Tunable Giant Rashba-type Spin Splitting in PtSe$_2$/MoSe$_2$ Heterostructure

TL;DR

This study demonstrates a giant, tunable Rashba-type spin splitting in PtSe₂/MoSe₂ heterostructures, with potential applications in spin field-effect transistors, achieved through first-principles calculations revealing strong interfacial spin-orbit coupling.

Contribution

The paper introduces a large, tunable Rashba spin splitting in a 2D heterostructure, supported by first-principles calculations, and proposes a novel spintronic device model utilizing this effect.

Findings

Spin splitting energy of 110 meV at the Γ point.

Generalized Rashba constant η_R as large as 1.3 eV·Å.

Effective tuning of η_R via biaxial strain and electric field.

Abstract

We report a giant Rashba-type spin splitting in two-dimensional heterostructure PtSe$_2$/MoSe$_2$ with first-principles calculations. We obtain a large value of spin splitting energy 110 meV at the momentum offset $k_0$=0.23 \AA$^{-1}$ around $\mathrm{\Gamma}$ point, arising from the emerging strong interfacial spin-orbital coupling induced by the hybridization between PtSe$_2$ and MoSe$_2$. Moreover, we find that the band dispersion close to valence band maximum around $\Gamma$ point can be well approximated by the generalized Rashba Hamiltonian $H(k_{||})=-\frac{\hbar^2 k_{||}^2}{2m}+c k_{||}+\alpha_R \vec{\sigma}\cdot(\vec{k}_{||} \times \vec{z})$. It is found that the generalized Rashba constant $\eta_R=c+\alpha_R$ in PtSe$_2$/MoSe$_2$ is as large as 1.3 eV$\cdot\text{\AA}$, and importantly $\eta_R$ can be effectively tuned by biaxial strain and external out-of-plane electrical field, presenting a potential application for the spin field-effect transistor. In addition, with the spin-valley physics at $\mathrm{K}/\mathrm{K}'$ points in monolayer MoSe$_2$, we propose a promising model for spin field-effect transistor with opto-valleytronic spin injection based on PtSe$_2$/MoSe$_2$ heterostructure.

Equations

$$H(k_{||})=-\frac{{\hslash }^2{k}_{||}^2}{2m}+c{k}_{||}+{\alpha }_R\vec{\sigma }\cdot ({\vec{k}}_{||}\times \vec{z}),$$

Full text

Finding materials with strong spin-orbit coupling (SOC) is at the heart of the research field of spin-orbitronics.Manchon et al. (2015) For example, the Datta-Das spin field-effect transistor (SFET) Datta and Das (1990) requires the channel material with large Rashba-type spin splitting and electrically tunable Rashba constantBychkov and Rashba (1984) to effectively modulate the electron current by steering the spin precession. The original Rashba Hamiltonian for two-dimensional (2D) electron gas is given by $H_{R}=\pm\frac{\hbar^{2}k_{||}^{2}}{2m}+\alpha_{R}\vec{\sigma}\cdot(\vec{k}_{||}\times\vec{z})$, in which the Rashba constant $\alpha_{R}$ represents the strength of SOC and $\vec{\sigma}$ the vector of Pauli matrices. In the past decades, considerable theoretical and experimental efforts have been devoted to finding Rashba SOC at interfaces or surfaces. The Rashba-type spin splitting was observed in III-V semiconducting heterostructure InGaAs/InAlAsNitta et al. (1997) with tiny Rashba constant ($\alpha_{R}$=$0.001$ eV$\cdot$Å). The surface states of heavy metals (e.g., Au,LaShell, McDougall, and Jensen (1996) Bi,Koroteev et al. (2004); Hirahara et al. (2006) Ir,Varykhalov et al. (2012) PbDil et al. (2008)) and topological insulator Bi2Se3King et al. (2011); Zhu et al. (2011) exhibited large Rashba-type spin splittings. In addition, surface alloys, by doping heavy atoms at the surface of metals (e.g., Bi/Ag(111)Ast et al. (2007)) or semiconductors (e.g., Bi/Si(111)Gierz et al. (2009) and Cs/InSb(110)Bindel et al. (2016)), provided an effective strategy to generate giant Rashba-type spin splitting. Typically, these observed Rashba-type surface states are accompanied with trivial surface states and spin degenerate bulk states, which hinders their practical applications. Recently, giant Rashba-type spin splitting is also observed in bulk BiTeIIshizaka et al. (2011) and GeTe.Di Sante et al. (2013); Liebmann et al. (2016) However, it is unclear whether these effects can survive when the bulk materials are downsized to a few atomic layers.Ma et al. (2014) After the discovery of graphene, 2D materials

