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Topological superconductors have attracted much research interest, because they were proposed to host non-abelian Ising Anyon Majorana zero modes and thus can be used to construct fault-tolerant topological quantum computers. This paper mainly reviews the electrical transport methods for detecting the presence of Majorana zero modes. First, the basic concepts of topological superconductivity, Majorana zero modes and non-Abelian statistics are introduced, followed by a summary of various schemes for implementing topological superconductivity. Then, the experimental methods for detecting topological superconductivity or Majorana zero modes by using low-temperature transport methods, including electron tunneling spectroscopy, Coulomb blockade spectroscopy and non-local conductance detection, which are widely used in superconductor/nanowire hybrid systems, are discussed. On the other hand, the measurements of the (inverse) AC Josephson effect and current (energy) phase relationships are also reviewed to identify Majorana zero modes in Josephson devices. Meanwhile, to deepen our understanding of Majorana zero modes, some mechanisms for explaining the experimental data observed in the above experiments are provided. Finally, a brief summary and outlook of the electrical transport methods of Majorana zero modes are presented.
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Keywords:
- topological superconductivity /
- Majorana zero modes /
- zero-bias conductance peak /
- Josephson effect
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图 1 二维手性拓扑超导和IQHE、二维螺旋拓扑超导和QSHE的对比示意图 (a) 二维手性拓扑超导和IQHE的对比示意图, 在这两个系统中, 时间反演对称性破缺, 其边缘态具有确定的手性; (b) 二维螺旋拓扑超导和QSHE的对比示意图, 这两个系统都有时间反演对称性和无能隙的螺旋边缘态, 在同一边缘上不同自旋极化的电子传播方向相反. 图中虚线表明拓扑超导体的边缘态是马约拉纳费米子, 其准粒子能谱中E＜0的部分是冗余的, 引自文献[95]
Figure 1. Schematic comparison of two-dimensional chiral topological superconductor (TSC), IQHE, two-dimensional helical TSC and QSHE. (a) Schematic comparison of two-dimensional chiral TSC and IQHE state. In both systems, time reversal symmetry is broken, and the edge states have a definite chirality. (b) Schematic comparison of two-dimensional time reversal invariant TSC and QSHE. Both systems maintain time reversal symmetry and have a helical pair of edge states, where opposite spin states counterpropagate. The dashed lines indicates that the edge states of the topological superconductors are Majorana fermions so that the E＜0 part of the quasi-particle spectra are redundant, adapted from Ref. [95].
图 2 MZMs非阿贝尔统计原理示意图 (a) γ1 环绕 γ2 逆时针旋转半圈; (b) γ1 环绕 γ3 逆时针旋转1圈; (c) 对两对MZMs进行编织操作的示意图
Figure 2. Schematic of the non-Abelian statistics principle of MZMs: (a) γ1 winds counterclockwise around γ2 by half a turn; (b) γ1 winds counterclockwise around γ3 by one full turn; (c) schematic of braiding two pairs of MZMs.
图 3 在超导复合结构中探测ZBCP (a) Sn/Bi2Se3复合结构中出现的ZBCP, 左、右两图分别为归一化的隧穿电导在不同温度和磁场下关于偏置电压Vbias的关系图, Δ1, Δ2标出了两个不同的能隙结构, 在低温及低场下能观察到显著的ZBCP, 引自文献[57]. (b) 在超导完全覆盖的纳米线器件中观测到的MZM迹象. 左图显示了Al完全覆盖的InAs半导体纳米线器件的假彩色扫描电子显微镜 (scanning electron microscope, SEM) 照片, 右图上方显示了隧穿电导dI/dV关于通过纳米线的外加磁通 (水平轴) 和源漏电压 (垂直轴) 的二维图, 揭示了零磁通位置的硬能隙, 以及一个磁通量子Φ0 位置处的能隙内的零能态, 下方显示了存在MZM情况的模拟结果, 引自文献[177]. (c) 第一个在超导/半导体纳米线结构中观察到ZBCP实验中所使用器件的SEM照片. 正常金属电极Au完全覆盖InSb纳米线, 而超导电极NbTiN则只覆盖其中一边. 下面的底门用1—4标记, 引自文献[47]. (d) 图 (c) 中器件在固定磁场情况下, dI/dV与磁场转角和偏置电压V的关系, 其中A图为当施加与纳米线平行的200 mT磁场时, ZBCP最大, 当磁场垂直于纳米线时, ZBCP消失, B图为在垂直于自旋轨道耦合场BSO的平面上施加150 mT的旋转磁场, ZBCP在所有角度都存在. 最上方的图像显示了与A图和 B图中相应颜色标识处的截线, 最右侧图像从上到下分别为: (i) B垂直BSO, 能隙打开, 解除费米面简并, 这是实现MZM的必要条件; (ii) B 平行BSO, 不同自旋能带垂直移动2EZ, 在这种配置中, 不会出现MZM; (iii) A图中旋转磁场示意图; (iv) B图中旋转磁场示意图, 引自文献[47]
