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Topological semimetals are exotic phases of quantum matter with gapless electronic excitation protected by symmetry. Benefitting from its unique relativistic band dispersion, topological semimetals host abundant quantum states and quantum effects, such as Fermi-arc surface states and chiral anomaly. In recent years, due to the potential application in topological quantum computing, the hybrid system of topology and superconductivity has aroused wide interest in the community. Recent experimental progress of topological semimetal-superconductor heterojunctions is reviewed in two aspects: 1) Josephson current as a mode filter of different topological quantum states; 2) detection and manipulation of topological superconductivity and Majorana zero modes. For the former, utilizing Josephson interference, ballistic transport of Fermi-arc surface states is revealed, higher-order topological phases are discovered, and finite-momentum Cooper pairing and superconducting diode effect are realized. For the latter, by detecting a.c. Josephson effect in Dirac semimetals, the 4π-periodic supercurrent is discovered. By all-electric gate control, the topological transition of superconductivity is obtained. Outlooks of future research on topological semimetal-superconductor heterojunctions and their application in Majorana braiding and topological quantum computing are discussed.
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Keywords:
- topological semimetals /
- Josephson effect /
- topological superconductivity /
- Majorana zero modes
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图 1 拓扑半金属能带结构 (a) 狄拉克点处的线性能带色散关系[10]; (b) 狄拉克, 外尔和绝缘相 (1)四重简并狄拉克点, (2)打破空间反演对称性, 分为4个外尔点, (3)塞曼场使狄拉克点沿磁场方向劈裂为两个手性相反的外尔点, (4)打破四重旋转对称性, 产生能隙[10]; (c)外尔半金属能带结构示意图, 外尔点(红点)在表面布里渊区(灰色)的投影连线形成费米弧表面态(黄线)[8]
Figure 1. Band structure of topological semimetals: (a) Linear band dispersion at Dirac points[10]; (b) Dirac, Weyl, and insulating phase (1) Fourfold degenerate Dirac point, (2) four Weyl points due to a small inversion breaking perturbation, (3) Dirac point splits into two Weyl points with opposite chirality along the direction of a Zeeman field, (4) gapped phase obtained by breaking the fourfold rotation symmetry[10]; (c) schematic of the band structure of the Weyl semimetal, the projection of Weyl points (red points) to the surface Brillouin zone (in gray) leads to Fermi-arc surface states (yellow line)[8].
图 2 狄拉克半金属Cd3As2费米弧表面态的法布里-珀罗干涉[133] (a)零电阻态的观测, 插图为约瑟夫森结的SEM图片; (b)微分电阻dV/dI随源漏电流Isd的演化; (c) dV/dI对栅压Vg和Isd的依赖, 中间深色区域为超导态, 其上边界反映了临界电流Ic随Vg的演化; (d) –1—0 V显著的Ic振荡; (e) 法布里-珀罗谐振腔示意图; (f) Ic(kF)振荡的FFT分析, 峰值F = 0.249对应周期ΔkF ~ 4.01 μm–1
Figure 2. Fabry-Pérot interferences of Fermi-arc surface states in Dirac semimetal Cd3As2[133]: (a) Observation of zero resistance, inset is an SEM image of the Josephson junctions; (b) differential resistance dV/dI as a function of the source-drain current Isd; (c) gate voltage Vg and Isd dependence of dV/dI. The central dark region represents the superconducting state, in which its upper boundary reflects the evolution of critical current Ic with Vg; (d) clear oscillations in –1—0 V; (e) sketch of the Fabry-Pérot resonator; (f) FFT analysis of the Ic(kF) oscillations, the peak F = 0.249 corresponds to a period ΔkF ~ 4.01 μm–1.
图 3 Cd3As2费米弧超导电流振荡[144] (a) Ic随Vg的超导拱形依赖; (b)多重Andreev反射; (c)平行磁场下Ic-B振荡, 插图为约瑟夫森结的光学显微镜图片, 标尺为2 μm; (d) 0.1 T磁场时的Ic-Vg及正常态电导GN-Vg演化, 狄拉克点处Ic取极大值; (e) Ic极大值随磁场增加向狄拉克点偏移; (f)费米面靠近(上图)和远离(下图)狄拉克点的费米弧
Figure 3. Fermi-arc supercurrent oscillations in Cd3As2[144]: (a) Superconducting dome of Ic-Vg; (b) multiple Andreev reflections (MAR); (c) Ic-B oscillations under parallel magnetic field, inset is an optical image of the Josephson junctions, scale bar is 2 μm; (d) evolution of Ic and normal conductance GN with varying Vg at B = 0.1 T; (e) the maximum of Ic shifts towards Dirac point with increasing B field; (f) the Fermi arcs for Fermi level close to (up panel) and away from (bottom panel) Dirac point.
