Abstract
The Rashba effect has recently attracted great attention owing to emerging physical properties associated with it. The interplay between the Rashba effect and the Zeeman effect, being produced by the exchange field, is expected to broaden the range of these properties and even result in novel phenomena. Here we predict an insulatortoconductor transition driven by the Rashba–Zeeman effect. We first illustrate this effect using a general Hamiltonian model and show that the insulatortoconductor transition can be triggered under certain Rashba and exchangefield strengths. Then, we exemplify this phenomenon by considering an Ag_{2}Te/Cr_{2}O_{3} heterostructure, where the electronic structure of the Ag_{2}Te monolayer is affected across the interface by the proximity effect of the Cr_{2}O_{3} antiferromagnetic layer with welldefined surface magnetization. Based on firstprinciples calculations, we predict that such a system can be driven into either insulating or conducting phase, depending on the surface magnetization orientation of the Cr_{2}O_{3} layer. Our results enrich the Rashba–Zeeman physics and provide useful guidelines for the realization of the insulatortoconductor transition, which may be interesting for experimental verification.
Introduction
The Rashba effect is a momentumdependent spin splitting of the energy bands driven by spinorbit coupling (SOC)^{1}. This phenomenon has recently stimulated vigorous research, owing to its potential application in spintronics^{2}. The Rashba effect occurs in material systems with broken spaceinversion symmetry such as surfaces^{3}, interfaces^{4}, and certain bulk materials^{5,6,7}. The SOC effect allows an electrical control of the spin degree of freedom that is interesting for device applications. For example, using the Rashba effect has been proposed for design of a spin fieldeffect transistor^{8}. Electrically switchable SOC parameter has been explored to design valleyspin valves^{9} and valleyspin logic gates^{10}. Furthermore, various physical phenomena such as currentinduced spin polarization^{11}, the spin Hall effect^{12}, and the spin galvanic effect^{13,14} have been inspired or reinvigorated.
The Rashbaaffected material systems may also exhibit a Zeeman effect. The Zeeman effect is characterized by the momentumindependent spin splitting of the energy bands and is associated with the interaction of spin with an external magnetic field or an exchange field. The Zeeman energy is typically ~0.1 meV/T and therefore is relatively small for applied external fields of a few Tesla or less. In contrast, the internal exchange fields arising from intrinsic magnetization^{15}, doped magnetic transition metals^{16} or ferromagnetic (FM)^{17}, antiferromagnetic (AFM)^{18}, and multiferroic^{19} insulator substrates owing to magnetic proximity effects^{20} can be sizable (~10^{2}–10^{3} T) and hence produce a nonnegligible Zeeman splitting of the energy bands.
The interplay between the Rashba and Zeeman effects (abbreviated below as the Rashba–Zeeman effect) is expected to produce novel features not found in pure Rashba or Zeemanaffected systems. For example, broken timereversal symmetry owing to the exchange field gives rise to the anomalous Hall effect^{21}. In addition, magnetically doped Rashba systems demonstrate entanglement of the spinorbit and magnetic orders. For example, the recent experimental work on Mndoped GeTe has demonstrated that the Rashba spin helicity can be altered by magnetization switching^{22}, and conversely, the magnetization can be reversed by polarization switching^{23} or by the currentinduced spinorbit torque^{24}. Moreover, the Rashba–Zeeman effect can also affect the quantum transport properties. For example, it has been demonstrated that in quantumpointcontact InSb nanowires with sizable Rashba SOC, the measured conductance plateau could be tuned from e^{2}/h to 2e^{2}/h by the magnetic field orientation^{25}.
In this work, we predict another striking phenomenon—an insulatortoconductor transition induced by the Rashba–Zeeman effect in a twodimensional (2D) system. The band structure of the 2D material can be controlled by the exchange field orientation, and under certain Rashba and exchange field strengths exhibits an insulatortoconductor transition. We explore this phenomenon for a realistic system—a monolayer of Ag_{2}Te deposited on an AFM Cr_{2}O_{3} (0001) substrate. Owing to broken inversion symmetry, an Ag_{2}Te monolayer exhibits a sizable Rashba band splitting, while the Cr_{2}O_{3} substrate has a robust surface magnetization coupled to the AFM order parameter and provides an exchange field affecting the electronic structure of the Ag_{2}Te through the proximity effect. Based on densityfunctional theory (DFT) calculations, we demonstrate that such a system can be driven into either insulating or conducting phase, depending on the boundary magnetization orientation of the Cr_{2}O_{3} layer.
