Insulator-to-conductor transition driven by the Rashba–Zeeman effect

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 insulator-to-conductor transition driven by the Rashba–Zeeman effect. We first illustrate this effect using a general Hamiltonian model and show that the insulator-to-conductor transition can be triggered under certain Rashba and exchange-field strengths. Then, we exemplify this phenomenon by considering an Ag2Te/Cr2O3 heterostructure, where the electronic structure of the Ag2Te monolayer is affected across the interface by the proximity effect of the Cr2O3 antiferromagnetic layer with well-defined surface magnetization. Based on first-principles calculations, we predict that such a system can be driven into either insulating or conducting phase, depending on the surface magnetization orientation of the Cr2O3 layer. Our results enrich the Rashba–Zeeman physics and provide useful guidelines for the realization of the insulator-to-conductor transition, which may be interesting for experimental verification.


INTRODUCTION
The Rashba effect is a momentum-dependent spin splitting of the energy bands driven by spin-orbit 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 space-inversion 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 field-effect transistor 8 . Electrically switchable SOC parameter has been explored to design valley-spin valves 9 and valley-spin logic gates 10 . Furthermore, various physical phenomena such as current-induced spin polarization 11 , the spin Hall effect 12 , and the spin galvanic effect 13,14 have been inspired or reinvigorated.
The Rashba-affected material systems may also exhibit a Zeeman effect. The Zeeman effect is characterized by the momentum-independent 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 non-negligible 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 time-reversal symmetry owing to the exchange field gives rise to the anomalous Hall effect 21 . In addition, magnetically doped Rashba systems demonstrate entanglement of the spin-orbit and magnetic orders. For example, the recent experimental work on Mn-doped 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 current-induced spin-orbit torque 24 . Moreover, the Rashba-Zeeman effect can also affect the quantum transport properties. For example, it has been demonstrated that in quantum-point-contact 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 insulator-to-conductor transition induced by the Rashba-Zeeman effect in a two-dimensional (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 insulator-to-conductor 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 density-functional 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. 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 vectorm 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 band-gap 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 single-band 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 band-gap 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 (mjjẑ), the band gap is opened, whereas for the exchange field parallel to the y axis (mjjŷ), 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. Formjjẑ, 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. Formjjŷ (ormjjx), 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 two-phase 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 insulator-to-insulator (phase I) or from conductor to conductor (phase III). For phase II, we observe a nontrivial insulator-to-conductor 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 insulator-to-conductor transition occurs. A smaller E 0 always yields the conductor phase, whereas larger E 0 requires the strong exchange field to induce the insulator-to-conductor transition. A larger Rashba parameter favors the transition for the system with larger E 0 . Supplementary Figure 2 shows that the insulator-to- The phase boundaries are determined by E gz = 0 or E gy = 0. The red star in c indicates a set of parameters corresponding to the band structure in b.
conductor 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 non-equal parameters (see Supplementary Note 2).

Electrical conductivity and anomalous Hall conductivity
The predicted insulator-to-conductor 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 À Á 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 orientationm. It is seen that around E F = 0, the conductivity is zero formjjẑ (red line in Fig.  2a), whereas the conductivity is nonzero formjjŷ 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 insulator-conductor 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 σ xx ðθÞ ¼ σ xx ð0Þ þ σ xx π=2 ð ÞÀσ xx ð0Þ ½ sin 2 θ (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 whenmjjẑ (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 insulator-conductor 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 Rashba-Dresselhaus SOC 32,33 . The analysis of the 2D systems, which exhibit these types of SOC shows that a similar insulator-to-conductor 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 insulator-toconductor 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 non-centrosymmetric corundum structure, Cr 2 O 3 is a magnetoelectric AFM insulator which belongs to the magnetic space group is R3 0 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 . Orbital-projected band structure indicates that the bands near the Fermi energy are mainly composed of the Ag-s, d and Te-p orbitals (see Supplementary Note 8), consistent with the previous results 26,34 . A band gap of~16 meV is observed, and the Zeeman-type spin splitting at the Γ point is 2Δ = 131 meV for the bottom of conduction bands. The Rashba-type SOC of the conduction bands is evident from the in-plane 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 in-plane magnetization, i.e., when mjjx (Fig. 5a) ormjjŷ (Fig. 5b), the band structure reveals a conducting phase characterized by electron and hole pockets crossing the Fermi energy. In contrast, for the out-of-plane magnetization (mjjẑ), 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 insulator-to-conductor transition does not occur. Also, it is noteworthy that the insulator-to-conductor 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 formjjx (mjjŷ). 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 formjjx can be expressed as . 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 À c k y < 0 À Á <E À c k y > 0 À Á , 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 formjjŷ can be explained in the same way. We note that the shapes of the electron pockets are different formjjx andmjjŷ owing to a higher k-order 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 insulator-to-conductor transition can occur under the condition E c < Δ þ m c α 2 c =2 h 2 . This condition is indeed satisfied, which is seen by plugging the fitted parameters into the above inequality.
Experimentally, the predicted insulator-to-conductor 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 perpendicular-to-plane 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 insulator-to-conductor 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 betweenmjjẑ andmjjŷ 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 insulator-to-conductor 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 insulator-to-conductor 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 insulator-to-conductor 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 on-site potentials via the Zeeman effect and SOC is absent.
In summary, we have predicted the insulator-to-conductor transition that can be triggered by the exchange field via the Rashba-Zeeman, Dresselhaus-Zeeman or Rashba-Dresselhaus-Zeeman 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 first-principles calculations. We hope that our work will enrich the Rashba-Zeeman physics and stimulate experimental studies of the predicted phenomenon.

DFT calculations
Our atomic and electronic structure calculations were performed using the projector-augmented 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 Hubbard-U correction U eff = 2 eV on Cr-d orbital 51 were adopted throughout. A 4 × 4 × 1 k-point grid for Brillouin zone integration was used for structural relaxation and a 10 × 10 × 1 grid was used for self-consistent electronic structure calculations. The optimized in-plane lattice constant of 4.86 Å for Ag 2 Te was found close to the optimized in-plane 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 in-plane 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 DFT-D3 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.