the film plane. The upper half part of (i) is the image of a 50 nm thick area, and the lower half part of (i) is the image of a wedge area, fabricated by the focused ion beam method, with a thickness from 10 nm (bottom) to 50 nm (middle). (j) Magnifie d view of (i). The color wheels represent the magnetization direction. Small white arrows represent the in-plane magnetization direction at each point.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 142401 (2020); doi: 10.1063/5.0024071 117, 142401-3 Published under license by AIP Publishingwhich a magnified view of a single skyrmion is shown in Fig. 3(h) ,c a n be clearly observed with the size of a single skyrmion as small as 60 nm. Figures 3(i) and3(j)show room-temperature (300 K) LTEM images with a magnetic field of 20 mT perpendicular to the film plane. The upper and lower half parts in Fig. 3(i) are images of a 50 nm thick area and a wedge area with a thickness from 10 nm (bottom) to 50 nm (middle), respectively. Both the 50 nm thick and wedge areas show both the helical magnetic structure and a low density of isolated sky- rmions (the Sk-II phase). It should be noted that the wedge area with a thickness lower than 50 nm hosts a slightly higher density of sky-rmions than the 50 nm thick area, which possibly signifies the stabili- zation of skyrmions at reduced dimensionality. Figure 3(j) shows that the size of the skyrmions is still ca. 60 nm, which indicates that the size of the skyrmions remained unaffected by the temperature and thick- ness, and is the same for the Sk-I phase and the Sk-II phase. The temperature-dependent behavior of skyrmion formation observed by the present LTEM study is consistent with the phase diagram[Fig. 3(f) ] derived from the Hall transport measurements. The prominent feature of our Fe 2/C0xPdxMo 3Nt h i nfi l m si st h e existence of two types of skyrmion phases, that is, the low-temperature Sk-I phase and the high-temperature Sk-II phase. The Sk-I phase, rep- resenting the large positive THE with the zero-field qTH, creates dense skyrmions that exist in a temperature range of 100 K down to low tem- perature. Above around 100 K, at which temperature the topological Hall resistivity changes its sign to negative, the Sk-II phase appears as the isolated skyrmion and survives at least up to 350 K (the highest measurement temperature here). Therefore, these two types of sky- rmion phases act in a completely different manner from the conven- tional topological skyrmion phase, which is stabilized by thermalfluctuation and confined to a narrow temperature near the Curie tem- perature. In addition, the Sk-I and Sk-II phases reveal quite different skyrmionic behavior. q max THshows a huge enhancement below 100 K with a reduction in R0, and even the skyrmion size is unaffected by the temperature. The qmax THvalue is estimated to be less than 0.013 lX/C1cm (nSkX<1,P<1) for the Sk-II phase at 300 K using Eq. (2),w h i c h matches THE data well. But for the Sk-I phase, qmax THis estimated to be at most 0.026 lX/C1cm ( nSkX¼1,P¼1) at 4 K, which suggests that the extra part of THE might be induced by different magnetic structures20 and that the Sk-I phase is complicated and beyond the simple sky-rmion picture. Recently, new types of skyrmion phases, that is, the dis- ordered skyrmion phases, were reported for Co xZnyMn z(xþy þz¼20) compounds with the b-Mn-type chiral structure, which are attributed to magnetic disorder and geometrical frustration in the 8 c sublattice and the 12 dsublattice.17,18The neutron and synchrotron diffraction study19on Co xZnyMn zindicated that the b-Mn-type mate- rials contain a large amount of atomic and magnetic disorder in the two sublattices, which strongly influences the formation and behavior o ft h es k y r m i o np h a s e s .I nt h ec a s eo fF e 2/C0xPdxMo 3N thin films, the flexible nature of the b-Mn structure gives rise to a large amount of atomic and magnetic disorders, including antisite disorder, in the two sublattices, and could stabilize disordered skyrmions below 350 K. In conclusion, epitaxial (110) thin films of Fe 2/C0xPdxMo 3Nc h i r a l magnets with the filled b-Mn-type structure were fabricated and examined by topological Hall resistivity and LTEM imaging. The exis- tence of the two types of skyrmion phases was identified in a wide temperature range up to well beyond room temperature, and the small size (ca. 60 nm) of the skyrmions remains almost constant over theentire temperature range examined. Further studies are necessary to elucidate the skyrmionic behavior and other physical properties ofthin films of the family of molybdenum nitrides. These results indicatethat limiting factors in terms of both operation temperature and mate- rial variation have been overcome, which could represent an important step toward the spintronic application of magnetic skyrmions. This work was supported by the Japan Society for the Promotion of Science (KAKENHI Grant Nos. 17K19054, 17H02737, 19K15445, and 20H02602), a research grant from the Kyosho Hatta Foundation and the JST-Mirai Program Grant (No. JPMJMI18G2). A part of thiswork was supported by the Nanotechnology Platform Program<Molecule and Material Synthesis >(No. JPMXP09S-19-MS-1099) of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010). 2J. Choi, S. Kang, S. W. Seo, W. J. Kwon, and Y. Shin, Phys. Rev. Lett. 111, 245301 (2013). 3S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012). 4M. Ezawa, Phys. Rev. Lett. 105, 197202 (2010). 5H. Y. Kwon, K. M. Bu, Y. Z. Wu, and C. Won, J. Magn. Magn. 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5.0020925.pdf

Appl. Phys. Lett. 117, 112401 (2020); https://doi.org/10.1063/5.0020925 117, 112401 © 2020 Author(s).Spin–orbit torque-induced multiple magnetization switching behaviors in synthetic antiferromagnets Cite as: Appl. Phys. Lett. 117, 112401 (2020); https://doi.org/10.1063/5.0020925 Submitted: 05 July 2020 . Accepted: 08 September 2020 . Published Online: 17 September 2020 Libai Zhu , Xiaoguang Xu , Mengxi Wang , Kangkang Meng , Yong Wu , Jikun Chen , Jun Miao , and Yong Jiang COLLECTIONS Paper published as part of the special topic on Spin-Orbit Torque (SOT): Materials, Physics, and Devices ARTICLES YOU MAY BE INTERESTED IN Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745 Spin current generation and detection in uniaxial antiferromagnetic insulators Applied Physics Letters 117, 100501 (2020); https://doi.org/10.1063/5.0022391 Concurrent magneto-optical imaging and magneto-transport readout of electrical switching of insulating antiferromagnetic thin films Applied Physics Letters 117, 082401 (2020); https://doi.org/10.1063/5.0011852Spin–orbit torque-induced multiple magnetization switching behaviors in synthetic antiferromagnets Cite as: Appl. Phys. Lett. 117, 112401 (2020); doi: 10.1063/5.0020925 Submitted: 5 July 2020 .Accepted: 8 September 2020 . Published Online: 17 September 2020 Libai Zhu, Xiaoguang Xu,a) Mengxi Wang, Kangkang Meng, Yong Wu,Jikun Chen, JunMiao, and Yong Jianga) AFFILIATIONS School of Materials Science and Engineering, University of Science and Technology Beijing, Beijing 100083, China Note: This paper is part of the Special Topic on Spin-Orbit Torque (SOT): Materials, Physics and Devices. a)Authors to whom correspondence should be addressed: xgxu@ustb.edu.cn and yjiang@ustb.edu.cn ABSTRACT This paper studies spin–orbit torque (SOT) switching behaviors in synthetic antiferromagnet (SyAF) structures of Ta/[Pt/Co] m/Ru/[Co/Pt] n, which are asymmetric between the upper multilayer (UML) and the bottom multilayer (BML). The SOT-induced magnetization switching loops show multiple transitions of switching orientations between clockwise and anticlockwise with an increasing in-plane magnetic field, determined by the effects of the Dzyaloshinskii–Moriya interaction from both the BML and UML in the different stacking structures.Moreover, the field-free SOT switching was observed in the structure of Ta/[Pt/Co] 3/Ru (0.5)/[Co/Pt] 4. It can be attributed to the horizontal component of magnetic moments in its UML acting as an equivalent field. Therefore, the SyAF structures could be potential candidates forthe future SOT-based spintronic devices. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020925 Electrically controlled magnetization switching in ferromagnets driven by spin–orbit torque (SOT) 1–4offers a new method to design magnetic memories and logic devices.5,6Compared with spin-transfer torque,7SOT enables the spin charge current generated by the spin Hall effect in a heavy metal (HM) film to flow into an adjacent ferro-magnetic metal (FM) film directly. Recent studies on SOT mainlyfocused on HM/FM heterostructures with perpendicular magnetic anisotropy. 1–3,8–11As shown in Fig. 1(a) , when applying a charge current along the þXdirection, a vertical spin current with spin polar- ization along the /C0Ydirection will accumulate at the HM/FM inter- face and exert torques on the magnetic moments. Usually, an externalmagnetic field is needed to break the inversion symmetry to realize the deterministic magnetization switching of the FM layer. 1–3The external magnetic field can be replaced by an equivalent field through severalapproaches, 8–10such as competing spin currents,11interlayer exchange coupling,12spin-current gradient,13and out-of-plane polarized spin current,14and, thus, a field-free SOT switching can be achieved. However, in HM/FM/HM sandwich stacks with the same sign of spin Hall angle in the upper and bottom HM films, competing spin cur- rents will exert opposite torques on the FM film.15,16The chirality of domain walls (DWs), caused by the Dzyaloshinskii–Moriya interac-tion (DMI) at the HM/FM interface, 17–19determines the direction of DW motion.15,16Therefore, the HM/FM interface with a thicker HMfilm dominates the sign of the DMI constant in the whole stacks.15 Moreover, when the single FM layer is replaced by a multilayer such as Co/Ni/Co, the thickness of the FM films can also influence thedominant interfaces. 16 Synthetic antiferromagnets (SyAFs), containing two antiferro- magnetically coupled ferromagnetic thin films separated by an ultra-thin Ru layer, are widely used in spin-valve 20and magnetic tunnel junction devices21as a reference layer22or a free layer.23Recently, SOT has been studied in several kinds of SyAFs, such as Pt/Co/Ru/ [Co/Pt]/Co/SiO 224with a symmetry upper multilayer (UML) structure and Pt/[Co/Pd/Co]/Ru/[Co/Pd/Co]25with a small spin Hall angle of Pd. In this case, due to the negligible SOT in the UMLs, the bottomPt/Co interface is dominant, and the SOT switching orientations onlychange from clockwise to anticlockwise when the applied magneticfield increases from H smtoHla.24,25Here, HsmandHlarepresent an external magnetic field smaller and larger than the exchange couplingfield, respectively. However, to date, the SyAF structures with non-negligible SOT in the UMLs have not been investigated yet, and itshould be important to study how the upper Co/Pt interface affectsthe switching behaviors. In this paper, we clarify the multiple magneti-zation switching behaviors in the SyAF structures in which the SOTswitching orientations can also change from anticlockwise to clockwisewith the external magnetic field increasing from H smtoHla. Appl. Phys. Lett. 117, 112401 (2020); doi: 10.1063/5.0020925 117, 112401-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplMoreover, a field-free SOT-induced magnetization switching can be achieved in the SyAF structures. A series of multilayers with different stacking structures of Ta/Pt/ [Pt (1)/Co (0.4)] 3/Ru (tRu)/[Co (0.4)/Pt (1)] n(units in nanometer) were deposited on Si/SiO 2substrates by magnetron sputtering with a base pressure better than 2 /C210/C05Pa, and the layout of the device is shown in Fig. 1(b) . For simplification, the periodic structures of [Pt (1)/Co (0.4)] 3and [Co (0.4)/Pt (1)] nwere labeled as bottom multilayer (BML) and UML, respectively, in which the subscript represents the repeating times of the periodic structures. We defined the samples in the study as follows. Samples A 3-n(tRu) represents Ta (2)/Pt (4)/BML 3/Ru (tRu)/UML n, and Sample B is Ta (2)/Pt (5)/[Co (0.4)/Pt (1)] 3. The multilayers were patterned into Hall bar devices with the charge channel width of 10 lm and the voltage channel width of 5lm by electron beam lithography and Ar ion milling technique. Finally, the electrodes of Ta (5)/Pt (80) were deposited by magnetron sputtering followed by a lift-off process. Figure 1(c) sketches the detailed measurement configuration and the definition of coordinatesystem used in this paper. To optimize the SyAF structures, the thickness of the Ru film was studied first by varying t Ruin Samples A 3–5(tRu). The hystere- sis (M–H) loops of Samples A 3–5(tRu) were measured in the out-of-plane direction, and the results are presented in Fig. S1 ofthesupplementary material .A l lt h e M–Hloops show typical mag- netization characters of SyAFs, indicating an antiferromagnetic coupling (AFC) between the perpendicularly magnetized BML 3 and UML n. However, the strength of AFC varies with tRu,w h i c h can be reflected by the saturation field HS.26As shown in Fig. 2(a) , theHSshows an oscillatory relationship with tRu,a n dt h efi r s t peak appears at about 0.5 nm, representing the strongest AFC state.Therefore, t Ruwas set as 0.5 nm in the following SyAFs, except for Sample A 3–5(0.8) for comparison. As shown in Fig. 2(b) , the anomalous Hall resistance ( RH)–H loop of Sample A 3–4(0.5) shows four stable resistive states, which is consistent with its M–Hloop. The sharp magnetization switching around 61.2 kOe indicates the opposite magnetization directions between the BML 3and the UML 4. When the external magnetic field islarger than 2 kOe, the AFC between the BML 3and the UML 4is bro- ken; consequently, the magnetization of all the FM films is in the same direction. As shown in Fig. 2(b) and Fig. S2 of the supplementary material ,S a m p l e sA 3–4(0.5), A 3–5(0.5), and A 3–6(0.5) have similar HS values of about 2 kOe, which indicates that the AFC strength is mainly determined by the thickness of the Ru film in the multilayers with the same bottom structure of Ta/Pt/BML 3. Moreover, to study the differ- ence between the Co/Pt periodic structure and the single FM film, themeasurements of the SOT-induced magnetization switching werecarried out on Sample B. As shown in Fig. S3 of the supplementary material , Sample B shows similar SOT switching behaviors with the typical HM/FM structures with a single FM film; 1,2therefore, the (Pt/Co) multilayers adopted in the BML 3or UML ncan be considered as a single FM film. SOT switching behaviors of the Hall bars were measured to clar- ify the contributions of the BMLs and the UMLs. The RH–Icurves of Sample A 3–4(0.5) are presented in Fig. 2(c) . The switching orienta- tions change from anticlockwise to clockwise when the magnetic fieldincreases from þ700 Oe ( H sm)t oþ3200 Oe ( Hla). To identify the crit- ical magnetic field for switching orientation changes and the magni- tude of the exchange coupling field,27,28the switching loops of Sample A3–4(0.5) under various magnetic fields between þ700 Oe and þ3200 Oe are presented in Fig. S4(a) of the supplementary material . When the magnetic field is þ2000 Oe, the SOT switching nearly disap- p e a r s .T h i sp h e n o m e n o nc a nb ee x p l a i n e da st h ee x c h a n g ec o u p l i n g field is completely compensated by the external magnetic field. Consequently, the net magnetic field acted on UML or BML cannotbreak its inversion symmetry. When the external magnetic field fur-ther increases, the SOT switching can be observed again. So, the direc-tion and magnitude of the external in-plane magnetic field can affect the switching orientations. The critical magnetic field and the FIG. 2. (a) Oscillatory relationship between the saturation field HSand the thickness of Ru film tRu. (b) Field dependence of the out-of-plane magnetization (red line) and the Hall resistance (black line) for Sample A 3-4(0.5). Mis normalized to the satura- tion magnetization MS. (c) RH–Iloops of Sample A 3-4(0.5) under different external magnetic field Halong the Xdirection. FIG. 1. (a) Illustrations of the geometric relation between current J, accumulated spin current js, and magnetization direction min the Pt/Co heterostructure. (b) Schematic of the stacking structure of the samples. (c) Measurement setup alongwith the definition of the coordinate system.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112401 (2020); doi: 10.1063/5.0020925 117, 112401-2 Published under license by AIP Publishingexchange coupling field are both about 2000 Oe. Moreover, as shown in Fig. S4(b) of the supplementary material ,S a m p l eA 3–6(0.5) has the similar critical magnetic field of about 2000 Oe, indicating that the AFC strength is mainly determined by the thickness of the Ru layer inthe multilayers with the same bottom structure of Ta/Pt/BML 3. The switching orientations of Samples A 3–5(0.5), A 3–6(0.5), and A3–5(0.8) are summarized in Table I , and the detailed RH–Iloops are presented in Fig. S2 of the supplementary material . Different from pre- vious reports,24,25the switching orientations in our samples change from both clockwise to anticlockwise and anticlockwise to clockwise,when the magnetic field increases from H smtoHla. The multiple tran- sitions of switching orientations must be caused by the periodicUML n, which is the key difference between our structures and the pre- vious SyAF-based SOT structures. Because of the non-negligible SOT in the UML n, the DMI originates from both the BML 3and UML n, which determines the direction of DW motion in whole stacks.Therefore, the competition between the DMI of the BML 3and UML n mainly decides the final net magnetic moments and, thereafter, theSOT switching orientations. 15,16Moreover, the SOT effect of the whole stacks is mainly dominated by the interface with a thicker HMlayer; 15,16therefore, the bottom Pt (5)/Co interface in our samples is t h em a i ns o u r c ec o n t r i b u t i n gt ot h eS O Te f f e c t . To further study the multiple switching behaviors, here we com- pare the SOT switching of Samples A 3–4(0.5), A 3–5(0.5), A 3–6(0.5), and A 3–5(0.8) under Hsm.I nt h i sc a s e , Hsmcan overcome the DMI field to change the relative velocity VRof DWs, and the high and low resistive states correspond to the up and down directions of the netmagnetic moments, respectively. 24,25Moreover, the BML 3and the UML nstill maintain AFC since Hsmis smaller than HS. Therefore, the BML 3and the UML ncan be considered as a single FM layer and the dominant interface determines the switching orientations. Theschematic diagram of the DW model under H smis shown in Fig. 3(a) . The directions of DWs in the BML 3and the UML nare aligned along theþXand/C0Xdirections, and VRBandVRUrepresent the VRof DWs in the BML 3and the UML n,r e s p e c t i v e l y .O w i n gt ot h eo p p o s i t e chirality of the DWs in the BML 3and the UML n,VRBandVRUhave opposite signs. However, the competition of the VRhas not been con- sidered in the systems with negligible SOT in the upper FM layer in previous studies,24,25in which the switching orientation only presents clockwise, like Sample A 3–5(0.8) in our study. As shown in Table I , Samples A 3–4(0.5) and A 3–5(0.5) show anticlockwise switching orien- tation, while Sample A 3–6(0.5) shows clockwise switching orientation. These phenomena can be attributed to the thickest FM films in the UML 6. As a result, the upper Co/Pt (1) interface is the dominant one in Sample A 3–6(0.5). On the contrary, the bottom Pt (5)/Co interface is dominant in Samples A 3–4(0.5) and A 3–5(0.5). Comparing Samples A3–5(0.5) and A 3–5(0.8) with the same UML 5, the switchingorientations present anticlockwise and clockwise, respectively. The dif- ference between these two samples is the AFC strength. With increasing external magnetic field, the switching behaviors vary dramatically for Samples A 3–4(0.5), A 3–5(0.5), A 3–6(0.5), and A3–5(0.8). The switching orientations of Samples A 3–4(0.5), A 3–5 (0.5), and A 3–5(0.8) change to opposite signs between HsmandHla compared with those under Hsm, while the switching orientation of Sample A 3–6(0.5) maintains the same sign. Hlais larger than the sum of the DMI field and HS; thus, the AFC between the BML 3and the UML nwill be broken. As shown in Fig. 3(b) , in this case, the magneti- zation directions of DWs in the BML 3and the UML nare both aligned along the þXdirection, while VRBandVRUare in the same direction. Therefore, the final magnetization directions of the BML 3and the UML nare down and up, respectively, and the switching orientation should only present anticlockwise, like Sample A 3–5(0.8) in our study. However, the switching orientation also present clockwise characterfor Samples A 3–4(0.5), A 3–5(0.5), and A 3–6(0.5). The key difference between these samples and Sample A 3–5(0.8) is the AFC strength, which should be the reason for the different switching behaviors. To FIG. 3. The domain structures and the domain wall movement directions under Hsm(a) and Hla(b), respectively. The yellow arrows represent the directions of magnetic moments.TABLE I. Extractive SOT switching orientations of Samples A 3-4(0.5), A 3-5(0.5), A 3-6(0.5), and A 3-5(0.8) under HsmandHla. Samples 2Hla 2Hsm Hsm Hla Dominate interface A3-4(0.5) Anticlockwise Clockwise Anticlockwise Clockwise Bottom Pt (5)/Co A3-5(0.5) Anticlockwise Clockwise Anticlockwise Clockwise Bottom Pt (5)/Co A3-6(0.5) Anticlockwise Anticlockwise Clockwise Clockwise Upper Co/Pt (1) A3-5(0.8) Clockwise Anticlockwise Clockwise Anticlockwise Upper Co/Pt (1)Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112401 (2020); doi: 10.1063/5.0020925 117, 112401-3 Published under license by AIP Publishingsum up, the multiple switching behaviors under HsmandHla, both the thickness of the UML nand the AFC strength can affect the dominant interface of the whole stacks. The bottom Pt (5)/Co interface deter- mines the switching orientation of the multilayer with a stronger AFC (reflected by the Ru layer thickness) and a relative thin UML amongthe series samples. Otherwise, the upper Co/Pt (1) interface determines the switching orientation. Besides the multiple SOT switching behaviors, a clear field-free SOT switching was observed in Sample A 3–4(0.5). As shown in Fig. 2(c) , the field-free switching orientation presents anticlockwise, corresponding to the up direction of the net magnetic moments. When the magnetic field changes from /C0700 Oe to /C050 Oe, the switching orientation changes from clockwise to anticlockwise and the SOT switching disappears at /C0200 Oe, which is consistent with the switching behaviors around zero field in conventional HM/FMstructures. 1,2However, when the magnetic field increases from 0 Oe to þ700 Oe, the switching behaviors present the same direction of clock- wise and the SOT switching disappears at þ50 Oe. There are several approaches to realize field-free SOT switching, such as competing spin currents11and interlayer exchange coupling.12In view of the coexis- tence of the two adjacent HM layers with the opposite spin Hall angle of the Ta/Pt structure and the interlayer exchange coupling in oursamples, we have performed the electrical measurement on Sample B and ferromagnetic resonance (FMR) measurement on Sample A 3–4 (0.5) to clarify the origin of the field-free SOT switching. As shown inFig. S3 of the supplementary material , field-free switching cannot be observed, from which we can exclude the possibility of competing spin currents. The FMR measurement results are presented in Fig. S5 of the supplementary material , and the resonance fields are extracted and summarized in Fig. 4(a) . It is obvious that the easy axis of Sample A 3–4(0.5) is not strictly perpendicular to the film plane. For Sample A 3–4(0.5), its UML has more ferromagnetic layers than BML. Therefore, the magnetization of the whole multi- layer is always consistent with that of the UML, no matter whetherthe UML and BML are antiferromagnetically coupled. Consequently, the UML has horizontal magnetic component along the X axis. Therefore, as shown in Fig. 4(b) , the horizontal compo- nent of magnetic moments in the UMLs exerts an equivalent field (H eff) on the BMLs via interlayer exchange coupling, and the field- free SOT switching can be observed consequently. In Fig. 2(c) , when the magnetic field is /C0200 Oe, the Heffcan be counteracted by the external field, and the SOT switching disappears. Therefore, the switching curves under /C0700 Oe and -50 Oe present opposite directions. When the magnetic field is þ50 Oe, the horizontal com- ponent of magnetic moments reduces. Consequently, the Heffcan- not overcome the DMI field in BML 3and the SOT switching disappears. As a result, the switching curves under 0 Oe and þ7 0 0O ep r e s e n tt h es a m ed i r e c t i o n . In summary, we have investigated the SOT-induced magnetiza- tion switching behaviors in the SyAF structures. The SOT switching loops show multiple transitions of switching orientations betweenclockwise and anticlockwise with the increase of the in-plane magnetic field. Because of the non-negligible SOT in the UMLs, the DMI effects f r o mt h ei n t e r f a c e so ft h eB M L sa n dt h eU M L sa l la f f e c tt h ec h i r a l i t y of the magnetic moments in the whole stacks. Therefore, the switching orientation is determined by the dominant interface, which is decided by both the thickness and AFC strength. Besides, the direction andmagnitude of the external in-plane magnetic field can also affect the switching orientations. Moreover, the field-free SOT switching wasobserved in the SyAF of Ta/[Pt/Co] 3/Ru (0.5)/[Co/Pt] 4, which derives from the horizontal component of magnetic moments in the UMLsacting as an equivalent field on the BMLs via interlayer exchange cou-pling. These results offer potential applications in the future high-efficiency spintronic devices. See the supplementary material for the hysteresis loops of Samples A 3–5(tRu), the anomalous Hall resistance loops of Samples A3–5(0.5), A 3–6(0.5), and A 3–5(0.8); the SOT switching loops of Samples A 3–4(0.5), A 3–5(0.5), A 3–6(0.5), A 3–5(0.8), and Sample B; and the ferromagnetic resonance curves of Sample A 3–4(0.5). This work was partially supported by the National Key Research and Development Program of China (No.2019YFB2005801), the Beijing Natural Science Foundation KeyProgram (Grant No. Z190007), the National Natural ScienceFoundation of China (Grant Nos. 51731003, 51971024, 51671019,51971027, 51927802, and 51971023), and the FundamentalResearch Funds for the Central Universities (No. FRF-MP-19-004). FIG. 4. (a) Polar curve of the relationship between the measurement degree cand the values of resonance field. Error bars are added. (b) Illustrations of the measure- ment degree c(in plane X–Z) along with the definition of the coordinate system. The yellow solid arrows represent the directions of the magnetic moments and theyellow dotted arrows represent the components of the magnetic moments in UMLalong the X and Z axes.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112401 (2020); doi: 10.1063/5.0020925 117, 112401-4 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 1L. Q. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 109, 096602 (2012). 2L. Q. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). 3I. M. Miron, K. Garello, G. Gaudin, P. J. Zermatten, M. V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011). 4S. Fukami, C. L. Zhang, S. Dutta Gupta, A. Kurenkov, and H. Ohno, Nat. Mater. 15, 535 (2016). 5S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln /C19ar, M. L. Roukes, A. Y. Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). 6S. Parkin, X. Jiang, C. Kaiser, A. Panchula, K. Roche, and M. Samant, Proc. IEEE 91, 661 (2003). 7D. C. Ralph and M. D. 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Mater. 16, 712 (2017). 14Y. Cao, Y. Sheng, K. W. Edmonds, Y. Ji, H. Z. Zheng, and K. Y. Wang, Adv. Mater. 32, 1907929 (2020). 15P. P. J. Haazen, E. Mure `, J. H. Franken, R. Lavrijsen, H. J. M. Swagten, and B. Koopmans, Nat. Mater. 12, 299 (2013). 16K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nat. Nanotechnol. 8, 527 (2013). 17I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958). 18T. Moriya, Phys. Rev. 120, 91 (1960). 19P. X. Qin, H. Yan, X. N. Wang, Z. X. Feng, H. X. Guo, X. R. Zhou, H. J. Wu, X. Zhang, Z. G. G. Leng, H. Y. Chen, and Z. Q. Liu, Rare Met. 39, 95 (2020). 20Y. Jiang, T. Nozaki, S. Abe, T. Ochiai, A. Hirohata, N. Tezuka, and K. Inomata, Nat. Mater. 3, 361 (2004). 21J. Hayakawa, S. Ikeda, Y. M. Lee, R. Sasaki, T. Meguro, F. Matsukura, H. Takahashi, and H. Ohno, Jpn. J. Appl. Phys., Part 2 45, L1057 (2006). 22A. D. Kent and D. C. Worledge, Nat. Nanotechnol. 10, 187 (2015). 23S. Yakata, H. Kubota, T. Sugano, T. Seki, K. Yakushiji, A. Fukushima, S. Yuasa, and K. Ando, Appl. Phys. Lett. 95, 242504 (2009). 24C. Bi, H. Almasi, K. Price, T. Newhouse-Illige, M. Xu, S. R. Allen, X. Fan, and W. G. Wang, Phys. Rev. B 95, 104434 (2017). 25P. X. Zhang, L. Y. Liao, G. Y. Shi, R. Q. Zhang, H. Q. Wu, Y. Y. Wang, F. Pan, and C. Song, Phys. Rev. B 97, 214403 (2018). 26J. Y. Zhao, Y. X. Wang, Y. Z. Liu, X. F. Han, and Z. F. Zhang, J. Appl. Phys. 104, 023911 (2008). 27Y. Sheng, K. W. Edmonds, X. Q. Ma, H. Z. Zheng, and K. Y. Wang, Adv. Electron. Mater. 4, 1800224 (2018). 28Y. Cao, A. Rushforth, Y. Sheng, H. Z. Zheng, and K. Y. Wang, Adv. Funct. Mater. 29, 1808104 (2019).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 112401 (2020); doi: 10.1063/5.0020925 117, 112401-5 Published under license by AIP Publishing

5.0019295.pdf

J. Chem. Phys. 153, 114305 (2020); https://doi.org/10.1063/5.0019295 153, 114305 © 2020 Author(s).Formation of NaCl through radiative association: Computations accounting for non-adiabatic dynamics Cite as: J. Chem. Phys. 153, 114305 (2020); https://doi.org/10.1063/5.0019295 Submitted: 22 June 2020 . Accepted: 30 August 2020 . Published Online: 18 September 2020 Magnus Gustafsson ARTICLES YOU MAY BE INTERESTED IN Decoherence-corrected Ehrenfest molecular dynamics on many electronic states The Journal of Chemical Physics 153, 114104 (2020); https://doi.org/10.1063/5.0022529 From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 JCP Emerging Investigator Special Collection 2019 The Journal of Chemical Physics 153, 110402 (2020); https://doi.org/10.1063/5.0021946The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Formation of NaCl through radiative association: Computations accounting for non-adiabatic dynamics Cite as: J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 Submitted: 22 June 2020 •Accepted: 30 August 2020 • Published Online: 18 September 2020 Magnus Gustafssona) AFFILIATIONS Applied Physics, Division of Materials Science, Department of Engineering Science and Mathematics, Luleå University of Technology, 97187 Luleå, Sweden a)Author to whom correspondence should be addressed: magnus.gustafsson@ltu.se ABSTRACT The radiative association (RA) rate constant is computed for the formation of the diatomic sodium chloride (NaCl) molecule in the temper- ature interval 1 K–30 K. At these temperatures, RA of NaCl through non-adiabatic dynamics is important. A scattering program has been implemented to carry out calculations of RA cross sections, accounting for coupled dynamics on the lowest ionic and the lowest neutral dia- batic1Σ+states. The study shows that the non-adiabatic treatment gives a cross section that exceeds that of conventional adiabatic dynamics by one to four orders of magnitude. The contribution to the RA rate constant from Na and Cl approaching each other in the A1Πstate has also been computed using an established quantum mechanical method. Ab initio data from the literature have been used for the potential energy curves, the diabatic coupling, and the electric dipole moments of NaCl. Published under license by AIP Publishing. https://doi.org/10.1063/5.0019295 .,s I. INTRODUCTION Radiative association (RA) as a mode of molecule formation is important in interstellar environments.1,2RA rate coefficients are difficult, and in many cases technically impossible, to measure in the laboratory.3Calculations of RA rate constants are of impor- tance, as they are used in modeling of interstellar environments. The formation of diatomic molecules through RA ( A+B→AB +̵hω) has been investigated much theoretically, and several meth- ods, based on both classical and quantum dynamics, have been established (see, e.g., Ref. 4). These studies are typically carried out ignoring couplings between different molecular states, other than the dipole coupling with the electromagnetic field. There are some exceptions. Quantum dynamical calculations of RA for tri- atomic molecules ( A+BC→ABC +̵hω) have been carried out for a few molecules,5–8in which case couplings of the BC rovibra- tional states, induced by the encounter with A, have been naturally included. Calculations of RA cross sections for the formation of CN, including spin–orbit and rotational couplings, have also been performed.9In this work, a method is implemented for quantum dynami- cal computation of RA cross sections and rate constants for systems with avoided crossings in the Born–Oppenheimer approximation. The scheme is applied to the formation of diatomic sodium chloride (NaCl) through the reaction Na(2S)+ Cl(2P)→NaCl(1Σ+)→NaCl(1Σ+)+̵hω, (1) which in the intermediate step experiences a crossing between the coupled ionic and atomic electronic states. To account for this, a scattering calculation on the diabatic ionic and atomic poten- tials, including the coupling between those states, is carried out. This provides the free (continuum) wave functions. The bound states are computed on the ionic potential, and the bound-free matrix elements of the (permanent ionic) dipole moment are com- puted simultaneously with the scattering calculation. The method is based on the work by Freed and co-workers,10–12and various fla- vors of it have been used for photodissociation of diatomic13,14and triatomic15,16molecules and for radiative association of triatomic molecules.5–8,17,18 J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The cross section for the adiabatic, single channel X→X reaction Na(2S)+ Cl(2P)→NaCl(X1Σ+)→NaCl(X1Σ+)+̵hω (2) is computed using an established quantum method4,19for compar- ison with that of reaction (1) in order to quantify the effect of non- adiabatic dynamics. In addition, the cross section and rate constant for radiative association through the reaction Na(2S)+ Cl(2P)→NaCl(A1Π)→NaCl(X1Σ+)+̵hω (3) are computed to complement those for reaction (1). Apart from reactions (1) and (3), the atoms could approach each other on one of the NaCl triplet states,3Σ+and3Π, which also correlate with ground state atomic Na and Cl. The triplet states, however, appear to be repulsive20,21and are not considered in this work. Further- more, spin–orbit and rotational couplings, which could potentially give rise to mixing of the molecular states (1Σ+,1Π,3Σ+, and3Π), have been ignored in this work. Spin–orbit and rotational couplings in radiative association of CN have been investigated before,9in which case the low energy cross section was affected some, but likely by much less than an order of magnitude. We will return to this approximation in Sec. V. NaCl has been observed in circumstellar envelopes of evolved stars22,23and in a star forming region.24This work, thus, serves two purposes. First, calculated rate constants can eventu- ally be used for modeling of the chemistry in interstellar envi- ronments. Second, the method presented for radiative associa- tion with non-adiabatic dynamics can be used for other similar systems. II. THEORY The following treatment is based on that for predissociation of the CO molecule as presented by Tchang-Brillet et al.14The pre- sentation here is rather brief, as there are only a few formulas that are different for the case of radiative association, and details of the method may be found in Ref. 14. In the present case, we have Na interacting with Cl in their low- est atomic and ionic1Σ+states. The corresponding diabatic potential curves, as functions of interatomic distance R, are denoted as V1(R) andV2(R) for the atomic and ionic states, respectively, and the dia- batic coupling between those states is V12(R). The potential energy curves and the coupling are shown in Fig. 1. Since the diabatic cou- pling is rather small, the curves are shown in the region just around the crossing in Fig. 2. Colliding Na and Cl according to reaction (1) may be described by partial wave expanded, radial, continuum wave functions F, which are solutions to the coupled Schrödinger equation d2 dR2F(R,E,J)+2μ ̵h2[EI−V(R,J)]F(R,E,J)=0, (4) where Iis the 2 ×2 unit matrix, V(R,J)=⎡⎢⎢⎢⎢⎣V1(R)+̵h2J(J+1) 2μR2 V12(R) V12(R) V2(R)+̵h2J(J+1) 2μR2⎤⎥⎥⎥⎥⎦(5) FIG. 1 . Diabatic NaCl(1Σ+) potential energies for the atomic, V1(R), and ionic, V2(R), states, respectively, vs internuclear separation, R. The diabatic coupling constant, V12(R), is also displayed. The inset shows the permanent dipole for the ionic state, D2(R). The data are taken from Ref. 25. is the effective potential matrix, Eis the asymptotic kinetic energy, Jis the dumbbell angular momentum, and μis the reduced mass of the NaCl complex. In this work, only energies below the dissociation limit of V2will be considered. For the open channel, the asymptotic boundary condition F1(R,E,J)/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne/leftr⫯g⊸tl⫯ne→ R→∞√ 2μ π̵h2Ksin(KR−Jπ/2 +ηJ,E)eiηJ,E(6) is imposed, where ηJ,Eis the scattering phase shift and K=√ 2μE/̵h is the wave number. The closed channel wave component, F2, is an exponentially decaying function for large R. FIG. 2 . The diabatic potential energies and the coupling constant, as in Fig. 1, are shown here around the crossing point at R= 18.37 bohrs. J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp As the coupled Schrödinger equation (4) is integrated, the bound-free matrix elements of the permanent ionic dipole, D2(R), τv′,J′,J(E)=∫∞ 0ϕv′J′(R)D2(R)F2(R,E,J)dR, (7) may be computed. This requires that the bound states, with wave functionsϕv′J′(R) and eigenenergies Ev′J′, on the ionic potential have been precomputed. The approximation that the bound states are unaffected by the non-adiabatic coupling V12has been applied in this work. Without that approximation, the non-adiabatic cou- pling is expected to mix the 1 and 2 components of the bound states, affecting the bound states that have energy close to the dis- sociation limit of the V1potential, in particular. A test calculation presented in Appendix D, however, shows that overall those vibra- tional states contribute very little to the RA cross section and the rate constant. Thus, the non-coupled bound state approximation seems reasonable. Finally, the radiative association cross section can be obtained as σ(E)=2 3h2 4πϵ0c3P 2μE∑ v′J′Jω3 Ev′J′SJ→J′∣τv′,J′,J(E)∣2, (8) whereϵ0is the permittivity of vacuum and cis the speed of light. The statistical weight P= 1/12 for reaction (1), i.e., Σ→Σtransitions in NaCl (the same as those for HF26), and the non-zero Hönl–London factors are SJ→J+1=J+ 1 and SJ→J−1=J. The angular frequency of the emitted photon is ωEv′J′= (E−Ev′J′)/̵h. The summation over bound states v′,J′in Eq. (8) runs over all bound states with Ev′J′<0. When the cross section is known, the thermal rate constant at temperature T, k(T)=(8 μπ)1/2 (1 kBT)3/2 ∫∞ 0Eσ(E)e−E/kBTdE, (9) can be computed. kBin Eq. (9) is the Boltzmann constant. We note that for reactions where the radiative association cross section has many sharp resonances, the energy integral in Eq. (9) is commonly evaluated using the Breit–Wigner method.4,27The idea is to separate the direct and resonance mediated contributions. The direct con- tribution (cross section base line) is typically computed using some method based on classical trajectories28,29and the resonances by semiclassical approximations such as those used in LEVEL.30For reac- tion (1), this separation requires a more elaborate treatment since it is mediated through Feshbach (multichannel) resonances rather than shape resonances. Feshbach resonances could, for example, be characterized using the time delay matrix (see, e.g., Refs. 5, 17, and 18). A treatment of the sort may be necessary for some situations to correctly account for the resonances. In this work, however, Eq. (9) is simply integrated numerically to give the rate constant. The test calculations of cross sections are presented in Appendix C, which indicate that all resonances appear to be found and that they are reasonably well resolved. III. COMPUTATIONAL DETAILS The diabatic potential energy curves, the diabatic coupling, and the permanent electric dipole moment used in the calculations ofthe cross section of reaction (1) are taken from the ab initio study by Giese and York.25The curves are displayed in Fig. 1, and the diabatic potential curves and the diabatic coupling are shown in the region of the crossing point in Fig. 2. The coupling is about 3.3 meV at the crossing. For calculations on adiabatic potentials, for reactions (2) and (3), the adiabatic potential energy VX(R) and permanent dipole moment DX(R) are taken from Ref. 25, and the adiabatic potential VA(R) and transition dipole moment DAX(R) are taken from Ref. 31. The data are displayed in Fig. 3. For the integrations of the coupled Schrödinger equation (4), the computer program COUPLE developed by Mies, Julienne, and Sando32was used. The program is designed to propagate multichan- nel scattering wave functions from short to large distances, where scattering boundary conditions are imposed. The program is also prepared for the propagation of bound-free integrals, such as those in Eq. (7). Subroutines for the specific case of radiative associa- tion, and for bound state computations, needed to be supplemented. In Appendix A, it is described how the current implementation in COUPLE was tested for an uncoupled case, to verify agreement with calculations done with an independent quantum dynamical code. The collision energy (asymptotic kinetic), E, is defined rela- tive to the asymptotic atomic potential. The ionic channel (with an asymptotic energy of ∼1.4 eV) is asymptotically closed, i.e., has negative collision energy, for all the scattering calculations. Should energies where the ionic channel is open be considered in future work, then a Coulombic asymptotic boundary condi- tion has to be imposed for that channel, rather than the boundary condition in Eq. (6). Coulombic boundary conditions are imple- mented in COUPLE and may be selected with a parameter in the code. The coupled Schrödinger equation (4) is integrated with a renormalized Numerov algorithm. The grid in the radial coordinate, FIG. 3 . Adiabatic NaCl interaction energies for the ground state, X1Σ+, and the first excited state, A1Π, respectively, vs internuclear separation, R. The inset shows the permanent and transition dipole moments for the ground state, DX(R), and the excited to the ground state, DAX(R), respectively. VXandDXare taken from Ref. 25. VAandDAXare taken from Ref. 31. J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp R, goes from R1= 3.1 to R2= 20 bohrs in steps of 0.002 bohr. A con- vergence test of the distance R2, at which the asymptotic boundary conditions [Eq. (6)] are imposed, is presented in Appendix B. For the calculations of the cross section in Eq. (8), the number of par- tial waves Jneeded for convergence depends on the collision energy, E. For example, Jfrom zero up to 30, 90, and 110 were sufficient at energies of 1 meV, 10 meV, and 30 meV, respectively. The bound states (with Ev′J′<0) are computed on the ionic potential, V2(R), using a discrete variable representation (DVR).33 The DVR grid in Rgoes from 3.1 bohrs to 20 bohrs in steps of 0.03 bohr. A formula provided in Ref. 33 is used to compute the bound wave function on the Numerov (free wave function) grid points so that the dipole matrix element (7) can be properly evaluated. IV. RESULTS The cross section for the formation of NaCl through the radia- tive association reaction (1) is shown vs collision energy in Fig. 4. It is computed with Eq. (8) using the method described in Secs. II and III. The baseline of the cross section grows from about 2.5 ⋅10−9 bohrs2at 10μeV to 2.1 ⋅10−5bohrs2at 30 meV. The cross section has a large number of resonances appearing as sharp peaks. The cross section for reaction (2), i.e., using the adiabatic approximation, is also shown in Fig. 4. The adiabatic cross section is more than one and four orders of magnitude too small at the lowest and highest energies, respectively. There is a distinct beat structure in the cross section for reac- tion (1) presented in Fig. 4. The resonances appear in clusters as the energy is increased. These appear roughly in the intervals 0.3 meV–0.5 meV, 1.5 meV–2.5 meV, 3 meV–4 meV, 5 meV– 6 meV, and around 8 meV. Above 10 meV, the clusters appear to overlap. To explain the beat structure, consider the energies of the resonant states, whose wave functions are largely localized on the ionic effective potential. The energies of those resonant states FIG. 4 . Cross sections for forming NaCl through reaction (1), computed using the method described in Sec. II (labeled QM), and reaction (2), computed using conventional single channel quantum dynamics (labeled conventional QM).4,19increase with increasing angular momentum, J. Now, consider the crossing energy between the atomic and ionic effective potential curves, which essentially provides the “doorway” from the contin- uum to the resonant states. That crossing energy also increases with increasing angular momentum, J, but this increase is slower than that of the resonant state energies, since it occurs for a larger distance (about 18 bohrs) than the expectation value of the dis- tance between Na and Cl in a resonant state. As a consequence, the optimal conditions for resonance mediated radiative association reoccur periodically with increasing energy and increasing angular momentum. Up to a collision energy of 1 meV, the energy resolution in the calculation is 5 μeV, for collision energies from 1 meV to 10 meV, it is 10μeV, and above that, it is 20 μeV. A test calculation with a 10 times higher energy resolution has been carried out for a few par- tial waves. The result is shown in Appendix C. Evidently, the chosen resolution appears to give satisfactory description of the resonances, which are broader and lower than those that have been seen in previ- ous studies with single channel dynamics.34,35This is fortunate since we will compute the rate constant by numerical integration of the quantum mechanical (QM) cross section according to Eq. (9). A fur- ther argument for computing the rate constant this way, rather than using the Breit–Wigner method,27is that there appears to be inter- ference between the resonances and the continuum. The interfer- ence is clearly visible in Fig. 5, where the cross section computed with Eq. (8) is displayed from 3 meV to 4 meV. Partial wave contribu- tions, i.e., omitting the Jsummation in Eq. (8), from J= 26 to J= 31 are also shown. Consider, for example, the contribution from J= 28. The resonance peak appears at 3.45 meV, and at energies just below this, the cross section dips to a minimum at 3.34 meV. The dip can only be explained by destructive interference, and the shape of the cross section vs energy is consistent with a Fano line shape.36Thus, it is not straightforward to describe the cross section with a Lorentzian resonance superimposed on a smooth background. The interfer- ence makes sense from a physical viewpoint since there is no clear FIG. 5 . Cross sections for forming NaCl through reaction (1), computed using the method described in Sec. II, and those partial wave contributions that give rise to the resonances in this energy interval. J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp separation between direct and resonance mediated radiative associ- ation in reaction (1), which is mediated through the non-adiabatic coupling alone. The bound states on the ionic potential, V2, with energies Ev′J′ >0 could possibly be used to estimate the positions of scattering resonances. This could provide help in the search for narrow res- onances, to avoid that they are missed in the scattering calculation. A test calculation for several partial waves showed, however, that the positions of the resonances are shifted by more than 1 meV relative to the bound states with Ev′J′>0. Thus, the bound states provide limited help in the search for scattering resonances. Possibly, cou- pled bound or pseudo-bound states (on V1andV2), where a wall or a complex absorbing potential is applied at the large Rboundary, would give better information. This has not been tested in this work, however. There is a large number of rovibrational states, v′,J′, on the ionic potential, V2. This makes the calculations time consum- ing since many bound-free matrix elements have to be computed according to Eq. (7). One apparent means to reduce the compu- tational cost is to try and include only those vibrational states, v′, that contribute significantly. It has been shown previously that, for radiative association with no change in the electronic state, for- mation of molecules in the uppermost few vibrational levels dom- inates.26Inspired by this observation, I limited the calculation of bound-free matrix elements in Eq. (7), and the v′summation in Eq. (8), to the highest 10 and 20 vibrational states, v′. The result from that test is shown in Appendix D, and it is clear that the high- est 10 or 20 vibrations are far from sufficient to reach convergence. Evidently, the coupled dynamics affects the scattering wave func- tion in a way that radiative transitions are efficient to many bound states, with a wide range of vibrational energies. Consequently, in the calculations presented in this work, allbound ( Ev′J′<0) rovibra- tional states have been included. The observation that the coupled dynamics makes transitions efficient to many more bound states than what is the case in adiabatic dynamics is likely the explanation for that the former cross section is so much larger than the latter (see Fig. 4). In Fig. 6, the cross section for reaction (3), A→X, com- puted with conventional quantum mechanical perturbation the- ory, is shown. For comparison, the cross section for reaction (1) is also shown. The A→Xcross section has no resonances since the initial potential curve, that for the A1Πstate, is purely repul- sive, and thus, it does not support any quasibound states. Fur- thermore, spin–orbit and rotational couplings are ignored in this work, in particular, between A1Πand other molecular states. The statistical weight and Hönl–London factors for the A→Xfor- mation are the same as those used for HF in Ref. 26. The A →Xcross section is larger than that for reaction (1) up to a collision energy of about 0.5 meV, above which the opposite holds. Figure 7 shows the rate constants for reactions (1) and (3) and the sum of them, which is the estimated total rate constant of radia- tive association of NaCl, for temperatures from 1 K to 30 K. The rate constants are computed by numerical integration of Eq. (9). The A →Xrate constant plays a role up to about 3 K. Otherwise, reaction (1) dominates. The total rate constant grows rapidly, from about 4⋅10−21cm3/s at the lowest of the temperatures to about 3 ⋅10−18 cm3/s at the highest. FIG. 6 . Cross sections for forming NaCl through reaction (1), as in Fig. 4, and reaction (3), computed using conventional single channel quantum dynamics.4,19 The total rate constant can be approximated using the three- parameter Kooij formula in two temperature intervals according to k(T)=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩4.015 81 ⋅10−13(T 300)3.521 56 e1.645 87/T, 1 K≤T≤2.5 K 6.5541 ⋅10−17(T 300)1.266 91 e−3.400 22/T, 2.5 K<T≤30 K (10) with k(T) in units cm3/s. The fit is generally within 5% of the computed points, and for all temperatures, it is well within 10%. FIG. 7 . Rate constant for the formation of NaCl through reaction (1) and reaction (3). The sum of those two rate constants is also displayed as total. J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp V. CONCLUSION A method is implemented for computations of radiative asso- ciation cross sections and rate constants for systems with coupled states. The method is tested for the formation of NaCl in low energy collisions of atomic Na and Cl in their ground states. This system has a non-adiabatic (vibronic) coupling between the ionic and neu- tral molecular states of symmetry1Σ+. The radiative association of NaCl is thus Feshbach resonance mediated. It takes place via reso- nant states that are embedded in the continuum. Such a process is often denoted as inverse predissociation.19,37,38 Inverse predissociation is typically handled by using the Breit– Wigner formula with parameters (lifetimes) for the predissociating state. That treatment is based on the separation of the direct process from the resonance mediated. The calculations of the rate constants presented here assume no such separation, and it is shown that the separation into direct and resonant contributions is not always pos- sible. In particular, the computed cross sections in this work show distinct interference effects. The radiative association cross section computed through non- adiabatic dynamics, reaction (1), is one to four orders of mag- nitude larger than that from adiabatic dynamics, reaction (2), in the considered energy interval. Apparently, the non-adiabatic treat- ment makes probable many more paths of molecule formation. Through test calculations, it is, indeed, noted that transitions to many bound rovibrational states contribute to the cross section in the non-adiabatic treatment. In the established (adiabatic) treatment of reactions in the absence of electronic transitions, on the other hand, it has been shown that only the highest rovibrational states contribute.26 Although the coupling investigated in this work is vibronic, the method is general enough to handle other couplings, such as spin–orbit and rotational couplings. In a proper treatment of radiative association of NaCl, all those couplings should be included simultaneously. This could potentially affect the results presented here. The influence from spin–orbit and rotational cou- plings that has been seen in previous work,9however, is signifi- cantly smaller than that from the vibronic coupling studied in this work. The regions of the circumstellar envelopes where NaCl has been found typically have temperatures in the range of 1000 K– 1500 K,22,23and where NaCl has been detected in Orion SrcI, it is of several hundred kelvins.24Thus, it is desirable to adapt the programs used here so that rate constants can be computed efficiently at higher temperatures. In the cross sections calculated in this work, some of the resonances are not perfectly resolved. Since the calculations are computationally costly, for higher energy studies, a scheme to find the resonances, e.g., by analysis of the time delay matrix, would be desirable. Then, a denser energy grid could be applied to the cross section calculations in the vicinity of resonances. ACKNOWLEDGMENTS Support from the Knut and Alice Wallenberg Foundation is acknowledged. Preliminary studies of radiative association of NaCl, using adiabatic dynamics, were carried out by R. K. Kathir in the initial phase of the present work. FIG. 8 . Cross sections for forming NaCl through reaction (1), computed using the method described in Sec. II, and reaction (2), computed using conventional single channel quantum dynamics4,19(solid blue line) and the adiabatic implementation inCOUPLE (red crosses). APPENDIX A: ADIABATIC APPROXIMATION In order to make sure that the implementation described in Sec. II is done correctly, it has been tested for adiabatic dynamics. In this case, the results can be verified through comparison with cross sections obtained with the established quantum mechanical perturbation theory.4,19 The test of the program is done with a flag in the input file, which causes the following modifications of the calculation. The potential V1in Eq. (5) is replaced by the adiabatic VXpotential, and the effective potential becomes a 1 ×1 matrix. The bound wave func- tions,ϕv′J′, and eigenvalues, Ev′J′, in Eqs. (7) and (8) are computed on the adiabatic potential, VX. The ionic dipole moment, D2, and the scattering wave function component, F2, in Eq. (7) are replaced by the adiabatic dipole moment, DX, and the (scalar) scattering wave function, F1, respectively. Figure 8 shows the result from this test. The adiabatic approx- imation implemented in the new program (red crosses) agrees perfectly with the conventional quantum mechanical perturbation theory (blue curve). APPENDIX B: CONVERGENCE WITH RESPECT TO THE UPPER LIMIT OF THE RADIAL COORDINATE Figure 9 shows how well the cross section converges with respect to R2, i.e., the point at which the asymptotic boundary con- dition [Eq. (6)] is imposed. The test shows that the values 18.5 bohrs and 18.7 bohrs are too small for an accurate description of the res- onances in partial wave J= 20. Calculations done with R2set to 19 bohrs and 20 bohrs agree almost perfectly, and the value that has been used in this work R2= 20.0 bohrs should thus be sufficiently big for converged scattering calculations. APPENDIX C: ENERGY RESOLUTION TEST In order to evaluate the energy resolution chosen in this work, test calculations have been carried out for three partial wave cross J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 9 . Cross section for partial wave J= 20 for forming NaCl through reaction (1), computed using the method described in Sec. II. Several distances, R2, at which the asymptotic boundary condition [Eq. (6)] is applied, are tested. sections [omitting the Jsummation in Eq. (8)], and the result is pre- sented in Fig. 10. Two of the partial waves, J= 20 and 30, are chosen randomly, while J= 43 was picked because it has one particularly sharp resonance around 2 meV. The resolutions for the cross sec- tions shown with symbols in Fig. 10 are 5 μeV and 10μeV at energies below and above 1 meV, respectively (the same as in the rest of this work). The resolutions for the solid lines are ten times higher, i.e., 0.5μeV and 1μeV at energies below and above 1 meV, respectively. The peaks of a few of the resonances are clearly better resolved at the higher resolution. The high resolution test reveals no missed res- onances, however, and the resolutions chosen in the calculations of this work appear to be sufficient for the computation of a reasonably accurate rate constant. FIG. 10 . Cross section for partial waves J= 20, 30, and 43 for forming NaCl through reaction (1), computed using the method described in Sec. II. The circles show results using the same energy resolution as in the calculations presented in Sec. IV. The solid black lines are computed with a ten times higher resolution. FIG. 11 . Cross section for forming NaCl through reaction (1), computed using the method described in Sec. II. Apart from the total cross section, the cross sections where only the upper 10 and 20 final vibrational states, v′, are included for each final rotational state, J′. APPENDIX D: LIMIT NUMBER OF INCLUDED BOUND STATES In Fig. 11, the effect of limiting the number of bound states in thev′summation in Eq. (8) is illustrated. The total cross section is displayed, together with those computed such that for each final rotational quantum number J′, the highest 10 and 20 bound states with vibrational quantum number v′andEv′J′<0 are included. For the lowest rotation, J′= 0, the upper 10 and 20 vibrational states have energies Ev′J′above −61 and −131 meV, respectively. The total number of vibrational states at J′= 0 is 222. It is clear that includ- ing many vibrational states is important to obtain an accurate result. For example, including the highest 10 vibrational states accounts for about 20% of the cross section at energies from 0.1 meV to 0.6 meV. At 10 meV, the 10 highest vibrations account for 0.1% of the cross section. DATA AVAILABILITY The data that support the findings of this study are available from the author upon reasonable request. REFERENCES 1J. F. Babb and K. P. Kirby, in The Molecular Astrophysics of Stars and Galaxies , edited by T. W. Hartquist and D. A. Williams (Clarendon Press, Oxford, 1998), p. 11. 2E. Herbst, Chem. Soc. Rev. 30, 168–176 (2001). 3D. Gerlich and S. Horning, Chem. Rev. 92, 1509 (1992). 4G. Nyman, M. Gustafsson, and S. V. Antipov, Int. Rev. Phys. Chem. 34, 385–428 (2015). 5F. Mrugała, V. Špirko, and W. P. Kraemer, J. Chem. Phys. 118, 10547–10560 (2003). 6M. Ayouz, R. Lopes, M. Raoult, O. Dulieu, and V. Kokoouline, Phys. Rev. A 83, 052712 (2011). 7T. Stoecklin, F. Lique, and M. Hochlaf, Phys. Chem. Chem. Phys. 15, 13818– 13825 (2013). J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 8T. Stoecklin, P. Halvick, H.-G. Yu, G. Nyman, and Y. Ellinger, Mon. Not. R. Astron. Soc. 475, 2545–2552 (2018). 9S. V. Antipov, M. Gustafsson, and G. Nyman, J. Chem. Phys. 135, 184302 (2011). 10Y. B. Band, K. F. Freed, and D. J. Kouri, J. Chem. Phys. 74, 4380–4394 (1981). 11S. J. Singer, S. Lee, K. F. Freed, and Y. B. Band, J. Chem. Phys. 87, 4762–4778 (1987). 12S. J. Singer, S. Lee, and K. F. Freed, J. Chem. Phys. 91, 240–249 (1989). 13F. Mrugała, Mol. Phys. 65, 377–389 (1988). 14W. L. Tchang-Brillet, P. S. Julienne, J. Robbe, C. Letzelter, and F. Rostas, J. Chem. Phys. 96, 6735–6745 (1992). 15R. W. Heather and J. C. Light, J. Chem. Phys. 78, 5513–5530 (1983). 16D. E. Manolopoulos and M. H. Alexander, J. Chem. Phys. 97, 2527–2535 (1992). 17F. Mrugała and W. P. Kraemer, J. Chem. Phys. 122, 224321 (2005). 18F. Mrugała and W. P. Kraemer, J. Chem. Phys. 138, 104315 (2013). 19J. F. Babb and A. Dalgarno, Phys. Rev. A 51, 3021 (1995). 20The National Institute of Standards and Technology, NIST Chemistry Web- Book, https://webbook.nist.gov/chemistry/, 2018. 21A. Hellman and M. Slabanja, arXiv:physics/0412013 [physics.chem-ph] (2004). 22J. Cernicharo and M. Guelin, Astron. Astrophys. 183, L10–L12 (1987), available at https://ui.adsabs.harvard.edu/abs/1987A%26A...183L..10C/abstract. 23S. N. Milam, A. J. Apponi, N. J. Woolf, and L. M. Ziurys, Astrophys. J. 668, L131–L134 (2007).24A. Ginsburg, B. McGuire, R. Plambeck, J. Bally, C. Goddi, and M. Wright, Astrophys. J. 872, 54 (2019). 25T. J. Giese and D. M. York, J. Chem. Phys. 120, 7939–7948 (2004). 26M. Gustafsson, M. Monge-Palacios, and G. Nyman, J. Chem. Phys. 140, 184301 (2014). 27R. A. Bain and J. N. Bardsley, J. Phys. B: At. Mol. Phys. 5, 277 (1972). 28D. R. Bates, Mon. Not. R. Astron. Soc. 111, 303 (1951). 29M. Gustafsson, J. Chem. Phys. 138, 074308 (2013). 30R. J. Le Roy, “LEVEL 8.0: A computer program for solving the radial schrödinger equation for bound and quasibound levels,” University of Waterloo Chemical Physics Research Report CP-663, 2007. 31Y. Zeiri and G. G. Balint-Kurti, J. Mol. Spectrosc. 99, 1–24 (1983). 32F. H. Mies, P. S. Julienne, and K. M. Sando, “A close coupling code,” private ommunication (1993). 33D. T. Colbert and W. H. Miller, J. Chem. Phys. 96, 1982–1991 (1992). 34S. V. Antipov, T. Sjölander, G. Nyman, and M. Gustafsson, J. Chem. Phys. 131, 074302 (2009). 35M. Gustafsson and G. Nyman, Mon. Not. R. Astron. Soc. 448, 2562–2565 (2015). 36U. Fano, Phys. Rev. 131, 259 (1963). 37P. S. Julienne and M. Krauss, in Molecules in the Galactic Environment , edited by M. A. Gordon and L. E. Snyder (John Wiley & Sons, New York, 1973), pp. 353–373. 38S. K.-M. Svensson, M. Gustafsson, and G. Nyman, J. Phys. Chem. 119, 12263– 12269 (2015). J. Chem. Phys. 153, 114305 (2020); doi: 10.1063/5.0019295 153, 114305-8 Published under license by AIP Publishing

5.0020720.pdf

J. Chem. Phys. 153, 154701 (2020); https://doi.org/10.1063/5.0020720 153, 154701 © 2020 Author(s).Influence of tungsten doping on nonradiative electron–hole recombination in monolayer MoSe2 with Se vacancies Cite as: J. Chem. Phys. 153, 154701 (2020); https://doi.org/10.1063/5.0020720 Submitted: 03 July 2020 . Accepted: 29 September 2020 . Published Online: 16 October 2020 Yating Yang , Marina V. Tokina , Wei-Hai Fang , Run Long , and Oleg V. Prezhdo COLLECTIONS Paper published as part of the special topic on Excitons: Energetics and Spatio-temporal DynamicsEXEN2020 ARTICLES YOU MAY BE INTERESTED IN Potential and pitfalls: On the use of transient absorption spectroscopy for in situ and operando studies of photoelectrodes The Journal of Chemical Physics 153, 150901 (2020); https://doi.org/10.1063/5.0022138 Use the force! Reduced variance estimators for densities, radial distribution functions, and local mobilities in molecular simulations The Journal of Chemical Physics 153, 150902 (2020); https://doi.org/10.1063/5.0029113 Probing the high-pressure viscosity of hydrocarbon mixtures using molecular dynamics simulations The Journal of Chemical Physics 153, 154502 (2020); https://doi.org/10.1063/5.0028393The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Influence of tungsten doping on nonradiative electron–hole recombination in monolayer MoSe 2 with Se vacancies Cite as: J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 Submitted: 3 July 2020 •Accepted: 29 September 2020 • Published Online: 16 October 2020 Yating Yang,1Marina V. Tokina,2 Wei-Hai Fang,1 Run Long,1,a) and Oleg V. Prezhdo2,a) AFFILIATIONS 1College of Chemistry, Key Laboratory of Theoretical and Computational Photochemistry of Ministry of Education, Beijing Normal University, Beijing 100875, People’s Republic of China 2Department of Chemistry, University of Southern California, Los Angeles, California 90089, USA Note: This paper is part of the JCP Special Topic on Excitons: Energetics and Spatio-Temporal Dynamics. a)Authors to whom correspondence should be addressed: runlong@bnu.edu.cn and prezhdo@usc.edu ABSTRACT Two-dimensional transition metal dichalcogenides (TMDs) are receiving significant attention due to their excellent electronic and optoelectronic properties. The material quality is greatly affected by defects that are inevitably generated during material synthe- sis. Focusing on chalcogenide vacancies, which constitute the most common defect, we use the state-of-the-art simulation methodol- ogy developed in our group to demonstrate that W doping of MoSe 2with Se vacancies reduces charge carrier losses by two mech- anisms. First, W doping makes the formation of double Se vacancies unfavorable, while it is favorable in undoped MoSe 2. Sec- ond, if a Se vacancy is present, the charge carrier lifetimes are extended in the W-doped MoSe 2. Combining ab initio real-time time-dependent density functional theory with nonadiabatic molecular dynamics, the simulations show that the nonradiative car- rier losses in the presence of Se vacancies proceed by sub-10 ps electron trapping and relaxation down the manifold of trap states, followed by a 100 ps recombination of trapped electrons with free holes. The electron–vibrational energy exchange is driven by both in-plane and out-of-plane vibrational motions of the MoSe 2layer. The atomistic studies advance our understanding of the influence of defects on charge carrier properties in TMDs and guide improvements of material quality and development of TMD applications. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020720 .,s I. INTRODUCTION Two-dimensional (2D) transition metal dichalcogenides (TMDs) (MX 2, M = Mo, W; X = S, Se, Te) have attracted tremen- dous interest due to their excellent physical and chemical proper- ties,1such as appropriate bandgaps within the infrared to the visible light region, ranging from 1.10 eV for MoTe 22to 1.99 eV for WS 2,3 coupled spin and valley physics for spintronics,4,5and high hydrogen evolution catalytic activities.6Based on these properties, 2D TMDs are appealing candidates for electronics,7,8optoelectronics,9,10 catalysis,6,11,12energy storage,13valleytronics,14etc. Compared with the most studied monolayer molybdenum disulfide (MoS 2),molybdenum diselenide (MoSe 2) possesses a narrower bandgap of around 1.55 eV,15matching well the solar spectrum, and a higher optical absorbance.16Monolayer MoSe 2forms a periodic 2D hon- eycomb lattice with covalently bonded Se–Mo–Se tri-layer atoms, insuring its mechanical strength. Each Mo atom is surrounded by six Se atoms, and each Se is bonded with three Mo atoms. The layered structure is held together by weak van der Waals interac- tions between layers. As a result, not only chemical vapor deposition (CVD) and molecular beam epitaxy (MBE)17–19but also liquid exfo- liation and micromechanical cleavage are available for synthesis of monolayer TMDs.20,21Among these methods, the CVD approach has the greatest potential for its low cost and large-scale production. J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 153, 154701-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The arrangement of atoms in real crystals cannot be as com- plete and regular as in the perfect structure, due to complicated for- mation conditions, and a balance between thermodynamic stability and synthesis kinetics. Various defects break the perfect symmetry and induce deviations from the fully periodic lattice. Vacancies are the most common form of point defects, which also include intersti- tials, substitutions, ad-atoms, and dopants. In particular, vacancies are common in CVD-grown samples of monolayer MoSe 2. Because removal of a Mo atom requires breaking of six Mo–Se bonds, while removal of a Se atom breaks only three bonds, it is much easier to form a Se vacancy (V Se) than a Mo vacancy under neutral condi- tions. Due to universality and functionality of vacancies, their study has gradually emerged as an important approach to understand and control optical, electrical, and magnetic properties of materials. For example, vacancies in TMDs can act as electron and hole traps,22–24 influence charge transport and optical properties,25and trigger mag- netism.26Hence, a comprehensive understanding of the types and concentration of vacancies is necessary in order to account for the contribution of vacancies to material properties and to develop tech- niques for eliminating detrimental vacancies that appear during synthesis. Doping constitutes an effective method to repair vacancies in monolayer TMDs. Doping with atoms that are isoelectronic to the native atoms is particularly favorable because it maintains the elec- tronic structure of the pristine material. Experiments have demon- strated that isoelectronic tungsten doping into monolayer MoSe 2 can effectively reduce the concentration of Se vacancies to 50% and enhance photoluminescence exciton lifetime by suppressing non- radiative electron–hole recombination.27The experimental obser- vation provides an important insight into Se vacancy repair, offer- ing guidance for promoting TMD performance. Understanding the atomistic mechanism underlying this effect is required for ratio- nal design of high quality TMD materials and devices. Such an understanding can be provided by ab initio quantum dynamics simulation of the evolution of charge carriers coupled to atomic motions. In this work, we use ab initio time-domain density functional theory (TDDFT)28combined with nonadiabatic molecular dynam- ics (NAMD)29to model nonradiative charge trapping and recombi- nation in a series of MoSe 2monolayers with different concentrations of Se vacancies and W dopants. The simulations mimic directly the time-resolved ultrafast pump–probe spectroscopy experiments.27 First, we calculate formation energies to demonstrate that W dop- ing of monolayer MoSe 2is energetically favorable. Second, we show that formation of a second Se vacancy in pristine MoSe 2 next to an existing vacancy is more favorable than formation of the first vacancy. In contrast, W doping reduces the likelihood offormation of the second Se vacancy, thereby reducing the overall vacancy concentration, in agreement with the experiment. Then, we study charge trapping and recombination and demonstrate that the overall mechanism is the same in both W-doped and undoped MoSe 2. Se vacancies create electron trap states that are popu- lated within 10 ps, and the trapped electrons recombine with free holes on a 100 ps timescale. However, the recombination in the most stable W-doped system with the Se vacancy is slower than the dynamics in pristine MoSe 2with Se vacancies. Therefore, we conclude that the enhancement of the charge carrier lifetimes in W- doped MoSe 2arises from a combination of reduced concentration of Se vacancies and reduced vacancy-mediated recombination rate. II. METHODS The NAMD simulations are performed using the decoherence induced surface hopping (DISH) method30as implemented in the PYXAID code.31,32The approach has been applied to study excited state dynamics in a broad range of systems, including black phos- phorous (BP),33a STi 3nanoribbon,34lead halide perovskites,35and TMD systems.36–38 The simulations cells for monolayer MoSe 2with Se vacancies and W dopants are shown in Fig. 1 and Fig. S1. A vacuum layer of 15 Å is applied perpendicular to the periodic direction to screen off interactions between adjacent layers. The geometry optimiza- tion, electronic structure, and adiabatic MD calculations are all per- formed using the Vienna ab initio simulation package (VASP).39The Perdew–Burke–Ernzerhof (PBE) electronic exchange–correlation functional40and projected-augmented wave pseudopotentials41are used. The energy cutoff for the plane-wave basis is 400 eV. The geometry optimization is carried out with a Γ-centered Monkhorst– Pack426×6×1 k-point mesh. The electronic structure is obtained with a much denser 11 ×11×1 k-point mesh. The HSE06 hybrid functional43is used to obtain accurate electronic proper- ties at the optimized geometries, Fig. S3, to compare with the PBE functional. The systems are fully relaxed until the calculated Hellmann– Feynman forces are less than 0.05 eV/Å. The systems are then heated to 300 K by repeated velocity rescaling. Then, 5 ps adiabatic MD trajectories are generated in the microcanonical ensemble with a 1 fs time step. The nonadiabatic couplings (NACs)s between the relevant states are computed along the trajectory, and the result- ing nonadiabatic Hamiltonian is iterated multiple times, in order to model the quantum dynamics of nonradiative charge trap- ping and recombination on a hundred of picoseconds timescale. 5000 geometries are used as the initial conditions for the DISH simulations. 1000 stochastic realizations of the DISH algorithm FIG. 1 . Top views of: (a) MoSe 2with two Se vacancies, denoted MoSe 2_2V Se, and (b) and (c) MoSe 2with one Se vacancy and a W atom in the place of a Mo atom, denoted W@Mo_1V Se_1 and W@Mo_1V Se_2, respectively. The structure in (b) has V Sesurrounded by three Mo atoms, while the structure in (c) has V Sesurrounded by two Mo atoms and one W atom. The Se vacancies are highlighted by the red circles. Additional considered structures are shown in Fig. S1. J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 153, 154701-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp are sampled for each initial geometry. The NAMD simula- tions are performed with the PYXAID software package.31,32 Additional simulation details are provided in the supplementary material. III. RESULTS AND DISCUSSION A. Geometric structure and formation energy The MoSe 2monolayer is represented by a 5 ×5 supercell. Removing two neighboring Se atoms from the prefect MoSe 2mono- layer forms the MoSe 2_2V Sesystem, leading to six different config- urations presented in Fig. S1 together with the corresponding total energies. Figure 1(a) presents the energetically favorable structure, in which the two Se vacancies are next to each other in the same hexagon, highlighted by the red circles. Replacing four Mo atoms by the same number of W atoms generates a 25% W doping concentra- tion in monolayer MoSe 2, consistent with the experiment.27Remov- ing one Se atom in the doped system forms the W@Mo_1V Se_1 and W@Mo_1V Se_2 systems, shown in Figs. 1(b) and 1(c), respectively. These two systems differ in the relative position of the Se vacancy with respect to the doped W atoms. The Se vacancy is surrounded by three Mo atoms in the W@Mo_1V Se_1 system, while it is sur- rounded by two Mo atoms and one W atom in the W@Mo_1V Se_2 system. First, we compute the formation energy for the defective MoSe 2 monolayers with and without W doping, E′=E+n1⋅μSe+n2(μMo−μW)−Epure - MoSe 2. Here, E′is the calculated formation energy. Erefers to the total energy of each defective structure. μSe,μW,μMoare the chemi- cal potentials of Se, W, and Mo atoms, respectively. The chemical potentials correspond to the energy of one atom in the bulk mate- rial. n1is the number of V Sein the systems. n2is the number of doped W atoms. Epure - MoSe 2is the energy of the pristine MoSe 2 monolayer without vacancy and dopant. The E f@defect column in Table I displays the formation energies for the different defective structures. The negative values indicate that introducing W into MoSe 2is energetically favorable, especially for the W@Mo_1V Se_1 system. The E f@V Secolumn in Table I represents the energy of remov- ing a single Se atom in each structure. Because V Seis the only defect in the MoSe 2_1V Sesystem, the E f@defect and E f@V Secolumns con- tain the same number, 2.474 eV. The MoSe 2_2V Sesystem contains two vacancies, and the E f@V Secolumn represents the energy of TABLE I . Formation energies (eV) of the defective structures (E f@defect) and of the Se vacancy (E f@V Se) in these structures. The corresponding geometries are shown in Fig. 1 and Fig. S1. Ef@defect (eV) E f@V Se(eV) MoSe 2_1V Se 2.474 2.474 MoSe 2_2V Se 4.939 2.465 W@Mo_1V Se_1 −2.334 2.456 W@Mo_1V Se_2 −2.285 2.504 W@Mo_2V Se_1 0.151 2.485formation of the second vacancy. The vacancy formation energy is computed by E′′=E2+μSe−E1. Here, E′′is the vacancy formation energy. E1andE2refer to the total energy of the structures before and after removal of a Se atom, for example, pristine MoSe 2and MoSe 2_1V Se.μSeis the chemical poten- tial of the Se atom, equal to the energy of one Se atom in bulk Se. The formation energy of the second Se vacancy in MoSe 2is slightly lower than the formation energy of the first vacancy, indicating that it is favorable for Se vacancies to pair up or form clusters. In contrast, the energy of formation of the second Se vacancy in W-doped MoSe 2 is higher than the formation energy of the first vacancy, compare the W@Mo_1V Se_1 and W@Mo_2V Se_1 systems. Therefore, the vacan- cies tend to be isolated in W-doped MoSe 2. Even though formation of an isolated V Seis more favorable in the doped system, the dop- ing prevents formation of multiple vacancies, thereby decreasing the overall vacancy concentration. To sum up, W doping of monolayer MoSe 2is energetically favorable. At the same time, W doping stops formation of the second VSe, explaining qualitatively the 50% reduction in the V Seconcen- tration detected experimentally.27Because formation of the second VSeis favored in pristine MoSe 2and prevented in doped MoSe 2, our NAMD analysis focuses on the MoSe 2_2V Se, W@Mo_1V Se_1, and W@Mo_1V Se_2 systems. B. Electronic structure Figure 2 shows a diagram of the energy levels involved in charge carrier trapping and recombination in pristine and W-doped MoSe 2 FIG. 2 . Energy diagram depicting the charge trapping and recombination dynam- ics in the MoSe 2monolayer in the presence of a Se vacancy with and without W doping. Absorption of a photon promotes an electron from the VB to the CB. Recombination can occur between the CBM and the VBM bypassing the mid-gap state, 1⃝. Alternatively, the photoexcited electron can get trapped by a mid-gap state, 2⃝. The trapped electron then recombines with the hole in the VBM, 3⃝. The thin gray line denotes the Fermi level. J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 153, 154701-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp monolayers with Se vacancies. The diagram is representative of all systems and corresponds most directly to systems with a single V Se. Absorption of a photon promotes an electron from the valence band (VB) to the conduction band (CB). The CB electron can recombine directly with the hole left in the VB through process 1⃝. VSecreates a pair of unoccupied states below the CB. The states act as electron traps providing an alternative recombination pathway, by which the electron is first trapped, process 2⃝, and then recombines with a free hole in the VB, process 3⃝. The rates and branching ratio of the different processes are determined by electron–vibrational inter- actions and can be elucidated by performing quantum dynamics simulations. Figure 3 demonstrates the projected density of states (PDOS) and the charge densities of the key states in the MoSe 2_2V Se, W@Mo_1V Se_1, and W@Mo_1V Se_2 structures. By removing two valences, each Se vacancy creates two electron trap states. The MoSe 2_2V Sesystem contains two deep and two shallow electron trap states, contributed by both Mo and Se atoms, Fig. 3(a). By comparison, the W@Mo_1V Se_1 and W@Mo_1V Se_2 systems have two deep trap states each. For W@Mo_1V Se_1, the PDOS shows a slight contribution of W to the trap states. The W contribu- tion to the trap states is higher in W@Mo_1V Se_2. As V Segets closer to a W atom (W@Mo_1V Se_2), the two trap states become more split in energy due to symmetry breaking. This fact is con- firmed by the band structure plots shown in Fig. S2. Figure S3 shows the PDOS calculated with the hybrid HSE06 functional for the MoSe 2_2V Se, W@Mo_1V Se_1, and W@Mo_1V Se_2 systems. The new results agree with the original PDOS obtained using the PBE functional. In particular, both HSE06 and PBE func- tionals predict the midgap states created by the vacancies and dopants in MoSe 2. Though the energy gaps for charge trapping andrecombination are underestimated by the PBE functional, the recombination rate has a weak dependence on the gap, compared to the dependence on the nonadiabatic coupling and coherence time. In reference to the experiment,27suppression of deep trap levels is considered to be the major factor prolonging nonradiative recombination lifetime, which remains to be proven by the quan- tum dynamics simulation. It is interesting to note that the energy gap between the VB maximum (VBM) and the deepest electron trap state is approximately the same for the systems with one and two Se vacancies. The charge densities of the key orbitals shown in the bottom part of Fig. 3 demonstrate that the VBM and the CB minimum (CBM) are delocalized over the whole MoSe 2, even in the presence of the vacancies and dopants. In contrast, the trap states are local- ized near the vacancy sites. Proximity of V Seto a W atom makes the trap states less localized, Fig. 3(c), because the trap states delocalize onto the dopant. This observation is in agreement with the PDOS, showing a larger contribution of W atoms to the trap states in the W@Mo_1V Se_2 system. Note that W@Mo_1V Se_1 is more stable than W@Mo_1V Se_2. The enhanced localization of the trap states in the more energetically favorable W@Mo_1V Se_1 system decreases the overlap of the trap state with the VBM, leading to a smaller NAC and a slower nonradiative charge recombination. We will elaborate on this part below. C. Nonradiative charge trapping and recombination dynamics Figure 4 shows evolution of the populations of the trap and ground state (VBM) of the electron that has been excited initially from the VBM to the CBM, in the three systems under detailed FIG. 3 . Projected density of states (PDOS) and charge densities of the key states involved in charge trapping and recombination in (a) MoSe 2_2V Se, (b) W@Mo_1V Se_1, and (c) W@Mo_1V Se_2. Two Se vacancies in MoSe 2_2V Secreate two deep and two shallow electron trap states. A single Se vacancy creates two deep trap states in both W@Mo_1V Se_1 and W@Mo_1V Se_2. The pairs of trap states are almost degenerate in energy and are localized around the Se vacancy. J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 153, 154701-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Electron trapping and electron–hole recombination dynamics in (a) MoSe 2_2V Se, (b) W@Mo_1V Se_1, and (c) W@Mo_1V Se_2. The black curve is the population of the VBM, and the other curves are the populations of different trap states. investigation. In general, the VBM is populated both via the direct transition from the CBM and via the intermediate trap states. The timescales reported in Fig. 4 and Table II are obtained by exponen- tially fitting the data. In particular, the rise and decay of the trap state populations are fitted separately to the generic exponential function, f(t)=a×exp(−t τ)+b. Figure 4 demonstrates that the trap states are populated on a few picosecond timescale, while the complete electron–hole recombination takes about 100 ps. If a charge is trapped in a midgap state, it is eliminated from the corresponding band and cannot conduct current. Therefore, sep- arate consideration of charge trapping and recombination is nec- essary. While an electron is trapped and cannot conduct current, the corresponding hole is still in the VB and acts as a current car- rier. Therefore, the current is reduced to 50% rather than to 0.The influence of trap states on current should exhibit a com- plex dependence on density of both defects and charge carriers. If defect density is low and charges are trapped fast, then the defects are filled, and the remaining charge carriers can no longer be trapped and continue to conduct. If charge carrier density becomes very high, then Auger-type processes, i.e., charge–charge scatter- ing, become important, and charge trapping and recombination can be driven by both coupling to phonons and interaction with other charges, which can accommodate the lost energy instead of phonons.44Auger processes require a significantly more complex simulation and are not considered at present. By providing the timescales for charge trapping, evolution within the defect state manifold, and charge recombination, the current atomistic study generates key mechanistic insights and input for phenomenologi- cal reaction–diffusion calculations of charge transport in nanoscale materials. The significant rise of the trap population curves indicates that trap-assisted charge recombination constitutes the dominant path- way. It is important to note that the relative contribution of the trap-assisted recombination, compared to the direct CBM-to-VBM pathway, is characterized by the maximum amplitude of the trap populations, rather than by the timescale associated with the trap population rise. In particular, the situation with a rapid but small rise of a trap state population and a slower population decay implies that the rate constant for the population decay is, in fact, larger than the rate constant for the population rise because the popula- tion maximum is low. This is the case for all but the lowest energy trap state. The population of trap 1 reaches 0.5–0.75, depending on the system, indicating that the electron rapidly progresses down the manifold of trap states and then more slowly hops to the VBM. This scenario can be understood by considering gaps between the CBM, the trap states, and the VBM. Smaller gaps correspond to faster transitions. Further analysis of the quantum dynamics data is supported by the values of the NAC and pure-dephasing/decoherence times between pairs of states. Electron–vibrational interaction can lead to inelastic and elastic scattering. The NAC magnitude character- izes the former process, while the pure-dephasing time describes the latter. Importantly, loss of coherence also influences inelastic electron–vibrational exchange. Rapid decoherence generally slows down quantum dynamics, while slow decoherence can accelerate transitions.45,46Currently, decoherence times are all much shorter than the corresponding transition times, Table II, corresponding TABLE II . Canonically averaged energy gaps, nonadiabatic couplings (NACs), decoherence times, and transition times characterizing electron–vibrational interactions during electron trapping and trap-assisted electron–hole recombination. Energy NAC Decoherence Time gap (eV) (meV) (fs) (ps) e-trapping (all CBM and trap pairs) MoSe 2_2V Se 0.66 3.33 9.9 6.1 W@Mo_1V Se_1 0.60 3.54 8.4 5.9 W@Mo_1V Se_2 0.74 3.85 7.3 5.7 e–h recombination (trap1–VBM) MoSe 2_2V Se 0.81 1.74 12.7 57 W@Mo_1V Se_1 0.86 0.91 9.4 83 W@Mo_1V Se_2 0.95 2.26 7.6 33 J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 153, 154701-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp to the former limit. All decoherence times are on the order of 10 fs and vary relatively little, compared to the NACs, which are on the order of 1 meV and differ by over a factor of 3, Table II. The NAC is proportional to the overlap of the two wavefunctions and is inversely proportional to the energy gap between the corre- sponding pairs of states. The more significant delocalization of the trap state charge densities in the W@Mo_1V Se_2 system [Fig. 3(c)], compared to W@Mo_1V Se_1, correlates with the larger NAC (Table II). Figure 5 presents a 2D map of the average absolute NAC val- ues between pairs of orbitals, with the x-axis and y-axis represent- ing the orbital index. The diagonal values in the plots are set to 0. Generally, couplings between neighboring states are larger, as reflected by the brighter spots next to the diagonal. The NACs between the nearly degenerate pairs of trap states, see the PDOS in Fig. 3, are the highest. All pairs of states are coupled, and therefore, a variety of recombination pathways operate simultane- ously. In particular, four scenarios are possible in the systems with two trap states, i.e., VBM–CBM, VBM–trap2–CBM, VBM–trap1– CBM, and VBM–trap2–trap1–CBM. Four trap states allow 16 path- ways, i.e., VBM–CBM, four pathways populating one trap state, four pathways populating three trap states, six pathways populat- ing two trap states, and a pathway populating all four traps. The above estimation of the number of pathways is obtained assum- ing transitions always happen down in energy, which is not the case for the pairs of nearly degenerate states, Fig. 3. Therefore, the actual number of pathways for electron relaxation is even larger. The quantum dynamics data in Fig. 4 demonstrate that sequential transitions down in energy constitute the dominant mechanism. Next, we analyze the phonon modes that couple to the elec- tronic subsystem. Figure 6 shows influence spectra, also known as spectral densities, computed as Fourier transform of the autocor- relation functions of the phonon-driven fluctuation of the energy gap between the VBM and the lowest energy trap state. The focus is on this pair of states because the transition between them con- stitutes the bottleneck of the nonradiative charge recombination in the systems under study. Since this is a fully atomistic calculation, electron–phonon coupling arising from all modes available in the system is considered explicitly. Both elastic scattering and inelas- tic scattering are considered and characterized by decoherence and NA coupling, respectively. Treated semi-classically, decoherence is a time-domain equivalent of the Franck–Condon and Huang–Rhys FIG. 6 . Spectral densities computed as Fourier transform of the autocorrelation functions of fluctuations of the energy gaps between trap1 and the VBM in (a) MoSe 2_2V Se, (b) W@Mo_1V Se_1, and (c) W@Mo_1V Se_2. factors. The peaks seen in the MoSe 2_2V Sestructure corre- spond to the Raman-active out-of-plane A 1′vibration of mono- layer MoSe 2at 240.27 cm–1.47The same peak is seen in W@Mo_1V Se_1. The main peak in W@Mo_1V Se_2, seen also in W@Mo_1V Se_1, arises from the in-plane E′′mode at 167.70 cm–1. The higher frequency peaks are overtones of the low frequency signals. Overall, the quantum dynamics simulation data show that W doping can both accelerate and slow down the non-radiative carrier recombination, depending on the relative arrangement of the W atoms with respect to the Se vacancy. However, the more stable arrangement makes the recombination slower compared to the undoped MoSe 2, 83 ps vs 55 ps. Interestingly, the energy gap between the VBM and the lowest energy electron trap state is nearly the same for systems with one and two Se vacancies (Fig. 3) and, in particular, parts (a) and (d) of Fig. S2. Importantly, W doping does not eliminate trap levels. Rather, W doping makes formation of the second Se vacancy less favorable, Table I. FIG. 5 . 2D visualization of the aver- aged absolute nonadiabatic coupling between pairs of orbitals involved in charge trapping and recombination in (a) MoSe 2_2V Se, (b) W@Mo_1V Se_1, and (c) W@Mo_1V Se_2. J. Chem. Phys. 153, 154701 (2020); doi: 10.1063/5.0020720 153, 154701-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp IV. CONCLUSIONS In conclusion, we have investigated charge carrier trapping and recombination dynamics in monolayer MoSe 2with Se vacan- cies and have studied the effects of W doping on these processes. Geometries, energies of doping and defect formation, and elec- tronic structure have been described by the standard ab initio DFT, while the quantum dynamics simulations have been carried out using the state-of-the-art methodology developed in our group and combining real-time TDDFT with NAMD. The calculations have demonstrated that W doping of MoSe 2is energetically favorable, while Se vacancy formation is unfavorable. Se vacancies are formed in experiments due to kinetic factors, since pristine MoSe 2is the most thermodynamically stable structure. Formation of a second Se vacancy next to an existing Se vacancy in pristine MoSe 2requires less energy than formation of the first vacancy. However, the oppo- site is true upon W doping. Therefore, W doping prevents for- mation of additional vacancies and reduces the overall vacancy concentration. A Se vacancy forms two electron trap states below the CB, regardless of whether MoSe 2is W-doped or not. The energy gap between the lowest energy electron trap state and the VBM remains approximately the same, independent of whether there is one or two Se vacancies next to each other, and whether or not MoSe 2is W-doped. The presence of multiple trap states creates an ensem- ble of pathways for electron–hole recombination. The dominant pathway involves rapid, sub-10 ps electron trapping and relaxation down the manifold of trap states, followed by a 100 ps recom- bination of the electron in the lowest energy trap state with the hole in the VBM. W doping can both increase and decrease the recombination time. However, the recombination increases for the most stable arrangement of the Se vacancy and the W dopant atoms. Therefore, W doping of MoSe 2carries two positive effects. It reduces Se vacancy concentration and prolongs charge carrier lifetimes in the presence of vacancies. The reported simulations expand our understanding of electron–hole recombination dynam- ics in defective MoSe 2and further guide synthesis and applications of high quality 2D TMD materials. SUPPLEMENTARY MATERIAL See the supplementary material for a description of the sim- ulation methodology; comparison of bond length changes due to excitation and thermal fluctuations; geometric structures of MoSe 2 monolayers containing two Se vacancies in various arrangements; band structures of the four systems discussed in the main text; and densities of states obtained with a hybrid functional. ACKNOWLEDGMENTS This work was supported by the National Science Founda- tion of China, Grant Nos. 21973006, 51861135101, 21688102, and 21590801. R.L. acknowledges the Recruitment Program of Global Youth Experts of China, the Fundamental Research Funds for the Central Universities, and the Beijing Normal University Startup Package. O.V.P. acknowledges funding from the U.S. Department of Energy, Grant No. DE-SC0014429.DATA AVAILABILITY The data that support the findings of this study are available from the corresponding authors upon reasonable request. REFERENCES 1X. Duan, C. Wang, A. Pan, R. 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5.0014865.pdf

Appl. Phys. Lett. 117, 083102 (2020); https://doi.org/10.1063/5.0014865 117, 083102 © 2020 Author(s).Electrically tunable high Curie temperature two-dimensional ferromagnetism in van der Waals layered crystals Cite as: Appl. Phys. Lett. 117, 083102 (2020); https://doi.org/10.1063/5.0014865 Submitted: 22 May 2020 . Accepted: 06 August 2020 . Published Online: 25 August 2020 Hua Wang , Jingshan Qi , and Xiaofeng Qian 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 Evolution of strong second-order magnetic anisotropy in Pt/Co/MgO trilayers by post- annealing Applied Physics Letters 117, 082403 (2020); https://doi.org/10.1063/5.0018924 Anisotropic domains and antiferrodistortive-transition controlled magnetization in epitaxial manganite films on vicinal SrTiO 3 substrates Applied Physics Letters 117, 081903 (2020); https://doi.org/10.1063/5.0016371Electrically tunable high Curie temperature two-dimensional ferromagnetism in van der Waals layered crystals Cite as: Appl. Phys. Lett. 117, 083102 (2020); doi: 10.1063/5.0014865 Submitted: 22 May 2020 .Accepted: 6 August 2020 . Published Online: 25 August 2020 Hua Wang,1Jingshan Qi,2,a) and Xiaofeng Qian1,a) AFFILIATIONS 1Department of Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, USA 2School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou 221116, People’s Republic of China a)Authors to whom correspondence should be addressed: qijingshan@jsnu.edu.cn andfeng@tamu.edu ABSTRACT Identifying intrinsic low-dimensional ferromagnets with high magnetic transition temperature and electrically tunable magnetism is crucial for the development of miniaturized spintronics and magnetoelectrics. Recently, long-range 2D ferromagnetism was observed in van der Waals crystals CrI 3and Cr 2Ge2Te6, however, their Curie temperature is significantly lowered when reducing down to monolayer/few layers. Herein, using renormalized spin-wave theory and first-principles electronic structure theory, we present a theoretical study of electrically tun-able 2D ferromagnetism in van der Waals layered CrSBr and CrSeBr semiconductors with a high Curie temperature of /C24150 K and a sizable bandgap. The high transition temperature is attributed to the strong anion-mediated superexchange interaction and a sizable spin-wave exci- tation gap due to large exchange and single-ion anisotropy. Remarkably, hole and electron doping can switch the magnetization easy axis from the in-plane to the out-of-plane direction. These unique characteristics establish monolayer CrSBr and CrSeBr as a promising platformfor realizing 2D spintronics and magnetoelectrics such as 2D spin valves and spin field effect transistors. Published under license by AIP Publishing. https://doi.org/10.1063/5.0014865 Achieving long-range magnetism at low dimensions and high temperature is of both fundamental and technological importance. 1–3 Particularly, intrinsic low-dimensional semiconducting ferromagnetswith high Curie temperature T c, large bandgap, and high carrier mobil- ity will help go beyond dilute magnetic semiconductors1and pave the way for the development of next-generation ultra-miniaturized, highlyintegrated spintronics and magneto-optoelectronics. 2,3However, the coexistence of ferromagnetic (FM) and semiconducting characteristicsin a single material is generally difficult, 3,4whereas achieving long- range magnetic ordering in two-dimensional (2D) materials is evenharder. According to the Mermin–Wagner theorem, 2D FM/antiferro- magnetic (AFM) order is prohibited by thermal fluctuations withintheisotropic Heisenberg model with continuous SU(2) symmetry. 5 Finite magnetic anisotropy such as exchange anisotropy and single-ion anisotropy becomes critical for establishing long-range magneticorder, e.g., in ultrathin metallic films decades ago. 6,7Long-range 2D FM order was also observed recently in van der Waals (vdW) insula-tors CrI 38(Tcof/C2445 K) and Cr 2Ge2Te69(Tcof/C2425 K), while their Tc is markedly lowered with decreasing layers. High Tcsemiconductingferromagnets are, thus, highly desirable,10which could further impact 2D multiferroics.11Recently, vdW layered chromium sulfur bromide (CrSBr) has attracted attention because its bulk, first synthesized50 years ago, was an AFM semiconductor. 12,13Several recent theoreti- cal studies14–17predicted that monolayer CrSBr and CrSeBr are FM semiconductors with much higher Tccompared with CrI 3and Cr2Ge2Te6,r a n g i n gf r o m /C24300 K to 160 K using the 2D Ising model,15 2D Heisenberg model without magnetic anisotropy,17and 2D Heisenberg model with single-ion anisotropy.16 Herein, based on first-principles density-functional theory (DFT),18,19we develop an anisotropic Heisenberg XYZ model includ- ing both single-ion anisotropy and exchange anisotropy and aspin-wave theory for monolayer CrSBr and CrSeBr and find thatmonolayer CrSBr and CrSeBr have a high T cof 168 and 150 K, respec- tively. Our results show that the combination of a large spin-waveexcitation gap and exchange constants leads to high T c.W ea l s of o u n d that monolayer CrSBr possesses a remarkable electrically tunable mag-netic ordering where the magnetization easy axis can be tuned fromthe in-plane to the out-of-plane by electrostatic doping. We provide amicroscopic mechanism for the origin of high T cand electrically Appl. Phys. Lett. 117, 083102 (2020); doi: 10.1063/5.0014865 117, 083102-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplcontrollable magnetism in monolayer CrSBr, which not only offers design rules for high-temperature FM semiconductors but also allowsfor realizing spin field effect transistors (spin FETs) in monolayer 2Dmaterials. 20 Electronic structure calculations were carried out using DFT as implemented in the Vienna Ab initio Simulation Package (VASP)21,22 with the projector-augmented wave method.23Electronic band struc- tures were calculated using HSE06 and PBE exchange-correlationfunctional. 24Hubbard U corrections were included in the DFT-PBE calculations with U eff¼U-J¼3 eV to account for the correlation effect from 3 dtransition metal.25For magnetic anisotropy, spin–orbit coupling (SOC) is taken into account at the full-relativistic level. Morecalculation details are included in the supplementary material . Bulk CrXBr (X ¼S, Se) are vdW layered crystals with a Pmmn orthorhombic space group, and their monolayer has a Pmmn space group [ Figs. 1(a)–1(d) ]. The optimized structure (see Table S1) agrees with experiment. The cleavage energy for CrSBr and CrSeBr is onlyabout /C240.3 J/m 2, less than 0.465 J/m2of graphene calculated using the same vdW correlation functional (see Fig. S1). Dynamical stability isverified by phonon dispersion (see Fig. S2) with the absence of imagi-nary modes. Thus, monolayer CrSBr and CrSeBr may be easily exfoli- ated from their bulk counterpart. 13The magnetic property of monolayer CrXBr is largely dependent on the local environment ofCr 3þ, which can be viewed as a distorted octahedron with each Cr3þ surrounded by four S2/C0(Se2/C0) and two Br/C0,a si l l u s t r a t e di n Fig. 1(e) . In an ideal octahedron, crystal field splitting breaks fivefold degeneratedorbitals into double-degenerate e gand threefold degenerate t2gorbi- tals. The presence of two types of anions further reduces it to C2v, which lifts doubly degenerate egto two a1orbitals and lifts triply degenerate t2gtob1,b2,a n d a2orbitals. Because spin pairing energy U p for transition from parallel spins on two orbitals to antiparallel spinson a single orbital is greater than crystal field splitting energy Dc,t h e parallel high spin state and, hence, ferromagnetism should be favored. To confirm the above analysis, we calculate the band structure with the HSE06 hybrid functional. As shown in Figs. 2(a) ,2(b),S 3 , and S4, monolayer CrSBr and CrSeBr exhibit a highly anisotropic elec- tronic structure and a bandgap of 1.66 eV and 0.78 eV, respectively. This large anisotropy is evidenced by a small effective hole mass of only /C240.11m0and a large hole mobility of /C24720 cm2V/C01s/C01(see Table S2).26,27The HSE06 bandgap is higher than that from DFT-PBE calculations (0.2 and 0.8 eV for CrSBr and 0.1 eV for CrSeBr)15–17due to the inclusion of short-range screened non-local Fock exchange interaction. Next, we investigate ferromagnetism in monolayer CrXBr.Using a 2 /C22/C21s u p e r c e l l( F i g .S 5 ) ,w efi r s tv e r i fi e dt h a tF Mo r d e r i n g is more stable than AFM ordering. Spin density (Fig. S6) mainly comes from Cr with /C243l Bper Cr, consistent with the high spin state of Cr3þ. In contrast, S/Se atoms carry small opposite spin moment and Br atoms carry smaller opposite spin moment as listed in Table S3 and Fig. S7 due to less (two) nearest neighboring Cr atoms for Br com- pared to those for S/Se. The FM coupling between Cr atoms originates from the superexchange interaction mediated by S (Se) and Br.28–30 To illustrate this, we classify the linking geometry between Cr3þions into two types. In type I, the neighboring octahedra have two common edge atoms S(Se) and Br, where the Cr–S(Se)–Cr and Cr–Br–Cr angleis/C2490 /C14,e . g . , aalong the abdiagonal and bandcalong the aaxis [Figs. 1(b) and1(c)]. In type II, the neighboring octahedra along the b axis have one common corner S (Se) with a Cr–S(Se)–Cr angle dof /C24160/C14[Fig. 1(d) ]. According to Goodenough–Kanamori–Anderson rules,30,31FM coupling is favored for 90/C14superexchange interaction between two magnetic ions with partially filled dshells, while AFM coupling is preferred for 180/C14superexchange interaction. As dsignifi- cantly deviates from 90/C14and 180/C14, the superexchange interaction along bexhibits competing FM and AFM coupling. In contrast, FM coupling along the aaxis and abdiagonal is strongly favored due to /C2490/C14a,b,a n d c, establishing the FM ground state. Single-ion anisotropy and exchange anisotropy are two impor- tant sources that contribute to magnetocrystalline anisotropy energy (MAE), which is crucial for establishing long-range 2D ferromagnetic order. MAE values are listed in Table S5. It shows high magnetic anisotropy with the easy axis along a(i.e., xdirection), distinct from CrI38and Cr 2Ge2Te69with the out-of-plane easy axis. We then build the corresponding spin Hamiltonian with the classical HeisenbergXYZ model, 32 H¼/C0X ijhiJ1S* i/C1S* j/C0X ijhihiJ2S* i/C1S* j/C0X ijhihihiJ3S* i/C1S* j /C0X iDxSx i/C0/C12/C0X iDySy i/C0/C12/C0X ijhikx 1Sx iSxj/C0X ijhihikx 2Sx iSxj /C0X ijhihihikx 3Sx iSxj/C0X ijhiky 1Sy iSy j/C0X ijhihiky 2Sy iSy j/C0X ijhihihiky 3Sy iSy j: (1) J1;2;3are isotropic Heisenberg exchange coupling constants for the first, second, and third nearest-neighbor (NN) spins, and Siis 3/2 for Cr3þ.DxandDyrefer to single-ion anisotropy along xandy.T h e rest refer to the exchange anisotropy. We calculate these parameters bymapping magnetic configurations (Fig. S5) to the above Hamiltonian, FIG. 1. Monolayer CrXBr and their structural properties. (a)–(d) Crystal structure of vdW layered crystals CrXBr (X ¼S, Se) in their monolayer form. (e) Crystal field splitting from an ideal octahedron with O hsymmetry to a distorted octahedron with C2vsymmetry.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 083102 (2020); doi: 10.1063/5.0014865 117, 083102-2 Published under license by AIP Publishinglisted in Table S5. With this Hamiltonian, we calculate the Curie tem- perature using the Monte Carlo (MC) method implemented in theVAMPIRE code 33using a supercell of 11 250 spins. The results are shown in Figs. 2(b) and2(f)and listed in Table S5. The critical expo- nent is obtained by fitting magnetization to the Curie–Bloch equation, yielding the TCof 168 K and 150 K and bof 0.41 and 0.42 for mono- layers CrSBr and CrSeBr, respectively. Tcfrom the Ising model is 590 K and 520 K, much higher than that from the Heisenberg model as expected. The calculated Curie temperatures are significantly higher than those of 2D CrI 38(45 K) and Cr 2Ge2Te69(20 K). Our present cal- culation is based on the anisotropic Heisenberg XYZ model with up to the third nearest neighbors, thus resulting in lower Tccompared to the recent theoretical prediction with the 2D Ising model15and 2D Heisenberg model without magnetic anisotropy.17The agreement with the prediction using the 2D Heisenberg model with single-ion anisot- ropy16indicates the importance and large influence of magnetic anisotropy. To better understand the physical origin of high Tc,w ea p p l yl i n - ear spin-wave approximation with the Heisenberg XXZ model32and arrive at a second quantization representation using Holstein–Primakoff transformation,34 Hspin/C0wave¼X ie0bþ ibi/C0J1SX ijhibþ ibj/C0J2SX ijhihibþ ibj /C0J3SX ijhihihibþ ibj: (2) The parameters are included in Table S6 of the supplementary material . The resultant spin-wave excitation gap D0at the Cpointfrom Eq. (2)yields 0.07 meV for CrSBr and 0.31 meV for CrSeBr. The 2D magnon dispersion is shown in Figs. 2(a) and2(d) for CrSBr and CrSeBr, respectively. We then estimate the Curie temperature Tcbased on renormalized spin-wave theory (RSWT),32 MTðÞ¼S/C01 NsNkX nX kexpMTðÞEnkx;ky/C0/C1 SkBT ! /C01"#/C01 ;(3) where n,Ns,a n d Nkrefer to the band index, number of spins per unit cell, and number of kpoints sampled in the first Brillouin zone. For the RSWT calculations, a dense k-point sampling of 200 /C2200/C21 was used for the BZ integration. The calculated TCvalues from RSWT are 150 K and 152 K for monolayer CrSBr and CrSeBr, respectively, in reasonable agreement with TCfrom MC simulations. In RSWT, the bosonic operator is kept up to the fourth order followed by a mean field approximation, which essentially scales/renormalizes the hoppingenergy and spin-wave gap. As shown in Figs. 2(d) and 2(h),T cis strongly dependent on spin-wave excitation gap D0;which is further determined by exchange and single-ion anisotropy. The underlying physical mechanism of FM order at finite temperature, thus, originates from magnetic anisotropy.35 High Tc2D ferromagnetism in monolayer CrSBr and CrSeBr opens up exciting opportunities. For example, externally controlled magnetism is highly desirable for magnetoelectrics. Recent works have shown that carrier doping can be introduced into monolayers by elec- tric gating to control magnetism.36–38In addition, the vdW gap in 2D magnetic layers induces giant tunneling magnetoresistance.39,40Here, we show that carrier doping can drastically change MAE in 2D CrSBr and switch the magnetization easy axis. As shown in Fig. 3(a) ,u n d e ra FIG. 2. Electronic structure and ferromagnetism in monolayer CrSBr and CrSeBr. (a) and (e) Electronic band structure with the HSE06 functional. Red (blue) i ndicates spin majority (minority). (b) and (f) Temperature dependent normalized magnetization using different theoretical models, including the Ising model, a nisotropic Heisenberg model, and RSWT. (c) and (g) Magnon dispersion with spin-wave excitation gap D0located at C. (d) and (h) Magnetic moment as a function of temperature and spin-wave excitation gap. Two dots indicate the corresponding Tcfor monolayer CrSBr and CrSeBr.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 083102 (2020); doi: 10.1063/5.0014865 117, 083102-3 Published under license by AIP Publishinghole doping of n>4/C21012/cm2, the out-of-plane magnetization becomes favored. In contrast, electron doping, though showing a simi-lar trend, has a relatively weak influence on the magnetization crystal-line anisotropy. Experimentally, it is possible to achieve a carrierconcentration of up to 10 13–1014/cm2in 2D materials; therefore, car- rier doping could be an effective strategy to control magnetization ordering in monolayer CrSBr. The doping-induced tunability of MAE can be understood from a perturbation theory analysis.41Given a pair of valence ( v)e i g e n s t a t e wvand conduction ( c)e i g e n s t a t e wc, their contribution to MAE is given by DEvc¼1 DvcHsoc vcx*ðÞ/C12/C12/C12/C12/C12/C122 /C0Hsoc vcz*ðÞ/C12/C12/C12/C12/C12/C122/C18/C19 ; (4) where Hsoc vcn*ðÞ¼hwvjHsocðn*Þjwciis the SOC matrix element and Dvc ¼ev/C0ec.Hsocn*ðÞ¼nr*/C1L*,w h e r e r*¼rx;ry;rz ðÞ are the 2 /C22 Pauli matrices, L*is orbital angular momentum operator, and nis the SOC strength. The spin and magnetic quantum number of orbital characters determine the sign of DEvc[Fig. 3(b) ], while the sum of DEvc in Eq. (4)over all valence-conduction pairs determine the MAE and easy axis. Cr contributes to both in-plane and out-of-plane magnetiza-tion, while S and Br favor the in-plane easy axis. Due to the higher atomic number of Br, its SOC strength is much larger ( /C243t i m e so fC r and 30 times of S); hence, Br has a stronger influence on MAE than Crand S. Consequently, the in-plane easy axis is preferred. Upon electrondoping, Cr- d x2/C0y2;"in the lowest conduction band becomes occupied ( F i g .S 8 ) ;t h u s , DEvcbetween conduction Cr- dx2/C0y2;"and valence Cr-dyz;"favors in-plane magnetization. Increasing occupation in Cr-dx2/C0y2;"with electron doping reduces the stability of in-planemagnetization. Upon hole doping, S- py;"and Br- py;"in the highest valence band become unoccupied, thereby reducing the stability of in- plane magnetization since DEvcbetween valence Br- py;"and conduc- tion Br- pz;"favors in-plane magnetization. Due to Br’s stronger SOC, hole doping has a larger impact on MAE than electron doping, reflected in the stiffer slope upon hole doping [see Fig. 3(a) ]. Hence, the easy axis can be more easily tuned by carrier doping at a critical hole concentration of 4 /C21012/cm2. The doping-modulated easy axis allows for realizing 2D spin- FETs.20A schematic of such a magnetoelectric device is proposed in Fig. 3(c) , where monolayer CrSBr is double-gated for carrier doping with two dielectric layers (e.g., hexagonal BN) to prevent direct tunnel- ing. Gate voltage controls the carrier concentration and changes easy axis upon hole doping, while longitudinal in-plane source-drain volt- age drives spin-dependent transport. Upon critical hole doping, the easy axis switches from the in-plane to out-of-plane. Thus, an in-plane FM/out-of-plane FM interface emerges between hole-doped and undoped regions. Strong scattering will take place at this hetero- magnetic interface with high resistance, while homo-magnetization below critical doping corresponds to a low-resistance state, thereby realizing an electrically controlled giant magnetoresistance effect by dynamically electrostatic doping. In summary, we presented a theoretical study of 2D ferromag- netic CrSBr and CrSeBr with the high Tcof 168 K and 150 K and sizable bandgap of 1.66 and 0.78 eV, respectively. Remarkably, the magnetization easy axis can be tuned from the in-plane to the out-of- plane by electrostatic doping. Monolayer CrSBr and CrSeBr ferromag- nets offer long-desired alternatives to dilute magnetic semiconductors and provide unprecedented opportunities for 2D spintronics such as spin valves and spin FETs. See the supplementary material for details on the calculation methods, crystal and electronic structure, Heisenberg XYZ model of monolayer CrSBr and CrSeBr. During the submission of this paper, we notice a recent experi- ment manuscript by Telford et al., which shows AFM-coupled CrSBr layers with individual layers being FM-ordered.42Our calculated HSE06 electronic gap of 1.66 eV agrees well with the measured gap of 1.5 eV60.2 eV. The measured N /C19eel temperature reaches 132 K, sug- gesting that the Tcof monolayer CrSBr very likely goes beyond 132 K. AUTHORS’ CONTRIBUTIONS H.W. and J.Q. contributed equally to this work. X.Q. and H.W. gratefully acknowledge the support from the National Science Foundation (NSF) under Grant No. DMR- 1753054 and Texas A&M University President’s Excellence Fund X-Grants Program, respectively, and acknowledge the advanced computing resources provided by Texas A&M High Performance Research Computing. J.Q. acknowledges the financial support from the National Natural Science Foundation of China (Project Nos. 11674132 and 11974148). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. FIG. 3. Electrical control of 2D ferromagnetism in monolayer CrSBr. (a) Energy of FM configurations in different magnetization directions as a function of carrier concentration n. (b) Contribution of each valence and conduction state pair to the in-plane/out-of-pane magnetization determined by the spin and magnetic quantumnumber of orbital characters in valence and conduction states. (c) A schematic of the 2D magnetoelectric device that can realize the giant magnetoresistance effect controlled by electrostatic doping.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 083102 (2020); doi: 10.1063/5.0014865 117, 083102-4 Published under license by AIP PublishingREFERENCES 1T. Dietl, Nat. Mater. 9, 965 (2010). 2I./C20Zutic´, J. Fabian, and S. Das Sarma, Rev. Mod. 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Dean, preprint arXiv:2005.06110 (2020).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 083102 (2020); doi: 10.1063/5.0014865 117, 083102-5 Published under license by AIP Publishing

5.0022702.pdf

J. Chem. Phys. 153, 124115 (2020); https://doi.org/10.1063/5.0022702 153, 124115 © 2020 Author(s).Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems Cite as: J. Chem. Phys. 153, 124115 (2020); https://doi.org/10.1063/5.0022702 Submitted: 22 July 2020 . Accepted: 10 September 2020 . Published Online: 29 September 2020 Gaurav Harsha , Thomas M. Henderson , and Gustavo E. Scuseria ARTICLES YOU MAY BE INTERESTED IN 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 OrbNet: Deep learning for quantum chemistry using symmetry-adapted atomic-orbital features The Journal of Chemical Physics 153, 124111 (2020); https://doi.org/10.1063/5.0021955 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 Wave function methods for canonical ensemble thermal averages in correlated many-fermion systems Cite as: J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 Submitted: 22 July 2020 •Accepted: 10 September 2020 • Published Online: 29 September 2020 Gaurav Harsha,1,a) Thomas M. Henderson,1,2 and Gustavo E. Scuseria1,2 AFFILIATIONS 1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA 2Department of Chemistry, Rice University, Houston, Texas 77005, USA a)Author to whom correspondence should be addressed: gauravharsha05@gmail.com ABSTRACT We present a wave function representation for the canonical ensemble thermal density matrix by projecting the thermofield double state against the desired number of particles. The resulting canonical thermal state obeys an imaginary-time evolution equation. Starting with the mean-field approximation, where the canonical thermal state becomes an antisymmetrized geminal power (AGP) wave function, we explore two different schemes to add correlation: by number-projecting a correlated grand-canonical thermal state and by adding correlation to the number-projected mean-field state. As benchmark examples, we use number-projected configuration interaction and an AGP-based perturbation theory to study the hydrogen molecule in a minimal basis and the six-site Hubbard model. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022702 .,s I. INTRODUCTION Thermal properties of many-body systems can be computed either in the canonical ensemble or in the grand-canonical ensem- ble. The choice of ensemble makes no practical difference in the final result in large systems. It does so, however, for a finite sys- tem. This is because the relative fluctuation in particle number in the grand-canonical ensemble scales as the inverse square root of particle number itself, i.e., √ ⟨N2⟩gc−⟨N⟩2gc ⟨N⟩gc∼1√ ⟨N⟩gc, (1) and vanishes in the limit ⟨N⟩gc→∞, where⟨⋯⟩gcdenotes the grand- canonical thermal expectation value. A wide range of methods are available to study the thermal properties of quantum systems within the grand-canonical ensem- ble, e.g., thermal Hartree–Fock,1,2perturbation theories,3–5path integral and Green’s function methods,6finite-temperature Quan- tum Monte Carlo (QMC),7–16density matrix renormalization group and density functional theory based methods,17–22as well as themore recently explored thermal equivalents of configuration inter- action and coupled cluster,23–34and algorithms for quantum com- puters.35–38 In contrast, canonical ensemble techniques are scarce and even fewer are suitable for efficient application to correlated electronic systems. One way to enforce a fixed number of particles is by intro- ducing a second Lagrange multiplier μ2for the fluctuation, in much the same spirit as the chemical potential μ1acts as a Lagrange mul- tiplier to fix the number of particles. That is, one can either define a generalization of the density operator as ρ=exp[−β(H−μ1(N−N0)−μ2(N2−N2 0))], (2) where the parameters μ1andμ2enforce the constraints, ⟨N⟩=N0and⟨N2⟩=N2 0, (3) or introduce corrections to the grand-canonical ensemble averages by subtracting contributions from wrong number sectors in the Hilbert space.39While this provides the convenience of using several available grand-canonical methods, such simultaneous optimization J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 153, 124115-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp problems can be numerically tedious as the optimized values of μ2 are generally very large and ideally infinite, something which has also been observed in spin-projection.40On the other hand, we can evaluate the ensemble averages in the appropriate number sector to begin with, e.g., in the minimally entangled typical thermal states algorithm,19,41canonical ensemble perturbation theory (PT),42and projection based techniques.43–46 For a wide variety of problems that involve isolated finite systems with a fixed number of particles, the canonical ensemble is more appropriate. Examples of such systems include molecules in a warm gaseous phase (of interest in geochemistry),47ultra- cold chemical systems,48,49quantum wires with number conserv- ing Majorana modes,50–52and superconductivity in small grain systems.53Besides, the canonical ensemble provides a potential computational advantage over grand canonical alternatives since it eliminates the need for finding the appropriate chemical poten- tial. Evidently, a robust and convenient framework to study the canonical-ensemble finite-temperature properties of finite many- body fermionic systems is desirable. In this manuscript, we leverage the thermofield dynamics54–57 to construct a number-projected thermal wave function, called the canonical thermal state, which provides an exact wave function rep- resentation of the canonical ensemble density matrix. It obeys an imaginary-time Schrödinger equation, which can be solved at vari- ous levels of approximation, and at the level of mean-field, it reduces to a number-projected Bardeen-Cooper-Schrieffer wave function, also known as the antisymmetrized geminal power (AGP) state.58 A similar number-projected BCS theory for the canonical thermal state was also proposed by the authors of Refs. 43–45. Mean-field description, however, misses out on a lot of important physics. Here, we provide a recipe to generalize correlated ground-state theories (e.g., perturbation theory, configuration interaction, and coupled cluster) to finite-temperature. Moreover, the identification of the mean-field state as an AGP allows us to exploit the newly developed tools for the efficient evaluation of the thermal expectation values via AGP density matrices.59We restrict our discussion to electronic systems, but generalization to other fermionic and bosonic systems is straightforward. II. THERMOFIELD DYNAMICS Thermofield dynamics is conventionally formulated for the grand-canonical ensemble, where it constructs a wave function rep- resentation of the thermal density operator by introducing a con- jugate copy of the original system such that the ensemble thermal averages can be expressed as an expectation value over the thermal state, ⟨O⟩=Tr(e−β(H−μN)O)=⟨Ψ(β)∣O∣Ψ(β)⟩ ⟨Ψ(β)∣Ψ(β)⟩, (4) where the thermal state | Ψ(β)⟩is given by ∣Ψ(β)⟩=e−β(H−μN)/2∣I⟩, (5a) ∣Ψ(0)⟩=∣I⟩=∏ p(1 +c† p˜c† p)∣−;−⟩. (5b)Hereβ,μ,H, and Nare the inverse temperature, chemical potential, the Hamiltonian, and the number operator, respectively. The iden- tity state ∣I⟩is the exact infinite-temperature thermal state and is an extreme BCS state with Cooper pairs formed by pairing physical par- ticles with the corresponding conjugate particles. The norm of the state gives the partition function. The product in Eq. (5b) runs over all spin-orbitals p, and |−;−⟩denotes the vacuum state for both the physical and conjugate systems. By its definition, the thermal state obeys imaginary-time evolution equations, one each for βandμ, ∂ ∂β∣Ψ(β)⟩=−1 2H∣Ψ(β)⟩, (6a) ∂ ∂μ∣Ψ(β)⟩=β 2N∣Ψ(β)⟩, (6b) where we have assumed that [ H,N] = 0, as ab initio electronic systems are number-conserving. Like the ground state, finding | Ψ(β)⟩exactly is possible only for very small systems with a few electrons, and suitable approximations are generally required. The simplest approximation is the mean-field approach, where His replaced with a one-body mean-field Hamilto- nian H0. In the basis where H0=∑pϵpc† pcp, the resulting mean-field thermal-state is a BCS state of the form ∣0(β,μ)⟩=e−β(H0−μN)/2∣I⟩, =∏ p(1 +e−β(ϵp−μ)/2c† p˜c† p)∣−;−⟩. (7) Higher order approximations are generally formulated with the mean-field state as the reference, ∣Ψ(β)⟩≃Ω(β,μ)∣0(β,μ)⟩, (8) which resembles the interaction picture approach. We exploited this theory in Refs. 30 and 31 to formulate finite-temperature versions of configuration interaction and coupled cluster theory. We rec- ommend these articles and references therein for further details of thermofield theory. III. CANONICAL ENSEMBLE THEORY The canonical ensemble thermal state can be constructed by projecting the grand-canonical state against the desired particle number N0, ∣Ψ(β)⟩c=PN0∣Ψ(β)⟩gc, (9) wherePN0projects | Ψ(β)⟩gconto the Fock-space with N0electrons. The particle-conserving property of Himplies that [H,PN0]=0, and the resulting canonical thermal state obeys an imaginary-time evolution equation analogous to its grand-canonical counterpart, d dβ∣Ψ(β)⟩c=−1 2H∣Ψ(β)⟩c. (10) Like the grand-canonical theory, a series of approximations can be introduced from a simple mean-field theory to higher order theories that add correlation effects on it. J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 153, 124115-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp A. Mean-field formalism The imaginary-time evolution equation can be integrated within the mean-field approximation, H≈H0. As for the grand- canonical theory, using an H0that carries no implicit tempera- ture dependence and working in a basis where it is diagonal, the mean-field state becomes ∣Ψ0(β)⟩c=PN0∣0(β,μ=0)⟩ (11a) =PN0∏ p(1 +e−βϵp/2c† p˜c† p)∣−;−⟩ (11b) =PN0∏ p(1 +ηpP† p)∣−;−⟩ (11c) =1 N0!(Γ† β)N0∣−;−⟩=∣ΨAGP(β)⟩, (11d) whereηp=e−βϵp/2, and we have identified P† p=c† p˜c† pas the pair– creation operator. As already noted, the un-projected product state in Eq. (11b) is a BCS state and its number-projected version is well known as AGP, with the geminal creation operator Γ† βdefined as Γ† β=∑ pηpP† p. (12) Identification of the mean-field state as an AGP is interesting and, with recent developments on the efficient evaluation of overlaps and expectation values, as well as geminal based correlated wave function theories,59–63provides a good starting point to include correlation effects. An improved mean-field description can also be obtained by optimizing both the energy levels ϵand the one-electron basis to find anH0that minimizes the Helmholtz free energy, in much the same way as Mermin’s thermal Hartree–Fock theory in Ref. 1, as discussed in Refs. 43–45. B. Correlated thermal state A plethora of approximate wave function methods are avail- able to study the ground-state properties of correlated electronic systems. As we have shown in Refs. 30 and 31, the thermofield formalism allows for a direct generalization of these methods to finite-temperature. Since physical electronic systems conserve the number of particles, i.e., [H,PN0]=0, we face two options while constructing a correlated approximation to the canonical thermal state: projection after correlation (PAC) and correlation after pro- jection (CAP). In PAC, we first construct an approximate grand- canonical thermal state by adding correlation on a broken-symmetry mean-field reference (thermal BCS in our case) and then perform the number-projection, ∣Ψ⟩≃PN0Ω(β)∣0(β)⟩,∣0(β)⟩=e−βH0/2∣I⟩. (13) The correlation operator Ω is built out of number non-conserving BCS quasiparticles,64,65and the un-projected part of the thermal state, Ω(β)|0(β)⟩, looks like a standard single-reference CI wave function, which simplifies the process of correlating the reference.In order to carry out the projection efficiently, we use an integral form for the projection operator,66–68i.e., PN0=1 2π∫2π 0dϕeiϕ(N0−N). (14) Computing matrix elements and overlaps in the presence of P involves the use of transition density matrices and can be compli- cated (see, e.g., Refs. 69–73). For CAP, we use the thermal AGP state in Eq. (11) as the reference and add correlation using a number- conserving wave operator, ∣Ψ⟩≃Λ(β)∣ΨAGP(β)⟩=Λ(β)PN0∣0(β)⟩. (15) Contrasting with CAP, the projection problem here is trivial but adding correlation becomes complicated. Both of these techniques have been explored extensively for ground-state methods.60–63,70,72–76Here, we discuss an example for each: a finite-temperature generalization of the number-projected CI, along the lines discussed by Tsuchimochi and Ten-no,70and an imaginary-time perturbation theory based on the thermal AGP as the reference, as explored in Refs. 60–62. 1. Projection after correlation The number-projected thermal CI state is parameterized as ∣Ψ(β)⟩=PN0et0(1 +T)∣0(β)⟩, (16) where |0(β)⟩is the thermal BCS state at inverse temperature β,t0 keeps track of the norm of the state (related to the grand potential), andTcreates quasiparticle excitations on the BCS, T=∑ pqtpqa† p˜a† q+1 4∑ pqrstpqrsa† pa† q˜a† s˜a† r+⋯. (17) The CI amplitudes can be determined in two different ways. One can compute them in the grand-canonical ensemble, as we have done in Ref. 30, and then perform a one-shot projection. This approach is generally known as projection after variation (PAV). Alternatively, the amplitudes can be computed in the presence of the projection operator by solving the imaginary-time evolution equation, referred to as variation after projection (VAP). VAP allows for more varia- tional freedom and thus performs better than PAV. Accordingly, we focus our attention on VAP hereafter. Substituting this CI ansatz into Eq. (10) and evaluating the overlaps of the resulting equation against the ground and excited BCS states, we get ∫2π 0dϕ⟨0(β)∣νeiϕ(N0−N)((1 +T)dt0 dβ+dT dβ)∣0(β)⟩ =∫2π 0dϕ⟨0(β)∣νeiϕ(N0−N)¯H∣0(β)⟩, (18) where ¯His the effective Hamiltonian ¯H=−1 2(H(1 +T)−(1 +T)H0) (19) andνtakes the values from {1,˜aqap,˜ar˜asaqap,. . .}to construct ground and excited BCS states for the bra. Both the amplitudes J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 153, 124115-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and the quasiparticle operators are functions of temperature; there- fore, theβ-derivative can be broken down into the derivative of the amplitudes and that of the operator parts, dT dβ=dampT dβ+dopT dβ. (20) We can rewrite Eq. (18) as a system of first-order ordinary dif- ferential equations (ODEs) that govern the evolution of the CI- amplitudes, ∑ μAνμ⋅∂tμ ∂β=Bν, (21) where Ais the overlap matrix, Aνμ=∫2π 0dϕ⟨ν(β)∣e−iϕ(N−N0)Lμ∣0(β)⟩, (22a) with Lμ={1 +T,μ=1 μ,μ∈{a† p˜a† q,a† pa† q˜a† s˜a† r}.(22b) The right-hand side vector Bνis given by Bν=∫2π 0dϕ⟨ν(β)∣e−iϕ(N−N0)R∣0(β)⟩, (23a) R=¯H−dopT ∂β. (23b) Here, we have used ν,μas a composite notation for the ground and excited quasiparticle states. Equation (21) can be integrated starting fromβ= 0, where T= 0 is the exact initial condition. 2. Correlation after projection For the correlation after projection method, a numerical inte- gration to perform the projection is not required as it uses a strictly number conserving state, the thermal AGP, as the reference. As an example for this approach, we consider the perturbation theory (PT), where we partition the Hamiltonian as H=H0+λV, where H0is the mean-field contribution and Vacts as a perturbation. The canonical thermal state can be expanded as a series in λ, ∣Ψ(β)⟩=∣Ψ0⟩+λ∣Ψ1⟩+λ2∣Ψ2⟩+⋯ (24a) =e−βH0/2(∣ϕ0⟩+λ∣ϕ1⟩+λ2∣ϕ3⟩+⋯). (24b) Substituting this form for | Ψ⟩in Eq. (10) and collecting terms at var- ious orders in λgive∂|ϕ0⟩/∂τ= 0, or equivalently | Ψ0⟩= |ΨAGP(β)⟩ for terms at O(λ0), and ∂ ∂τ∣ϕn⟩=−1 2eτH0/2Ve−τH0/2∣ϕn−1⟩ (25) forO(λn),n≥1. Integrating Eq. (25) yields perturbative cor- rections identical to those in a time-dependent interaction picture theory. We work in a basis where H0is diagonal. This allows us to integrate the equations analytically. Detailed notes on both theprojected CI and the AGP-based perturbation theory are available in the supplementary material. IV. IMPLEMENTATION DETAILS We use the ground-state Hartree–Fock eigenvalues to build H0, which, in turn, is used to define the mean-field reference state (thermal BCS for the projected CI and thermal AGP for the pertur- bation theory). We have used PySCF77to generate the Hartree–Fock eigenvalues and Hamiltonian matrix elements. One can also choose anH0that optimizes the free energy at any given β. While this may lead to a better thermal reference state, it makes the under- lying equations very complicated, and therefore, in this work, we use a fixed H0. This is also analogous to typical interaction picture theories. Nevertheless, to gauge the relevance of optimization, in Fig. 1 of the supplementary material, we compare the performance of ther- mal AGP with optimized and unoptimized η’s for various bench- mark systems which we study below. We note that for larger sys- tems, the optimization of η’s does not introduce any significant improvement, therefore justifying the use of an unoptimized H0. Both the projected CI [Eq. (21)] and perturbation theory [Eq. (25)] equations are integrated starting from β= 0, where the mean-field is exact and the correct initial condition is known. The cost for computing these equations is similar to standard projected quasiparticle or AGP-based CI, i.e., O(N6). While the PT2 cor- rections can be obtained by a straightforward integration of the underlying quantities along the imaginary-time axis, the projected CI amplitudes satisfy a set of linear ordinary differential equations (ODE). The exact solution of these ODE’s requires the inversion of the overlap matrix A, which is computationally expensive. More- over, Amay also have zero or near-zero eigen-modes. To avoid these issues, at each β-grid point, we solve for the derivative vector itera- tively using MinresQLP,78,79a robust algorithm for singular linear systems, and then use a fourth order Runge–Kutta method to per- form the integration. This adds an additional cost to the projected CI theory. In all the data presented below, we use a step size of Δβ = 0.005 or smaller to integrate the ODE in projected CI, which is sufficiently small to guarantee the convergence for the Runge–Kutta method (see Sec. V in the supplementary material). We also observe that the partial traces of higher rank terms in the CI operator are proportional to the lower-rank terms, e.g., CI with single and double excitations is equivalent to CI with just the double excitations. To avoid linear dependencies in the over- lap matrix, we keep only the highest rank terms in our truncated CI theory. The number projection in the projected CI equations is carried out numerically and converges rapidly as the number of grid points becomes greater than the number of spin-orbitals. V. RESULTS We apply the projected CI with double excitations (CID) as well as the second order perturbation theory (PT2) to small molecular and model systems to highlight the performance of these finite- temperature canonical ensemble methods against the exact bench- mark results. Figure 1 shows the error in the canonical-ensemble J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 153, 124115-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Error in the canonical-ensemble internal energy for (left) the hydrogen molecule in STO-3G basis with a bond length of 0.74 Å and (right) the half-filled six-site Hubbard model at U/t= 2, 6, computed using the projected BCS wavefunction (mean-field), AGP-based perturbation theory, and projected CI truncated at doubles. internal energy for the hydrogen molecule in the minimal STO-3G basis and at a bond length of 0.74 Å (left) and the six-site Hub- bard model with U/t= 2, 6 (right). The results compare the per- formance of projected thermal BCS or AGP (which is indicated in the plot by “mean-field”), PAV and VAP projected thermal CISD and CID, respectively, and AGP-based PT2. We use the ground- state spin-restricted Fock operator as our unoptimized H0for the hydrogen molecule and the Hubbard model with U/t= 2, and the spin-unrestricted Fock operator for U/t= 6. It is apparent that the mean-field approach misses out a lot of correlation, a part of which is recovered by CID and PT2. In fact, the VAP CID, like its ground- state analog and unlike the grand-canonical CISD in Ref. 30, is exact for a two-electron system like the hydrogen molecule and expect- edly outperforms the PAV approach. The second order perturbation theory, though not exact for the two-electron case, also improves upon the mean-field results. All the CI and PT results approach their appropriate ground-state counterparts in the zero temperature limit, i.e., the number-projected CID approaches ground-state CISD, and the AGP-based perturbation theory approaches ground- state perturbation theory as β→∞. In particular, we note that theAGP-based PT2 performs better than projected CI for U/t= 2 but does not introduce any significant improvement over the mean-field forU/t= 6. This, in fact, is analogous to the ground-state perfor- mance of these theories (see Fig. 2 and Table 1 in the supplementary material for the ground-state results). To highlight the merits of the projected CI theory over mean- field, as well as the distinction between canonical and grand- canonical ensemble properties, we plot the total internal energy (left panel) and the specific heat (right panel) for the six-site Hubbard model with U/t= 6 at half-filling in Fig. 2. We compare the mean- field theory, CISD for grand-canonical, and CID for the VAP pro- jected CI against the exact numerical results. We remind the reader that the grand-canonical mean-field state is a thermal BCS, which, upon number-projection, gives the canonical thermal state. We use the spin-unrestricted Fock operator to construct H0. We notice a striking difference in the behavior of the specific heat in the two ensembles. The two different peaks in the exact specific heat curves (shown in solid blue and red lines), which correspond to the spin and charge excitation energy scales, are more pronounced and distinct in the canonical ensemble. While the mean-field theory completely fails FIG. 2 . Comparison of total internal energies and specific heats for the half-filled six-site Hubbard model with U/t= 6 as a function of temperature. The mean-field, CI, and exact results highlight the difference between the grand-canonical (blue) and canonical (red) ensemble properties. J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 153, 124115-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Comparison of total internal energies and specific heats for the six-site Hubbard model with U/t= 2 and four electrons as a function of temperature. The mean-field, CI, and exact results highlight the difference between the grand-canonical (blue) and canonical (red) ensemble properties. to account for the spin-excitation peak, the projected CID performs better both qualitatively and quantitatively. We repeat this exercise for the hole-doped six-site Hubbard model with U/t= 2 and four electrons to further demonstrate the dif- ference between the two ensembles. We use the spin-restricted Fock operator to construct H0. The results are plotted in Fig. 3. Note that unlike the half-filled case, this hole-doped Hubbard model shows appreciably different results in the canonical and grand-canonical ensembles. This is because the half-filled Hubbard model corre- sponds to the lowest energy state in Fock space, and excitations to sectors with different particle numbers are high in energy and are effectively frozen out in the low-temperature limit so that the grand canonical ensemble becomes effectively canonical. This is not the case for the doped Hubbard model. Finally, we note that the low-temperature specific heat results in Fig. 3, for both the grand-canonical and the canonical CI, are noisy. We attribute this noise to two different sources: 1. The evolution of the CI amplitudes is carried out with respect to the inverse temperature β, and we compute the specific heat as Cv=−β2dE dβ. (26) Any error in the integration due to the finite step size would be amplified by a factor of β2. This explains the noise present in both the grand-canonical and the canonical CI. 2. Recall that for the projected CI, we solve a generalized linear equation [see Eq. (21)]. As we approach low temperatures (or largeβ), the number of near-zero modes in the overlap matrix Abecomes large, which leads to inconsistencies in the solution, further adding to the noise. VI. CONCLUSION We have presented a theory to generalize correlated ground- state wave function theories, namely, Hartree–Fock, perturbation theory, and CI, to study canonical ensemble thermal properties in fermionic many-body systems. In the low-temperature regime, where the canonical ensemble is most applicable, these methodsperform as well as their ground-state counterparts for the bench- mark problems studied. The ability to build both canonical and grand-canonical methods also signifies the robustness of ther- mofield theory for finite-temperature wave function methods. At zero temperature, one is generally required to go to much higher orders in CI or PT to obtain highly accurate results, and bet- ter alternatives, such as the coupled cluster theory and multi- reference methods, are generally preferred. While a number- projected formulation of the coupled cluster theory for the ground- state has been worked out in Ref. 73, the underlying equa- tions are complicated for a direct generalization to finite tem- peratures. Our work is a first step toward achieving finite- temperature analogs of such sophisticated techniques. It also estab- lishes a firm standing ground to build number-conserving finite- temperature Monte Carlo methods, something that has been rela- tively less explored in the QMC community. Most of the available thermal methods use an imaginary-time evolution starting from β= 0 or T=∞, while one is generally interested in low and inter- mediate temperature scales. A theory that uses ground-state or T= 0 as the starting point would not only be more practical but also allow us to systematically eliminate the inconsistencies in the projected CI evolution due to the near-zero modes in the overlap matrix. SUPPLEMENTARY MATERIAL Detailed equations for the projected-CID and AGP-based PT2, along with their derivations, are presented in the supplementary material. We also provide additional data comparing the optimized and the unoptimized thermal mean-field, ground-state limits of the thermal methods, and convergence of the Runge–Kutta method with respect to the step-size in the evolution of the projected-CI equations. ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Computational and Theoretical Chemistry Program under Award No. DE-FG02-09ER16053. G.E.S. acknowledges support as a Welch Foundation Chair (Grant No. C-0036). J. Chem. Phys. 153, 124115 (2020); doi: 10.1063/5.0022702 153, 124115-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1N. D. Mermin, Ann. Phys. 21, 99 (1963). 2J. Sokoloff, Ann. Phys. 45, 186 (1967). 3T. Matsubara, Prog. Theor. Phys. 14, 351 (1955). 4R. 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5.0009223.pdf