and their van der Waals heterostructuresAjayan, Kim, and Banerjee (2016); Novoselov et al. (2016); Kasai et al. (2018) offer new opportunities to search for materials with Rashba SOC.Manchon et al. (2015) For example, the Rashba effect was predicted in the monolayer LaOBiS2 Liu, Guo, and Freeman (2013) and BiSb Singh and Romero (2017). By stacking Bi2Se3 ultrathin film on MoTe2 substrate, Wang et at.Wang and Jeng (2017) predicted a wide-range Rashba electron gas. The Rashba effect was also predicted in other van der Waals heterostructures.Singh and Romero (2017); Zhang and Schwingenschlögl (2018) Despite these important progresses, challenges still exist in these predicted systems, such as the accompanying non-Rashba states, weak tunability of Rashba constant and small Rashba energy $E_{R}$ (for example, $8$ meV in Bi2Se3/MoTe2Wang and Jeng (2017), $5$ meV in BiSb/AINSingh and Romero (2017).). Therefore, further efforts are required to search for appropriate materials with tunable Rashba SOC and large spin splitting.

Very recently, the semiconducting T-phase transition metal dichalcogenide (TMDC) PtSe2 monolayer has attracted much attention due to its superior transport properties.Wang et al. (2015); Zhao et al. (2017) Moreover, Yao et al.Yao et al. (2017) observed the spin-layer locking phenomenaZhang et al. (2014); Yuan et al. (2019) induced by local dipole field in this material, which manifests the significant role of SOC. However, the spin degeneracy protected by inversion symmetry in PtSe2 impedes its applications to SFET. To lift the degeneracy the structural inversion asymmetry should be introduced, for example, by substitutional doping.Absor et al. (2018) However, we note that building van der Waals heterostructure is more favorable in experiment compared to the doping method. On the other hand, monolayer H-phase TMDCs, such as MoSe2, have been exfoliated successfully and also possess strong SOC.Chang et al. (2014); Reyes-Retana and Cervantes-Sodi (2016) Therefore, the heterostructure stacked by T-phase and H-phase TMDCs monolayer, with broken inversion symmetry, provides a promising platform for studying spin physics. In this work, we investigate the electronic properties of the heterostructure PtSe2/MoSe2 through first-principles calculations. We demonstrate that a generalized Rashba-type electron system can be formed and effectively tuned by exerting biaxial strain and external out-of-plane electrical field, providing a promising material for SFET application.

All the calculations are performed with density functional theory (DFT) by the projector-augmented wave (PAW) method implemented in the Vienna Ab initio Simulation Package (VASP)Kresse and Joubert (1999). The generalized gradient approximation as parameterized by Perdew, Burke, and Ernzerhof (PBE)Perdew, Burke, and Ernzerhof (1996) is employed. The kinetic energy cutoff $460$ eV and a $\mathrm{\Gamma}$-centered $6\times 6\times 1$ k-mesh in $2$D Brillouin zone have been used for the convergence of total energy with a criterion of 1$\times$10*-6* eV per atom. The SOC is not included during the geometry optimization but is added in the electronic structure calculations. The lattice constants of monolayer MoSe2 and PtSe2 are 3.286 Å and 3.780 Å, respectively. To minimize the artificial internal strain caused by lattice mismatch, the heterostructure is built from $\sqrt{3}\times\sqrt{3}\times 1$ supercell of PtSe2 and $2\times 2\times 1$ supercell of MoSe2, as shown in FIG.1 (I). A vacuum space of 20 Å is adopted to avoid the interaction between two periodic slabs along $[001]$ direction. The DFT-D3Grimme et al. (2010) method has been applied to simulate the van der Waals interaction between MoSe2 and PtSe2 (see TABLE S1 in the supplement for more structural information).

