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Electrical tuning of transport properties of topological insulator ultrathin films H. Li, J. M. Shao, H. B. Zhang and G. W. Yang* Considering that topological insulator (TI) ultrathin films (UTFs) provide an ideal platform for the transport measurement of topologically protected surface states, we have investigated the transport properties of the three-dimensional (3D) TI UTFs through an array of potential barriers. The 3D TI UTF was considered to be thin enough (5 nm) that the top and bottom surface states of the UTF can hybridize to create an energy gap at the Dirac point, which results in a hyperbola-like energy dispersion. It was found that the Klein tunneling effect disappears due to the interaction between the top and bottom surface states. By tuning the barrier strength or the incident energy, three kinds of transport processes can be realized, and the conditions of the transport processes were determined. The oscillatory characters of the transmission (conductance) spectra without a decaying envelope are due to the novel surface states of TIs, which are quite different from that observed for a conventional two-dimensional electron gas. For the structure consisting of two anti-parallel potential barriers, the conductance spectra exhibit a perfect on/off switching effect by tuning the barrier strength, which is favorable for electrically controllable device applications. In the case of a superlattice (SL) structure, due to the mini-gaps induced by the SL geometry, some additional resonant peaks and valleys can be observed in the transmission spectra, and similar characters are also
Received 1st November 2013 Accepted 10th December 2013
reflected in the conductance spectra. Owing to the Dirac characters of the charge carriers therein, the transmission (conductance) spectra never decay with increasing barrier strength, which is distinguished
DOI: 10.1039/c3nr05828j www.rsc.org/nanoscale
from that observed for semiconductor SLs. These findings were not only meaningful for understanding the basic physical processes in the transport of TIs, but also useful for developing nanoscaled TI-based devices.
1. Introduction A three-dimensional (3D) topological insulator (TI) is a new quantum state of matter with conducting surface states in the bulk charge excitation gap.1–4 Recently, some specic materials of 3D TIs have been theoretically predicted and experimentally conrmed in several topologically nontrivial systems including BixSb1x alloys,5–8 Bi2Se3, Bi2Te3 and Sb2Te3 crystals.9–11 The most intriguing and technologically important properties of 3D TIs originate from their unique surface states. In a 3D TI, the combination of the strong spin–orbit coupling (SOC) and the band inversion generates a time-reversal invariant topological order which guarantees the existence of surface states.4,12 Within a certain parameter range,1,2 these surface states consist of an odd number of spin-momentum locked Dirac cones and are robust against the effect of time-reversal invariant perturbations and electron–electron interactions.2 This unique electronic structure enables TIs to be promising materials in spintronics13 and fault-tolerant quantum computing.2,14
State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, Nanotechnology Research Center, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong, P. R. China. E-mail: [email protected]
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To realize the practical device applications of TIs, it is important to control and modulate the transport properties of the surface states. However, in all existing TI materials studied to date, the bulk states in reality are metallic due to inevitable doping such as substitutions and crystal lattice defects.10,11,15 Therefore, ideal transport measurements of topologically protected surface states remain difficult. One effective method to suppress the bulk conductance is to prepare TI samples in the form of an ultrathin lm (UTF). In contrast to the bulk sample, it is convenient to control the topological order by tuning the lm thickness,16,17 and modulate the chemical potential by gating the concentration of charge carriers. Moreover, UTFs allow one to directly reduce the bulk conduction18–23 due to their relatively large surface-to-volume ratio. In addition, from the application point of view, UTFs provide a standard platform for massive device integration at the nanoscale. In the light of the above, in this paper we study the transport properties of the 3D TI UTF through an array of potential barriers (see Fig. 1). From a practical point of view, such a potential prole can be induced either by an array of metallic contacts24,25 or by a pattern of parallel striped gates on the surface.26–37 In this work, we emphasize that the metallic contacts and the gating geometry have been extensively utilized for the electrostatic modulation (results in the potential prole)
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Fig. 1
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Schematics of the one-dimensional square-shaped potential barrier.
in the TI device,18,22,23,32,33 especially for the UTF structure.18,22,23 As recently reported by Zhang et al.,22 a gate tunable TI device can be fabricated by molecular-beam epitaxy (MBE) grown Bi2Se3 lms on SrTiO3 substrates with prepatterned electrodes. Owing to the large dielectric constant of SrTiO3 at low temperature, the Fermi level of the top surface states can be largely modulated by the gating geometry. Moreover, devices fabricated on prepatterned substrates by MBE (e.g. cleaved edges or the walls of chemically grooved substrates) can host automatically precise details, as proposed by H. L. Stormer et al.34 As a matter of fact, by means of MOCVD and MBE techniques, superlattices based on chalcogenides have been extensively studied since the 1990s for thermoelectric applications, and the Bi2Se3/Sb2Te3 superlattices with high interface quality were successfully fabricated on GaAs substrates.35,36 More recently, K. L. Wang et al.37 addressed that TI heterostructures and superlattices with smart geometry can be well fabricated by MBE. In this study, the 3D TI UTF is considered to be thin enough (L ¼ 5 nm without a special emphasis) that the top and bottom surface states can hybridize to generate an energy gap at the Dirac point. This hybridization results in a hyperbola-like energy dispersion,38–40 which results in the disappearance of the Klein tunneling effect. It is found that the transmission (conductance) spectra oscillate with respect to the barrier strength, but the amplitudes of the transmission and conductance never decay with increasing the barrier strength. Clearly, these features are quite different from those observed for conventional two-dimensional electron gas (2DEG). In a structure consisting of two anti-parallel barriers, a remarkable transmission (conductance) cusp is observed by tuning the barrier strength. This character can be utilized as a high performance electrical switch. For the superlattice (SL) structure, it is found that some additional resonant peaks and valleys are present in the transmission spectra due to the minigaps induced by the SL. The oscillatory characters of the transmission (conductance) spectra without a decaying envelope are attributed to the Dirac character of the charge carriers, which is in stark contrast to that observed for the semiconductor SLs. This paper is organized as follows. In Section 2 we introduce a theoretical model and basic formalism. Then, Section 3 presents numerical results and a detailed analysis. Finally, Section 4 presents our conclusion.
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2.
