Pure Bulk Orbital and Spin Photocurrent in Two-Dimensional Ferroelectric Materials

We elucidate light-induced orbital and spin current through nonlinear response theory, which generalizes the well-known bulk photovoltaic effect in centrosymmetric broken materials from charge to the spin and orbital degrees of freedom. We use two-dimensional nonmagnetic ferroelectric materials (such as GeS and its analogues) to illustrate this bulk orbital/spin photovoltaic effect, through first-principles calculations. These materials possess a vertical mirror symmetry and time-reversal symmetry but lack of inversion symmetry. We reveal that in addition to the conventional photocurrent that propagates parallel to the mirror plane (under linearly polarized light), the symmetric forbidden current perpendicular to the mirror actually contains electron flows, which carry angular momentum information and move oppositely. One could observe a pure orbital moment current with zero electric charge current. This hidden photo-induced orbital current leads to a pure spin current via spin-orbit coupling interactions. Therefore, a four-terminal device can be designed to detect and measure photo-induced charge, orbital, and spin currents simultaneously. All these currents couple with electric polarization $P$, hence their amplitude and direction can be manipulated through ferroelectric phase transition. Our work provides a route to generalizing nanoscale devices from their photo-induced electronics to orbitronics and spintronics.

Introduction. Bulk photovoltaic (BPV) effect, 1 which converts incident alternating optical field into direct electric current in centrosymmetric broken materials, has attracted tremendous attention during the past few decades for its easy manipulation and low energy cost. Comparing with conventional light-to-current conversion in a p-n junction between two semiconductors, BPV effect produces electric current everywhere light shines onto the material, which could significantly enhance the conversion efficiency and density. From physics point of view, BPV effect is a second order nonlinear optical effect, which includes two photons (with frequency and − ; absorption and emission) and an electron (with moving velocity ). 2 The BPV effect uses electron charge degree of freedom (DOF) to generate a biased electric potential in semiconductors, [3][4][5][6] which are serving as promising electronic devices. In order to further increase the information read/write kinetics and storage density, one may resort to new DOFs of electron, such as its spin angular momentum.
The study of the intrinsic spin and its induced magnetic moment is thus referred to as "spintronics", [7][8][9] which has been shown to hold unprecedented potential in the future miniaturized devices, especially in the field of quantum computing and neuromorphic computing. Roughly speaking, when the velocities of electrons in the spin up and spin down channels are different ( ↑ − ↓ ≠ 0), there is a collection motion of electron spin and leads to a nonzero spin current.
In addition to spin, another DOF that could produce angular momentum and magnetic moment is the electron orbital which describes the electron travelling around one or a few nuclei. It is an overlooked DOF because in most conventional materials, the orbital moment is significantly or completely quenched under strong and symmetric crystal field. However, when the symmetry and strength of crystal field are reduced, especially in low-dimensional materials, orbital DOF may play an important role in their magnetic properties, topological behaviors, and valleytronic features. [10][11][12][13] Similar as spintronics, this novel field is thus termed as "orbitronics", 14,15 which is predicted to further enhance information read/write speed significantly. If the electron velocities carrying different orbital magnetic moments are different (e.g., − ≠ 0), then one could also expect an orbital current, analogous to spin current. Note that such an orbital current has been predicted in the linear response Hall effect picture, [16][17][18] in addition to spin Hall effect 19,20 and valley Hall effect. 11,[21][22][23] In the current work, we predict that in addition to nonlinear BPV effect, there exists a hidden orbital current which carries colossal orbital moment when light shines onto two-dimensional (2D) nonmagnetic ferroelectric materials. We refer to this effect as bulk orbital photovoltaic (BOPV) effect, which is described by a second order nonlinear optical process. We use 2D ferroelectric group-IV monochalcogenide monolayers (GeS, SnS, GeSe, SnSe, GeTe, and SnTe) 24 and pure BOPV current to be measured in the -direction. Here pure orbital current means no electric charge current is mixed. We also show that the spin-orbit coupling (SOC) interaction could convert such BOPV current into spin DOF, namely, bulk spin photovoltaic (BSPV) current (also along ). [33][34][35][36] When circularly polarized light (CPL) is used, the conventional charge BPV current will be along , while the BOPV and BSPV currents flow along .  We calculate the nonlinear photo-conductivity coefficients explicitly. According to the second order Kubo response theory and within the independent particle approximation framework, 5,40 one can compute the complex nonlinear photoconductivity from its band structure via

