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Transport through quantum wells and superlattices on topological insulator surfaces

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 185007 (http://iopscience.iop.org/0953-8984/26/18/185007) 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 18/06/2017 at 18:36 Please note that terms and conditions apply.

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 185007 (9pp)

doi:10.1088/0953-8984/26/18/185007

Transport through quantum wells and superlattices on topological insulator surfaces J-T Song1, Y-X Li1, and Q-f Sun2, 3 1

  Department of Physics and Hebei Advanced Thin Film Laboratory, Hebei Normal University, Hebei 050024, People's Republic of China 2   International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, People's Republic of China 3   Collaborative Innovation Center of Quantum Matter, Beijing 100871, People's Republic of China E-mail: [email protected] Received 30 December 2013 Accepted for publication 14 March 2014 Published 22 April 2014 17

Abstract

We investigate electron transmission coefficients through quantum wells and quantum superlattices on topological insulator surfaces. The quantum well or superlattice is not constituted by general electronic potential barriers but by Fermi velocity barriers which originate in the different topological insulator surfaces. It is found that electron resonant modes can be renormalized by quantum wells and more clearly by quantum superlattices. The depth and width of a quantum well and superlattice, the incident angle of an electron, and the Fermi energy can be used to effectively tune the electron resonant modes. In particular, the number N of periodic structures that constitute a superlattice can further strengthen these regulating effects. These results suggest that a device could be developed to select and regulate electron propagation modes on topological insulator surfaces. Finally, we also study the conductance and the Fano factor through quantum wells and quantum superlattices. In contrast to what has been reported before, the suppression factors of 0.4 in the conductance and 0.85 in the Fano factor are observed in a quantum well, while the transport for a quantum superlattice shows strong oscillating behavior at low energy and reaches the same saturated values as in the case of a quantum well at sufficiently large energies. Keywords: surface states, topological insulator, quantum well, quantum superlattice (Some figures may appear in colour only in the online journal)

1. Introduction

three-dimensional (3D) TI materials [3] (Bi2Se3, for example) has raised hopes that practical 3D TIs can be produced and may also play a key role in future quantum computing. It has been found that the electrons in 3D TIs have very strong spin– orbit coupling and that the surface states can be described by a massless Dirac equation. More intriguingly, the directions of the spin angular momentum and linear momentum are tied to each other. Based on these fascinating properties, some significant transport behaviors of 3D surface states have been reported, such as the Majorana bound states at a magnet-superconductor interface [4], the anomalous magnetoresistance of a ferromagnet–ferromagnet junction [5], the

Over the last decade, new materials known as topological insulators (TIs) have attracted much research attention due to their novel properties. TIs are characterized as having insulating bulk states but topologically protected gapless states on their edges or surfaces [1, 2]. These gapless states are completely immune to all external influences as long as the system's nontrivial topology remains unchanged. The robust properties of gapless states endow the TIs with some promising possibilities with respect to quantum computing. Recently, the identification, using angle-resolved photoemission spectroscopy, of 0953-8984/14/185007+9$33.00

1

© 2014 IOP Publishing Ltd  Printed in the UK

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J. Phys.: Condens. Matter 26 (2014) 185007

In this paper, we investigate transport properties through quantum wells and quantum superlattices, as shown in figure 1(a). It should be borne in mind that the quantum well described here is a real space well, not a potential well. In this quantum well, the two vertical surfaces (regions II and IV in the y−z plane) have the same velocity vi, which is different from that of other regions in the x−y plane (regions I, III and V). The results indicate that the transmission amplitudes through this quantum well show distinct resonances but with the same cutoff angle of incidence as that in the nanostep junction [10]. For the quantum superlattice shown in figure 1(b), the transmission amplitudes are intriguing and many zero tunneling (perfect reflection) regions appear at some incidence angles. 2.  Model and method The low-energy effective Hamiltonian for Bi2Se3 can be represented by a four-band model [14]: H ( k ) = ϵ 0 ( k ) + M ( k ) σ0 τz + A1 σz τx + A2 k x σx τx + A2 k y σy τx , where ϵ 0 ( k ) = C + D1k z2 + D2 ( k x2 + k y2 ), M ( k ) = M − B1k z2 − B2 ( k x2 + k y2 ), and the Pauli matrixes σα and τα represent spin and orbital indices, respectively. In order to make a comparison with the future experimental results convenient, parameters appropriate to the 3D TI Bi2Se3 were adopted. The parameters A1 = 2.2 eVÅ, A2 = 4.1 eVÅ, B1 = 10 eVÅ2, B2 = 56.6 eVÅ2, C = −0.0068 eV, D1 = 1.3 eVÅ2, D2 = 19.6 eVÅ2, and M = 0.28 eV were determined for Bi2Se3 with ab initio calculations [14]. For regions I, III and V in figure 1(a), the surface states lie in the x−y plane and the 3D wave functions for these surface states are exponentially damped in the z direction with finite surface depth λ. We consider a 3D TI with dimensions much larger than λ. In this case, the effective surface Hamiltonian in regions I, III and V could be written straightforwardly as

