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Electric tuning of the surface and quantum well states in Bi2Se3 films: a first-principles study

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 395005 (http://iopscience.iop.org/0953-8984/26/39/395005) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 132.239.1.231 This content was downloaded on 15/04/2017 at 15:34 Please note that terms and conditions apply.

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 395005 (6pp)

doi:10.1088/0953-8984/26/39/395005

Electric tuning of the surface and quantum well states in Bi2Se3 films: a first-principles study Hong Yang1,2,3, Xiangyang Peng1,2, Wenliang Liu1,2, Xiaolin Wei1,2, Guolin Hao1,2, Chaoyu He1,2, Jin Li1,2, G Malcolm Stocks4 and Jianxin Zhong1,2 1

  Hunan Key Laboratory of Micro-Nano Energy Materials and Devices, Xiangtan University, Hunan 411105, People’s Republic of China 2   Laboratory for Quantum Engineering and Micro-Nano Energy Technology and Faculty of Materials and Optoelectronic Physics, Xiangtan University, Hunan 411105, People’s Republic of China 3   Department of Physics, Jishou University, Hunan 416000, People’s Republic of China 4   Materials Science & Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA E-mail: [email protected] and [email protected] Received 1 May 2014, revised 25 June 2014 Accepted for publication 11 July 2014 Published 28 August 2014 Abstract

Based on first-principles calculations in the framework of van der Waals density functional theory, we find that giant, Rashba-like spin splittings can be induced in both the surface states and quantum well states of thin Bi2Se3 films by application of an external electric field. The charge is redistributed so that the Dirac cones of the upper and lower surfaces become nondegenerate and completely gapless. Interestingly, a momentum-dependent spin texture is developed on the two surfaces of the films. Some of the quantum well states, which reside in the middle of the Bi2Se3 film under zero field, are driven to the surface by the electric field. The Rashba splitting energy has a highly non-linear dependence on the momentum and the electric field due to the large contribution of the high-order Rashba terms, which suggests complex spin dynamics in the thin films of Bi2Se3 under an electric field. Keywords: topological insulators, rashba effect, first-principles (Some figures may appear in colour only in the online journal)

1. Introduction

of the spintronic devices. These adsorbates are found to act as surface donors, producing an effective electric field near the surface and altering the surface and quantum well states. In device applications, an externally applied electric field is preferable because it is much more tunable and controllable than the adsorbate-induced effective field. As is known, an external electric field can modulate the phase transition [12] and change the surface states [13–15] in materials. It would be desirable if the Rashba spin splittings and surface Dirac cones in TIs could also be tuned by external electric field. There have been some preliminary experimental and theoretical indications of the feasibility of tuning the TIs electrically [8, 16–18]. However, the rich physics arising from the interplay between the electric field and strong spin–orbit coupling in TIs is still to be explored. A TI film has topological surface

Bi2Se3, as one of the second-generation strong 3D topological insulators (TIs), is the focus of current research. It has a relatively large bulk gap and topologically protected metallic surface states induced by large spin-orbit interactions [1, 2]. Due to the exceptional properties derived from its non-trivial band topology, it is expected to be an ideal material for applications in spintronics and quantum computing [3–5]. Recently, a series of experiments revealed that the adsorption of gas molecules [6, 7], alkaline atoms [8, 9], transition metal atoms [10] and heavy metal atoms [11] could induce a giant Rashba effect in Bi2Se3, with a magnitude one or two orders larger than that of conventional semiconductors, which shows great promise for room-temperature operation and miniaturization 0953-8984/14/395005+6$33.00

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J. Phys.: Condens. Matter 26 (2014) 395005

