Large and nonlinear electric field response in a two-dimensional ferroelectric Rashba material

1.   Introduction

The Rashba effect, which originates from structural inversion asymmetry (SIA), represents a typical spin–orbit coupling (SOC) phenomenon and offers a potent means for spin transport and electrical spin control^[1]. The critical feature of Rashba SOC is that moving electrons experience an effective magnetic field driving the precession of the spin orientation of the electrons^[2,3]. In recent decades, the Rashba effect has undergone extensive exploration in diverse systems, including oxide interfaces^[4], topological insulators^[5], semiconductor quantum wells^[6], alloy surfaces and heavy atoms^[7], perovskite-structured compounds^[8], and low-dimensional nanomaterials^[9]. In addition, Rashba effect-based spintronic devices have emerged, a prominent example of which is the spin field effect transistor (SFET)^[10]. Due to the Rashba SOC, electron spins precess around the effective magnetic field in the SFET^[11]. SFETs have been experimentally demonstrated in 2D semiconductor quantum well systems^[12,13].

To date, a variety of 2D Rashba semiconductors have been proposed, e.g., 2D AlBiS₃^[14], AlBi^[15], SrTiO₃ and LaAlO₃^[16], BiTeX (X = I and Br) monolayers^[17], and Bi/Ag/CoFeB trilayers^[18]. To design high-performance SFETs, Rashba materials must have a large Rashba constant α_R and a significant electric field response |∆α_R/∆E|^[19]. This phenomenon entails large spin splitting, particularly in the conduction band (CB) or valence band (VB). Moreover, it is crucial that the Rashba constant α_R exhibits a sensitive response to the electric field, allowing for modulation of the spin precession angle (θ) via the electric field, based on the equation Δθ = 2m*Δα_RL/ℏ², where L denotes the channel length of the SFET and m* represents the effective mass of the carriers. In previously reported representative Rashba materials such as T-RbPbI₃^[10], BiSb^[19], and MAPbI₃ (MA=CH₃NH₃)^[20], their Rashba constants vary linearly with the electric field, with |∆α_R/∆E| values of 0.44, 0.92, and 1.95 e·Å², respectively^[10,19,20]. Searching for 2D Rashba materials with stronger electric field responses remains a pending and challenging task.

Here, the transition metal phosphorus trichalcogenide FePS₃ is selected as the structural prototype, and Fe, P and S are replaced by other elements. Specifically, we replace Fe with nonmagnetic elements, namely, Al, Bi, Ga, In, Os, Ru, Sb, Sc, and Y; substitute group VI elements, including N, Sb, As, and Bi, for P; and replace S with the chalcogens Se and Te. Eventually, 2D YSbTe₃, which exhibits the largest electric field response and a large Rashba constant, is screened out compared with the literature. Moreover, 2D YSbTe₃ has also been shown to be a novel ferroelectric Rashba semiconductor, where the spin texture can be reversed through the inversion of ferroelectric polarization.

2.   Computational methods

The calculations in this study were conducted utilizing density functional theory (DFT) implemented in the Vienna ab initio simulation package (VASP)^[21]. The projector augmented wave (PAW) method^[22] and the generalized gradient approximation of the Perdew-Burke-Ernzerhof (GGA-PBE) functional were applied^[23]. The plane-wave cutoff energy was set to 500 eV. Lattice constants and atomic structures were fully relaxed until the convergence of energy and force was less than 10⁻⁶ eV and 0.01 eV·Å⁻¹, respectively. A 7×7×1 k-point sampling mesh grid was employed for the structure optimization and calculation of the electronic properties. The vacuum layer length was set to 20.0 Å in the z-direction. Considering van der Waals (vdW) interactions, the DFT-D3 method was employed^[24]. The investigation of Rashba spin splitting incorporated the effect of SOC. The electronic band structures were calculated using the HSE06 hybrid functional since the PBE functional generally underestimates the band gap^[25]. To investigate the dynamic stability of 2D YSbTe₃, the phonon spectrum was calculated with a 3×3×1 supercell via the finite difference method facilitated by the phonopy package interfaced with VASP^[26,27]. Additionally, the solid-state nudged elastic band (SS-NEB) method was used for calculating the ferroelectric polarization switching process^[28]. Ab initio molecular dynamics simulations were performed to investigate the thermal stability of 2D YSbTe₃. The simulation was controlled by a Nosé‒Hoover thermostat and executed for a 3×3×1 supercell with a time step of 1.0 fs at 500 K.