Figure 1.  Schematic diagram of a real space quantum well (a) and a quantum superlattice (b). The width and the depth of each well are Lx and Lz, respectively.

anisotropy of a spin-polarized scanning tunneling microscope [6], and the edge states of a TI surface junction [7, 8]. When two 3D TIs are brought close to each other and their surfaces touch each other along a line, a line junction is formed [9]. In this junction two two-dimensional surfaces governed by massless Dirac equations, with possibly different Fermi velocities v1 and v2, share a one-dimensional boundary. Calculated results indicate that the reflection and transmission amplitudes at the line junction show anomalous properties [8, 9] when the Dirac velocities v1 and v2 have different signs or amplitudes in the two touching TI surfaces. Further, when two half-infinite TI surfaces are connected by a mesoscopic side surface instead of a line junction, a so-called nanostep junction [10] is formed. It has been reported recently that in a nanostep junction with a vertical side surface the conductance 1 is suppressed by a factor of as compared to the conductance 3 of a junction between two coplanar surfaces with different electronic potentials [10]. Intriguingly, this suppression fac1 tor of is correlated with only the density of the resonance 3 tunneling states in the nanostep junction. The robust properties of edge and surface states against nonmagnetic disorders, which has been confirmed thoroughly [11, 12], has inspired researchers to consider whether some electronic devices with strong anti-interference and low power consumption could be fabricated using these robust edge or surface states. Much effort has been devoted to finding some feasible methods for producing such devices. The work on various junctions discussed above leads to a further question: What would happen if many nanostep junctions were arranged periodically, forming a superlattice in real space, as distinct from a superlattice of potential wells? The study of this question is very important for realizing the type of electronic device mentioned above, and would also help people to gain a better understanding of the properties of TI surface states.

Hxy = Exy + ℏvFxy ( k yσx − k xσy ), (1)

with the eigenstates

1 ⎡ 1 ⎤ ik y y ± i k x x e , Ψ±xy = ⎢ ± iθ ⎥ e (2) 2 ⎣ ∓ ie ⎦ D where Exy = C + 1 M , and ℏvFxy = A1 1 − ( D1 / B1 )2 is the B1 Fermi velocity in the x−y plane. The corresponding energy eigenvalues of the eigenstates above are E = Exy ±ℏvFxy k x2 + k y2 . Note that the incident angle θ is defined as θ = arctan( k y / k x ). For regions II and IV, the effective Hamiltonian is given by ⎛ A ⎞ Hyz = E yz − eVs + ℏv Fyz ⎜ σy 1 kx − σzk y ⎟ , (3) ⎝ A2 ⎠

with the eigenstates

⎡ ± icos( γ ) ⎤ ik y ±ik z 1 Ψ±yz = ⎢ ⎥e y e z , (4) 2(1 − sin( γ )) ⎣ 1 + sin( γ ) ⎦ D2 M , and ℏvFyz = A2 1 − ( D2 / B2 )2 is the B2 Fermi velocity in the y−z plane. The corresponding energy eigenvalues are E = Eyz − eVs ±ℏvFyz k y2 + ( A12 / A22 ) k z2 , and the angle γ = arctan( A2 k y / A1k z ). The +(−) label of the wave

where Eyz = C +

2

J Song et al

J. Phys.: Condens. Matter 26 (2014) 185007

(a)

(b)

(c)

(d)