Results and discussion
Hamiltonian model and phase diagram
We first illustrate the insulatortoconductor transition using a general Hamiltonian model. We consider a 2D direct bandgap semiconductor with the Rashba effect dominating at the band edges, affected by the exchange field arising from a magnetic insulator substrate (Fig. 1a). A singleband k·p Hamiltonian of this system can be written as follows:
Here, indices c and v indicate the bottom of the conduction band and the top of the valence band, respectively. The first term represents the kinetic energy with m_{c,v} being the electron effective mass, E_{c,v} is the band edge energy, the third term is the Rashba SOC with α_{c,v} being the Rashba parameters. The last term is the Zeeman term, where σ is the Pauli spin matrix, Δ is the exchange field, and the unit vector \({\hat{\mathbf m}}\) denotes the field orientation.
This model realistically describes certain types of 2D materials deposited on a magnetic insulator substrate. Specifically, there exist a handful of 2D direct bandgap semiconductors, such as Ag_{2}Te^{26}, BiSb^{27}, and LiAlTe_{2}^{28} monolayers whose electronic structure around the conduction band minimum and the valence band maximum can be well described by the singleband Rashba model. When these monolayers are deposited on a proper magnetic insulator substrate, their electronic structure is expected to be well captured by Eq. (1) provided that the effect of the substrate is dominated by the exchange field. The latter requirement entails a weak electronic hybridization between the 2D material and the substrate, which is expected to be valid for a sufficiently wide bandgap insulator.
Figure 1b shows the calculated electronic structure based on Hamiltonian (1) for typical parameters corresponding to the model. It is seen that the energy spectrum represents four bands (two conduction bands and two valence bands) whose appearance depends on the exchange field orientation. For the exchange field parallel to the z axis (\({\hat{\mathbf m}}{\hat{\mathbf z}}\)), the band gap is opened, whereas for the exchange field parallel to the y axis (\({\hat{\mathbf m}}{\hat{\mathbf y}}\)), the band gap is closed. Thus, by controlling the magnetization direction of the substrate it is possible to achieve a phase transition in the 2D system from the insulating state to the conducting state.
To elucidate this phase transition in more detail, we derive an analytic expression for the band gap. For \({\hat{\mathbf m}}{\hat{\mathbf z}}\), the band gap E_{gz} is given by (see Supplementary Note 1)
where E_{0} = E_{c} − E_{v} is the band gap in the absence of the Rashba–Zeeman effect. For \({\hat{\mathbf m}}{\hat{\mathbf y}}\) (or \({\hat{\mathbf m}}{\hat{\mathbf x}}\)), the band gap E_{gy} (or E_{gx}) reads
Note that in Eqs. (2) and (3), the negative sign of the band gap implies band inversion and thus no band gap.
Using Eqs. (2) and (3), we obtain the band gaps E_{gz} and E_{gy} depending on parameters of the model. Figure 1c, d shows the resulting phase diagrams in the (E_{0}, Δ) and (E_{0}, α) planes, respectively. It is seen that there are three distinctly different phases I, II, and III classified according to the sign of the band gaps E_{gz} and E_{gy}. The twophase boundaries (shown by black lines in Fig. 1c, d) are determined by E_{gz} = 0 or E_{gy} = 0. For phases I and III, both band gaps are positive (phase I) or negative (phase III), indicating the trivial phase transition from insulatortoinsulator (phase I) or from conductor to conductor (phase III). For phase II, we observe a nontrivial insulatortoconductor transition or via versa as a result of changing the exchange field orientation from the z axis to the y(x) axis. The illustration of this transition is revealed in the band structure of Fig. 1b, which corresponds to a set of parameters indicated by the red star in Fig. 1c.
