Solid State Communications 188 (2014) 45–48 Contents lists available at ScienceDirect Solid State Communications journal homepage: www.elsevier.com/locate/ssc Transport in a quantum spin Hall bar: Effect of in-plane magnetic field Fang Cheng a,b,n, L.Z. Lin b, D. Zhang b a b Department of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410004, China SKLSM, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, China art ic l e i nf o a b s t r a c t Article history: Received 30 May 2013 Received in revised form 18 September 2013 Accepted 25 February 2014 by S. Tarucha Available online 6 March 2014 We demonstrate theoretically that edge transport in quantum spin Hall bar can be controlled by in-plane magnetic fields. The in-plane magnetic field couples the opposite spin orientation helical edge states at the opposite edges, and induces the gaps in the energy spectrum. The hybridized electron wave functions ψ ↑ ðx; ky Þ of the edge states can be destroyed with increasing the in-plane magnetic fields. When the Fermi surface is located within this energy gap induced by the in-plane magnetic field, one can expect that the conductance of the edge states becomes e2/h. By tuning the magnetic field and Fermi energy, the edge channels can be transited from opaque to transparent. This switching behavior offers us an efficient way to control the topological edge state transport. & 2014 Elsevier Ltd. All rights reserved. Keywords: A. Topological insulator D. Electronic states D. In-plane magnetic field D. Quantum transport 1. Introduction Topological insulators (TIs), a strong spin–orbit coupling system, exhibit rich and fascinating physics, which have been investigated intensively both theoretically and experimentally [1–4]. The twodimensional (2D) TIs have been realized in HgTe quantum wells (QWs) and InAs/GaSb QWs [5,6] by tuning the thickness of the QWs or electric field [7–11]. HgTe is a narrow gap semiconductor with very strong spin–orbit interaction (SOI) [12]. Strong SOI inverts the band structure of HgTe, leading to a topological insulating phase. In this phase, HgTe possesses an insulating in the bulk with a gap separating the valence and conduction bands but with gapless helical edge states that are topologically protected by the timereversal symmetry [13,14]. The existence of the helical edge states in 2D TIs was proved by recent experiments [15,16]. The helical feature and suppressed backscattering render edge states an attractive platform for high mobility charge- and spin-transport devices. Since the topological edge states in 2D TI are protected by timereversal symmetry and robust against to backscattering, control of the edge states, e.g., switch on/off, is a challenging issue from the viewpoint of basic physics and potential device application. Recently, there have been a few proposals to control the edge state transport using a quantum point contact [17–20]. These electrical means can control the transport, magnetic properties and even quantum phase transition, and provide us an efficient way to control spin transport [11,20–22]. It is natural to ask if there is any other method to control the edge state transport? In this letter, we study the effect of in-plane magnetic field on the transport property of a quantum spin Hall (QSH) bar. The magnetic field can lead to a large Zeeman term because of the large g factor of HgTe material. The Zeeman term couples spin-up and spin-down electron and holes, and induces the gaps in the energy spectrum. Electrons with the opposite spin orientation at the opposite edges couple together due to in-plane magnetic fields. And the density distributions of hybridized electron wave functions ψ ↑ ðx; ky Þ become more localized in the center of the QSH bar, indicating destroy of the edge state. When the Fermi surface is located within this energy gap induced by the in-plane magnetic field, one can expect that there is only an edge state and the conductance of the edge states becomes one-half of the conductance quantum 2e2/h. The in-plane magnetic field can control the coupling between the edge states at opposite edges and between the topological edge state and the bulk state. Tuning the in-plane magnetic field, one can switch-on/off the edge channel in the finite width QSH bar system when the Fermi energy is the gap. This feature provides us an efficient means to control the edge state transport in QSH bars. 2. Theoretical model n Corresponding author at: Department of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410004, China. Tel.: þ 86 15807314598. E-mail address: [email protected] (F. Cheng). http://dx.doi.org/10.1016/j.ssc.2014.02.028 