Hidden orbital polarization in diamond, silicon, germanium, gallium arsenide and layered materials

It was previously believed that the Bloch electronic states of non-magnetic materials with inversion symmetry cannot have finite spin polarizations. However, since the seminal work by Zhang et al. [Nat. Phys. 10, 387-393 (2014)] on local spin polarizations of Bloch states in non-magnetic, centrosymmetric materials, the scope of spintronics has been significantly broadened. Here, we show, using a framework that is universally applicable independent of whether hidden spin polarizations are small (e.g., diamond, Si, Ge, and GaAs) or large (e.g., MoS2 and WSe2), that the corresponding quantity arising from orbital - instead of spin - degrees of freedom, the hidden orbital polarization, is (i) much more abundant in nature since it exists even without spin-orbit coupling and (ii) more fundamental since the interband matrix elements of the site-dependent orbital angular momentum operator determines the hidden spin polarization. We predict that the hidden spin polarization of transition metal dichalcogenides is reduced significantly upon compression. We suggest experimental signatures of hidden orbital polarization from photoemission spectroscopies and demonstrate that the current-induced hidden orbital polarization may play a far more important role than its spin counterpart in antiferromagnetic information technology by calculating the current-driven antiferromagnetism in compressed silicon.


INTRODUCTION
Electronic states at a given Bloch wavevector in non-magnetic materials with inversion symmetry are degenerate. Until recently, it was believed that there is no spatial spin distribution if averaged over these two spin-degenerate states. However, it has been found that even in centrosymmetric, non-magnetic crystals, the degenerate Bloch states can have local spin polarization if atoms are not at an inversion center. 1 Reference 1 reported that the lack of the local inversion symmetry at atomic sites leads to hidden, or site-dependent, spin polarization, expanding the scope of spintronics significantly, even to bulk materials with global inversion symmetry.
On the other hand, the orbital contribution to the magnetic moment of solids can be sizable (e.g., References 2 and 3) and even larger than the spin contribution. 4 The orbital magnetization becomes more important than the spin magnetization in some physical phenomena, e.g., current-induced magnetization 5 and the gyrotropic magnetic effect, 6 if the spin-orbit coupling (SOC) is weak. Additionally, the important role of orbital polarization in Rashba-split bands [7][8][9] and quantum anomalous Hall phases 10 of systems without inversion symmetry has been explored.
In this paper, we report the finding that the hidden, or sublattice-dependent, orbital polarization of Bloch states of centrosymmetric materials can be large (on the order of ħ) even without SOC by using the simplest, best-known materials, such as diamond, Si, and Ge, as examples. We describe that, in any non-magnetic, centrosymmetric material, including the aforementioned zinc-blende materials and layered materials such as MoS2 and WSe2, in which the hidden spin polarization is quite large, the hidden spin polarization is completely determined by the interband matrix elements of the site-dependent orbital angular momentum operator.
This finding, together with the fact that in materials with weak SOC the hidden spin polarization is small or absent, suggests that the hidden orbital polarization is a more fundamental quantity.
We show that the sublattice-dependent spin-orbital texture of centrosymmetric crystals is qualitatively different from that of non-centrosymmetric crystals and that the hidden orbital polarization can play an important role in current-induced magnetization 5 of both centrosymmetric materials and non-centrosymmetric materials such as GaAs. We then discuss the experimental evidence from photoemission spectroscopies and the technological implications of our findings in antiferromagnetic information technology using current-induced hidden orbital polarizations, which, according to our calculations, could be much more important than their spin counterpart.

MATERIALS AND METHODS
We calculated the electronic structures of diamond, Si, Ge and GaAs by using a tightbinding model including atomic s and p orbitals ( 3 * model) 11 and the on-site spin-orbit coupling term Δ SOC = ( ⋅ + ̅ ̅ ⋅ )/ℏ 2 , in which A and ̅ denote the two sublattices in the zinc-blende structure (see Figure 1b)

RESULTS AND DISCUSSION
First, we discuss the orbital polarization of diamond, whose SOC is negligible. If SOC is neglected, the spin-up and -down states have the same energy and orbital wavefunction. Figure   1b shows the local orbital polarization 〈 〉 = 〈 | | 〉 for the orbital part of a Bloch state, | 〉, corresponding to P in Figure 1a. ( Interestingly, the hidden orbital polarization can be large even when the total orbital angular momentum is quenched. Note that the total orbital angular momentum is a ground-state property of a crystal, whereas the hidden orbital polarization is a property of quasi-particle excitations and is a function of the Bloch wavevector and the band index. Even in a material where d orbitals of a transition metal element experience a strong octahedral crystal field and the (total) orbital angular momentum is quenched, for example, when the t2g bands are empty / half occupied / fully occupied, a quasi-particle state (either an electron or a hole) from the t2g bands can still have a large hidden orbital polarization.