Figure 3. Detecting ZBCP in superconducting hybrid structures. (a) ZBCP structure observed in Sn/Bi2Se3, the normalized tunnelling conductance as a function of bias voltage Vbias collected at different temperatures (left) and magnetic fields (right), with Δ1 and Δ2 indicating two different gap structures, a significant ZBCP can be observed at low temperatures and low fields, adapted from Ref. [57]. (b) MZM fingerprints in full-shell nanowires. Left image shows a false-color SEM image of a full superconducting shell surrounding an InAs semiconducting nanowire. Top right shows a color map of the tunnelling conductance dI/dV as a function of external magnetic flux (horizontal axis) through the nanowire and source-drain voltage (vertical axis), revealing a hard induced superconducting gap near zero magnetic flux and a gapped region with a discrete zero-energy state around one flux quantum Φ0 . Bottom right shows the simulation results for the presence of MZM, adapted from Ref. [177]. (c) SEM image of the device used in the first observation of ZBCP in superconductor/semiconductor nanowire structures. The normal metal electrode Au completely covers the InSb nanowire, while the superconducting electrode NbTiN covers only one side. The bottom gates are labelled with numbers 1 through 4, adapted from Ref. [47]. (d) The relationship between dI/dV and the angle of magnetic field and bias voltage at a fixed magnetic field. Fig. A shows rotating the magnetic field
$ \left| B \right| = 200{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mT}} $ in the plane of the substrate. The ZBCP is maximal when B is parallel to the nanowire and absent when B is perpendicular to the nanowire. Fig. B shows rotating the magnetic field$ \left| B \right| = 150{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{mT}} $ in the plane perpendicular to BSO. Now, the ZBCP exists at all angles. The top panels show the linecuts at angles with corresponding colors in Fig. A and Fig. B. The rightmost panels show, from top to bottom: (i) The gap opening lifts the Fermi surface degeneracy when B is perpendicular to BSO, which is a necessary condition for achieving MZM; (ii) when B is parallel to BSO, different spin bands shift vertically by 2EZ, MZM is absent; (iii) schematic of the rotation of B in Fig. A; (iv) schematic of the rotation of B in Fig. B, adapted from Ref. [47].
图 4 利用库仑阻塞谱探测MZM (a) 利用库仑阻塞谱探测MZM的原理; (b) 上图是电导dI/dVb 关于磁场
${B_{//} }$ 和栅极电压VPG的关系, 展示了2e周期向1e周期转变的过程; 下图展示了库仑峰间距So, e 随磁场${B_{//} }$ 的振荡, 引自文献[180]; (c) 类似 (b)图, 展示了随磁场出现的2e电子输运过程的宇称从偶宇称到奇宇称的转变, 插图为转变区的放大图, 引自文献[180]; (d) 左上图为利用超导全覆盖纳米线制作的包含6个具备独立栅极的超导岛器件的SEM照片; 左下图展示了电导dI/dV关于磁场${B_{//} }$ 和栅极电压VG的二维图, 右图为库仑峰振荡振幅A关于岛长度L的关系曲线, 满足指数关系$ A = {A_0}{{\text{e}}^{ - L/\xi }} $ , 引自文献[177]Figure 4. Using Coulomb blockade spectroscopy to detect MZM: (a) Principal of detecting MZM by Coulomb blockade spectroscopy. (b) Top panel shows the relationship between the conductance dI/dVb and the magnetic field
${B_{//} }$ and gate voltage VPG, displaying a transition from 2e-periodic Coulomb-peak to 1e-periodic Coulomb-peak. Bottom panel shows the Coulomb peak spacing So, e as a function of magnetic field${B_{//} }$ , exhibiting clear oscillations, adapted from Ref. [180]. (c) Similar to (b), the parity undergoes a transition from even to odd in the 2e-periodic electron transport process as the magnetic field increases. The inset shows a zoom view of the transition region, adapted from Ref. [180]. (d) Top left panel shows the SEM image of six superconducting islands with individual gates constructed on a single full-shell nanowire. Bottom left panel shows the conductance dI/dV as a function of magnetic field$ {B_\parallel } $ and gate voltage VG. Right panel shows the relationship between the oscillatory amplitude A and the island length L, which satisfies an exponential relationship$ A = {A_0}{{\text{e}}^{ - L/\xi }} $ , adapted from Ref. [177].