0038-1098 & 2014 Elsevier Ltd. All rights reserved. The total Hamiltonian for the system in the presence of an external in-plane magnetic field is H ¼ H0 þ HZ ; ð1Þ 46 F. Cheng et al. / Solid State Communications 188 (2014) 45–48 where the first term is the single-particle Hamiltonian of electron in HgTe QWs and the second term HZ is the Zeeman effect. The electron transport in the quasi-one-dimensional (Q1D) QSH bar is along the longitudinal y-direction. The four-band single-particle Hamiltonian H reads as 0 1 ϵk þ MðkÞ Ak g e μB B 0 B C ϵk MðkÞ 0 g h μB B C B Ak þ C; H¼B ð2Þ B g μB B 0 ϵk þ MðkÞ Ak þ C e @ A 0 g h μB B Ak ϵk MðkÞ where k ¼ ðkx ; ky Þ is the in-plane momentum of electrons, 2 2 ϵk ¼ C þ Vðx; yÞ Dðkx þ ky Þ with Vðx; yÞ being the confinement 2 2 potential, MðkÞ ¼ M Bðkx þky Þ, k 7 ¼ kx 7 iky , A, B, C, D, and M are the parameters describing the band structure of the HgTe/CdTe QW. g e=h denotes electron or hole g factor, respectively. μB is the Bohr magneton. B is the external transversal x-direction magnetic field. The transport property of a Q1D QSH bar can be obtained by discretizing the Q1D system into a series of in-plane stripes along the transport direction. Assuming a hard-wall in-plane confining potential, the traveling-wave-like or evanescent-wave-like eigenstates of the Schrödinger equation Hψ ¼ Eψ in a given region λ can be written as the form λ ψ λ ðx; yÞ ¼ expðiky yÞ∑χ λn φn ðxÞ; ð3Þ n where rffiffiffiffiffiffi 2 nπx sin φn ðxÞ ¼ W W ε (meV) in which W is the width of the lead, and the subband index n¼1,2,… N with N being the number of the basis function which is chosen to ensure the convergence of the energies of the low subbands near the Dirac point. fχ λn g ðλ ¼ L; RÞ are the expanded coefficients. The longλ itudinal wave vector ky and the eigenvector χ λn (n¼ 1,2,3,…) L n R By using scattering matrix theory, we can calculate the coefficients r m , t m in the left and right leads. Thus we can obtain the total conductance from the Landauer–Büttiker formula υRm 2 t j ; L m m;n υn RM G ¼ G0 ∑ 3. Numerical results and discussions In the case of a QSH bar in the absence of in-plane magnetic field, the finite size effect induces the overlap of the wave functions of the edge states localized at the opposite edges, and can open a minigap in the energy spectrum of the edge states at ky ¼ 0 (see Fig. 1(a)). The in-plane magnetic field couples the edge states at the opposite edges (see Eq. (1)), and induces a mass term for massless Dirac electrons in the edge states. Therefore the gaps in the energy spectrum increase with increasing the magnetic fields. 0 8 8 4 4 0 −0.02 0 ky (nm−1) 0.02 ð5Þ where G0 ¼ e2/h is the conductance unit, RM denotes the summation over all right-moving modes in the left and right leads, tm is the transmission coefficient where the electron incidents from the subband n in the left lead to be scattered into the subband m in the right lead, and υλm ¼ 〈^υ λm 〉 ¼ 〈∂H=∂ky 〉 are the group velocity of the electron in the subband m in the leads along the QSH bar, i.e., the yaxis direction. 4 0.02 ð4Þ mn 4 0 mn ψ R ¼ ∑ t m χ Rm;n eikm y φn ðxÞ: 8 −0.02 L ψ L ¼ eikI y ∑χ LI;n φn ðxÞ þ ∑ r m χ Lm;n e ikm y φn ðxÞ; 8 0 ε (meV) are determined from the generalized eigenvalue problem [20]. Assuming an electron injected from a given energy with wave vector kLI in the left lead, the wave functions in the left lead and the right lead can be written as 0 −0.02 −0.02 0 0.02 0 0.02 ky (nm−1) Fig. 1. The energy spectra with width W ¼ 200 nm under different in-plane magnetic fields (a) B¼ 0, (b) B ¼0.5 T (c) B¼ 1 T, and (d) B¼ 2 T. The parameters used in the calculation are A ¼ 364.5 meV, B ¼ 686 meV nm2, C¼ 0, D ¼ 512 meV nm2, M¼ 10 meV. F. Cheng et al. / Solid State Communications 188 (2014) 45–48 We plot the density distributions of edge states for a QSH bar with the width W¼200 nm for the different strengths of the in-plane magnetic fields. In the case of a QSH bar in the absence of in-plane magnetic fields, the two states of ψ ↑ ðx; 7 ky Þ (and ψ ↓ ðx; 7 ky Þ) are degenerate but localized at the opposite edges. The overlap between these two edge states, the finite size effect, can open a minigap (see Fig. 1(a)). Electrons with the opposite spin orientation at the opposite edges couple together due to in-plane magnetic fields, i.e., the off-diagonal elements in the Hamiltonian (see Eq. (1)). The wave functions of two spin directions are hybridized by the in-plane magnetic field. Therefore the densities of the hybridized wave functions ψ ↑=↓ ðx; ky Þ (and ψ ↑=↓ ðx; ky Þ) are symmetrically distributed