Generation and electric control of spin valleycoupled circular photogalvanic current in WSe 2 1. Electronic band structure of WSe 2 thin film and its spin texture with/without an external electric field SUPPLEMENTARY INFORMATION DOI: 10.1038/NNANO.2014.183 Hongtao Yuan, 1,2 Xinqiang Wang, 3,4 Biao Lian, 1 Haijun Zhang, 1 Xianfa Fang, 3,4 Bo Shen, 3,4 Gang Xu, 1 Yong Xu, 1 Shou-Cheng Zhang, 1,2 Harold Y. Hwang, 1,2* Yi Cui 1,2* To simulate the effect of an external electric field applied on the surface of WSe 2, ab initio calculations are carried out on a 3 unit-cell free-standing WSe 2 thin film with and without an electric field along the [001] direction. Supplementary Figs. 1a-c show the evolution of the band structures of the WSe 2 thin film before and after the application of an electric field. We can see that the conduction band minimum (CBM) is located around non-symmetric -points along the direction, which implies that the Fermi surface should appear first at -points and then -points when the higher energy bands are filled with electrons under an electric field. The spin textures of the lowest two conduction bands are shown in Supplementary Figs. 1d and S1e. The most important feature is that the spin texture is in-plane around the -point, and yet out-of-plane in the other region in the Brillouin zone. So in this system, whether there is isotropic Zeeman spin splitting or the vortical Rashba spin texture strongly depends on whether the region is near or in the Brillouin zone. This spin texture indicates the existence of valley-spin coupling, and also that the valley-dependent phenomena are spin dependent. Some important characteristic features of the bands are stressed here: 1) The spin degeneracy is broken under an electric field. 2) Compared with the in-plane spin polarisation around the zone centre (high symmetric -point) of the Brillouin zone, an out-of-plane Zeeman-type spin polarisation takes place at low symmetric -points (CBM sitting along - ) and -points (the corners of the Brillouin zone), in agreement with previous work 1 ; 3) Rashba-type in-plane spin polarisation at the Corresponding author: hyhwang@stanford.edu, yicui@stanford.edu. NATURE NANOTECHNOLOGY www.nature.com/naturenanotechnology 1
-point in conduction bands is extremely small and the lowest conduction band at the -point is much higher than the CBM (~1.5 ev), indicating the obtained photocurrent does not involve the Rashba effect at the -point; and 4) the energy splitting increases with increasing external electric field. Supplementary Figure 1. Evolution of band structure before and after applying an electric field. a-c, Band structure of a 3 unit-cell free-standing WSe 2 thin film with an external electric field E ex = 0 ev/å in (a), 0.03 ev/å in (b) and 0.05 ev/å in (c). d and e, Spin textures of the 1 st and 2 nd conduction subbands. Colour bar indicates the energy level of each position in k-space. The green arrow represents for the direction and magnitude of the spin. They split from one spin-degenerate band, and have the opposite spin texture. The spin texture around the centre of the Brillouin zone ( point) is in-plane while it is out-of-plane near and K points. 2
Supplementary Figure 2. Atomic orbital character of the states along to K (refer to Fig. 5a for notation) a-e, The band structure with the projection of all d orbitals marked by the red dots. f, The projected density of states. The projected band structure and density states of bulk WSe 2 are shown in Supplementary Fig. 2. The bands near the band gap most come from d orbitals of W. In particular, we note that the lowest conduction band consists of d xy, d x 2 y 2, d 2 2 orbitals only. These projections onto different orbitals are useful in constructing our theoretical models discussed below. 2. Optical setup for CPGE measurement As shown schematically in the experimental setup in Supplementary Fig. 3, a quartz λ/4 plate is used to convert the incoming linearly polarized light (near infrared laser radiation with a wavelength of 1064 nm) into circularly polarized light. A chopper is used to modulated the light with a fixed modulating frequency of 233 Hz, which frequency is chosen to avoid the background noise around 50 Hz in electrical measurements. Thus, the electrical signals with the same base frequency of the 3