In FIG.2 (I-III), we present the band structures for monolayer MoSe2, PtSe2, and their heterostructure without/with SOC. The Fermi level is set to valence band maximum (VBM) in all band structures. As shown in FIG.2 (I)-(II), the pristine monolayers MoSe2Zhu, Cheng, and Schwingenschlögl (2011) and PtSe2Wang et al. (2015) are direct and indirect semiconductors, respectively. When SOC is considered, the bands of monolayer MoSe2 (PtSe2) are split (not split) along $\mathrm{M}$-$\mathrm{K}$-$\mathrm{\Gamma}$ $k$-path due to the absence (presence) of inversion symmetry. Because the heterostructure PtSe2/MoSe2 is formed by the interlayer van der Waals interaction which is relatively weaker than intralayer bonding, therefore, the band structure of PtSe2/MoSe2 should preserve most of the characteristics of its constituent monolayers. As expected, the heterostructure PtSe2/MoSe2 retains the semiconducting nature with an indirect band gap $0.89$ eV when SOC is considered, as shown in FIG.2 (III). In addition, the heterostructure also preserves the large spin splitting (181 meV) at $\mathrm{K}$ point with a slight reduction in comparison with that in monolayer MoSe2 (186 meV as seen in FIG.2 (I)).

Interestingly, an emerging spin splitting around $\mathrm{\Gamma}$ point is observed, which does not appear in the monolayer MoSe2 and PtSe2. To unveil its physical origins, we firstly investigate the layer-projected band structure, as shown in FIG.2 (IV). The contribution from the layer MoSe2 and PtSe2 are indicated by red and blue circles, respectively. From the projected band structure, we find an important hybridization from two constituent monolayers at the valence band edge (VBE) around $\mathrm{\Gamma}$ point. Compared to previous studies of bilayer MoS2 and heterostructure MoSe2/MoS2Kang et al. (2013); Komsa and Krasheninnikov (2013); Su et al. (2016), in which the hybridization around $\Gamma$ point is dominated by Mo-$d$ orbitals (see FIG.S2 in the supplement.), the observed hybridization around $\Gamma$ point in PtSe2/MoSe2 displays a distinct feature: it is mainly contributed by Mo-$d$ orbitals in MoSe2 and Se-$p$ orbitals in PtSe2. Furthermore, to capture more details of the band splitting around $\mathrm{\Gamma}$ point, the band decomposed charge densities are plotted, as shown in FIG.2 (V-VII). For monolayer PtSe2 and MoSe2, the charge density around $\Gamma$ point has an equal contribution from top and bottom Se layers. Therefore, the total vertical electrical field acting on the electrons is canceled and no band splitting can be observed. However, for PtSe2/MoSe2 heterostructure as shown in FIG.2 (VII), the charge density around $\mathrm{\Gamma}$ point, mainly contributed by $d_{z^{2}}$ of Mo in MoSe2 and $p_{z}$ of Se in PtSe2, displays an asymmetric feature due to the interlayer hybridization, resulting in an effective vertical electric field to induce strong SOC at the interface. As for the band splitting at $\mathrm{K}$ point, almost no hybridization between two monolayers is found and the contribution is dominated by $d_{xy/x^{2}-y^{2}}$ orbitals of Mo atoms, as shown in FIG.2 (IV) and (VIII).

