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磁场中拓扑物态的量子输运

强晓斌 卢海舟

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磁场中拓扑物态的量子输运

强晓斌, 卢海舟

Quantum transport in topological matters under magnetic fields

Qiang Xiao-Bin, Lu Hai-Zhou
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  • 拓扑物态包括拓扑绝缘体、拓扑半金属以及拓扑超导体. 拓扑物态奇异的能带结构以及受拓扑保护的新奇表面态, 使其具有了独特的输运性质. 拓扑半金属作为物质的一种三维拓扑态具有无能隙的准粒子激发, 根据导带和价带的接触类型分为外尔半金属、狄拉克半金属和节线半金属. 本文以拓扑半金属为主回顾了在磁场下拓扑物态中量子输运的最新工作, 在不同的磁场范围内分别给出了描述拓扑物态输运行为的主要理论.
    Topological matters include topological insulator, topological semimetal and topological superconductor. The topological semimetals are three-dimensional topological states of matter with gapless electronic excitations. They are simply divided into Weyl, Dirac, and nodal-line semimetals according to the touch type of the conduction band and the valence band. Their characteristic electronic structures lead to topologically protected surface states at certain surfaces, corresponding to the novel transport properties. We review our recent works on quantum transport mainly in topological semimetals. The main theories describing the transport behavior of topological matters are given in different magnetic regions.
      通信作者: 卢海舟, luhz@sustech.edu.cn
      作者简介:
      卢海舟, 2007年在清华大学高等研究院获博士学位. 同年赴香港大学做博士后研究, 2012年晋升为研究助理教授. 2015年加入南方科技大学, 现为物理系和深圳量子科学与工程研究院教授. 主要从事凝聚态物理, 特别是量子输运理论的研究. 研究兴趣集中在利用量子场论方法研究拓扑和新奇材料中的量子态和量子输运. 系统地研究了二维和三维狄拉克电子的量子输运行为, 并应用于拓扑绝缘体/半金属/超导体, 二维层状材料等. 研究过的物理效应包括弱局域化、奇异负磁阻、量子振荡、强磁场量子极限、各类霍尔效应等, 在拓扑半金属中提出了三维量子霍尔效应的一种基于费米弧的新机制, 多个理论工作被实验广泛支持和应用
    • 基金项目: 国家重点基础研究发展计划(批准号: 2016YFA0301700)、国家自然科学基金(批准号: 11925402)、广东省自然科学基金(批准号: 2016ZT06D348)和深圳市科学技术创新委员会(批准号: ZDSYS20170303165926217, JCYJ20170412152620376)资助的课题
      Corresponding author: Lu Hai-Zhou, luhz@sustech.edu.cn
    • Funds: Project supported by the National Key R&D Program of China (Grant No. 2016YFA0301700), the National Natural Science Foundation of China (Grant No. 11925402), the Natural Science Foundation of Guangdong Province, China (Grant No. 2016ZT06D348), and the Science and Technology Innovation Commission of Shenzhen, China (Grant Nos. ZDSYS20170303165926217, JCYJ20170412152620376)
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  • 图 1  拓扑半金属的能带结构和贝里曲率 (a)拓扑半金属的能谱示意图, $ (k_x, k_y, k_z) $为波矢, $k_{/\!/}^2 = k_x^2+k_y^2$; (b)贝里曲率矢量场, 拓扑半金属的导带和价带在外尔点处接触, 且在该处存在一对单极子. 转载自文献 [56]

    Fig. 1.  The band structure and Berry curvature of the topological semimetal: (a) The energy spectrum of a topological semimetal,$ (k_x, k_y, k_z) $ is the wave vector, $k_{/\!/}^2 = k_x^2+k_y^2$; (b) the vector field of the Berry curvature. The conduction and valence bands of a topological semimetal touch at the Weyl nodes, and there is a pair of monopoles. Reproduced with permission from Ref. [56].

    图 2  在沿$ z $方向的磁场$ B $下, 外尔和狄拉克半金属的最小模型中沿$ k_{z} $色散的朗道能带. 转载自文献[74]

    Fig. 2.  The Landau energy bands along the $ k_{z} $ dispersion in the minimum model of Weyl and Dirac semimetals under the magnetic field $ B $ along the $ z $ direction. Reproduced with permission from Ref.[74].

    图 3  测量非线性霍尔效应的示意图. 转载自文献[82]

    Fig. 3.  Schematic of how to measure the nonlinear Hall effect. Reproduced with permission from Ref.[82].

    图 4  (a)半导体-超导体纳米线结构示意图[172-177], 两端可能存在一对马约拉纳束缚态; (b)−(d)杂化能随着磁场变化的振荡曲线. 红色和黑色曲线为实验数据[172], 蓝色为理论曲线. 转载自文献[84]

    Fig. 4.  (a) Schematic of the semiconductor-superconductor nanowire structure[172-177], its two ends may host a pair of Majorana bound states; (b)−(d) oscillation curves of hybridization energy vary with magnetic field. The red and black curves are experimental data adapted from Ref. [172]. The blue curves are the theoritical results. Reproduced with permission from Ref.[84].

    图 5  在无序(虚线)和电子-电子相互作用(波浪线)下, 计算3D外尔半金属电导率的费曼图[71,72,191,196-199], 有向直线代表格林函数. 转载自文献 [73]

    Fig. 5.  In the disorder (dashed lines) and electron-electron interaction (wavy lines), the Feynman diagram[71,72,191,196-199] of the conductivity of 3D Weyl semimetal, and the directed line represents the Green's function. Reproduced with permission from Ref [73].

    图 6  不同条件下的磁导$\delta\sigma^{\rm {qi}}(B)$对参数的依赖关系 (a) $\eta_{\rm I} = \eta_{*} = 0$ 时不同的相干长度$ l_{\phi} $; (b) $ \eta_{*} = 0 $时不同的$ \eta_{\rm I} $; (c)有限$ \eta_{*} $时不同的$ \eta_{\rm I} $; (d) $ \eta_{\rm I} $$ \eta_{*} $之间的差异, 其中$ \eta_{\rm I} $与能谷间散射相关, 而$ \eta_{*} $与能谷内散射相关. 虚线表示两个散射过程的相关性, $ \nu = \pm $是能谷指标. 转载自文献[73]

    Fig. 6.  The dependence of magnetoconductivity $ \delta\sigma^{\rm {qi}}(B) $ on parameters under different conditions: (a) Different coherence length $ l_{\phi} $ at $ \eta_{\rm I} = \eta_{*} = 0 $; (b) different $ \eta_{\rm I} $ at $ \eta_{*} = 0 $; (c) different $ \eta_{\rm I} $ at finite $ \eta_{*} $; (d) the difference between $ \eta_{\rm I} $ and $ \eta_{*} $, where $ \eta_{\rm I} $ is correlated with intervalley scattering and $ \eta_{*} $ is correlated with intravalley scattering. The dashed lines indicate the correlation between the two scattering processes. $ \nu = \pm $ is the valley index. Reproduced with permission from Ref.[73].

    图 7  3D拓扑半金属动量空间中的费米球, 其中位于原点的点表示单极子荷$ {\cal{N}} $ (a) $ P $表示从波矢$ {{k}} $到标记为(${{k}}_1, {{k}}_2, \cdots, {{k}}_n$)的中间态的背散射, $ P' $ 表示$ P $的时间反演; (b) $ P $$ P' $ 之间的相位差等效于在环路 $ {\cal{C}} = P+\bar{P} $周围累计的贝里相位. 转载自文献 [75]

    Fig. 7.  The Fermi sphere in 3D topological semimetal momentum space, where the dot at the origin represents monopole charge $ {\cal{N}} $: (a) $ P $ is the backscattering from the wave vector $ {{k}} $ to $ -{{k}} $ via intermediate states labeled as (${{k}}_1, {{k}}_2, \cdots, {{k}}_n$), $ P' $ represents the time-reversal of $ P $; (b) the phase difference between $ P $ and $ P' $ is equivalent to the Berry phase accumulated around loop $ {\cal{C}} = P+\bar{P} $. Reproduced with permission from Ref. [75].

    图 8  电导率$ \Delta\sigma $随温度$ T $变化的示意图. 选择$ c_{\rm {ee}} = c_{\rm {qi}} $, $ T_{\rm c} $是电导率随温度降低而下降的临界温度. 转载自文献 [73]

    Fig. 8.  The schematic diagram of conductivity $ \Delta\sigma $ changes with temperature $ T $. We choose $ c_{\rm {ee}} = c_{\rm {qi}} $, $ T_{\rm c} $ is the critical temperature at which the conductivity drops with temperature. Reproduced with permission from Ref. [73].

    图 9  (a)节线半金属的轮胎状费米面, 小半径$ \kappa $, 主半径$ k_0 $, 极向角$ \varphi $, 环面角$ \theta $; (b)对于短程杂质势导致在环形方向上产生从$ { k} $$ -{ k} $ 的相干背散射; (c)在长程杂质势作用下, 沿极向的$ \delta { k} $$ -\delta { k} $的散射, 此过程积累一个大小为$ \pi $的贝里相位. 转载自文献[147]

    Fig. 9.  (a) Torus-shaped Fermi surface of nodal-line semimetals, with minor radius$ \kappa $, major radius$ k_0 $, poloidal angle$ \varphi $, and toroidal angle $ \theta $; (b) a coherent backscattering from wave vector $ { k} $ to $ -{ k} $ around the toroidal direction for shortranged impurity potentials; (c) backscattering from wave vector$ \delta { k} $ to $ -\delta { k} $ along the poloidal direction under long-ranged impurity potentials.The process contributs a $ \pi $ Berry phase. Reproduced with permission from Ref.[147].

    图 10  不同相位相干长度$ l_\phi $下的磁导率, 短程极限(a)和长程极限(b)分别对应(33)式和(34)式. 转载自文献[147]

    Fig. 10.  The magnetoconductivity in the (a) short range limit Eq. (33) and (b) long range limit Eq.(34) for different phase coherence lengths $ l_\phi $. Reproduced with permission from Ref.[147].

