Tunable Electronic Properties and Large Rashba Splittings Found in Few-Layer Bi$_2$Se$_3$/PtSe$_2$ Van der Waals Heterostructures

We use first-principles calculations to show that van der Waals (vdW) heterostructures consisting of few-layer Bi$_2$Se$_3$ and PtSe$_2$ exhibit electronic and spintronics properties that can be tuned by varying the constituent layers. Type-II band alignment with layer-tunable band gaps and type-III band alignment with spin-splittings have been found. Most noticeably, we reveal the coexistence of Rashba-type spin-splittings (with large $\alpha_{\rm R}$ parameters) in both the conduction and valence band stemming from few-layer Bi$_2$Se$_3$ and PtSe$_2$, respectively, which has been confirmed by spin-texture plots. We discuss the role of inversion symmetry breaking, changes in orbital hybridization and spin-orbit coupling in altering electronic dispersion near the Fermi level. Since low-temperature growth mechanisms are available for both materials, we believe that few-layer Bi$_2$Se$_3$/PtSe$_2$ vdW heterostructures are feasible to realize experimentally, offering great potential for electronic and spintronics applications.


Introduction
The rise of graphene undoubtedly served as a paradigm shift towards two-dimensional (2D) materials providing a versatile platform for future technological revolution. [1][2][3] Layer-by-layer assembly of transition metal dichalcogenides (TMDCs) soon after also enabled intelligent design possibilities for van der Waals (vdW) heterostructures of all dimensions revealing numerous exotic phenomena. [4][5][6][7][8][9] Amongst a plethora of known 2D systems, few-layer PtSe 2 recently created substantial scientific interest due to the report of a range of properties, such as a layer-dependent band gap, 10 high room-temperature electron mobility, 11 large stretchability, 12 which could be used in countless applications demonstrated for photocatalysis 13 and optoelectronics. 14 Different growth mechanisms have been adopted for high-quality PtSe 2 synthesis on various substrates including Pt(111), 15 silicon, 16 sapphire 17 and bilayer graphene/6H-SiC (0001) 18 to list a few.
Recently, complementary metal-oxide-semiconductor compatible large-scale fabrication of a trilayer PtSe 2 MOSFET has been demonstrated with a current ON/OFF ratio approaching 1600 at 80 K hinting at improvements in 2D nanoelectronics. 19 Moreover, monolayer PtSe 2 has been shown to host helical spin-texture and show local dipole induced Rashba effect with spin-layer locking which is advantageous for electrically controllable spintronics devices. 20 Heterostructures consisting of vertically stacked PtSe 2 /MoSe 2 show type II band alignment and interface states originating from the strong-weak interlayer coupling of the constituent systems. 21 Three-dimensional (3D) topological insulator Bi 2 Se 3 is also a promising material for spintronic applications owing to room-temperature spin-polarized surface currents, 22,23 efficient charge-to-spin conversion via doping, 24 giant spin pumping and inverse spin Hall effects through ferromagnetic contacts. 25 Moreover, few-layer Bi 2 Se 3 has been shown to demonstrate interesting properties, e.g., thickness-modulated semiconducting behavior 26 in comparison to its topologically insulating (TI) bulk counterpart. 27,28 Several novel functionalities have been enabled using few-layer Bi 2 Se 3 (e.g., coexistence of topological order and superconductivity, 29 hedgehog spin texture and Berry phase tuning, 30 thermoelectrics, 31 and ultrafast carrier dynamics, 32 ). Furthermore, few-layer Bi 2 Se 3 also shows sizable Rashba-type spinsplittings due to substrate-induced structural inversion asymmetry. 26,[33][34][35] As both few-layer Bi 2 Se 3 and PtSe 2 demonstrate layer-dependent properties and have several distinctive features, it is therefore of great interest to theoretically study their vdW heterostructures (by varying the number of constituent layers) and search for possible synergy effects that could be utilized. Interestingly, both materials have low temperature growth mechanisms (up to 450 • C), thus, it is likely that such heterostructures can be accomplished experimentally. Motivated by this, we employ first-principles calculations and unfold tunable and sizeable type-II and type-III band alignments as well as several different spintronics features of few-layer Bi 2 Se 3 /PtSe 2 vdW heterostructures. We reveal the coexistence of Rashba-type spin-splittings in the conduction(valence) band originating from few-layer Bi 2 Se 3 (PtSe 2 ) due to inversion symmetry breaking and structural asymmetry. Our findings provide a promising pathway to manipulate the charge and spin degrees of freedom using carefully designed vdW heterostructures for the next-generation nanoscale electronic and spintronics devices.