图 4 铋(Bi)纳米线中的边缘超导电流[157] (a)钨(W)-Bi-W约瑟夫森结示意图; (b)面外磁场下的Ic-B振荡; (c) 面内磁场下的Ic-B振荡; (d) Bi纳米线中局域态密度的紧束缚计算
Figure 4. Edge supercurrent in bismuth (Bi) nanowires[157]: (a) Schematic of the tungsten (W)-Bi-W Josephson junction; (b) Ic-B oscillations under an out-of-plane magnetic field; (c) Ic-B oscillations under an in-plane magnetic field; (d) tight-binding calculation of the local density of states (LDOS) in bismuth nanowire.
图 5 Cd3As2中的高阶拓扑相[164] (a) 面外磁场下的Ic振荡, 插图为约瑟夫森结SEM图片, 标尺为2 μm; (b) Ic-B偏离夫琅禾费图样, 可被反对称SQUID模型拟合; (c) 超导电流密度Jc受限在一维通道; (d) 扣除光滑背景后的拍频振荡; (e) 对(d)做FFT, 得到两个振荡频率; (f) Cd3As2中高阶拓扑棱态示意图
Figure 5. Higher-order topological phase in Cd3As2[164]: (a) Ic oscillations under an out-of-plane magnetic field, inset is an SEM image of the Josephson junctions, scale bar is 2 μm; (b) Ic-B deviates from the Fraunhofer pattern and could be fitted by the asymmetric SQUID model; (c) the supercurrent density Jc is confined in 1D channels; (d) beating oscillations after subtracting a smooth background; (e) two frequencies obtained by applying FFT to (d); (f) schematic of the higher-order topological hinge states in Cd3As2.
图 6 外尔半金属Td-MoTe2中的边缘超导电流[165] (a) S1样品中的两种Ic-B振荡模式, 内侧夫琅禾费形的低频模, 外侧扇贝形的高频模; (b)低频模(红箭头)和高频模(蓝箭头)随磁场的演化; 较大面积样品S2 (c)和S6 (d)中则只有高频模; (e) 由振荡周期推算的面积与物理面积的标度关系; (f) 对称性破缺的弱激发态
Figure 6. Edge supercurrent in the Weyl semimetal Td-MoTe2[165]: (a) Two Ic-B oscillation modes in sample S1, the slow mode displays the inner Fraunhofer pattern, while the fast mode exhibits the outer scalloped boundary; (b) magnetic field evolution of the slow mode (red arrows) and fast mode (blue arrows); in large-area crystals S2 (c) and S6 (d), only the fast mode is visible; (e) scaling between the flux penetration area derived from the oscillation period and the physical area; (f) weak excitation branches with broken symmetry.
图 7 Cd3As2约瑟夫森结中的有限动量配对[192] (a) Nb-Cd3As2纳米线-Nb约瑟夫森结在平行磁场下的Ic振荡; (b) Ic随Vg和平行磁场的演化
Figure 7. Finite momentum pairing in Cd3As2 Josephson junctions[192]: (a) Ic oscillations with parallel magnetic field in Nb-Cd3As2 nanowire-Nb Josephson junctions; (b) evolution of Ic with Vg and parallel magnetic field.
图 8 Bi0.97Sb0.03约瑟夫森结中塞曼效应诱导0-π相变[193] (a) Bi0.97Sb0.03与Nb构成的非对称SQUID器件示意图; (b) 20 mK的锯齿形CPR; (c) 平行磁场诱导的0-π相变; (d) 不同平行磁场下的CPR; (e) 临界电流随平行磁场的振荡; (f) 塞曼效应诱导的有限动量配对
Figure 8. Zeeman-effect-induced 0-π transitions in Bi0.97Sb0.03 Josephson junctions[193]: (a) Schematic of the asymmetric SQUID made of Bi0.97Sb0.03 and Nb; (b) sawtooth-shaped CPR at 20 mK; (c) parallel-magnetic-field-induced 0-π transitions; (d) CPR at different parallel magnetic fields; (e) critical current oscillations with parallel magnetic field; (f) illustration of the Zeeman-effect-induced finite-momentum pairing.