J. Chem. Phys. 152, 194303 (2020); https://doi.org/10.1063/5.0009223 152, 194303 © 2020 Author(s).Identification of the Jahn–Teller active trichlorosiloxy (SiCl3O) free radical in the gas phase Cite as: J. Chem. Phys. 152, 194303 (2020); https://doi.org/10.1063/5.0009223 Submitted: 29 March 2020 . Accepted: 27 April 2020 . Published Online: 19 May 2020 Tony C. Smith , and Dennis J. Clouthier ARTICLES YOU MAY BE INTERESTED IN Mode-specific vibrational predissociation dynamics of (HCl) 2 via the free and bound HCl stretch overtones The Journal of Chemical Physics 152, 194301 (2020); https://doi.org/10.1063/5.0003652 Probing the ionization potentials of the formaldehyde dimer The Journal of Chemical Physics 152, 194305 (2020); https://doi.org/10.1063/5.0009658 Infrared photodissociation spectroscopy and anharmonic vibrational study of the HO 4+ molecular ion The Journal of Chemical Physics 152, 174309 (2020); https://doi.org/10.1063/5.0005975The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Identification of the Jahn–Teller active trichlorosiloxy (SiCl 3O) free radical in the gas phase Cite as: J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 Submitted: 29 March 2020 •Accepted: 27 April 2020 • Published Online: 19 May 2020 Tony C. Smith and Dennis J. Clouthiera) AFFILIATIONS Ideal Vacuum Products, LLC, 5910 Midway Park Blvd. NE, Albuquerque, New Mexico 87109, USA a)Author to whom correspondence should be addressed: djc@idealvac.com ABSTRACT The ˜A2A1–˜X2Eelectronic transition of the jet-cooled trichlorosiloxy (SiCl 3O) free radical has been observed for the first time in the 650– 590 nm region by laser induced fluorescence (LIF) detection. The radical was produced by a pulsed electric discharge through a mixture of silicon tetrachloride and oxygen in high pressure argon at the exit of a pulsed molecular beam valve. The LIF spectrum shows low frequency intervals, which we assign as activity in the normally forbidden degenerate v′ 5and v′ 6modes, indicative of a significant Jahn–Teller effect in the ground state. Single vibronic level emission spectra show level dependent spin–orbit splittings in the ground state and Jahn–Teller predictable variations depending on which upper state level is pumped. The measured lower state energy levels have been fitted to a Jahn– Teller model that simultaneously includes spin–orbit coupling and linear and quadratic multimode coupling. In SiCl 3O, the Jahn–Teller interaction predominates over spin–orbit effects. Published under license by AIP Publishing. https://doi.org/10.1063/5.0009223 .,s I. INTRODUCTION Silanes (SiX 4) and siloxanes (X 3Si–O–SiX 3) are important pre- cursor compounds in the industrial production of silicon thin films for semiconductors and high purity silica for the optical fibers used in telecommunications. In the process of precursor degradation, either by high temperature reactions or in plasmas, it has been shown theoretically1,2that a large variety of reactive intermediates are produced although few of these have been identified chemi- cally or spectroscopically. In particular, one class of such species, the siloxy free radicals (SiX 3O), is unknown, despite its direct relevance to several industrial processes. There are many motivations for spectroscopic studies of silicon-based reactive systems, due to the pivotal role they play in semiconductor growth processes. Many such species have been pos- tulated, but few have actually been identified or characterized. For example, Si xHyOzmolecules are thought to be intermediates in the oxidation of SiH 4and play an important role in the formation of Si, SiO, and SiO 2thin films and nanoparticles.2As pointed out by Hitchman and Jensen3in their review of chemical vapor deposition(CVD) processes, “despite the increased use of diagnostic proce- dures for monitoring CVD reactions, still relatively little is known about the nature of gas phase intermediates in most deposition pro- cesses, and even less is known about the kinetic parameters for the formation and consumption of those intermediates.” It is, there- fore, important to establish sensitive methods for detecting and characterizing reactive species to aid in experimental and theoreti- cal programs aimed at optimizing semiconductor growth processes. Such methods are most likely to be spectroscopic in nature, and future studies of the transient intermediates involved in semicon- ductor growth processes will rely on a well-established database of molecular constants, transition frequencies, transition intensities, and photophysical parameters. Of particular interest in the present work is the high tem- perature reaction of SiCl 4with O 2, an industrially important pro- cess that leads to the production of silicon dioxide aerosols and high-purity silica for the optical fibers used in telecommunications. The reaction appears to be exceedingly complex, especially in the 800–1000○C range, generating a multitude of chlorosiloxanes with the general formula Si xOyClz, which have been identified by mass J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp spectrometry4and limited matrix isolation studies.5,6Recent laser- induced fluorescence studies of the products of electric discharges in SiCl 4+ O 2mixtures in our laboratory sparked our interest in the chlorosiloxy radicals (SiCl 3O) as possible intermediates in sil- icon tetrachloride oxidation processes. The oxidation of silane is also of considerable interest, both as a fundamental model system, which is still poorly understood,2and as a method for the production of SiO 2thin films. Low-pressure chemical vapor deposition using SiH 4/O2/N2gas mixtures has been studied, and the film growth is thought to be originated by an unknown SiO xHyintermediate in the gas phase.2Whether the SiH 3O radicals participate in such reac- tions is unknown. In a similar fashion, fluorine-containing gases, often with added oxygen, are used in the plasma etching of Si, SiO 2, and SiN materials, and SiF 3O radicals may be formed under such conditions.7 In sharp contrast to the wealth of information in the litera- ture about Jahn–Teller active methoxy radicals,8virtually nothing is known about the corresponding small siloxy (SiX 3O) radicals. Neutralization reionization mass spectrometry experiments9have shown that SiH 3O can be made from the corresponding cation and that it is stable during the μs time scale of the experiment. The SiF 3O radical has been detected in electron spin resonance experiments,10 and the intermediacy of the SiCl 3O radical has been postulated to explain the results of gas phase FTIR studies of trichlorosilyl radical reactions.11None of these species have been detected or studied in the gas phase by spectroscopic techniques. By comparison with the methoxy radicals, we can anticipate that the ground state of the siloxy free radicals will be2E. However, the Jahn–Teller distortion will split this into a2A′and a2A′′compo- nent. In addition, these states will have spin–orbit splittings whose magnitude will depend on the extent of the Jahn–Teller quench- ing. The upper state will be2A1and does not, therefore, have any Jahn–Teller complications. The extent of activity of the non-totally symmetric ( e) vibrational modes (v′ 4- v′ 6) in the absorption [or laser induced fluorescence (LIF)] spectra depends again on the extent of the Jahn–Teller effect in the lower state. The presence of bands involving the emodes in LIF or emission spectra would be charac- teristic of the siloxy radicals with a significant Jahn–Teller effect in the ground electronic state. In the present work, we have conclusively identified the laser-induced fluorescence spectrum of the jet-cooled SiCl 3O free radical in the gas phase. Analysis of single vibronic level (SVL) emission spectra has shown that the ground state is subject to a significant Jahn–Teller effect, which is evident in the pronounced activity of the normally forbidden degenerate v′ 5and v′ 6modes. The measured lower state vibronic energy levels have been fitted with a Jahn–Teller model that includes spin–orbit coupling and linear and quadratic multimode coupling. The resulting param- eters have been used to simulate SVL emission spectra, which are found to agree satisfactorily with experiment, validating the analysis. II. EXPERIMENT It was found that a new LIF spectrum attributable to the SiCl 3O radical could be generated either using a precursor mixture of SiCl 4 and oxygen (60 Torr of each) in high pressure (120 psi) argon orby seeding the room temperature vapor pressure ( ∼7 Torr) of hex- achlorodisiloxane (Cl 3SiOSiCl 3, Gelest, 90%) in 50 psi of argon. As described in detail elsewhere,12,13a pulsed molecular beam valve (General Valve, series 9) injected the precursor mixture into a flow channel where an electric discharge between two stainless steel ring electrodes fragmented the precursor, producing the species of inter- est and a variety of other products. The reactive intermediates were rotationally and vibrationally cooled by free jet expansion into vac- uum at the exit of the pulsed discharge apparatus. A 1.0 cm long reheat tube14added to the end of the discharge apparatus increased production of the radicals and suppressed the background glow from excited argon atoms. Low resolution (0.1 cm−1) LIF spectra were recorded using a neodymium:yttrium aluminum garnet (Nd:YAG) pumped dye laser (Lumonics HD-500) excitation source. The fluorescence was collected using a lens and focused through appropriate longwave pass filters and onto the photocathode of a photomultiplier tube (RCA C31034A). The spectra were calibrated with optogalvanic lines from various argon- and neon-filled hollow cathode lamps. The laser-induced fluorescence and calibration spectra were digi- tized and recorded simultaneously on a homebuilt computerized data acquisition system. For emission spectroscopy, the band maxima of features in the LIF spectrum were excited by the dye laser, and the resulting fluorescence was imaged with f/4 optics onto the entrance slit of a 0.5 m scanning monochromator (Spex 500M). The pulsed fluo- rescence signals were detected with a gated CCD camera (Andor iStar 320T) and recorded digitally. The emission spectra were cal- ibrated to an estimated accuracy of ±1 cm−1using emission lines from an argon filled hollow cathode lamp. A 1200 line/mm grat- ing blazed at 750 nm was employed in this work, which gave a band pass of 29.9 nm with an 18 mm effective active area on the CCD. III. RESULTS AND ANALYSIS A.Ab initio calculations We have carried out a series of modest level ab initio calcula- tions to predict the properties of the ground and excited states of the SiCl 3O free radical using the Gaussian 09 program package.15 The2E C 3vground state is subject to the Jahn–Teller distortion, and by starting with a slightly lower symmetry geometry, we were able to converge on a2A′ground state with all positive frequen- cies using density functional (B3LYP)16,17and second order Moller– Plesset (MP2) theory using a triple zeta basis set with added dorbital polarization functions (6-311G∗). The ground state has an . . .a2 1e3 electron configuration ( C3vnotation), with the geometry distortion as shown in Fig. 1. The unpaired electron is almost completely local- ized on the oxygen atom. The geometric parameters, vibrational frequencies, and energies are reported in Table I. The ˜A2A1excited state involves the promotion of an a1sec- ond highest occupied molecular orbital (SHOMO) electron to the highest occupied molecular orbital (HOMO) of esymmetry. This state is not orbitally degenerate and, therefore, not subject to the Jahn–Teller distortion, so the symmetry is rigorously C3v. The elec- tron promotion involves a migration of unpaired electron density from the oxygen atom to the chlorine atoms although the majority J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . The molecular structures of the SiCl 3O free radical in the electronic excited and ground states. The distortion of the latter has been greatly exaggerated to show the Jahn– Teller symmetry breaking, which involves an elongation of one of the Si–Cl bonds and a rotation of the Si–O moiety off theC3axis. remains on the oxygen. The calculated properties of the excited state are summarized in Table I. The calculations indicate that the first electronic transition of SiCl 3O free radical should occur in the visible spectrum around 15 000 cm−1(∼670 nm) and that the ground state is subject to a sub- stantial Jahn–Teller distortion. On electronic excitation, only the ν5 andν6vibrational frequencies change significantly, and the geomet- ric parameters, with the exception of a ∼0.04 Å diminution of the Si–O bond length, are largely unaffected. B. The Jahn–Teller effect In this section, we summarize important aspects of the Jahn– Teller effect that are relevant to our analysis of the SiCl 4+ O 2 spectrum. These come from the literature following the general discussion of Herzberg18and the more detailed experimental and theoretical studies by Barckholtz and co-workers.8In a moleculewith degenerate modes of vibration, there is a vibrational angular momentum in the direction of the symmetry axis characterized by the quantum number li=vi,vi−2, . . ., 1 or 0. The levels of dif- ferent liand the same viwould be degenerate at the lowest level of approximation. In nondegenerate electronic states, these levels do show small splittings due to anharmonicity. However, in degener- ate electronic states, the vibrational angular momentum is coupled with the electronic orbital angular momentum to form a resultant vibronic angular momentum, which generally causes a substantial splitting of the degenerate vibrational levels. In linear (and bent molecules with less than a threefold axis of symmetry), this type of vibronic interaction is termed the Renner–Teller effect. In nonlinear molecules with a threefold or greater symmetry axis and a degener- ate electronic state, the vibronic coupling is termed the Jahn–Teller effect. The Jahn–Teller theorem proves that any state with a degen- erate electronic wave function is unstable in the most symmetric TABLE I . The ab initio parameters (MP2, with B3LYP in parentheses, the basis set is 6-311G∗in both cases) for28Si35Cl316O. Quantities without explicit units are in cm−1. Parameter ˜X2A′Ã2A1 r(Si–Cl) (Å) 2.017/2.014a(2.039/2.042) 2.014 (2.053) r(Si–O) (Å) 1.666 (1.653) 1.632 (1.605) <(Cl–Si–O) (deg) 110.6/103.7 (111.8/101.2) 108.9 (110.1) <(Cl–Si–Cl) (deg) 111.5/103.7 (111.5/108.8) 110.0 (108.9) ω1(Si–O symm. stretch, a1)b904 (872) 899 (903) ω2(Si–Cl symm. stretch, a1) 474 (443) 480 (422) ω3(Si–Cl 3umbrella, a1) 245 (223) 255 (236) ω4(Si–Cl asymm. stretch, e) 646/653 (595/604) 653 (562) ω5(Si–Cl 3rock, e) 251/258 (234/240) 305 (293) ω6(Si–Cl 2scissors, e) 100/178 (116/162) 179 (167) T0 . . . 16 468 (14 130) aIn the Jahn–Teller distorted ground state, the Si–Cl bond lengths and angles are not all equivalent. In each case, the first entry is the value for the two equivalent parameters and the second for the unique value. bThe symmetry labels apply strictly to the C3vsymmetry. In the Jahn–Teller distorted ground state, the “ e” modes are not quite degenerate, so two distinct frequencies are given. J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Schematic energy levels for the SiCl 3O free radical, showing the allowed transitions.conformation and will exhibit a spontaneous symmetry breaking that lowers the overall energy. The vibronic interaction splits the degeneracy of the vibrational levels so that there will be as many different vibronic levels as there are vibronic species for each vibra- tional level. The vibronic interaction compromises the goodness of the electronic orbital angular momentum quantum number Λ=±1 and the previously defined vibrational angular momentum quantum numbers lileading to the definition of a new first order Jahn–Teller quantum number, which, for molecules of C3vsymmetry, is j=l+Λ/2=1/2, 3/2, 5/2, 7/2, etc. (1) The states with j= 3/2, 9/2, . . .transform as a1ora2, while all others, j= 1/2, 5/2, 7/2, 11/2, . . ., transform as e.In a nondegenerate state, for example, the2A1excited state of the SiCl 3O radical, j=l. The relevant vibronic selection rule is then Δj=±1/2. The presence of an unpaired electron further complicates the situation. In an orbitally nondegenerate state, such as the ˜A2A1 state of SiCl 3O, spin–orbit coupling is usually very small, and its effect on the vibronic levels can be neglected. However, in orbitally degenerate states, spin–orbit coupling can be substantial, similar to that for the Π,Δ. . .electronic states of linear molecules. For large spin–orbit coupling, the Jahn–Teller instability is quenched, whereas small spin–orbit coupling has the effect of splitting some of the vibronic states into two spin–orbit components. The conventional notation is to designate the vibrational level, such as 5 1, and add a superscript + or −indicating whether the level is the upper or lower spin component, respectively, along with the jvalue in parentheses. FIG. 3 . Schematic potential energy surfaces (PES) for the Jahn–Teller distorted C3vground state of a species like the SiCl 3O free radical. The top panel shows the PES result- ing from the linear Jahn–Teller coupling, which produces a “Mexican hat” potential with a circular moat. The bottom panel shows the effect of linear plus quadratic Jahn–Teller coupling, which creates local minima and maxima around the moat. J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Thus, 5+ 1(j= 1/2) denotes the upper spin–orbit component of the j= 1/2 level of v′′ 5= 1. It is important to note that the eigenvectors are actually a linear combination of basis functions in the solu- tion of the Jahn–Teller problem and this notation, where appro- priate, only designates the leading term in the expansion. If the combining electronic states have the same spin multiplicity, then the ΔS= 0 selection rule is satisfied, and in SiCl 3O, transitions can occur from an upper state vibronic level to both spin–orbit components of a lower state vibronic level as long as the Δj=±1/2 selection rule is satisfied. The spin–orbit coupling in the Jahn–Teller active state is quantified by the spin–orbit coupling constant aζe, where the dimensionless ζeis the projection of the electronic orbital angular momentum on the symmetry axis and a(cm−1) is an approximate measure of the strength of the interaction of the unpaired electron with the individual nucleus. With this background, we show in Fig. 2 a schematic of a few of the relevant vibronic energy levels for the ˜A2A1and ˜X2Eelec- tronic states of the SiCl 3O free radical. The ν6vibration is doubly degenerate, and the Jahn–Teller and spin–orbit interactions in the ground state lead to four levels ( j=1/2and 3/2) with the vibrational label 6 1. Due to the Jahn–Teller coupling, the ground state 0 0level is actually an admixture of the vibrational basis functions of the degenerate modes, so both the excitation and emission spectra will exhibit features due to the Jahn–Teller active modes, subject only to theΔj=±1/2 restriction. Cold band absorption or LIF spectra from 0 0will then involve transitions to the totally symmetric lev- els (0 0, 11, 21, 31) with j= 0 and to levels involving the degenerate modes with j= 1 or 0. Single vibronic level emission spectroscopy is the key to understanding the lower state Jahn–Teller problem. As shown in Fig. 2, emission from the 00level ( j= 0) can only terminate on j=1/2levels, accessing only half of the 6 1vibrational states. However, emission from the 61level ( j= 1) is allowed to both FIG. 4 . The ˜A2A1–˜X2ELIF spectrum of the jet-cooled SiCl 3O free radical with vibronic assignments. The inset shows the spin splitting of the 00 0band. The neg- ative going spectrum is a Franck–Condon simulation of the expected absorption spectrum based on the ab initio vibrational frequencies and Cartesian displace- ment coordinates of the combining states (Table I). The simulation predicts a weak 21 0band that is clearly not evident in the experimental spectrum.j=1/2andj= 3/2 states, so a comparison of the two types of emission spectra can be used to differentiate between these levels. In addi- tion, observation of these differences can provide confirmation of LIF assignments. Finally, it is necessary to discuss the potential energy surfaces and the parameters that are used to define the Jahn–Teller coupling. The usual treatment of the problem results in a potential that can be approximated as the sum of a linear and a quadratic term.8The linear Jahn–Teller coupling distorts the molecule of nominally C3v symmetry, so that the potential for a single active mode has a global minimum at the bottom of the moat, as shown in Fig. 3, and is characterized by a dimensionless linear Jahn–Teller coupling con- stant Difor the ith mode. The previously defined quantum number j=l+Λ/2 is only strictly conserved for linear coupling. Quadratic coupling modifies the moat, so that it contains maxima and min- ima (3 in C3vsymmetry, see Fig. 3), and the extra stabilization is characterized by the dimensionless quadratic coupling constant Ki. In practice, one can use the Spin–Orbit Coupling and Jahn–Teller (SOCJT) program of Barckholtz and Miller19to fit the observed TABLE II . Approximate band centers (cm−1) and assignments [originating from the lower or (-) spin–orbit component in the ground state] of the features in the LIF spec- trum of the SiCl 3O free radical. The feature quoted in parentheses is the band center of the less intense spin–orbit component (where resolved), which is the transition from the upper or (+) level. Band center (cm−1) Assignment Comment 15 313.7 (15 303.2) 00 0 Spin–orbit = 10.5 15 477.3 (15 467.0) 61 0 ν′ 6= 163.6 15 551.0 (15 541.0) 31 0 ν′ 3= 237.3 15 606.4 (15 597.1) 51 0 ν′ 5= 292.7 15 639.2 (15 630.0) 62 0 61 0+ 161.9 15 715.3 (15 704.7) 31 061 0 61 0+ 238 15 769.0 (16 758.4) 51 061 0 61 0+ 291.7 15 804.3 (15 794.6) 63 0 62 0+ 165.1 15 843.7 (15 833.7) 31 051 0 31 0+ 292.7 15 901.1 (15 891.5) 52 0 51 0+ 294.7 15 934.1 51 062 0 62 0+ 294.9 16 006.6 31 051 061 0 31 051 0+ 162.9 16 063.3 52 061 0 52 0+ 162.2 16 094.7 51 063 0 63 0+ 290.4 16 186.3 53 0 52 0+ 285.2 16 226.2 11 0and 52 062 0 ν′ 1= 912.6 16 350.3? 53 061 0 53 0+ 164.0 16 389.6? 11 061 0 61 0+ 912.3 16 490.0 54 0 53 0+ 303.7 16 520.1 11 051 0 51 0+ 913.7 16 552.6 11 062 0 62 0+ 913.4 16 653.8 54 061 0 54 0+ 163.8 16 683.3 11 051 061 0 51 061 0+ 914.3 16 784.6 55 0 54 0+ 294.6 16 816.0 11 052 0 52 0+ 914.9 J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 5 . The experimentally observed rotational profiles of the61 0,62 0, and 63 0LIF bands of the SiCl 3O free radical. vibronic energy levels of a C3vJahn–Teller active state to obtain the Jahn–Teller and spin–orbit coupling parameters and validate the analysis by simulation of the observed absorption and emission spectra. C. LIF and emission spectra An intense LIF spectrum with an onset at 15 300 cm−1and extending to 16 900 cm−1, illustrated in Fig. 4, was discovered using the SiCl 4+ O 2precursor mixture. It was, subsequently, shown that exactly the same spectrum could be obtained using hexachlorodis- iloxane (Cl 3Si–O–SiCl 3) as the starting material. The spectrum com- mences with a strong 0–0 band at 15 313.5 cm−1, nearby strongbands +163.5 cm−1and +292.9 cm−1higher in energy, and then a wealth of weaker bands out to 16 900 cm−1. The strong bands show a distinct splitting with a weaker satellite 10–15 cm−1to lower energy, as shown in the inset of Fig. 4. In the absence of Jahn–Teller distortions, the absorption spec- trum for a2A1–2Eelectronic transition should consist of vibronic transitions involving only the totally symmetric vibrational modes. It is readily apparent from the low frequency intervals in the spec- trum (experiments in which the expansion conditions were changed show these are cold bands) that the e′modes must be involved, as anticipated for a Jahn–Teller active ground state. In fact, the ab initio vibrational frequencies (Table I, ω′ 6∼175 cm−1andω′ 5 ∼300 cm−1) immediately suggest assignments of 61 0(+163.5 cm−1) FIG. 6 . A comparison of the single vibronic level emission spectra of the SiCl 3O free radical for laser excitation of the LIF 00 0,61 0,51 0, and 31 0bands (stronger spin–orbit com- ponent in each case). The assignments and lower state jvalues for a few key features in the spectra are given. Ver- tical dashed lines link features with the same lower state. The spectra are plotted as displacement from the laser exci- tation wave number (cm−1), giving a direct measure of the relative ground state energy for each transition. J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp and 51 0(+292.9 cm−1) for the first two strong bands above the 0–0 band. A much weaker band at +238 cm−1(calc. = 252 cm−1) is iden- tified as 31 0. The prominence of the 51 0and 61 0bands in the spectrum suggests that the ground state Jahn–Teller effect is substantial. The pattern of bands repeats with fairly harmonic intervals throughout the spectrum so that most features can be assigned as involving v′ 3, v′ 5, and v′ 6, until a weak peak at 16 389 cm−1, +1076 cm−1above the 0–0 band. This interval is too large to be a ground state vibrational fundamental (see Table I) but can be assigned as v′ 1+ v′ 6(ab initio = 1070–1078 cm−1). Subtracting the experimental v′ 6= 163.6 cm−1 yields 11 0= 16 225.4 cm−1, coincident with a broad observed band we assigned as 51 061 0. It is likely that both vibronic transitions are buried under the profile of this band, and we have made this assign- ment in Fig. 4. The measured band centers, spin–orbit splittings, and assignments of the LIF spectrum are summarized in Table II. In the latter stages of the analysis, we attempted to simulate the LIF spectrum by calculating the Franck–Condon (FC) factors for all possible transitions in the harmonic approximation, using the pro- gram previously described.20,21The input was the B3LYP/6-311G∗ geometries, vibrational frequencies (see Table I), and mass-weighted Cartesian displacement coordinates from the ab initio output. If we constrain both states to have C3vsymmetry, we obtain a very sim- ple predicted absorption spectrum, which only involves the totally symmetric modes, as anticipated from the vibrational selection rules, with prominent 31 0, 11 0, and 12 0transitions. If we allow the lower state to undergo the Jahn–Teller distortion to the geometry shown in Table I, the resulting spectrum, shown as the downward-going trace in Fig. 4, is astonishingly similar to the experiment, clearly establish- ing that the species responsible for the observed LIF spectrum is the SiCl 3O free radical. Of course, the simulation does not include spin– orbit effects, so the small splittings observed experimentally are not evident, which also provides confirmation that we are dealing with a species with an unpaired electron. The FC simulation also provided strong evidence for the validity of our initial vibronic assignments of the LIF spectrum, including the overlap of 11 0and 52 062 0and the absence of activity in v′ 2and v′ 4. The rotational contours of the 61 0, 62 0, and 63 0bands are shown in Fig. 5, illustrating the complexity of the higher bands in the spec- trum. For a molecule as heavy as SiCl 3O, we would not expect any resolvable rotational structure at our resolution of 0.1 cm−1, even at our estimated rotational temperature of 15 K, and this is evidently the case. The three chlorine atoms (35Cl = 75.77%, 37Cl = 24.23%) should lead to more and more isotopic complica- tions in the band contours with additional quanta of v′ 6(the Si–Cl 2 scissors mode), whose ab initio fundamental frequency we find to be affected by 1–4 cm−1depending on which chlorine isotopologue one calculates. These changes in the band contour made it difficult to obtain precise band positions and vibrational intervals from the LIF spectra. The SVL emission spectra obtained from pumping the 00 0, 61 0, 51 0, and 31 0bands in the SiCl 3O LIF spectrum are compared in Fig. 6. It is immediately apparent that the 00and 61emission spectra exhibit the behavior predicted in Fig. 2, with new transitions down to 6 1 (j= 3/2) evident in the latter. A similar pattern is found in the 51 emission with transitions down to the 5 1(j= 3/2) levels. In sharp contrast, the 31emission spectrum is dominated by an intense tran- sition to a level at 233 cm−1, which is readily assigned as 3 1based on the ab initio vibrational frequencies (Table I) of 245 cm−1and223 cm−1. There is also a prominent feature (not shown) in the 0–0 band spectrum at a displacement of 894 cm−1with a spin splitting of 11 cm−1, which we assign as 10 1(ab initio v′′ 6= 904 cm−1and 872 cm−1). It is very clear that the observed rotational contours, isotopic splittings, vibrational intervals, Jahn–Teller splittings, and very good agreement between the observed and calculated Franck– Condon profiles of the LIF spectra are entirely consistent with our assignment of the carrier as the SiCl 3O free radical. D. Jahn–Teller analysis of the ground state vibronic energy levels The emission spectra (Fig. 6) provide us with the most infor- mation on the ground state vibronic energy levels of the SiCl 3O free radical. We have used the SOCJT program19to fit our data: the input TABLE III . Energies (cm−1), assignments, quantum numbers, and Obs–Calc residu- als (cm−1) for the ground state vibronic energy levels of the SiCl 3O free radical. Energy (Assign.) j n jΣ Obs–Calc 0(0− 0) 0.5 1 0.5 0.0 10(0+ 0) 0.5 1 −0.5 6.6 58(6− 1) 1.5 1 −0.5 9.4 89 (6+ 1) 1.5 2 −0.5 −1.3 155 0.5 2 −0.5 1.7 162 0.5 2 0.5 1.3 173 0.5 3 0.5 6.3 184 0.5 3 −0.5 3.5 210 (5− 1) 1.5 3 −0.5 3.3 239 (5+ 1) 1.5 4 −0.5 −2.6 245 0.5 4 −0.5 −3.4 250 0.5 4 0.5 −2.6 275 1.5 5 −0.5 3.3 290 0.5 6 0.5 −5.2 290 0.5 6 −0.5 −1.6 303 1.5 6 −0.5 8.0 317 0.5 7 0.5 0.8 317 0.5 8 −0.5 6.0 362 0.5 8 −0.5 −2.4 362 0.5 8 0.5 −3.5 380 1.5 9 −0.5 −1.4 380 1.5 10 −0.5 −0.5 393 1.5 11 −0.5 2.5 412 1.5 12 −0.5 −3.5 421 0.5 11 −0.5 1.4 453 0.5 12 −0.5 −1.7 453 0.5 12 0.5 −0.4 457 1.5 13 −0.5 2.4 463 0.5 14 −0.5 4.1 469 1.5 14 −0.5 4.9 506 0.5 17 −0.5 −2.8 527 0.5 19 −0.5 3.8 546 1.5 19 −0.5 −8.4 555 1.5 20 −0.5 1.2 562 0.5 21 −0.5 2.5 J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE IV . Comparison of the Jahn–Teller parametersaof the trichlorosiloxy and various methoxy free radicals.b Parameter SiCl 3O CH 3O CD 3O CF 3O CF 3S ω′′ e,5 251.0 1417 1070 600 536 D5 0.396 0.075 0.17 0.04 <0.01 K5 −0.162 −0.032 −0.03 . . . . . . ω′′ e,6 152.5 1065 825 465 320 D6 0.916 0.24 0.20 0.45 0.24 K6 −0.048 −0.14 −0.16 0.05 . . . aζe −183 −145 −145 −140 −360 εtotal 239 419 410 233 77 εSO total 203 370 367 203 0 aAll quantities in cm−1except for DandKwhich are dimensionless. bRef. 23. is the measured vibrational energy relative to the lowest 0- 0(j=1/2) level, the jvalue, nj,which an index identifying the njth eigenvalue of a given value of jandΣ, andΣ, the quantum number for the projec- tion of the total electron spin angular momentum on the symmetry axis. In this case, for j=1/2,Σ= +0.5 for the lower spin–orbit com- ponent and −0.5 for the upper spin–orbit component, whereas the j= 3/2 levels are not split by spin–orbit coupling, so only Σ=−0.5 levels were calculated. The energy levels to be fitted were only those involving the Jahn–Teller active modes ν5andν6; levels that con- tained one or more quanta of the totally symmetric vibrations were not included. The parameters to be varied in fitting the data were the har- monic vibrational frequency ωe,iof each Jahn–Teller active mode i, their linear and quadratic Jahn–Teller coupling constants Diand Ki, and the global spin–orbit coupling constant aζe. The fitting pro- cess was laborious as only one or two levels could be added to thedataset at a time, and the SOCJT program took 1–3 h to execute on a 3.4 GHz central processing unit (CPU) desktop computer. As shown in Fig. 2, absorption from the 0- 0(j=1/2) level can only populate j′= 0 (00, 11, 62, etc.) and j′= 1 levels (61, 51, etc.), which subsequently emit down to j′′=1/2orj′′=1/2and 3/2, respec- tively, severely limiting the range of observed levels in the ground state. As shown in Fig. 6, a few of the lowest levels [6- 1, 6+ 1(j= 3/2) and 5- 1, 5+ 1(j= 3/2)] were readily assigned and initially fitted. An examination of the eigenvector coefficients showed that the mixing was extreme, even for the very lowest levels, so that making vibra- tional assignments was generally not possible, and we had to resort to simply specifying the eigenvector with the quantum numbers j, nj, andΣ. Due to the substantial width of the emission features (7–10 cm−1) and the broadening due to the various chlorine iso- topologues, the transitions could not be measured to better than 1–2 cm−1, so we have truncated the measured energy levels to the nearest wave number. For ground state energies up to about 300 cm−1, the levels were fairly discrete and readily identified. Beyond that point, there were often multiple almost degenerate cal- culated energy levels, so the assignments were made on less secure grounds. We used three criteria for making assignments starting with the jvalue, where determinable, particularly for j= 3/2, which were identifiable by their absence from the 00and 31level emis- sion spectra. Second, bootstrapping up from a minimal set of well- defined levels, there had to be a good correspondence between observed and calculated values. Third, there had to be reasonable agreement between the observed and calculated intensities in the various emission spectra. After each least squares refinement, we checked these criteria before augmenting the dataset. In the majority of the emission spectra, there are few strong transitions with dis- placements greater than 600 cm−1, and assignments became very problematic beyond that point, due to the density of Jahn–Teller active levels and the extensive mixing of the basis states. FIG. 7 . A comparison of the experimental (upward going) and SOCJT simulated (downward going) single vibronic level emission spectra from the 5161and 52levels of the SiCl 3O free radical. The spectra are plotted as displace- ment from the laser excitation wave number (cm−1), giving a direct measure of the relative ground state energy for each transition. J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp In the final analysis, we fitted 35 ground state assignments, varying seven constants for an overall standard deviation of 4.0 cm−1, with the results shown in Tables III and IV. Care was taken to ensure that the basis set was large enough to guarantee reasonable convergence. The agreement between the observed and calculated emission spectra for the higher 5161and 52levels, which provide a stringent test of the analysis, is illustrated in Fig. 7. Recalling that only activity in the Jahn–Teller modes, ν5andν6, is simulated and that the observed energy levels have residuals ranging up to almost 10 cm−1, the agreement between the observed and calculated spectra is quite satisfactory. IV. DISCUSSION The results presented in this paper are the first detailed Jahn– Teller analysis of the electronic spectrum of a gas phase free radical as heavy as SiCl 3O. The presence of two low frequency Jahn–Teller active modes and the additional complication of the spin–orbit cou- pling make the spin-vibronic energy level structure in the ground state quite complex. Of the 29 levels involving v′′ 5and/or v′′ 6that we calculate in the first 400 cm−1, we have assigned 22, or 76%. Pos- sible transitions to the unassigned levels are overlapped by other transitions, either involving nondegenerate modes (such as v′′ 3) or assigned features calculated to be more intense. It is of interest to compare the Jahn–Teller effect in the trichlorosiloxy radical with that in the much better known methoxy radicals, as documented in Table IV. In all cases, the major Jahn– Teller active modes are v′′ 5and v′′ 6, although there is a slight contri- bution from v′′ 4in CH 3O and CD 3O. The siloxy linear Jahn–Teller parameters Diare much larger than those of the methoxy radicals, primarily due to the much smaller vibrational frequencies in the former. In the linear coupling approximation, the contribution of mode ito the depth of the moat relative to the symmetric configura- tion isε(1) i=Diωe,i, so for a fixed ε(1) i,Diincreases as the vibrational frequency decreases. Indeed, the data at the bottom of Table IV shows that the total SiCl 3O linear Jahn–Teller stabilization energy (239 cm−1) is less that of CH 3O and CD 3O but comparable to that of CF 3O. In the present case, both modes make a sizable contribution to the linear Jahn–Teller stabilization energy, with ε(1) 5= 99 cm−1 andε(1) 6= 140 cm−1, whereasε(1) 6= 209.3 cm−1clearly dominates in CF 3O. The significant contribution of both SiCl 3O modes is evident in the LIF (Fig. 4) and emission (Fig. 6) spectra, where transitions involvingν5andν6predominate. Spin–orbit coupling competes with Jahn–Teller coupling for the electronic orbital angular momentum: they involve the coupling of the electronic orbital angular momentum of the molecule to either the spin angular momentum or the vibrational angular momentum, respectively. Barckholtz and Miller8discuss two separate regimes: one where the spin–orbit coupling largely eliminates the Jahn–Teller distortion (when aζe>4Diωe,i) and the other where the molecule is distorted but the spin–orbit splitting is quenched (when aζe <4Diωe,i). The former is exemplified in the ground state of the CF 3S radical, where the inclusion of the spin–orbit coupling drops the Jahn–Teller stabilization to 0 cm−1(|aζe|= 360 cm−1>4D6ωe,6 = 307 cm−1) and there is no geometric distortion. The latter regime is more typical as shown in Table IV, where the spin–orbit coupling only slightly diminishes the total Jahn −Teller stabilization, including a small 15% effect in SiCl 3O.The measured splitting of the 6 1(j= 3/2) levels (see Fig. 4 and Table III) is 31 cm−1, much larger than the 10 cm−1splitting of the 00(j=1/2) levels. As discussed elsewhere,8this enhanced splitting of thej= 3/2 levels is caused by the presence of quadratic coupling in the Jahn–Teller modes. In this case, the total quadratic Jahn–Teller stabilization energy ε=D5ωe,5K5+D6ωe,6K6= 23 cm−1, about 10% of the linear contribution and rather smaller than the 39 cm−1total quadratic stabilization in the methoxy free radical.22This implies that the minima and maxima around the SiCl 3O moat are sepa- rated by only 2 ε(2) total= 46 cm−1. We also note that our spectrum fits just as well with positive or negative starting values of the quadratic constants Ki, whereas the signs are predominantly negative in the methoxy radicals (see Table IV). The primary effect of the sign of Kiis to affect the positions of the local minima and maxima around the moat. In addition, quadratic coupling breaks the degeneracy of thej= 3/2 levels into their a1anda2components, and the sign of Ki affects the relative energies of these levels. The spin–orbit coupling does not further split these levels, but rather mixes them together and shifts them. The extent of such mixing can be gauged from the relative intensities of the spin–orbit components. In the present case, the spin–orbit coupling is substantial and mixes the a1anda2levels of the lowest j= 3/2 levels so thoroughly that the spin–orbit com- ponents (6+ 1and 6- 1or 5+ 1and 5- 1) have very similar intensities in the dispersed fluorescence spectra. Since the signs of the Kiare inde- terminate in the SiCl 3O spectrum, we have elected to make them negative in accord with the results for the various methoxy radicals. A. Comparison with theoretical predictions We can compare the ab initio results in Table I to our experi- mental findings, to get an idea of the utility of such modest calcula- tions. For the excited state, we find that our LIF values for ν1= 912.6, ν3= 237.3,ν5= 292.7, and ν6= 163.6 cm−1are all within 2% of the B3LYP values, with generally slightly higher MP2 frequencies. Cer- tainly, these theoretical frequencies, particularly in conjunction with a FC simulation of the spectrum, can be used to distinct advantage to identify and assign the LIF spectrum. In this case, the FC simulation works because only the vibrational zero-point level in the ground state is involved and the excited state is nondegenerate. Although both are deficient by almost 1200 cm−1, the B3LYP and MP2 elec- tronic excitation energies nicely bracket the experimental T0value, providing further confirmation of the carrier of the spectrum. In the ground state, the only nondegenerate fundamentals we have assigned are ν1= 894 cm−1(904/872 cm−1) in the 0–0 band emission spectrum and ν3= 233 cm−1(245/223 cm−1) in the 31 0 band emission spectrum. The ab initio values given in parentheses (see Table I) are in good accord with the experimental assignments. The averages of the calculated frequencies for ν5= 246 cm−1and ν6= 139 cm−1are also in good accord with the harmonic basis func- tion frequencies ω5= 251.0 and ω6= 152.5 cm−1obtained in our Jahn–Teller analysis of the emission spectra. We can obtain a crude estimate22of the expected Jahn–Teller stabilization by comparing the ground state energy of SiCl 3O con- strained to C3vsymmetry with that obtained from the relaxed geom- etry given in Table I. The results are B3LYP = 260 cm−1and MP2 = 157 cm−1, both in general agreement with the fitted total lin- ear stabilization energy of 239 cm−1. Theory clearly shows that our analysis gives a Jahn–Teller stabilization which is quite reasonable. J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp V. CONCLUSIONS The trichlorosiloxy free radical has now been conclusively iden- tified and characterized in the gas phase. The jet-cooled SiCl 3O species has been studied by LIF and resolved emission techniques, and it has been shown that there is a substantial Jahn–Teller effect in the ground state. Thirty-five ground state spin-vibronic levels have been fitted to an overall standard deviation of 4.0 cm−1using a Jahn–Teller model that includes multimode activity in ν5andν6 and substantial spin–orbit coupling. The spin–orbit splittings in the lowest j= 3/2 levels are significantly larger than those of the 0 0level, indicating the necessity of including both linear and quadratic Jahn– Teller couplings. The spin–orbit coupling only slightly diminishes the Jahn–Teller stabilization in the SiCl 3O free radical, leading to pronounced activity in the ν′′ 5andν′′ 6Jahn–Teller active modes in the LIF and single vibronic level emission spectra. ACKNOWLEDGMENTS The authors thank Dr. Gretchen Rothschopf for her support with various aspects of this work and Dr. Brandon Tackett for early explorations of the SiCl 4+ O 2spectrum. They are grateful to Timo- thy Barckholtz and Terrance Codd for valuable advice concerning the SOCJT program. This research was funded by Ideal Vacuum Products. REFERENCES 1F. Ojeda, A. Castro-García, C. Gómez-Aleixandre, and J. M. Albella, J. Mater. Res.13, 2308 (1998). 2M. R. Zachariah and W. Tsang, J. Phys. Chem. 99, 5308 (1995).3Chemical Vapor Deposition. Principles and Applications , edited by M. L. Hitch- man and K. F. Jensen (Academic Press, London, 1993). 4M. Binnewies and K. Jug, Eur. J. Inorg. Chem. 2000 , 1127, and references therein. 5M. Junker, A. Wilkening, M. Binnewies, and H. Schnöckel, Eur. J. Inorg. Chem. 1999 , 1531. 6M. Junker and H. Schnöckel, J. Chem. Phys. 110, 3769 (1999). 7V. M. Donnelly and A. Kornblit, J. Vac. Sci. Technol., A 31, 050825 (2013). 8T. A. Barckholtz and T. A. Miller, Int. Rev. Phys. Chem. 17, 435 (1998) and references therein. 9R. Srinivas, D. K. Boehme, D. Suelzle, and H. Schwarz, J. Phys. Chem. 95, 9836 (1991). 10M. C. R. Symons, J. Chem. Soc., Dalton Trans. Inorg. Chem. 16, 1568 (1976). 11H. Niki, P. D. Maker, C. M. Savage, L. P. Breitenbach, and M. D. Hurley, J. Phys. Chem. 89, 3725 (1985). 12H. Harjanto, W. W. Harper, and D. J. Clouthier, J. Chem. Phys. 105, 10189 (1996). 13W. W. Harper and D. J. Clouthier, J. Chem. Phys. 106, 9461 (1997). 14D. L. Michalopoulos, M. E. Geusic, P. R. R. Langridge-Smith, and R. E. Smalley, J. Chem. Phys. 80, 3556 (1984). 15M. J. Frisch, G. W. Trucks, H. B. Schlegel et al. , GAUSSIAN 09, Revision A.02, Gaussian, Inc., Wallingford, CT, 2009. 16A. D. Becke, J. Chem. Phys. 98, 5648 (1993). 17C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). 18G. Herzberg, Molecular Spectra and Molecular Structure III. Electronic Spectra and Electronic Structure of Polyatomic Molecules (Van Nostrand, New York, 1966). 19T. A. Barckholtz and T. A. Miller, Program SOCJT downloaded from https:// www.asc.ohio-state.edu/miller.104/molspect/goes/software/socjt/index.html. 20D. S. Yang, M. Z. Zgierski, A. Bérces, P. A. Hackett, P. N. Roy, A. Martinez, T. Carrington, Jr., D. R. Salahub, R. Fournier, T. Pang, and C. Chen, J. Chem. Phys. 105, 10663 (1996). 21B. S. Tackett, Y. Li, D. J. Clouthier, K. L. Pacheco, G. A. Schick, and R. H. Judge, J. Chem. Phys. 125, 114301 (2006). 22T. A. Barckholtz and T. A. Miller, J. Phys. Chem. A 103, 2321 (1999). 23T. A. Barkholtz, M.-C. Yang, and T. A. Miller, Mol. Phys. 97, 239 (1999). J. Chem. Phys. 152, 194303 (2020); doi: 10.1063/5.0009223 152, 194303-10 Published under license by AIP Publishing

5.0022731.pdf

Phys. Plasmas 27, 113103 (2020); https://doi.org/10.1063/5.0022731 27, 113103 © 2020 Author(s).Propagation dynamics of an azimuthally polarized Bessel–Gauss laser beam in a parabolic plasma channel Cite as: Phys. Plasmas 27, 113103 (2020); https://doi.org/10.1063/5.0022731 Submitted: 23 July 2020 . Accepted: 22 October 2020 . Published Online: 11 November 2020 Rong-An Tang , Li-Ru Yin , Xue-Ren Hong , Ji-Ming Gao , Li-Hong Cheng , and Ju-Kui Xue Propagation dynamics of an azimuthally polarized Bessel–Gauss laser beam in a parabolic plasma channel Cite as: Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 Submitted: 23 July 2020 .Accepted: 22 October 2020 . Published Online: 11 November 2020 Rong-An Tang,1,a) Li-Ru Yin,1,a) Xue-Ren Hong,1 Ji-Ming Gao,1Li-Hong Cheng,2and Ju-Kui Xue1 AFFILIATIONS 1Key Laboratory of Atomic and Molecular Physics and Functional Materials of Gansu Province, College of Physics and Electronic Engineering, Northwest Normal University, Lanzhou 730070, People’s Republic of China 2School of Science, Guizhou University of Engineering Science, Bijie 551700, People’s Republic of China a)Authors to whom correspondence should be addressed :tangra79@163.com andyinlr96@163.com ABSTRACT The propagation dynamics of an azimuthally polarized dark hollow laser beam described by a first-order Bessel–Gauss laser beam in a para- bolic plasma channel is investigated by adopting the weakly relativistic limit. By using the variational method, the evolution equation of thering-beam radius is derived and the ring-beam width is proportional to and synchronous with the radius. It is found that the azimuthalpolarization can weaken the vacuum diffraction effect and the propagation dynamics of the dark hollow laser beam may be classified into three types, i.e., propagation with a constant ring-beam radius and width, or synchronous periodic defocusing oscillation, or synchronous periodic focusing oscillation. Their corresponding critical conditions and characteristic quantities, such as the amplitudes and spatial wave-lengths, are obtained. Further investigation indicates that, with the increase in the initial laser power or the ratio of initial ring-beam radiusto channel radius, the dark hollow beam may experience a process from synchronous periodic defocusing oscillation to constant propagation and then to synchronous periodic focusing oscillation, in which the corresponding amplitudes decrease sharply to zero (constant propaga- tion) and then increase gradually, while the spatial wavelength decreases continuously. The evolution type of this kind of dark hollow beamalso depends on its initial amplitude but is insensitive to the initial laser profile which, however, has a large influence on the spatial wave-length. These results are well confirmed by the numerical simulation of the wave equation. A two-dimensional particle-in-cell simulation ofan azimuthally polarized laser beam is performed finally and also reveals the main results. Published under license by AIP Publishing. https://doi.org/10.1063/5.0022731 I. INTRODUCTION Recently, dark hollow beams (DHBs) with zero central inten- sity have attracted much attention especially because of some ben-eficial characteristics, such as center phase singularity, a helical wavefront, a barrel-shaped intensity distribution, and carrying spin and orbital angular momentum and exhibiting spatial propa-gation invariance for some cases, 1w h i c he n a b l et h e mt ob ee x t e n - sively applied in laser–plasma interactions, such as the excitationof the plasma wave, 2the generation of the THz waves, bright gamma rays, harmonic waves, and attosecond electron packets,3–6 the simulated Brillouin backscattering and Raman scattering,7,8 interacting with a solid target to achieve homogeneous mm-size distribution of the plasma ionization degree,9and the accelerations of electrons, protons, and ions.10–12The DHBs can be generated through many experimental techniques and methods, for instance,the optical holographic method,13geometric optical method,14 mode conversion method,15etc. There are several kinds of DHBs, such as the dark hollow Gaussian beam, the Laguerre–Gaussian beam, the TEM 0lmode donut beam, the non-zero order Bessel beam, the non-zero order Mathieu beam, the LP 01mode hollow beam,1etc. Compared with others, the non-zero order Bessel beam and non-zero order Mathieu beam are not only vortex beams (i.e., carrying orbital angular momentum) but also non-diffractive at a certain distance.1,16In the current popular case of cylindrical symmetry for laser amplitude, the Mathieu beamalso degenerates into the Bessel beam. 1So, as an exact solution of the paraxial Helmholtz equation in the cylindrical coordinate, the Bessel beam has aroused interest in the field of the plasma.17–19In fact, the Bessel beam generated in experiment is not exact but a modified one with a modulation because the exact Bessel beam carries infinite power Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-1 Published under license by AIP PublishingPhysics of Plasmas ARTICLE scitation.org/journal/phpand has infinite distribution space. In the pioneering experiment for producing the Bessel beam,20one kind of modulation is hidden in the experimental devices, i.e., a circular slit and a lens, but its special math- ematical formula is not known. Following this work, a general and proper Gaussian modulation is proposed, with which the modifiedBessel beam is found to be a new type of solution of the wave equation and called the Bessel–Gauss beam. 21The non-zero order Bessel–Gauss beam,22,23which is a new kind of DHB with orbital angular momen- tum and nearly non-diffractive feature, can be easily generated in experiment by a resonator24and has already been applied in laser- based acceleration of electrons25and quantum information processes.26 As the basic process, many studies closely related to the DHBs propagating in plasmas have been carried out. The electric field profile of the non-zero order Bessel beam propagating in a plasma is studiedprimarily by considering the radiation absorption and non-linearity of the medium. 27The self-focusing of the TEM 01mode donut laser beam i np l a s m a si st h e nr e p o r t e d .28The propagation characteristics of Laguerre–Gaussian laser beams in plasmas are numerically investi- gated by using the finite-difference time-domain method.29By using the paraxial ray approach or the variational method, the evolution of the dark hollow Gaussian beam has also been studied in homogeneousplasmas, 30in magnetoplasmas,31and with considering the relativistic and ponderomotive non-linearities simultaneously,32,33and all found that the evolution type of the laser beam depends on the initial laser profile strongly besides other factors. Recently, the propagation char- acteristics of a dark hollow Gaussian beam in a tapered plasma chan- nel were studied.34The evolution dynamics and the self-compression of the dark hollow Gaussian laser pulse are also investigated inplasmas. 35,36 It can be first seen from the above studies that as a basic for other research and applications, the investigations for the propagation ofDHBs mainly focus on the dark hollow Gaussian beam in plas- mas. 30–36Except for the only three studies about the non-zero order Bessel beam,27the TEM 01mode donut laser beam28and the Laguerre–Gaussian laser beam,29we have not seen the investigations of DHBs of other types, for example, the non-zero order Bessel–Gauss beam, which is non-diffractive at a certain distance and has non-zero orbital momentum that can carry a large amount of information.Second, the polarization of DHBs in all the above investigations is only linear or circular and there is no work about the cylindrical vector polarization. In the past few years, cylindrically vector polarized beams, especially azimuthally and radially polarized beams, have drawn much attention theoretically and experimentally 37–40because of the polarization symmetry which leads them to be useful for improving the accelerating of particles,41stable trapping of par- ticles,42,43improving the efficiency of laser machining,44,45harmonic generation,46,47etc. The study of the azimuthally polarized first-order Bessel–Gauss beam shows that it is a closed-form solution of the azi- muthal paraxial scalar wave equation48and can be produced by the concentric-circle-grating surface-emitting semiconductor lasers.49–51 Third, the above studies focus on the uniform plasma background and no related work about the DHBs is in a preformed plasma channel except for a recent pure theoretical study about a dark hollow Gaussian beam.34In a uniform plasma, the laser beam cannot propa- gate a long enough distance if the laser power is less than a critical value. Therefore, the preformed plasma channel52–54,57,58,60is widelyused to overcome this difficulty. Nowadays, the preformed parabolic plasma channel is more popular and general in theory and experi- ment,54,57–63and is widely applied in optical guiding,60particle acceler- ation,61,62plasma wakefield generation,63etc. Finally, except for the primary work about the non-zero order Bessel beam,27we have not yet seen a theoretical study of DHBs in plasmas associated with a con- firmation by an important numerical simulation of the basic waveequation, not to mention a more powerful particle-in-cell (PIC) simu-lation. The numerical simulations, especially the PIC method which is closer to reality, 64have become more and more important in the cur- rent study of the laser–plasma interaction.4,6,10–12,65–67 Based on these considerations, we study the propagation of an azimuthally polarized first-order Bessel–Gauss laser beam in a pre- formed parabolic plasma channel theoretically and numerically here. Under the weakly relativistic condition and using the variationalmethod, 33,53–56,68we find that the azimuthal polarization can weaken the vacuum diffraction effect and the propagation dynamics of the ring-beam radius and width of the laser beam present the three evolu- tion types, i.e., propagation with the constant ring-beam radius andwidth, synchronous periodic defocusing oscillation, and synchronousperiodic focusing oscillation. The physical condition for each evolution type as well as the characteristic quantities, such as the amplitudes and spatial wavelength of the ring-beam radius and width, are obtained.The propagation dynamics, depending on the initial laser power,amplitude, and ratio of initial beam radius to channel radius closely, are well studied. For the effect of the initial laser profile, our study shows that, unlike the most familiar dark hollow Gaussian beamwhose evolution type depends on it strongly, the evolution type ofthe Bessel–Gauss laser beam is insensitive to it which, however, affects the spatial wavelength obviously here. These results are well verified by the numerical simulation of the wave equation. A two-dimensional (2D) PIC simulation for an azimuthally polarizedfirst-order Bessel–Gauss laser beam is also performed finally and reveals the same results as the theory. Besides these new findings, considering that the previous research studies on the propagationof DHBs in plasmas are basically pure theoretical and lack furtherverification, this theoretical work with numerical confirmation seems to be more important to improve the knowledge of the inter- action between DHB and plasmas. T h eo r g a n i z a t i o no ft h ep a p e ri sa sf o l l o w s .I nS e c . II, the basic wave equation and evolution equations of the ring-beam radius and width are derived. In Sec. III, the propagation behaviors with their conditions and characteristic quantities are obtained. Then, severalfactors affecting the propagation behaviors of the laser beam are inves-tigated. In Sec. IV, the main results are verified by the numerical simu- lation of the basic wave equation. In Sec. V,t h e2 DP I Cs i m u l a t i o ni s done. Finally, conclusions are given in Sec. VI. II. BASIC MODEL AND RING-BEAM RADIUS AND WIDTH We consider an azimuthally polarized laser beam propagating along the z-direction in a preformed plasma channel. So, the normal- ized vector potential of the laser field with a slowly varying complex envelope can be expressed as 39,48,69,70 aðr;z;tÞ¼1 2aðr;z;tÞ^/expiðk0z/C0x0tÞ ½/C138 þ c:c:; (1)Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-2 Published under license by AIP Publishingwhere aðr;z;tÞis normalized by m0c2=e,m0andeare the rest mass and the charge of electron, cis the speed of light in vacuum, aðr;z;tÞ refers to the complex amplitude, ^/is the unit vector in the azimuthal direction, k0is the laser wave number, x0is the laser frequency, and c.c. denotes the complex conjugate. In the case of azimuthal polariza- tion,jaj2’jaj2=2.Figure 1 depicts an azimuthally polarized Bessel–Gauss laser beam. For simplicity, the plasmas are assumed to be cold, collisionless, and underdense, and the channel is in the popular parabolic form54,57–63 npðrÞ¼n01þr2 r2 ch ! ; (2) where randrchare the transverse coordinate and the channel radius, respectively, and n0is a characteristic value of the electron density interacting with laser. Under the Coulomb gauge r/C1a¼0, starting from the Maxwell’s equations and the hydrodynamic equations, adopting theweakly relativistic limit (i.e., jaj 2/C281) and the approximation of underdense plasma (i.e., n=nc/C281, where nc¼m0x2 0=ð4pe2Þis the critical density), two coupling equations for an intense laser beampropagating in the parabolic plasma channel described by Eq. (2)are obtained, 54,57,71 r2/C01 c2@2 @t2/C18/C19 a¼x2 p c21þr2 r2 chþdn n0/C0jaj2 2 ! a; (3) @2 @t2þx2 p1þr2 r2 ch !"# dn n0¼c21þr2 r2 ch ! r2jaj2 2; (4) where cis the speed of light in vacuum, xp¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pn0e2=m0p is the plasma frequency, and dnis the transverse electron density perturba- tion arising from the spatial profile of the laser beam. Obviously, Eqs. (3)and(4)are the wave equations for the laser field and the perturbed electron density equation associated with excitation of wakefield.Then, using the long pulse limit (i.e., xps0/C291, where s0is the laser pulse width), employing the paraxial approximation (i.e., j@a=@zj /C28jk0aj) and slowly varying envelope approximation (i.e., j@a=@sj/C28jx0aj), the wave equation for aðr;z;tÞis given by48,54,69,72,73 r2 ?þ2ik0@ @z/C01 r2þk2 pjaj2 4/C0r2 ?jaj2 4/C0k2 pr2 r2 ch"# aðr;zÞ¼0;(5) where the disappearance of tis due to the slowly varying envelop approximation and the long pulse limit and kp¼xp=cis the plasma wave number. Equation (5)is remarkably similar to the usual paraxial equation for circular symmetry, with the exception of the term /C0a=r2, which results directly from the choice of azimuthal polarization.48The first two terms, the fourth term, and the last two terms on the left-hand side of Eq. (5)are associated with the effects of the vacuum dif- fraction, the relativistic self-focusing, and the preformed channel focusing, respectively. The fifth term relates to the effect of the ponder-omotive self-channeling, that is to say, the variation of the electron density dn n0’12k/C02 pr2?jaj2is caused by the ponderomotive force, which affects the laser in turn. The linear dispersion relation is x2 0¼x2 pþk2 0c2. In order to study the evolution of the laser beam, the variation method33,53–56,68is used to solve Eq. (5). Accordingly, the Lagrangian density of the system is as follows: L¼j r ?aj2þik0a@a/C3 @z/C0a/C3@a @z/C18/C19 þjaj2 r2þk2 pr2 r2 chjaj2 /C01 8@jaj2 @x/C18/C192 þ@jaj2 @y !22 43 5/C0k2 pjaj4 8: (6) For simplicity, we study the Bessel–Gauss laser beam with only one ring [as displayed in Fig. 1(a) ]. According to the knowledge of laser focusing characteristics in the plasma channel and Bessel–Gauss laser beam,74we give the following trial function of FIG. 1. The schematic diagram of the azimuthally polarized Bessel–Gauss beam propagating along the z-axis. (a) The transverse amplitude distribution of the vector potential. (b) The amplitude distribution along the radial distance. The normalized ring-beam radius ^Rrefers to the radial distance between the position of the maximal amplitude and the center of the laser beam. The normalized ring-beam width ^Wrrepresents the full width at 1 =eof the maximal value of the amplitude distribution.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-3 Published under license by AIP Publishingthe complex envelope a(r,z) for the Bessel–Gauss laser beam of order one approximately aðr;zÞ¼arzðÞ NjðÞJ1jr RðzÞhðjÞ/C18/C19 e/C0r2=R2zðÞh2jðÞ /C2eib zðÞr2þiuzðÞ(7) withj<3 resulting from the limitation of only one ring, where J1ð/C1Þ is the first-order Bessel function of the first kind, NðjÞ ¼J1ðj hðjÞÞe/C01 hðjÞ2is the normalized coefficient for J1ð/C1Þeð/C1Þin Eq. (7)and is displayed in Fig. 2 (the red curve), ar;R;b;uare real functions of the propagation distance z,arðzÞrefers to the real amplitude of the vector potential, R(z) represents the ring-beam radius of the Bessel–Gauss laser beam, i.e., the radial distance between the position of the maximal amplitude and the center of the laser beam [see Fig. 1(b)],b(z) is the spatial chirp parameter related to the wave-front cur- vature radius, uðzÞis the axis phase factor of the beam, and hðjÞis the maximum positive root of the equation ½2 hðjÞþhðjÞ/C138J1ðj hðjÞÞ /C0jJ0ðj hðjÞÞ¼0 and is displayed by the black curve in Fig. 2 . The ring- beam width WrðzÞof the Bessel–Gauss laser beam shown in Fig. 1(b) can be represented by R(z), i.e., WrðzÞ¼RðzÞHðjÞ; (8) where HðjÞ¼hðjÞðqmaxðjÞ/C0qminðjÞÞ.qmaxðjÞandqminðjÞare, respectively, the maximum and minimum positive solutions of the fol-lowing equation: J 1ðjqÞe/C0q2/C0NðjÞ=e¼0: (9) Equation (8)indicates that the ring-beam width WrðzÞis proportional to the radius R(z). By solving Eq. (9), the proportional coefficient HðjÞis shown in Fig. 2 (the blue curve). Equation (8)also presents the physical meaning of j, i.e., it plays a role in adjusting the ratio between WrðzÞandR(z). So we call jthe laser-profile factor in the following.Substituting Eq. (7)into Eq. (6)and integrating the Lagrangian density transversely, i.e., L¼Ð1 0Lrdr, the Lagrangian of the system is reduced, which is shown in the Appendix .T h e n ,u s i n gt h e Euler–Lagrange equation@ @z@L @ljz/C16/C17 /C0@L @lj¼0, where lj¼ar,R,b, uðj¼1;2;3;4Þ,a n d ljz¼@lj=@z, the motion equations are obtained @ @zarRðÞ¼0; (10) k0@R @z¼2bR; (11) k0@b @z¼2F1ðjÞ R4h4ðjÞ/C02b2/C0k2 p 2r2 ch/C02F3ðjÞa2 r R4h4ðjÞN2ðjÞ /C0F1ðjÞk2 pa2r 2F2ðjÞR2h2ðjÞN2ðjÞ; (12) k0@u @z¼/C0S1ðjÞ 2R2h2ðjÞþF3ðjÞS1ðjÞa2 r 2F1ðjÞR2h2ðjÞN2ðjÞ þ3S1ðjÞk2 pa2r 16F2ðjÞN2ðjÞ; (13) where F1ðjÞ;F2ðjÞ;F3ðjÞ,S1ðjÞare F1ðjÞ¼I0j2 4/C18/C19 þI1j2 4/C18/C19 I0j2 4/C18/C19 /C0I1j2 4/C18/C19 ; (14) F2ðjÞ¼j2I0j2 4/C18/C19 þI1j2 4/C18/C19/C20/C21 4X4jðÞe/C0j2 4; (15) F3ðjÞ¼84X1jðÞ/C04X2jðÞþX3ðjÞ/C2/C3 j2I0j2 4/C18/C19 /C0I1j2 4/C18/C19/C20/C21 ej2 4; (16) S1ðjÞ¼j2I0j2 4/C18/C19 þI1j2 4/C18/C19/C20/C21 I1j2 4/C16/C17 : (17) Equation (10) indicates the power conservation relation 1 2k0h2ðjÞI1j2 4/C16/C17 e/C0j2 4a2 rR2¼1 2k0h2ðjÞI1j2 4/C16/C17 e/C0j2 4a2 0R20,w h e r e a0and R0are the initial ( z¼0) amplitude and ring-beam radius of the Bessel–Gauss laser beam, respectively. Noting arR¼a0R0and combin- ing Eqs. (11)and(12), the normalized equation describing the evolution of the ring-beam radius of the Bessel–Gauss laser beam is given by @2^R @^z2¼F1jðÞ ^R3/C0F2jðÞh2jðÞN2jðÞp a2 0R0 rch/C18/C192 ^R /C0F1jðÞp ^R3/C0a2 0F3jðÞ N2jðÞ^R5; (18) where p¼k2 pa2 0R20h2ðjÞX4ðjÞ N2ðjÞj2½I0j2 4ðÞþI1j2 4ðÞ/C138ej2 4is the normalized laser power and R0=rchrepresents the ratio of initial ring-beam radius to channelFIG. 2. The relationship between the laser-profile factor jand the normalized coef- ficient NðjÞ(the red curve), the proportional coefficient HðjÞ(the blue curve), and the parameter hðjÞ(the black curve), respectively.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-4 Published under license by AIP Publishingradius. In obtaining Eq. (18), the normalization ^z!z=ZR; ^R!R=R0,p!P=Pchave been used, where ZR¼k2 0R20h2ðjÞ=2i s the Rayleigh length, Pc¼½F2ðjÞI1j2 4/C16/C17 e/C0j2 4/C138=k2 p/C2x2 4cmec2 e/C16/C172 is the critical power for relativistic self-focusing in plasmas. The terms on the right-hand side of Eq. (18) arise from vacuum diffraction, pre- formed channel focusing, relativistic self-focusing, and ponderomotive self-channeling. In Eq. (18), the contribution of azimuthal polarization [/C0a=r2in Eq. (5)] is contained in the first vacuum diffraction term F1ðjÞ=R3.S i n c e F1ðjÞ>0 and the contribution of azimuthal polari- zation is proportional to /C01=R3,t h et e r m /C0a=r2caused by azimuthal polarization in Eq. (5)plays a role in weakening the vacuum diffrac- tion of the laser beam. It can be seen from Eq. (18)thatjhas an influ- ence on the vacuum diffraction, relativistic self-focusing, ponderomotive self-channeling, and preformed channel focusing effects in the laser evolution. Specific analysis shows that the vacuumdiffraction, relativistic self-focusing, and ponderomotive self-channeling effects become stronger with the increase in j, while the preformed channel focusing effect becomes weaker. In addition, the influence of jon the preformed channel focusing effect is relatively larger than that on other effects. For the critical power P c, the term F2ðjÞI1j2 4/C16/C17 e/C0j2 4becomes larger with the increase in j,w h i c hl e a d st o the larger critical power Pc. The evolution of the normalized ring-beam radius ^Rcan be obtained by solving Eq. (18), and the corresponding normalized ring- beam width ^Wr, which is also normalized by R0,i st h e ng i v e nb y Eq.(8). III. PROPAGATION CHARACTERISTICS A. Evolution types as well as the corresponding physical conditions and characteristic quantities The nature of propagation and the variations of ring-beam radius and width can be obtained by analyzing the properties of the solutions of Eq. (18). As we can see, Eq. (18)is a second order differential equa- tion with the non-linear terms namely ^R/C03and^R/C05besides the linear term namely ^R. The properties of the solutions of this kind of equation can be analyzed and obtained by using the Sagdeev potentialmethod. 54The corresponding analysis54tells us that the behaviors of the laser beam depend on the following critical laser parameters pt,pm, and critical value ðR0=rchÞ/C3ofðR0=rchÞ: pt¼/C0a2 0F3jðÞ 2F1jðÞN2jðÞþ1þF2jðÞF3jðÞh2jðÞ F2 1jðÞR0 rch/C18/C192 /C0R0 F2 1jðÞNjðÞrch/C20 F2jðÞF3jðÞh2jðÞ /C2/C18 /C0a2 0F1jðÞF3jðÞþ2F2 1jðÞN2jðÞ þF2jðÞF3jðÞh2jðÞN2jðÞR0 rch/C18/C192/C19/C211=2 ; (19) pm¼/C0a4 0F3jðÞþa2 0F1jðÞN2jðÞ a2 0F1jðÞþF2jðÞh2jðÞN2jðÞR0 rch/C18/C192"# /C21 N2jðÞ; (20)R0 rch/C18/C19/C3 ¼a2 0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F1jðÞF3jðÞp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C03a2 0F2jðÞF3jðÞþ2F1jðÞF2jðÞN2jðÞp /C21 hjðÞNjðÞ: (21) Thus, for a collimated incident laser beam, the evolution types of the Bessel–Gauss laser beam in the plasma channel are obtained asfollows: (1) Constant radius and width. When the initial laser and plasma parameters satisfy p¼p mand R0=rch>ðR0=rchÞ/C3, the laser beam propagates with a constant ring-beam radius and widthin the plasma channel. (2) Synchronous periodic focusing oscillation of the radius and width. When p m<p<ptand R0=rch>ðR0=rchÞ/C3,t h er i n g - beam radius and width of laser beam oscillate periodically and synchronously between their own minimal and initial values.The corresponding spatial wavelength of the oscillation isgiven by K¼2ð 1 ^R2d^Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C02V^RðÞp ; (22) where ^R2¼( 1 4F2jðÞh2jðÞN4jðÞpR0 rch/C18/C192 /C2/C20 Npþ/C18 N2 p/C08a4 0F2jðÞF3jðÞh2jðÞN4jðÞpR0 rch/C18/C192/C191 2/C21)1 2 ; (23) is the minimum radius with Np¼2a2 0N2ðjÞF1ðjÞð1/C0pÞ /C0F3ðjÞa4 0andVð^RÞis the Sagdeev potential given by V^RðÞ¼F1jðÞ1/C0pðÞ 2^R2þF2jðÞh2jðÞN2jðÞp 2a2 0 /C2R0 rch/C18/C192 ^R2/C0F3jðÞa2 0 4N2jðÞ^R4/C0V0; (24) with V0¼F1jðÞ1/C0pðÞ 2þF2jðÞh2jðÞN2jðÞp 2a2 0R0 rch/C18/C192 /C0F3jðÞa2 0 4N2jðÞ:(25) The corresponding amplitudes of the ring-beam radius and width are, respectively, 1/C0^R2 (26) and HðjÞ1/C0^R2/C0/C1 : (27) F o rt h i sk i n do fp r o p a g a t i o nb e h a v i o r , a/C21a0in the process of the laser evolution. To prevent afrom exceeding the weakly relativistic limit in evolution, p<p/C3is required by using the power conservation relation a2 0^R2 0¼a2^R2and jaj2<1, wherePhysics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-5 Published under license by AIP Publishingp/C3¼F3jðÞ1þa2 0/C0/C1 /C02F1jðÞN2jðÞ 2N2jðÞ F1jðÞþF2jðÞh2jðÞN2jðÞR0 rch/C18/C192"# : (28) (3) Synchronous periodic defocusing oscillation of the radius and width. If p<pmfor any R0=rch, the ring-beam radius and width of laser beam oscillate periodically and synchronously betweentheir own maximal and initial values. The corresponding spatial wavelength of the oscillation is given by K¼2ð ^R2 1d^Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /C02V^RðÞp ; (29) where ^R2and Vð^RÞare given by Eqs. (23) and (24), respec- tively. Here, ^R2>1 represents the maximum radius. The amplitudes of the ring-beam radius and width are, respectively, ^R2/C01 (30) and HjðÞ^R2/C01/C0/C1 : (31) Besides the above three evolution types, the Sagdeev potential analysis54also displays other two types, but they are beyond the weakly relativistic limit and out of our discussion. Concretely, when p>pt for any R0=rchorpm<p/C20ptforR0=rch/C20ðR0=rchÞ/C3, the ring-beam radius and width of the Bessel–Gauss laser beam tend to be zero, i.e., catastrophic focusing of the laser beam is caused. Of course, this singu-larity never occurs because when the ring-beam radius and width are focused to be zero, the intensity increases infinitely which is obviously beyond the weakly relativistic limit. If p¼p tandR0=rch>ðR0=rchÞ/C3 orp¼pmandR0=rch<ðR0=rchÞ/C3, the ring-beam radius and width of the Bessel–Gauss laser beam are solitary waves. However, owing to thestrong focusing effect, the maximum peak of the vector potential is usually greater than 1 in propagation, which is also beyond the weakly relativistic limit. B. The parameter regions and factors affecting the parameter regions According to the above analysis, if the values of the initial ampli- tude a 0and laser-profile factor jare given, the evolution of the Bessel–Gauss laser beam is completely dependent on the initial laserpower pand ratio of initial ring-beam radius to channel radius R 0=rch. In order to gain a full view of the evolution types, the critical laser parameters pmandp/C3vsR0=rchare plotted in Fig. 3 according to Eqs. (19),(20),a n d (28), respectively. In this figure, the blue curve, which satisfies p¼pmforR0=rch>ðR0=rchÞ/C3, denotes the propagation with a constant ring-beam radius ^Rand width ^Wr. The black curve satisfy- ingp¼p/C3represents the weakly relativistic limit. The area p<p/C3we are concerned with is divided by the blue curve (related to the constant ^Rand ^Wr) into two parts corresponding to the synchronous periodic defocusing and focusing oscillations of ^Rand ^Wrrespectively (see Fig. 3). The blue curve for propagating with the constant ^Rand ^Wrlooks like a hyperbolic. That is, the laser initial power pdecreases sharply with the increase in the ratio R0=rchbefore R0=rch/C250:15 and decreases slowly with R0=rchafter R0=rch/C250:15. The blue curve tells us that, for the Bessel–Gauss laser beam with a high initial power, thecorresponding small ratio is required for keeping constant ^Rand ^Wr in propagation. On the contrary, for the Bessel–Gauss laser beam with a low initial power, a large ratio is needed. The two parameter partsdivided by the blue curve indicate that if porR 0=rchis somewhat larger (smaller) than that for constant ^Rand ^Wr,t h el a s e rr a d i u sa n d width will be synchronous periodic focusing (defocusing) oscillation. T h a ti sb e c a u s ew h e nt h ev a l u eo f pis given, the preformed channel focusing effect will become stronger with the increase in R0=rch,w h i c h will cause a phenomenon that the laser beam evolves from the syn- chronously periodic defocusing oscillation to the propagation of con-stant ^Rand ^W rand then to the focusing oscillation. Similarly, when the value of R0=rchis given, the effects of relativistic self-focusing and preformed channel focusing will both become stronger with the increase in p, which will also lead to the phenomenon. Now, we pay attention to the effects of the initial amplitude a0 and laser-profile factor jon the above parameter regions. Figures 4(a) and4(b)display the parameter regions with different a0andj, respec- tively. As can be seen from Fig. 4(a) ,w h e n a0becomes larger, the curve ofpmfor constant ^Rand ^Wrhas a relatively large upward shift around R0=rch¼0:15 but basically unchanges for R0=rch>0:60, while the curve of p/C3hardly changes for any R0=rch. As can be noted from Fig. 4(b), with the increase in j,t h ec u r v e so f pmandp/C3have very little upward shift. That is, jhas little effect on the parameter regions (or the evolution type), which is quite different from the results of the dark hollow laser beam in Refs. 30–33 and35where evolution types strongly depend on the initial laser profile. C. The effects of the parameters on the propagation behaviors To see the influence of the initial power pand ratio of initial ring-beam radius to channel radius R0=rchon the propagation behav- iors, the corresponding amplitudes and spatial wavelength of ring-beam radius ^Rand width ^W rvspandR0=rchare, respectively, plotted inFigs. 5(a1)–5(a2) andFigs. 5(b1)–5(b2) according to Eqs. (30),(31), (22),a n d (29),w h e r e a0¼0:3a n d j¼1:0. It should be mentioned FIG. 3. Parameter regions for the different evolution types. In all cases, a0¼0:3;j¼1:0.R0=rchðÞ/C3¼0:0 054 883 is obtained according to Eq. (21) and the eight points labeled (1)–(8) are selected for the numerical studies given in Sec. IVand the particle-in-cell simulation given in Sec. V.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-6 Published under license by AIP PublishingFIG. 4. Parameter regions for different initial amplitudes a0and laser-profile factors j. (a) shows pmandp/C3withj¼1:0 for a0¼0:2, 0.3, and 0.35. (b) Displays with a0¼ 0:3 forj¼0:2, 1.0, and 2.5. FIG. 5. The effects of the initial power pand the ratio R0=rchon the propagation behaviors under the weakly relativistic limit, where a0¼0:3 and j¼1:0. The amplitudes of the ring-beam radius ^Rand width ^Wrvs the initial power por the ratio R0=rchare displayed, respectively, by the red and blue curves in (a1) and (a2) and the variations of their spatial wavelengths are shown in (b1) and (b2). In (a1) and (a2), if ^R2/C01>0 and HðjÞð^R2/C01Þ>0, which means the synchronous periodic defocusing oscillation. If ^R2/C01<0 and HðjÞð^R2/C01Þ<0, which means the synchronous periodic focusing oscillation. Their absolute values represent the amplitudes of ^Rand^Wr. The black circles marked in (a1) and (a2) correspond to the propagation with constant ^Rand^Wr, i.e., the amplitudes are zero and the white circles marked in (b1) and (b2) are the dis- continuity point corresponding to the propagation with constant ^Rand^Wr. The remaining points labeled by squares, stars, and red circles represent the amplitudes and spatial wavelengths of ^Rand^Wr, respectively, and are the results of the following numerical simulations given in Sec. IV. For the numerical results, the errors of amplitudes and spa- tial wavelengths are 0.04 and 0.0002, respectively.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-7 Published under license by AIP Publishingthat we do not give the amplitudes directly but give ^R2/C01a n d HðjÞð^R2/C01Þinstead, which show not only the amplitudes by their absolute values but also the type of oscillation (synchronous periodic defocusing or focusing oscillations) by their plus and minus signs. Itc a nb es e e nt h a t ^Rand ^W rare the synchronous periodic defocusing oscillations with relatively large amplitudes and spatial wavelengthwhen porR 0=rchis small. As porR0=rchincreases, the amplitudes and spatial wavelength of ^Rand ^Wrdecrease sharply, when the spatial wavelength drops to 2.9363, the amplitudes decrease almost to zero,and then the laser beam enters the propagation with a constant ^Rand^W rwhen p¼0.0864 or R0=rch¼0:2 labeled by the black and white circles. As porR0=rchcontinues increasing, ^Rand ^Wrconvert to the synchronous periodic focusing oscillation. Their amplitudes increase slowly with porR0=rch, while the spatial wavelength continues decreasing slowly, following the value of the synchronous periodicdefocusing oscillation 2.9363. In order to see the effect of the laser-profile factor jon the ampli- tudes and spatial wavelength of ^Rand ^W r, we selected three sets of parameters ( p,R0=rch)w i t h a0¼0:3 labeled (5), (6), and (7) in Fig. 3 and plotted the corresponding amplitudes and spatial wavelengths vs j FIG. 6. The effect of the laser-profile factor jon the propagation behaviors under the weakly relativistic limit, where three sets of parameters ( p,R0=rch) we selected in (a1)–(a3) correspond to the points labeled by (5)–(7) inFig. 3 anda0¼0:3. The amplitudes of the ring-beam radius ^Rand width ^Wrvs the laser-profile factor jare dis- played, respectively, by the red and blue curves in (a1)–(a3) and the variation of their spatial wavelengths is shown in (b1)–(b3). The meanings of the positive and negative ^R2/C01 and HðjÞð^R2/C01Þas well as their absolute values, and the marked points are the same as those shown in Fig. 5 . For the numerical results, the errors of amplitudes and spatial wavelengths are 0.04 and 0.0002, respectively. Since the errors cannot be obviously displayed in (a2) and (b1)–(b3), the error bars are on ly added in (a1) and (a3).Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-8 Published under license by AIP PublishinginFigs. 6(a1)–6(a3) and6(b1)–6(b3) , respectively. It can be seen from Figs. 6(b1)–6(b3) that whether ^Rand ^Wrof the laser beam are the syn- chronous periodic defocusing oscillation, the synchronous periodic focusing oscillation or the oscillation near the propagation with the con- stant ^Rand ^Wr, their spatial wavelengths decrease obviously with the increase in jand display similar variations, i.e., they show an approxi- mately parabolic decrease when j<1b u tl i n e a rd e c r e a s ew h e n j>1. Figures 6(a1)–6(a3) show that the effect of jo nt h ea m p l i t u d e si sv e r y weak. Concretely, for the synchronous periodic defocusing (focusing) oscillation, their amplitudes increase (decrease) with jslightly. IV. NUMERICAL SIMULATION AND DISCUSSION In this section, Eq. (5)will be solved numerically to test the valid- ity of theoretical analysis and display the propagation characteristics directly. For this purpose, Eq. (5)is normalized as /C264i h2jðÞ@ @^zþ1 ^r@ @^r^r@ @^r/C18/C19 /C01 ^r2/C04F2jðÞp a2 0h2jðÞR0 rch/C18/C192 ^r2 /C01 4^r@ @^r^r@ @^rjaj2/C18/C19 þN2jðÞF2jðÞp a2 0h2jðÞjaj2/C27 /C2a^r;^zðÞ¼0;(32)where the evolution of the Bessel–Gauss laser beam is assumed to be axisymmetric and the normalization is similar to Eq. (18), i.e., ^r;^z, andpare normalized by the initial radius R0, Rayleigh length ZR,a n d critical laser power Pc, respectively. By using the initial profile of the Bessel–Gauss laser beam að^r;0Þ¼a0 NðjÞJ1ðj^r=hðjÞÞe/C0^r2=h2ðjÞ,w h e r e a0¼0:3 in this part, Eq. (32) is numerically solved by means of the central difference method with a fourth-order Runge–Kutta methodto advance along the propagation distance ^z. 72,73 To test the validity of the parameter regions shown in Fig. 3 ,w e selected three representative points labeled (1)–(3) corresponding to the propagation with the constant ^Rand ^Wr, synchronous periodic defocusing and focusing oscillations of ^Rand ^Wr, and the correspond- ing numerical simulation results are shown in Figs. 7(a1)–7(c1) , 7(a2)–7(c2) ,a n d 7(a3)–7(c3) , respectively. It can be seen that the evo- lution of the Bessel–Gauss laser beam is exactly consistent with theprediction of the theoretical parameter figure. Furthermore, we mea-sured the corresponding amplitudes and spatial wavelengths of ^Rand ^W r. For synchronous periodic defocusing oscillation, the amplitudes of^Rand ^Wrare 0.88 and 1.59, respectively, and their spatial wave- length is 5.40, which are very close to the theoretical values 0.850,1.606, and 5.451 according to Eqs. (30),(31),a n d (29), respectively. FIG. 7. The modulus of the laser slow-varying envelope jað^r;^zÞjon the plane ð^r;^zÞfora0¼0:3 and j¼1:0, where ^ris the radial coordinate. (a1)–(c1), (a2)–(c2), and (a3)–(c3) correspond to the points (1)–(3) labeled in Fig. 3 , where (b1)–(b3) and (c1)–(c3), displaying the changes of the radius ^Rand width ^Wrwith the propagation distance ^z, are extracted from (a1)–(a3), respectively.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-9 Published under license by AIP PublishingFor synchronous periodic focusing oscillation, the measured ampli- tudes of 0.64 and 1.04 and the spatial wavelength of 1.30 are also very close to the theoretical values 0.561, 1.060, and 1.295, respectively. These results confirm the theoretical formulas for amplitudes and spa-tial wavelength, i.e., Eqs. (30),(31),a n d (29). To check the effect of the initial laser power pon the propagation behavior, we set the ratio of initial ring-beam radius to channel radius R 0=rch¼0:2 and selected five points marked as (4)–(8) inFig. 3 ,a n dthe corresponding numerical simulation results are displayed in Fig. 8 . Under the parameters marked by point (4), it can be seen from Figs. 8(a1)–8(c1) that the Bessel–Gauss laser beam evolves with synchro- nous periodic defocusing oscillation, just as the prediction of theoreti-cal parameters as shown in Fig. 3 . With the increase in p,i ti s noted from Figs. 8(a2)–8(c2) that the amplitudes and spatial wavelength become smaller. Then, the laser beam enters the propaga- tion with constant ^Rand ^W rwhen pincreases to 0.0864 [shown in FIG. 8. The modulus of the laser slow-varying envelope jað^r;^zÞjon the plane ð^r;^zÞfor different pwitha0¼0:3 and j¼1:0, where ^ris the radial coordinate. (a1)–(c1), (a2)–(c2), (a3)–(c3), (a4)–(c4), and (a5)–(c5) correspond to the points (4)–(8) labeled in Fig. 3 , where (b1)–(b5) and (c1)–(c5), displaying the changes of the radius ^Rand width ^Wrwith the propagation distance ^z, are extracted from (a1)–(a5), respectively.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-10 Published under license by AIP PublishingFigs. 8(a3)–8(c3) ]. As pcontinues to increase, the evolution of the laser beam converts to the synchronous periodic focusing oscillation andthe corresponding amplitudes become larger, while the spatial wave- length continues becoming shorter too [shown in Figs. 8(a4)–8(c4) andFigs. 8(a5)–8(c5) ], following the value of the synchronous periodic defocusing oscillation. The values of the corresponding amplitudes and spatial wavelengths of ^Rand ^W rshown in Fig. 8 are measured and marked in Figs. 5(a1) and5(b1) by the squares, stars, and red circles, respectively. It is found that the amplitudes and spatial wave-lengths of ^Rand ^W rare in good agreement with the theoretical values in the range of error permission. Similarly, numerical simulation for the influence of the ratio R0=rchon propagation behavior is also done with p¼0.0864 for R0=rch¼0:085, 0.12, 0.2, 0.3, and 0.4 and the measured values of the corresponding amplitudes and spatial wave- lengths are marked in Figs. 5(a2) and5(b2) by the same symbols as before. It can be seen that the evolution of the laser beam with R0=rch is also consistent with the theory exactly. To test the influence of the laser-profile factor jon the amplitudes and spatial wavelengths of ^Rand ^Wrshown in Fig. 6 ,w es e l e c t e d j¼0:05, 0.5, 1.6, and 2.6 for numerical simulations. For the obvious synchronous periodic defocusing and focusing oscillations, we directly measured the values of the amplitudes and spatial wavelengths and then marked them by symbols, respectively, in Figs. 6(a1)–6(b1) andFigs. 6(a3)–6(b3) . For the oscillation near the propagation with constant ^R and ^Wr, our program cannot exceed the precision of one percent, the magnitude of the amplitudes for ^Rand ^Wr, in a limited simulation time. But for j¼1:6a n d2 . 6w i t hr e l a t i v e l yl a r g e ra m p l i t u d e so f ^Rand ^Wr, we can observe slight fluctuation and the spatial wavelengths can also be measured and marked by symbols as shown in Fig. 6(b2) .I ti s found that the amplitudes and spatial wavelengths of ^Rand ^Wrof the numerical simulation are in good agreement with the theory. V. PARTICLE-IN-CELL SIMULATION FOR THE LASER BEAM Although it is almost impossible for us to carry out a three- dimensional (3D) particle-in-cell (PIC) simulation for an azimuthallypolarized Bessel–Gauss laser beam in plasmas because of the huge amount of computation, we still find that 2D PIC simulation can reflect the above main conclusions. Here, by using the electromagneticmodel of the commercial software VSim (the relativistic, arbitrarydimensional, hybrid plasma, and beam simulation code from Tech-X Corporation), the PIC simulation is done in a cross section with /¼0 approximately for the azimuthally polarized laser beam, that is, the laser beam is polarized in the yd i r e c t i o na n de v o l v e si nt h e plane ( x,z), where zcorresponds to the direction of propagation and t h es i z eo ft h eg r i di n xand zdirections is 2 :198/C210 /C07ma n d 6:656/C210/C08m, respectively. For more stable propagation of the laser beam, the parabolic plasma channel nðxÞ¼n0½0:1þx2 r2 ch/C138with n0¼1:48 /C21025m/C03is adopted. A Bessel–Gauss laser beam with a wavelength k¼1064 nm, i.e., E¼a0x0mece/C01N/C01ðjÞexp½/C0x2=ðR2 0h2ðjÞÞ/C138 /C2J1½jx=ðR0hðjÞÞ/C138sinðxtÞ^y, will be injected into the channel at z¼0 and from time t¼0, where x0¼2pc=1064 nm ¼1:77/C21015s/C01is the frequency and k¼x0=cis the wave number of the laser beam. InFig. 9 , the propagation of the Bessel–Gauss laser beam in the parabolic plasma channel for different initial laser powers pis simu- lated, where the laser-profile factor j¼1:0, the ratio of initial ring- beam radius to channel radius R0=rch¼0:2, and the peak intensity ofthe Bessel–Gauss laser beam is up to 1 :09/C21017w=cm2(a0¼0:3). The parameters adopted in Figs. 9(a)–9(e) are basically the same as those marked by (4)–(8) inFig. 3 , respectively. It can be seen that the evolution type of the Bessel–Gauss beam is consistent with the predic-tion of the 3D theoretical parameter given by Fig. 3 and the variations FIG. 9. Contour plots of the electric field Eyfrom a two-dimensional planar simula- tion for the propagation of a Bessel–Gauss laser beam in a preformed parabolic plasma channel under different pwitha0¼0:3 and R0=rch¼0:2. The initial radius R0of the laser beam in (a)–(e) are 4 :183/C210/C06m, 5:477/C210/C06m;6:596 /C210/C06m, 1:000/C210/C05m, and 1 :225/C210/C05m, respectively.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-11 Published under license by AIP Publishingof the amplitudes and spatial wavelength of ^Rand ^Wrare also in good agreement with the 3D theory. The PIC simulation of the effect of the ratio R0=rchon the propagation behavior is also done, where the parameters we selected are the same as those shown in Fig. 5(a2) .I ti s found that the evolution type and the variations of the amplitudes andspatial wavelength are also consistent with the previous theory. Finally, we measured the corresponding amplitudes and spatial wavelength of ^Rand ^W rin the PIC simulations and listed them in Tables I andII with the theoretical values. It is interesting that the results of the 2D PIC simulations are almost identical to the theoretical 3D results. In this part, the influence of the initial amplitude a0on propaga- tion behavior is studied. The 3D theoretical study shown in Fig. 4(a) shows that the curve of pmfor constant ^Rand ^Wrhas a relatively large upward shift when a0becomes larger. That is, if a Bessel–Gauss laser beam propagates with constant ^Rand ^Wrfor a value of a0,i tw i l lb e the synchronous periodic defocusing oscillation with a larger a0and the synchronous periodic focusing oscillation with a smaller a0. Therefore, we performed PIC simulations with p¼0.087 and R0=rch¼0:2f o r a0¼0:35, 0.30, and 0.20, respectively, in Fig. 10 , where a0¼0:30 corresponds to the propagation with constant ^Rand ^Wr. It is found from Figs. 10(a) and10(c) that the Bessel–Gauss laser beam behaves as the synchronous periodic defocusing oscillation of ^R and ^Wrwith the larger a0¼0:35 and focusing oscillation with the smaller a0¼0:20 obviously. That is to say, the results of the 2D PIC simulations show that a0has a significant influence on the evolution type of the Bessel–Gauss laser beam and are consistent with those of the 3D theoretical analysis in Sec. III. The effect of the laser-profile factor jon the amplitudes and spatial wavelengths of the synchronous periodic defocusing and focusing oscillations is simulated, where the parameters we selected are the same as those shown in Figs. 6(a1) and6(a3) .I ti s found that the spatial wavelengths decrease obviously but the amplitudes change very little with jand the variations are consis- tent with the previous theory. The measured amplitudes andspatial wavelengths of ^Rand ^Wrare listed in Table III with the theoretical values. It can be also seen that the results of the 2D PIC simulations are almost identical to the theoretical 3D results, espe-cially the spatial wavelength. It is well known that the channel density will change after laser propagation, in order to understand the variations of the electron den-sity in the parabolic plasma channel directly, they are measured at dif- ferent positions and compared with the theory. It is found that the density decreases slightly around the place where the light intensity isrelatively large especially at x¼R, while it increases slightly around the place where the light intensity is relatively small especially at x¼0, which is caused by the ponderomotive force. The magnitudes of thevariations are about one percent at x¼Rand x¼0 and about one thousandth or less in other places, which are approximately consistent with the theory used in Sec. IIin Eq. (5), i.e., dn n0’12k/C02 pr2?jaj2for the ponderomotive self-channeling effect. VI. CONCLUSIONS In this paper, by taking the azimuthally polarized first-order Bessel–Gauss laser beam as an example, the propagation dynamics of the dark hollow laser beam under the weakly relativistic limit is inves- tigated in the parabolic plasma channel. Considering the effects of therelativistic self-focusing, ponderomotive self-channeling, and pre- formed channel focusing, the evolution equation of the ring-beam radius is obtained by the variational analysis, and the ring-beam widthis proportional to and synchronous with the radius. It is also foundthat the azimuthal polarization can weaken the vacuum diffraction effect. Using the Sagdeev potential method, we find that the evolution of this dark hollow laser beam can be classified into three types, i.e.,the propagation with a constant ring-beam radius and width, the syn- chronous periodic defocusing oscillation, and the synchronous peri- odic focusing oscillation. The physical condition of each type and theexpressions of the corresponding characteristic quantities, such as theamplitudes and spatial wavelength of the ring-beam radius and widthTABLE I. Comparison between theoretical and PIC results for different p, where K,A^R, and A^Wrepresent the spatial wavelength, amplitudes of ^R, and ^Wr, respectively, and DK;DA^R, andDA^Ware the difference values for K,A^R, and A^Wbetween the theory and PIC results, respectively. pKK D K A^R A^R DA^R A^W A^W DA^W Theoretical PIC Difference Theoretical PIC Difference Theoretical PIC Difference 0.035 4.608 4.507 0.101 0.619 0.673 0.054 1.170 1.099 0.071 0.060 3.523 3.449 0.074 0.219 0.196 0.023 0.413 0.430 0.0170.200 1.938 1.893 0.045 0.391 0.350 0.041 0.739 0.615 0.1240.300 1.590 1.522 0.068 0.542 0.445 0.097 1.023 0.901 0.122 TABLE II. Comparison between theoretical and PIC results for different R0=rch, where K,A^R, and A^Wrepresent the spatial wavelength, amplitudes of ^R, and ^Wr, respectively, andDK,DA^R, andDA^Ware the difference values for K,A^R, and A^Wbetween the theory and PIC results, respectively. R0 rchKK D K A^R A^R DA^R A^W A^W DA^W Theoretical PIC Difference Theoretical PIC Difference Theoretical PIC Difference 0.085 6.897 6.908 0.011 1.361 1.350 0.011 2.571 2.715 0.144 0.120 4.889 4.782 0.107 0.671 0.668 0.003 1.268 1.365 0.0970.300 1.964 1.908 0.056 0.337 0.272 0.065 0.637 0.540 0.0970.400 1.476 1.495 0.019 0.507 0.439 0.068 0.957 0.941 0.016Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-12 Published under license by AIP PublishingFIG. 10. Contour plots of the electric field Eyfrom a two-dimensional planar simula- tion with the propagation of a Bessel–Gauss laser beam in a preformed parabolic plasma channel for p¼0.087 andR0=rch¼0:2 with the decrease in the initial amplitude a0. (a) displays the synchronous periodic defocusing oscilla- tion of ^Rand ^Wr, where a0¼0:35 (the peak intensity up to 1 :484/C21017w=cm2) and the initial radius R0 is 5:653/C210/C06m. The parameters of (b) are the same with those of Fig. 9(c) dis- playing the propagation with constant ^R and^Wr. (c) shows the synchronous peri- odic focusing oscillation of ^Rand ^Wr, where a0¼0:20 (the peak intensity up to 4:840/C21016w=cm2) and R0 is 9:893/C210/C06m. TABLE III. Comparison between theoretical and PIC results for different j, where R0=rch¼0:20 and p¼0.060 and 0.200 corresponding to the initial laser power in the regions of the synchronous periodic defocusing and focusing oscillations, respectively. DK;DA^R, andDA^Ware the difference values for K,A^R, and A^Wbetween the theory and PIC results, respectively. j pKK D K A^R A^R DA^R A^W A^W DA^W Theoretical PIC Difference Theoretical PIC Difference Theoretical PIC Difference 0.05 0.06 3.977 3.844 0.133 0.215 0.189 0.026 0.408 0.374 0.034 0.20 2.189 2.168 0.021 0.393 0.399 0.006 0.744 0.610 0.134 0.50 0.06 3.856 3.774 0.082 0.215 0.207 0.008 0.408 0.414 0.006 0.20 2.123 2.083 0.040 0.393 0.399 0.003 0.743 0.591 0.152 1.60 0.06 2.953 2.827 0.126 0.236 0.222 0.014 0.440 0.419 0.021 0.20 1.625 1.619 0.006 0.382 0.328 0.054 0.715 0.581 0.134 2.60 0.06 1.980 1.980 0.000 0.291 0.334 0.043 0.524 0.498 0.026 0.20 1.089 1.101 0.012 0.354 0.303 0.051 0.639 0.528 0.111Physics of Plasmas ARTICLE scitation.org/journal/php Phys. Plasmas 27, 113103 (2020); doi: 10.1063/5.0022731 27, 113103-13 Published under license by AIP Publishingfor synchronous periodic defocusing and focusing oscillations, are obtained. The figure of parameter regions for the evolution types isplotted in the R 0=rch–pplane, where R0=rchandpare the ratio of initial ring-beam radius to channel radius and initial laser power, respectively.The effects of the initial laser amplitude and laser-profile factor on theparameter regions (or the evolution type) are studied. It is found thatthe parameter curve of the propagation with a constant ring-beamradius and width has a relatively large upward shift when the initialamplitude becomes larger, while the laser-profile factor has little influ-ence on the parameter regions (or the evolution type). The impacts of the initial laser power and the ratio on the propagation behaviors are also investigated. It is shown that the ring-beam radius and width arethe synchronous periodic defocusing oscillation with relatively largeamplitudes and spatial wavelengths when the initial laser power or theratio is small. With the increase in the initial laser power or the ratio,the amplitudes and spatial wavelength decrease sharply, when theamplitudes decrease almost to zero, and then the laser beam enters thepropagation with a constant ring-beam radius and width. As the initiallaser power or the ratio continues increasing, the ring-beam radius andwidth convert to the synchronous periodic focusing oscillation. Theiramplitudes increase slowly with the initial laser power or ratio, whilethe spatial wavelength continues decreasing slowly, following the valueof spatial wavelength for the synchronous periodic defocusing oscilla-tion. The investigation about the effect of the laser-profile factor on the amplitudes and spatial wavelength shows that the spatial wavelength obviously decreases with the increase in the laser-profile factor for boththe synchronous periodic defocusing and focusing oscillations, whilethe influence of the laser- profile factor on the amplitudes is very weak. The numerical simulation of the basic wave equation verifies the three kinds of propagation behaviors, the correctness of the parameterregions (or the evolution type), and the effects of the initial laserpower, ratio of initial ring-beam radius to channel radius, and laser-profile factor on the propagation behaviors. The 2D PIC simulation for the azimuthally polarized Bessel–Gauss laser beam in a preformed parabolic plasma channel is finally performed and also reveals these conclusions. It is interesting that the main characteristic quantitiessuch as the amplitudes and spatial wavelengths are almost consistentwith the 3D theoretical values approximatively. Considering the importance and wide application of the dark hollow lasers, for example, exciting the ring-shaped wake-field to trapthe ionized electrons in order to generate a high-quality hollow elec-tron beam, 10reducing the initial emission solid angle of the target nor- mal sheath acceleration source to accelerate the ions better,12and accelerating the elections more efficiently in vacuum with cylindrical vector polarization,41this work may not only provide a reference for other theoretical and experimental research on the laser-plasma inter-action, but also be useful in the field of the optical guiding, high har-monic generation, particle acceleration, etc. ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Nos. 11765017, 11865014, 11847304, and11764039), by the Scientific Research Project of Gansu HigherEducation (No. 2019B-034), by the Natural Science Foundation ofEducation Department of Guizhou Province of China (Grant No. Qianjiaohe-KY-[2017]301), and by the Science and Technology Project of Guizhou Province of China (Grant No. Qiankehe-LH-[2017]7008). APPENDIX: THE LAGRANGIAN OF THE SYSTEM Substituting Eq. (7) into Eq. (6) and integrating the Lagrangian density transversely, i.e., L¼Ð1 0Lrdr, the Lagrangian of the system is reduced to L¼1 8a2 rðzÞ N2ðjÞe/C0j2 4j2þj2b2ðzÞR4ðzÞh4ðjÞ/C2/C3 I0j2 4/C18/C19 þj2/C0j2b2ðzÞR4ðzÞh4ðjÞ/C2/C3 I1j2 4/C18/C19 /C26/C27 þ1 16k0a2 rðzÞ N2ðjÞR2ðzÞh2ðjÞe/C0j2 4/C2j2R2ðzÞh2ðjÞ@b @zI0j2 4/C18/C19 /C0I1j2 4/C18/C19/C20/C21 þ8@u @zI1j2 4/C18/C19 /C26/C27 þj2k2 pa2rðzÞR4ðzÞh4ðjÞ 32r2 chN2ðjÞe/C0j2 4I0j2 4/C18/C19 /C0I1j2 4/C18/C19/C20/C21 þa4 rðzÞ N4jðÞ/C02X1jðÞþ2X2jðÞ/C01 2X3jðÞ/C20/C21 /C01 8a4 rðzÞ N4jðÞk2 pX4jðÞR2ðzÞh2jðÞ; (A1) where I0ð/C1Þand I1ð/C1Þare the zeroth-order and first-order modified Bessel functions of the first kind respectively, X1jðÞ;X2ðjÞ; X3ðjÞ;X4ðjÞare X1ðjÞ¼ð1 0~r3e/C04~r2J4 1ðj~rÞd~r; (A2) X2ðjÞ¼ð1 0~r2e/C04~r2J3 1ðj~rÞjJ0ðj~rÞ/C01 ~rJ1ðj~rÞ/C20/C21 d~r; (A3)X3ðjÞ¼ð1 0~re/C04~r2J2 1ðj~rÞjJ0ðj~rÞ/C01 ~rJ1ðj~rÞ/C20/C212 d~r; (A4) X4ðjÞ¼ð1 0~re/C04~r2J4 1ðj~rÞd~r; (A5) where ~r¼r=RðzÞhðjÞ.Physics of Plasmas ARTICLE scitation.org/journal/php Phys. 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5.0019178.pdf

J. Chem. Phys. 153, 121102 (2020); https://doi.org/10.1063/5.0019178 153, 121102 © 2020 Author(s).Cooling molecular electronic junctions by AC current Cite as: J. Chem. Phys. 153, 121102 (2020); https://doi.org/10.1063/5.0019178 Submitted: 22 June 2020 . Accepted: 04 September 2020 . Published Online: 23 September 2020 Riley J. Preston , Thomas D. Honeychurch , and Daniel S. Kosov ARTICLES YOU MAY BE INTERESTED IN Structural and thermodynamic properties of hard-sphere fluids The Journal of Chemical Physics 153, 120901 (2020); https://doi.org/10.1063/5.0023903 Recursive evaluation and iterative contraction of N-body equivariant features The Journal of Chemical Physics 153, 121101 (2020); https://doi.org/10.1063/5.0021116 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/10.1063/5.0018930The Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Cooling molecular electronic junctions by AC current Cite as: J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 Submitted: 22 June 2020 •Accepted: 4 September 2020 • Published Online: 23 September 2020 Riley J. Preston, Thomas D. Honeychurch, and Daniel S. Kosova) AFFILIATIONS College of Science and Engineering, James Cook University, Townsville, QLD 4811, Australia a)Author to whom correspondence should be addressed: daniel.kosov@jcu.edu.au ABSTRACT Electronic current flowing in a molecular electronic junction dissipates significant amounts of energy to vibrational degrees of freedom, strain- ing and rupturing chemical bonds and often quickly destroying the integrity of the molecular device. The infamous mechanical instability of molecular electronic junctions critically limits performance and lifespan and raises questions as to the technological viability of single- molecule electronics. Here, we propose a practical scheme for cooling the molecular vibrational temperature via application of an AC voltage over a large, static operational DC voltage bias. Using nonequilibrium Green’s functions, we computed the viscosity and diffusion coefficient experienced by nuclei surrounded by a nonequilibrium ”sea” of periodically driven, current-carrying electrons. The effective molecular junc- tion temperature is deduced by balancing the viscosity and diffusion coefficients. Our calculations show the opportunity of achieving in excess of 40% cooling of the molecular junction temperature while maintaining the same average current. Published under license by AIP Publishing. https://doi.org/10.1063/5.0019178 .,s I. INTRODUCTION The basic building block for molecular electronics is the single- molecule junction: a molecule chemically linked to two leads. To date, the major achievements in molecular electronics have been in the development of a fundamental understanding of the quan- tum transport in single molecules and ways to control it. However, there is a key scientific challenge to be overcome before the com- mercial potential of these technologies can be realized—the lifetime of molecular devices is notoriously small.1–3The record lifetime achieved this year in a breakthrough experiment is still only 2.7 s,4 which is obviously much shorter than what is expected for feasible post-silicon technology. Electric current flowing from macroscopic leads through a sub-nanometer wide molecular constriction deposits significant amounts of power into the molecular junction. Moreover, the situation is exacerbated by the large molecule–metal contact resis- tance, typically 1 MΩ–100 MΩ, which comes from the mis- alignment of molecular levels and the leads’ Fermi energies. Molecular junctions behave like insulators and require a high volt- age bias for device operation. As a result, the significant opera- tional voltage bias of a few volts across the molecular length alongwith large electric current densities destroys the molecular device’s structural integrity through chemical bond rupture, large scale molecular geometry alteration, or electromigration of the lead inter- facial atoms. The physical mechanisms of current-induced molecular device breakdown have been comprehensively studied experimentally5–8 and theoretically.9–18Recent theoretical proposals to specially engi- neer energy dependence of the lead density of states10seem to lack practical appeal. The idea of using spin-polarized current to cool vibrations19,20is limited to the use of ferromagnetic leads, which makes it inapplicable to the vast majority of molecular electronic devices. Peskin et al. have recently proposed to use electrode plas- mon excitations in electrodes to reduce power dissipation in molec- ular junctions.21The possibility to reduce the heating by increasing the ambient temperature of the device has been recently demon- strated theoretically.22Despite all these efforts, a practical solution to the sensitivity of structural stability in molecular junctions remains elusive. Subsequently, we propose a new strategy to decrease the Joule heating in molecular junctions: the application of a sinusoidal voltage over the large DC voltage bias, which acts to reduce the effective vibrational temperature of the molecular junction. We set̵h= 1 in our derivations. J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 153, 121102-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp II. MODEL HAMILTONIAN The molecular junction is described by the general tunneling Hamiltonian H=HM+HML+HMR+HL+HR, (1) where HMis the Hamiltonian for the molecule and HLandHRare the Hamiltonians for the left and right leads, while HMLandHMRare for the interaction between the central region and the left and right leads, respectively. The molecule is modeled as a molecular orbital coupled to a single classical degree of freedom HM=ϵ(x)d†d+p2 2m+U(x). (2) Here,ϵ(x) is the energy of the molecular orbital. It is a function of the classical coordinate x, which along with the corresponding momentum pand potential U(x) describes the molecular geometry. Operator d†(d) creates (annihilates) an electron in the molecule. We assume in our calculations that the molecular orbital depends linearly on x, ϵ(x)=ϵ0+λx, (3) whereλis the coupling strength between the electronic and nuclear degrees of freedom. The classical potential is taken in the harmonic oscillator form U(x)=1 2kx2, (4) where kis the spring constant associated with the chemical bond of interest. Notice that x=x(t) is a stochastic, time-dependent vari- able that satisfies the Langevin equation with noise and viscosity obtained from nonequilibrium Green’s functions (NEGF) calcu- lations described below. The spin of electrons does not play any physical role here and will not be included explicitly in the equations. Both leads are modeled as macroscopic reservoirs of noninter- acting electrons, HL+HR=∑ kα=L,Rϵkα(t)a† kαakα, (5) where a† kα(akα) creates (annihilates) an electron in the single-particle state kof either the left ( α=L) or the right ( α=R) lead. The lead energy levels have a sinusoidal dependence on time due to an external AC driving with frequency Ω and amplitude Δα, ϵkα(t)=ϵkα+Δαcos(Ωt). (6) Additionally, the leads are also held at different static chemical potentialsμαat all times, and the difference between them corre- sponds to the applied DC voltage bias V=μL−μR. Both sinusoidal AC and static DC voltages are applied symmetrically in our calcula- tions: ΔL=−ΔRandμL=−μR, and the leads’ Fermi energies are set to zero. The coupling between the central region and the left and right leads is given by the tunneling interaction HML+HMR=∑ kα=L,R(tkαa† kαd+ h.c.), (7) where tkαis the tunneling amplitude between the leads’ single- particle states and the molecular orbital.III. NONEQUILIBRIUM VISCOSITY AND NOISE PRODUCED BY AC DRIVEN ELECTRONS Joule heating is a balancing process between the electronic time-dependent viscosity and the amplitude of the random force exerted by AC driven electrons. They play opposite roles: the viscos- ity deposits energy from nuclear vibrations back onto the electrons, while the random force dissipates the power of the electric current into nuclear motion. If one prevails, it results in the domination of cooling or heating processes. The time-dependent electronic viscosity and diffusion coeffi- cients produced by the AC driven electrons are computed using NEGF. This method is based on the time-separation solution of the Keldysh–Kadanoff–Baym equations for nonequilibrium Green’s functions and utilizing this to compute the force exerted by the elec- trons on the nuclear degrees of freedom and the time correlation of the dispersion of the force operator.17,23–26These derivations follow directly the derivations of the viscosity and diffusion for a system with time-dependent coupling to the leads, as discussed in Refs. 17 and 27, as long as the time-derivatives of the self-energies are not specified explicitly. Notice that in addition to the standard assump- tion that the dynamics of mechanical degrees of freedom are slow in comparison with the electron tunneling time, we have to assume that the rate of change of the leads’ energies due to the external AC driving is also smaller than the electronic tunneling time across the junction.28This means that the validity of our approach requires that the characteristic frequency of the nuclear motion and the AC driving frequency should both be smaller than the molecular level broadening due to the coupling to the leads. Under these separable timescale assumptions, the viscosity ξ and the diffusion coefficient Dcan be computed from the Keldysh– Kadanoff–Baym equation and are given by the following expressions (details of the derivations are shown in the supplementary material): ξ(t)=[ϵ′(t)]2 2∫dω 2πGR(t,ω)GA(t,ω)(GA(t,ω)−GR(t,ω)) ×∂ωΣ<(t,ω), (8) D(t)=[ϵ′(t)]2∫dω 2πG<(t,ω)G>(t,ω) +[ϵ′(t)]2∫dω 2π{δG<(t,ω)G>(t,ω) +G<(t,ω)δG>(t,ω)}. (9) The viscosity and diffusion coefficients depend on the advanced and retarded adiabatic Green’s functions, GA/R(t,ω)=[ω−ϵ(t)−ΣA/R(t,ω)]−1 , (10) G</>(t,ω)=GR(t,ω)Σ</>(t,ω)GA(t,ω), (11) which follow the adiabatically, instantaneously computed time- dependent trajectories of the mechanical degrees of freedom as well as the external AC driving of the leads. The diffusion coefficient also depends on the first order dynamical corrections to the lesser and greater Green’s functions due to the AC driving in the leads’ J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 153, 121102-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp self-energies, δG</>(t,ω)=i 2GR(t,ω)GA(t,ω)(GA(t,ω)−GR(t,ω)) ×∂tΣ</>(t,ω). (12) In all our calculations, we employ the wideband approxima- tion for the leads’ density of states and tunneling amplitudes, which results in the following expressions for the self-energies of the AC driven leads: Σ< α(t,ω)=iΓα∞ ∑ n=−∞fα(ω+nΩ/2)(−1)nJn(2Δαcos(Ωt) Ω), (13) Σ> α(t,ω)=−iΓα+Σ< α(t,ω), (14) ΣA α=i 2Γα;ΣR α=−i 2Γα, (15) where Γαis the standard level-broadening function due to the cou- pling to lead α,fα(ω) is the Fermi–Dirac distribution of electronic occupations in the αlead, and Jn(z) is the Bessel function of the first type. The total level broadening Γ=ΓL+ΓR (16) will be used as an energy scale to represent all model parameters used in the calculations. The expression for the viscosity (8) resembles the viscosity for a system with DC current;17,23–26however, it is defined here using Green’s functions with sinusoidally driven lead self-energies. The diffusion coefficient (9) consists of two terms: the first term is again the standard expression similar to what was used in DC current junctions17,23–26and the second term is new and arises from the dynamical corrections to the lesser and greater Green’s functions (again computed using sinusoidally oscillating self-energies). Figure 1 shows the viscosity and diffusion coefficient (aver- aged over a period of oscillation and also statistically averaged with respect to possible values of x). The viscosity and diffusion coeffi- cients are shown as a ratio to the corresponding DC (static) values for a given average voltage, and the DC calculations are performed using (8) and (9) and setting the amplitude of sinusoidal voltage modulation Δ= 0.17The lead temperature is set to 0.02 Γin all our calculations. At the center of the resonance region, application of the AC driving acts to decrease the diffusion coefficient while slightly increasing the viscosity. In our previous work17(also in agreement with the results of Subotnik et al.29), peaks in the friction occur when the molecular orbital energy aligns with either the left or right chem- ical potentials, since the electrons can deposit any amount of energy taken from the nuclear degrees of freedom to the leads via inelastic scattering to the available empty states above the chemical potential. This is in contrast to the diffusion that has contributions from all lead states in the resonant region. Applying this analysis to our sys- tem, we observe that the viscosity increase in the resonant region is a result of the lead chemical potentials being allowed to shift closer to the resonant level and inducing increased interaction between the nucleus and the high-energy electrons in the leads. However, the AC voltage has minimal effect on increasing the diffusion near the resonance. The growth of the viscosity relative to the diffusion FIG. 1 . Electronic viscosity (a) and diffusion coefficient (b) computed as functions of the AC driving frequency and the molecular orbital energy. Parameters used in calculations: λ2/k= 0.002 Γ,V= 2Γ/3, and Δ=Γ/3.Ωandϵ0are given in terms of Γ. coefficient results in an optimal transport regime in which the molecular junction is cooled relative to the static case. As we shift our resonant level to the edges of the resonant region, we observe a notable decrease in the viscosity upon application of the AC leads, relative to the DC case. In the static regime, the viscosity is maxi- mized here due to the alignment between the resonant level and the leads, which is broken upon application of an AC voltage. Resul- tantly, the AC driving acts to increase the junction temperature in this region. IV. EFFECTIVE TEMPERATURE Using viscosity ξand diffusion coefficient D, we define an effec- tive temperature of the molecular junction via analogy with the equilibrium fluctuation–dissipation theorem as given byD 2ξfor one- dimensional motion. There are two options in defining the effective temperature for a given frequency; we can calculate the instanta- neous effective temperature for each given time in a period of AC leads oscillation and then average over the period, given by J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 153, 121102-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp Tinst AC=Ω 2π∫2π/Ω 0dtD(t) 2ξ(t), (17) or alternately, we can first take Dandξto be time-averaged quan- tities over a period of AC oscillation and then calculate the effective temperature as Tave AC=∫2π/Ω 0dtD(t) 2∫2π/Ω 0dtξ(t). (18) In Fig. 2, we compare these options with average temperature data obtained from kinetic energy calculated via numerical Langevin simulations for the same parameters. In the interest of computa- tional efficiency, λ2/k=Γ/6, where all other parameters coincide with other results presented in this study. We observe Tave ACto be a far more accurate measure of the average nuclear temperature within the system for these parameters. Given that ξandDare each pro- portional to λ2, decreasing λ2/k(in line with the presented results in this paper) will only further improve the accuracy of the average calculation, since the molecule will react more slowly to temperature variations due to the AC leads oscillations. As such, we choose to use Tave ACas our measure for the effective temperature. Figure 3 shows the ratio TAC/TDCcomputed for various trans- port regimes. The AC temperature is compared to the static DC temperature TDCcomputed again as the ratio between the diffusion and friction coefficients, but now obtained using NEGF calculations for static leads.17Both temperatures are again statistically averaged over xand over a period of AC driving. As we have already deduced from the behavior of the viscosity and diffusion coefficient, cooling is observed in the central resonance transport regime, while heating is observed at the edges of resonant transport. The effect of cooling is more significant for slow AC driving [Fig. 3(a)] and is amplified if the amplitude of voltage driving is increased [Fig. 3(b)]. Figure 3(c) shows the case of a large static bias voltage, which also enables the consideration of large AC voltage amplitudes; as one observes, it enables a reduction to the effective temperature by as much as 80% for the chosen parameters, while the corresponding decrease to the average current is less than 20% for the same ϵ0. Figure 4 demonstrates the role of the chemical bond spring constant kand the coupling strength between the electronic popula- tion and the nuclear motion. These parameters are interconnected. FIG. 2 . Comparison of methods of effective temperature calculation with Langevin- simulated nuclear temperature results for the same parameters. Parameters used in calculations: λ2/k=Γ/6,Ω=Γ/15, V= 2Γ/3, and Δ=Γ/3.ϵ0is given in terms of Γ. FIG. 3 . Ratio of AC and DC molecular temperatures computed as functions of molecular orbital energy. (a) shows the results for different AC driving frequen- cies with Δ=Γ/3 and V= 2Γ/3. (b) shows the results for different amplitudes of AC voltage oscillations with Ω=Γ/15 and V= 2Γ/3. (c) shows the temperatures ratio, currents ratio, and cooling ratio defined by Eq. (19) for a higher DC voltage V= 5Γ/3 and Δ= 5Γ/6.λ2/k= 0.002 Γandϵ0is given in terms of Γ. The termλxin (3) results in the shift of the molecular orbital energy due to deviations away from the equilibrium nuclear posi- tion.λdescribes the magnitude of this shift, while kgoverns the range of variation in the xcoordinate. Therefore, λ2/kis an energy related quantity, which encapsulates both effects. As shown in Fig. 4, the cooling effects are observed in the resonance regime when J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 153, 121102-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp FIG. 4 . Ratio of AC and DC molecular temperatures as a function of λ2/kandϵ0. Parameters used in calculations: Ω=Γ/15, V= 2Γ/3, and Δ=Γ/3. Bothλ2/kand ϵ0are given in terms of Γ. λ2/k<0.2Γ. This means that this cooling phenomenon can be observed for systems with rigid chemical bonds or weak electron– nuclear coupling. In any other case, the deviations in the energy level due to nuclear motion may be large enough such that the level leaves this cooling region. The temperature change per se for a given average voltage may not be a complete measure of heating/cooling, since the AC driv- ing may simply produce a smaller current (averaged over the period of oscillation) relative to the corresponding DC voltage, resulting in less power dissipated over the molecule. It is illuminating for us to then consider the heating/cooling effects upon application of an AC driving, relative to the DC case at a given average current (but now a different average voltage). As such, the static lead electric current JDCis computed using the Landauer formula for static leads, and JACis the exact electric current (averaged over a period of oscilla- tion) computed using Jauho, Meir, and Wingreen NEGF theory for AC driven quantum transport.30Then, we introduce the following quantity called the ”cooling ratio”: η=TAC(J) TDC(J), (19) which provides a measure of the heating/cooling observed upon application of an AC driving for a given average current; η<1 means that the AC driving yields a lower temperature while allowing for the same average current. Figure 3(c) shows that the application of an AC driving allows for in excess of 40% cooling of the molecular junction while maintaining the same average current as in the DC case. V. CONCLUSIONS We have demonstrated that the application of an AC driv- ing in the leads’ voltage can result in a significant reduction to the power dissipation in a molecular junction, relative to the case of a large static voltage. The lifetime of a chemical bond is τlife ∼eEb/kT, where Ebis the energetic barrier for bond dissociation. One observes that the lifetime depends exponentially on the effectivetemperature T; therefore, even a moderate temperature reduction produces a colossal extension of the device lifetime. The observed effect is quite robust and does not require special fine-tailoring of the model parameters. Moreover, using a master equation derived in the time-averaged Born–Markov approximation and assuming that the driving period must be shorter than characteristic electron tunnel- ing time 1/ Γ, Peskin et al. demonstrated that the harmonically driven leads may reduce the vibrational temperature of the molecular junc- tion.21The approach of Ref. 21 is complementary in all respects to what we consider in this paper regarding transport and AC driv- ing regimes. This serves as a strong indication that the proposed effect is very robust and ubiquitous and may be applicable for vari- ous transport scenarios. Although the cooling was the main focus of our paper, it has not escaped our notice that depending upon the parameters, the sinusoidal driving of the leads may result in sig- nificant heating of the molecular junction. However, this may also allow for enhanced device functionality as this parameter-controlled heating may be utilized for current-induced selective bond break- ing and energy efficient single-molecule catalysis of chemical reactions. SUPPLEMENTARY MATERIAL See the supplementary material for detailed derivations of the main equations presented in the paper. DATA AVAILABILITY The data that support the findings of this study are available within the article. REFERENCES 1M. Tsutsui, M. Taniguchi, and T. Kawai, J. Am. Chem. Soc. 131, 10552 (2009). 2M. Tsutsui, M. Taniguchi, and T. Kawai, Nano Lett. 8, 3293 (2008). 3Z. Huang, F. Chen, P. A. Bennett, and N. Tao, J. Am. Chem. Soc. 129, 13225 (2007). 4C. R. Peiris, S. Ciampi, E. M. Dief, J. Zhang, P. J. Canfield, A. P. Le Brun, D. S. Kosov, J. R. Reimers, and N. Darwish, Chem. Sci. 11, 5246 (2020). 5H. Li, N. T. Kim, T. A. Su, M. L. Steigerwald, C. Nuckolls, P. Darancet, J. L. Leighton, and L. Venkataraman, J. Am. Chem. Soc. 138, 16159 (2016). 6H. Li, T. A. Su, V. Zhang, M. L. Steigerwald, C. Nuckolls, and L. Venkataraman, J. Am. Chem. Soc. 137, 5028 (2015). 7A. C. Aragonès, N. L. Haworth, N. Darwish, S. Ciampi, N. J. Bloomfield, G. G. Wallace, I. Diez-Perez, and M. L. Coote, Nature 531, 88 (2016). 8E.-D. Fung, D. Gelbwaser, J. Taylor, J. Low, J. Xia, I. Davydenko, L. M. Campos, S. Marder, U. Peskin, and L. Venkataraman, Nano Lett. 19, 2555 (2019). 9A. Erpenbeck, C. Schinabeck, U. Peskin, and M. Thoss, Phys. Rev. B 97, 235452 (2018). 10D. Gelbwaser-Klimovsky, A. Aspuru-Guzik, M. Thoss, and U. Peskin, Nano Lett. 18, 4727 (2018). 11J.-T. Lü, M. Brandbyge, and P. Hedegård, Nano Lett. 10, 1657 (2010). 12J.-T. Lü, M. Brandbyge, P. Hedegård, T. N. Todorov, and D. Dundas, Phys. Rev. B 85, 245444 (2012). 13L. Simine and D. Segal, Phys. Chem. Chem. Phys. 14, 13820 (2012). 14A. A. Dzhioev and D. S. Kosov, J. Chem. Phys. 135, 074701 (2011). 15R. Härtle and M. Thoss, Phys. Rev. B 83, 125419 (2011). 16A. A. Dzhioev, D. S. Kosov, and F. von Oppen, J. Chem. Phys. 138, 134103 (2013). J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 153, 121102-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsCOMMUNICATION scitation.org/journal/jcp 17R. J. Preston, V. F. Kershaw, and D. S. Kosov, Phys. Rev. B 101, 155415 (2020). 18G. Weick, F. Pistolesi, E. Mariani, and F. von Oppen, Phys. Rev. B 81, 121409 (2010). 19J. Brüggemann, S. Weiss, P. Nalbach, and M. Thorwart, Phys. Rev. Lett. 113, 076602 (2014). 20P. Stadler, W. Belzig, and G. Rastelli, Phys. Rev. Lett. 113, 047201 (2014). 21M. Kuperman, L. Nagar, and U. Peskin, Nano Lett. 20, 5531 (2020). 22R. Härtle, C. Schinabeck, M. Kulkarni, D. Gelbwaser-Klimovsky, M. Thoss, and U. Peskin, Phys. Rev. B 98, 081404 (2018).23N. Bode, S. V. Kusminskiy, R. Egger, and F. von Oppen, Beilstein J. Nanotechnol. 3, 144 (2012). 24W. Dou, G. Miao, and J. E. Subotnik, Phys. Rev. Lett. 119, 046001 (2017). 25W. Dou and J. E. Subotnik, Phys. Rev. B 96, 104305 (2017). 26W. Dou and J. E. Subotnik, Phys. Rev. B 97, 064303 (2018). 27W. Dou and J. E. Subotnik, J. Chem. Phys. 146, 092304 (2017). 28T. D. Honeychurch and D. S. Kosov, Phys. Rev. B 100, 245423 (2019). 29W. Dou and J. E. Subotnik, J. Chem. Phys. 148, 230901 (2018). 30A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev. B 50, 5528 (1994). J. Chem. Phys. 153, 121102 (2020); doi: 10.1063/5.0019178 153, 121102-6 Published under license by AIP Publishing