It is evident that there is a certain range of parameters for which the insulatortoconductor transition occurs. A smaller E_{0} always yields the conductor phase, whereas larger E_{0} requires the strong exchange field to induce the insulatortoconductor transition. A larger Rashba parameter favors the transition for the system with larger E_{0}. Supplementary Figure 2 shows that the insulatortoconductor transition is not only limited by equal Rashba parameters α_{c} and α_{v} and effective masses m_{c} and m_{v} but can also occur for nonequal parameters (see Supplementary Note 2).
Electrical conductivity and anomalous Hall conductivity
The predicted insulatortoconductor transition can be probed by measuring electrical conductivity. We calculate the conductivity σ_{xx} of the 2D system within the approximation of a constant relaxation time τ, as discussed in Supplemental Note 3. Figure 2a shows σ_{xx} as a function of Fermi energy E_{F}. In the absence of the exchange field, Δ = 0, σ_{xx} for conduction bands can be expressed analytically as follows (see Supplementary Note 3)
where \(E_R = m_c\alpha _c^2{\mathrm{/}}\left( {2\hbar ^2} \right)\) is the Rashba energy and \(\sigma _0 = e^2\tau E_R{\mathrm{/}}\left( {\pi \hbar ^2} \right)\) is the conductivity unit. This limiting case is shown in Fig. 2a by the dashed line. We see distinct energy dependent behaviors below and above the conduction band minimum. When Δ ≠ 0, σ_{xx} depends on the exchange field orientation \({\hat{\mathbf m}}\). It is seen that around E_{F} = 0, the conductivity is zero for \({\hat{\mathbf m}}{\hat{\mathbf z}}\) (red line in Fig. 2a), whereas the conductivity is nonzero for \({\hat{\mathbf m}}{\hat{\mathbf y}}\) in the whole energy range (blue line in Fig. 2a). For higher Fermi energy, σ_{xx} scales linearly with E_{F}, as expected from Eq. (4).
Figure 2b shows σ_{xx} as a function of azimuthal angle θ for φ = 90° and E_{F} = 0. The critical points for the insulatorconductor transition are around θ = 0.13π and θ = 0.87π. In the conducting phase (aqua color), σ_{xx} versus θ can be well described by \(\sigma _{xx}(\theta ) = \sigma _{xx}(0) + \left[ {\sigma _{xx}\left( {\pi /2} \right)  \sigma _{xx}(0)} \right]sin^2\theta\) (blue solid line), which is the conventional behavior known for anisotropic magnetoresistance^{29}. Overall, changing the magnetization orientation of the substrate reveals perfect anisotropy in the conductivity of the 2D system.
Probing the anomalous Hall conductivity σ_{xy} provides another way to explore the phase transition. We calculate σ_{xy} assuming that there is only an intrinsic contribution to the anomalous Hall conductivity^{30}. This contribution is determined by the Berry curvature as discussed in the Supplemental Note 4. Figure 2c shows the calculated σ_{xy} of the 2D system as a function of E_{F} for different angles θ (determined in Fig. 1a). It is seen that σ_{xy} is zero in the energy gap region when \({\hat{\mathbf m}}{\hat{\mathbf z}}\) (θ = 0° or θ = 180°). For θ = 45° or θ = 135°, the gap is closed and σ_{xy} is nonzero in the whole energy range.
Figure 2d shows σ_{xy} as a function of θ changing continuously from 0° to 180°. For E_{F} = 0 (blue circles and line in Fig. 2a), we observe the same critical points for the insulatorconductor transition at around θ = 0.13π and θ = 0.87π. This transition disappears for E_{F} = 0.05 eV (red circles and line in Fig. 2d) consistent with Fig. 2c. For any Fermi energy, there is a sign change in σ_{xy} at θ = 90°. This sign change is are explained by the properties of the Berry curvature Ω_{z}, as demonstrated in Supplementary Fig. 4.
The above approach can be expanded to other types of SOC, such as the Dresselhaus SOC^{31} or the RashbaDresselhaus SOC^{32,33}. The analysis of the 2D systems, which exhibit these types of SOC shows that a similar insulatortoconductor transition can be observed in both models (see Supplementary Note 5 for details).