Model and basic formalism
We start with a low-lying effective model for the 3D TI UTF, where the coupling between the top and bottom surface states generates a nite gap at the Dirac point (k ¼ 0). The relevant Hamiltonian is given by39 D1 p 2 B1 p2 2 2 s þ s sz ; H ¼ þ D k þ A k s k þ B k 2 2 x y y x z 2 L2 L2 (1) where sm(m ¼ x, y, z) is the Pauli matrix corresponding to the electron spin, sz ¼ 1 is the electron pseudospin index, and L is the thickness of the 3DTI UTF. Here the rst three terms are the Hamiltonian for the top and bottom surfaces with kx(ky) being the momentum, and k2 ¼ kx2 + ky2. The last term describes the coupling between the top and bottom surfaces. As a matter of fact, eqn (1) can be regarded as a Hamiltonian for the massive Dirac fermions.40 In this study, we choose Bi2Se3 as the prototype material and the corresponding model parameters take the ˚ , B1 ¼ 10 eV A ˚ 2, B2 ¼ following values:9 A2 ¼ 4.1 eVA 2 2 ˚ , D1 ¼ 1.3 eV A ˚ and D2 ¼ 19.6 eV A ˚ 2. 56.6 eV A Neglecting effects induced by the substrate, the two components of the electron pseudospin are decoupled,39,40 hereaer, we only focus on one component of pseudospin. To be specic and without loss of generality, we choose sz ¼ 1. To generate an electrostatic modulation of the transport properties, we apply a set of square-shaped potential barriers V(x, y) ¼ V(x) on the top of the 3D TI UTF, as schematically shown in Fig. 1. We stress here that, on the one hand, the squareshaped potential barrier approximation has been extensively used in similar systems, including semiconductor superlattices,41 graphene42–44 and topologically protected surface states.45 Regardless of their sharp boundary, the square-shaped potential barriers can be experimentally realized. As H. L. Stormer et al.34 proposed, the MBE technique can support the fabrication of abrupt, atomically precise potential barriers in two-dimensional (2D) systems, and they have successfully fabricated a 2D electron gas containing an atomically precise, lateral Kronig–Penney potential of 10.2 nm periodicity. On the other hand, we show that the results on the existence of strong transport modulations (e.g. the on/off switching effect as well as non-decaying features of the transmission and conductance with increasing the barrier strength and length) mainly originate from the energy gap and the strong SOC, which are intrinsic
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properties of the proposed system and cannot be altered by the external electrostatic potential. Therefore, a different choice of the potential prole would not inuence our conclusions. This point can be clearly conrmed by Fig. 4(a) and (d), Fig. 6(c) and (d) and Fig. 8(c), where one can nd that the transmission conguration is insensitive to structural parameters. Moreover, the validity of the square-shaped potential barrier approximation has been demonstrated by S. E. Ulloa et al.41 In their work about the transport properties of semiconductor superlattices, they proposed that the choice of the potential prole does not change the main results of the square-shaped approximation, and the possible differences from one period to another in real systems produce only minor variations in conductance when the potential strength is large enough. In addition, it is also found that the square-shaped approximation has been successfully used in graphene-based systems.43 For example, by the squareshaped approximation, the main results of Tworzydlo et al.43 were experimentally conrmed by R. Danneau et al.44 In our calculation, the width (along the y direction) of the sample, Ly, is assumed to be large enough (Ly [ Wb(Ww)) so that the edge effects can be safely neglected and the translational invariance in the y direction is preserved. These assumptions have been extensively applied to similar models based on graphene42,43 and TIs.45 Consequently, the eigenstates can be written as j(x, y) ¼ j(x)eikyy. Solving the eigenvalue equation {H + V(x, y)}j ¼ Ej we can obtain, for the constant V(x, y) ¼ V, the eigenvalue, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 D1 p 2 B 1 p2 2 2 Es ðkÞ ¼ V þ 2 þ D2 k þ s þ B k þ A2 2 k2 ; (2) 2 L L2
tering coefficients gn(n ¼ 1, 2, 3, 4) can be decided by matching the boundary conditions. As the Hamiltonian is a second-order differential operator, total wave functions in two adjacent regions should be matched via boundary conditions ensuring their continuity as well as the continuity of their rst derivatives. For convenience of notation, we dene a matrix consisting of the total wave function and its rst derivative in the jth region ( j is an integer)
where s ¼ +1(1) corresponds to the conductance (valence) band. According to eqn (2), the energy gap D ¼ 2B1p2/L2, which corresponds to the thickness of the 3D TI UTF. The corresponding eigenstates 1 eiðkx xþky yÞ ; (3) j k x ; ky ¼ fs ðkx Þ 8 9, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 30 nm regardless of ky, as denoted by the green curve in Fig. 2(a) and (b). In addition, the transmission takes a relatively large value and exhibits slight oscillation when V ¼ 0.05 eV (red curves in Fig. 2(a) and (b)). Importantly, the transmission exhibits a switching effect by tuning the barrier strength (red curve in Fig. 2(c)). This character is favorable for electrically controllable device applications. Similar features also can be obtained by modulating the incident energy E (Fig. 2(d)). When E changes from 0.034 eV to 0.045 eV in Fig. 2(d), the transmission disappears because of no propagating modes at the input end. In detail, in Fig. 2, we clearly show the difference of transport scenarios between the two cases. (i) The black dashed curve in Fig. 2(a) corresponds to a thick topological lm (L ¼ 50 nm) which does not hybridize, and the Klein tunneling effect can be clearly observed. This character is remarkably different from that observed for the hybridization case (see the other curves in Fig. 2(a), where the Klein tunneling disappears due to the hybridization gap). Although the hybridization does not break
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the time reversal symmetry, it offers additional backscattering channels so that the backscattering cannot be completely inhibited. (ii) The differences are also reected in the conductance spectra. Note that the on/off switching effect mainly results from the hybridization gap, and this phenomenon is absent in thick samples where the surface state is gapless. The physics of these transport scenarios can be understood from the band structures. Actually, three kinds of transport processes can be realized by tuning the barrier strength V (or equivalently, the incident energy E), i.e. the inter-band process (Fig. 3(a)), the intra-band process (Fig. 3(b)), and the tunneling process through the energy gap (Fig. 3(c)). We would like to emphasize that the last transport process mainly results from the energy gap, which is not realized in the system where energy gaps are absent. For the purpose of quantitatively understanding the transport properties, the conditions for the transport processes are concluded. For a xed incident energy E (E > (D1 + B1)p2/L2 or E < (D1 B1)p2/L2 to ensure a propagating incident mode), the inter-band process can be achieved when V > E + (B1 D1)p2/L2 for E > 0 (from conduction bands to valence bands) and V < E (B1 + D1)p2/L2 for E < 0 (from valence bands to conduction bands). According to the parameters utilized in Fig. 2 (except for Fig. 2(d)), the inter-band process (from conduction bands to valence bands) occurs when V > 0.094 eV. Owing to the electron-hole asymmetry (cf. eqn (2)) as well as the multiple-reection induced by the two potential barrier boundaries, the transmission exhibits sharp oscillations and forms Fabry–P´ erot resonances. In the case of incident energy E > 0, the intra-band process (between conduction bands) occurs when V < E (B1 + D1)p2/L2. When E < 0, the condition for the intra-band process (between valence bands) can be expressed as V > E + (B1 D1)p2/L2. In Fig. 2, the condition for the intra-band process is specied as V < 0.015 eV, and the red curves (V ¼ 0.05 eV and E ¼ 0.06 eV) in Fig. 2(a) and (b) correspond to the intra-band process. For a small barrier length (no more than the decay length of the evanescent modes, i.e. 1/|b|), the tunneling process can be realized when E (B1 + D1)p2/L2 < V < E + (B1 D1)p2/L2. In Fig. 2, the conditions for the tunneling process can be written as 0.015 eV < V < 0.094 eV. For a small barrier length Wb ( 1/|b|), as shown in Fig. 4(d) where the exponential decaying features of the tunneling process are clearly illustrated. Intriguingly, for the inter- and intra-band processes, the transmission just periodically oscillates with respect to the barrier strength V and the potential length Wb, but never decays with increasing the amplitudes of V (panel (c)) and Wb (panel (d)). The oscillatory features without a decaying envelope originate from the novel surface states of TIs. As a matter of fact, the physics of the charge carriers in the present model can be treated as massive Dirac fermions (eqn (1)). Although the energy gap is opened at the Dirac point, the spin and momentum of the surface states remain correlated due to the strong SOC. This character should be distinguished from the conventional 2DEG
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where electrons are controlled by the Schr¨ odinger equation, and the transmission exponentially decays with V and Wb.50 Fig. 5 shows the conductance spectra which offer experimentally measurable predictions. It is found that the conductance can be effectively controlled by electrical modulation. As expected (similar to the transmission), the conductance oscillates with the barrier strength without a decaying envelope (see the red and blue curves in Fig. 5(b)), which is apparently different from that observed for 2DEG.51 It is noteworthy that the conductance spectra exhibit an on/off switching effect when the barrier length Wb is large enough (e.g. Wb ¼ 120), this character is interesting for device applications such as the electrically controllable switch. We address here that the electrically controllable on/off switching effect mainly results from
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Fig. 5 Plots of the conductance G as a function of the incident energy E (panel (a)) and the potential strength V (panel (b)) of the single potential barrier structure, respectively. Red and blue curves correspond to the barrier lengths Wb ¼ 120 nm, and black curves denote Wb ¼ 5 nm. The film thickness L ¼ 5 nm.