    图 11  (a)磁导率 $ \delta\sigma\equiv\sigma(B)-\sigma(0) $与磁场$ B $的关系; (b) 电导率$ \sigma $ 与温度$ T $的关系. 转载自文献[72]

    Fig. 11.  The magnetoconductivity $ \delta\sigma\equiv\sigma(B)-\sigma(0) $; (b) conductivity $ \sigma $ vs temperature $ T $. Reproduced with permission from Ref. [72].

    图 12  理论计算的负磁阻与实验[152-154]的比较. 转载自文献[80]

    Fig. 12.  The comparison between the theoretical negative magnetoresistance and the experiments[152-154]. Reproduced with permission from Ref. [80].

    图 13  对于(1)式描述的外尔半金属, 数值(散点)和解析(实线)得到的频率$ F $的曲线 (a)固定$ M $对应不同的$ A $; (b)固定$ A $对应不同的$ M $. (c)固定$ E_M $ 不同的$ E_A $ 对应的相移$ \phi $的曲线. 曲线断裂是因为在拍频模式出现时, F$ \phi $无法拟合. 垂直虚线表示栗弗席兹点. 转载自文献[77]

    Fig. 13.  For the Weyl semimetal described in Eq. (1), the frequency $ F $ obtained by numerical (scatters) and analytical (solid curves): (a) Fixed $ M $ corresponds to different $ A $; (b) fixed $ A $ corresponds to different $ M $. (c) Fixed $ E_M $, for different $ E_A $ corresponds to the curve of phase shift $ \phi $. The curve breaks because $ F $ and $ \phi $ can not fit when beating patterns occur. The vertical dashed lines represents the Lifshitz point. Reproduced with permission from Ref.[77].

    图 14  (a) (17)式中节线半金属的节线(虚线环), 轮胎状和鼓形费米面, $ E_{\rm F} $是费米能, $ u $ 是模型参数; (b)轮胎状费米面在节线平面内的最大($ \alpha $)和最小($ \beta $)截面; (c)轮胎状费米面在节线平面外的最大($ \gamma $)和最小($ \delta $)截面. 转载自文献[79]

    Fig. 14.  (a) In the model of nodal-line semimetal Eq.(17), the nodal line (dashed ring), torus and drum Fermi surface, $ E_{\rm F} $ is Fermi energy, $ u $ is model parameter; (b) the maximum ($ \alpha $) and minimum ($ \beta $) cross sections of the torus Fermi surface; (c) the maximum ($ \gamma $) and minimum ($ \delta $) cross sections of the Fermi surface outside the nodal-line plane. Reproduced with permission from Ref. [79].

    图 15  (a)外尔半金属中的费米弧和体态的色散, $ k_{/\!/} $表示$ (k_x, k_y) $; (b)在$k_z\text-k_x$平面上$ y = L/2 $, $ E_{\rm F} = E_w $处的费米弧; (c)宽度为$ W $, 厚度为$ L $的外尔半金属板; (d)在$E_{\rm F} = E_w$处的费米弧(实线); (e)−(g)波函数在$ k_z = 0 $处沿$ y $轴的分布; (h) 3D量子霍尔效应中的朗道能级和边缘态; (i)单一表面的电子无法被y方向的磁场$ B $驱动完成一个完整的回旋运动. 转载自文献[78,283]

    Fig. 15.  (a) The energy dispersions of the Fermi arc and bulk states in a Weyl semimetal, $ k_{/\!/} $ stands for $ (k_x, k_y) $; (b) the Fermi arc at $ y = L/2 $ and $ E_{\rm F} = E_w $ in the $ k_z\text-k_x $ plane; (c) a Weyl semimetal slab with width $ W $ and thickness $ L $; (d) Fermi arc (solid) at $ E_{\rm F} = E_w $; (e)−(g) the distribution of wave function along $ y $-axis at $ k_z = 0 $; (h) the Landau levels and edge states in the 3D quantum Hall effect; (i) an electron in single surface could not be driven in a y-direction magnetic field $ B $ to perform a complete cyclotron motion. Reproduced with permission from Refs. [78,283].

    图 16  左图: 2D电子气在磁场中形成量子霍尔态. 中间图: 3D时朗道能级变为一系列2D的朗道能带. 右图:电荷密度波使朗道能带打开能隙, 使体态绝缘, 可以观察到3D量子霍尔效应. 转载自文献[87]

    Fig. 16.  Left: the quantum Hall state in 2D electron gas under magnetic field. Center: in 3D, the Landau levels turn to one dimensional Landau bands. Right: the charge density wave gap the Landau band, so that the bulk is insulating and the 3D quantum Hall effect can be observed. Reproduced with permission from Ref. [87].

    图 17  不同的势范围下, 外尔半金属在$ \hat{{z}} $方向磁场B中的纵向电导率$ \sigma_{zz} $和横向电导率$ \sigma_{xx} $. 转载自文献[76]

    Fig. 17.  The longitudinal conductivity $ \sigma_{zz} $ and transverse conductivity $ \sigma_{xx} $ of the Weyl semimetal in the $ \hat{{z}} $-direction magnetic field B under the different potential ranges. Reproduced with permission from Ref. [76].

    图 18  (a) 3D拓扑绝缘体的零场能谱($ k_x = k_y = 0 $); (b)在强磁场中, 费米能只穿过$ 0+ $朗道能带; (c)实验测得Pb1–xSnxSe的磁阻[226]; (d)理论计算出的磁阻. 转载自文献[81]

    Fig. 18.  (a) The zero field energy spectrum of 3D topological insulator ($ k_x = k_y = 0 $); (b) in a strong magnetic field, fermi energy $ E_{\rm F} $ can only crosses the $ 0+ $ Landau energy band; (c) the magnetoresistance of Pb1–xSnxSe in experiment[226]; (d) the theoretical calculated magnetoresistance. Reproduced with permission from Ref. [81].

    图 19  外尔半金属在垂直$ y $方向的磁场$ {{B}} $中的朗道能带, 其中$ k_{/\!/}\equiv k_x\sin\phi+k_z\cos\phi $为平行于$ {{B}} $的波矢, $\tan\phi = B_x/B_z$. 红色曲线是第0个朗道能带, 虚线是费米能. 转载自文献[70]

    Fig. 19.  The Landau energy band of Weyl semimetal in the magnetic field $ {{B}} $ perpendicular to the $ y $ direction, where $ k_{/\!/}\equiv k_x\sin\phi+k_z\cos\phi $ is the wave vector parallel to $ {{B}} $, $ \tan\phi = B_x/B_z $. The red curve is the $ 0 $th Landau energy band, and the dashed line represents the Fermi energy. Reproduced with permission from Ref. [70].

    图 20  实验与理论的比较 (a) $M_{/\!/}$(黑线); (b)$ M_{\rm T} $(黑线). 转载自文献[85]

    Fig. 20.  Comparison between experiments and theory for the $M_{/\!/}$ and $ M_{\rm T} $ of TaAs: (a) $M_{/\!/}$(black line); (b) $ M_{\rm T} $(black line). Reproduced with permission from Ref. [85].

    图 21  零磁场下ZrTe5的电阻率$ \rho(T) $(黑色)和塞贝克系数$ –S_{x x}(T) $(蓝色)的温度依赖曲线. 插图:实验测量示意图, $ B $是磁场, $ \nabla T $是温度梯度, $ a $, $ b $$ c $是晶轴. 转载自文献[86]

    Fig. 21.  Temperature dependence of the electrical resistivity $ \rho(T) $ (black) and Seebeck coefficient $ –S_{x x}(T) $ (blue) of ZrTe$ _5 $ at zero magnetic field. Inset: the measurement setup. $ B $ is the magnetic field and $ \nabla T $ is the temperature gradient. $ a, b$ and $c$ are crystallographic axes. Reproduced with permission from Ref. [86].

    图 22  (a)不同温度下对$ –S_{x x} $的强场测量; (b)不同温度下对 $ S_{xy} $的强场测量. 临界场$ B^{*} $附近$ S_{xy} $改变符号, $ –S_{x x} $收敛到零. 转载自文献[86]

    Fig. 22.  (a) High-field measurements of $ –S_{x x} $ at several temperatures; (b) high-field measurements of $ S_{xy} $ at several temperatures. Near the critical field $ B^{*} $, $ S_{xy} $ changes its sign and $ –S_{x x} $ converges to zero. Reproduced with permission from Ref. [86].

    表 1  对称类(正交、辛和幺正)[194]与弱局域化(WL)和弱反局域化(WAL)之间的关系[195]. 转载自文献 [88]

    Table 1.  The relation between the symmetry classes (orthogonal, symplectic and unitary) [194] and weak localization (WL) and anti-localization (WAL) [195]. Reproduced with permission from Ref. [88].

    正交 幺正
    时间反演 ×
    自旋旋转 × ×
    WL/WAL WL WAL ×
    下载: 导出CSV

    表 2  对于具有不同色散和维度的系统, (48)式中的相移$\phi$. $B_z$$B_{/\!/}$是节线平面内外的磁场. $\alpha$, $\beta$, $\gamma$, δ 对应于图14中费米面的截面. 转载自文献[79]

    Table 2.  For systems with different dispersion and dimensions, the phase shift ϕ in Eq. (48). $B_z$ and $B_{/\!/}$ are magnetic fields outside and inside the nodal-line plane. $\alpha$, $\beta$, $\gamma$, and $\delta$ correspond to the cross sections of Fermi surface in Fig. 14. Reproduced with permission from Ref. [79].

    系统 电子载流子 空穴载流子
    2D抛物线 –1/2 1/2
    3D抛物线 –5/8 5/8
    2D线性 0 0
    3D线性 –1/8 1/8
    磁场$B_z$中的节线 $-5/8(\alpha), 5/8(\beta)$ $5/8(\alpha), -5/8(\beta)$
    磁场$B_{/\!/}$中的节线 $-5/8(\gamma), 1/8(\delta)$ $5/8(\gamma), -1/8(\delta)$
    下载: 导出CSV

    表 3  从Cd3As2的实验中得到的相移$\phi_{\exp}$. 转载自文献[77]

    Table 3.  The phase shift $\phi_{\exp}$ obtained from the experiment of Cd3As2. Reproduced with permission from Ref. [77].

