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Tunning Spin Hall conductivities in GeTe by Ferroelectric Polarization

Controlling charge-spin current conversion by electric fields is crucial in spintronic devices, which can be realized in diatom ferroelectric semiconductor GeTe where it is established that ferroelectricity can change the spin texture. We demonstrated that the spin Hall conductivity (SHC) can be further tuned by ferroelectricity based on the density functional theory calculations. The spin texture variation driven by the electric fields was elucidated from the symmetry point of view, highlighting the interlocked spin and orbital degrees of freedom. We observed that the origin of SHC can be attributed to the Rashba effect and the intrinsic spin-orbit coupling. The magnitude of one component of SHC {\sigma}_xy^z can reach as large as 100 {\hbar}/e/({\Omega}cm) in the vicinity of the band edge, which is promising for engineering spintronic devices. Our work on tunable spin transport properties via the ferroelectric polarization brings novel assets into the field of spintronics.


I. Introduction
Spintronics is a multidisciplinary subject where the coupling of the spin and orbital degrees of freedom for electrons plays a vital role. The most magnificent demonstration of such a coupling is the conversion of charge current into spin current, i.e., spin Hall effect (SHE). Such a phenomenon has be measured in many different material systems, such as heavy metals [1] , semiconductors [2] , and interfaces between oxides such as LAO/STO [3] . Manipulation of the charge-spin conversion is one of the keys to future implementation of spintronic devices [4][5] . The controllable spin-orbit coupling leading to a gateable electric signal in LAO/STO has been observed by several groups, as reviewed by Han et. al [6] . The underlying mechanism is the change of the electron density and hence the occupation of different orbitals [7] .
Very recently, another material family was proposed to show electrically controllable spinorbit coupling based on ferroelectricity, called ferroelectric Rashba semiconductors (FERSC).
These materials include GeTe [8][9] , SnTe [10] and MX 2 [11] etc. Among them, GeTe, a narrow gap semiconductor, has drawn intensive attention due to a spectrum of interesting properties such as thermoelectricity [12] , ferroelectricity [13][14][15] , and electrical phase change [16] , realized within such simple stoichiometry and the two-atom rhombohedral unit cell [17] . Both experiments and theoretical calculation show that the strength of Rashba coupling in GeTe can be tailored by the electric polarization. Meanwhile, the spin texture can also be fine-tuned by the electric polarization as revealed by theoretical calculations [9] based on the density functional theory (DFT) and spin resolved ARPES [18] : The clockwise winding directions in the reciprocal space of the spin vectors can be reversed to anticlockwise. There are several other ferroelectric compounds which show the reversible spin texture controlled by electric polarization, such as HfO 2 [19] , BiTeI [20] , and GeTe/InP superlattice [21] . The controllable spin texture may lead to tunable spin life time, so that the proposed spin transistor [4] may be realized, as in the recently demonstrated ferromagnetic-free all-electric spin Hall transistor [5] . Although it is well established that the spin texture can be tuned by the polarization by the ARPES and DFT calculations, it is still not clear how the spin-charge current conversion is influenced. Such a tunable conversion is the crucial to utilize the effect in real devices. For instance, the spin-charge conversion in GeTe was observed by Rinaldi et al. [22] for the first time very recently, but the tuning of the conversion efficiency has not been reported yet.
In this work, we elaborate by ab initio calculations, as well as by general arguments based on the effect of electric polarization on the spin texture, that the electric fields can be applied to modulate the spin Hall conductivity (SHC), which is originated from both the Rashba effect caused by polarization and the bulk effect. It is observed that the bulk effect is only marginally influenced, while the magnitude of the Rashba effect near the band edge change significantly under ferroelectric distortions with even a sign change. Thus, it leads to tunable SHC, which opens the door for spintronics.

II. Calculation Details
GeTe crystallizes into the noncentrosymmetric rhombohedral structure (space group R3m, No. 160) with rhombohedral lattice constant a = 4.373 A and the angle α between the axis 57.76°.
Ferroelectric polarization is realized by the relative displacement of Ge and Te along the [111]direction which is chosen as the z-axis in our calculations. The lattice parameters are fully relaxed when the polarization is changed. The electronic structure was calculated by the psuedopotential plane wave method (PWscf) [34] within the generalized gradient approximation (GGA) parameterized by Perdew et al. [35] . The energy cut-off was set to 30 Ry and the Brillouzin zone integration was performed on a uniform k-grid with 20×20×20 k-points. The electronic structure analyse was carried out by the graphic interface VNL , which has the following form ,