the energy gap, which is absent in the gapless systems including graphene and thick TIs (L > Lc). Considering that the tunneling transport process becomes apparent (with certain E and V) when the barrier length Wb is comparable with the decay scale (1/|b|) of evanescent modes, the barrier length Wb should be large enough to get a high performance on/off switching effect. For example, L > 30 nm should be satised when E ¼ 0.06 eV and V ¼ 0.05 eV. B. Transmission and conductance through two potential barriers We now turn to consider a system consisting of two squareshaped potential barriers with the same barrier length Wb,
and separated by a distance Ww. The barrier strengths are denoted as V1 and V2, respectively. Considering that M1 ¼ M3 ¼ M5 and N1(x) ¼ N3(x) ¼ N5(x), we have U ¼ M11M2N21(Wb)M21M1N1(Wb)N11(Wb + Ww) M11M4N4(Wb + Ww)N41(2Wb + Ww) M41M1N1(2Wb + Ww). (15) The transmission and conductance can be obtained by combining eqn (9), (10), (13) and (15). The contour plots of transmission T for the structures consisting of two parallel (V1 ¼ V2) and anti-parallel (V1 ¼ V2) potential barriers are shown in Fig. 6. Compared to that of the
Fig. 6 Contour plots of transmission T of the two potential barrier structure with the thickness L ¼ 5 nm. The distance between the two potential barriers is chosen as Ww ¼ 50 nm. The other parameters are: (a) V1 ¼ V2 ¼ 0.12 eV and Wb ¼ 120 nm; (b) V1 ¼ V2 ¼ 0.12 eV and Wb ¼ 120 nm; (c) V1 ¼ V2 ¼ V, E1 ¼ 0.06 eV, ky ¼ 0 nm1; (d) V1 ¼ V2, E1 ¼ 0.06 eV, ky ¼ 0 nm1. The white areas in panels (a) and (b) denote the parts where a is imaginary.
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single potential barrier structure, the transmission spectra of the parallel barrier structure exhibit more resonant peaks in the inter- and intra-band processes (cf. panels (a) and (c)). This phenomenon results from the constructive interferences which are enhanced by increasing the number of potential barriers. For the anti-parallel potential barrier structure, the constructive interferences are weakened due to the asymmetry geometry. Consequently, the interference enhancement textures of the transmission spectra in the inter- and intra-band processes are not apparent (panel (b)). Interestingly, in stark contrast to that of the single potential barrier, the anti-parallel barrier structure exhibits a perfect transmission in a narrow range of V regardless of the barrier length, as shown in Fig. 6(d). And the transmission spectra of the anti-parallel barriers structure present stronger oscillatory features. These textures are also reected in the conductance spectra, as shown in Fig. 7(a) and (d). The conductance spectra show a remarkable cusp around V ¼ 0, especially for a large barrier length. This character can be utilized as an electrical switch with high performance. From Fig. 7, we also nd that the conductance spectra do not decay with increasing the amplitude of the barrier strength, which is apparently different from that of 2DEG where the conductance decreases with potential strength and sample length.51 The differences between the red and black curves imply that the charge carriers in the proposed structures are also involved in the tunneling transport process. C.
Superlattice
It is well known that the semiconductor superlattice (SL) geometry can effectively tailor band structures and induce mini-gaps.52,53 Consequently, the transport properties can be apparently
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modulated. It is natural to investigate the difference between the SLs based on the conventional semiconductor and the massive Dirac fermion system. To this end, in this subsection we consider a one-dimensional superlattice based on a 3DTI UTF. It can be divided into N identical units, and each unit consists of a squareshaped potential barrier with length Wb and a square-shaped well with length WW. Considering that M1 ¼ M2l+1, M2 ¼ M2l, N1(x) ¼ N2l+1(x), and N2(x) ¼ N2l(x) (where l ¼ 1, 2., N), then following the standard procedure of matching the boundary conditions at boundaries, we can dene the transfer matrix T ¼
N Y l¼1
U1 ðWl ÞU2 ðWb þ Wl Þ;
(16)
where Wl ¼ (l 1)(Wb + Ww), and U1(Wl) ¼ N11(Wl)M11M2N2(Wl),
U2(Wl) ¼ N21(Wl)M21M1N1(Wl) In doing so, we can rewrite eqn (13) as follows 0 1 0 1 t 1 B0C BrC B C ¼ T B C; @ te A @0A 0 re
(17)
(18)
The transmission can be obtained by solving eqn (18) numerically. As an example, we calculate the transmission and conductance through the structure consisting of 10 units, as shown in Fig. 8. On the one hand, the transmission spectra exhibit some
Fig. 7 Plots of the conductance G as a function of the incident energy E (panels (a) and (b)) and potential strength V (panels (c) and (d)) of the structure consisting of two potential barriers, respectively. Panels (a) and (c) correspond to parallel barriers and panels (b) and (d) for anti-parallel barriers. The inter-barrier distance Ww ¼ 50 nm and the film thickness L ¼ 5 nm.
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Fig. 8 Panels (a–c): contour plots of the transmission T of the superlattice structure consisting of 10 units, the film thickness L ¼ 5 nm. The distance between the two potential barriers is chosen as Ww ¼ 50 nm. Panels (a) and (b) V1 ¼ 0.12 eV and Wb ¼ 120 nm, the white areas in panel (a) denote the parts where a is imaginary; (c) V1 ¼ V2, E1 ¼ 0.06 eV, ky ¼ 0 nm1; (d) the conductance G as a function of barrier strength V with E1 ¼ 0.06 eV and Wb ¼ 120 nm.
additional resonant peaks and valleys compared to those for the single potential barrier structure (Fig. 2 and 4). From panels (a–c), sharp oscillations even in the intra-band process can be observed. These phenomena result from the mini-gaps induced by the SL geometry. On the other hand, because of the Dirac character of the charge carriers, the decaying envelope is absent in the transmission spectra, which is remarkably different from that observed for the semiconductor SLs.45 The conductance is shown in Fig. 8(d). In contrast to that of the single potential barrier structure, the conductance strongly oscillates with the barrier strength due to the mini-gaps. As expected (similar to the transmission), unlike that observed for semiconductor SLs,54 the envelope of the conductance oscillations does not decay with the barrier strength. Owing to the energy gap, the conductance spectra exhibit an on/off switching effect which is interesting for device application.
4. Conclusion In summary, we have investigated the transport properties of the massive Dirac fermions through an array of potential barriers based on a 3DTI UTF. The hybridization between the top and bottom surface states has been considered. It was found that the Klein tunneling effect disappears due to the interaction between the top and bottom surface states. Firstly, we considered the transmission and conductance of massive Dirac fermions through the single potential barrier structure. By tuning the barrier strength (or equivalently, the incident energy), three kinds of transport processes can be realized. The conditions for these transport processes were determined. The band structures were introduced to understand
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these transport scenarios. In the intra- and inter-band transport processes, it was found that the transmission (conductance) spectra oscillate with respect to the barrier strength, but the amplitudes of the transmission (conductance) never decay with increasing the barrier strength. The oscillatory features of the transmission (conductance) without a decaying envelope originate from the novel surface states of TIs, which is quite different from that observed for conventional 2DEG. When the barrier length is larger than the decaying scale of evanescent modes, an on/off switching effect of the conductance can be obtained by tuning the barrier strength (incident energy). This character is favorable for the electrically controllable device applications. Secondly, the transport properties of the structure consisting of two parallel and anti-parallel potential barriers were studied. Since the constructive interferences can be enhanced by increasing the number of potential barriers, some additional resonant peaks of the transmission spectra were observed for the parallel barrier structure compared to those for the single barrier structure. In the anti-parallel barrier structure, a remarkable transmission (conductance) cusp was observed by tuning the barrier strength, which can be utilized as a high performance electrical switch. Finally, a one-dimensional SL based on a 3D TI UTF was investigated. Compared to the single potential barrier structure, the transmission spectra exhibit some resonant peaks and valleys due to the SL geometry. Owing to the Dirac character of the charge carriers, the transmission (conductance) spectra never decay with increasing the barrier strength, which is quite different from that observed for semiconductor SLs. These ndings were not only meaningful for understanding the basic physical processes in the transport of TI, but also useful for developing nanoscaled TI-based devices.
