Wide-range ideal 2D Rashba electron gas with large spin splitting in Bi₂Se₃/MoTe₂ heterostructure

Abstract

An application-expected ideal two-dimensional Rashba electron gas, i.e., nearly all the conduction electrons occupy the Rashba bands, is crucial for semiconductor spintronic applications. We demonstrate that such an ideal two-dimensional Rashba electron gas with a large Rashba splitting can be realized in a topological insulator Bi₂Se₃ ultrathin film grown on a transition metal dichalcogenides MoTe₂ substrate through first-principle calculations. Our results show the Rashba bands exclusively over a very large energy interval of about 0.6 eV around the Fermi level within the MoTe₂ semiconducting gap. Such a wide-range ideal two-dimensional Rashba electron gas with a large spin splitting, which is desirable for real devices utilizing the Rashba effect, has never been found before. Due to the strong spin–orbit coupling, the strength of the Rashba splitting is comparable with that of the heavy-metal surfaces such as Au and Bi surfaces, giving rise to a spin precession length as small as ~10 nm. The maximum in-plane spin polarization of the inner (outer) Rashba band near the Γ point is about 70% (60%). The room-temperature coherence length is at least several times longer than the spin precession length, providing good coherency through the spin processing devices. The wide energy window for ideal Rashba bands, small spin precession length, as well as long spin coherence length in this two-dimensional topological insulator/transition metal dichalcogenides heterostructure pave the way for realizing an ultrathin nano-scale spintronic device such as the Datta–Das spin transistor at room-temperature.

Introduction

The idea of spin field-effect transistor (SFET) proposed by Datta and Das¹ opens up a route to spin information processing.² It exploits the Rashba effect, which describes lifting the spin degeneracy of electronic bands through the space inversion asymmetry.^3,4,5 In a heterojunction without reflection symmetry with respect to the operation z→−z, z is parallel to the interface normal, the spin–orbit coupling term can be expressed as H _R = a _R (k × $\hat{z}$)·σ, where σ is the Pauli matrix vector and α _R is the Rashba parameter representing the strength of the Rashba effect. The spin-splitting electronic bands, named Rashba bands (RBs), given from the Hamiltonian diagonalization can be expressed as⁶

$${E}_{\pm }({\bf{k}})=\frac{{\hbar }^{2}{k}^{2}}{2{m}^{\ast }}\pm {\alpha }_{{\rm{R}}}k,$$

(1)

where k is the magnitude of the electron wave vector and m* the electron effective mass. The in-plane spin polarizations of the “+” and “−” eigenstates are oppositely aligned and normal to the electron wave vector. The Rashba splitting has been observed in many two-dimensional (2D) systems such as the III–V semiconductor heterostructures,^7,8,9 the metal/semiconductor heterostructures,^{10, 11} and the heavy (high-Z) metal films or surfaces.^{12,13,14,15,16,17,18,19,20,20} The Rashba parameter of the heavy metal systems is generally one order of magnitude larger than that of the III–V semiconductor heterostructures due to the strong spin–orbit coupling of high-Z atoms. For example, the heavy-metal alloy Bi/Ag(111) exhibits a largest Rashba spin splitting of α _R ≈ 3.05 eVÅ.¹⁷

For a practical application of these RBs to spintronic devices, two requirements have to be satisfied: (i) a large Rashba splitting and (ii) an ideal Rashba electron gas (REG), i.e., except for the RB states, there is no other electronic states near the Fermi level. A large Rashba splitting mainly comes from the strong spin–orbit coupling of high-Z atoms, which are either metals or semimetals. Therefore, the two requirements, to some extent, are not easily satisfied at the same time. This is the reason why the Datta–Das spin transistor proposed in 1990,¹ which utilizes the Rashba splitting for spin information processing, is finally realized experimentally in 2015.²¹ For the III–V semiconductor heterostructures,^7,8,9 they satisfy the second requirement, but their Rashba splitting is about one order of magnitude smaller than that of the systems composed of high-Z atoms. For the heavy metal surfaces^{12,13,14,15,16,17} and metal thinfilms (quantum well) grown on metal substrates,^18,19,20 there are always some metal bulk states around the Fermi level. Similar difficulties are also observed in metal/semiconductor junctions^{10, 11, 22} as well as in topological insulators.^{23,24,25,26,27} Here, we note that the spatial separation between the wave functions of the topological surface states (TSSs) localized on the opposite sides of the Bi₂Se₃ film will deteriorate the performance of the SFET, which is operating through the interference of the wave functions between the two RBs.¹ In the extreme case that the wave functions of the TSSs localized on the opposite sides of the Bi₂Se₃ film are completely separate, the electron spins do not precess because the spin direction of the electrons on each side of the film aligns along the effective magnetic field.^{2, 5, 6} To our best knowledge, only the Te-terminated surface of the polar semiconductor BiTeX (X = I, Br, and Cl)^28,29,30 and the TiO₂-terminated surface of the SrTiO₃(001) crystal³¹ simultaneously satisfy the two requirements with the largest Rashba spin splitting of α _R ≈ 3.8 eVÅ in BiTeI.²⁸ However the energy interval of the ideal RBs in these bulk-based systems are ~0.25 eV at most.