By zooming in the two highest valence bands in FIG.3 (I) (marked by the black rectangular box in FIG.2 (III)), we note that our results around $\mathrm{\Gamma}$ point exhibit close resemblance to the Rashba-type spin splitting in semiconductor quantum wells and surfaces of heavy metals.Nitta et al. (1997); LaShell, McDougall, and Jensen (1996); Koroteev et al. (2004); Hirahara et al. (2006); Varykhalov et al. (2012); Ast et al. (2007); Gierz et al. (2009) To confirm that the observed band splitting is Rashba-type, we calculate the spin components $s_{x}$, $s_{y}$ and $s_{z}$ at a constant energy cut, indicated by the blue dotted horizontal line in FIG.3 (I). As illustrated in FIG.3 (II)-(IV), the spin polarizations are mainly in-plane: $s_{x}$ and $s_{y}$ are much larger than $s_{z}$. Because the blue dashed line is crossing the upper splitting band, the inner contours show the same helical spin polarization as the outer contour, as shown in FIG.3 (II) and (III). These spin projections display the iconic features of the Rashba effect. The small out-of-plane spin component $s_{z}$ in FIG.3 (IV) is owing to the small in-plane potential gradient, which is also responsible for the hexagonal shape of the outer contour.LaShell, McDougall, and Jensen (1996); Koroteev et al. (2004)

As one can find in FIG.3 (I), there are two key properties at VBE around $\Gamma$ point: (1) the shape of the band is very close to parabolic; (2) the spin texture is almost that given by original Rashba Hamiltonian. Therefore, a generalized Rashba Hamiltonian is proposed to capture the physics for energies close to VBE around $\Gamma$ point (from about $-$0.1 eV to 0 eV), as indicated by the red dashed line in FIG.3 (I),

[TABLE]

where the first two terms gives rise to the so-called ”sombrero hat” dispersionRybkovskiy, Osadchy, and Obraztsova (2014) (see the purple line around $\Gamma$ in FIG.2(III)). Here, $c$ and $\alpha_{R}$ characterize the strength of the spin-independent interaction with crystal field and Rashba-type SOC, respectively. Based on Eq.1 the states close to VBE are suitable for the application of SFET (see the supplement). We introduce a generalized Rashba constant $\eta_{R}$=$c$+$\alpha_{R}$ which determines the differential phase shiftDatta and Das (1990). Importantly, $\eta_{R}$ can still be estimated by $\eta_{R}$=$2E_{R}/k_{0}$ with $E_{R}$ the generalized Rashba energy and $k_{0}$ the momentum offset. From FIG.3 (I), we obtain $\eta_{R}$=1.3 eV$\cdot$Å by reading $E_{R}$=150 meV and $k_{0}$=0.23 Å*-1*. The obtained $\eta_{R}$ for PtSe2/MoSe2 heterostructure is one of the largest among the Rashba constants observed in previous studies.Nitta et al. (1997); LaShell, McDougall, and Jensen (1996); Koroteev et al. (2004); Hirahara et al. (2006); Varykhalov et al. (2012); Ast et al. (2007); Gierz et al. (2009) Moreover, the spin splitting energy at the momentum offset $k_{0}$ of VBM around $\Gamma$ point is giant, as large as 110 meV. In addition, by calculating the whole family of PtX2/MX2 (M=Mo, W; X=S, Se, Te) heterostructures, we find the PtSe2/MoSe2 is the most promising candidate for SFET application (see the supplement).