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Acknowledgements This work is supported by the State Key Laboratory of Optoelectronic Materials and Technologies of Sun Yat-sen University.
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References 1 M. Z. Hasan and C. L. Cane, Rev. Mod. Phys., 2010, 82, 3045. 2 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys., 2011, 83, 1057. 3 J. E. Moore, Nature, 2010, 464, 194. 4 L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2007, 98, 106803. 5 L. Fu and C. L. Kane, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 045302. 6 J. C. Y. Teo, L. Fu and C. L. Kane, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 045426. 7 D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nature, 2008, 452, 970. 8 D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava and M. Z. Hasan, Science, 2009, 323, 919. 9 H. J. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and S.-C. Zhang, Nat. Phys., 2009, 5, 438. 10 Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nat. Phys., 2009, 5, 398. 11 Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain and Z.-X. Shen, Science, 2009, 325, 178. 12 C. X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang and S.-C. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 045122. 13 D. Pesin and A. H. MacDonald, Nat. Mater., 2012, 11, 409. 14 L. Fu and C. L. Cane, Phys. Rev. Lett., 2008, 100, 096407. 15 D. Culcer, Phys. E, 2012, 44, 860. 16 Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, S.-Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-C. Ma and Q.-K. Xue, Nat. Phys., 2010, 6, 584. 17 T. Hirahara, Y. Sakamoto, Y. Saisyu, H. Miyazaki, S. Kimura, T. Okuda, I. Matsuda, S. Murakami and S. Hasegawa, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 165422. 18 L. He, F. Xiu, X. Yu, M. Teague, W. Jiang, Y. Fan, X. Kou, M. Lang, Y. Wang, G. Huang, N.-C. Yeh and K. L. Wang, Nano Lett., 2012, 12, 1486. 19 A. A. Taskin, S. Sasaki, K. Segawa and Y. Ando, Phys. Rev. Lett., 2012, 109, 066803. 20 N. Bansal, Y. S. Kim, M. Brahlek, E. Edrey and S. Oh, Phys. Rev. Lett., 2012, 109, 116804. 21 G. Yin, D. Wickramaratne and R. K. Lake, J. Appl. Phys., 2013, 113, 063707. 22 T. Zhang, N. Levy, J. Ha, Y. Kuk and J. A. Stroscio, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 115410. 23 T. Zhang, J. Ha, N. Levy, Y. Kuk and J. Stroscio, Phys. Rev. Lett., 2013, 111, 056803.
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24 B. Huard, N. Stander, J. A. Sulpizio and D. GoldhaberGordon, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 121402. 25 N. Stander, B. Huard and D. Goldhaber-Gordon, Phys. Rev. Lett., 2009, 102, 026807. 26 B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang and D. G. Gordon, Phys. Rev. Lett., 2007, 98, 236803. 27 J. R. Williams, L. Dicarlo and C. M. Marcus, Science, 2007, 317, 638. ¨ 28 B. Ozyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov and P. Kim, Phys. Rev. Lett., 2007, 99, 166804. 29 J. B. Oostinga, H. B. Heersche, X. Liu, A. F. Morpurgo and L. M. K. Vandersypen, Nat. Mater., 2008, 7, 151. 30 R. V. Gorbachev, A. S. Mayorov, A. K. Saychenko, D. W. Horsell and F. Guinea, Nano Lett., 2008, 8, 1995. 31 G. Liu, J. J. Velasco, W. Bao and C. N. Lau, Appl. Phys. Lett., 2008, 92, 203103. 32 H. Steinber, D. R. Gardner, Y. S. Lee and P. J. Herrero, Nano Lett., 2010, 10, 5032. 33 Y. Wang, F. Xiu, L. Cheng, L. He, M. Lang, J. Tang, X. Kou, X. Yu, X. Jiang, Z. Chen, J. Zou and W. L. Wang, Nano Lett., 2012, 12, 1170. 34 H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West and J. Spector, Appl. Phys. Lett., 1991, 58, 726. 35 T. C. Harman, P. J. Taylar, M. P. Walsh and B. E. LaForge, Science, 2002, 297, 2229. 36 R. Venkatasubramanian, T. Colpitts and B. ƠQuinn, Appl. Phys. Lett., 1999, 75, 1104. 37 L. He, X. Kou and K. L. Wang, Phys. Status Solidi RRL, 2013, 7, 50. 38 J. Linder, T. Yokoyama and A. Sudbø, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 205401. 39 W.-Y. Shan, H.-Z. Lu and S.-Q. Shen, New J. Phys., 2010, 12, 043048. 40 H.-Z. Lu, W.-Y. Shan, W. Yao, Q. Niu and S.-Q. Shen, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 115407. 41 S. E. Ulloa, E. Casta˜ no and G. Kirczenow, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 12350. 42 C. Bai and X. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 075430; M. Barbier, P. Vasilopoulos and F. M. Peeters, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 205415; L.-G. Wang and S.-Y. Zhu, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 205444. 43 J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz and C. W. J. Beenakker, Phys. Rev. Lett., 2006, 96, 246802. 44 R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo and P. J. Hakonen, Phys. Rev. Lett., 2008, 100, 196802. 45 S. Mondal, D. Sen, K. Sengupta and R. Shankar, Phys. Rev. Lett., 2010, 104, 046403; A. Yamakage, K.-I. Imura, J. Cayssol and Y. Kuramoto, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 125401; T. Yokoyama, Y. Tanaka and N. Nagaosa, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 121401.
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46 S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. 47 G. J. Xu, X. G. Xu, B. H. Wu, J. C. Cao and C. Zhang, J. Appl. Phys., 2010, 107, 123718. 48 H. C. Nguyen and V. L. Nguyen, J. Phys.: Condens. Matter, 2009, 21, 045305. 49 M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nat. Phys., 2006, 2, 620. 50 Z.-L. Ji and D. W. L. Sprung, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 56, 1045.
This journal is © The Royal Society of Chemistry 2014
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51 N. K. Patel, L. M. Moreno, M. Pepper, R. Newbury, J. E. F. Frost, D. A. Ritchie, G. A. C. Jones, J. T. M. B. Janssen, J. Singleton and J. A. A. J. Perenboom, J. Phys.: Condens. Matter, 1990, 2, 7247. 52 R. Tsu and L. Esaki, Appl. Phys. Lett., 1973, 22, 562. 53 F. Beltram, F. Capasso, D. L. Sivco, A. L. Hutchinson, S.-N. G. Chu and A. Y. Cho, Phys. Rev. Lett., 1990, 64, 3167. 54 P. Pereyra and E. Castillo, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 65, 205120.
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Electrical tuning of transport properties of topological insulator ultrathin films H. Li, J. M. Shao, H. B. Zhang and G. W. Yang* Considering that topological insulator (TI) ultrathin films (UTFs) provide an ideal platform for the transport measurement of topologically protected surface states, we have investigated the transport properties of the three-dimensional (3D) TI UTFs through an array of potential barriers. The 3D TI UTF was considered to be thin enough (5 nm) that the top and bottom surface states of the UTF can hybridize to create an energy gap at the Dirac point, which results in a hyperbola-like energy dispersion. It was found that the Klein tunneling effect disappears due to the interaction between the top and bottom surface states. By tuning the barrier strength or the incident energy, three kinds of transport processes can be realized, and the conditions of the transport processes were determined. The oscillatory characters of the transmission (conductance) spectra without a decaying envelope are due to the novel surface states of TIs, which are quite different from that observed for a conventional two-dimensional electron gas. For the structure consisting of two anti-parallel potential barriers, the conductance spectra exhibit a perfect on/off switching effect by tuning the barrier strength, which is favorable for electrically controllable device applications. In the case of a superlattice (SL) structure, due to the mini-gaps induced by the SL geometry, some additional resonant peaks and valleys can be observed in the transmission spectra, and similar characters are also
Received 1st November 2013 Accepted 10th December 2013
reflected in the conductance spectra. Owing to the Dirac characters of the charge carriers therein, the transmission (conductance) spectra never decay with increasing barrier strength, which is distinguished
DOI: 10.1039/c3nr05828j www.rsc.org/nanoscale
from that observed for semiconductor SLs. These findings were not only meaningful for understanding the basic physical processes in the transport of TIs, but also useful for developing nanoscaled TI-based devices.