A. Electronic Structure and Ferroelectricity
The Kohn-Sham band structure calculated using the experimental lattice constants is shown in Fig. 1, where also a comparison of the results with and without SOC is made.
Without SOC, the ferroelectric distortion leads to the direct band gap of about 0.66 eV near the L-point as depicted by the dashed blue curves. The value is only 0.05 eV higher than the experimental value reported by Park et al. [23] . We note that even in the absence of SOC, there are humps around the L-point due to the finite electric polarization. SOC leads to splitting (∆SOC) of the bands about 0.7 eV along the Γ−L line, which indicates that the SOC field is more or less collinear in this direction. However, the Rashba like splitting are only around the L-point, and mostly obvious at the VBM at L as shown by the solid red lines. This direction is in the direction of polarization in our coordinate system. The contributions of the SOC are mainly observed in the valence bands due to the fact that they are mostly originated from the Te 5p-orbitals, while the conduction bands are from the Ge 2p-orbitals. This is in contrast to a simple interpretation of resonant bonds, because the 5p-states of Te is much higher in energy than the 2p-states of Ge. The so-called "cross gap hybridization" was used to depict this phenomenon in the highly ionic compound [24][25] . This anomaly is related to the enhanced Born effective charge which in turn improves the bonding with distortion.  Fig.2(a) and (c). This contributions are reversed when the polarization is reversed from +P to −P. As a result, the winding directions of the spin vectors are reversed in the k-space as reported in previous works [9] due to the SOC. We

B. Spontaneous Polarization and Rashba Coupling
The ferroelectricity in GeTe was well explored and understood more than fifty years ago [26][27] . The soft optical phonon is the cause of the displacive movement of the diatomic compound. In GeTe, ferroelectric displacement of Te along [111] shows double well energy minimums at τ = 0.03 a0 away from the middle point the diagonal, where a0 is the lattice parameter. The potential barrier height is about 54 meV, which is twice as large as that of BaTiO3 and half the value of PbTiO3 [28] . This is reflected by the fact that the Curie temperature of GeTe is about 670 K, lying between those of BaTiO3 and PbTiO3. The spontaneous ferroelectric polarization of 67 µC/cm 2 is obtained by the Berry phase theory of polarization [29] . The Born effective charge (Z*) is estimated by the finite difference method from the expression where δu is the first-order change of the positions of Te and Ω is the volume of the unit cell.
We estimate the Z* is about 9.58 e from the data when u is below 0.1 A . This is a large value at the same scale of other ferroelectric compounds indicating the strong interactions of the orbitals between Te and Ge along the [111]-direction.
In ferroelectric materials, the electric field will induce displacement of the ions and the electronic structure will change accordingly as shown in Fig. 4. The nonlinearity and asymmetry is apparent when the electric field gets larger in the two different directions.
The Zener breakdown field EZb is estimated to be EZb = Eg/L ∼ 0.01a.u. = 51 MV/cm, where Eg is the band gap and L is the length of the unit cell in the direction of the electric field. In our calculations, the maximum field strength is about one order smaller than the breakdown field in order to avoid such an instability. As expected when the field is along The Rashba parameter αR, which is related to ER and kR by αR = 2ER/kR, is proportional to the electric field D  . From our calculations, the ER, kR and αR are estimated to be 0.24 eV, 0.10 A −1 and 4.8 eV A , respectively, when the field is not applied. The Rashba parameter αR is comparable with the experiments [30] and also previous theoretical calculations [9] .

C. SHC and its dependence on the ferroelectric polarization
Since the intrinsic SHC is determined fully by the band structure, it is compatible with the point group symmetry of the Hamiltonian. We therefore use the symmetry analysis  [22] reveals that the charge current produced by the spin current in GeTe is opposite to that found in Pt. As we know, the SHC calculated in Pt [32] at the Fermi level is 2200 ℏ/e(Ωcm) −1 , which is in good agreement with the experiments [33] . Thus the sign of SHC is in agreement with the experiment, however, its magnitude has not been measured yet in GeTe. The SHC as a function of the Fermi level is shown in Fig. 5. The SHC has the same signs for the electron and hole doping as seen in the figure. The maxima at the VBM and CBM are about -100 ℏ/e(Ωcm) −1 which are mainly due to the Rashba effect, in the sense that it is sensitively dependent on the electric polarization as can be seen in the following discussions. We see there is much larger SHC amounting 900 ℏ/e(Ωcm) −1 around 2 eV below the VBM. When comparing with the band structure in Fig. 1, we notice that there is a large SOC splitting (∆SOC) about 0.7 eV for the bands within this energy range. We suspect that the enhanced SHC is caused by the large splitting of the bands induced by SOC.
The dependence of SHC with respect to the electric polarization can be seen in Fig.6

IV. Conclusions
To summarize, our theoretical calculations show that the spin texture is closely related to the ferroelectric polarization. When the ferroelectric polarization is reversed, the spin winding direction is reversed accordingly. This is due to the change of the equivalent coordinate system upon polarization reversal. It should be quite general in FERSC as shown in several compounds. The electric fields can be applied to tune the Rashba parameters.
The SHCs have contributions from both the intrinsic SOC and Rashba effects while the latter can be easily tuned by the external electric fields. From the results we propose that the electric field is can be utilized to moderate the spin and electron transportation properties of FERSC materials. It provides a new dimension to design spintronic devices.