5.0015608.pdf

J. Chem. Phys. 153, 074311 (2020); https://doi.org/10.1063/5.0015608 153, 074311 © 2020 Author(s).Complete characterization of the 3 p Rydberg complex of a molecular ion: MgAr+. II. Global analysis of the A+ 2 and B+ 2+ (3 ) states Cite as: J. Chem. Phys. 153, 074311 (2020); https://doi.org/10.1063/5.0015608 Submitted: 29 May 2020 . Accepted: 19 July 2020 . Published Online: 20 August 2020 Matthieu Génévriez , Dominik Wehrli , and Frédéric Merkt ARTICLES YOU MAY BE INTERESTED IN Complete characterization of the 3p Rydberg complex of a molecular ion: MgAr+. I. Observation of the Mg( )Ar+ B+ state and determination of its structure and dynamics The Journal of Chemical Physics 153, 074310 (2020); https://doi.org/10.1063/5.0015603 Threshold photoelectron spectroscopy of the methoxy radical The Journal of Chemical Physics 153, 031101 (2020); https://doi.org/10.1063/5.0016146 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 Complete characterization of the 3 pRydberg complex of a molecular ion: MgAr+. II. Global analysis of the A+ 2Πand B+ 2Σ+(3pσ,π) states Cite as: J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 Submitted: 29 May 2020 •Accepted: 19 July 2020 • Published Online: 20 August 2020 Matthieu Génévriez, Dominik Wehrli, and Frédéric Merkta) AFFILIATIONS Laboratory of Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland a)Author to whom correspondence should be addressed: merkt@phys.chem.ethz.ch ABSTRACT We report a global study of the 3 pRydberg complex of the MgAr+molecular ion. High-resolution spectroscopic data on the two spin–orbit components of the A+electronic state were obtained by isolated-core multiphoton Rydberg-dissociation spectroscopy up to vibrational levels as high as v′= 29, covering more than 90% of the potential wells. Accurate adiabatic potential-energy functions of the A+and B+states, which together form the 3 pRydberg complex, were obtained in a global direct-potential-fit analysis of the present data and the extensive data on the B+state reported in Paper I [D. Wehrli et al. , J. Chem. Phys. 153, 074310 (2020)]. The dissociation energies of the B+state, the two spin–orbit components of the A+state, and the X+state of MgAr+are obtained with uncertainties (1 cm−1) more than two orders of magnitude smaller than in previous studies. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015608 .,s I. INTRODUCTION The complete experimental characterization of molecular Ryd- berg states necessitates the measurement of spectra of a large num- ber of states of different principal and angular-momentum quantum numbers and, for each of them, of a large number of rotational and vibrational levels, extending from the vibrational ground state to levels close to the dissociation threshold.1–10This task is particu- larly difficult for cations because of the low densities in which they can be produced compared to neutral species and the larger pho- ton energies required for electronic excitation. In addition, diatomic cations produced by conventional ion sources are, in most cases, in their vibrational ground state, which restricts the accessible vibra- tional levels of electronically excited states to those having significant Franck–Condon overlap with the ground vibronic state. For this rea- son, hardly any systematic studies of the Rydberg states of molecular cations have been reported so far. This situation is in stark con- trast to that prevailing for the ground and low-lying valence states of cations, for which many examples of detailed characterization by high-resolution spectroscopy are known; see, e.g., Refs. 11–20 and references therein.We showed recently, with the example of MgAr+, that state- selected cations can be prepared in a wide range of vibrational levels of the electronic ground state by photoexcitation of neutral molecules to high-lying Rydberg states.21For sufficiently large val- ues of the principal quantum number ( n≥100) or orbital-angular- momentum quantum number of the Rydberg electron, the ionic core of the Rydberg molecule can be considered as isolated from the Rydberg electron and behaves as the bare ion. With the method of isolated-core multiphoton Rydberg-dissociation (ICMRD) spec- troscopy,21the ionic core is first prepared in a selected vibrational level of the electronic ground state and subsequently photodissoci- ated in a multiphoton process. ICMRD spectroscopy opens up the possibility of systematically studying a broad range of vibrational levels belonging to electronically excited states of molecular ions in a background-free manner. As a demonstration, we have recorded spectra of the 3 pσB+(v′) state of MgAr+starting from the ground vibrational level ( v′= 0) and extending to the dissociation limit and beyond.22The present article extends this study to the 3 pπcom- ponent, which consists of the two Ω = 1/2 and Ω = 3/2 spin–orbit components of the A+state. J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The four lowest electronic states of MgAr+follow the struc- ture of the 3 sand 3 porbitals of the Mg+ion and thus possess a Rydberg character in the sense of Mulliken23(see Fig. 2 in Ref. 22). The 3 sσX+ 2Σ+ground state correlates with the Ar(1S0) + Mg+(2S1/2) dissociation asymptote, whereas the 3 pcomplex consists of three excited electronic states correlating with the Ar(1S0) + Mg+(2P1/2,3/2 ) dissociation limits. At short internuclear distances, two of these states can be regarded as the two spin–orbit components of the 3pπA+ 2ΠΩ′=1/2,3/2 state, and the third can be regarded as the 3 pσ B+ 2Σ+state. They will be denoted as A+ 1/2, A+ 3/2, and B+below. The low-lying vibrational levels ( v′≤14) of the two spin–orbit com- ponents of the A+state were first observed in the pioneering work of Duncan and co-workers,24–26and the rotational constant of the v′= 5 state was determined in a subsequent high-resolution study.27 The spectra of the A+(v′= 0, 1, 5) ←X+(v′′= 3) vibrational bands with a partially resolved rotational structure were also observed recently by ICMRD spectroscopy.21In parallel, the A+state of MgAr+was investigated theoretically in Refs. 22 and 28–35. The rovibrational structure of the B+state has first been measured in the study reported in Paper I.22 The present article reports high-resolution spectroscopic data on the A+state of MgAr+obtained by ICMRD spectroscopy. Vibra- tional levels as high as v′= 29 have been observed, and the rotational structure of the vibrational bands could be partially resolved up to v′= 28. The data cover more than 90% the potential wells of each of the two spin–orbit components of the A+state. Combined with the data reported on the B+state in Paper I,22which cover its entire potential depth, the new data provide complete and detailed infor- mation on the 3 pRydberg complex of MgAr+. From the experimen- tal results, high-accuracy adiabatic potential-energy functions were determined for the three electronic states using a direct-potential- fit technique36combined with a model describing the spin–orbit interaction at the relevant internuclear distances. A similar approach was previously used to describe the ground and first electronically excited states of the homonuclear rare-gas dimers.37–41 II. EXPERIMENT The experimental setup and the technique of ICMRD spec- troscopy used in the present work were described in detail in Refs. 21 and 42. Briefly, a molecular beam containing rotationally and vibrationally cold MgAr molecules is first formed in a pulsed laser- ablation supersonic-expansion source. MgAr molecules are, for their vast majority, in the metastable Mg(3 s3p)Ar a3Π0(v= 0) state, and their rotational temperature is estimated to be of the order of 10 K for the data presented below. After passing through a 3- mm-diameter skimmer located 8 cm downstream from the source, the molecular beam enters the photoexcitation chamber, where it is intersected at right angles by the beams of two pulsed dye lasers pumped by the second harmonic of a Nd:YAG laser. The first laser pulse, corresponding to the frequency-tripled output of the first dye laser and with a wave number tunable in the range from 39 000 cm−1to 39 500 cm−1, excites the neutral MgAr molecules from the metastable a3Π0state to high-lying Rydberg states with principal quantum numbers n≃120 lying just below a specific X+ 2Σ+(v′′) ionization threshold of MgAr. The second laser pulse, corresponding to the frequency-doubled output of the sec- ond dye laser and with a wave number ˜ν2tunable in the range from31 000 cm−1to 35 800 cm−1, resonantly excites the ionic core from the selected X+(v′′) state to a vibrational level ( v′) of either of the two spin–orbit components of the A+excited state. Further photoab- sorption leads to the dissociation of the molecule into a ground-state neutral Ar atom and a Mg atom in a high Rydberg state.21 The interaction between the MgAr molecules and the laser pulses occurs within a 5.8-cm-long stack of five equally spaced and resistively coupled cylindrical electrodes. A small electric field of ≃1.7 V/cm is applied across the stack immediately after photoexci- tation and photodissociation and displaces the ions produced by the laser pulses, called prompt ions, away from the interaction region without affecting the atoms and molecules excited to Rydberg states. After≃5μs, a pulsed extraction field of 130 V/cm is applied to the stack to field ionize the Rydberg atoms and molecules and accelerate all positively charged particles into a linear time-of-flight region, at the end of which they are detected by a micro-channel-plate detec- tor. The spatial separation between the prompt ions and ions gener- ated by pulsed-field ionization prior to extraction makes it possible to distinguish them by their time of flight. By recording the yield of Mg+ions produced by delayed pulsed field ionization as a func- tion of the wave number of the second laser, the resonance-enhanced multiphoton dissociation of the MgAr+ion core via the A+state can be recorded in a background-free manner. III. EXPERIMENTAL RESULTS We have recorded the spectra of the transitions from the X+(v′′) state to the A+ Ω′(v′)state of24MgAr+with Ω′= 1/2 and 3/2 andv′= 0–29. For each v′, the value of the vibrational quantum number v′′of the initial state was chosen so that the correspond- ing Franck–Condon factor was large enough for the transition to be observed. The assignment of the observed bands to the given values of v′andv′′is based on the isotopic-shift analysis carried out in previous studies.21,25,43The ability to prepare the ground- state molecular ion in a broad range of selected (ro)vibrational states ( v′′= 0–9 in the present case) is a property of the isolated- core multiphoton Rydberg-dissociation technique and is essen- tial for observing vibrational states of the A+excited electronic state with v′as high as 29. In contrast, ion-beam-based tech- niques are typically limited to the rotational and vibrational levels of the ground electronic state of the ion that are thermally popu- lated in the ion source, e.g., v′′= 0–1 in the case of MgAr+,24,25 which limits the accessible vibrational levels of the A+state tov′≤14.24,25 Figures 1 and 2 show the high-resolution spectra of the A+ 1/2(v′=17)←X+(v′′=3)and A+ 3/2(v′=17)←X+(v′′ =3)bands, respectively. Their rotational structure consists of four branches associated with transitions from the rotational levels of the X+(2Σ+) state, described by Hund’s coupling case (b) and labeled by the quantum number N′′associated with the total angular momentum without spin, to rotational levels of the A+(2ΠΩ′) states, described by Hund’s coupling case (a) and labeled by the quantum number J′associated with the total angular momentum. In the case of the transition to the A+ 1/2(v′=17)level, shown in Fig. 1, the rotational structure is fully resolved except in the region of the band- head of the ΔJ′N′′=J′−N′′=−0.5 branch between 34 779.9 cm−1 and 34 781.5 cm−1. As in our previous work,21the positions of these J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Measured and calculated spectra of the A+ 1/2(v′=17)←X+(v′′=3) band of24MgAr+. The assignment bars give the positions of the transitions to levels of the A+state of successive values of J′. The red arrow indicates the position of theA+ 1/2(v′=17,J′=1/2)←X+(v′′=3,N′′=2)transition. FIG. 2 . Measured and calculated spectra of the A+ 3/2(v′=17)←X+(v′′=3) band of24MgAr+. The assignment bars give the positions of the transitions to levels of the A+state of successive values of J′. The red arrow indicates the calculated position of the hypothetical transition A+ Ω′(v′=17,J′=1/2)←X+(v′′=3, N′′=2). This transition is not observed, indicating that Ω′= 3/2.individual lines were determined by fitting them with Gaussian func- tions with a full width at half maximum of 0.1 cm−1. The origin of the band ˜νv′v′′and the rotational constants B′′ v′′andB′ v′of the initial and final states, respectively, were obtained from the line positions in a least-squares fit based on the standard expression44,45 ˜ν=˜νv′v′′+B′ v′[J′(J′+ 1)−Ω′2]−B′′ v′′N′′(N′′+ 1). (1) At the low values of N′′andJ′relevant in the present experiment (N′′≲11,J′≲25/2), the effects of the centrifugal distortion on the positions of the rotational levels are negligible. The 1σstatistical uncertainties of the band origins determined in the fit are Δ˜νv′v′′=0.1 cm−1. They represent the standard devi- ation of values extracted from several independent spectra recorded under identical experimental conditions.21The systematic uncer- tainties of the band origins are 0.04 cm−1and correspond to the absolute accuracy of the wave-number calibration. The 1 σstatisti- cal uncertainties of the rotational constants are those resulting from the least-squares fits. The spectra displayed in the lower panel of Figs. 1 and 2 were calculated using the fit results and the rotational line strengths for transitions from a state described by Hund’s cou- pling case (b) to a state described by Hund’s coupling case (a).45The initial rotational states were assumed to be thermally populated at a temperature of 7 K. The agreement between the experimental and simulated spectra is excellent, with the exception of the intensities of the members of the ΔJ′N′′=−1.5 branch, which appear underes- timated in the simulation. Possible reasons for this discrepancy are discussed in Ref. 21. The assignment of the bands to particular values of v′andv′′ is known.21,25,43However, their attribution to Ω′= 1/2 or Ω′= 3/2 relied until now on the assumption that the energetic ordering of the two spin–orbit components of the A+molecular state is regular, as in the case of the 3 pj(j= 1/2, 3/2) states of Mg+, and therefore that the levels with Ω′= 1/2 are lower in energy than the corresponding ones with Ω′= 3/2.27The spectra of the A+ 1/2(v′=17)←X+(v′′=3)and A+ 3/2(v′=17)←X+(v′′=3)bands shown in Figs. 1 and 2 unam- biguously verify this assumption. For Ω′= 1/2, the lowest possible value of J′is 1/2, and the line corresponding to the transition from N′′= 2 to J′= 1/2, marked by a red arrow in Fig. 1, confirms that the final electronic state of the transition is indeed an Ω′= 1/2 state. For Ω′= 3/2, J′≥3/2, and the transition from N′′= 2 to J′= 1/2 can- not be observed. The wave number at which one would expect such a transition for an Ω′= 1/2 final state was calculated using Eq. (1) and is marked by the red arrow in Fig. 2. The absence of a line at this position in the experimental spectrum confirms the attribution to the Ω′= 3/2 final state. The band origins ˜νv′v′′determined from least-squares fits of the experimental spectra recorded from the X+ground-state levels with v′′= 3, 6, 7, and 9 to A+levels up to v′= 29 are listed in Table I, along with the rotational constants of the A+ 1/2,3/2(v′)states up to v′= 28. In some cases, only a few isolated rotational lines could be identi- fied. To increase the reliability of the fit results, B′′ v′′was fixed to the weighted average of the values determined from other bands (see Table II). The 1 σstatistical uncertainty of B′ v′was also increased to 0.01 cm−1in these cases. In the case of v′= 29, only prominent band- heads were observed, which did not permit the determination of B′ 29 for either of the two spin–orbit components. The band origins were obtained in least-squares fits of the simulated rotational contours of J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Observed band origins ˜νv′v′′of the A+ Ω′(v′)←X+(v′′)transitions. The term values T′(obs) v′ of the A+ Ω′(v′)lev- els are also given, along with the difference between these values and those extracted from available experimental data25,27 (ΔT′ v′(1)) and those calculated using the potential-energy functions determined in this article ( ΔT′ v′(2)=T′(obs) v′−T′(calc) v′ ). The measured rotational constants B′ v′(obs)of the A+ Ω′(v′)levels are also compared to the results obtained from these potential-energy functions ( ΔB′ v′=B′ v′(obs)−B′ v′(calc)). All values are in cm−1. The values in parentheses represent one standard deviation in units of the last digit. v′Ω′v′′˜νv′v′′ T′(obs) v′ ΔT′ v′(1)ΔT′ v′(2)B′(obs) v′ ΔB′ v′ 0 1/2 3 31 153.72 (10) ... ... ... 0.1950(30) 0.0001 0 3/2 3 31 230.71 (10) 76.99(14) 1.09 0.16 0.1920 (40)−0.0026 1 1/2 3 31 419.76 (10) 266.04(14)−1.06 0.08 0.1900 (10)−0.0024 1 3/2 3 31 496.07 (10) 342.35(14)−1.95 0.03 0.1910 (20)−0.0011 2 1/2 3 31 679.25 (10) 525.53(14) 0.13 0.02 0.1899 (7) 0.0001 2 3/2 3 31 755.17 (10) 601.45(14) 1.35 0.05 0.1900 (20) 0.0005 3 1/2 3 31 932.39 (10) 778.67(14) 0.67 0.03 0.1890 (20) 0.0018 3 3/2 3 32 007.78 (10) 854.06(14) 0.16 −0.01 0.1890 (10) 0.0021 4 1/2 3 32 179.12 (10) 1025.40(14) 2.20 0.08 0.1841 (8)−0.0005 4 3/2 3 32 254.03 (10) 1100.31(14) 1.31 0.02 0.1837 (2)−0.0005 5 1/2 3 32 419.32 (10) 1265.60(14) 6.20 0.08 0.1813 (2)−0.0006 5 3/2 3 32 493.79 (10) 1340.07(14) 3.67 0.04 0.1818 (2) 0.0002 6 1/2 3 32 653.05 (10) 1499.33(14) 6.73 0.13 0.1800 (30) 0.0008 6 3/2 3 32 727.04 (10) 1573.32(14) 6.62 0.07 0.1790 (20) 0.0002 7 1/2 3 32 880.17 (10) 1726.45(14) 5.15 0.12 0.1750 (10)−0.0014 7 3/2 3 32 953.68 (10) 1799.96(14) 4.86 0.03 0.1760 (20)−0.0001 8 1/2 3 33 100.66 (10) 1946.94(14) 4.54 0.09 0.1730 (20)−0.0006 8 3/2 3 33 173.62 (10) 2019.90(14) 5.30 −0.10 0.1743 (10) 0.0010 9 1/2 3 33 314.54 (10) 2160.82(14) 6.42 0.09 0.1695 (10)−0.0012 9 3/2 3 33 387.06 (10) 2233.34(14) 5.84 −0.10 0.1717 (10) 0.0013 10 1/2 3 33 521.58 (10) 2367.86(14) 6.86 −0.05 0.1684 (10) 0.0006 10 3/2 3 33 593.81 (10) 2440.09(14) 6.49 −0.09 0.1684 (10) 0.0009 11 1/2 3 33 722.03 (10) 2568.31(14) 6.61 −0.04 0.1653 (10) 0.0004 11 3/2 3 33 793.78 (10) 2640.06(14) 6.26 −0.12 0.1653 (6) 0.0007 12 1/2 7 33 602.61 (10) 2761.97(20) 6.87 −0.03 0.1637 (5) 0.0019 12 3/2 7 33 673.92 (10) 2833.28(20) 6.68 −0.11 0.1630 (10) 0.0015 13 1/2 7 33 789.43 (10) 2948.79(20) 6.59 −0.01 0.1562 (8)−0.0025 13 3/2 7 33 860.29 (10) 3019.65(20) 7.95 −0.11 0.1582 (2)−0.0002 14 1/2 7 33 969.32 (10) 3128.68(20) 4.48 −0.04 0.1540 (20)−0.0016 14 3/2 7 34 039.72 (10) 3199.08(20) 6.08 −0.17 0.1570 (30) 0.0017 15 1/2 3 34 455.40 (10) 3301.68(14) ... −0.01 0.1519 (7)−0.0005 15 3/2 3 34 525.47 (10) 3371.75(14) ... −0.05 0.1530 (8) 0.0010 16 1/2 3 34 621.38 (10) 3467.66(14) ... −0.02 0.1491 (7) 0.0001 16 3/2 3 34 691.06 (10) 3537.34(14) ... −0.04 0.1492 (5) 0.0005 17 1/2 3 34 780.38 (10) 3626.66(14) ... 0.02 0.1455 (2)−0.0002 17 3/2 3 34 849.64 (10) 3695.92(14) ... −0.02 0.1448 (4)−0.0006 18 1/2 3 34 932.30 (10) 3778.58(14) ... 0.03 0.1417 (2)−0.0005 18 3/2 3 35 001.21 (10) 3847.49(14) ... 0.04 0.1420 (4) 0.0001 19 1/2 3 35 077.15 (10) 3923.43(14) ... 0.07 0.1376 (10)−0.0010 19 3/2 3 35 145.69 (10) 3991.97(14) ... 0.09 0.1380 (5)−0.0003 20 1/2 7 34 901.78 (10) 4061.14(20) ... 0.08 0.1362 (4) 0.0012 20 3/2 7 34 969.91 (10) 4129.27(20) ... 0.06 0.1348 (2) 0.0001 21 1/2 7 35 032.34 (10) 4191.70(20) ... 0.06 0.1327 (3) 0.0014 21 3/2 7 35 100.24 (10) 4259.60(20) ... 0.16 0.1302 (10)−0.0008 22 1/2 6 35 227.25 (10) 4315.20(24) ... 0.12 0.1273 (5)−0.0001 22 3/2 6 35 294.82 (10) 4382.77(24) ... 0.21 0.1280 (10) 0.0008 23 1/2 6 35 343.73 (10) 4431.67(24) ... 0.28 0.1237 (10) 0.0002 23 3/2 6 35 410.79 (10) 4498.74(24) ... 0.16 0.1239 (10) 0.0007 J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . (Continued .) v′Ω′v′′˜νv′v′′ T′(obs) v′ ΔT′ v′(1)ΔT′ v′(2)B′(obs) v′ ΔB′ v′ 24 1/2 6 35 452.75 (10) 4540.70(24) ... 0.10 0.1195 (4) 0.0000 24 3/2 6 35 519.78 (10) 4607.72(24) ... 0.18 0.1191 (5)−0.0001 25 1/2 6 35 554.84 (10) 4642.79(24) ... 0.05 0.1161 (10) 0.0008 25 3/2 6 35 621.67 (10) 4709.61(24) ... 0.15 0.1154 (4) 0.0003 26 1/2 7 35 578.56 (10) 4737.92(20) ... 0.07 0.1096 (10)−0.0015 26 3/2 7 35 645.15 (10) 4804.51(20) ... 0.08 0.1097 (2)−0.0012 27 1/2 9 35 536.79 (10) 4825.82(24) ... −0.17 0.1080 (8) 0.0013 27 3/2 9 35 603.45 (10) 4892.48(24) ... −0.03 0.1080 (20) 0.0014 28 1/2 9 35 617.88 (10) 4906.91(24) ... −0.32 0.1040 (10) 0.0018 28 3/2 9 35 684.62 (10) 4973.65(24) ... −0.16 0.1050 (10) 0.0028 29 1/2 9 35 691.99 (20) 4981.02(30) ... −0.63 ... ... 29 3/2 9 35 759.01 (20) 5048.04(30) ... −0.43 ... ... the bands to the experimental spectra after fixing the values of B′ 29 to the values (0.099 cm−1for both Ω′= 1/2 and 3/2) obtained by extrapolation using the standard expression44 B′ v′=Be−αe(v′+1 2)+βe(v′+1 2)2 . (2) The values of Be,αeandβehad been fitted beforehand to the exper- imental values of B′ v′withv′in the range from 0 to 28, and their numerical values are listed in the supplementary material. The rota- tional constant of the v′′= 9 initial state was fixed to 0.105 cm−1, corresponding to the weighted mean of the values of B′′ 9obtained from bands with v′≤28. The uncertainties of the origins of the v′= 29−v′′= 9 bands are 0.2 cm−1, which is twice as large as the uncertainties of the other band origins. To obtain the vibrational term values of the observed A+lev- els, we first determined the difference in energy T′′ v′′−T′′ 3between the various vibrational levels of the initial X+state used to record TABLE II . Rotational constants B′′ v′′of the X+(v′′) states and observed combination differences T′′ v′′−T′′ 3with respect to the v′′= 3 vibrational level, determined from the band origins ˜νv′v′′of A+ Ω′(v′)←X+(v′′)transitions. All values are in cm−1. The values in parentheses represent one standard deviation in units of the last digit. v′′v′Ω′˜νv′v′′ T′′ v′′−T′′ 3 3 11 3/2 33 793.78(10) 7 11 3/2 33 480.70(10) 313.08 (14) 7 25 3/2 35 550.25(10) 6 25 3/2 35 621.68(10) 241.67 (20) 7 21 1/2 35 032.34(10) 9 21 1/2 34 902.67(10) 442.75 (20) v′′B′′ v′′ B′′ v′′43 3 0.127 27(9) 0.128 6 0.116 33(30) 0.116 7 0.111 32(11) 0.112 9 0.106 01(30) 0.103the spectra ( v′′= 6, 7, 9) and the v′′= 3 level. For that purpose, we recorded the spectra of transitions from at least two different initial levels v′′to the same final level v′and determined the corre- sponding ground-state combination differences, reported in Table II along with the respective rotational constants B′′ v′′. The present values of the energy differences agree within the error bars with those determined from the PFI-ZEKE photoelectron spectrum of the X+←a3Π0photoionization transition.43The values of the rotational constants B′′ v′′of the X+vibrational states were obtained as weighted averages of the values determined in the fits of individual bands. The present values for B′′ v′′are in good agreement with the values derived from the potential-energy function of the X+state reported in Ref. 43. Using the energy differences T′′ v′′−T′′ 3discussed above, the term values of the levels of the A+state are obtained from the band origins ˜νv′v′′using T′ v′=˜νv′v′′+(T′′ v′′−T′′ 3)−˜ν03, (3) as listed in Table I, where they are compared with the values reported by Pilgrim et al.25The two sets of term values differ by up to ∼6 cm−1, and the differences increase with v′. IV. MODEL POTENTIALS The present data for the A+state of MgAr+extend up to v′= 29 and cover more than 90% of the total depth of the interac- tion potentials. Combined with the data reported for the B+state,22 which extend from the vibrational ground state to the dissociation limit, they provide a complete and detailed picture of the (3 pσ, 3pπ) Rydberg complex of MgAr+and can be used to derive accurate adiabatic potential-energy functions describing the three electronic states associated with the 3 pcomplex in a global fit. For that purpose, the treatment of the spin–orbit interaction between the three states is necessary. At large internuclear distances, the three states of the 3 pcom- plex converge to the two Mg+(3p1/2,3/2 ) + Ar(1S0) dissociation limits (see Fig. 4 in Sec. V below). In this region, the spin–orbit interaction dominates over other interactions and the states are well described by Hund’s angular-momentum-coupling case (c). They are labeled by an index specifying their dissociation limit (II for Mg+(3p1/2) and J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp III for Mg+(3p3/2)) and by Ω′, the quantum number correspond- ing to the projection of the total electronic angular momentum onto the internuclear axis. This nomenclature yields the symbols II(1/2), III(1/2), and III(3/2) for the three electronic states of the 3 pcom- plex. At shorter internuclear distances, the spin–orbit interaction becomes weaker than electrostatic interactions so that the quan- tum numbers ΛandΣassociated with the projections of the orbital and spin angular momenta onto the internuclear axis become good quantum numbers. The three electronic states are well described by Hund’s coupling cases (a) and (b) and are designated A+ 2Π1/2, A+ 2Π3/2, and B+ 2Σ+, respectively. The different symbols associated with the three states of the 3 pcomplex at different internuclear dis- tances Rand their connection to the nomenclature A+ 1/2, A+ 3/2B+ used in this article are summarized in Table III. A transition between Hund’s coupling cases (a) or (b) and Hund’s coupling case (c) takes place as the internuclear distance increases and results in a progressive mixing of the2Σand2Π characters of the states with Ω′= 1/2, caused by the increasing importance of the spin–orbit interaction. The Ω′= 3/2 state remains unaffected and retains its2Π3/2Hund’s case (a) character at large Rvalues. Consequently, in order to obtain the adiabatic potential- energy functions of all three electronic states at all internuclear dis- tances R, a correct description of the evolution of the spin–orbit coupling with Ris required. Because the 3 pcomplex of MgAr+is well isolated from other electronic states, the potential-energy functions can be described reliably using a model originally developed by Cohen and Schnei- der.46This model was used, for example, to study the transition from Hund’s coupling case (a) to Hund’s coupling case (c) in the elec- tronic states of the 6 pcomplex in Au–rare-gas dimers47and in the low-lying electronic states of the rare-gas dimer ions.37–41In the lat- ter case, it is necessary to further take into account the R-dependence of the spin–orbit coupling constant.40,41In the model, the spin– orbit interaction can be written, within the adiabatic approximation and using Hund’s coupling case (a) basis functions, in the following matrix form: ⎛ ⎜⎜⎜ ⎝VΣ(R)a(R)√ 20 a(R)√ 2VΠ(R)−a(R) 20 0 0 VΠ(R)+a(R) 2⎞ ⎟⎟⎟ ⎠. (4) The eigenvalues of this matrix are the adiabatic potential-energy curves of the three electronic states [II(1/2), III(1/2), and III(3/2)] of the 3 pcomplex. a(R) is the (positive) spin–orbit constant, and VΣ(R) andVΠ(R) are the potential-energy functions of the electronic states without spin–orbit interaction. Along the matrix diagonal, one rec- ognizes the potential-energy functions of the2Σ+ 1/2,2Π1/2, and2Π3/2 TABLE III . Summary of the various denominations of the three electronic states of the 3pcomplex of MgAr+in dependence of the internuclear distance R. State Small R Large R A+ 1/22Π1/2 II(1/2) A+ 3/22Π3/2 III(3/2) B+ 2Σ+III(1/2)states, respectively, the latter two being split by the spin–orbit inter- action a(R). The off-diagonal matrix elements couple the two states with Ω′= 1/2 and are responsible for the transition to Hund’s cou- pling case (c) at large internuclear distances, where the potential energies VΣandVΠare almost degenerate. Consequently, the elec- tronic states II(1/2) and III(1/2) possess mixed ΣandΠcharacter, as discussed above. In the separated-atom limit, the splitting between the II(1/2) and III(1/2, 3/2) states predicted by the model is3 2a(R) and must be equal to the 3 p1/2−3p3/2spin–orbit splitting of Mg+ (91.57 cm−148), which implies that a(R→∞) = 61.05 cm−1. The R- dependence of a(R) isa priori unknown. However, it was found to be well described by Morse-type functions in previous studies of the rare-gas-dimer ions.40,41The functional form for a(R) used in the present study is described below. In the case of the B+state, the transition from Hund’s coupling case (b) to Hund’s coupling case (c) is directly visible in the high- resolution spectra of the B+(v′)←X+(v′′= 7) transitions recorded for increasing v′.22Thev′= 1 and v′= 14 bands are shown as exam- ples in the upper and lower panels of Fig. 3, respectively. The rota- tional structure of the v′= 1 band is typical of transitions between twoΣstates, characterized by the rotational quantum numbers N′′ andN′, respectively, and consists of overlapping P ( ΔN′N′′=−1) and FIG. 3 . Measured spectra of the B+(v′= 1)←X+(v′′= 7) (top) and B+(v′= 14) ←X+(v′′= 7) (bottom) bands of24MgAr+. The assignment bars give the positions of the transitions to levels of the B+state of successive values of either N′for v′= 1, described by Hund’s coupling case (b), or J′forv′′= 14, described by Hund’s coupling case (c). J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp R (ΔN′N′′=+1) branches. These two branches are expected to over- lap when the rotational constant of the final state is much smaller than that of the initial state, as in the present case (see Ref. 22 for a complete simulation of the band). The spectrum of the v′= 14 band consists, instead, of four distinct rotational branches, which indicates the departure of the B+state from Hund’s coupling case (b). The rotational structure of the band can be well described by the four rotational branches that characterize transitions from a state described by Hund’s coupling case (b) to a state described by Hund’s coupling case (c). The branches are labeled ΔJ′N′′and connect initial states with integer values of the total-angular-moment-without-spin quantum number N′′to final states with half-integer values of the total-angular-momentum quantum number J′. The potential-energy functions VΣ(R) and VΠ(R) in Eq. (4) can be represented by functions of the form VΛ(R)=AΛe−2βΛ(R)(R−RΛ e)−CΛe−βΛ(R)(R−RΛ e)+uΛ LR(R), (5) analogous to the double-exponential/long-range (LR) model poten- tial described by Le Roy.36AΛand CΛare obtained from the equilibrium distance RΛ eand the potential-well depth DΛ eusing AΛ=DΛ e+uΛ LR(RΛ e)+1 βΛ(RΛe)duΛ LR dR∣ R=RΛ e, (6) CΛ=2DΛ e+ 2uΛ LR(RΛ e)+1 βΛ(RΛe)duΛ LR dR∣ R=RΛ e, (7) and the function βΛ(R) is given by βΛ(R)=BΛ+nΛ ∑ i=1aΛ i⎛ ⎝R5−RΛ ref5 R5+RΛ ref5⎞ ⎠i , (8) with RΛ e,DΛ e,BΛ, and aΛ ibeing adjustable parameters of the model. In the present study, we fixed RΣ ref=6.3a0for the Σstate and RΠ ref=4.3a0for the Πstate, corresponding to values at which the least-squares fits rapidly converged. Starting from nΛ= 1, the number nΛof terms of the sum in Eq. (8) was systematically increased until agreement between the measured and calculated vibrational term values and rotational constants was reached. We found that setting nΣ= 2 and nΠ= 3 was sufficient to reproduce the experimental results within experimental accuracy. The long-range behavior of the potential is described by the function uΛ LR(R)[see Eq. (5)], which accounts for the dominant electrostatic interactions between Mg+(3p) and Ar, uΛ LR(R)=−αAr 2R4ϕ(4) Λ(R)−CΛ 6 R6ϕ(6) Λ(R). (9) The damping functions ϕ(m) Λ(R)progressively switch off electro- static interactions at short internuclear distances, where covalent and repulsive forces dominate. We use the damping functions ϕ(m) Λ=[1−exp(−3.95ρΛR m−0.39(ρΛR)2 √m)]m , (10) similar to the one reported by Douketis et al.49(see also Ref. 50). The system-dependent range-scaling parameters ρΛ(Λ= 0, 1) rep- resent two additional adjustable parameters of the model. In Eq. (9), αAris the static electric polarizability volume of the Ar atom ( αAr = 11.077 a0351), and−αAr/2r4accounts for the charge – induced- dipole interaction in atomic units ( Eh). The term −CΛ 6/r6combines three electrostatic interactions: (i) the charge – induced-quadrupole interaction, (ii) the permanent-quadrupole – induced-dipole inter- action, and (iii) the dispersion interaction. These interactions must be accounted for to correctly describe the interaction potentials of the electronic states of the npcomplexes of metal–rare-gas dimers (see, e.g., Refs. 31 and 52). The only value of C6available in the lit- erature is the estimate of Le Roy29for the Π(Λ= 1) state, which does not include interaction (ii), and no value is available for the Σ (Λ= 0) state. We have therefore re-evaluated the values of the CΛ 6 TABLE IV . Parameters describing the potential-energy functions of the A+and B+states of MgAr+according to Eqs. (4), (5), (8), (10), and (12). The parameters in the upper part of the table were obtained in a weighted least-squares fit to the experimental term values and rotational constants. The parameters in the lower part of the table were left fixed during the least-squares fit. All values are in atomic units. The weighted root-mean-square deviation of the fit was 1.3. VΣ(R) VΠ(R) a(R) Parameter Value Parameter Value Parameter Value RΣ e 8.352 a0 RΠ e 4.526 3 a0 H 1.01 ×10−5Eh DΣ e 0.001 156 3 Eh DΠ e 0.025 672 7 Eh S 6.38 a0−1 BΣ 0.516 4 a0−1BΠ 0.985 02 a0−1I 0.728 a0 ρΣ 0.7 a0−1ρΠ 0.374 2 a0−1 aΣ 1 0.05 aΠ 1 0.071 6 aΣ 2 0.22 aΠ 2 0.013 8 aΠ 3 0.082 4 RΣ ref 6.3 a0 RΠ ref 4.3 a0 CΣ 6 47.44 a06Eh CΠ 6 225.59 a06Eh αAr 11.077 a3 0αAr 11.077 a3 0 J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp coefficients by independently calculating the contribution of each interaction according to CΛ 6=α(2) Ar 2+ 3Θ(Λ) zzαAr+Cdisp 6(Λ). (11) In Eq. (11), α(2) Ar=50.21 a5 0is the quadrupole polarizability of Ar,53 andCdisp 6(Λ)is theΛ-dependent dispersion coefficient, calculated to be 205.5 Eha6 0forΛ= 0 and 108.9 Eha6 0forΛ= 1.54Θ(Λ) zzis the per- manent quadrupole moment of Mg+(3p) for a given value of Λ, i.e., for a fixed value of the projection of the orbital angular momentum of the valence electron of Mg+onto the internuclear axis. For Λ= 1, we used the permanent quadrupole moment Θ(1) zz=2.756 ea02, the absolute value of which was calculated by Mitroy and Zhang.54From the value of Θ(1) zz, we deduce that Θ(0) zz, the quadrupole polarizability of Mg+(3p) forΛ= 0, is−5.512 ea02(see, e.g., Ref. 55). Using Eq. (11), we obtain the values for the coefficients CΛ 6listed in Table IV. The spin–orbit coupling constant in Eq. (4) is described by the Morse-potential-type function a(R)=H[1−e−S(R−I)]2 −H+aMg+ SO, (12) where aMg+ SO=2 3⋅91.57 cm−1is the 3 p1/2−3p3/2spin–orbit cou- pling constant in Mg+.48The parameters H,S, and Iin Eq. (12) are adjustable. In total, there are 16 adjustable parameters in the model ( RΛ e, DΛ e,BΛ,ρΛ,aΛ 1,...aΛ nΛ,H,I, and S, with Λ= 0, 1) that describe the three potentials Vi(R) (i= 1–3) corresponding to the eigenvalues of Eq. (4). The energies associated with the rotationless vibrational lev- els of each potential are calculated by numerically solving the nuclear Schrödinger equation [−1 2μd2 dR2+Vi(R)]ψi,v(R)=Ei,vψi,v(R) (13) with a Legendre–Gauss–Lobatto discrete-variable-representation (DVR) technique.56–59The rotational constants Bi,vof each vibronic energy level are calculated from the expectation value of the 1/ R2 operator, as described in Ref. 43. Optimal values of the 16 model parameters are determined in a weighted nonlinear least-squares fit of the calculated energies and rotational constants to the corresponding experimental values, which comprise 154 data points in total. Initial values for these parameters were obtained by fitting the model potential-energy functions to the ab initio potential-energy curves of the A+and B+ states reported in Paper I.22Successive iterations were carried out during which only the parameters of either the VΠ,VΣ, or a(R) functions were varied. Once a satisfactory set of parameters was obtained, all the parameters were optimized simultaneously in a global fit. In a last step, the parameters of the VΣpotential were slightly re-optimized to reproduce the maxima and minimum of the photodissociation yield recorded above the dissociation threshold of the B+state (see Ref. 22). The relative photodissociation cross section was calculated from the B+adiabatic potential using a Legendre– Gauss–Lobatto finite-element DVR method with exterior complex scaling57,60and under the assumption that the electronic transition moment is independent of Rover the relevant range of internuclear distances. We made sure that this re-optimization had no influenceTABLE V . Observed band origins ˜νv′v′′of the B+(v′)←X+(v′′= 7) transitions, taken from Ref. 22, term values T′(obs) v′ of the B+-state levels with respect to the A+ 1/2(v′=0)ground vibrational level, and deviations ΔT′ v′(2)=T′(obs) v′−T′(calc) v′ of the term values calculated using the potential-energy function determined in this article. The measured rotational constants B′ v′(obs)of the B+(v′) states are com- pared to the results obtained from the potential-energy function ( ΔB′ v′=B′ v′(obs) −B′ v′(calc)). All values are in cm−1. The values in parentheses represent one standard deviation in units of the last digit. v′˜νv′v′′ T′(obs) v′ ΔT′ v′(2)B′(obs) v′ ΔB′ v′ 0 36 148.43 (20)5307.79(26)−0.30 0.0608 (10) 0.0030 1 36 181.34 (20)5340.70(26) 0.10 0.0550 (20)−0.0019 2 36 211.78 (20)5371.14(26) 0.05 0.0543 (10)−0.0015 3 36 240.31 (20)5399.67(26) 0.33 0.0521 (10)−0.0018 4 36 265.99 (20)5425.35(26) 0.13 ... ... 5 36 289.44 (20)5448.80(26) 0.16 ... ... 6 36 310.23 (20)5469.59(26) 0.02 ... ... 7 36 328.53 (20)5487.89(26)−0.14 0.0420 (10)−0.0019 8 36 344.53 (20)5503.89(26)−0.16 0.0388 (10)−0.0018 9 36 358.28 (20)5517.64(26)−0.08 0.0359 (10)−0.0015 10 36 369.52 (20)5528.88(26)−0.29 0.0358 (10) 0.0019 11 36 379.13 (20)5538.49(26)−0.10 0.0324 (10) 0.0014 12 36 386.73 (20)5546.09(26)−0.09 0.0293 (10) 0.0018 13 36 392.74 (20)5552.10(26)−0.10 0.0212 (10)−0.0028 14 36 397.68 (20)5557.04(26) 0.18 0.0203 (10)−0.0006 15 36 401.25 (20)5560.61(26) 0.20 0.0187 (10) 0.0015 16 36 403.92 (20)5563.28(26) 0.23 0.0190 (20) 0.0044 17 36 405.95 (40)5565.31(43) 0.35 ... ... 18 36 407.28 (40)5566.64(43) 0.35 ... ... 19 36 407.80 (40)5567.16(43)−0.02 ... ... 20 36 408.25 (40)5567.61(43)−0.12 ... ... 21 36 408.85 (40)5568.21(43) 0.15 ... ... on the accuracy of the potential-energy function in the region rele- vant for the bound states. This final step ensured that the potential describing the B+state is accurate for both bound states and contin- uum states corresponding to kinetic energy releases up to EKER/(hc) ≃300 cm−1. The optimal parameter values are presented in Table IV. The calculated term values and rotational constants corresponding to these parameters are compared to the experimental data obtained for the A+and B+states in Tables I and V, respectively. The vast majority of the calculated results are in agreement with experimental values within the error bars. The value of the weighted root-mean- square deviation61is 1.3, which indicates that the model adequately describes the experimental data. V. DISCUSSION The potential-energy functions corresponding to the optimal values of the parameters are depicted in Fig. 4, where the posi- tions of the measured vibrational energy levels are also indicated. The levels of the A+ 1/2state are not shown for clarity. The A+ 1/2and A+ 3/2states are much more strongly bound ( De= 5612.1 cm−1and 5626.6 cm−1, respectively) than the B+state ( De= 277.0 cm−1). The J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . Potential-energy curves of the first three electronic states of24MgAr+. The curve for the X+state was taken from Ref. 43. The curves of the A+ Ω′=1/2,3/2 and B+states were determined in the present work. The vibrational levels of the X+,43A+ Ω′=3/2, and B+22states that were observed experimentally are shown as horizontal lines. The inset shows the bound part of the B+potential-energy curve on an expanded scale. equilibrium distances Reof the A+ 1/2and A+ 3/2states (4.524 a0and 4.528 a0, respectively) are also significantly shorter than that of the B+state (8.289 a0). The present values of both Deand Reare in good agreement with the ab inito calculations reported in Paper I,22 which predict potential depths of De= 5615.2 cm−1, 5629.3 cm−1, and 253.2 cm−1for the A+ 1/2, A+ 3/2, and B+states, respectively, and equilibrium distances of Re= 4.528 a0, 4.530 a0, and 8.560 a0. The reason for the large differences between the A+and B+ states has already been discussed22,24,25and can be qualitatively understood when considering the alignment of the Mg+(3p) orbital with respect to the internuclear axis. In the case of the A+state, the axis of the Mg+(3pπ) orbital is perpendicular to the internu- clear axis and leaves the Ar atom exposed to an unscreened Mg2+ core. At large R, this translates into an attractive interaction between the permanent quadrupole moment of Mg+(3p) and the induced dipole moment of Ar. The other dominant electrostatic interactions (charge – induced-dipole and charge – induced-quadrupole) areindependent of the orbital alignment. In the case of the B+state, the axis of the pσorbital is parallel to the internuclear axis and almost perfectly screens, for the Ar atom, half of the charge of the Mg2+ core. At large R, this translates into a repulsive induced-dipole – permanent-quadrupole interaction. Moreover, because of the differ- ent alignment of the porbital, the spatial overlap of the electronic clouds of Mg+and Ar, resulting in repulsive forces, is important at larger internuclear distances for the B+state than for the A+ state. The evolution of the two spin–orbit components of the A+state from Hund’s coupling case (a) to Hund’s coupling case (c) is well illustrated by the evolution of the Ω′= 1/2−Ω′= 3/2 spin–orbit splitting with v′, as shown in Fig. 5. At low v′values, the vibrational wave function is located near the equilibrium distance Re(A+ Ω′), and in this region, the A+ 2Π1/2state is hardly mixed with the B+ 2Σ+state because the corresponding potential-energy functions lie energeti- cally far apart. In this case, the A+ 1/2−A+ 3/2splitting is simply a(R) [see diagonal elements in Eq. (4)]. The observed low- v′splittings differ from a(R→∞) = 61.05 cm−1because the spin–orbit con- stant a(R) increases as Rdecreases. At higher v′, the vibrational wave function extends to larger internuclear distances and is influenced by the A+ 2Π1/2–B+ 2Σ+mixing, or equivalently by the transition from Hund’s case (a) to Hund’s case (c). The spin–orbit splitting increases and reaches, asymptotically,3 2a(R→∞)=91.57 cm−1, the spin– orbit splitting of the Mg+ion. The numerical values of all spin–orbit splittings determined in this work are listed in the supplementary material. Precise values of the dissociation energies D0of the A+and B+ states can be determined from the model potentials. They are com- pared in Table VI to the values available in the literature. Because the experimental data for the B+state extend up to the dissociation limit and those for the A+states cover more than 90% of the total poten- tial wells, the uncertainties on the D0values derived from the model potentials are small and estimated to be of the order of 1 cm−1, which FIG. 5 . Spin–orbit splitting between the A+ 1/2(v′)and A+ 3/2(v′)levels determined from the experimental (open circles) and calculated (crosses) term values. The limit of3 2a(R→∞)=91.57 cm−1is shown by the dashed gray line. It cor- responds to the atomic Mg+(3p1/2–3p3/2) spin–orbit splitting and is asymptotically reached for high vibrational levels. J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE VI . Dissociation energies D0(in cm−1) of the X+, A+, and B+electronic states of24MgAr+and of the a3Π0−state of24MgAr determined from the experimental data. In Ref. 29, the value of D0(X+) was derived from D0(A+) using a thermochemical cycle. This D0(X+) value has been updated using the most recent experimental value of the electronic origin of the A+←X+transition (see Refs. 21 and 43). The values in parentheses represent one standard deviation in units of the last digit. State D0(cm−1) Reference A+ 1/25476.7(10) This work 5445(165) Reference 29 A+ 3/25491.5(10) This work 5460(165) Reference 29 B+260.2(10) This work X+1246.0(12) This work 1254(60) Reference 43 1214(165) Reference 29 1180(80) Reference 62 1237(40) Reference 63 a3Π0−167.6(11) This work 160(40) Reference 63 represents an improvement of more than two orders of magnitude in the uncertainties of D0(A+ 1/2,3/2)over previously available values.29 The present D0values for the two spin–orbit components of the A+state are in agreement within error bars with the values deter- mined by Le Roy29in a near-dissociation-expansion analysis of the v′= 0–14 term values measured by Pilgrim et al.24,25They are sig- nificantly larger than the spin–orbit-averaged value of 2983.39 cm−1 calculated by Gaied et al. using a pseudopotential approach.34The present D0values are in very good agreement with the ab initio val- ues of 5479.4 cm−1and 5493.7 cm−1reported in Paper I for the Ω′ = 1/2 and 3/2 components, respectively.22No previous values for the dissociation energy of the B+state D0(B+) were available in the liter- ature before its observation in Ref. 22. The D0(B+) value of 260.2(10) cm−1reported here is in reasonable agreement with the ab initio value of 237.7 cm−1reported in Ref. 22. The dissociation energy D0(X+) of the X+ground state of MgAr+can be calculated to be 1246.0(12) cm−1from the disso- ciation energy D0(B+) of the B+state using the thermochemical cycle, D0(X+)=˜ν07+T′′ 7+D0(B+)−T(Mg+, 3p3/2). (14) In Eq. (14), the wave number ˜ν07of the B+(v′= 0)←X+(v′′= 7) transition is taken from Ref. 22, the term value T′′ 7of the X+(v′′ = 7) level is taken from Ref. 43, and T(Mg+, 3p3/2) = 35 760.88 cm−1 is the term value of the 3 p3/2state of Mg+.48The present value of the dissociation energy of the X+state is compared in Table VI to other experimental values or values determined from experimental data. Our new value is in excellent agreement with the most recent ones but much more precise. It is also in agreement with the latest ab ini- tiovalue of 1238 cm−1obtained by Tuttle et al.35and the value of 1235 cm−1calculated by Gaied et al.34The dissociation energy of the a3Π0−metastable state of the neutral MgAr molecule can be determined in a similar way using D0(a)=˜ν07+˜ν′ 70+D0(B+)−˜ν(Mg+(3s1/2−3p3/2)) −Ei(Mg(3s3p3P0))/(hc), (15) where Ei(Mg(3 s3p3P0)) is the ionization energy of the 3 s3p3P0state of atomic Mg and ˜ν′ 70is the wave number of the X+(v′′= 7)←a (v′′′= 0) ionizing transition64reported in Ref. 43. The present value of 167.6(11) cm−1forD0(a) is in close agreement with the value of 160(40) cm−1reported by Massick and Breckenridge and esti- mated from the results of Birge–Sponer extrapolations and ab initio calculations.63 In Paper I,22we showed that high-lying vibrational levels of the B+state ( v′≥7) decay predominantly by spin–orbit predisso- ciation into the Mg+(2P1/2) + Ar(1S0) continuum associated with the Ω′= 1/2 spin–orbit component of the A+state (see inset in Fig. 4).22 This effect can be studied theoretically with the model presented in Sec. IV by diagonalizing the part HSOof the Hamiltonian in Eq. (4) corresponding to the spin–orbit interaction between Ω′= 1/2 states in a basis of diabatic Hund’s-coupling-case-(a) vibronic functions |v′,2Λ′ 1/2⟩. In doing so, we neglect heterogeneous perturbations. The matrix elements are ⟨v,2Σ1/2∣HSO∣v′,2Σ1/2⟩=0, (16) ⟨v,2Π1/2∣HSO∣v′,2Σ1/2⟩=∫dR(χv Π(R))∗a(R)√ 2χv′ Σ(R), (17) ⟨v,2Π1/2∣HSO∣v′,2Π1/2⟩=−∫dR(χv Π(R))∗a(R) 2χv′ Π(R). (18) The vibrational wave functions χv Σandχv Πare obtained by solving the nuclear Schrödinger equation (13) for the potentials VΣandVΠ, respectively. The basis set must include both bound and continuum vibrational wave functions, which we calculate using a Legendre– Gauss–Lobatto finite-element DVR technique with exterior com- plex scaling.57,60Diagonalization of the large complex-symmetric diabatic matrix yields complex eigenvalues with, for the metastable vibrational levels of the B+state ( v′= 7–21), a real part equal to the energy of the level Ev′and an imaginary part equal to −Γv′/2, where Γv′is the predissociation width. The predissociation widths are found to lie in the range from 0.03 cm−1to 0.0003 cm−1, depend- ing on v′, which corresponds to lifetimes between 18 ns and 1.8 μs. These values agree with the experimental observations that (i) the broadening of individual lines in the measured spectra resulting from predissociation is equal to or smaller than the laser linewidth of 0.1 cm−1and (ii) the predissociation rate is faster than the time scale of the experiment ( ∼5μs).22The values of the calculated widths Γv′ are listed in the supplementary material. The positions of the predis- sociative vibrational levels of the B+state ( v′≥7) calculated using either the adiabatic potential or the method just described differ by less than 0.25 cm−1, i.e., by less than the experimental uncertain- ties of the measured term values. Lower-lying levels ( v′= 0–6) differ by at most 0.4 cm−1. Therefore, non-adiabatic corrections to the adiabatic vibrational energies are small, as expected from the fact that the weighted root-mean-square deviation of the fit ( ≃1.3) of the potential-energy functions described in Sec. IV is close to unity. J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp VI. CONCLUSION We reported a global study of the 3 pRydberg complex of the MgAr+molecular ion. Using the ICMRD technique, high-resolution spectroscopic data were obtained for the two spin–orbit components of the A+state, up to vibrational levels as high as v′= 29. Com- bined with the data reported on the B+state in Paper I,22they give a complete and detailed picture of the 3 pRydberg complex. Potential- energy functions were extracted from the experimental energies and rotational constants in a direct-potential-fit approach that included theR-dependent spin–orbit interaction between the three states of the complex. These potential-energy functions reproduce the posi- tions of all observed vibrational levels of the A+ 1/2, A+ 3/2, and B+states to within less than 0.5 cm−1. The dissociation limits of the B+state and of the two spin– orbit components of the A+state were determined from the fitted potential-energy functions and have uncertainties of only 1 cm−1, i.e., they are more precise compared to those obtained in previ- ous studies by more than two orders of magnitude.29Dissociation energies with significantly reduced uncertainties could also be deter- mined for the X+ 2Σ+state of MgAr+and the a3Π0−state of MgAr. Finally, the non-adiabatic effects caused by the spin–orbit cou- pling between the B+and A+states were investigated theoretically and provided results in excellent agreement with the experimental ones.22 The 4 s, 3d, and 4 pRydberg complexes of MgAr+lie ener- getically about 30 000 cm−1above the 3 pcomplex. Their inves- tigation by ICMRD from the A+or B+states is underway. The 4sand 3 dcomplexes lie in the same region of energy as charge- transfer states correlated with the Mg(1S0) + Ar+(2P3/2,1/2 ) dis- sociation asymptotes. The investigation of these charge transfer states by ICMRD may be possible by detecting the pulsed field ionization of the Ar fragment in high Rydberg states instead of the Mg fragment in high Rydberg states as in the present work. Such an investigation would enable the characterization of the effects of the charge-transfer interaction on the Rydberg states of molecular ions. Exciting the molecular ion even further, we expect that the ground electronic state of the doubly charged MgAr2+ion can be reached and investigated by pulsed-field-ionization zero- kinetic-energy (PFI-ZEKE) photoelectron spectroscopy following the procedure we recently tested with Mg+.42This procedure is gen- eral and can be applied to molecular cations other than MgAr+ in future work. The investigation of doubly charged molecular ions by high-resolution photoelectron spectroscopy would permit the exploration of the diversity and complexity of the mecha- nisms responsible for the stability or instability of such systems, as illustrated in Fig. 1 of Paper I22and discussed by Schröder and Schwarz.65 SUPPLEMENTARY MATERIAL See the supplementary material for tables listing the molecu- lar constants of the A+and B+states, the v-dependent spin–orbit splittings and spin–orbit-averaged term values of the A+ Ωstates, and the calculated nonadiabatic spin–orbit-predissociation widths of the vibrational levels of the B+state.ACKNOWLEDGMENTS We thank J. A. Agner and H. Schmutz for technical support and M. Reiher and S. Knecht for fruitful discussions. 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A 100, 032517 (2019). 60B. Simon, Phys. Lett. A 71, 211 (1979). 61See, e.g., Ref. 43 for the definition of the weighted root-mean-square deviation used in the present work. 62H. Hoshino, Y. Yamakita, K. Okutsu, Y. Suzuki, M. Saito, K. Koyasu, K. Ohshimo, and F. Misaizu, Chem. Phys. Lett. 630, 111 (2015). 63S. Massick and W. H. Breckenridge, Chem. Phys. Lett. 257, 465 (1996). 64The wave numbers of the X+ 2Σ+(v′′= 7)←a3Π0(v= 0) ionizing transitions of24MgAr,25MgAr, and26MgAr reported in Table 1 (third column) of Ref. 43 must be corrected from 39 341.1(5) cm−1, 39 334.7(5) cm−1, and 39 328.8(5) cm−1, respectively, to 39 340.5(5) cm−1, 39 334.1(5) cm−1, and 39 328.2(5) cm−1. All other table entries are not affected. 65D. Schröder and H. Schwarz, J. Phys. Chem. A 103, 7385 (1999). J. Chem. Phys. 153, 074311 (2020); doi: 10.1063/5.0015608 153, 074311-12 Published under license by AIP Publishing

cjcp2006088.pdf

Chin. J. Chem. Phys. 33, 595 (2020); https://doi.org/10.1063/1674-0068/cjcp2006088 33, 595 © 2020 Chinese Physical Society.Photodynamics of methyl-vinyl Criegee intermediate: Different conical intersections govern the fates of syn/anti configurations Cite as: Chin. J. Chem. Phys. 33, 595 (2020); https://doi.org/10.1063/1674-0068/cjcp2006088 Submitted: 09 June 2020 . Accepted: 23 July 2020 . Published Online: 10 November 2020 Ya-zhen Li , Jia-wei Yang , Lily Makroni , Wen-liang Wang , and Feng-yi Liu COLLECTIONS Paper published as part of the special topic on Special Issue of 16th National Chemical Dynamics Symposium ARTICLES YOU MAY BE INTERESTED IN Experimental and theoretical studies of the doubly substituted methyl-ethyl Criegee intermediate: Infrared action spectroscopy and unimolecular decay to OH radical products The Journal of Chemical Physics 152, 094301 (2020); https://doi.org/10.1063/5.0002422 Quantum dynamical investigation of the simplest Criegee intermediate CH 2OO and its O–O photodissociation channels The Journal of Chemical Physics 141, 134303 (2014); https://doi.org/10.1063/1.4894746 Electronic spectroscopy of methyl vinyl ketone oxide: A four-carbon unsaturated Criegee intermediate from isoprene ozonolysis The Journal of Chemical Physics 149, 244309 (2018); https://doi.org/10.1063/1.5064716CHINESE JOURNAL OF CHEMICAL PHYSICS VOLUME 33, NUMBER 5 OCTOBER 27, 2020 ARTICLE Photodynamics of Methyl-Vinyl Criegee Intermediate: Dierent Conical Intersections Govern the Fates of Syn/Anti Congurationsy Ya-zhen Li ,Jia-wei Yang ,Lily Makroni ,Wen-liang Wang ,Feng-yi Liu∗ Key Laboratory for Macromolecular Science of Shaanxi Province, School of Chemistry & Chemical Engineering, Shaanxi Normal University, Xi'an 710062, China (Dated: Received on June 9, 2020; Accepted on July 23, 2020) Methyl vinyl ketone oxide, an unsaturated four-carbon Criegee intermediate produced from the ozonolysis of isoprene has been recognized to play a key role in determining the tro- pospheric OH concentration. It exists in four congurations ( anti-anti,anti-syn,syn- anti, and syn-syn) due to two dierent substituents of saturated methyl and unsaturated vinyl groups. In this study, we have carried out the electronic structure calculation at the multi-congurational CASSCF and multi-state MS-CASPT2 levels, as well as the trajectory surface-hopping nonadiabatic dynamics simulation at the CASSCF level to reveal the dif- ferent fates of syn/anti congurations in photochemical process. Our results show that the dominant channel for the S 1-state decay is a ring closure, isomerization to dioxirane, dur- ing which, the syn(CO) conguration with an intramolecular hydrogen bond shows slower nonadiabatic photoisomerization. More importantly, it has been found for the rst time in photochemistry of Criegee intermediate that the cooperation of two heavy groups (methyl and vinyl) leads to an evident pyramidalization of C3 atom in methyl-vinyl Criegee inter- mediate, which then results in two structurally-independent minimal-energy crossing points (CIs) towards the syn(CO) and anti(CO) sides, respectively. The preference of surface hopping for a certain CI is responsible for the dierent dynamics of each conguration. Key words: Trajectory surface hopping nonadiabatic dynamics, Criegee intermediate, Dioxirane, Photoisomerization, Minimal-energy crossing point I. INTRODUCTION Isoprene (2-methyl-1,3-butadiene) is one of the most abundant volatile organic compound in the atmo- sphere [1, 2]. Its reaction with ozone has been re- ported as a major tropospheric removal pathway that plays a signicant source of OH in the tropospheric oxidation cycle especially during daytime and win- ter months [3]. Depending on the position of O 3 attacking at the two C=C bonds of isoprene ( i.e., C1C2 or C3 C4), two isomers of C 4Criegee inter- mediate, namely a methyl-vinyl ketone oxide (MVK- OO, (CH 2=CH)(CH 3)COO) and a methacrolein oxide (MVCR-OO, CH 2=C(CH 3)CHOO), with total branch- ing of 23% and 19%, respectively, are formed [4, 5]. The more abundant MVK-OO, also known as methyl-vinyl Criegee intermediate (MVCI), is predicted to have four congurations with respect to the orientations around C3C4 and C3=O1 bonds (FIG. 1), of which each has considerable abundance [5]. Therefore, the for- yPart of the special issue on \the Chinese Chemical Society's 16th National Chemical Dynamics Symposium". Author to whom correspondence should be addressed. E-mail: fengyiliu@snnu.edu.cnmation, interconversion and atmospheric reactions of MVCI have drawn great attention from chemists [6{8]. Until now, the electronic structure of the sim- plest Criegee intermediate (SCI) is elusive, therefore, the substitution eect has shown signicant dierence based on whether it is saturated or unsaturated sub- stituents. It is well understood that the unimolecular reaction of SCI is in favor of ring-closure to a three- membered-ring structure (dioxirane) while against hy- drogen atom transfer to terminal oxygen [9{11]. How- ever, the asymmetrical substitution on SCI by a sat- urated methyl group (-CH 3) forming syn- and anti- CH3CHOO has shown signicantly dierent reaction pathways in anti/synconguration [11{15]. More ev- idently, the addition of unsaturated substituents sug- gests a totally dierent channel. A recent theoreti- cal calculation demonstrated that in vinyl-Criegee in- termediate (VCI), both the C OO ring closure and H transfer are suppressed, while another ring-closure channel to either a four- or ve-membered-ring prod- uct becomes dominant [11]. In addition to the thermal reactions, the photochemistry of Criegee Intermediates (CIs) also shows strong substituent dependence. The high-level dynamically-weighted complete active space self-consistent eld (DW-CASSCF)/MRCI-F12 calcula- tions by Guo and coworkers suggested that the SCI has DOI:10.1063/1674-0068/cjcp2006088 595 c⃝2020 Chinese Physical Society596 Chin. J. Chem. Phys., Vol. 33, No. 5 Ya-zhen Li et al. two spin allowed photo-dissociation branches, i.e., H2CO(X1A1)+O(1D) and H 2CO(a3A′′)+O(3P), on the second excited (S 2) state [16]. Our electronic-structure calculations and nonadiabatic dynamic simulations on SCI, methyl-Criegee intermediate (MCI) and VCI re- vealed a rst excited (S 1) state involved process that led to dioxirane, in addition to the known O O bond cleav- age. For each conformer/conguration of MCI and VCI, the ring tension, hydrogen bond and conjugation caused by the dierent orientations of substituents, govern the relative nonadiabatic transition probabilities and prod- uct yields [17, 18]. In MVCI, there are both methyl and vinyl groups bonded to the carbon site, therefore it provides a unique opportunity to comparatively identify the roles of saturated and unsaturated substituents in a sin- gle system. Recently, experimental and theoreti- cal studies combined by Lester et al . has reported that the thermal reactions of MVCI are conguration- dependent. The syn(CO)-MVCI produces OH radi- cal through a hydrogen-atom-transfer mechanism, while theanti(CO) conguration is predicted to produce OH via a dioxole pathway, and the former has a much lower barrier height than the latter [15]. All these ndings show non-negligible contribution from both saturated and unsaturated substituent eects in MVCI. This syn- ergistic eect observed from this study is a quite dis- similar atmospheric fate from other relevant congura- tions. Therefore, for the unidentied photochemistry of MVCI, we have carried out the electronic struc- ture calculations at the state-averaged (SA)-CASSCF and multi-state second-order perturbation theory (MS- CASPT2), along with the CASSCF-level trajectory surface-hopping (TSH) [21, 22] nonadiabatic molecular dynamics, to reveal the potential energy surface (PES) and photodynamics of all congurations of MVCI. The contribution of the saturated and unsaturated sub- stituents is expected to be well understood. II. COMPUTATIONAL METHODS A. Electronic structure calculations Electronic structure calculations are carried out at the SA3-CASSCF [19] and MS3-CASPT2 [20] levels of theory, respectively, using def2-SVP and def2-TZVP basis set [23]. An active space consists of the 2s or- bital of the middle O atom, the lone-pair orbital on terminal O atom, σ/σ∗orbitals of O O bond, and all the ve π/π∗orbitals, as well as the electrons dis- tributed on them, namely, the CAS(12e, 9o), are used (see FIG. S1 in supplementary materials). First, four ground-state (S 0) congurations of MVCI, i.e., the anti- anti,anti-syn,syn-anti, and syn-synMVCI (for con- venience, which are respectively abbreviated as AA, AS, SA and SS, in the following discussion), are optimized at the SA-CASSCF/def2-SVP, MS-CASPT2/def2-SVP and MS-CASPT2/def2-TZVP levels, respectively, and their corresponding excitation energies to the low-lyingexcited states are calculated. To get a systematic under- standing of the unimolecular process, one-dimensional (1D) potential energy curves (PECs) and 2D-PESs are constructed with a series of constrained optimiza- tions along the relevant reaction coordinates at the SA3-CASSCF/def2-SVP level. Moreover, the nona- diabatic processes are further validated by searching for the S 1/S0-minial energy crossing points (CIs) at the CASSCF level. The transition states (TSs) for COO ring-closure on the ground-state are optimized using the PBE0 density functional with a def2-SVP basis set. Their relative energies are then rened by MS3-CASPT2 single-point calculation. All CASSCF and MS-CASPT2 calculations are carried out using the MOLCAS program [24], and DFT calculations are ob- tained in Gaussian 09 [25]. B. Trajectory surface-hopping dynamics Trajectory surface-hopping (TSH) molecular dynam- ics simulations, combined with the SA3-CASSCF/def2- SVP quantum-chemistry calculations, are employed to gain a better understanding of photodynamics of MVCI. Here, trajectory dynamics simulation of all the four con- gurations are carried out to investigate the substitu- tion eects, even though the calculated Boltzmann pop- ulations (Table S1 in supplementary materials) show very low percent for SS. For each conguration, 300 ini- tial conditions are respectively generated on the base of the SA3-CASSCF optimized geometry and frequen- cies, then a Wigner distribution [26] is generated. Both the S 1and S 2states are initially populated based on our previous calculations [17], while the triplet process is ex- cluded due to the negligible spin-orbital coupling eects [27]. Energy windows for each state are set individually considering their dierent excitation energies (see Table S2 in supplementary materials). Finally, about 100 tra- jectories are successfully simulated for each of the four congurations from either S 1or S 2state (in detail: 108 for AA, 99 for AS, 73 for SA and 88 for SS from S 1state; and 130 for AA, 131 for AS, 131 for SA and 130 for SS from S 2state). The trajectories are submitted with a time step of 0.5 fs and run up to 400 fs. During the sim- ulation, the \SHARC dynamics" of diagonal potentials and vectorial nonadiabatic couplings [28] are used. The decoherence correction is considered using an energy dierence-based correction (EDC) scheme [29], and the standard SHARC surface hopping scheme is used to de- scribe the surface hopping. TSH dynamic simulations are performed by using the SHARC program [30, 31], combined with CASSCF calculations implemented in the MOLPRO package [32]. III. RESULTS AND DISCUSSION A. Electronic structures and PESs FIG. 1 presents the four congurations of MVCI with key geometrical parameters optimized at the MS3- DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 33, No. 5 Photodynamics of Methyl-Vinyl Criegee Intermediate 597 FIG. 1 The four S 0-state congurations of MVCI (where A and S stand for anti and syn, respectively, the rst A/S indicate the anti/synconguration with respect to the ori- entations of the vinyl group and terminal O atom beside C3O1 bond; the second one corresponds to that with re- spect to C3 C4 bond). Geometries optimized at the MS3- CASPT2/def2-TZVP level, along with the important bond lengths (in A) are shown on the left, and denitions of key (dihedral) angels are shown on the right panel. CASPT2/def2-TZVP level (structures with more pa- rameters, and those optimized at the DFT-PBE0/def2- SVP, CASSCF/def2-SVP and MS3-CASPT2/def2-SVP levels are shown in FIG. S2). It is seen that all con- gurations are planar, indicating an extended conju- gation moiety caused by the vinyl substituent. Com- pared with the geometry of SCI calculated at the same level of theory (in which R(CO)=1.288 A and R(OO)=1.330 A), all congurations of MVCI show a systematically elongated O O bond (1.347 1.370 A), suggesting a larger zwitterionic character in the ground state. Also, in both SA and SS congurations that con- tain the same syn(C3O1) skeleton, a short O2


H distance is observed. The hydrogen-bond-like interac- tion shortens the O O bond (with respect to those in AA and AS, which share the same anti(C3O1) skele- ton), and is expected to stabilize the syn(CO) cong- urations. Meanwhile, it is found that in SA and SS con- gurations, ring tension is also more signicant, which is re ected by the large bond angles around the C O bond ( i.e.α,β, and γ, as dened in FIG. 1). The hydrogen bond and ring tension play opposite roles in stabilizing the congurations, as being found in the VCI [18]. As a result, the MS-CASPT2/def2-TZVP calcula- tions suggest that the relative energies of 0.0, 0.6, 1.3, and 3.7 kcal/mol for AA, AS, SA and SS congurations, respectively, are well consistent with Lester's compu- tational results at the ANL0-B2F//B2PLYP-D3/cc- pVTZ level (in which the relative energies are 0.00, 1.76, 2.57, and 3.05 kcal/mol) [15]. The order of relative sta-bilities is in a sharp contrast to that of VCI (in which the relative stabilities are SA SS>AA>AS at the MS- CASPT2/def2-SVP level) [18], caused by the substitu- tion of the more steric methyl group. These dierences observed between VCI and MVCI suggest their dierent fates in photo-induced reactions. The MS-CASPT2/def2-TZVP calculated excita- tion energies of the four congurations to the S 1 (n!π∗) state are in the range of 51.3 55.4 kcal/mol (516558 nm), falling into visible region. The S 2exci- tation, with a π!π∗character in 79.4 84.5 kcal/mol (i.e., 339360 nm), is comparable with the experimen- tal values (350 381 nm) [33]. We attempt to optimize all possible minima on the S 1and S 2-state PESs, while only SS conguration in S 1state can be found by MS- CASPT2/def2-TZVP and only SA conguration can be found by MS-CASPT2/def2-SVP (both S 1-SA and S 1- SS can be optimized at the CASSCF level). From the optimized S 1-SA and S 1-SS structure, we observed sig- nicant elongation of the C O bond (1.385 and 1.387 A in S 1-SA and S 1-SS at the MS-CASPT2 level, respec- tively) compared with that in the ground state, there- fore the C O bond shows a strong tendency to rotate (i.e., O2 atom rotates out of the molecular plane). Such a tendency actually exists in all congurations of MVCI, as being found in SCI and substituted CIs [17, 18]. FIG. 2 shows the S 1-PECs constructed by a series of constraint geometry optimizations along the dihedral angle ϕ(see FIG. 1 for denition). Ideally, rotations around the C3 C4 and C4 C5 bonds are both possi- ble, while they can be excluded due to either high bar- rier heights (see FIG. S3 in supplementary materials) or insignicance for distinguishable congurations. As seen in FIG. 2, the C O rotary PECs show a downhill trend from the Frank-Condon (FC) point of AA and AS conguration to the S 1/S0degenerate region, suggest- ing a spontaneous decay to possible conical intersections (CIs); while for FC-SA and FC-SS, very mild barriers are observed, therefore the accessing of CIs from these congurations is comparatively less straightforward. At the bottom of each PECs, a pair of CIs is optimized at the CASSCF level. For instance, as seen in FIG. 2(a), two CIs, namely, CI A−and CI A+lie on the left and right of 90◦, towards the SA and AA side, respec- tively. In addition to the dihedral angle ϕ, another ma- jor dierence between them are the pyramidalization angle τ, which re ects the intensity of C3 popping out of the C4 C6O1 plane. More specically, CI A−and CIA+show a positive and negative pyramidalization an- gle (+6.8◦and6.4◦), respectively. The ndings here are dierent from that in MCI and VCI (where only one CI is found on the S 1-PEC, that is, the photoisomerization from synandanti con- former/conguration shares the same CI), while being similar to the situation in cis-trans photoisomerization of alkenes [34{36]. Clearly, the three heavy groups (i.e., methyl, vinyl and peroxy groups) bonded on C3 atom increase C3's pyramidalization intensity, thus the DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical Society598 Chin. J. Chem. Phys., Vol. 33, No. 5 Ya-zhen Li et al. FIG. 2 The CASSCF/def2-SVP computed S 1-state energy proles along the C3 O2 rotary coordinate ( i.e., dihedral angle ϕ) (a) between SA and AA and (b) between SS and AS, FCs are shown in red triangles, fully optimized S 1-SA and S 1-SS are shown in red circles; (c) S 1-SA, S 1-SS and the CIs optimized on the same level, with important geometrical parameters. The CIs are named as follows: subscript S and A stand for the conguration around the unchanged C3 C4 bond, and +/ indicates the dihedral angle ϕlarger or smaller than 90, respectively. S1/S0crossing point bifurcates into two geometrically dierent ones. Due to their inequivalent geometries, the two CIs are expected to play dierent roles in the syn(CO)!anti(CO) and anti(CO)!syn(CO) photoisomerization processes, which will be veried by TSH dynamics. The 2D S 1-PESs along the dihedral angle ϕand C3O2 distance (FIG. S4 in supplementary materi- als), demonstrate the above-mentioned S 1-state de- cay pathways. After S 1!S0nonadiabatic transi- tion, there exist three possible channels, including the anti(CO)/ syn(CO) isomerization, ring closure to dioxi- rane, and internal conversion (IC) back to the initial conguration in S 0state. Because of the lowest rela- tive energies of dioxirane (more than 21 kcal/mol lower in energy than the respective MVCI congurations, see FIG. S3 in supplementary materials), we can predict the highest ring-closure yields. In the S 1/S0-CI!S0- dioxirane process, obvious shortening of C O bond (from 2.1A to 1.3A) and continued decrease ofbond angle α(from 95◦to60◦) are seen (FIG. S2 in supplementary materials), and will be further observed in the results of TSH dynamics. B. S 1-state photodynamics of MVCI The trajectories starting from S 1state for all cong- urations show an ultrafast transition to S 0state. The averaged S 1lifetimes for the AA, AS, SA and SS con- gurations, 278.4, 279.7, 486.5, and 337.5 fs (FIG. S5), respectively, are consistent with that observed in VCI [18]. The hydrogen-bonded structures (SA and SS) show slower decay towards the perpendicular CI region, due to the time spent in breaking the hydrogen bond. Similarly, as predicted above in electronic-structure cal- culations, in SA and SS, the existed S 1-state minima at the FC region trap the trajectories and slow down their further decay (see FIG. 2 and discussion in section III A). The hopping events of S 1trajectories, characterized by the geometric features of the hopping points, as DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 33, No. 5 Photodynamics of Methyl-Vinyl Criegee Intermediate 599 FIG. 3 Distributions of the hopping points near the S 0/S1-CIs for the trajectories starting from the S 1state of (a) AA, (b) AS, (c) SA and (d) SS congurations, following the dihedral angle ϕand pyramidalization angle τ. The CASSCF/def2-SVP optimized CIs are shown in blue and red \+" symbols, respectively. shown in FIG. S6 (distributions for various coordi- nates), can be classied into three types: the majority (63%71%) corresponds to the S 1!S0hop caused by the CO bond rotation, which takes place in the vicin- ity of the CASSCF-optimized CIs. Correspondingly, the hopping points are densely distributed around di- hedral angle ϕ=90◦. The second type is the S 1!S0hop occurring, in a few cases (1% 12%), at the FC region, due to the relatively low S 1excitation energy. And the third one takes place along the O O elongating path, featured by a long O O distance ( >1.7A), which then leads to either S 1!S0or S1!S2hop (1% 5%). Each of the three types of hops forwards the trajectory to dier- ent branches and thus generates dierent products, i.e., ring closure to dioxirane (or syn-anti isomerization to the opposite conguration that shares the same type of hopping points as ring closure), restoring to initial con- guration, and forming a methyl-vinyl ketone+singlet O atom, respectively. FIG. 3 illustrates the distributions of the hopping points for the dominant type ( i.e., hops in the vicin- ity of CI) of MVCI, along ϕandτ. It is interesting to nd that, for trajectories starting from dierent cong- urations, the hopping points favor dierent CIs. Most of the hopping points of AA and AS (with an initial ϕof 180◦) are distributed on the right side of ϕ=90◦, close to CI A+and CI S+, respectively; while those of SA and SS congurations (in which ϕis initially 0◦), on the contrary, majorly fall on the left or on top of 90◦, thus slightly favoring the CI A−and CI S−. Directly compar-ing the distributions in FIG. 3 (a) and (c) (and that in FIG. 3 (b) and (d)) makes the fact more notable, that is, the hops of MVCI show a clear preference for the CI at the \reactant" side. Again, it is dierent from what has been observed in VCI, in which all hops, from either the synoranti conguration, are evenly distributed around the only CI between each pair of congurations [18]. The dramatic change of the hopping events from VCI to MVCI, emphasizes the heavy-group eect caused by methyl substituent. The photodynamics after the nonadiabatic decay takes place mainly in the ground state and is thus straightforward. As summarized in FIG. 4 (a detailed analysis of branching ratios based on various geometri- cal parameters are illustrated in FIG. S7 in supplemen- tary materials), from the CI, two isomerization chan- nels are discovered. In one case, the C O bond con- tinuously rotates from 0◦to 180◦(or reversely, from 180◦to 0◦), asyn-to-anti (oranti-to-syn) isomeriza- tion around the C O bond can be achieved; in the other case, if the C OO bending mode is involved, then a ring closure (indicted by the shortened C3 O2 bond and decreased angle α) to dioxirane is expected. In principle, a third channel, namely, the internal conver- sion to initial conguration is possible but not observed in all simulations. For all congurations, the ring clo- sure is the dominant channel in all congurations (the yields are 66%, 67%, 55% and 47 for AA, AS, SA and SS, respectively), which may be attributed to the much lower relative energy of dioxirane. The time spent in the DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical Society600 Chin. J. Chem. Phys., Vol. 33, No. 5 Ya-zhen Li et al. FIG. 4 The major photoisomerization channels of S 1-state MVCI started from (a) AA (in black) and AS (in purple), (b) SA and SS congurations. Yields for each process rela- tive to the total successful trajectories are shown in light-red and purple background. The relative energies (in kcal/mol) of minima are calculated at the MS3-CASPT2/def2-TZVP level, whereas those of the S 0-TSs and CIs are calcu- lated at the MS-CASPT2/def2-TZVP//PBE0/def2-SVP and MS-CASPT2/def2-TZVP//CASSCF/def2-SVP level, respectively. ring-closure photoisomerization process is in a range of 120280 fs. The yields of the syn-anti photoisomer- ization and O O bond dissociation products, are rela- tively small (4% 16% for the former and 1% 5% for the later), which are thus not discussed in detail. The good consistency between the S 1-PECs and TSH dynamics, conrms the novel nding of photo-induced ring closure of MVCI to dioxirane. More importantly, the newly found preference of hopping for certain CIs, reveals the importance of heavy-group eect on the pho- toisomerization processes. C. S 2-state photodynamics of MVCI FIG. 5 shows the variations of O O distance in the S2-trajectories with respect to the simulation time. It is FIG. 5 Geometrical evolution of trajectories in four congu- rations of (a) AA, (b) AS, (c) SA and (d) SS, initiated from the S 2state. Trajectories in red and black are that of O O dissociation and unreactive one during the simulation time, respectively. seen, 86%, 86%, 82% and 89% of all successful trajecto- ries in AA, AS, SA and SS congurations that started from S 2state, respectively, undergo an ultrafast O O dissociation channel, which is nished in the initial 50 fs of the simulation. Also, during the O O elongation, the molecules do not always maintain its planar struc- ture, as seen in FIG. S8 in supplementary materials, a mild torsion around C O bond ( ϕvaries in a range of 30◦) are observed. Furthermore, due to the known degeneracy in ground- and low-lying excited states of CIs [16, 17], in MVCI, 12%, 21%, 24% and 12% of AA, AS, SA and SS trajectories, respectively, eventually de- cay to S 0state within 300 fs (FIG. S9). Majority of the S2!S1and S 2!S0hops, as shown in FIG. S10 in sup- plementary materials, occur in 50 fs, with an elongated OO bond length of 2.0A, while the rest (mainly the S1!S0hop) takes place with an even longer O O dis- tance. The TSH dynamics well replicate the previous high-level calculations on the dissociation channel [16]. IV. CONCLUSION In this study, we report the multi-congurational CASSCF and MS-CASPT2 electronic-structure calcu- lations and CASSCF trajectory surface hopping nona- diabatic dynamic simulations on the photoinduced uni- DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical SocietyChin. J. Chem. Phys., Vol. 33, No. 5 Photodynamics of Methyl-Vinyl Criegee Intermediate 601 molecular processes of an atmospherically important Criegee intermediate, namely, the methyl-vinyl Criegee intermediate. Important ndings and conclusions are drawn as below: (i) In MVCI, the substitutions by methyl and vinyl groups play a cooperative role in altering the relative stabilities of the four congurations, in which the hy- drogen bond between the vinyl H and terminal O atoms decreases whereas the ring tension introduced by the same group increases the relative energies of the syn- anti andsyn-syncongurations. Thus, it causes that all four congurations are in considerable abundance, none of which can be neglectable during thermal or photo-induced reactions. (ii) The dominant channel for the S 1-state decay cor- responds to a ring closure process to dioxirane, which is much more favored compared to other channels. Dur- ing the photoisomerization, the hydrogen bond caused by the vinyl group signicantly slows down the nonadi- abatic photoisomerization of the syn(CO) ( i.e.,syn- synandsyn-anti) congurations. (iii) More importantly, during the S 1photoisomer- ization, the introducing of the methyl group in MVCI (with respect to VCI) increases the intensity of the pyramidalization of C3 atoms, which results in two structurally independent CIs towards the syn(CO) and anti(CO) sides, respectively. Nonadiabatic transitions in photodynamics of the syn(CO) and anti(CO) congurations show dierent preferences to- wards a certain CI, which has been reported for the rst time in Criegee-intermediate chemistry. The ndings here are expected to not only deepen the knowledge on the photochemical processes of the Criegee intermediate, but also inspire the rethinking of the \old" concept of substitution eect, that is, how a usually thought \inert" methyl group would dramati- cally change the nonadiabatic transition and photody- namics of a conjugated molecule. Supplementary materials: CASSCF and MS- CASPT2-optimized geometries, PESs, and details of dynamic simulations, as well as the Cartesian coordi- nates of key structures are provided. V. ACKNOWLEDGMENTS This work was supported by the the National Natural Science Foundation of China (No.21873060 and No.21473107), the Fundamental Research Funds for the Central Universities (No.GK201901007, No.2018CBLY004). [1]R. A. Rasmussen, J. Air. Pollut. Control. Assoc. 22, 537 (1972).[2]A. B. Guenther, R. K. Monson, and R. Fall, J. Geophys. Res. 96, 10799 (1991). [3]S. M. Aschmann and R. Atkinson, Environ. Sci. Tech- nol.28, 1539 (1994). [4]K. T. Kuwata, L. C. Valin, and A. D. Converse, J. Phys. Chem. A 109, 10710 (2005). [5]T. B. Nguyen, G. S. Tyndall, J. D. Crounse, and A. P. Teng, K. H. Bates, R. H. Schwantes, M. M. Coggon, L. Zhang, P. Feiner, D. O. Milller, K. M. Skog, J. C. Rivera-Rios, M. Dorris, K. F. Olson, A. Koss, R. J. Wild, S. S. Brown, A. H. Goldstein, J. A. de Gouw, W. H. Brune, F. N. 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Seijo, Comp. Mater. SCI. 28, 222 (2003). [25]M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical Society602 Chin. J. Chem. Phys., Vol. 33, No. 5 Ya-zhen Li et al. A. Petrone, T. Henderson, D. Ranasinghe, V. G. Za- krzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr.; J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Keith, T. Staroverov, R. Kobayashi, J. Normand, K. Raghavachari, A. Ren- dell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. 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DOI:10.1063/1674-0068/cjcp2006088 c⃝2020 Chinese Physical Society

5.0028042.pdf

J. Appl. Phys. 128, 173103 (2020); https://doi.org/10.1063/5.0028042 128, 173103 © 2020 Author(s).Effect of excessive Cs and O on activation of GaAs(100) surface: From experiment to theory Cite as: J. Appl. Phys. 128, 173103 (2020); https://doi.org/10.1063/5.0028042 Submitted: 02 September 2020 . Accepted: 20 October 2020 . Published Online: 05 November 2020 Yijun Zhang , Kaimin Zhang , Shiman Li , Shan Li , Yunsheng Qian , Feng Shi , Gangcheng Jiao , Zhuang Miao , Yiliang Guo , and Yugang Zeng ARTICLES YOU MAY BE INTERESTED IN Use of carrier injection engineering to increase the light intensity of a polycrystalline silicon avalanche mode light-emitting device Journal of Applied Physics 128, 173104 (2020); https://doi.org/10.1063/5.0020113 Evolution of the electrical characteristics of Cu(In,Ga)Se 2 devices with sodium content Journal of Applied Physics 128, 173102 (2020); https://doi.org/10.1063/5.0025183 In situ investigation of hot-electron-induced Suzuki−Miyaura reaction by surface-enhanced Raman spectroscopy Journal of Applied Physics 128, 173105 (2020); https://doi.org/10.1063/5.0023623Effect of excessive Cs and O on activation of GaAs (100) surface: From experiment to theory Cite as: J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 View Online Export Citation CrossMar k Submitted: 2 September 2020 · Accepted: 20 October 2020 · Published Online: 5 November 2020 Yijun Zhang,1,a) Kaimin Zhang,1 Shiman Li,1 Shan Li,1 Yunsheng Qian,1Feng Shi,2Gangcheng Jiao,2 Zhuang Miao,2Yiliang Guo,3and Yugang Zeng4 AFFILIATIONS 1School of Electronic and Optical Engineering, Nanjing University of Science and Technology, Nanjing 210094, China 2Science and Technology on Low-Light-Level Night Vision Laboratory, Xi ’an 710065, China 3North Night Vision Technology Co., Ltd, Nanjing 211106, China 4Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China a)Author to whom correspondence should be addressed: zhangyijun423@126.com ABSTRACT The surface Cs –O activation process directly determines quantum efficiency and stability of negative-electron-affinity photocathodes. To investigate the effects of excessive Cs and O supply on activation and to explore a more effective Cs –O activation recipe, Cs –O activation experiments of GaAs(100) photocathodes are carried out based on the current-driven solid Cs and O dispensers. By a comparison of differ- ences in activation photocurrent, quantum efficiency, and photocurrent decay, it is found that the recipe of excessive O and non-excessiveCs is not suitable for activating GaAs photocathodes, while the recipe of continuous and completely excessive Cs along with intermittentand non-excessive O can achieve the most excellent photoemission performance, including the highest quantum efficiency in the long-wave threshold region and best stability under intense light irradiation after activation. Furthermore, this improved activation recipe with the least Cs–O alternating cycles is easier to operate. Combined with density functional calculations and dipole layer model, it is found that the activation recipe of completely excessive Cs and non-excessive O can form effective dipoles to the greatest extent, and avoid the directinteraction between As atoms and O atoms to form As –O–Ga oxides on the GaAs(100) reconstructed surface. Published under license by AIP Publishing. https://doi.org/10.1063/5.0028042 I. INTRODUCTION Due to the excellent properties including high quantum effi- ciency, large current density, low dark emission, high polarizability,and centralized electron energy distribution, negative-electron-affinity(NEA) GaAs photocathodes have been widely used in many fields such as high-performance low-light image intensifiers, spin-polarized electron sources, low-voltage scanning electron microscopes, andphoton-enhanced thermionic energy converters. 1–4In practical appli- cations, quantum efficiency and stability are important indicatorsto evaluate the performance of the NEA GaAs photocathode, so improving the quantum efficiency and emission stability of GaAs p h o t o c a t h o d e sa tt h es a m et i m eh a sb e e nt h ef o c u si nt h ef i e l do fcathode R&D so far. 5–8 The quantum efficiency and stability of GaAs photocathodes are closely related to the cesium (Cs) –oxygen (O) activation process.9–11 Nowadays, Cs –O activation recipes of GaAs photocathodes can be mainly divided into two types: yo –yo activation recipe with analternately intermittent Cs supply and O supply, and co-deposition activation recipe with a continuous Cs supply and intermittent O supply.12–14Compared with the traditional yo –yo activation recipe, the co-deposition activation recipe has been gradually preferred to activate GaAs photocathodes due to the virtues of easier operationand equally excellent photoemission performance. 15Based on the Cs–O co-deposition recipe, Miao et al studied the effects of different ratios of Cs/O on the performance of GaAs photocathodes and foundthat more Cs flux could contribute to the enhancement of cathodestability. 16Togawa et al. proposed an improved yo –yo activation recipe, named as “Nagoya activation recipe, ”which can improve the quantum efficiency of cathodes through maximizing the Cs flux.17 Based on the co-deposition activation recipe, Zhang et al. realized the computer-controlled activation of GaAs photocathodes using thesolid oxygen dispensers instead of gaseous oxygen, so as to achievethe desired symmetry of the photocurrent curve shape. 11Although the activation process of GaAs photocathodes has been extensivelyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-1 Published under license by AIP Publishing.studied, the effects of excessive Cs flux and excessive O flux on the cathode performance during the Cs –O co-deposition activation process require further experimental verification. Meanwhile, theproblem of frequent switch of a Cs or O source in the process of thewhole Cs –O activation process is not conducive to the implementa- tion of computer-controlled automatic activation, especially when the gaseous oxygen is controlled by a leak valve. 18Therefore, the activa- tion process of GaAs photocathodes still needs further optimization. In this paper, by using the soli d Cs and O dispensers and the co-deposition activation recipe, th e effect of excessive Cs and O supply on the activation of the reflection-mode GaAs photocathode is investi- gated by experiments in terms of activating photocurrent, quantum efficiency, and photocurrent decay. Inspired from the “Nagoya activa- tion recipe, ”an improved activation recipe was proposed, which can achieve higher long-wave quantum efficiency and better emissionstability, with fewer alternating cycles and an easier operating nature at t h es a m et i m e .C o m b i n e dw i t ht h ee x p e r i m e n t a lr e s u l t s ,f i r s t - p r i n c i p l e s calculations based on the density functional theory (DFT) were usedto qualitatively explain the adsorption of excessive Cs and non-excessive O on the GaAs(100) reconstructed surface. II. EXPERIMENTAL The reflection-mode GaAs cathode samples used in the activa- tion experiments were all cleaved from the same 2-in.-diameterGaAs wafer grown by metalorganic vapor phase epitaxy, and thesize was 11 × 11 mm 2. The epilayers including a 0.5- μm-thick p-type Ga 1−xAlxAs buffer layer and a 1.0- μm-thick p-type GaAs emission layer were grown on a low-defect GaAs(100) substrate. Inthe Ga 1−xAlxAs buffer layer, the zinc-doping concentration was 1×1 019cm−3and the Al composition varied linearly from 0.9 to 0, while in the GaAs emission layer, the zinc-doping concentration changed from 1 × 1019cm−3reduced to 1 × 1018cm−3in a quasiex- ponential form.19The cleaved samples were first degreased in the acetone and ethanol by ultrasonic cleaning, and then etched withHF acid solution followed by the HCl –isopropanol solution, which has more advantages in removing oxides and carbon contamina- tions. 20Before loaded into the vacuum, the wet chemical etched samples were rinsed in de-ionized water and dried with nitrogen.After that, the samples were transferred to the preparation chamberwith a base pressure of of 1 × 10 −7Pa and were annealed at 450 °C for 10 min to obtain the As-stabilized (2 × 4) reconstruction surface.21After the sample temperature dropped to room tempera- ture, Cs –O activation was performed. The schematic diagram of the Cs –O activation setup is shown in Fig. 1 . During the activation process, a 12 V/100 W halogen lamp via the optical fiber was used as the electron-excited light source, and the commercial solid Cs dispensers from SAES Getters and self-developed solid O dispens-ers packaged in nickel containers were used as the activationsources. 11Through a multi-information online measurement and control system, the flux of the Cs source and O source could be adjusted by controlling the applied direct current. Meanwhile, pho- toelectrons generated by the illuminating light were collected by thering anode under a 200 V biased voltage, and the photocurrentchanges were recorded online by the computer. In order to investigate the effects of excessive Cs and O supply on the activation performance of GaAs photocathode during theactivation process, four GaAs cathode samples numbered by sample 1 to sample 4, cleaved from the same epitaxial wafer, were activated by four different Cs –O activation recipes. The four groups of Cs –O activation experiments were labeled as A, B, C, and D. In all activation experiments, the Cs source current was 4.0 A and Osource current was 1.7 A. The processes of the activation experi-ments corresponding to samples 1, 2, 3, and 4 were as follows: A. Activation experiment A When the photocurrent reached the peak for the first time, the O source was introduced, while the Cs source was still contin- ued. After that, a new peak of the photocurrent appeared and whenthe photocurrent dropped to 85% of the new peak due to the exces-sive supply of O flux, the O source was stopped. When the photo-current rose to another peak again caused by the Cs flux, the O source was introduced again. This procedure was repeated until the photocurrent peak no longer increased. B. Activation experiment B After the photocurrent reached the peak for the first time and then dropped to 85% of the peak due to the excessive supply of Csflux, the O source was introduced, while the Cs source was stillcontinued. Until a new peak of photocurrent appeared, the O source was stopped. Then, when the photocurrent rose to another new peak and immediately dropped to 85% of the new peak, the Osource was introduced again. This procedure was also repeateduntil the photocurrent peak no longer increased. C. Activation experiment C The “Nagoya activation recipe ”was used. 17After the photo- current reached the peak for the first time and then dropped to a certain minimum value with a very slow decay rate due to the completely excessive supply of the Cs flux, the Cs source was FIG. 1. Schematic diagram of the Cs –O activation setup.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-2 Published under license by AIP Publishing.stopped and the O source was introduced. When the photocurrent reached a new peak, the O source was stopped and the Cs source was introduced to drop the photocurrent again. This procedure wasalso repeated until the photocurrent peak no longer increased. D. Activation experiment D Differing from the “Nagoya activation recipe, ”the Cs source was supplied all along. After the photocurrent reached the peak for the first time and then dropped to a certain minimum value with a very slow decay rate due to the completely excessive supply of theCs flux, and the O source was introduced while the Cs source wasstill continued. When the photocurrent reached a new peak, the Osource was stopped and the photocurrent dropped to the certainminimum value again. This procedure was also repeated until the photocurrent peak no longer increased. After Cs –O activation, the quantum efficiency curves and the photocurrent decay curves were tested by the multi-informationonline measurement and control system. In the photocurrent decayprocess after activation, the samples were still illuminated with the same 12/100 W white halogen light source and the vacuum pres- sure was around 2 × 10 −7Pa. III. RESULTS A. Activation photocurrent comparison In the four groups of Cs –O activation experiments, the Cs source current, O source current, and photocurrent were recordedonline, as shown in Fig. 2 . In the Cs –O activation process, the FIG. 2. Cs–O activation processes of the four GaAs cathode samples corresponding to (a) activation experiment A, (b) activation experiment B, (c) activation ex periment C, and (d) activation experiment D, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-3 Published under license by AIP Publishing.changes in the vacuum pressure in the Cs –O activation process for the four GaAs cathode samples are given in Fig. S1 in the supplementary material . For experiments A and B, the co-deposition activation recipe was based on the continuous Csand intermittent O. The difference between them lies in the Osupply. In experiment A, the Cs supply is non-excessive, while the O supply is excessive. In experiment B, the Cs supply is excessive, while the O supply is non-excessive. From Figs. 2(a) and2(b),i ti s seen that, according to the two Cs –O activation processes, the Cs–O alternation time is relatively longer in the early stage, and with the number of alternating cycles increases, the Cs –O alternat- ing cycle time gets shortened. During the alternating activation process, it takes a period of time for the photocurrent to rise afterthe O source is introduced, and with the decrease in the Cs –O alternating cycle time, this phenomenon becomes more obvious.The reason is that the release of oxygen molecules requires a certain warm-up time after the current-driven O source is turned on. By comparison of the activation results in Figs. 2(a) and2(b),i t is found that the final photocurrent of activation experiment Bwith excessive Cs is much higher than that of activation experimentA with excessive O. Although activation experiment A has more alternating cycles, the increase in the photocurrent after Cs –O alternating activation is not so high. For activation experiment A,the photocurrent exhibits a large increase after the O source isintroduced for the first time, while in the following alternating acti- vation process, the photocurrent only increases when the O source is stopped and Cs is supplied alone. When both Cs and O sourcesare supplied, the photocurrent increases slightly and then decreasesquickly. It is noted that the case of photocurrent change in activa-tion experiment B is completely different. After each time the O source is turned on, the photocurrent increases significantly and increases slightly when the O source is stopped and only Cs flux issupplied. Two groups of activation experiments C and D are designed to investigate whether excessive Cs can be advantageous to the Cs–O activation. The activation results of the traditional “Nagoya activation recipe ”are shown in Fig. 2(c) . The activation photocur- rent of activation experiment C is higher than those of activationexperiments A and B. The activation recipe of experiment D isimproved on the basis of activation experiment C. In the activation experiment D, Cs supply is continuous, and the O supply is inter- mittent. This activation recipe obtains a higher final photocurrentas same as that of activation experiment C, and the number of acti-vation cycles is much less than activation experiments A and B. Although activation experiments C and D have less Cs –O alter- nating cycles, the photocurrent decay time and average alternatingcycle time is extended each time when Cs is excessive, and the total activation time of activation experiments C and D is longer. From Figs. 2(c) and 2(d), it is found that when the Cs source is completely excessive, the activation with the continuous Cs fluxand intermittent O flux can be obtained the same high photocur-rent as well as the “Nagoya activation recipe. ”Compared with acti- vation experiment C, the activation experiment D has the higher photocurrent peak after the first O introduction and fewer Cs –O alternate cycles, thereby, the total activation time is shortened.Besides, as for the activation recipe D, only the O source needs tobe switched while the Cs source is kept on, so this improved activa- tion recipe is easier to operate. The parameters of the Cs –O activation process corresponding to the four GaAs cathode samples are listed in Table I . It is seen that, for the four samples, the photocurrent starts to increase atabout 20 min after the initial Cs supply, and the time when the photocurrent reaches the first Cs peak is nearly coincident, indicat- ing that the Cs deposition rates and amounts in the initial Cs acti-vation experiments are approximately the same. The less Cs –O alternating cycles in activation experiments C and D for samples 3and 4 indicate that a completely excessive Cs flux during activation can be applied to reduce the number of alternating cycles, and the average Cs –O alternating cycle time is also extended. By comparing the final photocurrent, it is found that sample 1 in activation exper-iment A with non-excessive Cs flux and excessive O flux has the lowest photocurrent, and sample 4 in activation experiment D in which the Cs flux is continuous and completely excessive has thehighest photocurrent. Compared with sample 3, sample 4 has lessactivation alternating cycles and shorter total activation time.Therefore, the activation recipe D of continuous and completely excessive Cs along with intermittent O is more conducive to achieve high photocurrent. B. Quantum efficiency comparison Quantum efficiency is an important parameter for evaluating the photoresponse capability of the photocathode under the irradi- ation in the wavelength range of interest. Through the multi-information online measurement and control system, the quantumefficiency curves of the four Cs –O activated GaAs photocathode samples in the region of 400 –1000 nm were tested in situ , which are shown in Fig. 3 . From the cutoff wavelengths, it is known that the surface of the four GaAs cathode samples shows the NEA state through Cs –O activation. Sample 4 in activation experiment D has the highest spectral responsivity, while sample 1 in activation experiment A TABLE I. Cs–O activation process parameters of the four GaAs cathode samples. SampleFirst Cs peak time (min)First Cs peak photocurrent ( μA)Cs–O alternating cyclesAverage alternating cycle time (min)Final photocurrent (μA)Total activation time (min) 1 34 6.0 16 2.9 63.5 84 2 33 8.0 13 3.1 85.6 783 30 7.4 6 19.0 90.3 166 4 32 8.1 4 20.0 91.5 138Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-4 Published under license by AIP Publishing.has the lowest spectral responsivity. By comparison of quantum efficiency curves between samples 1 and 2, it is found that sample 2 activated with excessive Cs flux and non-excessive O flux can obtain a higher quantum efficiency in the long-wave responseregion than sample 1 activated with non-excessive Cs flux andexcessive O flux. For samples 3 and 4 both with complete excessiveCs flux activation, the quantum efficiency of sample 4 with contin- uous Cs and intermittent O activation is higher in the long-wave response region than that of sample 3 with alternately intermittentCs–O activation. Therefore, it is inferred that, through the excessive Cs flux activation, the surface barrier of GaAs photocathode can belowered to be more favorable for the escape of low-energy electrons excited by long-wave light. In a word, the activation recipe of con- tinuously and completely excessive Cs flux can effectively improvethe quantum efficiency of GaAs photocathodes, especially in thelong-wave threshold region. C. Stability comparison The photocurrent decay under intense white light irradiation was executed immediately after the quantum efficiency test, and the results are shown in Fig. 4(a) . Meanwhile, the changes of vacuum pressure in the decay process for the four cathode samplesare seen in Fig. S2 in the supplementary material . Because of the poor vacuum pressure at the level of 10 −7Pa and the intense white light irradiation of 100 lx, the photocurrent decay rates of the four cathode samples are relatively fast. Usually, the decay of photocur- rent or quantum efficiency at room temperature can be fitted bymeans of single exponential lifetime analysis. 22However, the simu- lated curve in this simple form is not consistent with our experi- mental data. In this case, two forms of fitting formula regarding the photocurrent decay under intense white light irradiation areproposed empirically, which are expressed as I(t)¼I1e/C0atþI2e/C0btþI3, (1) I(t)¼I1e/C0at/C0ct2þI3, (2) where I1and I2are defined as the dominant photocurrent and sec- ondary photocurrent, respectively, I3is the relatively stable photo- current after decay, ais defined as the dominant decay coefficient, and band care defined as the secondary decay coefficient. Note that no single fitting formula can be applicable to the four samples at the same time, and it is found that Eq. (1)is applicable to samples 1 and 2, while Eq. (2)is applicable to samples 3 and 4. The reason for the difference in the fitting formula is not clear, butit should be related to the adsorption of Cs on the surface. In Eqs. (1)and (2), the smaller the decay coefficient, the better the emission stability. In addition, the dominant decay coefficient a FIG. 3. Quantum efficiency curves of the four GaAs cathode samples. FIG. 4. Stability test results of the four GaAs cathode samples. (a) Linear ordi- nate and (b) logarithmic ordinate.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-5 Published under license by AIP Publishing.plays a more important role than secondary decay coefficient band c. By comparison of photocurrent decay curves, it is seen that, under the same environmental conditions, samples 3 and 4 havebetter emission stability, while samples 1 and 2 have worse emis- sion stability. In addition, it is worth noting that the photocurrent of samples 1 and 2 in experiments A and B decays from the begin-ning, while the photocurrent of samples 3 and 4 in experimentsC and D first rises to the another peak value, then decays withtime. Because Cs sources are both continuously supplied in activa- tion experiments A and B, the advantage in emission stability of sample 2 with excessive Cs supply and non-excessive O supply isnot obvious. In activation experiments C and D, Cs supply iscompletely excessive, and the difference lies in whether the Cssource supply is continuous or not in the whole activation process. The experimental photocurrent decay curves of the four GaAs cathode samples are well fitted by using Eqs. (1)and(2), as shown inFig. 4(b) , wherein the ordinate scale is in the logarithmic form to highlight the fitting consistency. Besides, the first ten minutes ofphotocurrent increase for samples 3 and 4 are ignored for a better fit. The fitted parameters of the photocurrent decay curves for the four samples are listed in Table II . It is seen from Table II that the dominant decay coefficient afor samples 3 and 4 is approximately half of that for samples 1 and 2. Compared with sample 1, sample 2 has an equal dominant decay coefficient aand a slightly larger secondary decay coefficient b. Nevertheless, the inappreciable dif- ference in secondary decay coefficient for samples 1 and 2 can beignored. Among the four samples, sample 4 has the minimumdominant decay coefficient a, which means the best emission stability for sample 4. Compared with sample 3 with the intermit- tent Cs supply, sample 4 with the continuous Cs supply has asmaller dominant decay coefficient aand a smaller secondary decay coefficient c. Therefore, it can be inferred that the time of Cs overdose is vital to the cathode stability after activation. The exces- sive Cs flux is beneficial to the formation of a more stable Cs –O activation layer on the GaAs surface, and the orderly and robustsurface dipoles can weaken the damage of vacuum residual gas onthe surface and reduce the decay rate of cathode photoemissioncapability. IV. DISCUSSION The above experimental results demonstrate that the GaAs cathode sample activated with non-excessive Cs and excessive O has the lowest final photocurrent and near-infrared response, and the quickest photocurrent decay rate under intense white lightirradiation after activation, whereas the case of the GaAs cathode sample activated with completely excessive Cs and non-excessive O is quite different. This phenomenon indicates that the improvedactivation recipe can make the adsorbed Cs –O activation layer more stable and more effective to reduce surface work function.According to the double dipole layer model proposed by Su et al. , 23 the NEA surface of activated GaAs photocathode is formed by the formation of the GaAs –O–Cs dipole layer and Cs+–O−2–Cs+dipole layer. Therefore, the surface barrier of the GaAs cathode can have ablocking effect on the escape of low-energy photoelectrons excitedby near-infrared photons, while the height and width of the surface barrier are closely related to the effective adsorption of Cs –Oo n the cathode surface. It can be inferred that the activation recipewith excessive O flux and non-excessive Cs flux can cause O atomsto interact with the GaAs surface to form more oxides, which willconstruct a higher and wider surface barrier to hinder the decrease of the work function. In order to confirm this conjecture, it is nec- essary to study the effect of different situations of Cs –O adsorption on the GaAs surface from the atomic level by first-principlescalculation. Considering that all cathode samples activated in the experi- ments are (100)-oriented p-type GaAs, and the GaAs(100)- β 2(2 × 4) reconstruction surface is the most stable and most concernedsurface. 24–29Besides, the As-stabilized GaAs(100)-(2 × 4) reconstruc- tion surface was observed after treatment in HCl –isopropanol solu- tion and subsequent annealing at 410 ∼480 °C in ultrahigh vacuum.21In this case, a seven-layer GaAs(100)- β2(2 × 4) reconstruc- tion surface model consisting of four layers of As atoms and threelayers of Ga atoms was built, wherein there were 28 As atoms, 21 Gaatoms, and 1 Zn atom in the fourth layer substituted for 1 Ga atom as the p-type dopant. In this case, the doping ratio is 1/50, namely, the doping density is in the order of magnitude of 10 20cm−3.I nt h e surface structure, the reconstruction phase contains two As dimersin the top layer and one As dimer in the third layer. Besides, thepseudo-hydrogen atoms are situated at the bottom to saturate the dangling bonds. As for the GaAs(100)- β 2(2 × 4) reconstruction surface, researchers have found that As –O–Ga oxides from the bonding of one O atom, one As atom in the top As dimer, and oneadjacent Ga atom with the dangling bond is more stable than As – O–As or As vO oxides from the bonding of one O atom and two dimer As atoms in the first layer or third layer. 27,28In order to understand the mechanism of Cs adsorption, Cs –Oa d s o r p t i o na n d GaAs oxides on the surface properties, different absorption models,such as Cs adsorption, Cs –O adsorption, Cs-bonded O adsorption, 2Cs-O adsorption, and 2Cs adsorption on the clean GaAs (100)- β 2(2 × 4) reconstruction surface were built, as shown in Fig. 5 , where Cs atoms and O atoms are all located nearby the As atoms inthe top As dimer and adjacent Ga atoms with the dangling bond inthe second layer. In Fig. 5(d) ,t h es t a b l eA s –O–Ga oxide from the bonding of one As atom in the top As dimer and one adjacent Ga atom with the dangling bond in the second layer was preferred as anexample. By using the CASTEP software package, first-principles cal-culations based on DFT were performed on these models. In the cal-culations, the exchange correlation energy interaction was treated by the generalized-gradient-approximation (GGA) Perdew –Burke – Ernzerhof (PBE) functional, and the Broyden –Fletcher – Goldfarb-Shanno (BFGS) algorithm was used for geometryTABLE II. Fitted parameters of the photocurrent decay curves for the four GaAs cathode samples. Sample I1(μA) I2(μA) I3(μA) ab c 1 50.2662 7.0898 0.0500 0.0541 0.0152 N/A 2 80.1761 11.8661 0.0800 0.0541 0.0158 N/A3 93.6352 N/A 2.7101 0.0297 N/A 4.0 × 10 −5 4 97.4396 N/A 2.8632 0.0277 N/A 2.2 × 10−5Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-6 Published under license by AIP Publishing.optimization. The ultra-soft pseudopotential plane wave cutoff energy of 400 eV and the convergence accuracy of 2 × 10−6eV/atom were adopted, and the K-point mesh grid in the form of Monkhorst –Pack was set as 6 × 3 × 1. In the optimization process, the top four layers above were allowed to fully relax and the remain-ing layers below were constrained and always fixed in the ideal posi-tion. The vacuum layer with a thickness of 15 Å was used to avoid mirror interaction between the two periodic slab surfaces. The calculation results of work function corresponding to these six surface models are presented in Fig. 6 . It can be seen that work function of the clean GaAs(100)- β 2(2 × 4) reconstruction surface is 4.7 eV. When one Cs atom is adsorbed on the surface, i.e., the Cs surface density is 0.78 atoms/nm2, the work function decreases due to the orbital electron transfer between Cs atom andsurface As atoms. When one O atom locates below the Cs atomclose to the Ga atom with a dangling bond in the second layer, thework function does not decrease; instead, it slightly increases to 3.86 eV. When the O atom below the Cs atom forms a bond with the As atom in the top As dimer and the adjacent Ga atom withthe dangling bond, the surface work function further rises. If thereare two Cs atoms above the O atom, the surface work functionchanges remarkably and drops to 2.9 eV, which could be related to the interaction of Cs atoms, O atoms, and surface Ga/As atoms. When only two Cs atoms are adsorbed on the surface, i.e., the Cssurface density is 1.56 atoms/nm 2, the surface work function further decreases. This change in the surface work function indi- cates that surface Cs/O ratio has a significant effect on the emission ability of GaAs photocathodes.Mulliken charge population analysis can reflect atomic charge distribution, charge transfer, and chemical property. Figure 7 gives the respective value of Mulliken charge population of atoms at dif- ferent sites, where all the considered atoms marked as Ga1 ∼Ga4, FIG. 5. Top and side views of the six surface models before geometry optimization corresponding to (a) clean, (b) Cs adsorption, (c) Cs –O adsorption, (d) Cs-bonded O adsorption, (e) 2Cs –O adsorption, and (f) 2Cs adsorption, respectively. FIG. 6. Work function of the GaAs(100)- β2(2 × 4) reconstruction surface with different adsorption models.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-7 Published under license by AIP Publishing.As1∼As6, Cs1, Cs2, and O are located on the GaAs(100)- β2(2 × 4) reconstruction surface as shown in Fig. 5 . A larger negative value indicates that more electrons are obtained, while a larger positive value indicates that more electrons are lost. As for the clean GaAs (100)- β2(2 × 4) reconstruction surface, a Ga –As bond exists in the form of covalent bonds, and Ga atoms are positively charged, whileAs atoms are all negatively charged, because the electronegativity ofAs atoms is greater than that of Ga atoms. When one Cs atom marked as Cs1 is adsorbed at the position shown in Fig. 5(b) , the positive charge population of Ga1 increases, and the absolute valueof negative charge population of As1 and As2 increases more evi-dently, indicating that the adsorbed Cs atom prefers to interactwith As1 and As2 in the top As dimers, and charge transfer occurs among them. When one O atom is adsorbed above the Ga1 atom as shown in Fig. 5(c) , the charge population of O becomes negative, and the charge population of As1 adjacent to the O atom changesfrom negative to positive, and the positive value of Ga1 increased significantly, indicating that the O atom interacts with Ga1 and As1, and the charges of the O atom are transferred to Ga1 and As1at the same time. When one As –O–Ga bond from the O atom, Ga1 atom and As1 atom is formed, as shown in Fig. 5(d) , the Mulliken charge population distribution of this model are quite consistent with that of the model shown in Fig. 5(c) , indicating that the O atom located at this position is indeed easy to form bonds withGa1 and As1, and the formed As –O–Ga oxide is very stable. If one additional Cs atom marked as Cs2 is added above the O atom asshown in Fig. 5(e) , the positive charges of Ga1 and As1 decrease evidently, while the positive charges of Ga2, Ga3, and Ga4 increase, and the negative charges of As2 and As3 also increase. This varia-tion in Mulliken charge population indicates that the newly addedCs2 atom reduces the interaction among O, Ga1, and As1 andfurther reduces the charge transfer caused by oxidation. Meanwhile, the Cs2 atom can cause the charge population redistri- bution of other Ga/As atoms and Cs1 atom, leading to the newcharge transfer between atoms. If there is no O atom adsorbed onthe surface as shown in Fig. 5(f) , the As atoms in the top As dimer get more negative charge population, which means that the charge transfer increases between Cs1 atom and As atoms in the in thetop As dimers. The top and side views of Cs –O adsorption sites after geometry optimization on the GaAs(100)- β 2(2 × 4) reconstruction surface are shown in Fig. S3 in the supplementary material . As presented in Figs. S3(e) and S3(f) in the supplementary material , when two Cs atoms are adsorbed, the position of Cs1 atom changes greatly and is far away from the orig-inal position because of the repulsion between two Cs atoms.When one O atom is below the Cs1 atom, the O atom can restrain the separation of Cs1 atom and the Cs2 atom can reduce the inter- action among O, Ga1, and As1. In contrast to the 2Cs –O adsorp- tion model with the oxidation effect, the 2Cs adsorption model hasa lower work function due to its stronger Cs +–As−dipoles. Therefore, the excessive Cs and non-excessive O are beneficial to the form of effective dipoles. For the Cs –O alternating activation recipe with non-excessive Cs supply and excessive O supply, effective dipoles can be consti-tuted to reduce the work function at the beginning through the interaction between O atoms and Cs atoms. However, in this acti- vation process, more O atoms will easily interact with surface Gaatoms and As atoms, especially interact with As atoms in the topAs dimer and Ga atoms with dangling bonds to form As –O–Ga oxides, which construct the surface barrier that hinder the escape of photoelectrons. The As –O–Ga oxides are disadvantageous to the reduction of the work function and the formation of Cs +–As− dipoles consisting of Cs atoms and the top As dimers. In the subse- quent Cs –O alternation process, only when the O flux is stopped and the Cs flux is introduced alone, the newly added Cs atoms can reduce the interaction between O atoms and surface Ga/As atoms and generate new charge transfer, so as to eliminate the inhibitoryeffect of oxides on the reduction of the work function. While forthe Cs –O alternating activation recipe with excessive Cs supply and non-excessive O supply, the increased coverage of Cs atoms can reduce the probability of O atoms directly bonding with Ga atoms and As atoms. Meanwhile, the adsorption probability of O atoms isalso improved with the increased coverage of Cs atoms on thesurface, which allows more O atoms to be adsorbed on the GaAs surface. 30Due to the catalysis effect of surface Cs atoms, O 2mole- cules are easily dissociated into O atoms, and O atoms are smaller FIG. 7. Mulliken charge population of atoms at different sites on the GaAs(100)- β2(2 × 4) reconstruction surface.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 173103 (2020); doi: 10.1063/5.0028042 128, 173103-8 Published under license by AIP Publishing.in size and thus can enter into Cs atoms easily. The migrated O atoms will cause Cs atoms to dissociate again,31which helps to form more Cs+–O−2-Cs+dipoles and further reduce the surface work function. Furthermore, as for the Cs –O alternating activation recipe with completely excessive Cs supply and non-excessive Osupply, the increment of photocurrent peak after the first O intro- duction can be larger. In addition, each alternation cycle can maxi- mize the interaction of more O atoms with more Cs atoms to formdipoles and reduce the charge transfer probability between Asdimers and O atoms to avoid the appearance of As –O–Ga oxides as much as possible. From the photocurrent changes in activation experiments C and D, it is seen that the continuous Cs supply can further reduce the Cs –O alternating cycles compared to the inter- mittent Cs supply, which indicates that more Cs +–O−2–Cs+dipoles are formed in each alternating cycle of activation experimentD. Therefore, in order to improve the photoemission performance of GaAs cathodes, the direct interaction between As atoms in the top As dimers and adsorbed O atoms should be avoided to formAs–O–Ga oxides. In this way, more Cs +–As−dipoles along with Cs+–O−2–Cs+dipoles can be formed during the Cs –O activation process. Finally, it should be pointed out that, we only consider the Cs adsorption and Cs –O adsorption nearby the top As dimer and the adjacent Ga atom with the dangling bond in the second layer. Infact, the cases regarding Cs and Cs-O adsorption are complicated because of the numerous adsorption sites and increased adatom coverage. In recent work by Karkare et al, 32Cs adsorption on the Ga-terminated GaAs(100)-(4 × 2) reconstructed surface was investi-gated by DFT, and the calculated work function reduction with thechange in the Cs surface density is different from our results, which could be ascribed to the different reconstructed surface and adsorption site. As we know, different adsorption sites includingT2, T2 0, T3, T30, T4, T40,D ,D0, and H on the GaAs(100)- β2(2 × 4) surface have a significant impact on surface work function andadsorption energy. 24,25Moreover, the annealing temperature can affect the variation of reconstructed phases on the GaAs(100) surface.21,26In addition to the As-terminated β2(2 × 4) surface, some other As-terminated GaAs(100) reconstructed phases such asα(2 × 4), α 2(2 × 4), β(2 × 4), and β3(2 × 4), and Ga-terminated GaAs(100) reconstructed phases such as α(4 × 2), α2(4 × 2), β (4 × 2), β2(4 × 2), β3(4 × 2) can be also modeled to investigate the effect of Cs –O adsorption on GaAs surface activation.33 V. CONCLUSION Based on the current-driven solid Cs dispensers and O dis- pensers, different Cs –O activation recipes of p-type GaAs(100) photocathodes were investigated. By comparing differences in pho-tocurrent, quantum efficiency curve, and photocurrent decayamong samples using different activation recipes, the effect ofexcessive Cs supply and O supply on photoemission performance of GaAs photocathodes were clarified by experiments. In combina- tion with density functional calculations and dipole layer model,Cs–O adsorption on the GaAs(100)- β 2(2 × 4) surface was studied theoretically, and the effects of excessive Cs and O on this recon- structed surface were qualitatively analyzed. The results indicate that the activation recipe adopting continuous and completelyexcessive Cs supply along with intermittent and non-excessive O supply can obtain higher long-wave response capability and better emission stability. More importantly, the properties of lessalternating cycles and easier operation will provide a technical wayfor realizing the computer-controlled automatic activation of high-performance GaAs photocathodes. SUPPLEMENTARY MATERIAL See the supplementary material for the changes of vacuum pressure in the Cs –O activation process and in the decay process for the four GaAs cathode samples shown in Figs. S1 and S2,respectively. The top and side views of Cs-O adsorption sites onthe GaAs(100)- β 2(2 × 4) reconstruction surface after geometry opti- mization are presented in Fig. S3. ACKNOWLEDGMENTS The project was supported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 61771245 and61301023) and the Science and Technology on Low-Light-Level Night Vision Laboratory Foundation of China (Grant No. J20150702). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1K. Chrzanowski, Opto-Electron. Rev. 21, 153 –181 (2015). 2S. Karkare, L. Boulet, L. Cultrera, B. Dunham, X. Liu, and W. Schaff, Phys. Rev. Lett. 112, 097601 (2014). 3H. Morishita, T. Ohshima, M. Kuwahara, Y. Ose, and T. 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5.0014947.pdf