The 2D materials, such as the V–VI semiconductors M₂N₃ (M = Bi, Sb; N = Te, Se) and the transition metal dichalcogenides XY₂ (X = Mo or W; Y = S, Se, or Te), have attracted tremendous interests in recent years. These materials have been demonstrated its capability to be integrated into electronic devices and open a new perspective in device development.^{32, 33} In this work we investigate the electronic structure of a quintuple layer (1QL)-Bi₂Se₃ on a one-trilayer (1TL) MoTe₂ through first-principle calculations. As shown in Fig. 1 and will be discussed later, a very good lattice match between the Bi₂Se₃ and the MoTe₂ films can be realized in a quite small supercell. We find, in a wide energy range (~0.6 eV), except for the RB states, there is no other electronic states within this MoTe₂ semiconducting gap. The Rashba parameter is about one order of magnitude greater than that of the III–V semiconductor heterostructures.^7,8,9 This is, to our knowledge, the first report of an ideal 2D REG over a very wide energy window with a large Rashba splitting in ultrathin 2D materials. We find the RB states are mainly composed of Bi₂Se₃ orbitals and evolves into the TSSs as the thickness of the Bi₂Se₃ film increases. We also calculate the electronic structures of 1QL-Bi₂Se₃/nTL-MoTe₂, n = 1–4, heterostructures, which also exhibit the features of ideal 2D REG and large Rashba splittings. This shows that the RB states are robust and not deteriorated by the presence of a MoTe₂ substrate or a MoTe₂-terminated substrate. Although we only consider the Bi₂Se₃/MoTe₂ heterostructure in this work, similar behavior is expected in some of the other M₂N₃/XY₂ heterostructures.

Fig. 1

The optimized lattice of the 1QL-Bi₂Se₃/1TL-MoTe₂ heterostructure. a Top view of the Bi₂Se₃ part. b Top view of the MoTe₂ part. c Side view of the whole system. The supercell used in the calculations is denoted in a and b by the black solid line, in which the Bi₂Se₃-$(\sqrt{3}\times \sqrt{3})$ R30°/MoTe₂-(2 × 2) system is incorporated. The shaded regions in a and b denote, respectively, the Bi₂Se₃ and the MoTe₂ 1 × 1 unit cell. The corresponding Brillouin zone is shown in (d)

Results and discussion

The band structures of 1QL-Bi₂Se₃/1TL-MoTe₂ are shown in Fig. 2. The energy zero is set at the the Fermi level. The Bi₂Se₃ (MoTe₂) components and the spin polarizations of the electronic states are represented by the colors of the data points. As can be seen, the RBs are mainly composed of the Bi₂Se₃ orbitals, while the lower energy bands are mainly composed of MoTe₂ orbitals. The spin splitting of the RBs induced by the interface electric field extends over a very wide energy interval of ~0.6 eV ranging from ~0.04 eV, the K-point energy of the MoTe₂ band, to about 0.67 eV. More importantly, there is no other electronic states within this MoTe₂ semiconducting band gap. This wide-range ideal RBs are surely promising for real applications. The Rashba parameter can be approximately estimated by α _R = 2E _R/k ₀, in which E _R is the Rashba energy and k ₀ the wave number offset as indicated in Fig. 3a.¹⁷ In the present case, the deduced Rashba parameter is α _R = 0.71 eVÅ, which is about one order of magnitude larger than those of the cases of III–V semiconductor heterostructures α _R ~ 0.07 eVÅ,^7–9 and comparable with those of the cases of heavy metals such as Au(111) surface,¹² α _R ≈ 0.36 eVÅ, and Bi(111) surface,¹⁵ α _R  ≈ 0.56 eVÅ.