1. Introduction A three-dimensional (3D) topological insulator (TI) is a new quantum state of matter with conducting surface states in the bulk charge excitation gap.1–4 Recently, some specic materials of 3D TIs have been theoretically predicted and experimentally conrmed in several topologically nontrivial systems including BixSb1x alloys,5–8 Bi2Se3, Bi2Te3 and Sb2Te3 crystals.9–11 The most intriguing and technologically important properties of 3D TIs originate from their unique surface states. In a 3D TI, the combination of the strong spin–orbit coupling (SOC) and the band inversion generates a time-reversal invariant topological order which guarantees the existence of surface states.4,12 Within a certain parameter range,1,2 these surface states consist of an odd number of spin-momentum locked Dirac cones and are robust against the effect of time-reversal invariant perturbations and electron–electron interactions.2 This unique electronic structure enables TIs to be promising materials in spintronics13 and fault-tolerant quantum computing.2,14
State Key Laboratory of Optoelectronic Materials and Technologies, Institute of Optoelectronic and Functional Composite Materials, Nanotechnology Research Center, School of Physics and Engineering, Sun Yat-sen University, Guangzhou 510275, Guangdong, P. R. China. E-mail: [email protected]
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To realize the practical device applications of TIs, it is important to control and modulate the transport properties of the surface states. However, in all existing TI materials studied to date, the bulk states in reality are metallic due to inevitable doping such as substitutions and crystal lattice defects.10,11,15 Therefore, ideal transport measurements of topologically protected surface states remain difficult. One effective method to suppress the bulk conductance is to prepare TI samples in the form of an ultrathin lm (UTF). In contrast to the bulk sample, it is convenient to control the topological order by tuning the lm thickness,16,17 and modulate the chemical potential by gating the concentration of charge carriers. Moreover, UTFs allow one to directly reduce the bulk conduction18–23 due to their relatively large surface-to-volume ratio. In addition, from the application point of view, UTFs provide a standard platform for massive device integration at the nanoscale. In the light of the above, in this paper we study the transport properties of the 3D TI UTF through an array of potential barriers (see Fig. 1). From a practical point of view, such a potential prole can be induced either by an array of metallic contacts24,25 or by a pattern of parallel striped gates on the surface.26–37 In this work, we emphasize that the metallic contacts and the gating geometry have been extensively utilized for the electrostatic modulation (results in the potential prole)
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Fig. 1
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Schematics of the one-dimensional square-shaped potential barrier.
in the TI device,18,22,23,32,33 especially for the UTF structure.18,22,23 As recently reported by Zhang et al.,22 a gate tunable TI device can be fabricated by molecular-beam epitaxy (MBE) grown Bi2Se3 lms on SrTiO3 substrates with prepatterned electrodes. Owing to the large dielectric constant of SrTiO3 at low temperature, the Fermi level of the top surface states can be largely modulated by the gating geometry. Moreover, devices fabricated on prepatterned substrates by MBE (e.g. cleaved edges or the walls of chemically grooved substrates) can host automatically precise details, as proposed by H. L. Stormer et al.34 As a matter of fact, by means of MOCVD and MBE techniques, superlattices based on chalcogenides have been extensively studied since the 1990s for thermoelectric applications, and the Bi2Se3/Sb2Te3 superlattices with high interface quality were successfully fabricated on GaAs substrates.35,36 More recently, K. L. Wang et al.37 addressed that TI heterostructures and superlattices with smart geometry can be well fabricated by MBE. In this study, the 3D TI UTF is considered to be thin enough (L ¼ 5 nm without a special emphasis) that the top and bottom surface states can hybridize to generate an energy gap at the Dirac point. This hybridization results in a hyperbola-like energy dispersion,38–40 which results in the disappearance of the Klein tunneling effect. It is found that the transmission (conductance) spectra oscillate with respect to the barrier strength, but the amplitudes of the transmission and conductance never decay with increasing the barrier strength. Clearly, these features are quite different from those observed for conventional two-dimensional electron gas (2DEG). In a structure consisting of two anti-parallel barriers, a remarkable transmission (conductance) cusp is observed by tuning the barrier strength. This character can be utilized as a high performance electrical switch. For the superlattice (SL) structure, it is found that some additional resonant peaks and valleys are present in the transmission spectra due to the minigaps induced by the SL. The oscillatory characters of the transmission (conductance) spectra without a decaying envelope are attributed to the Dirac character of the charge carriers, which is in stark contrast to that observed for the semiconductor SLs. This paper is organized as follows. In Section 2 we introduce a theoretical model and basic formalism. Then, Section 3 presents numerical results and a detailed analysis. Finally, Section 4 presents our conclusion.
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2.
Model and basic formalism
We start with a low-lying effective model for the 3D TI UTF, where the coupling between the top and bottom surface states generates a nite gap at the Dirac point (k ¼ 0). The relevant Hamiltonian is given by39 D1 p 2 B1 p2 2 2 s þ s sz ; H ¼ þ D k þ A k s k þ B k 2 2 x y y x z 2 L2 L2 (1) where sm(m ¼ x, y, z) is the Pauli matrix corresponding to the electron spin, sz ¼ 1 is the electron pseudospin index, and L is the thickness of the 3DTI UTF. Here the rst three terms are the Hamiltonian for the top and bottom surfaces with kx(ky) being the momentum, and k2 ¼ kx2 + ky2. The last term describes the coupling between the top and bottom surfaces. As a matter of fact, eqn (1) can be regarded as a Hamiltonian for the massive Dirac fermions.40 In this study, we choose Bi2Se3 as the prototype material and the corresponding model parameters take the ˚ , B1 ¼ 10 eV A ˚ 2, B2 ¼ following values:9 A2 ¼ 4.1 eVA 2 2 ˚ , D1 ¼ 1.3 eV A ˚ and D2 ¼ 19.6 eV A ˚ 2. 56.6 eV A Neglecting effects induced by the substrate, the two components of the electron pseudospin are decoupled,39,40 hereaer, we only focus on one component of pseudospin. To be specic and without loss of generality, we choose sz ¼ 1. To generate an electrostatic modulation of the transport properties, we apply a set of square-shaped potential barriers V(x, y) ¼ V(x) on the top of the 3D TI UTF, as schematically shown in Fig. 1. We stress here that, on the one hand, the squareshaped potential barrier approximation has been extensively used in similar systems, including semiconductor superlattices,41 graphene42–44 and topologically protected surface states.45 Regardless of their sharp boundary, the square-shaped potential barriers can be experimentally realized. As H. L. Stormer et al.34 proposed, the MBE technique can support the fabrication of abrupt, atomically precise potential barriers in two-dimensional (2D) systems, and they have successfully fabricated a 2D electron gas containing an atomically precise, lateral Kronig–Penney potential of 10.2 nm periodicity. On the other hand, we show that the results on the existence of strong transport modulations (e.g. the on/off switching effect as well as non-decaying features of the transmission and conductance with increasing the barrier strength and length) mainly originate from the energy gap and the strong SOC, which are intrinsic