states near its two surfaces, and quantum well states (QWSs) in the middle. The two surfaces become asymmetric in the electric field. It is not clear if the topological surface states will be preserved or not or how the two asymmetric surfaces’ responses to the electrical field will differ. The Dirac fermions in TIs have a unique spin texture. One would also ask whether such a spin texture will remain unchanged or be broken under the electric field. The charge of the quantum well states will be redistributed, and might appear in the surface. It is still to find out how the Rashba splitting varies with the magnitude of the electric field. The experiments have shown that the band gap and topological phase of TI films have an interesting thickness dependence. It is important to find out if there is also a thickness-dependence response of TI films to the electrical fields. In this paper, based on van der Waals density functional theory (vdW-DFT), we studied the effects of external electric fields on the surface states and quantum well states in Bi2Se3 thin films of various thicknesses. After the application of an electric field, it is found that giant Rashba spin splittings are induced, and are tunable by the strength of the field. The spin texture is altered and becomes momentum dependent. Interestingly, some quantum well states, which reside well inside Bi2Se3 film when there is no external field, are driven to one of the surfaces of the film by electric field. The high-order Rashba terms are found to have considerable contribution to the Rashba spin splitting. The surface Dirac cones of the two surfaces of ultrathin Bi2Se3 films, which are degenerate with small gaps under zero field, become nondegenerate and gapless. One surface is electron rich, while the other surface is electron deficient. As a result, the Dirac cone of the former is well below the Fermi level, whereas that of the latter is above the Fermi level. Our results suggest that it is promising to realize the electrical tuning of the TI-based devices.

Figure 1.  The atomic conventional structure of bulk Bi2Se3. The green (small) and purple (big) spheres denote the Se and Bi atoms, respectively.

experiments and previous calculations [26]. Recently, the long-range, many-body effects were taken into account in the vdW corrections [29, 30], the implementation of which could lead to further refinement. In the study of Bi2Se3 films, a vacuum layer of about 15 Å is used to decouple the adjacent atomic slabs in the supercells. The surface Brillouin zone is sampled by a 9  ×  9 × 1 Γ centered k-point mesh. The electric field is applied by addition of an artificial dipole layer in the middle of the vacuum, as implemented in VASP [31], and the atomic structures are fully relaxed under the electric field. The implicit direction of the electric field is along the c-axis from the upper to the lower surface of the film.

2.  Computational details In the framework of vdW-DFT [19, 20], the calculations are performed by using the Vienna Ab initio Simulation Package (VASP) with Perdew-Burke-Ernzerhof generalized gradient approximation (GGA-PBE [21–23]) and van der Waals (vdW) corrections (optPBE-vdW [24–26]). The cutoff energy for the plane-wave expansion is set to be 340 eV. We have tested that 340 eV gives well-converged results, and larger cutoff energy almost makes no difference. The electron-ion interaction is described by projected augmented wave (PAW) potentials [27]. The spin orbit coupling (SOC) is taken into account. As shown in figure 1, Bi2Se3 has a rhombohedral layered structure, and its crystal is formed by stacking quintuple layers (QLs) along its (0001) direction, with weak vdW interactions between neighboring QLs. Inside a QL, there are five atomic layers ordered as Se(1)-Bi-Se(2)-Bi-Se(1). During structural relaxation, all atoms of Bi2Se3 are allowed to move until the forces on them are less than 0.01 eV/Å. The relaxed lattice parameters of bulk Bi2Se3 (a = 4.153 Å, c = 28.3 Å) agree well with the corresponding experimental values (a = 4.138 Å, c = 28.64 Å [28]). The calculated height of QL and the separation between neighboring QLs are also consistent with the

3.  Results and discussions We first study the electronic structures of Bi2Se3 films, taking vdW interaction into consideration. To identify surface bands, a wave function at a given energy band and momentum (with the momentum parallel to the surface) is projected onto spherical harmonics that are nonzero within some radius around each ion. Here, a criterion is applied based on a critical percentage of the projections of the wave functions. A state is distinguished as a surface state when its projection onto the upper surface (topmost QL) or lower surface (bottommost QL) exceeds the critical percentage, which is set to be 50%, 45% and 40% for 3, 4 and 6 QLs Bi2Se3 films in our calculations [32]. The solid triangles and open circles in figures  2 and 3 denote the upper and lower surface states, respectively. Figure 2(a) shows the calculated band structure in the vicinity of Γ point for a 3QL-Bi2Se3 film. The states of the upper and lower surfaces are degenerate owing to the inverse symmetry of the thin film. The band gap at Γ point ΔΓ is calculated to be 2

