⇦Chinese Physics Letters, 2021, Vol. 38, No. 7, Article code 077303 Constructing Low-Dimensional Quantum Devices Based on the Surface State of Topological Insulators Tian-Yi Zhang (张天一)1, Qing Yan (闫青)2,3, and Qing-Feng Sun (孙庆丰)2,3,4* Affiliations 1School of Physical Science and Technology, Soochow University, Suzhou 215006, China 2International Center for Quantum Material, School of Physics, Peking University, Beijing 100871, China 3Collaborative Innovation Center of Quantum Matter, Beijing 100871, China 4CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China Received 29 March 2021; accepted 10 May 2021; published online 3 July 2021 Supported by the National Key R&D Program of China (Grant No. 2017YFA0303301), the National Natural Science Foundation of China (Grant Nos. 11921005 and 11574007), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB28000000), and Beijing Municipal Science and Technology Commission, China (Grant No. Z191100007219013).
*Corresponding author. Email: sunqf@pku.edu.cn Citation Text: Zhang T Y, Yan Q, and Sun Q F 2021 Chin. Phys. Lett. 38 077303    Abstract We propose a new method to construct low-dimensional quantum devices consisting of the magnetic topological insulators. Unlike previous systems based on locally depleting two-dimensional electron gas in semiconductor heterojunctions, magnetization provides a simpler and rewriteable fabrication way. The motion of electrons can be manipulated through the domain wall formed by the boundary between different magnetic domains. Here, three devices designed by local magnetization are presented. For the quantum point contact, conductance exhibits quantized plateaus with the increasing silt width between two magnetic domains. For the quantum dot, conductance shows pronounced peaks as the change of gate voltage. Finally, for the Aharonov–Bohm ring, conductance oscillates periodically with the external magnetic field. Numerical results show that the transport of these local magnetization systems is identical to that of the previous systems based on depleting two-dimensional electron gas, and the only difference is the approach of construction. These findings may pave the way for realization of low-power-consumption devices based on magnetic domain walls. DOI:10.1088/0256-307X/38/7/077303 © 2021 Chinese Physics Society Article Text The discovery of topological insulators (TIs) is a milestone of condensed matter physics and induces intense interest in studying topologically non-trivial materials.[1,2] The phase transition of these systems has nothing to do with the symmetry breaking (the conventional routine to identify different phases) but the topological change of the electronic state. The magnetic TI, specifically, is a special member of the big TI family. By introducing magnetism into a thin TI system, the time-reversal symmetry is broken and the quantum anomalous Hall effect occurs without the external magnetic field.[3,4] The insulating bulk of magnetic TI can be classified by a non-trivial Chern number,[5,6] which characterizes the chiral edge states (CESs) according to the bulk-edge correspondence. Up to date, the quantum anomalous Hall effect was experimentally realized in Cr-doped (Bi,Sb)$_2$Te$_3$ films,[7–9] V-doped (Bi,Sb)$_2$Te$_3$ films,[10,11] and the intrinsic MnBi$_2$Te$_4$ films,[12–14] which present CESs propagating along the edge with low-dissipation. With the rapid development of electronic devices, many scientific discoveries have been realized. For example, quantum point contact (QPC) is a system consisting of two depletion areas with a narrow slit in the middle, of which the width determines the conductance.[15] A quantum dot (QD) is an artificial system that can be filled with electrons, and one could measure the electronic properties by connecting voltage probes to this reservoir.[16,17] Moreover, the Aharonov–Bohm ring (AB-ring) is made of two beam splitters penetrated by a magnetic field. The transmission coefficient oscillates with the field because of the phase interference. Typically, such devices are based on GaAs–AlGaAs heterojunctions where a thin two-dimensional electron gas layer forms at the interface between GaAs and AlGaAs.[18] Such a system can be locally depleted by an electric field tuned by changing the negative voltages to metal gate electrodes on top of the heterostructure, and the place where the electrons could go is shaped by the gate structure. However, this fabrication is relatively sophisticated, and the shape of the gate electrodes stays unchangeable once the device has been made. A convenient, flexible way to build effective quantum devices is in demand. Very recently, using the tip of a magnetic force microscope, the magnetic domain and the magnetization direction can be arbitrarily written in Cr-doped (Bi,Sb)$_2$Te$_3$.