Three dimensional topological insulators are insulators with topologically protected surface states that can have a high band velocity and high mobility at room temperature. This suggests electronic applications that exploit these surface states, but the lack of a band gap poses a fundamental difficulty. We report a first principles study based on density functional theory for thin Bi2Se3 films in the context of a field effect transistor. It is known that a gap is induced in thin layers due to hybridization between the top and bottom surfaces, but it is not known whether it is possible to use the topological states in this type of configuration. In particular, it is unclear whether the benefits of topological protection can be retained to a sufficient degree. We show that there is a thickness regime in which the small gap induced by hybridization between the two surfaces is sufficient to obtain transistor operation at room temperature, and furthermore, that the band velocity and spin texture that are important for the mobility are preserved for Fermi levels of relevance to device application.

Three dimensional topological insulators (3D TI)1,2 are materials with a bulk band gap but metallic surfaces due to topologically protected bands crossing the gap. These states are spin-textured due to time reversal symmetry; this suppresses the non-magnetic scattering. This spin texture and the fact that these states typically have a near linear (Dirac) dispersion with high velocity can lead to high mobility, even at room temperature. In the case of Bi2Se3, it has been shown that even though these are surface states they can dominate conduction in films3 and even in bulk.4 Importantly for potential electronics applications, Bi2Se3 has a single Dirac cone surface state dispersion, a relatively large bulk band gap for a TI,5 a strong spin texture even at room temperature,6,7 and furthermore its topological surface states are robust against shrinking the lateral scale down to the few nm range.8 The use of layered materials in transistors can enable new types of devices such as very short channel transistors.9 

A key challenge is the fact that the topological surface states are gapless, similar to the gapless band structure of graphene. This impedes the fabrication of conventional field effect devices. It is, however, known that small gaps can be induced by interaction between the top and bottom surfaces of ultrathin 3D TI films. This includes films of Bi2Se3, and that spin texture is retained over some energy range.6,10–23 In this regard, FETs using 3D TI thin films and nanowires have been demonstrated to show a gate-tunable conductance,3,24–29 although not necessarily as a consequence of topological surface states (note that Bi2Se3 is a semiconductor, so field effect devices based on the bulk behavior can be made). The present focus is on the possibility of a transistor in which the conducting state is based on the spin textured Dirac cone, with its high band velocities, enabling small, fast devices. In particular, there is a thickness for Bi2Se3 where the hybridization of top and bottom surface states is strong enough to open a gap adequate for transistor operation but at the same time, not so strong as to substantially impact the spin texture at the relevant Fermi energy positions. To connect with the experiment, for bulk He et al.3 reported that the surface states can conduct so well that they dominate bulk, and in thin (100 nm) layers, the surfaces state conduction is retained.24 However, only an insulating behavior was observed in reported experiments on ultrathin samples with thicknesses where a gap can occur.25 In the present work, we address the question of whether this is an intrinsic problem or alternately whether it is possible to use the surface states in the regime where a gap is opened by hybridization between the top and bottom surfaces. We examine the thickness dependent gap opening in the context of electronic application, specifically a field effect device. The analysis is based on first principles calculations of the band structure of Bi2Se3 for different film thicknesses, followed by a transport analysis as a function of temperature and Fermi level. We also calculate the spin-texture, and find that the surface conduction band is strongly spin textured over the whole bulk gap region. We find that two quintuple layer (QL) Bi2Se3 has the potential as an electronic material, even at room temperature.

We performed density functional calculations using both the linearized augmented planewave (LAPW) method30 as implemented in the WIEN2K code,31 and also the projector augmented wave (PAW) method with the VASP code,32,33 and standard PAW settings. We cross-checked the results and found a good agreement. Experimental lattice constants34 of bulk Bi2Se3 were used with relaxed internal coordinates using local density approximation (LDA) functional. We used the LDA for these relaxations, because it provides a convenient way of obtaining a realistic structure, while standard generalized gradient approximations (GGAs) often do not give realistic structures for van der Waals bonded materials.35–37 Once the structures were determined, we used the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)38 for the electronic structure calculations. We used LAPW sphere radii of 2.5 bohr for both Bi and Se along with a high basis set cut-off parameter of RminKmax = 9. We used a k-point sampling of 10 × 10 in-plane for the total energy calculations and a k-mesh of 32 × 32 for the density of states (DOS). We used a vacuum thickness of 8 Å in the slab calculations and found this to be converged. We used the BoltzTraP code39 to obtain transport integrals. A denser 40 × 40k-point grid was used for these calculations. Spin-orbit coupling was included in all the calculations except structure relaxations. The spin-textured band structures were obtained with the PyPROCAR code based on VASP calculations.