DFT results for Ag_{2}Te/Cr_{2}O_{3}
Next, we discuss a possible realization of the insulatortoconductor transition in a realistic system, namely a monolayer of Ag_{2}Te deposited on a magnetic Cr_{2}O_{3} (0001) substrate. A buckled Ag_{2}Te monolayer possesses a 2D hexagonal lattice of the P6mm symmetry and has a sizable band gap of 150 meV^{26}. Owing to broken inversion symmetry, an Ag_{2}Te monolayer exhibits Rashba band splitting with a large Rashba parameter of 3.84 eV Å^{34} (see Supplementary Note 6 for details). In a noncentrosymmetric corundum structure, Cr_{2}O_{3} is a magnetoelectric AFM insulator which belongs to the magnetic space group is R\(\bar 3^\prime\)c′. It exhibits the surface magnetization, which is an intrinsic property of all magnetoelectric antiferromagnets^{35,36}. This magnetization is electrically switchable with an AFM order parameter of Cr_{2}O_{3}, as has been demonstrated in experiment^{37}. The exchange coupling between Cr_{2}O_{3} and Ag_{2}Te across the interface in the Ag_{2}Te/Cr_{2}O_{3} structure is mediated by the surface magnetization through the proximity effect. The recent work has shown that a topological phase of graphene can be tuned by magnetization orientation in a graphene/Cr_{2}O_{3} system^{38}.
Figure 3a shows the atomic structure (produced using the VESTA software^{39}) of the Ag_{2}Te/Cr_{2}O_{3}(0001) system consisting of monolayer Ag_{2}Te and Cr_{2}O_{3} substrate composed of 6 and 12 atomic layers of O and Cr, respectively. The magnetic moments of Cr atoms in Cr_{2}O_{3} are aligned parallel in the (0001) plane and antiparallel along the (0001) direction. The Cr_{2}O_{3} surface is terminated by a single layer of Cr, which has the lowest surface energy^{40}. As seen from Fig. 3b, the interface atomic configuration has one Te atom and one Ag atom located at the hollow (H) sites and another Ag atom located at the top (T) site of the Cr_{2}O_{3} (0001) surface. This atomic configuration is among three possible highly symmetric atomic structures which have the lowest energy (see Supplementary Note 7).
Figure 4a shows the calculated band structure of Ag_{2}Te/Cr_{2}O_{3} for magnetization parallel to the z axis. It is noteworthy that the bands near the Fermi energy arise predominantly from the Ag_{2}Te layer, suggesting weak electronic hybridization between Ag_{2}Te and Cr_{2}O_{3}. Orbitalprojected band structure indicates that the bands near the Fermi energy are mainly composed of the Ags, d and Tep orbitals (see Supplementary Note 8), consistent with the previous results^{26,34}. A band gap of ~16 meV is observed, and the Zeemantype spin splitting at the Γ point is 2Δ = 131 meV for the bottom of conduction bands. The Rashbatype SOC of the conduction bands is evident from the inplane spin texture shown in Fig. 4b, c.
Next, we investigate the effect of magnetization orientation on the electronic band structure. Figure 5 shows the calculated results. It is seen that for the inplane magnetization, i.e., when \({\hat{\mathbf m}}{\hat{\mathbf x}}\) (Fig. 5a) or \({\hat{\mathbf m}}{\hat{\mathbf y}}\) (Fig. 5b), the band structure reveals a conducting phase characterized by electron and hole pockets crossing the Fermi energy. In contrast, for the outofplane magnetization (\({\hat{\mathbf m}}{\hat{\mathbf z}}\)), a band gap of 16 meV opens and the system is driven into an insulator phase (Fig. 5c). Thus, the Ag_{2}Te/Cr_{2}O_{3} system can be either a conductor or an insulator depending on the surface magnetization direction of Cr_{2}O_{3} substrate.
We note that in the above calculations, the polarization of Ag_{2}Te was pointing downward. For polarization pointing upward, the buckling height between Ag and Te layers is suppressed and the bond length between Te and Cr increases, suggesting that the Rashba effect and Zeeman field are both suppressed. The insulatortoconductor transition does not occur. Also, it is noteworthy that the insulatortoconductor transition is Cr_{2}O_{3} thickness independent, as expected from the exchange field arising from the magnetic proximity effect at the interface.