    文献 $\phi_{\rm{exp}}$ $\phi_{\rm{Weyl}}$ $\phi_{\rm{Dirac}}$
    [50] 0.06 — 0.08 –0.94 — –0.92 –5/8
    [51] 0.11 — 0.38 –0.89 — –0.62 –5/8
    [54] 0.04 –0.96 –5/8
    下载: 导出CSV

    表 4  节线半金属的相移$\phi$. $\alpha, \beta, \gamma, \delta$图14中的极值截面. 转载自文献[79]

    Table 4.  The phase shift $\phi$ of the nodal-line semimetal. $\alpha, \beta, \gamma, \delta$ are the extremal cross sections in Fig. 14. Reproduced with permission from Ref. [79].

    贝里相位 最大/
    最小
    电子 空穴
    $\alpha$ 0 最大 $ -1/2+0-1/8 = -5/8 $ +5/8
    $\beta$ 0 最小 $-1/2+0+1/8 = -3/8 \leftrightarrow 5/8$ –5/8
    $\gamma$ 0 最大 $ -1/2+0 - 1/8 = - 5/8 $ +5/8
    δ π 最小 $-1/2+\pi/2\pi+1/8 = 1/8$ –1/8
    下载: 导出CSV
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出版历程
  • 收稿日期:  2020-06-15
  • 修回日期:  2020-07-29
  • 上网日期:  2021-01-15
  • 刊出日期:  2021-01-20

磁场中拓扑物态的量子输运

  • 1. 南方科技大学物理系, 深圳量子科学与工程研究院, 深圳 518055
  • 2. 深圳市量子科学与工程重点实验室, 深圳 518055
  • 通信作者: 卢海舟, luhz@sustech.edu.cn
    作者简介:
    卢海舟, 2007年在清华大学高等研究院获博士学位. 同年赴香港大学做博士后研究, 2012年晋升为研究助理教授. 2015年加入南方科技大学, 现为物理系和深圳量子科学与工程研究院教授. 主要从事凝聚态物理, 特别是量子输运理论的研究. 研究兴趣集中在利用量子场论方法研究拓扑和新奇材料中的量子态和量子输运. 系统地研究了二维和三维狄拉克电子的量子输运行为, 并应用于拓扑绝缘体/半金属/超导体, 二维层状材料等. 研究过的物理效应包括弱局域化、奇异负磁阻、量子振荡、强磁场量子极限、各类霍尔效应等, 在拓扑半金属中提出了三维量子霍尔效应的一种基于费米弧的新机制, 多个理论工作被实验广泛支持和应用
    基金项目: 国家重点基础研究发展计划(批准号: 2016YFA0301700)、国家自然科学基金(批准号: 11925402)、广东省自然科学基金(批准号: 2016ZT06D348)和深圳市科学技术创新委员会(批准号: ZDSYS20170303165926217, JCYJ20170412152620376)资助的课题

摘要: 拓扑物态包括拓扑绝缘体、拓扑半金属以及拓扑超导体. 拓扑物态奇异的能带结构以及受拓扑保护的新奇表面态, 使其具有了独特的输运性质. 拓扑半金属作为物质的一种三维拓扑态具有无能隙的准粒子激发, 根据导带和价带的接触类型分为外尔半金属、狄拉克半金属和节线半金属. 本文以拓扑半金属为主回顾了在磁场下拓扑物态中量子输运的最新工作, 在不同的磁场范围内分别给出了描述拓扑物态输运行为的主要理论.

English Abstract

    • 拓扑半金属的导带和价带在节点(节线)处接 触[1,2], 节点处低能激发的准粒子(外尔/狄拉克费米子)具有线性的能带色散关系, 非常类似于三维(3D)的石墨烯模型. 大量的理论[3-13] 和实验观测[14-25] 均给出了相关材料存在的证据.

      拓扑物态由于具有独特的能带结构和表面态行为导致了许多新颖的输运现象[26-68]. 本文基于在Front. Phys.发表的两篇论文[69,70], 以拓扑半金属为主回顾了有关拓扑物态中量子输运的最新工作[71-87], 相关的综述文章也可以参考[88-93]. 依据磁场强度的大小本文的结构安排如下: 第2节, 介绍用于描述外尔、狄拉克、节线半金属及拓扑绝缘体的相关模型; 第3节, 介绍非线性霍尔效应的最新进展, 并给出半导体-超导体纳米线中马约拉纳振荡的理论描述; 第4节, 总结拓扑半金属和拓扑绝缘体中的弱局域化和弱反局域化理论; 第5节, 给出贝里曲率与负磁阻之间的关系, 介绍平面霍尔效应的最新工作; 第6节, 讨论拓扑半金属中量子振荡的相移规则, 给出两种实现3D量子霍尔效应的新机制; 第7节, 总结量子极限下的输运理论, 包括背散射禁止导致的电阻下降, 外尔费米子的湮灭, 外尔半金属中的不饱和磁化现象, 以及反常热电行为; 第8节给出评论和展望.

    • 外尔半金属的最小模型为[94]

      $ H = A(k_x{{\sigma}}_x+k_y{{\sigma}}_y)+M ( k_w^2-{{k}}^2){{\sigma}}_z, $

      其中$ {{\sigma}}_{x, y, z} $是泡利矩阵, $ {{k}} = (k_{x}, k_{y}, k_{z}) $是波矢, $ A $, $ M $, $ k_w $是模型参数. 两个能带在$ (0, 0, \pm k_w) $处相交(图1).

      图  1  拓扑半金属的能带结构和贝里曲率 (a)拓扑半金属的能谱示意图, $ (k_x, k_y, k_z) $为波矢, $k_{/\!/}^2 = k_x^2+k_y^2$; (b)贝里曲率矢量场, 拓扑半金属的导带和价带在外尔点处接触, 且在该处存在一对单极子. 转载自文献 [56]

      Figure 1.  The band structure and Berry curvature of the topological semimetal: (a) The energy spectrum of a topological semimetal,$ (k_x, k_y, k_z) $ is the wave vector, $k_{/\!/}^2 = k_x^2+k_y^2$; (b) the vector field of the Berry curvature. The conduction and valence bands of a topological semimetal touch at the Weyl nodes, and there is a pair of monopoles. Reproduced with permission from Ref. [56].

      在两个节点$ (0, 0, \pm k_w) $附近, $ H $约化为两个独立的局部模型

      $ H_{\pm} = {{M}}_{\pm}\cdot{{\sigma}}, $

      其中 $ {{M}}_{\pm} = \left(A\widetilde{k}_{x}, A\widetilde{k}_{y}, \mp2 M k_w\widetilde{k}_{z}\right) $, $ (\widetilde{k}_{x}, \widetilde{k}_{y}, \widetilde{k}_{z}) $ 为外尔点处的有效波矢. (2)式同时定义了具有相反手性的两种外尔费米子[94], 对于单一手性粒子, 自旋与动量严格相关. 对于无能隙的外尔费米子, 手性即螺旋度可以定义为$ h = {{p}}\cdot{{\sigma}}/p $, 其具有${\pm} 1$两个本征值.

      贝里曲率[95]$ \varOmega({{k}}) = \nabla_{{k}}\times{{A}}({{k}}) $可以给出体系的拓扑性质, 其中$ {{A}}({{k}}) = {\rm{i}}\left\langle u({{k}})\right|\nabla_{{k}}\left|u({{k}})\right\rangle $ 为贝里联络. 对于外尔半金属的+能带 $\left|u({{k}})\right\rangle = [\cos(\varTheta/2), \sin(\varTheta/2){\rm{e}}^{{\rm{i}}\varphi}]^{\rm{T}}$(其中$ \cos\varTheta\equiv{\cal{M}}_{\bf{k}}/E_{+} $, $ \tan\varphi\equiv k_{y}/k_{x} $). 对单个外尔点做曲面积分, 结果表明在$ \pm k_w $处产生了符号相反的拓扑荷$ \mp{\rm{sgn}}(M) $, 对应动量空间中的一对“磁单极子”.

      在离散情况下, 陈数定义为在任意可定向闭曲面上正方格子的贝里曲率通量之和[96]. 这里取具有轮胎面拓扑结构的连续参数空间, 原因是某些物理参数空间实际上确实具有这种拓扑结构, 且对应的陈数具有物理意义. 对于2D晶格的布里渊区, 其中晶格动量$ (k_x, k_y) $, $ (k_x+2\pi, k_y) $$(k_x, k_y+ 2\pi)$是等价的. 自然地, 连续情况下对贝里曲率通量的求和被整个参数空间$ {\cal{P}} $上对贝里曲率的面积分所代替

      $ n_{\rm c} = -\frac{1}{2\pi}\int_{{\cal{P}}}\varOmega\,{\rm{d}}k_x{\rm{d}}k_y, $

      由于其可以解释为离散陈数的连续极限, 所以(3)式具有和离散陈数相同的性质, 即连续陈数是一个规范不变的整数.

      对于外尔半金属, 若给定$ k_z $, 陈数可以表征$k_{x}\text-k_{y}$平面上的拓扑性质[97],

      $ n_{\rm{c}}(k_{z}) = -\frac{1}{2}\Big\{{\rm{sgn}}[M(k_w^2 -k_{z}^{2})]+{\rm{sgn}}(M )\Big\}, $

      $ -k_w < k_{z} < k_w $时, 陈数$n_{\rm c}(k_{z}) = -{\rm{sgn}}(M)$, 否则$n_{\rm c}(k_{z}) = 0$[4]. 根据体边对应关系, 非零陈数对应于与$ k_z $相关的一组边界态(费米弧)[4].

      为了得到费米弧表面态的解, 在$ y = 0 $处应用开放边界条件, 得到表面态的色散[76,94]

      $ E_{\rm{arc}}(k_x,k_z) = {\rm{sgn}}(M)Ak_x, $

      对应的波函数[98,99]

      $ \varPsi_{k_x,k_z}^{\rm{arc}}({{r}}) = C {\rm{e}}^{{\rm{i}}k_{x}x+{\rm{i}}k_{z}z} \left[\begin{array}{c} {\rm{sgn}}(M) \\ 1 \end{array}\right] ({\rm{e}}^{\lambda_1y}-{\rm{e}}^{\lambda_2y}), $

      C为归一化因子, $\lambda_{1, 2} \!=\! \dfrac{A}{2 |M|} \!\mp \! \sqrt{(A/2 M)^{2}-\varDelta_k}$, $ \varDelta_k = k_w^2-k_x^2-k_z^2 $. 存在费米弧时有$ \lambda_1\lambda_2 > 0 $, 因此$ \varDelta_k > 0 $, 所以费米弧表面态的解严格限制在一个圆内$ k_x^2+k_z^2 < k_w^2 $.