incident circularly polarized light can be obtained from the samples, which can be directly extracted by using standard AC lock-in techniques. The obtained photocurrent is expressed by j Csin2 Lsin4 A, where j Csin 2 is the circular polarisation contributed current, y CPGE j Csin 4 is the linear polarisation contributed current, and A is the parameter which can be LPGE described by other components in the Fourier series expansion. The same setup has been used for successfully measuring the circular photogalvanic effect for various Rashba systems based on group III-V semiconductors. 2 All measurements were performed in the regime with linear response of the photocurrent as a function of laser power, which was observed to extend at least up to ~0.5 W. Based on the reported absorption coefficient of WSe 2 sample (10 5 cm -1 ) for 800 nm laser, 3 the corresponding penetrate depth is about 100 nm when the intensity of the radiation inside WSe 2 falls to 1/e of its original value. Considering the fact that the longer wavelength laser with a smaller absorption coefficient normally gives a longer penetrate depth, the penetrate depth of the 1064 nm laser will be larger than 100 nm. Although the laser can have the chance to interact with bulk carriers, there is no inversion symmetry breaking in the bulk (Supplementary Fig. 4) and thus no CPGE current can be generated from bulk. Therefore no contribution to the CPGE current originates from the bulk. Supplementary Figure 3. Schematic diagram of the CPGE measurement setup and the laser power dependence of the total photocurrent. 4
3. Transfer characteristics of WSe 2 EDLT for CPGE current As shown in the transfer characteristics (sheet conductance xx as a function of gate voltage V G ) in Supplementary Fig. 4a, ambipolar transport of WSe 2 EDLTs with liquid gating can be achieved at room temperature. One can clearly see a U-shape current response in the DS -V G plot, which suggests that electrons are accumulated at the interface when a positive V G is applied, while holes accumulate at a negative V G. The relative large OFF current around zero-bias originates from the residual n-type (electron) conduction in the bulk crystal. Reducing the thickness of WSe 2 flakes can reduce the OFF current and further increase the ON-OFF ratio of the devices. 4 However no influence (no improvement) on the generation of the CPGE current is observed since the inversion symmetry breaking only occurs at the surface of the flakes, with no bulk contribution. To avoid any potential electrochemical reaction, we limit the bias within ±2 V, which is well within the chemical potential window (± 3 V for DEME-TFSI). Supplementary Figure 4. (a). Transfer characteristic and ambipolarity of typical WSe 2 EDLTs. (b). Surface band bending and carrier distribution profile near WSe 2 surface with ionic gel gating. As shown in (b), due to the surface band bending and the potential drop near WSe 2 surface (blue curve), there is the inversion symmetry breaking under external perpendicular electric field near the surface while no inversion symmetry breaking in bulk. 5
Supplementary Figure 5. Bias dependent CPGE current in WSe 2 EDLTs with electron accumulation. Electric field modulation of photo current j y (green circles), CPGE current j CPGE (red dots) and LPGE current j CPGE (blue dots) in WSe 2 EDLTs at varied gate voltage V G from 0 V to 1.1 V. The solid thin green line is the fitting cure with the expression j C sin 2 L sin 4 A. y Note that the observed CPGE photocurrent originates from the WSe 2 surface in a 2D form (from the 2DEG inside the quantum well caused by the surface band bending), as shown in Supplementary Fig. 4b. The 2DEG quantum well only has the thickness less than 1 nm (depending on how large is the applied gate bias). As we have addressed in the main text, the generation of CPGE photocurrent requests the broken inversion symmetry. In bulk WSe 2, there is NO inversion symmetry breaking and therefore the bulk is impossible to generate any CPGE photocurrent. Only near the WSe 2 surface with the applied perpendicular external electric field from the transistor, where there is a surface band bending and resulting inversion symmetry breaking, we can generate the CPGE photocurrent and further control it with a transistor configuration. Importantly, the modification of SOC with an external perpendicular electric field provides a simply way to control the CPGE spin photocurrent. Supplementary Figs 5a-5l show the light polarisation dependent j y, obtained in a WSe 2 EDLT at 6
varied external bias V G from zero to 1.1 V. As clearly seen, j CPGE dramatically increases with V G from tens of pa to thousands of pa (Supplementary Fig. 3m and 3n), unambiguously indicating an electric modulation of the CPGE photocurrent. 4. Valley optical selection rules and generation of pure CPGE valley current in WSe 2 without SOC In the main text we show under the approximation that the six valleys have the same carrier densities, the total current is ( (, ) (, ) ), where is the transition amplitude at, and is proportional to the increment of the electron velocity non-zero only if, and positively correlated to the carrier density. We now prove the current is. We can calculate explicitly:,, (, )(, ) (, )(, ) (,, ) Therefore, in order to have a non-vanishing current, one must have, and. We exhibit in this section a simpler 3D analysis following the main text. The total current consists of the contributions of the transition from band onto several higher bands, the amplitude of which is denoted by. As is shown in the main text, through the first principle calculations and symmetry analysis, we find the initial state, 2 d xy d x 2 y 2 d 2, where d x,y, is short for the Bloch state 2 d, x,y, with. We now consider the transition to band, which has a wave function, 2 d xy d x 2 y 2 d 2. 7