Figure 5d–f show 3D plots of the band structure for different magnetization orientations. The Fermi contours, which are shown in insets of Fig. 5d, e, suggest that the hole pocket is nearly a circle centered around the Γ point (red lines), whereas the electron pocket (blue lines) appears in the k_{y} < 0 (k_{x} > 0) quadrant for \({\hat{\mathbf m}}{\hat{\mathbf x}}\) (\({\hat{\mathbf m}}{\hat{\mathbf y}}\)). This behavior can be well explained by the Rashba–Zeeman model proposed earlier. According to Eq. (1), around the conduction band minimum, the dispersion along the k_{y} direction for \({\hat{\mathbf m}}{\hat{\mathbf x}}\) can be expressed as \(E_c^  = \hbar ^2k_y^2{\mathrm{/}}2m_c + E_c  \sqrt {\left( {\alpha _ck_y + {\Delta} } \right)^2}\). Here Δ is negative owing to a higher band energy for spin up than for spin down (inset in Fig. 4a). Thus, we have \(E_c^  \left( {k_y\, <\, 0} \right) < E_c^  \left( {k_y\, > \,0} \right)\), indicating that the band energy at k_{y} < 0 is lower than that at k_{y} > 0, which yields the band branch at k_{y} < 0 crossing the Fermi energy. The appearance of the electron pocket at k_{x} > 0 for \({\hat{\mathbf m}}{\hat{\mathbf y}}\) can be explained in the same way. We note that the shapes of the electron pockets are different for \({\hat{\mathbf m}}{\hat{\mathbf x}}\) and \({\hat{\mathbf m}}{\hat{\mathbf y}}\) owing to a higher korder contribution.
Our k·p model can be used to describe the DFT calculated band structure of Ag_{2}Te/Cr_{2}O_{3}. Using Supplementary Eq. (1), we fit the conduction band around the Γ point (red line in Fig. 5c). The fitting yields the following Hamiltonian parameters: m_{c} = 0.5 m_{0}, α_{c} = 1.89 eV Å, E_{c} = 0.03 eV. According to Supplementary Eqs. (5) and (12), the insulatortoconductor transition can occur under the condition \(E_c \,<\, {\Delta} + m_c\alpha _c^2{\mathrm{/}}2\hbar ^2\). This condition is indeed satisfied, which is seen by plugging the fitted parameters into the above inequality.
Experimentally, the predicted insulatortoconductor transition can be observed in the Ag_{2}Te/Cr_{2}O_{3} (0001) heterostructure where a 180° AFM domain wall is formed in Cr_{2}O_{3} between two domains with a uniform perpendiculartoplane Néel vector pointing in opposite directions. In this case, in the domain wall region, the continuous rotation of the surface magnetization in Cr_{2}O_{3} results in the formation of a conducting phase of Ag_{2}Te, whereas within the domains, Ag_{2}Te remains insulating (semiconducting). Thus, the enhancement of the electrical conductivity is expected in the domain wall region.
It is noteworthy that the predicted insulatortoconductor transition in Ag_{2}Te/Cr_{2}O_{3} may only be observed in the low temperature regime owing to the small band gap of ~16 meV (see Supplementary Note 9). Even though, as seen from Supplementary Fig. 10b, a sizable difference in conductivity between \({\hat{\mathbf m}}{\hat{\mathbf z}}\) and \({\hat{\mathbf m}}{\hat{\mathbf y}}\) does exist even at room temperature.
In addition to the Ag_{2}Te/Cr_{2}O_{3} system, there are other potential candidates with different 2D materials and magnetic substrates to explore the insulatortoconductor transition. For example, the above mentioned BiSb^{27} and LiAlTe_{2}^{28} monolayers have a smaller band gap and giant Rashba parameters. The magnetic insulator materials can be extended to FM EuO^{41} and CrI_{3}^{42,43}, and ferrimagnetic YIG^{44}. In comparison with AFM Cr_{2}O_{3}, these magnetic insulators have an advantage of controlling their magnetization by an external magnetic field.