The transport functions and coefficients are obtained using the relaxation time approximation to Boltzmann transport theory based on the first-principles electronic structure. We used the constant scattering time approximation (CSTA), which assumes that the energy dependence of the scattering rate (τ−1) is small compared to the energy dependence of the electronic structure.

The rhombohedral bulk Bi2Se3 crystal structure consists of five-atomic-layer building blocks, known as quintuple layers (QLs), van der Waals stacked along c. Each QL has three Se layers and two Bi layers in the sequence of Se1-Bi-Se2-Bi-Se1 (“1” and “2” are symmetry inequivalent). We studied the thin film structures from 1 QL to stacks of 5QLs. A two-QL slab crystal structure as used in our calculations is shown in Fig. 1.

FIG. 1.

Structure of 2QL Bi2Se3.

FIG. 1.

Structure of 2QL Bi2Se3.

Close modal

The evolution of the surface band dispersion of Bi2Se3 thin films as a function of slab thickness using the PBE GGA is shown in Fig. 2. As seen, the interaction between opposite surfaces leads to the opening of a gap at the Dirac point (Γ point). The gap values decrease rapidly with increasing film thickness, which has been known and confirmed by angle-resolved photoemission spectroscopy (ARPES).13,41 As shown in Fig. 3, our band gaps at Γ are somewhat smaller than those reported in the prior experimental work.13 Prior GW calculations show a similar level of agreement but in the opposite direction.40 We note that the gap value is determined by the electronic coupling of the top and bottom surfaces, which has an exponential dependence due to the bulk band gap. Therefore it can be expected to be sensitive to details, especially the bulk band gap and the exact structure. This may offer an opportunity for tuning, e.g., by changing the substrate, which can then influence the structure, especially the exact van der Waals spacing between the lowest two QL.

FIG. 2.

Surface band dispersions of Bi2Se3 thin films.

FIG. 2.

Surface band dispersions of Bi2Se3 thin films.

Close modal
FIG. 3.

Calculated Γ point surface band gap values of Bi2Se3 thin films with different numbers of Bi2Se3 layers. The black triangles and red diamonds are the experimental13 and GW results,40 respectively. The dashed lines are guides to the eye.

FIG. 3.

Calculated Γ point surface band gap values of Bi2Se3 thin films with different numbers of Bi2Se3 layers. The black triangles and red diamonds are the experimental13 and GW results,40 respectively. The dashed lines are guides to the eye.

Close modal

In any case, we focus on the 2QL structure due to the reasonable band gap (∼0.13 eV), which is not so small as to be washed out by kT at room temperature, and not so big as to be comparable to the bulk gap. The band structure of 2QL Bi2Se3 is shown in Fig. 4. Isoenergy cuts of the band structure are shown in Fig. 5. It is important to note that Dirac cone in Bi2Se3 is close to the top of the bulk valence band, and not mid-gap, and that this feature is preserved in the film. The importance of this is that if excess electrons are introduced, they will occupy the upper part of the Dirac cone for a large energy range, before the Fermi level, EF, reaches the bulk conduction band minimum. We consider therefore the case of adding electrons, which in a device would consist of gate voltages that bring electrons into the channel. We treat the band structure as rigid and begin by considering the variation of EF from the gap to the edge of the conduction band (CBM-1) above the bulk gap crossing state, i.e., the band coming from the bulk conduction band. This is at 0.47 eV relative to the valence band maximum (VBM) as shown in Fig. 4. This defines the range of Fermi level that can be reasonably considered for a device exploiting the topological band. The corresponding carrier concentration range is up to 2.7 × 1013 cm−2 for 300 K and 2.4 × 1013 cm−2 for 100 K at CBM-1. The relation between carrier concentration and EF at 100 K and 300 K is shown in Fig. 6. The position of the conduction band minimum (CBM) for the topological band is indicated by the dashed line. The carrier concentrations for this value of EF are 2.3 × 1011 cm−2 at 100 K and 6.8 × 1011 cm−2 for 300 K, respectively.

FIG. 4.

Surface band structure of a 2QL Bi2Se3 film.

FIG. 4.