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Figure 2. (a)–(c) The band structures of the 3QL-Bi2Se3 film under the electrical field of 0, 0.01, and 0.02 V/Å, respectively. The states

of the upper and lower surfaces of the film are marked by solid triangles and open circles, respectively. Q1 and Q2 indicate the two QWSs in the conduction bands of 3QL-Bi2Se3, whereas M1 the QWS is in the valence bands. (d) The zoomed view of S1 in (b). The inset is the energy contours in the Brillouin zone at E=0.28 eV, on which the in-plane spin polarization is indicated by arrows. The hexagon is the surface Brillouin zone of Bi2Se3 film. The energy contours and spin polarization vector along them at E = 0.1 eV are very similar to those at E = 0.28 eV, and are not shown for brevity. The Rashba energy and momentum offset of S1 state near the Γ point ES1 and KS1 are indicated. (e) The charge densities of the quantum well states Q2 in (a), (b) and (c). (f) The average planar electrostatic potentials (APEPs) of the 3QL-Bi2Se3 film. The black solid line and red dash line are APEPs under external electric fields of 0 and 0.02 V/Å, respectively. The arrow indicates the direction of the applied field. The number gives the workfunction under different fields. (g) The Rashba energy ES1, momentum offset kS1, and Rashba parameter αS1 versus the magnitude of the external electric field. (h) The Rashba energy splitting versus the wave vector. The blue triangles, red circles and black squares denote the splitting of the band S1 in (b) and (c) and the band Q2 in (c), respectively. The blue, red and black dotted lines are the linear fits of energy splitting with respect to small k.

the surfaces of the film in zero field [figure 2(e)]. As the thickness of Bi2Se3 film increases to 4 and 6QLs, the ΔΓ decreases to 0.065 and 0.017 eV, respectively. Therefore, the gap seems to persist, though small, in the films of finite thickness, and the expected gapless Dirac cone is not easy to realize. Although it has been shown that by application of compressive strain along the c-axis, the gap can be closed [33], and it is preferable to find a more effective and convenient means to close the gap between the Dirac cones of thin Bi2Se3 films. Figure 2(b) shows Rashba-like band splittings of bands S1 and S2 of 3QL-Bi2Se3 when Eext = 0.01 V/Å. There is still a gap near the Fermi level. A close-up view of the S1 band is displayed in figure  2(d). The inset in it shows the energy contours in the Brillouin zone at the level of E = 0.1 and 0.28 eV, and the in-plane spin polarization vectors along the two circular contours are indicated. One can find that the in-plane spin vector is perpendicular to the k vector and tangent to the circles, exhibiting opposite chirality along the two circles. However, there is a significant distinction from the conventional Rashba effect [34] regarding the

0.0047 eV for the unrelaxed 3QL-Bi2Se3 film, with an interQL separation of 2.253 Å, whereas ΔΓ and inter-QL separation become 0.12 eV and 2.41 Å after full atomic relaxation with vdW corrections. The gap of the relaxed structure is much closer to the corresponding experimental value 0.138 eV [18], which indicates that the atomic relaxation has a large influence on the electronic structures of TI films and has to be taken into consideration when the electric field is either zero or nonzero. To see the effect of the vdW correction on the electronic structure, we also calculated band structure of the above relaxed structure without vdW corrections. It is found that there is no appreciable change in the band structure and the gap is reduced only by 0.02 eV, indicating that in the case of Bi2Se3, vdW correction modifies the band structure mainly through the correction of the geometry. Above the lowest conduction band S1 and below the highest valence band S2 [figure 2(a)], there are two QWSs, Q1 and Q2, and one QWS, M1, in the vicinity of the Γ point, respectively, in agreement with the previous calculations [9] and experiments [18]. It can be seen that the QWSs are located in the middle instead of on 3

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Figure 3. (a) and (b): the band structures of the 4QL- and 6QL-Bi2Se3 films under zero field, respectively. (c) and (d): the band structures of the 4QL- and 6QL-Bi2Se3 films under the external electrical field of 0.015 and 0.005 V/Å, respectively. The states of the upper and lower surfaces of the film are marked by solid triangles and open circles, respectively. Q1, Q2 and Q3 indicate the three QWSs in the conduction bands of 4QL-Bi2Se3, and M1 and M2 are the two QWSs in the valence bands. Q5 and M4 indicate the highest and the lowest QWS in the conduction bands and valence bands of 6QL-Bi2Se3 film, respectively.