3.   Results and discussion

3.1   Geometry structures and stability

Fig. 1a illustrates that 2D YSbTe₃ possesses a noncentrosymmetric structure with P31m symmetry. The optimized lattice parameters for YSbTe₃ are a = b = 7.56 Å. The formation energy (E_f) of YSbTe₃ is calculated to be −2.02 eV per formula unit using E_f(YSbTe₃) = E(YSbTe₃)−E(Y)−E(Sb)−3E(Te), where E(YSbTe₃), E(Y), E(Sb), and E(Te) indicate the energies of 2D YSbTe₃, bulk Y, Sb and Te in their most stable phases, respectively. This formation energy is on the same order of magnitude as that of 2D CsSnCl₃ (−2.28 eV per formula unit)^[10] synthesized experimentally^[29], indicating the good stability of this ternary compound. The phonon spectrum shown in Fig. 1b presents no imaginary frequency, which affirms the dynamic stability of the 2D YSbTe₃. To further assess its thermal stability, ab initio molecular dynamics simulations were performed. As illustrated in Fig. S1, the structure of 2D YSbTe₃ can be retained at 500 K, with both the temperature and total energy fluctuating around their equilibrium values.

Figure  1.  (a) Top and side views of the structure of 2D YSbTe₃. The Y, Sb, and Te atoms are represented by blue, red, and yellow balls, respectively. (b) Phonon spectrum of YSbTe₃.

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3.2   Electronic structures

Then, we investigated the electronic properties of YSbTe₃. Fig. 2a and b depict the electronic band structures of YSbTe₃, which were calculated by using the HSE06 and HSE06 + SOC methods, respectively. The 2D YSbTe₃ demonstrates the characteristics of a nonmagnetic semiconductor, featuring an indirect bandgap of 1.19 eV without considering the SOC. The band gap with the SOC is slightly smaller, decreasing to 1.04 eV. Furthermore, the valence band of 2D YSbTe₃ displays significant Rashba spin splitting with lifted spin degeneracies, i.e., the VB (red) and VB−1 (blue) bands are differentiated by momentum-dependent energy splitting ∆E=2α_Rk induced by the Rashba SOC. The time-reversal symmetry of the system makes the VB and VB−1 bands contact the Γ-point. The electronic band structures computed using the PBE and PBE+SOC methods are also shown in Fig. S2 and are similar to those of the HSE06 and HSE06 + SOC results. Fig. 2c, d illustrates the spin textures of VB and VB−1 in the hexagonal reciprocal space. The inner branch VB−1 exhibits a clockwise spin texture, while the outer branch VB shows a counterclockwise spin texture. Notably, the VB exhibits significant Rashba splitting, and the Rashba constant α_R is determined to be 1.54 eV·Å according to the equation α_R=2E_R/k_R (see the Supporting information for more details of the formula derivation for the Rashba model).

Figure  2.  (a, b) Band structures of 2D YSbTe₃ calculated with HSE06 and HSE06+SOC, respectively. VB and VB−1 are shown in red and blue, respectively. The characteristic spin texture structures of (c) VB and (d) VB−1 calculated with the HSE06 + SOC functional. The Fermi levels are set to zero.

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3.3   Tunable Rashba spin splitting

To investigate the electric field response |∆α_R/∆E|, we applied external positive and negative electric fields perpendicular to the atomic plane of 2D YSbTe₃. It is worth noting that the VB keeps a typical Rashba splitting under a positive electric field, whereas it is not completely split when subjected to a negative electric field (Fig. S3). The Rashba constant (α_R) presents a nonlinear relationship with increasing positive electric field (E), as depicted by the red line in Fig. 3. The nonlinear relationship between E and α_R is characterized by an S-shaped curve, i.e., a gradual increase, a steep increase, and a slow increase as E increases. Consequently, the electric field response |∆α_R/∆E| also changes with the electric field (the blue line in Fig. 3). When E reaches 0.13 V·Å⁻¹, |∆α_R/∆E| attains a peak value of 4.80 e·Å², surpassing that of other Rashba semiconductors reported in previous literature^[10,19,20].

Figure  3.  The relationship between the Rashba constant and the electric field (red line and black dots, left y-axis) and the corresponding electric field response of the Rashba constant (|∆α_R/∆E|) (blue line, right y-axis) for 2D YSbTe₃.