Appl. Phys. Lett. 117, 080502 (2020); https://doi.org/10.1063/5.0014947 117, 080502 © 2020 Author(s).Perspective on the pressure-driven evolution of the lattice and electronic structure in perovskite and double perovskite Cite as: Appl. Phys. Lett. 117, 080502 (2020); https://doi.org/10.1063/5.0014947 Submitted: 22 May 2020 . Accepted: 09 August 2020 . Published Online: 26 August 2020 Nana Li , Qian Zhang , Yonggang Wang , and Wenge Yang COLLECTIONS This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Microwave engineering for semiconductor quantum dots in a cQED architecture Applied Physics Letters 117, 083502 (2020); https://doi.org/10.1063/5.0016248 Epitaxial stabilization of rutile germanium oxide thin film by molecular beam epitaxy Applied Physics Letters 117, 072105 (2020); https://doi.org/10.1063/5.0018031 Light induced electron spin resonance properties of van der Waals CrX 3 (X = Cl, I) crystals Applied Physics Letters 117, 082406 (2020); https://doi.org/10.1063/5.0010888Perspective on the pressure-driven evolution of the lattice and electronic structure in perovskite and double perovskite Cite as: Appl. Phys. Lett. 117, 080502 (2020); doi: 10.1063/5.0014947 Submitted: 22 May 2020 .Accepted: 9 August 2020 . Published Online: 26 August 2020 Nana Li, Qian Zhang, Yonggang Wang, and Wenge Yanga) AFFILIATIONS Center for High Pressure Science and Technology Advanced Research (HPSTAR), Shanghai 201203, People’s Republic of China a)Author to whom correspondence should be addressed: yangwg@hpstar.ac.cn ABSTRACT Perovskite ABO 3as one of the most common structures has demonstrated great structural flexibility and electronic applications. Evolving from perovskite, the typical double perovskite A 2BB0O6has two element species (B/B0), where the ordered arrangements of BO 6and B0O6 octahedron provide much more tunability. Especially, by applying external pressure, the energetic order between different phases inperovskite and double perovskite materials can be notably modified with more fascinating physical properties. However, it is still a challenge to propose a general model to explain and predict the high-pressure structures and properties of various perovskites and double perovskites due to their flexibility and complexity. In this perspective, we will discuss pressure effects on the crystalline structure and electronic configu-rations in some perovskites and double perovskites. We then focus on a prediction method for the evolution of the lattice and electronicstructure for such materials with pressure. Finally, we will give a perspective on current challenges and opportunities for controlling and opti-mizing structural and electronic states of a given material for optimized functionalities. Published under license by AIP Publishing. https://doi.org/10.1063/5.0014947 Perovskite-type (Pv) oxides ABO 3and double perovskite-type (dPv) oxides A 2BB0O6possess a wide variety of both scientifically and commercially physical properties depending on the choice of the con- stituting elements.1–8The ideal ABO 3perovskite structures have a cubic symmetry, consisting of a framework of corner-sharing BO 6 octahedra with the A-type cation in each resulting cub-octahedral interstice and forming AO 12dodecahedron, as shown in Fig. 1(a) . Double perovskite-type oxide A 2BB0O6is the evolution of ABO 3in which the B site is occupied with two different atoms B/B0,a n dB O 6 and B0O6octahedral alternated with A atoms occupying the interstitial spaces, as shown in Fig. 1(b) . The ordered arrangements of BO 6and B0O6octahedron provide much more tunability on physical properties. For example, the double perovskite Sr 2FeMoO 6shows the room- temperature magnetoresistance, promising for the development of ordered perovskite magnetoresistive devices that are operable at roomtemperature. 9Lead free double perovskite La 2NiMnO 6is a new prom- ising material for photovoltaic applications.10 To accommodate different sizes of A and B/B0cations, most Pv and dPv present flexible structures by rotation or tilting of the B(B0)O6 octahedrons,11–16as shown in Fig. 1 . The stability of the Pv and dPv structures for the different A and B/B0constituents is commonlydiscussed in terms of the Goldschmidt’s tolerance factor ( sas defined below):11,17 sPv¼rAþrO ðÞ =21=2rBþrO ðÞ/C16/C17 ; (1) sdPv¼rAþrO ðÞ =21=2rBþrB0 ðÞ =2þrO/C0/C1/C16/C17 ; (2) where r A,rB,rB0, and r Oare the ionic radii of the respective ions calcu- lated from Shannon’s ionic radii.18Ideal cubic perovskites usually hold svalue between 0.95 and 1.0 such as CaSiO 3(sPv¼0.99) at room tem- perature and ambient pressure (RTAP).13When 0.75 <sPv<0.95, the Pv is distorted to orthorhombic, such as REFeO 3(RE¼rare earth, sPv/C240.90).14There are also various hexagonal non-Pv phases known with the ABO 3stoichiometry and sPv>1.15,16For dPv, the situation gets more complicated. The structure shows a huge difference evenwith a similar s dPvvalue. For instance, sdPvis 0.9789 for Ba 2YIrO 6with a cubic structure, while the sdPvis 0.9792 for Sr 2ZnWO 6with a tetrag- onal structure at RTAP.19,20In some cases, the dPv has the cubic struc- ture at RTAP when the sPv>1. For example, sdPvis 1.0380 and 1.0351 for ideal cubic Ba 2MgWO 6and Ba 2ZnWO 6, respectively.21It was caused by the multiple element combinations at A and B/B0sites. Appl. Phys. Lett. 117, 080502 (2020); doi: 10.1063/5.0014947 117, 080502-1 Published under license by AIP PublishingApplied Physics Letters PERSPECTIVE scitation.org/journal/aplThe distortions from the cubic symmetry give rise to changes in the physical properties that are important for the Pv and dPv applica-tions. 22,23For example, the intrinsic TiO 6local bonding distortion and octahedral tilting in CaTiO 3perovskite-based materials can modify intermediate energy states within the bandgap and associated photolu-minescence emission profile. 22Also, the high- Tcferrimagnetism in Ca2FeOsO 6was driven by lattice distortion, which represents complex interplays between spins and orbitals.23Besides, as the predominant phase of the lower mantle and the most abundant mineral in Earth,the study on the orthorhombic (Mg,Fe)SiO 3perovskite under extreme pressure and temperature leads to the discovery of the post-perovskitestructure, which is considered as one of the most important events atthe Earth’s core–mantle boundary. 24,25 The lattice distortion can be induced by many means. Among them, the external high pressure has been considered as a cleaner toolcompared to other methods since it acts only on interatomic distances,which in turn modifies the material’s mechanical and electronic prop-erties. 26–38For example, the pressure-induced magnetoelectric phase transition and the largest ferroelectric polarization among spin-drivenferroelectrics were reported in TbMnO 3.29Pressure-induced polymor- phism and piezochromism were also studied in double perovskiteMn 2FeSbO 6.37KNbO 3transformed from a ferroelectric orthorhombic Cm2mphase to another ferroelectric tetragonal P4mm phase at 7.0 GPa and then to a paraelectric cubic Pm3mphase at about 10.0 GPa.39Additionally, pressure can also redistribute the charge and melt the charge ordering via the structural phase transition. For exam-ple, at ambient pressure, BiNiO 3crystallizes in the insulating phase (space group: P-1) with two ordered Bi valence (Bi3þ,B i5þ)a n dN i2þ valence occupation (Bi3þ 0.5Bi5þ 0.5Ni2þO3). At high pressure, by melt- ing the charge ordering and disproportionation, the high-pressurephase (space group: Pbnm ) turns to be metallic with different valence state (Bi 3þNi3þO3).38,40What’s more, in earth science, the structural properties of mantle phases, such as post-perovskite (Mg,Fe)SiO 3 under high pressure, are very important for understanding the enig-matic seismic features observed in the Earth’s lower mantle down tothe core–mantle boundary. 41–44The ferromagnesian silicate [(Mg,Fe)SiO 3] with nominally 10 mol. % Fe was found unstable under 95–101 GPa and 2200–2400 K and dissociated into an Fe poororthorhombic phase and an Fe rich hexagonal phase, which suggests that the lower mantle may contain previously unidentified major phase.43 The structures of the Pv and dPv typically evolve in two ways under high pressure without breaking the polyhedron framework: tilt- ing of the B(B0)O6octahedra and polar cation displacement inside the B(B0)O6or AO 12polyhedra.45–51For example, BiFeO 3undergoes a phase transition at 3 GPa from a rhombohedrally distorted perovskite to a distorted monoclinic structure by the superimposition of tilts and cation displacements, and subsequent structural phase transition above 10 GPa from the distorted monoclinic structure to the nonpolar orthorhombic Pnma structure, as characterized by the cation displacements.50,51 For perovskite, there are several rules proposed to predict and explain the structural behavior under high pressure.52–59Based on the bond valence mechanism, Zhao et al. proposed that the AO 12dodeca- hedra are expected to be significantly more compressible than the BO 6 octahedral in orthorhombic perovskites, with both A and B cationshaving the formal charge þ3 (3:3 perovskites) and the higher- symmetry structure should be expected at high pressure, but for perov- skites with a þ2 cation at the A site and a þ4 cation at the octahedral B site (2:4 perovskites), they are predicted to become more distorted with pressure. 54Am a j o r i t yo fA B O 3perovskites with non-magnetic elements at B site follows Zhao’s rule.60–64For example, the distortion in the CaO 12and SnO 6polyhedra and the octahedral SnO 6tilting is attributed to the less compressible SnO 6octahedron than CaO 12 dodecahedron in site in CaSnO 3(2:4 perovskite).63Also, they gave a general rule based on the ratios of the compressibility (M A/MB)o ft h e AO 12and BO 6polyhedra.54If M A/MB>1, a transition to a higher- symmetry phase is expected under high pressure, whereas the opposite should occur when M A/MB<1. For LaAlO 3,w h i c hh a sM A/MB>1, its AlO 6octahedra are therefore more compressible than the LaO 12 sites and it undergoes a rhombohedral-to-cubic phase transitionaround 14 GPa. 60 There are some exceptions like the strongly correlated systems that have strong electronic configuration effects under high pressure, such as Jahn–Teller (JT) systems.65–72For example, LaMnO 3with an orthorhombic structure shows a complex behavior caused by the delo- calization of electron states, which suppresses the JT effect of the MnO 6octahedron but is insufficient to make the system metallic under high pressure.67It undergoes a transition to an unknown phase around 70 kbar because of the closing of the JT gap: the ionic Mn3þ species disappear and the system evolving toward a metallic-like phase is reached.70Another research also found that the total removal of the local JT distortion would occur only for pressures around 30 GPa in LaMnO 3, where metallization is reported to take place.71Besides, multiferroic material BiFeO 3,which has the strong tilt and polar dis- tortions at room temperature, also exhibits a complex phase transition at high pressure caused by the changes in octahedron tilts and dis- placements of Bi3þand Fe3þcations.50,51,72Through absorption crystal-field spectroscopy, the linear redshift of both 4T1and 4T2Fe3þ bands was induced by high pressure, consistent with the compression of the FeO 6octahedron under pressure.72Also, Fe3þoff-center dis- placements in FeO 6still persist in the high-pressure phase ( Pnma )f o r BiFeO 3.72In addition, REFeO 3perovskites (RE ¼rare earth) show a spin transition of Fe3þfrom high spin (HS) to low spin (LS) accompa- nied by a large volume collapse.68,73Therefore, the structure FIG. 1. The schematic representation of the crystal structure for perovskite (a) and double perovskite (b).Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 080502 (2020); doi: 10.1063/5.0014947 117, 080502-2 Published under license by AIP Publishingconfigurations in strongly correlated perovskites will be more com- plex caused by the correlation between the lattice and electronicstructures under high pressure. How about dPvs, what’s the general rule to predict the structural change for such materials under high pressure? Based on the elementspecies, we divide dPvs into two types: One is the “weakly correlatedelectronic” (WCE) dPv in which the electronic structure does notstrongly affect the lattice, and B(B 0)O6octahedra and AO 12polyhedra play a major role in determining distortions of the cation sites; theother is the “strongly correlated electronic” (SCE) dPv in which thelattice is strongly coupled with the spin, orbital, and valence. In thispart, we try to explore whether there is a general rule to guide thestructural change for WCE dPv under high pressure. Because of the distribution in the valence state, a majority 4:8 dPv with a þ2 alkali metal cation at the A site and a total þ8c a t i o na t the octahedral B and B 0sites are WCE dPv. Several high-pressure works have reported on such WCE dPv.74–83Basically, WCE dPvs tend to transform to a lower symmetry structure with the lattice dis-tortion under high pressure, as shown in Fig. 2(a) . For example, Ba 2YTaO 6undergoes a structural phase transition from cubic to tetragonal with the onset of an octahedral tilting distortion about the c axis under high pressure.74Also, Ba 2BiTaO 6transforms from rhombo- hedral with out-of-phase tilts about the [111] axis to the monoclinic structure with the [110] axis tilting around 4 GPa.75Experimental and theoretical studies on Sr 2CrReO 6show a phase transition from cubic to tetragonal structure around 9 GPa.76Besides, Sr 2CoWO 6,Sr2CaWO 6,a n de v e nP b 2CoTeO 6all show the structural phase transi- tion toward a lower symmetry under high pressure.77–79In our previ- ous work, we studied the structural phase transition of Sr 2ZnWO 6 under high pressure.80It also turns to a lower symmetry from mono- clinic to triclinic at 9 GPa induced by the increasing of the distortionin the Zn(W)O 6octahedron. Studies on other 4:8 WCE dPvs under high pressure have the similar trend to lower symmetry and more dis- tortion of lattice.80–83 As summarized in Fig. 2(b) ,S r 2ZnTeO 681and Sr 2NiWO 6 (unpublished) remain stable up to 31 GPa with their ambient tetrago- nal structure. Both Ba 2MgWO 6and Sr 2MgWO 6also remain in their original phase under high pressure.82Let us take a look at the effect of one B site radius in Sr 2BWO 6series: among Sr 2MgWO 6,S r 2ZnWO 6, Sr2CoWO 6,a n dS r 2CaWO 6,M g2þhas the smallest radius 0.72 A ˚, while Zn2þ,C o2þ,a n dC a2þh a v eal a r g e rr a d i u sa s0 . 7 4A ˚,0 . 7 5A ˚,a n d 1.00 A ˚,r e s p e c t i v e l y .18Sr2BWO 6(B¼Zn, Co, and Ca) all undergo a structural phase transition at a pressure below 13 GPa, but Sr 2MgWO 6 remains in the ambient structure up to 31 GPa. We can conclude thatthe smaller B ion can keep the ambient structure sustainable to higher pressure. There has no enough data to compare the effect of the A site radius. However, we adopt the tolerance factor ( s dPv)d e fi n e di nE q . (2)to quantitatively check the structure stability under pressure, in which we can simultaneously consider both A and B site radii. We draw a high-pressure structure diagram of some WCE double perov- skites with their sdPv,a ss h o w ni n Fig. 2(b) . The larger the value of sdPv, the more stable the structure under high pressure. In contrast, with the smaller value of sdPv,t h el a t t i c ei sm o r ep r o n et os h o wt h e structural phase transition to low symmetry under high pressure. The smaller ion radius at the A site with small sdPvproduces more space for the distorted B(B0)O6octahedron under high pressure. For the smaller ion radius at the B(B0) site with a large sdPvvalue, it is easier to compress the B(B0)O6octahedron and keep the lattice stable under high pressure. Therefore, the smaller the ion radius at the A site andthe larger the ion radius at the B(B 0) site, the much easier it is for such double perovskites to show a structural phase transition to lower sym-metry under high pressure. In contrast, the larger the ion radius at the As i t ea n dt h es m a l l e rt h ei o nr a d i u sa tt h eB ( B 0) site, the crystal struc- ture of such double perovskite is easier to remain stable. For perovskite predicting models at high pressure, there are not only qualitative but also quantitative method, which shows the detailed distortion of perovskite structures as a function of pressure. There are some excellent works that have been successfully applied to some per-ovskites, such as CaTiO 3,C a S n O 3, and MgSiO 3.54,84For WCE dPv, we try to build a simple model for quantitatively investigating the lattice evolution at high pressure, only using ambient-pressure crystal struc- ture data and the unit-cell parameters. First, we introduced a modifiedfactor: the local instability index (LII), which is derived from the bondvalence sums at the cation sites alone: 84 LII¼DVAðÞ2þDVBðÞ2þDVB0ðÞ2 3/C20/C211=2 ; (3) where DVA,DVB,a n dDVB0a r et h ed i f f e r e n c eo ft h eb o n dv a l e n c e sums at the cation sites for the fractional atomic coordinates of a model structure at pressure Pand the ambient-pressure structure at ambient pressure.84DVA,DVB,a n dDVB0can be obtained by SPuDS program.85 Here, we used “fixed coordinate” model to calculate the LIIval- ues under high pressure for such dPv, which are only based upon the FIG. 2. (a) The schematic representation of the lattice distortion for WCE dPv under pressure. (b) Structure evolution of certain WCE double perovskites under pressurewith different tolerance factors s dPv.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 080502 (2020); doi: 10.1063/5.0014947 117, 080502-3 Published under license by AIP Publishingknowledge of the ambient-pressure structure and the unit-cell parame- ters at high pressures. The LIIvalues at high pressure for some WCE dPv are shown in Fig. 3(a) .O b v i o u s l y ,t h e LIIvalues all become greater with increasing pressure for such dPv whether the samples have phasetransition or not. Hereupon, we define a new factor “compromiseinstability index” ( CII), which is derived from the difference of the bond valence sums in A and B/B 0sites, CII i¼VBicalcðÞ /C0VBoxðÞ/C0/C12þVB0icalcðÞ /C0VB0oxðÞ/C0/C12 /C04/C3VAicalcðÞ /C0VAoxðÞ/C0/C12; (4) where Vi(ox) is the formal valence (equal to its oxidation state) and Vi(calc) is the calculated bond valence sum for the A, B, and B0ions atpressure Pi.Vi(calc) can be obtained by SPuDS program85using the “fixed coordinate” model. Then, we gave the CIIvalues with pressure for those WCE dPvs, as shown in Figs. 3(b) and3(c). The compromise instability index CII is more inclined to decrease (d CII/dP<0) for WCE dPv, which has no structural phase transition under high pressure; in contrast, CIIis more inclined to increase (d CII/dP>0) for WCE dPv, which has structural phase transition under high pressure. For Ba 2MgWO 6, Sr2MgWO 6,a n dS r 2NiWO 6in which the crystal structure is stable within the pressure range of the study, CIIdecrease with increasing pressure. However, CIIincreases with pressure for Sr 2ZnWO 6and Ba2YTaO 6, in which the structural phase transition occurs around 10 GPa and 5 GPa, respectively. For Sr 2CoWO 6, it shows two structural phase transitions around 2 GPa and 12 GPa. Although CIIdecreases in the intermediate pressure range of two phase transitions, it still increases near the phase transition point. Therefore, the compromiseinstability index CIIcan be used as an important parameter to judge whether there is phase transition or not for WCE dPv at high pressure. In the strongly correlated electronic (SCE) Pv and dPv in which the lattice is strongly coupled with the spin, orbital, or valence, the high-pressure behaviors become more complicated. 86–91It will induce more significant physical properties such as the charge transfer, insula- tor-to-metal transition, magnetic transition, and even non-fermi liquidbehavior. For instance, Ba 2PrRu 0.8Ir0.2O6shows an unusual lattice change under high pressure, in which a structural phase transition from monoclinic to tetragonal is driven by the charge transfer from Pr3þto Pr4þwith pressure.91Besides, compression on Sr 2FeOsO 6 drives an unexpected transition from the antiferromagnetic to ferri- magnetic order, accompanied by the lattice stable up to 56 GPa.92,93It was caused by the increase in the crystal-field splitting at Os5þsites rather than by bending of the Fe–O–Os bonds.92,93What’s more, the magnetic transition temperature is largely and slightly enhanced for Ba2FeMoO 6and Sr 2FeMoO 6, respectively.86Sr2FeMoO salso shows a metal–insulator transition caused by the compression of the unit cell around 2.1 GPa.87Especially, PrNiO 3shows an insulator to metal tran- sition with the transforming to a non-Fermi-liquid phase with pres-sure, which is caused by both lattice and spin fluctuations where the transition temperature T IM¼TNis terminated.94The similar phe- nomenon was also discovered in other RENiO 3(RE¼rare earth) at high pressure.95 The transition metals, such as Fe3þ/Fe2þand Mn3þin SCE sys- tems, show novel and interesting high-pressure behaviors because of the correlation among the lattice, spin, and orbital, which leads tostrong modification on the magnetic, electronic, optical, and other properties in the corresponding systems. Especially, the pressure- induced spin transition occurred from high spin to low spin for such transition metal largely correlates with the lattice and electronic prop- erties. For instance, in LaFeO 3, the spin transition of Fe3þaccompa- nied by the lattice collapse occurs at around 50 GPa.67Concurrent of the HS to LS transition and lattice collapse during structural phase transition has been observed in other systems.96–98Also, they show some interesting pressure-induced spin crossover phenomenon that have not been discovered yet in perovskite and double perovskite oxides. For instance, the spin crossover of Fe2þcan induce the super- conducting in FePSe 3under high pressure.96What’s more, a high-to- low spin crossover of Fe2þin CuFeS 2is manifested along with the structural phase transition and a surprising n-type semiconductor to FIG. 3. (a) The change of LIIvalues of Ba 2MgWO 6,B a 2YTaO 6,S r 2CoWO 6, Sr2NiWO 6, and Sr 2ZnWO 6under high pressure. (b) is the pressure dependence of CIIfor Ba 2MgWO 6,S r 2MgWO 6, and Sr 2NiWO 6. (c) is the pressure dependence of CIIfor Sr 2ZnWO 6,B a 2YTaO 6, and Sr 2CoWO 6.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 080502 (2020); doi: 10.1063/5.0014947 117, 080502-4 Published under license by AIP Publishingp-type semiconductor transition with pressure.98The study and pre- d i c t i o no nt h ep r e s s u r e - i n d u c e ds p i nc r o s s o v e ro fd P vo x i d e sc a ng i v e an opportunity to produce such unusual electronic states and interest- ing physical properties. In this part, we mainly discuss the correlation between the lat- tice and spin configurations in SCE dPv under high pressure. In general, the spin crossover in these transition metals is caused by thed-orbital splitting with pressure. We gave a schematic d-orbital splitting diagram of Fe3þin the FeO 6octahedra under high pres- sure, as shown in Fig. 4(a) . The net spin magnetic moment of tran- sition metals in the B(B0)O6octahedron in SCE double perovskites is mainly controlled by the competition between the crystal-field splitting Dcf(favorite for the LS state) and the intra-atomic Hund’s exchange term J(favorite for the HS state). Dcfis sensitive to exter- nal pressure, which will promote the crystal field splitting and theninduce the spin transition of such ions and finally strongly influ- ence the electronic and magnetic properties of SCE dPv. For Sr 2FeMoO 6, a spin crossover of Fe ion from high-spin to low-spin state is found accompanied by a transition from the ferrimagnetichalf-metallic to nonmagnetic semiconductor state. 87,88Anothertheoretical study on La 2VMnO 6shows that Mn3þexperiences a transition from the high spin state ( t2g3eg1) to low spin state ( t2g4), making La 2VMnO 6a half-metallic ferrimagnet accompanied by t h ev o l u m ec o l l a p s eu n d e rs o m ec r i t i c a lp r e s s u r e .89 To date, most studies of double perovskites have only one transi- tion metal involved in spin transition. How about the interplay of double-spin crossover with two transition metals? Is there a possible charge transfer involving in the double-spin crossover to minimize theoverall energy? To check all these questions, we studied the high- pressure structure of La 2FeMnO 6, and monitored the spin and valence status of Fe and Mn at the B-site.99Unlike LaMnO 3and LaFeO 3,67,68,100La2FeMnO 6shows a double-spin crossover behavior under high pressure. The strong coupling between Fe and Mn leads to a combined valence/spin transition: Fe3þ(S¼5/2)!Fe2þ(S¼0) and Mn3þ(S¼2)!Mn4þ(S¼3/2), with an isostructural phase transition under high pressure, as shown in Fig. 4(b) . However, in La 2FeMnO 6, the randomly distributed Fe/Mn leads to a “diffuse pressure-induced phase transition,” a structural transfor-mation over a broad pressure range, without a well-defined critical pressure point. So both lattice and spin transitions spread out over a large pressure range with different configurations at each pressure stage in La 2FeMnO 6. This phenomenon is not conducive to the practi- cal application of such a material because of the broad pressure range for the structural transformation. To avoid this problem, we can design an ordered B/B0site in dPv and choose different transition metal combinations such as Fe/Co, Co/Mn, and Co/Cr that may undergo a double spin transition, and the pronounced spin–orbit cou- pling between two transition metals may provide more options for designing novel spintronic materials with tailored properties.101–103 For instance, the theoretical studies on Sr 2FeCoO 6show that Co4þ/ Fe4þin the high spin states can lead to its metallicity and ferromagne- tism.101Then, the spin transition of Co4þ/Fe4þwill give a huge change in their electronic and magnetic properties with pressure. Besides, the system will become more complex if the B(B0)O6octahedra involve orbitally degenerate transition metal ions (B/B0:C u2þ,C r2þ,F e2þ, Mn3þ,N i3þ,o rC o3þ). The JT coupling induces low-symmetry B(B0)O6distortions, which are eventually responsible for the striking properties related to both the orbital ordering and JT distortion.102For instance, JT distortions of the MnO 6octahedra in La 0.85MnO 3/C0dwill induce the reduction in the metal–insulator transition temperature at high pressure.103Also, the coherence length of the JT distortions in La3/4Ca1/4MnO 3induced a structural modification with high pres- sure.104In RENiO 3, the localized electrons of Ni undergo a cooperative JT distortion but stronger Ni–O bonding in alternate NiO 6/2octahedra creates molecular eorbitals within the more strongly bonded clus- ters.105All these induce the approach to crossover to itinerant-electron behavior from the localized-electron side.105Besides, high-pressure research on Cu2þand Mn3þshowed that JT distortion reduced upon compression and is eventually suppressed at pressures above 20 GPa.67,102–106If we combine the transition metals in B/B0,w h i c h contains both JT distortion and spin crossover, it will induce amazingphysical properties at high pressure. For instance, the intermediate spin states of Mn 3þ/Co3þcan lead to JT distortions where the JT ion will have a single egelectron and the double degeneracy of egstate will be lifted with pressure. This will lead to itinerant behavior of the single electron, which will contribute to the magnetic and transport properties.107,108 FIG. 4. (a) A schematic d-orbital splitting diagram for Fe3þin the FeO 6octahedra with pressure. (b) The high-pressure behavior of La 2FeMnO 6,99LaFeO 3,68and LaMnO 3.67For LaMnO 3, reproduced with permission from Loa et al. , Phys. Rev. Lett. 87, 125501 (2001).67Copyright 2001 American Physical Society. For LaFeO 3, reproduced with permission from Xu et al. , Phys. Rev. B 64, 094411 (2001).68 Copyright 2001 American Physical Society. For La 2FeMnO 6, reproduced with per- mission from Li et al. , Phys. Rev. B 99, 195115 (2019).99Copyright 2019 American Physical Society.Applied Physics Letters PERSPECTIVE scitation.org/journal/apl Appl. Phys. Lett. 117, 080502 (2020); doi: 10.1063/5.0014947 117, 080502-5 Published under license by AIP PublishingHere, we established a general rule among the tolerance factor s, compromise instability index CIIand the stability of the crystal struc- ture in the 4:8 WCE double perovskites under high pressure. The cor- relation among the lattice and spin in some SCE dPvs was discussed comprehensively. For Pv and dPv with polar non-centrosymmetric, ferroelectrics has been extensively studied for potential applications in room-temperature pyroelectric infrared detectors and electronic and pho- tonic devices. 109–111Currently, the pressure effect is still not quite clear. For instance, what is the major contribution from the octahedral tilting and the polar cation displacements with pressure, and whether the ferroelectricity can be indeed enhanced or reappeared at high pres- sure for such materials? Further research is needed on the evolution of the lattice correlated with the electronic structure for such Pv and dPv under high pressure. This review only involves two types of structures: ABO 3or A2BB0O6. Oxygen-deficiency in Pv and dPv could provide another way to modify the structure and valence status, and their pressure responses. One typical oxygen-deficient perovskite A 2B2O5differs in the ordering patterns of anion vacancies. These compounds crystallize in the brownmillerite-type (B-type) structure consisting of alternating layers of corner-sharing BO 6octahedra and corner-sharing BO 4tetra- hedral, as shown in Fig. 5(a) . Under high pressure, the B-type structure prefers to transform to the perovskite-type (P-type) structure because of the breaking and forming of B–O–B bonds in the octahedral and adjacent tetrahedral B–O layers of the brownmillerite-type structure. For example, Sr2Fe2O5shows a structural phase transition from the B-type structure (space group Ibm2) to a tetragonal perovskite-like structure with oxygen-deficient at 15 GPa, while SrFeO 3remains in its cubic structure up to 56 GPa.112–114One can directly synthesize the tetragonal perovskite-like structure of Sr 2V2O5with a large volume press under high pressure and temperature conditions (unpublished). The tetrago- nal phase of Sr 2V2O5can sustain a pressure of up to 40 GPa. Another high-pressure study on Ca 2Fe2O5shows that it remains B-type struc- tures up to 10 GPa.115We predict that it will transform to the perovskite-like structure with a higher pressure. However, further experiments are called in to confirm these predictions. The equations of state for CaFeO 2.5, SrVO 2.5,S r F e O 2.5,a n dS r F e O 3are displayed in Fig. 5(b) for comparison. Double perovskite can be derived to other forms such as AA03B4O12-type and AB 2/3B01/3O3-type according to the different atomic ratios in the A or B site. For example, the A-site-ordered dou- ble perovskite LaCu 3Fe4O12adopts a cubic structure, where the Cu ions at the A site make the square-planar AO 4units and the Fe ions at the B site form corner-sharing BO 6octahedra.116,117Under external pressure, a phase transition occurs along with a significant volume col- lapse and charge transfer between Cu and Fe, which leads to a surpris- ing electric/magnetic property change from an antiferromagnetic insulating state to a paramagnetic metallic state.117In3Cu2VO 9with AB2/3B01/3O3type adopts the honeycomb-lattice structure, which con- sists of alternating layers of InO 6octahedra, and Cu2þand V5þions in the trigonal–bipyramidal coordination.118In the new cuprate La4Cu3MoO 12, the Cu and Mo are coordinated by O in the corner- sharing trigonal bipyramids that are sandwiched between layers of lan- thanum cations.119In these specific derivative double perovskites, the transition metals can occupy the lattice in very rich ways, which mayinduce more advanced mechanical and electronic properties under high pressure. 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5.0017778.pdf

Appl. Phys. Lett. 117, 122406 (2020); https://doi.org/10.1063/5.0017778 117, 122406 © 2020 Author(s).Ultrafast demagnetization of ferromagnetic semiconductor InMnAs by dual terahertz and infrared excitations Cite as: Appl. Phys. Lett. 117, 122406 (2020); https://doi.org/10.1063/5.0017778 Submitted: 09 June 2020 . Accepted: 02 September 2020 . Published Online: 22 September 2020 E. A. Mashkovich , K. A. Grishunin , H. Munekata , and A. V. Kimel ARTICLES YOU MAY BE INTERESTED IN Monitoring laser-induced magnetization in FeRh by transient terahertz emission spectroscopy Applied Physics Letters 117, 122407 (2020); https://doi.org/10.1063/5.0019663 All-optical probe of magnetization precession modulated by spin–orbit torque Applied Physics Letters 117, 122403 (2020); https://doi.org/10.1063/5.0020852 Static and dynamic origins of interfacial anomalous Hall effect in W/YIG heterostructures Applied Physics Letters 117, 122405 (2020); https://doi.org/10.1063/5.0019235Ultrafast demagnetization of ferromagnetic semiconductor InMnAs by dual terahertz and infrared excitations Cite as: Appl. Phys. Lett. 117, 122406 (2020); doi: 10.1063/5.0017778 Submitted: 9 June 2020 .Accepted: 2 September 2020 . Published Online: 22 September 2020 E. A. Mashkovich,1,a) K. A. Grishunin,1,2 H.Munekata,3 and A. V. Kimel1 AFFILIATIONS 1Radboud University, Institute for Molecules and Materials, 6525 AJ Nijmegen, The Netherlands 2MIREA—Russian Technological University, Moscow 119454, Russia 3Laboratory for Future Interdisciplinary Research of Science and Technology, Tokyo Institute of Technology, Yakohama, Kanagava 226-8503, Japan a)Author to whom correspondence should be addressed: e.mashkovich@science.ru.nl ABSTRACT Subpicosecond pumping of ferromagnetic semiconductor InMnAs in the ranges of intra- and interband electronic transitions can result in an efficient demagnetization of the medium up to 60% of the initial magnetic moment. Here, we report about the efficiency of ultrafastdemagnetization by a duo of terahertz and infrared pulses that trigger intra- and interband electronic transitions, respectively. Varying theintensities of the pulses and the delay between them, we study the degree of demagnetization caused by the pulse duo. It is shown that theresult of the excitation does not depend on the pulse sequence. Our findings indicate that both intra- and interband electronic transitions result in ultrafast demagnetization of the semiconductor via the very same mechanism, which evolves at a ps timescale. Independent of the origin of the electronic transition, ultrafast demagnetization is a result of a temperature increase in the free charge carriers (holes). Published under license by AIP Publishing. https://doi.org/10.1063/5.0017778 Recent trends show that 7% of global energy consumption will be spent on data storage centers. 1Magnetism is still the cheapest and the most reliable way to store information. The ability to control magne- tism in the fastest and the most efficient way is the key to further progress in information technologies. Controlling the magnetic stateof media with the help of femtosecond laser pulses facilitates the fastestand least dissipative switching of magnetization between stable bitstates. 2 For several decades, magnetic semiconductors have been consid- ered as promising candidates for the optical control of magnetism. Thediscovery of the Ruderman–Kittel–Kasuya–Yosida (RKKY) mecha-nism of the exchange interaction mediated by conduction electronsnaturally led to speculations about the optical control of magnetism insemiconductors by injecting photocarriers via electronic transitionsfrom the valence to the conduction band. 3The development of femto- second lasers dramatically changed the field of optical control of mag-netism. 4–8It was discovered that femtosecond excitation of free carriers, i.e., intraband electronic transitions, in magnetic metals is able to launch ultrafast demagnetization on a subpicosecond timescale9and even to reverse magnetization without any magnetic fields.10In our recent paper,11we have studied spin dynamics induced in ferromagnetic semiconductor InMnAs by terahertz (THz) and infra-red (IR) pulses, which triggered intraband and interband electronic transitions, respectively. The experiments revealed that THz and IR pump pulses induce ultrafast demagnetization at nearly similar time-scales, but THz pulse does it 100 times more energy efficiently thanthe IR counterpart. An increase in the intensity of the THz pump ledto a saturation of the demagnetization. The maximum change of themagnetization component normal to the sample plane was about 40%and 60% for the THz and IR excitations, respectively. One may expectthat a combination of the THz and the IR pumping should lead to100% demagnetization of the ferromagnetic semiconductor. However,such a demagnetization would mean that the ultrafast effects of theTHz pulses and the IR pulses on the magnetization are mutually inde-pendent and saturate due to different reasons. Aiming to reveal mutual dependencies between the effects of THz and IR pumping on ferromagnetism in InMnAs, we studied ultrafast demagnetization of InMnAs triggered by the duo of one THz and one IR pulse. Varying the intensities of the pulses and the delaybetween them, we studied the degree of the demagnetization caused Appl. Phys. Lett. 117, 122406 (2020); doi: 10.1063/5.0017778 117, 122406-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplby the pulse duo. It is shown that if the intensity of the first pulse is large enough to reach the saturation value of 60%, the second pulse does not change the magnetization any further independently of the sequence of the THz and the IR pulses in the duo. If the intensities of the pulses are well below the values corresponding to the saturation,the result of the excitation also does not depend on the pulse sequence. Our findings indicate that both intra- and interband electronic excita- tions result in ultrafast demagnetization of the semiconductor via the very same mechanism evolving at a ps timescale. Independent of the origin of the electronic excitation, ultrafast demagnetization is a resultof a temperature increase in the free charge carriers (holes). In the experiment, we use a Ti:sapphire amplifier to generate the 1 kHz sequence of light pulses at the central wavelength of 800 nm with the energy of 7 mJ per pulse and pulse duration of 100 fs. A beam splitter divides the beam with a ratio of 40/60%. The beam with pulseswith the energy of 3 mJ was used to tune the wavelength of light with the help of an OPA (optical parametric amplifier). These pulses are used as the IR pump. The rest of the split beam (4 mJ) is taken to gen- erate the THz pump. For this purpose, we used a tilted front technique and optical rectification in lithium niobate crystals. 12,13Also, using a beam splitter, a part of that beam (tens of lJ) at the wavelength of 800 nm was taken to be used as a probe. The probe and both thepumps were focused on the sample. To control the time delay between the IR pump, the THz pump, and the probe, we employed two mechanical delay stages in the paths of the probe and the IR pump. The THz electric field was calibrated using a thin GaP crystal and the Pockels effect. 14The maximum electric field achieved for the THz pump was estimated to be around 500 kV/cm, which corresponded to the THz magnetic field with the amplitude of 160 mT. The centralwavelength of the IR pump was tuned in the range of 1150 nm (pho- ton energy 1.1 eV) to 1550 nm (photon energy 0.8 eV). Using two wire-grid polarizers, we controlled the strength of the THz electric field. A set of neutral density filters attenuated the fluence of the IR pump. For fine adjustments of the intensity, we employed a combina-tion of a half-wave plate and a Glan–Taylor polarizer. After the THz and IR pulses launch spin dynamics in the medium, the magnetic state of the latter was monitored using the probe beam, the magneto-optical Kerr effect (MOKE), and a conventional balance detection scheme for measurements of polarization rotation. The ferromagnetic order of Mn 2þmagnetic moments in InMnAs is mediated by the exchange coupling between the spins of holes and Mn2þions.15The studied p-doped InMnAs thin film on the GaSb/GaAs substrate was produced by low-temperature molecular beam epitaxy followed by low-temperature annealing favoring the high Curie temperature (T C¼65 K).16,17InMnAs is a direct bandgap semiconductor. Figure 1(a) shows the schematic electronic band struc- ture of the material. The bandgap energy and hole concentration are 333 meV and 1.5 /C21020cm/C03, respectively. According to previous studies, electronic transitions launched by the low-energy photon of the THz pump can be attributed to impurity band transition18,19and/ or heavy-hole, light-hole, and split-off valence band transitions.20,21 However, one cannot exclude that the THz pulse simply pumps intra- band transitions, resulting in a heating of free charge carriers in the semiconductor (holes). The sample was in a cold-finger cryostat, and it was cooled down well below T C. The sample has an easy-axis type of magnetic anisotropy with the axis along the normal to the sample. It means that without any applied magnetic field, the equilibriumorientation of the magnetization is perpendicular to the sample surface. The THz and IR pumps were at normal incidence with respect to thesample plane. The angle of incidence for the probe beam was around 2 /C14. Although the external magnetic field was applied at an angle of 15/C14to the sample surface, the MOKE in the experimental geometry with aprobe beam at the nearly normal incidence is mainly sensitive to themagnetization component normal to the sample surface. 22Figure 1(b) shows the MOKE measured as a function of the applied magnetic fieldat temperature T ¼6K . T h e e x t e r n a l m a g n e t i c fi e l d H extas high as 200 mT closes the hysteresis loop and saturates the magnetization, whereh 0is the amplitude of the static MOKE. In time-resolved pump-pump- probe measurements, we employed a data acquisition card (DAQ),23 allowing us to deduce the MOKE signals induced by the THz pumponly, IR pump only, and dual-pump, respectively. The sketch of theexperimental setup is shown in Fig. S1 of the supplementary material . It is known that both IR and THz pulses efficiently launch spin dynamics in the very same InMnAs sample. 11The dynamics starts with a fast demagnetization. Tuning the orientation and the strengthof the applied magnetic field, it is possible to achieve that the demag-netization is followed by a precession of the net magnetization at thefrequency of the ferromagnetic resonance. In our experiments, the FIG. 1. (a) Schematic InMnAs electronic band structure with respect to pump and probe photon energies. (b) Static magneto-optical Kerr effect measured at the wavelength of 800 nm and at T ¼6 K as a function of the applied magnetic field Hext. The field was applied at the angle of 15/C14with respect to the sample surface. (c) Dynamics of the magneto-optical Kerr effect induced solely by the THzpump (blue circles), solely by the IR pump (red triangles), and by both the THz and IR pumps arriving simultaneously (gray squares). The central wavelength and the fluence of the IR pump were k¼1150 nm (photon energy 1.1 eV) and I IR¼1 mJ/cm2, respectively. The electric field of the THz pump was E THz ¼200 kV/cm. The measurements were performed at T ¼6 K and the external mag- netic field H ext¼230 mT. (inset) The maximum value of the magneto-optical Kerr effect h[see panel (c)] vs the THz pump electric field (blue circles) and the IR pump fluence. The measurements were performed for two wavelengths of the IRpump— k¼1150 nm (red triangles) and k¼1550 nm (red rhombuses), respec- tively. The IR measurements were performed at T ¼6 K, while the data shown on the inset for the case of the THz pump correspond to T ¼27 K.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122406 (2020); doi: 10.1063/5.0017778 117, 122406-2 Published under license by AIP Publishingprecession is not very much pronounced [see Fig. 1(c) ]. In the figure, h i st h em a x i m u mo ft h ep u m p - i n d u c e dM O K Ea n dt h ed e g r e eo ft h e demagnetization is defined as h/h0. The experiment at T ¼27 K revealed that an increase in the strength of the THz field eventually results in an onset of saturation of the demagnetization around 350 kV/cm [ Fig. 1(d) ]. From previous studies, it is known that a simi- lar dependence is expected at T ¼6 K (see the supplementary material , Fig. S2). Contrary to the measurements at the pump wavelength of 2500 nm (photon energy: 0.5 eV), pumping the very same sample withpulses at the wavelength of 1150 nm (photon energy: 1.1 eV) or 1550 nm (photon energy: 0.8 eV) reveals a saturation of the ultrafast demagnetization at 1.5 mJ/cm 2and 7 mJ/cm2, respectively [see the inset in Fig. 1(c) ]. This trend is consistent with the data reported in Ref.11, where for the pump at the wavelength of 2500 nm (photon energy: 0.5 eV), no saturation was observed for pump fluences up to 30 mJ/cm2. Figure 1(c) shows dynamics of the magneto-optical Kerr effect at the THz electric field and the IR fluence set below the regime of satura- tion (E THz¼200 kV/cm and I IR¼1 mJ/cm2). If the sample is excited by the THz or IR pulse only, the magneto-optical signal rapidly decreases and the decrease corresponds to ultrafast demagnetization of InMnAs. The demagnetization is followed by a recovery of the magne- tization on a timescale of 600 ps. If the THz and the IR pulses arrive simultaneously, the dynamics is slightly different. However, the degree of the demagnetization h/h0in the case of the dual pump is by far lower than the sum of the demagnetizations caused by the IR and the THz pulses alone. Repeating similar measurements for the THz and IR pumps arriving simultaneously at the sample, but for the THz electric field above the saturation threshold E THz¼400 kV/cm, we reveal that the dynamics is practically not affected by the IR pump pulse [see Fig. 2(a)]. If the IR fluence (I IR¼8m J / c m2) and the THz electric field (ETHz¼500 kV/cm) are higher than the corresponding saturation val- ues, the maximum photo-induced change does not exceed the level r e a c h e db yas i n g l ep u m pe x c i t a t i o n[ s e e Fig. 2(b) ]. Finally, we decrease the THz electric field down to E THz¼200 kV/cm, w h i l ek e e p i n gt h efl u e n c eo ft h eI Rp u l s ea t8m J / c m2.T h er e s u l t s shown in Fig. 2(c) reveal that in this case, the dynamics practically follows that triggered by a single IR pump and hardly feels the THz pump pulse. Hence, the experiments show that dual pump excita- tion does not allow exceeding the maximum degree of demagneti- zation achieved in single pump experiments. These findings strongly imply that the ultrafast laser-induced demagnetizations of InMnAs caused by THz and IR pumping are mutually dependent and the threshold known from single-pump experiments cannot be exceeded. One should keep in mind, however, that the relaxa- tion of the MOKE after the first pump depends on the strength of the latter. The higher the IR fluence or THz electric field, the slower the MOKE relaxes. To reveal if the electronic excitations in the THz and IR spectral ranges play in the ultrafast demagnetization distinct roles, we performed dual-pump excitation with mutually delayed THz and IR pump pulses. The THz pump electric field and the IR pump fluence were set below the corresponding saturation thresholds. Dynamics of the magneto-optical Kerr effect lunched by the IR pump arriving 60 ps prior to the THz pump and vice versa are plotted in Figs. 3(a) and 3(b), respectively. It is seen that themaximum value of the magneto-optical Kerr effect induced by duos does not depend on the sequence of the pulses. In each case, the effect caused by the second pulse depends on the recovery of the medium after excitation with the first pulse. The dynamics triggered by the dual pulse excitation support the hypothesis that THz and IR electronic excitations eventually result in ultrafast demagnetization of ferromagnetic semiconductor InMnAs due to the very same mechanism. For instance, according to Ref. 24, the degree of the demagnetization is determined by laser-induced heating of holes and our work strongly supports this mechanism. Although several interband transitions are expected in the spectral range of the THz pulse,18–21it is safe to assume that the interaction of the pulse with a heavily p-doped semiconductor is described by the Drude theory25and the light-matter interaction is dominated by the interaction of the electric field of light with holes. The energy of the THz photon is fully used to increase the kinetic energy (i.e., tempera- ture) of holes, while a substantial part of the energy of the IR photons is used to trigger interband transitions and, thus, increase the potential energy of electrons without any contribution to ultrafast heating of FIG. 2. Dynamics of the magneto-optical Kerr effect induced solely by the THz pump (blue circles), solely by the IR pump (red triangles), and by both the THz andIR pumps arriving simultaneously (gray squares) at H ext¼230 mT and T ¼6K . The central wavelength of the IR pump was k¼1150 nm. The panels correspond to data obtained with the following THz electric fields and IR pump fluences: (a) ETHz¼400 kV/cm and I IR¼1m J / c m2;( b )E THz¼500 kV/cm and I IR¼8m J / c m2;( c ) ETHz¼200 kV/cm and I IR¼8m J / c m2.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122406 (2020); doi: 10.1063/5.0017778 117, 122406-3 Published under license by AIP Publishingholes. We also note that the IR fluence, at which ultrafast demagneti- zation saturates, is lower at higher pump photon energy [see the insetofFig. 1(c) ]. Finally, we would mention that contrary to Ref. 26, we failed to achieve 100% demagnetization of InMnAs. It can be due to the lower doping concentration in our sample (p ¼1.5/C210 20cm/C03) compared to the one in Ref. 26(p¼4/C21020cm/C03). According to Ref. 24,a decrease in the doping concentration in InMnAs decreases theefficiency of the angular momentum transfer from or to spins ofMn-ions. In conclusion, using a duo of THz and IR pulses, we studied if intra- and interband electronic transitions play distinct roles in ultra- fast demagnetization of InMnAs. The degree of the demagnetizationdoes not depend on the sequence of the IR and THz pulses. Thesefindings imply that both intra- and interband electronic excitationsresult in ultrafast demagnetization of the semiconductor via the very same mechanism evolving at a ps timescale. For instance, ultrafast demagnetization can be due to a temperature increase in the free holesand depend only on the amount of heat deposited into the free holesbut not on the way of the deposition. See the supplementary material for the details of the experimental setup and dependence of the maximum of the THz-induced MOKE on the THz electric field. The authors thank S. Semin and Ch. Berkhout for technical support. This work was supported by the Netherlands Organizationfor Scientific Research (NWO) and Russian Foundation For BasicResearch 18-02-40027. H.M. acknowledges partial support for the present work from KAKENHI (JSPS) 18H03878. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1N. 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May, and B. W. Wessels, “Optical properties of Mn-doped InAs and InMnAs epitaxial films,” Physica B 344, 379 (2004). 19S. Ohya, I. Muneta, Y. Xin, K. Takata, and M. Tanaka, “Valence-band structure of ferromagnetic semiconductor (In,Ga,Mn)As,” P h y s .R e v .B 86, 094418 (2012). 20M. A. Meeker, B. A. Magill, G. A. Khodaparast, D. Saha, C. J. Stanton, S. McGill, and B. W. Wessels, “High-field magnetic circular dichroism in ferro-magnetic InMnSb and InMnAs: Spin-orbit-split hole bands and g factors,”Phys. Rev. B 92, 125203 (2015). 21M. Bhowmick, T. R. Merritt, G. A. Khodaparast, B. W. Wessels, S. A. McGill, D. Saha, X. Pan, G. D. Sanders, and C. J. Stanton, “Time-resolved differentialtransmission in MOVPE-grown ferromagnetic InMnAs,” Phys. Rev. B 85, 125313 (2012). 22A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials , 1st ed. (CRC Press, 1997). FIG. 3. Dynamics of the magneto-optical Kerr effect induced solely by the THz pump (blue circles), solely by the IR pump (red triangles), and by the dual-pump (gray squares) at H ext¼230 mT and T ¼6 K. The THz electric field, the central wavelength of the IR pump, and the fluence of the latter were set toE THz¼200 kV/cm, k¼1150 nm, and I IR¼1 mJ/cm2, respectively. The THz pump delay sTHzand the IR pump delay sIRare shown near the corresponding curves. (a)sTHz¼60 ps and sIR¼0 ps. (b) sTHz¼0 ps and sIR¼60 ps. The timescale begins from the probe and the first pump pulse overlap.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122406 (2020); doi: 10.1063/5.0017778 117, 122406-4 Published under license by AIP Publishing23J. Lu, X. Li, Y. Zhang, H. Y. Hwang, B. K. Ofori-Okai, and K. A. Nelson, “Two- dimensional spectroscopy at terahertz frequencies,” Top. Curr. Chem. 376,6 (2018). 24Ł. Cywi /C19nski and L. J. Sham, “Ultrafast demagnetization in the sp-d model: A theoretical study,” Phys. Rev. B 76, 045205 (2007).25K. Hirakawa, A. Oiwa, and H. Munekata, “Infrared optical conductivity of In1-xMn xAs,” Physica E 10, 215 (2001). 26J. Wang, C. Sun, J. Kono, A. Oiwa, H. Munekata, Ł. Cywi /C19nski, and L. J. Sham, “Ultrafast quenching of ferromagnetism in InMnAs induced by intense laserirradiation,” Phys. Rev. Lett. 95, 167401 (2005).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 122406 (2020); doi: 10.1063/5.0017778 117, 122406-5 Published under license by AIP Publishing

5.0019602.pdf

AIP Conference Proceedings 2269 , 030108 (2020); https://doi.org/10.1063/5.0019602 2269 , 030108 © 2020 Author(s).Facile hydrothermal synthesis of zinc cobalt sulfide nanosheets with enhanced electrochemical performance as supercapacitor electrode materials Cite as: AIP Conference Proceedings 2269 , 030108 (2020); https://doi.org/10.1063/5.0019602 Published Online: 12 October 2020 M. Dakshana , S. Meyvel , M. Silambarasan , P. Sathya , M. Malarvizhi , R. Ramesh , and S. Prabhu ARTICLES YOU MAY BE INTERESTED IN NiMoO 4 nanorods supported on nickel foam for high-performance supercapacitor electrode materials Journal of Renewable and Sustainable Energy 10, 054101 (2018); https:// doi.org/10.1063/1.5032271Facile Hydrothermal Synthesis of Zinc Cobalt Sulfide Nanosheets with Enhanced Electrochemical Performance as Supercapacitor Electrode Materials M. Daksha na1,b), S. Meyvel1, a), M. Silambarasan2, P. Sathya3, M. Malarvizhi1, R. Ramesh4, and S. Prabhu4 1Department of Physics, Chikkaiah Naicker College, Erode, Tamilnadu, India , 638 004 2Department of Physics, Vivekanandha College of Arts and Sciences for Women, Tiruchengode, Tamilnadu, India, 637 205 3Department of Physics, Salem Sowdeswari College, Salem, Tamilnadu, India , 636010 4Department of Physics, Periyar University, Salem, Tamilnadu, India , 636011 Corresponding author: a)meyvelphd @gmail. com b)mdakshu @gmail.com Abstract. Ternary transition metal sulfides are pro pitious electrode materials caused by low electronegativity, high conductivity and higher electrochemical activity in comparison to transition metal oxides and single metal sulfides. In this work, Zinc Cobalt Sulfide (ZCS)nanosheets have been successfully prepared through a simple, c ost-effective hydrothermal method and characterized by X - Ray Diffraction (XRD), Field Emission Scanning Electron Microscope (FE-SEM) with Energy Dispersive X - Ray Analysis (EDX), Transmission Electron Microscope (TEM), and X -Ray Photoelectron Spectroscopy (XPS). The supercapacitor performance of ZCS nanomaterial electrodes was done by electrochemical measurements and ZCS nanosheets revealed a clear enhancement in specific capacity and cycling stability. Electrochemical impedance spectroscopy showed a much lower charge transfer resistance of the samples. These results proved that the ZCS nanosheets may be utilized as capable electrode materials for next generation energy storage devices. INTRODUCTION Energy storage/conversion is considered as a prime facto r in determining the sustainable growth for human lives [1] concerned with rapid depletion of fossil fuels and increasing environment pollution . This coerces the researchers to start up renewable energy storage/conversion systems, which has a ability to hold high specific power and energy . Based on energy storage mechanisms , electrochemical capacitors are classified into electrical double -layer capacitors (EDLCs) and pseudocapacitors. Electrical energy stored via electrostatic accumulation of charges in- between electric double layer electrode/ electrolyte interfaces comes under EDLC, while in pseudocapacitors energy is stored by means of reversible faradaic reactions at the electrode surface, which results in high specific capacitance [2]. Among , various electrode materials ternary transition metal sulfides is a distinct faradaic type supercapacitor material [3] provides high electrochemical active sites via fast reversible redox reactions. In particular, z inc cobalt sulfide is a new class of ternary metal sulfides offers a n optical band gap and high electrical conductivity compare to binary metal sulfides and ZnCo 2O4. Sulfide pact very low electro negativity compare to oxygen, hence it offers highly stable and flexible architecture [4]. In this w ork, Zinc Cobalt Sulfide (ZCS) is prepared via simple hydrothermal approach. XRD analysis confirmed the existence of cubic phase Zn0.76Co0.24S followed by TEM microstructural studies which reveal ed the presence of nanosheets like structure with a polycryst alline characteristics . The information of valence state of ZCS is given by International Conference on Multifunctional Materials (ICMM-2019) AIP Conf. Proc. 2269, 030108-1–030108-6; https://doi.org/10.1063/5.0019602 Published by AIP Publishing. 978-0-7354-2032-8/$30.00030108-1XPS analysis. The surface area and porosity measurements are studied using nitrogen adsorption/desorption analysis. The electrochemical characterization displays the pseudocapactio r behavior with a notable specific capacitance of 705.55 Fg-1 at a current density of 2Ag-1 endowed with a cyclic stability of 3500 cycles with capacitance retention of 87.6% at a current density of 10Ag-1. EXPERIMENTAL In this typical process, 0.2mmol of Zn(NO 3)26H 2O and 0.4mmol of Co(NO 3)26H 2O were dissolved in 50ml of distilled water under constant stirring for 1 hour to achieve a homogenous solution, later 0.8 mmol of C 2H5NS (sulfur source) g et dissolved in 30 ml of ethanol and added in drop wise ma nner to the above homogenous solution . After a constant stirring for 4 hours the resultant soluti on was transferred into Teflon -coated stainless autoclave. The autoclave was sealed and heated at 160° for 24 hours in a hot air oven. Then , it was allowed to cool down naturally to room temperature. The as obtained black precipitate was isolated via centrifugation (Hamilton Bell v6500 Vanguard centrifuge, 2500 RPM for 15 minutes) and rinsed thoroughly with deionized water and ethanol by a number of times to era dicate the residuals . It was dried in a vacuum oven at 100°C for 7 hours followed by fine grinding and resultant Zn0.76Co0.24S nanoparticles were labeled as NZ. Material characterization Philips XPERT -PRO (Cu -Kα λ=1.5418Å) instrument were used to determ ine the X -Ray Diffraction (XRD) Pattern. Transmission electron microscope (TEM) images and their corresponding selected area electron diffraction (SAED) patterns were investigated through JEM -2100 (JEOL, JAPAN). The elemental chemical states were monitored by X -Ray photoelectron spectr a (XPS) which was recorded using XPS Ultra axis instrument (Krato s Analytical). The nitrogen adsorption -desorption studies were performed using a BETSORP MAX (Microtrac BEL, Japan) to examine the surface area and porous nature of the sample. In order to perceive the pseudocapacitive properties , the electrochemical measurements were executed with the help of CHI 6 60E electrochemical workstation via a three -electrode cell frame work in 3.0M KOH electrolyte solution with a potential range 0 -0.6V Vs (Ag/Agcl). Framing of Electrode Activated working electrode is composed of 80 :15:5 prepare d sample, carbon black, PVDF (Polvinylidene Fluoride) respectively and required amount of NMP were added to mold a homogenous slurry , the consequential slurry was incrusted on the nickel foam (1x1 cm2) followed by drying at vacuum oven at 80°C for 5 hours. RESULTS AND DISCUSSI ONS X-Ray Diffraction Analysis XRD was done to determine structur e, crystall ine nature , phase and chastity of as-obtained samples. The diffractogram were depicted in Fig.1.Diffraction p eaks were observed at 2θ = 28.66°, 33.25°, 47.69°, 56.60° and 77.07 ° were crystal planes of cubic phase Zn 0.76Co0.24S(JCPDS 00 -047-1656). Analogous diffraction patterns were observ ed by Hao Tong et al., [4] in synthesis of Zinc cobalt sulfide nanosheets as high-performance electrode for supercapacitors. The calculated lattice constant value is 5.3918 Aº. The diffraction pattern of the representative powdered samples was found to be absolutely distinct from binary metal sulfides, this evince d the absence of single metal sulfides like ZnS and CoS. 030108-2 FIGURE 1 . X-ray diffraction pattern of Zn 0.76Co0.24S nanoparticles NZ TEM and HRTEM Studies Under TEM the (NZ) 2D - nanosheets appears to be transparent ( Fig.2 (a)) in the size range of less than 100 nm leads to ultra -thin characteristics. This structure offers a very high surface area and a abundant open space among each nanosheets for electrolyte incursion w hich in turn creates more active sites for faradaic reactions, results in enhancement of electrochemical performance [1,4] . Further, the HRTEM image (Fig.2 ( b)) displays the lattice fringes with d-spacing value as 0.259nm, which coincides with the theoretical in terplanar cubic plane (200) of Zn0.76Co0.24S and Fig.2 (b) inset displays its corresponding Selected Area Electron Diffraction Patt ern (SAED) , which authenticates its polycrystalline nature. The presence of Zn , Co, S elements existed in the representative sample were confirmed E nergy Dispersive Spectroscopy (Fig.2 (c)). (a) (b) (c) FIGURE 2 . (a) TEM micrograph (b) HRTEM image (c) EDS spectrum of Zn 0.76Co0.24S nanoparticles NZ X-Ray Photoelectron Spectroscopy XPS spectr a used to explore the elemental composition and chemical states of Zn 0.76Co0.24S. From Zn 2p spectrum displayed in Fig.3(a ), it is significant to mention that the binding energy values of Zn 2p 3/2 at 1045.3 eV and Zn 2p 1/2 at 1022.5eV were equivalent to the previously reported values of Zn2+[3,5]. High resolution XPS spectrum of Co element labeled in Fig.3(b ) unveil the existen ces of a low energy band of Co 2p 3/2 at 778.7e V and a high energy band of Co 2p 1/2 at 793.9e V. The obtained values were compatible with the previously reported values by Xiao.J et al., [2]. The spin -orbit splitting value of Co 2p3/2 and Co 2p 1/2 is over 15e V, indicating the concurrence of Co2+ and Co3+[2,6] . Fig.3 (c) presents the XPS spectrum of S energy region, the component at 161.5e V is the typical metal -sulfur bond [6]. Despite that the peak at 169.6e V corresponds to the shake -up satellite. Based on the XPS analysis, the surface of Zn 0.76Co0.24S nanosheet s holds a composition of Zn2+, Co2+, Co3+ and S2- and these results acknowledge with the phase of Zn 0.76Co0.24S. 030108-3 (a) (b) (c) (b) FIGURE 3 . (a-c) XPS spectra and chemical states of Zn 0.76Co0.24S Nitrogen (N 2) Adsorption -Desorption Measurements Nitrogen (N 2) adsorption -desorption measurements were carried out to understand the insight of surface area and pore size distribution in NZ sample as shown in the Fig.4 . The isotherm profile comments under type IV and hysteresis loops were observed in the range of 0.6 -1.0 P/P 0, manifest the mesoporous nature of the prepared samples, it can be further conf irmed by the Barret -Joyner -Halenda (BJH) pore size distribution (inset of fig.4.) results shows that most of the pores were lie in the range of 2 -10 nm . The calculated pore sizes using BJH method was centered at 1.84 nm considered as an ideal pore size structure for fast ion diffusion. The sample has an appreciable specific surface area of 6.45 m2 g-1 due to its ultra-thin 2-D nanosheets architecture. The incorporation of large surface area and mesoporosity makes a large contribution in redox reactions related energy storage. FIGURE 4. N2 adsorption -desorption measurements (inset) BJH pore size distribution curve of Zn 0.76Co0.24S Electrochemical Performance Figure .5 (a) CV curves at different scan rats display a pair of redox current peaks of a typical pseudocapacitance behavior and reversible faradaic reactions between electrode material and alkaline electrolyte i.e. M -S/M-S-OH where M refers to Zn or Co. As scan rates increases the anodic peaks shift to higher potential while cathodic peaks shift lower potential due to fast charge and discharge rate [4,7,8] . The symmetrical GCD curves (Fig.5 (b)) suggests the fast reversibili ty of the 2D -nanosheets (NZ) and plateau shaped GCD curves acknowledge the pseudocapacitive property which is mainly influenced by the redox reactions of the sa mple [9]. The electrode exhibits a high specific capacitance of 705.55 F/g at 2 A/g which is superior than the previously reported values [10] because unique 2D - nanosheets like structure offers a good structural stability over long term charge/ discharge pr ocess at various current densities [4] As shown in the F ig.5(c) as the current density increases, specific capacitance decreases due to inadequate utilization of the active material with high current densities, limited diffusions and migration of OH- ions in the electrolyte solution [11]. 030108-4 (a) (b) (c) FIGURE 5. (a) CV at different scan rates (b ) Galvanostatic Charge/Discharge curves at different current densities (c) Current density Vs Specific Capacitance plot At high current density of 10A/g , the electrode subjected to continuous charge /discharge cycles of 3500 cycles and capac itance retention of 87.6% of its initial capacitance afte r 3500 cycles was achieved as indicated by Fig.6 (a). This put forward the complete activation process of the electro -active material and outstanding cyclic stability due to structural stability of th e electrode material. Practical applications of NZ electrode material depends on its energy and power density as displayed via Ragone plot (Fig.6 (b)) which offers a suitable energy density of 15.67 W h kg-1 with a power density of 447.71 W kg-1[12]. (a) (b) FIGURE 6 . (a) Cyclic stability for 3500 cycles at high current density 10 A g-1 (b) Ragone Plot of the prepared sample EIS analysis was conducted in the frequency range 0.01Hz to 100 kHz at an open -circuit potential of amp litude 5mV. Nyquist plot (Fig.7 .) composed of two sections a semicircle at a high frequency region attribu ted to the faradaic charge transfer resistance (Rct) and the straight line in the low frequency region corresponds to the capacitive behavior. Rct value of NZ is 0.02 Ω, this low Rct value suggests that it is a suitable candidate for supercapacitor applica tions [9]. FIGURE 7. EIS plot of Zn0.76Co0.24S 030108-5CONCLUSION A simple low-cost hydrothermal method has been adopted for the preparation of Zn 0.76Co0.24S nanoparticles. The obtained product holds 2D-thin nanosheets like structure which provides an appreciable specific surface area and its corresponding valence states were acknowledged by XPS analysis. Electrochemical characterization reveals the pseudo capacitive nature with its high specific capacitance of 705.55F/g at a current density of 2A/g. Moreover, they provide an outstanding cyclic stability over 3500 cycles at a current density of 10 A/g with capacitance retention of 87.6%. Remarkable electrochemical performance is achieved by the nanosheets like morphology which provides abundant electro active sites for redox reactions, it reduces the length of ion diffusion results in complete utilization of electro-active material and creates the electronic paths for fast electron transport this strengthen the electrochemical kinetics. The above results suggest that Zn 0.76Co0.24S nanosheets are promising alternative for enhancing electrochemical performance in supercapacitor applications. REFERENCES 1. Zhang X, Z heng Y, Zheng W, Zhao W, Chen D, J. Mater. Sci, 52, 5179 -87 (2017) . 2. Xiao J , Wan L, Yang S, Xiao F, Wang S, Nano Lett, 14, 831-8 (2014) 3. Mao X, Wang W, Wang Z, ChemPlusChem , 82, 1145 -52 (2017) 4. Tong H, Bai W, Yue S, Gao Z, Lu L, Shen L, Dong S, Zhu J, He J, Zhang X , J. Mater. Chem. 4, 11256 - 63 (2016) . 5. Cheng C, Zhang X, Wei C, Liu Y, Cui C, Zhang Q, Zhang D, Ceram. Int , 44, 17464 -72 (2018) . 6. Gong X, Cheng J, Ma K, Liu F, Zhang L, Zhang X, Mater. Chem . Phys. 173, 317-24 (2016) . 7. Chen W, Xia C, Alshareef HN. ACS nano . 8, 9531- 41 (2014). 8. Li D, Gong Y, Pan C, Sci. Rep. 6, 29788 (2016) . 9. Beka LG, Li X, Xia X, Liu W, Diam. Relat. Mater. 73, 169-76 (2017) . 10. Yang J, Zha ng Y, Sun C, Guo G, Sun W, Huang W , Yan Q, J. Mater. Chem A . 3, 11462 -70 (2015) . 11. Zhu T, Zhang G, Hu T, He Z, Lu Y, Wang G, Guo H, Luo J, Lin C, Chen Y, J. Mater. Sci , 51, 1903 -13 (2016) . 12. Tang J, Ge Y, Shen J, Ye M, Chem. Commun , 52, 1509- 12 (2016). 030108-6

5.0021421.pdf

J. Appl. Phys. 128, 135703 (2020); https://doi.org/10.1063/5.0021421 128, 135703 © 2020 Author(s).Structural and optical properties of transparent, tunable bandgap semiconductor: α-(AlxCr1−x)2O3 Cite as: J. Appl. Phys. 128, 135703 (2020); https://doi.org/10.1063/5.0021421 Submitted: 10 July 2020 . Accepted: 11 September 2020 . Published Online: 05 October 2020 Ravindra Jangir , Velaga Srihari , Ashok Bhakar , C. Kamal , A. K. Yadav , P. R. Sagdeo , Dharmendra Kumar , Shilpa Tripathi , S. N. Jha , and Tapas Ganguli ARTICLES YOU MAY BE INTERESTED IN Full InGaN red light emitting diodes Journal of Applied Physics 128, 135704 (2020); https://doi.org/10.1063/5.0016217 Publisher’s Note: “Individual contribution of electrons and holes to photocarrier-induced bandgap renormalization in intrinsic bulk GaAs” [J. Appl. Phys. 128, 115704 (2020)] Journal of Applied Physics 128, 139901 (2020); https://doi.org/10.1063/5.0030785 Efficient anharmonic lattice dynamics calculations of thermal transport in crystalline and disordered solids Journal of Applied Physics 128, 135104 (2020); https://doi.org/10.1063/5.0020443Structural and optical properties of transparent, tunable bandgap semiconductor: α-(Al xCr1−x)2O3 Cite as: J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 View Online Export Citation CrossMar k Submitted: 10 July 2020 · Accepted: 11 September 2020 · Published Online: 5 October 2020 Ravindra Jangir,1,a) Velaga Srihari,2 Ashok Bhakar,1,3C. Kamal,1,3 A. K. Yadav,2P. R. Sagdeo,4 Dharmendra Kumar,5Shilpa Tripathi,2S. N. Jha,2,3 and Tapas Ganguli1,3 AFFILIATIONS 1Raja Ramanna Centre for Advanced Technology, Indore 452013, Madhya Pradesh, India 2Bhabha Atomic Research Centre, Mumbai 400085, Maharashtra, India 3Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India 4Indian Institute of Technology, Indore 453552, Madhya Pradesh, India 5School of Study in Electronics, Pt. Ravishankar Shukla University, Raipur 492010, Chattisgarh, India a)Author to whom correspondence should be addressed: ravindrajangir@rrcat.gov.in andrjangir.cat@gmail.com ABSTRACT Detailed structural and optical properties of α-(Al xCr1−x)2O3(0≤x≤1) synthesized by the solid state reaction method have been investi- gated. Single phase α-(Al xCr1−x)2O3with space group R /C223c is obtained for the full composition range of 0 ≤x≤1. Variations in the lattice parameters aand chave been determined. Lattice parameter cfollows Vegard ’s law, while the lattice parameter ashows a clear deviation with a bowing parameter of −0.035 Å. This behavior of the lattice parameters of α-(Al xCr1−x)2O3with xis explained in detail by studying the local structure. Extended x-ray absorption fine structure spectroscopy shows a reduction in the values of Cr –O bond lengths with com- position x. Optical absorption measurements of α-(Al 1−xCrx)2O3for 0≤x≤1 show a large bandgap tunability of 1.9 eV (from 3.4 eV to 5.3 eV). The photoemission spectroscopy data and the analysis of partial density of states obtained from first principles electronic structurecalculations suggest that the valence band maxima is mainly composed of Cr 3d levels, which hybridize with the O 2p levels. Increased con- tribution of O 2p partial density of states is observed with Al substitution, which is expected to enhance p-type carrier conduction in the α-(Al xCr1−x)2O3system as compared to the parent α-Cr 2O3system. Thus, the large bandgap, its tunability in the UV region, and the predicted enhancement of p-type conductivity in the α-(Al xCr1−x)2O3system make it a potential candidate for application in UV based photo-detectors and transparent electronics. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021421 I. INTRODUCTION Optical transparency (E g> 3.1 eV) and electrical conductivity are, in general, mutually exclusive properties in a single material.Transparent conducting oxides (TCOs) are a unique class of mate-rials that have both these properties in a single material and, there-fore, have applications in flat screen displays, light emitting devices, thin film transistors, and solar cells. 1–3Most of the studied TCOs such as In 2O3, ZnO, SnO 2, etc., are n-type with their high conduc- tivity because of highly dispersive conduction band (consisting ofcation “ns”states) and bandgap more than 3.1 eV. 4In these TCOs, the valence band maximum (VBM) that consists of O 2p states is very less dispersive (high hole effective mass) and suffers from for- mation of polaronic holes that are localized at oxygen sites. Highelectronegativity of oxygen hinders shallow acceptor formation in these materials,5which makes it very difficult to develop good quality p-type TCOs from these material systems. To solve this issue,Hosono and co-workers proposed the concept of hybridizing the O2p orbitals with the cation ’s d or s orbitals also referred to as “chemi- cal modulation of the valence band ”(CMVB). 6The majority of p-type conducting oxides have been reported to have conductivity using this concept of hybridization between the d state of cationsand 2p states of oxygen in the valence band. 7,8Initially, various Cu based material systems such as CuAlO 2, LaCuOS, Mg-doped LaCuOSe, etc., have been explored.7–11They have either reduced conductivity or reduced optical transparency because of their lowbandgap. Recently, α-Cr 2O3(with a bandgap of 3.4 eV12)h a sb e e nJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-1 Published under license by AIP Publishing.proposed as a p-type transparent conducting oxide (TCO),13,14and research on Cr3+based oxides has gained interest because of its use- fulness in the future design of high-performance p-TCOs.5,15,16 α-Cr 2O3is a highly insulating material because of its localized d levels. p-type conductivity in α-Cr 2O3and Cr3+based materials has been increased by applying various approaches. p-type conduc- tivity upto 28 S cm−1and 4 S cm−1by doping α-Cr 2O3with Mg2+ and Ni2+, respectively, has been reported.17–19In addition, Ni2+ doping also helps in increasing the delocalization of holes in the VB, which leads to a lower effective mass.20Instead of cation doping, anion doping has also been investigated to improve the p-type conductivity. It is found that substitution of oxygen with sulfur induces a curved VBM while preserving the transparency. Itcan also overcome the issues of the flat valence band maximum(VBM) in Cr 2O3.21Cr based ternary perovskite material system like Sr doped LaCrO 3has been reported to have p-type conductivity up to 15 S cm−1.22,23Among all the delafossite oxides, the highest elec- trical conductivity ( ∼217 S cm−1) has been reported in Mg2+substi- tuted p-type CuCrO 2.24,25Apart from p-type conductivity, figure of merit (FOM) of the TCOs depends on the transparency in thevisible region. Therefore, bandgap tunability provides a way to increase the visible region transparency (or FOM) of the TCOs. Bandgap tunability also creates the possibility of use in UV basedtransparent electronic and photovoltaic applications. p-type CuGa xCr1−xO2system has shown a bandgap tunability of 0.46 eV from 3.09 to 3.56 eV. It has been shown that its transparency in the visible region increases up to 80% with the increase in thebandgap. 26–28In this direction, optical bandgap tunability from 3.41 eV to 3.66 eV in Ag-based p-type delafossite AgCr 1−xMg xO2 and 3.41 –3.60 eV in Ni doped α-Cr 2O3has also been reported.25,29 Bandgap tunability of Ga substituted α-Cr 2O3has been reported from∼3.4 eV to 3.95 eV, which is one of the highest in α-Cr 2O3 based composite materials.30 Immiscibility, of the constituent materials, restricts the tuning of the bandgap in several cases. For example, only 10% Ni2+and 45% Ga3+atom substitution in α-Cr 2O3is possible which limit the tunability of these ternary systems.30,31However, full miscibility of theα-Cr 2O3–α-Al2O3material system is reported for the complete composition range32,33and the optical bandgap of α-Cr 1.71Al0.29O3 is reported up to 4.35 eV.34However, the detailed investigation on structural and optical properties of this material system is absent in the literature. As the α-(Al xCr1−x)2O3material system has signifi- cant potential for applications in the field of transparent electronicdevices, it is important to study the properties of this material system. In this work, α-(Al xCr1−x)2O3was synthesized by the solid state reaction method. Synchrotron x-ray diffraction (XRD) mea-surements confirm full miscibility and R /C223c crystal structure of the compound. The basal plane lattice parameter shows a clear devia-tion from Vegard ’s law as a function of Al concentration ( x), whereas out of plane lattice parameter varies linearly according to Vegard ’s law. The variation of lattice parameters with x, in the α-(Al xCr1−x)2O3material system, is still a matter of debate.35,36In this work, a detailed understanding of variations in lattice parame-ters “a”and “c”has been developed by studying micro-strain and local structure behavior of this material system using XRD and EXAFS measurements. Optical measurements reveal large tunabil-ity (∼1.9 eV) in the bandgap from 3.4 eV to 5.3 eV, which is thelargest in any Cr based composite material system reported until now. Bowing in the bandgap with the Al composition is also observed. The nature of the density of states and the effect of Al 3+ substitution on the valence band have been probed using densityfunctional theory (DFT) based calculations, and the results arecompared with valence band photoemission spectroscopy (PES) measurements. II. EXPERIMENTAL AND COMPUTATIONAL DETAILS Conventional solid-state reaction method was used to prepare α-(Al xCr1−x)2O3(0≤x≤1) solid solutions by taking Al 2O3 (99.999%, Alfa Aesar) and Cr 2O3(99.97%, Alfa Aesar) as starting materials. The powders of the starting materials were taken in dif- ferent molar ratios to vary the Al ( x) concentration and then homogenized using a mortar and pestle for half an hour. It wasfurther mixed using planetary ball mill (Fritsch make) for 6 h at 150 rpm speed. The calcination of the powders was carried out at 1400 °C for 24 h in air. The calcined powder was grounded usingthe mortar and pestle. A small amount of binder was mixed in thepowder before pelletization. Pellets of 15 mm diameter were pre-pared at 8 tons of pressure. Sintering of all the pellets was done at 1500 °C for 12 h. The structural quality of α-(Al xCr1−x)2O3samples were examined at the Extreme Conditions X-ray Diffraction(ECXRD) beamline (BL-11) Indus-2 synchrotron source. The mea-surements were carried out using a wavelength of ∼0.629 Å. Two-dimensional (2D) diffraction data were collected using MAR345 image plate area detector. NIST standards of CeO 2and LaB 6were used to calibrate the sample to detector distance and the wavelength of the x-ray beam. Calibration and conversion/integra-tion of 2D diffraction data to 1D intensity vs 2 θformat were carried out with the help of FIT2D software. 37EXAFS measure- ments at Cr K-edge in transmission mode were performed at scan- ning EXAFS beamline (BL-09) of the Indus-2 synchrotron source.ATHENA software was used for background removal and data nor-malization. 38The fitting of the normalized EXAFS data was carried out in ARTEMIS of Demeter software package. Supra55 Carl Zeiss Field Emission Scanning Electron Microscope (FESEM) was usedto investigate the microstructure and the average grain size distribu-tion of the samples. Al and Cr compositions were confirmed in thetwo samples of different compositions using energy dispersive spectroscopy (EDS) measurements (shown in the supplementary material ). The optical bandgap of α-(Al xCr1−x)2O3was measured using diffuse reflectance spectroscopy (DRS). The measurementswere carried out in the wavelength range of 200 –800 nm using Cary-60 UV-VIZ-NIR spectro-photometer having the Harrick Video Barrelino diffuse reflectance probe. The beam spot size (diameter) on the sample was around 1.5 mm. PES measurementswere carried out on the samples at the Indus-1 beamline BL-03, inthe angle integrated mode, using a photon energy of 50 eV (10 eVpass energy) at a typical spectral resolution of 150 meV. To understand the experimental results, we have also carried out spin-polarized electronic structure calculations based ondensity functional theory (DFT) 39by using Vienna Ab-initio Simulation Package (VASP)40,41in the projector augmented wave (PAW) method framework. We have used the generalized gradient approximation (GGA) given by Perdew, Burke, and ErnzerhofJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-2 Published under license by AIP Publishing.(PBE)42for exchange-correlation. We have also carried out the cal- culations using the DFT + U method, where U is the on-site Coulomb interaction between the electrons of d-orbitals of Cratoms and choose value of U = 1 –4 eV. The energy cut-off of plane waves (basis set) is 400 eV. We use the Monkhorst –Pack scheme with the k-mesh of 16 × 16 × 16 for Brillouin zone integration. The convergence criteria for energy in self-consistent field (SCF) cycles are chosen to be 10 −6eV. The calculations have been performed for the corundum structure of α-Cr 2O3with R /C223c space group which contains 30 atoms in a unit cell, with the Cr atoms in the anti-ferromagnetic configuration along c-direction. In the calculations of the composite system α-(Al xCr1−x)2O3, we have substituted Cr atoms by Al atoms taking into consideration of different possiblegeometrical arrangements between the substituted atoms. III. RESULTS AND DISCUSSION Room temperature Synchrotron XRD data for α-(Al xCr1 −x)2O3solid solutions with 0 ≤x≤1 are shown in Fig. 1 . All Bragg peaks were indexed with a single rhombohedral (R /C223c) phase for all Al compositions in the range of 0 ≤x≤1, and no additional peaks are observed. The position of all Bragg peaks shifts monotonicallytoward higher 2 θangle with increasing Al composition. This indi- cates that unit cell parameters decrease with increasing x.F o r clarity, the enlarged view of (104) and (110) Bragg peaks are shown in the inset of Fig. 1 . In this crystal system, the (110) peak corre- sponds to the basal plane lattice parameter ( a) and the variation in the Bragg peak position of the peak with Al concentration isdirectly correlated with the lattice parameter a. The peak position of (110) shifts continuously to higher 2 θwith the Al composition, which corresponds to a decrease in the lattice parameter aof α-(Al xCr1−x)2O3solid solutions. Similarly, the peak position of FIG. 1. Synchrotron XRD data of α-(Al xCr1−x)2O3samples, taken from the ECXRD beamline in Indus-2 at a wavelength of ∼0.6318 Å, show the single phase structure for full composition range. Inset shows variation in the (104)and (110) Bragg peak positions as a function of x. FIG. 2. Rietveld refinement of synchrotron XRD data for the Al composition of (a)x= 0.0, (b) x= 0.50, and (c) x= 1.0.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-3 Published under license by AIP Publishing.(104) shifts continuously to higher 2 θ. These results, thus, qualita- tively confirm that both the lattice parameters aand cdecrease with an increase in Al composition ( x). Rietveld refinement43of the XRD patterns was carried out using Fullprof software44for accurate lattice parameter and atomic position determination. Rietveld refinement fitting of α-(Al xCr1 −x)2O3(for x= 0, 0.5, and 1.0) is shown in Fig. 2 . The variation of lattice parameters ( aand c) with Al concentration xis plotted in Fig. 3 . The variation of the in-plane lattice parameter ( a) with x deviates from Vegard ’sl a w45and is fitted using the quadratic equa- tion, a(x)¼xaAl2O3þ(1/C0x)aCr2O3/C0ba XRDx(1/C0x), (1) where aAl2O3(aCr2O3) corresponds to the lattice parameter aofα-Al2O3(α-Cr 2O3).ba XRDis the lattice bowing parameter, and its value is estimated to be −0.035 Å. The lattice parameter c,however, follows Vegard ’sl a w , c(x)¼xcAl2O3þ(1/C0x)cCr2O3, (2) where cAl2O3(cCr2O3) corresponds to the lattice parameter cof α-Al2O3(α-Cr 2O3). The micro-strain in a system is a prominent factor, which influences the lattice bowing parameter in the crystal system, and FIG. 3. Variation of (a) lattice parameter “a”and (b) lattice parameter “c”of α-(Al xCr1−x)2O3as a function of Al composition ( x) (filled blue circles) obtained from the Rietveld refinement of XRD data. Solid lines (red color) show fitting of the experimental data with (a) the quadratic equation (Vegard ’s law with bowing parameter) and (b) the linear equation (Vegard ’s law). FIG. 4. Variation of volume average crystallite size and micro-strain with the Al composition ( x) is shown in the upper part of the figure and the lower part of the figure shows FESEM images of α-(Al xCr1−x)2O3samples for the composi- tions of x= 0, 0.5, 0.7, and 1.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-4 Published under license by AIP Publishing.XRD line profile analysis is a commonly used method to determine the microstructural (size-strain) parameters of a sample.46XRD data of NIST SRM 660b (LaB 6) were used for preparing the instru- ment resolution file (IRF) using FullProf software as reported byRodr ıguez-Carvajal and Roisnel. 47The Thompson –Cox –Hastings (TCH) peak profile function48,49was used during the Rietveld refinement of XRD data of all α-(Al xCr1−x)2O3samples for extract- ing the microstructural effects (crystallite size and microstrain)due to samples, and the detailed strategy and mathematicalprocedure used for isotropic microstructural analysis are similar toreferences. 50,51The resulting volume average crystallite size and microstrain values are plotted in Fig. 4 . In the case of α-Cr 2O3 (x= 0), the peak widths are almost comparable with the widths of the SRM 660b except for a little broadening at the tails. Theestimated volume average apparent crystallite size is approximately820 (±50) nm with negligible microstrain effect. In the samples corresponding to x=0–0.5, the broadening of XRD peaks is mainly concentrated at the tails as compared to instrumentalbroadening. In Al rich composition, i.e., from x= 0.6 to 1.0, the XRD peak widths are also significantly broadened. For this compo-sition range, the estimated volume average apparent crystallite size saturates at about 60 (±4) nm, while micro-strain shows an increasing trend with x. This implies that for Al rich compositions, the crystallite size is controlled by the chemistry of Al 2O3.46 FESEM images of α-(Al xCr1−x)2O3samples for the compositions ofx= 0, 0.5, 0.7, and 1 are shown in Fig. 4 . Nature of the grain distribution confirms the polycrystalline nature of the samples. FESEM results also show that the grain size decreases with Al com-position, which is in line with the results obtained from XRD dataanalysis. FIG. 5. (a) Bonding of O1 and O2 (red color balls) with Al/Cr ions (blue color balls) in the ball and stick model of the unit cell of α-(Al xCr1−x)2O3, (b) –(d) variations in the bond lengths Cr/Al-O1 and Cr/Al-O2 and bond angle θO1-Cr/Al-O2 are shown as a function of Al composition ( x), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-5 Published under license by AIP Publishing.It is observed that bowing is present only in the lattice param- eter a. Bowing in the lattice parameters is mainly related to the local structure in the system. To find out the reason behind thedifferent behavior of lattice parameters aand c, the variations of Cr/Al –O bond lengths and bond angles obtained from Rietveld refinement have been investigated in detail. In the R /C223c phase of α-(Al xCr1−x)2O3, there are two types of oxygen atoms (O1 and O2), which are categorized based on their different bond distancesfrom the Cr/Al atom [shown in Fig. 5(a) ]. The ball and stickmodels of α-(Al xCr1−x)2O3, shown in this work, are drawn using VESTA software.52Variations in the bond lengths Cr/Al-O1 and Cr/Al-O2 and bond angle θO1-Cr/Al-O2 are plotted as a function of Al composition ( x)i n Figs. 5(b) –5(d). It is well established that XRD only gives the average values of Cr –O and Al –O bond lengths and bond angles. This implies that variation in the parameters should be linear as a function of composition if the bond lengths and bond angles remain same as in the constituent binary com-pounds, e.g., α-Cr 2O3andα-Al2O3, and this is shown by “line ”in FIG. 6. (a) The ball and stick model of the α-(Al xCr1−x)2O3unit cell, (b) bonding of O1 and O2 with the Al/Cr ions is shown in the zoomed portion, and (c) and (d) varia- tions in the bond angles Ψ1andθ1and bond length d 16are shown as a function of Al composition ( x), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-6 Published under license by AIP Publishing.all the figures. Bond length dCr/Al-O2 and bond angle θO1-Cr/Al-O2 show bowing from linearity as a function of Al composition ( x). This indicates that the bond lengths and/or bond angles of Cr –O and Al –Oi n α-(Al xCr1−x)2O3are different from respective constit- uent compounds. These variations directly affect the lattice parame-ters of the material system. Figures 6(a) and6(b) show the ball and stick model of α-(Al xCr1−x)2O3. Some of the Cr/Al atoms are num- bered for ease in explanation. From Fig. 6(b) , we see that the lattice parameter ais equal to d17(distance between Cr/Al-1 and Cr/Al-7), which is equal to 2 d16sin(Ψ1/2). Distances d16and d67 are equal in magnitude and the perpendicular drawn from Cr/Al-6 divides the angle Ψ1into two equal parts. Distance d16(and d67)i s directly related to Cr/Al-O1 and Cr/Al-O2 bond lengths and bondangle θ 1. Bond angle θ1also depends on the Cr/Al-O1 and Cr/Al-O2 bond lengths and bond angle θO1-Cr/Al-O2 .F r o m Fig. 6(b) , it is, thus, clear that the lattice parameter ais directly affected by the change in the local structure. Figures 6(c) and6(d) show the variation in the angles ( Ψ1,θ1) and distance ( d16) with the Al composition ( x). There is an upward bowing in all the three parameters, which govern the bowing in the value of the latticeparameter “a”. Figure 7 shows that the lattice parameter cis a sum of two distances d 12(distance between Cr/Al-1 and Cr/Al-2) and d23 (distance between Cr/Al-2 and Cr/Al-3). This implies that the nature of change in the value of cis governed by both these distances. d 12depends on the bond length Cr/Al-O1 and bond angle θ2shown in Fig. 7(a) . There is a downward bowing in d12 and is plotted in Fig. 7(b) . The distance d23depends on the dis- tances d25and d35and angle Ψ2,which are dependent on the angles θ3andθ4and Cr/Al-O1 and Cr/Al-O2 bond lengths [Fig. 7(a) ]. The upward bowing in d23, observed from Rietveld fitting, is shown in the inset of Fig. 7(c) . From Figs. 7(b) and7(c), it can be seen that bowing in distances d12and d23are opposite with almost equal magnitudes (inset in the figures44). As thelattice parameter cis the sum of these two distances, the bowing cancels out and cshows linear behavior with Al composition ( x). From the discussion, it is demonstrated in the present case(for the R /C223c crystal structures) that origin of bowing in “a”and “c”are quite different. It is further possible that both the lattice parameters can have different bowing depending on specific local structure changes. The bandgap of α-(Al xCr1−x)2O3(0≤x≤1) samples was determined using diffuse reflection spectroscopy (DRS) at roomtemperature. The measured DRS spectrum was modified to equiva- lent absorption using Kubelka –Monk (KM) formalism. 53,54The KM function F(R1) can be shown as K S¼(1/C0R1)2 2R1;F(R1), (3) where R1¼Rsample /Rstandard .Rsample and Rstandard correspond to diffuse reflectance of the sample and standard (BaSO 4), respec- tively. The parameters KandSare Kubelka –Monk absorption and scattering functions, respectively. KM scattering coefficient Sis a constant with respect to wavelength, when the material scatters light in a perfectly diffuse manner. In that condition, the value of Kequals to absorption coefficient (2 α).Figures 8(a) and 8(b) show theF(R1)v s hvplots for α-(Al xCr1−x)2O3from x=0 t o x= 0.9, respectively. Four absorption peaks are observed in all the plots ofFigs. 8(a) and 8(b), which are related to various transitions of α-Cr 2O3. These absorption peaks A, B, C, and D, observed at ∼2.06 eV, ∼2.68 eV, ∼3.45 eV, and ∼4.96 eV respectively, are assigned to transitions from4A2g→4T2g,4A2g→4T1g, Cr t 2g →eg*, and O 2p→Cr d* states, respectively.55–59From Figs. 8(a) and8(b), we find that the absorption peaks A, B, and C continu- ously shift toward high energy with Al composition. Absorptionpeak “C”(∼3.40 eV), which originates from the d –d splitting, cor- responds to the bandgap of α-Cr 2O3. Variation of the bandgap (position of peak “C”) with Al compositions ( x) is plotted in Fig. 8(c) . The bandgap increases systematically from 3.4 eV to 5.3 eV as a function of xand is fitted with a second order polynomial, EOptical gap (eV)¼3:4þ0:3xþ2:1x2: (4) This equation shows a large variation of ∼1.9 eV in the bandgap with Al composition going from x=0 t o x=1 . T o t h e best of our knowledge, this variation is the largest among allthe Cr based transparent oxide materials investigated so far. Large bandgap tuning indicates a huge potential for the application of this material system in the field of UV transparent devices. A rela-tively smaller bandgap tuning ( ∼1.1 eV) has been shown in the Ni xMg 1−xO material system. Density functional theory calculation of the Ni xMg 1−xO electronic structure has showed that the bandgap of this material system is mainly derived from Ni 3d e g and t 2g levels, which create localized states inside the MgO bandgap and the bandgap decreases with the increase in Ni con-centration. 60These observations are similar to our case. Therefore, unlike semiconductor alloys in which bandgap is caused by periodic potential of the lattice, the bandgap bowing in α-(Al xCr1−x)2O3(0≤x≤1) is similar to the Ni xMg 1−xO material system. In most of the early transition metal oxide semiconductors, bandgap is governed by transitions between d –d levels (Mott insu- lators). In transition metal ions, separation between dlevels depends on the intensity of the crystalline field surrounding theion. When a transition metal ion is replaced by another ion, itleads to distortions in the ordered average crystal lattice. These dis- tortions result in the change of hybridization of metal 3d with O 2p levels, (or intensity of the crystalline field surrounding the ion)which reflects in a change in the bandgap. Bandgap shifts towardhigher or lower energy values with the increase or decrease in thecrystalline field, respectively. Thus, the changes in the bandgap are directly associated to local distortions in the ordered average crystal lattice. One of the important parameters which affect the crystallinefield is the distance between the transition metal and surroundingligands. The magnitude of the crystalline field is directly related tocrystal field splitting ( Δ 0), which is determined by the following formula,61 Δ0¼10Dq¼ZLe2 6R5r4/C10/C11 ¼Qr4hi R5, (5)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-7 Published under license by AIP Publishing.FIG. 7. (a) The ball and stick model of the α-(Al xCr1−x)2O3unit cell with two zoomed portions is shown to illustrate the bonding of O1 and O2 with the Al/Cr ions, (b) and (c) variations in bond lengths d 12and d 23are shown as a function of Al composition ( x), respectively, and (d) and (e) show deviation from the linearity in bond lengths d 12 and d 23as a function of Al composition ( x), respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-8 Published under license by AIP Publishing.where Qis a constant, ZLeis the ligand ’s charge, Ris the separation of the ligand from the cation, and ris coordinate of the electron in the polar coordinate system. As the local structure about the Cr atom is crucial in determin- ing both bowing in the lattice parameter and the optical properties mentioned above, we have carried out room temperature EXAFSmeasurements about the Cr K-edge for the α-(Al xCr1−x)2O3 (0≤x≤1). EXAFS data, in real space co-ordinates (R), are shown in Fig. 9(a) . For clarity of presentation, all the curves are shifted verti- cally. Normalized absorption data χ(k) were Fourier transformed to get the EXAFS data in real space co-ordinates χ(R). From the geometric structure of α-(Al xCr1−x)2O3[Fig. 5(a) ], it is clear that Cr ions are surrounded by six oxygen atoms. Based on the value of bond lengths, the Cr atom is bonded with two types of oxygen (O1 and O2). First peak [ Fig. 9(a) ], which corresponds to bothCr-O1 and Cr-O2 bond lengths, was fitted to determine the Cr-O1/O2 bond lengths. Fittings for the Al composition of 0.5 are shown in Fig. 9(b) as a representative data. The value of an amplitude reduction factor ( S02) was determined from α-Cr 2O3data and kept same for all the fittings and was found to be 0.8. Inindividual samples, parameter E 0was kept same for all the paths and its value was between −0.42 ± 0.73 and −1.78 ± 2 eV, and the R-factor of the fitting varied in the range of 0.007 to 0.02. Figure 9(c) shows variation in the nearest neighbor ( NN) Cr-O1 and Cr-O2 bond distances with Al composition ( x). Nearest neighbor (NN) Cr-O1 and Cr-O2 bond distances decrease with Al composi- tion, which leads to a contraction of the octahedral sites. This contraction of the bond lengths about the octahedral sites [term “R” in Eq. (6)] leads to the observed blue shift in crystal field splittingand, consequently, the increase in the energy of the transitions oftheA, B, and Cfeatures in the optical absorption data with an increase in Al concentration x. The observed non-linear decrease in the NNCr-O1 and Cr-O2 bond distances with x,a l s od i r e c t l y affects the lattice parameters, and is also one of the reasons(other being the microscopic strain) for observed bowing in thelattice parameters. The valence band of α-Cr 2O3andα-(Al xCr1−x)2O3was probed using the PES experiment. The angle integrated PES mea-surements were carried out with the Indus-I synchrotron radia-tion source at BL-03. Both α-Cr 2O3andα-AlCrO 3samples were insulating in nature; hence, significant amount of charging effect was present in the binding energy of PES data. An electron floodgun (OmniVac make, model PS-FG 100 with the parametersvoltage: −8 V ,c u r r e n t :0 . 4 m A )w a su s e dt oe l i m i n a t et h ec h a r g - ing effect and to improve the data quality. Only relative changes in the valence band spectra which contain the information related to interaction between O 2p and Cr 3d states wereprobed. The absolute values of the Fermi level in the two spectracould be different due to the cha rging effect. The results are shown in Figs. 10(a) and10(b) forα-Cr 2O3andα-(Al xCr1−x)2O3 (x= 0.5) (henceforth referred to as α-AlCrO 3), respectively. The band at around 1.6 eV ( D1) arises primarily from Cr 3d levels with a small contribution from the O 2p levels. The second band centered at around 5 –6e V( D2) arises primarily from O 2p levels with some mixture of Cr 3d levels.19We discuss these assign- ments in further details, using our first principles calculations mentioned subsequently. By comparing the spectra fromα-Cr 2O3andα-AlCrO 3, we find that the gap between D1a n d D2 bands decreases in α-AlCrO 3as compared to α-Cr 2O3.As i g n i f i - cant change in the shape of the D2 band is observed in α-AlCrO 3, which clearly indicates valence band modifications due to Al substitution. In order to understand and substantiate our experimental results, DFT based spin-polarized electronic structure calcula- tions have been performed for α-Cr 2O3andα-(Al xCr1−x)2O3 material systems. Results of calculated total and partial DOS of Cr 3d and O 2p states for α-Cr 2O3and α-(Al xCr1−x)2O3 (x= 0.5) systems are summarized in Fig. 11 . The position of the Al 2p states for α-AlCrO 3is also shown in the figure. Corresponding to the PES data, the features D1a n d D2 have been indicated in the calculated DOS also. It is clear that thestates close to the Fermi level in the valence band of α-Cr 2O3 FIG. 8. (a) and (b) Variations of the Kubelka –Munk function with energy for α-(Al xCr1−x)2O3for different Al compositions and peaks are assigned to different transition levels of α-Cr2O3and (c) optical bandgap variation as a function of Al composition ( x). Schematic diagrams of optical transitions of α-Cr 2O3are shown in the inset.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-9 Published under license by AIP Publishing.andα-AlCrO 3a r ep r i m a r i l yd o m i n a t e db y3 do r b i t a l so ft h eC r atom with an admixture of 2p orbitals of O atom. There are afew important observations from the theoretical calculations of the valence band of α-Cr 2O3and α-AlCrO 3, which are as follows: (a) there is an increase in O 2p partial density of statesrelative to the Cr3d states in α-AlCrO 3as compared to α-Cr 2O3in the feature D1, (b) there is only a very negligible presence of the density of states from Al in the feature D1i n α-AlCrO 3, and (c) the separation between D1a n d D2 bands decreases in α-(Al xCr1−x)2O3as compared to α-Cr 2O3.W eh a v e also carried out similar calculations by employing the DFT + Umethod with different U values. The trend in the variation ofthe separation between D1 and D2 is similar, and the results are given in the supplementary material .From the theoretical calculation, it is evident that the valence band of α-Cr 2O3is dominated by Cr 3d and O 2p states with small amount of mixing between them. This hybridization of Cr 3d and O 2p states is important for p-type conductivity in this system (spe- cifically in the D1 band), thereby making α-Cr 2O3a suitable candi- date for the wide gap p-type semiconducting material. Increasedmixing of O 2p and Cr 3d orbitals increases the delocalization ofhole states in the valence band, and this leads to improvement in p-type conductivity. 17Our results clearly indicate that this hybridi- zation is enhanced in α-(Al xCr1−x)2O3as compared to α-Cr 2O3. The reduction in Cr-O1 and Cr-O2 bond lengths with an increaseinx, as observed from EXAFS data analysis, results in an increase in the overlap of the Cr and O orbitals, thereby resulting in an increase in observed hybridization of the Cr 3d and O 2p bonds. FIG. 9. (a) EXAFS data of α-(Al xCr1 −x)2O3in real space co-ordinates (R) at Cr K-edge for different Al composi- tion ( x). For clarity in presentation, EXAFS spectra are shifted vertically,(b) fittings in both χ(k) and χ(R) are shown for composition x=0 . 5 0 a t C r K-edge, and (c) variation of Cr-O1 and Cr-O2 bond distances as a func-tion of Al composition findout from theEXAFS data fittings.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-10 Published under license by AIP Publishing.Preliminary impedance measurements were carried out to measure the ac conductivity of both the α-Cr 2O3andα-AlCrO 3samples using the Wayne Kerr 6505B impedance analyzer. AC conductivityis calculated from the impedance by the formula σ= d*cos( θ)/A*Z, where Z is impedance of the sample, θis the phase angle, A is the area of contact on the sample, and d is sample thickness. From the impedance measurements, it has been observed that the ac conductivity of α-AlCrO 3increases around one order of magnitude as compared to α-Cr 2O3,which indicates the positive effects of enhancement of mixing of both Cr 3d and O 2p levels(shown in supplementary material ). Detail studies will be a part of a future work. FIG. 11. T otal and partial density of states (Cr 3d, O 2p and Al 3p) for (a) α-Cr2O3and (b) α-(Al xCr1−x)2O3(x= 0.5) systems, obtained from DFT based electronic structure calculations. Energies of states are given with respect to theFermi level. FIG. 10. Valence-band spectra of (a) α-Cr2O3and (b) α-(Al xCr1−x)2O3taken at angle resolved photoemission spectroscopy (ARPES) beamline installed atIndus-1 synchrotron radiation source. Incident photon energy was 50 eV during measurements.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 135703 (2020); doi: 10.1063/5.0021421 128, 135703-11 Published under license by AIP Publishing.IV. CONCLUSIONS In summary, single phase α-(Al xCr1−x)2O3(0≤x≤1) was synthesized by using the solid state reaction method. Variation ofthe lattice parameter cwith xfollows Vegard ’s law, while the varia- tion of the lattice parameter awith xshows a deviation from Vegard ’s law with the bowing parameters of −0.035 Å. This devia- tion is related to the micro-structural effects and local structuraldistortion, as observed by detailed Rietveld analysis of the XRDdata. EXAFS measurements indicate a reduction in the Cr –O bond length with increasing Al content. Optical absorption measure- ments show a large bandgap tunability from 3.4 eV to 5.3 eV inα-(Al 1−xCrx)2O3for 0≤x≤1. Micro-structural effects (crystallite size and micro-strain), extracted using Thompson –Cox –Hastings peak profile function of the diffraction data, show a reduction in the crystallite size with the increase in the micro-strain as a func- tion of composition x. Valence band photoemission spectroscopy and first principles based electronic structure calculations indicatethat the energy levels close to the valence band maximum are pri-marily composed of Cr 3d states with a small admixture of O 2p states and the relative percentage of O 2p states increases with Al substitution. This indicates that Al substitution in α-Cr 2O3[i.e., in α-(Al xCr1−x)2O3] increases the hole delocalization, thus having the potential for application as a transparent p-type semiconductor forthe applications in UV based transparent electronics, photodetec- tors, light sources, etc. SUPPLEMENTARY MATERIAL See the supplementary material for the EDS results, total and partial density of states obtained from DFT + U based electronicstructure calculations, and ac conductivity of the studied α-Cr 2O3 andα-AlCrO 3samples. ACKNOWLEDGMENTS The authors would like to thank Mr. S. V. Nakhe, Director, Materials Science Group, RRCAT for his constant encouragement and support. C. Kamal acknowledges the Scientific ComputingGroup, Computer Division, RRCAT for providing help andsupport in running the codes. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. Sheng, G. Fang, C. Li, S. Xu, and X. Zhao, Phys. Status Solidi A 203, 1891 –1900 (2006). 2P. Qin, G. Fang, Q. He, N. Sun, X. Fan, Q. Zheng, F. Chen, J. Wan, and X. Zhao, Sol. Energy Mater. Sol. Cells 95, 1005 –1010 (2011). 3Z. C. Wang, J. Miao, M. Yang, R. H. Zhao, Y. Wu, X. G. Xu, and Y. Jiang, J. Alloys Compd. 723, 311 –316 (2017). 4S. C. Dixon, D. O. Scanlon, C. J. Carmalt, and I. P. Parkin, J. Mater. Chem. C 4, 6946 –6961 (2016). 5K. H. L. Zhang, K. Xi, M. G. Blamire, and R. G. Egdell, J. Phys. Condens. Matter 28, 383002 (2016).6H. Kawazoe, M. Yasukawa, H. Hyodo, M. Kurita, H. Yanagi, and H. Hosono, Nature 389, 939 –942 (1997). 7O. F. Schirmer, J. Phys. Condens. Matter 18, R667 –R704 (2006). 8D. O. 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5.0021838.pdf

Appl. Phys. Lett. 117, 114001 (2020); https://doi.org/10.1063/5.0021838 117, 114001 © 2020 Author(s).Nanoassembly technique of carbon nanotubes for hybrid circuit-QED Cite as: Appl. Phys. Lett. 117, 114001 (2020); https://doi.org/10.1063/5.0021838 Submitted: 15 July 2020 . Accepted: 06 September 2020 . Published Online: 17 September 2020 T. Cubaynes , L. C. Contamin , M. C. Dartiailh , M. M. Desjardins , A. Cottet , M. R. Delbecq , and T. Kontos COLLECTIONS Paper published as part of the special topic on Hybrid Quantum Devices HQD2021 This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Black phosphorus field effect transistors stable in harsh conditions via surface engineering Applied Physics Letters 117, 111602 (2020); https://doi.org/10.1063/5.0021335 Strong interface-induced spin-charge conversion in YIG/Cr heterostructures Applied Physics Letters 117, 112402 (2020); https://doi.org/10.1063/5.0017745 Phononic bandgap and phonon anomalies in HfN and HfN/ScN metal/semiconductor superlattices measured with inelastic x-ray scattering Applied Physics Letters 117, 111901 (2020); https://doi.org/10.1063/5.0020935Nanoassembly technique of carbon nanotubes for hybrid circuit-QED Cite as: Appl. Phys. Lett. 117, 114001 (2020); doi: 10.1063/5.0021838 Submitted: 15 July 2020 .Accepted: 6 September 2020 . Published Online: 17 September 2020 T.Cubaynes,a)L. C. Contamin, M. C. Dartiailh, M. M. Desjardins, A.Cottet, M. R. Delbecq, and T. Kontosb) AFFILIATIONS Laboratoire de Physique de l’Ecole normale sup /C19erieure, ENS, Universit /C19e PSL, CNRS, Sorbonne Universit /C19e, Universit /C19e Paris-Diderot, Sorbonne Paris Cit /C19e, 75005 Paris, France Note: This paper is part of the Special Issue on Hybrid Quantum Devices. a)Electronic mail: tino.cubaynes@kit.edu b)Author to whom correspondence should be addressed: takis.kontos@ens.fr ABSTRACT A complex quantum dot circuit based on a clean and suspended carbon nanotube embedded in a circuit quantum electrodynamic (cQED) architecture is a very attractive platform to investigate a large spectrum of physics phenomena ranging from qubit physics to nanomechanics.We demonstrate a carbon nanotube transfer process allowing us to integrate clean carbon nanotubes into complex quantum dot circuits inside a cQED platform. This technique is compatible with various contacting materials such as superconductors or ferromagnets. This makes it suitable for hybrid quantum devices. Our results are based on eight different devices demonstrating the robustness of thistechnique. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021838 One challenge for the investigation of quantum phenomena is the fabrication of systems that are sufficiently decoupled from their environment such that they manifest their quantum nature. Thisdecoupling usually comes at the cost of low tunability of the system parameters. From this perspective, a suspended and clean carbon nanotube embedded in a circuit, hence allowing large tunability of its parameters, and with the “active” part of the system confined to the suspended section of the nanotube, is very attractive. 1 Because it can be suspended, a carbon nanotube can be engi- neered to be arbitrarily far from any interface, which is known to be o n eo ft h em a i ns o u r c e so fc h a r g en o i s e .2The cleanliness of the nano- tube also plays a key role since it ensures a very low amount of charge fluctuators. These two points suggest a long electronic coherence time in such carbon nanotube-based circuits. Besides, there is also the possi- bility to tailor the phonon spectrum of the suspended nanotube via gate voltages.3–5In the case of a qubit, this means that one can tune the relaxation time due to electron-phonon interaction.6The nanotube can also be connected with different types of metals, making this plat- form interesting for studying hybrid circuits.7 Recently, various carbon nanotube transfer techniques have been developed.1,8–13The common idea is to grow carbon nanotubes on a separate substrate and then transfer one of the nanotubes onto the circuit at the final step of the process. Depending on the growth andthe transfer conditions, several of these works have demonstrated the clean nature of the transferred nanotube,1,10,11up to the observation of a 1D Wigner crystal.14 This technique has the advantage to provide suspended car- bon nanotubes and is now largely used in nanomechanics experi- ments.15–17The suspended nature of the nanotube is also desirable to limit charge noise in quantum dot circuits. Also, because the fabrication of the circuit and the synthesis of the carbon nanotubes are now completely independent, there is a great flexibility in the circuit fabrication process. More recently, this technique has been adapted to the cQED platform11,13by growing carbon nanotubes on a fork-like chip.9,12 Here, we present a fabrication technique inheriting from these recent advances. The nanotubes are grown on a substrate containing a comb of 48 cantilevers allowing us to do a pre-selection of the trans-ferred nanotube. In addition to producing clean suspended carbon nanotube-based circuits embedded in a cQED platform, we show that this technique is compatible with a superconductor and a ferromagnetas metallic contacts, making it suitable for many types of hybrid quan- tum devices. Our results are based on eight devices with different geometry and contact materials. The process includes two distinct samples: the growth chip, on which carbon nanotubes are grown, and the circuit chip. For the Appl. Phys. Lett. 117, 114001 (2020); doi: 10.1063/5.0021838 117, 114001-1 Published under license by AIP PublishingApplied Physics Letters ARTICLE scitation.org/journal/aplgrowth chip, we use a commercial cantilever chip with 48 cantilevers and a spacing of 30 lm between cantilevers as depicted in Fig. 1(a) . Carbon nanotubes are grown all over the sample, using a standard CVD-growth recipe based on CH 4feedstock gas and using Mo-Fe nanoparticles as a catalyst.18Depending on the catalyst distribution, several carbon nanotubes will be suspended between adjacent cantile- vers. The large number of cantilevers is particularly useful to integrate multiple nanotubes on the same circuit chip. Prior to the transfer of acarbon nanotube to the circuit chip, pictures of the tips of all thecantilevers are taken using a Scanning Electron Microscope (SEM) atlow acceleration voltage (2 kV). While common belief is that SEM observation induces the deposition of the hydrocarbon layer onto the nanotube, 19hence should introduce disorder in the electronic spec- trum of the nanotube, we have found that it is still possible to obtainclean transport spectra using such a brief observation. One possibleexplanation for this low contamination of suspended carbon nano- tubes by the e-beam exposure is the fact that most of the hydrocarbon is on the surface of the substrate, and thus, suspended nanotubes aremuch less affected by e-beam induced diffusion of hydrocarbon. 20 This first step allows us to pre-select the isolated carbon nanotubes.Images of carbon nanotubes suspended in between adjacent cantilevers are shown in Fig. 1(a) . Between the localization of carbon nanotubes and the stamping process, the cantilever chip isstored in a vacuum chamber (pressure: 5 /C210 /C07mbar), to minimizethe exposition time of nanotubes to the ambient atmosphere down to approximately 10 min. On the circuit chip, the niobium microwave cavity and the circuit electrodes are fabricated using e-beam lithography as in previous work.21In addition, two trenches on both sides of the cir- cuit are etched by reactive ion etching using SF 6gas. This process allows us to etch 10–15 lm deep trenches, which are used to ensure a proper contacting of the nanotube to the circuit electrodes [see Figs. 1(c) and1(d)]. The transfer of the carbon nanotube to the circuit chips is realized in a dedicated vacuum chamber with a base pressure of 5/C210–7mbar. A schematic of this chamber is presented in Fig. 2(c) . The circuit chip is placed on a rotary arm, so that it can be either in an horizontal position for the transfer of the nanotubes or in a vertical position, facing an argon gun, in order to clean the surface of the contact electrodes, as well as remove unwanted nanotubes. The growth chip is fixed with a tilt angle of 45/C14with respect to the horizontal plane, to ensure a good visibility from thetop view, while maintaining a small footprint of the circuit chip. The position of the growth chip is controlled by piezo-motors and micro-manipulators. There is also the possibility to isolate the growth chip from the main chamber, in order to protect the carbon nanotubes during the cleaning of the circuit chip with the argon gun or during the replacement of the circuit chip. The growth chip is lowered using the piezo-motor stage, and the contact is detected by monitoring the current between the two external contact electrodes (at V bias¼0.5–1.5 V) and the two inner contacts, as depicted in Fig. 1(c) . After the contact, the resistance of the circuit typ- ically ranges between 10 M Xand 100 G X. Similar to ref,1,22the two external sections of the nanotube are cut by driving a large current through it (typically between 10 lAa n d2 0 lA). During this opera- tion, all the other contacts and gates are set to a floating potential, to avoid accidental cutting of the central section. Two examples of I–Vcurves corresponding to the cutting of the nanotube are shown in the inset of Fig. 2(b) . The two curves show very similar features, such as the current and the bias voltage at which the nanotube is cut. This shows that the nanotube has almost identical properties at two differ- ent sections a few lm away from each other, attesting the cleanliness of the transferred nanotube. One advantage of this cutting procedure is that it acts as a local annealing of the interface between the contact metal and the nanotube, drastically lowering the contact resistance, which can reach values below 1 M X(seeTable I ). The cutting step also allows us to distin- guish between a single nanotube, which displays a single current drop [seeFig. 2(b) ], and a bundle of nanotubes, which display multiple cur- rent drops. At this stage, it is possible to further improve the contact quality by driving a high current in the central section of the device, being careful not to exceed the cutting current. If the contacted nano-tube turns out to be a bundle or display unwanted characteristics, we remove it using the Ar gun. To obtain a suspended nanotube over the gate array, we found that the spacing of the contact electrode should not exceed ten times the height difference between contact and gate electrodes. As a last characterization study of the nanotube at room temper- ature, we measure the gate dependence of the current in the central section, in order to differentiate between a small gap, a semiconductingor a metallic nanotube. Finally, during the transfer of the circuit chip FIG. 1. (a) Optical photograph of the growth chip. It is composed of 48 cantilevers, which are visible on the bottom edge of the chip. Scale bar: 500 lm. The zoom-in on the cantilevers shows isolated carbon nanotubes suspended in between two adjacent cantilevers. Scale bar: 10 lm. (b) False color scanning electron micro- graph of the circuit chip (device 5). The circuit, constituted of the gate electrodes(in the dark blue) and the contact electrode (in yellow), is sandwiched in betweenthe two trenches. Scale bar: 50 lm.Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 114001 (2020); doi: 10.1063/5.0021838 117, 114001-2 Published under license by AIP Publishinginto the cryostat, all the electrodes are grounded and the sample is maintained under a nitrogen atmosphere or a low vacuum. The results discussed in the following are based on 8 different devices with various geometries and contact metals. The measure-ments have been performed at a base temperature of 20 mK (250 mKfor device 8). We present in Fig. 3 a single dot stability diagram in the Coulomb blockade regime. The continuous evolution of the contact transparency as the gate voltage is swept and the absence of chargejumps are indicative of the clean environment of the nanotube. The tunability of the circuit parameters has also been observed in double quantum dot circuits. Figure 4 presents stability diagrams ofthree double quantum dots devices with different carbon nanotube electronic behaviors: small-gap in Fig. 4(a) and semiconducting in Figs. 4(b) and4(d). The transition from double quantum dot behavior to a large single dot behavior is also visible in Fig. 4(c) ,s h o w i n ga g a i n the weak influence of disorder on the confinement potential, which isinstead dominated by electrostatic gating. The control over the interdot coupling was also demonstrated via cavity transmission measurements, where the transition from the resonant to the dispersive regime wasobserved (see the supplementary material ). In the resonant regime, the resonator not only is a readout circuit but can also be used for manipu- lation and coupling of qubits (spin or topological). The capability to FIG. 2. Principle of the transfer technique. (a) Schematics of the carbon nanotube transfer process. The cantilevers (in purple) are lowered until the carbo n nanotube makes the connection between external and internal contacts (in orange) of the circuit chip. The nanotube is then cut on both sides of the circuit by successi vely biasing only one of the two external contacts. (b) False color scanning electron micrograph of the region where the nanotube is integrated (Device 1). The carbon nanotub e (in black) is suspended over a gate array (in dark blue). Scale bar: 1 lm. Insets: current as a function of the bias voltage applied on the left external contact (bottom-left inset) and on the right external contact (top-right inset). (c) Schematic of the vacuum chamber used for the transfer of the carbon nanotube. The argon gun on the left side of the chambe r is used for cleaning the circuit chip or to remove unwanted carbon nanotubes. The growth chip (in purple) can be locked-up in the bellows chamber on the right side of the cham ber, for protecting the nanotubes during ventilation of the main chamber or the use of the argon gun. TABLE I. Extensive characterization of eight samples. RT resistance tunnel rates C1=2ptunnel rates C2=2pCharging energy Contact metal Geometry Quality factor (M X) (GHz) (GHz) (meV) Device 1 Au 3 gates 7400 0.08 0.5 16.5 1 Device 2 Au 3 gates 16000 7 0.3 253 7 Device 3 PdNi(25 nm)/Pd(4 nm) 5 gates 4000 2 … … …Device 4 PdNi(25 nm)/Pd(4 nm) 5 gates … 0.8 0.5 0.5 2Device 5 PdNi(25 nm)/Pd(4 nm) 5 gates … 0.23 0.5 67 1.5Device 6 PdNi(25 nm)/Pd(4 nm) 5 gates 4000 0.55 … … 2.2–2.4Device 7 Nb(45 nm)/Pd(10 nm) 3 gates … 0.2 33.6 3.7 /C210 /C041.0 Device 8 Nb(45 nm)/Pd(10 nm) 1 gate … 0.25–1.0 7 80 2.4Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 114001 (2020); doi: 10.1063/5.0021838 117, 114001-3 Published under license by AIP Publishingtune an electronic transition in the carbon nanotube to be resonant with the microwave cavity is, therefore, crucial for cQED applications. Importantly, we also applied this technique to multiple contact- ing materials. Table I gathers information on the eight samples investi- gated in this article, including the coupling rates obtained with various materials. Using this transfer technique on ferromagnetic PdNi(25 nm)/Pd(4 nm) contacts allowed us to induce local polariza- tion of electronic spin states in the nanotube and to couple it to cavityphotons. 23This technique can also be applied to superconducting con- tacts. In Fig. 3 , we observe a non-closing of Coulomb diamonds and shift of their apex characteristic of a superconducting contact on aquantum dot 24(see Fig. S3 of the supplementary for a zoom on a spe- cific Coulomb diamond). The size of the superconducting gap can be extracted from the distance between the apices. We find Dof the orderof 0.35–0.4 meV with large coupling rates (7 GHz and 80 GHz), using Nb(45 nm)/Pd(10 nm) contacts. We believe that the cleanliness of our quantum dot circuit and the compatibility with various contacting materials are partially due tothe fact that the transfer process is performed under vacuum, whichresults in a cleaner nanotube-metal interface. Transferring a carbon nanotube on top of circuit electrodes has proven to be a very efficient technique to integrate clean and sus-pended carbon nanotubes into complex circuit designs with the possi-bility to be part of a cQED platform. Here, we show that this approachcan also be adapted to hybrid circuits containing superconducting and ferromagnetic materials using commercial cantilevers, hence enlarging the scope of this technique. 25–30 See the supplementary material for extended data for the reso- nant case and its tunability. The theory of cavity-double quantum dotcoupling is briefly explained, and the corresponding measurementsare shown in Fig. S2. A close-up on specific Coulomb diamonds ofFig. 3 is also presented in Fig. S3 together with a cut at V G¼/C00:48V, which shows the superconducting gap. The devices have been made within the consortium Salle Blanche Paris Centre. The authors gratefully acknowledge helpfrom Jos /C19e Palomo, Aur /C19elie Pierret, and Michael Rosticher. This work was supported by the ERC Starting Grant “CirQys,” the ERCProof of Concept grant “QUBE,” the ANR “FunTheme,” and theQuantera grant “SuperTop.” DATA AVAILABILITY The authors declare that the main data supporting the findings of this study are available within this article. Extra data are available fromthe corresponding authors upon reasonable request. REFERENCES 1J. Waissman, M. Honig, S. Pecker, A. Benyamini, A. Hamo, and S. Ilani, “Realization of pristine and locally tunable one-dimensional electron systems in carbon nanotubes,” Nat. Nanotechnol. 8, 569–574 (2013). 2T. Sharf, J. W. Kevek, T. DeBorde, J. L. Wardini, and E. D. Minot, “Origins of charge noise in carbon nanotube field-effect transistor biosensors,” Nano Lett. 12, 6380–6384 (2012). FIG. 3. Differential conductance measured in device 7 vs bias voltage VSDand gate voltage VGat B¼0 T. The fact that the Coulomb diamonds do not close at VSD¼0 V is a manifestation of the superconducting contacts (see Table I ). This 2D map is spanning over a large range of gate voltages in the Coulomb blockade regime (more than 30 Coulomb diamonds) and shows continuous evolution of the contact transparency, which indicates an electrostatic control of the coupling rates. From the contrast of the cou- lomb diamonds, one can notice that in the gate voltage region: /C00.7 V <VG</C00.5 V, we have C1<C2, then for /C00.4 V <VG</C00.3 V, we have C1’C2, and finally for /C00.2 V <VG</C00.1 V, we have C1’C2. This observation further emphasizes the electrostatic control of the coupling rates. FIG. 4. Double quantum dot stability diagram measured on three devices. (a) Device 5 displays a narrow-gap behavior, and one can see the four different chargedistributions depending on whether the two dots are filled with electrons or holes.Carbon nanotubes (b) (Device 4) and (d) (device 6) are semiconducting, and only the electron–electron filling distribution is visible in the gate-gate plane. (c) For other gate voltage parameters of device 6, we observed a smooth transition from a singledot behavior (parallel lines in the top right region) to a double quantum dot behavior(anti-crossing in the bottom left region).Applied Physics Letters ARTICLE scitation.org/journal/apl Appl. Phys. Lett. 117, 114001 (2020); doi: 10.1063/5.0021838 117, 114001-4 Published under license by AIP Publishing3V. Sazonova, Y. Yaish, H. €Ust€unel, D. Roundy, T. A. Arias, and P. L. McEuen, “A tunable carbon nanotube electromechanical oscillator,” Nature 431, 284–287 (2004). 4B. Witkamp, M. Poot, and H. S. J. van der Zant, “Bending-mode vibration of asuspended nanotube resonator,” Nano Lett. 6, 2904–2908 (2006). 5B. Lassagne, D. Garcia-Sanchez, A. Aguasca, and A. Bachtold, “Ultrasensitive mass sensing with a nanotube electromechanical resonator,” Nano Lett. 8, 3735–3738 (2008). 6A. Cottet and T. Kontos, “Spin quantum bit with ferromagnetic contacts for circuit QED,” Phys. Rev. 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Sch €onenberger, “Clean carbon nano- tubes coupled to superconducting impedance-matching circuits,” Nat. Commun. 6, 7165 (2015). 12S. Blien, P. Steger, A. Albang, N. Paradiso, and A. K. H €uttel, “Quartz tuning- fork based carbon nanotube transfer into quantum device geometries,” Phys. Status Solidi B 255, 1800118 (2018). 13S. Blien, P. Steger, N. H €uttner, R. Graaf, and A. K. H €uttel, “Quantum capaci- tance mediated carbon nanotube optomechanics,” Nat. Commun. 11, 1636 (2020). 14I. Shapir, A. Hamo, S. Pecker, C. P. Moca, €O. Legeza, G. Zarand, and S. Ilani, “Imaging the electronic Wigner crystal in one dimension,” Science 364, 870–875 (2019). 15A. Benyamini, A. Hamo, S. V. Kusminskiy, F. von Oppen, and S. Ilani, “Real-space tailoring of the electron-phonon coupling in ultraclean nanotubemechanical resonators,” Nat. Phys. 10, 151–156 (2014). 16I. Khivrich, A. A. Clerk, and S. Ilani, “Nanomechanical pump-probe measure- ments of insulating electronic states in a carbon nanotube,” Nat. 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Collins, “Engineering carbon nanotubes and nanotube circuits using elec- trical breakdown,” Science 292, 706–709 (2001). 23T. Cubaynes, M. R. Delbecq, M. C. Dartiailh, R. Assouly, M. M. Desjardins, L. C. Contamin, L. E. Bruhat, Z. Leghtas, F. Mallet, A. Cottet, and T. Kontos, “Highly coherent spin states in carbon nanotubes coupled to cavity photons,” npj Quantum Inf. 5, 47 (2019). 24L. E. Bruhat, J. J. Viennot, M. C. Dartiailh, M. M. Desjardins, T. Kontos, and A. Cottet, “Cavity photons as a probe for charge relaxation resistance and photon emission in a quantum dot coupled to normal and superconducting continua,” Phys. Rev. X 6, 021014 (2016). 25M. Mergenthaler, A. Nersisyan, A. Patterson, M. Esposito, A. Baumgartner, C. Sch€onenberger, G. A. D. Briggs, E. A. Laird, and P. J. Leek, “Realization of a carbon-nanotube-based superconducting qubit,” preprint arXiv:1904.10132 (2019). 26M. M. Desjardins, L. C. Contamin, M. R. Delbecq, M. C. Dartiailh, L. E.Bruhat, T. Cubaynes, J. J. 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Lett. 117, 114001 (2020); doi: 10.1063/5.0021838 117, 114001-5 Published under license by AIP Publishing