Fig. 2

Electronic structures of the 1QL-Bi₂Se₃/1TL-MoTe₂ junction, where a the Bi₂Se₃ and the MoTe₂ components, b–d spin polarization along the x, y and z directions are represented by the color of the dots. The yellow-shaded region in a indicates the energy window (over 0.6 eV) for ideal Rashba bands without any other electronic states

Fig. 3

a The zoom-in of the electronic structure of the 1QL-Bi₂Se₃/1TL-MoTe₂ heterostructure near the Γ point. The Rashba energy E _R, the wave number offset k ₀, and the WND between the two RBs Δk are indicated. b The WND between the two RBs Δk and the corresponding spin precession length L as functions of energy. The inset shows the coherence length l _c and the ratio of the l _c to the spin precession length as functions of energy

In a Datta–Das SFET, when electrons in the channel move a distance d, the electron spins precess by an angle of Δkd, where Δk is the wave number difference (WND) between the two RBs as indicated in Fig. 3a.¹ Therefore the spin flips when the electron wave propagates over the precession length L = π/Δk. As shown in Fig. 3b, the WND Δk decreases monotonically with increasing energy, while the spin precession length L increases approximately linearly with energy. Due to the strong spin–orbit coupling, the strength of the Rashba splitting is comparable with that of the heavy-metal surfaces such as Au and Bi surfaces, giving rise to a spin precession length as small as 10 nm. In comparison with the micrometer scale Datta–Das spin transistor realized in 2015,²¹ the large spin splitting and small precession length in this ultrathin 1QL-Bi₂Se₃/1TL-MoTe₂ heterostructure pave the way to achieve the spintronic device at the nanoscale. Similar behavior is also expected in different ultrathin TI/TMD heterostructures, providing high flexibility for real applications.

For the ideal 2DEG model, the RBs are perfectly parabolic as shown in Eq. (1). Since the WND and the spin precession length L are energy-independent, or equivalently the Rashba spin splitting is linear-in-k, all the electron spins precess coherently by the same angle during transport. However, in practical cases, the Rashba band dispersion deviates from the ideal parabolic form, the WND as well as the spin precession length are thus energy-dependent. This will cause spin precession incoherency of the electric current in a SFET and make the device ineffective. The ideal Rashba dispersion shown in Eq. 1 has two features, the linear-in-k Rashba spin splitting and the energy-independent WND. It would be more meaningful to investigate the deviation from the energy-independent WND than the deviation from the linear-in-k Rashba spin splitting because the Datta–Das transistor is operating through the wave-function interference between the two RB states of the same energy with different wave numbers.¹ To quantitatively describe the spin coherency of conduction electrons, which mainly distribute over the thermal energy range k _B T around the Fermi level, we define the coherence length l _c as

$${l}_{{\rm{c}}}(E,T)=\left|\frac{1}{\Delta k(E)-\Delta k(E+{k}_{{\rm{B}}}T)}\right|\mathrm{.}$$

(2)

The coherence length l _c is the electron propagation distance over which the difference of the precession angles between any two electrons within the energy range k _B T is smaller than unity, i.e., 57°. Here, for simplicity, we assume the magnitude of Δk varies monotonically with energy in an energy range k _B T. The room-temperature (k _B T = 25.9 meV) coherence length l _c obtained by Eq. (2) and the ratio of the l _c to the spin precession length L are plotted in the inset of Fig. 3b. As can be seen, the coherence length is about 0.1 μm, and at least several times larger than the spin precession length of about 10 nm, guaranteeing the coherency of all the electrons transporting over the precession length.