Figure 2.  Transmission as a function of incident angle. (a) for a quantum well, (b) for a quantum superlattice with N = 10, (c) for a nanostep junction, and (d) for a planar junction. Other parameters are set to Lx = Lz = 50 nm and E = 0.28 eV.

function indicates right (left) traveling carriers in regions I, III and V, and downwards (upwards) traveling carriers in region II and IV. Here, Vs represents the gate voltage used to tune the Fermi energy of the y−z plane and is set to 0.03 V in the following calculations. For an electron incident from left to right on a quantum well, the wave functions in all regions are given by

factor which follows by integrating over the Fermi energy and incident angle reads as

∫∫ ET [1 − T ][ fL ( E ) − fR ( E )]2 cos θ dE dθ . F= (7) ∫∫ ET [ fL ( E ) − fR ( E )]cos θ dE dθ Here, fi(E) is the zero-temperature Fermi distribution function with i = L, R.

⎧ ΨI (x≤ 0,   z= 0) = Ψ+xy + r Ψ−xy, ⎪ + − ⎪ ΨII ( x= 0,   0 ≤ z≤ L z ) = a Ψyz + b Ψyz, ⎪ + ⎨ ΨIII (0 ≤ x≤ L x,   z= L z ) = c Ψxy + d Ψ−xy, (5) ⎪ + − ⎪ ΨIV ( x= L x,   0 ≤ z≤ L z ) = eΨyz + f Ψyz, ⎪ ΨV (x≥ L x,   z= 0) = t Ψ+xy . ⎩

3.  Results and discussion We first calculate the transmission probability through the real space TI quantum well, as shown in figure 2(a). We can see from figure 2(a) that the transmission coefficient for the quantum well oscillates as a function of the incident angle, with many resonant modes having a transmission coefficient T = 1. Because of the mismatch of the finite energies and velocities on different TI surfaces, a cutoff angle of incidence appears, with critical angle ⎛ v xy E − Exy ⎞ ⎟, such that the transmission is θc = arcsin ⎜ Fyz ⎝ v F E − E yz + eVs ⎠ suppressed completely when the incident angle is larger than the critical angle. In this case, perfect reflection occurs. On the other hand, for the case of normal incidence, the transmission coefficient is exactly 1 and perfect transmission occurs. Interestingly, the minimum of the transmission coefficient varies non-monotonically as a function of the incident angle, which is different from the case of the nanostep junction shown in figure 2(c) and the planar junction shown in figure 2(d). For comparison, the transmission coefficients for the nanostep junction [10] are shown in figure 2(c). A nanostep

Note that the parameters r, a  −  f and t are undetermined coefficients. Given that there is no scattering source in regions I and V, the parameters r and t actually represent the reflection and transmission amplitudes. Using energy conservation and the momentum conservation along the y direction, the coefficients in equation (5) can be obtained using the transfer matrix method and the continuity of the wave function at the boundaries [13]. Then, the transmission probability can be calculated straightforwardly as T = T(E, E sin θ) = |t|2. Based on the Landauer–Bu¨ttiker formula [13], the ballistic conductance in this system can be written as



π /2

G ( E ) = G0 T ( E , E sin θ )cos θ dθ , (6) −π /2

2

where G0 = 2e ELy/(π h) is taken as the conductance unit and Ly≫ Lx(Lz) is the sample size along the y direction. The Fano 3

J Song et al

J. Phys.: Condens. Matter 26 (2014) 185007

1 (a)

(b)

(c)

(d)

(e)

(f)

Transmission

0.8 0.6 0.4 0.2 0

Transmission

0.8 0.6 0.4 0.2

0 0 0.1 0.2 0.3 0.4 θ/π

0 0.1 0.2 0.3 0.4 θ/π

0 0.1 0.2 0.3 0.4 0.5 θ/π

Figure 3.  Transmission for a quantum well as a function of the incident angle for different size wells. (a) Lx = 200 nm, Lz = 50 nm,

(b) Lx = 200 nm, Lz = 100 nm, (c) Lx = 200 nm, Lz = 200 nm, (d) Lx = 20 nm, Lz = 200 nm, (e) Lx = 50 nm, Lz = 200 nm, and (f) Lx = 100 nm, Lz = 200 nm. Other parameters are the same as in figure 2.