5.0011134.pdf

APL Mater. 8, 091113 (2020); https://doi.org/10.1063/5.0011134 8, 091113 © 2020 Author(s).Strong spin-dephasing in a topological insulator-paramagnet heterostructure Cite as: APL Mater. 8, 091113 (2020); https://doi.org/10.1063/5.0011134 Submitted: 17 April 2020 . Accepted: 24 August 2020 . Published Online: 24 September 2020 Jason Lapano , Alessandro R. Mazza , Haoxiang Li , Debangshu Mukherjee , Elizabeth M. Skoropata , Jong Mok Ok , Hu Miao , Robert G. Moore , Thomas Z. Ward , Gyula Eres , Ho Nyung Lee , and Matthew Brahlek ARTICLES YOU MAY BE INTERESTED IN Recent advancements in the study of intrinsic magnetic topological insulators and magnetic Weyl semimetals APL Materials 8, 090701 (2020); https://doi.org/10.1063/5.0015328 Growth and characterization of ferromagnetic Fe-doped GaSb quantum dots with high Curie temperature APL Materials 8, 091107 (2020); https://doi.org/10.1063/5.0017938 Catalog of magnetic topological semimetals AIP Advances 10, 095222 (2020); https://doi.org/10.1063/5.0020096APL Materials ARTICLE scitation.org/journal/apm Strong spin-dephasing in a topological insulator-paramagnet heterostructure Cite as: APL Mater. 8, 091113 (2020); doi: 10.1063/5.0011134 Submitted: 17 April 2020 •Accepted: 24 August 2020 • Published Online: 24 September 2020 Jason Lapano,1 Alessandro R. Mazza,1 Haoxiang Li,1 Debangshu Mukherjee,2 Elizabeth M. Skoropata,1 Jong Mok Ok,1 Hu Miao,1 Robert G. Moore,1 Thomas Z. Ward,1 Gyula Eres,1 Ho Nyung Lee,1 and Matthew Brahlek1,a) AFFILIATIONS 1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 2Center for Nanophase Materials Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA a)Author to whom correspondence should be addressed: brahlekm@ornl.gov ABSTRACT The interface between magnetic materials and topological insulators can drive the formation of exotic phases of matter and enable function- ality through the manipulation of the strong spin polarized transport. Here, we report that the transport processes that rely on strong spin- momentum locking in the topological insulator Bi 2Se3are completely suppressed by scattering at a heterointerface with the kagome-lattice paramagnet, Co 7Se8. Bi 2Se3–Co 7Se8–Bi 2Se3trilayer heterostructures were grown using molecular beam epitaxy, where magnetotransport measurements revealed a substantial suppression of the weak antilocalization effect for Co 7Se8at thicknesses as thin as a monolayer, indicat- ing a strong dephasing mechanism. Bi 2−xCoxSe3films, in which Co is in a non-magnetic 3+state, show weak antilocalization that survives to higher than x= 0.4, which, in comparison with the heterostructures, suggests that the unordered moments of Co2+act as a far stronger dephasing element. This work highlights several important points regarding coherent transport processes involving spin-momentum locking in topological insulator interfaces and how magnetic materials can be integrated with topological materials to realize both exotic phases and novel device functionality. ©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/10.1063/5.0011134 .,s Topology has entered condensed matter physics and mate- rials science as a classification scheme that distinguishes materi- als based on a single number, the topological invariant.1In the case of Z2topological insulators (TIs),2this invariant is deter- mined by the ordering of the energy states of the band struc- ture. At the boundary between three-dimensional (3D) materi- als from different topological classes, new two-dimensional (2D) singly degenerate energy states can emerge that possess linear Dirac- like dispersion and exhibit strong spin-momentum locking. These states can be modified by breaking the time-reversal symmetry, which gives many routes to control their properties and the elec- tronic structure. Specifically, controllably opening a gap at the Dirac point (DP) can give rise to other exotic phases of topolog- ical matter, particularly the quantized anomalous Hall phase,3–7 as well as the states that were envisioned in the realm of parti- cle physics, for example, axion phases,8which can be realized inthe condensed matter platforms. Due to these unique properties, this broad class of magnetic and topological materials is extremely promising for the fundamental studies of the exotic phases of mat- ter, as well as applications ranging from spintronics to quantum computing.9–12 Numerous studies have focused on magnetically doping TIs to stabilize a ferromagnetic ground state13–17that has resulted in the observation of the quantum anomalous Hall phase.6Combining TIs and magnetic materials as heterostructures offers an alternate route over bulk doping. This creates a unique means of tuning the mate- rial as coupling between a ferromagnet and a TI can be achieved without excessive scattering off random magnetic impurities,18as well as enabling the creation of distinct interfaces at the top and bottom surfaces, which has been used to access axionic phases.19–21 Specific questions remain regarding the fundamental scattering pro- cesses at interfaces between TIs and magnetic systems, particularly APL Mater. 8, 091113 (2020); doi: 10.1063/5.0011134 8, 091113-1 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm how strong spin-momentum locking is modified in the presence of magnetic scattering centers and global magnetic ground states. Such questions are especially relevant for understanding the inter- faces with the magnetic materials that are electrically insulating, for example, yttrium iron garnet22and Cr 2Ge2Te6/CrGeTe 3,23,24as well as metallic interfaces, such as permalloy and Pt.25Here, we report that the spin-momentum-locked transport in the TI Bi 2Se3 is completely suppressed by scattering at the heterointerface with the kagome-lattice paramagnet, Co 7Se8. Despite the time-reversal symmetry being fully intact, these results highlight the sensitivity of the scattering processes involving spin-momentum locking at inter- faces with strong moments. This dramatic effect raises the question whether the topological properties are fundamentally compromised at the Co 7Se8interfaces as well as the two remote interfaces between Bi2Se3and the surface and the substrate. To answer this, we show that the topology is intact at the surface by the in situ angle-resolved photo-emission spectroscopy (ARPES). This work highlights the nontrivial interplay among the magnetic and topological materials, which can be used to realize both exotic phases and novel device functionality. Key to this study is utilizing molecular beam epitaxy (MBE; see the supplementary material for the experimental methods) growth to create atomically sharp interfaces with the TI; here, Bi 2Se3and the kagome paramagnetic material Co 7Se8. Bi 2Se3is a prototypical topological insulator having a relatively large bandgap of 0.3 eV and a simple band structure with a single spin-polarized surface Dirac cone at the Γpoint that is well separated from the 3D bulk bands.26–28 As shown in the structural models in Fig. 1(a), the unit cell of Bi 2Se3 is composed of three monolayers (ML) called quintuple layers (QL) that are roughly 1 nm thick. Co 7Se8has a monoclinic structure with a unit cell height of 0.52 nm and 2 Co ML per unit cell. The Co 7Se8 unit cell is characterized by an ordered-vacancy derivative of the hexagonal CoSe compound, in which 1/8 Co atoms are missing, creating alternating planes of close-packed hexagonal and kagome lattice structures,29as shown in Fig. 1(b). This yields Co with mixed2+and 3+valence states and a low-carrier density metal. Previous studies on the bulk powder samples of Co 7Se8reported paramag- netism, and in nano-crystalline powders, trace amounts of Co or other Co–Se impurity phases can contribute to a weak ferromag- netic signature.29–31However, the MBE growth typically results in highly uniform and homogeneous systems.32,33As shown in Fig. S1 of the supplementary material, Co 7Se8films exhibit paramagnetism with a saturation moment close to the expectation of 5/7 μB≈0.71 μBbohr magnetons per Co atom, which corresponds to 5 Co2+with 1μBand 2 Co3+with zero moment. As such, these respective mate- rials enable the application of strong magnetic disorder on the TI without breaking the time reversal symmetry, which allows a cleaner view of the mechanisms driving the spin polarized transport. The trilayer structures used here were composed of a 7 QL Bi 2Se3bot- tom layer, a variable thickness Co 7Se8layer ranging in the number of monolayers from n= 1–23 ML, and a 7 QL Bi 2Se3top layer, as schematically shown in Fig. 1(c) along with a corresponding scan- ning transmission electron microscopy (STEM) image in Fig. 1(d). This structure was chosen since the bottom Bi 2Se3layer served as a necessary nucleation layer for Co 7Se8, and the top Bi 2Se3layer preserved the inversion symmetry between the Co 7Se8interfaces. Since Bi 2Se3is in the regime where the bulk bands are occupied and Co7Se8is also metallic, both the top and the bottom surface states can interact with Co 7Se8through the free electrons in the bulk state of Bi 2Se3. As such, this heterostructure enables probing both the fundamental nature of spin-momentum locking and the scattering processes, as well as the topological character via ARPES on the top surface of Bi 2Se3. To understand the global structure of the trilayer heterostruc- tures, x-ray diffraction (XRD) studies were performed to probe crys- tallinity, morphology, and interfacial character. The 2 θ–θscans of the parent materials, Bi 2Se3, and Co 7Se8, as well as the trilayer het- erostructures for various Co 7Se8thicknesses are shown in Fig. 2(a). The Bi 2Se3sample was 14 nm, Co 7Se8was 10 nm, and for the tri- layer heterostructures, the Co 7Se8thickness ranged from n= 1–23 FIG. 1 . Materials and experimental schematic. (a) Crystal structure of TI Bi 2Se3showing the layered hexagonal structure consisting of three repeated quintuple layer (QL) units of Se–Bi–Se–Bi–Se, three of which make the unit cell. (b) Co 7Se8is characterized by an ordered vacancy structure forming a kagome network of Co atoms in layer 1 and a hexagonal structure in layer 2. This leads to a mixed valence of 2+and 3+states, which are shown at the bottom decorating the layers as red arrows and blue circles, respectively. The structure is formally monoclinic, however, the ⟨1¯10⟩can be thought of as the pseudohexagonal ⟨010⟩direction, showing the hexagonal arrangement between the two unit cells. [(c) and (d)] Schematic of the trilayer structure (c) and accompanying scanning high-angle annular dark field (HAADF) scanning transmission electron microscopy (STEM) image (d) for a Bi 2Se3(7 QL)/Co 7Se8(23 ML)/Bi 2Se3(7 QL) trilayer sample. APL Mater. 8, 091113 (2020); doi: 10.1063/5.0011134 8, 091113-2 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm FIG. 2 . X-ray diffraction (XRD) measurements and the simulations of trilayer het- erostructures and parent materials. (a) XRD 2 θ-θscans of the parent materials Bi2Se3and Co 7Se8, as well as trilayer heterostructures with the Co 7Se8thick- ness ranging from n= 1–23 monolayers (ML). The Bi 2Se3peaks are marked by square-symbols, the Co 7Se8peaks are marked by circles, and the Al 2O3peaks are marked by asterisks. (b) Results from simulated structures, including single- layer Bi 2Se3, bi-layer Bi 2Se3with differing c-axis lattice parameters labeled “lightly mixed” and “heavily mixed” (the expected splitting at large 2 θis labeled “Δ2θ”), as well as a Bi 2Se3heterostructure with spacing equivalent to 8 ML of Co 7Se8 labeled “ideal 8 ML” (see the supplementary material). X-ray diffraction of the doped samples can be found in Fig. S2. ML. The 2 θ–θscan of Bi 2Se3shows the 003 mseries of peaks ( m is an integer), which are highlighted by square-symbols. The peaks due to the Al 2O3substrate are marked with asterisks. For the Bi 2Se3 peaks, Laue oscillations due to coherent scattering off the top and bottom interfaces can be seen about the most intense peaks, which indicates that the films are extremely flat. The Co 7Se8film shows only the 001 and 002 reflections, as marked by circles. The lack of Laue oscillations indicate that the films are slightly rougher or, since the intensity of the oscillations scale with the intensity of the reflection, they are below the scan’s detection threshold due to the relative weakness of Co 7Se8001 and 002 peaks. The latter point is highlighted by comparing the 0012 reflection of Bi 2Se3at around 2θ≈37○, which has a similar magnitude at the 002 peak of Co 7Se8. Neither of these reflections show the Laue oscillations, which indi- cates that the films are likely of similar flatness. The 2 θ–θscansof the trilayer samples with the Co 7Se8thickness ranging from n = 1–23 ML, reveal a complex array of features. For the single mono- layer, the data look very similar to those of Bi 2Se3, only with some of the reflections slightly distorted and with weak oscillations that are superimposed on the main peaks. With an increase in the Co 7Se8- thickness, the Bi 2Se3peaks are further distorted, and these unusual oscillations become more prominent. Moreover, at 4 ML, the Co 7Se8 peaks clearly emerge and by 23 ML they are relatively sharp, which indicate that a significant portion of scattering emanates solely from Co7Se8. We further investigated the source of these intensity oscilla- tions utilizing the XRD simulations of the heterostructures, which, depending on n, exhibits two key features (1) strong interference at a low 2θ(15○–20○) and (2) weak or no interference at a higher 2 θ (45○–50○). These observations are fundamentally rooted in coher- ent interference for which there are two plausible origins (see the supplementary material and Ref. 34 for additional details). For both cases, when the films are thick, interference should qualitatively occur if reflections from both the parent materials are close in 2 θ. This occurs since the reflected x-rays can interfere constructively or destructively; away from these regions, no coherent scattering should be seen. In the first scenario, two pristine Bi 2Se3layers with sharp interfaces are separated by a thickness equivalent to n monolayers of Co 7Se8, representing an ideal structure. As shown in Fig. 2(b), the result is plotted for the n= 8 condition labeled “ideal 8 ML.” This accounts for the interference off the various interfaces and clearly reproduces the strong interference around 2 θ ≈15○–20○, as well as the weaker interference observed around 2 θ ≈45○–50○. In the second scenario, we assume interdiffusion of Co into Bi 2Se3and perhaps Bi into Co 7Se8which likely occurs asym- metrically if the Bi 2Se3/Co 7Se8interface is different from the subse- quent Co 7Se8/Bi2Se3interface.33,35This would be accompanied by a modification of the out-of-plane lattice parameter in the various lay- ers,36thus creating, for example, peak broadening or multiple Bi 2Se3 reflections very close in 2 θdepending on the degree of lattice dis- tortion and cobalt interdiffusion. However, interference should only occur at lower 2 θvalues where the overlap of the reflections would be maximum and would give way to a splitting of the peaks at higher 2θvalues. This is shown in the simulation in Fig. 2(b) labeled “lightly mixed” and “heavily mixed,” where the intensity is calculated for a bilayer structure composed of two Bi 2Se3of slightly different out- of-plane lattice parameters. For the lightly mixed case, a small (1%) variation in the lattice constant from bulk Bi 2Se3reproduced the reflections at high 2 θvalues but not the strong splitting at lower 2θvalues. For the heavily mixed sample, which had a larger (5%) variation in the lattice constant, there is some interference at low 2θvalues but transitions to a large degree of peak splitting at high 2θvalues. Moreover, the surface sensitive x-ray photoelectron spec- troscopy (XPS) was carried out on the trilayer with n= 8 ML and the doped samples and is shown in Fig. S4. No cobalt signature could be seen on the surface of the trilayer sample, confirming that there is a minimal diffusion of Co into the Bi 2Se3layers and that the sur- face is free of Co impurities. As such, this scenario can be eliminated. Considering this and returning to the experimental data in Fig. 2(a), comparing the trilayer structures to the simulated data reveals that only the ideal case closely matches with the experimental data at both high and low 2 θvalues, confirming the global structural quality of the trilayer heterostructures. APL Mater. 8, 091113 (2020); doi: 10.1063/5.0011134 8, 091113-3 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm Now that the structural quality has been confirmed, we move on to magnetotransport measurements. Conductance vs magnetic field is shown in Fig. 3(a) for the trilayer samples in comparison to the 14 nm Bi 2Se3(0 ML) film measured at 2 K. With an increase in the magnetic field, the conductance of the Bi 2Se3sample shows a sharp drop; at higher fields, the conductance transitions to a weak B2dependence, where Bis the magnetic field, which is the character- istic of the free-electron response. For the trilayer heterostructures, the sharp drop in the conductance is substantially suppressed even for the samples with the Co 7Se8thickness as thin as a single mono- layer, and by 4 ML, the conductance exhibits solely a B2dependence. The transport in TIs in a low magnetic field is characterized by a sharp cusp-like feature centered around B= 0 T.37The origin of this feature, called weak antilocalization (WAL), is due to the magnetic field suppressing coherent backscattering. In TIs, specifically, and, more generally, 2D systems with a strong spin–orbit coupling, WAL is due to a reduction in the probability to backscatter, which is driven by the accumulation of phases of the opposite sign for clockwise and counter-clockwise backscattering loops.38The application of a small magnetic field suppresses this effect, which manifests as a sharp drop in the conductance. The WAL effect can be quantified by fitting the experimental data using the Hikami–Larkin–Nagaoka (HLN) model for the change in the conductance, ΔG(B)=α(e2 2πh)[ln(B0 / B)−Ψ(1 2+B0 / B)], (1) in whichΨ(x) is the digamma function, his Planck’s constant, e is the electron charge, and the two free parameters are α, whichquantifies the number of 2D channels (we have redefined the pref- actor relative to the original HLN formula such that α→−1/2α, and thus, the number of channels corresponds to a positive inte- ger), and B0 /, the dephasing magnetic field.39The dephasing mag- netic field is related to the phase coherence length l0 /by the following equation: B0 /=–h 4el0 /2. (2) The data were fit using the HLN model with an additional B2 term to account for the free electron response. The fits are shown in Fig. 3(a) as the solid lines for Bi 2Se3and the trilayer with n= 1. The resulting l0 /andαare plotted vs Co 7Se8thickness in Figs. 3(b) and 3(c), respectively. For larger thicknesses, the low-field kink was absent, and therefore, we took both l0 /andαto be zero indicating that the transport processes involving spin-momentum locking are absent. For the single monolayer sample, WAL is still apparent, yet substantially suppressed. This implies that there is enough Co2+to be the dominating source of dephasing, yet not all electrons that encounter the interface with the single monolayer undergo such scattering; this could be either intrinsic, all Co are in a 2+valence state, yet the probability for an electron to encounter a Co2+is not 100%, or extrinsic, where some of the Co atoms are in a 3+ valence state due to slight intermixing thus reducing the probabil- ity of dephasing, or a combination of the two. Nevertheless, with an increase in the thickness the probability to encounter a Co2+ increases and WAL is completely quenched. This is surprising since the transport for TIs should be immune from such scattering events so long as the time reversal symmetry is intact. This immunity is FIG. 3 . Conductance vs magnetic field for the trilayer het- erostructures and doped samples. [(a) and (d)] Change in conductance vs magnetic field for the trilayer heterostruc- tures (a) and for Bi 2−xCo xSe3(d) where the data are shown as symbols and fit is the solid lines. Curves are offset for clarity. [(b), (c), (e), and (f)] Extracted α(number of conduc- tance channels) and lø(phase coherence length) param- eters from the HLN model are plotted for the trilayers [(b) and (c)] and Bi 2−xCo xSe3[(e) and (f)], respectively. The sin- gle Bi 2Se3layer is represented by the black square symbol, the trilayer by blue triangles, and Bi 2−xCo xSe3by the red circles. The dashed lines are guides to the eye. APL Mater. 8, 091113 (2020); doi: 10.1063/5.0011134 8, 091113-4 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm demonstrated by the magnetoconductance data shown in Fig. 3(d). These data were taken from Bi 2−xCoxSe3, where xranged from 0.07 to 0.50. For these films, it is assumed that Co takes a non-magnetic 3+ valence state and predominately replaces Bi3+. Over this range, the WAL effect can be seen clearly to survive to the highest doping range as the cusp in the conductance in the low field regime is maintained. Fitting to the HLN function shows that αis nominally constant and lϕdrops relative to the undoped materials. To understand the data for both the trilayer structures and the doped samples, it is instructive to compare the length scales for the scattering processes. For the trilayer samples, scattering off the Co 7Se8layer provides complete dephasing; thus, the min- imum length scale is then set by the Bi 2Se3thickness. Physically, this implies that all backscattering loops that contribute to the WAL effect contain at least one scattering event off the Co 7Se8 layer. For Bi 2Se3, where both the bulk and the surface states con- tribute to the WAL effect, scattering off the surfaces does not provide a dephasing mechanism since WAL is observed from the thick-limit of hundreds of nanometers down to the thin limit where surface scattering dominates the resistance.37,40,41Moreover, in Bi 2Se3/(Bi 1−xInx)2Se3/Bi2Se3trilayer heterostructures, there is a strong dependence on the doping levels and thickness of the (Bi 1−xInx)2Se3.42In contrast to this, in Bi 2−xCoxSe3, the length scale is approximately set by the average spacing of the dopants, which is of the order of a few nanometers for x≈0.1. Therefore, scat- tering due to Co defects should occur at a significantly higher rate than scattering off the Co 7Se8interfaces in the trilayer heterostruc- tures. Yet, WAL is clearly observed for the highest doped samples.It is emphasized that the Co should be in a 3+valence state with no net moment. This highlights the extreme sensitivity of the topo- logical insulators to the scattering processes involving the magnetic defects. As the WAL is so strongly suppressed, the question arises whether the trilayer system is topological. To answer this question, in situARPES was performed on a trilayer sample with a Co 7Se8thick- ness of 8 ML at a low temperature ( <10 K). The spectrum of this sample is shown on the left of Fig. 4(a). From this, the linearly dis- persive topological surface band is visible. This, and the degenerate Dirac point (DP), can be more clearly seen by processing the spec- trum with the curvature method described in Ref. 43, as shown on the right-hand side of Fig. 4(a) and the equipotential map at various energies shown in Fig. 4(b). To further illustrate the gapless nature of the DP, energy distribution curves (EDCs) are plotted in Fig. 4(c). The red curve indicates the EDC at kx= 0, which shows a single peak at∼300 meV that corresponds to the DP position in the 2D spectrum in Fig. 4(a). However, the ARPES measurement only probes the top surface of the trilayer heterostructure. The electronic structure of the interface between Co 7Se8and Bi 2Se3remains uncertain. This, how- ever, motivates the future question regarding the nature of the states that form at the interfaces of Co 7Se8. The strong magnetic moments of the Co will certainly interfere with the spin polarized states, but strong hybridization among the Co 7Se8states and the topological surface states may also inhibit its formation entirely. Since this inter- face is well below the escape depth for photoelectrons used here, this question will have to be addressed in the future in combination with first principles calculations. FIG. 4 . Angle-resolved photo-emission spectroscopy (ARPES) spectrum and equipotential maps for a trilayer heterostructure with 8 ML Co 7Se8thickness. (a) ARPES spectrum cut across the Γpoint along the kxmomentum direction. Left panel is the raw APRES spectrum while the right panel is the second derivative of the spectrum to enhance the peak intensity of the dispersion.43(b) Equipotential maps taken above and below the Dirac point, which is 300 meV below the Fermi level. (c) Energy distribution curves (EDCs) of the ARPES spectrum in panel (a). The red curve indicates the EDC across the Dirac point (DP) at kx= 0. APL Mater. 8, 091113 (2020); doi: 10.1063/5.0011134 8, 091113-5 © Author(s) 2020APL Materials ARTICLE scitation.org/journal/apm To conclude, we have shown that the coherent transport pro- cesses that rely on strong spin-momentum locking are completely suppressed in the topological material Bi 2Se3by embedding an epitaxial layer of a kagome-paramagnetic material, Co 7Se8, using MBE growth. Scattering off the magnetic Co2+in Co 7Se8is the primary source of the strong spin dephasing. This is in stark con- trast to Bi 2−xCoxSe3where such processes are found to survive the non-magnetic disorder due to Co3+even in doping levels higher than x= 0.4. In situ ARPES measurements show that, despite the absence of the spin polarized transport, the topological band structure on the top surface remains intact. Taken with a broader prospective, the current measurements have deep implications. One of the biggest impacts comes from device functionality that relies on spin-polarization. The topological materials are supposed to be immune to disorder that obeys time reversal symmetry. Although Co7Se8is not magnetic, it does have a net moment that causes com- plete spin dephasing of the charge carriers. As such, this demon- strates that in a device, the materials need to be carefully chosen to avoid anything with a net moment—specifically, paramagnetic substrates with a net moment should be avoided for the growth of topological materials. Alternatively, if control over spin dephas- ing is desired, strong moment materials, such as Co 7Se8, can be incorporated epitaxially with TIs. Further questions regarding the source of dephasing in topological systems necessitate the under- standing and quantifying the dependence of the scattering processes on the details of the interfaces. As an example, metallic materi- als with a net moment, like Co 7Se8, may be more susceptible to suppressing transport processes that rely on spin-momentum lock- ing than insulating materials with net moments. As such, proposals for realizing novel topological phases necessitate interfacing mag- netic metals, insulators, or both, highlighting the significance of these questions. This work brings about a broader view of trans- port processes that originate from spin-momentum locking in the topological materials, which is significant for realizing novel phases of matter at topological interfaces and for engineering topological devices. See the supplementary material for experimental methods and x-ray diffraction data, details regarding the simulations, and x- ray photoemission spectroscopy, which includes Refs. 34, 36, 44, and 45. This work was supported by the U.S. Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (transport, structural characterization, and MBE growth), as part of the Computational Materials Sci- ence Program (part of transport), and by the Laboratory Directed Research and Development Program of Oak Ridge National Labo- ratory, managed by UT-Battelle, LLC, for the U. S. Department of Energy (ARPES measurements). The electron microscopy work was conducted as a user project at the Center for Nanophase Materials Sciences, which is a U.S. DOE Office of Science User Facility. This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy. The United States government retains and the publisher, by accepting the article for publication, acknowledges that the United States government retains a non-exclusive, paid-up, irre- vocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for UnitedStates government purposes. The Department of Energy will pro- vide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/ downloads/doe-public-access-plan). DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010). 2J. E. Moore and L. Balents, Phys. Rev. B 75, 121306 (2007). 3C.-X. Liu, S.-C. Zhang, and X.-L. Qi, Annu. Rev. Condens. Matter Phys. 7, 301 (2016). 4F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). 5C. X. Liu, X. L. Qi, X. Dai, Z. Fang, and S. 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J. Vac. Sci. Technol. A 38, 063207 (2020); https://doi.org/10.1116/6.0000567 38, 063207 © 2020 Author(s).Photoluminescence, thermoluminescence, and cathodoluminescence of optimized cubic Gd2O3:Bi phosphor powder Cite as: J. Vac. Sci. Technol. A 38, 063207 (2020); https://doi.org/10.1116/6.0000567 Submitted: 19 August 2020 . Accepted: 02 October 2020 . Published Online: 16 October 2020 Mogahid H. M. Abdelrehman , Robin E. Kroon , Abdelrhman Yousif , Hassan A. A. Seed Ahmed , and Hendrik C. Swart Photoluminescence, thermoluminescence, and cathodoluminescence of optimized cubic Gd2O3:Bi phosphor powder Cite as: J. Vac. Sci. Technol. A 38, 063207 (2020); doi: 10.1116/6.0000567 View Online Export Citation CrossMar k Submitted: 19 August 2020 · Accepted: 2 October 2020 · Published Online: 16 October 2020 Mogahid H. M. Abdelrehman,1,2 Robin E. Kroon,1 Abdelrhman Yousif,1,2Hassan A. A. Seed Ahmed,1,2 and Hendrik C. Swart1,a) AFFILIATIONS 1Department of Physics, University of the Free State, Bloemfontein ZA 9300, South Africa 2Department of Physics, University of Khartoum, Box 321, Omdurman 11115, Sudan a)Electronic mail: Swart@ufs.ac.za ABSTRACT Cubic Gd 2−xO3:Bixphosphor powders were prepared with a combustion method and the effect of different annealing temperatures and dopant concentration on the photoluminescence (PL), thermoluminescence (TL), and cathodoluminescence (CL) were investigated. A single-phase cubic crystal structure with the Ia /C223 space group was formed. The average crystallite size increased and decreased, respectively, with an increased annealing temperature and an increased Bi3+doping concentration. Absorption bands at 250, 275, and 315 nm were observed due to 4f-4f transitions of the Gd3+ions and at 260, 335, and 375 nm due to the excitation of Bi3+ions. The emission was obtained from two centers associated with the substitution of the Gd3+ions with Bi3+ions at the two different sites in the crystal lattice of Gd 2O3 (with a point symmetries C 2and S 6). The TL glow curves of the UV-irradiated samples showed a low temperature peak at about 364 K and a high temperature peak at 443 K for all the samples. The surface and CL stability during electron irradiation was monitored. The CLemission of the Gd 2O3:Bi was stable after removal of surface contaminants. The phosphor might be usable for solid state lighting and displays due to its broad blue-green emission. Published under license by AVS. https://doi.org/10.1116/6.0000567 I. INTRODUCTION Rare earth oxides have proved to be excellent host materials for many luminescent activator ions as they have outstanding optical,mechanical, chemical, and electronic properties. 1Phosphors based on these materials can be used in devices such as white light emitting diodes, cathode ray tubes, high definition televisions, medical diag-nostic equipment, catalysts, and night glowing panels, guidinghighway traffic and to enhance the efficiency of solar cells. 2Among suitable rare earth oxide hosts having the cubic crystal structure, Gd2O3has the largest cation radius,3even though the ionic radius of VI-coordinated Gd3+(93.8 pm) is still slightly less than that of Bi3+ (103 pm).4Gd2O3has received considerable attention for optoelec- tronics, data storage, sensors, and display applications.5It has a high thermal conductivity (0.1 W cm−1K−1).6Since the phonon energies of cubic rare earth oxides decrease as the lattice constant increase,7 Gd2O3also has lower phonon energies than other cubic rare earth oxide hosts, which reduces the probability of detrimentalphonon-assisted non-radiative transitions. Although often grouped with the cubic rare earth oxides, Gd 2O3may also be found in the monoclinic phase under ambient conditions.8Its phase may depend on the synthesis conditions, e.g., the type of fuel used during com- bustion synthesis.9This is because Gd 2O3lies near the controversial transition between the cubic rare earth oxides and those with larger ionic radii having a monoclinic structure (or, for even greater ionic radii, hexagonal structure).10,11Figure 1 shows the crystal structure of cubic Gd 2O3as drawn using the VESTA program.12The cubic phase of Gd 2O3has a space group Ia /C223 and forms crystals with two types of Gd sites, each with a coordination number of 6 but with dif- ferent coordination geometries.13In the unit cell are 8 sites that are octahedrally coordinated with six oxygen atoms and another 24 sites that have four shorter and two longer bonds with O. Emission from undoped Gd 2O3is not usually reported, although Zatsepin et al.14attributed emission at 315 nm to Gd3+ ions associated with intrinsic defects, while numerous peaks in theARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-1 Published under license by A VS.ultraviolet and visible range have been recorded in isolated studies.9 Following some pioneering work,15–17there has been renewed interest in Bi doped Gd 2O3over the past decade. Some of the current interest originates from the prospect of using Bi as a codo-pant to improve the luminescence of other activators in Gd 2O3, e.g., Sm3+,18Tb3+,19Eu3+,19,20Nd3+,21,22and Yb3+,23although there is also significant interest in the luminescence from the Bi3+ions themselves. The luminescence from Bi3+ions is highly dependent on the host and also depends on the coordination characteristics ofthe activator site. 24Bi3+ions have a ground state electronic configu- ration of 6s2corresponding to one level (singlet1S0), while the excited 6s6p configuration is comprised of four levels: a triplet (3P0,3P1,and3P2) and singlet (1P1).25Absorption bands of Bi3+are usually attributed to1S0→3P1and1S0→1P1transitions,25although the latter may often lie at very short inaccessible wavelengths. Theposition and the splitting of the emitting 3P1level is sensitive to the surrounding lattice, resulting in the band profile and peak wave- length of Bi3+emission varying appreciably in different hosts. The luminescence of Bi3+in cubic Gd 2O3thin films has been reported to consist of a blue emission band peaked at 420 nm when excitedat 372 nm and a wide blue-green emission band with maximum at 502 nm when excited at 328 nm. 26Broad emissions covering both these bands were observed from electroluminescent devices fabri-cated with Gd 2O3:Bi thin film emitting layers.26Cubic Gd 2O3 features two types of Gd3+sites, both with coordination number of 6 but with different coordination geometries, i.e., where two O2− ions are missing either along the body (S 6) or a face (C 2) diagonal of a cube surrounding a Gd3+ion. When Bi3+occupies a S 6site having inversion symmetry in cubic Y 2O3, its Stokes shift is less compared to when it occupies a less symmetric C 2site having more neighboring ions to one side than the other27and the same effect is apparent for Gd 2O3.23The peak position for the blue-green emis- sion has also been reported at slightly longer wavelengths of515 nm, 20520 nm,23and 542 nm.21The luminescence of Bi3+ions in monoclinic Gd 2O3was investigated by Zou et al.5and was found to consists of a single emission peak, which varied only slightly from 449 –458 nm although the excitation was varied considerablybetween 319 and 370 nm. Recently, Zhang et al.28identified superbroad near-infrared luminescence between 800 and 1400 nm from mixed phase bismuth doped Gd 2O3(predominantly mono- clinic) which had been reduced, in addition to visible blue lumines-cence at 478 nm. Although infrared emissions have beenestablished to occur for bismuth ions reduced to lower oxidation states than Bi 3+,29it was instead suggested that the infrared emis- sion originated from Bi3+ions substituting Gd3+ions in monoclinic Gd2O3for which the coordination environment had been affected by oxygen vacancies. Thermoluminescence (TL) is a technique forstudying the defects in both insulators and semiconductor materi- als: when a sample is exposed to an ionizing source such as ultra- violet radiation or electron, proton, neutron, gamma, and ion beamradiation, the generated free electrons and holes can be trapped bydefects in the material. The consequent heating of the material canrelease some of these trapped carriers and recombination may then produce a TL signal. 30 Electron beam degradation studies have been reported for bismuth doped oxide hosts such as SrO25and La 2O331that demon- strated stability during electron irradiation. Also, electron beamdegradation has been reported by Swart et al. 32for Gd 2O2S:Tb3+ thin films, demonstrating that during prolonged electron bombard- ment some Gd 2O3phase was formed on the surface that helped to stabilise the cathodoluminescence (CL). In this context, a furtherstudy of Gd 2O3:Bi to include its CL and electron beam stability would be useful. In this work the photoluminescence (PL), TL, and CL of the optimized cubic Gd 2O3:Bi produced using the combus- tion method is presented and its CL degradation investigated. II. EXPERIMENT A. Sample preparation Powder samples of the host Gd 2O3as well as doped Gd 2−xO3: Bix(x = 0.0005, 0.0009, 0.002, 0.003, 0.004, and 0.006) phosphor were prepared by using urea-nitrate solution combustion synthesis. The concentration values of Bi in Gd 2O3reported were derived from the precursor mixture used. Gadolinium nitrate hexahydratewas used as the host precursor and urea for the fuel, since this hasbeen shown to produce cubic phase materials. 10All chemicals were purchased from Sigma Aldrich and used as received. To produce the host, the precursors were combined stoichiometrically accord- ing to the reaction 2 Gd(NO 3)3/C16H2Oþ5NH 2CONH 2 !Gd2O3þ5CO 2þ22H 2Oþ8N2 (1) by dissolving 0.01 mol of Gd(NO 3)3⋅6H2O and 0.025 mol of H2CONH 2in 50 ml of de-ionized water under stirring using a magnetic agitator. Bismuth nitrate pentahydrate was substituted forsome gadolinium nitrate hexahydrate to produce doped samples. The solution was heated at 100 °C in a beaker for 2 h to obtain a mixed aqueous homogeneous solution with no suspended particles.As the water content decreased, the solution gradually changed intoa transparent gel. The gel was placed in a furnace preheated to a fixed temperature of 600 °C to undergo combustion and produce the desired material. The authentic temperature of a combustion FIG. 1. Unit cell of cubic Gd 2O3.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-2 Published under license by A VS.reaction is generally much lower than the calculated theoretical as a result of various factors including the heat evolution of a large amount of gas, incomplete combustion of fuels, and loss of heat byradiation. It took about 5 min for the combustion process to com-plete, resulting in a foam-like crispy end products. The productswere allowed to cool to room temperature and were gently ground into fine powders using an agate mortar and pestle. Such as in many cases of combustion, synthesis needs to be followed by a cal-cination step to remove excess organic impurities from theas-synthesised powders. The samples were unformed in composi-tion and therefore portions of these as-prepared samples were also annealed in air at various temperatures (800, 900, 1000, 1100 and 1200 °C, respectively) for 2 h with an initial increasing heating rateof 5 °C/min starting from room temperature up to the requiredtemperature for each sample to be formed. In the combustionprocess, the combustion temperature is affected by the chemical composition of the mixture and also maybe the concentration of the dopant. A possible reason could be a stronger bond betweentransition metal oxides and the fuel depending on the chemicalcompositions, stronger or weaker metal-to-fuel interactions arepresent in the combustion mixture, leading to a stronger or weaker redox reaction between the fuel and the oxidant. Therefore, the combustion procedure must be adapted according to the chemicalcomposition of the product and to the concentration of the constit-uents, to avoid a too high ignition temperature and a too intense/ slow combustion. In our samples, Eq. (1) , the stoichiometric com- position mixture was calculated in a way that the total oxidizingand reducing valencies of the oxidizer and fuel were balanced andthe release of energy was maximized for each reaction. B. Characterization The Gd 2−xO3:Bixpowders were characterized using a Bruker D8 Advance powder diffractometer, a JEOL JSM-7800F scanning electron microscope (SEM), a Lambda 950 UV-vis diffuse reflec- tance (DR) spectrophotometer with a spectralon integrating sphereaccessory. An Edinburgh Instruments FS5 spectrophotometer witha 150 W CW ozone-free xenon lamp as an excitation source wasused for PL measurements. TL glow curves of the samples excited with 254 nm UV radiation for 10 min were recorded at a heating rate of 5 K/s using a TL 10091 NUCLEONIX instrument. The CLspectra were measured using both a Gatan MonoCL4 accessoryfitted to the JEOL JSM-7800F system as well as in a PHI 549 Augerelectron spectrometer using an Ocean Optics PC2000 spectrometer. Auger electron spectroscopy (AES) was performed simultaneously with CL measurements to monitor the relationship between CLdegradation and surface reactions during prolonged electronbombardment. III. RESULTS AND DISCUSSION A. Structure and morphology Figure 2(a) shows the XRD patterns of Gd 2−xO3:Bix = 0.002 annealed at various temperatures between 800 and 1200 °C. The XRD patterns are in agreement with the JCPDS standard No. 43-1014 of the cubic phase of Gd 2O3. The structure of cubic Gd 2O3 has been formed at the lowest annealing temperature and has notchanged with the increasing temperature. No monoclinic Gd 2O3or extra phases were observed and no shift occurred in the diffraction angles, which means that there was no change in the structure withthe increasing annealing temperatures of the Gd 2O3matrix. The average crystallite size was calculated by the Scherrer equation byusing the 2 θfull width at half maximum of the high-intensity dif- fraction peak. 33The calculated average crystallite sizes were 45.5, 47.1, 49.3, 53.7, and 54.3 nm for annealing at 800, 900, 1000, 1100,and 1200 °C, respectively. A slight increase in the average crystallitesize with increasing annealing temperature was observed.Figure 2(b) shows the XRD patterns of Gd 2−xO3:Bixpowders with x varying from 0 to 0.006 annealed at 1000 °C. The addition of Bi in the Gd 2O3matrix did not change the structure, as might be expected due to the similar ionic radii of Bi3+and Gd3+given earlier. A decrease in crystallite size with an increase in the Bi con-centration was observed, i.e., 51.6 nm for the undoped sample, 51.2 nm for x = 0.0005, 50.3 nm for x = 0.0009, 49.3 nm for x = 0.002, 49.1 nm for x = 0.003, 48.7 nm for x = 0.004, and 47.8 nmfor x = 0.006. The slight difference in the size of the radii may havecaused a strain field or stress that disturbed the grain growthprocess or some of the dopant might have formed a diffusion barrier at the grain boundaries. Figure 3 shows SEM images of the Gd 2−xO3:Bix = 0.002 phos- phor after annealed at the various temperatures. Figure 3(a) shows that the sample annealed at 800 °C has a non-uniform morphology of flake-like particles. Then when annealed at 900 °C, irregular microstructures were observed as in Fig. 3(b) . For an annealing temperature of 1000 °C, Fig. 3(c) shows a homogeneous micro- structure has formed with small grains that were highly agglomer-ated with an open porosity, where diffusion and growth as well as agglomeration to form bigger particles with size greater than 100 nm has occurred. When the temperature was increased to1100 °C, Fig. 3(d) displays irregular crystallites formed with larger particles having spherical and more longitudinal shapes. Thesample annealed at the highest temperature of 1200 °C shown in Fig. 3(e) has bigger spherical particles grown by diffusion with a diameter close to 400 nm. These changes in the morphology of thecubic Gd 2O3annealed at different temperatures prepared by solid- state reaction were also considered by Tamrakar et al.6who reported the increase in agglomeration of the smaller nanoparticles into bigger particles with an increasing temperature. Figure 4 shows the SEM images for the samples of Gd 2−xO3: Bixdoped at different concentrations, after annealing at 1000 °C. There is no systematic change in the morphology. B. Diffuse reflection spectra and bandgap calculations Figure 5(a) shows the DR measurements for the Gd 2O3host sample, as prepared and annealed at 800 and 1000 °C. The spectracontain several absorption bands in the near UV region. Thestrong band at about 227 nm corresponds to inter-band transitions of the Gd 2O3matrix.34The absorption bands at 275 and 315 nm can be attributed to the 4f-4f optical transitions of the Gd3+ion from the ground8S7/2to the excited6IJand6PJstates, respec- tively.35The weak absorption band near 250 nm is probably due to another 4f-4f transition of the Gd3+ion from8S7/2to6DJ, which is similar to that observed in other rare-earth doped Gd 2O3ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-3 Published under license by A VS.systems.36,37The DR data (R ∞) can be transformed into values proportional to absorbance by using the Kubelka –Munk equation38 F(R1)¼(1/C0R1)2/(2R1): (2) The optical bandgap energy was then calculated using the well-known Tauc law relation39 αhν¼C(hν/C0Eg)m, (3) where F(R∞) was substituted for the absorption coefficient ( a),hv is the incident photon energy, Cis a constant, Egis the optical bandgap energy, and m= ½ for a direct bandgap material such as Gd2O3. Plotting [ F(R∞)hν]2against hνand extending a linear fit to the energy axis as shown by Fig. 5(b) allowed the determination of Egas 5.09 for the as prepared sample and 5.15 eV and 5.18 eV for the samples annealed at 800 and 1000 °C, respectively. The optical bandgap increased when the annealing temperature increased andwas similar to the value of 5.09 eV from the literature. 35 Figure 5(c) shows the DR spectra of doped samples (x = 0.002) annealed at various temperatures. All the samples showed an absorption edge around 227 nm, which corresponds to the opticalbandgap of Gd 2O3. There are another three bands, located around 260, 335 and 375 nm, which were the same as the maximum PLluminescence excitation wavelengths as shown in Sec. III C ; there- fore, these bands are due to the excitation transitions of Bi 3+ions. The strength of these absorption bands increased when the temper-ature was increased to 1100 °C and then decreased at a higher tem-perature, to become almost similar to the host. This may be due toa loss in Bi 3+as volatile species at the higher annealing tempera- tures. Figure 4(d) shows the DR measurements for the various doped samples. The same four absorption band edges wereobserved and those associated with Bi 3+increased with the doping concentration as expected. C. Photoluminescence properties Figure 6(a) shows the effect of the annealing temperature on the PL excitation and emission spectra for Gd 2−xO3:Bix = 0.002 samples annealed at various temperatures. The spectra show the luminescence excited at 375 nm with the emission in the bluerange centered at 418 nm. The PL intensity increased with theannealing temperature up to 1000 °C and then decreased, as shown by the inset. The reason for the intensity increase up to 1000 °C may be the increased crystallinity of the samples. Zou et al. 5 FIG. 2. XRD patterns of (a) Gd 2−xO3:Bix = 0.002 after it was annealed at different temperatures. (b) Gd 2−xO3:Bixfor different doping concentrations after annealing at 1000 °C.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-4 Published under license by A VS.reported that the emission intensity of monoclinic Gd 2O3:Bi phos- phors was greatly enhanced by calcination at higher temperatures. The decrease in the intensity above 1000 °C may be due to the high volatility of Bi which causes it to vaporize and therefore leave thesample surface. Jafer et al. 39reported that the emission intensity of the luminescence of Y 2O3:Bi phosphor was strongly affected by the reaction temperatures and that a decrease in the luminescence intensity occurred due to the fact that the majority of the Bi ionsevaporated from the sample surfaces as volatile species. The effect of the Bi 3+concentration on the emission of the Gd2−xO3:Bixphosphors annealed at 1000 °C was also investigated. The PL excitation and emission for x = 0.0005, 0.0009, 0.002, 0.003, 0.004, and 0.006 are shown in Fig. 6(b) . The spectra show the emis- sion in the blue range centered at 418 nm when excited at 375 nm.For our samples, Fig. 6(b) shows that the PL intensity of the sample having x = 0.003 was the maximum. Bi 3+has excited states of3P0,3P1,3P2, and1P1in sequence of increasing energy, while the ground state electronic configuration[Xe] 4f 145d106s2has a single level1S0. Excitation peaks can occur for transitions from1S0to3P1,3P2and1P1which are usually denoted as A, B, and C, respectively.26The transitions1S0→3P0and 1S0→3P2are spin forbidden, while spin –orbit coupling of the1P1 and3P1levels means that the1S0→3P1transition is generally observed.26Fukada et al.26reported that intense blue PL emissions were observed from all Bi-activated Gd 2O3cubic phosphor thin films post annealed at a high temperature. The PL emission from the optimum Gd 2−xO3:Bix = 0.003 phosphor powder annealed at1000 °C was evaluated under 260, 335, and 375 nm excitation as shown in Fig. 7(a) , which were also the wavelengths of the absorp- tion bands noted earlier for DR spectra of the Bi3+doped samples. Either blue emission with a single broad peak at around 418 nm orblue-green emission consisting of broad peaks from 400 to 600 nmwas observed when exciting at 375 and 335 nm, respectively. For green emission at 505 nm, two excitation bands were observed in the 200 –400 nm range, with maxima at 260 and 335 nm. For blue emission at 418 nm three excitation bands were observed in the200 –400 nm range with the maxima at 260, 335, and 375 nm. When excited by 375 nm, the only emission was the blue emission at 418 nm, while when excited by 260 nm the same blue peak at 418 was observed with a low intensity. If excited at 335 nm UVlight, a similar emission was obtained as a shoulder with anotherextra broad green emission centered around 505 nm. 26The excita- tion band centered at 335 nm is a result of the excitation of the Bi3 +ion in the C 2site, while the band centered at 375 nm is a result of the excitation of the Bi3+ion in the S 6site.22The luminescence is similar to that for Y 2O3:Bi reported by Zou et al.22Due to the high sensitivity of the Bi to its environment, the emission spectra aredependent on the position of the Bi 3+in the Gd 2O3matrix. The blue and green emissions are ascribed to the Bi3+under S 6and C 2 symmetry, respectively. The energy level schemes of free Bi3+ions and the split energy levels under S 6and C 2symmetry are shown in Fig. 7(b) . The emission band at 418 nm can be assigned to the tran- sition3Eu→1Agof Bi3+at the S 6site. Thus, the excitation bands of the blue emission centered at 260 and 375 nm can be assigned to FIG. 3. SEM images of the Gd 2−xO3:Bix = 0.002 samples after annealed at the different temperatures for (a) 800 °C, (b) 900 °C, (c) 1000 °C, (d) 1100 °C, and (e) 1200 °C.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-5 Published under license by A VS.the transitions1Ag→3Auand1Ag→3Euof Bi3+ion at the S 6site, respectively. The green emission band centered at 505 nm can beascribed to the 3B→1A transition of Bi3+at the C 2site, while the transitions1A→3A and1A→3B of the Bi3+ion under C 2symmetry are responsible for the excitation bands at 330 –345 nm monitored at the green emission of 505 nm. We therefore interpret the shortwavelength shoulder of the broad Gd 2O3:Bi emission excited at 335 nm in Fig. 7(a) as originating from the upper3A→1A transition of Bi3+at the C 2site, with the emission at 505 nm (reported at 542 nm by Awater and Dorenbos40) therefore being associated not with a charge transfer emission, but originating from the3B→1A transition of Bi3+at the C 2site. Some overlap of the excitation bands of Bi3+at the S 6and C 2sites may also allow a partial contri- bution of emission at 418 nm from Bi3+at the S 6site to the shoulder observed in the emission excited at 335 nm. Lee et al.41report that the PL spectrum of cubic Y 2O3:Bi which was very similar to our emissions spectra from cubic Gd 2O3:Bi with a slight difference in wavelength positions with the same S 6and C 2symme- try in the structure. The report revealed two Bi3+emission bands, a narrow band centered at 409 nm and a broad band centered at 490 nm, originating from S 6and C 2sites that Bi3+ions may occupy within the Y 2O3host lattice. The emission at 409 nm showed two excitation bands, one centered at 330 nm and another at 390 nm.For the 490 nm emission, two excitation bands were also observed centered at 330 nm and 345 nm. Scarangella et al. 42reported that the PL spectrum of cubic Y 2O3:Bi thin films have wavelength posi- tions with S 6and C 2symmetry in the structure. When the emission wavelength is fixed at 406 nm, the two excitation peaks at 334 nm and 368 nm which associated to the1S0→3P1transition, where the 3P1excited state splits in the doublet [3Au,3Eu] typical of Bi3+in FIG. 4. SEM images of Gd 2−xO3:Bixsamples doped with different concentrations of Bi3+(x) (a) 0, (b) 0.0005, (c) 0.0009, (d) 0.002, (e) 0.003, (f) 0.004, and (g) 0.006, were annealed at 1000 °C for 2 h in air.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-6 Published under license by A VS.the S 6site. Therefore under 368 nm, corresponding to the1S0→3P1 (3Eu) transition of Bi3+in the S 6site, the sharp blue emission at 406 nm. When the emission wavelength is fixed at 500 nm, thebroad and asymmetric excitation band peaked at 330 nm it can beascribed to the same transition 1S0→3P1, where3P1splits in the triplet [3A,3B,3B] owing to the presence of the crystalline field and the occurrence of spin-orbit coupling when Bi3+is in the C 2site. The location of the cation within the host material is very impor-tant as it determines the optical and physical properties of the finalmaterial. But Y 2O3and Gd 2O3have similar crystal structures and therefore Bi3+doped in them has the same energy level structure for PL although they have different wavelengths of the excitation/emission bands. Thus, the cause for the changes in the emission intensity may be due to changes in the Bi3+when introduced into the host material due to lattice distortions caused by the differencein ionic radius between the Y 3+ion (90 pm), Gd3+ion (93.8 pm) and the Bi3+ion (103 pm),29because Bi3+ions are very sensitive to the surrounding environment, resulting in the different peaks wave- length of Bi3+emissions in different hosts. The tabulation of Bi3+ emissions in different hosts by Awater and Dorenbos40includes that Gd 2O3:Bi (considering the C 2site) excited at 347 nm has emis- sion bands at 425 and 542 nm which they attributed to the A band and charge transfer (CT) emission, respectively, with a similar scheme for the other Bi3+doped cubic oxides Y 2O3,L u 2O3and FIG. 5. DR spectra of (a) Gd 2O3and (b) Tauc plot to determine the bandgap. (c) DR spectra of Gd 2−xO3:Bix = 0.002 after different annealing temperatures, and (d) DR spectra of Gd 2−xO3:Bixsamples annealed at 1000 °C.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-7 Published under license by A VS.Sc2O3. In contrast, Scarangella et al.43used temperature dependent measurements of Bi3+doped Y 2O3excited at 325 nm to motivate that the two observed emission wavelengths, 430 nm and 500 nm,originated from the upper and lower levels of the crystal field split 3P1energy level of Bi3+ions at the C 2symmetry site.The Commission Internationale de l ’Eclairage (CIE) chroma- ticity diagram44was used to illustrate the color of the Gd 2−xO3: Bix = 0.003 phosphor under excitation at 260, 335, and 375 nm and is shown in Fig. 8 drawn using the Vista program. The color coordi- nates for the emitting phosphor are situated in the blue-green FIG. 6. PL excitation and emission spectra of (a) Gd 2−xO3:Bix = 0.002 produced using different annealing temperatures. The inset shows the maximum PL intensity of 418 nm as a function of annealing temperature, (b) Gd 2−xO3:Bixphosphor for different concentrations of Bi3+annealed at 1000 °C. The inset shows the maximum PL intensity (at 418 nm) as a function of Bi3+concentration. FIG. 7. (a) Emission and excitation spectra of optimized sample. (b) The energy level schemes of Bi3+ion. (i) Free Bi3+ion. (ii) The split energy levels under S 6symmetry. (iii) The split energy levels under C 2symmetry.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-8 Published under license by A VS.region and have significant variation with excitation wavelength. This is in contrast to monoclinic Gd 2O3:Bi which shows little variation of its emission wavelength when the excitation wavelengthis varied. 5D. Thermoluminescence studies TL measurements are used to investigate the defect levels in the bandgap of insulating or semiconducting materials. The traplevels react differently to different heating rates, due to the different characteristics of trapped charge carriers. 32Glow curves giving the intensity of light as a function of time or temperature may be usedto determine the traps created or activated in an emitting materialthat has been exposed to ionizing radiation. TL glow peak tempera-ture and intensity depend on various parameters such as the nature of the sample such as the structure of the sample, including the impurity content and crystallinity. Some factors influenced themeasurements, such as heat treatment given to the sample beforeirradiation, the nature of the ionizing radiation, the amount of irra-diation (dose), temperature at which the TL measurements are made, the time interval between the measurements, the environ- ment condition of the sample during TL measurements, the type ofdetector and the heating rate. 45 Figure 9(a) shows the TL glow curves of Gd 2−xO3:Bix = 0.002 which had been annealed at various temperatures, after exposure to UV radiation for 15 min. The TL was recorded at a heating rate of 5K s−1. Two broad peaks with maxima at about 365 K and 443 K were obtained for all the samples. These peaks are due to the pres-ence of V −centeres and oxygen vacancies46and no shift in the maximum peak temperatures was observed. The presence of oxygen vacancies causes some electronic states to appear above and below the Fermi level in the case of the bulk electronic structure forthe cubic Gd 2O3phases. So, maybe Bi3+make strong distortions of the periodic distribution of the electronic densities of states, because of the localized point-defects of the crystalline lattice. However, as shown in Fig. 9(a) , the structure of the TL glow curves is strongly affected by the annealing temperature. The maximumintensity was obtained for the sample annealed at 800 °C. With an FIG. 8. CIE coordinates for emission of optimized sample at different excitation wavelengths of 375 nm, 260 nm, and 335 nm. FIG. 9. (a) TL glow curves of Gd 2−xO3:Bix = 0.002 after different annealing temperatures. The inset shows the variation of the TL glow peak intensity at 364 K as a function of annealing temperature. (b) TL glow curves of Gd 2−xO3:Bixfor different concentrations of Bi3+annealed at 1000 °C. The inset shows the variation of the TL glow peak intensity at 364 K as a function of Bi3+concentration.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-9 Published under license by A VS.increase of the annealing temperature to 900 °C, the height of the low and main temperature peaks started to decrease, which may be attributed to the existence of trapping levels at different energiesbelow the conduction band. 47Then at 1000 °C, the height of the main peak increased again which may be attributed to the dissolu-tion of the defect complexes. Then, when the annealing tempera- ture was increased at 1100 and 1200 °C the height of the low and main temperature peaks starts to decrease which may be attributedto the increasing effect on the sensitivity of lower concentration ofBi in the Gd 2O3matrixes as a result of volatilization at high tem- peratures. The inset of Fig. 9(a) shows the variation of the TL glow peak intensity at 364 K as a function of annealing temperatures. It presented an irregular arrangement of the TL glow peak intensitywith annealing temperature, which means that there is a correlationbetween the TL area and the amount of energy released during thecombustion (heat of combustion). TL glow curves of undoped Gd 2O3and Gd 2−xO3:Bix (x = 0.0005, 0.0009, 0.002, 0.003, 0.004, and 0.006) annealed at 1000 °C and exposed to UV rays for 10 min are shown in Fig. 9(b) . For the undoped sample, the TL intensity was very small, but itincreased with Bi 3+concentration and reached a maximum at x = 0.002. It then decreased after further increasing of Bi3+concen- tration, as shown in the inset of Fig. 9(b) . When the Bi concentra- tion was increased, it caused an increase in the concentration of therecombination centers. Yousif et al. 30and Lyu and Dorenbos48 reported that Bi3+may act as both an electron and hole trap and Bi3+also can act as a deep electron trapping center in Y 2O3and YPO 4. In our samples, when increasing the Bi concentration maybe that is lead to a restructuring of traps in the Gd 2O3host crystal lattice was caused by the incorporation of Bi ions into the host lattice. This resulted into deeper traps, which suggest that the incorporation of Bi3+ions was responsible for the creation of addi- tional electron traps.30 To understand the TL glow curve behavior of the Gd 2−xO3: Bixit is important to evaluate the kinetic parameters. A glow-curve deconvolution procedure (GCD) was applied to the data analysis using the TLANAL program.49The TL light intensities for the first, second, and general order kinetic processes have been given byRandall –Wilkins, Garlick –Gibson, and May Partridge, 50respec- tively, and are expressed as I(t)¼nse/C0E/KT, (4) I(t)¼(n2/N)se/C0E/KT, (5) I(t)¼nbse/C0E/KT, (6) where Eis the activation energy for the TL process or trap depth, k is the Boltzmann constant, t is the time, Tis the absolute tempera- ture, sis a constant pre-exponential frequency factor (or attempt to escape frequency), n is the concentration of trapped electrons at time t, bis the kinetic order with values typically between 1 and 2, and Nis the total trap concentration. The integration ofEqs. (7) –(9)for a linear heating rate β, are, respectively,50,51 I(t)¼n0sexp (/C0E/KT) exp/C0s/βðT T0exp/C0E/KT/C18/C18/C19 dT/C18/C20/C21 , (7) I(t)¼n2 0s,exp (/C0E/KT)exp/C0s/βðT T0exp/C0E/KT/C18/C18/C19 dT/C18/C20/C21 , (8) I(t)¼s,,n2 0exp(/C0E/KT)1þs,,(b/C01)/βðT T0exp/C0E/KT/C18/C18/C19 dT/C18/C20/C21 (/C0b/(b/C01)) , (9) where βis the heating rate, T0is the initial temperature, n0is the initial number of filled traps, and the parameter s,,¼s,n(b/C01) 0.52 Two samples with Bi concentrations x = 0.002 and 0.003 annealed at 1000 °C were chosen since these have the maximum TL intensityand optimum PL emission and the results are presented in Fig. 10 andTable I . The samples showed two broad glow peaks and each glow peak consisted of more than one peak, five peaks for both samples x = 0.002 and 0.003, peaks located at 356, 380, 417, 447,and 480 K were fitted into the TL glow curve for both samples,which indicated that more than one trap was responsible for eachglow peak. The determined TL peaks that corresponded to the order of kinetics, trap depth (activation energy) values are tabulated inTable I . The similarity of certain peaks in the nanopowders samples suggests that the defects responsible for the traps may besimilar while having slightly different energy depths may be belowthe conduction band due to synthesis conditions and/or structural disorder. In Table I , it is noticed that the strong intensity peaks were attributed to the second-order kinetics which indicated thatthe retrapping probability was greater. 30The calculated frequency factors for all deconvoluted peaks are tabulated in Table I . The obtained values are closed to the normal lattice vibration in the materials.53The densities of traps were found to be a maximum for peak 5 for samples (0.002) and peak 1 for sample (0.002). Fromthese results, it was concluded that the trap depth and frequencyfactors are reasonably good for Gd 2O3:Bi under UV exposure and the high trap density peak may be suitable for dosimeter studies. E. Cathodoluminescence properties Figure 11 shows the CL spectra for the samples measured with the Gatan system connected to the SEM for an electron energy of5 keV in a vacuum of the order of 10 −5–10−6Torr. Figure 11(a) shows the CL spectra of Gd 2−xO3:Bix = 0.002 annealed at various temperatures. Similar to the PL emission, the intensity first increased for the samples annealed up to 1000 °C and thendecreased, as shown by the inset of Fig. 11(a) . The effect of doping concentration on the CL of Gd 2−xO3:Bixphosphors annealed at 1000 °C is given in Fig. 11(b) . The inset shows the variation of intensity as a function of Bi concentration and the sample doped with x = 0.003 has the highest CL intensity, as for the PL emission.The CL consists of two emission bands: a blue band centered at418 nm and a green band centered at 505 nm were observed for all the samples. As indicated in PL studies the blue and green emis- sions are ascribed to Bi 3+under S 6and C 2symmetry, respectively,ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-10 Published under license by A VS.suggesting that the electron beam did not change the electron energy level configuration or transitions of the activator ion in thephosphor. The broadening at longer wavelengths in the CL emis-sion may be due to the large energy difference as well as the differ-ent mechanisms for the excitation. High energy electrons would excite the Bi 3+ions at both sites in the host lattice. Abdelrehman et al.25reported that CL excitation sources energy have the ability to excite Bi3+ions as well as the host lattice. Scarangella et al.41 reported that CL excitation sources for Y 2O3:Bi thin films under electrical pumping have the ability to excite Bi3+ions at both lattice sites C 2and S 6excitation energy. There was no shift in the main peak position at 418 nm between the PL and CL maximum inten-sity positions in our samples which is due to no change the elec-tron energy level configuration of the Bi 3+ion in the phosphor. In Y2O3:Bi thin films, a slight peaks redshift (<10 nm) in the peak position between the PL and CL was reported,41which was attrib- uted to the presence of charge effects and local oxidationmechanisms in the Bi3+since the energy position of Bi is strongly dependent on the ion environment. F. Surface analysis and CL degradation Figure 12 shows the CL peak intensities of the Gd 2−xO3: Bix = 0.003 sample annealed at 1000 °C at 418 and 505 nm as a func- tion of electron dose in (a) a vacuum with a base pressure of1.3 × 10 −8Torr for around 100 h and (b) an oxygen partial pressure of 1.1 × 10−7Torr for around 40 h, excited using an electron beam of 2.5 keV energy in the Auger system. The CL spectra are shown by the insets before and after the degradation process. The intensityof the peak at 505 nm is higher than that at 418 nm, which differsfrom the spectra recorded for the same sample using the Gatansystem in the SEM at 5 keV. Since both these electron energies are orders of magnitude above the bandgap of the host material, it seems improbable to consider the differences in the CL spectra as FIG. 10. Glow curve deconvolution of (a) Gd 2−xO3:Bix = 0.002 and (b) Gd 2−xO3:Bix = 0.003 annealed at 1000 °C. TABLE I. Trapping parameters obtained from the TL glow curves of the Gd 2−xO3:Bix = 0.002 and the Gd 2−xO3:Bix = 0.003 samples. Sample Peak Peak position (K) Order of kineticsActivation energy (eV)Frequency factor (s−1)Trap density (arb. units) Gd2−xO3:Bix = 0.002 1 356 2 1.1 6.4 × 1083.3 × 106 2 380 2 1.2 5.7 × 1083.8 × 106 3 417 2 1.3 1.3 × 1093.4 × 106 4 447 2 1.3 3.9 × 1093.6 × 106 5 480 2 1.5 8.8 × 1094.5 × 106 Gd2−xO3:Bix = 0.003 1 356 2 1.1 6.6 × 1082.1 × 106 2 380 2 1.2 1.3 × 1092.2 × 105 3 417 2 1.3 5.9 × 1099.4 × 105 4 447 2 1.4 4.2 × 1091.2 × 106 5 480 2 1.5 2.9 × 1091.3 × 106ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-11 Published under license by A VS.due to different excitation processes or the interaction volume. However, there are very large differences in the beam currents anddiameters which may influence the excitation dynamics and the rel-ative abundance of phonons for the sample excited by the two systems. Further investigation of this for Gd 2−xO3:Bixand other Bi-doped oxide phosphors is planned. By simultaneously monitor-ing the CL intensity and Auger peak-to-peak heights (APPHs) overtime for 100 h in a vacuum and 33 h in O 2, the CL degradation andsurface chemical changes of the phosphor could be directly corre- lated. For vacuum Fig. 12(a) , the CL intensity for the peak at 418 nm reduced at around 150 C/cm2and stabilized thereafter, while the peak at 505 nm has slightly reduced up to an irradiation of about 450 C/cm2and then stabilized. With O 2inFig. 12(b) , taking into account the smaller values of electron dose on thehorizontal scale it is clear that the degradation rate was slightlyfaster, where the CL intensity for the peak at 418 nm reduced up to FIG. 11. CL emission spectra for (a) Gd 2−xO3:Bix = 0.002 obtained after annealing at different temperatures. The inset shows the maximum CL intensity as a function of annealing temperatures. (b) Gd 2−xO3:Bixphosphor for different concentrations of Bi3+annealed at 1000 °C. The inset shows the maximum CL intensity as a function of Bi3+concentration. FIG. 12. CL intensity as a function of electron dose exposure in (a) vacuum and (b) backfilled with oxygen. The insets represent the CL spectra before and after degradation.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-12 Published under license by A VS.50 C/cm2and then stabilized, while the peak at 505 nm slightly reduced up to an irradiation of about 150 C/cm2and then stabi- lized. A nonluminescent surface layer, which formed due to theelectron-stimulated surface chemical reactions (ESSCRs) afterremoval of the C and Cl from the surface maybe was responsiblefor the CL degradation. So, this layer it could be a layer with high defect density quenching luminescence, or maybe the surface layer becomes depleted of Bi or the Bi forms clusters in this layer whichin either case would reduce the Bi 3+emission. The peak positions at 418 and 505 nm and the shape of the CL spectra remained thesame before and after degradation, with only a small decrease in the intensity, especially in the vacuum where it decreased more than in O 2. The CL intensity in O 2decreased more quickly in thebeginning and then it settled continuously so that at the end of the measurements it was only about 70% of the initial value. Figures 13(a) and 13(b) show the Auger spectra of the Gd2−xO3:Bix = 0.003 sample annealed at 1000 °C before and after electron beam exposure in (a) vacuum and (b) backfilled withoxygen. The presence of the elements of Gd 2O3, namely, Gd at the lower energy range of 50 –170 eV and O at 510 eV54were con- firmed, but Bi was not observed due to its low concentration. Cland C were detected (at 188 and 272 eV, respectively) before degra-dation and were attributed to adventitious impurity species on thesurface due to handling and exposure to atmospheric pressure. The corresponding Auger peak-to-peak heights (APPHs) as a function of electron dose are presented in Figs. 13(c) and13(d) . During the FIG. 13. Gd2−xO3:Bix = 0.003 powder before and after electron-beam bombardment: AES spectra in (a) vacuum and (b) backfilled with oxygen, APPHs in (c) vacuum and (d) backfilled with oxygen.ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-13 Published under license by A VS.period of electron beam exposure, the C and Cl Auger signals decreased or were almost eliminated as these elements were con- verted to volatile species (such as CO and CO 2),55which indicated that the contaminants were only present on the sample surface. Ingeneral, ESSCRs occur when the electron beam dissociates themolecular O 2and other residual gases that adsorb on the surface from molecular species to atomic/ionic species that will than react with C and Cl to form volatile compounds (CO x,C H 4, etc.).56The surface impurities (Cl and C) reduced at a faster rate in the O 2 atmosphere compared to the vacuum base pressure because thereaction of O 2with the adventitious species occurred at a higher rate to form volatile compounds for the irradiated sample.57Swart et al.58reported that the removal of carbonates from the surface of phosphors during degradation may occur due to ESSCRs. After300 C/cm 2, no further degradation reactions occurred because the APPH signals remained constant after the ESSCR mechanism removed Cl and C completely as shown in Fig. 13 . The Auger signals associated with the host elements (Gd and O) are expectedto increase when C was removed from the surface, as reported forthe degradation of oxides component such as SrO:Bi, 25ZnO55,58 and La 2O3:Bi.31Fig. 13(c) for vacuum shows an increase in the APPH of Gd and O after an electron dose of about 30 C/cm2, when most of the Cl and C contaminations have been removed from thesurface. Figure 13(d) APPH in O 2shows an increase in the decay rates of the Gd and O peaks very fast during oxygen uptake after an electron dose of about 30 C/cm2indicating oxygen-induced more segregation of Gd which maybe was then creating derivativesof Gd 2O3. Swart et al.57report that during the degradation of ZnS, the Zn increased during oxygen uptake, indicated oxygen-inducedsegregation of Zn which was then converted into either ZnO or ZnSO 4. Also, Swart et al.59reported that a thin SrO surface layer formed during electron bombardment of SrAl 2O4:Eu2+,Dy3+in oxygen pressure. The CL degradation can also be explained by theESSCRs as reported by Swart et al., 59Hasabeldaim et al.,58 Abdelrahman et al.,25and Jaffar et al.31Abdelrehman et al.25 reported that some changes in the surface character and the pres- ence of defects played a crucial role in the CL emission and degra-dation of SrO:Bi. The presence of surface contamination such as Cand Cl initially severely affected the CL degradation. Also maybethe CL emission intensity degrades due to the formation and possi- bly diffusion of point defects during electron beam irradiation. The sample became stable after some initial effects when C and Cl wereremoved under long term electron bombardment, which makes itan excellent candidate for applications in field emission displays. IV. CONCLUSION Gd 2−xO3:Bixpowders were successfully synthesized and char- acterized. The XRD results showed that the samples were success-fully prepared by the combustion method and that the single cubiccrystal structure phase with an Ia 3 space group was formed for the different annealing temperatures and different concentrations of Bi3+. The average crystallite size and bandgap increased slightly with increasing annealing temperature, while there was a decreasein the crystallite size with an increase in the Bi doping concentra- tion. The DR spectra for the undoped sample showed two bands located around 275 and 315 nm that were attributed to the 4f-4foptical transitions of Gd 3+, while for the doped samples three bands located around 260, 335, and 375 nm were observed due to the excitation transitions of Bi3+ions into the different sites (C2and S 6). The optimum annealing temperature for the maximum PL and CL intensity of the Gd 2−xO3:Bixwas 1000 °C. The intensity first increased due to increased crystallinity of samples and then decreased for the highest annealing temperatures of 1100 and 1200 °C due to a decrease in the dopant concentrationdue to the formation of volatile species. The optimum Bi 3+concen- tration for the maximum PL intensity of Gd 2−xO3:Bixwas found to be at x = 0.003, while for the higher Bi3+doping levels concen- tration quenching occurred for the PL intensity. The PL and CL results showed that the phosphor has two emission bands centeredat 418 nm for blue emission and a third band centered at 505 nmfor the green emission. The excitation spectra showed 3 bands thatcorrespond to the 1S0→3P1transitions in the Bi3+ion in the two different sites (S 6and C 2). The excitation to the two main levels at 335 and 375 nm resulted in blue emission for the S 2site and only broad green emission for the C 2site. TL glow curves of the UV-irradiated Gd 2O3:Bi samples exhibited second-order kinetics. Two broad peaks with maxima at about 365 K and 443 K in the TL intensities for the different temperature ’s dopants in the Gd 2O3was observed. The TL kinetic parameters were calculated using the glowcurve shape method. The CL intensity for both the blue and greenpeaks only slightly reduced during prolonged electron bombard- ment. A nonluminescent surface layer, which formed due to the ESSCR after removal of C and Cl from the surface, was responsiblefor the CL degradation. Except for the initial degradation, Gd 2O3: Bi powder was found to be stable under electron irradiation in boththe base vacuum and back-filled O 2environments. From these results, it can be inferred that the phosphor is an excellent candi- date for application in field emission displays and quite suitable forapplications in high radiation environments. ACKNOWLEDGMENTS The authors express their sincere thanks to the South African Research Chairs Initiative of the Department of Science andTechnology and the National Research Foundation of South Africa (No. 84415). The financial assistance from the University of the Free State, South Africa is highly recognized. REFERENCES 1H. Guo, Y. Li, D. Wang, W. Zhang, M. Yin, L. Lou, and S. Xia, J. Alloys Compd. 376, 23 (2004). 2S. K. Singh, K. Kumar, and S. B. Rai, J. Appl. Phys. 106, 093520 (2009). 3E. 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M. Mothudi, Phys.B 407, 1664 (2012).ARTICLE avs.scitation.org/journal/jva J. Vac. Sci. T echnol. A 38(6) Nov/Dec 2020; doi: 10.1116/6.0000567 38,063207-15 Published under license by A VS.

1.5131467.pdf

J. Appl. Phys. 128, 124101 (2020); https://doi.org/10.1063/1.5131467 128, 124101 © 2020 Author(s).Structural, ferromagnetic, electrical, and dielectric relaxor properties of BaTiO3 and CoFe2O4 bulk, nanoparticles, and nanocomposites materials for electronic devices Cite as: J. Appl. Phys. 128, 124101 (2020); https://doi.org/10.1063/1.5131467 Submitted: 14 October 2019 . Accepted: 31 August 2020 . Published Online: 25 September 2020 Syed Adnan Raza , Saif Ullah Awan , Shahzad Hussain , Saqlain A. Shah , Asad M. Iqbal , and S. Khurshid Hasanain ARTICLES YOU MAY BE INTERESTED IN 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/10.1063/5.0015281 First-principles study on effects of local Coulomb repulsion and Hund's coupling in ferromagnetic semiconductor CrGeTe 3 Journal of Applied Physics 128, 123901 (2020); https://doi.org/10.1063/5.0015566 Coherent superconducting qubits from a subtractive junction fabrication process Applied Physics Letters 117, 124005 (2020); https://doi.org/10.1063/5.0023533Structural, ferromagnetic, electrical, and dielectric relaxor properties of BaTiO 3and CoFe 2O4bulk, nanoparticles, and nanocomposites materials forelectronic devices Cite as: J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 View Online Export Citation CrossMar k Submitted: 14 October 2019 · Accepted: 31 August 2020 · Published Online: 25 September 2020 Syed Adnan Raza,1,2 Saif Ullah Awan,3,a) Shahzad Hussain,4Saqlain A. Shah,5Asad M. Iqbal,2 and S. Khurshid Hasanain2 AFFILIATIONS 1Centro Brasileiro de Pesquisas Físicas (CBPF), 22290-180 Rio de Janeiro, Brazil 2Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan 3Department of Electrical Engineering, NUST College of Electrical and Mechanical Engineering, National University of Sciences and Technology (NUST), Islamabad 44000, Pakistan 4Department of Physics, COMSATS University Islamabad (CUI), Islamabad 44000, Pakistan 5Department of Physics, Forman Christian College (University), Lahore 54590, Pakistan a)Author to whom correspondence should be addressed: saifullahawan@ceme.nust.edu.pk andullahphy@gmail.com ABSTRACT Barium titanate (BaTiO 3, BTO) and cobalt ferrite (CoFe 2O4, CFO) nanoparticles, bulk, and nanocomposites samples were synthesized at optimized parameters using the chemical route. Structural studies revealed that all the samples showed a single-phase structure. The value of activation energy in the case of nanocomposites was 975 meV, while it was 1.58 eV and 1.0 eV for BTO and CFO-nanoparticles, respectively. We observed that the saturation magnetization and remanence of the CFO-nano sample were three times greater than the0.3CFO –0.7BTO nanocomposite. At 10 kHz, the dielectric constants are measured 4500 (BTO-bulk), 1550 (BTO-nano), 820 (CFO-bulk), 275 (CFO-nano), and 375 (BTO –CFO nanocomposites) for a various sample of series. We found at 10 kHz, the transition from ferroelectric to paraelectric in the case of BTO-nano (T c= 363 °C) and CFO-nano (T c= 212 °C), while nanocomposite BTO –CFO initial change phases from ferroelectric to anti-ferroelectric (relaxor behavior) at T d= 312 °C and then from anti-ferroelectric to paraelectric (T m> 400 °C). Similarly, in the case of CFO-bulk, we noticed T m= 204 °C (ferroelectric to anti-ferroelectric) and T m= 314 °C (anti-ferroelectric to para- electric). Overall, we concluded from these studies and data that nanocomposite 0.7BaTiO 3–0.3CoFe 2O4and CFO-bulk sample showed relaxor behavior as well along with the transformation change from ferroelectric to paraelectric. While BTO-nano and CFO-nano and BTO- bulk showed their transformation direct from ferroelectric to paraelectric. These simultaneous ferromagnetic and ferroelectric (dielectric) measurements confirmed the presence of multiferroic properties in our nanocomposite as well as CFO-nano and CFO-bulk systems. Theseproposed materials may be useful for ferroelectric and data storage devices. Published under license by AIP Publishing. https://doi.org/10.1063/1.5131467 I. INTRODUCTION Multiferroic materials possess coexisting bifunctionality of ferroelectric and ferromagnetic properties, e.g., magnetizationand polarization order. 1Over recent years, researchers have shown a huge interest in the field of multiferroics with various other orders, e.g., ferrotoroidicity and ferroelasticity.1–3In a fewmultifunctional materials, both magnetic and electrical polariza- tions can be optimized providing sufficient opportunities for devices ’manipulations. This coupling can permit the direct control of the magnetization by an electric field and vice versa.2 Ferromagnetics (FM) and ferroelectrics (FE) having different cou- pling orders,3either macroscopically through grain boundariesJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-1 Published under license by AIP Publishing.magnetostriction or microscopically via exchange striction,4make them potential materials for applications in memories, sensors, actuators, and spintronics.5,6They also offer great opportunities to study essential aspects of spin-lattice coupling effects. In reality,managing the magnetization of FM materials with electric fields viathe interaction called magneto-electric (ME) coupling may lead to the construction of electrically addressable magnetic memory devices. 7Unfortunately, very few materials possess an electrical polarization and a net magnetization simultaneously. Instead, mostelectric and magnetic based multiferroics, e.g., bismuth ferrite(BiFeO 3, BFO), is antiferromagnetic with ferroelectric behavior where the two multiferroic orders are coupled and this type of oxide system shows multiferroic properties.8There are very few natural multiferroic ME systems that are both FM and FE in thesame phase. The multiferroic composites of diverse compoundsconsist of the FM phase (also piezomagnetic) and the FE phase (also piezoelectric). Through the stress intervention, there is a cou- pling between the two order parameters, i.e., a magnetic fieldinduces a distortion of the piezomagnetic phase, which in turn dis-torts the piezoelectric phase in which an electric field is produced.The ME effect is extrinsic in this case since this effect is not dem- onstrated by any of the component phases on their own. Multifunctional ME materials are potential candidates for industrial use since the multiferroic coupling allows the intercon-nection between magnetic and electric fields. New memory devices have been proposed that will be electrically written and magneti- cally read based on ME materials. 1The ME composition can be synthesized as single or complex phase materials by coalescing ofmagnetostrictive and piezoelectric properties. Composites of MEmaterials can yield a giant response while the single-phase materi- als exhibit weak ME coupling. The ME coupling in the composite materials depends strongly on their microstructure. 2,3Many inher- ent preparation problems as atomic diffusion and undesirablechemical reactions between phases can affect the interface, modify-ing its chemical and structural properties and thus lowering the coupling. 4Composite ME multiferroic materials, which contained FE and FM phases, show much superior ME results compared tosingle-phase materials. In the composites, the ME outcome is natu-rally produced during strain occurring under a functional magneticor electric field at the boundary between the two components. 5 Nanostructured materials are particularly important where a highconcentration of boundaries may improve the ME coupling. Relaxors, also known as “relaxor ferroelectric ”materials, sug- gested an extensive variety of valuable properties that make them elegant for numerous extraordinary technological devices and applications. 6,7These relaxor ferroelectrics can be used in sensors, actuators, and resonant wave devices such as radio-frequency filtersdue to high piezoelectric effects, can be used in non-volatile memo-ries due to ferroelectric hysteresis behavior of relaxor, are used in capacitors due to high permittivity, used in infra-red detectors due to high pyroelectric coefficients, used in optical switches due tostrong electro-optic effects, and used in electric-motor overloadprotection circuits due to anomalous temperature coefficients ofthe resistivity. 6–8 The ferroelectricity is often observed in the perovskite struc- tured materials, e.g., BTO (barium titanate BaTiO 3). The ferroelec- tricity itself is thoroughly related to the intrinsic disturbancesassociated with the perovskite structures. As a typical ferroelectric material, BTO is also used for positive temperature coefficient resis- tors, multilayer ceramic capacitors, high-density optical datastorage, piezoelectric devices, ultrasonic transducers, thermistors,and electro-optics 9–12applications. There are two main classes of cubic ferrites, spinel ferrites, and garnet ferrites. The spinel ferrites are further divided into three types on the bases of sites ’occupa- tion: (i) normal spinel ferrites, (ii) inverse spinel ferrites, and (iii)mixed spinel ferrites. Cobalt ferrite (CoFe 2O4, CFO) belongs to the family of spinel ferrites. This material (CFO) is a hardmagnetic material with high coercivity and magnetization that makes CFO suitable for magnetic recording applications, 13–15spin- tronics devices, composite multiferroic hetero-structures,16and magneto-elastic devices.17As we discussed above, BTO is a strong ferroelectric material with large piezoelectricity, and CFO is ferri-magnetic with large magnetostriction. Composites of BTO –CFO combine the ferroelectricity of BTO and ferrimagnetism of CFO simultaneously. A significant benefit of BTO –CFO composites is the spinodal disintegration of this dual system, which avoids reac-tion between the ingredients during the elevated temperaturemethod. It is well-recognized that ferroelectric properties of BTO and ferromagnetic properties of BTO depend on particle size. Likewise, one should imagine the size effect on the ME coupling inthe composites. 18 The advanced nanocomposite system (CFO –BFO)19and the magneto-dielectric system (e.g., nanocomposite system CFO – BFO)20can be invented as an innovative material with multiferroic properties. These ME systems are frequently mixtures that have allthe possible properties of their parent FE and FM materials.Moreover, these systems have bifunctional properties not existing in material with a single order parameter. The exploration of these materials is motivated by the vision of monitoring spin motion bythe applied voltage and electric charge by applied magnetic fieldsand expending this to build novel procedures of multifunctionaldevices. 21In previous reports, diverse roots of ferroelectricity have been proposed, e.g., lone-pair electron effects, charge collection, geometrical prevention, octahedral distortion, magneticallyattracted ferroelectricity, strain mediation, etc. On the other side, amagnetic phenomenon is associated with assembling of unpairedspins of electrons in inadequate ionic orbitals, that is, it initiates from the moderately occupied d-orbitals. Single-phase multiferroic is rarely reported due to the contradictions revealed above betweenthe conventional mechanisms in ferroelectric materials that wantbare d-subshells and development of net magnetic dipole spins that are affected due to incomplete d-subshells. Otherwise, the construc- tion of non-natural multiferroics provides the opportunity ofadapting the properties using two diverse complexes, one being fer-roelectric and the other being ferromagnetic. The purpose of thismethod is to produce systems that demonstrate properties of the parent mixtures and their coupling. 22,23 Due to the coexistence of both FM and FE in the same mate- rial, it is predictable to display either these properties or couplingof these two properties in a single material. In this paper, we havestudied a multiferroic composite since the technology demand of multiferroic materials for device applications. For this purpose, we choose a well-known FE material BTO with large piezoelectric andstrong magnetic material CFO with the composition of 70% andJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-2 Published under license by AIP Publishing.30%, respectively. When a magnetic field is applied to the compos- ite, there is stress generated by the CFO due to its ferromagnetic properties. Such stress can create an electric field in the BTO dueto its piezoelectricity. The reverse process is also possible in whichan electric field applied to the composite produces stress due to thepiezoelectric response of the BTO. The stress induces a magnetic field in the CFO due to its magnetostriction. The aim of this is to investigate the structural, ferromagnetic, electrical, and dielectricproperties of the bulk and nanoparticles of BTO and CFO for com-parison with the multiferroic nanocomposite BTO –CFO system. Our main objective is to study and investigate the variation of dielectric constant with increasing the temperature at specific lower (10 kHz) and higher (400 kHz) frequency ranges for varioussamples. From these dielectric measurements, we were able toinvestigate the Curie temperature (T c) from ferroelectric to para- electric transitions for the variety of samples and compare the T c values for bulk, nanoparticles, and nanocomposites. II. EXPERIMENTAL PROCEDURE We prepared the ceramics (bulk), nanoparticles, and nano- composite samples of BTO and CFO systems with solid-state and with co-precipitation techniques. For the synthesis of barium tita- nate (BaTiO 3) bulk (onward call it BTO-bulk sample), powders (BaCO 3and TiO 2) were mixed and grinded in an agate pestle with mortar. Mixed powders were then heated at 1150 °C. This processwas performed in three steps, i.e., grinding for 2 h, heat treatment for 4 h, then the same was repeated with 2 h grinding, and finally with 1 h of grinding. After calcination at 1150 °C, the powder waspressed in disk-shaped pallets of 13 mm diameter and 0.6 mm – 0.75 mm thickness, and for this purpose, we used the hydraulicpress at a pressure of 5 tons/in. 2for 1 h. These pallets were then sintered at 1350 °C for 4 h. After sintering, the pallets were polished on both sides and silver paste was applied on both sides to be usedas electrodes for electrical measurement. A sample of ceramic cobalt ferrites CoFe 2O4bulk (onward call it CFO-bulk) was prepared by the conventional sintering method. Ceramic powders of cobalt oxide (Co 3O4) and iron oxide (Fe 2O3) were used in this reaction. Both reactants were mixed with the helpof pestle and mortar for 2 h. Mixed powders were then heated at950 °C. This process was done in two steps, i.e., grinding for 2 h and then heat treatment of 4 h .Finally, the powder of CoFe 2O4was pelletized in disk-shaped pallets of 13 mm diameter and 0.6 mm – 0.75 mm thickness, and for this purpose, we used th hydraulicpress at a pressure of 5 tons/in. 2for 1 h .Then final sintering was performed at 1350 °C for 4 h. In the present work, we have successfully synthesized barium titanate (BaTiO 3) nanoparticles (onward call it BTO-nano) using the oxalate route. Initially, the stoichiometric amounts of titaniumisopropoxide Ti(OC 3H7) 3.8152 ml solution were prepared. Then, the 0.5M solution of oxalic acid H 2C2O4.2H 2O 3.1516 g/50 ml was prepared in isopropanol. After mixing in a single flask, both solu- tions of titanium isopropoxide and oxalic acid were heated at 60 °Cunder constant magnetic stirring for 1 h. Barium chloride BaCl 2 (aq) was used as a precipitating agent. The 0.5 M solution of barium chloride 3.0533 g/ml was prepared in distilled water. After attaining a specified temperature of 60 °C, the precipitating agentsolution was added dropwise at a very slow rate. The solution was heated at 60 °C under constant magnetic stirring until the solution got converted into a viscous gel and was allowed to cool at roomtemperature. To obtain (Ba, Ti) oxalate precursor, the cooled gelwas dried in an oven at 100 °C overnight. The oxalate precursormaterial was calcined at 450 °C for 2 h. Before and after calcination, we noted the mass of the sample powder approximately 30% less than the original and the lost mass was due to the removal of theoxalate from the sample powder. The nanoparticles of cobalt ferrites CoFe 2O4(onward call it CFO-nano) were synthesized by the co-precipitation method. First of all, we took a salt solution of 0.2 M cobalt chloride (CoCl 2.6H 2O, 1.185 g/25 ml) and 0.4 M ferric chloride (FeCl 3.6H 2O, 2.705 g/25 ml). Both salt solutions were made in dis- tilled water, and as a result, we get ionized salts. The molar ratio ofthe reactants was taken as 0.2 M and 0.4 M because the product (CoFe 2O4) contains Co:Fe as 1:2. Then these independent solutions were mixed with one another. The measured pH value for differentsamples ranged between 2 and 3 (acidic region). Now the precipi-tating agent (NaOH) with a molar ratio of 3 M (3 g/25 ml) wasadded into the salt solution. The pH value was again checked. The pH value jumped to a basic range (>7) with the inclusion of the precipitating agent, and the precipitates formed were continuouslyand vigorously stirred for half an hour so that pH got stabilized.These precipitates were heated till 70 °C and when this temperature was reached, it was maintained with ±1.5 °C variations. After 45 min, stirring and heating were turned off simultaneously andwashing of these precipitates was carried by ethanol and two timeswith distilled water, and in each washing, these precipitates werecentrifuged at 2000 rpm. Furthermore, for preparing the nanocomposite sample of desire composition [xCoFe 2O4+( 1–x) BaTiO 3; x = 0.3], initially, we prepared the CFO and BTO-nanoparticles as their detail synthe-sis procedure is mentioned above, then just mix them for 30% CFOand 70% BFO. Finally, the nanocomposite of BaTiO 3–CoFe 2O4 (onward call it BTO –CFO nanocomposite) was synthesized with the composition of 70% barium titanate (BaTiO 3) and 30% cobalt ferrite (CoFe 2O4). We optimized the composition value as we wanted to study the effects of cobalt ferrite in barium titanate; thatis why we choose 70% of BTO and 30% of CFO. To get the required composition, we made the synthesis in two steps: first, pre-annealed both samples at 450 °C to remove oxalate, thenmixed both samples and grinded for 2 h, then final annealing at700 °C, then made the pellets of 13 mm diameter at a pressure of 5 tons, and then used for further characterizations as discussed in Secs. III A –III E. X-ray diffractometer (XRD), model JDX-11, of Joel Company Ltd., Japan, worked at 40 kV and 30 mA and has been used for theverification of the crystal structure. This is a powder type x-ray dif- fractometer and x rays are CuK αwith a wavelength of 1.5406 Å. The data were performed from 2 θ= 20° –80°, the increment in the angle was 0.02°, and the stay time was 0.2 s per step. A vibratingsample magnetometer (VSM) of Model BHV-50 of Riken DenishCompany Ltd., Japan, was used for the measurement of magnetiza- tion vs the applied magnetic field. This is a very fine type magnetic moment vs the applied filed (M –H) curve tracer in which a mag- netic sample is vibrated at a fixed small amplitude at a frequencyJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-3 Published under license by AIP Publishing.(30 Hz), and the magnetization strength is found directly from the magnitude of electromotive force induced in the coil in the vicinity with the sample. Dielectric measurements as a function of tempera-ture were carried out for nanoparticles, bulk, and nanocompositesamples. The pellets were polished with silver paste on both surfa-ces to be used as a dielectric cell placed inside the self-made two probe system. We examined the capacitance and dissipation factor well above room temperature. The DC resistivity, AC conductivity,and dielectric properties of all samples were measured by using theWayne Kerr 4275 LCR Meter Bridge. The dielectric constant wasmeasured from the capacitance at a different frequency range of 100 Hz –1 MHz at different temperatures using the LCR meter bridge. We did the measurements at high temperature ranges from25 °C to 390 °C by using a potential difference to the heater. III. RESULTS ANALYSIS A. Structural characterization Rietveld refinement fitted results of XRD patterns of BTO-bulk, CFO-bulk, BTO and CFO-nanoparticles, and nanocom-posite samples are shown in Fig. 1 . The BTO systems exhibited a tetragonal structure with P4mm symmetry while CFO systems showed cubic crystal with Fd3m symmetry at room temperature. We observed highly intense and narrow peaks for bulk systems,whereas for nanosystems (nanoparticles and nanocomposites),broad diffraction peaks were observed. All the samples were phasepure as confirmed from the Rietveld refinement of the XRD data. It has been observed that the major reflection is from the (110) plane at 2θ= 31.61 for the case of BTO system (bulk and nano), whereas the diffraction peak at 2 θ= 35.65 corresponding to the (311) plane is the major reflection for the CFO system (bulk and nano). Peak positions of BTO systems are observed at 2 θ= 22.25, 31.61, 38.95, 45.03, 50.93, 56.21, 65.83, 70.33, 74.83, and 79.17 which correspond to the (100), (110), (111), (200), (210), (211),(220), (300), (310), and (311) planes, respectively. Peak positions ofCFO systems are found at 2 θ= 30.21, 35.65, 37.15, 43.33, 53.16, 57.15, and 62.73 which correspond to the (220), (311), (222), (400), (422), (511), and (440) planes, respectively. The crystallite size ofBTO and CFO-nano systems was 52 nm and 26 nm, respectively, asdetermined after Rietveld refinement of the data. All the structuralinformation (lattice parameters, unit cell volume, crystal density, particle size, etc.) extracted via Rietveld fitting has been given in Table I . The goodness of fit (Chi-square) values are close to one for all samples, which show that the data are very well fitted. Thelattice parameters (a =b=3.994 Å and c = 4.028 Å) of the tetrago- nal crystal structure of BTO-bulk and nanosystems and lattice parameters of cubic CFO (a = b = c = 8.382 Å) systems bulk and nanosystems are very close to the values reported in literaturestudies. 24,25 The Rietveld fitting of the XRD pattern of the 70%BaTiO 3– 30%CoFe 2O4nanocomposite system confirmed the presence of constituent phases (BTO and CFO) and the structural information extracted via Rietveld fitting is given in Table I . We noticed that the diffraction peaks of the BTO system were dominant compared tothe CFO system. This is probably due to the high content (70%) and the densely packed structure of the BTO system. We observed broad width peaks in the composite sample, probably indicatingthe nanostructured system and few 24weak peaks of the ferrite phase were barely observed due to the background noise. No impu- rity peaks were observed in the nanocomposites sample. Thezoomed-in view of diffraction peaks for both BTO-nano andbulk systems in the 2 θrange of 30° –33° and 73° –78° are given in Figs. 2(a) and 2(b). These XRD patterns of nanoparticles system show that the peak position was shifted toward higher values of 2 θ, and the peak intensity was decreased 2/3 times of that bulk.Moreover, we noticed that the peaks (103), (301), and (310) whichcould be seen separately in the case of BTO-bulk merged to form abroad single peak in the case of nanoparticles. This indicates that the crystal structure has transformed from tetragonal to pseudo- cubic or almost cubic. B. Ferromagnetic properties The magnetization of CFO-nano and BTO –CFO nanocompo- site was recorded at room temperature (300 K) and low temperature (75 K) in an applied field of ±30 kOe. The DC magnetization mea-surements (M vs H) of the CFO-nano sample at room temperatureshown in Fig. 3(a) confirmed the s-shaped hysteresis behavior of ferromagnetism. From these data, we extracted the saturation mag- netization (M S) to be 60.477 emu/g. We found that this value of saturation magnetization is somewhat less than the reported valuefor bulk CFO which is 80.80 emu/g. 26This change in the saturation magnetization may arise due to the surface effects and the sizedependence of magnetization in nanoparticles. 5,27The measured values of coercivity (H C) and remanence of the CFO-nano sample are 894 Oe and 22.193 emu/g, respectively. The M/H data of theBTO –CFO nanocomposite at 300 K has been presented in Fig. 3(a) that established ferromagnetic hysteresis behavior. In the magne-tization loops, we clearly noticed the decrease in the M Sbetween pure CFO-nano and the (0.7BTO –0.3CFO) nanocomposite sample. This is because the ratio of cobalt ferrite in the nanocom-posite is about 30%. The coercivity represents the magnetic fieldneeded to demagnetize the magnetic material. The lowered coer- civity of the nanocomposite indicates that it can be demagnetized at a low applied field. Another t echnologically important parame- ter for recording materials and permanent magnets is the rema-nence magnetization (M r). It is the measure of magnetization when the applied field is zero during the field reversal toward a negative direction. Again, we observed a lower value of M r/Msin t h ec a s eo f( 0 . 7 B T O –0.3CFO) nanocomposite indicating that the composite is a softer ferrimagnet. We observed in Fig. 3(a) that in the composite data there is a smaller remanence magnetization ascompared to the pure cobalt ferrite data. This smaller remanence magnetization is due to the smalle r quantity of cobalt ferrite in the composite material. The values of the coercivity, remanence,and saturation magnetization of different materials are given inTable II . Figure 3(b) shows the M vs H data obtained at 75 K for CFO-nano and BTO –CFO nanocomposite samples. We noted from Table II that M s,Hc, and M rvalues were decreased strongly of the nanocomposite sample as compared to pure CFO-nano at 75 K.We noticed a drastic change in the hysteresis loops from room tem- perature to low temperature (75 K). The calculated values of M s, Hc, and M rwere increased at 75 K as compared to 300 K. TheJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-4 Published under license by AIP Publishing.FIG. 1. Rietveld refinement fitting results of the x-ray powder diffraction patterns of (a) BTO-bulk sintered at 1300 °C for 4 h and (b) CFO-bulk samples sinte red at 1300 °C for 4 h. (c) BTO-nano, (d) CFO-nano, and (e) BTO –CFO nanocomposite sintered at 700 °C for 2 h showing the observed pattern (stars in green color), the best fit Rietveld profiles (red solid lines), reflection markers (vertical bars), and difference plot at the bottom (blue solid lines).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-5 Published under license by AIP Publishing.saturation magnetization increased by about 8% –10%. However, the hysteresis showed a drastic increase in coercivity from 894 Oeto 11 868 Oe for the pure CFO-nano sample, while it increasesfrom 712 Oe to 12 182 Oe for the nanocomposite with varying thetemperature from 300 K to 75 K. Hence, it is clear that the increase in the coercivity is not due to the nanocomposite nature and follows the behavior of anisotropy and thermal activation in nano-particles. 28The area of the hysteresis loops is the measure of the energy needed to magnetize or demagnetize, we observed that theloops area increased at low temperature (75 K) which means that our sample becomes a very hard magnetic material at the low tem- perature range. We noticed from the comparison of the magnetiza-tion data for 77 K and room temperature that the coercivityincreased by almost an order of magnitude both in the case of CFO-nano and nanocomposite as shown in Figs. 3(a) –3(b). The coercivity is H c= 894 Oe at room temperature for the CFO-nanoparticles and is 712 Oe for the nanocomposite, while it is11 868 Oe for pure CFO-nano and 12 182 Oe for the composite at75 K. This change is occurred due to the drastic increase in the anisotropy constant at low temperatures due to which the magnetic spins of the nanoparticles cannot be aligned in the direction of theexternal magnetic field. 29 We can also compare the change in the magnetic moment between the CFO-nano and BTO –CFO nanocomposite samples. Since the composite has 0.3 g of cobalt ferrite per 1 g of the com- posite, the net moment in the composite of 19.218 emu/g convertsto about 63 emu/g of the ferrite component. This value compareswell with that of the pure CFO-nano sample (60.47 emu/g). For thedata of composite at 75 K, the net magnetic moment is equal to 20.95 emu/g, which converts to about 20.95/0.3 = 67 emu/g. For the CFO-nano, the value is 65.54 emu/g and agrees within error. Theremanence magnetization is 6.735 emu/g. However, since the com-posite has 0.3 g of cobalt ferrite, therefore, 6.735/0.3 is equal to 22.45 emu/g of the CFO. For the composite at 75 K, the net moment is 17.94 emu/g, and the cobalt ferrite moment is given byTABLE I. Structural parameters of BTO-bulk, CFO-bulk, BTO-nano, CFO-nano, and BTO –CFO nanocomposite samples obtained from XRD data using Rietveld refinement fitting method. Parameters BTO-bulk CFO-bulk BTO-nano CFO-nanoBTO –CFO nanocomposite BTO CFO Lattice parameters ( Ǻ) a = b 3.994 934 0.000 0768.382 921 0.000 2874.010 168 0.000 4718.379 272 0.000 4104.007 726 0.000 4398.375 532 0.005 075 C 4.028 451 0.000 1408.382 921 0.000 2874.005 076 0.000 8138.379 272 0.000 4104.023 495 0.001 2568.375 532 0.005 075 Volume ( Ǻ3) 64.292 0.002589.096 0.06164.407 0.005588.327 0.08664.625 0.014587.081 1.067 Particle size (nm) …… 52 26 37 Formula weight 238.462 1872 217.163 1968.263 227.907 1750.213Crystal density (g/cm 3) 6.159 5.277 5.599 5.555 5.856 5.750 χ21.146 1.301 1.389 1.357 1.199 wRP 0.1329 0.1011 0.1109 0.1025 0.0695 Rp 0.1023 0.0763 0.0863 0.0793 0.0554 FIG. 2. Comparison of the XRD patterns of BTO-bulk and BTO-nano for (a) (310), (103), and (301) peaks and (b) for (110) peak positions.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-6 Published under license by AIP Publishing.17.94/0.3 = 59 emu/g. We can conclude that the cobalt ferrite con- tinues to contribute the full value of its magnetic moment in thecomposite both in saturation magnetization and remanencemagnetization.C. DC resistivity studies The high temperature DC resistivity ( ρ) measurements were performed in the range of 300 K –700 K. The variation of tempera- ture against DC resistivity for BTO-bulk, CFO-bulk, BTO-nano, CFO-nano, and BTO –CFO nanocomposite has been demonstrated inFigs. 4(a) and4(b), respectively. The decreasing behavior of DC resistivity with the temperature is evident but a maximum atT∼350 K can also be predicted clearly from data. This indicates that for lower temperatures, the conductivity has a metallic trend. We observed from measured data that the resistivity increases with decreasing temperature for other samples, i.e., both bulk and nano-particles systems of CFO and BTO as usually expected for any non-metallic system (as reported in Ref. 32). Such a trend of decreasing resistivity with increased temperature is attributed to the presence of additional carriers generated by temperature. In bulk, nanoparti- cles and composite systems of CFO and BTO, the formation ofoxygen vacancies after annealing at such sufficiently high tempera-tures as discussed above is obvious, 30which act as n-type carriers. FIG. 3. Magnetization loops for CFO-nano and 0.3CFO –0.7BTO nanocomposite at (a) 300 K and (b) 75 K, respectively. TABLE II. Different magnetic parameter of different samples. Sample H c(Oe) M s(emu/g) M r(emu/g) M r/Ms CFO-Nano 300 K 894 60.47 22.19 0.366 CFO –BTO 300 K 712 19.218 6.73 0.350 CFO-Nano 75 K 11 868 65.54 56.205 0.857 CFO –BTO 75 K 12 182 20.205 17.94 0.887 FIG. 4. DC resistivity of (a) BTO-bulk and CFO-bulk and (b) BTO-nano, CFO-nano, and BFT –CFO-nanocomposite samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-7 Published under license by AIP Publishing.These trends, to some extent, have been indicated in the AC con- ductivity and dielectric measurements as well that temperature plays a role to excite the conducting carriers. To quantify thesechanges, we have determined the variation of the activation energyof the carriers in these samples. These activation energies will cor-respond to the energy required to activate the carriers from their bound states into conducting states. The increase in the conductiv- ity of the ferrite itself is a consequence of the hopping mechanismbetween Fe +2and Fe+3ions, which is thermally activated and rises with temperature. We plotted the ln( ρ) vs 1/T for BTO-bulk, CFO-bulk, BTO-nano, and CFO-nano demonstrated in Figs. 5(a) –5(d), respec- tively. The plot of ln( ρ) against 1/T graph is quite linear over an extended portion and typically shows a change in slope at a partic-ular temperature (T p). This temperature may be corresponding to the temperature where the dielectric constant showed a strong var- iation. The resistivity data were fitted using the Arrhenius equation ρ=ρoexp(E a/KbT).31The activation energy (E a) is derived by fitting the resistivity data by applying the Arrhenius equation. Theextracted values of T pfor BTO-bulk, CFO-bulk, BTO-nano, and CFO-nano are 625 K, 390 K, 500 K, and 434 K, respectively. Similarly, the obtained values of E afor BTO-bulk and BTO-nano are 974 meV and 1.462 eV, while for CFO-bulk and CFO-nano are587 meV and 1005 meV, respectively. Figure 5(e) shows the varia- tion of resistivity with 1/T for the BTO –CFO nanocomposite sample and the trend is quite linear till the temperature of 417 K that is the particular temperature for this sample. The semicon-ducting behavior is evident for all temperatures above 350 K. Fromthe Arrhenius plot, we obtained E a= 984 meV values for the BTO – CFO nanocomposite sample. It is clear that the value of the activation energy of the nano- composite sample is very close to that of the CFO-nano sample.The resistivity of CFO-nano is 1 eV, while that of the nanocompo-site is 984 meV and that of the BTO-nanoparticles is 1.46 eV.Hence, the lower resistance path offered by the CFO-nanoparticles dominates the conduction, and the dc current mostly bypasses the BTO-nanoparticles. D. AC conductivity measurements The AC conductivity ( σ ac) measurements of all samples were carried out to study the dielectric properties. AC conductivity of each sample was obtained by using the dielectric constant ( ε0) and dielectric loss tan( δ) given as σac=ε0εoωtan(δ),32where ω is the angular frequency and εothe vacuum permittivity. Figures 6(a) –6(e)show plots of σacas a function of temperature (T) for the entire series of samples at low frequency (10 kHz) and high frequency (400 kHz). We observed that plots show very consistentbehavior at high and low frequencies; there is a slow initial increasefollowed by a rapid exponential-like increase. However, in the case ofthe high-frequency data, that increasing trend is very rapid. We also notice that the value of conductivity in the case of high-frequency data is in general almost twice that of the low-frequency data. Forexample, we found that the maximum value of conductivity in thecase of a high-frequency range is 0.17 ( Ωm) −1, while that for the low-frequency range is 0.28 ( Ωm)−1for the CFO-bulk sample. As shown in Fig. 6(d) , the conductivity of the CFO-nanoparticles ’sample shows a large upturn at T ∼245 °C. Interestingly, this tem- perature coincides well with the peak temperature in the dielectric constant of the CFO-nanoparticles ’sample (will be discussed in Sec.III E ). It is, therefore, probably that the rapid increase of the conductivity at T ∼245 °C is related to the decline of the dielectric constant. The decline of the dielectric constant due to an increase in conductivity is a well-understood effect.33The increase in the con- ductivity of the ferrite itself is a consequence of the electron hoppingbetween Fe +2and Fe+3ions, which are a thermally activated process and increase with temperature.32,34 Another interesting observation from Fig. 6(a) reflects the measured data of σacas a function of temperature for BTO-bulk at 400 kHz high frequency. These plots show a quite interestingbehavior with the effects of the ferroelectric transition beingreflected in the ac conductivity. For the higher frequency data,there is a very sharp peak near the ferroelectric (T c) of 120 °C, while for the lower frequency data of 10 kHz, there is a very slight bump at the same temperature. This phase transition of theBTO-bulk sample from the tetragonal to the cubic structure willhelp during the measurements of dielectric properties of thissystem. We also noticed that the maximum value of the conductiv- ity is larger for the low-frequency data ∼10 −3(Ωm)−1and is sig- nificantly lower ∼10−4(Ωm)−1for the higher frequency. This is consistent with the hopping conduction becoming more difficult athigher frequencies. For the BTO-nanoparticles ’sample, σ acvs T plot has been drawn in Fig. 6(c) at two variant frequencies. This graph shows very consistent behavior at high and low frequencies;there is a slow initial increase followed by a rapid exponential-likeincrease. We again noticed that the value of conductivity in thecase of high frequency is quite small compared to that for the low- frequency data. This may be the reason that the rapid fluctuations minimize the conductivity values at a higher frequency range. The AC conductivity as a function of temperature for the BTO –CFO nanocomposite sample at 10 kHz and 400 kHz in Fig. 6(e) , plots show very consistent behavior at high and low fre- quencies. We found that there is a slow initial increase followed by a rapid, exponential-like increase. However, in the case of the low-frequency data, that increasing trend is very rapid. We also noticedthat the maximum value of conductivity in the case of high fre-quency is 0.020 ( Ωm) −1and for the low-frequency data, it is 0.010 ( Ωm)−1for the nanocomposite sample. The AC conductivity measurements as a function of frequency were taken at room temperature (300 K) with the help of a home-made two probe system and the LCR meter. Figure 7(a) reflects σ ac as a function of frequency data obtained at 25 °C for CFO-bulk and CFO-nano samples. It can be seen that both plots show veryconsistent behavior at low and high frequencies, there is decreasingbehavior with increasing frequency. We also noticed that themaximum value of conductivity in the case of the CFO-nanoparticles ’sample is 1.8 × 10 −7(Ωm)−1and for the case CFO-bulk, it was 1.0 × 10−8(Ωm)−1. The room temperature σacas a function of frequency for BTO-bulk and BTO-nano is demon-strated in Fig. 7(b) . It can be seen that plots show very consistent behavior at low frequencies, there is an increasing behavior with an increasing frequency range. However, in the case of the BTO-bulk, a peak was observed at a high frequency. We also noticed that themaximum value of conductivity in the case of nanoparticles data isJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-8 Published under license by AIP Publishing.FIG. 5. Activation energy of (a) BTO-bulk, (b) CFO-bulk, (c) BTO-nano, (d) CFO-nano, and (e) BFT –CFO-nanocomposite samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-9 Published under license by AIP Publishing.1.8 × 10−5(Ωm)−1and for the bulk, it was 1.1 × 10−4(Ωm)−1 showing the weakening conduction for the case of the nanoparticles as compared to the bulk due to the enhanced inter-particle bound- aries in the nanoparticles. Figure 7(c) shows the σacas a function of frequency for the BTO –CFO nanocomposite sample at room temperature. The general trend is an increase in the AC conductivity with a fre-quency similar to the trend of the BTO-nano sample. However, we noted that the conductivity values are enhanced compared to bothBTO and CFO nanoparticle samples suggesting improved AC con- ductivity in the nanocomposite system. We also noted the presenceof the peak in σ acvs log (f) at high frequency, which indicates that for high enough frequencies, the conductivity of the composite decreases. We noted that neither the CFO-nano nor theBTO-nano, which together comprise the composite, show any suchmaximum with frequency. Hence, it is suggested that this peakmay be a consequence of the magneto-electric coupling in the nanocomposite sample. FIG. 6. AC conductivity measured for (a) BTO-bulk at 10 kHz and 400 kHz, (b) CFO-bulk at 10 kHz and 400 kHz, (c) BTO-nano at 10 kHz and 400 kHz, (d) CFO-nano at 10 kHz and 400 kHz, and (e) BTO –CFO-nanocomposite at 10 kHz and 400 kHz samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-10 Published under license by AIP Publishing.E. Dielectric properties Dielectric measurements have been taken in two different ways by using the principle of a parallel plate capacitor. (1)Measuring the values of capacitance and dissipation factor as a function of temperature (keeping frequency constant). (2) Measuring the values of capacitance and dissipation factor as afunction of frequency (keeping temperature constant). The dielec- tric constant (permittivity) of the sample was calculated by usingthe relation ε¼ Cd εoA, where Cis the capacitance of our sample, dis the thickness of pellets, Ais the area of the sample occupied between the plates of the capacitor, and εois the permittivity of free space which is equal to 8.85 × 10−12m−3kg−1s4A2. The dielectric constant is a complex quantity which shows the interac-tion of matter with an external electric field. The energy stored inthe matter is measured by the real part of the dielectric constant. The dissipation of energy in the system by the alternating AC field is measured by the imaginary part of the dielectric constant. We obtained the real part of the dielectric constant as a func- tion of temperature at a constant frequency for the BTO-bulksample at two different 10 kHz (A) and 400 kHz (B) frequencies as plotted in Fig. 8(a) . We observed that both, at low 10 kHz and high 400 kHz frequencies, data show identical behavior with very sharpand well-defined peaks at ∼122 °C. We observed the expected ferroelectric-to-paraelectric phase transition known as Curie tem- perature (T c) at 120 °C for the case of the BTO-bulk sample as reported in the literature.35The dielectric constant values are FIG. 8. Dielectric constant vs temperature response of (a) BTO-bulk and (b) BTO-nano at 10 kHz and 400 kHz, respectively. FIG. 7. Variation of AC conductivity with log f (frequency) of (a) CFO-bulk, CF-nano, (b) BTO-bulk, BTO-nano, and (c) BTO –CFO-nanocomposite at 25 °C.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-11 Published under license by AIP Publishing.almost five times lower for the higher frequency as compared with the lower one. As the dielectric constant is a magnitude determin- ing the capability of a material to accumulate electrical energy inan electric field, so the higher values obtained ( ∼4500) here at 10 kHz as compared to lower vales (950) at 400 kHz. Similar behav-ior has been discussed by various authors 36as arising from the finite relaxation of the dipoles. To see the dielectric response with respect to temperature variation, the BTO-nano sample was heatedfrom room temperature to 390 °C with a fixed frequency of 10 kHz(peak-A) and 400 kHz (peak-B) in the presence of an applied ACsignal of 50 mV. The dielectric response was recorded in the heating process with the help of a home-made two probe system. The plot for the real part of the dielectric constant as a function oftemperature for the case of the BTO-nano sample is shown inFig. 8(b) . Both data sets show a very similar general trend with a sharp peak, sharp decline on either side of the peak. We observed that the real part of the dielectric constant tends to increases with the increasing temperature of the sample, which is typically ferro-electric susceptibility behavior in the region T < T c,χ= 1/(T c–T). Note that the y axis has two different scales for the BTO-nanosample. Both data show a single peak of temperature transition (T c) from ferroelectric to paraelectric at 363 °C and 345 °C at 10 kHz and 400 kHz frequencies, respectively. The peak “B”for 400 kHz is relatively sharper and its magnitude of the dielectric constant (102)is lesser than the data in peak “A”for 10 kHz, i.e., 1520. The high- frequency data have a lower value of the dielectric constant as com- pared to the low-frequency data. We noted that in the literature,different values of the T chave been reported.35It is suggested that the higher temperature transition in the case of the BTO-nano maycorrespond to another structural phase, e.g., the hexagonal phase or the surface tetragonal phase. The reduction in dielectric constant with decreasing grain size has also been reported previously. 37,38 The real part of the dielectric constant for both BTO-nano and BTO-bulk samples as a function of temperature at 10 kHz ismore interesting. The dielectric constant as a function of tempera- ture curves [ Figs. 8(a) and8(b)] reflects the grain size dependence of the permittivity έ, as reported by various authors. 39The highest values of dielectric έ(T) were observed at Curie point (T c), i.e., fer- roelectric to paraelectric transition temperature which is 122 °C forthe case of BTO-bulk and 363 °C for the case of BTO-nano samples. We noticed from the above curves that by comparing the BTO-bulk with BTO-nano, the values of the dielectric constantdecreased from 4500 to 1520, while the values of Tc increased from122 °C to 363 °C and narrow sharp peak nature becomes broader. Many authors have discussed the size dependence of the structural properties in the case of BTO systems. It has been reported that forparticle sizes less than a specific size, known as critical size, BTOloses its room-temperature tetragonal structure. Sometimes, theyhave been shown to have an orthorhombic crystal structure or cubic structure at room temperature, depending upon the synthesis techniques. 39–41 We measured the real part of the dielectric constant at a low frequency of about 10 kHz (B) and high frequency 400 kHz (A) forthe CFO-bulk sample as presented in Fig. 9(a) . We observed that data “A”for 10 kHz show two clear peaks at 204 °C and 314 °C, respectively. From earlier reported work, it is exposes that the polarnano regions are made due to the compositional disorder in thecrystal structure due to the discrepancy of ionic radii of diverse atoms. As in the present sample CoFe 2O4, various ions with differ- ent ionic radii (Co2+,F e3+, and O2−) are present in the CFO crystal sites, which leads to the establishment of nanosize areas of polari-zation (basically called as nano-polar region) 42and, hence, affect the dielectric performance of the samples. For data “A”in peak-1, the nano-polar region would be stationary at the lower temperature but the dielectric constant enhances with the upsurge in tempera- ture. The rising temperature is as much as high to start the biggerpolar region converted into the nano-polar region in the samplesfor reducing the thermal energy and attained a very highest dielec-tric constant due to the presence of a very large number of nano- polar territories. In the nano-polar regions, the electric dipoles are obligatory very less energy to be polarized. So, the highest dielectricconstant is detected between peak-1 and peak-2. Hence, during theformation of the nano-polar region, 42,43the dielectric constant starts to decrease, which is clearly observed in Fig. 9(a) with the peak-2 region. Due to this effect, the 1st peak is observed, which FIG. 9. Dielectric constant vs temperature response of (a) CFO-bulk and (b) CFO-nano at 10 kHz and 400 kHz, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-12 Published under license by AIP Publishing.not a phase transition, but it is a relaxor type behavior at T d∼204 ° C( T dknown as depolarization temperature42,43) of the CFO-bulk sample at 10 kHz. Generally, in the literature, the phase transitionfrom ferroelectric to anti-ferroelectric materials is the formationand existence of relaxor behavior in the system. After minimizingthe energy of the system, the dielectric constant upturns again with enhancing temperature and achieved higher dielectric constant due to the presence of suitable amount of nano-polar regions. Finally,the increasing temperature is that much higher, the ordered electricdipoles become a disorder due to excess of thermal energy, and thedielectric constant suddenly decreases. The second peak (T m= maximum dielectric transition temperature42,43) is occurred due to the transition from the anti-ferroelectric to the paraelectricphase transition and peak lie at 325 °C. Two dielectric inconsisten-cies are evident; one is an obvious big peak T d∼204 °C, corre- sponding to the transition from ferroelectric phase to an anti-ferroelectric phase, whereas the 2nd is a T mat 314 °C, which corresponds to the transition from an anti-ferroelectric to a para-electric phase as reported previously for other M-type leadHexaferrite 44and [(PbBa)Zr, Sn, Ti)]O 345systems. The data “B”for 400 kHz shows the single peak of transition from ferroelectric to paraelectric at around 244 °C with no indication of the second peak. The dielectric constant value in the case of data “A”is approximately 800, while that of “B”is about 80. The drop of data “A”is very sharp toward low temperature below the second peak, while it is slow for peak “A.” We measured the real part of the dielectric constant for the case of CFO-nano with respect to the temperature as shown inFig. 9(b) for both at low and high-frequency ranges. The low fre- quency (10 kHz) data correspond to well-defined peak-A at a posi- tion about 252 °C correspond to T c, while a sharp raise between 300 °C and 250 °C than relatively slow decline. The high frequency(400 kHz) data of peak “B”shows a very sharp raise of a peak for obtaining transition temperature (i.e., T c∼209 °C) and then a gradual decline to low temperature. The decline in the case of “B” is significantly less than the case “A.”The comparative study of both nano and bulk of CFO at 10 kHz was conducted, for thispurpose, we increased the temperature of the sample from room tohigh temperature (<400 °C) at a constant frequency. We observedthe CFO-nano have a single transition peak, while the CFO-bulk system displayed two separate transitions as shown in Figs. 9(a) and 9(b). Furthermore, to verify the behavior observed for the CFO-nanoparticles, we annealed the nanoparticles for up to 700 °Cfor 3 h to increase the grain size. The plot for the real part of the dielectric constant is given in Fig. 10 . We expected that if we annealed the nanoparticles at high temperature, the grain sizewould become large and the dielectric response of nanoparticleswould show bulk-like behavior. In the CFO-nano annealed sample,there is the highest strong peak at higher temperature T m∼306 °C (anti-ferroelectric to paraelectric transition) and another less strong peak at lower temperature T d∼210 °C (transition known as relaxor behavior, i.e., ferroelectric to anti-ferroelectric transition) forf= 10 kHz, and the nature of this double peak curve is similar as reported previously for other dielectric relaxor 42–45systems. These CFO-nano annealed sample and CFO-bulk sample (peak at Tm= 306 °C and T d= 217 °C) establishing the similarity of the CFO-nano annealed nanoparticles and bulk. There is someadditional structure in the region between T = 200 °C and 250 °C. Thus, the nanoparticles annealed at T ∼700 °C are consistent with the behavior of the bulk. By comparing the T cvalues with the annealed temperature (i.e., sizes of particles) of CFO-bulk (1350 °C), CFO-nano (450 °C), and CFO-nano (700 °C) at 10 kHz measurements from Figs. 9(a) , and9(b), and 10showed another interesting result. The CFO-nano (450 °C) sample has only a single peak-1 at T cvalue 250 °C, while CFO-nano (700 °C) sample has peak-1 at 217 °C, and the CFO-bulk (1350 °C) sample has transition temperature peak-1 at 204 °C. From these results, we may argue that as the annealing tem-perature increased (i.e., in other words, particle size increased), thetransition occurs at lower transition temperature values. However, remarkably another peak-2 start to appear as the annealing temper- ature increased from 700 °C to 1350 °C for CFO samples. The dielectric constant as a function of temperature for (0.7BaTiO 3–0.3CoFe 2O4) nanocomposite at low and high frequen- cies has been investigated as presented in Fig. 11(a) . The dielectric constant measured for this nanocomposite at 10 kHz was 375, while this amount decreased to 300 as we increased the frequencyto 400 kHz level. Higher frequencies usually reduced the dielectricconstant due to the more fluctuations, the dipoles dispersed andmisaligned. Another innovative work, we found two transition peaks indication at a low-frequency response of nanocomposite data as few authors reported in other oxide systems. 42–45The data inFig. 11(a) obtained at 10 kHz (low frequency) related to curve “A”shows the prominent peak at about 312 °C known as T d(tran- sition from ferro-to-anit ferroelectric), while due to the lack of facility in our apparatus (as our apparatuses work only up to 400 °C), we did not obtain the exact position of T mpeak but obvi- ously our data reflect that T mlies >400 C. This is a clear indication that phase transition from anti-ferroelectric to paraelectric occurs beyond 400 °C so our composite sample may be a good candidate for electric devices as its T mlies away from room temperature. The FIG. 10. Dielectric constant vs temperature response at 10 kHz of CFO-nano (annealed at high temperature).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-13 Published under license by AIP Publishing.lower frequency data also show a small long range hump developed around 130 °C which may be related due to the presence of BTO inthe nanocomposite sample as we obtained a similar transition signal in the BTO-bulk system. However, we found a single broad peak at about 377 °C for 400 kHz (high frequency) curve of thenanocomposite sample. All the data have been verified by repeatingthe experiment. The data can also be compared for the CFO-nano and BTO-nano and CFO –BTO nanocomposite samples as shown in Fig. 11(b) at a frequency of 10 kHz. The CFO-nano sample corre- sponds to curve “A”shows a peak at about 260 °C while data “B” for BTO-nano show a peak at about 363 °C. The indication of twoprominent peaks for the nanocomposite sample as discussed above has been compared here with CFO-nano and BTO-nano. From these comparative data, we found that the BTO-nano sample hasthe highest dielectric constant values as compared to otherCFO-nano and 0.7BaTiO 3–0.3CoFe 2O4sample. The transitions from ferroelectric to paraelectric in the case of BTO-nano (Tc= 363 °C), CFO-nano (T c= 212 °C) are transitions fromferroelectric to paraelectric while BTO –CFO nanocomposite (Tm> 400 °C) has a transition from ferroelectric to anti-ferroelectric to paraelectric. Another advanced result, we found from the BTO – CFO nanocomposite sample, a relaxor behavior exists. This relaxorbehavior also found in the CFO-bulk sample at 10 kHz as well. So,here we may argue that relaxor behavior in the nanocomposites ’ sample may arise due to the presence of CFO content in the 0.7BaTiO 3–0.3CoFe 2O4sample. In the CFO-nano sample, we did not observe any relaxor behavior; however, we found relaxor behav-ior in the nanocomposite sample as shown in Fig. 11(b) . IV. DISCUSSION Multiferroics have been properly defined as materials that show more than one primary ferroic order parameter (ferromagne-tism, ferroelasticity, ferroelectricity, and ferrotoroidicity) at thesame time. 46However, many researchers of this field consider materials as multiferroic only if they exhibit coupling between the physical parameters. On the other hand, the multiferroic systemscan be extended to include non-primary order parameters, such asantiferromagnets or ferrimagnetism. 47–49The coexistence of mag- netization and electric polarization might allow an additional degree of freedom in the design of novel devices such as actuators, transducers, and storage devices. Other potential applicationsinclude multiple state memory elements, in which data are storedboth in the electric and the magnetic polarizations, or novelmemory media which might allow writing of the ferroelectric data bit, and reading of the magnetic field generated by association. 50 Barium titanate (BTO) is a perfect perovskite oxide-based material having structure resemblance as a set of BO 6octahedra arrangement in a cubic configuration with the Ba2+ions situated in the places among the octahedra and the Ti4+ions conquering the center of the octahedra. Perovskite materials have a common empirical formula of ABO 3.51 Moreover, a unit cell of CFO has an inverse spinel struc- ture,52,53with lattice parameter (a o) of 8.39 Å, consists of eight formula units, with 8 Co2+occupying half of the octahedral sites (16d) and the other eight sites are occupied with Fe3+ions which are aligned anti-ferromagnetically and remaining eight Fe3+ions occupy the tetrahedral sites via super-exchange interactions medi-ated by oxygen ions. 54,55We have synthesized the multiferroic nanocomposite of BaTiO 3–CoFe 2O4with the composition of 70% barium titanate and 30% cobalt ferrite. The single-phase crystalstructure of all samples was verified by the XRD. By using Scherer ’s formula, the average particle sizes are obtained as 27 nm for CFOand 49 nm for BTO-nanoparticles. Magnetic studies (M/H) have been carried out for both pure CFO-nanoparticles and CFO – BTO-nanocomposite at room temperature (300 K) and low temper-ature (75 K). The saturation magnetization increased by about8%–10% for lower temperatures. However, the hysteresis showed a drastic increase in coercivity from 894 Oe to 11 868 Oe for pure cobalt ferrite; while it increased from 712 Oe to 12 182 Oe for the nanocomposite. Hence, it is clear that the increase in the coercivityis not due to the composite nature and follows the behavior ofanisotropy and thermal activation in nanoparticles. Ferroelectric material such as BTO undergoes a paraelectric phase near the Curie temperature (T c). At the Curie temperature, FIG. 11. Dielectric constant vs temperature response of (a) CFO –BTO nano- composite at 10 kHz and 400 kHz and (b) comparison of BTO-nano, CFO-nano, and CFO –BTO nanocomposite at 10 kHz.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-14 Published under license by AIP Publishing.the ferroelectric material has a maximum value of dielectric cons- tant ( ε) due to the change in the structure of the BTO. While in the paraelectric phase, the dielectric constant decreases with theincrease in temperature obeying the Curie –Weiss law. 36,39At any higher temperature (i.e., greater than the Curie temperature, Tc),due to sufficient thermal energy, it allows the Ti 4+ions to move randomly from one position to another, so there is no fixed asym- metry. A large dipole moment is formed in the presence of theapplied field by the octahedral structure Ti 4+ions but there is no longer spontaneous alignment of dipoles. In this symmetric config-uration, the material is paraelectric. At temperatures below T c, the structure changes from cubic to tetragonal symmetry with the Ti4+ ion in an off-center position giving rise to a net dipole moment. The crystallographic phase change is associated with the BTO unitcell. 56,57When the temperature of the BTO is changed, the crystallographic dimensions ’changes are occurring. The large temperature of the dielectric constant is occurred because of the distorted octahedral are coupled together, there is a very largespontaneous polarization. The BTO has a rhombohedra structurewhen T < −90 °C, then the orthorhombic structure in the tempera- ture range of −90 °C < T < 5 °C. The tetragonal system at 5 °C < T < 120 °C, the cubic phase shifting occurred in the tempera- ture range when T > 120 °C. 58,59 In order to investigate the high temperature electrical proper- ties (<400 °C), both dielectric constant and DC resistivity measure- ments have been done on samples. The AC conductivity of all samples shows an increasing behavior with increasing temperaturebut the bulk BTO sample shows a peak indication near at T ∼120 °C, which corresponds to the transition temperature of theferro-to-para transition. The maximum value of AC conductivity 0.30 ( Ωm) −1was seen in the case of bulk CFO at 400 kHz data. The DC resistivity of all samples confirms the semiconducting/insulating behavior of systems. The activation energy (E a) is derived by fitting the resistivity to an Arrhenius equation. The value of theactivation energy in the case of nanocomposite was 975 meV; however, this value was 1.58 eV and 1.0 eV for BTO and CFO-nanoparticles, respectively. Hence, the lower resistance pathoffered by the nanoferrite dominated the conduction, and the dccurrent mostly bypassed the nano BTO. The dielectric constant ofbulk BTO shows a sharp peak at T ∼120 °C which corresponds to the ferro-to-para transition but in the case of nanoparticles BTO shows a broad peak at T ∼360 °C. Our results showed that the dielectric constant first increased with temperature up to a peak value know as T cand then dropped to a relative small value. Broadly, three factors play a contribution to the temperature dependence of a dielectric constant: (1) thedecrease in the number of polarizable particles per unit volume asthe temperature increases, which is a direct result of the volumeexpansion, (2) the increase of the macroscopic polarizability due to the volume expansion, and (3) the temperature dependence of the macroscopic polarizability at a constant volume. The temperaturedependence of the dielectric constant is determined by the polari-zation feature of BTO-nano and the conducting or hopping featureof CFO. We note that the peak appearing in the dielectric response of the CFO-nano sample at T c∼252 °C is to be related to the effects of conductivity. It is evident that the decline of the dielectricconstant for T > 252 °C is related to the rapid increase of theconductivity, which is a well-understood effect. The increase in the conductivity of the ferrite itself is a consequence of the hopping mechanism between Fe 2+and Fe3+ions which is thermally acti- vated and rises with temperature. The dielectric constant in thecase of bulk CFO shows two well-defined peaks at T d= 204 °C and Tm= 314 °C but in the case of CFO-nano shows a single peak at Tc∼244 °C, no indication of the second peak. We may argue that in the case of CFO-bulk, the phase transformations occurs fromferroelectric to anti-ferroelectric (relaxor) to paraelectric naturewhile in CFO-nano phase transformations occurs direct from ferro-electric to paraelectric nature. The dielectric constant of the CFO – BTO nanocomposite shows a peak at T d∼312 °C for 10 kHz data, while there is a local minimum at about 350 °C, after which itshows another clear rise to the higher temperature. So, the overallwe found that T m> 400 °C, while the 400 kHz data of nanocompo- site sample show a single peak at T c∼377 °C. However, unlike the BTO-nano, the nanocomposite sample shows an upturn in the value of the dielectric constant at higher temperatures. This mayindicate the presence relaxor behavior in nanocomposite samples,i.e., the phase transition from anti-ferroelectric to paraelectric. We observed that nanoparticlkes of the CoFe 2O4system are ferromagnetic at room temperature (RT) while BaTiO 3showed week ferromagnetic or paramagnetic at RT, while nanoparticles ofBaTiO 3are showed ferroelectric order at RT. As our detailed study reflects that there is dielectric phase transition from ferroelectric to paraelectric at different T cvalues for different samples including those contains the content of BFO. CFO-nano and bulk both areparaelectric at RT; however, we observed transitions in these bothsamples, while the inclusion of CFO with BTO for nanocompositecombination also altered the dielectric T cvalues of the BTO-nano sample. Obviously, ferroelectricity based on the well alignment of electric dipoles even in the absence of the applied electric field,more sound aligned electric dipoles induced higher electric polari-zation. But as the alignment of electric dipoles (in the case of theBTO system) disturbed by increasing temperature as shown in our results, the ferroelectric phase changed to paraelectric phase. So here, the alignment of electric order is dependent on temperature.Furthermore, T cconfirmed the extent level of ferroelectricity in any substance, so this was our main goal to identify the T cvalue for various samples to investigate the nature of the dielectric systems. Also, the magnetic diploes play a major role for the collection of saturation magnetization. So, by applying the magnetic field, wenoticed that the unpaired electron of the Fe and Co are coupledmutually and finally the overlapping of the magnetic spin interac- tions of these electron induced magnetic polarization. We found that the CFO-nano system showed the highest magnetic order thanother systems as due to the well coupling of the spin of unpairedelectrons; however, we did not notice such type of strong ferromag-netic interactions in the BTO system. It is obvious that the electric dipole due their iconicity differ- ence creates electric polarization and the spin interactions ofunpaired electron generate magnetic polarization. So, in our case,the CFO system is best ferromagnetic while the BTO systemshowed higher ferroelectricity (T c∼363 °C) than others. As, the electric polarization disturbed due to temperatures, so as more strong ferroelectric material can sustain its polarization till achiev-ing the higher temperatures, so their Tc will be higher thanJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 124101 (2020); doi: 10.1063/1.5131467 128, 124101-15 Published under license by AIP Publishing.CFO-nano and bulk systems. In the case of nanocomposite 0.7BaTiO 3–0.3CoFe 2O4sample, the saturation magnetization decreased as compared to the CFO-nano sample while the dielec-tric Tc values increased beyond 400 °C. These simultaneous ferro-magnetic and ferroelectric (dielectric) measurements confirmed thepresence of multiferroic properties in our nanocomposite as well as CFO-nano and CFO-bulk systems. V. CONCLUSIONS In this study, we report the structural changes, electrical trans- port properties in the BaTiO 3, CoFe 2O4, and 0.7BaTiO 3– 0.3CoFe 2O4nanocomposite materials. In summary, the ceramic and nanocomposite samples were synthesized by using a chemical and simplified solid-state reaction process. The XRD data of nano-composite shows that both the CFO and BTO phases are polycrys-talline without any extra phases. The coercivity increases and thesaturation magnetization decreases with the increase of BTO content at lower temperatures. The DC resistivity of all samples confirms the semiconducting/insulating behavior of the systems.The activation energy shows that in the composite, the lower resist-ance path offered by the nanoferrite dominated the conduction,and the dc current mostly bypasses the BTO-nano. The increase in the conductivity of the ferrite itself is a consequence of the hopping mechanism between Fe 2+and Fe3+ions, which is thermally acti- vated and rises with temperature. We found that at 10 kHz, in thecase of BTO-nano (T c= 363 °C) and CFO-nano (T c= 212 °C) are transitions from ferroelectric to paraelectric while BTO –CFO nanocomposite initially change phases from ferroelectric to anti-ferroelectric (relaxor) at T d= 312 °C and then from anti- ferroelectric to paraelectric (T m> 400 °C). Similarly, in the case of CFO-bulk, we noticed T m= 204 °C and T m= 314 °C. Overall, we inferred from these studies and data that nanocomposite 0.7BaTiO 3–0.3CoFe 2O4and CFO-bulk sample showed relaxor behavior as well along with the transformation change fromferroelectric to paraelectric, while BTO-nano and CFO-nano andBTO-bulk showed their transformation directly from ferroelectric to paraelectric with increasing temperature. Hence, the transition temperature is determined by the polarization feature of BTO andthe conducting or hopping feature between Fe +2and Fe+3ions of CFO. At 10 kHz, the dielectric constants are measured 4500(BTO-bulk), 1550 (BTO-nano), 820 (BTO-nano), 275 (CFO-nano), and 375 (BTO –CFO nanocomposites) for various sample of the series. Hence, the temperature dependence of the dielectric cons-tant is determined by the polarization feature of nano BTO and theconducting or hopping feature of CFO. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. 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Chaos 30, 103115 (2020); https://doi.org/10.1063/5.0020775 30, 103115 © 2020 Author(s).Bifurcations and chaos in a Lorenz-like pilot- wave system Cite as: Chaos 30, 103115 (2020); https://doi.org/10.1063/5.0020775 Submitted: 03 July 2020 . Accepted: 29 September 2020 . Published Online: 14 October 2020 Matthew Durey ARTICLES YOU MAY BE INTERESTED IN Chaos in Cartan foliations Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 103116 (2020); https:// doi.org/10.1063/5.0021596 Assessing observability of chaotic systems using Delay Differential Analysis Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 103113 (2020); https:// doi.org/10.1063/5.0015533 Spontaneous transitions to focal-onset epileptic seizures: A dynamical study Chaos: An Interdisciplinary Journal of Nonlinear Science 30, 103114 (2020); https:// doi.org/10.1063/5.0021693Chaos ARTICLE scitation.org/journal/cha Bifurcations and chaos in a Lorenz-like pilot-wave system Cite as: Chaos 30, 103115 (2020); doi: 10.1063/5.0020775 Submitted:3July2020 ·Accepted:29September2020 · PublishedOnline:14October2020View Online Export Citation CrossMark Matthew Dureya) AFFILIATIONS DepartmentofMathematics,MassachusettsInstituteofTechn ology,Cambridge,Massachusetts02139,USA a)Author to whom correspondence should be addressed: mdurey@mit.edu ABSTRACT A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating fluid bath, guided by its self-generated wave field. This hydrodynamic pilot-wave system exhibits a vast range of dynamics, including behavior previously thought to be exclusive to the quantum realm. We present the results of a theoretical investigation of an idealized pilot-wave model, in which a particle is guided by a one-dimensional wave that is equipped with the salient features of the hydrodynamic system. The evolution of this reduced pilot-wave system may be simplified by projecting onto a three-dimensional dynamical system describing the evolution of the particle velocity, the local wave amplitude, and the local wave slope. As the resultant dynamical system is remarkably similar in form to the Lorenz system, we utilize established properties of the Lorenz equations as a guide for identifying and elucidating several pilot-wave phenomena, including the onset and characterization of chaos. Published under license by AIP Publishing. https://doi.org/10.1063/5.0020775 An oil droplet may “walk” along the surface of a vertically vibrat- ing fluid bath, propelled by its self-generated wave field. The droplet possesses a “path memory” of its prior trajectory, encoded within its accompanying guiding wave. This hydrodynamic pilot- wave system has been shown to exhibit many dynamical prop- erties previously thought to be exclusive to the quantum realm, including unpredictable tunneling, orbital quantization, and the emergence of wavelike statistics. We present a theoretical study of an idealized pilot-wave system that is endowed with the funda- mental features of the walking-droplet problem. In this reduced framework, we identify and rationalize several pilot-wave phe- nomena, paying particular attention to the onset and form of chaotic behavior arising in the long-path-memory limit. I. INTRODUCTION A millimetric oil droplet may bounce on the surface of a vertically vibrating fluid bath,1generating subcritical, quasi- monochromatic Faraday waves at each impact. The decay time of the Faraday waves increases with the vibrational acceleration, thus increasing the longevity and propulsive influence of waves gener- ated at prior droplet impacts. Beyond a critical threshold of the vibrational forcing, the horizontal wave force acting on the droplet exceeds the drag force resisting motion; the axisymmetric bouncingdestabilizes to a horizontal “walking” state, in which the droplet is propelled across the surface of the bath by the slope of the waves generated over all prior impacts.2The droplet is thus endowed with a “path memory” of its past trajectory,3encoded within its accompa- nying guiding or “pilot” wave field.4This hydrodynamic pilot-wave system exhibits a vast range of periodic and chaotic dynamics, including behavior previously thought to be peculiar to the quan- tum realm, such as tunneling,5,6orbital quantization,7–12and the emergence of wavelike statistics.9,13–15 Just beyond the walking threshold, the droplet motion may typically be characterized in terms of a supercritical pitchfork bifur- cation, according to which the walking speed increases with the square root of the distance to the walking threshold.16In this walking state, the droplet typically bounces in synchrony with the accom- panying Faraday waves2,16and is hence referred to as a resonant walker.17The amplitude of the accumulated Faraday wave field typ- ically increases with the vibrational forcing, resulting in steeper waves and thus an increase in the droplet speed.2,16,17As the vibra- tional forcing is further increased, the droplet’s vertical motion may destabilize to chaotic bouncing,18resulting in fluctuations of the droplet’s horizontal speed.17 To analyze the speed and stability of resonant walkers, Ozaet al.19developed a theoretical model for a droplet’s two- dimensional horizontal motion, time-averaging the periodic verti- cal motion of the droplet and the waves. This stroboscopic model Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-1 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha describes the droplet motion in terms of an integro-differential equation, encapsulating the droplet’s path memory within an inte- gral over the droplet’s prior trajectory. This model exhibits the experimentally observed supercritical pitchfork bifurcation at the onset of motion and robustness of the walking state to in-line and lateral perturbations. To further investigate the dynamics of the stroboscopic tra- jectory equation,19Durey et al.20considered a parametric general- ization of the stroboscopic model, an idealized system in which a particle is guided by a wave field endowed with the fundamental properties of the walking-droplet problem. Within this generalized pilot-wave framework,4it was shown that steady particle propulsion destabilizes in the long-path-memory limit via a subcritical Hopf bifurcation.20Beyond this critical threshold, chaotic and periodic dynamics arise in different regions of the parameter space when the particle is confined to a line. The chaotic motion resembles a ran- dom walk: The particle is at rest for sustained periods during which its pilot wave accumulates before the particle is rapidly propelled to a neighboring minimum of its pilot wave. Notably, an ensem- ble of similarly prepared systems in this chaotic regime exhibits asymptotic diffusion. While this study20demonstrated the richness of the strobo- scopic pilot-wave model,19analysis was limited due to the com- plexity of the integro-differential equation governing the particle motion. With the aim of providing a precise characterization of the system’s chaotic behavior and to identify any previously missed bifurcations and periodic states, we seek a simplified model to pro- vide a convenient framework for a tractable mathematical analysis. We thus consider an idealization of the walking-droplet problem, in which a particle is propelled by its one-dimensional, monochro- matic pilot wave. We recast this reduced pilot-wave system as a three-dimensional dynamical system whose form is strikingly simi- lar to the Lorenz equations.21Using the established properties22,23of the Lorenz equations as a guide, we then explore the dynamics of this pilot-wave system, rationalizing the onset and form of the parti- cle’s chaotic motion arising in the long-path-memory limit, opening up exciting new vistas of pilot-wave dynamics. This paper is arranged as follows: In Sec. II, we formulate the stroboscopic pilot-wave model and recast the system in a form sim- ilar to the Lorenz equations. We then demonstrate in Sec. IIIthat this pilot-wave system in one spatial dimension exhibits the same dynamical features as its two-dimensional counterpart,19lending credence to our one-dimensional approach. In Sec. IV, we system- atically demonstrate the existence of a subcritical Hopf bifurcation and a homoclinic bifurcation,23the latter of which gives rise to tran- sient chaos. In Sec. V, we rationalize the form of the chaotic attractor in terms of a Lorenz-like map whose form we relate to the particle’s asymptotic diffusion.20 II. THE HYDRODYNAMIC PILOT-WAVE SYSTEM Based on resonant walkers in the hydrodynamic system, we consider an idealized model for rectilinear particle motion, in which a particle is propelled by the local slope of its guiding monochro- matic wave field and resisted by drag. Specifically, we consider a particle of mass mwith position xp(t)at time t, where the particleexerts a periodic vibrational force over a period TF, generating sym- metric standing waves about the particle’s instantaneous position. When time-averaging over one vibration period, the particle evolves according to m¨xp(t)+D˙xp(t)= −F∂xh(xp(t),t), (1) where Dis the drag coefficient, Fis the time-averaged vibrational force, h(x,t)is the time-averaged pilot wave, and dots denote differ- entiation with respect to t.17 By modifying the developments of Oza et al.,19we pose that h(x,t)may be expressed as the superposition of monochromatic, sinusoidal waves centered on the particle trajectory. The waves decay exponentially in time over a time scale TM, which governs the longevity of the particle’s path memory. Under these assumptions, we obtain h(x,t)=A TF/integraldisplayt −∞cos(k(x−xp(s)))e−(t−s)/TMds, (2) where Ais the amplitude of the wave generated at each instant in time and λ=2π/kis the wavelength of the sinusoidal pilot wave. The pilot-wave system (1)–(2) was explored by Molá ˇcek24in the case of particle motion confined by a harmonic potential. The model of Oza et al.19describing a pilot wave in two spa- tial dimensions with particle motion confined to a line20is nearly identical to (1)and(2); the cosine in (2)is simply replaced by J0, the Bessel function of the first kind with order zero. This wave form arises from the axisymmetric, quasi-monochromatic standing wave field generated by each droplet impact. Projecting onto the particle’s line of motion yields h(x,t)=A TF/integraldisplayt −∞J0(k(x−xp(s)))e−(t−s)/TMds. (3) To distinguish between these two pilot-wave forms, we describe (1) and(2)as the cosine model and describe (1)and(3)as the Bessel model. The analysis of the cosine model presented herein is con- siderably simpler than that of the Bessel model,19,20allowing for a detailed investigation into pilot-wave dynamics. We non-dimensionalize (1)and(2)by scaling x∼k−1,xp∼ k−1,t∼T0=(DTF/FAk2)1/2, and h∼AT0/TF. By defining the dimensionless inertial coefficient κ0=m/DT0and the decay rate µ=T0/TM>0, we obtain the dimensionless pilot-wave system, κ0¨xp(t)+˙xp(t)= −∂xh(xp(t),t), (4a) h(x,t)=/integraldisplayt −∞cos(x−xp(s))e−µ(t−s)ds. (4b) Notably, projecting honto the particle trajectory19yields the integro- differential trajectory equation, κ0¨xp(t)+˙xp(t)=/integraldisplayt −∞sin(xp(t)−xp(s))e−µ(t−s)ds, where the particle’s path memory is encoded within the integral. As this integro-differential equation is challenging to analyze, we instead seek a low-dimensional, temporally local form of the pilot- wave system (4). Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-2 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha A. Reduction to a three-dimensional dynamical system We proceed to recast the pilot-wave system (4)as a three- dimensional dynamical system governing the evolution of the parti- cle velocity, v(t)=˙xp(t), the local wave amplitude, a(t)=h(xp(t),t), and the local wave slope, b(t)=∂xh(xp(t),t). Specifically, we use (4b)to express the sinusoidal wave field, h(x,t), as h(x,t)=a(t)cos(x−xp(t))+b(t)sin(x−xp(t)), where a(t)andb(t)are defined as a(t)=/integraldisplayt −∞cos(xp(t)−xp(s))e−µ(t−s)ds, (5a) b(t)= −/integraldisplayt −∞sin(xp(t)−xp(s))e−µ(t−s)ds. (5b) Forµ >0, we differentiate (5)with respect to t, yielding (6b)–(6c) below. Then, upon substituting vandbinto(4a), we recast the pilot- wave system (4)as κ0˙v+v+b=0, (6a) ˙a−vb+µa=1, (6b) ˙b+va+µb=0. (6c) This dynamical system, which is symmetric under the mapping (v,a,b)/mapsto→(−v,a,−b), has two nonlinear terms, arising from expressing hin the particle frame of reference. Furthermore, the source term in (6b)corresponds to particle-induced wave genera- tion, prompting particle propulsion. B. Comparison in form to the Lorenz system For positive parameters σ,ρ, and β, the Lorenz system21 describing the evolution of x(t),y(t), and z(t)takes the form ˙x=σ(y−x), (7a) ˙y=x(ρ−z)−y, (7b) ˙z=xy−βz. (7c) We first note that both the pilot-wave system (6)and the Lorenz system (7)have precisely two quadratic terms. In fact, it is readily verified that recasting the Lorenz system (7)in terms of the variables ¯v(t)=x(t),¯a(t)=ρ−z(t),¯b(t)= −y(t), yields the following form: σ−1˙¯v+¯v+¯b=0, (8a) ˙¯a−¯v¯b+β¯a=βρ, (8b) ˙¯b+¯v¯a+¯b=0. (8c) By associating the parameters σ−1↔κ0,β↔µ, and restrict- ingρ=β−1, we observe that (8)is almost identical to the pilot-wave system (6). In fact, only the dissipation terms differ: In the pilot- wave system (6), the wave amplitude, a, and the wave slope, b, havethe same dimensionless dissipation rate, µ; in the Lorenz system (8), the variables ¯aand¯bhave different dissipation rates, specifically β and 1, respectively. As such, while we might anticipate analogies between the pilot-wave system and the Lorenz system, the extent of this similitude is unclear. Nevertheless, we utilize established properties of the Lorenz system as a guide to elucidate pilot-wave phenomena. In particular, we characterize the onset and form of the chaotic behavior arising in the pilot-wave system in the long-path- memory limit, µ/lessmuch1, corresponding to small βand large ρin the Lorenz system. III. THE ONSET OF PARTICLE PROPULSION We proceed to explore the onset of particle propulsion, demon- strating that the dynamical transitions predicted by the cosine model asµare decreased are identical to those arising in the Bessel model.19 Moreover, these transitions are analogous to the bifurcations aris- ing in the Lorenz system as ρis increased (for βandσfixed).21,23 Specifically, we find that the rest state destabilizes via a supercrit- ical pitchfork bifurcation as µis decreased through unity, giving rise to stable particle propulsion at a constant speed. As µis fur- ther decreased, the influence of prior perturbations increases; as a consequence, the response of the steady propulsion state to pertur- bations transitions from overdamped to underdamped oscillations, eventually destabilizing in the long-path-memory limit. A. The stability of the rest state Forµ >0, the pilot-wave system (6)has a fixed point (v,a,b)=(0,µ−1, 0), corresponding to the rest state of the particle. As depicted in Fig. 1(a), the particle rests on a peak of the sinu- soidal wave field; as µis decreased, the wave amplitude beneath the particle increases, resulting in an increase of the wave force acting on the particle when the system is perturbed. While the rest state may appear to be unstable when regarding the wave field as a static potential, the particle-induced wave generation deems the pilot wave a dynamic potential, stabilizing the rest state for sufficiently strong wave dissipation. FIG. 1.The pilot wave, h, arising for (a) the stable rest state ( /Gamma1= −0.3) and (b)stableparticlepropulsion( /Gamma1=0.5),where /Gamma1=1−µ.Thedotsdenotethe particleposition.Thearrowin(b)denotesthedirectionofp articlepropulsion. Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-3 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha We proceed to show that the rest state is globally stable for allµ >1: all trajectories in the phase space approach the rest state ast→ ∞ . We construct a Lyapunov function23E(v,a,b)satisfying E≥0 for all (v,a,b)and˙E≤0 along trajectories, with equality if and only if the system lies at the rest state (0,µ−1, 0). A candidate Lyapunov function is a paraboloid whose minimum is (0,µ−1, 0), where the coefficients of the paraboloid are chosen so that ˙E≤0 along trajectories. A suitable Lyapunov function is, therefore, E(v,a,b)=1 2/bracketleftBig κ0v2+µ(a−1/µ)2+µb2/bracketrightBig , (9) where differentiation of (9)with respect to tyields dE dt= −/bracketleftbig (v+b)2+(µ2−1)b2+(µa−1)2/bracketrightbig along trajectories given by (6). As˙E≤0 for µ >1, with ˙E=0 only at the rest state, we conclude that the rest state is a global attractor, and no other fixed points or periodic states exist for µ >1. This global stability is a new theoretical result consistent with experimental observations of bouncing droplets for weak vibrational forcing.2 To demonstrate that the rest state is locally unstable for µ <1, we consider the linearization of (6)about the fixed point (v,a,b)= (0,µ−1, 0). The evolution of such a perturbation is determined by the eigenvalues, s∈C, satisfying (s+µ)/bracketleftbig κ0s2+(µκ 0+1)s+/parenleftbig µ−µ−1/parenrightbig/bracketrightbig =0. Asµdecreases through unity, the constant term in square brack- ets changes sign and the rest state destabilizes, transforming from a stable node into a saddle point, and sustained particle motion ensues. B. A Stuart–Landau equation for the onset of propulsion To describe the propulsive particle dynamics arising for µ <1, we introduce the parameter /Gamma1=1−µ, where /Gamma1=0 corresponds to the onset of sustained particle propulsion and /Gamma1=1 corresponds threshold at which the wave dissipation vanishes (analogous to the Faraday threshold in the hydrodynamic system).4,25InAppendix A , we derive a Stuart–Landau equation governing the particle velocity just beyond the onset of sustained particle propulsion. By denot- ing/Gamma1=ε2and considering 0 < ε/lessmuch1, the particle velocity satisfies v(t)∼εV(τ)+O(ε2), where V(τ)evolves over the slow timescale τ=ε2taccording to dV dτ=2 1+κ0/parenleftbigg V−1 2V3/parenrightbigg . (10) We deduce that, just beyond the onset of sustained particle propul- sion, the particle slowly evolves toward the steady propulsion speed |V| =√ 2, corresponding to |v| ∼√ 2/Gamma1+O(/Gamma1). This behavior is characteristic of a supercritical pitchfork bifurcation, as reported for the walking-droplet system.16 C. Steady particle propulsion For 0 < /Gamma1 < 1, we seek a pair of solutions corresponding to steady particle propulsion at speed ueither to the left or right. Itfollows from (6)that these fixed points are v= ±u,a=µ,b= ∓u, where u=/radicalbig 1−µ2=√ 2/Gamma1−/Gamma12is the steady propulsion speed, which increases monotonically with /Gamma1and satisfies u→1 as /Gamma1→1. The corresponding pilot wave, h, is presented in Fig. 1(b), where h(x,t)=cos(x−xp(t)+θ),xp(t)=vt+constant, and θ=arcsin v. As the particle speed increases, the particle descends from the peak of its pilot wave when /Gamma1=0 toward a zero of has /Gamma1→1. Moreover, the pilot wave has a peak immediately behind the particle.19 We infer from the Stuart–Landau equation (10)that steady particle propulsion is stable for 0 < /Gamma1/lessmuch1. However, it remains to determine the asymptotic linear stability of this state for all 0 < /Gamma1 < 1. Upon considering small perturbations from each of this pair of fixed points and linearizing (6), we find that the asymptotic behav- ior of a perturbed trajectory is governed by the eigenvalues, s∈C, satisfying κ0s3+(2µκ0+1)s2+(κ0+µ)s+2(1−µ2)=0. (11) The roots of this cubic polynomial are computed numerically for κ0>0 and 0 < /Gamma1 < 1, with results presented in Fig. 2 . We charac- terize the particle’s response to perturbations in terms of the eigen- value (or a pair of complex-conjugate eigenvalues) with a maximum real part. Consistent with the form of the Stuart–Landau equation (10), all three eigenvalues are real and negative for 0 < /Gamma1/lessmuch1, cor- responding to an attracting node in the phase space: all velocity perturbations are overdamped and decrease monotonically to zero. As/Gamma1is increased, two real eigenvalues collide and a pair of complex-conjugate eigenvalues are born. When the collision is between the two dominant eigenvalues (corresponding to κ0<2.54), the velocity perturbation becomes underdamped, giving FIG.2.Regimediagramdelineatingtheresponseofthesteadypropu lsionstate to small perturbations, as predicted by the dominant eigenv alue computed from Eq.(11). As/Gamma1is progressively increased, the response transitions from over- tounder-dampedoscillations,beforedestabilizingat /Gamma1=/Gamma1c(κ0)forallκ0>0. Thedashedcurvedenotestheonsetofsubdominantoscillatio ns. Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-4 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha rise to an attracting focus-node in the phase space. In contrast, for κ0>2.54, the two subdominant eigenvalues collide, resulting in a persistence of overdamped oscillations. However, as /Gamma1is further increased, the real part of the pair of complex-conjugate eigenvalues grows, eventually surpassing the real eigenvalue, yielding a qual- itative shift from overdamped to underdamped oscillations. The longevity of prior perturbations continues to increase with /Gamma1, result- ing in an instability of the steady propulsion state at /Gamma1=/Gamma1c(κ0), beyond which each of the pair of fixed points is saddle-foci. IV. THE ROUTE TO CHAOS We proceed to rationalize the onset of chaos as /Gamma1is increased. Our analysis reveals a subcritical Hopf bifurcation23and evidence of a Lorenz-like series of homoclinic explosions, giving rise to infinitely many unstable periodic and aperiodic trajectories.22 A. The onset of instability: A subcritical Hopf bifurcation To characterize the onset of instability of the steady propulsion state at /Gamma1=/Gamma1c, we first note that, as the constant term in (11)is positive for 0 < µ < 1, the steady propulsion state cannot destabi- lize via a real eigenvalue. We hence seek a root of (11)of the form s=iωcforµ=µc=1−/Gamma1c, where i is the imaginary unit. We con- siderωc>0 without loss of generality. By substituting into (11)and grouping together real and imaginary parts, we obtain −ω2 c(2µcκ0+1)+2(1−µ2 c)=0, −κ0ω2 c+κ0+µc=0, from which it follows that ωc=√1+µc/κ0and µc=1 8κ0/parenleftbigg −1−2κ2 0+/radicalBig 1+20κ2 0+4κ4 0/parenrightbigg . Forµ < µ c(or/Gamma1 > /Gamma1 c), steady propulsion is unstable, taking the form of a saddle-focus in the phase space: trajectories approach each fixed point monotonically in the direction of the stable man- ifold and spiral away in the plane of the unstable manifold. Notably, the asymptotic form of this instability curve is µc∼κ0forκ0/lessmuch1 andµc∼(2κ0)−1forκ0/greatermuch1, indicating that steady propulsion destabilizes for all κ0>0. To explore the dynamics in the vicinity of this instability threshold, we perform a multiple-scales analysis, introducing the slow timescale τ=ε2tfor a small parameter 0 < ε/lessmuch1, where µ=µc+ςε2andς∈ {−1, 1}. We note that ς= ±1 denotes the regimes in which steady propulsion is stable ( +) or unstable ( −). As this methodology is standard for analyzing Hopf bifurcations23,26 and the calculations are lengthy, we simply outline the procedure in Appendix B . We obtain that the particle velocity near the fixed point (u,µ,−u)is v(t)∼vc+ε[V(τ)eiωct+c.c.]+O(ε2), where vc=/radicalbig 1−µ2 cand the complex amplitude, V(τ), evolves according to the complex Stuart–Landau equation, dV dτ= −ςγ1V+γ2|V|2V. (12)The coefficients γ1,γ2∈Care defined in Appendix B . To determine trajectories arising near the instability threshold, we first denote γ1=r1+ic1andγ2=r2+ic2, where rjandcjare real and depend on κ0. Then, by writing V(τ)=R(τ)ei/Phi1(τ), it follows from (12)that the amplitude, R≥0, and the phase, /Phi1∈R, evolve according to dR dτ= −ςr1R+r2R3, d/Phi1 dτ= −ςc1+c2R2. Consistent with the linear stability analysis, we have r1>0, while we find that r2>0 for all κ0>0. In the regime of stable steady propulsion ( ς=1), the ampli- tude, R(τ), has a non-zero steady state, R0=√r1/r2, which is unsta- ble. In this state, the phase, /Phi1(τ)=/Phi10τ+constant, is linear (with /Phi10= −c1+c2r1/r2) and the particle evolves on a small-amplitude, unstable limit cycle with velocity v(t)∼vc+2R0√µ−µccos(/Omega1t)+O(ε2), where the angular frequency is /Omega1=ωc+(µ−µc)/Phi10. In contrast, when steady propulsion is unstable ( ς= −1), no additional states emerge in the vicinity of the fixed point; conse- quently, the nonlinear system (6)approaches a distant attractor. This behavior corresponds to a subcritical Hopf bifurcation.20,23This analysis provides the foundation for identifying a homoclinic bifur- cation arising for /Gamma1 < /Gamma1 c, which ultimately gives rise to the onset of chaos. B. The onset of pulsation: A homoclinic bifurcation To determine the emergence of this pair of unstable limit cycles as /Gamma1is increased, we explore the behavior of the bifurca- tion branch subtending from the subcritical Hopf bifurcation. We observe dynamics indicative of a homoclinic bifurcation, analogous in form to the homoclinic bifurcation arising in the Lorenz system.22 To track the trajectory carved through the phase space by each unstable limit cycle as /Gamma1is decreased from /Gamma1c, we represent the limit cycle as a Fourier series with period T=2π/ω. The imple- mentation of this numerical method is detailed in Appendix C . As presented in Fig. 3 , the speed variations grow as /Gamma1is decreased, and the limit cycle passes increasingly close to the saddle point (v,a,b)=(0,µ−1, 0)(the rest state analyzed in Sec. III A). More- over, the period, T, increases as /Gamma1decreases, diverging at /Gamma1=/Gamma1h (where 0 < /Gamma1 h< /Gamma1 c) with T∼ |log(/Gamma1−/Gamma1h)|as/Gamma1→/Gamma1h. This lim- iting behavior is characteristic of a homoclinic bifurcation.23For /Gamma1/greaterorsimilar/Gamma1h, the particle pulsates, moving slowly for an extended period during which its pilot wave accumulates, before being rapidly pro- pelled to the next minimum of its pilot wave whence the process repeats. This pulsation is presented in Fig. 3(d) . InFig. 4 , we present the computed bifurcation curve, /Gamma1h(κ0), at which a homoclinic bifurcation arises. The distance /Gamma1c−/Gamma1his approximately constant, but it appears that /Gamma1h→1 in the singular limitκ0→0 of gradient-driven particle motion. To prove that no limit cycles exist in this limit, we set κ0=0 in(6), yielding v= −b Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-5 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 3.Evidence of a homoclinic bifurcation at /Gamma1=/Gamma1h∼0.6507 for κ0=1. (a) and (b) Bifurcation diagrams of the stable (solid curv es) and unsta- ble (dashed–dotted curves) steady propulsion state and the unstable periodic state. The red curves denote the multiple-scales approximat ion of the subcriti- cal Hopf bifurcation (valid for 0 < /Gamma1c−/Gamma1/lessmuch1). The periodic state ceases to exist for /Gamma1 < /Gamma1 h, denoted by the dotted vertical line. (a) The upper and lower speed bounds of the period state. The colored ticks at the top o f the panel cor- respond to the values of /Gamma1appearing in (c). (b) The period, T, of the unstable limitcycleandtheperiodarisingfromtheasymptoticlinea rstabilityofthesteady propulsionstate.Inset:Theperiod, T,withalogarithmichorizontalaxis,demon- strating the characteristic scaling T∼ |log(/Gamma1−/Gamma1h)|as/Gamma1→/Gamma1h. (c) Sample phaseportraitsforthevaluesof /Gamma1correspondingtotheticksin(a).As /Gamma1→/Gamma1h, the limit cycle passes near the saddle point (v,˙v)=(0,0). (d) The evolution of the pilot wave (blue curves) and the particle position (red d ots) over two periods oftheunstablelimitcyclearisingfor /Gamma1=0.6513. and ˙a=f1(a,b)≡1−b2−µa, (13a) ˙b=f2(a,b)≡b(a−µ). (13b) Asb=0 is a separatrix and the system (13)is symmetric under the mapb/mapsto→ − b, we need to consider only the region b>0. To prove that no nonconstant periodic solutions of (13)exist in this region, we apply the Bendixson–Dulac Theorem23with the auxiliary function φ(a,b)=1/b. Specifically, we compute ∂a(φf1)+∂b(φf2)= −µ/b, which is negative for b>0. We conclude that limit cycles cease to exist for κ0=0; the homoclinic bifurcation curve thus satisfies /Gamma1h(κ0)→1 asκ0→0, buttressing the behavior evident in Fig. 4 . C. Toward a series of homoclinic explosions This study has revealed the “birth” of a pair of unstable limit cycles from a homoclinic bifurcation as /Gamma1is increased; might this pilot-wave system exhibit more limit cycles, and, if so, from whatFIG.4.Thebifurcationcurve, /Gamma1h(κ0),atwhichthenumericallycomputedhomo- clinic bifurcation arises (red crosses), superimposed on t he stability diagram of thesteadypropulsionstate(see Fig.2). kind of bifurcation does each limit cycle arise? In the Lorenz sys- tem, the homoclinic bifurcation in fact gives rise to infinitely many unstable periodic and aperiodic trajectories and is thus termed a “homoclinic explosion.”22Indeed, there is a series of these homo- clinic explosions as ρis increased, giving rise to yet more periodic and aperiodic trajectories. Given the similarity between our system and the Lorenz system, we anticipate that the pilot-wave system (6)might also exhibit a homoclinic explosion at /Gamma1=/Gamma1hand addi- tional homoclinic explosions as /Gamma1is increased. Proving the existence of such bifurcations is beyond the scope of the present work, but we proceed to find numerical evidence supporting this claim. The ramifications of this bifurcation structure on the onset of chaos are discussed in Sec. IV D. To provide evidence suggestive of a homoclinic explosion at /Gamma1=/Gamma1h, we first consider simulated stable limit cycles arising for /Gamma1 > /Gamma1 c. Using the algorithm presented in Appendix C , we then track these limit cycles as /Gamma1is decreased (with κ0fixed), where, at some point along the bifurcation branch, these limit cycles destabi- lize. As shown in Fig. 5(a) , multiple limit cycles are born precisely at/Gamma1=/Gamma1h. This multiplicity is suggestive of a homoclinic explosion. Furthermore, we find limit cycles born from a second homoclinic bifurcation, arising at /Gamma1=/Gamma1/prime h> /Gamma1 h, presenting the possibility of a Lorenz-like sequence of homoclinic explosions. As portrayed in Figs. 5(b) –5(d), the velocity evolution of these unstable limit cycles close to a homoclinic bifurcation is similar to the pulsation pre- sented in Fig. 3(d) , although the particle direction may alternate periodically. D. Transient chaos and coexisting attractors For/Gamma1/greaterorsimilar/Gamma1h, we observe evidence of transient chaos: as pre- sented in Fig. 6(a) , the particle moves erratically between the unsta- ble limit cycles (those born from homoclinic explosions) for a finite amount of time, before approaching one of the two stable fixed Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-6 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG. 5.Evidence suggestive of a series of homoclinic explosions ar ising for κ0=0.4. (a) The period, T, of three unstable limit cycles, each born from a homoclinicbifurcationas /Gamma1isincreased.Theblueandblackbranchesarebornat /Gamma1h∼0.6336.ThebluebranchvanishesatthesubcriticalHopfbifu rcationarising at/Gamma1c∼0.764. The purple branch is born from a secondary homoclinic b ifurca- tion at /Gamma1/prime h∼0.7587. (b)–(d) The velocity pulsation of each limit cycle ar ising at T=50,presentedovertwoperiods. points (the steady propulsion states) in an unpredictable manner. The average transient time increases with /Gamma1, before diverging at a critical threshold, /Gamma1a< /Gamma1 c(where, typically, /Gamma1c−/Gamma1a/lessorsimilar0.02). For /Gamma1a< /Gamma1, a strange attractor arises, coexisting with the stable propul- sion state, as is demonstrated in Fig. 6(b) forκ0=5. As a conse- quence, there is hysteresis between attractors as /Gamma1is increased or decreased. In the chaotic regime, the particle exhibits lateral oscilla- tions over a distance comparable to the wavelength of the pilot wave. As/Gamma1is further increased, the strange attractor collapses to a sta- ble limit cycle for κ0=5 [see Fig. 6(c) ] for which lateral oscillations persist. This transition to chaos is analogous to that arising in the Lorenz system when ρis increased.22,23,27Moreover, coexisting attractors also arise at the onset of chaos in the walking-droplet system.28Informed by these developments, we discover coexist- ing attractors and transient chaos arising in the Bessel model, as presented in the supplementary material . A more comprehensive exploration of the system’s transient chaos and coexisting attractors is left for future consideration. V. THE FORM OF THE CHAOTIC ATTRACTOR In regimes in which a chaotic attractor arises, the parti- cle exhibits a jittering motion with velocity pulsations similar to those observed in the vicinity of a homoclinic bifurcation (see the supplementary material ). Indeed, due to the unpredictable changes in the particle direction, this chaotic jittering resembles a random walk on short timescales and is diffusive as t→ ∞ .20As presented inFIG. 6.Transient chaos and coexisting attractors arising at κ0=5 and /Gamma1/lessorsimilar/Gamma1c=0.905. All simulations were initialized with a=b=0 andv=v0. (a)Transientchaosat /Gamma1=0.8821,with v0=0.01(bluecurve)and v0=0.005 (redcurve).(b)and(c)Coexistingattractorsinitialized withv0=0.1(bluecurve) andv0=0.9(redcurve):(b)Approachtosteadypropulsionandachao ticattrac- torat/Gamma1=0.89and(c)approachtosteadypropulsionandaperiodicatt ractorat /Gamma1=0.90. Fig. 7(a) , the chaotic attractor typically exhibits two “wings,” arising due to the symmetry of the pilot-wave system (6)under the mapping (v,a,b)/mapsto→(−v,a,−b). Moreover, in certain parameter regimes, the chaotic attractor has the characteristic butterfly-like form of the Lorenz attractor,21as is evident when projecting the chaotic attrac- tor into the (v,a)-plane. We proceed to characterize the precise form of the chaotic jittering in terms of a Lorenz-like map and discuss the implications of the short timescale dynamics on the asymptotic diffusion. To determine underlying order in the emergent chaotic dynamics, we adopt the approach taken by Lorenz21and determine the times t1<t2<· · ·at which the local wave amplitude, a(t), is maximal, denoting an=a(tn). As presented in Fig. 7(b) , in certain parameter regimes, the data points (an,an+1)approximately lie on a curve (up to a fine-scale, fractal structure), suggesting the dynam- ical form an+1=f(an). To elucidate the structure arising from the cusps of f, we color-code each data point in the (an,an+1)-plane by the signed distance, /Delta1n=sgn(vnvn+1)/vextendsingle/vextendsinglexn+1−xn/vextendsingle/vextendsingle, traveled by the particle in the time interval [ tn,tn+1], where xn=xp(tn),vn=v(tn), and sgn is the signum function. Here, /Delta1n>0 (red shading) corresponds to a continuation of the particle direction at times tnandtn+1, while /Delta1n<0 (blue shading) corre- sponds to a reversal in the direction. This color-coding reveals the approximate functional form /Delta1n=g(an), where changes in the sign of/Delta1narise at the cusps of f(an). This beguiling relationship suggests that the particle’s chaotic dynamics may be idealized by the iterative Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-7 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha map, xn+1−xn=sgn(xn−xn−1)g(an),an+1=f(an), a form to be explored in greater detail elsewhere. We proceed to lend insight into the dependence of the chaotic pilot-wave dynamics on the system parameters. In Fig. 7 , we present the chaotic attractor and the resulting Lorenz-like map for three dif- ferent values of κ0, with /Gamma1=0.92 fixed. For κ0=0.7, the map, f, exhibits three cusps: one cusp arises at a negative value of an, whilethe other two cusps arise for positive values of an. The system has a tendency for changes in the particle direction, as only values of ansufficiently close to the cusps correspond to /Delta1n>0. Moreover, there is a propensity for positive peaks in the local wave amplitude (an>0). However, an<0 arises when an−2is sufficiently close to one of the positive cusps. In this case, an−1is then large and posi- tive and prompts a change in the particle direction at time tn. The following peak ( an<0) yields a small and negative value of /Delta1n: the distance traveled by the particle between times tnandtn+1is short FIG.7.Chaoticpilot-wavedynamicsfor /Gamma1=0.92with κ0=0.7(leftcolumn), κ0=0.9(middlecolumn),and κ0=1.1(rightcolumn).(a)Thechaoticattractorinthephase space(bluecurves).Theblackcurvescorrespondtoprojecti onsofthechaoticattractoronto(fromlefttoright)the (a,b)-,(v,a)-,and(v,b)-planes.(b)TheLorenz-likemap forthesuccessivemaxima, an,ofthelocalwaveamplitude, a,color-codedbythesigneddistance, /Delta1n.Redandblueshadingdenotesarespectivecontinuationora change intheparticledirectionfromtime tntotn+1.Thewhitelineis an+1=an.Theyellowlinesformasamplecobweb.(c)Asampleparticlet rajectory,xp(t),wheretheyellowcurve correspondstothetimeintervalhighlightedbythecobwebi n(b).Markersarecolor-codedbythevalueofthemaximumwav eamplitude, an,arisingattime tn. Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-8 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha FIG.8.Evolutionofthemean-squareddisplacement, D0(t),forκ0=0.7(blue), κ0=0.9 (red), and κ0=1.1 (black). The form D0(t)∼tast→ ∞indicates asymptotic diffusion. The mean-squared displacement was co mputed by aver- aging over N0=25000 realizations, each initialized with xn(0)=a(0)=b(0) =0andv(0)sampledfromastandardnormaldistribution. and there is another reversal in the particle direction. Finally, the fixed points of f[for which an=f(an)] correspond to periodic tra- jectories in the phase space, all three of which are unstable in the current parameter regime. We note, however, that stable periodic trajectories arise for smaller values of κ0. When increasing κ0from 0.7 to 0.9, the outward spirals on the two wings of the chaotic attractor may be tighter: the manifestation of this change is the extension of fcloser toward the origin. Fur- thermore, the largest of the three cusps no longer exists, resulting in the loss of two (unstable) fixed points of the map, f. However, a second negative cusp emerges and the interval of forbidden values ofandecreases in width. We also see the emergence of isolated data points, which one might expect to correspond to further structure of the map, arising for infrequent dynamical behavior. Forκ0=1.1, there is a wide range of anbetween the largest two cusps for which /Delta1n>0; hence, the particle is more prone to continue its motion in the same direction. To characterize the effect of this qualitative change in the particle motion on the long-time dynamics, we consider the mean-squared displacement, D0(t)=1 N0N0/summationdisplay n=1(xn(t)−xn(0))2, where xn(t)is the particle position computed over N0realizations of similarly prepared systems. As evidenced in Fig. 8 , the mean- squared displacement has the form D0(t)∼tast→ ∞ . This long- time behavior is indicative of asymptotic diffusion of the particle position.20Moreover, the asymptotic diffusion coefficient increases asκ0is increased between 0.7, 0.9, and 1.1. It would thus appear that the form of the map, f, is correlated with the asymptotic dif- fusion coefficient since one might expect less frequent changes in the particle direction to increase the asymptotic diffusivity. How- ever, a detailed study of the dependence of the asymptotic diffusion coefficient on the system parameters remains the focus of future work. Finally, we note that for κ0larger than that considered in Fig. 7 , the approximation of a one-dimensional map describing the evolution of anbecomes inadequate, and one must seek a morecomplex projection of the chaotic attractor in order to rationalize its form. Nevertheless, the relationship between anand/Delta1nper- sists, which presumably is a fundamental feature of the chaotic random-walk-like dynamics. VI. DISCUSSION We have characterized the onset and form of steady propul- sion and chaotic behavior arising in a reduced pilot-wave sys- tem, in which a particle is propelled by its monochromatic, one- dimensional guiding wave. As the rate of wave dissipation is decreased, steady particle motion arises via a supercritical pitchfork bifurcation. At sufficiently long path-memory, we observe transient chaos, coexisting periodic and chaotic attractors, and the instability of steady propulsion via a subcritical Hopf bifurcation. Analogous transitions from steady to chaotic dynamics also arise in the Lorenz equations,21forming a new correspondence between pilot-wave hydrodynamics and the Lorenz system. We have provided strong evidence of a homoclinic bifurca- tion arising as /Gamma1is increased, beyond which a pair of unstable limit cycles emerge. Indeed, it appears that this homoclinic bifurcation is in fact part of a Lorenz-like series of homoclinic explosions: each homoclinic explosion gives rise to infinitely many unstable periodic and aperiodic trajectories,22culminating in transient chaos and the emergence of a chaotic attractor. This study, therefore, provides a theoretical rationale for the homoclinic-like bifurcations, coexist- ing attractors, and Shil’nikov chaos reported in prior numerical28–31 and experimental18,28studies of walking droplets. Furthermore, we note that this route to chaos is markedly different in form to con- fined orbital pilot-wave dynamics8,9,32–34for which period-doubling cascades and the Ruelle–Takens–Newhouse scenario have been reported.35 Our study has also revealed that, in certain parameter regimes, the chaotic attractor has the characteristic butterfly-like form of the Lorenz attractor. Furthermore, random-walk-like dynamics and asymptotic diffusion arise at long path-memory.20While the asymp- totic diffusion of the particle motion was rationalized elsewhere,20 the simplicity of the cosine model allows for a far deeper investiga- tion of the short timescale chaotic dynamics. This chaotic behavior was rationalized in terms of a Lorenz-like map with a beguilingly simple structure: each peak of the local wave amplitude, a, deter- mines the following distance and direction traveled by the particle and the next peak of a. As such, one can construct a reduced itera- tive map to describe the random-walk-like dynamics in far greater detail than a simple stochastic model.20 To determine the influence of the form of the wave kernel on the pilot-wave dynamics, we compare the behavior described by the cosine and Bessel models. The onset of particle motion is the same in both models, as is the transition from overdamped to underdamped speed oscillations as /Gamma1is increased, leading to the destabilization of the steady propulsion state via a subcritical Hopf bifurcation.19,20Furthermore, both models exhibit transient chaos and coexisting attractors. However, the form of the chaotic attractor is more complex for the Bessel model, and the approxima- tion of the dynamics by a one-dimensional map is inadequate (see thesupplementary material ). These disparities are a likely conse- quence of the infinite-dimensional phase space of the Bessel model.20 Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-9 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha We, therefore, conclude that, while the pilot-wave dynamics are qualitatively insensitive to the form of the wave kernel for steady particle motion, the differences between the two models become more pronounced when the dynamics are chaotic. In the vicinity of the subcritical Hopf bifurcation ( /Gamma1/lessorsimilar/Gamma1c), we observe speed oscillations similar in form to those observed in experiments;18,28yet, these oscillations are unstable in both the cosine and Bessel models. This disparity may be attributed to phys- ical effects not described by these two idealized pilot-wave systems, such as variations in the droplet’s vertical motion18or the finite con- tact time of the droplet and its pilot wave.28Our study has also revealed the emergence of periodic velocity pulsations in the vicin- ity of the homoclinic bifurcation ( /Gamma1/greaterorsimilar/Gamma1h). It remains, however, the subject of future work to fully explore the manifestation of the stable limit cycles arising at long path-memory. Simulations have revealed stable pulsations, as well as zig-zag motion characterized by rapid particle oscillations superimposed on a net drift. Further character- ization of the myriad limit cycles and their comparison in form to those arising in the Lorenz equations, the Bessel model, and walking- droplet experiments18,28remains the subject of future investigation. Of particular interest is the limit in which inertial effects dominate (κ0/greatermuch1), a regime inaccessible in the walking-droplet system.36 In the walking-droplet problem, chaotic pilot-wave dynamics typically take the form of chaotic switching between unstable peri- odic orbits8,9,13,14,33,37or irregular motion akin to jittering.31,33,38–40 However, underlying structure similar to that elucidated herein might also arise for other rectilinear pilot-wave configurations, pro- viding new insight into the mechanisms underpinning observed quantum-like behavior, such as unpredictable tunneling between submerged cavities.5,6Moreover, a similar approach might ratio- nalize confined droplet dynamics in the long-path-memory limit, such as the form of the emergent wavelike statistics in a rectangular corral41or the mean-pilot-wave potential arising for particle motion confined by a harmonic potential.40This abundance of dynamical behavior emphasizes the richness of pilot-wave hydrodynamics as a burgeoning class of dynamical system. SUPPLEMENTARY MATERIAL See the supplementary material for an implementation of the algorithm presented in Appendix C , movies of the chaotic pilot- wave dynamics, and simulations of the Bessel model. ACKNOWLEDGMENTS I am grateful to Rubén Rosales for an informative discussion about homoclinic bifurcations. APPENDIX A: THE STUART–LANDAU EQUATION GOVERNING THE ONSET OF PROPULSION By defining ε=√ /Gamma1with 0 < ε/lessmuch1, we consider the evolution ofx=(v,a,b)Tin the vicinity of the rest state, x0=(0, 1, 0)T, by posing the multiple-scales expansion, x(t)∼x0+∞/summationdisplay n=1εnxn(t,τ),where τ=ε2tis the slow timescale and xn=(vn,an,bn)T. We sub- stitute this ansatz into (6)and identify each xnthrough a series of solubility conditions. At O(εn), we obtain a system of the form ∂txn+M0xn=Fn, where the singular matrix M0= κ−1 00κ−1 0 0 1 0 1 0 1 arises from the linearization of the rest state at /Gamma1=0. By defining the row vector xl=(κ0, 0,−1)satisfying xlM0=0, it follows from ∂txn+M0xn=Fnthat the solubility condition for bounded xnis that the solution of ∂t(xlxn)=xlFnis bounded as t→ ∞ . At each power of ε, we neglect the contribution to xlFnof terms that decay exponentially on the fast timescale, t, as these terms correspond to decay in xlxn. AtO(ε), we obtain F0=0, which is identical to the result of the linear stability analysis. As xr=(1, 0,−1)Tsatisfies M0xr=0, we conclude that x0(τ)=V(τ)xr, where an evolution equation for the particle velocity, V(τ), is determined at O(ε3). Similarly, consid- eration of the O(ε2)terms yields F2=(0, 1−V2, 0)T. By a similar procedure to O(ε), we consider only the long-lived contribution to the solution x2(τ)=(U(τ), 1−V2(τ),−U(τ))T, where the cor- rection, U, to the particle velocity can be determined at O(ε4). By repeating this procedure at O(ε3), the solubility condition gives rise to the Stuart–Landau equation for V(τ)stated in (10). APPENDIX B: THE STUART–LANDAU EQUATION DESCRIBING THE SUBCRITICAL HOPF BIFURCATION Following from Sec. IV A , we define x=(v,a,b)andxc =(vc,µc,−vc)as one of the two steady propulsion states at µ=µc and pose the multiple-scales expansion, x(t)∼xc+∞/summationdisplay n=1εnxn(t,τ). We substitute this ansatz into (6)and solve for xnthrough construc- tion of solubility conditions defined below. At O(εn), we obtain an equation of the form ∂txn+Mcxn=Fn≡∞/summationdisplay m=−∞F(m) neimωct, (B1) where Mc= κ−1 0 0κ−1 0 vcµc−vc µcvcµc arises from the linearization of (6)about x=xcforµ=µc. We neglect terms that decay exponentially on the fast timescale, t. We note that Mchas two neutrally stable eigenvalues, ±iωc, and one neg- ative real eigenvalue. To obtain a bounded solution for xn, we require thatFnhas no secular terms; specifically, we pre-multiply (B1)by a row vector, xl, satisfying i ωcxl+xlMc=0from which we derive the solubility condition xlF(1) n=0. We now consider xl=(1,αl,βl)and define the column vec- torxr=(1,αr,βr)Tsatisfying i ωcxr+Mcxr=0, where αl,αr,βl, Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-10 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha andβrare readily determined. By following the procedure outlined above, we obtain x1(τ)=V(τ)xratO(ε)and then we compute x2at O(ε2). Finally, the solubility condition at O(ε3)yields the complex Stuart–Landau equation, c0dV dτ= −ςc1V+c2|V|2V, where the complex coefficients are c0=1+αlαr+βlβr, c1=αlαr+βlβr+ αlβr−βlαr −βl αl T M−1 c 0 µc −vc , and c2= αlβ∗ r−βlα∗ r −βl αl T (2iωcI+Mc)−1 0 βr −αr +2 αlβr−βlαr −βl αl T M−1 c 0 Re[βr] −Re[αr] . Equation (12)follows by defining γ1=c1/c0andγ2=c2/c0. APPENDIX C: NUMERICAL COMPUTATION OF LIMIT CYCLES To seek limit cycles of (6), we define the Fourier transform, Fn[f], of a T-periodic function, f(t), as ˆfn=Fn[f(t)]≡1 T/integraldisplayT 0f(t)e−inωtdt, (C1) where ω=2π/T. The inverse transform is then defined as f(t)=F−1[ˆfn](t)≡∞/summationdisplay n=−∞ˆfneinωt. We proceed to derive a system of equations for the period, T, and the Fourier coefficients ˆvn=Fn[v],ˆan=Fn[a], and ˆbn=Fn[b]. We first apply Fnto(6a), from which we deduce that ˆbn= −(1+inωκ0)ˆvn. (C2) Then, by applying Fnto(6b)and(6c), eliminating ˆbnin favor of ˆvnusing (C2), and applying the convolution theorem to the Fourier transform of a product, we obtain the following system of equations forˆvnandˆanfor all n∈Z: (µ+inω)ˆan+∞/summationdisplay m=−∞(1+imωκ0)ˆvmˆvn−m=δn0, (C3a) (µ+inω)(1+inωκ0)ˆvn−∞/summationdisplay m=−∞ˆvmˆan−m=0, (C3b) where δnmis the Kronecker delta. To close the system and introduce an equation for the angular frequency, ω, we recall that the system exhibits temporal invariance. We thus choose the reference timet=0 as a time when v(t)has an extremum, specifically ˙v(0)=0, corresponding to the condition ∞/summationdisplay n=−∞nˆvn=0. (C4) For numerical implementation, we truncate the system (C3) and(C4) by setting ˆvn=ˆan=0 for |n|>N, where Nis a pre- scribed number of frequencies required to accurately describe the limit cycle. To solve (C3) and(C4) for the 4 N+3 unknowns ω,ˆv−N,. . .,ˆvNandˆa−N,. . .,ˆaNfor given /Gamma1 < /Gamma1 candκ0>0, we employ a Newton method. For the results presented in Figs. 3 and4, we first consider /Gamma1/lessorsimilar/Gamma1c, where, as an initial guess for the Newton method, we use the results of the Stuart–Landau equation (12)describing the velocity evolution close to the subcritical Hopf bifurcation. For the results presented in Fig. 5 , we instead use a sim- ulated stable periodic trajectory as the initial guess. In both cases, we then use numerical continuation to track the resultant bifurcation branch when decrementing /Gamma1. As the period, T=2π/ω, appears to asymptote at some unknown value of /Gamma1, it is easier to prescribe the oscillation period, T, and instead treat /Gamma1as an unknown. For numerical results presented in Figs. 3 and4, the Fourier truncation was set to be N=150, which was sufficient to ensure that |v±N|<10−6for all cases considered. (Indeed, only such large val- ues of Nwere necessary for large T.) Given this truncation of the Fourier modes, each root of (C3) and(C4) was computed within a tolerance of 10−10in the maximum-value norm. The bifurcation curve was tracked until the angular frequency, ω, decreased to 0.05, from which the corresponding value of /Gamma1was estimated to satisfy 0< /Gamma1−/Gamma1h<10−4. This method, while allowing for insight near the homoclinic bifurcation, has two primary limitations. First, this method requires a sufficiently good initial guess for each Fourier mode; an alternative method that side-steps this issue is to use return maps, as presented by Sparrow.22Second, we can only compute finite-period trajecto- ries close to the homoclinic bifurcation and not the homoclinic loop itself. However, such a computation is not necessary to provide an accurate estimation of /Gamma1hbased on the local asymptotic behavior ofT. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1J. Walker, “Drops of liquid can be made to float on the liquid. What enab les them to do so?,” Sci. Am. 238, 151–158 (1978). 2Y. Couder, S. Protière, E. Fort, and A. Boudaoud, “Walking and orbiting droplets,” Nature 437, 208 (2005). 3A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, “Inform a- tion stored in Faraday waves: The origin of a path memory,” J. Fluid Mech. 674, 433–463 (2011). 4J. W. M. Bush, “Pilot-wave hydrodynamics,” Annu. Rev. Fluid Mech. 47, 269–292 (2015). 5A. Eddi, E. Fort, F. Moisy, and Y. Couder, “Unpredictable tunneling of a classical wave-particle association,” Phys. Rev. Lett. 102, 240401 (2009). 6A. Nachbin, P. A. Milewski, and J. W. M. Bush, “Tunneling with a hydr odynamic pilot-wave model,” Phys. Rev. Fluids 2, 034801 (2017). Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-11 PublishedunderlicensebyAIPPublishing.Chaos ARTICLE scitation.org/journal/cha 7E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y. Couder, “Path-memory induced quantization of classical orbits,” Proc. Natl. Acad. Sci. U.S.A. 107, 17515–17520 (2010). 8S. Perrard, M. Labousse, M. Miskin, E. Fort, and Y. Couder, “Self-organ ization into quantized eigenstates of a classical wave-driven particle ,”Nat. Commun. 5, 3219 (2014). 9D. M. Harris and J. W. M. Bush, “Droplets walking in a rotating fram e: From quantized orbits to multimodal statistics,” J. Fluid Mech. 739, 444–464 (2014). 10A. U. Oza, D. M. Harris, R. R. Rosales, and J. W. M. Bush, “Pilot-wave dynamics in a rotating frame: On the emergence of orbital quantization,” J. Fluid Mech. 744, 404–429 (2014). 11M. Labousse, A. U. Oza, S. Perrard, and J. W. M. Bush, “Pilot-wave dynam ics in a harmonic potential: Quantization and stability of circular orbit s,”Phys. Rev. E 93, 033122 (2016). 12M. Durey and P. A. Milewski, “Faraday wave-droplet dynamics: Dis crete-time analysis,” J. Fluid Mech. 821, 296–329 (2017). 13D. M. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. W. M. Bush, “Wavel ike statistics from pilot-wave dynamics in a circular corral,” Phys. Rev. E 88, 011001 (2013). 14A. U. Oza, Ø. Wind-Willassen, D. M. Harris, R. R. Rosales, and J. W . M. Bush, “Pilot-wave hydrodynamics in a rotating frame: Exotic orbits,” Phys. Fluids 26, 082101 (2014). 15P. J. Sáenz, T. Cristea-Platon, and J. W. M. Bush, “Statistical proje ction effects in a hydrodynamic pilot-wave system,” Nat. 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Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Westview Press, 2001).24J. Molá ˇcek, “Bouncing and walking droplets: Towards a hydrodynamic pilot- wave theory,” Ph.D. thesis (Massachusetts Institute of Technology, 2013). 25A. U. Oza, “A trajectory equation for walking droplets: Hydrodynamic p ilot- wave theory,” Ph.D. thesis (Massachusetts Institute of Technology, 2014). 26J. K. Kevorkiab and J. D. Cole, Multiple Scale and Singular Perturbation Methods , Applied Mathematical Sciences Vol. 114 (Springer, New York, 1996). 27J. A. Yorke and E. D. Yorke, “Metastable chaos: The transition to sus tained chaotic behavior in the Lorenz model,” J. Stat. Phys. 21, 263–277 (1979). 28V. Bacot, S. Perrard, M. Labousse, Y. Couder, and E. Fort, “Multistabl e free states of an active particle from a coherent memory dynamics,” Phys. Rev. Lett. 122, 104303 (2019). 29A. Rahman and D. Blackmore, “Neimark-Sacker bifurcations and evi dence of chaos in a discrete dynamical model of walkers,” Chaos Solitons Fractals 91, 339–349 (2016). 30A. Rahman and D. Blackmore, “Interesting bifurcations in walkin g droplet dynamics,” Commun. Nonlinear Sci. 90, 105348 (2020). 31M. Hubert, S. Perrard, M. Labousse, N. Vandewalle, and Y. Couder, “T un- able bimodal explorations of space from memory-driven deterministic dynamics,” Phys. Rev. E 100, 032201 (2019). 32S. Perrard and M. Labousse, “Transition to chaos in wave memory dynam ics in a harmonic well: Deterministic and noise-driven behavior,” Chaos 28, 096109 (2018). 33T. Cristea-Platon, P. J. Sáenz, and J. W. M. Bush, “Walking droplets i n a circular corral: Quantisation and chaos,” Chaos 28, 096116 (2018). 34N. B. Budanur and M. Fleury, “State space geometry of the chaotic pilot-wa ve hydrodynamics,” Chaos 29, 013122 (2019). 35L. D. Tambasco, D. M. Harris, A. U. Oza, R. R. Rosales, and J. W. M. Bu sh, “The onset of chaos in orbital pilot-wave dynamics,” Chaos 26, 103107 (2016). 36A. U. Oza, R. R. Rosales, and J. W. M. Bush, “Hydrodynamic spin states,” Chaos 28, 096106 (2018). 37S. Perrard, M. Labousse, E. Fort, and Y. Couder, “Chaos driven by interf ering memory,” Phys. Rev. Lett. 113, 104101 (2014). 38T. Gilet, “Dynamics and statistics of wave-particle interac tions in a confined geometry,” Phys. Rev. E 90, 052917 (2014). 39T. Gilet, “Quantumlike statistics of deterministic wave-par ticle interactions in a circular cavity,” Phys. Rev. E 93, 042202 (2016). 40M. Durey, P. A. Milewski, and J. W. M. Bush, “Dynamics, emergent s tatistics, and the mean-pilot-wave potential of walking droplets,” Chaos 28, 096108 (2018). 41D. M. Harris, “The pilot-wave dynamics of walking droplets in confin ement,” Ph.D. thesis (Massachusetts Institute of Technology, 2015). Chaos30,103115(2020);doi:10.1063/5.0020775 30,103115-12 PublishedunderlicensebyAIPPublishing.