Next, we investigate the tunability of electronic properties of PtSe2/MoSe2 heterostructure under biaxial strain and external electric field. At first, we impose biaxial strains from $\delta$=$-$1.5% (compression) to $+1.5\%$ (tension) on the heterostructure, in which the strength of the strain is defined as $\delta=(a-a_{0})/a_{0}$ with $a_{0}$ the original lattice constant. For different strains, the electronic structures are calculated after all atoms are fully relaxed. In particular, we investigate the generalized Rashba constant $\eta_{R}$ and the relative energy shift $\Delta{E}=E^{\Gamma}_{VBM}-E^{K}_{VBM}$ between the VBMs around $\mathrm{\Gamma}$ point and at $\mathrm{K}$ point, as shown in FIG.4 (I). It is found that the generalized Rashba constant $\eta_{R}$ (in blue squares) is changed from 1.62 eV$\cdot$Å to 1.00 eV$\cdot$Å as $\delta$ is changed from $-$1.5% to +1.5%. In the meanwhile, $\Delta{E}$ can be increased to $+0.1$ eV by applying $+1.5\%$ tensile strain. Thus, imposing tensile strain provides an efficient approach to generate an available energy interval lived by only Rashba-type states. When applying an external electric field from $-$1.5 V/nm to 1.5 V/nm along $\vec{z}$ direction, the $\eta_{R}$ and $\Delta{E}$ are reduced at the same time, as shown in FIG.4 (II), which means the external electric field also can promote the VBM near $\mathrm{\Gamma}$ point above that at $\mathrm{K}$ point. Furthermore, the generalized Rashba constant $\eta_{R}$ exhibits a notable change (24%) from 1.46 eV$\cdot$Å to 1.18 eV$\cdot$Å when the external electric field is changed from $-$1.5 V/nm to 1.5 V/nm, indicating that the spin precession can be effectively tuned with a gate voltage. Since the difference of spin precession angle is given byDatta and Das (1990); Chuang et al. (2015) $\Delta{\theta}=2m\Delta{\eta_{R}}L/\hbar^{2}$ ($\Delta{\eta_{R}}$ represents the change of the generalized Rashba constant, and $L$ the distance between spin injector and detector), the appreciable tunability of $\eta_{R}$ is crucial for SFET application. Note that the spin precession angle is independent of energy,Datta and Das (1990) the upper split band (from $-$0.1 eV to 0 eV) around $\Gamma$ point can be utilized to realize the SFET with a light hole doping. The competing states from $K$ point for the upper split band can be reduced by tensile strain.

Finally, based on the tunable giant Rashba-type spin splitting in PtSe2/MoSe2 heterostructure and the valley physics from H-phase monolayer MoSe2, we propose a SFET with PtSe2/MoSe2 as illustrated in FIG.4 (III). Similar to the SFET designed by Luo et al.,Luo et al. (2017) the valley polarization at $\mathrm{K}/\mathrm{K}^{\prime}$ points in MoSe2 is utilized to achieve spin injection, the spin manipulation is achieved with Rashba SOC in PtSe2/MoSe2 and finally a ferromagnetic contact is used to detect the spin. We estimate the minimum channel length (7.1 nm) of the SFET with two different electric fields ($\pm$1.5 V/nm) under $1.5\%$ tensile strain. 111The change of the effective mass $m$ is minor, we use the average value $m=0.81m_{e}$ in our estimation. Since monolayer PtSe2 Wang et al. (2015) and MoSe2 Chang et al. (2014) have already been exfoliated and proven to be air-stable at room temperature, it should be feasible to prepare PtSe2/MoSe2 heterostructure in experiment.

In summary, we investigate the electronic properties of the PtSe2/MoSe2 heterostructure through density functional theory calculations. A giant Rashba-type spin splitting (110 meV at the momentum offset $k_{0}$=0.23 Å*-1* around $\mathrm{\Gamma}$ point) near Fermi level is predicted, arising from the emerging strong interfacial SOC. A generalized Rashba Hamiltonian is introduced to describe the Rashba-type physics at VBE around $\Gamma$ point. We demonstrate that both biaxial strain and external electrical field can be utilized to achieve a 2D Rashba electron system. Furthermore, the generalized Rashba constant $\eta_{R}$ in PtSe2/MoSe2 is as large as 1.3 eV$\cdot\text{\AA}$, and can be tuned effectively by the external electrical field. Combining the tunable giant Rashba-type spin splitting around $\mathrm{\Gamma}$ point in PtSe2/MoSe2 and the valley-physics from H-phase monolayer MoSe2, we propose a potential application for SFET.

supplementary material

The supplementary material contains four sections: (I) the structure information of all heterostructures; (II) the generalized Rashba Hamiltonian; (III) the orbital hybridization properties of different heterostructures; (IV) the estimation of channel length.

Acknowledgements.

Q.Z. thanks the financial support from NSFC with grant No.11704121, Y.K. thanks the support from NSFC with grant No.11874265 and ShanghaiTech start-up. The authors thank HPC platform of ShanghaiTech University for providing computational facility.