[19] It has been realized that the boundary between magnetic TI domain with different topological properties forms a magnetic domain wall (DW),[20] which behaves like a waveguide of the low energy CESs, and whose chiral transport was observed in the experiment. One appealing advantage of this writing technique is that the DW in the system can be easily erased and rewritten with the change of magnetic field, creating the opportunity to construct quantum devices. Because the magnetic moments open the energy gap on the surface of 3D TIs, in low-energy cases, the transport behaviors are dominated by the CES, so electrons can only propagate along the DW and the area without magnetic moments.[21] Thus, we construct several quantum devices by regulating magnetization on the surfaces of TIs. In this Letter, we study the transport of several electron devices consisting of some magnetic DWs. We begin with the 2D Dirac Hamiltonian describing the surface states of 3D TIs. To clearly see the role of the DW, we build a two-terminal QPC, where two magnetic domains are divided by a non-magnetic slit. The conductance plateaus appear because the number of transport channels decreases with the decreasing slit width. If two regions are in the same magnetization direction, the conductance drops to zero as the slit closes. In the opposite-direction case, conductance stays finite even if the slit is closed, since a chiral edge mode always resides on the DW. Next, we build a QD to study the electrons confined in visibly discrete levels. A small area without magnetization is surrounded by a magnetization area so electrons can only come into the dot from a DW, building a well in three dimensions. The conductance shows peaks with the increasing gate voltage, which is evidence of the discrete energy levels. Finally, to study the phase interference of electrons, we introduce an AB-ring system. A ring made by DWs is penetrated by a magnetic flux with electrons moving along the arms and collecting phases. The results show the conductance oscillates periodically, which is very consistent with the theoretical analysis. The Hamiltonian we use during the calculation is as follows. The surface states of 3D TIs can be described by a 2D modified Dirac Hamiltonian[22] $H=\sum_k \varPsi_{{\boldsymbol k}}^† H(k) \varPsi_{{\boldsymbol k}}$, with $$ H(k)=\hbar \nu_{_{\scriptstyle \scriptscriptstyle\rm F}} (k_y \sigma_x-k_x \sigma_y)+\frac{1}{2}ak^2\sigma_z W +M_z \sigma_z,~~ \tag {1} $$ where $k^2=k_x^2+k_y^2$, $\nu_{_{\scriptstyle \scriptscriptstyle\rm F}}$ is the Fermi velocity, $\sigma_x,\sigma_y,\sigma_z$ are the Pauli matrices, $k_x,k_y$ are the momentum, and $a$ is the lattice constant. Here $\varPsi_{{\boldsymbol k}}=[\psi_{{\boldsymbol k} \uparrow},\psi_{{\boldsymbol k} \downarrow}]^{\rm T}$ is a two-component electron annihilation operator, $\uparrow$ and $\downarrow$ denote electrons with up-spin and down-spin; $k^2 \sigma_z Wa/2$ is the Wilson term avoiding the fermion doubling problem.[22–24] $M_z \sigma_z$ is the magnetic moment. The transport of the CESs at the DW depends on the configuration of the DW,[25,26] and here we set magnetic vectors pointing to the $z$-direction for simplicity. The sign of $M_z$ determines the upward ($M_z>0$) and downward ($M_z < 0$) magnetization. For the numerical calculation, we discretize the Hamiltonian in Eq. (1) into a lattice version,[27–29] $$\begin{aligned} H={}&\sum_{n,m}[c_{n,m}^† T_0 c_{n,m}+(c_{n,m}^† T_x c_{n+1,m} \\ &+c_{n,m}^† T_y c_{n,m+1}+{\rm H.c.})], \end{aligned}~~ \tag {2} $$ where $T_0=(2\,W/a+M_z)\sigma_z$, $T_x=-(W/2a)\sigma_z+({i}\hbar \nu_{_{\scriptstyle \scriptscriptstyle\rm F}}/2a)\sigma_y$, $T_y=-(W/2a)\sigma_z-({i}\hbar \nu_{_{\scriptstyle \scriptscriptstyle\rm F}}/2a)\sigma_x$; $c_{n,m}$ and $c_{n,m}^†$ are the annihilation and creation operators at site $(n,m)$, respectively. It is expedient to rewrite this Hamiltonian into a dimensionless form with the following substitutions: $E_0=\hbar \nu_{_{\scriptstyle \scriptscriptstyle\rm F}}/2a$, $W=w \cdot \hbar \nu_{_{\scriptstyle \scriptscriptstyle\rm F}}$, $M_z=m_z \cdot \hbar \nu_{_{\scriptstyle \scriptscriptstyle\rm F}}/2a$. Thus operators are reduced to $T_0/E_0=(4w+m_z)\sigma_z$, $T_x/E_0=-w\sigma_z+{i}\sigma_y$, and $T_y/E_0=-w\sigma_z-{i}\sigma_x$. Coefficient $w$ from the Wilson term is an auxiliary parameter, and hereon we fix it to 0.8. The magnitude of magnetization can be adjusted by changing the dimensionless parameter $m_z$.