Next, we show that in non-magnetic, centrosymmetric materials, the hidden spin polarization is a physical quantity completely determined by the site-dependent orbital angular momentum. When SOC is absent, it is apparent that a hidden spin texture cannot exist in these materials; since the electron potential does not depend on the spin, all bands are spin-degenerate, and each Bloch state cannot have a spatially inhomogeneous spin distribution. Conversely, we showed that there can be a large hidden orbital polarization even when SOC is absent. When there is SOC, the spin-up and spin-down bands mix with each other, but they remain degenerate due to PT symmetry. We define the spin or orbital polarization of each band as the average of the expectation values of the two degenerate states. 1 Let | ⟩ = | 〉 ⊗ | 〉 be spin-degenerate eigenstates of the Hamiltonian without SOC, where | 〉 is the orbital part and | 〉 is the spin part. In our model, in which SOC is taken into account by Δ SOC = ( ⋅ + ̅ ⋅ )/ℏ 2 , we can express the local spin polarization 〈 〉 avg = −〈 ̅ 〉 avg in terms of the matrix element of the site-dependent orbital angular momentum operator using first order perturbation theory: (1) Here, is the projection operator onto sublattice β, σ is the Pauli spin matrix, and is the energy of the state | 〉 when SOC is absent. In the third equality of Equation 1, we have used , and (iv) | ⟩ is equal to | ⟩ up to a phase factor (recall that | ⟩ is the orbital part of the wavefunction). We can then calculate the hidden spin polarization from the site-dependent orbital angular momentum operator using Equation 1, one of our key results.
It is straightforward to extend Equation 1 and calculate higher-order terms in a regime where SOC is not small; even in this regime, the interband matrix elements of the orbital angular momentum operator determine the hidden spin polarization. Additionally, Equation 1 can be easily extended to materials with more than two atoms per unit cell or to cases involving d or higher-l orbitals. 15 Figure 2. The local spin-orbital texture ( = 0) at sublattice A of the Bloch states of diamond obtained by using C = 4 meV (the physical value for diamond) and C = 1 eV. Discussion 3). However, in some centrosymmetric materials, the hidden spin polarization can be nearly fully polarized; 1 even in this case, our claim that the orbital polarization determines the spin polarization is valid. It is noteworthy that the hidden spin polarization shown in Figure   2 is almost identically reproducible by Equation 1, and the lowest-order result in Equation 1 holds for a wide range of SOC strengths up to C = 1 eV.
Interestingly, the directions of spin and orbital polarizations are exactly opposite each other ( Figure 2). It is difficult to find a simple reason for this (anti-)alignment because Equation 1 expresses the hidden spin polarization in terms of the off-diagonal matrix elements of , rather than the diagonal ones. However, we can understand this behavior in some limited cases as explained in Supplementary Discussion 2. As mentioned previously, this result can also be obtained by using higher-order perturbation theory only with respect to SOC; in hindsight, expanding the above equation, we know that these higher-order terms should coincide term by term with In contrast with MoS2 or WSe2, in which the two subsystems comprising the unit cell are weakly coupled, the two sublattices of diamond or silicon are strongly coupled to each other; hence, the typical energy separation between energy bands is on the order of the nearestneighbor hopping integral (a few eVs) and is much larger than the SOC. Remarkably, we can now understand, from the same principles, why the hidden spin polarization in diamond, silicon or germanium is very small and why that in MoS2 or WSe2 is very large.   (Figures 1 and 2).
Comparing the upper spin-split band (Figures 5a-5b) and the lower spin-split band (Figures   5c-5d), we note that, except near the or axes, the orbital polarizations of the upper and lower bands are approximately the same because the SOC mixes only the spin-up and -down bands together; its magnitude is smaller than the energy distance from those bands to other adjacent bands.