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properties of the proposed system and cannot be altered by the external electrostatic potential. Therefore, a different choice of the potential prole would not inuence our conclusions. This point can be clearly conrmed by Fig. 4(a) and (d), Fig. 6(c) and (d) and Fig. 8(c), where one can nd that the transmission conguration is insensitive to structural parameters. Moreover, the validity of the square-shaped potential barrier approximation has been demonstrated by S. E. Ulloa et al.41 In their work about the transport properties of semiconductor superlattices, they proposed that the choice of the potential prole does not change the main results of the square-shaped approximation, and the possible differences from one period to another in real systems produce only minor variations in conductance when the potential strength is large enough. In addition, it is also found that the square-shaped approximation has been successfully used in graphene-based systems.43 For example, by the squareshaped approximation, the main results of Tworzydlo et al.43 were experimentally conrmed by R. Danneau et al.44 In our calculation, the width (along the y direction) of the sample, Ly, is assumed to be large enough (Ly [ Wb(Ww)) so that the edge effects can be safely neglected and the translational invariance in the y direction is preserved. These assumptions have been extensively applied to similar models based on graphene42,43 and TIs.45 Consequently, the eigenstates can be written as j(x, y) ¼ j(x)eikyy. Solving the eigenvalue equation {H + V(x, y)}j ¼ Ej we can obtain, for the constant V(x, y) ¼ V, the eigenvalue, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 D1 p 2 B 1 p2 2 2 Es ðkÞ ¼ V þ 2 þ D2 k þ s þ B k þ A2 2 k2 ; (2) 2 L L2
tering coefficients gn(n ¼ 1, 2, 3, 4) can be decided by matching the boundary conditions. As the Hamiltonian is a second-order differential operator, total wave functions in two adjacent regions should be matched via boundary conditions ensuring their continuity as well as the continuity of their rst derivatives. For convenience of notation, we dene a matrix consisting of the total wave function and its rst derivative in the jth region ( j is an integer)
where s ¼ +1(1) corresponds to the conductance (valence) band. According to eqn (2), the energy gap D ¼ 2B1p2/L2, which corresponds to the thickness of the 3D TI UTF. The corresponding eigenstates 1 eiðkx xþky yÞ ; (3) j k x ; ky ¼ fs ðkx Þ 8 9, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 30 nm regardless of ky, as denoted by the green curve in Fig. 2(a) and (b). In addition, the transmission takes a relatively large value and exhibits slight oscillation when V ¼ 0.05 eV (red curves in Fig. 2(a) and (b)). Importantly, the transmission exhibits a switching effect by tuning the barrier strength (red curve in Fig. 2(c)). This character is favorable for electrically controllable device applications. Similar features also can be obtained by modulating the incident energy E (Fig. 2(d)). When E changes from 0.034 eV to 0.045 eV in Fig. 2(d), the transmission disappears because of no propagating modes at the input end. In detail, in Fig. 2, we clearly show the difference of transport scenarios between the two cases. (i) The black dashed curve in Fig. 2(a) corresponds to a thick topological lm (L ¼ 50 nm) which does not hybridize, and the Klein tunneling effect can be clearly observed. This character is remarkably different from that observed for the hybridization case (see the other curves in Fig. 2(a), where the Klein tunneling disappears due to the hybridization gap). Although the hybridization does not break
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the time reversal symmetry, it offers additional backscattering channels so that the backscattering cannot be completely inhibited. (ii) The differences are also reected in the conductance spectra. Note that the on/off switching effect mainly results from the hybridization gap, and this phenomenon is absent in thick samples where the surface state is gapless. The physics of these transport scenarios can be understood from the band structures. Actually, three kinds of transport processes can be realized by tuning the barrier strength V (or equivalently, the incident energy E), i.e. the inter-band process (Fig. 3(a)), the intra-band process (Fig. 3(b)), and the tunneling process through the energy gap (Fig. 3(c)). We would like to emphasize that the last transport process mainly results from the energy gap, which is not realized in the system where energy gaps are absent. For the purpose of quantitatively understanding the transport properties, the conditions for the transport processes are concluded. For a xed incident energy E (E > (D1 + B1)p2/L2 or E < (D1 B1)p2/L2 to ensure a propagating incident mode), the inter-band process can be achieved when V > E + (B1 D1)p2/L2 for E > 0 (from conduction bands to valence bands) and V < E (B1 + D1)p2/L2 for E < 0 (from valence bands to conduction bands). According to the parameters utilized in Fig. 2 (except for Fig. 2(d)), the inter-band process (from conduction bands to valence bands) occurs when V > 0.094 eV. Owing to the electron-hole asymmetry (cf. eqn (2)) as well as the multiple-reection induced by the two potential barrier boundaries, the transmission exhibits sharp oscillations and forms Fabry–P´ erot resonances. In the case of incident energy E > 0, the intra-band process (between conduction bands) occurs when V < E (B1 + D1)p2/L2. When E < 0, the condition for the intra-band process (between valence bands) can be expressed as V > E + (B1 D1)p2/L2. In Fig. 2, the condition for the intra-band process is specied as V < 0.015 eV, and the red curves (V ¼ 0.05 eV and E ¼ 0.06 eV) in Fig. 2(a) and (b) correspond to the intra-band process. For a small barrier length (no more than the decay length of the evanescent modes, i.e. 1/|b|), the tunneling process can be realized when E (B1 + D1)p2/L2 < V < E + (B1 D1)p2/L2. In Fig. 2, the conditions for the tunneling process can be written as 0.015 eV < V < 0.094 eV. For a small barrier length Wb ( 1/|b|), as shown in Fig. 4(d) where the exponential decaying features of the tunneling process are clearly illustrated. Intriguingly, for the inter- and intra-band processes, the transmission just periodically oscillates with respect to the barrier strength V and the potential length Wb, but never decays with increasing the amplitudes of V (panel (c)) and Wb (panel (d)). The oscillatory features without a decaying envelope originate from the novel surface states of TIs. As a matter of fact, the physics of the charge carriers in the present model can be treated as massive Dirac fermions (eqn (1)). Although the energy gap is opened at the Dirac point, the spin and momentum of the surface states remain correlated due to the strong SOC. This character should be distinguished from the conventional 2DEG
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where electrons are controlled by the Schr¨ odinger equation, and the transmission exponentially decays with V and Wb.50 Fig. 5 shows the conductance spectra which offer experimentally measurable predictions. It is found that the conductance can be effectively controlled by electrical modulation. As expected (similar to the transmission), the conductance oscillates with the barrier strength without a decaying envelope (see the red and blue curves in Fig. 5(b)), which is apparently different from that observed for 2DEG.51 It is noteworthy that the conductance spectra exhibit an on/off switching effect when the barrier length Wb is large enough (e.g. Wb ¼ 120), this character is interesting for device applications such as the electrically controllable switch. We address here that the electrically controllable on/off switching effect mainly results from
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Fig. 5 Plots of the conductance G as a function of the incident energy E (panel (a)) and the potential strength V (panel (b)) of the single potential barrier structure, respectively. Red and blue curves correspond to the barrier lengths Wb ¼ 120 nm, and black curves denote Wb ¼ 5 nm. The film thickness L ¼ 5 nm.
the energy gap, which is absent in the gapless systems including graphene and thick TIs (L > Lc). Considering that the tunneling transport process becomes apparent (with certain E and V) when the barrier length Wb is comparable with the decay scale (1/|b|) of evanescent modes, the barrier length Wb should be large enough to get a high performance on/off switching effect. For example, L > 30 nm should be satised when E ¼ 0.06 eV and V ¼ 0.05 eV. B. Transmission and conductance through two potential barriers We now turn to consider a system consisting of two squareshaped potential barriers with the same barrier length Wb,
and separated by a distance Ww. The barrier strengths are denoted as V1 and V2, respectively. Considering that M1 ¼ M3 ¼ M5 and N1(x) ¼ N3(x) ¼ N5(x), we have U ¼ M11M2N21(Wb)M21M1N1(Wb)N11(Wb + Ww) M11M4N4(Wb + Ww)N41(2Wb + Ww) M41M1N1(2Wb + Ww). (15) The transmission and conductance can be obtained by combining eqn (9), (10), (13) and (15). The contour plots of transmission T for the structures consisting of two parallel (V1 ¼ V2) and anti-parallel (V1 ¼ V2) potential barriers are shown in Fig. 6. Compared to that of the
Fig. 6 Contour plots of transmission T of the two potential barrier structure with the thickness L ¼ 5 nm. The distance between the two potential barriers is chosen as Ww ¼ 50 nm. The other parameters are: (a) V1 ¼ V2 ¼ 0.12 eV and Wb ¼ 120 nm; (b) V1 ¼ V2 ¼ 0.12 eV and Wb ¼ 120 nm; (c) V1 ¼ V2 ¼ V, E1 ¼ 0.06 eV, ky ¼ 0 nm1; (d) V1 ¼ V2, E1 ¼ 0.06 eV, ky ¼ 0 nm1. The white areas in panels (a) and (b) denote the parts where a is imaginary.