Clearly, we have, 2 and, 2. This means that is invariant under symmetry operation. Also we have, 2 and, 2, 2, so is invariant under as well. However, is a vector, therefore, changes sign under, and, changes sign under. This means the only non-vanishing amplitude component is,x. This means we have,x x,x x, where is the argument of,x. By the condition for producing current we derived above, this channel produces no net current. Following similar analysis, we can prove that the only non-vanishing amplitudes from, 2 to, 2 and to, 2 are, 2 and,y respectively. Similarly, all these channels have no contributions to the net current. The CPGE is therefore absent without external electric field. When a perpendicular electric field x is applied, as we have shown in the main text, band and mix to form two new states, 2 x, 2 and, 2, 2 x, 2. For the transition from band to band, since band satisfies, 2, we still have,y. However, we now have,x,x, and,, 2, x. Further, under symmetry x, we have x, 2, x, 2 and x, 2. By applying a x symmetry operation, it is easily shown that,x,x is real while,, is purely imaginary. We then have,x, y. For the polarized light applied in our experiment,, [( x y ) ( x y ) ], and it is straightforward to show(,, ) y, where circular polarisation corresponds to angles. We then arrive at the final expression for the total current in this channel: 8
(,x, ) ( ( ) ( )) x,x, x,x, 2, where ( x y ) is the in-plane unit vector perpendicular to the incident light, and is a coefficient for band. Adding contributions of transition from band does not change the form of the net current. The final current can then be written as x, where the coefficient is given by,x, 2. As one can see, though the system is only three-fold rotational symmetric, the current produced is always perpendicular to the incident light, and the magnitude is independent of the direction of the incident light. Note that the newly arising non-vanishing amplitude,, x comes from the symmetry breaking of by the electric field, which is a purely orbital effect, and SOC is not required at all. We then expect an electric field controlled purely orbital galvanic effect to arise in materials with similar crystal structures but without significant SOC, like graphene, silicene, germanene or stanene. 5. Estimation of the magnitude of CPGE photocurrent In this section we give an estimation of the magnitude of the CPGE photocurrent from theory. In the last section we assumed a coefficient for the CPGE current produced by optical transitions from band to. It can be determined in the following way. According to the Fermi Golden rule, the optical transition probability from state to (with the photon momentum neglected) at a valley is 5 9
If the density of states of band at the valley is, the transition probability to band is simply Suppose the life time of an electron in the conduction band is, the electron density accumulated in band will be. At room temperature, a typical relaxation time is around. With these facts, we can estimate the photocurrent produced at the valley as 2, which gives The density of state can be estimated using the 3D carrier density as. In our experiment, a typical 2D carrier density is m, trapped in thickness l m near the surface, which gives a 3D carrier density l m. The photon energy in experiment is. As an order of magnitude estimation, we can take. The transition amplitude defined as m, where the momentum expectation can be estimated from uncertainty principle as l. We find the photocurrent l m x x For the light adopted in experiment, the intensity is around m mm m, and the current density produced is x m, where x represents the mix of the two bands and due to the inversion symmetry breaking, as we have shown above. The area of the current production is around mm m m, and therefore the total current is around x. A reasonable estimation of the mixing fraction is x, which leads to a current of order, in 10