The spin texture of GaAs in Figure 5  These features can be explained as follows. When the SOC is neglected, the spin-up anddown bands are degenerate and share the common orbital wavefunction | 〉 . Within degenerate perturbation theory, the effect of the SOC is described by diagonalizing Δ SOC = ( Ga Ga + As As ) ⋅ /ℏ 2 in the two-dimensional Hilbert space spanned by the spin-up and spin-down states. (We set Ga = 0.12 eV and As = 0.28 eV. 18 ) Therefore, if there is no other degeneracy, the direction of the spin polarization of one spin-split band is parallel to 〈 |[ Ga Ga + As As ]| 〉 (we will denote a unit vector aligned in this direction as ̂) and the spin polarization of the other spin-split band points in the opposite direction. We define |↑;̂〉 and |↓;̂〉 as the spinors whose spin quantization axes are parallel to and anti-parallel to ̂, respectively. Then, the wavefunctions of the spin-split bands are | 〉 ⊗ |↑;̂〉 and | 〉 ⊗ |↓;̂〉. Therefore, the spin is nearly fully polarized in each spin-split band, and the spin polarizations at Ga atoms and As atoms are parallel to each other.
We can further understand the direction of the spin polarization of each spin-split band.
Since 〈 Ga 〉 is anti-parallel to 〈 As 〉 and both the orbital polarization and the atomic SOC of As are larger than those of Ga, ̂ is parallel to 〈 As 〉 . Hence, the spin of the electronic states in the upper spin-split band, at both sublattices, aligns with 〈 As 〉 , and that in the lower spin-split band anti-aligns with 〈 As 〉 ( Figure 5). This behavior is different from the hidden spin polarization in centrosymmetric materials, in which the spin polarizations at the two sublattices are opposite to each other.
In addition, in GaAs or other non-centrosymmetric materials, if we decrease the strength of the SOC, the spin polarization of a spin-split band does not change appreciably because the eigenvectors of the full Hamiltonian are independent of the scaling of the spin-orbit interaction Hamiltonian in the small SOC limit. This behavior is different from the case of the hidden spin polarization in centrosymmetric materials, in which the magnitude scales linearly with the strength of the SOC in the same limit (Equation 1 and Figure 2).
Despite the fact that GaAs lacks inversion symmetry, its transport properties are effectively determined by the average of the spin-split bands depending on the level of impurity and temperature. For this reason, the j=3/2 Luttinger model 12 is commonly adopted in studying the transport properties of GaAs (e.g., see Reference 19). Although each spin-split band of GaAs is nearly fully spin polarized (Figures 5a-5d), when we average the spin polarization over the two spin-split bands, the spin polarization is very much reduced, but the orbital polarization is almost invariant upon averaging (Figures 5e and 5f). The averaged spin and orbital polarizations at As atoms (Figure 5f) are similar to the hidden spin and orbital polarizations at sublattice A in diamond (Figure 2b). In all cases, including diamond with a very large SOC of C = 1 eV Recently, spin-polarized photocurrents were measured from bulk WSe2, 20 a non-magnetic, centrosymmetric material. The results confirm the hidden spin polarization and the hidden orbital polarization, as the former is generated from the latter. Moreover, the hidden orbital polarization in materials with a small SOC can also be observed by measuring the spinintegrated photocurrents because it is not the spin polarization but the orbital polarization that determines the coupling between electrons and photons. Provided that the final state is well approximated by s-like states, the hidden orbital polarization also manifests itself in the circular dichroism of a non-magnetic, centrosymmetric material.
We now discuss the technological implications of our findings. When an electric current is applied to a centrosymmetric material, non-equilibrium, site-dependent orbital and spin magnetization can be generated. The current-induced magnetization is antiferromagnetic due to the nature of the hidden orbital and spin polarizations, and its direction depends on the direction of the current. 21 Antiferromagnetic spintronic devices, in which a current generates sublattice-dependent spin-orbit torques and changes the magnetic state of a material, have several advantages over conventional spintronic devices based on ferromagnetism. Since the total magnetic moment of an antiferromagnet is zero, antiferromagnetic devices are largely insensitive to the external environment and do not introduce magnetic crosstalk. Additionally, they operate much faster than ferromagnetic devices. 22 The concept of hidden orbital polarization established here should be considered in properly predicting the site-dependent magnetism because, as we have shown, the spin polarization of a Bloch state could be much smaller than the orbital polarization in many materials (e.g., see Figure 2 and Figures 5e and 5f). Moreover, even in materials with weak SOC, the hidden orbital polarization can be used in antiferromagnetic information storage and processing because of the exchange interactions between localized, hidden orbital moments. 25 To illustrate the idea that the current-induced hidden orbital polarization can play a more important role than the hidden spin polarization, we looked into the current-driven antiferromagnetism of silicon under a 2 % uniaxial compressive strain along the [001] direction, achievable in real experiments. 28,29 (Because silicon has many point group symmetries, an electric current in silicon does not generate site-dependent magnetization; however, a strain can result in current-induced magnetization by breaking some symmetries. 21,22 ) Although silicon may not be the best material for antiferromagnetic information technology applications, it is one of the simplest and most well-known materials, a good candidate for supporting our hypothesis.