Computational method
We have performed density functional theory (DFT) calculations using the projector augmented wave method 36,37 as implemented in the Vienna Ab-initio Simulation Package . 38 For the exchange-correlation potential, we have used the non-local optB86b-vdW density functional 39,40 to account for the van der Waals interactions between the layers as success- Figure 1: Side views of (a) few-layer Bi 2 Se 3 , and (b) PtSe 2 (black lines depict unit cell) alongside (c) 1QL Bi 2 Se 3 /1L PtSe 2 as a representative vdW heterostructures considered in this study. The interlayer distance is given for each case. Pt, Se, and Bi atoms are shown in grey, green, and purple, respectively. fully employed before for similar systems. 41,42 The plane-wave cutoff energy was set to a sufficiently large value of 420 eV. A gamma-centered Monkhorst-Pack 4 × 4 × 1 k-mesh was used for the structural relaxation whereas for the band structure calculations, the Brillouin zone integration was performed using a dense 7 × 7 × 1 k-mesh. Moreover, to compute 2D spin-textures, we set up a 2D k-mesh (k x × k y : 15 × 15) centered at the gamma-point (k z = 0). For the iterative solution of the Kohn-Sham equations, we ensured the total energy to converge until the change is below 10 −6 eV and residual forces on the atoms to decline to less than 10 −3 eV/Å. Since our systems contain heavy elements, the effects of spin-orbit coupling (SOC) were taken into account in the band structure and density of states (DOS) calculations. The heterostructures were modeled using a 15Å thick vacuum layer in the out-of-plane direction to avoid periodic images interactions. Finally, the PyProcar python library 43 and Matplotlib graphics package 44 were used for pre-and post-processing of the data and plotting.
We have used DFT to simulate single-and few-layer Bi 2 Se 3 and PtSe 2 and their heterostructures, as shown in Figure 1(a-c). Each Bi 2 Se 3 slab consists of five atomic sheets (termed a quintuple layer (QL)) having Se-Bi-Se-Bi-Se atoms held together by covalent bonding, whereas a Pt-atom is covalently sandwiched between two Se-atom-layers in a PtSe 2 slab with 1T-phase trigonal geometry. The lowest-energy structure for a representative 1QL Bi 2 Se 3 /1L PtSe 2 heteroestructure is given in Figure 1(c), for which different lateral stackings were carefully inspected before arriving at this configuration. We note that interlayer interactions between PtSe 2 and Bi 2 Se 3 are homogeneous meaning Bi or Se atoms do not prefer particular sites on PtSe 2 . For six different lateral stacking configurations, shown in supplementary Figure ??(a-f), the energy differences fall in the of range 0 to 10 meV from which the minimum energy configuration is adopted for calculations.
We first determine the electronic properties of pristine few-layer Bi 2 Se 3 and PtSe 2 (see The CB of few-layer PtSe 2 however also involves equally mixed Se(p)-Pt(d) states much alike few-layer Bi 2 Se 3 . The reason for the differences between 1(Q)L and 2(Q)L is that although the strong interlayer binding is of van der Waals type, the resulting physisorption leads to induced moments and Pauli repulsion of the electron density (contracted in the layers) that effects the properties since the layers are squeezed together compared to a monolayer. 45 Unlike other TMDCs such as MoS 2 , layer-tunable band gaps and mixed atomic hybridization in both few-layer Bi 2 Se 3 and PtSe 2 influences the observed novel features described above, as will be discussed in the following section.
We have considered four different heterostructures combining few-layer Bi 2 Se 3 and PtSe 2 , denoted nQL Bi 2 Se 3 /nL PtSe 2 (see Table 1), for which we report the structural and energetic properties. The binding energy per PtSe 2 is calculated through equation (1),  Heterostructure and PtSe 2 showing relatively weaker interaction than for the constituent systems.
For the 1QL(2QL) Bi 2 Se 3 /1L PtSe 2 heterostructure, including the effect of SOC (see   Figure ??(a)). Comparing to the case without SOC (Figure 3(a,c)), the SOC significantly affects the magnitude of the band gaps in the vdW heterostructures much alike pristine 1QL(2QL) Bi 2 Se 3 (see supplementary Figure ??(ad)). For the 1QL(2QL) Bi 2 Se 3 /1L PtSe 2 heterostructures the energy gap shrinks with the type-II band alignment when going from 1QL to 2QL, which is in line with the change of the CB in pristine Bi 2 Se 3 in 1QL and 2QL (c.f. Figure 2(a,b)). In the context of theoretical and display Rashba-type spin-splittings in the CB originating from few-layer Bi 2 Se 3 . We note that these splittings resemble to those observed in graphene/TI vdW heterostructures resulting in gate-tunable spin-galvanic effects at room temperature. 48 To confirm this, we set up a Γ-centered 2D k-mesh along the xy-plane and plot the fixed-energy contours of spin components s x , s y and s z as shown in Figure 4(a) alongside the transformation of the CB with respect to the SOC effect. The typical Rashba-split bands along the momentum-axis with large(small) in-plane(out-of-plane) spin components, respectively, support our findings.