cpl-38-7-077303-fig1.png
Fig. 1. Band structure and CESs of local magnetization TI. (a) The illustration of magnetic TIs extends along the $x$-direction. Here $+M_z$ and $-M_z$ indicate upward (red) and downward (blue) spontaneous magnetization, respectively. The diagram on the left (right) depicts two magnetization slabs with the same (opposite) direction leaving a non-magnetization slit (green) in the middle. The latter forms a topological DW and the CES is denoted by the red arrow. (b) Linear dispersion of non-magnetization area. The width of the slab is $100a$, and parameters we use are $w=0.8$ and $m_z=0$. [(c), (d)] The band structures of the left and right panels in (a), respectively. A pair of CESs appears in the gap in (d), suggesting the appearance of DW. Relevant parameters are $(w,m_z)=(0.8,1.1)$ and $(w,m_z)=(0.8,-1.1)$ for (c) and (d), respectively.
To see how the topological DW works, we study the spectrum of the Hamiltonian. Figure 1(a) shows that two slabs of magnetic TIs extend along the $x$-direction and have a finite width in the $y$-direction. Two magnetization areas are separated by a non-magnetic slit. Red (blue) color denotes the upward (downward) magnetization, and green color denotes the area without magnetization. In the left slab, the magnetic vectors are in the same direction, and in the right slab they are in the opposite directions. In the calculation, the band structures are calculated numerically from the Hamiltonian in Eq. (2), and the open boundary condition is used along the $y$-direction. As the slab is translational invariant along the $x$-direction, the momentum $k_x$ is a good quantum number. The energy spectrum clearly shows that the magnetization opens an energy gap in the bulk and leads to the existence of CES on the DW. Figure 1(b) shows the band diagram of the system without magnetization, namely, the pure green area with $m_z=0$. A Dirac cone appears at $k_x=0$, and we could view the green part as a metal. Figures 1(c) and 1(d) show the band structure of the same and opposite magnetization direction, respectively. We can see from Fig. 1(c) that the energy gap appears because of the magnetization, but with no chiral mode in the gap. However, for the opposite direction case in Fig. 1(d), the presence of CESs residing on the DW arises from the discontinuous change of topological number at the slit. The propagation direction of the CES is marked in the right panel of Fig. 1(a), of which the partner in the opposite direction is on the edge. Since the chiral partner always propagates on the edge, we keep leading away from the upper and lower edges to avoid the detection of this state. In this way we can measure the correct property of the devices which comes from the center of the surface. According to the above discussion, it is ready to present the QPC system. As shown in Figs. 2(a) and 2(b), in the system with a slab width $L_y$, two magnetization areas are separated by $L_0$ in the leftmost or rightmost region, and they come closer in the middle, leaving a small slit.[15] With low incident energy, modes cannot propagate across the magnetic area, most of which would be reflected but some would penetrate through the slit. To detect the conductance of the QPC, we set two rectangle (colored black) leads beside the QPC separate by a distance $L_x$. The length of the leads are $90a$, which is enough to avoid the measurement of the boundary state. By using the non-equilibrium Green's function method, the transmission coefficient can be calculated from $T_{\alpha\beta}(E)=\mathrm{Tr}[\varGamma_{\alpha} G^r \varGamma_{\beta} G^a]$,[30,31] with the incident energy $E$. Here $\alpha, \beta$ are the indices of the lead, Green's function $G^r(E)=[G^a]^†=1/[E- H_{\rm cen} -\sum_{\alpha}\varSigma_{\alpha}^r]$, the $\varGamma_{\alpha}={i}[\varSigma_{\alpha}^r-\varSigma_{\alpha}^a]$, and $H_{\rm cen}$ being the Hamiltonian of the central region. The retarded self-energy $\varSigma_{\alpha}^r$ due to the coupling to the leads can be calculated numerically.