图 5 非局域电导探测 (a) 用于测量电导矩阵的三端器件配置, 两端正常电极连接到中央接地的超导区域, 其中的电势可以通过栅极控制, 引自文献[186]; (b) 在
$ L \gg \xi $ 下, 能量空间中可能的散射过程示意图, 引自文献[182]; (c) 电导矩阵关于磁场B和偏置电压VL, R的二维图, 引自文献[183]; (d) 电导矩阵关于磁场B和偏置电压VT, B的二维图, 引自文献[184]Figure 5. Non-local conductance spectroscopy measurement: (a) Three-terminal device configuration. Two normal electrodes are connected to central grounded superconducting region, where the potential can be controlled by the gates, adapted from Ref. [186]. (b) Illustrations of possible scattering processes in energy space when
$ L \gg \xi $ , adapted from Ref. [182]. (c) The conductance matrix as a function of magnetic field B and bias voltage VL, R, Adapted from Ref. [183]. (d) Two-dimensional plot of the conductance matrix as a function of magnetic field B and bias voltage VT, B, adapted from Ref. [184].
图 6 约瑟夫森结的能量 (电流) 和相位的关系以及Landau-Zener 跃迁机制示意图 (a) 常规约瑟夫森结的能量相位关系 (上方) 和电流相位关系 (下方) 示意图, 呈现出2π周期性, 透射率分别为D = 1 (橙色曲线) 和D = 0.6 (蓝色曲线); (b) 拓扑约瑟夫森结的能量相位关系 (上方) 和电流相位关系 (下方) 示意图, 呈现出4π周期性, 透射率分别为D = 1 (橙色曲线) 和D = 0.6 (蓝色曲线); 在发生零次 (绿色实线) 和两次 (蓝色虚线) LZT后, 能量 (c) 和电流 (d) 关于相位φ的关系, (c) 中灰色和红色区域分别对应于绝热和非绝热演化过程, 绿色实线与蓝色虚线分别代表绝热极限 (呈2π周期性) 和非绝热极限 (呈4π周期性), 引自文献[74]
Figure 6. Energy (current) vs. phase relationship and LZT in Josephson junctions: (a) The energies and the currents in the conventional Josephson junction as functions of the phase difference φ for D = 1 (orange) and D = 0.6 (blue); (b) the energies and the currents in the topological Josephson junction as functions of the phase difference φ for D = 1 (orange) and D = 0.6 (blue); energy (c) and current (d) vs. phase φ after zero (green solid line) and two (blue dashed line) LZTs occurred, in (c), the gray and red regions correspond to adiabatic and non-adiabatic evolutions, respectively, the green solid line (2π periodicity) and blue dashed line (4π periodicity) represent the evolution in the adiabatic and the non-adiabatic limits, adapted from Ref. [74].
图 7 交流约瑟夫森效应的探测 (a) HgTe约瑟夫森结器件的假彩色SEM照片, 蓝色为被绝缘层覆盖的HgTe, 紫色为Al电极, 黄色为栅极. 并联电阻RS使结电压保持稳定. 通过测量跨越结和RI的电压, 可以直接得到V和I. 射频信号通过偏置器进入到放大电路中, 最后进入频谱分析仪. 引自文献[65]. (b) 拓扑约瑟夫森结辐射功率关于直流电压V和探测频率fd的二维图, 依次可以看到
$ {f_{\text{J}}}/2 $ ,$ {f_{\text{J}}} $ 和$ 2{f_{\text{J}}} $ 的发射线, 引自文献[65]. (c) 欠阻尼约瑟夫森结的发射线, 引自文献[192]. (d) 利用片上探测器探测约瑟夫森结的辐射信号, 从左到右依次为: (i) 放置在3个栅极上的纳米线约瑟夫森结的SEM照片, 插图展示了结的假彩色SEM照片, 其中外延Al壳以青色突出显示; (ii) 纳米线约瑟夫森结 (蓝色框) 和检测结 (红色框) 之间耦合电路的光学照片; (iii) 两个并联的约瑟夫森结构成的微波探测器的SEM照片, 引自文献[68]. (e) 探测器的跨导dIPAT/dVNM关于探测器电压VDET和纳米线电压VNW的二维图, 图中标出了理论预言的2e (绿色虚线) 和1e (红色虚线) 电子行为的峰位, 在磁场B = 650 mT处观察到了1e电子发射行为 (橙色实线), 引自文献[68]Figure 7. Detection of AC Josephson effect. (a) False-color SEM image of HgTe Josephson junction device. Blue represents HgTe covered by an insulating layer, purple represents Al electrodes, and yellow represents the gate. The shunt resistor RS enables stable voltage bias. The measurement of the voltage across the junction and RI is directly yields V and I. The rf signal enters the amplification circuit through the bias T circuit and finally enters the spectrum analyzer, adapted from Ref. [65]. (b) The power collected from the topological Josephson junction as a function of the DC voltage V and the detection frequency fd. The emission lines of