      外尔半金属的单节点有效模型为

      $ H = v {{k}} \cdot { \sigma}, $

      该模型等价于方程(2). 价带的旋量波函数

      $ |{ u}_+(k,\theta,\varphi)\rangle = \left[\begin{array}{c} \sin\dfrac{\theta}{2} \\ -\cos\dfrac{\theta}{2}{\rm{e}}^{{\rm{i}}\varphi} \\ \end{array} \right], $

      其中 $ \cos\theta\equiv k_z / k $, $ k = \sqrt{k_x^2+k_y^2+k_z^2} $, 得到贝里联络为$ {{A}} = (0, 0, \cos^2(\theta/2)/(k\sin\theta)) $. 由此得到贝里曲率为$ - 1/(2 k^2) \hat{e}_k $, 求得单极子荷$ {\cal{N}} = -1 $. 在另一个手性相反的外尔点处, 有效哈密顿量 $H = -v{{k}}\cdot {{\sigma}}$, 单极子荷为1. 因此, 对于双节点模型总单极子荷为零[26].

      $ z $方向的磁场中, 外尔半金属的能谱被量子化为一组沿$ k_{z} $色散的一维(1D)朗道能带(图2(a)). 考虑沿$ z $ 方向施加磁场$ {{{B}}} = (0, 0, B) $, 选择朗道规范$ {{A}} = (-yB, 0, 0) $. 得到本征能量为[74,100-102]

      图  2  在沿$ z $方向的磁场$ B $下, 外尔和狄拉克半金属的最小模型中沿$ k_{z} $色散的朗道能带. 转载自文献[74]

      Figure 2.  The Landau energy bands along the $ k_{z} $ dispersion in the minimum model of Weyl and Dirac semimetals under the magnetic field $ B $ along the $ z $ direction. Reproduced with permission from Ref.[74].

      $ \begin{split} &E_{k_{z}}^{\nu\pm} = \omega/2\pm\sqrt{{\cal{M}}_{\nu}^{2}+\nu\eta^{2}},\ \nu\geqslant 1, \\ &E_{k_{z}}^{0} = \omega/2-M_{0}+M_{1}k_{z}^{2},\ \ \nu = 0, \end{split} $

      其中 $ \omega = 2 M /l_{B}^{2} $, $ \eta = \sqrt{2}A/l_{B} $, 磁长度定义为 $l_{B} = \sqrt{\hbar/(e|B|)}$. 由于色散关系不是$ k_{x} $的显函数, 不同的$ k_{x} $ 代表了单位$x{\text{-}}y$平面内的朗道简并度$N_{\rm L} = 1/(2\pi l_B^2) = eB/h$.

    • 对于狄拉克半金属, 由于时间反演和反演对称性的保护, 狄拉克点是多重简并的[8]. 其模型可以用方程(1)中的$ H({{k}}) $ 及其时间反演 $ H^*(-{{k}}) $描述[60,61,103-105], 其中星号表示复共轭. 一个简单的模型如下[76]:

      $ { H} = A(k_{x}{{\alpha}}_{x}+k_{y}{{\alpha}}_{y})+M(k_w^{2}-k^{2}){{\beta}}, $

      其中狄拉克矩阵为$ {{\alpha}}_{x} = {{\sigma}}_{x}\otimes{{\sigma}}_{x} $, $ {{\alpha}}_{y} = {{\sigma}}_{x}\otimes{{\sigma}}_{y} $, $ { \beta} = {{\sigma}}_z\otimes{{\sigma}}_0 $, $ {{\sigma}}_0 $$ 2\times 2 $ 的单位矩阵. $ \hat{{z}} $ 方向的表面态由两个具有相反自旋和相反有效速度的分支组成. 通过幺正变换将模型写成对角形式

      $ { H} = \left[\begin{array}{cc} H({{k}}) & 0\\ 0 & H^{*}(-{{k}}) \end{array}\right], $

      狄拉克点是四重简并的[94], 对于特定的$ k_z $两个狄拉克点之间存在拓扑不变量, 根据体边对应关系, 每个陈数对应一个边界态, 两个节点之间存在关于$ k_z $的一组边界态. 这些边界态的色散构成了在费米面附近连接狄拉克点的费米弧, 这是拓扑狄拉克半金属的关键特征之一.

    • 外尔半金属中的每个外尔点都具有1或–1的单极子荷. 在双外尔半金属中, 单极子荷为2或–2 [6,106,107]. 对于外尔半金属或双外尔半金属的单一能谷, 最小模型为

      $ { H} = \left[ \begin{array}{cc} \chi v_z \hbar k_z & v_{{/\!/}}(\hbar k_{+})^{\cal{N}} \\ v_{{/\!/}} (\hbar k_{-})^{\cal{N}} & -\chi v_z \hbar k_z \\ \end{array} \right], $

      其中$ k_{\pm}\! =\! k_x\pm {\rm{i}} k_y $, $ \chi\! =\! \pm 1$是能谷指标, $ {{k}} $为外尔点处的动量. $ {\cal{N}} = 1, 2 $分别对应单和双外尔半金属. 色散关系为$ E_{{k}} = \pm\sqrt{v_z^2 \hbar^2 k_z^2+v_{{/\!/}}^2 (\hbar^2 k_x^2+\hbar^2 k_y^2)^{{\cal{N}}}} $. 在能谷$\chi = +1$处导带的本征态为

      $ |{{k}}\rangle = \left[\begin{array}{c} \cos (\theta/2)\\ \sin (\theta/2)\exp(-{\rm{i}} {\cal{N}} \varphi)\\ \end{array}\right], $

      其中$\cos \theta\equiv v_z k_z/E_{{k}}$, $ \tan \varphi\equiv k_y/k_x $. 通过在包围外尔点的任意费米球$ \varSigma $ 上对贝里曲率积分得到单极子荷

      $ \frac{1}{2\pi} \int_{\varSigma} {\rm{d}} {{S}}\cdot {{\varOmega}} = \pm {\cal{N}}, $

      等式右边 ± 对应 ± 能谷.

    • 节线半金属[108-111]的导带和价带在动量空间交叉形成一个闭合环[112-115], 并形成类似鼓形的表面态[116,117]. 当对称性破缺时, 节线半金属可能变为狄拉克半金属, 拓扑绝缘体或表面陈绝缘体[118]. 理论预测了不少存在这种奇异拓扑性质的材料[119-139], 部分已被实验观测所证实[140-146].

      节线半金属可以用一个双能带有效模型来描述[147],

      $ { H} = \hbar\lambda(k_x ^2+k_y ^2-k_0 ^2){ {\tau}}_x+\hbar v k_z { {\tau}}_y, $

      其中 $ {{k}} = (k_x, k_y, k_z) $是波矢, ${ {\tau}}_{x, y}$对应于双带赝自旋空间的泡利矩阵, 两个带在$ k_x ^2+k_y ^2 = k_0 ^2 $以及$ k_z = 0 $时相交. 如果费米能满足$ E_{\rm F}\ll\hbar \lambda k_0 ^2 $, 通过变量代换$k_x = (k_0+\kappa \cos \varphi){\rm {cos}}\theta, k_y = (k_0+\kappa {\rm {cos}} \varphi) \times{\rm {sin}} \theta,\; k_z = \kappa {\rm {\sin}} {\varphi/\alpha}$(其中$v_0 = 2\lambda k_0,\; \alpha = v/v_0$, $ v $$ v_0 $分别为沿着$ z $方向和x-y平面内的有效速度), 哈密顿量(15)式可以线性化为更为简单的形式:

      $ { H} = \hbar v_0 \kappa(\cos \varphi{ {\tau}}_x +\sin \varphi { {\tau}}_y). $

      导带的色散为$ \varepsilon_k = \hbar v_0\kappa $, $ \kappa $为轮胎状费米面的小圆半径, 费米能以下的态密度为$\rho_0 = {\cal{K}}E_{\rm F} /(\alpha \hbar^2 v_0^2)$, 其中$ {\cal{K}} = 2\pi k_0 $为节线环的周长.

      对于更一般的四带模型, 节线半金属的哈密顿量可以进一步写为[124,143]

      $ { H} \!=\! \big\{\big[\hbar^2( k_x^2+ k_y^2)/2 m-u\big]{ {\tau}} _3 \!+\! \lambda k_z { {\tau}} _1\big\}\otimes { {\sigma}} _0,$

      其中$ \lambda $, $ m $, 和$ u $ 是模型参数[124,143]. 哈密顿量的本征值为$E_{\pm} = \pm \sqrt{[{\hbar^2(k_x^2+k_y^2)}/{(2 m)}-u]^2+\lambda^2 k_z^2}$. 当$ u $为正时, 两个能带在零能处相交, 其半径为$ \sqrt{2 mu}/\hbar $. 当$ E_{\rm F} < u $ 时(对应于载流子密度较低的情况[140,141]), 费米面形成一个轮胎面, 当$ E_{\rm F} > u $时, 费米面演化成鼓状结构. 模型(17)式具有镜面反射对称性[124,143], 除此之外, 节线也可由其他对称性保护[110,113,121,123,140,141,148].

    • 3D拓扑绝缘体可以用 $ {{k}}\cdot { p} $哈密顿量来描述[94,149,150]:

      $ { H}_0 \!=\! { C}_{{k}}\!+\! \left[\!\! \begin{array}{cccc} M_{{k}} & 0 & {\rm{i}} V_n k_z & -{\rm{i}} V_{\perp} k_-\\ 0 & M_{{k}} & {\rm{i}} V_{\perp} k_+ & {\rm{i}} V_n k_z\\ -{\rm{i}} V_n k_z & -{\rm{i}} V_{\perp} k_- & -M_{{k}} & 0\\ {\rm{i}} V_{\perp} k_+ & -{\rm{i}} V_n k_z & 0 & -M_{{k}} \end{array} \!\!\right], $

      其中$ M_{{k}} = M_0+M_{\perp}(k_x^2+k_y^2)+M_z k_z^2 $, $C_{{k}} = C_0+ C_{\perp}(k_x^2+k_y^2)+C_z k_z^2$, $ C_i $, $ M_i $$ V_i $ 是模型参数. 当$ M_0 M_\perp < 0 $, $ M_0 M_z < 0 $时该模型描述3D强拓扑绝缘体[94]. 该模型能有效地描述拓扑表面态[97,98,151]并解释拓扑绝缘体中的负磁阻现象[80,152-155].