Noteworthy is the fact that the predicted insulatortoconductor transition is different from the conventional metal–insulator transitions driven by structural distortions, magnetic ordering, and electron correlations via Peierls, Mott, and Slater mechanisms^{45}. Within the proposed mechanism, neither the structural distortions nor magnetostructural transitions or electron correlations are essential. The proposed mechanism is also different from the recently predicted insulatortoconductor transition in van der Waals spin valves^{46}. For the latter, gap closing or opening at the Dirac point is due to a change of the onsite potentials via the Zeeman effect and SOC is absent.
In summary, we have predicted the insulatortoconductor transition that can be triggered by the exchange field via the Rashba–Zeeman, DresselhausZeeman or RashbaDresselhausZeeman effect in 2D/FM or 2D/AFM systems and demonstrated its possible realization for a realistic Ag_{2}Te/Cr_{2}O_{3} heterostructure using firstprinciples calculations. We hope that our work will enrich the Rashba–Zeeman physics and stimulate experimental studies of the predicted phenomenon.
Methods
DFT calculations
Our atomic and electronic structure calculations were performed using the projectoraugmented wave method^{47,48} implemented in the Vienna ab initio simulation package^{49}. An energy cutoff of 400 eV for the plane wave expansion, generalized gradient approximation^{50} for the exchange and correlation functional with HubbardU correction U_{eff} = 2 eV on Crd orbital^{51} were adopted throughout. A 4 × 4 × 1 kpoint grid for Brillouin zone integration was used for structural relaxation and a 10 × 10 × 1 grid was used for selfconsistent electronic structure calculations. The optimized inplane lattice constant of 4.86 Å for Ag_{2}Te was found close to the optimized inplane lattice constant of 4.97 Å (experimental value 4.95 Å^{52}) for bulk Cr_{2}O_{3}, so that the lattice mismatch was ~2%. We fixed the inplane lattice constant to be 4.95 Å in all our calculations. The atomic coordinates were fully relaxed in the absence of SOC with the force tolerance of 0.01 V/Å. The DFTD3 method with Becke–Jonson damping was used to include the van der Waals corrections^{53}. A vacuum region of >20 Å along the z direction was imposed in the supercell calculations.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The related codes are available from the corresponding authors upon reasonable request.
References
 1.
Rashba, E. Properties of semiconductors with an extremum loop. 1. Cyclotron and combinational resonance in a magnetic field perpendicular to the plane of the loop. Sov. Phys. Solid State 2, 1109–1122 (1960).
 2.
Manchon, A., Koo, H. C., Nitta, J., Frolov, S. M. & Duine, R. A. New perspectives for Rashba spinorbit coupling. Nat. Mater. 14, 871–882 (2015).
 3.
LaShell, S., McDougall, B. A. & Jensen, E. Spin splitting of an Au(111) surface state band observed with angle resolved photoelectron spectroscopy. Phys. Rev. Lett. 77, 3419 (1996).
 4.
Caviglia, A. D. et al. Tunable Rashba spinorbit interaction at oxide interfaces. Phys. Rev. Lett. 104, 126803 (2010).
 5.
Ishizaka, K. et al. Giant Rashbatype spin splitting in bulk BiTeI. Nat. Mater. 10, 521–526 (2011).
 6.
Di Sante, D., Barone, P., Bertacco, R. & Picozzi, S. Electric control of the giant Rashba effect in bulk GeTe. Adv. Mater. 25, 509–513 (2013).
 7.
Tao, L. L. & Wang, J. Straintunable ferroelectricity and its control of Rashba effect in KTaO_{3}. J. Appl. Phys. 120, 234101 (2016).
 8.
Datta, S. & Das, B. Electronic analog of the electrooptic modulator. Appl. Phys. Lett. 56, 665 (1990).
 9.
Tao, L. L. & Tsymbal, E. Y. Twodimensional spinvalley locking spin valve. Phys. Rev. B 100, 161110(R) (2019).
 10.