Surface band structure of a 2QL Bi2Se3 film.

Close modal
FIG. 5.

Isoenergy cuts of the 2QL band structure. The figure is the supercell electronic structure, displayed with a slight tilt from the c-axis to make the sections visible. The valence bands (top) are shown in blue and the conduction bands (bottom) in red. Note that they consist of very two dimensional sections confirming the adequacy of the vacuum layer thickness, and that there are no pockets near the valence or conduction band edges except those shown in the band structure plot of Fig. 4.

FIG. 5.

Isoenergy cuts of the 2QL band structure. The figure is the supercell electronic structure, displayed with a slight tilt from the c-axis to make the sections visible. The valence bands (top) are shown in blue and the conduction bands (bottom) in red. Note that they consist of very two dimensional sections confirming the adequacy of the vacuum layer thickness, and that there are no pockets near the valence or conduction band edges except those shown in the band structure plot of Fig. 4.

Close modal
FIG. 6.

Calculated carrier concentration with respect to Fermi energy at 100 K and 300 K of 2QL Bi2Se3.

FIG. 6.

Calculated carrier concentration with respect to Fermi energy at 100 K and 300 K of 2QL Bi2Se3.

Close modal

Within the diffusive relaxation time approximation, the electrical conductivity is given as σ = (σ/τ) × τ, where τ is the relaxation time, or inverse scattering rate. It is possible to calculate σ/τ directly from the electronic structure as a function of carrier concentration and temperature. Specifically, we used the constant scattering time approximation. This assumes the relaxation time to be constant and takes the energy dependence of the band structure as the essential factor in the energy dependence of the conductivity in the transport formula.39,42 The scattering rate may be sensitive to the Fermi level,43–45 but calculating this would require a detailed knowledge of the energy dependent scattering mechanisms including different sources of scattering in combination. For instance, based on phase space, one might infer a scattering rate proportional to the density of states (DOS). We show the DOS of the 2QL Bi2Se3 in Fig. 7, where we can see a flat region near CBM. This suggests a DOS and scattering rate that is weakly dependent on energy from just above the band edge to the onset of the first bulk conduction band. Other potential scattering mechanisms such as impurity scattering, as may be important for graphene,43–45 may be important but are not intrinsic and could at least in principle be controlled. More generally, both phase space and scattering matrix elements determine the resistivity. The featureless density of states and the spin texture similar to bulk suggest that the present approximation, while necessarily simple, may be reasonable.

FIG. 7.

Calculated total and projected density of states of 2QL Bi2Se3 using PBE with spin-orbit coupling.

FIG. 7.

Calculated total and projected density of states of 2QL Bi2Se3 using PBE with spin-orbit coupling.

Close modal

Figure 8(a) shows the relation between σ/τ and carrier concentration for 300 K. σ/τ is very low for EF in the gap, even at 300 K. The key characteristic of a FET device is the controllable electrical conductivity based on the applied gate voltage. For a device, the contrast between the ON and OFF states is important. In Fig. 8(b), we show the calculated conductivity ratio, which is evaluated from (σ/τ)(σ/τ)min with respect to the EF at both 100 K and 300 K. We find a signal ratio of 24 for 300 K and approximately 10 000 for 100 K at the point where the EF is at the middle in between CBM and CBM-1, as indicated by the black dotted line. Higher ratios can be obtained by driving EF closer to the CBM-1 level. The maximum conductivity ratio is approximately 40 for 300 K. The slope of the linear dispersing band near CBM from 0.23 eV to 0.43 eV is calculated to be 3.27 eV Å using E(k)/k, corresponding to a band velocity of 5 × 105 m/s.

FIG. 8.

Calculated σ/τ versus carrier concentration at 300 K (a) and σ/τ ratio with respect to Fermi energy (b) at 100 K and 300 K for 2QL Bi2Se3.

FIG. 8.

Calculated σ/τ versus carrier concentration at 300 K (a) and σ/τ ratio with respect to Fermi energy (b) at 100 K and 300 K for 2QL Bi2Se3.

Close modal

The above is based on the band shapes, including the approximate linear dispersion of the conduction band, but does not take account of the spin texture. It is therefore a pessimistic scenario. For EF in the gap, the spin texture is lost and therefore one may expect a significant electron phonon and non-magnetic scattering. To the extent that the conduction band keeps the topological character, these sources of scattering will be suppressed, and the ON state conductivity increased.