junction can be realized from figure 1(a) when only regions I, II and III are retained. The transmission coefficient through the nanostep junction is given by [10]

junction is larger than those in the quantum well and the nanostep junction, which should be attributed to the different energy positions of the Dirac points for different TI surfaces. One of the most important points of focus in this paper is the quantum superlattice structure which consists of periodic quantum wells or nanostep junctions, as shown in figure 1(b). Because there exist more line junctions in the quantum superlattice than in the quantum well or step junction, the effect of the mismatch in the Fermi velocities between the horizontal and vertical surfaces is amplified, which could lead to a complete prohibition of some transmission modes at some angles. This effect is clearly shown in figure 2(b), where, for instance, an exact zero transmission occurs at some incident angles but which, however, correspond to a nonzero transmission in ­figure 2(a). This coherent transmission characteristic in quantum superlattices is very similar to the case of the Fabry–Perot interference in optics. Recently, the coherent transmission analogue to Fabry–Perot interference has also been observed in other quantum periodic systems [16–18]. This quantum superlattice structure can therefore probably be used to control the transmission of incident electrons, depending on their incident angle, wave length or energy. We have also studied the size effect of a quantum well on the transmission coefficients. On increasing Lz from 50 nm to 200 nm in figures 3(a)–(c), two main changes can be observed. One is that the number of resonant states increases gradually as the value of Lz increases. The other is that the envelope curve consisting of all transmission minimums is considerably modified and also drops lower with increasing Lz. These

cos 2θ cos 2γ , Tstep ( θ ) = (8) 2 2 cos θ cos γ cos 2 (k zLz ) +sin 2 (k zL z ) ⎛ v yz E − Exy ⎞ sin θ ⎟. The transmission with γ = arcsin ⎜ Fxy ⎝ v F E − E yz + eVs ⎠ coefficient through a planar junction [15]4, which corresponds to a velocity barrier in the TI x−y surface, is also given in figure 2(d). As a function of the incident angle, the transmissions for the quantum well, the nanostep junction and the planar junction have the same cutoff angle because there is the same velocity mismatch between the horizontal and vertical surfaces. Furthermore, for all cases, perfect transmission occurs at zero incident angle and the transmission coefficient is very close to 1 at relatively small incident angles. When the carriers tunnel through a quantum well or a nanostep junction, they first propagate in the x−y plane and then change into the y−z plane. Once the propagation plane changes, some carriers are transmitted into edge states along the nanostep line in the y  direction [8, 10]. Comparing figures 2(a), (c) and (d), we can see clearly that the transmission coefficient in the planar   For a planar junction, in all three regions the surface states lie in the x−y plane and a top gate is applied to the middle region, the transmission for a 2 ⎤ ⎡ planar junction is Tplanar ( θ ) = 1 / ⎢ cos 2 (kx L) +sin 2 (kx L) [1-sin ( θ )sin ( γ )] ⎥. ⎣ cos 2 ( θ )cos 2 ( γ ) ⎦

4

4

J Song et al

J. Phys.: Condens. Matter 26 (2014) 185007

1

(a)

(b)

(c)

(d)

Transmission

0.8 0.6 0.4 0.2 0

Transmission

0.8 0.6 0.4 0.2 0 0

0.1

0.2

0.3

0.4

0

θ/π

0.1

0.2

0.3

0.4

0.5

θ/π

Figure 4.  Transmission for superlattices of different sizes as a function of incident angle. (a) N = 2, (b) N = 5, (c) N = 20, and (d) N = 50. Other parameters have the values: Lx = Lz = 50 nm and E = 0.28 eV.