Tao, L. L., Naeemi, A. & Tsymbal, E. Y. Valleyspin logic gates. Phys. Rev. Appl. 13, 054043 (2020).
 11.
Edelstein, V. M. Spin polarization of conduction electrons induced by electric current in twodimensional asymmetric electron systems. Sol. State Commun. 73, 233 (1990).
 12.
Sinova, J. et al. Universal intrinsic spin Hall effect. Phys. Rev. Lett. 92, 126603 (2004).
 13.
Ivchenko, E. L. & Pikus, G. E. New photogalvanic effect in gyrotropic crystals. JETP Lett. 27, 604 (1978).
 14.
Ganichev, S. D. Spingalvanic effect and spin orientation by current in nonmagnetic semiconductors. Int. J. Mod. Phys. B 22, 113–114 (2008).
 15.
Jiang, P., Li, L., Liao, Z., Zhao, Y. X. & Zhong, Z. Spin direction controlled electronic band structure in two dimensional ferromagnetic CrI_{3}. Nano Lett. 18, 3844–3849 (2018).
 16.
Cheng, Y., Zhang, Q. & Schwingenschlögl, U. Valley polarization in magnetically doped singlelayer transitionmetal dichalcogenides. Phys. Rev. B 89, 155429 (2014).
 17.
Yang, H. X. et al. Proximity effects induced in graphene by magnetic insulators: firstprinciples calculations on spin filtering and exchangesplitting gaps. Phys. Rev. Lett. 110, 046603 (2013).
 18.
Xu, L. et al. Large valley splitting in monolayer WS_{2} by proximity coupling to an insulating antiferromagnetic substrate. Phys. Rev. B 97, 041405 (2018).
 19.
Ibrahim, F. et al. Unveiling multiferroic proximity effect in graphene. 2D Mater. 7, 015020 (2020).
 20.
Žutić, I., MatosAbiague, A., Scharf, B., Dery, H. & Belashchenko, K. Proximitized materials. Mater. Today 22, 85–107 (2019).
 21.
Culcer, D., MacDonald, A. & Niu, Q. Anomalous Hall effect in paramagnetic twodimensional systems. Phys. Rev. B 68, 045327 (2003).
 22.
Krempasky, J. et al. Entanglement and manipulation of the magnetic and spin–orbit order in multiferroic Rashba semiconductors. Nat. Commun. 7, 13071 (2016).
 23.
Krempaský, J. et al. Operando imaging of allelectric spin texture manipulation in ferroelectric and multiferroic Rashba semiconductors. Phys. Rev. X 8, 021067 (2018).
 24.
Yoshimi, R. et al. Currentdriven magnetization switching in ferromagnetic bulk Rashba semiconductor (Ge, Mn)Te. Sci. Adv. 4, 9989 (2018).
 25.
Kammhuber, J. et al. Conductance through a helical state in an Indium antimonide nanowire. Nat. Commun. 8, 478 (2017).
 26.
Ma, Y., Kou, L., Dai, Y. & Heine, T. Twodimensional topological insulators in group11 chalcogenide compounds: M_{2}Te (M = Cu, Ag). Phys. Rev. B 93, 235451 (2016).
 27.
Singh, S. & Romero, A. H. Giant tunable Rashba spin splitting in a twodimensional BiSb monolayer and in BiSb/AlN heterostructures. Phys. Rev. B 95, 165444 (2017).
 28.
Liu, Z., Sun, Y., Singh, D. J. & Zhang, L. Switchable outofplane polarization in 2D LiAlTe_{2}. Adv. Electron. Mater. 5, 1900089 (2019).
 29.
McGuire, T. & Potter, R. Anisotropic magnetoresistance in ferromagnetic 3d alloys. IEEE Trans. Magn. 11, 1018–1038 (1975).
 30.
Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall effect. Rev. Mod. Phys. 82, 1539 (2010).
 31.
Tao, L. L., Paudel, T. R., Kovalev, A. A. & Tsymbal, E. Y. Reversible spin texture in ferroelectric HfO_{2}. Phys. Rev. B 95, 245141 (2017).
 32.