agreement with the experiment. It is however hard to make the calculation exact, because both the mixing fraction relaxation time cannot be determined precisely from numerical calculations (ab initio). x and the 6. CPGE valley current for WSe 2 2D system without considering SOC In the experiment the surface band bending together with electron confinement occurs within 1-2 nm at WSe 2 surface in 2D form and the photocurrent is also generated from such a 2DEG system, so the above 3D bulk analysis no longer holds strictly. 6 Here we show that the above conclusions also validate for the thin WSe 2 system in 2D form (take the few layer thin films case as examples), and the intrinsic physics remains the same. Obviously, WSe 2 thin films break the symmetry because of the lack of translation symmetry in direction. Besides, if the number of layers is even, there is also no reflection symmetry. Therefore, reconsideration is needed for even and odd layers separately. Before giving a microscopic analysis, we give a phenomenological explanation. Odd-layer thin films still have the mirror symmetry. Though the even layer thin films do not have this symmetry, they have inversion symmetry. As we have shown, a phenomenological description of the CPGE is given by, where, takes space indices,. Under inversion,,, which means that. Similarly under mirror reflection, one also has. Therefore, is ensured by either mirror symmetry or inversion symmetry, namely, CPGE is always absent for thin films without external electric field. We analyze first the case of even-layer WSe 2 thin films without external electric field x. Though both the and symmetries are lost, their combination x is still a symmetry, which represents a rotation about the axis through the center (see Fig. 5c in the main text). Since x, all the electron states at (in the 2D Brillouin zone) can be labelled by index x. We consider a rather general transition from a subband to another subband at the valley. Under the rotation x the vector potential transforms as x x, y y, 11
. As a consequence, if and have the same x index, only,x can be nonzero, and the photocurrent produced vanishes as is shown above. If and have opposite x indices,,x while, and, can be nonzero. However, due to the symmetry x, the amplitudes, and, must have the same complex argument (namely, can be simultaneously real). This means, and the photocurrent still vanishes. There is therefore no CPGE current for even-layer films in the absence of the external electric field. Now we turn to the case of odd-layer thin films without the external electric field x. In this case the material lacks the symmetry, but the reflection symmetry about the central layer remains. The electron states at (in the 2D Brillouin zone) are therefore still characterized by the index. Suppose we again consider the transition from a subband to another at. If the two bands have opposite indices, only, can be non-vanishing, and according to our previous discussion the photocurrent produced is zero. If subbands and have the same index, one finds, and both,x and, can be nonzero. Further, the x symmetry shows that if one choose,x to be real,, is purely imaginary, which means,x,y. One may then expect a photocurrent to arise. However, the rotational symmetry about axis again protects the photocurrent from being nonzero. To see this, one first notice that the component of,, is given by, independent of the azimuth angle. According to the formula for the photocurrent, we find ( (, ) (, )),x,y, since the sum of vectors is zero ( symmetry). So similarly, in the absence of the external electric field, there is no CPGE current for odd-layer thin films. 12
When the external electric field x is present, the discussion can also be made for 2D films, and one can verify the form of the CPGE current is the same as that written down in the main text. 7. Valley optical selection rule considering the SOC and spin-coupled valley current in WSe 2 The spin-orbital coupling term take the form, where the angular momentum of d-orbitals is determined as ( ), x (, y ) ( ), under basis d x 2 y 2, d xy, d x, d y, d 2 2. Without the external electric field, the spin degeneracy is ensured by time reversal symmetry and the inversion symmetry x. This is because the combined operation does not change the momentum but reverses the spin. However, due to the introduction of SOC, electron states may no longer be the eigenstates of the symmetry, and the index in states like d xy are no longer exact. Through first principle calculations, we can obtain the orbital components and spin directions of each band (Supplementary Fig. 1 and Fig. 2). For the lowest conduction band, first principle calculations show two eigenstates at, 2,, 2 and, 2,, where and are the up and down eigenstates of, and they are related to each other through the symmetry. Provided the orbital components of the wave functions of the bands are those as shown in the main text, their spin will be along the direction. This is because for band consisting of orbitals d x 2 y 2, d xy, d 2 2, it is easily shown that the only non-vanishing orbital angular momentum is, which is only coupled to. The spin quantization axis is therefore the axis. Similar results apply to higher conduction bands. 13