The non-trivial spin texture feature in Bi2Se3 surface states has been observed in both experiments and theoretical calculations.6,14,41,46 We confirm this character in 2QL Bi2Se3, and find that the spin texture persists over the entire CBM up to the onset of CBM-1. This is based on analysis of the band structures. The 2QL slab that we use is centrosymmetric. Therefore the bands retain a 2-fold degeneracy with the spin-orbit. The bands come from states on the top and bottom surfaces, which need to be separated to discuss the spin texture. We do this by projecting onto the top-most Bi atom in the slab and adding the contributions from the two bands. The spin-textured band structures around Γ point including high symmetry directions along Γ-K (kx) and Γ-M (ky) are shown in Fig. 9. As shown, the projection of the spin vector shows a larger magnitude in the in-plane (xy plane) direction than the out-of-plane (z) direction. This indicates that the spin lies almost completely in the plane, in accordance with the prior work,47 where it was found that the z-component of the spin is only about 1/40 of the in-plane component. The most striking feature is the momentum-dependent spin configuration. Systematic mapping of the surface spin texture in energy and momentum space using spin-resolved, angle-resolved photoemission spectroscopy (SR-ARPES) has shown a wave-vector-dependent net spin polarization in thin Bi2Se3 films.6,41,46 The important feature is that we find the spin texture to be strong over the entire energy range up to the CBM-1 onset, meaning that the suppression of scattering due to spin texture is expected to be maintained as EF is swept up to this point.

FIG. 9.

Spin-textured band structures of 2QLs Bi2Se3 around the Γ point along Γ-K (kx) and Γ-M (ky) directions. The spin texture is calculated for each spin component (i.e., Sx, Sy, and Sz). The y and z spin component along ky is zero, thus we do not show them in this figure. Red and blue represent the opposite spin directions.

FIG. 9.

Spin-textured band structures of 2QLs Bi2Se3 around the Γ point along Γ-K (kx) and Γ-M (ky) directions. The spin texture is calculated for each spin component (i.e., Sx, Sy, and Sz). The y and z spin component along ky is zero, thus we do not show them in this figure. Red and blue represent the opposite spin directions.

Close modal

An important question is whether the spin texture in a 2QL slab is significantly reduced relative to thicker slabs and bulk. We find that in fact over most of the energy range there is no significant reduction. We calculated the spin-texture for 3QL Bi2Se3 slab which has a much smaller gap and hybridization between the surface in the same way and found a similar behavior, as shown in Fig. 10. Specifically, we used the same method as treated for 2QL Bi2Se3 where the top-most Bi atom is considered with both contributions from the 2-fold degenerate bands. The y-axis was taken to be the ratio between the spin projection along x direction with respect to the total spin of all the atoms. The energy zero is set to the CBM and the Γ-K line is shown. The figure shows that except very close to the band edge where the dispersion is parabolic, the hybridization between the top and bottom surfaces does not significantly affect the spin texture. This is a key result because it means that the intrinsic features leading to the high mobility of the bulk surface state are retained down to 2QL.

FIG. 10.

Spin ratio between the top-most Bi layer and the total spin of all the atoms with respect to energy. Both 2QL and 3QL Bi2Se3 along Γ-K are presented for comparison, and the CBM is set at energy zero.

FIG. 10.

Spin ratio between the top-most Bi layer and the total spin of all the atoms with respect to energy. Both 2QL and 3QL Bi2Se3 along Γ-K are presented for comparison, and the CBM is set at energy zero.

Close modal

In summary, we investigate thin film Bi2Se3 in the context of possible application in a transistor that uses the topological surface states as a conduction channel. We find that a 2QL slab structure has a small but usable band gap, this gap is located at the bulk valence band edge, which leaves a sufficient energy range to obtain an ON state with the Fermi level in the topological band, and the spin texture of the topological state is retained over a wide energy range. The results suggest that performance could be obtained even at 300 K. We note that this is a highly flexible system, that is expected to be tunable, for example, by interaction with the substrate and capping layers. One aspect will be finding substrates that increase the electronic coupling of the lowest two QL to increase the gap, which would help room temperature performance. It will be of interest to further explore this system experimentally to determine the channel conductivity as a function of the Fermi level. It will also be important to investigate the dependence on substrates, capping layers, and other experimental conditions.

This work was supported by the Department of Energy, Office of Basic Energy Sciences through the MAGICS center, Award No. DE-SC0014607.

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