charge distribution of the states on the contours. The level E = 0.28 eV cross the upper-surface bands (indicated by solid triangles in figure 2(d)), whereas the level E = 0.1 eV cross the bands of both upper and lower surfaces (indicated by solid triangles and open circles, respectively, in figure 2(d)). Therefore, the states on the two circular contours at E = 0.28 eV are located at the upper surface. In contrast, the states on the inner and outer contours at E = 0.1 eV are respectively distributed on the lower and upper surfaces of the film instead of on the same surface, leading to a spatial separation. Since the electrons with E = 0.28 eV have larger momentum than those with E=0.1 eV, the spin texture on the surfaces has an interesting momentum dependence, which

indicates that the response of the states on Dirac cone is very complex. Some of them have larger charge distribution on the upper surface, and some on the lower surface. We can estimate the Rashba parameter by α ≈ 2ER / k 0 [34], where ER and k 0 are the Rashba energy and momentum offset (see figure 2(d)), respectively. The Rashba energy and momentum offset of S1 state near the Γ point are ES1 = 0.0219 eV and kS1 = 0.028 Å–1, respectively, yielding a large Rashba parameter αS1≈ 1.564 eVÅ, which is one order of magnitude larger than that of the typical semiconductor heterostructures. Such a large Rashba splitting is very desirable for the application of a spin field effect transistor at room temperature. In agreement with the experiments [7], the valence bands 4

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0.01 V/Å to 0.03 V/Å, as shown in figure 2(g). The momentum offset increases with the magnitude of the electric field. Usually the Rashba energy will increase with the external field. However, our calculations show that it first grows for the electric field between 0.01 and 0.02 V/Å, and then goes down for a larger electric field. The Rashba parameter reaches the maximum value of about 1.9 eVÅ at Eext=0.015 V/Å. In figure  2(h), we plot the Rashba energy splitting (the energy difference between a split band at a given k point) of the bands S1 with respect to the wave vector for 3QL-Bi2Se3 film under a different external electric field. The Rashba energy splitting first grows almost linearly with small k, in agreement with the standard picture of Rashba splitting. However, after k exceeds a certain value, the splitting stops growing and starts dropping. This is because the high-order Rashba terms become more and more prominent, leading to a large deviation from the linear relation between Rashba energy splitting and the wave vectors as the momentum of the electrons increases. Because the Bi in Bi2Se3 film is a high-Z (Z is the atomic number) atom with strong relativistic effect, the contribution of high-order Rashba terms to Rashba effect is significant [36]. We suppose that this property can be generalized to other TIs with heavy atoms. For the QWS Q2, we found similar dependence on the electric field and the considerable contribution of high-order Rashba terms when k was increased. In order to exploit the Rashba bands, some form of doping is usually necessary. We studied a 2×2 3QL-Bi2Se3 film with one Se vacancy, which has the effect of n-doping. It was found that under the electric field of 0.01 V/Å, the Fermi level was moved up to Q1 state and the features of the bands were basically kept. Importantly, the large Rashba spin splitting remained, suggesting that the Rashba effect in Bi2Se3 was robust and would easily be preserved in various environments. We also studied the 4 and 6QL-Bi2Se3 under the external field, as shown in figure  3. The thicker films, which have a smaller gap in zero field, were more sensitive to the external field. A smaller field is needed to close the gap in Dirac cones, as indicated in figures 2 and 3. It can be seen that the Dirac cones of the upper and lower surfaces are split and become gapless. The QWSs have large Rashba splitting, and some of them have significant distribution at the lower surface, as indicated by the open circles in figure 3.