Results and
This is based on a three band model that includes band-, , and . The phenomenological carrier lifetime is taken to be 0. . The latter is odd under mirror operation. We thus prove that BOPV and BSPV currents only occur along the -direction under or -LPL, consistent with previous analysis. In the long relaxation time approximation, one could demonstrate that LPL irradiation yields the BPV shift current and the CPL illumination gives an injection current for time-reversal symmetric systems. However, for the BSPV and BOPV, we find that the LPL induced photocurrent is injection-like, which is proportional to the relaxation time . The CPL, on the other hand, induces shift-like current (see Supporting Information for detailed discussions). 42 Now we apply Eq. (2) to compute nonlinear photo-conductivity in 2D nonmagnetic ferroelectric materials. Taking monolayer GeS as an example (Figure 2a), we calculate its LPL induced BOPV conductivity. In practice, the BZ integration in Eq. (2) is where is the total volume of simulation supercell and is the weight of each k-point. In the 3D periodic boundary condition, the supercell of 2D materials contains artificial vacuum space along , whose contribution needs to be eliminated. According to previous works, we rescale this result by using an effective thickness of 2D materials (taken to be 0.6 nm), which is estimated by the layer-tolayer distance when these 2D materials are van der Waals stacked into bulk. Thus, we can rescale the photo-conductivity by = ℎ/ , where and ℎ are the supercell calculated conductivity and the supercell lattice constant along , respectively. 43,44 This makes the second order conductivity of 2D materials consistent with conventional quantities of 3D bulk materials. In the following, we will report the values. As shown in Figure 2b, one sees that consistent with our previous symmetry analysis, where is eigenvalue of band-at momentum , and and represent the conduction and valence bands, respectively. The integral is taking in the first BZ.
According to Sokhotski-Plemelj formula, jDOS represents the resonant band transition between band-and band-in Eq. (2). One could see that ( = 2.83 eV, ) is mainly contributed around the ± and ± points in the BZ. We next plot the real part of the integrand of Eq.  Figure 3b), which show that they locate similarly as in the BOPV conductivities, and the ℳ symmetry still retains. Note that these 2D ferroelectric monolayers possess four electron valleys in the first BZ (near ± and ± ). Hence, we show that the photocurrent is mainly contributed from these valleys, which may provide promising physical properties among orbitronics, spintronics, and valleytronics. According to solid state physics theory, orbital moment in a bulk material is usually strongly quenched by the symmetric crystal field, so that it is the spin polarization that mainly contributes to the total magnetic moment. Hence, the orbital moment contribution is omitted in most cases. However, here we find that BOPV conductivity is generally much larger than that of BSPV. According to Eq. (2), the dominate interband contribution is a two-band transition, namely, | 〉 = | 〉, and the | 〉 band lies on the other side of the Fermi level (hence ≠ 0). We will limit our discussion on this two-band model. Thus, the difference between BOPV and BSPV conductivity can be understood by comparing ⟨{ , }⟩ and ⟨{ , }〉 for the low energy bands (near Fermi level), which is determined by the velocity and orbital/spin texture. In order to illustrate it more clearly, we plot the k-space distribution of orbital and spin angular momentum (⟨ ⟩ and ⟨ ⟩) of the highest valence band (VB) and the second highest This is because that the orbital texture is determined by the crystal field once the material forms, and changes marginally under SOC. On the other hand, the Rashbatype spin splitting yields that ⟨ ⟩ flips its sign between the VB and VB−1 at each k.
We also plot the spin and orbital angular momentum distributions of the lowest two conduction bands in Supporting Information, and similar results can be seen. The velocity texture distributions on VB and VB−1 are also similar (but not identical) ( Figure 4c). We plot ⟨{ , }⟩ and ⟨{ , }⟩ of bands near the Fermi level in Supporting Information. From all these evidences, we show that the crystal field determined orbital responses are similar at the Rashba splitting bands, while their contributions to the spin responses are opposite (but not completely cancelled due to small velocity distribution difference). Therefore, the BSPV conductivity is usually much smaller than that of BOPV. to note that the band dispersion will be significantly changed under very strong SOC, so that such linearity may not hold when is very big.
In order to further understand the mechanism of BOPV and BSPV photoconductivities, we artificially tune the SOC interaction = ⋅ /ℏ strength by multiplying a pre-factor ∈ [0,1] . Here = 0 turns off the SOC, and = 1 indicates full SOC. We find that the BOPV (and BPV) conductivity marginally changes under different (see Supporting Information). However, the BSPV conductivity linearly reduces to zero from = 1 (full SOC) to = 0 (no SOC), as shown in Figure 4d. This clearly demonstrates that the BOPV effect is ubiquitous even without SOC since orbital texture originates from crystal field, while SOC is crucial for BSPV effect in these nonmagnetic systems. Therefore, we could conclude that the BOPV effect arises when crystal is formed, and then it leads to BSPV effect through a finite SOC interaction ( ∝ ⋅ ). Similar relation can also be seen in the orbital and spin Hall effects. 16 Note that very strong SOC may not necessarily imply further enhanced BSPV, as the band dispersion would be significantly affected.
For the ferroelectric materials, one could easily apply external (electrical, mechanical, and optical) fields to modulate its polarization (for example, from to − ). The transition barrier between different ferroic orders is usually a high symmetric geometry, which is centrosymmetric and not electrically polarized ( = 0). We now examine the BOPV and BSPV photo-conductivity under different electric polarizations.
In Figure 5 we plot the polarization dependent BOPV and BSPV photo-conductivities.
One clearly observes that all these conductivities diminishes at = 0 state. This is consistent with symmetry analysis, ; ( ) = − ; (− ), where is inversion symmetry operator (angular momenta and are invariant under ). We also note that when the polarization flips (corresponding to a 180°-rotation from to − ), the photo-conductivities reverse their flowing direction while keeping same magnitudes.
If a 90°-rotation occurs, these conductivities also rotate 90°, flowing along the ±direction. Hence, one could control the ferroic polarization order to manipulate the BOPV and BSPV photocurrents, as well as the BPV effect. We now calculate the BOPV and BSPV conductivities for other analogues, namely, monolayers GeSe, SnS, SnSe, GeTe, SnTe, and Bi. Note that even though the monolayer Bi is a single elemental material, Peierls instability occurs due to strong s and p orbital hybridization, which leads to charge transfer within each atomic layer.
Thus, the monolayer Bi also shows in-plane ferroelectricity and fascinating optical properties. All these BOPV and BSPV photo-conductivity results are shown in Figure   6. For the BOPV conductivity, we observe clear similarities for all these systems, because their electronic band structure can be described by the same low energy model. 30 By comparing the main peaks in Figure 6a and 2b, we find that (for the group IV-VI systems) when the system is composed by small cation and large anion, the BOPV photo-conductivity shows larger peak values (over 10,000 ℏ ). Hence, the monolayer GeTe shows largest photocurrent responses, while the orbital photoconductivity of monolayer SnS is smallest. However, the BSPV does not have such similarity as the SOC interaction strength (proportional to Z 4 ) determines its responses.   (SC)