$ {f_{\text{J}}}/2$ ,$ {f_{\text{J}}} $ and$ 2{f_{\text{J}}} $ can be seen in order, adapted from Ref. [65]. (c) Emission lines of an underdamped Josephson junction, adapted from Ref. [192]. (d) Detection of the Josephson junction radiation signal using an on-chip detector. From left to right: (i) SEM images of nanowire Josephson junctions placed on three gates. The inset shows a false-color SEM image of the junction, with the epitaxial Al shell highlighted in cyan. (ii) Optical image of the coupling circuit between the nanowire Josephson junction (blue box) and the detection junction (red box). (iii) SEM image of the microwave detector composed of two parallel Josephson junctions, adapted from Ref.[68]. (e) Transconductance dIPAT/dVNM as a function of the detector voltage VDET and the nanowire voltage VNW. The dashed lines indicate the theoretically predicted peak positions of the 2e (green) and 1e (red) electron behavior. 1e electron emission behavior (orange solid line) is observed at a magnetic field of B = 650 mT, adapted from Ref. [68].
图 8 探测逆交流约瑟夫森效应 (a) 在三维拓扑绝缘体HgTe中观察到的分数约瑟夫森效应. 左图为微波功率f = 2.7 GHz时, 电压分布 (数据点按电压分组) 关于微波电流Irf和偏置电压V的二维图, 右边的直方图清晰地显示第1级台阶受到显著抑制, 引自文献[63]. (b) 在频率frf = 0.953 GHz时的微分电阻dV/dI关于功率Prf和电流I的二维图, 引自文献[67]. 20 mT 时在平行磁场 (c) 和垂直磁场 (d) 下观测到的电压分布关于微波功率P和偏置电压I的二维图, 40 mT 时在平行磁场 (e) 和垂直磁场 (f) 下观测到的电压分布关于微波功率P和偏置电压V的二维图, 引自文献[71]
Figure 8. Detection of the inverse AC Josephson effect. (a) Fractional AC Josephson effect observed in a three-dimensional topological insulator HgTe. Left panel shows a 2D plot of voltage distribution (data points grouped by voltage) as a function of microwave current Irf and bias voltage V at a fixed frequency f = 2.7 GHz, with the first step significantly suppressed. Right panel is a histogram showing the same result, adapted from Ref. [63]. (b) Differential resistance dV/dI as a function of power Prf and current I at a frequency of frf = 0.953 GHz, adapted from Ref. [67]. Voltage distribution as a function of microwave power P and bias voltage V under parallel (c) and perpendicular (d) magnetic fields of 20 mT. Voltage distribution as a function of microwave power P and bias voltage V under parallel (e) and perpendicular (f) magnetic fields of 40 mT, adapted from Ref. [71].
图 9 电流相位关系的测量 (a) 在Nb/3D-HgTe/Nb中测量到的CPR, 相比完全对称的正弦形式 (黑色实线) 有一定的倾斜, 引自文献[197]. (b) 利用扫描SQUID显微镜测量CPR的原理图. 分别显示了励磁线圈 (field coil)、拾取回路 (pickup loop)和感生电流I, 引自文献[197]. (c) 利用极不对称SQUID测量CPR的原理图. 其中γ表示参考结的相位, δ表示待测结的相位, Ib为激励电流, 引自文献[201]. (d) 利用单结耦合SQUID测量CPR的原理图. 电流I在结中激励起超流, 并在环中产生磁通. 产生的磁通通过磁通变换器耦合到SQUID, 引自文献[203]
Figure 9. Measurements of current-phase relationships. (a) Measured CPR in Nb/3D-HgTe/Nb, exhibiting a slight forward skewness compared to the perfectly symmetric sinusoidal form (black solid line). Adapted from Ref. [197]. (b) Schematic of measuring CPR using a scanning SQUID microscope. The field coil, pickup loop, and induced current I are shown. Adapted from Ref. [197]. (c) Schematic of measuring CPR using a highly asymmetric SQUID. Here, γ represents the phase of the reference junction, δ represents the phase of the measured junction, and Ib is the excitation current, adapted from Ref. [201]. (d) Schematic of measuring CPR using a single junction coupled to a SQUID. The current I excites a supercurrent in the junction and generates magnetic flux in the loop. The generated flux is coupled to the SQUID through a flux transformer. Adapted from Ref. [203].