have multiple M-shaped features, which are different from the standard single-particle-like Rashba dispersion. When the field is turned up to 0.02 V/Å, the splittings of S1 and S2 are appreciably increased, as shown in figure 2(c). Notably, the gap at the Fermi level is vanishing. One can see that segments in bands S1 and S2, indicated by open circles and solid triangles, actually form a Dirac cone localized at the lower surface and upper surface, respectively. Therefore, the splitting of bands S1 and S2 can be viewed as a result of the splitting of the Dirac cones of the upper and lower surfaces. The Dirac cone of the upper surface, indicated by solid triangles, is shifted down considerably into the valence bands. In contrast, the Dirac cone of the lower surface, indicated by open circles, is moved a little above the Fermi level. It is found that the Dirac cones of the upper and lower surfaces become gapless. Although the Dirac cones are expected to be gapless on the surface of Bi2Se3, it is found, as discussed above, that a gap remains in the thin films of finite thickness without an electric field. Therefore, truly gapless Dirac cones can be realized on the surfaces of a few-QL Bi2Se3 film by application of an electric field. We also calculated the workfunction of the two surfaces. The average planar electrostatic potentials (APEPs) along the c-axis of a 3QL-Bi2Se3 film under external fields of 0.0 and 0.02 V/Å are shown in figure  2(f), respectively. When Eext = 0, the workfunctions of the upper and lower surfaces are found to have an identical value of 5.52 eV, comparing well with the experimental results [35]. For Eext = 0.02 V/Å, the upper (lower) surface has a smaller (larger) workfunction of 5.32 (5.71) eV (figure 2(f)), indicating that the upper (lower) surface has a larger (smaller) electron chemical potential. Namely, there are more electrons near the upper surface. Therefore, the application of an external electrical field pointing from the upper to the lower surface is equivalent to doping electrons and holes to the upper and lower surfaces, respectively. As a result, the Dirac point of the lower surface is higher in energy than that of the upper surface. Such equivalent electron and hole doping on the two surfaces is desirable because it is easily tunable by the electric field. As shown in figure 2(b) and (c), the quantum well states (QWSs) Q1 and Q2 also have considerable Rashba splitting. The QWS Q2 is bulk-like in zero field with a charge distribution mainly in the middle of the 3QL-Bi2Se3, as depicted in figure  2(e). It has been discussed above that the total charge density has a larger distribution at the upper surface. Therefore, one would expect that the charge distribution of Q2 states would move to the upper surface when the electric field is pointing from the upper to the lower surface of the Bi2Se3 film. However, figure 2(b), (c) and (e) clearly show that Q2 states are mainly distributed at the lower surface instead of the upper surface. Increasing the intensity of the field will make Q2 states more and more surface-like, with more and more charge concentrated at the lower surface [figure 2(e)]. For the QWS M1 in the valence band, it can be seen in figure 2(b) and (c) that it is also driven from the inside into the uppermost QL. In order to study the relation between Rashba splitting and the magnitude of the electric field, we calculated the Rashba energy, momentum offset, and Rashba parameter of the band S1 of 3QL-Bi2Se3 film under the electric field ranging from

4. Conclusion In conclusion, we have investigated the Bi2Se3 films under electric fields by using first-principles calculations based on vdW-DFT. It is found that Rashba-like spin splitting of surface states and QWSs in Bi2Se3 films are induced by the external electric field. The Dirac cones of the opposite surfaces of Bi2Se3 film are split and shifted in opposite directions. The zero field energy gap in the Dirac cones is closed by the electric field. The spin texture on the Dirac cones becomes momentum dependent. Some of the quantum well states are driven by the external field to distribute in the surface region in ultra-thin films, and the tendency of surface distribution of the quantum well states becomes more 5

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apparent as the field increases. The Rashba spin splitting in the Bi2Se3 films can be modulated by the magnitude of the external electric field. The high-order Rashba terms are found to have large contribution, so the Rashba spin splittings in Bi2Se3 films are highly non-linearly related to the momentum and electric field, which is an indication of complex spin dynamics under the electric field. Acknowledgements The authors acknowledge the support of the National Natural Science Foundation of China (Grant No. 11074211, No. 11274265, No. 11464013, No. 11204262 and No. 11274262), National Basic Research Program of China (Grant No. 2012CB921303), Furong Scholar Program of Hunan Provincial Government, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 10A118), the Project supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20124301120006), the US Department of Energy (DOE), Materials Sciences and Engineering Division, Office of Basic Energy Sciences (G M S), and the Oak Ridge Institute for Science and Education (ORISE) HERE Program (J Z). References [1] Hsieh D et al 2009 Nature 460 1101 [2] Xia Y et al 2009 Nature Phys. 5 398 [3] Moore J E 2010 Nature 464 194 [4] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 [5] Qi X-L and Zhang S-C 2010 Phys. Today 63 33 [6] Benia H M, Lin C, Kern K and Ast C R 2011 Phys. Rev. Lett. 107 177602 [7] Bianchi M, Hatch R C, Mi J, Iversen B B and Hofmann P 2011 Phys. Rev. Lett. 107 086802 [8] King P D C et al 2011 Phys. Rev. Lett. 107 096802 [9] Zhu Z H et al 2011 Phys. Rev. Lett. 107 186405

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Electric tuning of the surface and quantum well states in Bi2Se3 films: a first-principles study.

Based on first-principles calculations in the framework of van der Waals density functional theory, we find that giant, Rashba-like spin splittings ca...
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