      对于本征磁性拓扑绝缘体, 例如MnBi2Te3, 具有面外反铁磁序, 需要额外引入[156]

      $ { H}_{X} = -m_{0} \sin ^{n}\left(\frac{\pi}{d} z\right) {{\sigma}}_{z} \otimes { \tau}_{0}, $

      其中$ m_0 $是层内反铁磁序的强度, $ d $是MnBi2Te3样品7元层的厚度, $ { \tau}_0 $$ 2\times2 $的单位矩阵, 这一模型可以解析求解, 为理解其他的本征磁性拓扑物态提供理论基础.

    • 霍尔效应由于本质上与对称和拓扑的深刻联系 [95,157]已然成为凝聚态物理的基本研究范式. 所有已知的可测霍尔效应都需要磁场或磁性掺杂来破缺时间反演对称性[157-159]. 最近, 通过测量霍尔电导的零频或倍频分量对低频驱动电场的响应(图3)发现了一种新的霍尔效应[160], 即非线性霍尔效应, 其特别之处在于不需要破缺时间反演对称性, 而是破缺反演对称性. 这一新现象的发现为探索新奇物态的对称性和拓扑性质开辟了新的途径. 在文献[82]中, 我们研究了非线性霍尔效应和二维(2D)系统能带特征之间的关系. 由于非线性霍尔响应与贝里偶极子相关[160], 通过分析倾斜有质量狄拉克费米子的一般模型, 我们发现倾斜能带在反交叉和能带反转附近存在非常强的贝里偶极子, 可以给出明显的非线性霍尔信号.

      图  3  测量非线性霍尔效应的示意图. 转载自文献[82]

      Figure 3.  Schematic of how to measure the nonlinear Hall effect. Reproduced with permission from Ref.[82].

      当施加一个振荡电场$ {{E}}(t) = {\rm{Re}}\{{\cal{E}}{\rm{e}}^{{\rm{i}}\omega t}\} $时, 响应电场的电流可以分解为直流分量和倍频分量$ J_a = {\rm{Re}}\{J_a^{(0)}+J_a^{(2)}{\rm e}^{{\rm i}2\omega t}\} $, 这里$ J_a^{(0)} = \chi _{abc}^{(0)}\varepsilon_b\varepsilon_c^* $$ J_a^{(2)} = \chi _{abc}^{(2)}\varepsilon_b\varepsilon_c $. 对于时间反演对称的系统, 非线性霍尔系数为[160]

      $ \chi_{a b c}^{(0)} = \chi_{a b c}^{(2)} = \varepsilon^{a c d} D_{b d} e^{3} \tau /\left[2 \hbar^{2}(1+{\rm{i}} \omega \tau)\right], $

      其中$ e $为电子电量, $ \tau $为动量弛豫时间, 在一般实验条件下$ \omega\tau\ll 1 $可忽略. 贝里偶极子定义为

      $ D_{b d} = -\sum\limits_{i} \int \frac{{\rm d}^{d} k}{(2 \pi)^{d}} \frac{\partial \epsilon_{{k}}^{i}}{\partial k_{b}} \varOmega_{i {{k}}}^{d} \frac{\partial f_{\rm {eq}}}{\partial \epsilon_{{k}}^{i}}, $

      其中${ \varepsilon}{abc}$为全反对称张量, $ a, b, c, d\in\{x, y, z\} $, $ f_{\rm {eq}} $为费米分布函数, $ \varOmega_{i{{k}}}^a $为贝里曲率. 注意在时间反演下$\varOmega_{i}^{a}(\!-\!{{k}}) \!=\! \!-\!\varOmega_{i}^{a}({{k}})$ 以及$\partial \epsilon_{{k}}^{i} / \partial\left(-k_{b}\right) \!=\! \!-\!\partial \epsilon_{{k}}^{i} / \partial k_{b}$, 即(21)式在时间反演对称的体系中不为0.

      由(21)式知, 为了给出不为0的贝里偶极子, 需要各向异性的能带结构, 通过在2D有质量狄拉克模型[94]中引入倾斜项来实现这一点,

      $ { H}_{d} \!=\! t k_{x} \!+\! v(k_{y} { \sigma}_{x}+\eta k_{x} {\sigma}_{y}) \!+\! (m / 2-\alpha k^{2}) {\sigma}_{z}, $

      其中$ (k_x, k_y) $为波矢, ${ \sigma}_{x, y, z}$为泡利矩阵, $ \eta = \pm 1 $, $ m $是能隙, $ t $使狄拉克锥沿$ x $方向倾斜即破缺了反演对称性. 如果没有倾斜($ t = 0 $), 贝里曲率将对称地集中在能带边缘, 因此没有贝里偶极子. 当狄拉克锥倾斜时, 贝里曲率不再对称分布, 贝里偶极子随着倾斜的增加而增加, 并在狄拉克锥完全倾斜$ (t > v) $的临界点$ (t = v) $附近达到最大值. 利用这一模型我们计算了双层 WTe2中的非线性霍尔响应[82], 给出的结果与最近实验[161,162]中的非线性霍尔响应及其角度依赖性高度一致, 这将启发更多关于非线性霍尔效应的相关实验和理论.

      除了由倾斜能带引起的贝里偶极子导致的本征非线性霍尔效应以外, 无序在其中也起着极其重要的作用. 在文献[83]中我们系统地计算了由无序导致的非线性霍尔效应, 并给出了霍尔电导的表达式. 不同于一般的线性霍尔效应, 由于非线性霍尔效应的机制需要费米能量切过能带, 主要的物理发生在费米面上, 这就导致即使在领头阶无序也扮演着非常重要的角色. 更重要的是, 我们提出了非线性霍尔效应的标度律, 这有助于识别不同机制的贡献, 解释未来实验中温度和厚度的依赖性. 需要指出的是, 无序是如何以一种特定的形式对非线性霍尔信号做出贡献的, 目前尚不清楚, 这也是未来研究的重点.

    • 在半导体-超导体纳米线中的马约拉纳零能模[163-168]总是成对出现, 并位于纳米线的两端. 由于在真实纳米线中的有限尺度效应, 马约拉纳束缚态的能量是杂化的. 杂化能量$ E_0 $因为是塞曼能量、化学势以及纳米线线长的函数而出现振荡[169-171], 称为马约拉纳振荡. 然而实验中观察到$ E_0 $作为磁场的函数[172-177]其振荡与理论预测的截然相反[170].

      我们通过假设沿纳米线的自旋轨道耦合具有阶梯状分布[84] (图4(a))解释了实验中的振荡模式[172-177] (图4(b)图(d)). 这种自旋轨道耦合的阶梯状分布的假设是合理的, 因为栅极施加的是非均匀的静电势, 而自旋轨道耦合又取决于垂直于纳米线的静电场[178-180]. 此外, 由于超导体和半导体之间的屏蔽效应以及功函数的失配, 超导体的存在可以极大地改变纳米线中的静电场[181]. 因此, 纳米线从覆盖有超导体的部分到没有超导体的部分(隧道势垒区), 自旋轨道耦合的非均匀性是很好的理论假设, 但这一点被大多数理论所忽略, 导致了对马约拉纳振荡的理论理解困难.

      图  4  (a)半导体-超导体纳米线结构示意图[172-177], 两端可能存在一对马约拉纳束缚态; (b)−(d)杂化能随着磁场变化的振荡曲线. 红色和黑色曲线为实验数据[172], 蓝色为理论曲线. 转载自文献[84]

      Figure 4.  (a) Schematic of the semiconductor-superconductor nanowire structure[172-177], its two ends may host a pair of Majorana bound states; (b)−(d) oscillation curves of hybridization energy vary with magnetic field. The red and black curves are experimental data adapted from Ref. [172]. The blue curves are the theoritical results. Reproduced with permission from Ref.[84].

      为了验证其中的物理图像, 使用阶梯状自旋轨道耦合进行计算. 纳米线的哈密顿量为

      $ { H} \!=\! \left[\!\frac{p_{x}^{2}}{2 m^{*}}\!-\!\mu(x)\!-\!\frac{1}{2 \hbar}{ \sigma}_{y}\left\{\alpha(x), p_{x}\right\}\!\right]\! { \tau}_{z}+V_{Z} { \sigma}_{x}+\varDelta { \tau}_{x}, $

      其中$ L, m^{*}, p_{x} = -{\rm{i}} \hbar \partial_{x}, \varDelta $以及 $ V_{Z} = g_{\rm{eff}} \mu_{\rm B} B / 2 $分别为纳米线的长度、 有效电子质量、动量算符、有效配对能以及由$ B $引起的塞曼能量. $ g_{\rm{eff}} $$ \mu_{\rm B} $为有效$ g $因子和玻尔磁子. 自旋轨道耦合具有如下形式(图1(a)中的绿色曲线):

      $ \alpha(x) = \frac{A}{2}\left[\tanh \left(\frac{x-x_{\rm L}}{\lambda_{\rm L}}\right)+\tanh \left(\frac{x_{\rm R}-x}{\lambda_{\rm R}}\right)\right]+\alpha_{0}, $

      其中$A, \alpha_0, x_{\rm {L/R}}, \lambda_{\rm {L/R}}$ 为描述自旋轨道耦合随空间分布变化的参数. 在实际实验中, 当改变栅极电压时, 参数会相互影响[181-183], 超导体的存在也会引起参数的变化[184,185]. 通过对角化晶格上的$ H $, 得到能谱和波函数, 其中最低的能量即束缚态杂化能量为$ E_0 $.