图 10 能量相位关系的探测 (a) Bi2Te3约瑟夫森三结的假彩色SEM照片. 在Bi2Te3薄片表面制作了一个由两个超导回路 (蓝色) 连接的约瑟夫森三结. 使用两个半匝线圈在回路中施加局部磁通. Au电极用于检测三结的中心和末端处的ABSs, 引自文献[205]. (b) 在T = 0.25 K处测量得到的dV/dIb关于电流IR和IL (施加磁通) 的二维图, 测量结果大体上和理论预测的约瑟夫森三结相图吻合, 引自文献[205]. (c) 利用隧穿结测量能量相位关系的原理图. 超导环 (蓝色) 构成SQUID, 激励电流IIN用作超流干涉 (衍射) 测量, 电压VT用作隧穿测量, 引自文献[208]. (d) Bi2Te3约瑟夫森结的假彩色SEM照片. Pb作为超导电极, Pd电极用来探测点接触谱, A、B和C分别代表结的两端和中心, 引自文献[204]. (e) 红色和黑色曲线为接近π相位时的接触电阻随电流Ib 的变化, 蓝线为正常态电阻, 可以观察到~95%的峰高变化, 反映了能隙的关闭. 引自文献[204]. (f) 态密度 (density of states, DOSs) 关于能量E和磁场
$ {{\varDelta }}{B_ \bot } $ 的二维图, 图像显示了几条离散的ABSs, 引自文献[208]Figure 10. Detection of energy-phase relationships: (a) False-color SEM image of a Josephson trijunction fabricated on the surface of Bi2Te3 with two superconducting loops (in blue). Local magnetic flux is applied to the loops using two half-turn coils. Au electrodes are used to detect the ABSs at the center and ends of the junction. Adapted from Ref. [205]. (b) The dV/dIb as a function of the currents IR and IL (for applying magnetic flux) measured at T = 0.25 K, in good agreement with the theoretically predicted Josephson trijunction phase diagram. Adapted from Ref. [205]. (c) Schematic of measuring energy-phase relationships using a tunnel junction. The superconducting loop (blue) forms a SQUID, and the excitation current IIN is applied to observe supercurrent interference (diffraction), while the voltage VT is applied for tunneling spectroscopy. Adapted from Ref. [208]. (d) False-color SEM image of a Josephson junction in Bi2Te3. Pb is used as the superconducting electrode, and Pd electrodes are used to detect point-contact spectra at points of A, B and C, which correspond to the ends and center of the junction, adapted from Ref. [204]. (e) Contact resistance as a function of current Ib near the π phase. The blue line represents the normal state resistance. The peak height change ratio of ~95% from 147 Ω (red) to 8 Ω (black) reflects the closing of the energy gap, adapted from Ref. [204]. (f) Density of states (DOSs) as a function of energy E and perpendicular magnetic field
$ {{\varDelta }}{B_ \bot } $ , showing several discrete ABSs, adapted from Ref. [208].表 1 超导体的对称性和相应的拓扑不变量[7]
Table 1. Symmetries and corresponding topological invariants of superconductors[7].
Class TRS PHS CS d = 1 d = 2 d = 3 Spinful or Spinless SC D 0 +1 0 $ Z_2^{{\gamma _{{\text{geom}}}}} $ Z(TKNN) 0 Spinful SC with TRS DIII –1 +1 1 $ Z_2^{{\gamma _{{\text{geom}}}}/2} $ $ Z_{2}^{{\text{(KM)}}} $ Z(3dW) Spinful SC with SU(2)-SRS C 0 –1 0 0 2Z(TKNN) 0 Spinful SC with SU(2)-SRS+TRS CI +1 –1 1 0 0 2Z(3dW) Spinful SC with TRS BDI +1 +1 1 Z(1dW) 0 0 注: TRS (time reversal symmetry, 时间反演对称性), PHS (particle hole symmetry, 粒子空穴对称性), CS (chiral symmetry, 手性对称性), SRS (spin rotation symmetry, 自旋旋转对称性), γgeom(几何相位), Z(TKNN) (TKNN不变量), $ Z_{2}^{{\text{(KM)}}} $ (Kane-Mele不变量),
Z(1dW) (一维缠绕数), Z(3dW) (三维缠绕数).
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