图 9 狄拉克半金属NiTe2中的约瑟夫森二极管效应[200] (a) 约瑟夫森结与面内垂直磁场By示意图; By = 20 mT的非互易临界电流
${I_{{\text{c}} + }} \ne \left| {{I_{{\text{c}} - }}} \right|$ (b)和整流效应(c); (d)$\Delta {I_{\text{c}}}$ 随By和温度的演化; (e) dV/dI-Bz干涉图案随平行磁场Bx的演化; (f) 计算的Ic-Bz干涉图案随平行磁场Bx的演化Figure 9. Josephson diode effect in a Dirac semimetal NiTe2[200]: (a) Schematic of a Josephson junction with in-plane perpendicular magnetic field By; non-reciprocal critical current
${I_{{\text{c}} + }} \ne \left| {{I_{{\text{c}} - }}} \right|$ (b) and rectification effect (c) with By = 20 mT; (d) dependence of$\Delta {I_{\text{c}}}$ on By and temperature; (e) evolution of the interference pattern of dV/dI-Bz due to the parallel magnetic field Bx; (f) calculated interference pattern of Ic-Bz evolving with Bx.
图 10 Cd3As2表面态承载的4π周期超导电流[236] (a), (b) 微波频率分别为6.7 GHz和2 GHz, 微分电阻随电流和微波功率的演化; (c) n = 1 Shapiro台阶的消失; (d), (e) 分别从(a)和(b)提取的Shapiro台阶宽度随功率的演化; (f)模拟的低频下n = 0台阶对微波功率的响应, 占比7%的4π周期超导电流可以产生明显的剩余电流
$I_0^{k = 1}$ Figure 10. 4π-periodic supercurrent carried by surface states of Cd3As2[236]: (a), (b) Differential resistance as a function of current and microwave power under irradiation frequency of 6.7 GHz and 2 GHz, respectively; (c) missing of the n = 1 Shapiro step; (d), (e) extracted power evolution of Shapiro step sizes from (a) and (b), respectively; (f) simulated response of n = 0 step to microwave power under low irradiation frequency, a 7% admixture of 4π-periodic supercurrent could generate a clear residual supercurrent
$I_0^{k = 1}$ .
图 11 Cd3As2纳米线约瑟夫森结中栅压调控的拓扑超导相变 (a) 栅压调控表面态拓扑相变[192]; (b)平行磁场中, 3个栅压下的归一化Ic-B曲线[192]; (c) 300 mT平行磁场中dV/dI关于Idc和Vg的函数图[192]; (d) dV/dI随栅压和平行磁场的变化[192]; 正栅压(e)和负栅压(f)下Shapiro台阶宽度随电压V和射频功率的演化[236]
Figure 11. Topological transition of superconductivity in Cd3As2 nanowire Josephson junctions by gate control: (a) Topological transition of surface states by tuning gate voltages[192]; (b) normalized Ic-parallel magnetic field B at three gate voltages[192]; (c) dV/dI as a function of Idc and Vg under a parallel magnetic field of 300 mT[192]; (d) evolution of dV/dI with Vg and parallel magnetic field B[192]; Shapiro step size as a function of voltage V and radio frequency power at positive (e) and negative (f) gate voltages[236].
图 12 纳米线Y型结构的Majorana零能模编织[255] (a) 理想的Majorana编织, 从真空中产生两对Majorana零能模(γi), 各一个被编织后, 每对Majorana零能模融合为一个费米子(ci), 始末状态可以定义一个拓扑量子比特的
$\left| 0 \right\rangle $ 和$\left| 1 \right\rangle $ ; (b)纳米线Y型结构中两个Majorana零能模的交换操作, 一次完整的编织需要进行两次这样的交换; (c) 栅压调控Majorana零能模的实验构型, Majorana零能模(γ)位于纳米线中拓扑超导(红色)和平庸(浅灰)区域的边界Figure 12. Braiding Majorana zero modes in a nanowire Y-junction[255]: (a) Ideal Majorana braiding, two pairs of Majorana fermions (γi) are created from the vacuum, and one from each pair is braided, after this process, each pair of Majorana fermions fuses to form a complex fermion (ci), initial and final states could be defined as the
$\left| 0 \right\rangle $ and$\left| 1 \right\rangle $ of a topological qubit; (b) sequence of moves for exchanging one from each pair of Majorana zero modes in a nanowire Y-junction, this exchange must be carried out twice to perform a complete braid; (c) schematic experimental setup for manipulating Majorana fermions by gate control, Majorana zero modes (γ) locate at the boundary between the topological regions (red) and trivial regions (light gray). -
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