      图4(b)图4(d)中的蓝色曲线分别给出了使用三组模型参数的数值结果. 数值结果与实验保持了很高的一致性, 不仅在衰减振幅方面, 而且还包括最低能量交叉(图4(b)图4(c))、反交叉(图4(d))以及磁场中振荡周期的增加(图4(c)). 注意到我们的结果是一般性的, 不依赖于具体的参数.

      文献[84]进一步分析了这种阶梯状的自旋轨道耦合也能导致安德列夫束缚态光谱的衰减振荡. 对于由马约拉纳束缚态影响的库仑阻塞[186-190], 预测峰间距通过$ \pi/2 $的相移与峰高相关联, 这在最近的实验中是不明确的, 但依然可以用阶梯状自旋轨道耦合来解释. 期望这一结果将激发更多的工作来重新检验这种通常存在于真实实验装置中的非均匀自旋轨道耦合效应.

    • 弱(反)局域化是存在于无序金属[191]中由量子干涉导致的输运现象. 如果量子干涉的修正为正, 即产生弱反局域化现象, 如果为负则产生弱局域化现象. 无论是弱局域化还是弱反局域化本质上均取决于体系的对称性(表1). 外尔费米子单一能谷的低能描述为$H = \pm\hbar v_{\rm F}{{\sigma}}\cdot{{k}}$, 只具有时间反演对称性. 拓扑半金属中已广泛观察到弱反局域化现象[41-43,46,55,56,60,61]. 这一节介绍处理这一问题的基本方法[73]和主要结论, 最近的相关研究[192,193]利用相似的方法系统地研究了更一般的3D狄拉克材料中的弱(反)局域化行为, 给出了与实验非常符合的理论结果.

      正交 幺正
      时间反演 ×
      自旋旋转 × ×
      WL/WAL WL WAL ×

      表 1  对称类(正交、辛和幺正)[194]与弱局域化(WL)和弱反局域化(WAL)之间的关系[195]. 转载自文献 [88]

      Table 1.  The relation between the symmetry classes (orthogonal, symplectic and unitary) [194] and weak localization (WL) and anti-localization (WAL) [195]. Reproduced with permission from Ref. [88].

      图5总结了用于研究量子干涉和相互作用引起的弱局域化和反局域化的费曼图[73]. 在计算中, 对电导率有贡献的主要有三项, 包括半经典的电导率(图5(a))、量子干涉修正(图5(b))以及电子-电子相互作用修正(图5(c)).

      图  5  在无序(虚线)和电子-电子相互作用(波浪线)下, 计算3D外尔半金属电导率的费曼图[71,72,191,196-199], 有向直线代表格林函数. 转载自文献 [73]

      Figure 5.  In the disorder (dashed lines) and electron-electron interaction (wavy lines), the Feynman diagram[71,72,191,196-199] of the conductivity of 3D Weyl semimetal, and the directed line represents the Green's function. Reproduced with permission from Ref [73].

      图6给出了理论计算得到的主要结果. 可以看出, 在没有能谷间散射的情况下, 对于较大的$ l_{\phi} $(相位相干长度), 磁导$\delta\sigma^{\rm {qi}}(B)$ 为负且与$ \sqrt{B} $ 成正比, 显示了3D外尔费米子的弱反局域化特征. 这种 $ -\sqrt{B} $ 的相关性与实验非常符合[41,42]. 随着$ l_{\phi} $变短, 从$ -\sqrt{B} $$ -B^{2} $的变化是很明显的, 最终$ \delta^{qi}(B) $$ l_{\phi} $和平均自由程$ l_e $相等时消失, 因为该系统不再处于量子干涉状态而进入半经典扩散状态.

      图  6  不同条件下的磁导$\delta\sigma^{\rm {qi}}(B)$对参数的依赖关系 (a) $\eta_{\rm I} = \eta_{*} = 0$ 时不同的相干长度$ l_{\phi} $; (b) $ \eta_{*} = 0 $时不同的$ \eta_{\rm I} $; (c)有限$ \eta_{*} $时不同的$ \eta_{\rm I} $; (d) $ \eta_{\rm I} $$ \eta_{*} $之间的差异, 其中$ \eta_{\rm I} $与能谷间散射相关, 而$ \eta_{*} $与能谷内散射相关. 虚线表示两个散射过程的相关性, $ \nu = \pm $是能谷指标. 转载自文献[73]

      Figure 6.  The dependence of magnetoconductivity $ \delta\sigma^{\rm {qi}}(B) $ on parameters under different conditions: (a) Different coherence length $ l_{\phi} $ at $ \eta_{\rm I} = \eta_{*} = 0 $; (b) different $ \eta_{\rm I} $ at $ \eta_{*} = 0 $; (c) different $ \eta_{\rm I} $ at finite $ \eta_{*} $; (d) the difference between $ \eta_{\rm I} $ and $ \eta_{*} $, where $ \eta_{\rm I} $ is correlated with intervalley scattering and $ \eta_{*} $ is correlated with intravalley scattering. The dashed lines indicate the correlation between the two scattering processes. $ \nu = \pm $ is the valley index. Reproduced with permission from Ref.[73].

    • 对于单一能谷连接费米球上从$ {{k}} $$ -{{k}} $的背散射中间态的每条路径(在图7(a)中标记为$ P $), 均在原点处包含了单极子荷, 并且存在一个对应的时间反演$ P' $. 量子干涉是由两个时间反演路径$ P $$ P' $之间的相位差决定的, 该相位差为沿$ P $$ \bar P\equiv -P' $形成的环路累积的贝里相位[95,196,200-202], 即从$ -{{k}} $$ {{k}} $的对应路径:

      图  7  3D拓扑半金属动量空间中的费米球, 其中位于原点的点表示单极子荷$ {\cal{N}} $ (a) $ P $表示从波矢$ {{k}} $到标记为(${{k}}_1, {{k}}_2, \cdots, {{k}}_n$)的中间态的背散射, $ P' $ 表示$ P $的时间反演; (b) $ P $$ P' $ 之间的相位差等效于在环路 $ {\cal{C}} = P+\bar{P} $周围累计的贝里相位. 转载自文献 [75]

      Figure 7.  The Fermi sphere in 3D topological semimetal momentum space, where the dot at the origin represents monopole charge $ {\cal{N}} $: (a) $ P $ is the backscattering from the wave vector $ {{k}} $ to $ -{{k}} $ via intermediate states labeled as (${{k}}_1, {{k}}_2, \cdots, {{k}}_n$), $ P' $ represents the time-reversal of $ P $; (b) the phase difference between $ P $ and $ P' $ is equivalent to the Berry phase accumulated around loop $ {\cal{C}} = P+\bar{P} $. Reproduced with permission from Ref. [75].

      $ \gamma = \oint_{\cal{C}} {\rm{d}} {{l}} \cdot {{A}} = \pi {\cal{N}}. $

      值得注意的是, 该贝里相位仅取决于单极子荷, 而不是特定的环路形状[75]对于双外尔半金属, 单极子荷$ {\cal{N}} = 2 $, 贝里相位为$ 2\pi $. 在$ 2\pi $的贝里相位下, 时间反演散射态之间产生相长干涉, 从而导致弱局域化效应. 但是, 对于单外尔半金属, 单极子荷为$ {\cal{N}} = 1 $且贝里相位为$ \pi $, 导致弱反局域化效应. 这里对贝里相位的分析与前文提到的对称性分类得到的结果是一致的[203]. 由于贝里相位是单极子荷产生的贝里曲率场的直接结果, 因此我们在弱(反)局域化效应与单极子荷$ {\cal{N}} $的奇偶性之间建立了直接的联系.

      通过计算最大交叉图得到对双外尔半金属电导率的修正. 最大交叉图的核心计算可以表达为粒子-粒子关联, 称为Cooperon, 对双外尔半金属其为[75]

      $ C_{{{k}}_1,{{k}}_2}\approx \frac{\hbar }{2\pi N_{\rm F} \tau ^2}\frac{{\rm{e}}^{{\rm{i}}2(\varphi_2-\varphi_1)}}{ D_1\left(q_x^2+q_y^2\right)+D_2 q_z^2}, $

      其中$ {{q}} = {{k}}_1+{{k}}_2 $是Cooperon波矢, $ {{k}}_1 $$ {{k}}_2 $分别为初态和末态的波矢, $ \varphi_1 $$ \varphi_2 $为相应波矢的方位角, $ D_1 = 8 \tau E_{\rm F} v_{{/\!/}} /(3 \pi) $$ D_2 = \tau v_z^2 $是扩散系数, $ N_{\rm F} $是态密度, $ \tau $是动量弛豫时间.

      ${{q}} \!\to\! 0$, 即$ {{k}}_1 \!=\! -{{k}}_2 $时, Cooperon发散, 成为背散射的最主要贡献. 在此极限下, $ \varphi_2 \!=\! \varphi_1 \!+\! \pi $(其中$\hbar k_x = \sqrt{k \sin \theta }\cos \varphi,~ \hbar k_y = \sqrt{k \sin \theta }\sin \varphi,~ 2\hbar m v k_z$=$ k \cos \theta $, 并通过设置$\varphi \to \varphi\!+\! \pi$$\theta\to \pi\!-\!\theta$得到$ -{{k}} $):

      $ C_{{{k}},{ q}-{{k}}}\approx +\frac{\hbar }{2\pi N_{\rm F} \tau ^2}\frac{1}{ D_1\left(q_x^2+q_y^2\right)+D_2 q_z^2}. $

      注意(27)式前的正号, 其对应于弱局域化效应, 即产生正的磁导, 这是双外尔半金属弱局域化的另一个特征. 在低温极限下$ l_{\phi}\gg l_B \gg l_z $, 磁导$\delta \sigma^ {\rm{qi}}_{zz}(B) \propto \sqrt{B}$, 在极限$ l_B\gg l_\phi $ 以及 $ l_B\gg l_z $时, $ \delta \sigma^ {\rm{qi}}_{zz}(B) \propto B^2 $.