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single potential barrier structure, the transmission spectra of the parallel barrier structure exhibit more resonant peaks in the inter- and intra-band processes (cf. panels (a) and (c)). This phenomenon results from the constructive interferences which are enhanced by increasing the number of potential barriers. For the anti-parallel potential barrier structure, the constructive interferences are weakened due to the asymmetry geometry. Consequently, the interference enhancement textures of the transmission spectra in the inter- and intra-band processes are not apparent (panel (b)). Interestingly, in stark contrast to that of the single potential barrier, the anti-parallel barrier structure exhibits a perfect transmission in a narrow range of V regardless of the barrier length, as shown in Fig. 6(d). And the transmission spectra of the anti-parallel barriers structure present stronger oscillatory features. These textures are also reected in the conductance spectra, as shown in Fig. 7(a) and (d). The conductance spectra show a remarkable cusp around V ¼ 0, especially for a large barrier length. This character can be utilized as an electrical switch with high performance. From Fig. 7, we also nd that the conductance spectra do not decay with increasing the amplitude of the barrier strength, which is apparently different from that of 2DEG where the conductance decreases with potential strength and sample length.51 The differences between the red and black curves imply that the charge carriers in the proposed structures are also involved in the tunneling transport process. C.
Superlattice
It is well known that the semiconductor superlattice (SL) geometry can effectively tailor band structures and induce mini-gaps.52,53 Consequently, the transport properties can be apparently
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modulated. It is natural to investigate the difference between the SLs based on the conventional semiconductor and the massive Dirac fermion system. To this end, in this subsection we consider a one-dimensional superlattice based on a 3DTI UTF. It can be divided into N identical units, and each unit consists of a squareshaped potential barrier with length Wb and a square-shaped well with length WW. Considering that M1 ¼ M2l+1, M2 ¼ M2l, N1(x) ¼ N2l+1(x), and N2(x) ¼ N2l(x) (where l ¼ 1, 2., N), then following the standard procedure of matching the boundary conditions at boundaries, we can dene the transfer matrix T ¼
N Y l¼1
U1 ðWl ÞU2 ðWb þ Wl Þ;
(16)
where Wl ¼ (l 1)(Wb + Ww), and U1(Wl) ¼ N11(Wl)M11M2N2(Wl),
U2(Wl) ¼ N21(Wl)M21M1N1(Wl) In doing so, we can rewrite eqn (13) as follows 0 1 0 1 t 1 B0C BrC B C ¼ T B C; @ te A @0A 0 re
(17)
(18)
The transmission can be obtained by solving eqn (18) numerically. As an example, we calculate the transmission and conductance through the structure consisting of 10 units, as shown in Fig. 8. On the one hand, the transmission spectra exhibit some
Fig. 7 Plots of the conductance G as a function of the incident energy E (panels (a) and (b)) and potential strength V (panels (c) and (d)) of the structure consisting of two potential barriers, respectively. Panels (a) and (c) correspond to parallel barriers and panels (b) and (d) for anti-parallel barriers. The inter-barrier distance Ww ¼ 50 nm and the film thickness L ¼ 5 nm.
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Fig. 8 Panels (a–c): contour plots of the transmission T of the superlattice structure consisting of 10 units, the film thickness L ¼ 5 nm. The distance between the two potential barriers is chosen as Ww ¼ 50 nm. Panels (a) and (b) V1 ¼ 0.12 eV and Wb ¼ 120 nm, the white areas in panel (a) denote the parts where a is imaginary; (c) V1 ¼ V2, E1 ¼ 0.06 eV, ky ¼ 0 nm1; (d) the conductance G as a function of barrier strength V with E1 ¼ 0.06 eV and Wb ¼ 120 nm.
additional resonant peaks and valleys compared to those for the single potential barrier structure (Fig. 2 and 4). From panels (a–c), sharp oscillations even in the intra-band process can be observed. These phenomena result from the mini-gaps induced by the SL geometry. On the other hand, because of the Dirac character of the charge carriers, the decaying envelope is absent in the transmission spectra, which is remarkably different from that observed for the semiconductor SLs.45 The conductance is shown in Fig. 8(d). In contrast to that of the single potential barrier structure, the conductance strongly oscillates with the barrier strength due to the mini-gaps. As expected (similar to the transmission), unlike that observed for semiconductor SLs,54 the envelope of the conductance oscillations does not decay with the barrier strength. Owing to the energy gap, the conductance spectra exhibit an on/off switching effect which is interesting for device application.
4. Conclusion In summary, we have investigated the transport properties of the massive Dirac fermions through an array of potential barriers based on a 3DTI UTF. The hybridization between the top and bottom surface states has been considered. It was found that the Klein tunneling effect disappears due to the interaction between the top and bottom surface states. Firstly, we considered the transmission and conductance of massive Dirac fermions through the single potential barrier structure. By tuning the barrier strength (or equivalently, the incident energy), three kinds of transport processes can be realized. The conditions for these transport processes were determined. The band structures were introduced to understand
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these transport scenarios. In the intra- and inter-band transport processes, it was found that the transmission (conductance) spectra oscillate with respect to the barrier strength, but the amplitudes of the transmission (conductance) never decay with increasing the barrier strength. The oscillatory features of the transmission (conductance) without a decaying envelope originate from the novel surface states of TIs, which is quite different from that observed for conventional 2DEG. When the barrier length is larger than the decaying scale of evanescent modes, an on/off switching effect of the conductance can be obtained by tuning the barrier strength (incident energy). This character is favorable for the electrically controllable device applications. Secondly, the transport properties of the structure consisting of two parallel and anti-parallel potential barriers were studied. Since the constructive interferences can be enhanced by increasing the number of potential barriers, some additional resonant peaks of the transmission spectra were observed for the parallel barrier structure compared to those for the single barrier structure. In the anti-parallel barrier structure, a remarkable transmission (conductance) cusp was observed by tuning the barrier strength, which can be utilized as a high performance electrical switch. Finally, a one-dimensional SL based on a 3D TI UTF was investigated. Compared to the single potential barrier structure, the transmission spectra exhibit some resonant peaks and valleys due to the SL geometry. Owing to the Dirac character of the charge carriers, the transmission (conductance) spectra never decay with increasing the barrier strength, which is quite different from that observed for semiconductor SLs. These ndings were not only meaningful for understanding the basic physical processes in the transport of TI, but also useful for developing nanoscaled TI-based devices.
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Acknowledgements This work is supported by the State Key Laboratory of Optoelectronic Materials and Technologies of Sun Yat-sen University.