at the two sides. The in-plane magnetic field can control the coupling between the topological edge state and the bulk states. Fig. 2(b)–(d) shows clearly that the density distributions of the hybridized wave functions ψ ↑ ðx; ky Þ become more localized in the center of the QSH bar with the increase of inplane magnetic field B, indicating destroy of the edge state. The variation of the conductance as a function of the Fermi energy is shown in Fig. 3 for the different strengths of the in-plane magnetic fields. In the absence of an in-plane magnetic field, the conductance in a QSH bar shows a perfect plateau 2e2/h in the bulk gap because of the two 1D spin-resolved conducting channels at the edges, and displays a minigap near the Dirac point (see black curve in Fig. 3). The width of the gap in the conductance plateau is determined by the width of the QSH bar, i.e., the overlap between the wave functions of the edge states. The minigap of the conductance plateau can be changed significantly by applying an in-plane magnetic field. The Zeeman term couples spin-up and spin-down electron and holes (see Eq. (1)), and induces a mass term for massless Dirac electrons in the edge states. Therefore the gaps in the plateau increase with increasing the magnetic fields. There is an odd number of the conductance quantum e2/h in the spectrum. When the Fermi surface is located within this energy gap induced by the in-plane magnetic field, the conductance becomes e2/h, which indicates that the conductance plateau comes from hybridized ψ ↓ ðx; ky Þ topological edge states. The topological edge states of the hybridized electron wave functions ψ ↑ ðx; ky Þ are destroyed because of the in-plane magnetic field. More interestingly, one can also 47 switch on/off this edge current by changing slightly the in-plane magnetic fields when the Fermi energy locates at the minigap. The magnetic field dependence of the QSH edge current also shows the interesting switching on/off feature of edge current (see Fig. 4(a)). Increasing the strength of the in-plane magnetic field, a crossover from the transmitting case to the opaque case occurs. Clearly, the crossover takes place at different magnetic fields for different Fermi energies. It means that the crossover can also be controlled by tuning the Fermi energy for a fixed magnetic field. Fig. 4(b) shows the conductance as a function of the width of the QSH bar for a fixed Fermi energy and three different strengths of the in-plane magnetic field. From this figure, we can see that the transition from the zero conductance to the plateau G0 or 2G0 depends not only on the width of the system, but also on the strength of the in-plane magnetic field. In the vicinity of the gap where the tunneling is forbidden, the critical width of the system depends sensitively on the in-plane magnetic field. Increasing the width of the QSH bar or decreasing the strength of the in-plane magnetic field will lead to weakening of the inter-edge coupling. Both the finite size and the in-plane magnetic field modify the Fig. 3. (Color online) The Fermi energy dependence of the conductance G, where the black line is for B¼ 0, the red line for B¼ 0.5 T and the blue line for B¼ 1 T. Fig. 2. (Color online) The density distribution of the edge states for a QSH bar with width W¼ 200 nm under different in-plane magnetic fields: (a) B ¼0, (b) B¼ 0.5 T, (c) B¼ 1 T, and (d) B ¼2 T. The red solid line corresponds to wave function ψ ↓ ðx; ky Þ ðψ ↑ ðx; ky ÞÞ, and black dashed line to ψ ↑ ðx; ky Þ ðψ ↓ ðx; ky ÞÞ at ky ¼ 0:01 nm 1 . 48 F. Cheng et al. / Solid State Communications 188 (2014) 45–48 Fig. 4. (Color online) The conductance as a function of (a) the strength of the in-plane magnetic field for two different Fermi energies EF ¼ 6 meV (the black solid line), 7 meV (the red dashed line). The width of the system is a fixed value W¼ 200 nm. (b) The width of the system for a fixed EF ¼ 6 meV and different strengths of magnetic field B¼ 0 (the black solid line), 0.5 T (the red dashed line), and 1 T (the blue dashed-dot line). overlap of the wave functions of the edges states localized at the opposite edges. Therefore the critical width for the blocking of the edge channels can be tuned by in-plane magnetic fields. The crossover from the zero conductance to the plateau G0 or 2G0 occurs very sharply, indicating a perfect switching effect. References [1] [2] [3] [4] [5] 4. Conclusions In summary, we demonstrate theoretically that the topological edge state transport through a QSH bar can be manipulated using in-plane magnetic fields. 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