Using the Rashba Hamiltonian for 2D-electron gas, where k = (k x , k y , 0) and m * being the in-plane momentum and effective mass of electron, α R is the Rashba parameter, #» σ is the vector of Pauli matrices and #» z is the out-of-plane unit vector. Taking E R as the energy difference between the CB minimum and band crossing at the Γ-point and k 0 as the momentum offset, the Rashba parameter for a parabolicdispersion is approximated by α R = 2E R /k 0 , whereas E R =h 2 k 2 0 /2m * and k 0 = m * α R /h 2 .
For 1QL(2QL) Bi 2 Se 3 /1L PtSe 2 vdW heterostructures, we obtain E R = 4.8 meV  On the other end, looking at the topmost VB of the band structure coming from 1L PtSe 2 , it is seen turning into a "Mexican hat" shape (compare Figure 3 s y and s z in Figure 4(b). One may also distinguish between Rashba-type spin-splittings of the CB and hedgehog-like band-splittings of the VB by looking at the fixed-energy contours of the spin-components in both Figure 4(a) and 4(b) right-side, respectively.
We also considered the effect of increasing the PtSe 2 thickness, i.e., few-layer Bi 2 Se 3 /2L PtSe 2 , for which the band structures show type-III band alignment as displayed in Figure   5(a-d) and supplementary Figure ??(b). Interestingly, both the CB and VB simultaneously show Rashba-type spin-splittings with each band being located on the separate constituent materials. The broken inversion symmetry is valid as for 1QL(2QL) Bi 2 Se 3 /2L PtSe 2 (i.e., top and bottom constituent layers experience different charge environment). Approximating the band dispersion around the Γ-point by Eq. 2, Table 2 Table ??). Moreover, the α R values for the CB also show the formation of a Rashba electron-gas over a large energy interval. Table 2: Rashba spin-splitting parameters of few-layer Bi 2 Se 3 /PtSe 2 vdW heterostructures. E R is energy difference between the CB/VB minimum/maximum and band crossing at the Γ-point, k 0 is the momentum shift and α R is the Rashba parameter.
Heterostructure To better describe these Rashba spin-splittings and associated spin-textures, Figure 6(a) shows the bands changeover under the influence of SOC alongside the definition of E R and k 0 for the VB. Also, the total and atom projected DOS in Figure 6(b) display significant changes in the orbital hybridization within the Bi 2 Se 3 and PtSe 2 layers around the Fermi level as compared to Figure 2, which is responsible for large SOC induced spin-splittings. We clarify Fermi level hybridize and give rise to spin-splittings. In order to confirm the Rashba-shift along the momentum axis, we also give fixed-energy contour plots of spin-components s x , s y , and s z which corroborates our findings and reveal in-plane spin-components with minuscule out-of-plane contributions for both the CB and the VB as shown in Figure 6(c). Comparing spin-splittings and Rashba-energies in the CB and the VB, we observe energy anisotropy in the spin texture similar to the case of graphene/TI heterostructure. 51 Since the optB86b-vdW functional is known to underestimate the band gap compared to experiment (and the use of a more accurate hybrid functional, such as HSE06, is prohibited due to the size of the models), it is anticipated that the transformation of type-II to type-III band alignment will occur at larger PtSe 2 thickness than 2L, which makes it possible to access the novel spintronics features predicted for these vdW heterostructures with few-layer Bi 2 Se 3 . Furthermore, low-temperature synthesis techniques are available for both systems, which also favors the formation of stacked vdW heterostructures.

Conclusion
In summary, we employed density functional theory calculations to discuss layer-dependent

Competing Interests
The Authors declare no competing financial or non-financial interests.

Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.  Figure S1: (a-f) Different lateral stacking configurations considered to arrive at the minimum energy configuration. The total energies fall in the range of 0 to 10 meV for these stackings.   Figure S3: Band structures of (a,b) 1QL Bi 2 Se 3 , and (c,d) 2QL Bi 2 Se 3 (without SOC (left,blue) and with SOC (right,red)). Figure S4: Band structures of (a,b) √ 7 × √ 7 1QL Bi 2 Se 3 , and (c,d) 3 × 3 1L PtSe 2 (without SOC (left,blue) and with SOC (right,red)). Figure S5: Atomic orbital-projected band structures of (a,b) 1QL Bi 2 Se 3 /1L PtSe 2 and (c,d) 1QL Bi 2 Se 3 /2L PtSe 2 vdw heterostructures, respectively. SOC is incorporated in all cases.