observations can be explained in a straightforward manner by noting the fact that the increase of Lz on the one hand generates more resonant states, each with the form sin (kzLz) or cos (kzLz), and on the other hand makes electron resonant tunneling more difficult because of the mismatch between the surface eigenfunctions of neighboring regions. The increase in the number of the resonant states leads to an increase in the density of resonant transmission peaks, whereas the drop in the transmission minimums can be ascribed to the mismatch of the electron velocities between neighboring surfaces. Similarly, as Lx increases from 20 nm to 100 nm (shown in figures 3(d)–(f)), the number of the surface resonant modes in region III gradually becomes larger and thus the resonant peaks also become more and more dense. However, it should be pointed out explicitly that the resonant transmission peak, e.g. θ≈0.22π in figure 3(d) or θ≈0.12π in figure 3(e), not only originates from the modulation of the resonant tunneling in region III, but also should be attributed to the combined modulation of the resonant surface states in regions II, III and IV. Note that the resonant transmission peak with T = 1 appears at θ≈0.22π in figure 3(d), whereas a transmission minimum at this incident angle which is far less than 1 is found in ­figure 3(e). A similar feature can also be observed at θ≈0.12π by comparing figures 3(d) and (e). In fact, this phenomenon also exists in figures 3(a)–(c), though it is a little obscure. The angle shift of the transmission maximums in different cases is strong proof that these resonant transmission peaks should

be attributed to the combined resonant effect of surface states in different regions, and is not simply the additive effect of all the regions taken separately. To sum up, the quantum well is much more distinct in the modulation of the electron resonant tunneling than in the planar and nanostep junctions. Another major concern is how many periodic structures for a quantum superlattice, namely the number N, are required to constrain the electron transmission to only those with the desired properties. Naturally, no matter how many line junctions there are in a superlattice structure, the wave functions in neighboring x−y and y−z planes can always be written as ⎧ ΨI (x− y  plane) = a i Ψ+xy + bi Ψ−xy, ⎨ (9) + − ⎩ ΨII (y− z  plane) = ci Ψyz + d i Ψyz .

Using the transfer matrix method and the continuity of the wave functions at the boundaries, the coefficients ai, bi, ci and di, where i represents the i-th line junction, can be computed. In figure 4, we detail the transmission coefficient in four different cases with N = 2, 5, 20 and 50. With N = 2 in figure  4(a), the modulation due to the superlattices is not obvious, and some transmission minimums are still approximately equal to what was seen in the case of the quantum well shown in figure 2(a). When N increases to 5, the effect of the superlattice on the tunneling electrons comes into play and some angles of nearly zero transmission can be found for some incident angle ranges in figure 4(b). Increasing the 5

J Song et al

J. Phys.: Condens. Matter 26 (2014) 185007

1 (a)

(b)

Transmission

0.8 0.6

Nanostep Junction:

Nanostep Junction:

Lz=50nm

Lz=100nm

0.4 0.2 0 (c)

(d) Quantum Well:

Transmission

0.8

Quantum Well:

Lx=50nm Lz=50nm

0.6

Lx=100nm Lz=100nm

0.4 0.2 0 (e)

Transmission

0.8 0.6 0.4

(f)

Quantum Superlattice:

Quantum Superlattice:

Lx=50nm; Lz=50nm; N=30.

Lx=100nm; Lz=100nm; N=30.

0.2 0

0

0.1

0.2

0.3

0.4

0

θ/π

0.1

0.2

0.3

0.4

0.5

θ/π

Figure 5.  Transmission for nanostep junction, quantum well and quantum superlattice at E = 0.0336 eV (red line) and E = 0.0616 eV

(blue line).

number of periodic structures to N = 10 in figure 2(b), N = 20 in figure 4(c) and N  =  50 in figure 4(d), the renormalization effect becomes stronger, which can be seen in the steep changes from the transmission minimum 0 to maximum 1, e.g. θ≈0.085π. Thus, a relatively small number of structures can result in a strong modulation of the electron states and, based on our simulations above, if the number N of periods of the structure is N = 10 ∼ 20, a superlattice can tune the electron resonant modes effectively, which makes the fabrication of a useful quantum superlattice experimentally feasible. In the numerical calculations above, the Fermi energy always has the same value E = 0.28 eV. A subsequent question

arises: What happens to the transmission coefficient when the Fermi energy changes? In figure 5, we map the transmission coefficient for both a quantum well and a quantum superlattice at two distinct Fermi energies. It is very interesting to find that in figure 5(a), for the case of a nanostep junction, the transmission coefficient for E = 0.0336 eV (red line) shows a large value over most of the angular range, whereas the transmission is almost zero for E = 0.0616 eV (blue line), except at very small angles. This phenomenon is also observed for a thicker step junction with Lz = 100 nm in figure 5(b), with the transmission coefficient somewhat smaller because of the greater difficulty in matching electron states on the x−y and y−z planes. 6

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J. Phys.: Condens. Matter 26 (2014) 185007

Figure 6.  Two dimensional transmission diagram with respect to E′ = E − Exy and θ for a nanostep junction, a quantum well and a quantum superlattice. Other parameters are the same as those in figure 5.