The x, y, and z components of spin polarization are shown in Fig. 2b, c, and d, in which the Γ–K direction is along the x direction and the Γ–M direction along the y direction as indicated in Fig. 1. For a free-standing Bi₂Se₃ film, the crystal structure has inversion symmetry and belongs to $P\overline{3}m1$ $({D}_{3d}^{3})$ space group. When the MoTe₂ film is incorporated, i.e., Bi₂Se₃/MoTe₂ heterostructure, the symmetry is reduced to P3 $({C}_{3}^{1})$ with broken inversion symmetry. As a result, the spin degeneracies of the Bi₂Se₃ branches are lifted, forming the conduction Bi₂Se₃ RBs with helical spin polarizations as evidenced by Fig. 2b and c. While the out-of-plane spin polarizations of the RBs in Fig. 2d are nearly zero around the Γ point and otherwise increases due to the warping effect. As for the top valence bands given from the 1TL-MoTe₂, the band dispersion as well as the spin polarization (Fig. 2) are similar to the standard free standing transition-metal dichalcogenides cases.³⁴

The spin texture of the RBs of energies 0.1, 0.3, and 0.5 eV is shown in Fig. 4a. The band dispersion is nearly isotropic below ~0.3 eV, and it exhibits significant hexagonal warping around 0.5 eV. The in-plane spin polarization (arrow) of the inner RB and the outer RB circulates the energy contour oppositely, keeping the overall time-reversal symmetry. While the out-of-plane spin polarization (color of the arrow) warps within ±30% with zero values at K points due to the reflection symmetry about the x–z plane in the Bi₂Se₃ layer. As shown in Fig. 4b, the magnitude of the in-plane spin polarization (solid line) is nearly a constant for electrons of the same energy (in between the dotted lines), and much greater than that of the azimuthal average of the out-of-plane spin polarization (dashed line). For the inner (outer) RB near the Γ point, the maximum values is about 70% (60%), which is comparable with that of the Bi₂Se₃ TSSs reported in refs. 35 and 36. It decreases slowly to about 50% as the energy increases to 0.5 eV. These less-than-unity spin polarization is due to the spin–orbit entanglement, by which the spin polarization contributed from different orbitals could cancel each other.^{35, 36}

Fig. 4

a Spin texture of the 1QL-Bi₂Se₃/1TL-MoTe₂ heterostructure. The in-plane spin-polarization directions S _|| are indicated by the arrows. The out-of-plane spin polarizations S _⊥ are represented by the color of the arrows. b The azimuthal average of the magnitude of the spin polarization. The blue (red) lines are those of the inner (outer) Rashba branch, and the solid (dashed) lines are those of the in-plane (out-of-plane) component. The maximum and minimum magnitudes of the in-plane spin polarization are indicated by the dotted lines

Figure 5a shows the optimized lattice structure of the 2QL-Bi₂Se₃/1TL-MoTe₂ heterostructure, in which the two Bi₂Se₃ QLs stack in the usual ABC fashion. Bi₂Se₃ is theoretically predicted^{37, 38} and experimentally observed^{37, 39} to be a three-dimensional topological insulator with a single Dirac cone. Therefore, through increasing the Bi₂Se₃ thickness, we can find the Dirac-like surface states as shown in Fig. 5b, in which there are two Dirac cones with the Dirac points at energies −29 and −212 meV. The two Dirac cones correspond to surface states on the interface side and the surface side of the Bi₂Se₃ film.^{27, 40} The higher energy interface Dirac state (inner RB) shows strong hybridization between the Bi₂Se₃ and the MoTe₂ orbitals. While the lower energy surface Dirac state (outer RB) is mainly composed of the Bi₂Se₃ orbitals. Due to the finite size effect,^{41, 42} the coupling between the interface states and the surface states opens a small gap of ~30 meV. For a thicker Bi₂Se₃ film (≥6QL), the interface states and the surface states evolve into gapless and spatially separate TSSs. This would cause the SFET losing its ability to modulate the electric current because each of the surface and interface states has electron spin aligned with the effective magnetic field on different side of the Bi₂Se₃ film. Further details about the thickness dependence and the charge density profiles of the RBs can be found in S6 of the Supplementary Information. Moreover additional Bi₂Se₃ QWSs emerge with band edge at 238 meV due to the increasing film thickness. Further restricted by the MoTe₂ valence band maximum of 38 meV at K, the energy range of the ideal 2D REG is suppressed to only ~0.2 eV. Similar features (i.e., the Dirac cones and the QWSs) can be found in 3QL-Bi₂Se₃/1TL-MoTe₂ and 2QL-Bi₂Se₃/2TL-MoTe₂ heterostructures as shown in Figure S3 of the Supplementary Information. Figure 5c shows the corresponding WND and the spin precession length as functions of energy where the unshaded region indicate the energy range of the ideal 2D REG. They are comparable with those of 1QL-Bi₂Se₃/1TL-MoTe₂. The inset shows the room-temperature coherence length. In the energy region from 0.16 to 0.21 eV, the coherence length is more than 50 times longer than the spin precession length.