Conclusion.
We predict robust and pure bulk photovoltaic currents in the carrier orbital and spin degrees of freedom. Using nonmagnetic 2D ferroelectric materials (GeS and its analogues) as exemplary materials, we show that the mirror symmetry forbidden BPV conductivity actually contains hidden electron motions, which carries orbital moment flow with zero net electric charge current. Under SOC interaction, the photo-induced orbital current could convert into spin current. Both of these currents are perpendicular to conventional BPV electric current, so that they can be purely and exclusively detected and used. When ferroic order switches, the photo-conductivities rotate their directions accordingly. We summarize such polarization and light dependent photovoltaic effects in Table 1. Our prediction of pure BOPV and BSPV effects can be easily detected and observed in experiments, and may provide potential ultrafast spintronic and orbitronic applications of 2D in-plane ferroelectric materials, in addition to their electronic features, especially when a four-terminal device is applied.

Methods.
We use first-principles density functional theory to calculate the geometric, electronic, and optical properties of 2D monolayer GeS and analogous systems, as implemented in the Vienna ab initio simulation package (VASP). 45 The conductivities, we fit the electronic structure by atomic orbital tight-binding model in atomic orbital basis set (s and p orbitals), as implemented in the Wannier90 package, 49,50 and the optical conductivities are integrated on a denser k-mesh of (901×901×1) grid.
The convergence of k-grid density is carefully tested. As for the estimate of orbital angular momentum contributed intra-atomically, we use | Corresponding authors: * J.Z.: jianzhou@xjtu.edu.cn