For the quantum wells in figures 5(c) and (d), the above conclusion still holds true except for small corrections to the transmission coefficients at some angles. Generally speaking, compared to a nanostep junction, a quantum well makes electron transmission more difficult and intuitively the transmission coefficient should be reduced. However, it can be clearly seen that the transmission coefficient for E = 0.0336 eV (red line) in figure 5(c) shows an unexpected larger value than that in figure 5(a) at large transmission angles. This originates from the characteristic resonant states between different surfaces of the quantum well, which is also observed and discussed in figure 3. Doubtlessly, the quantum superlattice makes the transmission coefficient for E  =  0.0336  eV (red line) oscillate as the transmission angle varies and prohibits non-resonant

transmissions to a large extent. More intriguing is the observation that in figure 5(f) the transmission coefficient for E = 0.0336 eV (red line) exhibits many resonant tunneling modes, which are not seen in the cases of the nanostep junction and the quantum well. Note also that the transmission is completely absent for E = 0.0616 eV (blue line) in figure 5(f). In general, all these phenomena can be illustrated by the characteristic structure of the quantum superlattice in figure 2(b). In order to clarify the above observations, we plot a twodimensional diagram of the transmission as functions of E′ = E − Exy and θ in figure 6. From the setting parameters Exy = 0.0296 eV, Eyz = 0.0902 eV, and eVs = −0.030 eV in ­figure 6, it is not difficult to obtain from figure 6 that the minimum transmission in figure 5 corresponds to E = Eyz−eVs, 7

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J. Phys.: Condens. Matter 26 (2014) 185007

1 0.8 0.6 0.4

Lx=Lz=50nm Lx=Lz=100nm

0.4

0.3

(b)

0.1

0

0.05 0.1 E’(eV)

0.15

0.2

Lx=Lz=100nm

0.3

0.2

−0.05

Lx=Lz=50nm

0

Fano factor

0

Fano factor

0.6

0.2

0.2

0 −0.1

(a)

0.8

G/G0

G/G0

1

(a)

0.2 0.1 0 −0.1

0.25

(b)

−0.05

0

0.05 0.1 E’(eV)

0.15

0.2

0.25

Figure 8.  Conductance (a) and Fano factor (b) for quantum superlattices as a function of the Fermi energy with E′ = E − Exy. Note that the number of quantum wells in the quantum superlattice is N = 10.

Figure 7.  Conductance (a) and Fano factor (b) for a quantum well as a function of the Fermi energy with E′ = E − Exy.

which is the position of the Dirac point in the y−z plane, and that the maximum corresponds to the case of E = Exy, which is the position of the Dirac point in the x−y plane. In other words, a perfect resonant transmission is obtained when Fermi energy is located near the Dirac point of the x−y plane, but transmission is almost entirely absent for Fermi energy near the Dirac point of the y−z plane due to a zero state density and mismatch of electron states between different surfaces. In addition, we can observe in figure 6 that the transmission states are modulated and renormalized gradually when changing from nanostep junctions, through quantum wells, to quantum superlattices. Furthermore, with increasing values of Lz or Lx, more resonant tunneling modes appear and a subtle renormalization can also be realized in this case. Another interesting characteristic of all the plots in figure 6 is the asymmetric distribution of the transmission coefficients with respect to E′  =  0. This can be ascribed to the inversion of the electron velocity, which depends on whether the Fermi energy is located below or above the Dirac point. In other words, a perfect resonant transmission near the Dirac point in the x−y plane and an absent transmission near the Dirac point in the y−z plane can be easily achieved by varying the Fermi energy and the transmission angle. This finding may be highly helpful for experimentally regulating and controlling the TI surface electron states. We next analyze the conductance and the Fano factor integrated over all angles. The zero-temperature conductance and the Fano factor for a quantum well are shown in figure 7. As a function of Fermi energy E′, the conductance for Lx = Lz = 50 nm oscillates, reaches a maximum at E′≈ 0 near the Dirac point of the x−y plane and a minimum at E′≈ 0.0336 eV near the Dirac point of the y−z plane, and then saturates with the value G = 0.6G0 at sufficiently large energies. The saturated conductance is suppressed due to the existence of the nanostep junction edge states [8, 10]. Obviously, when the conductance reaches its maximum and minimum, the Fano factor has, respectively, a minimum

