In recent years, there have been increasing interests in phenomena emerging from relativistic electrons in a solid. The spin-orbit interaction couples spin degrees of freedom with electronic orbits, therefore providing a link between the spin-space and the crystalline lattice. Generally considered as a “weak” interaction—and consequently neglected for long-time—spin-orbit coupling (SOC) is nowadays regarded as a rich source for novel physics. Among the wealth of recently discovered intriguing phenomena, SOC leads to topologically non-trivial insulating states [1, 2], characterized by the concomitant presence of a bulk band gap and symmetry-protected conducting surface states, or it can give rise to Dresselhaus and Rashba effects (i.e., k-dependent spin-splitting in the band structure) in materials which lack inversion symmetry [3, 4]. We recall that, according to the (k-linear) Rashba effect, [3, 4] the (otherwise spin-degenerate) dispersion of free-like electrons, with momentum k and effective mass m* subject to a potential gradient (i.e., an electric field E_z) and in the presence of SOC, splits into two branches for oppositely spin-polarized states (labeled with + or −), with energies: E±(k)=ℏ2k22m*±αRk where the so called Rashba parameter α_R (proportional to λE_z, λ being the spin-orbit constant) represents the strength of the Rashba effect. The Rashba momentum k_R = |m^*|α_R/ ħ² quantifies the mutual shift of the split bands. As for Rashba-like effects, the research has so far mostly focused on materials surfaces or interfaces (i.e., where the inversion symmetry is intrinsically broken), because of their potential applications in the field of semiconductor spintronics, aiming at an all-electric control of spin transport in novel devices. Furthermore, it has been recently noticed that topological insulators and non-centrosymmetric materials displaying Rashba effects share some similarities, due to the common key ingredient, SOC. In this framework, a major breakthrough was provided by BiTeI, the first known non-centrosymmetric semiconductor showing huge Rashba-like spin-splitting in its bulk (not surface) band-structure, [5, 6] driven by the presence of a non-trivial ordering of the bands near the Fermi surface [7]. As a matter of fact, BiTeI has been predicted to undergo a topological transition under pressure [8], pointing toward possible connections between topology and Rashba physics in the same (class of) materials.

An interesting class of bulk materials lacking inversion symmetry—a prerequisite for Rashba effects to arise—is represented by ferroelectrics, i.e., materials displaying, below a certain critical temperature, a long-range dipolar order with a permanent ferroelectric (FE) polarization switchable by an electric field. Novel functionalities could be exploited in semiconductor spintronics in the presence of the coexistence and possible coupling between ferroelectricity and SOC-induced Rashba effects. Indeed, the potential relevance of this phenomenology has just started to be discussed in our recent theoretical work, [9] where the reversal of ferroelectric polarization in GeTe was theoretically predicted to cause a full reversal of the spin texture (i.e., of the Rashba parameter) and where a Datta-Das spin-transistor architecture was proposed in order to exploit this peculiar property. We here recall our main theoretical findings and address the reader to our original publication [9] for full details.

GeTe is probably the “simplest” known ferroelectric [10–12]: below a Curie ordering temperature of about 720 K, it shows Ge and Te ions displaced from their ideal rocksalt sites along the [111] direction, as well as a rhombohedral distortion (space group R3m, No. 160). We remark that the type of ferroelectricity in GeTe is strongly debated between a displacive-like and an order-disorder-transition [13]. For our specific purposes, this issue is, however, not so crucial, as we consider the situation below the Curie temperature with ideally-perfect collective FE order. We performed first principles calculations via density functional theory (DFT), within the Generalized Gradient Approximation (GGA-PBE) [14], including spin-orbit-coupling and using the VASP code [15], for ferroelectric GeTe. We show the calculated band structure in Figures 1A,B: a spin-splitting occurs for electrons/holes with k-vectors belonging to the Brillouin zone plane perpendicular to the direction of ferroelectric polarization (i.e., around the Z point and specifically along the Z-A and Z-U lines). This Rashba splitting is indeed anomalously large for holes in the valence band maximum (VBM), opening the way to its exploitation in spintronic devices operating at room temperature.