Fig. 5

a Side view of the 2QL-Bi₂Se₃/1TL-MoTe₂ heterostructure. b Band structures of the 2QL-Bi₂Se₃/1TL-MoTe₂. The Bi₂Se₃ and the MoTe₂ components are represented by the color of the dots. The yellow-shaded region indicates the energy region for ideal RB states. c The WND between the two RBs Δk of the 2QL-Bi₂Se₃/1TL-MoTe₂ heterostructure and the corresponding spin precession length L as functions of energy. The inset shows the coherence length l _c and the ratio of the l _c to the spin precession length as functions of energy. The shaded regions indicate the energy regions where the other states (high-energy quantum well states or the MoTe₂ states near the K point) are involved. d–f are the counterparts of a–c, respectively, for the case of the 1QL-Bi₂Se₃/2TL-MoTe₂ heterostructure

To investigate the substrate effect, we consider the case of 1QL-Bi₂Se₃/2TL-MoTe₂. The optimized structure is illustrated in Fig. 5d, in which the two MoTe₂ TLs are in the 2H stacking sequence. Shown in Fig. 5e, the MoTe₂ band at Γ is higher than that at K and is 123 meV higher than the RB minimum. Similar valence-band maximum shift from K to Γ upon increasing TMD film thickness has been reported.⁴³ The resultant energy window for the ideal RBs, the spin precession length, as well as the coherence length remain comparable with those of the 1QL-Bi₂Se₃/1TL-MoTe₂ case. We also carry out similar calculations for 1QL-Bi₂Se₃/3TL-MoTe₂ and 1QL-Bi₂Se₃/4TL-MoTe₂ (see Supporting Information Fig. S1) and obtain similar results. This indicates that these RBs are robust and nearly free from the influence of the substrate if the 1QL-Bi₂Se₃ film is grown on the MoTe₂ or MoTe₂-terminated substrate.

In summary, we demonstrate that ideal 2D REG can be established in a 1QL-Bi₂Se₃ film grown on the MoTe₂ or MoTe₂-terminated substrate. Our calculations show a very wide energy range (~0.6 eV) for the TI-RBs exclusively around the Fermi level within the TMD semiconducting gap. The large spin splitting in the RBs is comparable with that of other heavy metal surface states, leading to a spin precession length as small as ~10 nm. The maximum in-plane spin polarization is about 70% (60%) for the inner (outer) RB, similar to the typical TI ones. For 1QL-Bi₂Se₃/nTL-MoTe₂, n = 1–4 studied in this work, we find the RBs very robust against the variation of the thickness of the MoTe₂ film. While increasing Bi₂Se₃ film thickness would suppress the energy window of the ideal REG and the performance of the SFET. The room-temperature coherence length of all the systems studied in this work is at least several times longer than the spin precession length. Thus the REG in our proposed ultrathin TI/TMD heterostructure will propagate coherently over the spin precession length, achieving the room-temperature 2D Datta–Das spin transistor at the nano-scale.

Methods

The calculations are performed in the framework of density functional theory by using Vienna ab-initio simulation package with the Perdew-Burke-Ernzerhof generalized gradient approximation.^44,45,46 The electron ion interaction is described by projected augmented wave potentials.⁴⁷ The spin–orbit coupling is taken into account. Because of the lattice mismatch between the Bi₂Se₃ and MoTe₂ films, we adopt supercell calculations. As shown in Fig. 1, the supercell contains 1QL of Bi₂Se₃-$(\sqrt{3}\times \sqrt{3})$ R30° and 1TL of MoTe₂-(2 × 2). The resulting lattice mismatch is 2%. A 7 × 7 × 1 Γ-centered Monkhorst-Pack k-mesh is used for the integration of Brillouin zone.⁴⁸ The cutoff energy for plane wave basis is set as 400 eV. All of the atomic positions as well as the lattice are fully relaxed until the magnitude of the force acting on all atoms is less than 0.005 eVÅ⁻¹ and the total energy converges within 10⁻⁷ eV. The van der Waals interactions based on the DFT-D3 method with Becke-Jonson damping^{49, 50} are included in the structure relaxation. The optimized interlayer spacing between the Bi₂Se₃ and the MoTe₂ layer is 3.33 Å.