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References 1 M. Z. Hasan and C. L. Cane, Rev. Mod. Phys., 2010, 82, 3045. 2 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys., 2011, 83, 1057. 3 J. E. Moore, Nature, 2010, 464, 194. 4 L. Fu, C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2007, 98, 106803. 5 L. Fu and C. L. Kane, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 045302. 6 J. C. Y. Teo, L. Fu and C. L. Kane, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 045426. 7 D. Hsieh, D. Qian, L. Wray, Y. Xia, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nature, 2008, 452, 970. 8 D. Hsieh, Y. Xia, L. Wray, D. Qian, A. Pal, J. H. Dil, J. Osterwalder, F. Meier, G. Bihlmayer, C. L. Kane, Y. S. Hor, R. J. Cava and M. Z. Hasan, Science, 2009, 323, 919. 9 H. J. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang and S.-C. Zhang, Nat. Phys., 2009, 5, 438. 10 Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava and M. Z. Hasan, Nat. Phys., 2009, 5, 398. 11 Y. L. Chen, J. G. Analytis, J.-H. Chu, Z. K. Liu, S.-K. Mo, X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. Zhang, I. R. Fisher, Z. Hussain and Z.-X. Shen, Science, 2009, 325, 178. 12 C. X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang and S.-C. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 045122. 13 D. Pesin and A. H. MacDonald, Nat. Mater., 2012, 11, 409. 14 L. Fu and C. L. Cane, Phys. Rev. Lett., 2008, 100, 096407. 15 D. Culcer, Phys. E, 2012, 44, 860. 16 Y. Zhang, K. He, C.-Z. Chang, C.-L. Song, L.-L. Wang, X. Chen, J.-F. Jia, Z. Fang, X. Dai, W.-Y. Shan, S.-Q. Shen, Q. Niu, X.-L. Qi, S.-C. Zhang, X.-C. Ma and Q.-K. Xue, Nat. Phys., 2010, 6, 584. 17 T. Hirahara, Y. Sakamoto, Y. Saisyu, H. Miyazaki, S. Kimura, T. Okuda, I. Matsuda, S. Murakami and S. Hasegawa, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 165422. 18 L. He, F. Xiu, X. Yu, M. Teague, W. Jiang, Y. Fan, X. Kou, M. Lang, Y. Wang, G. Huang, N.-C. Yeh and K. L. Wang, Nano Lett., 2012, 12, 1486. 19 A. A. Taskin, S. Sasaki, K. Segawa and Y. Ando, Phys. Rev. Lett., 2012, 109, 066803. 20 N. Bansal, Y. S. Kim, M. Brahlek, E. Edrey and S. Oh, Phys. Rev. Lett., 2012, 109, 116804. 21 G. Yin, D. Wickramaratne and R. K. Lake, J. Appl. Phys., 2013, 113, 063707. 22 T. Zhang, N. Levy, J. Ha, Y. Kuk and J. A. Stroscio, Phys. Rev. B: Condens. Matter Mater. Phys., 2013, 87, 115410. 23 T. Zhang, J. Ha, N. Levy, Y. Kuk and J. Stroscio, Phys. Rev. Lett., 2013, 111, 056803.
3136 | Nanoscale, 2014, 6, 3127–3137
Paper
24 B. Huard, N. Stander, J. A. Sulpizio and D. GoldhaberGordon, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 121402. 25 N. Stander, B. Huard and D. Goldhaber-Gordon, Phys. Rev. Lett., 2009, 102, 026807. 26 B. Huard, J. A. Sulpizio, N. Stander, K. Todd, B. Yang and D. G. Gordon, Phys. Rev. Lett., 2007, 98, 236803. 27 J. R. Williams, L. Dicarlo and C. M. Marcus, Science, 2007, 317, 638. ¨ 28 B. Ozyilmaz, P. Jarillo-Herrero, D. Efetov, D. A. Abanin, L. S. Levitov and P. Kim, Phys. Rev. Lett., 2007, 99, 166804. 29 J. B. Oostinga, H. B. Heersche, X. Liu, A. F. Morpurgo and L. M. K. Vandersypen, Nat. Mater., 2008, 7, 151. 30 R. V. Gorbachev, A. S. Mayorov, A. K. Saychenko, D. W. Horsell and F. Guinea, Nano Lett., 2008, 8, 1995. 31 G. Liu, J. J. Velasco, W. Bao and C. N. Lau, Appl. Phys. Lett., 2008, 92, 203103. 32 H. Steinber, D. R. Gardner, Y. S. Lee and P. J. Herrero, Nano Lett., 2010, 10, 5032. 33 Y. Wang, F. Xiu, L. Cheng, L. He, M. Lang, J. Tang, X. Kou, X. Yu, X. Jiang, Z. Chen, J. Zou and W. L. Wang, Nano Lett., 2012, 12, 1170. 34 H. L. Stormer, L. N. Pfeiffer, K. W. Baldwin, K. W. West and J. Spector, Appl. Phys. Lett., 1991, 58, 726. 35 T. C. Harman, P. J. Taylar, M. P. Walsh and B. E. LaForge, Science, 2002, 297, 2229. 36 R. Venkatasubramanian, T. Colpitts and B. ƠQuinn, Appl. Phys. Lett., 1999, 75, 1104. 37 L. He, X. Kou and K. L. Wang, Phys. Status Solidi RRL, 2013, 7, 50. 38 J. Linder, T. Yokoyama and A. Sudbø, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 205401. 39 W.-Y. Shan, H.-Z. Lu and S.-Q. Shen, New J. Phys., 2010, 12, 043048. 40 H.-Z. Lu, W.-Y. Shan, W. Yao, Q. Niu and S.-Q. Shen, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 115407. 41 S. E. Ulloa, E. Casta˜ no and G. Kirczenow, Phys. Rev. B: Condens. Matter Mater. Phys., 1990, 41, 12350. 42 C. Bai and X. Zhang, Phys. Rev. B: Condens. Matter Mater. Phys., 2007, 76, 075430; M. Barbier, P. Vasilopoulos and F. M. Peeters, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 80, 205415; L.-G. Wang and S.-Y. Zhu, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 205444. 43 J. Tworzydlo, B. Trauzettel, M. Titov, A. Rycerz and C. W. J. Beenakker, Phys. Rev. Lett., 2006, 96, 246802. 44 R. Danneau, F. Wu, M. F. Craciun, S. Russo, M. Y. Tomi, J. Salmilehto, A. F. Morpurgo and P. J. Hakonen, Phys. Rev. Lett., 2008, 100, 196802. 45 S. Mondal, D. Sen, K. Sengupta and R. Shankar, Phys. Rev. Lett., 2010, 104, 046403; A. Yamakage, K.-I. Imura, J. Cayssol and Y. Kuramoto, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 83, 125401; T. Yokoyama, Y. Tanaka and N. Nagaosa, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 121401.
This journal is © The Royal Society of Chemistry 2014
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Published on 03 January 2014. Downloaded by Northeastern University on 28/10/2014 14:09:00.
Paper
46 S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, 1995. 47 G. J. Xu, X. G. Xu, B. H. Wu, J. C. Cao and C. Zhang, J. Appl. Phys., 2010, 107, 123718. 48 H. C. Nguyen and V. L. Nguyen, J. Phys.: Condens. Matter, 2009, 21, 045305. 49 M. I. Katsnelson, K. S. Novoselov and A. K. Geim, Nat. Phys., 2006, 2, 620. 50 Z.-L. Ji and D. W. L. Sprung, Phys. Rev. B: Condens. Matter Mater. Phys., 1997, 56, 1045.
This journal is © The Royal Society of Chemistry 2014
Nanoscale
51 N. K. Patel, L. M. Moreno, M. Pepper, R. Newbury, J. E. F. Frost, D. A. Ritchie, G. A. C. Jones, J. T. M. B. Janssen, J. Singleton and J. A. A. J. Perenboom, J. Phys.: Condens. Matter, 1990, 2, 7247. 52 R. Tsu and L. Esaki, Appl. Phys. Lett., 1973, 22, 562. 53 F. Beltram, F. Capasso, D. L. Sivco, A. L. Hutchinson, S.-N. G. Chu and A. Y. Cho, Phys. Rev. Lett., 1990, 64, 3167. 54 P. Pereyra and E. Castillo, Phys. Rev. B: Condens. Matter Mater. Phys., 2002, 65, 205120.
Nanoscale, 2014, 6, 3127–3137 | 3137