As shown in Figures 1C,D, the switch of the ferroelectric state - obtained by switching the Ge-Te relative displacements—implies that the spin-expectation values—both in plane and out-of-plane (cfr. Figure 1F)—correspondingly switch their direction. Crucial for applications is therefore that switching the ferroelectric polarization causes a full reversal of the spin texture, as the spin direction in each sub-band changes by 180° upon reversal of polarization, thereby allowing its electrical control. Strong hexagonal “warping” effects (cfr. Figure 1E) as well as the presence of a significant out-of-plane spin-polarization are consistent with the deviation of the results from a linear Rashba model (according to a simple parabolic band-structure) and can be interpreted by including cubic terms in k in the Rashba Hamiltonian [9].

As previously mentioned, the strength of the Rashba effect in GeTe can be quantified through several parameters: the k-space shift k_R, the Rashba energy E_R and the “Rashba parameter” α_R. By neglecting the third order terms in the nearly-free-electron approximation [9], the Rashba parameter can be related to E_R and k_R by α_R = 2 E_R/k_R. The parameters are estimated for Rashba-splitted bands along the Z-A direction in k-space, in which the spin-splittings are larger. k_R is evaluated as the Rashba-induced momentum offset of the VBM or conduction band minimum (CBM) with respect to the high-symmetry point, Z. E_R is calculated as the difference between the VBM or CBM estimated at k_R and the corresponding energy values at the Z point. For GeTe, we calculated k_R = 0.094 Å⁻¹, E_R is estimated as 227 meV and 120 meV (in turn leading to α_R equal to 4.9 eVÅ and 2.5 eVÅ) for VBM and CBM, respectively. We remark that the choice of the DFT exchange-correlation functional does not seem to strongly affect the Rashba-related quantitites: [9] for example, when using accurate non-local hybrid-functionals according to the Heyd-Scuseria-Erzenhof (HSE) approach [16], one obtains for the VBM E_R = 187 meV, k_R = 0.088 Å⁻¹ and α_R = 4.2 eVÅ (although the energy band gap at the Z-point increases from 0.74 eV in GGA to 0.96 in HSE), so that variations with respect to GGA can be quantified on the order of 10%.

In order to show a comparison with other Rashba-like systems, we report in Table 1 the relevant parameters for different materials: our values show that the bulk Rashba effect in GeTe is indeed gigantic. In particular, we briefly discuss BiTeI, with a (non-centrosymmetric) hexagonal crystal structure (P3m1) consisting of alternating layers of Bi, Te, and I ions. On the basis of angular-resolved photoemission spectroscopy (ARPES), [5, 6] magnetotransport, optical spectroscopy, [25] as well as density-functional calculations (including SOC), [5, 7] the BiTeI polar structure and the large spin-orbit interaction give rise to a giant Rashba-like spin splitting in the bulk conduction and valence bands of BiTeI, also reflected in a giant and robust spin-splitting at its surface. Similar results were also found for related compounds, such as BiTeBr [19, 20]. However, the presence of bistability granted by GeTe ferroelectricity represents a remarkable improvement over the Rashba phenomenology observed in BiTeI and BiTeBr. The power of accurate first-principles simulations joint to modeling capabilities based on symmetry-considerations [9] is therefore evident in this field and constitute grounds for a deep understanding of ferroelectrics with strong Rashba effect.

In summary, we have proposed a brand new spin-electric coupling mechanism deriving from the interplay between spin-orbit-driven Rashba effect and ferroelectricity: in GeTe, the prototype of Ferroelectric Rashba Semiconductors (FERSC), the spin-texture is controlled and switched via an electric field. As such, FERSC therefore represent breakthroughs in different aspects of science and technology: (i) From the fundamental point of view, FERSC offer playground for new and completely unexplored microscopic understanding of “cross-coupled” spin-electric effects; (ii) From the materials science point of view, FERSC represent a novel class of multifunctional materials, and, as such, there is room for their development, accompanied by a deeper understanding and optimization; (iii) From the technological point of view, the FERSC phenomenology might be exploited in prototypes of an entirely new generation of semiconductor spintronic devices, with a remarkable integration of logic functionalities and non-volatility.