[32,33] The conductance $G$ can be calculated from the Landauer–Büttiker formula,[34,35] $G_{\alpha\beta}(E)=(e^2/h)T_{\alpha\beta}(E)$. Here we only consider that the electrons propagate in the top surface and the side surfaces have been neglected. The contributions of the side surfaces can be eliminated by the lead L/R away from side surfaces, or by the magnetization or by the bottom surface grounded in the experiment, although the side surfaces are gapless and important in usual.[36] In the calculation, we set $E=E_0$, $m_z=1.1$, and $(L_x,L_y,L_0)=(200a,100a,80a)$. Figures 2(c) and 2(d) show the dependence of conductance $G$ on the slit width $d$. With the increase of $d$, the conductance shows several plateaus with values close to $n e^2/h$ with the integer $n$. Since every transverse mode (or sub-band) contributes to a quantized conductance, every time the incident energy intersects a sub-band, the conductance would climb to an upper plateau. This analysis is quite identical to Figs. 2(c) and 2(d), where both the same-direction magnetization (c) and the opposite-direction magnetization (d) show the quantized conductance plateaus. However, there appears to be a slight difference. Indeed, in the same-direction case [Fig. 2(c)], the conductance becomes zero as the slit width diminishes to zero, but in the opposite-direction case [Fig. 2(d)], $G_{\rm LR}$ (represents the conductance from right to left) stays $e^2/h$ under the same circumstance. The remaining nonzero conductance even the slit is entirely disappeared is a concrete evidence of the CES travel on the DW. As mentioned above, the different direction of magnetization would form CESs propagating on the DW; because of the unique one-way characteristic, the state only travel from the right to the left. As a comparison, in Fig. 2(d) we draw the conductance curve with $G_{\rm RL}$ versus the slit width $d$, which is identical to the same-direction case since no CES contributes the transmission.
cpl-38-7-077303-fig2.png
Fig. 2. The transport property of QPC. [(a), (b)] Schematic diagram of the magnetic QPC with the same (a) and opposite (b) magnetization of edge slabs. In the center of QPC, two slabs come closer, forming a narrow slit with width $d$ in the middle. Two leads marked L and R on two sides of the QPC are used to measure the conductance. Parameters are $L_x=200a$, $L_y=100a$ and $L_0=80a$. [(c), (d)] The conductance dependence on the slit width of the system (a) and (b), respectively.
In addition, notice that the plateaus of conductance are slightly less than the quantized number $(ne^2/h)$. That is because of the reflection and scattering at the boundary of the magnetization area. However, the transmission probability of CES is strictly equal to 1 because it is robust against interferences. In the following, this will be used to build a one-way waveguide. Next, we use the local magnetization method to build a QD system. QD is an electronic well that constrains electrons in three dimensions, so the energy spectrum splits into many discrete levels. The number of energy levels is proportional to the size of the dot. If we take the electron–electron interaction into account, a Coulomb blockade would appear, that is, one electron in a dot repulses another to enter the dot. The current of the dot, as a result, shows peaks separated by equal distance with the change of gate voltage. Nevertheless, the Hamiltonian we consider in Eq. (2) is a single-electron Hamiltonian, and here we only focus on the discrete levels of the dot and neglect the Coulomb interaction.
cpl-38-7-077303-fig3.png
Fig. 3. Transport of the QD system. (a) Schematic diagram of the QD. Four leads marked 1 to 4 are placed on the DW for measuring purposes. Electrons flow in from the black leads and out from the white leads. Different propagate directions of the CESs are denoted by white arrows. Relevant parameters are $(L_x,L_y,L_0)=(50a,50a,30a)$. [(b), (c), (d)] The conductance $G$ vs gate voltage $V_{\rm g}$ of the QD. The larger the size, the greater the number of conductance peaks.