      根据文献[73,75]的理论结果, 我们提出了一个公式来拟合3D弱(反)局域化引起的磁导:

      $ \delta\sigma^ {\rm{qi}}_{zz} = C^ {\rm{qi}}_1\frac{B^2\sqrt{B}}{B_{\rm{c}}^2+B^2}+C^ {\rm{qi}}_2\frac{B_{\rm{c}}^2 B^2}{B_{\rm{c}}^2+B^2}, $

      其中拟合参数$ C^ {\rm{qi}}_{1} $$ C^ {\rm{qi}}_{2} $为正对应于弱局域化, 为负则对应于弱反局域化. 临界场$ B_{\rm{c}} $与相位相干长度$ l_\phi $有关($ B_{\rm{c}}\sim \hbar/(el_{\phi}^2) $), 相位相干长度与温度的关系近似为$ l_\phi \sim T^{-p/2} $, 即$ B_{\rm{c}} \sim T^p $, 其中$ p $由退相干机制(如电子-电子相互作用($ p = 3/2 $)或电子-声子相互作用($ p = 3 $))决定. 将该公式应用于TaAs的实验, 通过拟合磁导, 发现$ p\approx 1.5 $ [61].

      存在相互作用的情况下, 对于外尔费米子的一个能谷, 电导率随温度的变化可以概括为

      $ \Delta\sigma(T) = c_{\rm {ee}}T^{1/2}-c_{\rm {qi}}T^{p/2}, $

      其中$ c_{\rm {ee}} $$c_{\rm {qi}}$ 都是正的参数. (29)式描述了由相互作用引起的弱局域化和由干涉引起的弱反局域化之间的竞争(图8). 临界温度$T_{\rm{c}} \!=\! (c_{\rm {ee}}/p\cdot c_{\rm {qi}})^{2/(p\!-\!1)}$因为 $ c_{\rm {ee}}c_{\rm {qi}} \!>\! 0 $, 这意味着只要$ p \!>\! 1 $, 总有一个临界温度低于这个温度, 电导率就会随着温度的降低而下降. 对于3D中已知的退相干机制, $ p $ 总是大于1[191]. 通过一组典型参数的拟合, 发现$T_{\rm{c}}\!\approx\! 0.4— 10^{6}$ K[73].

      图  8  电导率$ \Delta\sigma $随温度$ T $变化的示意图. 选择$ c_{\rm {ee}} = c_{\rm {qi}} $, $ T_{\rm c} $是电导率随温度降低而下降的临界温度. 转载自文献 [73]

      Figure 8.  The schematic diagram of conductivity $ \Delta\sigma $ changes with temperature $ T $. We choose $ c_{\rm {ee}} = c_{\rm {qi}} $, $ T_{\rm c} $ is the critical temperature at which the conductivity drops with temperature. Reproduced with permission from Ref. [73].

      能谷间的散射和关联也会导致弱局域化, 如图6(b)所示, 随着$ \eta_{\rm I} $的增加, 负的$ \delta\sigma^{\rm {qi}} $被抑制. 此外, 图6(c)显示, 当$ \eta_{\rm I}+\eta_{*} = 3/2 $时, 磁导可以变为正, 即局域化趋势. 因此, 强的能谷间散射和关联的结合将增强无序外尔半金属的局域化倾向并可能导致金属-绝缘体相变[39].

    • 模型(15)具有两个基本的对称性[147], 即${\cal{T}}_1 H({{k}}){\cal{T}}_1^{-1} \!=\! H(\!-{{k}})$${\cal{T}}_2 H(\delta{{k}}){\cal{T}}_2^{-1} \!=\! H(\!-\delta{{k}})$, 其中$ {\cal{T}}_1 = K $, $ {\cal{T}}_2 = {\rm{i}}\tau_y K $为时间反演, $ k $是复共轭算符, $ \delta {{k}} = \kappa (\cos \varphi, \sin \varphi/\alpha) $. 由表1可知, $ {\cal{T}}_1 $$ {\cal{T}}_2 $分别对应于正交和辛对称类, 分别导致弱局域化和弱反局域化(图9(b)图9(c)). 简单起见, 考虑低能激发即$ E_{\rm F} $相对较小, 费米面的主半径满足$k_0 \gg 1/r_{\rm{sc}}\gg \kappa$, $ r_{\rm{sc}} $为散射势的范围. 此时不同的散射态$ {{k}} $之间可以用一个单一的参数$ \theta $来描述, 即$u_{{{k}}} = u(\theta) \!=\! u_0 f_{\varDelta}(\theta)$, 其中$f_{\varDelta}(x) = \varTheta(x+ \varDelta)\varTheta(-x+ \varDelta)$, $ \varTheta(x) $为阶跃函数. 当$ \varDelta = \pi $ 时为短程散射极限, $ \varDelta\rightarrow 0 $时为长程散射极限.

      图  9  (a)节线半金属的轮胎状费米面, 小半径$ \kappa $, 主半径$ k_0 $, 极向角$ \varphi $, 环面角$ \theta $; (b)对于短程杂质势导致在环形方向上产生从$ { k} $$ -{ k} $ 的相干背散射; (c)在长程杂质势作用下, 沿极向的$ \delta { k} $$ -\delta { k} $的散射, 此过程积累一个大小为$ \pi $的贝里相位. 转载自文献[147]

      Figure 9.  (a) Torus-shaped Fermi surface of nodal-line semimetals, with minor radius$ \kappa $, major radius$ k_0 $, poloidal angle$ \varphi $, and toroidal angle $ \theta $; (b) a coherent backscattering from wave vector $ { k} $ to $ -{ k} $ around the toroidal direction for shortranged impurity potentials; (c) backscattering from wave vector$ \delta { k} $ to $ -\delta { k} $ along the poloidal direction under long-ranged impurity potentials.The process contributs a $ \pi $ Berry phase. Reproduced with permission from Ref.[147].

      与第4.1节类似, 在文献[147]中我们利用费曼图方法计算了电导率的量子干涉修正, 计算中主要包括了三个领头阶的图[195,196]. 对于零温电导率, 总的量子修正为

      $ \sigma_z = s\eta_z^2 \frac{e^2 \hbar}{4\pi V^2}\sum\limits_{k,k'} v_{k}^z v_{k'}^z G_{k}^{\rm R} G_{k}^{\rm A} G_{k'}^{\rm R} G_{k'}^{\rm A} C_{k,k'}, $

      其中$ s = 2 $为自旋简并度, $ \eta_z = 2 $是对速度$ v_{k}^z $的顶点修正. 推迟(R)和超前(A)格林函数利用一阶波恩近似求解得到, 即$G_{k}^{{\rm R}, {\rm A}}(\omega) = 1/[\omega -\varepsilon_{k} \pm {\rm{i}}\hbar/(2\tau_{\rm{e}})]$, $ \tau_{\rm{e}} = 2\hbar /(n_i \rho_0 \tilde{u}^2) $ 是弹性散射的弛豫时间, 其中$\tilde{u}^2 \!=\! \displaystyle \int_0^{2\pi}\, {\rm{d}}\theta |u(\theta)^2 |$, $ C_{k, k'} $是最大交叉图的Cooperon[204]. 在短程散射极限下$ \varDelta = \pi $, 最主要的散射路径为$(k _1, k_2, \cdots, k_n)$及其时间反演$(-k_n, \cdots, -k_2, -k_1)$(图9(b)), 需要指出的是, 这些路径没有包围节线, 即不会积累贝里相位. 直流极限下($ \omega\rightarrow 0 $)[147]

      $ C_{k,k'} = \frac{\gamma}{2\tau_{\rm{e}}}\frac{1}{D_{xy}Q_{xy}^2+D_z Q_z^2}, $

      其中$Q_{xy} \!=\! \sqrt{Q_x^2+Q_y^2}$, $D_{xy} \!=\! v_0^2\tau_{\rm{e}}/4$$D_z \!=\! \alpha^2 v_0^2\tau_e$分别是x-y平面和$ z $方向的扩散系数. (31)式说明节线半金属在短程极限下存在类似于普通各向异性金属中的3D弱局域化行为.

      长程极限时$ \varDelta\rightarrow 0 $, 量子干涉修正主要由路径$(\delta k_1, \delta k_2, \cdots, \delta k_n)$$(-\delta k_n, \cdots, -\delta k_2, -\delta k_1)$所贡献(图9(c)), 此路径由于包围节线会积累一个大小为$ \pi $的贝里相位. 类似地可以得到[147]

      $ C_{\delta k,\delta k'} = \frac{\gamma}{2\tau_{\rm{e}}}\frac{f_{\varDelta}(\theta-\theta'){\rm{e}}^{-{\rm{i}}(\varphi-\varphi')}}{4D_{xy}q_{xy}^2\cos^2(\theta_q-\theta)+D_zq_z^2}, $

      其中${{q}} \!=\! \delta {{k}}-\delta {{k}}'$, $q_{xy} \!=\! \sqrt{q_x^2+q_y^2}$. (32)式包括2个$ \theta $依赖的因子, $ \varDelta\rightarrow 0 $$ f_{\varDelta}(\theta-\theta') $项确保$ \theta = \theta' $, 同时发散项$ \cos^2(\theta_q-\theta) $$ \theta_q-\theta = \pi/2 $ 时为0, 即散射只能发生在同一极向的平面上. 可见在长程极限下节线半金属表现出2D的量子扩散行为.