5.0028253.pdf

J. Chem. Phys. 153, 184702 (2020); https://doi.org/10.1063/5.0028253 153, 184702 © 2020 Author(s).Exploring reactivity and product formation in N(4S) collisions with pristine and defected graphene with direct dynamics simulations Cite as: J. Chem. Phys. 153, 184702 (2020); https://doi.org/10.1063/5.0028253 Submitted: 07 September 2020 . Accepted: 25 October 2020 . Published Online: 10 November 2020 Reed Nieman , Riccardo Spezia , Bhumika Jayee , Timothy K. Minton , William L. Hase , and Hua Guo ARTICLES YOU MAY BE INTERESTED IN An Organized Collection of Theoretical Gas-Phase Geometric, Spectroscopic, and Thermochemical Data of Oxygenated Hydrocarbons, C xHyOz (x, y = 1, 2; z = 1–8), of Relevance to Atmospheric, Astrochemical, and Combustion Sciences Journal of Physical and Chemical Reference Data 49, 023102 (2020); https:// doi.org/10.1063/1.5132628 Study of entropy–diffusion relation in deterministic Hamiltonian systems through microscopic analysis The Journal of Chemical Physics 153, 184701 (2020); https://doi.org/10.1063/5.0022818 QCT calculations of O 2 + O collisions: Comparison to molecular beam experiments The Journal of Chemical Physics 153, 184302 (2020); https://doi.org/10.1063/5.0024870The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Exploring reactivity and product formation in N(4S) collisions with pristine and defected graphene with direct dynamics simulations Cite as: J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 Submitted: 7 September 2020 •Accepted: 25 October 2020 • Published Online: 10 November 2020 Reed Nieman,1,2 Riccardo Spezia,3 Bhumika Jayee,2Timothy K. Minton,4,a) William L. Hase,2,b) and Hua Guo1,a) AFFILIATIONS 1Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA 2Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409, USA 3Laboratoire de Chimie Théorique, Sorbonne Université, UMR 7616 CNRS, 4 Place Jussieu, 75005 Paris, France 4Ann and H. J. Smead Department of Aerospace Engineering Sciences, University of Colorado, Boulder, Colorado 80303, USA a)Authors to whom correspondence should be addressed: tminton@colorado.edu and hguo@unm.edu b)Deceased. ABSTRACT Atomic nitrogen is formed in the high-temperature shock layer of hypersonic vehicles and contributes to the ablation of their thermal pro- tection systems (TPSs). To gain atomic-level understanding of the ablation of carbon-based TPS, collisions of hyperthermal atomic nitrogen on representative carbon surfaces have recently be investigated using molecular beams. In this work, we report direct dynamics simulations of atomic-nitrogen [N(4S)] collisions with pristine, defected, and oxidized graphene. Apart from non-reactive scattering of nitrogen atoms, various forms of nitridation of graphene were observed in our simulations. Furthermore, a number of gaseous molecules, including the experimentally observed CN molecule, have been found to desorb as a result of N-atom bombardment. These results provide a foundation for understanding the molecular beam experiment and for modeling the ablation of carbon-based TPSs and for future improvement of their properties. Published under license by AIP Publishing. https://doi.org/10.1063/5.0028253 .,s I. INTRODUCTION Entry of space vehicles into Earth’s atmosphere at high veloci- ties generates a thin, high-temperature shock layer above the vehi- cle’s leading surface, typically a thermal protection system (TPS) material. Within this shock layer, temperatures as high as 10 000 K may be reached, leading to dissociation of O 2and N 2.1,2The result- ing O and N atoms diffuse through the boundary layer and react with the TPS surface. If the TPS is ablative, reaction products are transported back into the flow, which, in turn, affects the chemi- cal state of the boundary layer.3The degradation, or ablation, of the TPS depends on the coupled gas-surface and gas-phase pro- cesses that add heat to the TPS and result in gas-surface reac- tions at high temperatures. To address the challenges of atmo- spheric entry, predictive models are used to guide TPS and vehicle design. The fidelity of such models is improved by increasing theunderstanding of the molecular-level chemistry under relevant extreme conditions. Many TPS materials are based on carbon (e.g., carbon–carbon composite) or become carbonized through charring (e.g., pyroly- sis of a phenolic ablator), so a clear understanding of oxidation and nitridation reactions on carbon is important for many TPS systems.4,5As a result, there has been strong recent interest in experimental and theoretical studies of collisions of the relevant gaseous species with carbon-based materials such as highly ori- ented pyrolytic graphite (HOPG) and vitreous carbon.6–11While the reactions between O 2and O with carbon surfaces have been extensively studied,6,8,9,11,12fewer studies have been conducted on nitrogen reactions on carbon. Nevertheless, there have been some investigations of the effect of nitrogen atoms on carbon surfaces, many of which were J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp conducted in the context of TPS ablation. N atoms were reported by McCarroll and McKee to etch graphite surfaces anisotropically above 1000 K, leaving characteristic hexagonal pits.13These pits are similar to those found with O-atom etching, suggesting that the two species could react through similar reaction mechanisms at sur- face defects.6However, no chemistry was observed with impinging molecular nitrogen. The ablation of a graphite surface by N atoms was also reported by Suzuki et al. , who found that for the range of surface temperatures ( TS), 1351 K–2184 K, the probability of reaction increased from 1.4 ×10−3to 3.2 ×10−3.14Another study of mass-loss measurements from samples of purified graphite ( TS = 873 K–1373 K) exposed to N atoms showed reaction probabili- ties of 0.2–9.8 ×10−3.15To obtain information on the mechanism of the mass loss, products resulting from such reactions have also been detected using various techniques.16–18It was found that the dominant product was C 2N2, which might have been produced via condensation of the nascent CN radicals produced by the reaction of atomic N with the carbon surface. However, there has been a lack of quantitative data on the CN formation.19 In addition to its relevance to TPS chemistry, nitridation of graphene is also of considerable interest in materials science.20The motivation here is to create N-doped graphene materials that have desired semiconducting properties.21One such approach is to bom- bard graphene with neutral or charged atomic nitrogen in plasmas, which causes reactions of these species with the carbon network.22,23 However, the detailed mechanisms of these processes are still poorly understood.24 In order to gain mechanistic insights into the nitridation of carbon, detailed gas-surface scattering investigations have recently been performed. Molecular beam experiments by Murray and Minton studied the scattering of hyperthermal N(4S) atoms and N2molecules with vitreous carbon at high temperatures between 1023 K and 1923 K.10The molecular beam contained N and N 2 with translational energies of 110 kcal/mol and 193 kcal/mol, respec- tively. They found that N 2scattered non-reactively, while some of the N(4S) reacted with the surface to form CN with an Arrhenius activation energy of 49.5 kcal/mol. It was concluded that the CN is formed by a Langmuir–Hinshelwood mechanism as CN was found to desorb in thermal equilibrium with the surface. Subsequent work by Murray et al. with a much higher flux of lower-energy N atoms (∼8 kcal/mol) is more relevant to the conditions in the shock layer on the leading edge of hypersonic vehicles and is thus more rele- vant for the modeling of TPS chemistry. This study also observed the CN reaction product and further refined the CN pathway activa- tion energy to 41.1 kcal/mol. These authors also reported evidence for the formation of N 2from recombination of N atoms. Again, it was believed that N 2was produced via a Langmuir–Hinshelwood mechanism as the translational energy and angular distributions of the molecular product were consistent with thermal desorption. Overall, N atoms were found to be less than 5% as reactive with the surface as O atoms when the surface temperature was below 1900 K.11 To understand the microscopic details of the aforementioned experimental observations, it is necessary to carry out theoreti- cal investigations of the reactive scattering. Although non-reactive scattering of N 2from carbon surfaces has been investigated,25,26 there have been few chemical dynamics investigations that allow the breaking and formation of chemical bonds. The only exception was arecent direct dynamics simulation, in which up to 100 N atoms were directed to a pristine graphene sheet in order to understand nitrogen doping resulting from nitrogen plasma exposure.27With incident energies of 23 kcal/mol, 46 kcal/mol, and 92 kcal/mol, atomic nitro- gen was found to react with the graphene sheet and release various molecular species including N 2, CN, CN 2, and C 2N2. These incident energies are higher than desired for modeling TPS chemistry. In this work, we report direct dynamics studies of the reac- tive scattering of hyperthermal atomic nitrogen in the ground elec- tronic state [N(4S)] on a variety of pristine, defected, and oxidized carbon surfaces, simulated by using the corresponding graphene sheets. The modeling of N-atom reactive scattering from oxidized graphene is of particular significance as it is relevant to air-carbon ablation chemistry, and no experiment has yet been performed that probed N-atom reactions on an oxidized carbon surface. These cal- culations, which use an efficient semi-empirical tight-binding den- sity functional theory (TB-DFT), allow not only the simulation of non-reactive events but also breaking and formation of chemical bonds. The choice of this approximate DFT method is primarily due to its efficiency, which is important for large systems and for long-time trajectories. The aim here is to explore various reaction channels under conditions similar to the aforementioned molec- ular beam experiments10,11and to quantify the reactivity of each channel. To this end, the chemical dynamics calculations were per- formed for atomic N scattering from model graphene surfaces at a relevant experimental temperature (1375 K) including a pristine, non-defected structure [model 1, Fig. 1(a)], a single carbon vacancy (SV) [model 2, Fig. 1(b)], and two oxygenated structures based on the SV surface [models 3 and 4, Figs. 1(c) and 1(d), respectively]. The graphene surfaces are modeled with periacene, and its mod- ified analogs are modeled with artificial edge constraints. Various gas-phase species as well as surface species have been found as a result of the reactive scattering, and the reaction mechanisms are discussed. II. METHODS A. Electronic structure calculations Direct dynamics simulations were performed in which the energy and forces are computed on the fly using the semi-empirical self-consistent charge density functional tight-binding (SCC-DFTB) method.28The SCC-DFTB (DFTB for short) method is an approx- imate density functional theory (DFT) method based on a second- order expansion of the Kohn–Sham total energy in terms of the charge density fluctuation relative to a reference density. It is much more efficient than the conventional DFT method as the Hamiltonian, which contains only terms for nearest neighbors (tight-binding), is parameterized semi-empirically. This method has been successfully employed in many applications, ranging from materials29to biological systems.30Indeed, it was also used in the aforementioned work by Moon et al. in a direct dynamics simula- tion of N-atom scattering from a graphene surface.27In our calcu- lations, the third-order correction31,32was utilized with the 3ob-3-1 Slater–Koster parameter set33and the Lennard-Jones (LJ) dispersion model.34 Because of the semi-empirical nature of the SCC-DFTB method, it is important to assess its accuracy and reliability in J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 1 . Periacene (5a, 6z) graphene models: (a) pristine (model 1), (b) sin- gle carbon vacancy (SV) defect (model 2), (c) SV defect with one oxygen atom (model 3), and (d) SV defect with two oxygen atoms (model 4). describing the systems of our interest. To this end, a detailed com- parison has been performed here against the conventional DFT method with the three-parameter Becke–Lee–Yang–Parr (B3LYP) functional,35,36with the 6-31G∗basis set.37In addition, the D3 dis- persion correction38with Becke–Johnson damping (D3BJ)39was used to handle the dispersion corrections. Fermi smearing was uti- lized to allow for quick convergence of the self-consistent processes. The quartet spin state was selected in all calculations. In such com- parisons, potential energy curves were generated first by optimizing geometry with fixed C–N or C–CN bond distances at several inter- vals with the B3LYP-D3BJ method. The DFTB3-LJ/3ob-3-1 energies were then determined at these geometries for comparison. Hydro- gen atoms were kept fixed to simulate the geometry restrictions of a larger graphene sheet. All DFT calculations were performed with the Gaussian 16 program,40and the DFTB calculations performed with the DFTB+ program.41 B. Direct dynamics simulations Although there have recently been several theoretical studies of atomic and molecular scattering from carbon surfaces, many were restricted to non-reactive events.25,26,42–45These studies used empir- ical force fields that may or may not have an accurate descriptions of the reactive events. To explore the chemical channels in atomic- nitrogen collisions with graphene, we chose to use a direct dynamics approach,46in which the energy and forces for the nuclear motion are computed on the fly. Such approaches have been successfully used by several groups recently on scattering dynamics involving HOPG.9,47–49Here, because of the relatively large number of atoms (and electrons), we have chosen to take advantage of the efficient DFTB Hamiltonian.Direct chemical dynamics simulations were carried out using the VENUS chemical dynamics program50externally interfaced to the DFTB+ program. Namely, the atomic positions of the whole sys- tem provided by VENUS were read by the DFTB+ program and used to calculate energies, which were then used by VENUS to generate the next trajectory steps. The atomic forces were deter- mined numerically using finite differencing (step size of 0.01 atomic units). In this way, it is possible to use all the quasi-classical tra- jectory features present in VENUS (for initial condition generation and for the propagation of an ensemble of trajectories) with the DFTB method. The VENUS/DFTB+ interface can be obtained upon request. In the work reported here, the DFTB3-LJ/3ob-3-1 method was used for all the on-the-fly direct dynamics simulations. The cal- culations were performed with graphene models based on peri- acene (pristine and modified with representative defects) and the ground state N(4S) atom. Initial conditions utilized the options in the VENUS program package for gas-surface interactions. The mass of every hydrogen atom at the edge of the periacene molecule was set to 1000 amu in order to simulate the restrictions present in a larger graphene sheet. The surface normal modes were populated from a Boltzmann distribution with TS= 1375 K. The algorithm for selecting the initial conditions for a gas-surface collision has been described in an earlier work,51and the coordinates are displayed in Fig. 2(a). The initial position and momentum of the N(4S) atom were chosen with respect to a surface plane and defect site. The sur- face plane was defined by three carbon atoms, NN1, NN2, and NN3, at the edge of the periacene sheet [Fig. 2(b)]. The reaction site was defined by displacing the origin of the surface plane, determined by NN1, by distances RX, RY, and RZ. The initial atomic coor- dinates and momenta were chosen randomly from the Boltzmann J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . Schematic for (a) the coordinate definitions used for initial conditions, and (b) carbon atoms, NN1, NN2, NN3, (red circles) define the surface plane in model 1 (these atom choices were made in the same way for all models). distribution at the surface temperature. Five coordinates were used to characterize the initial position and orientation of the N(4S) veloc- ity vector with respect to the surface: b,θ,φ1,φ2, and s. The coordi- nates, bandθ, designate an aiming point on the surface. The impact parameter, b, sets the distance from the defect site for the graphene sheet (models 1–4) and varies from 0 Å to 3 Å in intervals of 0.5 Å. For each value of b, 50 trajectories were calculated. The angle, φ1, was chosen randomly from a uniform distribution from 0○to 360○. The angle, θ, which sets the angle of the velocity vector of the N(4S) atom projectile relative to the surface normal, was fixed at 0○. The ini- tial collision energy of the projectile used to impact the surface was 14.9 kcal/mol and 110 kcal/mol, designed to simulate collision veloc- ities for hypersonic and low-Earth-orbital vehicles, respectively. (We note in passing that the latter could induce electronic excitations, which are not included in our theoretical model.) The angle, φ2, of the N(4S) atom projectile was chosen randomly between 0○and 360○ but was entirely redundant in normal collisions, having no effect on the reaction dynamics. The separation between the graphene sheet and the nitrogen atom, s, was chosen to be 10 Å, sufficiently large to ensure no interaction. All trajectories were integrated in time with the sixth order symplectic integration algorithm,52,53utilizing a time step of 0.5 fs and total simulation time of 3 ps. A trajectory is considered non- reactive if the nitrogen atom impacts the surface and leaves with- out forming a bond or altering the surface in a permanent way within the simulation time, which is confirmed with visualization of the trajectory. Occurrence of novel surface group functionalization, alteration, and gaseous product formation was also ascertained from trajectory visualization. III. RESULTS A. Assessment of the SCC-DFTB model A (5a, 6z) periacene was used to model a larger graphene sheet such as that utilized in a previous dynamics study with O 2.9Full optimization of the periacene with DFT and DFTB methods leads to root mean square difference (RMSD) values between identicalatomic centers that never exceed 0.058 Å for any structure (Fig. S1), suggesting that the DFTB method provides an excellent geometric description of the model graphenes. For the interaction of an N(4S) atom with pristine graphene (model 1), two routes were found from the direct dynamics simula- tions: N bonding to a single carbon center or N bonding to two car- bon centers at a C–C bond. In the first case of an N atom bonding to a single carbon atom, the optimized B3LYP-D3BJ/6-31G∗potential energy curve (PEC) features a barrier and a minimum for the bonded structure, which are 11.0 kcal/mol and −14.2 kcal/mol, respectively, with respect to the dissociation limit (Fig. S2 of the supplementary material), respectively. The minimum features an sp3-like carbon connected with the nitrogen. Along this PEC, the bonding carbon atom can be seen to rise smoothly out of the graphene plane as the N atom approaches the surface and makes the C–N bond (Fig. S3). The corresponding DFTB3-LJ/3ob-3-1 energy along the DFT PEC fol- lows the same trend, but the barrier and minimum are both higher in energy, suggesting underbinding. In the second N + graphene route of N approaching a C– C bond, the nitrogen atom bonds with both carbon atoms and the corresponding PEC is shown in Fig. S4. The potential min- imum at the DFT optimized geometry is below the dissociation limit by about 20.4 kcal/mol. The corresponding DFTB binding energy is at 8.4 kcal/mol above the dissociation limit. The barrier energy of about 15.5 kcal/mol found with DFT at a C–N sepa- ration of 2.1 Å is larger in DFTB at 24.0 kcal/mol. The bonding carbon atoms again smoothly rise out of the plane as the nitro- gen atom approaches the sheet (Fig. S5). This comparison further suggests that DFTB underestimates the binding of nitrogen with graphene. The PEC of a product molecule, CN, leaving an oxygenated sur- face in model 3 was also examined. In this case, a nitrogen atom would have bonded to a carbon atom at the inserted oxygen defect, pulled out of the surface, breaking the C–CN bond and leaving the surface as CN. The potential minimum for CN bonded to a car- bon at the defect site is −104.7 kcal/mol with DFT and is higher in energy, −90.2 kcal/mol, with DFTB (Fig. S6). No potential barrier is observed along the PEC. Selective structures from the PEC in Fig. S7 J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp show the dissociation of CN. As with the other cases, the bonding carbon attaching CN to the sheet falls back into the plane as the CN leaves the sheet (Fig. S8). Analysis of the CN bond distance shows that it does not vary greatly with respect to the C–CN fixed bond distance, decreasing by only about 0.01 Å compared to the free CN at the dissociation limit, 1.178 Å (Fig. S9). This compares well to the literature value for a CN triple bond of 1.157 Å.54 Finally, the PEC for NO leaving model 4 after an impinging N atom reacts with the carbonyl oxygen is found in Fig. S10. A 17.9 kcal/mol barrier is found at a C–ON distance of 1.9 Å, while the potential minimum is located above the complete dissociation limit by 7.28 kcal/mol and a C–ON distance of 1.39 Å. DFTB under binds by about 19.3 kcal/mol at the DFT minimum energy struc- ture. The fully optimized DFTB structure finds the minimum to be 19.0 kcal/mol with a C–ON distance of 1.36 Å. The bonding carbon atom (Fig. S11) goes back into the plane as NO leaves forming the model 3, single oxygen insertion defect. Based on these comparisons, we conclude that DFTB provides qualitatively correct interaction patterns for nitrogen interaction with graphene but underestimates the interaction energy. B. Reactive scattering 1. Model 1—Pristine graphene sheet Most trajectories in model 1 with a translational energy of 14.9 kcal/mol (Fig. 3, red) result in the nitrogen atom rebound- ing from the surface without reacting. The velocity distribution for the scattered nitrogen atoms is shown in Fig. 4. It is clear that the scattered N atoms exchanged much energy with the surface, mostly losing energy to the surface but with a small fraction gaining some energy. The average kinetic energy of scattered N is 9.42 kcal/mol, and the corresponding velocity distribution compares reasonably well with a Maxwell–Boltzmann distribution at 4750 K, much higher than the surface temperature in the experiment.10 The reactivity is quite small (only five trajectories react in total) and is mostly found with large impact parameters, b, nearing the edge of the periacene model, but never increases beyond a prob- ability [ Pr(b)] of 0.06 (3/50 trajectories) for a given bvalue. The only reactive event observed for model 1 at this incidence energy is nitrogen insertion into the graphene sheet breaking a C–C bond, as shown in Fig. 3. In addition to five such reactive trajectories, there are 11 additional trajectories that react with the surface, with N inserting into the sheet, but finally ejecting from the surface within 3 ps simulation time. In light of this, it is possible that these reactive trajectories contributing to the calculation of Pr(b) could eventually be ejected from the sheet with a longer simulation time if energy dis- sipation does not remove the energy in the vicinity of the impact site. Increasing the translational energy of the N(4S) atom to 110 kcal/mol is found to enhance the reactivity (Fig. 3, blue). The probability of reaction for a given bvalue increases slightly to a maximum of 0.12 (six trajectories). The character of the reactive trajectories does not change though, producing nitrogen insertions into the graphene sheet. The number of trajectories where the nitro- gen atom projectile reacts with the surface by inserting, but then is ejected before the total 3 ps passes, increases greatly to 110 of the total 350 trajectories. This accounts for the greater reactivity for the FIG. 3 . Probability of reaction, Pr(b), with respect to the impact parameter, b, from DFTB3-LJ/3ob-3-1 direct dynamics simulations of model 1. FIG. 4 . Velocity distribution of scattered N(4S) atoms from DFTB3-LJ/3ob-3-1 direct dynamics simulations of model 1 with N(4S) translational energy set to 14.9 kcal/mol. Maxwell–Boltzmann distributions shown for 1923 K and 4750 K. 14.9 kcal/mol trajectories at b= 3.0 Å: more of those are likely to stay bonded to the surface during the simulation time, whereas the N atom with 110.0 kcal/mol of incidence energy is much more likely to leave the surface after reacting. This is the only other change in reactivity associated with the higher translational energy N(4S) atom impacting the pristine graphene sheet. At both incidence energies, the nitrogen atom is not observed to undergo significant diffusion on the graphene surface. 2. Model 2—SV graphene The SV defect of model 2 displays increased reactivity com- pared to the pristine sheet of model 1. It has been well established J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp that the SV defect structure is more reactive compared to pristine graphene, owing to the presence of dangling bonds.55,56The prob- ability of reaction at low incidence energy (14.9 kcal/mol, Fig. 5, black) for values of bfrom 0.0 Å to 1.0 Å is either 1.0 or very close. The number of reactive trajectories then decreases with increas- ingbas the number of trajectories resulting in N-atom scattering increases. The trajectories at 14.9 kcal/mol incidence energy display a variety of different nitrogen functionalized configurations in addi- tion to N insertion between into a C–C bond [Fig. 5(c)]. For val- ues of b= 0.0 Å, 0.5 Å, and 1.0 Å, almost all of the trajectories result in the N atom inserting into the SV defect [Fig. 5(a)]. Asexpected, the probability of this process decreases with bthereafter. Beginning with values of b≥1.5 Å, a singly bonded CN group becomes another major configuration when the N atom bonds to a carbon at the defect and pulls it out of plane [Fig. 5(b)]. In addition, N-atom insertions at the defect site can also result in Stone–Wales (SW) [Fig. 5(d)] or square (SQ) configurations [Fig. 5(e)]. Gaseous CN species are also observed coming off the surface, leading to a double carbon vacancy after forming in a similar way to the singly bonded CN group [Fig. 5(f)]. Increasing the incidence translational energy to 110.0 kcal/mol decreases the probability of reaction for b= 0.0 Å and 0.5 Å because the incoming N atom can pass through the SV defect without FIG. 5 . (Top left) Probability of reaction, Pr(b), with respect to the impact parameter, b, from direct dynamics simulations of model 2 with N(4S) translational energy set to 14.9 kcal/mol (red) and 110.0 kcal/mol (black). Probabilities for the two primary N-functionalization products of the 14.9 kcal/mol (middle left) and 110.0 kcal/mol (bottom left) simulations. Typical configurations of these major reaction scenarios are shown in (a) N insertion in SV defect, (b) singly bonded CN group, (c) N insertion in C–C bond, (d) SW insertion, (e) SQ insertion, and (f) CN product formation. J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp reacting or scattering. For larger values of b, the probability of all other reactive configurations increases. A difference is seen for the singly bonded CN and CN product trajectories for small bvalues. The N atom bonds to a carbon atom at the SV defect site and pulls it out of plane as it passes through the vacancy. These trajectories result in functionalization of the opposite side of the sheet. There is a limitation in the current model’s ability to fully char- acterize this reaction with 110.0 kcal/mol of incidence energy as the N atoms are found to pass through the SV defect into vacuum. In an extended system with additional layers or in a real bulk sample, theN atoms would encounter another layer with which to react or from which to rebound. In the case of rebounding, the N atom could then react with the top layer from underneath. Reactions of rebounding N atoms could play a role in the pristine case described earlier in this work and would certainly impact the results of the defect structures discussed below. 3. Model 3—SV graphene with one oxygen atom The presence of an oxygen atom at the SV defect of graphene has a large effect on the reactivity of the N(4S) atom, as illustrated in FIG. 6 . (Top left) probability of reaction, Pr(b), with respect to the impact parameter, b, from direct dynamics simulations of model 3 with N(4S) translational energy set to 14.9 kcal/mol (red) and 110.0 kcal/mol (black). Probabilities for the two primary N-functionalization products of the 14.9 kcal/mol (middle left) and 110.0 kcal/mol (bottom left) simulations. Typical configurations of these major reaction scenarios are shown in (a) for the singly bounded CN group, (b) the doubly bonded CN group at the defect site, (c) N insertion into the C–C bond, (d) N sticking on surface, and (e) replacement of O by N. J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Fig. 6 (red and black columns for incidence energies of 14.9 kcal/mol and 110 kcal/mol, respectively). At the lower incidence energy, reac- tivity of the sheet toward the nitrogen atom decreases with increas- ingb(increasing distance from the defect). For trajectories aiming directly at the defect ( b= 0.0 Å), Pr(b) is the largest at 0.8 and smoothly decreases to 0.18 for trajectories aiming furthest from the defect ( b= 3.0 Å). No gaseous species were found to leave the graphene sheet though the nitrogen impact functionalized the surface resulting in two primary products at the defect site: a singly bonded CN group [Fig. 6(a)] and a doubly bonded CN group [Fig. 6(b)]. The former is found to be the major product and forms after the nitrogen atomattaches to a carbon atom at the oxygen defect, which sticks out of the plane, though it still remains bonded to the graphene sheet in all cases. Conversion between the two forms has been observed. Com- paring the black and green columns in Fig. 6 (top left and middle left plots), this reaction is found to account for almost all of the reactions at each impact parameter, b. The formation of a CN bond has a Pr(b) smaller than that of the attached CN group but still has a probabil- ity of greater than 0.1 for b= 0.5 Å, 1.0 Å, and 1.5 Å. This reaction occurs in a similar way to the attached CN group, but the nitrogen atom only inserts in the CO bond and does not pull a carbon atom out of plane. The reaction probability for this product decreases to zero by b= 2.5 Å. FIG. 7 . (Top left) probability of reaction, Pr(b), with respect to the impact parameter, b, from direct dynamics simulations of model 4 with N(4S) translational energy set to 14.9 kcal/mol (red) and 110.0 kcal/mol (black). Probabilities for the two primary N-functionalization products of the 14.9 kcal/mol (middle left) and 110.0 kcal/mol (bottom left) simulations. Typical configurations of these major reaction scenarios are shown in (a) for the singly bounded CN group, (b) N insertion, (c) gaseous NO product formation, (d) gaseous CO product formation, (e) gaseous NCO product formation, and (f) N sticking on surface. J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Increasing the incidence energy of the impinging N(4S) atom to 110 kcal/mol increases Pr(b) significantly, especially for trajectories with b>0.0 Å (Fig. 6, red). As was found for the 14.9 kcal/mol sim- ulations, most trajectories result in the singly bonded CN group, but this group stays attached to the graphene for the duration of the 3 ps integration time. In contrast to the 14.9 kcal/mol case, at large values ofb(1.5 Å–3.0 Å), a small number of trajectories (eight) is found to have a gaseous CN molecule ejected from the graphene sheet. These gaseous CN molecules are always formed from the attached CN groups. It is possible that the simulation time is insufficient to capture more of these CN groups detaching from the surface. A greater variety of nitrogen-surface functionalization is found with incident atomic N with 110.0 kcal/mol of translational energy (Fig. 6). The major configurations are presented in Figs. 6(a)–6(c). The primary surface functionalization remains the attached CN group as mentioned before, but many minor reaction pathways were observed and at higher probabilities than those at the lower transla- tional energy. Additional surface events such as N-atom insertions into a C–C bond and N-atom bonding to a single carbon center on the surface (N stick) were found. At small values of b, in a small number of trajectories, the incoming nitrogen atom may displace the oxygen atom defect forming a three-fold coordinated nitrogen dopant. 4. Model 4—SV graphene with two oxygen atoms Sampling model 4 with N(4S) atoms with 14.9 kcal/mol of inci- dence energy shows a rather different reactivity profile compared to the reactions with model 3 (Fig. 7, top left, black). Pr(b) values are much lower especially for small values of baiming at or near the defect site. The mechanism governing this involves the carbonyl oxygen atom playing a strong directing role, knocking away the incoming nitrogen atom as it approaches the defect site. Away from the defect site, this carbonyl oxygen plays less of a role in reducing thePr(b) values, and the reactivity becomes similar to that of model 3. The reactions that do occur (Fig. 7 middle left) are mostly N inser- tions into a C–C bond [Fig. 7(b)] or pulling a carbon atom out of the plane as a singly bonded CN group [Fig. 7(a)] as seen in the previous models. The latter mechanism serves to increase the size of the defect as well by damaging the surface. No gaseous product molecules are ejected from the surface. Increasing the incidence energy to 110.0 kcal/mol greatly increases the reaction probability at all values of b(Fig. 7, red). While the carbonyl oxygen still directs the incoming N(4S) projectile as discussed above, the increase in energy results in greater probabil- ity of reaction closer to the center of the defect (small values of b). The probability of reaction increased with increasing bin contrast to model 3. An important contribution to the increased values of Pr(b) comes from the variety of gaseous product molecules, which decreases with bvalues. Several gaseous products were observed to be ejected from the graphene sheet after the impact of the nitrogen atom projec- tile (Fig. 7, bottom left). Especially at values of b= 0.0 Å, 0.5 Å, and 1.0 Å, the increase in reaction probability comes primarily from the production of NO [Fig. 7(c)]. The reaction mechanism is Eley– Rideal (ER) recombination involving the direct reaction of gas phase species (N) and a surface species (O). The nitrogen atom strikes the carbonyl oxygen atom, and rather than being directed away from the sheet as was observed for 14.9 kcal/mol incidence energy, NO formsand is immediately ejected from the graphene surface. CO also forms [Fig. 7(d)] at larger bvalues after the nitrogen atom inserts into a C– C bond involving the carbonyl. The C–CO bond is then cleaved to form a C–N–C insertion at the SV defect in the sheet, while the CO is ejected [Fig. 7(e)]. Another major product is NCO, which is formed in much the same way as CO, but instead of CO leaving the surface, the nitrogen atom bonds to the CO, before the C–NCO bond breaks and NCO is ejected from the surface. Two minor products, oxygen atom produced by N–O replacement and CN, are also formed but at very low probabilities. IV. DISCUSSION The lack of reactivity in model 1 underscores the chemical inertness of pristine graphene. This point has been recognized in several previous studies.11,15As discussed above, the N–C bond is minimally stable in DFTB relative to the asymptote, which is con- sistent with the simulation results. The underbinding of N in DFTB relative to the DFT results suggests that the nitrogen atom should perhaps be significantly more sticky and reactive than revealed in the current simulations. More quantitative simulations might have to await future investigations within the more reliable (and more costly) DFT framework. In addition, we note that these chemisorbed nitrogen species could eventually encounter another surface N species and react to form N 2via a Langmuir–Hinshelwood mech- anism, which will then desorb. This could account for the observed N2products in the recent experiment.11Such events might take a long time, however, and are thus beyond the limit of the current direct dynamics simulations. However, the experimentally observed N2product is very likely to be both internally and translationally hot due to the large exothermicity of the reaction. The fact that the N2was found to have a Maxwell–Boltzmann distribution at the sur- face temperature11supports the notion that the scattered/desorbed species undergo further collisions with the rough carbon surface that was used in the experiment. Despite significant energy loss to the surface, interestingly, the scattered atomic nitrogen from defect-free graphene (model 1) pos- sesses a kinetic energy distribution that is markedly hotter than the experimental surface temperature of 1923 K.11As discussed above, the majority of the N-atom scattering trajectories are direct, without chemisorption on the graphene surface, and as such, N is unlikely to equilibrate with the surface in a single collision. This could be an artifact of the underbinding in DFTB or the small periacene model used to simulate graphene. On the other hand, we note that inelastic scattering of N atoms on vitreous carbon with 110 kcal/mol of inci- dence energy also showed high translational energy characteristic of impulsive, or direct, scattering.10The experiment with 8.2 kcal/mol N atoms incident on a vitreous carbon surface found that non- reactive N atoms desorbed in a Maxwell–Boltzmann distribution at the surface temperature, but the surface in this experiment was very rough, and it is thus likely that the scattered N atoms suffered many collisions before escaping to the gas phase, leading to scatter- ing angle randomization and significant energy transfer, ultimately yielding scattering dynamics that are characteristic of thermal des- orption. This speculation could be verified experimentally by using pristine HOPG instead of vitreous carbon. Extensive nitridation of the graphene surface was found in all models, but the precise functionalization differs. Our simulations J. Chem. Phys. 153, 184702 (2020); doi: 10.1063/5.0028253 153, 184702-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp suggest chemisorption at a bridge site, insertion into the C–C bond, the formation of CN moiety, and the insertion into the defect. Some of the motifs have been reported in a recent simulation of N col- lisions with graphene.27These surface species underscore the wide range of bonding the nitrogen can have with the carbon surface. The ultimate fate of these species was not explored in the current study because of the long simulation time required. We plan to examine the diffusion and reaction barriers of these species in future work, which will shed light on the degradation mechanisms of carbon surfaces. Our simulations found gaseous CN formed in models 2, 3, and 4 with graphene defects. The CN moiety is strongly bound with graphene, as shown in Fig. S6, with a binding energy of 104.7 kcal/mol and 90.2 kcal/mol at the DFT and DFTB levels of the- ory, respectively. Since the potential energy curve did not exhibit any barrier, the activation energy of 41 kcal/mol determined in the recent experiment11could be due to thermal desorption of CN. However, the current calculated binding energy is significantly larger than the experimental value, and this disagreement could conceivably stem from the highly defected carbon surface. This speculation of course requires further investigation, particularly by calculating the corre- sponding free-energy change for the CN bond in different defect sites of graphene. Our simulations further predicted several other gas phase molecules, such as NO, CO, NCO, and O, that have not been observed in experimental studies. NO, for example, represents a major product channel in model 4. However, NO is only formed at the higher incidence energy, so its relevance to the gas-surface chem- istry on the TPS of a hypersonic vehicle in dense air might not be a significant concern. Unfortunately, this species was not investigated in the recent experiments, so the theoretical prediction has yet to be confirmed. V. CONCLUSIONS This work reports an exploratory study of the physical and chemical processes initiated by the collisions of energetic nitro- gen atoms in their ground electronic state with pristine graphene and its modified forms, aiming to understand the underlying reac- tion dynamics related to recent experiments. The collision dynamics was computationally simulated with a model graphene sheet, using a direct dynamics method based on a semi-empirical electronic Hamiltonian. Apart from non-reactive scattering of the impinging N atom, several reactive events have been discovered. The nitro- gen atom was found to stick to the graphene surface, insert into the C–C bond and defect, functionalize the graphene to form vari- ous surface species, and form products that desorb from the surface. In particular, the experimentally observed scattered N atoms and CN molecules were confirmed. Furthermore, several other gaseous species were predicted. The detailed information provides insight into the chemistry in the shock layer at the gas-surface interface above the TPS, which is invaluable for the design of TPSs for re-entry vehicles. Despite the wealth of information provided by the simulations, it is also recognized that a complete and quantitative understanding of the chemical processes is still far from being realized. The semi- empirical method used here is shown to be reasonable in describing bond forming and bond breaking, but it is far from quantitative.Improvements are sorely needed to provide a more reliable char- acterization of the gas-surface interactions, presumably by using the more expensive and more accurate density functional theory. The modeling of the graphene sheet using periacene is another poten- tial shortcoming of the current model, which might provide a poor description of the energy transfer process between the projectile and the surface, and totally ignores energy exchange between the graphene layers. These energy dissipation processes are expected to impact reactivity and dynamics in a significant way. A more real- istic model is needed to simulate both scattering and energy trans- fer. Experimentally, a better characterization of the carbon surface is needed to compare with theoretical simulations. Work in these directions is underway in our groups. SUPPLEMENTARY MATERIAL See the supplementary material for comparison of the results calculated from DFT and DFTB methods. ACKNOWLEDGMENTS This work was supported by the Air Force Office of Scien- tific Research (Grant No. FA9550-18-1-0413 to H.G.), the Robert A. Welch Foundation (Grant No. D-0005 to W.L.H.), Texas Tech Uni- versity (to W.L.H.), University of Colorado (to T.K.M.), and ANR DynBioReact (Grant No. ANR-14-CE06-0029-01 to R.S.). The com- puter time from the High Performance Computational Center at Texas Tech University and the Center for Advanced Research Com- puting at the University of New Mexico is gratefully acknowledged. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1P. R. Mahaffy, M. Benna, T. King, D. N. Harpold, R. Arvey, M. Barciniak, M. Bendt, D. Carrigan, T. Errigo, V. Holmes, C. S. Johnson, J. Kellogg, P. Kimvi- lakani, M. Lefavor, J. Hengemihle, F. Jaeger, E. 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5.0024020.pdf

J. Chem. Phys. 153, 144705 (2020); https://doi.org/10.1063/5.0024020 153, 144705 © 2020 Author(s).The effects of oxygen-induced phase segregation on the interfacial electronic structure and quantum efficiency of Cs3Sb photocathodes Cite as: J. Chem. Phys. 153, 144705 (2020); https://doi.org/10.1063/5.0024020 Submitted: 05 August 2020 . Accepted: 17 September 2020 . Published Online: 09 October 2020 Alice Galdi , William J. I. DeBenedetti , Jan Balajka , Luca Cultrera , Ivan V. Bazarov , Jared M. Maxson , and Melissa A. Hines COLLECTIONS Paper published as part of the special topic on Special Collection in Honor of Women in Chemical Physics and Physical ChemistryWCP2020 ARTICLES YOU MAY BE INTERESTED IN Near atomically smooth alkali antimonide photocathode thin films Journal of Applied Physics 121, 044904 (2017); https://doi.org/10.1063/1.4974363 Surface photovoltage spectroscopy observes junctions and carrier separation in gallium nitride nanowire arrays for overall water-splitting The Journal of Chemical Physics 153, 144707 (2020); https://doi.org/10.1063/5.0021273 Probing the temperature profile across a liquid–vapor interface upon phase change The Journal of Chemical Physics 153, 144706 (2020); https://doi.org/10.1063/5.0024722The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp The effects of oxygen-induced phase segregation on the interfacial electronic structure and quantum efficiency of Cs 3Sb photocathodes Cite as: J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 Submitted: 5 August 2020 •Accepted: 17 September 2020 • Published Online: 9 October 2020 Alice Galdi,1 William J. I. DeBenedetti,2 Jan Balajka,2 Luca Cultrera,1 Ivan V. Bazarov,1 Jared M. Maxson,1and Melissa A. Hines2,a) AFFILIATIONS 1Cornell Laboratory for Accelerator-Based Sciences and Education and Department of Physics, Cornell University, Ithaca New York 14853, USA 2Department of Chemistry and Chemical Biology, Cornell University, Ithaca New York 14853, USA 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: Melissa.Hines@cornell.edu ABSTRACT High-performance photocathodes for many prominent particle accelerator applications, such as x-ray free-electron lasers, cannot be grown in situ . These highly reactive materials must be grown and then transported to the electron gun in an ultrahigh-vacuum (UHV) suitcase, during which time monolayer-level oxidation is unavoidable. Thin film Cs 3Sb photocathodes were grown on a variety of substrates. Their performance and chemical state were measured by x-ray photoelectron spectroscopy after transport in a UHV suitcase as well as after O2-induced oxidation. The unusual chemistry of cesium oxides enabled trace amounts of oxygen to drive structural reorganization at the photocathode surface. This reorganization pulled cesium from the bulk photocathode, leading to the development of a structurally complex and O 2-exposure-dependent cesium oxide layer. This oxidation-induced phase segregation led to downward band bending of at least 0.36 eV as measured from shifts in the Cs 3 d5/2binding energy. At low O 2exposures, the surface developed a low work function cesium suboxide overlayer that had little effect on quantum efficiency (QE). At somewhat higher O 2exposures, the overlayer transformed to Cs 2O; no anti- mony or antimony oxides were observed in the near-surface region. The development of this overlayer was accompanied by a 1000-fold decrease in QE, which effectively destroyed the photocathode via the formation of a tunnel barrier. The O 2exposures necessary for degrada- tion were quantified. As little as 100 L of O 2irreversibly damaged the photocathode. These observations are discussed in the context of the rich chemistry of alkali oxides, along with potential material strategies for photocathode improvement. Published under license by AIP Publishing. https://doi.org/10.1063/5.0024020 .,s INTRODUCTION Many prominent particle accelerator applications, such as x-ray free electron lasers and electron-ion colliders, require high bright- ness electron beams produced by optical excitation of a photocath- ode. In this context, high brightness means that the photocathodes must produce a large density of electrons all moving in the same direction (i.e., with the same momentum) upon irradiation. High beam brightness places stringent requirements on the photocathode. If the emitted electrons are to move with the same momentum, they must all be excited from one specific pointin the photocathode’s Brillouin zone, which requires threshold photoexcitation from an oriented, crystalline material. To avoid laser-induced electron heating and multiphoton photoemission,1,2 which would lead to excitation of a range of momenta, a semicon- ducting photocathode must be used. In semiconductors, electron– electron scattering is only important for excitation energies more than twice the band gap. As a result, semiconductors generally have higher quantum efficiency (QE) near the threshold and much smaller laser-induced electron heating than metals. To minimize the lateral divergence of the electron beam (i.e., the mean transverse energy of the beam), the electrons must be excited from a band J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp with a low effective mass.3Finally, to ensure that the electron beam is not deflected by stray electric fields as it leaves the surface, the photocathode must be atomically smooth and have a uniform work function down to the nanometer scale.4,5An ideal photocathode must also have high quantum efficiency and operate at a convenient wavelength, ideally in the visible or near-infrared spectrum. Alkali antimonides, such as Cs 3Sb and KCs 2Sb, have emerged as a promising class of next-generation photocathodes; however, many challenges remain in the growth and characterization of ori- ented, epitaxial alkali antimonide films with morphologically and chemically homogeneous surfaces. Like all materials that emit elec- trons easily, the alkali antimonides are easily oxidized and, thus, highly reactive. These materials are typically grown in a dedicated facility with few in situ diagnostics for monitoring growth, so many growers adjust their reactant fluxes to maximize the quantum effi- ciency (QE) of the growing photocathode at a convenient wave- length. Nevertheless, it has not been confirmed that the brightest photocathodes have the highest QE. After growth, the films must be transported to the electron gun in an ultrahigh-vacuum (UHV) suit- case, during which monolayer-level (or worse) oxidation is unavoid- able. In this study, we characterize the chemical composition of photocathodes grown using this protocol and study the effects of oxidation on a prototypical alkali antimonide, Cs 3Sb. Surface chemistry can have a profound effect on photocathodes even though near-threshold photoemission is not a surface process in semiconductors. The relevant length scales and energies for typ- ical∼100-nm-thick Cs 3Sb photocathodes are illustrated in Fig. 1. When the photocathode is irradiated with the above-threshold light (e.g., 530 nm or 2.34 eV), low energy electrons with a long atten- uation lengths ( ∼26 nm calculated) are generated within ∼40 nm of the surface (see the supplementary material for length scale cal- culations). Some of these photoelectrons travel to the surface. If FIG. 1 . The relevant length scales for these experiments are set by the attenuation length of the near-threshold (low energy) photoelectrons, high-energy core elec- trons for x-ray photoemission spectroscopy (XPS) in normal and glancing (70○) detection, and the characteristic length of band bending in Cs 3Sb.the surface is pristine, electrons with an energy greater than the Cs3Sb work function can escape into the vacuum. In contrast, an overlayer, such as a cesium oxide, will lead to the formation of a heterojunction. If the overlayer has a smaller work function than the semiconductor, the overlayer may donate electrons to the semi- conductor, becoming itself positively charged. This charge transfer can lead to the downward band bending in the semiconductor, an acceleration of the emitted electrons, and a lower energy thresh- old for photoemission. If the overlayer has a larger work function than the semiconductor, the semiconductor may donate electrons to the overlayer. This charge transfer may lead to upward band bending in the semiconductor, a deceleration of the emitted elec- trons, and a higher energy threshold for photoemission. In Cs 3Sb, band bending has a characteristic length scale of ∼1 nm at room temperature. This study uses x-ray photoemission spectroscopy (XPS) to monitor the surface chemistry of Cs 3Sb photocathodes. The high energy photoelectrons generated by core-level excitation have a much shorter mean free path ( ∼1.5 nm) than the near-threshold photoelectrons. Depending on the emission angle of the detected photoelectrons, XPS probes chemical composition within ∼0.5 nm (70○emission) or ∼1.5 nm (normal emission) of the photocathode surface. The unusually rich oxidation chemistry of cesium plays an important role in the performance of Cs 3Sb photocathodes. Cs metal has a relatively low work function of 1.95 eV,6which is compara- ble to that of Cs 3Sb (2.05 eV).7Upon oxidation, cesium forms at least 9 distinct compounds ranging in stoichiometry from Cs 7O to CsO 3.8When a metallic Cs film reacts with ∼1 L of O 2at low temper- atures, a superoxide with a 1.3 eV work function, assigned to Cs 11O3, first forms.9Additional O 2exposure ( ∼2.5 L) leads to the devel- opment of Cs 2O2, a peroxide, which has a 1.0 eV work function.9 This unusual behavior has been explained in terms of quantum con- finement.10–12Qualitatively, Cs 11O3can be envisioned as superoxide clusters embedded in the Cs metal. The superoxide clusters con- fine the free electrons to the metallic region, thereby, raising the Fermi energy of this “cluster metal” with respect to the free metal. With more O 2exposure, a true oxide, assigned to Cs 2O, with a work function of ∼2.6 eV forms.9This behavior is summarized in Fig. 2. Although the oxides of cesium do have well-defined stoi- chiometries and structures, many syntheses—including syntheses in ultrahigh vacuum (UHV)—lead to mixed-phase materials of vary- ing compositions and properties. For example, the work function of cesium oxides synthesized from Cs and O 2are dependent on the initial thickness of the initial Cs film.9,13Similarly, bulk Cs 2O is reported to have a work function of 6.2 eV,14which is much larger than that reported for UHV syntheses.9 We show that this unusual chemistry of cesium oxides enables trace amounts of oxygen to drive structural reorganization at the surface of Cs 3Sb photocathodes. This reorganization pulls cesium from the bulk photocathode, leading to the development of a struc- turally complex and O 2-exposure-dependent cesium oxide layer on the photocathode surface. This oxidation-induced phase segrega- tion leads to downward band bending of at least 0.36 eV. At low O2exposures, the surface develops a low work function cesium suboxide overlayer that has little effect on quantum efficiency. At somewhat higher O 2exposures, the overlayer transforms to cesium J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 2 . The oxidation of cesium leads to first a decrease and then an increase in the work function. Some cesium oxides have a lower work function than Cs 3Sb, others larger. oxide, Cs 2O; no antimony or antimony oxides are observed in the near-surface region. The development of this overlayer is accom- panied by a 1000-fold decrease in quantum efficiency, which effec- tively destroys the photocathode. We show that this QE decrease is explained by the formation of an oxide tunnel barrier. We also quantify the O 2exposures necessary for significant changes in pho- tocathode performance, showing that a 100 L exposure irreversibly damages the photocathode. EXPERIMENTAL Cs3Sb photocathodes were grown on five different single crys- tal substrates: twice on Si(100), and once each on Al 2O3(10¯10), rutile TiO 2(001), and cubic SiC (3C–SiC) (100). Cs 3Sb has a cubic struc- ture, and all of these substrates have a 90○angle between their in-plane lattice parameters. Two of the substrates, TiO 2(001) and SiC(100), have a good lattice match to Cs 3Sb, whereas the others do not. The choice of the substrate played a minor role in the results reported here ( vide infra ). The Si(100) substrates were prepared using standard cleaning protocols followed by a NH 4F etching procedure that yields a near- atomically flat, H-terminated surface.15,16The other substrates were sequentially sonicated in acetone and methanol for 15 min, then rinsed with ultrapure H 2O (Millipore). Cs3Sb photocathodes were grown by simultaneous dosing from shuttered effusive metal sources loaded with Sb (99.9999%, Alfa Aesar) and Cs (99.5%, STREM Chemicals) at a substrate tempera- ture of 70○C in a high vacuum deposition module (base pressure ∼3×10−9mbar) similar to that previously reported.17The sample stoichiometry was controlled by first measuring the Cs flux using a quartz crystal microbalance, then opening the Sb 4source18and measuring the Sb flux by the mass rate difference. Typical fluxes were 1–1.3 ×1013and 3 ×1012atoms cm−2s−1for the Cs and Sb4sources, respectively. During growth, the sample was biased at −18 V and illuminated with a mechanically chopped 504 nm diode laser; lock-in techniques were used to monitor the drain current (i.e., the photocurrent). The Sb source temperature was adjusted by a few degrees during growth to maximize the photocurrent and, thus, theQE. Growth was terminated when the photocurrent was saturated and stable with the source shutters closed. The photocathodes were transported to a different building for chemical analysis using a custom ultrahigh vacuum suitcase (base pressure ∼10−10mbar) pumped by a compact 100 l/s non- evaporative getter—8 l/s ion pump (SAES Getters). During transfer, mechanical vibrations caused pressure spikes of ∼10−9mbar. The suitcase was then attached to a pre-baked load lock chamber, and the sample was transferred to an ultrahigh vacuum chamber (base pressure∼2×10−10mbar). The photoresponse of the cathode could be monitored in both chambers and during transfer. After growth, the full spectral response of the photocathodes was determined before transfer by measuring QE as a function of the wavelength using a monochro- mated lamp source. In the other chambers, QE was measured using a picoammeter, laser diodes, and a biased coil placed near the sample. X-ray photoemission spectroscopy (XPS) was performed on electrons collected at both 0○and 70○from the surface normal after excitation with Mg K αx rays. A Tougaard baseline was removed from all spectra.19The Cs 3 dspectrum consists of two very similar spin–orbit doublets, so only the spectrum of the more intense 3 d5/2 transition is reported. The Sb 3 dspectrum also consists of two nearly identical spin–orbit doublets; however, the O 1 sspectrum overlaps with the Sb 3 d5/2transition. To resolve the O 1 sspectrum, the Sb 3d5/2transition has been digitally subtracted from the reported spec- tra using the known intensity ratio (2:3) and measured splitting (9.35 eV) between the Sb 3 d3/2and Sb 3 d5/2transitions. The accu- racy of this procedure was verified using the spectrum of unoxidized Sb metal as shown in the supplementary material. RESULTS QE and spectral response Figure 3 shows the typical spectral response of a Cs 3Sb pho- tocathode. The QE of the as-grown photocathodes was typically FIG. 3 . Representative spectral response of the Cs 3Sb photocathode grown on TiO 2(001) (line) after growth and (markers) after transport to the analysis chamber in the UHV suitcase. J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp 3%–7% at 530 nm, and the photoemission threshold was ∼700 nm. After the photocathodes were transferred to the analysis chamber in the UHV suitcase, a small decrease in QE at short wavelengths was observed. Chemical state of transferred photocathodes XP survey spectra of the photocathodes after transfer, shown in the supplementary material, displayed Sb, Cs, and O transitions with only trace amounts of C. Representative high-resolution spec- tra of the Cs 3 d5/2(purple), Sb 3 d3/2(blue), and O 1 s(red) transitions for two different photocathodes grown on two different substrates and transferred in two different runs are shown in Fig. 4. The spec- tra in the right panel were obtained at glancing (70○) detection and are more surface sensitive, whereas the spectra in the left panel were obtained at normal detection and probe somewhat deeper into the photocathode. All spectra have the same eight fiducial energies denoted by dotted vertical lines. Systematic deviations between these energies and the observed maxima are discussed later in the section on interfacial band bending. All samples displayed an intense Sb 3 d3/2transition near 535.4 eV that we attributed to Sb3−in Cs 3Sb in agreement with pre- vious researchers.20–22Some spectra displayed a distinct transition near 539.5 eV, which we attributed to an antimony oxide. Further assignment of this transition to Sb5+(e.g., CsSbO 3, Sb 2O5) or Sb3+ (e.g., Sb 2O3) is uncertain. In agreement with Liu et al. ,23we found that the measured chemical shift of this oxide with respect to Sb0 was more consistent with the formation of either the more ener- getically stable Sb 2O5or with CsSbO 3as suggested by Bates et al.20 Finally, spectra taken in glancing detection sometimes displayed a high binding energy shoulder, which we attributed to the formation of Sb−or Sb0species. These energies are denoted by vertical dotted lines in Fig. 4 and are summarized in Table I. All samples displayed an intense Cs 3 d5/2transition in the range 725.3 eV–726.3 eV, which we attributed to Cs+in Cs 3Sb and cesiumoxides. Previous researchers studying elemental and oxidized Cs films have assigned the Cs+and Cs0transitions to 724.95 eV and 726.25 eV, respectively.6,13These energies are indicated by verti- cal dotted lines in Fig. 4. Our assignment was not based primarily on binding energy. First, the presence of an intense Sb3−transi- tion implies the presence of a cationic species, which can only be Cs+. Second, we ruled out the possibility of significant Cs0con- tributions because metallic Cs displays a characteristic, almost tri- angular lineshape due to intense plasmon losses.6,13Additionally, we observed very weak Cs 5 stransitions in survey spectra (supple- mentary material). Bates et al.24have previously demonstrated that the ratioed areas of the Cs 5 s/Cs 5 ptransitions are inversely corre- lated with photocathode performance as the Cs 5 stransition is due to Cs0from photocathode degradation. We observed loss features from 728 eV to 733 eV, which we attributed to plasmon excita- tions; oxidation reduced the intensity of these features ( vide infra ). Similar losses have been observed in Cs 3Sb22,24and Cs 2Te25photo- cathodes, with stronger losses being correlated with higher quality photocathodes.22 The presence of a Cs0multilayer on the surface can be ruled out based on the desorption kinetics of Cs multilayers, as the low melting point of Cs (28.4○C) leads to a short residence time at the tempera- tures relevant to our experiments. The desorption of Cs multilayers from metal [Cu(110)26] and semiconductor [TiO 2(110)27] surfaces has been studied using temperature programmed desorption. The desorption from these two surfaces display similar kinetics in the multilayer regime,27where the primary interaction is between Cs atoms. Using a Redhead analysis (supplementary material),28we calculated the rate of desorption from a Cs bilayer at our growth temperature of 70○C to be 8.4 monolayers/s (ML/s), which cor- responds to a characteristic (1/ e) lifetime of 0.12 s for the outer monolayer. The rate of desorption from a Cs bilayer during trans- port, analysis or storage at 20○C was a calculated as 0.074 ML/s, which corresponds to a characteristic lifetime of 14 s for the outer monolayer. FIG. 4 . Representative XP spectra of as- transferred Cs 3Sb photocathodes grown on [(a) and (b)] Si(100) and [(c) and (d)] Al 2O3(10¯10) taken with normal (left) and glancing (right) detection. The Cs 3d5/2, Sb 3 d3/2, and O 1 stransitions are shaded purple, blue, and red, respec- tively. The dotted vertical lines represent the fiducial energies discussed in the text and summarized in Table I. The relative intensities of the two spectra in each set are arbitrarily scaled. J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp TABLE I . Summary of fiducial energies denoted by dotted vertical lines in XP spectra. Energy (eV) Species Energy (eV) Species 527.4 O2−537.6 Sb0 529.6 O 22−539.5 Sb5+ 531.6 O 2−725.0 Cs+ 535.4 Sb3−726.2 Cs0 The rate of desorption of a Cs0monolayer cannot be estimated without further understanding of the near-surface structure. This is particularly relevant in the case of cesium oxides where metastable de-excitation spectroscopy has shown that Cs 11O3consists of a sur- face Cs layer with buried oxygen.11In this case, the surface cesium is bound to the underlying oxygen and presumably has different desorption kinetics than a Cs multilayer. Finally, all samples displayed some degree of oxidation, as indi- cated by the O 1 stransitions shaded in red in Fig. 4, which are attributed primarily to cesium oxides. Cesium oxidizes to form superoxides (O 2−), peroxides (O 22−), and oxides (O2−). These species have characteristic O 1 sbinding energies, which were assigned by Jupille et al.29by comparison of valence and core level photoemission spectra and also studied by others.9,13,30The surfaces of the as-transferred photocathodes displayed intense superoxide transitions at ∼531.6 eV, and some photocathodes displayed a small peroxide transition at ∼529.6 eV. Most photocathodes also showed a small oxide transition at ∼527.4 eV. These three transitions are often assigned to the compounds Cs 11O3, Cs 2O2, and Cs 2O; however, the superoxide and peroxide species are rarely phase pure.29 The O 1 sbands are almost exclusively due to cesium oxides, as can be inferred from the relative areas of the O 1 sand the antimony oxide (539.5 eV) transitions and the >7:1 relative cross sections for Sb 3dand O 1 sphotoionization.31 Taken together, these spectra show that the as-transferred pho- tocathodes are Cs 3Sb, but have a predominantly cesium superox- ide surface coating. The formation of this coating is accompanied by partial reduction of antimony to Sb−or Sb0in the near-surface region and a small amount of antimony oxidation. This process pro- duces a vertically heterogeneous photocathode, as shown by the sig- nificant differences between the 0○and 70○spectra. In the absence of a structural model, this vertical heterogeneity precluded quantitative compositional analysis of the photocathode. Effect of trace O 2on photocathodes To understand the effects of trace oxidation during and after transfer, we dosed one photocathode with O 2(g) while monitor- ing the chemical and photophysical effects, as shown in Fig. 5. The photocathode had an initial QE of 2.6%, which dropped to 2.1% after transfer in the UHV suitcase. The initial chemical composition of the photocathode was similar to those in Fig. 4. A 1 Langmuir (1 L) O 2exposure had a small effect on QE, but led to significant increases in the amount of antimony and cesium oxides. This treat- ment also removed the small shoulder attributed to Sb−and Sb0on the as-transferred photocathode. Exposure to an additional 5 L of FIG. 5 . The effects of increasing O 2exposure on a Cs 3Sb photocathode grown on Al 2O3(10¯10) with an initial QE = 2.6% at 530 nm. (a) As transferred using the UHV suitcase (QE = 2.1% at 504 nm), (b) after 1 L of O 2(g), the QE dropped to 1.95%, (c) after an additional 5 L of O 2(g), the QE dropped to 0.69% at 504 nm, and (d) after an additional 94 L of O 2(g), the QE dropped to 0.0012% at 504 nm. All spectra were measured at glancing detection (70○detection). The dotted vertical lines represent the fiducial energies discussed in the text and sum- marized in Table I. The relative intensities of the two spectra in each set are arbitrarily scaled. O2decreased the QE to one-quarter of its original value and further increased the antimony and cesium oxides. The nature of the cesium oxides also changed significantly. After transfer or 1 L O 2exposure, superoxide and peroxide species dominated. After a 6 L exposure, cesium oxide began to dominate. The photocathode became effectively unusable (QE = 0.0012%) after only 100 L of O 2. At this stage, a thick layer of cesium oxide cov- ered the surface, and no antimony species were detectable in either glancing or normal detection XPS. The XP spectra were angle inde- pendent and, thus, vertically homogeneous within the XPS sampling depth, which enabled chemical quantification. The integrated areas of the Cs 3 d5/2and O 1 stransitions were consistent with the pro- duction of Cs 2.5O, a Cs-rich oxide surface layer; however, the large J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 6 . The best fit of the quantum efficiency of the photocathode in Fig. 5 to Eq. (1) is consistent with an effective O 2exposure of 0.96 ±0.33 L in transport. The inset shows the first four measurements on a logarithmic vertical scale. (11:1) difference in photoionization cross sections for Cs 3 d5/2and O 1smakes this quantification imprecise. Taken together, Figs. 4 and 5 show that O 2drove structural reorganization at the surface of Cs 3Sb photocathodes. This reorga- nization pulled cesium from the bulk photocathode, leading to the development of a structurally complex and O 2-exposure-dependent cesium oxide layer on the photocathode surface. At low O 2expo- sures, the surface develops a low work function, potentially benefi- cial cesium suboxide overlayer that contained both superoxide and peroxide species. The migration of Cs to the photocathode surface was accompanied by some oxidation of Sb that was initially on the photocathode surface. There was no indication of the Sb migration. The initial cesium suboxide overlayer had only a small effect on QE. At somewhat higher O 2exposures, the overlayer transformed into a thicker and much higher work function cesium oxide, Cs 2O, which completely buried the initial surface. The observed QE loss with an increase in the O 2exposure was consistent with the development of a nominally Cs 2O tunnel barrier on the surface. Cs 2O is an n-type semiconductor with an experimen- tal bandgap of 2 eV–3 eV.14Oxidized Cs films produce Cs 2O with a work function of 2.59eV–2.7532eV. In the initial stages of oxidation, the thickness of the oxide layer is expected to grow linearly with O 2 exposure. In this regime, the QE of the photocathode as a function of O 2exposure is predicted to be QE(ΛO2)=QE 0e−α(Λtransport +ΛO2), (1) where QE 0is the initial QE after growth, Λtransport is the effective O2exposure during photocathode transport, ΛO2is the additional O2exposure after transport, and αis a constant proportional to the square root of the barrier height. The best fit to Eq. (1) is shown in Fig. 6 and suggests that the photocathode received an effective O 2 exposure of 0.96 ±0.33 L during transport. Interfacial band bending and binding energies The Cs 3Sb photocathodes displayed significant variations in Cs 3d, Sb 3 d, and O 1 sbinding energies as shown by the spectra in FIG. 7 . Best fit line to measured Cs+3d5/2and Sb3−3d3/2binding energies using (hollow red markers) 70○detection and (solid blue markers) 0○detection on five different Cs 3Sb photocathodes grown on different substrates. The relative binding energy shifts are measured with respect to the lowest observed Cs+3d5/2and Sb3−3d3/2binding energies. Figs. 4 and 5. The observed energies varied from photocathode to photocathode and were systematically larger for the Cs 3 dtransi- tions. A shift in Cs 3 dbinding energy with photocathode aging was also reported by Martini et al.22 The chemical origin of the binding energy shifts is suggested by the data in Fig. 7, which show that the measured Cs+3d5/2and Sb3−3d3/2binding energies were linearly correlated. This correlation was inconsistent with sample charging, as the Cs 3 d5/2binding ener- gies shifted 62% more than the Sb3−3d3/2binding energies. Sample charging would have shifted both energies by the same amount. The binding energy shifts were also systematically larger (more pos- itive) for spectra measured in the more surface-sensitive geome- try (hollow red markers) than those measured with normal detec- tion (solid blue markers). This shows that the binding energy shifts were depth dependent. Both trends, the linear scaling and the depth dependence, were independent of the substrate; replicate growths on Si(100) displayed different shifts. Finally, these shifts could not be explained by changes in the sample work function, as XPS measures binding energies with respect to the common Fermi energy of the sample and spectrometer. XPS core level energies are independent of the sample work function. The observed binding energy shifts are consistent with the development of a p-nheterojunction at the interface between the p-type Cs 3Sb photocathode and the n-type cesium oxide overlayer as qualitatively sketched in Fig. 8. When charge is transferred from the overlayer to the substrate, an interfacial potential is created that shifts all bands down to larger binding energy. This band bending is depth dependent, with larger shifts near the surface and smaller shifts away from the surface. This depth dependence explains why larger shifts were observed for the Cs+3d5/2transition, which was primarily from the surface oxide layer, than the Sb3−3d3/2transition, which was primarily from the buried Cs 3Sb interface. Further support for this model comes from the oxidation- induced narrowing of the Cs+3d5/2linewidths observed in Fig. 5. J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 8 . Qualitative sketch of band bending at the interface between the (red) p- type Cs 3Sb photocathode and the (blue) n-type cesium oxide overlayer for a (left) thin oxide and (right) thick oxide. The distance scale is qualitative. The vertical arrows represent core-level excitations at varying depths beneath the surface. The color of the arrows represents the depth-dependent sensitivity of XPS. For thin oxides, the energies of the detected electrons are depth dependent and, thus, heterogeneously broadened. The five as-transferred photocathodes had a mean Cs+3d5/2 linewidth of 2.15 ±0.06 eV. Figure 5 shows that this linewidth nar- rowed by 15% upon oxidation of the photocathode. Band bending is an interfacial phenomenon as sketched in Fig. 8. After a 100 L O2exposure, the Cs 3Sb/Cs 2O interface was beyond the XPS detec- tion range, as shown by the complete absence of Sb transitions in Fig. 5(d). Because of this thick oxide layer, the detected Cs+ 3d5/2photoelectrons were homogeneously affected by the interfacial charge transfer as sketched in Fig. 8. All the photocathodes displayed downward band bending. Accurately measuring the magnitude of the bending is complicated by the possibility of some surface charging. Based on the correla- tion in Fig. 7, surface charging could account for no more than 62% of the observed variations in the Cs+3d5/2binding energies. There was a 0.95 ±0.21 eV deviation between our observed Cs+3d5/2 binding energies measured in glancing detection and the average value of 724.95 eV binding energy measured on oxidized cesium.6,13 The relatively large standard deviation in this measurement is due to sample-to-sample variability, which likely arises from the com- plicated nature of the cesium oxide layer. From this deviation, the downward band bending in our photocathodes was at least 0.36 eV and possibly larger. This energy is somewhat larger than the 0.175 eV band bending estimated from simulations of measured near- threshold Cs 3Sb photocathode performance.33Since the substrate appeared to have little influence on the chemical structure of thephotocathodes, additional experiments on more highly conducting substrates (e.g., highly doped Si, polished metal) would enable a more accurate measurement. This model also predicts that the best estimate of the “true” (unshifted) binding energies for the species in our photocath- odes would be obtained from the photocathode with the smallest observed binding energies. In other words, band bending and sur- face charging both increase the measured binding energy. Thus, the photocathode with the smallest binding energies is the least per- turbed by these effects. This least perturbed Cs 3Sb photocathode in our experiments is represented by the solid blue diamond in Fig. 7; this photocathode was grown on Si(100) and measured in normal detection. The XP spectra for this photocathode are shown in Fig. 4(a). Consistent with this prediction, the observed binding energies on this photocathode are in good agreement with the fidu- cial energies (dotted vertical lines) extracted from previous publi- cations. The Sb 3 d3/2transition from Sb3−in Cs 3Sb was observed at 535.4 eV on this photocathode, which is in good agreement with 535.6 eV average energy observed from Refs. 21 and 22. The Cs 3 d5/2 transition from Cs+in Cs 3Sb was observed at 725.3 eV on this pho- tocathode, which is in good agreement with the 724.95 eV average energy observed from Refs. 6 and 13. DISCUSSION From the standpoint of practical photocathode development, our most important finding is the extreme sensitivity of Cs 3Sb to oxidation. In comparison, a Cs-activated K 3Sb photocathode with a nominal surface composition of Cs 2.5K0.5Sb was∼30 times more resistant to oxidation by O 2than our Cs 3Sb photocathodes.21The QE of potassium cesium antimonide (K xCs3−xSb) photocathodes increases by almost an order of magnitude with an increase in the Cs content;34however, oxidation resistance would improve the photo- cathode lifetime. For example, Cs 3Sb photocathodes operated under high extraction electric field conditions (86 MV/m), in the PHIN photoinjector at CERN, had a lifetime of only 29 h.35Is there an optimum composition for long lived, high brightness photocath- odes? The chemical origin of the oxidation resistance is not currently understood. Computational studies of the thermodynamics and sta- bility of the K xCs3−xSb system, which are beginning to emerge,34 may provide clues to this behavior. Although better UHV suitcases and transfer protocols could be developed, the real issue is that the brightness of a photoin- jector is proportional to the extraction voltage.36As a result, high performance electron guns are designed with extraction fields of ∼100 MV/m, which typically precludes the high pumping speeds needed for ultrahigh vacuum operation. They are also prone to rf breakdown (arcing), which produces pressure bursts. Strategies for protecting these very fragile materials are needed.37,38 The development of an interfacial potential is not necessarily detrimental to photocathode performance; however, the heterogene- ity of the surface oxide is concerning. The work function of cesium oxides is very sensitive to the oxidation state as shown in Fig. 2. If the oxide is laterally heterogeneous, the brightness of the emitted electron beam may be degraded by lateral electric fields.4,5On the other hand, the rate of surface oxidation is often affected by the work function, as first explained by Mott and Cabrera,39because electron J. Chem. Phys. 153, 144705 (2020); doi: 10.1063/5.0024020 153, 144705-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp transfer to O 2is involved.40As a result, low work function regions may be more rapidly oxidized, which could serve to dampen lat- eral variations. Nanoscale measurements of the photocathode work function and measurements of the mean transverse energy of emit- ted electrons during photocathode aging are both needed to further understand these issues. Finally, an atomic-scale understanding of Cs 3Sb photocath- odes in operating environments is only beginning to emerge. Recent ab initio calculations41of the reaction of Cs 3Sb with small molecules, such as O 2and CO, in the low surface coverage limit reproduce some of the observed changes in performance (e.g., work func- tion changes). For example, this study suggested that O 2may react with the surface both molecularly and dissociatively. Neverthe- less, the heterogeneity of the interfacial layer—in particular, the presence of at least three O-containing species—and the paucity of experimental data at low surface coverage limit our current understanding. CONCLUSIONS Cs3Sb thin film photocathodes grown on a variety of sub- strates displayed initially high quantum efficiencies. Transport in a UHV suitcase exposed the photocathodes to an effective O 2expo- sure of∼1 L. After transport, the photocathodes had developed a thin surface layer that consisted primarily of cesium superoxide; however, some cesium peroxides and oxides were also detected in this layer. This oxidation caused a small decrease in QE and at least 0.36 eV of downward band bending. Further oxidation of the photocathodes with O 2(g) led to decreasing performance, which was quantitatively explained by the development of a cesium oxide tunnel barrier. After exposure to only 100 L of O 2, the photo- cathode was irreversibly damaged. This extreme reactivity was sig- nificantly larger than that observed on a potassium cesium anti- monide photocathode, which suggests one pathway for material improvements. SUPPLEMENTARY MATERIAL See the supplementary material for additional supporting data and figures. DEDICATION This paper is dedicated to all the female chemical physicists and physical chemists who tunneled through the barriers in their paths to shine brightly! ACKNOWLEDGMENTS This work was supported by the Center for Bright Beams, a Science and Technology Center funded by the National Science Foundation under Award No. PHY-1549132. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.REFERENCES 1J. Maxson, P. Musumeci, L. Cultrera, S. Karkare, and H. Padmore, “Ultra- fast laser pulse heating of metallic photocathodes and its contribution to intrinsic emittance,” Nucl. Instrum. Methods Phys. Res., Sect. A 865, 99–104 (2017). 2J. K. Bae, I. Bazarov, P. Musumeci, S. Karkare, H. Padmore, and J. Maxson, “Brightness of femtosecond nonequilibrium photoemission in metallic photocath- odes at wavelengths near the photoemission threshold,” J. Appl. Phys. 124, 244903 (2018). 3B. L. Rickman, J. A. Berger, A. W. 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5.0021485.pdf