Stroppa, A. et al. Tunable ferroelectric polarization and its interplay with spin–orbit coupling in tin iodide perovskites. Nat. Commun. 5, 5900 (2014).
 33.
Tao, L. L. & Tsymbal, E. Y. Persistent spin texture enforced by symmetry. Nat. Commun. 9, 2763 (2018).
 34.
NoorAAlam, M., Lee, M., Lee, H. J., Choi, K. & Lee, J. H. Switchable Rashba effect by dipole moment switching in an Ag_{2}Te monolayer. J. Phys. Condens. Matter 30, 385502 (2018).
 35.
Andreev, A. F. Macroscopic magnetic fields of antiferromagnets. JETP Lett. 63, 758–762 (1996).
 36.
Belashchenko, K. D. Equilibrium magnetization at the boundary of a magnetoelectric antiferromagnet. Phys. Rev. Lett. 105, 147204 (2010).
 37.
He, X. et al. Robust isothermal electric control of exchange bias at room temperature. Nat. Mater. 9, 579–585 (2010).
 38.
Takenaka, H., Sandhoefner, S., Kovalev, A. A. & Tsymbal, E. Y. Magnetoelectric control of topological phases in graphene. Phys. Rev. B 100, 125156 (2019).
 39.
Momma, K. & Izumi, F. VESTA 3 for threedimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).
 40.
Wysocki, A. L., Shi, S. & Belashchenko, K. D. Microscopic origin of the structural phase transitions at the Cr_{2}O_{3} (0001) surface. Phys. Rev. B 86, 165443 (2012).
 41.
Lukashev, P. V. et al. Spin filtering with EuO: insight from the complex band structure. Phys. Rev. B 85, 224414 (2012).
 42.
Huang, B. et al. Layerdependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).
 43.
Paudel, T. R. & Tsymbal, E. Y. Spin filtering in CrI_{3} tunnel junctions. ACS Appl. Mater. Int. 11, 15781–15787 (2019).
 44.
Cherepanov, V., Kolokolov, I. & Lvov, V. The saga of YIG: Spectra, thermodynamics, interaction and relaxation of magnons in a complex magnons. Phys. Rep. 229, 81–144 (1993).
 45.
Imada, M., Fujimori, A. & Tokura, Y. Metalinsulator transitions. Rev. Mod. Phys. 70, 1039 (1998).
 46.
Cardoso, C., Soriano, D., GarcíaMartínez, N. A. & FernándezRossier, J. Van der Waals spin valves. Phys. Rev. Lett. 121, 067701 (2018).
 47.
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953 (1994).
 48.
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758 (1999).
 49.
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio totalenergy calculations using a planewave basis set. Phys. Rev. B 54, 11169 (1996).
 50.
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
 51.
Íñiguez, J. Firstprinciples approach to latticemediated magnetoelectric effects. Phys. Rev. Lett. 101, 117201 (2008).
 52.
Finger, L. W. & Hazen, R. M. Crystal structure and isothermal compression of Fe_{2}O_{3}, Cr_{2}O_{3}, and V_{2}O_{3} to 50 kbars. J. Appl. Phys. 51, 5362 (1980).
 53.
Grimme, S., Ehrlich, S. & Goerigk, L. Effect of the damping function in dispersion corrected density functional theory. J. Comp. Chem. 32, 1456 (2011).
Acknowledgements
This research was supported by the National Science Foundation through the E2CDA program (grant ECCS1740136) and the Semiconductor Research Corporation (SRC) through the nCORE program. Computations were performed at the University of Nebraska Holland Computing Center.
Author information
Affiliations
Contributions
L.L.T. and E.Y.T. conceived the project. L.L.T. carried out numerical calculations. Both authors discussed the results and wrote the manuscript.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Tao, L., Tsymbal, E.Y. Insulatortoconductor transition driven by the Rashba–Zeeman effect. npj Comput Mater 6, 172 (2020). https://doi.org/10.1038/s41524020004410
Received:
Accepted:
Published:
Further reading

Polarization tunable Rashba effect in 2D LiAlTe2
Applied Physics Letters (2021)

Perspectives of spintextured ferroelectrics
Journal of Physics D: Applied Physics (2021)