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To explore why the relationship between the electric field E and Rashba constant α_R of YSbTe₃ is nonlinear, we analyzed the orbital composition of YSbTe₃ under electric fields of 0 V·Å⁻¹ and 0.3 V·Å⁻¹ according to the density of states (DOS), as displayed in Fig. S4. We find that the DOS under electric fields of 0 V·Å⁻¹ and 0.3 V·Å⁻¹ are nearly identical. In both cases, the VB bands are mainly attributed to the p_y and p_z orbitals of the Te element. Therefore, the variation in the electric field has no effect on the orbital composition. On the other hand, we investigated the correlation between the band gap (E_g) and the electric field of YSbTe₃ (Fig. S5) and compared it with the relationship between the Rashba constant and the electric field. It is clear that the variation trend of E_g with respect to E is the opposite of that of α_R, i.e., a nonlinear, inverse S-shaped curve is formed. As previously reported, α_R is negatively correlated with E_g^[30,31], indicating that as the band gap decreases, the Rashba constant increases. Therefore, the nonlinear electric field response of the Rashba constant originates from the nonlinear variation in the band gap of YSbTe₃ with the electric field. To understand the nonlinear behavior of the band gap, the charge density distributions of the VB and CB were analyzed. As shown in Fig. S6a, the VB of YSbTe₃ is primarily contributed by the Te atoms in the upper surface layer, while CB is mainly constructed by the Te atoms in both the upper and lower surface layers. Thus, the center of charge for the VB is located at a greater position than that for the CB. When a positive electric field is applied, the electrostatic potential energy of the VB is greater than that of the CB. That is, with increasing electric field, the energy level of the VB moves upward relative to the CB level, resulting in a linear reduction in the band gap (note that the electrostatic potential energy is a linear function of the electric field). However, such a conclusion is based on the assumption that the charge distributions of the VB and CB remain unchanged under an external electric field. In fact, we observe significant charge transfer between the upper and lower Te atoms. Moreover, such charge transfer is nonlinear with respect to the electric field (Fig. S6b). This may be the reason for the nonlinear variation in the band gap in 2D YSbTe₃. We note that nonlinearity in the band gap variation with the electric field also occurs in bilayer MoSi₂N₄, WSi₂N₄, and graphene^[32,33].

3.4   Ferroelectric properties

To date, 2D ferroelectric materials have garnered great interest, given that bulk ferroelectric materials tend to lose their ferroelectric polarization below a critical thickness due to the depolarizing electrostatic field and finite size effect. The calculation of the double-well potential with two degenerated ferroelectric states indicates the ferroelectric characteristics of YSbTe₃ (Fig. 4). The two ferroelectric states exhibit opposing ferroelectric polarization directions, which can be interconverted through atomic position deformations, specifically the upward or downward movement of Sb atoms. The switching energy barrier is determined to be 0.33 eV per formula unit, with the transition state (TS) being the paraelectric phase. The phonon dispersion for the paraelectric phase (P$\stackrel{-}{3}$1m) is shown in Fig. S7. Clearly, there is an imaginary, soft optical mode (λ) present in the phonon spectrum, while the ferroelectric phase has no soft mode (Fig. 1). As the temperature decreases below the critical temperature, the paraelectric phase evolves into the ferroelectric phase, and the soft mode is stabilized by symmetry breaking. In such a paraelectric state, where Sb atoms are symmetrically distributed in the plane of Y atoms, YSbTe₃ lacks the Rashba effect, as evidenced by the absence of clockwise and counterclockwise rotations in the spin texture of the VB (Fig. 4). Importantly, the spin textures of the two ferroelectric states are found to be opposite, i.e., the direction of the spin texture is reversed along with ferroelectric polarization switching. In addition, the electric polarization of 2D YSbTe₃ is calculated to be 2.91×10⁻¹² C·m⁻¹, which is comparable to that of other 2D ferroelectric materials in the literature (see Table S1). Thus, 2D YSbTe₃ is identified as a ferroelectric Rashba material.

Figure  4.  Double-well potential vs. atomic displacement along the z-axis. The energy of the ferroelectric state is taken as a reference. The VB spin textures of two degenerated ferroelectric states and one transition state (TS) are displayed.

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4.   Conclusions

In summary, YSbTe₃ is theoretically confirmed as a potential candidate for a 2D ferroelectric Rashba material. On the one hand, 2D YSbTe₃ shows a nonlinear relationship between the electric field and the Rashba constant α_R. The electric field response parameter |∆α_R/∆E| for YSbTe₃ is determined to be 4.80 e·Å², the largest among all 2D Rashba semiconductors, which is highly promising for designing highly integrated SFETs. However, YSbTe₃ is also ferroelectric. Ferroelectric polarization switching can reverse the Rashba spin texture.