The QD system constructed by local magnetization is shown in Fig. 3(a). The same as before, the slab extends along the $x$-direction. Far from the left and right, the area consists of three parts, the magnetization direction of each part along the $y$-direction is $+z/-z/+z$, respectively. Two DWs separated by $L_0$ form under this arrangement. At low energy (we set the incident energy to 0) only the CES can propagate along the wall with the direction marked by white arrows. In the middle of the device, a square area without magnetization with size $L_d \times L_d$ is surrounded by four magnetization areas. Four leads marked by 1, 2, 3, 4 connect the system on the top of the DW. The electrons come from lead 1, flow into the QD, and flow out of the dot from lead 2 or lead 4. Due to the chirality of state, electrons cannot propagate from lead 1 to lead 3, and we have the relation $G_{21}+G_{41}=1$, $G_{31}=0$. It is sufficient to treat $G_{41}$ as the representation of the transport property of the system. The same calculations have been processed in the QD. Parameters we chose are $m_z=1.4$, $(L_x,L_y,L_0)=(50a,50a,30a)$. By adjusting the gate voltage $V_{\rm g}$ where we tune the on-site energy in the QD region, the conductance $G_{41}$ (proportional to the transmission coefficient from lead 1 to lead 4) shows peaks at certain $V_{\rm g}$ [see Figs. 3(b)–3(d)], suggesting the existence of discrete energy levels. At a low gate voltage, the transmission probability from lead 1 to lead 4 is between 0 and 1, at these gate voltages, once an electron enters the QD from lead 1, it randomly comes out to lead 4 or lead 2. With the increasing $V_{\rm g}$, $G_{41}$ abruptly decreases to 0, and then suddenly goes up to 1. At high $V_{\rm g}$, many peaks appear, with the peak height equal to 1, indicating full transmission. In the valleys between the peaks, the conductance vanishes, electrons cannot propagate from lead 1 to lead 4. As we change the size of the QD, the number of peaks increases, illustrated in Figs. 3(c) and 3(d). The physical explanation of the calculation results is given as follows. The Hamiltonian we use has a Dirac-like dispersion [see Fig. 3(b)]. The 2D density of states (DOS) linearly increases with the energy. In the low-energy range ($0\sim 3E_0$), the DOS is very small, which causes the small number of energy levels, and the gap between levels is relatively large. Since the spread of the state is large in the energy space, the transmission probability is therefore about 0.5 and the incident electron from the lead 1 has almost the same probability to leads 2 and 4. In the high energy range, levels become denser. Once the voltage lies near a state, the conductance reaches a peak. On the other hand, if voltage does not match any state, the conductance naturally becomes very small. The transport property of two magnetization devices (QPC, QD) is identical to previous quantum devices in 2DEG but can be realized easily by the writing technique. To see if the same fabrication can work in the presence of a magnetic field, we introduce an AB-ring system [see Fig. 4(a)]. As the diagram shows, the structure of the AB-ring is similar to QD, and the only difference is that the center area of QD is replaced by a downward magnetization area. There are two ways for an electron to go from lead 1: getting into the ring or going across the small gap between two DWs. The reason why this would happen is the spatial distribution of the wave packet. The probability of jumping across the slit is proportional to the packet width. After choosing one way to go, the electron propagates along the lower arm and then makes another choice: going straight to lead 4 or cross the gap. All possible ways are marked in Fig. 4(a) with white arrows. Meanwhile, we apply a small magnetic field perpendicular to the plane piercing the ring with adjustable magnetic flux. In numerical calculation, the effect of the perpendicular magnetic field is introduced by adding a phase term $\phi_{n,n+1}=\int_n^{n+1} \boldsymbol{A} \cdot \mathrm{d}\boldsymbol{l}/\phi_0$ to $T_x$ in Eq. (2), where $\boldsymbol{A}=(-By,0,0)$ is the vector potential for a magnetic field $B$ parallel to the $z$-direction, and $\phi_0=\hbar/e$. The magnetic field is expressed in terms of $\phi$ with $\phi \equiv a^2B/\phi_0$. The parameters we choose are $(L_x,L_y,L_0,L_r)=(50a,50a,30a,19a)$, and $m_z=1.4$. The numerical results are shown in Fig. 4(c). The conductance $G_{41}$ oscillates periodically with $\phi$. As we change the incident energy $E_{\rm f}$ ($0.2E_0$, $0.6E_0$ and $1.0E_0$), the minimum value of conductance changes. With a larger $E_{\rm f}$, the oscillation amplitude increases.
cpl-38-7-077303-fig4.png
Fig. 4. Transport of the AB-ring. (a) Schematic diagram of the AB-ring. The device is nearly the same as QD, but the center non-magnetization area is replaced by a magnetization area with a magnetic field $\varPhi$ in the middle. Parameters are $(L_x,L_y,L_0,L_r)=(50a,50a,30a,19a)$. (b) Propagate trajectories of electrons. (c) Numerical results of the conductance $G_{41}$ vs magnetic flux $\phi$. The conductance shows an oscillating behavior, and the amplitude increases in larger incident energy $E_{\rm f}$. Black dashed lines are the analytic results from Eq. (4). (d) Relevant coefficients calculated from theoretical analysis. Larger incident energy $E_{\rm f}$ results in larger transmission amplitude $r$ and smaller oscillate frequency $\omega$.