When an out-of-plane external electric field x is applied, the inversion symmetry is lost, and the spin degeneracies will be lifted. As we have discussed in the main text, bands and will mix to form new states, 2 x, 2 and, 2 x, 2. Take state, 2 as an example, this induces an energy term, 2, 2 x y. The two spin states of band then split in energy, and obtain an opposite spin y component. We denote these two states as, 2, and, 2,. (The spin degeneracy of band is also slightly lifted due to a mixing with band, which we neglect here for simplicity.) The total current at includes the contributions of both, 2, and, 2,, namely 2 2, 2,. Due to the spin splitting, 2, is no longer equal to 2,. The spin current at can be defined as 2, 2, 2,. The spin direction of the current is seen to be perpendicular to the current. Since the splitting of band proportional to, we should have 2, 2 x. is The above conclusion also applies for the total current and the total spin current, namely x. The spin current has the same direction as that of the total current, and has an in-plane spin polarisation perpendicular to. Since SOC does not change the band structure much (from ab initio), we expect the spin current to be much smaller than the total current (probably by one order of magnitude). 8. A brief interpretation of the LPGE photocurrent in WSe 2 Our experiment have also detected a LPGE photocurrent in additional to the CPGE photocurrent. In general, the LPGE has a very different mechanism from that of the CPGE. 7 Different from CPGE, LPGE originates from the dissipative scatterings of electrons in the material, analogous to the origin of the longitudinal resistance. Due to its complexity, here we give a brief interpretation of the LPGE observed in our experiments. LPGE can be described by the phenomenological formula 8 14
( ), where,, can take,,. Under inversion changes sign, so LPGE can arise only in noncentrosymmetric materials. Thus, without the external electric field x breaking the inversion symmetry, the LPGE photocurrent cannot occur. Further, under the time-reversal symmetry, also changes sign, unlike the coefficients of CPGE which are time-reversal invariant. Analogous to resistivity that also breaks time-reversal symmetry, simply to say, LPGE is a dissipative phenomenon sensitive to the electron scatterings, disorders, and so on. This indicates LPGE and CPGE have very different mechanisms. Microscopically, LPGE arises due to the asymmetric electron distribution in the presence of scattering and relaxation. This can be very roughly interpreted through the Boltzmann equation: [ ], where is the electrical field of the light, is the relaxation time due to scattering, and represents the in-equilibrium electron distribution function in the absence of the incident light beam. If one expands, one finds, ( ), where is the group velocity of the electron state. When averaged with respect to time, and therefore, but and. This second order asymmetric electron distribution is then responsible for the LPGE current, 15
This integration involves integrations such as, which is non-zero only for noncentrosymmetric materials. We emphasize this interpretation is only qualitative and far from exact. For a more complete calculation, one has to include the electron distribution functions of both the valence bands and the conduction bands. Furthermore, the Boltzmann approximation may have already broken down under the fast oscillating electrical fields of the light, and one has to refer to the field theory methods. References 1. Yuan, H. T., Bahramy, M. S., Morimoto, K., Wu, S. F., Nomura, K., Yang, B. J., et al. Zeeman-Type Spin Splitting Controlled by an Electric Field. Nature Phys. 2013, 9(9): 563-569. 2. Yin, C. M., Yuan, H. T., Wang, X. Q., Liu, S. T., Zhang, S., Tang, N., et al. Tunable Surface Electron Spin Splitting with Electric Double-Layer Transistors Based on InN. Nano Lett. 2013, 13(5): 2024-2029. 3. Frindt, R. F. The optical properties of single crystals of WSe 2 and MoTe 2. J. of Phys. and Chem. of Solids, 1963, 24(9): 1107 1108. 4. Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V., Kis, A. Single-layer MoS 2 Transistors. Nature Nanotechnol. 2011, 6(3): 147-150. 5. Sakurai, J. J. Advanced Quantum Mechanics. Addison-Wesley, 1967. 6. Schuller, J. A., Karaveli, S., Schiros, T., He, K. L., Yang, S. Y., Kymissis, I., et al. Orientation of Luminescent Excitons in Layered Nanomaterials. Nature Nanotechnol. 2013, 8(4): 271-276. 7. Sturman, B. I., Fridkin V. M. The Photovoltaic and Photo-refractive Effects in Noncentrosymmetric Materials. Gordon and Breach Science Publishers, 1992. 8. Ganichev, S. D., Prettl, W. Spin Photocurrents in Quantum Wells. J. Phys. Condens. Mat. 2003, 15(20): R935-R983. 16