J. Appl. Phys. 128, 170902 (2020); https://doi.org/10.1063/5.0021485 128, 170902 © 2020 Author(s).A perspective on MXenes: Their synthesis, properties, and recent applications Cite as: J. Appl. Phys. 128, 170902 (2020); https://doi.org/10.1063/5.0021485 Submitted: 14 July 2020 . Accepted: 25 October 2020 . Published Online: 05 November 2020 Konstantina A. Papadopoulou , Alexander Chroneos , David Parfitt , and Stavros-Richard G. Christopoulos ARTICLES YOU MAY BE INTERESTED IN Magnetization dynamics of nanoscale magnetic materials: A perspective Journal of Applied Physics 128, 170901 (2020); https://doi.org/10.1063/5.0023993 Machine-learning predictions of polymer properties with Polymer Genome Journal of Applied Physics 128, 171104 (2020); https://doi.org/10.1063/5.0023759 Linking information theory and thermodynamics to spatial resolution in photothermal and photoacoustic imaging Journal of Applied Physics 128, 171102 (2020); https://doi.org/10.1063/5.0023986A perspective on MXenes: Their synthesis, properties, and recent applications Cite as: J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 View Online Export Citation CrossMar k Submitted: 14 July 2020 · Accepted: 25 October 2020 · Published Online: 5 November 2020 Konstantina A. Papadopoulou,1 Alexander Chroneos,1,2 David Parfitt,1 and Stavros-Richard G. Christopoulos1,a) AFFILIATIONS 1Faculty of Engineering, Environment and Computing, Coventry University, Priory Street, Coventry CV1 5FB, United Kingdom 2Department of Materials, Imperial College London, London SW7 2BP, United Kingdom a)Author to whom correspondence should be addressed: ac0966@coventry.ac.uk ABSTRACT Since 2011, after the discovery of new ceramic two-dimensional materials called MXenes, the attention has been focused on their unique properties and various applications, from energy storage to nanomedicine. We present a brief perspective article of the properties ofMXenes, alongside the most recent studies regarding their applications on energy, environment, wireless communications, and biotechnol- ogy. Future needs regarding the current knowledge about MXenes are also discussed in order to fully understand their nature and overcome the challenges that have restricted their use. Published under license by AIP Publishing. https://doi.org/10.1063/5.0021485 I. INTRODUCTION MXenes are a new type of material that was first discovered in 2011 after chemically etching a MAX phase. 1The first MXene was discovered1when the A-layer of a MAX phase was removed leaving behind two-dimensional (2D) flakes with the general formulaM nþ1XnTx(n¼1, 2, 3), where Tstands for a surface termination, more commonly fluorine (F), hydroxyl (OH), or oxygen (O) atoms.2MXenes, therefore, are a family of 2D crystalline materials with theoretically infinite lateral dimensions but atomically thinthickness, whose terminations, i.e., the surface atoms, make themexhibit different properties. The first MXene discovered was the titanium carbide (Ti 3C2) at Drexel.1Since then, the field has grown with studies of MXenes trying to shed light on their properties and applications. Useful inenergy storage, electromagnetic shielding, biology, and environ-mental applications like potable water, MXenes are promising can- didates for substituting current materials. It is the scope of this paper to provide an overview of the synthesis and applications ofMXenes, as well as a description of the latest discoveries in thefield. In Sec. II, methods used for obtaining MXenes from the cor- responding MAX phases are described, while Secs. IIIandIVrefer to MXenes ’properties and some of their most recent applications, respectively, discussing their promise regarding energy storage andconversion. Finally, in Sec. V, we conclude this Perspective articlewith a description of the gaps in the current knowledge, and issues that need to be addressed in order for MXenes to be widely used. II. METHODS TO OBTAIN MXENES A MAX phase is a layered structure of carbides and nitrides with the general formula M nþ1AX n(n¼1, 2, 3), where Mis an early transition metal (groups 3 –7 in the periodic table), Ais an element belonging in the A-group of the periodic table, and Xis either carbon (C) or nitrogen (N)3–5(see bottom panel of Fig. 1 ). The most common practice to obtain MXenes is by wet chemicaletching of atomic layers from a multi-layered MAX phase, consid-ering that, in a MAX phase, the layer-to-layer bonding is muchweaker than the intralayer one. Initially, the MAX phase is soaked in an acid that destroys the bonds between the transition metal and the A element. Experiments determined 6that MXenes obtained via wet chemical etching exhibit higher electronic conductivity and havefewer atomic defects. However, the procedure is only useful for carbon-based MXenes as it fails to remove the A-layer from nitride- based MAX phases. The first etchant used was hydrofluoric acid (HF), 2,7,8a mate- rial that is considered dangerous to the environment. Therefore, the need arose to find different etchants. A safer mixture of hydro- chloric acid (HCl) with lithium fluoride (LiF) started being used inJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-1 Published under license by AIP Publishing.2014.9The problem remained though, because HF gases are pro- duced in situ .2The issue was solved by the development of different methods to bypass the use of HF. These methods are discussedbelow. A. Urea glass route Maet al. , 10in 2015, synthesized Mo 2C and Mo 2N MXenes from the metal precursor MoCl 5using the urea glass route. They added ethanol into the precursor, thus causing its reaction with the alcohol in order to form Mo-orthoesters. Solid urea was thenadded to the solution and the mixture was stirred until the urea was fully solubilized. The gel-like result was then heated at 800/C14C under a N 2gas flow, and afterward calcinated. Silvery black powders were obtained. Maet al.10performed x-ray diffraction (XRD) on the samples, and no other crystalline side phases, e.g., MoO x, were observed, a fact that meant that the powders had high purity. B. Chemical vapor deposition Also, in 2015, Xu et al.11developed a chemical vapor deposi- tion (CVD) process to obtain MXenes, using methane as the source for C and a Cu foil above a Mo foil as the substrate. They experi-mented at a temperature above 1085 /C14C. During CVD, the decom- position of chemicals on the surface of the substrate leads todeposition, from the vapor phase, of films of materials. The high temperature used, allowed the Cu foil to melt so that a Mo –Cu alloy was formed at the liquid Cu/Mo interface. This resulted inMo atoms diffusing to the surface of the liquid Cu, thus formingMo 2C crystals after reacting with the C atoms produced by the decomposition of methane. However, the produced material, although it had MXene-like structure, was in fact a 2D transitionmetal carbide with a larger area than the thus far produced nano-sheets that reach up to 10 μm. 11 Xuet al.11obtained 2D ultrathin α-Mo 2C crystals a few nano- meters thick and with lateral sizes larger than 100 μm. They were able to control the thickness via the concentration of methaneused. The resulted MXene-like material was free of defects, and itssuperconductivity was stable when the material came in contactwith air for a few months. 11 C. Molten salt etching The first time the molten salt etching method was used for MXenes was in 2016.12Urbankowski et al.12mixed Ti 4AlN 3 powder with a fluoride salt mixture in a 1:1 mass ratio and heated the mixture at 550/C14C for 30 min. They found five different fluoride phases containing Al, but the absence of Ti-containing fluorides,confirmed the selectivity of etching. To dissolve the Al-containing fluorides, they used diluted sulfuric acid (H 2SO4) and the etching products were then removed by washing with de-ionized water andthen centrifugation and decanting. The XRD pattern showed thatalmost all the fluoride salts were removed and that the resultingpowder contained Ti 4N3Tx, where T stands for either OH or F surface terminations, and still unetched Ti 4AlN 3. To further decompose the layered Ti 4N3TxMXene, the powder was mixed with tetrabutylammonium hydroxide (TBAOH), which was thenremoved by washing with de-ionized water and then centrifugedand decanted. De-ionized water was added to the remaining powder, which was then probe sonicated and centrifuged. The result was finally filtered and the smaller unlayered Ti 4N3Txflakes (T¼OH, F) were collected. The resulted MXene showed more atomic defects than the one obtained through HF-etching, which reduces the value of breaking strength of the material. FIG. 1. The primitive cell of a M 3AC 2MAX phase17(top left panel), alongside the primitive cell of the resulted in M 3C2MXene17(top right panel). The blue spheres represent the M atoms, the red the A atoms, and the brown the Catoms. The bottom panel indicates the location, in the periodic table, of the com- ponents of a MAX phase.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-2 Published under license by AIP Publishing.D. Hydrothermal synthesis in an aqueous NaOH solution Liet al.13reported in 2018 a method to obtain Ti 3C2Tx MXene from the Ti 3AlC 2MAX phase, assisted by sodium hydroxide (NaOH). In this method, the hydroxide anions (OH/C0) attack the Al layers, which results in the oxidation of the Alatoms. The Al hydroxides Al(OH) /C0 4produced are then dissolved in alkali and the exposed Ti atoms are terminated by OH or O. However, the process allows for new Al hydroxides to beformed that are confined into the lattice from the Ti layers andcannot further react anew with the OH /C0. The problem waseliminated with the application of a series of hydrothermal tem- peratures and concentrations of NaOH water solutions under an argon atmosphere. The MXenes obtained through this hydrothermal procedure have more OH and O terminations than their HF-etching counter-parts, and this significantly enhances their performance as supercapacitors. E. Electrochemical synthesis at room temperature Yang et al. , 14in 2018, first proposed an electrochemical method without the use of F for the delamination of Ti 3C2in a binary aqueous electrolyte. They constructed a two-electrode system with bulk Ti 3AlC 2as anode and cathode. Only the anode underwent the etching process and produced Ti 3C2Tx. To avoid the fact that the etching only takes place on the surface15and allow the electrolyte ions to diffuse into the deeper layers of the anode,they used a mix of 1M NH 4Cl and 0.2M tetramethylammonium hydroxide (TMA /C1OH) with a pH .9. Applying a low potential of þ5 V, the bulk anode was gradually delaminated. Afterward, the sediment and suspended powders of Ti 3C2Txwere ground and transferred into 25% w/w TMAOH in order to get individual sheets of Ti 3C2Tx. The resulted MXenes exhibited an electrical conductivity similar to that of the ones produced after the use of HF or HCl/LiF. 9However, the electrochemical method appears to be the most promising one regarding the overall etching yield, as up to 60% of the bulk material can be transformed into Ti 3C2Tx.14 InFig. 1 , we see the Ti 3AlC 2MAX phase alongside the Ti 3C2 resulted bare MXene where the Al layers have been removed. In Fig. 2 , the terminated Ti 2CT2and Ti 3C2T2MXenes16are showed. Finally, Fig. 3 shows the timeline of the etching methods used to produce MXenes until 2017.18 FIG. 2. Side and top views of crystal structures of a 2D MXene monolayer: (a) Ti2CT2and (b) Ti 3C2T2. Republished with permission from Bai et al. , Royal Soc. Chem. 6(42) (2016). Copyright 2016 Clearance Center, Inc. FIG. 3. Timeline of the etching methods used to produce MXenes since the first discovery in 2011, and up to 2017. Reprinted with permission from Peng et al. , Chem 5 (1), 18 –50 (2019). Copyright 2019 Elsevier.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-3 Published under license by AIP Publishing.III. MXene PROPERTIES A. Mechanical properties Depending on the surface terminations, the mechanical properties of an MXene may show significant differences. Baiet al. 16determined for Ti 2Ca n dT i 3C2that there is a stronger interaction between the O terminations and Ti atoms than in the F or OH terminated MXenes. In addition, the O-terminatedMXenes have very high stiffness, and this is attributed tothe bonding strength between Ti –O ,w h i c hi sh i g h e rt h a ni nt h e Ti–OH and Ti –F cases. Magnuson et al. 19also found that the surface groups withdraw charge from the Ti –C bonds and weaken them. In particular, they found that the Ti-C bond is longer inTi 2C-T xthan in Ti 3C2-Tx, a fact they suggest could affect the elastic properties of the materials. The authors also suggested thatthe modification of the bond strength can be used to optimize the elasticity. In addition, Zha et al. 20suggested that O-terminated MXenes should be the first choice for applications regardingstructural materials, supercapacitors, and so on, due to theirlarger mechanical strength. Regarding the strain MXenes can withstand, Chakraborty et al. 21showed that boron (B) doped Ti 2CO 2, where B substitutes the C atoms, exhibits higher critical strain (by about 100%) com-pared to the bare MXene. This fact is attributed to a weaker Ti –B bond when compared to the Ti –C bond. Theoretically, Ti 2CO 2-based composites have been found to withstand large strain under uniaxial and biaxial tension.22 Moreover, Yorulmaz et al.23by calculating the Young modulus, noted that, for the carbides, the MXene gets stiffer as themass of the transition metal M increases. This result could not be replicated for the nitrides. The vast majority of carbide-based MXenes are considered mechanically stable, which is not the case for nitride-basedMXenes. In general, single MXene flakes are unstable in environ-ments that contain oxygen and water. 24,25However, they are relatively stable in water where oxygen has been removed or in dry air. MXenes, in general, have lower strength and stiffness than gra- phene. Despite this fact, molecular dynamics calculations have pre-dicted high enough rigidity for Ti 2C, Ti 3C2, and Ti 4C3MXenes, so that applied surface tension is not enough to overcome it.26–28In Fig. 4 , we can see the bending rigidity of the Ti nþ1CnMXenes, gra- phene multilayers, and MoS 2as a function of their thickness. It is clear that the thicker MXene nanoribbons have higher values of bending rigidity. The exact value depends on a variety of reasons, such as temperature, and size and shape of the samples. Finally, surface-modified MXenes were also studied early on in 2013 by Enyashin and Ivanovskii,29in particular, MXenes termi- nated with methoxy OCH 3groups. Their stability was examined by estimating their formation energies ΔH, where it was indicated that a negative value of ΔHis energetically favorable in order to have more stable products. In addition, by calculating the relative totalenergies, ΔE tot,o fT i 2C(OCH 3)2and Ti 3C2(OCH 3)2, it was found that if the methoxy groups are placed above the hollow sites between the three neighboring carbon atoms within the MXene layer (see Fig. 1 ), the structure is more stable than when placing the methoxy group above a Ti atom.The great potential of MXenes for applications in structural composites30remains to be explored, provided that bulk quantities can be produced. B. Electronic properties Perhaps, the most important electronic property of MXenes is their high electronic conductivity. Most bare MXenes and themajority of those with surface terminations exhibit a metallic behavior. In recent years, an attempt is being made to increase the MXene metallic conductivity, although the first discoveredTi 3C2Tx, which is also the most studied one,31is still the most conductive. The focus has been on developing new M nþ1Xnchemistries that will result in higher conductivity by controlling the surface ter-minations. However, the studies made in that regard lack experi-mental validation. 32A different approach to affect MXene conductivity is cation or organic-molecule intercalation where, for stacked materials, the resistance of the device can be increased by over an order of magnitude.32,33Figure 5 shows the temperature dependent electrical conductivity of various Mo-based MXeneswhere there is a rapid increase above 500 K. Zhang et al. , 34in 2018, showed that OH-terminated MXenes exhibit nearly free electron states, located outside the surface atoms and parallel to the surface. These states concentrate in regionswhere the positive charge is the highest and provide almost perfecttransmission channels for electron transport. Moreover, duringtheir experiments, O-terminated MXenes, for example, Ti nþ1CnO2, exhibited lower electrical conductivity than the ones terminated with F and OH, that is, Ti nþ1CnF2, and Ti nþ1Cn(OH) 2, respectively. Despite the fact that the simulated non-terminated MXenes are all metallic, upon surface functionalization, some of them become semiconducting35with a bandgap as large as 0 :194 eV FIG. 4. The bending rigidity of the Ti nþ1CnMXenes, graphene multilayers, and MoS 2as a function of their thickness h, cubed. Reprinted with permission from Borysiuk et al. , Comput. Mater. Sci. 143, 418 –424 (2018). Copyright 2018 Elsevier.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-4 Published under license by AIP Publishing.(W2CO 2). Due to electron transfer from the transition metal to the electronegative surface terminations, the transition metal exhibitslower density of states (DOS) at the Fermi level. 36,37At the moment, no MXene semiconductors have been experimentally real- ized, although many are theoretically predicted taking under con- sideration a wide bandgap for all surface terminations.37 In addition, there have been cases where simulated MXenes were predicted to be topological insulators.38,39In this category of materials, their edge states are conducting but they exhibit insulat- ing properties on the surface.40Currently, in all of the studies on topological behavior of MXenes, only O- or F-terminated oneshave been taken under consideration. None of the insulatorMXenes though have been observed experimentally, considering the fact that only ones with mixed terminations have been practi- cally obtained. Control of the surface terminations is once againthe key for obtaining this kind of materials. In general, processes like doping, 41strain application,42stack- ing,43and alloying44can control a material ’s electronic structure. MXenes can withstand a lot of strain (see Subsection III A ), so strain engineering has attracted considerable attention.45For example, Yu et al.46found an indirect-to-direct bandgap transition in Ti 2CO 2,Z r 2CO 2, and Hf 2CO 2when applying biaxial strains 4%, 10%, and 14%, respectively. Most of the studies regarding the electrical properties of the Ti3C2MXene have focused on modifying the surface terminations via thermal treatment.47,48However, it was determined49that when calcination temperature is increased to 800/C14C, the Ti 3C2nano- sheets collapse and the 2D nanostructure is destroyed. An additional method to manipulate the electrical properties of an MXene is by using reinforcing materials. An example is chito-san, 50a kind of fiber obtained from the exoskeleton of shellfish and insects, where the reinforced MXene exhibits gradient increase of its electrical resistivity as the amount of chitosan increases incrementally.C. Magnetic properties Materials with strong and controllable magnetic moments are important to applications of spintronic devices. Despite the varietyof MXenes, the ground states of the majority of them, bare or oth- erwise, are non-magnetic. This is attributed to the strong covalent bond between the transition metal and the X element. 35Some bare MXenes though have been predicted to be intrinsically magnetic.These include Cr 2C51and Ti 2N52(ferromagnetic), and Cr 2N53and Mn 2C54(anti-ferromagnetic). The magnetism in the case of MXenes can be55 (1) intrinsic properties of the transition metal, (2) defects in monolayers, and (3) surface terminations. As stated before, the most common MXenes, including Ti 3C2,a r e non-magnetic; however, there are cases where different magnetic structures were reported. For example, Dong et al.56determined that Ti 2MnC 2TxMXenes are ferromagnetic in the ground states, regardless of the nature of the surface terminations. Ti 2MnC 2Tx belongs to a new family of ordered double-transition-metalMXenes 57,58, which have one or two layers of a transition metal sandwiched between the layers of another one. In 2018, Sun et al.59 studied double-transition-metal MXenes with Ti atoms as the central layer and showed that different terminations and cationconfigurations lead to a variety of magnetic orders and propertiesdifferent from the single transition metal carbides known thus far. In 2017, reports about a novel MAX phase with in-plane chemical order were made. 60,61These compounds were called i-MAX and have the general formula ( M2=3M0 1=3)2AX.F r o m i-MAX phases came the i-MXenes after etching the A-elementatoms (e.g., W 4=3C and Nb 4=3C62,63). Prior to that, Zhu et al.64had found for Nb 2C a theoretical in-plane lattice constant value in agreement with the experimental one and showed that the energeti-cally favorable site for Li adsorption is located on top of C. In2020, Gao and Zhang, 65turned their attention to the magnetic properties of i-MXenes with a general formula ( M2=3M0 1=3)2X.I n particular, by performing density functional theory (DFT) calcula-tions, they examined the cases where M 0, i.e., the dopant element, is magnetic and one of the transition metals except Tc. Out of319 i-MXenes, they found 62 cases that they were magnetic. In addition, six can change their magnetic configuration, depending on their geometries (rectangular or hexagonal), which in turn needa strain to be applied. Therefore, the magneto-crystalline anisotropyof i-MXenes can be enhanced. InFig. 6 , we can see the classification of the magnetic ground states for the 319 i-MXenes. An extensive theoretical study of the magnetic properties of nitrogen-based MXenes, which have an extra electron per unit cellwhen compared to carbon-based ones, was carried out by Kumaret al. 66The authors suggested a simple model to predict the mag- netic behavior of M 2NT 2based on the assumption that only elec- trons occupying the non-bonding d-orbitals can contribute to magnetism. By applying their model to M 2NT 2MXenes with tran- sition metals belonging to the third period of the periodic table, they identified five nitride MXenes (Mn 2NF2,M n 2NO 2, Mn 2N(OH) 2,T i 2NO 2, and Cr 2NO 2) with robust ferromagnetic FIG. 5. The temperature dependent electrical conductivity of various Mo-based MXenes. The increase in conductivity results in a decrease in the resistivity.Reprinted with permission from Kim et al. , Chem. Mater. 29(15), 6472 –6479 (2017). Copyright 2017 American Chemical Society.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-5 Published under license by AIP Publishing.ground states. Curie temperatures for all terminations were also well above the room temperature, a fact that made the magneticallyordered phases stable. In 2018, Bandyopadhyay et al. 67investigated the point defect formation mechanisms in MXenes. They found that few of thedefective MXenes acquire a magnetic nature because of unpairedelectrons in the spin split d-orbitals. Therefore, they suggested that intrinsic point defects can be used to modify the magnetic proper- ties of MXenes. Moreover, Scheibe et al. , 55in 2019, investigated the influence the different terminations have on the magnetic properties ofTi 3C2Tx. They found that the bare TiC samples exhibit paramag- netic behavior, similarly to the samples with F or S-based termina- tions, but if the sample contained two kinds of terminations, then its behavior was shifted to ferromagnetic/paramagnetic. Therefore,the magnetic properties of Ti 3C2Txcan be altered via modulation of the surface terminations. Another example of the impact of the terminations is described in Ref. 68. When Cr 2C is terminated by F, H, OH, or Cl groups, it will be transformed from a ferromagnetichalf metal into an antiferromagnetic semiconductor. The potential use of MXenes in spintronics makes the methods to control their magnetism a rapidly advancing field. D. Optical properties Regarding optics, various interesting features of MXenes have been demonstrated in the past few years. These include optical transparency, plasmonic behavior, and efficient photothermal con-version. The ability of MXenes to interact with light in multipleways has had a significant impact on the research community. 69 Once again, it is the surface terminations that play the most signifi-cant part in determining a material ’s optical properties.Berdiyorov, 70in 2016, investigated the role of surface termina- tions on the optical properties of Ti 3C2T2(T¼F, O, OH) MXenes and compared the results to the bare MXene Ti 3C2. Using the dielectric constant, Berdiyorov calculated the refractive index, n, and the extinction coefficient, k, for all samples and found that at low photon energies, E, the terminations reduced n, while at higher E(.1 eV), nwas enhanced. The surface terminations also resulted in the reduction of kfor most of the photon energy range. In addi- tion, at even larger photon energies ( E.5 eV), the terminations resulted in stronger absorption as compared to the bare MXene.Finally, in the ultraviolet region of the spectrum (5 eV,E,10 eV), all surface terminations resulted in larger reflectivity as compared to the bare MXene. Halim et al. , 71by studying Ti 2CTxand Nb 2CTxfilms, showed a different approach to control the optical properties of MXenes isto change the nature of the transition metal. They found that the Nb-based films showed an increase in absorption toward the infra- red spectral range. This behavior significantly differed from that ofthe corresponding MAX phase Nb 2AlC, a fact that was not observed for the Ti-based films. Halim et al.71suggested that the difference in the optical properties lay within the electronic config- uration of the transition metals (Ti is in group 4 and Nb is in group 5 of the periodic table) and their bonding with the C atomsand surface terminations. In 2017, Li et al. 72showed that MXenes exhibit higher capabil- ity to absorb light than carbon nanotubes (CNTs). In addition, Jiang et al. ,73in 2018, examined the optical non-linearity of Ti3C2Txusing excitation sources with various wavelengths (800 nm, 1064 nm, 1550 nm, and 1800 nm). They found absorptionresponse two orders of magnitude larger than other processes but the refractive index was comparable to that of graphene. Finally, for Ti 2C, Ti 2N, Ti 3C2, and Ti 3N2, for E,1 eV, the reflectivity has been found to reach 100%, which means that these materials havethe ability to transmit electromagnetic waves. 74 In general, it is believed that both the linear (e.g., absorption) and nonlinear (e.g., refractive index) optical properties of MXenes are highly dependent on the energy structures (e.g., energybandgap, direct/indirect bandgap, topological insulators, etc.). 74 IV. APPLICATIONS Since they were first discovered in 2011, MXenes have received significant attention. Studies have revealed their potential applica- tions in energy storage, optoelectronics, spintronics and catalysis, and even in environmental and biological issues. Below, we sum-marize the most recent applications, with a particular focus onenergy storage. A. Energy storage and conversion Energy storage and conversion has been a key scientific chal- lenge for years, since the materials used for these applications need to be light, flexible, conductive, and have large surface-to-mass ratio [specific surface area (SSA) (m 2g/C01)].75Carbon black has SSA,900 m2g/C01,76while carbon nanotubes (CNTs) have SSA¼100/C01000 m2g/C01.77A sheet of graphene, on the other hand, has theoretically very large surface-to mass ratio, SSA¼2630 m2g/C01,77a value that is similar to that of the activated FIG. 6. Classification of the magnetic ground states for the 319 i-MXenes. Mag = Magnetic, NM = Non-magnetic, AFM = Antiferromagnetic, FM = Ferromagnetic, Noncol = Noncollinear Magnetic Structure. “NM to Mag ” and “AFM to FM ”mark the compounds whose ground state changes by impos- ing hexagonal geometry. Republished with permission from Gao et al. , Royal Soc. Chem. 12(10) (2020). Copyright 2020 Clearance Center, Inc.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-6 Published under license by AIP Publishing.carbon.78Other 2D materials that can be used for energy storage are the transition metal dichalcogenides (TMDs) like WS 2, in com- bination with graphene.79 MXenes are a new class of 2D crystals that can be used in energy applications. Compared to materials like graphene, MXeneshave much smaller specific surface area. For example, experimen- tally produced Ti 3C2Txhas SSA up to 66 m2g/C01and V 2CTxup to 19 m2g/C01,80while theoretically Ti 3C2Txcan reach up to 496 m2g/C01.R e n et al. ,81in 2016, managed to produce porous Ti3C2TxMXenes (p-MXenes) using a transition metal salt and acid treatment. They determined that the p-Ti 3C2TxMXene had SSA increased from 19 :6 (pristine Ti 3C2Tx)t o9 3 :6m2g/C01. Furthermore, a high specific capacity of /C251250 mAhg/C01was revealed for p-Ti 3C2Tx/CNT electrodes. Graphene electrodes, on the other hand, when chemically modified, can reach/C251200 mA h g /C01,82while Hassoun et al. ,83in 2014, showed that electrodes based on Cu-supported graphene nanoflakes ink can reach specific capacities of /C251500 mA h g/C01. Both these values are close or even larger than the corresponding ones for the MXenes. Regarding volumetric capacity, functionalized graphene shows results up to 200 F g/C01,78,84,85while for MXenes, researchers were able to achieve for Ti 3C2320 F g/C01as early as in 2014.86 At present, Li-ion batteries do not operate satisfactorily in large- scale operations like sustaining a clean power grid.87Intercalation pseudocapacitance,88which occurs through bulk redox reactions with ultrafast ion diffusion,89,90has risen as an alternative chemistry for advanced electrochemical energy storage devices. The intercalationpseudocapacitors have many benefits, considering the limited expo-sure of the surface to the electrolyte. This leads to a decrease of theirreversible surface reactions due to electrolyte decomposition, without affecting the charge storage. 91MXenes pose as good materi- als for these pseudocapacitors as their morphology minimizes thesurface, which interacts with the electrolytes, and their really smallinterlayer spacing makes for fast ion intercalation. In 2012, shortly after their discovery, Naguib et al. 92studied the MXene ’sT i 2C use as anode for lithium-ion batteries. They found that Ti 2C had reversible capacity five times higher than the corresponding MAX phase Ti 2AlC, exhibiting a stable capacity of 225 mA h g/C01. They suggested that the results were promising for the use of MXenes as Liþintercalation electrodes in lithium-ion batteries.92Since then, other batteries have been examined, for example, sodium-ion, pottasium-ion, and lithium-sulfur.93–95 In 2015, Zhu et al.64investigated the Li adsorption in mono- layer Nb 2C and Nb 2CX2MXenes and found that the most energeti- cally favorable site for Li adsorption was on top of the C atom. They also showed that, since Li forms clusters on Nb 2C(OH) 2and LiF forms clusters on Nb 2CF2and Nb 2CO 1,5F0,5, OH and F termi- nations must be avoided in battery applications. Moreover, Nb 2C was found to be the most promising electrode material for Li-ion batteries because of its low diffusion barrier. In addition, Zhu et al. ,96in 2016, replaced OH, F, and O ter- minations with S and predicted high electric conductivities andcycling rates for Li-ion batteries using Zr-based MXenes. The life-time of the battery was also found not affected by by-products like dendrites. More recently, in 2019, Shukla et al. 97studied S-terminated MXenes (V 2NS2and Ti 2NS2) as anodes for Li/Na ion batteries.They found that multilayer ion intercalation is possible, and that the materials manifest high capacity, especially for the Li case. Regarding ionic diffusion, the authors showed that the lowerenergy barriers result in ultrafast diffusion. Furthermore, in thesame year, Zhang et al. 98showed that MXenes can be used as a conductive binder for Si electrodes in order to decrease the inactive volume of the electrode and have better battery performance. Moreover, the Ti 3C2TxMXene was found to be useful as a reinforc- ing material for polymers such as polyvinyl alcohol (PVA), wherethe strong films fabricated can be used as electrodes forsupercapacitors. 99,100 In addition, Wang et al.101studied Ti 3C2MXenes as the S cathode host for LiS batteries. They showed that the bare MXenecannot be used directly for this purpose due to decomposition.However, S and O terminated Ti 3C2can achieve high performance. In 2020, Li et al.102investigated the use of chalcogenated Ti3C2as an anode material for Li-ion batteries. Chalcogenation gave the Ti 3C2T2,T¼O, S, Se, and Te, MXene enhanced Li-ion accessibility and mobility while Ti 3C2S2and Ti 3C2Se2MXene elec- trodes yielded a theoretical Li-storage capacity of 462 :61 mA h g/C01 and 329 :32 mA h g/C01, respectively. The authors noted that the fab- rication of MXenes with larger interlayer spacing could achieve higher Li-storage capacity. To summarize, if we were to compare graphene with MXenes as materials for anodes for Li-ion batteries, despite the fact that gra- phene has larger SSA and higher capacity, we should also take into consideration that graphene electrodes do not allow fast enoughdiffusion for energy applications that require high power output. 103 Furthermore, graphene ’s low lithiation potential enables the forma- tion of lithium dendrites, which are flammable and limit the cycling stability of the battery cells.104,105In contrast, MXenes have higher lithiation potential and wider interlayer spacing,1,106which allows for fast ion intercalation. Add to that fact graphene ’s limited commercial use, mainly due to its expensiveness, MXenes seemcurrently the most promising materials for battery applications. B. Other applications Apart from energy storage devices, Sarycheva et al. 107reported the use of MXenes for antenna applications. In particular, theyconstructed the first radio-frequency (RF) MXene devices for wire- less communication and showed that MXene antennas need lower thickness in order to work than those of the best-known metals,enabling ultra-thin and transparent wireless devices. Blanco et al. 108determined that MXene nano-sheets showed unique catalytic properties for the hydrodeoxygenation (i.e., a process to remove oxygen) of guaiacol in order to make it more stable. In addition, in Ref. 109, MXenes were studied as energy-efficient alternatives to the Haber –Bosch process, which is used in the production of ammonia. To this end, Mo 2TiC 2was examined as an electrocatalyst and the authors proved through the analysis of Gibbs free energy, that it could be a new, effective mate- rial for ammonia synthesis. Apart from the aforementioned environmental uses, MXenes have also been used for the removal of heavy metals from water. For example, Shahzad et al.110developed a system to remove copper from water using Ti 3C2TxMXenes. XRD analysis showedJournal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-7 Published under license by AIP Publishing.that copper reacts strongly with the surface terminations of the MXenes, resulting in surface oxidation. MXenes have also been found useful in biological applica- tions. Szuplewska et al.111examined the bio-compatibility of Ti2NT xtoward human skin malignant melanoma cells and human breast cancer cells. They determined that the multi- layered Ti 2NT xshows higher toxicity toward cancerous cells when compared to normal ones, thus revealing the potential useof MXenes in biotechnology and nanomedicine. Moreover,MXenes have been used in various bioimaging techniques 112 such as fluorescence microscopy, where MXene ’sf l u o r e s c e n c e properties are enhanced by attaching a fluorescence species to their surface,113in vitro bioimaging using quantum dots,114,115 and x-ray computed tomography (CT) where MXenes are used as imaging contrast elements.116 –118 The properties of MXenes also make them suitable materials for sensors. For instance, Liu et al.119proposed a nitrite bio-sensor containing hemoglobin (Hb) immobilized on Ti 3C2, which was used to detect the presence of nitrite in water samples. They pro-posed Ti 3C2MXenes as suitable candidates for enzyme immobili- zation, as they can provide a safe micro-environment for the proteins. In addition, Rakhi et al.120developed a glucose bio-sensor based on an Au/MXene nanocomposite to detect glucose in thepresence of other electroactive substances. Crystalline Au nanopar-ticles where placed on the surface of Ti 3C2TxMxenes and an improved electrical conductivity was found, making the Au/MXene composite a good enzyme immobilization matrix. Another study in2017 by Muckley et al. 121proposed the use of MXenes as sensors for water and humidity in air, while Ma et al.122in the same year constructed MXene-based sensors to detect human activities such as swallowing, coughing, joint-bending. In 2019, Montazeri et al.123devised MXene/GaAs/MXene photodetectors and showed that they have higher responsivity,quantum efficiency, and dynamic range than the current Ti/Au – GaAs –Ti/Au devices, proving that MXenes are also useful in the optoelectronics and microelectronics industries. Finally, MXenes ’s t r u c t u r ea l l o w sf o rd i s t a n c eb e t w e e na t o m i c layers and changes in conductivity when external pressure isapplied. 124Tan et al.124used MXene-based pressure sensors and, by designing an optoelectronic spiking afferent nerve, they were able to detect Morse code, braille, object movement, and handwritten words,showing promise for human-machine interaction technologies. InFig. 7 , we can see the summarized applications of MXenes beyond energy storage, as presented in this paper. V. SUMMARY AND FUTURE DIRECTIONS In the present paper, we provided an overview of the methods for the synthesis of MXenes from the corresponding MAX phases, alongside the advantages and disadvantages of each method. In addition, we described the mechanical, electronic, magnetic, andoptical properties of MXenes. The ability to manipulate theMXenes ’terminations led to materials with large mechanical strength, high conductivity, intrinsic magnetism, strong absorption, and large reflectivity. In addition, we outlined the most recent applications of MXenes, focusing on energy storage, which aresummarized in Fig. 7 . The latest results show that MXenes composed of transition metal elements of group 6 of the periodic system (Cr, Mo, W) are the most promising ones for superconductivity, at least in the caseof Mo and W. 125In addition, Luo et al. ,126in 2020, have, for the first time, examined the tensile behavior of Ti 3C2Tx. They found that the number of defects influences significantly the overall mechanical properties of the films, a fact that is reflected by the dependency of their tensile strength, elastic modulus, and fracturestrain on thickness. FIG. 7. Summary of the most recent applications of MXenes beyond energy storage, as referenced in this paper.Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-8 Published under license by AIP Publishing.Numerous scientific reports are published yearly regarding the MXenes ’possible applications. However, there are several gaps in the current knowledge that need to be addressed. Despite the amount of theoretical studies regarding their use in energy storage devices, the charge storage mechanism of theMXenes is not fully understood yet, neither are the effects of their different compositions on their performance as electrodes. Moreover, MXenes ’stability and durability are still important chal- lenges. MXene nanosheets degrade quickly when in contact withwater and oxygen, making their use in various applicationsdifficult. 127 Another significant problem is the fact that, for many MXenes studied through computational techniques, there is no precursoryMAX phase, 25for example, W 2C. The attention should be turned into the synthesis of new layered materials that could be used forthe production of MXenes. In that respect, a way to access more MXene compositions is by the synthesis of quaternary MAX phases. Quaternary MAX phases lead to the formation of MAXphases with elements (for example, bismuth) that are not presentin the synthesis of ternary MAX phases. 128 –131In addition, the ter- minations and their different characterizations after synthesis need to be examined, alongside the methods for achieving uniform ter- minations. Bare MXenes have not yet been produced, despite beingpredicted as metallic conductors. Moreover, the studies of the inter-action of the MXenes with light are at a very early stage. 73The non- linear optical phenomena regarding MXenes are still unexplored for the materials to be widely used in photonic applications.Studies on the toxicity of MXenes are also limited. Currently, there are more than 30 MXene composites and more are being discovered daily. MXenes have demonstrated poten- tial as next-generation materials in various fields, although they are mainly produced at laboratories and have small yield. If large-scaleproduction was to be achieved, then MXenes would have a largerimpact on a commercial scale. Dozens more MXene structures havebeen examined at a theoretical level using computational methods, contributing to the rapid advances in the field. When it comes to environmental applications, many MXenes and MXene-based composites have been proposed for processeslike water purification, desalination, etc. 132Furthermore, the removal of pollutants from the environment is still a challenge, which could be resolved by further studies on the MXenes.133 Biomedicine is another field that has shown it can be bene- fited from the use of MXenes, with studies on cancer treatment,bio-sensors, neural electrodes, dialysis, and theranostics 87,134being a research topic that has significantly expanded during the past two years.135 MXenes have also started taking over from other nanomateri- als like graphene in electromagnetic applications, for example, RFantennas 107and electromagnetic interference (EMI) shielding. The production, storage, and utilization of fuels to combat the ever-increasing energy consumption globally is also an issue thatcan be addressed via MXenes, due to their physical and chemicalproperties. Currently, Li-ion batteries are the largest power sourcein both small portable electronics, and electric vehicles. 133 However, with problems such as safety, limited natural lithiumresources, and production cost, 136,137there is a need for the devel- opment of non-lithium batteries.Perhaps, the biggest challenge yet, though, is the experimental validation of all the theoretical studies, and the verification of the predicted properties of MXenes. ACKNOWLEDGMENTS The authors acknowledge support from the International Consortium of Nanotechnologies (ICON) funded by Lloyd ’s Register Foundation, a charitable foundation which helps to protect life and property by supporting engineering-related education,public engagement, and the application of research. DATA AVAILABILITY Data sharing is not applicable to this article as no new data were created or analyzed in this study. REFERENCES 1M. Naguib, M. Kurtoglu, V. Presser, J. Lu, J. Niu, M. Heon, L. Hultman, Y. Gogotsi, and M. W. Barsoum, Adv. 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Soc. 135, 1167 (2013).Journal of Applied PhysicsPERSPECTIVE scitation.org/journal/jap J. Appl. Phys. 128, 170902 (2020); doi: 10.1063/5.0021485 128, 170902-11 Published under license by AIP Publishing.

5.0022891.pdf

AIP Advances 10, 095029 (2020); https://doi.org/10.1063/5.0022891 10, 095029 © 2020 Author(s).Structural, electronic, and transport properties of Co-, Cr-, and Fe-doped functionalized armchair MoS2 nanoribbons Cite as: AIP Advances 10, 095029 (2020); https://doi.org/10.1063/5.0022891 Submitted: 24 July 2020 . Accepted: 02 September 2020 . Published Online: 29 September 2020 M. DavoodianIdalik , and A. Kordbacheh 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 Spectromicroscopic measurements of electronic structure variations in atomically thin WSe 2 AIP Advances 10, 095027 (2020); https://doi.org/10.1063/5.0018639 Molecular dynamics study on scattering characteristics of nitrogen molecules from platinum surface by molecular beam method AIP Advances 10, 095028 (2020); https://doi.org/10.1063/5.0018905 Origin of band inversion in topological Bi 2Se3 AIP Advances 10, 095018 (2020); https://doi.org/10.1063/5.0022525AIP Advances ARTICLE scitation.org/journal/adv Structural, electronic, and transport properties of Co-, Cr-, and Fe-doped functionalized armchair MoS 2nanoribbons Cite as: AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 Submitted: 24 July 2020 •Accepted: 2 September 2020 • Published Online: 29 September 2020 M. DavoodianIdalika) and A. Kordbacheh AFFILIATIONS Materials Simulation Laboratory, Department of Physics, Iran University of Science and Technology, 1684613114 Tehran, Iran a)Author to whom correspondence should be addressed: majiddavoodian@chmail.ir ABSTRACT Using density functional theory, the structural, electronic, and transport properties of N, O, and F edge functionalized armchair molybde- num disulfide (AMoS 2) nanoribbons (NRs) substituted with Cr, Fe, and Co impurity atoms were investigated. The near edge position of functionalized AMoS 2NRs is preferred to substitute the impurity atoms, and all the structures are energetically stable. The bandgap of the structures is dramatically changed with 1% of the impurity metal atoms. In addition, multiple negative differential region phenomena exist with the substitution of these three metal impurities, and the peak to valley ratio of substituted NRs is more than that of unsubstituted nanoribbons. ©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/10.1063/5.0022891 .,s INTRODUCTION Ever since the emergence of graphene in 2004, 2D materials have been proven to have stable existence in real-life environments and to be easily prepared via micromechanical cleavage (MMC), paving the way to investigate low-dimensional materials (LDMs).1 In addition to exhibiting many markedly different properties with regard to their bulk counterparts due to edge effects and quantum confinement (QC),2–4LDMs are also reportedly technically used in nanoelectronics.5–18 Nanoribbons (NRs) (a quasi-1D 2D material) have been widely studied experimentally and theoretically. Their special structural and electronic properties are highly efficient in potentially imple- menting novel technologies in biological and chemical sensors, as well as nanoelectronics1,19,20and optoelectronics.21,22 Even though it has only recently emerged and surveyed, much attention has been paid to monolayer (ML) MoS 2, thanks to its applicability in 2D nanodevices.23–28As a direct bandgap semicon- ductor (1.8 eV bandgap),24ML-MoS 2is synthesizable simply via lithium–ion intercalation or by Scotch tape.23,25–29It is also geared to the needs of quasi-1D NRs by a hybrid chemical–electrochemical route.30,31Their edge passivation, geometry, and size affect theassociated electronic properties.31All MoS 2NRs have the same hon- eycomb structure and exhibit intriguing dimensionality effects.32,33 Recently, 1D MoS 2has undergone synthesis in various experi- ments.34–37Armchair MoS 2NRs (AMoS 2NRs) are nonmagnetic semiconductors whose bandgap shifts from indirect to direct with an increase in ribbon width.38Zigzag MoS 2NRs (ZMoS 2NRs) are mag- netic metals with ferromagnetic (FM) edge states.39Electronic prop- erties and the tunable energy bandgap must be taken into account for extensive application of MoS 2NRs in electronic devices. The manip- ulation of NR edges leads to a modified electronic structure, typically more receptive than their bulk counterparts, providing an effective means to tailor them to specific applications.40 Research also suggests that electronic properties of MLs can be effectively modulated in multiple pathways, including edge mod- ifications,41heteroatom doping,42–44the introduction of topologi- cal defects,45,46and application of external magnetic and electric fields.14,47 Edge modification and structure affect the physicochemi- cal properties of MoS 2NRs, causing them to display many cap- tivating properties.48–54For example, AMoS 2NRs turn into anti- ferromagnetic (AFM) or ferromagnetic (FM) semiconductors (AFMSs or FMSs) if they undergo various edge hydrogenation AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv modifications, contributing to their use in electronics and spintron- ics.48According to Xiao et al. ,49hole mobility could clearly materi- alize a tenfold increase in pristine ribbons when AMoS 2NR edges ended with F or H atoms. As predicted by Pan and Zhang29and Wang et al. ,39regarding conductivity, ZMoS 2NRs could be a semi- conductor ( p-type or n-type) or semimetal through edge structure control. This paper will investigate the electronic, structural, and trans- port properties of X–Y–substituted Co, Cr, and Fe atoms in edge- terminated AMoS 2NRs (X, Y = N, O, and F) using density func- tional theory (DFT). The remainder of the paper is structured as follows: Section II demonstrates how structures are considered. Sec- tion III examines the electronic properties. Finally, Sec. IV discusses transport properties. METHODOLOGY Electronic structure calculations were conducted using the SIESTA package,55and norm-conserving fully relativistic pseudopo- tentials of the Troullier–Martins type56,57in a fully separable form (Bylander–Kleinman) were employed for the whole species. The above-mentioned calculations were then checked by WIEN2k based on the spin polarized full-potential linearized augmented planewave (FLAPW) plus the local orbital method.58The supercell method was applied with periodicity in the z-direction. A 20 Å vacuum space was introduced to terminate and decouple ribbon-periodic image inter- actions. To consider electron exchange-correlation potential, the Perdew–Burke–Ernzerhof (PBE)59formulation of the generalized gradient approximation (GGA) with double-zeta polarized (DZP) basis sets was chosen. The mesh cutoff was set to 526 Ry, and the Monkhorst–Pack was allocated to k-sampling with a 1 ×1×18 grid, utilized for relaxation calculations. A 1 ×1×54 grid was also employed for state density calculations. Self-consistent calculations were terminated in case of a >1×10−6eV energy difference between the two steps. The Gaussian smearing method was also utilized with a 0.01 Ry smearing width. Conductivity and electron transfer were calculated by the quan- tum transport code GOLLUM,60which uses a SIESTA-derived quasiparticle Hamiltonian system, employed to calculate quantum transport within the framework of the Landauer–Büttiker formal- ism. Thus, the overlap and DFT Hamiltonian matrices were analyzed by SIESTA upon processing by employing the GOLLUM package. This code has been formulated based on the equilibrium transport theory, according to which branches, the central scattering region, and electrodes are determined. Transmission coefficients are cal- culated without requiring independent self-consistent calculations for density matrices. This method offers a couple of advantages, including being time-saving and computational resource-saving. ELECTRONIC AND STRUCTURAL PROPERTIES The unit cell for an X and Y edge-terminated armchair MoS 2 nanoribbon (AMoS 2NR–X–Y, X, Y = N, O, and F) substituted by one impurity atom (AMoS 2NR–X–Y–Im, Im = Cr, Fe, or Co) is shown in Fig. 1. Here, Zand ydenote the ribbon direction and the direction width, respectively. The number of dimer lines along the ribbon width is set to 9. One-dimensional periodic boundary conditions (1D PBCs) are used along the z-axis. The unit cell for FIG. 1 . Unit cell of an edge terminated armchair MoS 2nanoribbon substituted by impurity atoms (AMoS 2NR–X–Y–Im). Dashed lines show the unit cell of an edge terminated armchair MoS 2nanoribbon without impurity (left: top view; right: side view). the AMoS 2NR–X–Y is depicted by dashed lines. Hydrogenation is indeed passivation of edges mainly applied to atoms whose edges are terminated.61The rate of edge hydrogenation during the passivation process is determined by factors such as hydrogen concentration, pressure, and temperature. Various AMoS 2NR edge hydrogenations are achievable and controllable during the passivation process. For instance, each edge atom of either S or Mo may be terminated with/without an H atom, or each edge atom of S and Mo may be terminated with no or two H atoms.23,62However, stability may be prevented in the above-mentioned chemical functionalization due to fairly weak Mo–H and S–H bonds.63 This study aims to develop a DFT-based first-principle calcula- tions approach to evaluate the transport and electronic properties of edge-passivated AMoS 2NRs substituted by 3d transition metal (TM) atoms. Here, ribbon edge passivation and combination are performed by F, H, N, and O atoms. Each edge atom of S and Mo is terminated by an H and X/Y atom (X, Y= N, O, and F), respectively. As indicated in Fig. 1, different edge atoms of Mo can be terminated by different X/Y atoms, illustrated as homo- and FIG. 2 . Inequivalent positions to substitute the impurity atom. AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv TABLE I . Scaled energy of inequivalent substitution positions for different impurity atoms (Cr, Fe, and Co) in F-functionalized AMoS 2NRs. Position 1 2 3 4 5 Scaled energy-Fe (eV) 0 0.564 2.231 2.867 2.903 Scaled energy-Cr (eV) 0 3.345 255 4.482 314 4.572 058 4.592 721 Scaled energy-Co (eV) 0 1.381 191 2.474 97 4.249 862 4.448 76 hetero-edge AMoS 2NRs, namely, AMoS 2NR–X–X and AMoS 2NR– X–Y, respectively. Unlike hydrogen-passivated AMoS 2NRs and pure MoS 2NRs, AMoS 2NRs apparently exhibit greater stability when each edge atom of Mo and S is terminated by non-metallic (NM) atoms (i.e., F, N, and O) and an H atom, respectively, as seen in AMoS 2NR–X–Y.4 To analyze the impact of dopant atoms, the AMoS 2NR(9,3) was constructed as the model system, where “9” is the number ofdimer lines along the ribbon width, e.g., 9-AMoS 2NR (Fig. 1), and “3” implies three primitive unit cells of the 9-AMoS 2NR to prevent dopant interactions in adjacent supercells. To see if impurity Co, Cr, and Fe atoms can be substituted in AMoS 2NRs, Mo atoms of AMoS 2NRs can be substituted by the system consisting of 99 atoms and impurity atoms, outlined as the AMoS 2NR–X–Y–Im system, where Im represents Fe (Co, Cr). As displayed in Fig. 2, five non-equivalent impurity-atom doping sites in the 9-AMoS 2NR were taken from edge to center. AMoS 2NR–X–Y–Im would be notably distinct from AMoS 2NR–Y– X–Im. The per atom formation energy of impurity-doped systems can be calculated as follows: ΔHf=(Etot−nxEx−nyEy−nHEH−nimEim−nMoEMo−nSES)/i. In the equation above, Etotrepresents the total energy of the impurity-doped AMoS 2NR–X–Y. EH,Eim,EMo,Es, and Ex,yare the total energies of H- and impurity-doped (Fe, Co, or Cr) and free Mo, TABLE II . The band gap, formation energy per atom, bond length of impurity atoms and the S atom (inner and outer S atoms), and bond length of the impurity atom and the near edge atom. Impurity Formation Bond length Bond length Bond length atom Edge Structure Band gap energy Im-S_inner Im-S_outer Im–X AMoS 2NR–N–N–Cr 0.42 −4.57 2.34 2.4 1.59 CrHomo-edge AMoS 2NR–O–O–Cr 0.1 −5.29 2.4 2.43 2.16 AMoS 2NR–F–F–Cr 0.3 −4.6 2.4 2.43 1.92 Hetero-edgeAMoS 2NR–O–F–Cr 0.06 −4.95 2.4 2.43 2.15 AMoS 2NR–F–O–Cr 0.38 −4.96 2.4 2.43 1.93 AMoS 2NR–O–N–Cr 0.4 ∗ −4.94 2.4 2.43 2.15 AMoS 2NR–N–O–Cr 0.51 −4.94 2.34 2.4 1.59 AMoS 2NR–F–N–Cr 0.16 −4.59 2.4 2.43 1.92 AMoS 2NR–N–F–Cr 0.59 ∗ −4.58 2.34 2.4 1.59 AMoS 2NR–N–N–Fe 0.3 −4.59 2.26 2.28 1.55 FeHomo-edge AMoS 2NR–O–O–Fe 0.07 −5.31 2.43 2.44 2.14 AMoS 2NR–F–F–Fe 0.3 −4.62 2.46 2.45 1.86 Hetero-edgeAMoS 2NR–O–F–Fe 0.56 −4.96 2.43 2.43 2.15 AMoS 2NR–F–O–Fe 0.3 −4.98 2.46 2.45 1.88 AMoS 2NR–O–N–Fe 0.26 −4.96 2.43 2.44 2.15 AMoS 2NR–N–O–Fe 0.32 −4.95 2.26 2.28 1.55 AMoS 2NR–F–N–Fe 0.22 −4.61 2.46 2.45 1.88 AMoS 2NR–N–F–Fe 0.31 −4.6 2.26 2.28 1.55 AMoS 2NR–N–N–Co 0.33 −4.61 2.28 2.27 1.97 CoHomo-edge AMoS 2NR–O–O–Co 0.09 −5.31 2.42 2.4 2.13 AMoS 2NR–F–F–Co 0.5 −4.62 2.43 2.42 1.83 Hetero-edgeAMoS 2NR–O–F–Co 0.07 −4.97 2.41 2.4 2.12 AMoS 2NR–F–O–Co 0.3 −4.98 2.43 2.42 1.89 AMoS 2NR–O–N–Co 0.2 −4.96 2.42 2.4 2.13 AMoS 2NR–N–O–Co 0.38 −4.98 2.28 2.27 1.97 AMoS 2NR–F–N–Co 0.15 −4.61 2.43 2.43 1.89 AMoS 2NR–N–F–Co 0.29 −4.63 2.27 2.27 1.97 Pristine AMoS 2NR 0.52 −4.7 . . . . . . . . . AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . Band structure of the AMoS 2NR– F–F–Fe (left) and AMoS 2NR–F–F (right). S, X, or Y (X, Y = N, O or F) atoms in the AMoS 2NR–X–Y, respec- tively. nH,ns,nim,nMo, and nx,yare the number of H, S, impurity, Mo, and X(Y) atoms in the ribbon, respectively, and iis the total num- ber of atoms in the structure. Table I, for example, demonstrates the associated scaled formation energy for the Co-, Cr-, and Fe-doped AMoS 2NR–F–F in inequivalent sites. Table I shows a successive monotonic increase in “formation energy” for substitution sites 1–5 for structures substituted by asingle impurity atom. That is, it is highly possible that instead of atoms on interior sites, the Fe (Cr, Co) atom would replace near- edge Mo atoms. The first position has minimum formation energy as bonding between host and impurity atoms at edges can be entirely relaxed. The fifth position has maximum formation energy as the relaxation of the above bonds is restricted. Accordingly, outer- most cations tend to be substituted by impurities, translating to an edge effect with reduced formation energy when impurity atoms FIG. 4 . Band structure of the AMoS 2NR–X–Y–1Fe structures. AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv are closer to the edge, like other NRs, such as graphene and sil- icene.64–66The substitutional behaviors of Co, Cr, or Fe atoms in the AMoS 2NR–X–Y or even AMoS 2NR have scantily undergone experimentation, though cations at AMoS 2NR–X–Y edges would predictably, from a thermodynamic point of view, be substituted by the atoms above. Taking these favored sites, the angle and length of near- substituted-atom Mo–Mo (Mo–S, Mo–X, S–S, S–X, and X–X) bonds remarkably change as compared to the bond length and angle of the unsubstituted AMoS 2NR–X–Y. For instance, bond lengths of Fe–S (2.44 Å) and Fe–O (2.14 Å) are a bit shorter and longer than those of Mo–S (2.50 Å) and Mo–O (1.73 Å), respectively. The interatomic spacing of H–H and O–O bonds changes from 2.12 Å to 1.50 Å and 5.5 Å to 5.2 Å, respectively. The edge atom of O causes a ≈23% geometric deformation (Table II). Table II shows the bandgap, per- atom energy formation, the distance between impurity atoms and the S atom (inner and outer S atoms), and the distance between the impurity atom and the near-edge atom. A comparison has been made between the band structures of the Fe-doped AMoS 2NR–F–F at the substitution position with max- imum stability and the unsubstituted AMoS 2NR–F–F in Fig. 3 (the band structure of a pristine AMoS 2NR is presented in Fig. S13 in the supplementary material).Unlike pristine AMoS 2NR–X–Ys, the bandgap of Fe-substituted AMoS 2NR–X–Ys have clearly diminished. As shown in Fig. 3, the bandgaps of the AMoS 2NR–F–F–Fe and AMoS 2NR–F–F are 0.3 eV and 0.4 eV, respectively. According to Fig. 4 and sup- plementary material figures, Co- and Cr-doped structures have shown to have been affected similarly by a reduction in the bandgap. The AMoS 2NR–X–Y valence band (VB) is raised by MoS2-hybridized Fe- (Co-, Cr-) impurity bands, resulting in the reduced bandgap. Since the number of valence electrons is higher in Fe (3 d64s2) than in Mo (4 d55s1), the former would be capa- ble of injecting additional electron carriers into MoS 2, causing doping effects of the n-type. In a similar vein, MoS 2will be transformed into an n-type semiconductor by Co (3 d74s2) impu- rity doping, acknowledged by the results of the density of states (DOSs), suggesting a raised Fermi level by these 3 d-TM-impurity doping. Based on the DOSs of substituted impurities in functionalized AMoS 2NRs, metal impurities are greatly spin-polarized. No state at the Fermi level is occupied by a spin-up and spin-down elec- tron. Even though impurity-doped AMoS 2NR–X–Y structures have maintained their semi-conductivity, near-Fermi-level DOS distribu- tion has been vigorously regulated by impurities, unlike unsubsti- tuted AMoS 2NR–X–Ys (Fig. 5). FIG. 5 . Total density of states of homo-edge terminated AMoS 2NRs substituted by Cr, Fe, and Co. AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 6 . Left and right electrodes with a scattering region of the AMoS 2NR–X–X–Im. In a nutshell, AMoS 2NR–X–Y–Im energy gaps fall within the range 0.01 eV–0.59 eV (indirect or direct), smaller than those of unsubstituted AMoS 2NR–X–Ys, contributing to improved local chemical affinity, rendering them beneficial for electronic devices. TRANSPORT PROPERTIES Using a double-probe-system model, current–voltage (I–V), transmission spectra, and transport properties of Co, Cr, and Fe sub- stitution in a homo-edge-terminated AMoS 2NR–X–X (X = F, O, N) have been examined.Electrode interactions were eliminated by the presence of three pristine unit cells of the AMoS 2NR–X–X in the scatter section. As observed in Fig. 6, the system consists of a scatter section and double semi-infinite electrodes, namely, right and left electrodes. The bias voltage was in the range of 0 V–3.0 V. This study has explored the I–V properties of the AMoS 2NR– X–X–Im, the results of which suggested the occurrence of nega- tive differential region (NDR) phenomena in the AMoS 2NR–X–Y or bare AMoS 2NR. The NDR peak-to-valley ratio (PVR) (Ip/Iv) for bare-, F-, N-, and O-edge-terminated AMoS 2NRs has also been found to be around 1.66, 3.11, 2.33, and 11.35, respectively.4Inter- estingly enough, the NDR phenomenon is also manifest regard- ing impurity substitution in edge-terminated AMoS 2NRs, although the NDR Ip/Iv has been converted to different quantities. For instance, the Ip/Iv of Cr substitution in F- and N-edge-terminated AMoS2NRs is ∼100. The I–V characteristic exhibited a 0.5 eV threshold in the AMoS 2NR–N–N and Co-/Fe-substituted AMoS 2NR–O–O, which is raised to ≈1 V for Cr substitution in F- and O-edge-terminated AMoS 2NRs due to the absence of state density at the Fermi level (Fig. 5). The increase in the current value was associated with the voltage applied beyond the threshold. An NDR phenomenon occurs when an increased voltage leads to a decrease in the current. Based on Fig. 4, the bandgap structure is modified as a result of substitution by the aforesaid metal atoms and edge termination by the NM AMoS 2NR-bare. The current threshold, FIG. 7 . I–V characteristic of the homo-edge functionalized AMoS 2NR–X–X–Im. AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-6 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 8 . Current–voltage and transmis- sion spectra at three different voltages (b) V = 0.25, (c) V = 0.75, and (d) V = 1.25 of AMoS 2NR–N–N–Co. therefore, shifts for terminated and substituted NRs. Nevertheless, the I–V curve shown in Fig. 7 depicts the NDR phenomena, notably in reconstructed MoS 2NRs.4,44,52,61 A transmission spectrum plot at a changing bias voltage is depicted in Fig. 8, elucidating the NDR phenomenon. Here, the transmittance spectra and I–V characteristic of a two-probe AMoS 2NR–N–N–Co system are illustrated. The former was calcu- lated at three bias voltages (i.e., V 1= 0.25, V 2= 0.75, and V 3= 1.25). Due to the semi-conductive behavior of the AMoS 2NR–N–N–Co, a small transmittance band exists at the Fermi level at high volt- ages. The transport window looms due to elevated voltage, causing the disappearance of the transmission platform at 1.25 V, result- ing in the NDR in the I–V curve and for other impurity-substituted homo-edge-terminated MoS 2NRs. An increased bias voltage causes the transmission platform to vanish at various bias voltages for Cr (Fe, Co) substitution in homo- edge-terminated AMoS 2NRs, leading to the NDR in I–V curves. Charge carriers are typically transported along the edge of NRs. Additionally, the band structure of bare AMoS 2NRs changes as charge carriers of metallic impurities scatter. Hence, novel transport properties, including multiple NDR phenomena, emerge in homo- and hetero-edge-functionalized structures, potentially applicable in state-of-the-art digital systems. CONCLUSION Using density functional theory, the structural, electronic, and transport properties of different structures of substitution Cr, Fe, and Co in functionalized AMoS 2NR were studied. The edges of the AMoS 2NR were terminated with H, N, O, and F atoms, and one Mo was replaced by one impurity atom. The preferred position to substitute the impurity (Cr, Fe, and Co) atoms was found to bethe Mo atom near the edge of the nanoribbons. All the structures were energetically stable. The energy bandgap of the AMoS 2NR–X– Y reduces with the substitution of these impurity atoms and ranges from 0.01 eV to 0.59 eV, direct to indirect. Using the Green’s func- tion method, the transport properties were studied and multiple NDR phenomena were observed with substitution of these impu- rity atoms although the Ip/Iv ratio was increased dramatically in comparison with that of unsubstituted structures. SUPPLEMENTARY MATERIAL Band structures and total density of states of the homo- and hetero-edge terminated armchair MoS 2nanoribbon substituted with impurity atoms (Cr and Co) are presented in the supplementary material. DATA AVAILABILITY The data that support the findings of this study are available within the article and its supplementary material. REFERENCES 1K. S. Novoselov, Science 306, 666 (2004). 2Y.-H. Zhang, W.-J. Ruan, Z.-Y. Li, Y. Wu, and J.-Y. Zheng, Chem. Phys. 315, 201 (2005). 3C. Berger, Science 312, 1191 (2006). 4M. DavoodianIdalik, A. Kordbacheh, and F. Velashjerdi, AIP Adv. 9, 035144 (2019). 5M. Farahani, T. S. Ahmadi, and A. Seif, J. Mol. Struct.: THEOCHEM 913, 126 (2009). 6K. S. Novoselov, A. K. Geim, S. 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AIP Advances 10, 095029 (2020); doi: 10.1063/5.0022891 10, 095029-8 © Author(s) 2020

5.0022160.pdf

AIP Advances 10, 095021 (2020); https://doi.org/10.1063/5.0022160 10, 095021 © 2020 Author(s).Enhancement of spin signals by thermal annealing in silicon-based lateral spin valves Cite as: AIP Advances 10, 095021 (2020); https://doi.org/10.1063/5.0022160 Submitted: 17 July 2020 . Accepted: 02 September 2020 . Published Online: 21 September 2020 N. Yamashita , S. Lee , R. Ohshima , E. Shigematsu , H. Koike , Y. Suzuki , S. Miwa , M. Goto , Y. Ando , and M. Shiraishi COLLECTIONS Paper published as part of the special topic on Chemical Physics , Energy , Fluids and Plasmas , Materials Science and Mathematical Physics This paper was selected as an Editor’s Pick ARTICLES YOU MAY BE INTERESTED IN Numerical modeling and experimental validation of passive microfluidic mixer designs for biological applications AIP Advances 10, 105116 (2020); https://doi.org/10.1063/5.0007688 Catalog of magnetic topological semimetals AIP Advances 10, 095222 (2020); https://doi.org/10.1063/5.0020096 Numerical investigations of trajectory characteristics of a high-speed water-entry projectile AIP Advances 10, 095107 (2020); https://doi.org/10.1063/5.0011308AIP Advances ARTICLE scitation.org/journal/adv Enhancement of spin signals by thermal annealing in silicon-based lateral spin valves Cite as: AIP Advances 10, 095021 (2020); doi: 10.1063/5.0022160 Submitted: 17 July 2020 •Accepted: 2 September 2020 • Published Online: 21 September 2020 N. Yamashita,1,a) S. Lee,1 R. Ohshima,1 E. Shigematsu,1H. Koike,2Y. Suzuki,3 S. Miwa,3,b) M. Goto,3 Y. Ando,1and M. Shiraishi1 AFFILIATIONS 1Department of Electronic Science and Engineering, Kyoto University, Kyoto, Kyoto 615-8510, Japan 2Advanced Products Development Center, TDK Corporation, Ichikawa, Chiba 272-8558, Japan 3Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan a)Author to whom correspondence should be addressed: yamashita.naoto.64r@st.kyoto-u.ac.jp b)Present address: Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan. ABSTRACT The effect of thermal annealing on spin accumulation signals in silicon (Si)-based lateral spin devices is investigated. The annealing is carried out after fabrication of the spin devices, which allows us to directly compare the spin-related phenomena before and after annealing. The magnitude of non-local four-terminal signals ( ΔVnl) at room temperature is increased more than two-fold after annealing at 300○C for 1 h. The channel length dependence of ΔVnland the Hanle signals reveal that the spin polarization of the ferromagnetic contact is increased by the annealing. In contrast, the spin diffusion length and spin lifetime in the Si channel do not change. ©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/10.1063/5.0022160 .,s INTRODUCTION Silicon (Si) has great innovative potential for spintronic appli- cations because it enables good spin coherence of the carriers due to the weak spin–orbit interaction and lattice inversion symmetry.1,2In addition to these advantages, the good compatibility of Si-based spin devices with the present electronics industry has motivated tremen- dous efforts to investigate the spin transport properties in Si.3–16 Several spintronic devices based on Si have been proposed,17–20and some have been demonstrated even at room temperature.11,12,21–24 However, from the point of view of practical use, the demonstra- tion of spin-related phenomena is not sufficient; other requirements, e.g., the heat or chemical tolerances of the device, the operation endurance, and the disturbance stability of device operation, should also be satisfied. In particular, investigations of the thermal tolerance of devices are strongly desired because thermal treatments are com- mon processes in the fabrication of integrated circuits. Even after the fabrication of an integrated circuit, namely, during post-processing, post-annealing processes such as soldering are also necessary to mount the circuit on a printed circuit board.For a spintronic device, thermal treatments are employed to improve the spin and magnetic properties.25–31For instance, enhancement of the spin polarization of a tunneling current after thermal annealing has been reported in a magnetic tunnel junction. Because thermal treatment has a variety of possible effects on Si- based spintronic devices, such as the formation of silicides, the intro- duction of strain into the Si channel, and the activation/deactivation of donors and acceptors, and owing to the enhancement of spin polarization, a detailed investigation of the thermal annealing effect is strongly desired. While systematic investigations of only a degen- erate Si-based spin device exist,29an investigation of non-degenerate Si is quite important for its practical application. In this study, we investigate the effect of thermal annealing on the spin accumulation voltage in lateral spin valves based on a non-degenerate Si at room temperature. The magnitude of the spin accumulation signals detected by the non-local four-terminal (NL- 4T) configuration is increased by more than two-fold after annealing at 300○C. Hanle effect measurements and an investigation of the current–voltage characteristics are carried out to reveal the origin of the enhancement of the spin accumulation signals. AIP Advances 10, 095021 (2020); doi: 10.1063/5.0022160 10, 095021-1 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv EXPERIMENTAL Figure 1(a) shows a schematic of the lateral spin valve devices used in this study. The devices consisted of an n-type non-degenerate Si channel, Fe (12.4 nm)/CoFe (0.6 nm)/MgO (0.8 nm)/ n+-Si (20 nm) ferromagnetic contacts (F1 and F2), and FIG. 1 . Schematic image of the measured devices and the internal temperature of the furnace during annealing. (a) Schematic image of the Si-based spin transport device. The two Fe/CoFe/MgO electrodes F1 and F2 were fabricated on n+-Si as a spin detector and injector, respectively. (b) Time evolution of the temperature during thermal annealing measured by a thermocouple adjacent to the sample.two nonmagnetic ohmic contacts (M1 and M2). The devices were fabricated on a silicon-on-insulator substrate with a 100-nm-thick Si(100) layer/200-nm-thick SiO 2layer/625-μm-thick Si(100) sub- strate. Phosphorus (P) atoms were doped into the Si layer by the ion implantation technique to fabricate the n-Si and n+-Si layers. Then, a rapid thermal annealing was carried out for their activation. The dopant concentration in the Si channel measured by secondary- ion mass spectroscopy showed a small distribution perpendicular to the plane in the range of 1 ×1017to 2×1018cm−3, indicating a non-degenerate Si. The dopant concentration of the 20-nm-thick n+-Si layer, employed to suppress the width of the depletion layer at the ferromagnetic contacts, was 5 ×1019cm−3. The ferromagnetic contacts F1 and F2 composed of 12.4-nm-thick Fe/0.6-nm-thick CoFe/0.8-nm-thick MgO tunnel barriers were fabricated by electron beam deposition in an ultrahigh vacuum. While the width of F1 was designed to be 0.2 μm, that of F2 was designed to be 1.2 μm for the measurements of spin accumulation signals. Other samples with the same structure and different width of F2 (0.8 μm, 1.6μm, and 2.0μm) were also prepared for the measurements of the channel resistance. The widths of F2 of these samples were 0.8 μm, 1.6μm, and 2.0μm. The center-to-center distance between F1 and F2, L, var- ied from 1.6 μm to 2.2μm. Electron beam lithography and Ar ion milling were employed to fabricate F1 and F2 and to remove the n+- Si layer in the channel region. Finally, the Au/Al ohmic contacts M1 and M2 were fabricated. The spin accumulation signals and Hanle effect signals were measured in the NL-4T configuration. The interface resistances of F1 and F2 and the conductivity of the Si channel were evaluated from the current–voltage characteristics in three- and four-terminal con- figurations, respectively. The above measurements were performed before and after annealing. The devices were annealed at ∼300○C for 1 h in a vacuum. We employed a vacuum furnace because of the uniform distribution of temperature in the sample compared to other methods, such as rapid thermal annealing. Although annealing under a magnetic field is a common way to improve the quality of ferromagnetic materials, we carried out the annealing without exter- nal magnetic field to investigate the thermal tolerance of the devices. Figure 1(b) shows the time evolution of the temperature measured by a thermocouple placed adjacent to the sample. The pressure was kept below 1 ×10−4Pa during annealing to prevent oxidization. A thermocouple was placed adjacent to the sample to measure its tem- perature during annealing. The temperature was raised to 300○C within 30 min. After reaching the set temperature, the temperature settings were kept for 1 h. Then, the sample was cooled to the ambi- ent temperature by natural cooling in a vacuum, which took ∼4 h. Figure 1(b) shows the time evolution of the temperature measured by the thermocouple. The temperature was kept between 290○C and 295○C for at least 55 min. RESULTS We first investigated the annealing effect on the interface resistances of F1 and F2 and the conductivity of Si. The voltage drop, V3T, at F1 and F2 as a function of the bias current Iis shown in Figs. 2(a) and 2(b). The current–voltage configurations are shown in the insets of Figs. 2(a) and 2(b). The black and red lines show the results before and after annealing. Nonlinear I–V3T AIP Advances 10, 095021 (2020); doi: 10.1063/5.0022160 10, 095021-2 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 2 . Effects of annealing on the electrical properties of spin valve devices. Volt- age drop at the ferromagnetic contact, V3T, of (a) F1 and (b) F2 as a function of the charge current I. The black (red) solid lines represent the measured I–V3Tcurves before (after) thermal annealing. The insets in (a) and (b) show the average RA products of the four devices at I= 0.0 mA and 0.3 mA, respectively. (c) Channel length dependence Lchof the resistance measured by the four-terminal method. The black (red) dots represent the values measured before (after) thermal anneal- ing. The solid lines represent the result of the linear fitting. The inset shows the conductivity of Si estimated by the linear fitting of Rch.characteristics were obtained for both F1 and F2, indicating the contribution of the tunneling current. The interface resistance was slightly increased by thermal annealing, especially for F2. A simi- lar behavior, namely, a slight increment in the interface resistance after annealing, was observed for all devices. Here, we focus on the resistance area products of F1 ( RA 1) and F2 ( RA 2) atI= 0 mA and 0.3 mA, respectively, because these are the conditions for spin trans- port studies, which will be discussed later. While RA 1atI= 0 mA did not change after annealing, as shown in the inset of Fig. 2(a), RA 2at I= 0.3 mA slightly increased from 17 kΩ μm2to 18 kΩμm2[the inset of Fig. 2(b)]. The resistance of the Si channel was measured through the four-terminal method to estimate the conductivity of Si. Figure 2(c) shows the resistance ( Rch) of the Si channel as a function of the channel length Lch. We employed the nearest edge-to-edge distance between F1 and F2 as Lchto avoid the shunting effect of the n+- Si layer underneath F1 and F2. The conductivity of the Si channel (σ), evaluated by linear fitting of the data, was 2.0 ×103S/m and 0.9×103S/m before and after annealing, respectively. The modula- tion ofσby thermal annealing was pronounced compared to that of RA 1andRA 2. The possible origin of this considerable reduction in σ will be discussed in the Discussion section. Although several methods exist for investigating the spin diffu- sion length of the Si channel, λSi, and the spin polarization of F1 and F2,5,12,32we employed the NL-4T method, the current–voltage con- figuration of which is shown in the inset of Fig. 3(a). We employed F1 and F2 as a spin detector and injector, respectively, because no significant difference in spin signals is generally obtained even for the F1 spin injector and F2 spin detector. A current of 0.3 mA was applied from F2 to M2, and the non-local voltage at F1 was mea- sured with reference to M1. The external magnetic field for the mea- surements of the spin accumulation signals was applied along the y-direction, which was used to control the magnetic configurations of F1 and F2. For the NL-4T method, because the spin accumula- tion voltage underneath F1 (spin detector) is mainly generated by the diffusion of the pure-spin current from F2 (spin injector), we can use simple analyses to evaluate the spin-related parameters. The spin accumulation signals before annealing are shown as black dots in Fig. 3(a). The center-to-center distance, L, between F1 and F2 was 1.6μm. A clear rectangular hysteresis signal with steep volt- age changes corresponding to magnetization reversal was obtained. The magnitude of the spin accumulation signal ( ΔVnl) was ∼10μV atI= 0.3 mA. A clear rectangular signal was obtained even after annealing at 300○C, shown by the red dots in Fig. 3(a), indicat- ing a heat tolerance up to 300○C. Note that ΔVnlafter annealing was 24μV, more than two-fold that before annealing. In short, we confirmed not only the thermal tolerance of the spin-related func- tion but also its improvement by thermal annealing. We obtained a similar enhancement in ΔVnlfor devices on another Si substrate, the thermal annealing of which was carried out on different occa- sions to check the repeatability of the annealing effect. Therefore, we conclude that the enhancement of ΔVnlis a general feature for our Si-based spin devices. To investigate the mechanism of the enhancement of ΔVnlby thermal annealing in detail, the L-dependence of ΔVnlwas investi- gated.ΔVnlas a function of Lbefore and after annealing is shown in Fig. 3(b). An increase in ΔVnlby annealing was observed for all devices. We used an analytical solution of the one-dimensional spin AIP Advances 10, 095021 (2020); doi: 10.1063/5.0022160 10, 095021-3 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 3 . Spin signals and results of the analysis. (a) Measured non-local signal before and after annealing. The circles represent the measured value, and the lines are guides to the eye. The open (filled) circles represent the non-local voltage when the magnetic field was swept from a negative (positive) value to a positive (negative) value. The background voltage proportional to | By| was subtracted. A schematic image of the current–voltage configuration is shown in the inset. (b) Channel length, L, dependence of the magnitude of the non-local spin signals, ΔVnl. The open circles are the experimental results, and the lines are the curves fitted by Eq. (1). (c) Spin polarization of the tunnel barrier, β, and (d) the spin diffusion length of the Si channel, λSi, estimated from the L-dependence of ΔVnl.drift–diffusion equation33,34to evaluate λSi, whereΔVnldepends on the spin polarization of the tunnel barrier ( β),λSi,σ,RA 1, and RA 2. ΔVnlis expressed as ΔVnl=β2J {GN+1 2(Gi1+Gi2)+Gi1Gi2 4GN}exp(L λSi)−Gi1Gi2 4exp(−L λSi), (1) where GN=σ λSiis the spin conductance of Si and Gi1(2)=1 RA 1(2)is the spin conductance of the tunnel barrier of F1(F2). We assumed identicalβvalues for F1 and F2 and negligible spin resistance of the ferromagnetic metal. The experimentally obtained values of RA 1, RA 2, andσshown in Fig. 2 were used for the analysis. The fit- ting, shown by the black and red lines, yields β= 4.0 ±0.3% and λSi= 0.76 ±0.03μm before annealing and β= 6.0 ±3.1% and λSi= 0.61 ±0.17μm after annealing, respectively, which are summa- rized in Figs. 3(c) and 3(d). These values are consistent with previous studies.35Although the error bar of βafter annealing was relatively large because of an outlier for L= 1.8μm, the enhancement of β was confirmed for several sets of devices (see the supplementary material). The Hanle effect measurement was carried out to investigate the spin lifetime, τ, in the Si channel. While the current–voltage con- figuration was the same as that of the NL-4T method, the applied magnetic field was along the z-direction, as shown in the inset of Fig. 4(a). Three samples with different L(1.8μm, 2.0μm, and 2.2μm) were used for the Hanle measurements. Figure 4(a) shows the typi- cal Hanle signals before (black dots) and after (red dots) the thermal annealing. We showed a difference in non-local voltage, VP nl−VAP nl, between the parallel and antiparallel magnetic configurations. The magnitude of VP nl−VAP nlis confirmed to be consistent with ΔVnlin Fig. 3(b) and to increase after the thermal annealing. The solid lines show the curve fitting of the one-dimensional model by using the following equation:36 VP nl(Bz)−VAP nl(Bz) I =±S0∫∞ 0√ 1 4πDtcos(ωt)exp(−t τ)exp(−L2 4Dt)dt, (2) where Bzis the magnetic flux density along the z-direction, S0is the constant that determines the signal amplitude, ω=gμBBz/̵his the Larmor frequency, g= 2 is the gfactor for the electrons, μB is the Bohr magneton, and̵his the Dirac constant. Here, the effect of the conductivity mismatch is included into S0. Therefore, the anal- yses by Eqs. (1) and (2) are consistent. The fitting curves obtained using Eq. (2) well reproduced the experimental results. Figures 4(b) and 4(c) show τandλSi(=√ Dτ), respectively, for several devices, obtained from the fitting of Eq. (2). The obtained values are con- sistent with previous studies,6,21,36and no significant changes in λSi andτwere observed after annealing. To investigate the upper limit of the thermal tolerance, anneal- ing at 350○C was also carried out. However, we did not success- fully observe any spin accumulation signals after annealing (see the supplementary material). The interface resistance was drasti- cally decreased and showed almost linear I–V3Tcharacteristics. The AIP Advances 10, 095021 (2020); doi: 10.1063/5.0022160 10, 095021-4 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv FIG. 4 . Hanle signals and results of the analysis. (a) Hanle signals, i.e., difference inVnlbetween parallel and antiparallel configurations as a function of Bzbefore and after thermal annealing. The circles represent the measured values, and the solid lines represent the curves fitted by Eq. (2). Comparison of (b) the spin lifetime, τ, and (c)λSibefore and after annealing estimated from the curve fitting of Eq. (2) for several devices with different L. annealing at 350○C might induce the creation of pinholes in the tun- nel barrier or an enhancement of the leakage current at the edge of the electrodes, resulting in a reduction in β. DISCUSSION Hereafter, we discuss the mechanism of the enhancement of ΔVnlby annealing. We found that annealing mainly induces theenhancement of β. A similar increase in the spin polarization of the ferromagnetic contacts after annealing was reported in the CoFe/MgO/CoFe magnetic tunnel junction, where an improve- ment in the crystallinity of MgO was obtained.25Because MgO was deposited at ambient temperature in this study, the crystallinity of MgO might be poor in the as-grown samples. Therefore, annealing might improve the crystallinity of the MgO layer for ferromagnetic metal/MgO/Si structures. The changes in the interface resistance of the MgO barrier shown in Figs. 2(a) and 2(b) are relatively small compared to that of the spin accumulation signal. This behavior is consistent with that in the magnetic tunnel junction in previous studies, where an improvement in crystallinity and spin polariza- tion of the MgO layer was successfully achieved without a signif- icant change in the interface resistance.25We also confirmed that the small increment of the interface resistances themselves, which was expected to suppress the spin resistance mismatch, did not contribute significantly to the enhancement of ΔVnlin our device. The reduction in σalso causes the enhancement of ΔVnl, as shown in Eq. (1). Considerable modulation of σeven at a relatively low annealing temperature is unexpected because the typical tem- perature for thermal treatment in the fabrication process of com- mercial Si devices is much higher. Here, we discuss possible origins for the unexpected modulation. According to the Drude model, a reduction in σindicates reductions in the carrier mobility and/or the carrier concentration. The former case is generally induced by the modulation of a stress–strain in the Si channel. We employed the AlN capping layer to induce a horizontal stress–strain in the Si channel.37The stress might be modulated after the annealing. The other possibility is a reduction in the carrier concentration. The dopants are reportedly deactivated by long-term and relatively low-temperature annealing38corresponding to the conditions in this study. In this case, a small change in λSiandτis reasonable because the deactivation process does not change the mobility significantly, as phonon scattering is dominant in our Si channel at room tem- perature. Here, we also emphasize that the reduction in σis not a key factor in the enhancement of ΔVnlbecause we obtained a sim- ilar enhancement of ΔVnlby annealing with increased σin other devices, in contrast to the behavior in Fig. 2(c) (see the supplemen- tary material). Therefore, we conclude that the enhancement of βis the general behavior of annealing and that a change in σshows a more complicated behavior. CONCLUSION In conclusion, we have studied the effect of thermal annealing on the lateral spin valves based on non-degenerate Si. The magni- tude of non-local four-terminal signals ( ΔVnl) at room temperature was increased by more than two-fold after annealing at ∼300○C. The spin polarization of the ferromagnetic contact was increased by ther- mal annealing, whereas the spin diffusion length and spin lifetime in the Si channel did not change. The conductivity of the Si channel was also modulated by annealing, which in some cases contributed to the enhancement of ΔVnl. While several ways to increase the spin signal exist, such as the employment of highly spin-polarized ferromag- netic metals, the thermal treatments after fabrication of the devices reported in this study are simple and versatile ways to improve the device properties of Si-based spin devices. AIP Advances 10, 095021 (2020); doi: 10.1063/5.0022160 10, 095021-5 © Author(s) 2020AIP Advances ARTICLE scitation.org/journal/adv SUPPLEMENTARY MATERIAL See the supplementary material for the details of the theo- retical model and the results of another sample and a different temperature. ACKNOWLEDGMENTS N.Y. acknowledges the support from the Japan Society of the Promotion of Science (JSPS) Research Fellow Program (Grant No. 20J22776). A part of this study was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan; Grant-in-Aid for Sci- entific Research (S) “Semiconductor Spincurrentronics” Grant No. 16H06330; and Grant-in-Aid for Scientific Research (B) Grant No. 19H02197. N.Y.S.L. acknowledge the JSPS Research Fellowship. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1B. Huang, D. J. Monsma, and I. Appelbaum, Phys. Rev. Lett. 99, 177209 (2007). 2B. Huang, H.-J. Jang, and I. Appelbaum, Appl. Phys. Lett. 93, 162508 (2008). 3I. Appelbaum, B. Huang, and D. J. Monsma, Nature 447, 295 (2007). 4B. T. Jonker, G. Kioseoglou, A. T. Hanbicki, C. H. Li, and P. E. Thompson, Nat. Phys. 3, 542 (2007). 5M. Kameno, Y. Ando, T. Shinjo, H. Koike, T. Sasaki, T. Oikawa, T. Suzuki, and M. Shiraishi, Appl. Phys. Lett. 104, 092409 (2014). 6S. Lee, F. 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5.0016007.pdf

AVS Quantum Sci. 2, 031702 (2020); https://doi.org/10.1116/5.0016007 2, 031702 © Author(s).Vectorial light–matter interaction: Exploring spatially structured complex light fields Cite as: AVS Quantum Sci. 2, 031702 (2020); https://doi.org/10.1116/5.0016007 Submitted: 01 June 2020 . Accepted: 21 August 2020 . Published Online: 18 September 2020 Jinwen Wang , Francesco Castellucci , and Sonja Franke-Arnold COLLECTIONS Paper published as part of the special topic on Special Topic: Optical Micromanipulation of Quantum Systems This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Interactions of Vectorial Light with Matter Scilight 2020 , 381105 (2020); https://doi.org/10.1063/10.0002028 Quantum mechanics with patterns of light: Progress in high dimensional and multidimensional entanglement with structured light AVS Quantum Science 1, 011701 (2019); https://doi.org/10.1116/1.5112027 Nonclassical light and metrological power: An introductory review AVS Quantum Science 1, 014701 (2019); https://doi.org/10.1116/1.5126696Vectorial light–matter interaction: Exploring spatially structured complex light fields Cite as: AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 Submitted: 1 June 2020 .Accepted: 21 August 2020 . Published Online: 18 September 2020 Jinwen Wang,1,2 Francesco Castellucci,1 and Sonja Franke-Arnold1,a) AFFILIATIONS 1School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 2Shaanxi Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University, Xi’an 710049, China Note: This paper is part of the special topic on Optical Micromanipulation of Quantum Systems. a)Author to whom correspondence should be addressed :Sonja.Franke-Arnold@glasgow.ac.uk ABSTRACT Research on spatially structured light has seen an explosion in activity over the past decades, powered by technological advances for generating such light and driven by questions of fundamental science as well as engineering applications. In this review, the authors highlight their work on the interaction of vector light fields with atoms, and matter in general. This vibrant research area explores the full potential of light, with clear benefits for classical as well as quantum applications. 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/10.1116/5.0016007 TABLE OF CONTENTS I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 II. SPATIALLY STRUCTURED VECTOR FIELDS . . . . . . . 2 III. DICHROISM AND BIREFRINGENCE . . . . . . . . . . . . . . 3IV. PROPAGATION IN A NONLINEAR MEDIUM . . . . . 4V. MODE CONVERSION OF VECTORIAL LIGHT . . . . . 5 VI. ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND MEMORIES . . . . . . . . . . . . . . . 7 VII. LIGHT–MATTER INTERACTION UNDER STRONG FOCUSSING . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 VIII. CONCLUSIONS AND OUTLOOK . . . . . . . . . . . . . . . . 9 I. INTRODUCTION Considering the vector nature of light is relevant in all physical sys- tems that are affected by interference in its many guises, including diffrac-tion, scattering, and nonlinear processes, and by interaction with matter that has a polarization-sensitive symmetry. Being able to design the polar- ization structure of optical beams and even of individual photons opensnew opportunities in the classical as well as the quantum regime. Thisincludes the study of topological phenomena, 1,2the conversion between phase and polarization singularities,3,4and the interaction between orbital angular momentum (OAM) and spin angular momentum (SAM),5as well as technological advances in polarimetry and ellipsometry, sensingand focusing beyond the conventional diffraction limit. 6,7Classical vector fields can mimic quantum behavior in their cor- relation between polarization and spatial degrees of freedom and carry an increased information content, compared to homogeneously polar- ized light.8Both of these features make them interesting candidates for multiplexing in communication and information systems.9,10Aw i d e range of research and review articles provides information on the gen- eration, properties, as well as classical and quantum applications of vector beams.11–13 In this review article, we examine our current work on the inter- action of vector light with matter, specifically with atomic gases. This is a young research area, and the majority of experimental research falls within the semi-classical regime. The interaction of structured light with matter is, however, relevant for all elements of quantum information networks—for the transfer, storage, and manipulation of high-dimensional quantum information. This includes passive pro- cesses, such as the propagation of light through turbulence or density fluctuations, the effect of dichroism and birefringence, and the desired or unwanted mode conversion due to linear and nonlinear interactions. Atoms, on the other hand, are active optical elements: while the complex vector structure of a light beam is modified by its interaction with an atomic medium, the atomic populations and coherences are correspondingly modified by the optical beam, effectively entangling optical with atomic structures. Atomic interactions can be drastically AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-1 VCAuthor(s) 2020. AVS Quantum Science REVIEW scitation.org/journal/aqsenhanced in the vicinity of atomic resonances, leading to significant nonlinear effects, with a response that can furthermore be altered by external magnetic fields. Atomic transitions are intrinsically sensitive to the vector nature of light. The dominant effect in light–atom interactions, the electric dipole interaction, explicitly depends on the alignment of the optical field with the atomic dipole, affecting selection rules and transition strengths. Light polarization therefore plays an important role in the preparation, manipulation, and detection of atomic states. Light polar- ization affects incoherent processes via optical pumping, as well as coherent parametric processes involving several atomic transitions. The first few decades of exploring the light–matter interaction have almost entirely been restricted to the study of homogeneously polarized light, or “scalar” light—and indeed polarization structures tend to play a little role in linear paraxial optics. The most prevalent, but maybe also the least recognized use of spatially varying polariza- tion, is Sisyphus (or polarization gradient) cooling, honored in the 1997 Nobel Prize. In this case, the optical polarization is modulated along the beam propagation direction generated by a pair of counter- propagating laser beams. This generates a modulated AC Stark shift which, in conjunction with optical pumping, allows the dissipation of energy from the atomic motion to the optical field. While this is a well-understood process, of benefit to any experimenter wishing to obtain lower atomic temperatures from a magneto-optical trap, it is likely still hiding some secrets. In this review article, however, we will concentrate on transverse polarization structures in light and describe their interaction with atomic as well as other nonlinear media. We start with a brief descrip- tion of vector light fields in Sec. IIand discuss in Sec. IIIhow they exert optical dichroism and birefringence in matter and in the propa- gation of vector beams through nonlinear media. The next two sec- tions concentrate on situations where the vector light interacts coherently with atomic or other media, allowing parametric processes which facilitate mode conversion (Sec. V) and modify the medium itself (Sec. VI). We briefly discuss the effects of vector fields under strong focusing in Sec. VII,b e f o r et h eS e c . VIII. II. SPATIALLY STRUCTURED VECTOR FIELDS Historically, structured light referred to a modulation of the intensity profile of a light beam, implemented by amplitude filters. This was later extended to include tailored, complex amplitude pro- files, which can be conveniently implemented with programmable dif- fractive elements including spatial light modulators (SLMs),14digital micromirror devices (DMDs),15–17or various alternative methods.18 The most prominent examples of such spatially tailored complex lightare the orbital angular momentum (OAM) modes, e.g., Laguerre–Gauss (LG) 19,20or Bessel modes21,22and Ince–Gaussian vec- tor modes with elliptical symmetry.23,24 Light, or at least a propagating paraxial light beam, is a transverse vector field with two independent polarization components. In order to fully tailor vector fields, one needs to control the complex ampli- tudes of each polarization component separately.25,26Doing so gener- ates spatially varying polarization states, including radial, azimuthal, spiral,27and hybrid polarization,28,29as well as full Poincar /C19e beams30–34and custom-designed polarization modes.35–44 A paraxial vectorial light beam can be written as:~uð~r?Þ¼uhð~r?Þexp iUhð~r?Þ ½/C138 uvð~r?Þexp iUvð~r?Þ ½/C138 ! ; (1) where uh;vð~r?ÞandUh;vð~r?Þare the spatial amplitudes and phases of the horizontal and vertical polarization components respectively, and ~r?¼ðx;yÞis the transverse position. In the following section, we will omit the explicit position dependence for clarity. We can rewrite this equation to accentuate the physical interpretation as: ~uð~r?Þ¼uexp iU½/C138cosðHÞexp iDU½/C138 Þ sinðHÞ ! : (2) Here u¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2 hþu2 vp is the position-dependent complex amplitude of the light and U¼Uvis the overall spatially varying phase. The third term is the local polarization vector, with an ellipticity determined bythe differential phase, DU¼U h/C0Uv, and an orientation against the horizontal axis given by H, which can be found from the amplitude ratio: tan H¼uv=uh:We note that instead of decomposition into a linear polarization basis, we could have chosen a rotational basis, or infact any other orthonormal basis set. The local polarization of such beams varies from point to point and is usually measured by deter- mining the associated Stokes parameters. We have at our disposal a plethora of methods to generate struc- tured vector fields. These fall into two broad categories: the generation of specific and arbitrary modes. The former includes the use of bire-fringent or dichroic materials within the laser cavity as active ele- ments 45–47or outside as passive elements,25,48s- and q-plates,49,50 plasmonic metasurfaces,18,51and Fresnel cones.52,53These methods tend to be highly efficient, but the spatial modes are restricted to aspecific 2D subset within the infinite-dimensional spatial state space, usually with the same amplitude but different phase profiles. A typical example would be the generation of modes /expðiuÞ^r /C0 þexpð/C0i/Þ^rþ,w h e r e udenotes the azimuthal angle and ^r6are left- and right-handed polarization states. Polarization optics then allow manipulations within this subset, generating radial, azimuthal, hybrid,and spiraling polarization states. The resulting modes are in general not eigenmodes of propagation and experiments have to be performed in the appropriate imaging plane. Generating arbitrary structured vector beams, instead, requires the independent design of the complex amplitudes of the horizontal and vertical polarization components (or any other independent polarization components), without disturbing the transverse coher-ence of the light field. This can be achieved by placing programmable devices like SLMs or DMDs within interferometers. 15,54–57These tech- niques allow on-demand and real-time structuring of arbitrary vectorlight fields, 58limited only by the spatial and temporal resolution of the beam-shaping device. The drawback of these methods is that they operate at low efficiencies, and hence are more suited for classical lightbeams rather than single photons. Figure 1 illustrates the wide range of polarization structures, showing a selection of vector beams generated in our lab using a DMD-based generation method described in a previous publication. 15 In subsection (a), we explain the color scheme used throughout this review to depict polarization profiles. Figure 1(b) shows a subsection of the mode family built from the Laguerre–Gauss modes LG‘ p¼LG0 0 and LG20. The obtained polarization profiles, incidentally, are identical to the more familiar radial, azimuthal, and hybrid polarizationAVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-2 VCAuthor(s) 2020resulting from superpositions of LG61 0, but with a difference in the overall phase of exp ð/C0iuÞ: the depicted beams carry a net OAM of /C22h per photon, in contrast to the balanced case without OAM. Transformations within this mode family can easily be realized by mir- ror reflections and wave retarders. The polarization mode set of Fig. 1(b), in combination with homogeneous right and left circular states, forms a rotationally symmetric set of unbiased basis, with applications in alignment-free quantum communication.59Figure 1(c) shows two examples of Poincar /C19e beams, each containing the complete range of polarizations across their profiles, effectively mapping the polarization Poincar /C19e sphere onto the transverse beam profile. The particular examples shown here are the mode LG1 0^hþLG/C01 1^v(top) and LG2 0^h þLG/C03 1^v(bottom). III. DICHROISM AND BIREFRINGENCE Many materials, crystals, fluids, and also atomic media, can affect the polarization of light, either by optical dichroism or birefringence. The former stems from polarization-dependent absorption and the lat- ter from polarization-dependent dispersion, and each may discrimi- nate between linear or circular polarizations. Traditionally, these effects are observed for homogeneously polarized light beams; but of course they become even more interesting for polarization structured light. As absorption and dispersion are linked via the Kramers–Kronig relationship, dichroism and birefringence are connected, which is especially noticeable for near-resonant excitation in atomic media. Birefringence and dichroism can occur naturally in crystals or be induced by external forces. Crystals that naturally feature such an anisotropy are widely used to fabricate many kinds of polarization optics, including polarizers and wave retarders, and many publications describe the propagation of complex vector beams in anisotropic crystals.60–69Of particular importance for the manipulation of vector light fields are liquid crystals, which form the building blocks of SLMs and q-plates. The spatially varying birefringence of q-plates can be used as an interface to convert between spin and orbital states, for classical b e a m sa sw e l la sf o ri n d i v i d u a lp h o t o n s .70The flexibility of this approach has recently been beautifully demonstrated by generating atunable two-photon quantum interference of vector light, measured by observing the Hong–Ou–Mandel dip. 71 Atoms are not naturally anisotropic, but they can become polari- zation sensitive in external fields, most notably by magneto-optical effects and by optical pumping in strong probe light. These effects are well understood and utilized in atomic magnetometry72–75and polari- zation selective absorption spectroscopy.76–79 Experiments based on optical pumping usually use a strong pump laser to induce a spin alignment of the atomic medium, which is then tested with a co- or counter-propagating weak probe laser. The polarization of the strong pump determines the quantization axis and hence the spin alignment of the atoms, which modifies the interaction with the probe beam.80Linear polarization along the quantization axis drives ptransitions, while linear polarization perpendicular to the quantization axis drives superpositions of rþandr/C0transitions, where their phase difference is determined by the orientation of the linear polarization within the perpendicular plane.81 One of the first attempts at connecting the polarization structure of an optical beam to the spatial profiles of atomic spin alignments was published in a single-author paper in 2011.82Experiments were performed in a moderately heated vapor cell of85Rb, with a right hand circularly polarized Gaussian pump beam (i.e., ^r/C0) and a much (two orders of magnitude) weaker co-axial counter-propagating vector vor- tex beam (VVB), driving the D2 line from F ¼2!F0¼2. Atoms were optically pumped into the magnetic sublevel mF¼/C02;thereby FIG.1 .Vector beams. (a) Polarization Poincar /C19e sphere and its associated colormap, linking each Stokes vector to a unique color. (b) Experimental images of radial, azimuthal, and two hybrid polarizations. These polarization structures, together with homogeneous right and left circular polarizations, can be used as an alt ernative mutually unbiased vector basis. (c) Two examples of experimental Poincar /C19e beams.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-3 VCAuthor(s) 2020enhancing the transmission of the probe, where the local helicity of the probe matched that of the pump and reduced where its helicity was opposite. It is important to remember, of course, that optical polarization is denoted with respect to the beam’s propagation direc- tion, while atoms respond to the optical helicity, defined with respect to a set quantization direction. The optically pumped atoms effectively display light-induced circular dichroism and behave like a circular polarizer. Homogeneous pump beams result in homogeneous dichroism, whereas polarization-structured pump beams display a varying degree of dichroism, as illustrated in Fig. 2 . If the role of the pump and probe are reversed, a strong vector vortex beam will pump atoms into differ- ent magnetic sublevels depending on its local polarization direction, which then can be tested with a homogeneously polarized probe,82 transferring spatially resolved information from the pump to theprobe. Similar mechanisms can also be realized in more sophisticated level structures, with the pump and probe addressing different atomic transitions via the Doppler effect. This was demonstrated, e.g., on the D2 line of 87Rb using the crossover transition signal between F¼1 !F0¼1a n d F¼1!F0¼0.83,84By combining a pump beam with a (quasi) uniform amplitude and a spatially varying polarization profile and a probe with spatially varying amplitude and homogeneous polarization, this configuration can be used for spatially resolved opti- cal information selection.85The transverse profile of a probe beam with uniform circular polarization was encoded by an SLM with an image of spatially separated numbers. This probe was passed through an atomic vapor pumped by a strong hybrid polarized beam. The dif- ferent pieces of information written in the probe beam could then befiltered by rotating the probe beam polarization with respect to the pump beam polarization distribution. Finally, mixed linear or circular dichroism can also be tested by using vector vortex beams for both pump and probe beams. 86The spatial polarization profile of vector vortex beams leads to spatially varying dichroism of the atoms and provides the opportunity for spatially tailored manipulation oflight–atom interactions, which furthermore could be modulated by varying the detuning and intensity balance between the pump and theprobe. It was shown that this pump–probe technique provides a direct tool for acquiring both SAM and OAM of structured light, 87analo- gous to projective measurements using a combination of waveplatesand polarizers. The advantage of this technique is that the extractedpart still maintains the original polarization and the vortex phase. Inother words, a circular polarizer or filter can be produced by manipu-lating the polarized state of atoms through a pump field. This schemealso provides the possibility of developing atom-optical devices and integrated devices based on atoms for projective measurements. An alternative method to induce anisotropy within atomic media is via external magnetic fields. A field along the quantization axis shifts the Zeeman sublevels, changing the resonance frequencies for opposite circular polarized light components. The resulting difference in refrac-tive indices leads to a relative phase shift between right and left circularpolarization components, 72causing effective birefringence for near- resonant light, which can be observed as Faraday rotations. Faradayeffects in polarization structured light fields 88allow a high degree of freedom in controlling circular birefringence and circular dichroism ofthe atomic medium, enabling, e.g., the demonstration of an all-opticalisolator for radially polarized light. Overall, the combination of the mature research of hot and cold atomic vapors with today’s highly versatile generation of structuredvector light fields offers new opportunities in light–matter interactions,shaping the spin alignment of the atomic medium and in turn the polarization structure of a transmitted optical beam. IV. PROPAGATION IN A NONLINEAR MEDIUM Atomic gasses can be understood as a nonlinear medium, where the response of the atoms to the optical field generates a complexpolarizability, in turn acting as a complex refractive index for the pass-ing optical field and causing absorption and dispersion. This can betested with a probe beam, as discussed in the Sec. III, but this also FIG.2 .Dichroism in an atomic medium induced by a vectorial pump field. If atoms are exposed to a strong pump beam with spatially varying polarization (left), o ptical pumping differs locally. This induces spatially varying atomic population distributions (center) which can be probed by a weak uniform beam (right). In area s where the helicity of the probe matches that of the pump transmission when enhanced, whereas areas with the opposite helicity are absorbed—effectively realizing spatially d ependent absorption spectroscopy.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-4 VCAuthor(s) 2020changes the propagation of the light field that has induced these effects in the first place. If the light is detuned far enough, absorption can be neglected, and the atoms can be treated as a two-level system, with a saturable self-focusing nonlinearity (exhibiting Kerr lensing). In this regime, the atoms act as a passive nonlinear medium, similar to nonlinear crystals,such as nematic liquid crystals. 89,90The propagation of vector light through the medium can then be described by two coupled nonlinear equations, one for each polarization component, with Kerr lensing dueto intensity gradients affecting the phase of each polarization compo- nent and saturation of the atomic transition coupling the polarization amplitudes. For light with sufficiently high power compared to the saturation intensity, the resulting susceptibilities (especially the refractive index) can lead to higher order nonlinear optical effects, which will modulate the characteristics of the incident light such as intensity profiles andpolarization distributions. In experimental situations where a balance between dispersion and lensing is achieved, spatial solitons may form. Spatial solitons can be realized in nematic liquid crystals or materialswith thermal nonlocal nonlinearity as well as in atomic media, all sup- ported by similar theoretical frameworks. Early theoretical investigations 91–93predicted that vector vortex solitons exhibit more stable propagation for much larger distances than the corresponding scalar vortex solitons, as the long-range nonlocalnonlinear response provides a mechanism for vortex stabilization. This was experimentally confirmed in nematic crystals, 94by comparing the propagation of scalar vortex beams (exp ðiuÞ^rþ) and vector vortex beams without net angular momentum (exp ðiuÞ^rþþexpð/C0iuÞ^r/C0). An alternative approach for increasing the soliton stability is the use of multiple co-propagating fields, distinguished either by their c o l o ra n d / o rb yt h e i rs p a t i a lp r o fi l e .S u c ht w o - c o l o rv e c t o rv o r t e xs o l i -tons typically consist of an incoherent superposition of a vortex com- ponent, which is by itself unstable in a nonlinear medium, and a spatial soliton. The highly nonlocal refractive potential induced by thespatial soliton prevents the breakup of the vortex beam, leading to guided propagation with reduced defragmentation, as proposed in Refs. 95–97 and experimentally confirmed in nematic crystals. 98Very recently, these studies were generalized for higher-charge vortex soli- tons and vector vortex solitons and demonstrated in lead glass with strongly thermal nonlocal nonlinearity.99 Similar nonlinear effects can be realized in atomic media, where the interplay between self-focusing and intensity-dependent diffraction can lead to the formation of self-trapped light beams. This was simu- lated and observed experimentally for vector beams propagatingthrough a heated Rb vapor cell. 100The investigation showed that homogeneously polarized light beams fragment earlier than vector vor- tex beams and Poincar /C19e beams. Fragmentation is known to be induced by azimuthal modulation instabilities, which due to the self-focusing effect quickly escalate. For beams, where the polarization structure changes across the beam profile, interference effects are less drastic. The interaction of intense vectorial laser light with matter gives rise to a wide range of nonlinear effects, shaping the behavior of the material and the resulting light propagation.101While similar effects would be expected for atomic media, so far most experiments havebeen performed using liquids, crystals, or fibers. Nonlinear interactions may not only affect the spatial amplitude, and hence beam stability, upon propagation, but also the relative phasebetween the different spatial modes of its vector components. It has been shown theoretically that propagation in a nonlinear (self-focus- ing) medium can induce a cross-phase modulation, leading to an effec-tive polarization rotation as analyzed 102,103and observed in the linear104and nonlinear regimes,105respectively. The third-order non- linear susceptibility was studied in anisotropic Barium Fluoride crys-tals 106and a large amount of literature is available on the collapse of optical vector fields.107–109 The eigenmodes of optical fibers, including gradient-index, step- index, and hollow-core photonic crystal fibers, are vector fields,110pro- viding the natural choice for efficient communication links111,112and multiplexing.113,114Recent experiments have demonstrated quantum cryptography based on the hybrid entanglement of polarization and orbital angular momentum in a (graded index) vortex fiber115and over an air-core fiber.116 The nonlinearity of optical fibers leads to mode coupling and modification of the polarization profile upon propagation, which needs to be taken into account for mode division multiplexing of clas- sical and quantum communication. Experimental studies have dem- onstrated birefringence, dispersion, and intermodal nonlinear interactions, such as Raman scattering and four-wave mixing, of cylin-drically polarized modes in fibers. 117–119These features can be used for nonlinear quantum squeezing, permitting the creation of continuous-variable hybrid-entangled states.120 The highly nonlinear interaction provided by the gas-filled hol- low-core fiber is particularly suitable to achieve ultra-short pulses in the few, single, or even sub optical cycle regime. Typically, a light pulse is spectrally broadened in the hollow-core fiber and then temporally compressed. Radially polarized vector beams are particularly suited to the fiber geometry. Their propagation dynamics has been numericallystudied 121and experimentally observed,122and compression into the few-cycle regime has been demonstrated using a krypton-filled hol- low-core fiber, while still maintaining its radially polarized nature.123 Alternatively, high power vector vortex beams can be generated by amplifying low power vector vortex beams in nonlinear gain media, e.g., single crystal fiber amplifiers124,125or by energy transfer via stimu- lated Raman/Brillouin scattering medium.126 V. MODE CONVERSION OF VECTORIAL LIGHT We have already discussed how optical vector fields can be modi- fied through linear and nonlinear effects when propagating through an atomic medium with a given susceptibility. We have also seen that the susceptibility set by the interaction with one light field can affectthe propagation of another (usually weaker) field. In these situations, each of the involved light fields interacts with the atomic, or crystalline, medium through incoherent processes. It is however also possible todrive parametric transitions, where multiple optical fields interact coherently with an atomic gas or other nonlinear medium, and the fol- lowing two sections are dedicated to this mechanism. Multiple-wave-mixing is a nonlinear process that coherently combines the amplitudes of some input fields to generate one or more new optical fields, mediated by the nonlinear susceptibility of themedium and determined by the mode overlap of the participating beams. In this context, the Gouy phase associated with wavefront cur- vature and the modenumber of the participating optical fields playimportant roles. As a consequence, the generated signal is rigidly con- strained by phase matching conditions and is extremely sensitive toAVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-5 VCAuthor(s) 2020the polarization of the light. This makes multiple-wave-mixing an excellent candidate to implement mode conversion between spatial modes and to provide an interface for quantum networks. An important example of nonlinear effects is harmonic genera- tion, including second harmonic generation (SHG) and third har- monic generation, converting two or three photons of a pump beam into a photon at double/triple the pump frequency. A first attempt at atheoretical analysis of SHG of vectorial beams was published in 2002, 127investigating, in particular, the behavior of polarization singu- larities. For scalar vortex fields, it can be shown that phase matching is associated with angular momentum conservation, so that the charges of the input photons’ phase singularities combine, generating higher order angular momentum modes in the harmonic. For vector vortex fields, the situation is more interesting: a different phase evolution of the individual polarization components generates a polarization profile that may vary with propagation. Moreover, focused vector fields acquire an axial polarization component. This makes it more difficult to establish phase matching throughout the medium, and the effi- ciency of harmonic generation depends crucially on the vector struc- ture. This was theoretically investigated for specific examples of polarization profiles128–130and was observed131in a ZnSe crystal. It is interesting to note that polarization always plays an implicit role in multi-wave-mixing processes, as the crystal symmetry deter- mines the required polarization relationship between the input and generated beams. For type II nonlinear crystals operated in a collinear setup, the orthogonally polarized input modes can be understood as the spatial components of a vector vortex beam. The resulting multi- mode coupling has been investigated in detail for vector beams with opposite and equal OAM in the horizontal and vertical beam compo- nents, resulting in selection rules for the azimuthal and radial mode indices.132While not emphasized by the authors, this experiment demonstrated the conversion of a vector beam to a homogeneously polarized beam. Studying the effect of polarization structures in the fundamental beam on SHG was also shown to reveal varying spatial modes, while wiping out the polarization structure from the funda- mental beam.133 Many recent publications investigate the conversion of vector vortex beams, from fundamental infrared to visible harmonic frequen- cies. The challenge here is to maintain the inhomogeneous polariza- tion structure of the fundamental mode. This can be achieved in two complementary approaches: either by using two cascading type I crys- tals,134–136each addressing one polarization component of the beam at a time, or interferometrically, by separating the polarization compo- nents and performing independent SHG. The latter has been demon- strated in a Mach–Zehnder configuration,137but is more commonly performed in Sagnac interferometers.138–140The two above- mentioned experimental setups are visually represented in Fig. 3 . These experiments have shown the versatility of converting a wide range of vector beams, including polarization singularites, Poincar /C19e beams, and arbitrary polarization patterns. Especially, the more recent work is characterized by high fidelity transfer of the vectorial mode structures; and while demonstrated so far on classical beams only, the methods should persist in the quantum regime. Similar concepts apply also to higher-order harmonic generation, with the attractive potential of generating extreme ultraviolet (XUV) vector beams. This has been theoretically studied,141,142and experi- mentally realized in a gas jet141,143and with solid targets,144generatingthe capability of engineering XUV sources that utilize vector structures as an additional degree of freedom. While SHG in atomic samples is forbidden due to selection rules, third harmonic generation as well as other four-wave-mixing (FWM)processes are possible. Especially, FWM experiments based on quasi-resonant transitions benefit from the high efficiency of atom–light interactions, compared to other nonlinear processes. Nonlinear effects in atoms are not intrinsically polarization-sensitive, unless the atomicspin alignment is dictated by an external field or a more subtle inter-ference between Clebsch–Gordon coefficents. 145It is however possible to employ, once again, Sagnac interferometers to treat both polariza- tion components independently in separate FWM processes, beforerecombining the conjugate scalar beams to a conjugate vector beam, asrealized using a rubidium vapor. 146 As a curiosity, we also mention the indirect exploitation of vector beams for multiple wave mixing:147passing a vector vortex beam through a polarizer results in a characteristic petal pattern with aneven number of intensity lobes. When focused into a rubidium vaporcell, they obtain different wavevectors and undergo multiple wave mix-ing. If individual lobes within the input pattern are blocked, they are partially regenerated through the nonlinear interaction in the atomic medium. Phase conjugation of vector vortex beams has also been realized in photorefractive crystals, 148,149demonstrating the “healing” of polar- ization defects, as well as in stimulated parametric downconversion.150 The latter provides a convenient approach of generating a conjugate FIG.3 .Mode conversion of a vector field, (a) using orthogonal SPDC crystals, sub- sequently addressing each polarization component and (b) in a Sagnac interferom- eter with counterclockwise propagating orthogonal vector components, eachconverted individually before recombination (the pump beam is omitted for clarity).AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-6 VCAuthor(s) 2020vector beam from a vectorial probe used as a seed in real time, with potential applications in aberration-free imaging. Finally, we mention an experiment generating multicolor concentric vector beams, byusing cascaded four-wave mixing in a glass plate pumped by two intense vector femtosecond pulses, 151combining the manipulation of temporal, spectral, spatial, and polarization degrees of freedom. VI. ELECTROMAGNETICALLY INDUCED TRANSPARENCY AND MEMORIES In this section, we will continue to discuss the parametric light–- matter interaction, but place more emphasis on the structure inducedin the atoms. In electromagnetically induced transparency (EIT) and coherent population trapping (here we will use EIT to refer to both mechanisms, which differ only in the relative intensity of the involved beams), atoms are transferred into dark states, which are determined by the interplay between the driving optical fields, and indeed theirvector properties. This can be exploited for atomic memories, where the information transferred to and stored in the atoms is retrieved at a later time. EIT relies on the coherent interaction of light and atoms, render- ing the medium transparent for a resonant probe beam when simulta- neously exposed to an additional control beam. During this process, atoms are decoupled from the electromagnetic fields and populate a so-called dark state. The process can be formed by optically couplingpump and probe beams with atomic hyperfine levels or Zeeman suble- vels (Hanle type resonances). The anomalous dispersion associated with the “transparency window” has been exploited for the generationof slow and stopped light, and related techniques have led to EIT- based quantum memories. 152,153Recent research has demonstrated operation at room temperature, including EIT-based.154Quantum memories are a highly active research field, and here we only concen- trate on a small subsection that is currently of the most relevance toapplications with vector light. Initial quantum memories dealt with information encoded in the polarization of a photon, but also spatial mode structures or imagescan be stored—with phase vortices (i.e., angular momentum modes) presenting a favored basis system. A theoretical proposal to transfer and store an optical vortex in a Bose–Einstein condensate (BEC) was presented in 2004. 155Experimentally, the storage of optical vortices was demonstrated in atomic media at various temperatures and forclassical 156–160and quantum memories.161–166Image memories pro- vide an essential capability for high-dimensional quantum memories and quantum information applications. A different approach to high dimensionality was demonstrated by coupling a memory for photonic polarization qubits to spatially separated output channels into multiple spatially separate photonic channels.167Recent experiments are pushing the number of accessible spin wave modes. The simultaneous storage of up to 60 independentatomic spin-wave modes in Rb vapor 168and 665 spin-wave modes in a cold atomic ensemble has been demonstrated with a Raman mem- ory,168encoding modes determined by the photonic wavevector. A similar multiplexed atomic memory system has very recently been combined with an optical cavity, coupling spin-wave excitations withdifferent spatial profiles to the cavity photons 169via super-radiant enhancement. Before reviewing the storage of vector light, we will highlight different parametric processes that couple optical polariza- tion to atoms.An intriguing application of a vector vortex beams to atoms allowed the measurement of the rotational Doppler shift,170over a decade after it was first predicted.171The rotational Doppler shift that an atom experiences if exposed to the twisting phasefronts of an LG beam is dwarfed by a linear Doppler shift along the beam propagation axis. The researchers nevertheless managed to observe the rotational Doppler broadening of the Hanle EIT signal on the D1 line of87Rb in a room-temperature vapor cell, by using a superposition of two per- fectly aligned LG fields with opposite topological charges and orthogo-nal circular polarizations (i.e., a vector beam of the form expð/C0i‘uÞ^r þþexpði‘uÞ^r/C0), which exactly cancels axial and radial contributions to the Doppler broadening. The ingenious idea of this experiment is to use a narrow linewidth Hanle EIT signal as a back- ground to show the influence of the atomic rotation Doppler fre- quency shift induced by the topological charge ‘. While this work did not explicitly consider the spatial profiles or polarization distributions, it still took unique advantages of VVBs. In 2015, our group has demonstrated a spatially resolved EIT by exposing cold Rb atoms to VVBs with an azimuthally varying polari- zation and phase structure,172typically with a radial polarization pro- file exp ð/C0i‘uÞ^rþþexpði‘uÞ^r/C0,w i t h ‘up to 200. The principle of the experiment is shown in Fig. 4 . The left- and right-handed circular polarization components with opposite OAM in a single laser beam provide both the probe and the control for the EIT transition. The atomic system is closed by a weak transverse magnetic field, making the atoms sensitive to the phase difference between the complex exci- tation amplitudes. The atoms are pumped into spatially varying atomic dark states, leading to an angular variation of the opacity of the medium. The interaction allows coupling between an external mag-netic field and a polarization structured optical field via the atomic spin alignment, 173where the atomic transparency is set by the angle between the local polarization direction and the external magnetic field direction,174,175as shown in Fig. 4 . A generalization to more compli- cated EIT systems has been proposed,176replacing the magnetic cou- pling with additional optical transitions. Related effects have also been observed in a warm vapor by using a single hybrid vector beam cos ð‘uÞ^rþþsinð‘uÞ^r/C0, which contains alternating segments of right and left polarized light, separated by lin- ear polarization, and an external magnetic field in the direction of the beam propagation.177The transmission of the circular light compo- nents depended strongly on the strength and direction of the magnetic field, while linearly polarized light was always absorbed. This may be interpreted as the build-up of atomic coherence through the circular polarized light. The effect prevailed when two orthogonal polarizations were provided in the form of independent homogeneous beams, and the observed transparency depended on the spatial separation between the beams. This suggests that the free flight of atoms between the two beams provides a pathway for dynamically building quantum coher- ence similar to an adiabatic following or stimulated Raman adiabatic passage.178,179An alternative interpretation, following the discussion in Sec. III, may associate the transparency with optical pumping, deter- mined by the local polarization structure and resonance conditions between the Zeeman shift atomic levels “smeared out” by the thermal motion of the atoms. We will now return to the realization of quantum memories for vector vortex beams. While work on general image memories may ultimately prove to be more advantageous in accessing highAVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-7 VCAuthor(s) 2020dimensional spaces, the storage of vector light offers access to the hybrid entanglement of polarization and angular momentum, andprovides fundamental insight into the nature of light–matter interac- tions. The first storage and retrieval of vector vortices at the single- photon level (with attenuated coherent light) was realized for a multi-plexed ensemble of laser-cooled Cs atoms, using an intermediate con-figuration between EIT and off-resonant Raman schemes. 180The polarization structure was generated with a q-plate. The authors fol- lowed a method, familiar from various other approaches discussed, e.g., in Sec. V, of treating each polarization component separately. The vector beam was split with a calcite beam displacer into its constituentpolarization components. Each polarization component was pairedwith its own (homogeneous) coupling laser with the proper polariza-tion to store the components individually in the cold atom cloud, and the retrieved signals were recombined with a second beam displacer. The overall storage and retrieval efficiency was 26% for 1 ls, and the fidelity of the atomic memory was shown to be close to one, clearlyexceeding classical benchmarks for memory protocols. More recently, an EIT-based memory was realized in a warm 85Rb vapor, which is an intrinsically simpler system without additional cool-ing systems and beams. 181The vector beam, a superposition of different LG modes in the two polarization components, was generated via a Sagnac interferometer incorporating a vortex phase plate and split intoits linear polarization components before being stored in the vapor cell.The storage and retrieval efficiency was almost 30% for 1 ls. Projection measurements showed that both the spatial structure and phase infor-mation were preserved during storage. Quantum state tomography measurements were used to calculate the fidelities for the various VVBs. The obtained fidelities of the storage scheme in warm vapor satisfy thecriterion of the quantum no-cloning theorem, offering potential for theconstruction of a versatile vortex-based quantum network. Besides memories, quantum networks require various other ele- ments, including sources of photon pairs simultaneously entangled intheir polarization and spatial degrees of freedom. Entangled vectorbeams combine true quantum entanglement between locally separated particles with nonclassical correlations, or contextuality, between dif-ferent degrees of freedom within the wavefunction of each particle. To the best of our knowledge, sources of entangled vector states have not yet been realized with atomic systems, but initial experimentshave demonstrated just this based on spontaneous parametric down-conversion (SPDC) in b-barium borate (BBO) crystals. 182The system works by initially generating a polarization-entangled Bell state follow- ing the usual entanglement generation, ðjhi1jvi2/C0jvi1jhi2Þ=ffiffiffi 2p ,a n d passing photon 1 and photon 2 through different q-plates. The modeprofile generated by a given q-plate depends on the input polarization,and a rotation of the input polarization imposes a local phaseshift ofthe output polarization. A change from horizontal to vertical inputsrelates to a change from radial to azimuthal polarization (or their higher order equivalent). The action of the q-plate therefore transfers entanglement between homogeneous polarization states to entangle-ment between vectorial polarization states. The authors also showedthat linear local geometric transformations can generate a full Bell set,and that the states are indeed entangled and violate the Clauser-Horne-Shimony-Holt inequality. Alternatively, rather than introducing vector properties to already entangled photons, the procedure can be reversed: the possibility ofdirectly converting a vector beam via SPDC into hybrid entangledphoton pairs has been suggested 183and realized.184The nonlinear properties of the v2crystal can be considered for the independent polarization components of the vectorial pump beam, but specific care must be taken with phase-matching. The process of phase conjugationof vector beams was investigated theoretically for stimulated paramet-ric downconversion. 185 The entanglement between vector fields is not the only option when operating in a state space, defined by polarization and spatialmodes. Hyperentanglement has been demonstrated between the polar-ization of one photon and the OAM of its twin, 186between the polari- zation of one photon and a vector state of its twin,187and the FIG.4 .Vector fields, in combination with an external magnetic field, resulting in spatially varying atomic transparency. The orthogonal circular polarize d components of a vector field (shown left) drive different transitions in an atom (center). The resulting absorption image (right) shows transparency where the local polari zation direction is aligned with the transverse magnetic field BT.AVS Quantum Science REVIEW scitation.org/journal/aqs AVS Quantum Sci. 2, 031702 (2020); doi: 10.1116/5.0016007 2, 031702-8 VCAuthor(s) 2020connections between entanglement and contextuality have been ana- lyzed in quantum and classical settings.188It is even possible to prepare the vector state of one photon by operations on its remote twin—a vectorial analog to ghost imaging.189 Quantum features of hybrid entanglement have been successfully employed for several protocols, including quantum key distribution190 and quantum cryptography over an outdoor free-space link.191 VII. LIGHT–MATTER INTERACTION UNDER STRONG FOCUSSING When vectorial light fields interact with matter, one can distin- guish two different regimes, which we call the inhomogeneous and the anisotropic regime. In the former, the wavefunction of the atom is small compared to the spatial extent of the vector field, and individual atoms sample the local (quasi) homogeneous polarization of the light.Instead, in the anisotropic regime, an individual atom responds to the spatially varying polarization field. This regime is particularly interest- ing for strongly focused vector light, with its three-dimensional polari- zation structure, e.g., by positioning an atom or ion at the center of a focused polarization vortex. The same ideas may be generalized to crystals or other matter. Experimental work on atoms and nonlinear crystals is so far mainly situated within the inhomogeneous regime, responding to local polarization of a beam or of the spatial wavefunction of a photon, and all work described so far in this review falls into this category. Experiments with plasmons, nanowires, and individual ions and mole- cules instead can and have been performed in the anisotropic regime, as discussed below. These research fields are bound to benefit from the possibility to shape the 3D polarization profile. 192It is interesting to note that such polarization profiles, including Moebius strips, can be reconstructed by detecting scattering off nanoparticles.193Initial experiments have demonstrated polarization nanotomography by using the functional nanomaterial itself as a sensor.194 Generally when describing the interaction between light and atoms, and specifically for all the studies described so far, it is enough to describe the light–matter interaction in the dipole approximation to predict the value of the observables of the system. Considering instead an atom positioned at the center of a vortex beam (where there is no intensity), the dipole approximation is not satisfied.195The authors devised, instead, a so-called Poincar /C19e gauge, i.e., a gauge form of the vector potential which includes the orbital charge of the field and eval- uates explicitly the axial field component. This is further explored by proposing the use of vector light fields to generate specific electric and magnetic field components at the focus of a vector light field.196At the focus of an azimuthally polarized (electric) field, without net OAM, the axial component of the electric field vanishes, whereas the axial component of the accompanying magnetic field has a finite value. The opposite applies when focusing a radial polarized light field. This allows one, in principle, to induce a magnetic dipole interaction—a regime that is usually dwarfed by the dominant electric dipole interaction. The strong focusing limit has been explored experimentally in interactions with single molecules.197In this work, molecules with a fi x e dd i p o l em o m e n tw e r eu s e dt op r o b et h ea x i a lc o m p o n e n to ft h e light field and vice versa. It was demonstrated that the orientation of single molecules could be efficiently mapped out in three dimensions by using a radially polarized beam as the excitation source. Similarly,the role of longitudinal light fields was also explored for ions, examin- ing the interaction with focused VVBs195and the excitation of mag- netic dipole transitions at optical frequencies.198Another quantum system investigated under strongly focused vector light is nanowires. It was shown, for example, that SHG from oriented nanowires is mostefficient when driven by polarization along the growth axis, 199which can be excited by illumination with strongly focused radially polarized light. Another route to access the anisotropic interaction with vector light is to address the expanded wavefunction of Bose condensates. Early experiments have shown that scalar vortices can be generated in quantum degenerate gasses, containing quantized angular momentum and exhibiting persistent currents.200Vortices can also be created in spinor BECs. The atoms of spinor BECs are in a superposition of inter- nal quantum states, making its wave function a vector, with topologi-cal analogies to vector vortex fields. The generation and properties of such vector vortices are the subject of ongoing investigations. 201–207 VIII. CONCLUSIONS AND OUTLOOK The ability to generate and manipulate the vector nature of light offers new opportunities in designing light–matter interactions. This isrelevant in all situations, where the symmetry of the medium differen- tiates between orthogonal polarization components, whether due to intrinsic dichroism or birefringence or due to asymmetries induced by external electromagnetic fields, including the vector light itself. We have seen that, so far, many experiments deal with vector light one vector component at a time, whether for conversion between vector modes or for their storage. Other processes, however, operate moredirectly on a vectorial level, e.g., when different transitions within an atom are accessed simultaneously by the corresponding polarization components of light. One may assume that future years will see further investigations of vectorial light–matter interactions, exploring both the inhomogeneous and anisotropic nature of vector vortices as well as generic vector fields. Current advances in the generation and detection of vector fields via miniaturization and integration for photonics devices, 208–210in combination with new approaches based on machine learning,211,212 provide a new platform for technological exploitation of vectorial light–matter interactions. Photonic crystal slabs and metasurfaces offer a rich environment, suitable to explore topological effects on the nano- scale and to facilitate efficient spin–orbit interactions.213–217 The study of complex vector fields is not restricted to photons, as is shown by recent advances in vector neutron218and electron beams.219,220 We strongly suspect that vector fields have a long—and poten- tially twisted—future. ACKNOWLEDGMENTS The authors are grateful for discussions with Adam Selyem on the wider research area of vectorial light matter interaction, to Alison Yao and Gian-Luca Oppo for the scientific exchange onmode propagation, and to Aidan Arnold for proofreading the manuscript. F.C. and S.F.-A. acknowledge the financial support from the European Training Network ColOpt, which is funded by the European Union (EU) Horizon 2020 program under the Marie Sklodowska-Curie Action, Grant Agreement No. 721465. 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5.0023546.pdf