The effect of strain is simulated within our tight-binding model using Harrison's universal scaling method. 30 Following Reference 5, we obtain the non-equilibrium occupation factor ′ of a Bloch state | ⟩ by considering the change from the equilibrium Fermi-Dirac occupation factor = FD ( k ) of each Bloch state with the energy eigenvalue k : where denotes the scattering lifetime of charge carriers, e the absolute value of the charge of an electron, and the applied electric field. The current-induced, site-dependent magnetization at the A sublattice, , is then given by where B is the Bohr magneton. As in Reference 5, we assumed that the spin g-factor of electrons is 2. Figure 6 shows the calculated contributions of the orbital and spin polarizations to the induced magnetization of strained, hole-doped silicon at sublattice A per unit strength of the electric field as a function of the doping concentration . The scattering lifetime at each np is extracted from the measured mobility data 31 by using the Drude model.
Clearly, the orbital contribution to the current-induced antiferromagnetism is much larger than the spin contribution. Additionally, the induced magnetization at each site of silicon can be larger than the total induced magnetization of Cr2O3, the most well-known magnetoelectric material, with a 0 / value of approximately 1 ps/m. 32 Again, we are not claiming that compressed silicon is the best material for antiferromagnetic information technology exploiting the hidden orbital polarization; larger current-induced antiferromagnetism is expected in lowersymmetry materials. However, our proof-of-concept calculations illustrate that it is a worthwhile research direction to search for materials with large hidden orbital polarizations useful in antiferromagnetic information technology, irrespective of the size of the spin-orbit coupling. This result shows that investigating the effect of the hidden orbital polarization on antiferromagnetic information storage and processing is an important and promising theoretical and experimental future research direction.

CONCLUSIONS
In conclusion, we have shown that even in centrosymmetric, non-magnetic materials, there can exist large site-dependent, hidden orbital polarizations. In centrosymmetric group IV materials such as diamond, Si, and Ge, the hidden spin polarization is very small, but the hidden orbital polarization is on the order of ħ. We have also found, using a general perturbative scheme that is applicable not only to diamond, Si, and Ge (with small hidden spin polarizations) but also to layered materials such as MoS2 and WSe2 with hidden spin polarizations close to the maximum value, that the hidden spin polarization is completely determined by the sitedependent orbital angular momentum in general centrosymmetric, non-magnetic materials. If the energy distance between nearby bands is comparable to or smaller than the atomic spin orbit coupling, the hidden spin polarization is large. In the case of zinc-blende materials, this energy difference (nearest-neighbor hopping) is a few eV, and in the case of transition-metal dichalcogenides, this energy difference (interlayer hopping) is a few tens of meV. In any case, however, first-order or higher-order perturbative theory with respect to the SOC connects the hidden spin polarization to site-dependent orbital angular momenta. By comparing the strength of the SOC and the interlayer hopping constant in MoS2 and WSe2, we have shown that the hidden spin polarization in transition metal dichalcogenides can be significantly reduced by applying a pressure. Our study also illustrates that site-dependent orbital polarizations play an important role in current-induced magnetization of both centrosymmetric materials and noncentrosymmetric materials such as GaAs. We have discussed the experimental signatures of the hidden orbital polarization in centrosymmetric materials in both spin-resolved and -integrated photoemission spectroscopies. We have also calculated the current-driven antiferromagnetism in compressed silicon and have shown that an appreciable amount of orbital (antiferro-)magnetization can be induced even when the spin counterpart is negligible, demonstrating the potentially important role of hidden orbital polarizations in antiferromagnetic information technology. Because there are more degrees of freedom in orbital polarization than in spin polarization, the hidden orbital polarization may lead to new discoveries in physics.

CONFLICT OF INTEREST
The authors declare no conflict of interest.