F  ≈  0 and maximum value F  =  1/3 more or less, which is three times smaller than for a Poisson process. By increasing the Fermi energy, the Fano factor reaches the saturated value, F = 0.15. When Lx or Lz is increased to 100 nm, the main characteristics of the conductance are not changed but the minimum is closer to the ideal value 0 and the maximum closer to 1. Similarly, no distinct change is found for the Fano factor except for a reduction in the amplitude of the oscillation. Thus, the quantum well can suppress the maximum conductance and Fano effect with suppression factors of 0.4 and 0.85, respectively. Finally, the transport behavior of the quantum superlattice structures is shown in figure 8. Because the saturated conductance and Fano factor at very high energies have approximately equal values to those for the quantum well, we plot the conductance and the Fano factor only in the same low energy region as in figure 7. Comparing with the results for a quantum well in figure 7, the conductance is suppressed overall but the peaks and valleys become more clearly marked, which should be ascribed to the multiple regulation of propagating modes by the quantum superlattice. For example, the conductance for Lz = Lx = 50 nm (red line) shows stronger oscillating behavior than in the case of the quantum well shown in figure 7. The oscillation behavior becomes denser because more resonant tunneling modes are induced. The suppression phenomenon and strong oscillation are also manifested in the plot of the Fano factor, as shown in figure 8(b). Almost all peaks in figure 8(b) become sharper. It is worth mentioning that a peak in the Fano factor always corresponds to a valley in the conductance and that a valley in the Fano factor is accompanied by a peak in the conductance as may be seen in figure 8(a). As an example, at E′≈0.0336 eV, the transmission processes are suppressed completely and the conductance reaches zero, though at this energy the Fano factor shows a very sharp peak and reaches as high as 0.34 at maximum, which is bigger than that in 8

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Acknowledgments

either the step junction or the quantum well. These intriguing characteristics of the conductance and Fano factor are expected to be observed experimentally for the quantum well and superlattice.

This work was supported by the NSFC under grant nos 11047131(J.T.S.), 11274364(Q.F.S.), NSF-Hebei Province under grant nos A2013205168 (J.T.S.) and A2014205005 (Y.X.L.), and NBRP of China under grant no. 2012CB921303 (Q.F.S.).

4. Summary We have studied the transport properties of electrons through a planar junction, a step junction, quantum wells and quantum superlattices in real space, all of which are constructed on topological insulator surfaces. In contrast to the planar junction, the transmission coefficients in the other three structures show a clear oscillating behavior, similar to the Fabry–Perot interference in optics, when plotted as a function of the incident angle of the electrons. This behavior can be ascribed to interference effects between the different line junctions separating the planes that make up the structure. In addition, all minimums of the oscillating transmission coefficients constitute an envelope curve for a step junction, though this pattern is considerably modified in quantum wells and quantum superlattices. This is because the selection rule for resonant transmission modes is modified when more line junctions are included in the transport process. In particular, it is found that the width or depth of the quantum well, the electron incident angle and the Fermi energy can significantly modify the transmission coefficient. This effect is further amplified in the quantum superlattice structure. Moreover, a suppression factor 0.4 on the maximum conductance and 0.85 on the maximum Fano effect are obtained in quantum wells and are approximately of the same size as for quantum superlattices. Based on these observations, there would appear to be great promise for achieving control of electrons with respect to incident angle, wave length or energy in topological insulator surfaces. The results in this study ­provide us with deeper understanding about the transport properties of topological insulator surfaces, and are also instructive regarding the experimental realization of electronic manipulation on the topological insulator surface.

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Transport through quantum wells and superlattices on topological insulator surfaces.

We investigate electron transmission coefficients through quantum wells and quantum superlattices on topological insulator surfaces. The quantum well ...
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