J. Appl. Phys. 128, 125702 (2020); https://doi.org/10.1063/5.0023546 128, 125702 © 2020 Author(s).Unusual conduction mechanism of n- type β-Ga2O3: A shallow donor electron paramagnetic resonance analysis Cite as: J. Appl. Phys. 128, 125702 (2020); https://doi.org/10.1063/5.0023546 Submitted: 30 July 2020 . Accepted: 06 September 2020 . Published Online: 22 September 2020 H. J. von Bardeleben , and J. L. Cantin ARTICLES YOU MAY BE INTERESTED IN A review of Ga 2O3 materials, processing, and devices Applied Physics Reviews 5, 011301 (2018); https://doi.org/10.1063/1.5006941 Trapping of multiple H atoms at the Ga(1) vacancy in β-Ga2O3 Applied Physics Letters 117, 142101 (2020); https://doi.org/10.1063/5.0024269 Oxygen annealing impact on β-Ga2O3 MOSFETs: Improved pinch-off characteristic and output power density Applied Physics Letters 117, 133503 (2020); https://doi.org/10.1063/5.0021242Unusual conduction mechanism of n-type β-Ga 2O3: A shallow donor electron paramagnetic resonance analysis Cite as: J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 View Online Export Citation CrossMar k Submitted: 30 July 2020 · Accepted: 6 September 2020 · Published Online: 22 September 2020 H. J. von Bardelebena) and J. L. Cantin AFFILIATIONS Institut des NanoSciences de Paris, Sorbonne Université, UMR 7588 au CNRS 4, place Jussieu, 75005 Paris, France a)Author to whom correspondence should be addressed: vonbarde@insp.jussieu.fr ABSTRACT We have investigated the conduction mechanism in n-type, Si doped β-Ga 2O3bulk samples and evidenced carrier dynamics in the GHz frequency range at room temperature by electron paramagnetic resonance (EPR) spectroscopy. The Si shallow donor EPR and conductionelectron spin resonance (CESR) spectra show an unusual temperature dependence of the linewidth and line shape, which reveals a variable range hopping conduction and donor clustering. The temperature dependence of the EPR signal intensity can be fitted with two thermally activated processes with energies of 4 meV and 40 meV in the below and above 40 K temperature range. The value of 40 meV is attributedto the ionization energy of the Si shallow donor, indicating that hopping proceeds via the conduction band. Above T = 130 K and up toroom temperature, the conduction electron spin resonance (CESR) can be observed with a decreasing linewidth of ΔB < 1 G, which indicates negligible spin flip scattering. To illustrate the unusual behavior of the shallow donor in Ga 2O3, we have analyzed the hydrogen shallow donor in ZnO, for which we observe a different “classical ”behavior, characterized by donor localization below 40 K and thermal ionization in the conduction band above T = 90 K. In ZnO, the CESR can only be observed in a small temperature range at 90 K due to excessive linebroadening for higher temperatures. Published under license by AIP Publishing. https://doi.org/10.1063/5.0023546 INTRODUCTION β-Gallium oxide is an emerging semiconductor material, 1 which due to its high bandgap and suitable n-type conductivity, is well suited for microelectronic devices requiring high breakdownvoltages. β-Ga 2O3can be obtained as a bulk material2,3or in the form of epitaxial layers4grown by various techniques such as pulsed laser deposition (PLD), MBE, metalorganic chemical vapor deposition (MOCVD), and LPCVD. Contrary to standard semicon-ductors such as Si, SiC, or GaAs, β-Ga 2O3has a low symmetry, monoclinic crystal structure, giving rise to two nonequivalent Gasites and three nonequivalent oxygen lattice sites. Like many oxide wide bandgap semiconductors, Ga 2O3can be easily made n-type conductive. Doping with Si, Ge, or Sn has been predicted andshown to introduce shallow donors with ionization energies of 30 – 40 meV and high donor doping levels up to 10 19cm−3and room temperature conductivities of 10−2Ω−1cm−1have been achieved.4–7 The room temperature mobilities are in the range of 10 to 100 cm2V−1s−1with highest values in homoepitaxial layers. Sn andSi donors have been predicted to occupy preferentially different lattice sites and octahedral and tetrahedral Ga sites, respectively.5,6 As the lowest conduction band is composed of 4 sorbitals of octahedral Ga, the lattice site of shallow donors might have aninfluence on the conductivity properties. The conduction mechanism in n-type β-Ga 2O3has been investi- gated earlier,7–12but some issues remain to be investigated in more detail. In one recent study of highly doped Sn doped β-Ga 2O3with the donor concentration above the Mott limit, a variable rangehopping (VRH) conductivity for impurity band conduction has beenreported. 11In this work, we have studied the conduction mechanism in n-type bulk samples, where the carrier concentration was belowthe Mott limit for metallic conduction. We had shown recently 13in a correlated electrical transport and EPR study of ε-Ga 2O3that the conduction mechanism in Si doped epitaxial layers with a carrierconcentration of 2.8 × 10 18cm−3—close to but below the Mott transition —is characterized by variable range hopping. Variable range hopping conductivity is quite unusual for a crystalline semiconductorJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-1 Published under license by AIP Publishing.and generally a property of amorphous semiconductors. To investi- gate whether this process is related to the particular multidomain microstructure of the ε-polytype, we have undertaken a similar study in monocrystalline, n-type bulk β-Ga 2O3,w h i c hd o e sn o th a v ea n y microstructure. We have equally addressed the issue of negative Uproperties, recently evoked 10f o rt h eS id o n o ri n β-Ga 2O3. Both neutral shallow donors and conduction electrons are spin S = 1/2 centers, thus electron paramagnetic resonance spectroscopyis an adequate tool for the study of both. At low temperatures, theelectron is localized on the donor and the g-tensor, line shape, andlinewidth, as well as the spin concentration, can be determined. For a localized donor, a Gaussian line shape is expected due to inhomo- geneous broadening induced by the hyperfine interaction with theGa nuclei. In the case of delocalization, the hyperfine interaction isaveraged out, and the line shape becomes Lorentzian. In this work,we have focused on the temperature dependence of the EPR parame- ters. They provide information about the delocalization of the elec- trons, their dynamics, and their emission in the conduction band. To demonstrate the singularity of the conduction mechanism in Ga 2O3, we have performed a similar investigation for a shallow donor in n-type ZnO, which on the contrary shows a “classical ”behavior. EXPERIMENTAL DETAILS The bulk samples were commercially purchased from Tamura Comp. (Japan); they were grown by the edge-defined film-fed(EFG) technique 2,3and even though not intentionally doped, are n-type conducting due to Si contamination. The room temperature carrier concentration was 2 × 1017cm−3. No additional thermal treatment was performed. The samples were (010) b-plane orientedand of rectangular shape with edges along [102] and [ ⊥102]. The thickness was 400 μm and the EPR sample dimensions was 5×5m m 2. To verify the influence of electrical compensation and conductivity on the EPR spectra, we have equally investigated oneproton-irradiated sample, in which the carrier concentration wasreduced. This sample was irradiated at room temperature with protons of kinetic energy E = 12 MV at a fluence of 1 × 10 16cm−2. For this energy, the proton will completely cross the sample with apredicted end-of-range region at 450 μm and no hydrogen will be incorporated. Due to the formation of Ga vacancies, which aredeep acceptors, the increased compensation will reduce the neutral donor concentration. The irradiated sample is still n-type conduc- tive and the neutral donor resonance is observed at T = 4 K. Then-type bulk ZnO samples were grown by vapor phase epitaxy. Thedominant shallow donor was hydrogen with a bulk concentrationof 7 × 10 17cm−3. Transport measurements have been performed on this sample by Look (Ohio State University). The sample size was 5 × 5 × 0.5 mm3and it was (0001) plane oriented. The EPR measurements were performed with a Bruker X-band spectrometer in the temperature range from T = 4 K to T = 300 K. Foreach scan, the microwave frequency was also monitored in order to determine the g-values with a precision of Δg = 0.0002. The principal values and axis orientation of the g-tensor were obtained from theangular variation of the donor resonance in three lattice planes anddeduced in a standard procedure. The EPR line shape was analyzed with the EPR simulation program of Grachev. 14The linewidth ΔBi s defined as the peak-to-peak width of the first derivative line.EPR RESULTS IN β-Ga 2O3 The as-received samples show only one anisotropic single line spectrum, which can be observed in the whole temperature rangefrom 4 K to 300 K ( Figs. 1 and 2). At T = 4 K, we observe the neutral shallow donor, which has an electron spin of S = 1/2 and at T = 300 K the electron spin resonance of conduction electrons. AtT = 4 K, the EPR spectrum has a linewidth of 50 G and aLorentzian shape. The observation of a large linewidth of ∼50 G is not unexpected for localized electrons as the presence of 100% Ga nuclei with spin I = 3/2 gives rise to large inhomogeneous broaden- ing. Inhomogeneous broadening normally produces a Gaussianline shape, which is not observed here. The observed shape is FIG. 1. (a) Low temperature (T = 4 K) EPR spectrum (black circle) of the Si donor in as-received bulk β-Ga 2O3for B// band the simulation with Lorentzian and Gaussian shapes. (b) Low temperature (T = 4 K) EPR spectrum (black circle) of the Si donor in proton irradiated bulk β-Ga 2O3for B// band the line- shape simulation with Lorentzian and Gaussian shapes.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-2 Published under license by AIP Publishing.Lorentzian, an indication of dynamic effects already operating at temperatures as low as T = 4 K. If the carrier concentration is reduced by particle irradiation, the EPR spectrum is changed into a“normal ”Gaussian shape [ Fig. 1(b) ]. From the angular depen- dence of the resonance fields ( Fig. 3 ) in three lattice planes,its spin S = 1/2 and g-tensor have been determined: g xx=1 . 9 5 7 5 , gyy=1 . 9 6 0 3 , a n d g zz= 1.9630 with the principle axes zzaligned with the baxis and the xxand yyaxes lying in the b-plane with xxat an angle of 23° from the caxis. The g-values are close to those previously reported for shallow donors in β-Ga 2O3and con- duction electrons ( Table I ).8–10 FIG. 2. EPR spectra of the CESR in β-Ga 2O3at T = 300 K for a variation of the magnetic field between the baxis (red spectrum) and [ −201] (blue spectrum). FIG. 3. CESR spectra: g-factor for a variation of the magnetic field between the baxis and [ −201]; T = 300 K. TABLE I. EPR parameters of the shallow donor and conduction electrons in β-Ga 2O3: principal values of the g-tensor and the orientation αof the principal axes xx, yy in the crystal b-plane; the zzaxis is parallel to the crystal baxis. Modelg-tensor principal values: g xx,gyy,gzz Sample Reference T = 4 K T = 300 K Neutral donor D° T=4K1.9603 1.95751.9622 α= 23±3°EFG grown As grownThis work Conduction electrons T = 300 K1.9606 1.95781.9628 α= 23±3°EFG grown As grownThis work Conduction electrons T = 300 K1.9616 1.9590 1.9635 α= 27°±3Float zone growth Verneuil methodBinet Donor cluster Oxygen vacancy T = 300 K1.960 1.958 1.962 α= 24°Float zone growth Non int. dopedYamaga et al. 8 Donor (DX) T = 300 K/80 K1.9606 1.9577 1.9630 α= 23°1.9606 1.9577 1.9623 α= 23°Bulk plus 1100 °C annealSon et al.10Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-3 Published under license by AIP Publishing.When the temperature is increased from T = 4 K, the g-values vary only marginally with temperature, and the orientation of the principal axes is not changed. The linewidth however decreasesrapidly. In the range between T = 4 K and T = 40 K, it decreasesdrastically from 50 G to <1 G [ Figs. 4(a) and 4(b)] and the line shape becomes asymmetric, an indication of increasing conductiv- ity of the sample. The asymmetric Dyson-like lineshape is charac- terized by a small asymmetry A/B ratio of <1.6, whereas formetallic conduction, higher values of A/B > 2 should be observed.A and B are the amplitudes of the positive and negative lobes ofthe first derivative line. The comparison with the EPR parameters in the proton irradiated sample [ Fig. 4(b) ] shows only small differ- ences in the absolute value of the linewidth and their temperaturedependence. The only notable difference is the line shape, whichnow is Gaussian at low temperatures. The fact that the donor line-width is changing at a temperature as low as T = 4 K is surprising; for example, the linewidth of deep centers such as transition metalions is temperature independent. The mechanism leading to a reduction of the EPR linewidth is motional narrowing, which will average out the superhyperfine interactions, when the time scale ofmotion is sufficiently small, i.e., if the condition Δν τ > 1, where Δν is the linewidth in frequency units and τis the correlation time, corresponding to the inverse of the hopping frequency. For a line- width of 50 G, the hopping frequency must be higher than GHz to lead to a reduction of the linewidth. To get further insight into the dynamic processes of the donor electrons, we have tested whether the temperature variation of thelinewidth can be simulated by a variable range hopping process 15 such as previously demonstrated for shallow donors in ε-Ga 2O3.13 InFig. 5 , we have plotted the results in a so-called Mott plot, i.e., ln (ΔB) vs T−1/4, the expression for three-dimensional variable range hopping (VRH). This expression has been derived for the temperature variation of the electrical conductivity σ, σ¼σ00exp/C0T0 T/C18/C19 1 4"# , (1) where σ00is a prefactor and T 0is a material parameter.11 Assuming the same expression for the linewidth, we obtain ΔB¼ΔB00exp/C0T0 T/C18/C19 1 4"# : (2) The motionally narrowed EPR line will be of Lorentzian shape with a width ωp2τcwith ωpthe low temperature linewidth in FIG. 5. Mott plot of the linewidth ln( ΔB) as a function of temperature T−0,25in the as-received sample. FIG. 4. T emperature dependence of (a) EPR spectra for B//b, (b) peak-to-peak linewidth ΔB of the as grown (red circles) and irradiated (black squares) samples.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-4 Published under license by AIP Publishing.frequency units. Its width follows the same law as the correlation time τc, which is correlated with the hopping frequency. The plot ln(ΔB) vs T−1/4shows two linear regions T = 4 K, T = 40 K and T = 40 K, T = 130 K, which are in agreement with the model ofVRH conduction. The parameter T 0is defined as T0¼β kBD(E F)a3, (3) and its value can be obtained from the slope in Fig. 5 : T0=8×1 05K. Assuming a localization length equal to the effective Bohr radius a B= 1.9 nm and a numerical constant β= 18, we find a density of states near the Fermi level of D(E F)=4×1 019eV−1cm−3. The concentration of localized states at T = 100 K is thus n(E F)¼1/4D(E F)kBT(T/T 0)/C01/4¼1:6/C21018cm/C03, and the average hopping length at T=100 K is 6.6 nm. However, given the thermal ionization energy of the Si shallow donor of 31 meV7,12and a concentration of only 1017cm−3, one should a priori expect carrier localization below T = 40 K and delocalization above this temperature. However, the drastic reduction of the linewidth already in the low temperature range of T < 40 K indicates a more complex situation. We havereported 13in a previous study of Si shallow donors in epitaxial ε-Ga 2O3that donor clustering is a possible explanation of this phe- nomenon. It might give rise to close neighbor hopping at low tem- peratures and variable range hopping at T > 40 K between donor clusters. The temperature dependence of the EPR linewidth cannotbe analyzed above 130 K, as the linewidth is already close to theintrinsic one, determined by lifetime broadening. At temperatures above T = 130 K, we will have the coexistence of delocalized electrons and conduction electrons thermally emitted in the conduction band. The observation of conduction electronspin resonance (CESR) is possible and has been studied in all mainsemiconductors such as Si, 16GaN,18or SiC.17In all cases due to lifetime broadening, the CESR lines becomes rapidly (within some K) unobservable above the temperature, for which thermal ioniza-tion becomes dominant. In the case of the Si shallow donor inβ-Ga 2O3, the observation of the CESR at temperatures higher than T = 100 K is unexpected and the monotonous decrease of the line- width is opposite to the findings in most semiconductor materials. To illustrate this phenomenon, we have performed a similar studyfor the hydrogen shallow donor and CESR in ZnO, as shownbelow. We have equally analyzed the temperature dependence of the EPR signal intensity I. A plot of IvsTis shown in Fig. 6 .W e observe a monotonous variation of Iwith temperature with no indication of a negative U behavior, which would have introduceda maximum at an intermediate temperature. These results are dif-ferent from those reported by Son et al. 10We have analyzed the temperature dependence in a standard model, which considers Curie type paramagnetism and thermally activated emission of thedonor electrons ( Fig. 7 ). The Curie type paramagnetism decreases the EPR signal intensity as I∼1/Tin the high temperature approxi- mation and applies to the case of localized electrons. For the thermal emission of the donor electrons, we have considered twoactivation energies, I I0¼1þX ikie/C0Ei 2kBT/C16/C17 /C01 , (4) where E iis the thermal activation energy of process i with prefactor Ki. As shown in Fig. 7 , the experimental data can be well fitted with two activation energies, E1¼4 meV and K 1¼12, E2¼40 meV and K 2¼1550 : FIG. 7. Logarithm of the signal intensity I vs 1000/T and fit (red squares) with Eq.(4). FIG. 6. EPR signal intensity as a function of temperature in β-Ga 2O3.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-5 Published under license by AIP Publishing.The second activation energy is in fair agreement with the shallow donor value (36 ± 5) meV as reported before.7,12The smaller activation energy of E 1= 5 meV in the low temperature range has been observed also for other shallow donor systems, butits origin has not been identified; we tentatively assign it to thethermal activation energy for hopping conduction, Zno:H : To illustrate the particularity of the shallow donor properties and conduction electron properties in β-Ga 2O3, we have also inves- tigated the case of ZnO by EPR spectroscopy. This sample is n-typeconductive due to doping with the hydrogen shallow donor(Fig. 8 ). The EPR spectra can be observed in the range from T = 4 K to T = 90 K only ( Fig. 9 ). The EPR spectrum of the H donor has an axial symmetry with two principal values of the g-tensor due tothe hexagonal symmetry of the ZnO lattice. The values are g / /c- = 1.9570 and g⊥c= 1.9554 at T = 4 K. At T = 4 K, the shape of the EPR line is Lorentzian and the peak-to-peak linewidth is 2.5 G (Fig. 8 ). The linewidth for the localized donors in ZnO is much smaller than in the case of Si in β-Ga 2O3due to the small abun- dance of nuclei with non-zero nuclear spin (67Zn, I = 3/2, 4.1%). The linewidth is slightly decreasing up to T = 50 K with a hyperlor-entz line shape in a small temperature range at (25 ± 5)K. For T > 60 K, when the electrons are thermally emitted into the con- duction band, the linewidth is rapidly increasing and the observa-tion of the EPR signal becomes impossible for temperatures ofT > 90 K due to excessive line broadening ( Fig. 9 ). As the line shape is still symmetric and not Dyson-like, the decrease of the EPR signal amplitude is not due to the reduced microwave absorp-tion, which does occur when the skin depth becomes inferior thanthe sample thickness. The temperature dependence of the linewidth is opposite to the case of β-Ga 2O3. As shown in Fig. 10(b) ,a“Mott plot ”has nophysical significance; but its variation can be fitted with a power law [ Fig. 10(a) ]. Such strong broadening of the EPR line, when the electrons are moving in the conduction band, has been observedin many semiconductors and is generally attributed to phononscattering. As the g-values of the conduction electrons are closeto those of the shallow donor, their EPR lines are often undistinguishable. We have equally analyzed the temperature dependence of the EPR signal intensity in the range from T = 4 K to T = 80 K(Fig. 11 ) within the same model of Curie type paramagnetism and thermal emission of the donor electrons in the conduction band. 19 Again the data can be well fitted with two activation energies(Fig. 12 ). The fit gives the values of E 1¼7 meV, E2¼40 meV : These values are close to those reported previously for the hydrogen donor in ZnO in samples with comparable donor con-centrations of 10 17cm−3.19 DISCUSSION EPR studies of donors and/or conduction electrons in β-Ga 2O3have been reported before,8–10,20–22but the issue of the conduction mechanism has not been analyzed sufficiently. Theearly results can be classified in two phases. In the 1990s, EPRstudies have been undertaken on undoped, melt grown, “labora- tory”crystals, which without intentional doping showed already extremely high electron concentrations of >10 19cm−3. The origin of the high electron concentration was at that time attributed to thepresence of oxygen vacancies 20–22believed to be shallow donors and the EPR spectra were associated with conduction electron spin resonance. More recent results have shown that the donor FIG. 8. EPR spectra of the H donor in ZnO for B//c and B ⊥c at T = 4 K; blue circles experiment, lines simulation with Lorentzian and Gaussian lineshapes. FIG. 9. T emperature dependence of the EPR spectra in ZnO between T = 4 K and T = 90 K for B//c.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-6 Published under license by AIP Publishing.responsible for this high electron concentration cannot be an oxygen vacancy.5Later, Yamaga et al.8reported an EPR study of n-type conducting, laboratory, bulk samples grown by the float zone technique. They were non-intentionally doped and showed a RT conductivity of 20 Ω−1cm−1. They observed a single line aniso- tropic EPR spectrum with g-values typical for donor defects. Theynoticed already a strong temperature dependence of EPR linewidth,which they attributed to the effect of motional narrowing for T > 50 K, but the conduction mechanism was not analyzed. Our results indicate that conduction is determined by three-dimensionalvariable range hopping with two different activation energies. Themodel retained by us is that the donors are not randomly distribu- ted but forming local clusters in these non-intentionally doped bulk samples. Within this model, low temperature hopping occurswithin the clusters and at high temperatures hopping between dif- ferent clusters becomes possible. A recent study by terahertz spec- troscopy on comparable bulk samples has equally concluded on thepresence of carriers moving in localized potential minima at tem-peratures below T = 50 K. 23Interestingly, an activation energy of E = 5 meV has been reported similar to our results. The observation of the CESR with an extremely small line- width below ΔB = 1 G must be related to a nonstandard spin- phonon interaction. It is a quite unique feature different from thecase of silicon, 16silicon carbide,17GaN18or as shown here ZnO, where phonon scattering rapidly increases the CESR linewidth with increasing temperature. Binet et al.24have attributed the strongly reduced linewidth to the conduction band structure of β-Ga 2O3. They showed the lowest conduction band to be composed of 4 s FIG. 10. (a) EPR linewidth of the H shallow donor in ZnO as a function of tem- perature, black squares experiment, red line simulation with a power law depen-dence for T > 50 K, (b) tentative Mott plot of the linewidth: ln(width) vs T−0.25, experiment (black squares), the lines are only guides for the eye. FIG. 11. ZnO:H, EPR signal intensity as a function of temperature; at T = 25 K a hyper-Lorentz lineshape is observed; the lines are only guides for the eye. FIG. 12. ZnO:H, log (EPR Signal) vs 1000/T; black squares experiment, red squares fit with Eq. (4).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-7 Published under license by AIP Publishing.states of octahedral Ga, having a quasi-one-dimensional property along the baxis. According to these authors, the conduction elec- trons are moving along the octahedral chains, which are separatedthrough an energy barrier by the tetrahedral Ga chains. Finally, we want to comment on the DX properties of Si shallow donors. In a recent EPR study on donors in bulk β-Ga 2O3, Son et al.10have proposed a DX model. The term “DX center ”has been used previously for donors in AlGaAs alloys,25where from a particular Al composition on a deep donor state becomes the equi-librium configuration of the Si donor. The samples investigated inRef. 10were Si doped at a concentration of 7.1 × 10 17cm−3but showed surprisingly no neutral donor EPR spectrum in the as-grown state. Only after an annealing at 1100 °C in N 2gas flow, the samples became conductive and the EPR spectrum of a neutraldonor was observed. These authors concluded that the shallowdonor, supposed to be Si, is a negative-U center, similar to DX centers in AlGaAs. The DX like character of the Si donor, which has been questioned 7later on the basis of transport measurements in Si doped n-type thin films and is also not confirmed by ourresults. For more details on negative U properties, see the discus-sions in Refs. 7and10concerning Ga 2O3. In conclusion, we have shown that EPR spectroscopy of the shallow donor can be efficiently used to investigate the conductionproperties in β-Ga 2O3. As EPR is a volume sensitive technique, it is “free”from surface conductivity effects, which often complicate the transport measurements in oxide semiconductors. We have shown that carrier transport in β-Ga 2O3occurs via a VRH process even in samples with carrier concentrations below the Mott transition. Thecarrier dynamics already observed at the lowest temperature evi-dence that the donor distribution in the bulk samples is not uniform but gives rise to cluster formation. The small linewidth of the CESR might be attributed to a quasi-one-dimensional characterof the lowest conduction band. ACKNOWLEDGMENTS We thank D. C. Look from the Ohio State University for the characterization and supply of the ZnO sample and critical readingof the manuscript. We thank equally A. Parisini from the Parma University for helpful discussions. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. REFERENCES 1S. J. Pearton, J. Yang, P. H. Cary, F. Ren, J. Kim, J. Tadjer, and M. A. Mastro, “A review of Ga 2O3materials, processing, and devices, ”Appl. Phys. Rev. 5, 011301 (2018). 2A. Kuramata, K. Koshi, S. Watanabe, Y. Yamaoka, T. Masui, and S. Yamakoshi, “High quality β-Ga 2O3single crystals grown by edge defined film-fed growth, ” Jpn. J. Appl. Phys. 55, 1202A2 (2016). 3Y. Yao, Y. Ishikawa, and Y. Sugawara, “X-ray diffraction and Raman characteri- zation of β-Ga 2O3single crystal grown by edge-defined film-fed growth method, ”J. Appl. Phys. 126, 205106 (2019).4S. Rafique, L. Han, A. T. Neal, S. Mou, J. Boeckl, and H. Zhao, “Towards high mobility heteroepitaxial β-Ga 2O3on sapphire dependence on the substrate off-axis angle, ”Phys. Status Solidi A 215, 1700467 (2017). 5J. B. Varley, J. R. Weber, A. Janotti, and C. G. Van de Walle, “Oxygen vacancies and donor impurities in β-Ga 2O3,”Appl. Phys. Lett. 97, 142106 (2010). 6S. Lany, “Defect phase diagram for doping of Ga 2O3,”APL Mater. 6, 046103 (2018). 7A. T. Neal, S. Mou, S. Rafique, H. Zhao, E. Ahmadi, J. S. Speck, K. T. Stevens, J. D. Blevins, D. B. Thomson, N. Moser, K. D. Chabak, and G. H. Jessen, “Donors and acceptors in β-Ga 2O3,”Appl. Phys. Lett. 113, 062101 (2018). 8M. Yamaga, E. G. Vílora, K. Shimamura, N. Ichinose, and M. Honda, “Donor structure and electrical transport mechanism in β-Ga 2O3,”Phys. Rev. B 68, 155207 (2003). 9E. G. Víllora, K. Shimamura, T. Ujiie, and K. Aoki, “Electrical conductivity and lattice expansion of β-Ga 2O3below room temperature, ”Appl. Phys. Lett. 92, 202118 (2008). 10N. T. Son, K. Goto, K. Nomura, Q. T. Thieu, R. Togashi, H. Murakami, Y. Kumagai, A. Kuramata, M. Higashiwaki, A. Koukitu, S. Yamakoshi, B. Monemar, and E. Janzén, “Electronic properties of the residual donor in unintentionally doped β-Ga 2O3,”J. Appl. Phys. 120, 235703 (2016). 11Z. Kabilova, C. Kurdak, and R. Peterson, “Observation of impurity band con- duction and variable range hopping in heavily doped (010) β-Ga 2O3,”Semicond. Sci. Technol. 34, 03LT02 (2019). 12K. Irmscher, Z. Galazka, M. Pietsch, R. Uecker, and R. Fornari, “Electrical properties of β-Ga 2O3single crystals grown by the Czochralski method, ”J. Appl. Phys. 110, 063720 (2011). 13H. J. von Bardeleben, J. L. Cantin, A. Parisini, A. Bosio, and R. Fornari, “Conduction mechanism and shallow donor properties in silicon doped β-Ga 2O3 thin films: An electron paramagnetic resonance study, ”Phys. Rev. Mater. 3, 084601 (2019). 14V. Grachev, see www.visual-epr.com for“Visual EPR. ” 15N. F. Mott, “Conduction in non-crystalline materials, ”Philos. Mag. 19, 835 (1969). 16J. H. Pifer, “Microwave conductivity and electron spin resonance linewidth of heavily doped Si:P, ”Phys. Rev. B 12, 4391 (1975). 17D. V. Savchenko, “Electron spin resonance study of heavily nitrogen doped 6H-SiC crystals, ”J. Appl. Phys. 117, 045708 (2015). 18W. E. Carlos, J. A. Freitas, M. A. Khan, D. T. Olson, and J. N. Kuznia, “Electron-spin-resonance studies of donors in wurtzite GaN, ”Phys. Rev. B 48, 17878 (1993). 19D. M. Hofmann, A. Hofstaetter, F. Leiter, H. Zhou, F. Heneker, B. K. Meyer, S. B. Orlinskii, J. Schmidt, and P. G. Baranov, “Hydrogen: A relevant shallow donor in zinc oxide, ”Phys. Rev. Lett. 88, 045504 (2002). 20E. Aubay and D. Gourier, “Bistability of the magnetic resonance of conduction electrons in gallium oxide, ”J. Phys. Chem. 96, 5513 (1992). 21E. Aubay and D. Gourier, “Magnetic bistability and Overhauser shift of con- duction electrons in gallium oxide, ”Phys. Rev. B 47, 15023 (1993). 22L. Binet and D. Gourier, “Origin of the blue luminescence of β-Ga 2O3,” J. Phys. Chem. Solids 59, 1241 (1998). 23N. Blumenschein, C. Kadlec, O. Romanyuk, T. Paskova, J. F. Muth, and F. Kadlec, “Dielectric and conducting properties of unintentionally and Sn doped β-Ga 2O3studied by terahertz spectroscopy, ”J. Appl. Phys. 127, 165702 (2020). 24L. Binet, D. Gourier, and C. Minot, “Relation between electron band structure and magnetic bistability of conduction electrons in β-Ga 2O3,”J. Solid State Chem. 113, 420 (1994). 25T. N. Morgan, “Theory of the DX center in Al xGa1−xAs and GaAs crystals, ” Phys. Rev. B 34, 2664 (1986).Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 125702 (2020); doi: 10.1063/5.0023546 128, 125702-8 Published under license by AIP Publishing.

5.0019634.pdf

AIP Conference Proceedings 2269 , 030079 (2020); https://doi.org/10.1063/5.0019634 2269 , 030079 © 2020 Author(s).Structural and thermal properties of Ni- doped MoS2 synthesized via one-step hydrothermal method Cite as: AIP Conference Proceedings 2269 , 030079 (2020); https://doi.org/10.1063/5.0019634 Published Online: 12 October 2020 Niharika Chourasiya , Arun Kumar Singh , Akshkumar Verma , and Ashish Verma Structural and Thermal Properties of Ni -doped MoS 2 Synthesized via One -Step Hydrothermal Method Niharika Chourasiyaa), Aru n Kumar Singh, Akshkumar Verma and Ashish Verma b) Dr. Harisingh Gour Vishwavidyala ya, 470003,Sagar, (M.P.),India a) Corresponding author: niharika1ch@gmail.com b) vermaashish31@rediffmail.com Abstract. In the present work, Nickel (Ni) doped molybdenum disulfide (MoS 2) has been synthesi zed via a facile one step hydrothermal method using ammonium molybdate tetrahydrate [(NH 4)6Mo 7O24.4H 2O] , thioacetamide [CH 3CSNH 2] and nickel metal powder as precursors. The structural properties of the synthesized material were characterized by Powder X-Ray Diffraction (XRD), Fourier Transform Infrared Spectroscopy (FTIR). Indexing of the peaks in the XRD plot of the material indicates the formation of 2H - MoS 2 crystal structure. There is not much difference between 2H - MoS 2 XRD plot and Ni:MoS 2 XRD whi ch confirms the successful Ni doping in the molybdenum sites of MoS 2. Presence and percentage of doped nickel in the sample has been determined by Energy Dispersive X -Ray Analysis (EDX) characterization for the elemental analysis. Scanning Electron Microsc opy (SEM) has been used for morphological studies. Crystallite size and lattice strain of the prepared sample has been obtained by the Scherrer method and W -H plot method respectively. Capacitance performance of the material has been studied using electroc hemical workstation.For the analysis of thermal properties, Differential Scanning Calorimetery (DSC) and Thermo -Gravimetric Analysis (TGA) techniques have been used, which shows that the material has good thermal stability. INTRODUCTION Layered transition metal dichalcogenides of groups IVB, VB and VIB consist of structurally and chemically a well-defined family of compounds whose physical and chemical properties have stimulated a considerable amount of interest during the past few years [1 -3].These transit ion metal dichalcogenides and their intercalation compounds are useful materials in the field of light harvesting [4], high temperature -high pressure lubricants [3] and in solar energy conversion [5] etc. Furthermore, MoS 2 materials have an especially expl icit spin –orbit splitting property, which is considered to be highly suitable for spintronics applications [6, 7]. Theoretical studies have revealed that diverse magnetic orderings can be realized in transition metal doped MoS 2 [8-10]. The magnetic propert ies of Mn-, Fe-, Co-, Nb-, and Zn- doped MoS 2 have been studied by density functional theory calculations [11 –13]. These studies have shown that the electronic structures and magnetic properties of MoS 2 can be modified by transition metal (TM) dopants, whi ch may lead to a new class of dilute magnetic semiconductors (DMS) in the engineering field [14,15]. Recently, room -temperature ferromagnetism has been observed in Ni -incorporated ZnO nanostructures [16]. Therefore, it is of interest to investigate the pro perties of nickel doped MoS 2 nanostructures. Particle size and crystal morphology are important aspects in these applications, which have attracted the attention of researcher on the synthesis of nanocrystalline Ni:MoS 2. The particle size and morphology of material varies with the route adopted for synthesis. Hydrothermal synthesis route is a facile, feasible, cost effective and produces high yield. Therefore in this work Ni -doped MoS 2 have been synthesized by hydrothermal method. The structural properties of the as synthesized material have been investigated by XRD and FTIR. The X -ray peak analysis has been carried out for estimating the crystallite size and lattice strain of Ni:MoS 2 nanosheets based on W -H-plot method. Thermal stability of the material has been investigated by DSC/TGA and capacitance performance has been investigated by electrochemical measurements. International Conference on Multifunctional Materials (ICMM-2019) AIP Conf. Proc. 2269, 030079-1–030079-6; https://doi.org/10.1063/5.0019634 Published by AIP Publishing. 978-0-7354-2032-8/$30.00030079-1EXPERIMENTAL DETAILS Materials Ni-doped MoS 2 (Ni:MoS 2) nanosheets were synthesized using Ammonium Molybdate Tetrahydrate [(NH 4)6Mo 7O24.4H 2O], Thiourea [CH 3CSNH 2] and Nickel (Ni) metal powder as precursors. All the precursors used were of analytical grade and were used without further processing. Synthesis of Ni:M OS2 Nanosheets The Ni:MoS 2 nanosheets were synthesized by hydrothermal method as reported by Dezhi Wang et. al. [17] with few modifications. In the typical process, 0.04 gm Nickel powder was mixed in 50 ml DI water and sonicated for 2 hours in a bath sonicater . In the meanwhile, 1.76 gm ammo nium molybdate tetrahydrate [(NH 4)6Mo 7O24.4H 2O] was mixed in 25 ml DI water and stirred well for 30 min using a magnetic stirrer. In parallel to this, 1.88 gm thioacetamide [CH 3CSNH 2] was mixed in 25 ml DI beaker in a separate beaker and stirred well for 3 0 minutes on another magnetic stirrer. The three solutions were then mixed together and sonicated again for 1 hour. Now the reaction mixture was transferred into a 150 ml Teflon lined stainless steel hydrothermal autoclave and heated in a furnace for 24 ho urs at 220 ̊C. The autoclave in the furnace was maintained as such till it got cooled to room temperature naturally. After getting cooled to room temperature, the black precipitate was washed with water and ethanol many times using a centrifuge and dried i n a vacuum oven at 60 ̊C overnight, Fig. 1 shows the schematic of the synthesis process. FIGURE 1.Schematic diagram to explain the synthesis process. CHARACTERIZATION METHODS The XRD measurements were carried out using Bruker D8 Advance X -ray Diffractome ter in the diffraction angle range from 10 to 80 with Cu K -alpha X -Rays (λ=0.154 nm). Fourier Transform Infrared Spectroscopy (FT - IR) measurement was performed in the range 700 cm-1-4000 cm-1 without KBr using Bruker FTIR Spectrophotometer. DSC and TGA th ermograms were obtained from NETZSCH STA 449F1 analyzer. Scanning Electron Microscopy analysis has been carried out using FESEM –FEI NOVA NANO SEMTM 450, 30kV. Electrochemical studies were performed using Metrohm Autolab PGSTAT128N Workstation. 030079-2RESULT AND DISCUSSIONS Structural and Thermal Studies The XRD pattern of the synthesized Ni:MoS 2 nanosheets was obtained. All the XRD peaks were matched with the ICDD card No. 65 -7025 and has been indexed by hexagonal 2H -phase of MoS 2 as shown in Fig. 2. The peak broadening in the XRD pattern is the clear indication of the presence of nanocrystals in the sample. FIGURE 2(a). XRD patterns of Ni:MoS 2 FIGURE 2(b). W-H Plot of Ni-MoS 2 FIGURE 2. XRD analysis of Ni:MoS 2 The crystallite size of the materials has been obtained using the Scherrer equation. The average crystallite size is found to be 3.3235nm.Williamson -Hall method has been applied for the determination of the lattice strain. The lattice strain (ɛ) in the materia l has been found to be 3.97 and the crystallite size obtained from the W -H plot is 2.3nm which is approximate to the value obtained from Scherrer equation. FIGURE 3(a). SEM micrograph of Ni-MoS2 FIGURE 3(b). EDX mapping of Ni-MoS 2 The SEM micrographs s hown in Fig. 3(a) represents the formation of nanosheets. The EDX mapping shown in Fig. 3(b) shows the presence of molybdenum, sulphur, nickel, carbon and oxygen elements. The presence of carbon in the mapping is due to the carbon coating for measurement a nd oxygen is being considered to be an impurity. The presence of Nickel in the EDX mapping confirms that it is present in the material. Thus we can say that Nickel is successfully doped in MoS 2. FTIR spectroscopy is used to study the interaction between di fferent species and changes in chemical composition of the mixture [18]. As shown in the FTIR spectrum Fig.4, the characteristic peaks at 3431 cm-1 and 1630 cm-1 are ascribed to the stretching vibration and bending vibration of water, respectively [19]. Th e peak at ~3756cm-1 is due to the O -H stretching [20]. 030079-3 FIGURE 4. FTIR spectra of Ni:MoS 2 FIGURE 5. DSC/TGA thermograms of Ni:MoS 2 The synthesized material has been analyzed for its thermal stability using DSC/TGA thermograms, Fig.5. The TGA thermograms show that the only observable mass change of 7.30 % has been observed between the temperature range of 275 oC to 375 oC and an exothe rmic peak has been observed in DSC curve in the same temperature range. The 77.99 % residual mass of the material at 499.5 oC has been observed. Thus the thermal analysis shows that the material has good thermal stability. Capacitance performance Cyclic Voltammetry (CV) is a basic electrochemical tool to determine the specific capacitance of supercapacitor’s electrode materials [21]. The CV measurement has been performed using three electrode setup at scan rates ranging from 5 mV/s to 100 mV/s in 1M H 2SO 4 aqueous electrolyte solution. Figure6 shows that the CV curves are nearly rectangular, however some distortions have been observed which is due to stainless steel substrate in electrode configuration. FIGURE 6. The Cyclic voltammetry (CV) cu rve of Ni:MoS 2. FIGURE 7. Nyquist plot of Ni:MoS 2. Electrochemical impedance spectroscopy (EIS) of Ni:MoS 2 electrode has been performed in 1M H 2SO 4 aqueous electrolyte solution in the frequency range from 1 Hz to 10 kHz at 0. 2 V AC potential. Figure 7 shows the Nyquist plots of Ni:MoS 2 divided into two parts, a low frequency regions and a high frequency region. At higher frequency Nyquist plot, the plot between real (Z’) and imaginary (Z”) part of impedance gives the value of cell resistance which came out to be ~9.6 Ω. At low frequency Ni:MoS 2 has steeper slope which shows capacitive nature of the material and it is better confirmation for the calculated Specific Capacitance (C s) of CV results. 030079-4Table 1 shows specific capacitance (C s) of Ni:MoS 2 at different scan rates. Ni:MoS 2 has been found to have a maximum specific capacitance (C s) ~ 5.4 F/g at 5 mV/s scan rate. Graphical representation of specific capacitance, Fig.8 shows that there is decrease in specific capacitance C s with increase in scan rate. Table 1. Specific capacitance of Ni:MoS 2 at different scan rates. S.No. San rate (mV) Specific capacitance Cs(F/g) 1. 5 5.4 2. 10 4.1 3. 20 2.9 4. 30 2.4 5. 50 1.9 6. 100 1.5 FIGURE 8. Graphical representation of specific capacitance of Ni:MoS 2 at different scan rates. CONCLUSION The structural, morphological, thermal properties and capacitive performance of Ni doped MoS 2 have been studied in this work. It has been observed that slight modification in the synthesis process have lead to the change in morphological properties of the material. The material has good thermal stability and has maximum capacitance at the scan rate 5mV.The capacitance decreases on increase in the scan rate. Content of Ni dopant in MoS 2 can be varied and its effect on its properties can be further examined. ACKNOWLEDGEMENT Authors are grateful to UGC-DAE Consortium, Indore, M.P., India for XRD characterization, Dr. Harisingh Gour Vishwavidyalaya, Sagar, M.P., India, for financial support during the research, Department of Chemistry of the institute for FTIR facility and Sophisticated Instrumentation Center of the university for DSC/TGA and SEM, EDX and Electrochemical characterizations. REFERENCES 1. F. Jellinek, Ark. Kemi, 20, 447, (1963). 2. J.A. Wilson and A. D. Yoffe, Adv. Phys. 19, 169, (1969). 3. "Preparation and crystal growth of Materials with Layered Structures" (Reidel, Dordrecht, The Netherlands, 1976- 1979). 4. H. Wang, C. Li, P. Fang, Z. Zhang and J. Z. Zhang, Chem. Soc. Rev. , 47, 6101, (2018) . 5. H. Tributsch, Electroehem. Soc. 125, 1086, (1978). 6. B. Radisavljevic, M. B. Whitwick, and A. Kis, ACS Nano 5, 9934 (2011). 7. Z. Y. Zhu, Y. C. Cheng, and U. Schwingenschlogl, Phys. Rev. B 84, 153402 (2011). 8. X. J. Zhang, W. B. Mi, X. C. Wang, Y. C. Cheng, and U. Schwingenschlögl, Sci. Rep. 4, 7368 (2014). 9. W. S. Yun and J. D. Lee, Phys. Chem. Chem. Phys. 16, 8990 (2014). 10. A. N. Andriotis and M. Menon, Phys. Rev. B 90, 125304 (2014). 11. Y. Cheng, Z. Zhu, W. Mi, Z. Guo, and U. Schwingenschlogl, Phys. Rev. B 87, 100401 (2013). 12. A. Ramasubramaniam and D. Naveh, Phys. Rev. B 87, 195201 (2013). 13. K. Dolui, I. Rungger, C. Pemmaraju, and S. Sanvito, Phys. Rev. B 88, 075420, (2013). 14. P. Tao, H. H. Guo, T. Yang, and Z. D. Zhang, J. Appl. Phys. 115, 054305, (2014). 15. H. L. Zheng, B. S. Yang, D. D. Wang, R. L. Han, X. B. Du, and Y. Yan, Appl. Phys. Lett. 104, 132403 (2014). 030079-516. P. Satyarthi, S. Ghosh, B. Pandey, P. P. Kumar, C. L. Chen, C. L. Dong, W. F. Pong, D. Kanjilal, K. Asokan, and P. Srivastava, J. Appl. Phys. 113, 183708 (2013). 17. D. Wang, X. Zhang, Y. Shen and Z. Wu, RSC Adv. , 6, 16656- 16661, (2016) 18. Sudhasree S et al. Toxicological & Environmental Chemistry 96(5): 743-754 (2014). 19. Z. Guo, Q. Ma, Z. Xuan, F. Du and Y. Zhong, RSC Adv. , 6, 16730, (2016). 20. Nakamoto, Kazuo, Infrared and Raman spetra of inorganic and coordination compounds, p.170, Wiley, (1997). 21. Y. Zhu, S. Murali, M. D. Stoller, K. J.Ganesh, W.Cai, P. J. Ferreira ,A. Pirkle, R. M. Wallace, K. A. Cychosz, M. Thommes, D. Su, E. A. Stach and R. S. Ruoff, Science 332, 1537 –1541 (2011). 030079-6

1.5140243.pdf

J. Chem. Phys. 153, 084120 (2020); https://doi.org/10.1063/1.5140243 153, 084120 © 2020 Author(s).Excitation energies from thermally assisted-occupation density functional theory: Theory and computational implementation Cite as: J. Chem. Phys. 153, 084120 (2020); https://doi.org/10.1063/1.5140243 Submitted: 26 November 2019 . Accepted: 12 August 2020 . Published Online: 31 August 2020 Shu-Hao Yeh , Aaditya Manjanath , Yuan-Chung Cheng , Jeng-Da Chai , and Chao-Ping Hsu ARTICLES YOU MAY BE INTERESTED IN From orbitals to observables and back The Journal of Chemical Physics 153, 080901 (2020); https://doi.org/10.1063/5.0018597 Electronic structure software The Journal of Chemical Physics 153, 070401 (2020); https://doi.org/10.1063/5.0023185 Efficient yet accurate dispersion-corrected semilocal exchange–correlation functionals for non-covalent interactions The Journal of Chemical Physics 153, 084117 (2020); https://doi.org/10.1063/5.0011849The Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Excitation energies from thermally assisted-occupation density functional theory: Theory and computational implementation Cite as: J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 Submitted: 26 November 2019 •Accepted: 12 August 2020 • Published Online: 31 August 2020 Shu-Hao Yeh,1,2Aaditya Manjanath,1 Yuan-Chung Cheng,2 Jeng-Da Chai,3,4,a) and Chao-Ping Hsu1,a) AFFILIATIONS 1Institute of Chemistry, Academia Sinica, Taipei 11529, Taiwan 2Department of Chemistry, National Taiwan University, Taipei 10617, Taiwan 3Department of Physics, National Taiwan University, Taipei 10617, Taiwan 4Center for Theoretical Physics and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan a)Authors to whom correspondence should be addressed: jdchai@phys.ntu.edu.tw and cherri@sinica.edu.tw ABSTRACT The time-dependent density functional theory (TDDFT) has been broadly used to investigate the excited-state properties of various molecular systems. However, the current TDDFT heavily relies on outcomes from the corresponding ground-state DFT cal- culations, which may be prone to errors due to the lack of proper treatment in the non-dynamical correlation effects. Recently, thermally assisted-occupation DFT (TAO-DFT) [J.-D. Chai, J. Chem. Phys. 136, 154104 (2012)], a DFT with fractional orbital occupations, was proposed, explicitly incorporating the non-dynamical correlation effects in the ground-state calculations with low computational complexity. In this work, we develop TDTAO-DFT, which is a TD, linear-response theory for excited states within the framework of TAO-DFT. With tests on the excited states of H 2, the first triplet excited state (13Σ+ u) was described well, with non- imaginary excitation energies. TDTAO-DFT also yields zero singlet–triplet gap in the dissociation limit for the ground singlet (11Σ+ g) and the first triplet state (13Σ+ u). In addition, as compared to traditional TDDFT, the overall excited-state potential energy surfaces obtained from TDTAO-DFT are generally improved and better agree with results from the equation-of-motion coupled-cluster singles and doubles. Published under license by AIP Publishing. https://doi.org/10.1063/1.5140243 .,s I. INTRODUCTION Over the past decades, Kohn–Sham density functional the- ory (KS-DFT)1,2has been extensively used in the prediction of various ground-state (GS) properties of solids as well as finite- sized molecules.3–5Its time-dependent (TD) extension, known as time-dependent density functional theory (TDDFT),6–8has been a popular approach for computing excited-state properties, includ- ing the absorption and emission spectra,9photochemical reac- tions,10dynamics,11and energy and electron transfer,12due to its low computational cost and the availability of a plethora of computer codes in this area. The one-to-one correspondence between the TD density and the TD external potential wasrigorously demonstrated by Runge and Gross in 1984 in their the- orem.6The linear-response framework was further introduced,7,8 which brought forth a paradigm shift in the simulation of exci- tations of quantum systems from a density-functional perspec- tive13–15and is the main reason behind the popularity of this method. However, conventional TDDFT is derived from ground-state (GS) KS-DFT, which is a single-determinant-based method. As a result, it can fail to describe the excited-state phenomena heav- ily governed by non-dynamical (or static) correlation, such as photochemistry processes involving photoinduced bond breaking, and problems associated with conical intersection.7,9,16,17A pro- totypical example is the bond dissociation process of the H 2 J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-1 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp molecule. It is known that the excitation energy of the low- est triplet state of H 2, computed using conventional TDDFT,9 would become imaginary beyond a H–H bond distance of 1.75 Å, a phenomenon arising from a spin symmetry-breaking solution in the ground state,18,19a typical characteristic of nondynami- cal correlation effects. In contrast, in wavefunction-based meth- ods, the (nearly) degenerate determinants are considered on an equal footing when performing a self-consistent field (SCF) calcu- lation, and this is the basis of multi-configuration (MC) SCF or complete active space (CAS) SCF-based methodologies. However, these methods can be prohibitively expensive for large systems as their computational cost scales factorially with the size of active space. KS-DFT with proper exchange energy functionals may rea- sonably model systems with non-dynamical correlation, albeit at the expense of enormous computation efforts. For example, the works by Becke20,21and the works by Kong and co-workers22–24 demonstrated parametric functionals that need to be solved self- consistently within the single-determinant framework. Although these works significantly improved the bond dissociation trends of simple diatomic molecules, compared to the Hartree–Fock the- ory, they still deviate appreciably at the bond dissociation limit compared to a full configuration interaction (FCI) calculation.22,23 Moreover, the SCF associated with these functionals adds to the computational effort that can scale dramatically with the size of molecules. On the other hand, various approaches have been developed to cope with the non-dynamical correlation effects without the high computational cost of an exact exchange functional. The CAS- DFT model is one such method,25wherein some amount of cor- relation has been accounted for by a density functional calcula- tion. As a result, the dynamical correlation associated with the MC representation of the system might be “double counted.”26,27 To mitigate this issue, the multi-configuration pair-density func- tional theory26,27and multi-configuration range-separated DFT28,29 were developed. While the former utilizes the so-called on-top pair-density functional , the latter separates the electron interac- tion operator into short- and long-range parts, which are treated with DFT and wavefunction theory, respectively. Although the idea of using such a “hybrid” scheme seems to be an attrac- tive prospect,26–29they can be computationally demanding for increasing system sizes because of the initial generation of MC wavefunctions. Another category of computational methods exists, which can cope with non-dynamical correlation with the additional advan- tage that they are low-cost methods. They include the spin– flip, ionization-potential, and electron-affinity based approaches, which are aimed to start with a high-spin, with 1-less or 1- more electron single-determinant references such that the non- dynamical correlation problem is minimal.30–32These approaches require a well-balanced treatment of the orbitals in the refer- ence, and they can offer high-quality solutions in many cases. However, the requirement of balanced treatment of orbitals in the reference is not always feasible, and thus, applications are limited. In this regard, the thermally assisted-occupation density functional theory (TAO-DFT)33was developed by Chai in 2012 to alleviate the formidable challenge of balancing the computationalcost and simultaneously incorporating the non-dynamical correla- tion effects with reasonable accuracy. In contrast to traditional KS- DFT, the underlying principle of TAO-DFT is in the usage of frac- tional orbital occupations according to a given fictitious temperature (θ) to effectively incorporate the different electronic configurations of a system. This approach ensures that some “excitations” in the form of fractional populations of electrons in the low-lying virtual orbitals are considered along with the GS of the system, similar to a multi-determinant expansion of the wavefunction. The inclusion of fractional occupancies is a computationally cheaper alternative to a multi-determinant expansion for accounting non-dynamical corre- lation effects. As a result, TAO-DFT has a computational cost similar to that of KS-DFT, which is O(N3–4). In TAO-DFT, the entropy contribution [e.g., see Eq. (26) of Ref. 33] can reasonably capture the non-dynamical correlation energy of a system, which was dis- cussed and numerically investigated in Ref. 33, even when the sim- plest local density approximation (LDA) exchange-correlation (XC) energy functional is used. The XC energy functionals at the higher rungs of Jacob’s ladder, such as the generalized-gradient approxima- tion (GGA),34global hybrid,35and range-separated hybrid35,36XC energy functionals, can also be employed in TAO-DFT. Moreover, a self-consistent scheme that determines the fictitious temperature in TAO-DFT has been recently proposed to improve the perfor- mance of TAO-DFT for a wide range of applications.37Since TAO- DFT is similar to KS-DFT in computational efficiency, TAO-DFT has been recently adopted for the study of the electronic proper- ties of various nanosystems with pronounced radical nature.36,38–45 In particular, the electronic properties (e.g., singlet–triplet energy gaps, vertical ionization potentials, vertical electron affinities, fun- damental gaps, and active orbital occupation numbers) of linear acenes and zigzag graphene nanoribbons (i.e., systems with polyrad- ical character) obtained from TAO-DFT33–35,38have been shown to be in reasonably good agreement with those obtained from other accurate electronic structure methods, such as the particle–particle random-phase approximation (pp-RPA)46XC energy functional in KS-DFT, the density matrix renormalization group (DMRG) algorithm,47,48the variational two-electron reduced density matrix (2-RDM) method,49,50and other high-level methods.51–54 II. GROUND-STATE REFERENCE: TAO-DFT In TAO-DFT,33the electron density is represented by the ther- mal equilibrium density of an auxiliary system of Nenon-interacting electrons at a fictitious temperature θ(in energy units), ρ(r)=∑ ifiϕ∗ i(r)ϕi(r). (1) Here, fi(a value between 0 and 1) is the fractional occupation num- ber of the ith orbitalϕiand is given by the Fermi–Dirac distribution function, fi={1 + exp[(εi−μ)/θ]}−1, (2) whereμis the chemical potential for electrons and is determined by ∑ifi=Nefor a given θ, orbital energies { εi}, and total electron num- berNe. This choice for the fractional occupation function and the corresponding one-particle density matrix has been extensively used in other methods such as finite-temperature DFT (FT-DFT)55and J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-2 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp floating occupation molecular orbital-complete active space config- uration interaction (FOMO-CASCI).56With this assisted occupa- tion number and generalized density expression, the total ground- state energy functional can be written as EG[ρ]=TTAO[{fi,ϕi}]+Vext[ρ]+EKS Hxc+Eθ[ρ], (3) where TTAO is the kinetic free energy functional of non-interacting electrons [equivalent to Aθ sas defined in Eq. (24) of Ref. 33], Vext[ρ] is the energy functional of the external potential (or nuclei potential), EKS Hxcis the sum of Hartree and XC energy functionals in KS-DFT, and Eθis theθ-dependent energy functional.33Alternatively (to the original derivation33), from Eq. (3), upon performing the functional derivatives with respect to the orbitals ( ϕi), we can also obtain the SCF equations in TAO-DFT, [−1 2∇2 r+vext(r)+vKS Hxc(r)+vθ(r)]ϕi(r)=εiϕi(r), (4) where vext,vKS Hxc, andvθare the potentials (or functional derivatives) of corresponding energy functionals (i.e., Vext[ρ],EKS Hxc, and Eθ[ρ], respectively) in Eq. (3), and { ϕi} and {εi} are the TAO orbitals and orbital energies, respectively, which can be solved self-consistently through SCF. The algorithm is similar to KS-DFT, with the only differences being the vθ(r) term in the Hamiltonian and the determi- nation of chemical potential μ, making this approach attractive and easy in implementation. We have provided a variational perspective of TAO-DFT in Appendix A, which complements the derivation in Ref. 33. III. EXCITED-STATE THEORY: TDTAO-DFT A. Mathematical formalism In the present work, we propose TDTAO-DFT, which is a time-dependent linear-response theory for TAO-DFT, allowing excitation energy calculation using Casida’s formulation,8within the framework of TAO-DFT. In TDTAO-DFT, the TD density is given by ρ(r,t)=∑ pfpϕ∗ p(r,t)ϕp(r,t), (5) whereϕp(r,t) are the TD orbitals (for the fictitious particles) and fpare the corresponding fractional occupation numbers, which are assumed to be time -independent , and their values are taken from those obtained from the corresponding ground-state TAO-DFT calculation [Eq. (1)]. In order to facilitate the map- ping between the original interacting system of electrons mov- ing under the influence of a TD external potential and the auxiliary system of non-interacting particles, an action varia- tional principle in TAO-DFT should be established. Following the variational principle, the TD effective potential for the non- interacting TAO system can be partitioned into the following parts: vTAO eff(r,t)=vext(r,t)+vTAO Hxcθ[ρ](r,t), (6)where vTAO Hxcθis the functional derivative of the Hxc θ-action, which contains the time-dependent Hartree potential, exchange- correlation potential, and θpotentials for the fractional occupation. Further details are included in Appendix B1 accompanying this work. Similar to conventional TDDFT, with the equality connecting the effective potential and the functional derivative of TD action, an equation of motion for TDTAO-DFT can be expressed as i∂ ∂tϕp(r,t)=[−1 2∇2 r+vext(r,t)+vTAO Hxcθ[ρ](r,t)]ϕp(r,t) =ˆF(t)ϕp(r,t). (7) We note that vTAO Hxcθ[ρ](r,t)is also a TD generalization of the potential associated with the Hartree, exchange, correlation, and θ- functionals in GS TAO-DFT. The equation of motion is reformu- lated in terms of the one-particle density matrix P(t),9 i∂ ∂tP(t)=[F(t),P(t)], (8) where F(t), the time-dependent “Fock matrix,” is the matrix rep- resentation of the one-particle operator ( ˆF) in Eq. (7). The general time-evolution of the state of a system is given by P(t)=P○+δP(t) (9) and F(t)=F○+δVext(t)+δFHxcθ[P](t), (10) where P○andF○denote the initial conditions for solving Eq. (8) andδP(t),δVext(t), and FHxcθ[P](t)are the time-dependent changes in the matrices of density, external field, and Hartree-exchange- correlation potential in matrix representation interaction, respec- tively, in the system. The initial state (at t=t0) is commonly con- sidered to be the unperturbed GS of the system for convenience. In terms of the GS TAO orbitals, P○ pq=δpq⋅fp,F○ pq=δpq⋅εp. (11) If the electronic eigenspectrum of a system is desired, the ampli- tude of the change in the external field | δVext(t)| is assumed to be infinitesimally small.6–9It is therefore suitable to consider a linear- response relation between δFHxcθ[P](t) andδP(t). Using the GS TAO orbital basis, this can be obtained as δFHxcθ rs(t)=∑ pq∫dτ(δFHxcθ rs(t) δPpq(τ))δPpq(τ). (12) Employing the time-domain Fourier transformations δPqr(ω)=∫dte−iωt[δPqr(t)], (13) δVqr(ω)=∫dte−iωt[δVqr(t)], (14) J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-3 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp δFHxcθ rs δPpq(ω)=∫dte−iω(t−τ)[δFHxcθ rs(t) δPpq(τ)], (15) one could recast Eq. (8) into ∑ q⎡⎢⎢⎢⎢⎣F○ pq⋅δPqr(ω)−δPpq(ω)⋅F○ qr +⎛ ⎝δVpq(ω)+∑ st⎛ ⎝δFHxcθ pq δPst(ω)⎞ ⎠⋅δPst(ω)⎞ ⎠P○ qr −P○ pq⎛ ⎝δVqr(ω)+∑ st⎛ ⎝δFHxcθ qr δPst(ω)⎞ ⎠⋅δPst(ω)⎞ ⎠⎤⎥⎥⎥⎥⎦ =ω⋅δPpr(ω) (16) by neglecting all second-order (or higher) terms. Upon invok- ing the GS definitions in Eq. (11) and assuming all δVpq(ω) to be infinitesimally small, the corresponding working equation becomes (εp−εr)δPpr(ω)−(fp−fr)⎡⎢⎢⎢⎢⎣∑ st⎛ ⎝δFHxcθ pr δPst(ω)⎞ ⎠δPst(ω)⎤⎥⎥⎥⎥⎦ =ω⋅δPpr(ω). (17) A conventional linear-response relation [which is the inverse of Eq. (12)]7,57gives the TD density–density response function. The details of this derivation are provided in Appendix B2. Similar to conventional TDDFT, we apply the adiabatic approximation to the xc θ-kernel (i.e., the xc θ-kernel is assumed to be frequency-independent),8,9,58 δFHxcθ pr δPst(ω)≈δFHxcθ pr δPst⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪⌟⟨rro⟪⟪⟩r⟪P○ ≈∫d3rd3r′ϕ∗ r(r)ϕp(r)𝕗Hxcθ(r,r′)ϕt(r′)ϕ∗ s(r′) =(rp∣𝕗Hxcθ∣ts). (18) The working equation would be reduced to an eigenvalue equation, ∑ st[(εp−εr)⋅δps,st−(fp−fr)(rp∣𝕗Hxcθ∣ts)]⋅ΩR k,st=ωk⋅ΩR k,pr, (19) where ΩR pr=δPprand kdenotes the kth eigenvalue. This can be represented in the matrix form as Casida’s equation,8 ⎛ ⎝ˆA ˆB ˆB∗ˆA∗⎞ ⎠(X Y)=ωk(ˆI ˆ0 ˆ0−ˆI)(Xk Yk), (20) where Xk,pr=ΩR k,p>r,Yk,rp=ΩR k,p<rdenotes upward and downward transitions, respectively. The coupling matrices are defined asApr,st=(εp−εr)δpsδrt+Bpr,ts, (21) Bpr,st=−(fp−fr)(rp∣𝕗Hxcθ∣st). (22) These matrices are similar in form to those derived from con- ventional Casida’s equation, which most TDDFT works are based on.9,59However, we consider the fractional occupation number dif- ference ( Δf) pre-factor in Eq. (20), which is equivalent to the original Casida’s equation in Ref. 8. It is to be noted that the occupation num- bers are explicitly sourced from GS TAO. In Eq. (19), the superscript Rin ΩRimplies that the eigenvectors obtained are the right eigenvec- tors. Using the density–density response function (Appendix B2), an eigenvalue-like equation that is complementary to that in Eq. (19) can be derived. The details are included in Appendix B3. B.Idempotency in TDTAO-DFT In KS theory, an idempotent one-electron density matrix ( PP =P)9is derived from the single-determinant ansatz of the wave- function, so for any first-order changes in the one-electron density matrix, P○⋅δP+δP⋅P○−δP=0, (23) which when represented in terms of KS orbitals becomes (np+nq−1)⋅δPpq=0, (24) where { np} are the integer occupation numbers (either 0 or 1). Within this particular condition, the conventional Casida’s scheme allows transitions between only occupied ( ni= 1) and virtual ( na= 0) orbitals. On the other hand, due to fractional occupation numbers, the one-electron density matrix in TAO-DFT violates this idempo- tency condition for nonvanishing θ. Therefore, a relaxed condition in terms of TAO orbitals is proposed as (fp+fq−1)⋅δPpq∝θ, (25) where the KS limit of TDTAO-DFT is recovered for θ→0. This condition implies that transitions with fp+frtending to 1 would be dominant. These transitions require one of the pand qorbitals to be strongly occupied , 1/2≤fr≤1, with the other weakly occu- pied, 0≤fr<1/2. More details on the relaxed idempotency condition for TDTAO-DFT can be found in Appendix C accompanying this work. IV. COMPUTATIONAL DETAILS We implement this formalism in the development version of Q-Chem 5.2.60All numerical results are calculated with the cc- pVDZ basis set, which was determined by performing a comprehen- sive convergence test of different sets. The two-electron integrals are evaluated with the standard quadrature Euler–Maclaurin–Lebedev grid (50 194),61consisting of 50 Euler–Maclaurin62radial grid points and 194 Lebedev63angular grid points. V. H 2BOND DISSOCIATION USING TDTAO-DFT We demonstrate how some of the challenges plaguing TDDFT are rectified with our method through the GS bond dissociation J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-4 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp process of the H 2molecule. This system has been studied exten- sively for many years using a plethora of methods. Successfully capturing the mechanism of bond dissociation within the frame- work of DFT has been elusive owing to the lack of incorporation of non-dynamical correlation effects. Within TAO-DFT, however, this challenge was resolved by choosing an appropriate θof 40 mhartree.33,34It was further shown that, at the bond dissociation limit, the multi-reference character was more pronounced.33,34 In TDDFT, one encounters the challenge of imaginary frequen- cies (i.e., excitation energies) for the triplet states that occurs in most of the results obtained from adiabatic local density approx- imation (ALDA) functionals (kernels).18,19,64This issue is related to the symmetry breaking where the difference in spin densities (i.e.,ρα−ρβ) is not equal to zero for a large interatomic distance. In other words, the unrestricted (asymmetric) solution obtained using KS-DFT becomes lower in total energy than the restricted (symmetric) one, as demonstrated by Casida et al. using a two- level model.19TAO-DFT significantly rectifies this issue for a large enoughθvalue.33 Figure 1 shows the potential energy surface (PES) of the first triplet excited state (13Σ+ u) for H 2bond dissoci- ation using TDTAO-DFT and TDDFT ( θ= 0 mhartree). The TDDFT results show imaginary frequencies beyond the H–H bond distance of ∼1.5 Å. This is attributed to a poor ground- state reference, as mentioned previously, due to the lack of incor- poration of the non-dynamical correlation effects beyond this bond distance. In addition, this phenomenon is observed in TDTAO-DFT simulations for θ= 0 mhartree, 10 mhartree, 20 mhartree, and 30 mhartree. However, for θ≥40 mhartree, the imaginary-frequency issue is resolved. We also note here that the requirement for a real-value 13Σ+ u excitation energy mandates a higher threshold value for θthan FIG. 1 . Potential energy surface of the first triplet excited state (13Σ+ u) computed using TDDFT ( θ= 0 mhartree) and TDTAO-DFT with the Perdew–Burke–Ernzerhof (PBE) XC-functional, cc-pVDZ basis set, and gradient expansion approximation (GEA) version of the Eθfunctional34for TAO calculations. The inset shows a zoomed-in view for the large bond-distance regime.that obtained through a self-consistent scheme,37which is around 15.5 mhartree. While a lower θvalue is needed to describe the ground-state bond dissociation curves, our observation indicates that a higher θvalue is needed for excitation properties and an opti- mal determination scheme for θremains to be developed. One such direction is to include the excited-state information in the post-SCF variational scheme similar to that outlined in Eq. (9) of Ref. 56. Another advantageous aspect of TDTAO-DFT is that the energy of the first triplet excited (13Σ+ u) state in the dissociation limit correctly approaches the GS singlet energy. Figure 2 shows the singlet–triplet (11Σ+ g–13Σ+ u) vertical gap as a function of H–H bond dissociation computed using ground-state TAO-DFT, coupled– cluster singles and doubles (CCSD), and TDTAO-DFT. To compute the 11Σ+ g–13Σ+ ugap at the ground-state level (in order to mitigate the problem of imaginary frequencies in TDDFT), it was recom- mended to use the unrestricted ground-state SCF formalism for H2and other small molecules.22,23,64However, this does not guar- antee the convergence of the energy of the 13Σ+ ustate to that of the 11Σ+ gstate at the bond dissociation limit for H 2for TAO-DFT (Fig. 2). This gap may violate the covalent nature of the3Σ+ ustate, where the energies of covalent states 13Σ+ uand GS (11Σ+ g) should be the same at the bond dissociation limit.18In other words, at this limit, the electrons are located in the 1 sorbitals of the corre- sponding atoms and are, therefore, isolated enough with respect to one another. This gap increases with θdue to the increase in the energy of 13Σ+ uand a simultaneous decrease in the energy of 11Σ+ g FIG. 2 . The energy gap as a function of the H–H bond distance (in Å) between the singlet ground state (11Σ+ g) and the first excited triplet state (13Σ+ u) calculated using TDTAO-DFT and unrestricted TAO-DFT with the PBE XC-functional and GEA θ- functional. The equation-of-motion–coupled cluster singles doubles (EOM-CCSD) results are presented as a benchmark. The cc-pVDZ basis set was employed for all calculations. The inset shows a zoomed-in view for the large bond-distance regime. J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-5 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 3 . Potential energies of (a) singlet and (b) triplet excited states, computed using TDTAO-DFT (with θ= 40 mhartree and GEA θ-functional), EOM-CCSD, and conventional TDDFT. (c) shows only the1Σ+ gstates. (d) is a zoomed-in region showing the avoided crossing between two EOM-CCSD states that are not completely captured by either TDTAO-DFT or conventional TDDFT. The orange shaded regions in (a), (c), and (d) indicate portions of the EOM-CCSD curves that have double excitation character. DE in (a) signifies that the CCSD state is double excitation in nature. PBE is selected as the XC-functional for all DFT calculations, and cc-pVDZ is selected as the basis set for all calculations. (thisθ-dependent decrease is also observed for the total energy of 13Σ+ ucalculated with TDTAO-DFT in Fig. 1). On the other hand, the trend obtained for TDTAO-DFT (Fig. 2) is in excellent agree- ment with that obtained using the equation-of-motion coupled- cluster singles and doubles (EOM-CCSD) method or observed in experiments.65EOM-CCSD is used here as a benchmark method since it is equivalent to FCI for a two-electron system such as H 2. For the sake of completeness, we also computed the PESs of other excited states for H 2. The lowest six singlet and triplet excited states in TDTAO-DFT and TDDFT are demonstrated with low- lying PESs from EOM-CCSD in Figs. 3(a) and 3(b). The overall feature of singlet and triplet states from TDTAO-DFT is in excel- lent agreement with the EOM-CCSD results, except for the charge- transfer state (11Σ+ u) and the missing states with double excitation character [purple curve with unfilled squares and curves highlighted in orange with unfilled diamonds in Figs. 3(a) and 3(b)]. We specu- late that the problem with the 11Σ+ ustate could be due to the usage ofthe simple adiabatic approximation to the xc θ-kernel6,19,66,67as well as the time-independent occupation numbers in our formalism.68–70 The missing CCSD double excited states also indicate the inability of TDTAO-DFT to capture the avoided crossing between the first two1Σ+ gexcited states [orange shaded regions as shown in Figs. 3(a), 3(c), and 3(d)]. A more detailed investigation is certainly required for resolving these challenges. VI. RELATIONSHIP BETWEEN θAND IMAGINARY FREQUENCIES: A QUALITATIVE DESCRIPTION We perform a detailed analysis of the PESs with different θ values to acquire more insight about the qualitative relationship betweenθand the imaginary roots. Two molecular systems were chosen for this analysis, H 2and N 2, and their S–T gaps are shown in Fig. 4. The problem of imaginary frequencies is fixed with TDTAO- DFT for a suitable choice of θ, irrespective of the system under con- sideration, thereby indicating its versatility. However, we note that J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-6 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp FIG. 4 . S–T gap of (a) H 2and (b) N 2with the bond distance and different θ (in mhartree) values, calculated using TDTAO-DFT with the PBE XC-functional, cc-pVDZ basis set, and GEA version of the Eθfunctional.34The filled symbols indicate potential energy surfaces without any imaginary frequencies. θis a system-dependent quantity and a robust algorithm is needed to ascertain it. Based on the optimal choice of θ, we observe that the S–T gap vanishes at the bond dissociation limit for N 2[Fig. 4(b)], similar to that in H 2[Fig. 4(a)]. This is also in agreement with experiments.71 VII. CONCLUDING REMARKS In summary, a time-dependent linear-response theory for pre- dicting excited-state properties based on the TAO-DFT framework, TDTAO-DFT, is proposed. This theory takes advantage of TAO- DFT, where the spin-symmetry-breaking problem of orbitals in ground-state SCF is resolved. As a result, TDTAO-DFT provides a correct description of low-lying triplet excited states, without imag- inary energies, at the bond dissociation limit for a molecule. This was demonstrated through the dissociation curve of the hydrogen molecule, in which a reasonable lowest triplet state (13Σ+ u) is cap- tured by TDTAO-DFT, but is not so, for TDDFT. Additionally,TAO-DFT (with a large fictitious temperature θ) may produce an erratic gap between the 13Σ+ uand ground states at the dissocia- tion limit, which is resolved by TDTAO-DFT. The PESs for higher excited states of stretched H 2are also improved significantly as compared to TDDFT. SUPPLEMENTARY MATERIAL The supplementary material includes additional results and the numerical data presented in this work. AUTHORS’ CONTRIBUTIONS S.-H.Y. and A.M. contributed equally to this work. ACKNOWLEDGMENTS C.-P.H. acknowledges support from Academia Sinica and the Investigator Award (Grant No. AS-IA-106-M01) and the Ministry of Science and Technology of Taiwan (Project No. 105-2113-M- 001-009-MY4). J.-D.C. acknowledges support from the Ministry of Science and Technology of Taiwan (Grant No. MOST107-2628-M- 002-005-MY3) and National Taiwan University (Grant No. NTU- CDP-105R7818). A.M. acknowledges additional financial support from the Academia Sinica Distinguished Postdoctoral Fellowship . This work also benefited from discussions facilitated through the National Center for Theoretical Sciences, Taiwan. APPENDIX A: A VARIATIONAL PERSPECTIVE OF TAO-DFT In this the section, we briefly present the derivation of the TAO- DFT KS-like equations based on an alternative, variational principle. The same variational approach is also employed in the derivation of the linear-response theory, which will be presented in Appendix B of this work. According to the partition of energy functional,33,34the func- tional derivative of the total energy functional can be expressed as δE[ρ] δϕi(r)=δTTAO s δϕi(r)+δVext δϕi(r)+δEHxcθ[ρ] δϕi(r), (A1) where TTAO s is the kinetic (free) energy functional, and Vext+EHxcθ is the energy associated with the effective potential. The explicit derivative of the kinetic (free) energy functional would be δTTAO s δϕ∗ j(r′)=δ δϕ∗ j(r′)(∑ ifi∫drϕ∗ i(r)ˆtϕi(r) +θ{∑ ifilnfi+(1−fi)ln(1−fi)}) =fj⋅ˆtϕj(r′)+∑ i⎡⎢⎢⎢⎣δfi δϕ∗ j(r′)⋅∫drϕ∗ i(r)ˆtϕi(r)⎤⎥⎥⎥⎦ +θ∑ i⎡⎢⎢⎢⎣{lnfi−ln(1−fi)}⋅δfi δϕ∗ j(r′)⎤⎥⎥⎥⎦, (A2) J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-7 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp where ˆt=−∇2/2 andδϕj(r′)/δϕ∗ j(r′)=0. Similarly, the derivatives of the energy term associated with external potential as well as the Hxcθenergy term are, respectively, δVext δϕ∗ j(r′)=∫dr′′δρ(r′′) δϕ∗ j(r′)⋅δ δρ(r′′)[∫drvext(r)ρ(r)] =∫drvext(r)δρ(r) δϕ∗ j(r′) =fj⋅vext(r′)ϕj(r′) +∑ i⎛ ⎝δfi δϕ∗ j(r′)⋅∫drvext(r)ϕ∗ i(r)ϕi(r)⎞ ⎠(A3)and δEHxcθ[ρ] δϕ∗ j(r′)=∫drδρ(r) δϕ∗ j(r′)⋅δEHxcθ[ρ] δρ(r) =∫drδρ(r) δϕ∗ j(r′)⋅vHxcθ[ρ](r) =fj⋅vHxcθ(r′)ϕj(r′) +∑ i⎡⎢⎢⎢⎣δfi δϕ∗ j(r′)∫drvHxcθ[ρ](r)ϕ∗ i(r)ϕi(r)⎤⎥⎥⎥⎦. (A4) Combining the three terms above, an explicit expression of the total energy functional is derived, δE[ρ] δϕ∗ j(r′)=δTTAO s δϕ∗ j(r′)+δVext δϕ∗ j(r′)+δEHxcθ[ρ] δϕ∗ j(r′) =fj⋅ˆtϕj(r′)+∑ i⎡⎢⎢⎢⎣δfi δϕ∗ j(r′)⋅∫drϕ∗ i(r)ˆtϕi(r)⎤⎥⎥⎥⎦+θ∑ i⎡⎢⎢⎢⎣(lnfi−ln(1−fi))⋅δfi δϕ∗ j(r′)⎤⎥⎥⎥⎦+fj⋅vext(r′)ϕj(r′) +∑ i⎡⎢⎢⎢⎣δfi δϕ∗ j(r′)⋅∫drvext(r)ϕ∗ i(r)ϕi(r)⎤⎥⎥⎥⎦+fj⋅vHxcθ(r′)ϕj(r′)+∑ i⎡⎢⎢⎢⎣δfi δϕ∗ j(r′)⋅∫drvHxcθ[ρ](r)ϕ∗ i(r)ϕi(r)⎤⎥⎥⎥⎦ =fj⋅[ˆt+vext(r′)+vHxcθ[ρ](r′)]ϕj(r′)+∑ iδfi δϕ∗ j(r′)⋅{θln(fi 1−fi)+∫drϕ∗ i(r)[ˆt+vext(r)+vHxcθ[ρ](r)]ϕi(r)}. (A5) Enforcing the normalization conditions for both density and orbital functions, a Lagrangian is introduced, L[ρ]=E[ρ]−∑ ij[λij∫drϕ∗ i(r)ϕj(r)−δij]−μ[∫drρTAO(r)−Ne], (A6) where {λij} andμare Lagrange multipliers. Considering the functional derivative with respect to orbital functions δL[ρ] δϕ∗ j(r′)=fj⋅[ˆt+vext(r′)+vHxcθ[ρ](r′)]ϕj(r′)+∑ iδfi δϕ∗ j(r′)⋅{θln(fi 1−fi) +∫drϕ∗ i(r)[ˆt+vext(r)+vHxcθ[ρ](r)]ϕi(r)}−∑ iλji⋅ϕj(r′)−μ∑ iδfi δϕ∗ j(r′)−μfj⋅ϕj(r′) =fj⋅[ˆt+vext(r′)+vHxcθ[ρ](r′)]ϕj(r′)+∑ i{εi−θ[1 θ(εi−μ)]}⋅δfi δϕ∗ j(r′)−∑ iλji⋅ϕi(r′)−μ∑ iδfi δϕ∗ j(r′)−μfj⋅ϕj(r′) =fj⋅[ˆt+vext(r′)+vHxcθ[ρ](r′)]ϕj(r′)+μ∑ iδfi δϕ∗ j(r′)−∑ iλji⋅ϕi(r′)−μ∑ iδfi δϕ∗ j(r′)−μfj⋅ϕj(r′) =fj⋅[ˆt+vext(r′)+vHxcθ[ρ](r′)]ϕj(r′)−(λjj+μfj)⋅ϕj(r′)+i≠j ∑ iλji⋅ϕi(r′). (A7) and using the variational condition δL[ρ]/δϕ∗ j(r′)=0, one obtains [ˆt+vext(r′)+vHxcθ[ρ](r′)]ϕj(r′)=(f−1 jλjj+μ)⋅ϕj(r′)+f−1 ji≠j ∑ iλji⋅ϕi(r′). (A8) J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-8 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Note that the second term in Eq. (A7) indicates that it is necessary to introduce the entropy term θ[∑ifilnfi+ (1−fi)ln(1−fi)] to the kinetic functional in order to preserve the correct variational prop- erty such that the derivative terms arising from vextandvHxcθ[last terms in Eqs. (A3) and (A4)] are compensated. With a canonical orbital assumption (because the orbitals are orthonormal to one another), the equation can be recast into an eigenvalue equation, similar to a KS-like equation, ˆhTAO[ρ](r)ϕi(r)=εi⋅ϕi(r), (A9) where ˆhTAO=ˆt+vext+vHxcθandεi=f−1 iλii+μ. APPENDIX B: DETAILED DERIVATION OF LR-TDTAO-DFT 1. Variational principle for TAO action functional and TD effective potential Starting from the action variational principle72and its modified form,73we have the general definitions of action functionals for a physical system, A[ρ]=∫τ 0dt⟨Ψ(t)∣(∂ ∂t−ˆH)∣Ψ(t)⟩, (B1) B[ρ]=A[ρ]+∫dt∫drvext(r,t)ρ(r,t), (B2) δB[ρ] δρ(r,t)=vext(r,t)+i⟨Ψ[ρ](τ)∣δΨ[ρ;τ] δρ(r,t)⟩, (B3) where Ψ[ρ;τ] represents the wavefunction at time tandτdenotes the upper bound of the time integral. For a TDTAO system, the definition of universal action func- tionals can be written similarly, following that of the conventional TDDFT scheme,73 δATAO[ρ] δρ(r,t)=i⟨ΨTAO[ρ](τ)∣δΨTAO[ρ;τ] δρ(r,t)⟩, (B4) BTAO[ρ]=ATAO[ρ]+∫τ 0dtdrveff(r,t)ρ(r,t). (B5) The TD effective potential for TAO can be expressed as veff(r,t)=δBTAO[ρ] δρ(r,t)+i⟨ΨTAO[ρ](τ)∣δΨTAO[ρ;τ] δρ(r,t)⟩. (B6) One can define the difference between the two functionals as AHxcθ[ρ]=BTAO[ρ]−B[ρ], (B7) which is the TAO extension of Hartree-exchange-correlation func- tionals. Summarizing the equations above, similar to TDDFT, one can recast the effective potential in TAO asvHxcθ(r,t)=δAHxcθ[ρ] δρ(r,t) =veff(r,t)+i⟨ΨTAO[ρ](τ)∣δΨTAO[ρ;τ] δρ(r,t)⟩ −vext(r,t)−i⟨Ψ[ρ](τ)∣δΨ[ρ;τ] δρ(r,t)⟩. (B8) 2. Density–density response function Here, we show that the linear-response equation can also be constructed inversely , δρ(rt)=∫dr′dt′χTAO s(rt,r′t′)δvTAO eff(r′t′), (B9) where χTAO s(rt,r′t′)≡δρTAO(r,t) δvTAO eff(r′,t′)(B10) is the density–density response function for a non-interacting TAO system. With the density expression in terms of TD orbitals, one obtains χTAO s(rt,r′t′)=δρTAO(r,t) δvTAO eff(r′,t′)=δ[∑pfpϕ∗ p(r,t)ϕp(r,t)] δvTAO eff(r′,t′) =∑ pfpδ[ϕ∗ p(r,t)ϕp(r,t)] δvTAO eff(r′,t′) =∑ pfp[δϕ∗ p(r,t) δvTAO eff(r′,t′)ϕ○ p(r,t) +ϕ○∗ p(r,t)δϕp(r,t) δvTAO eff(r′,t′)], (B11) whereϕ○ p(r,t)and its complex conjugate represent the evolution of the TD orbitals in the absence of any TD perturbation (i.e., the TD external field). Applying the first-order perturbation theory, the TD orbital functions in a TD external field can be described by the equation ϕp(r,t)=[1−i∫t 0dt′r≠p ∑ rsϕ○ r(r)e−iεr(t−t′) ×[∫dr′ϕ○∗ r(r′)δvTAO eff(r′,t′)ϕ○ s(r′)]e−iεst′ ×∫drϕ○∗ s(r)]ϕ○ p(r) =ϕ○ p(r)−i e−iεrt∫t 0dt′∑ r,r≠pϕ○ r(r) ×[∫dr′ϕ○∗ r(r′)δvTAO eff(r′,t′)ϕ○ p(r′)]e−i(εp−εr)t′ , (B12) and the corresponding orbital response functions are expressed explicitly in terms of initial orbitals (ground-state TAO orbitals) and orbital energies, J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-9 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp δϕp(r,t) δvTAO eff(r′,t′)=−iΘ(t−t′)e−iεrt∑ r,r≠pϕ○ r(r)ϕ○∗ r(r′)ϕ○ p(r′)e−i(εp−εr)t′ , δϕ∗ p(r,t) δvTAO eff(r′,t′)=iΘ(t−t′)eiεrt∑ r,r≠pϕ○∗ r(r)ϕ○ r(r′)ϕ○∗ p(r′)ei(εp−εr)t′ .(B13) Combining Eqs. (B11) and (B13), the time-domain non-interacting response function in TDTAO-DFT can be evaluated as follows: χTAO s(rt,r′t′)=∑ pfp⎧⎪⎪⎨⎪⎪⎩⎡⎢⎢⎢⎢⎣iΘ(t−t′)eiεrt∑ r,r≠pϕ○∗ r(r)ϕ○ r(r′)ϕ○∗ p(r′)ei(εp−εr)t′⎤⎥⎥⎥⎥⎦ϕ○ p(r,t) +ϕ○∗ p(r,t)⎡⎢⎢⎢⎢⎣−iΘ(t−t′)e−iεrt∑ r,r≠pϕ○ r(r)ϕ○∗ r(r′)ϕ○ p(r′)e−i(εp−εr)t′⎤⎥⎥⎥⎥⎦⎫⎪⎪⎬⎪⎪⎭ =ir≠p ∑ prfpΘ(t−t′){ϕ○ p(r)ϕ○∗ r(r)ϕ○ r(r′)ϕ○∗ p(r′)ei(εr−εp)(t−t′)−ϕ○∗ p(r)ϕ○ r(r)ϕ○∗ r(r′)ϕ○ p(r′)e−i(εr−εp)(t−t′)}. (B14) Performing a Fourier transformation, the corresponding frequency- domain expression becomes χTAO s(r,r′,ω)=r≠p ∑ prfp{ϕ○ p(r)ϕ○∗ r(r)ϕ○ r(r′)ϕ○∗ p(r′) ω−(εr−εp)+iη −ϕ○∗ p(r)ϕ○ r(r)ϕ○∗ r(r′)ϕ○ p(r′) ω+(εr−εp)+iη} =r≠p ∑ pr(fp−fr)ϕ○∗ p(r′)ϕ○ r(r′)ϕ○∗ r(r)ϕ○ p(r) ω−(εr−εp)+iη, whereη→0. (B15) We note that there are no self-transition terms in both Eqs. (B15) and (B12) since every TD orbital is considered as an orthonormalized function at any given instant of time. As a result, an explicit response function for a non-interacting reference system (TAO system) is obtained, and the resulting expression is similar to the conventional TDDFT.7 3. Alternative path to Casida’s equation Recall the partition of effective potential57 δvTAO eff(r,ω)=δvext(r,ω)+∫dr1δρ(r1,ω)⋅𝕗Hxcθ(r,r1,ω) =δvext(r,ω)+δvHxcθ[ρ](r,ω), (B16) where 𝕗Hxcθis the Fock matrix defined in Eq. (17). Since an infinitesi- mal external field change is considered [ δvext(r1,ω)→0],8,9Eq. (B9) can be recast into δρ(r,ω)=∫dr1∫dr2χTAO s(r,r1,ω)δρ(r2,ω)⋅𝕗Hxcθ(r1,r2,ω). (B17)If∫dr𝕗Hxcθ(r′,r,ω)is operated on both sides of the equation, one obtains an iterative formula δvHxcθ(r,ω)=∫dr1∫dr2𝕗Hxcθ(r,r1,ω) ×χTAO s(r1,r2,ω)δvHxcθ(r2,ω). (B18) Recalling the explicit expression of the non-interacting response function in Eq. (B15), Eq. (B18) can be reformulated into δvHxcθ rs(ω)=q≠p ∑ pq∫dr∫dr1ϕ○∗ r(r)ϕ○ s(r)ϕ○∗ q(r1)ϕ○ p(r1) ×𝕗Hxcθ(r,r1,ω)⎡⎢⎢⎢⎢⎣(fp−fq)⋅δvHxcθ pq(ω) ω−(εq−εp)+iη⎤⎥⎥⎥⎥⎦, (B19) whereδvHxcθ rs(ω)=∫drϕ○∗ r(r)ϕ○ s(r)δvHxcθ(r,ω)is the Hxcθpoten- tial projected on the single-particle basis set. Similar to the deriva- tion in the main manuscript, the two-electron integral is defined as follows: (rs∣𝕗Hxcθ(ω)∣pq)≡∫drϕ○∗ r(r)ϕ○ s(r) ×∫dr1ϕ○∗ q(r1)ϕ○ p(r1)𝕗Hxcθ(r,r1,ω). (B20) With a rescaling factor [ ω−(εs−εr) +iη], an iterative equation in a finite basis set is obtained, [ω−(εs−εr)]⋅ΩL rs(ω)=q≠p ∑ pq[(rs∣𝕗Hxcθ(ω)∣pq)(fp−fq)]⋅ΩL pq(ω), (B21) where ΩL rs(ω)≡δvHxcθ rs(ω) ω−(εs−εr)+iη. (B22) J. Chem. Phys. 153, 084120 (2020); doi: 10.1063/1.5140243 153, 084120-10 Published under license by AIP PublishingThe Journal of Chemical PhysicsARTICLE scitation.org/journal/jcp Within ALDA, the corresponding eigenvalue equation would be ∑ pq[(εq−εp)δqsδpr−(rs∣𝕗Hxcθ∣pq)(fq−fp)]ΩL k,pq=ωk⋅ΩL k,rs, (B23) where kdenotes the kth eigenvalue. We note that this eigenvalue equation is not exactly the same as Eq. (19) in the main text. How- ever, because of the transpose relation between the two matrices, they will generate the same eigenspectra. APPENDIX C: RELAXED IDEMPOTENCY CONDITION In conventional TDDFT, transitions between orbitals are pre- selected by the idempotency condition,9which is derived from a single-determinant assumption, and can be formulated as (fp+fq−1)δPpq=0, (C1) where Ppqis a matrix element of transition density matrices. This condition leads to the result that only transitions between occupied and virtual orbitals would contribute to a physical (single) excita- tion. On the other hand, since the single-determinant assumption is removed from TAO-DFT, we consider an alternative invariant assumption based on the recurrence relation of the derivative of Fermi function ∂fp ∂εp=−(fp−f2 p)/θ (C2) or in the matrix representation ∂P0 ∂F0=−(P0−P2 0)/θ, (C3) where P0−P2 0on the left-hand side implies a relaxed idempotency feature of TAO one-particle density matrix. In other words, instead of equating to zero, P0−P2 0is associated with another constant, θ⋅∂P0/∂F0. To employ the relaxed condition in the excited-state TAO, we further assume that the simple partial derivative form would be preserved in the TD extension of ∂P0/∂F0. Recall the total functional derivative of the density matrix δ(P−P2)=δP−P0⋅δP−δP⋅P0, (C4) and combine it with Eq. (C2), δPpq−fp⋅δPpq−δPpq⋅fq=−(fp+fq−1)δPpq =−θ⋅δ[∂P ∂F] pq, (C5) whereδ[∂P/∂F]pqis not an explicit derivative and is assumed as an infinitesimal constant. 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5.0015993.pdf

J. Appl. Phys. 128, 083901 (2020); https://doi.org/10.1063/5.0015993 128, 083901 © 2020 Author(s).Charge density waves beyond the Pauli paramagnetic limit in 2D systems Cite as: J. Appl. Phys. 128, 083901 (2020); https://doi.org/10.1063/5.0015993 Submitted: 31 May 2020 . Accepted: 02 August 2020 . Published Online: 24 August 2020 Alex Aperis , and Georgios Varelogiannis 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 ARTICLES YOU MAY BE INTERESTED IN Antiferromagnetic spintronics Journal of Applied Physics 128, 070401 (2020); https://doi.org/10.1063/5.0023614 Piezoresponse force microscopy imaging and its correlation with cantilever spring constant and frequency Journal of Applied Physics 128, 084101 (2020); https://doi.org/10.1063/5.0013287Charge density waves beyond the Pauli paramagnetic limit in 2D systems Cite as: J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 View Online Export Citation CrossMar k Submitted: 31 May 2020 · Accepted: 2 August 2020 · Published Online: 24 August 2020 Alex Aperis1,a) and Georgios Varelogiannis2 AFFILIATIONS 1Department of Physics and Astronomy, Uppsala University, P.O. Box 516, SE-75120 Uppsala, Sweden 2Department of Physics, National Technical University of Athens, GR-15780 Athens, Greece Note: This paper is part of the Special Topic on 2D Quantum Materials: Magnetism and Superconductivity a)Author to whom correspondence should be addressed: alex.aperis@physics.uu.se ABSTRACT Two-dimensional materials are ideal candidates to host Charge Density Waves (CDWs) that exhibit paramagnetic limiting behavior, similar to the well-known case of superconductors. Here, we study how CDWs in two-dimensional systems can survive beyond the Pauli limitwhen they are subjected to a strong magnetic field by developing a generalized mean-field theory of CDWs under Zeeman fields that includes incommensurability, imperfect nesting, and temperature effects and the possibility of a competing or coexisting Spin Density Wave (SDW) order. Our numerical calculations yield rich phase diagrams with distinct high-field phases above the Pauli limiting field.For perfectly nested commensurate CDWs, a q-modulated CDW phase that is completely analogous to the superconducting Fulde –Ferrell –Larkin –Ovchinnikov (FFLO) phase appears at high fields. In the more common case of imperfect nesting, the commensurate CDW ground state undergoes a series of magnetic-field-induced phase transitions first into a phase where commensurate CDW and SDW coexist and subsequently into another phase where CDW and SDW acquire a q-modulation that is, however, distinct from the pure FFLO CDW phase. The commensurate CDW + SDW phase occurs for fields comparable to but less than the Pauli limit and survives above it.Thus, this phase provides a plausible mechanism for the CDW to survive at high fields without the need for forming the more fragile FFLOphase. We suggest that the recently discovered 2D materials like the transition metal dichalcogenides offer a promising platform forobserving such exotic field-induced CDW phenomena. Published under license by AIP Publishing. https://doi.org/10.1063/5.0015993 I. INTRODUCTION Magnetic fields are detrimental to superconductivity because they tend to break the Cooper pairs by coupling either to the orbital motion or to the spin of the electrons. 1Usually, the superconducting upper critical field is limited by the orbital effect; however, the lattercan be minimized for thin film geometries and generally in two-dimensional systems. 2In such a case, the superconductor is para- magnetically, or Pauli, limited and the upper critical field for conven- tional superconductors can be estimated by the simple BCS relationH P¼Δ0=ffiffiffi 2p withΔ0the zero temperature and magnetic field value of the superconducting gap.3,4For decades, it is known that super- conductors can exceed the Pauli limit via a phase transition to a modulated state with Cooper pairs that acquire a finite momentum, which is driven by the external field, the so-called Fulde –Ferrell – Larkin –Ovchinnikov (FFLO) phase.5,6Charge Density Waves (CDWs) are quantum states of matter that are characterized by the freezing of the conduction electron charge density into a periodic modulation pattern below a critical temperature.7When the spin density becomes modulated instead, one speaks of a Spin Density Wave (SDW) state.8Charge/spin density waves are frequently encountered in the phase diagrams of correlated materials where they may compete or coexist with super- conductivity and, hence, they have been investigated thoroughly over the past few decades.9–12Due to the fact that the nesting properties of the Fermi surface are enhanced with low dimensionality, these states are most typical for quasi-one-dimensional systems such as, e.g., the Bechgaard salts.13However, density waves have also been found to occur in many two-dimensional systems like chro- mium films,14tellurides,15and transition metal dichalcogenides (TMDs).16,17Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-1 Published under license by AIP Publishing.Up to now, the effect of strong magnetic fields on CDW/SDW states has been a subject of studies mostly in the context of 1D organic materials,18where it has been shown that the coupling of the magnetic field to the electron ’s motion can give rise to field-induced charge19,20and spin21,22density waves. However, other field-induced phases that are unrelated to orbital effects have been observed experimentally beyond 1D in diverse systems such as, e.g., the phases accompanied by metamagnetic transitions inURu 2Si223–25and the manganites.26,27 Since CDWs are spin singlet condensates, they can in princi- ple exhibit Pauli limiting behavior in complete analogy to super- conductors.18Usually, the paramagnetic critical field of a CDW corresponds to magnetic field values of tenths of Teslas given thatthe critical temperature of such condensates is quite large. Thequasi-1D organic salt (Per) 2Au(mnt) 2is a rare case of a CDW mate- rial with a relatively low Tcand, as a result, with a Pauli limit that lies in the experimentally accessible range around 37 T. In this system, a transition to a new CDW phase for H.HPwas indeed observed in which a weaker CDW gap coexists with normal stateregions. This phase survives for magnetic field values that are wayabove the theoretical Pauli limit of the material, and, therefore, it was identified as the first example of a FFLO phase observed in a charge density wave. 28The interpretation of this phase as a FFLO CDW relies on the dominance of the Zeeman effect;28however, this picture is obscured by the presence of the competing orbital effect, which is generally imposed by the quasi-1D nature of the system.19,29 In order to reach to the unambiguous observation of this exotic phase, it would be desirable to be able to minimize the orbitaleffects. Similar to the case for FFLO superconductors, optimal exper-imental conditions for this purpose could be achieved by applying a magnetic field in the plane of a purely two-dimensional CDW metal. 2In this respect, the recently synthesized single and few layer atomically thick TMDs could offer a promising route to tackle thisproblem. 30These novel 2D materials can display enriched properties as compared to their bulk counterparts and thus they have emerged as a testbed for the fundamental understanding of their archetypical CDWs31–35and the coexisting superconductivity.30,36,37For example, many TMDs at their monolayer limit exhibit an intricate spin –orbit coupling that fixes the electron spins perpendicular to the plane.30 As a result, this mechanism gives rise to so-called Ising superconduc- tivity, which has been observed to survive under strong in-plane magnetic fields beyond the Pauli limit.38–40Interestingly, recent experiments have provided evidence of q-modulated superconductiv- ity of the FFLO type for strong fields in monolayer H /C0NbS 2,41 while FFLO superconductivity has also been predicted for bilayer TMDs.42Therefore, these materials appear as suitable candidates for probing CDWs beyond the paramagnetic limit and possibly identify-ing the formation of the FFLO CDW phase or, as we show here,other field-induced CDW phases. Motivated by the above experimental picture, here we revisit the problem of estimating the impact of strong magnetic fields on theCDW state. It has been pointed out that CDW and SDW states gen-erally coexist when particle –hole asymmetry and ferromagnetism are present. 43However, in the majority of previous studies, these two states are considered separately with a few exceptions as in, e.g., 1D systems,19despite the fact that many CDW materials are strongly correlated and in fact can host SDW phases as well.10,44,45Here, we generalize the study of cases where CDW/SDW can coexist under Zeeman fields to any system dimensionality by for- mulating a suitable effective mean-field theory that takes intoaccount both these states on the same footing. In order to includethe possibility of FFLO states, we extend previous works 43,46by allowing the possibility for our considered CDW/SDW states to acquire an incommensurate modulation, i.e., our analysis fully includes incommensurate density wave ordering. As a concretecase, we numerically solve our model self-consistently for a single-band two-dimensional system. For perfectly nested 2D systems, wefind that at low temperature and for magnetic fields above the Pauli limit, the CDW undergoes a first order phase transition into a q-modulated FFLO CDW, which is exactly analogous to the FFLO phase of superconductors. Interestingly, when the perfect nestingcondition of the underlying Fermi surface is not satisfied, we findthat instead of a FFLO CDW phase, the system undergoes a transi- tion into a phase where the commensurate CDW coexists with com- mensurate SDW order. Remarkably, this phase becomesenergetically favorable already for fields below the Pauli limit and itsurvives for field strengths above it. Our calculations show that thisCDW + SDW phase can survive for even higher fields by undergoing a subsequent phase transition into a q-modulated CDW + SDW phase that bears many similarities with an FFLO state although themodulation wavevector qappears to be constant with the field strength. Overall, our findings reveal a rich phase diagram for 2D CDW systems under an in-plane magnetic field and offer qualitative predictions for our observed high-field states that could be tested,e.g., in two-dimensional TMDs or other 2D systems. II. METHOD In this section, we will first present our generalized mean-field theory of coexisting charge/spin density waves, 46extended to include in our study the possibility of incommensurate density waves and deviations from perfect nesting. Next, we will discussqualitatively the FFLO CDW state in the small- qmodulation limit within arguments based on the fermiology of the system. In the lastpart of the section, we provide a discussion on the method and the toy model that we use for numerical calculations. A. Theory of coexisting CDW and SDW orders under Zeeman fields Our starting point is the generalized one-band Hamiltonian that describes interacting electrons in the presence of an externalZeeman magnetic field, H tot¼H0þHdwwith H0¼X k,σξk/C0σμBH ðÞ cy kσckσ, (1) where ξkis the electron energy dispersion, c(y) kσare electron annihi- lation (creation) operators at momentum kand spin index σ, and μBHis the Zeeman spin splitting due to a magnetic field Hchosen parallel to the ^z-axis (henceforth, we set μB¼1). Scattering processes in the particle –hole channel with exchanged momentum Qare described by the following effectiveJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-2 Published under license by AIP Publishing.four-fermion interaction Hamiltonian: Hdw¼/C01 2X k,k0X s1,s2,s3,s4cy k,s1ckþQ,s2eVcy k0þQ,s3ck0,s4, (2) with spin indices si¼"#. The interaction potential eVcan be further separated into spin singlet and spin triplet parts,47 eV¼Vw k,k0þQ,kþQ,k0^σ0s1,s2^σ0s3,s4þVm k,k0þQ,kþQ,k0~σs1,s2~σs3,s4, (3) with ~σ¼(^σ1,^σ2,^σ3) and ^σithe Pauli matrices. The effective inter- action potentials Vw(m)act in the charge (spin) density wave channel and thus can mediate particle –hole ordering, respectively. These potentials can arise from the interplay between various degrees of freedom in metals, like, e.g., the electron –phonon interaction (after phonons are integrated out) and the Coulomb interaction. Theirmicroscopic origin is not important for the phenomena that wepredict here and we, therefore, choose to keep the discussion as generic as possible by not adopting any specific microscopic mecha- nism. 31Within mean-field theory the interacting Hamiltonian of Eq.(2)can be decoupled in different CDW (W) and SDW (M) channels by introducing the generalized order parameters, Wk,kþQ,s1,s2¼X k0X s3,s4Vw k,k0þQ,kþQ,k0 /C2^σ0s1,s2^σ0s3,s4hcy k0þQ,s3ck0,s4i, (4) Mk,kþQ,s1,s2¼X k0X s3,s4Vm k,k0þQ,kþQ,k0 /C2~σs1,s3~σs2,s4hcy k0þQ,s3ck0,s4i, (5) with charge/spin modulation wavevector Q. We further focus here on conventional and isotropic CDW and SDW order parameters by assuming the interaction kernels in the above as momentum inde-pendent and choose the SDW polarization parallel to that of theapplied magnetic field. With these considerations, we are left withthe following two order parameters: W¼X k0,σVwhcy k0,σck0þQ,σi, (6) M¼X k0,σVmσhcy k0,σck0þQ,σi, (7) and the resulting mean-field Hamiltonian reads43 H¼X k,σξk/C0σH ðÞ cy kσckσ/C01 2WX k,σcy kσckþQσþH:c:/C16/C17 /C01 2MX k,σσcy kσckþQσþH:c:/C16/C17 þW2 VwþM2 Vm: (8) The second and third term in the above are the mean-field Hamiltonians of CDW and SDW, respectively, in analogy to the BCStheory and the last two terms that arise from the decoupling process can be understood as the energy barrier that the system has to over- come in order for condensation to be energetically favorable [see,e.g., Eq. (13) below]. With the above considerations, Eq. (8)can be compactly rewritten with the use of the following spinor: ζy k,σ¼1ffiffiffi 2p(cy kσ,cy kþQσ), (9) and the 2 /C22 basis of ^ρiPauli matrices as H¼X k,σζy k,σ(γk^ρ3þδk/C0W^ρ1/C0σM^ρ1/C0σH)ζk,σ, (10) where the last two terms in Eq. (8)are omitted for now. In the above, we have decomposed the electron energy disper- sion of Eq. (8)into two terms, ξk¼γkþδk, with the functions γk andδkgiven by the relations γk¼ξk/C0ξkþQ 2,δk¼ξkþξkþQ 2: (11) Recalling the nesting condition ξk¼/C0ξkþQ,γkcan be understood as the nested part of the band structure, whereas δkas a term mea- suring deviations from perfect nesting. For the case of a commen-surate wavevector, i.e., Q¼Q 0with 2 Q0a reciprocal wavevector, γkis antisymmetric, and δkis symmetric with respect to Q0-translations. In this special case, γkis a particle –hole symmetric term and δkmeasures particle –hole asymmetry in the system.43,48,49 The Hamiltonian in Eq. (10) is quadratic and can be diagonal- ized by means of a fermionic Bogoliubov transformation that yields the four quasiparticle energy dispersions, Eσ+(k)¼δk/C0σH+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi γ2 kþ(WþσM)2q : (12) The free energy of the system can be found from F¼/C0TlnZ whereZis the fermionic partition function. We find F¼W2 VwþM2 Vm/C0T 2X k,σX +ln 1 þe/C0Eσ+(k) T/C16/C17 : (13) By minimizing the above free energy with respect to our order parameters, i.e., taking ϑF=ϑW¼0 and ϑF=ϑM¼0, we arrive at the following set of coupled self-consistent equations: W¼VwX k,σWþσM 2Eσþ(k)/C0Eσ/C0(k) ½/C138/C2nFEσ/C0(k) ½/C138 /C0 nFEσþ(k) ½/C138 ðÞ , (14)Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-3 Published under license by AIP Publishing.M¼VmX k,σMþσW 2Eσþ(k)/C0Eσ/C0(k) ½/C138/C2nFEσ/C0(k) ½/C138 /C0 nFEσþ(k) ½/C138 ðÞ : (15) The above equations have the interesting feature that on their right-hand-side, there exist terms that are not proportional to theorder parameter of the left-hand-side. They, thus, differ from the typical BCS equations that one would have obtained if the CDW/ SDW orders were not studied on the same footing, i.e., taken sepa-rately. Setting M¼0 on the right-hand-side of Eq. (15), one can observe that for W=0, the SDW order parameter on the left-hand-side can be nonzero if additionally δ k=0 and H=0. The same holds for the CDW case if we set W¼0 on the right-hand-side of Eq. (14) and assume M=0, instead. Therefore, we see that CDW or SDW ordering can be induced in a systemwhere one of them exists in the presence of finite δandH. In this sense, these four terms form a pattern of coexisting conden- sates 43,48,49and this property will be pivotal in understanding our numerical results presented below. As a crosscheck, one can showthat for δ k¼0, Eqs. (14) and (15) coincide with those obtained previously from a Green function approach.46 Equations (14) and(15) can be solved iteratively to determine the corresponding values of W,M. This was done previously in the case of Q¼Q0and δk¼0.46Here, the theory of coexisting CDWs/SDWs is extended to include incommensurability effects byallowing Qto be determined by minimizing either the free energy of Eq. (13) or the free energy difference between the condensed and the normal state, δF¼W2 VwþM2 Vm/C0T 2X k,σX +ln1þe/C0Eσ+(k)=T 1þe/C0ϵσ+(k)=T, (16) with ϵσ+(k) the normal state energy dispersions corresponding to setting W¼M¼0 in Eq. (12). In practice, we will use Eq. (16) since this allows to avoid cases where the condensed state is a local free energy minimum and the global minimum is achieved in thenormal state. For completeness, the induced magnetization of the system can be found by the relation M¼/C0ϑF=ϑH, which yields M¼μB 2X k,σσnF(Eσ,/C0(k))þnF(Eσ,þ(k)) ½/C138 : (17) The corresponding induced ferromagnetic (FM) splitting, which is measured in units of energy, is found from eH¼MHas eH¼H 2X k,σσnF(Eσ,/C0(k))þnF(Eσ,þ(k)) ½/C138 : (18) Equations (17) and (18) admit as input the self-consistently obtained W,M, and Qvalues. For W¼M¼0 and sufficiently large magnetic fields so that the medium is fully polarized, Eq. (18) yields eH¼h, whereas it gives eH,Hin all other situations, as it should.It is worth pointing out that in the special case where the ordering wavevector Q¼Q0is commensurate, it is possible to work in the folded Brillouin Zone (BZ). However, since here noprior assumption regarding the commensurability of the CDW/SDW orders is made, all k-sums are taken in the full (unfolded) BZ, instead. B. Qualitative discussion of the FFLO CDW state Before proceeding with the numerical solutions to our model, we will first provide a heuristic discussion on the mechanism ofFFLO CDW formation by examining how the density wave wave-vector Qcan be affected due to changes in the topology of the underlying Fermi surface. For a weakly coupled density wave system where the momentum dependence of the effective interac- tions is not essential to the resulting electron –hole pairing, Eq. (6) implies that the CDW order is maximized when Qis such that it satisfies the general nesting condition, ξ k,σ¼/C0ξkþQ,σ, (19) for as many k-points in the Brillouin zone as possible. In the above ξk,σ¼ξk/C0σHas in Eq. (1). Without loss of generality, we can make further progress by writing Qas Q¼Q0þq, (20) where Q0is a commensurate wavevector as discussed previously and qmeasures possible deviations from incommensurability. Next, we substitute Eq. (20) into Eq. (19) and Taylor expand both sides of Eq. (19) around q¼0. Keeping only O(q) terms, we have ξkþξkþQ0/C02σH¼/C0q/C1(∇kξkþQ0): From Eq. (11) and the related discussion and assuming for simplic- ity that the Q0-symmetric term is independent of k, i.e., δk¼/C0μ with μthe chemical potential, we arrive at the relation q¼2(μþσH) υFx¼(μþσH)α, (21) where α¼2 υFx,υFis the Fermi velocity and x¼cosθwith θthe angle between qand the Fermi wavevector kF. The above result provides a qualitative estimate of qin the limiting case of a constant DOS at the Fermi level or in the case of a one-dimensionalsystem. 19For two-dimensional systems like the ones we are inter- ested in here, ξkcan generally be quite anisotropic in momentum space. As a result, ∇kξk=υFand the optimal choice of α depends on the BZ direction with the highest DOS near the Fermilevel and should be obtained numerically by employing the theorypresented in Sec. II A. The simplifications used in this section are useful to reach to a qualitative understanding, and more realistic situations are discussed in Sec. III, where numerical results are presented. Equation (21) is the CDW analog to the celebrated FFLO result for the case of superconductors. 5,6Similar to the supercon- ducting case, it states that it is possible for the particle –hole pairsJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-4 Published under license by AIP Publishing.of a CDW state to acquire an extra q-modulation, which is linearly proportional to the external magnetic field. In other words, it is possible for a CDW to become incommensurate in order to surviveat high enough magnetic fields. The basic difference with thesuperconducting case is that jqjhere is a function of spin, thus allowing for a possible phase between the charge density of each spin species, and concomitantly the induction of a spin density wave. 19One can see this effect clearer if we write down the equa- tions for the modulation of charge and spin in real space, ρc(r)/X σcos ( Q0þ(μþσH)α)r ½/C138 ¼2 cos ( Q0þαμ)r ½/C138 cos ( αH)r ½/C138 ,(22) ρs(r)/X σσcos [( Q0þ(μþσH)α)r] ¼2 cos ( Q0þαμ)rþπ 2hi cos ( αH)r/C0π 2hi : (23) From the above equations, one immediately observes that for H=0, the modulated part of the spin density, ρs, is nonzero, sig- naling the induction of a SDW. As seen from Eqs. (21) –(23) terms that destroy the perfect nesting condition of Eq. (19), like, e.g., a chemical potential, may also lead to incommensurate CDWs, as isgenerally expected. The relations in Eqs. (22) and (23) constitute the generalization of the so-called double cosine phase that hasbeen discussed in 1D systems. 50 We note here that the situation discussed in this section con- cerns a system where only CDW ordering is assumed in contrast tothe more complete theory that we developed in Sec. II A, where both CDW and SDW orders are included on equal footing. In thisrespect, the mechanism of SDW induction due to the incommensu- rate FFLO CDW that is implied by Eq. (23) is different from the field-induced coexistence of CDW + SDW states that we discussedin relation to Eqs. (14) and (15). For example, this difference can be observed from the fact that Eq. (23) gives an induced SDW even when μ¼0, whereas Eqs. (14) and(15) indicate that μ=0 is nec- essary for this to happen. As we will show below, our direct numer- ical solutions verify the latter physical picture. C. Details for the exact numerical solution to the model In this section, we describe the procedure for the numerical solution to the model introduced in Sec. II A. Given a specific electron energy dispersion, a magnetic field strength and interac- tion potentials, Eq. (8)contains three unknowns: the density wave gaps W,Mand the ordering wavevector Q. Our method for obtaining an exact solution to Eq. (8)consists of simultaneously minimizing the free energy difference given by Eq. (16) with respect to the gaps W,M and the optimal wavevector Q.I n s t e a d of working with Q, we decompose it as in Eq. (20) and, noting thatQ 0is fixed by the choice of the underlying band structure, we are left with qas the unknown wavevector instead. For our calcula- tions, we assume an electron energy dispersion given by a square lattice tight-binding (TB) model with nearest neighbor hoppingenergy, t, and chemical potential μ, ξk¼/C0tcoskxþcosky/C0/C1 /C0μ: (24) For this dispersion, Q0¼(π,π)=a,w i t h athe lattice constant (here a¼1). Inclusion of longer-range hopping, i.e., to next- nearest neighbors etc., is allowed by our theory. Such terms would contribute to δksince they generally lead to imperfect nesting. Therefore, they would generally promote the coexistence ofCDW + SDW under applied magnetic fields if, of course, they arenot so strong so as to destroy the CDW ground state altogether.Here, δ k¼/C0μis chosen for simplicity as discussed below. In all our calculations, we set t¼1 and vary temperature, T, magnetic field strength, H, and μfor a given choice of Vw(m). All quantities are measured in units of t. Numerical solutions are achieved by employing a parallelized numerical code that iteratively solves the set of coupled self-consistent equations (14) and(15) on a6 4/C264k-grid in the full BZ for different values of the wavevec- torqthat are taken from a 32 /C232q-grid in the irreducible wedge of the BZ. In this way, for each set of parameters ( T,H,μ), we cal- culate W,Mand the corresponding free energy at every q. The physical solution that is kept is the one that minimizes δF. Given the numerical complexity of the involved calculations, wechose to restrict our tight-binding model only to nearest neighborsso that the optimal q(when it is found to be nonzero) always forms a 45 0angle with Q0, i.e., it always points at the Van Hove points where the DOS is maximal. Technically, this allows us to focus only on determining the amplitude, q, of the wavevector q. The final density wave solution that is obtained is a superposition of harmonics with Q¼(+Q0+q,+Q0) and Q¼(+Q0,+Q0+q). III. RESULTS AND DISCUSSION We have repeated the computational procedure of Sec. II C for several sets of ( Vw,Vm) values and found that depending on the relative strength of these potentials, the ground state solution can either be a CDW or a SDW state, as expected. Focusing on caseswith a commensurate CDW as ground state solution, there exists awide range of values for the ratio V w=Vm, where the phase transi- tion phenomena that we report in this section can be triggered by varying ( T,H,μ). As a general trend, the system becomes more susceptible to such phenomena as the ratio Vw=Vmapproaches unity due to the interplay between CDW and SDW orders becom-ing more pronounced. As a representative example, here we reportresults for ( V w¼1:5,Vm¼1:2). This choice corresponds to a system with a CDW ground state but enhanced effective interac- tions in the SDW channel. For example, such a situation couldarise in a material where the electron –phonon interaction is sufficient to drive the system to a CDW instability but Coulombinteractions are nevertheless pronounced, as is typical for two- dimensional systems. Figure 1(a) shows the calculated H–Tphase diagram of a CDW insulator for μ¼0. For sufficiently low fields and high tem- perature or high enough temperature and low fields, the CDW order gives way to the normal state through a second order phase transition (marked by solid lines). At low temperatures, theJournal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-5 Published under license by AIP Publishing.transition from this commensurate CDW to the normal state becomes first order (marked by dashed lines). Had we not includedincommensurability effects in our theory, this would have been atypical phase diagram of a Pauli limited spin singlet condensate similar to, e.g., that of an s-wave superconductor. However, in our case, we find that at very low temperatures, a first order transitionfrom commensurate to incommensurate CDW takes place. Thelatter phase is continuously suppressed for higher fields and disap- pears through a second order phase transition in accordance with what is expected for a FFLO CDW phase. 5,19To gain more insight on this phase, we show in Fig. 1(b) the calculated zero-temperature, magnetic field dependence of all order parameters in our theory. In the same figure, the amplitude of thecalculated incommensuration wavevector normalized by the respec-tive value of the commensurate wavevector, q=Q 0(black star symbols) is shown. The left axis in this figure measures the order parameter value (normalized by W0) and the right axis the q=Q0 ratio. The values of the x axis are normalized as H=W0, where W0 is the CDW gap value for T¼0 and H¼0. We find that there exists a range of magnetic field values where the CDW (green line)becomes incommensurate with Q¼Q 0þqandq/H. The transi- tion to this phase is first order and occurs at HP/C250:65W0, which is quite close to the expected Pauli limit for systems with isotropicFermi surfaces, H iso P/C250:707W0;3,4thus, the deviation of our result from the latter value is attributed to the anisotropic underlyingband structure. The high-field CDW phase survives for field strengths above the Pauli limit H c¼0:86W0.HPbefore it is destroyed via the second order phase transition. For H,HP, the CDW is commensurate and has a full gap over the Fermi surface.This can be deduced by the fact that the induced ferromagneticsplitting (red line) is zero in this region. In the CDW qphase, the modulated CDW order allows for a coexistence of gapped CDW and normal state regions. The latter are polarized by the fieldleading to a finite ferromagnetic splitting. The above features are inperfect agreement with the predictions from the FFLO theory for s-wave superconductors; 2,5thus, in this low temperature –high field region, the solution is a typical FFLO CDW phase as discussedqualitatively in Sec. II B. However, in contrast to Eq. (23), which predicts the emergence of an accompanying SDW in this phase, forthe case of μ¼0, we find that the SDW order is absent (blue line). This is despite the fact that interactions in the SDW channel are included in the theory. The phase diagram of Fig. 1(a) is significantly changed when μ=0. Here, we focus on the case where μ¼0:1. This value is small enough so as not to influence the ground state CDW solu- tion, i.e., W 0(μ¼0:1)¼W0(μ¼0) and concomitantly the expected value of HPis the same as for μ¼0. Results for μ¼0:1 are presented in Figs. 2(a) and 2(b), where the notations follow those of Figs. 1(a) and1(b). As shown in Fig. 2(a) , new transitions appear, which notably involve a finite SDW order. As discussed in Sec.II A, the presence of the SDW order can be understood by the fact that the order parameters W,Malways coexist when μ=0 and H=0, thus forming a pattern of coexisting CDW + SDW condensates.43,48,49A consequence of this mechanism is that at suf- ficiently high temperatures where thermal quasiparticle excitations above the CDW gap become possible, the presence of the appliedmagnetic field results in the induction of a weak SDW order, whichis labeled as CDW + wSDW phase in Fig. 1(a) . This phase appears as a smooth crossover, which is indicated by the dashed-dotted line inFig. 1(a) . Much more interesting is the low temperature –high field part of the phase diagram that exhibits cascades of newmagnetic-field-induced transitions. In this case, as the fieldincreases, the system undergoes a first order phase transition from commensurate CDW order to a phase where commensurate CDW and SDW coexist. In the CDW + SDW phase, Wand Mhave similar magnitudes as can be seen from Fig. 2(b) . For zero T, this FIG. 1. Results for μ¼0. (a) Calculated H–Tphase diagram. Second and first order phase transitions are marked with solid and dashed lines, respectively. Both axes are normalized by W0, which is the value of the CDW gap for T¼H¼0. The FFLO CDW phase is indicated with CDW qsince there qis finite as shown below. (b) Left axis: calculated magnetic field dependence of theorder parameters W(green), eH(red), and M(blue), normalized by W 0,a t T¼0. Right axis: calculated magnetic field dependence of the amplitude of the incommensurability wavevector qnormalized by the amplitude of the commen- surate wavevector, Q0, shown with black symbols. In both (a) and (b), the Pauli and the upper critical field are indicated by HPandHc, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-6 Published under license by AIP Publishing.transition takes place at Hc1/C250:53W0, a value that is almost 20% lower than the expected Pauli limiting field. Notably, the reduced Hc1makes this phase more easily accessible by experiment as com- pared to the FFLO CDW case. This coexisting CDW + SDW phaseis also a superposition of gapped and normal state regions, similar to the FFLO CDW phase as can be inferred by the build up of finite ferromagnetic splitting [see Fig. 2(b) ]. These gapless Fermisurface portions could manifest in experiments as resistance drops when the external magnetic field sweeps across the CDW + SDW phase transition, similar to the transport anomalies observed, e.g.,in TaS 2.51However, since a large fraction of the carriers is frozen into the CDW and SDW condensates, the anticipated resistancedrop will be less than what one would observe in the normal state of the metal. The CDW + SDW phase survives for fields up to H c2/C250:79W0, thus already surpassing the Pauli limit by 18%. These characteristics resemble those of the FFLO CDW phase.However, our field-induced CDW + SDW phase is markedly differ-ent; it exhibits no q-modulation; therefore, it is commensurate , and the transition out of this phase which takes place at H c2is also first order. In fact, from Fig. 2(b) , one can see that this phase appears to be bounded by two metamagnetic transitions at H c1and H c2. Remarkably, Fig. 2(a) shows that depending on the tempera- ture, the transition at Hc2may either be to the normal state or to yet another ordered phase. At low Tand for H.Hc2, the system enters into another coexistence phase where both CDW and SDWare modulated with an additional wavevector q. By acquiring this incommensurate modulation, the CDW state survives for upto even higher magnetic fields and eventually disappears at H c/C250:89W0through a second order transition [e.g., see Figs. 2(a) and2(b)]. This FFLO phase differs from the one found for μ¼0 where there is no SDW order. Moreover, as can be seen inFig. 2(b) , there is no clear linear correlation between qandH.I n the commensurate coexistence phase, the energy gaps W,Mare almost half of the W 0value. In contrast, in the modulated coexis- tence phase, the gaps are an order of magnitude smaller than W0. Therefore, this latter phase could be particularly difficult to observeexperimentally. This phase is also expected to be fragile against the presence of impurities and, therefore, very clean samples would be required for its experimental detection. It is also worth noting thatforV m!Vw, the pure FFLO phase becomes less energetically favorable as compared to the commensurate coexistence phase,even when μ!0. 46 The results of Fig. 2 show that in CDW systems with imper- fectly nested Fermi surfaces, the commensurate CDW + SDW phaseis energetically stable over a much wider range of temperatures andmagnetic field strengths as compared to the q-modulated CDW + SDW phase. In addition, by entering into the former phase, a CDW can exceed the Pauli limit without forming the more fragile FFLO phase. This means that in CDW insulators that survivebeyond the Pauli limit, the high-field phase can be a mixture ofCDW + SDW orders without any additional modulation of the density wave wavevector as compared to that of the ground state. Our findings also indicate that the mechanism for the coexis- tence of CDW and SDW inside the high-field phase of CDWsystems is not simply due to incommensurability effects that aredriven by q=0 as Eq. (23) implies and as was suggested in the case of 1D systems. 19Instead, the driving mechanism for the induction of such a phase is more general and is due to deviationsfrom perfect nesting in the underlying band structure that enforcesthe coexistence of CDW and SDW orders in the presence of a mag-netic field, as discussed in Sec. II A and previously. 43These devia- tions, here expressed as μ=0, are always present when the term δkof Eq. (11) is nonzero, a situation that is actually the most common in real 2D systems. FIG. 2. Results for μ¼0:1. (a) Calculated H–Tphase diagram. Second and first order phase transitions are marked with solid and dashed lines, respec- tively. Both axes are normalized by W0, which is the value of the CDW gap for T¼H¼0. The dotted-dashed line marks a crossover region where a very weak SDW order parameter (wSDW) is finite due to thermal excitations abovethe CDW gap. CDW + SDW indicates the coexistence phase of commensurate charge and spin density waves, CDW qþSDW qindicates the coexistence of FFLO CDW and SDW phases. (b) Left axis: calculated magnetic field depen-dence of the order parameters W(green), eH(red), and M(blue), normalized by W 0,a tT¼0. Right axis: calculated magnetic field dependence of the ampli- tude of the incommensurability wavevector qnormalized by the amplitude of the commensurate wavevector, Q0, shown with black symbols. In both (a) and (b), the critical fields of the transition into the commensurate CDW + SDW phase,the FFLO CDW + SDW phase, and the upper critical field are indicated by H c1, Hc2, andHc, respectively.Journal of Applied PhysicsARTICLE scitation.org/journal/jap J. Appl. Phys. 128, 083901 (2020); doi: 10.1063/5.0015993 128, 083901-7 Published under license by AIP Publishing.It is easy to observe that the only momentum dependent quan- tities entering in Eqs. (14) and(15) are the poles of Eq. (12).T h e form of these poles does not depend explicitly on the choice of aspecific underlying TB model but it is dictated by the properties ofour assumed density wave order parameters. In this respect, andsimilarly to the single- Qcase that we assume here, multi- QCDW and SDW orders can coexist under the application of Zeeman fields when the corresponding δ kis nonzero, as well. Additionally, the same mechanism can lead to the coexistence of CDWs and SDWs inmulti-band systems. Therefore, the magnetic-field-induced phenom-ena that we report here are not specific to the square lattice TB model, which is chosen here for simplicity. Material specific applica- tions of our theory are out of the scope of the present work and areleft for future investigations. The above described mechanism relies on the dominance of the paramagnetic (Zeeman) effect over the orbital effect; therefore, it is most relevant in two-dimensional materials where these condi- tions can be satisfied by applying the magnetic field in-plane.Moreover, in two-dimensional systems Coulomb interactions aregenerally enhanced due to the reduced dimensionality; 52thus, the tendency for SDW ordering is also enhanced. Our choice of com- parable values for the effective potentials Vw(m)is in line with this general picture. In fact, several 2D TMDs have been shown toexhibit such competing interactions 53,54and it has also been pro- posed that magnetism and CDW order are closely related.45,55,56 Among them, the strongly correlated CDW systems 2H /C0NbSe 256 and 1T /C0TaSe 257appear as plausible platforms for the experimen- tal observation of our predicted high-field phases. An experimentalplatform for the realization of our predicted phases that complieswith our here used TB model are the Rare-Earth (RE) tellurides, which are well-known single-Q CDW systems 58and in fact some members exhibit antiferromagnetism as well.59,60Progress in exfoli- ating RE-tellurides to the ultrathin limit has been achieved justrecently. 61,62 IV. CONCLUSIONS In conclusion, we have presented a generalized mean-field theory that allows to study the effect of applied magnetic fields inPauli limited two-dimensional CDW systems while fully includingincommensurability, imperfect nesting and temperature effects, and the possibility for a competing/coexisting SDW order. Our numeri- cal solutions showed that the magnetic field –temperature phase diagram of such systems can contain several different phasesdepending on the nesting properties of the underlying Fermisurface and the interplay between CDW and SDW ordering and revealed two different mechanisms that could allow the CDW to survive beyond the Pauli paramagnetic limit. For systems withperfect nesting, this can happen through a transition to a FFLOCDW phase that is completely analogous to the superconductingcase. For imperfectly nested systems near the Pauli critical field, the FFLO CDW is unstable against a phase where commensurate CDW and SDW coexist. This phase can appear below the Pauli limit andcan, therefore, be observable for lower fields than H P. Interestingly, in this phase of coexisting CDW + SDW the system is not fully gapped so that a finite magnetization develops similar to the FFLO phase. 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