To explain the dependence of conductance on the magnetic field, we propose a theoretical analysis. To simplify the problem, we assume the structure to be fully symmetrical. Suppose that the reflection amplitude across the gap is $r$, thus the transmission amplitude of going straight is $t$, satisfying $r^2+t^2=1$. Without a magnetic field, the electron acquires dynamical phase shift $\chi$ when traveling along each arm. In the presence of the magnetic field, the electron adds a $\varPhi/2$ term in the phase shift, where the term $\varPhi$ expresses the magnetic flux penetrating the whole ring. These discussions are illustrated in Fig. 4(b). We then consider how to calculate the transmission probability from lead 1 to lead 4. One feasible trajectory is just to start from lead 1, go along the lower arm, and then reach lead 4, which can be expressed as $t{e}^{{i}(\chi+\varPhi/2)}t$. The other feasible trajectory is to start from lead 1, go along the lower arm, then jump to the upper arm, go along the upper arm, jump to the lower arm, and finally reach lead 4. In other words, the latter is the same as the former with an extra closed loop, so the amplitude can be written as $t{e}^{{i}(\chi+\varPhi/2)}r{e}^{{i}(\chi+\varPhi/2)}r{e}^{{i}(\chi+\varPhi/2)}t$. The other trajectories are contained with more loops, and by adding them all, we attain the final transmission amplitude from lead 1 to lead 4: $$\begin{aligned} t_{14}={}&t {e}^{{i}(\chi+\varPhi/2)}t+t {e}^{{i}(\chi+\varPhi/2)}r{e}^{{i}(\chi+\varPhi/2)}r{e}^{{i}(\chi+\varPhi/2)}t\\ &+\cdots \\ ={}&{e}^{{i}(\chi+\varPhi/2)}t^2 \sum_{k=0}^{\infty} [{e}^{{i}(2\chi+\varPhi)}r^2]^k \\ ={}&\frac{{e}^{{i}(\chi+\varPhi/2)}t^2}{1-{e}^{{i}(2\chi+\varPhi)}r^2}. \end{aligned}~~ \tag {3} $$ Assuming $\varPhi=\omega \phi$, we derive the transmission coefficient[37] $$ T_{41}=\frac{(1-r^2)^2}{1+r^4-2r^2 \cos (\omega \phi+ 2\chi)}.~~ \tag {4} $$ The theoretical result in Eq. (4) is perfectly identical to the numerical result in Fig. 4(c). To further find out the relation between analytical parameters and $E_{\rm f}$, we calculate relevant coefficients in Eq. (4) [see Fig. 4(d)]. As the incident energy $E_{\rm f}$ increases, the transmission amplitude $r$ increases. In our analysis, the flux in the whole ring is supposed to be proportional to the area of the ring, which yields $\varPhi=(L_r/a)^2\phi$, and $\omega=\omega_0 \equiv (L_r/a)^2$, but actually the frequency $\omega$ decreases. To explain this, we must consider the spatial distribution of the wave packet. As the Fermi energy gets higher, the width of the packet broadens. The broadening of the width increases the probability to cross the gap of two domain walls, so $r$ increases. In addition, the center of the packet gradually deviates from the domain wall, and the effective area of the ring decreases, which induces the decline of $\omega$. In summary, by the virtue of local magnetization, we propose several schemes of quantum devices. For QPC, the conductance plateau characterizes the dependence of the number of sub-bands on the width of the non-magnetization slit. For QD, the peaks of conductance show the discrete energy levels of a quantum well surrounded by magnetization area. For the AB-ring, the results undoubtedly manifest the accumulation of the electron phase traveling in the ring under a perpendicular magnetic field. Our calculation results can be viewed as the guidance of device manufacture, and a deep understanding of magnetic topological insulators. 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