      引入磁场后, 退相干机制抑制了无序诱导的电导率量子修正[204]. 在短程极限下得到磁导率为

      $ \begin{split} \sigma_z^{\rm S}(B) =\;& -\!\frac{s\eta_z^2\alpha e^2}{(2\pi)^2h}\!\left[\!\Psi\left(l_B^2/l_e^2+\frac{1}{2}\right)\bigg/l_e\right.\\ &-\Psi\left(l_B^2/l_{\phi}^2+\frac{1}{2}\right)\bigg/l_{\phi}\\ &\left.-\int_{1/l_{\phi}}^{1/l_e}\,{\rm d}x \Psi\left(l_B^2x^2+\frac{1}{2}\right) \right], \end{split}$

      类似地在长程极限下

      $ \begin{split} \sigma_z^{\rm L}(B) =\;& \frac{s\eta_z^2{\cal{K}}\alpha e^2}{16\pi^2h}\int_0^{2\pi}\,\frac{{\rm{d}}\theta}{2\pi}\left[ \Psi\left(\frac{l_B^2}{l_e^2\alpha\mid \sin\theta\mid}+\frac{1}{2}\right)\right.\\ &\left.-\Psi\left(\frac{l_B^2}{l_{\phi}^2\alpha\mid \sin\theta\mid}+\frac{1}{2}\right) \right], \\[-15pt] \end{split}$

      其中$ \Psi(x) $为双伽马函数. 图10给出了(33)式和(34)式对磁导率的理论预测. 短程极限下弱局域化效应导致正的磁导(图10(a)), 而在长程极限下的弱反局域化效应导致了负的磁导(图10(b)). 实验中可以通过对温度的调节来控制相位相干长度$ l_\phi $的大小以匹配不同的曲线, 在低温及$ B\sim0.1 $—1.0 T时, 由于$ l_\phi\!\sim\! 100 $ nm—1 μm, $ l_B \!\sim\! 10 $ nm, 即$l_{\phi} \!\gg\! l_B$. 通常, 与3D扩散相比, 2D扩散会导致更大的弱局域化或弱反局域化效应. 此外, 节线半金属的3D扩散行为可以视为大量的2D有效子系统, 导致磁导率的显著提高[205]. 实际上, 在图10中可以观察到, $ \sigma_z^{\rm L}(B) $$ \sigma_z^S({ B}) $大3个数量级, 这表明弱反局域化效应在节线半金属的长程散射极限下是个非常强的特征.

      图  10  不同相位相干长度$ l_\phi $下的磁导率, 短程极限(a)和长程极限(b)分别对应(33)式和(34)式. 转载自文献[147]

      Figure 10.  The magnetoconductivity in the (a) short range limit Eq. (33) and (b) long range limit Eq.(34) for different phase coherence lengths $ l_\phi $. Reproduced with permission from Ref.[147].

    • 基于费曼图方法我们计算了拓扑绝缘体磁掺杂表面态的磁导[71], 磁掺杂的引入以两种方式影响了体系的性质, 首先磁性杂质导致的平均场在狄拉克点处打开能隙, 其次平均场的局部涨落以随机方式散射表面态电子. 经过计算发现, 除了无能隙狄拉克费米子产生的弱反局域化效应, 能隙的打开导致了额外的弱局域化项, 随着能隙的进一步增加将使体系展现出完全的弱局域化.

      零温磁导公式为

      $ \begin{split} \;& \sigma(B) =\\ & \sum\limits_{i = 0,1} \!\frac{\alpha_{i} e^{2}}{\pi h}\bigg[\!\Psi\bigg(\frac{l_{B}^{2}}{l_{\phi}^{2}} \!+\! \frac{l_{B}^{2}}{l_{i}^{2}} \!+\! \frac{1}{2}\bigg)-\ln \bigg(\frac{l_{B}^{2}}{l_{\phi}^{2}} \!+\! \frac{l_{B}^{2}}{l_{i}^{2}}\bigg)\!\bigg], \\[-15pt]\end{split}$

      其中$ \Psi $是双伽马函数,

      $ \begin{split} &\alpha_{1} = -\frac{\eta_{v}^{2}\left(1+2 \eta_{H}\right)}{2\left(1+\dfrac{1}{g_{0}}+\dfrac{1}{g_{2}}\right)}, \quad\\ &l_{1}^{-2} = \frac{g_{1}}{2 l^{2} \sin ^{2} \theta\left(1 \!+\! \dfrac{1}{g_{0}} \!+\! \dfrac{1}{g_{2}}\right)}, ~~\alpha_{0} \!=\! \dfrac{\eta_{v}^{2}\left(1+2 \eta_{H}\right)}{2\left(\dfrac{1}{g_{1}}+1\right)},\\ & l_{0}^{-2} = \dfrac{g_{0}}{2 l^{2} \sin ^{2} \theta\left(\dfrac{1}{g_{1}}+1\right)},\\[-19pt] \end{split} $

      其中$1/l^2 \equiv 1/l_{\rm e}^2+1/l_{\rm m}^2$, $l_{\rm e}$是平均自由程, $l_{\rm m}$是磁性杂质的散射长度. 在没有磁性杂质的情况下$ \alpha_0 = 0, \alpha_1 = -1/2 $, 即给出弱反局域化的磁导特征. 对于有限的能隙, 由于$ \alpha_0 $$ \alpha_1 $的符号相反, 前者导致弱局域化, 后者导致弱反局域化, 两者呈现出竞争的行为.

      对非磁性掺杂的拓扑绝缘体的电子输运实验展现出非常大的矛盾[206-209], 磁导率随磁场的增加而下降, 即展现出反局域化的行为(图11(a)红色曲线). 然而, 当温度降低时, 测得的电导率呈对数下降, 显示出普通无序金属中弱局域化的典型特征(图11(b)蓝色曲线). 为了引入温度的效应, 我们考虑电子-电子相互作用对电导率的修正[72]:

      图  11  (a)磁导率 $ \delta\sigma\equiv\sigma(B)-\sigma(0) $与磁场$ B $的关系; (b) 电导率$ \sigma $ 与温度$ T $的关系. 转载自文献[72]

      Figure 11.  The magnetoconductivity $ \delta\sigma\equiv\sigma(B)-\sigma(0) $; (b) conductivity $ \sigma $ vs temperature $ T $. Reproduced with permission from Ref. [72].

      $ \sigma^{\rm {ee}} = \frac{e^{2}}{\pi h}\left(1-\eta_{\varLambda \varGamma} F\right) \ln \frac{2 l_{\rm e}^{2}}{l_{{\rm T}}^{2}}-\frac{e^{2}}{\pi h} \eta_{\varGamma} F \psi\left(\frac{1}{2}+\frac{l_{{\rm T}}^{2}}{l_{B \phi}^{2}}\right), $

      其中$ 1 / l_{\phi i}^{2} \equiv 1 / l_{\phi}^{2}+1 / l_{i}^{2} $ 以及 $1 / l_{B \phi}^{2} \equiv- \big(1 / (2 l_{B}^{2})+ 1 / l_{\phi 1}^{2}\big) / (2 \alpha_{1})$. 相位相干长度$ l_{\phi} $正比于$ T^{-p / 2} $, $ T $是温度, 参数$ p $可从实验中得到[207,209]. $l_{\rm T} \!\equiv\! \sqrt{D \hbar / (2 \pi k_{\rm B} T)}$是热扩散长度, 其中 $ k_{\rm B} $ 是玻尔兹曼常数, 扩散系数 $D = l_{\rm{e}} v \sin \theta \sqrt{\dfrac{\left(1+\cos ^{2} \theta\right)} {\left(1+3 \cos ^{2} \theta\right)}}$, $\cos \theta \equiv \dfrac{\varDelta} { 2 E_{\rm F}}$, $ v $$ \varDelta $是狄拉克模型的参数. 对于狄拉克模型, 相互作用的屏蔽因子

      $ F = \frac{2}{\pi} \frac{\arctan \sqrt{1 / x^{2}-1}}{\sqrt{1-x^{2}}}, \quad x \equiv \frac{8 \pi \varepsilon_{0} \varepsilon_{\rm r} \gamma \sin \theta}{e^{2}}, $

      $ \varepsilon_0 $是真空介电常数, $ \varepsilon_{\rm r} $是考虑晶格离子和价电子影响的相对介电常数, $ \gamma = v\hbar $. 电导率公式是温度$ T $和磁场$ B $的函数, 并且依赖于狄拉克模型参数$ \varDelta/(2 E_{\rm F}) $$ \gamma $, 以及与样品相关的参数$ l_e, l_\phi, F $. (38)式明确阐明了由相互作用主导的电导率的温度依赖性, 而磁导率主要由量子干涉贡献, 解决了拓扑绝缘体输运实验上的矛盾.

    • 在拓扑半金属中非平庸的贝里曲率可以将外部磁场与电子速度耦合, 产生随着磁场的增加而增长的额外导电性[28,29]. 由于它与成对外尔点之间的手征电荷转移相关, 负磁阻也被认为是手征反常的一个特征[26,210,211]. 许多拓扑半金属材料中都观察到了负磁阻现象[41-43,46,55-57,60,61,63,66,68]. 除了在纵向磁场下产生随磁场增大的手征电流以外, 最近的理论发现, 外尔半金属中可以存在平面霍尔效应[212,213], 即磁场平行于电场时横向有电荷积累即产生霍尔电导率$ \sigma_{yx} $. 不同于经典的霍尔效应, 平面霍尔效应不是由洛伦兹力导致的, 因此吸引了很多理论和实验的关注[214-216]. 文献[217]研究了普通外尔半金属和倾斜外尔半金属中的平面霍尔效应, 相比于前者, 倾斜外尔半金属的纵向电导率$ \sigma_{xx} $和霍尔电导率$ \sigma_{yx} $均给出了不同的磁场$ B $依赖关系.

      我们从电子的半经典运动方程开始[95,218-220]:

      $ \begin{split} &\dot{{r}} = {{v}}-\dot{{k}}\times {{\varOmega}}_{{k}}, \\& \hbar\dot{{k}} = -e{\bf{E}}-e\dot{{r}}\times {{{B}}}, \end{split} $

      其中$ {{v}} = \ \partial \epsilon_{{k}}/\hbar \partial {{k}} $是费米速度. 第一个方程中的第二项表明, 存在电场时, 电子可以获得与能带的贝里曲率成正比的反常速度. 这种反常速度是许多输运现象的原因. 迭代方程(39)得到[28]

      $\begin{split} & \dot{{r}} = D_{{{k}}}^{-1} \left[{{v}}+ \frac{e}{\hbar} {{E}}\times {{\varOmega}}_{{k}}+\frac{e}{\hbar} ({{\varOmega}}_{{k}}\cdot{{v}}){{B}} \right], \\ &\hbar\dot{{k}} = D_{{{k}}}^{-1} \left[-e{{E}}- e{{v}}\times {{B}}-\frac{e^2}{\hbar}( {{E}}\cdot {{B}}){{\varOmega}}_{{k}} \right], \end{split} $

      其中$D_{{{k}}} = 1+ (e/\hbar){{B}}\cdot {{\varOmega}}_{{k}}$是对相空间体积的修正项, 包含$ {{\varOmega}} $ 的各项分别导致了反常霍尔效应[221-223]、手征磁效应[36]、以及负磁阻[28,29].

      由于电导率是流-流关联(图5(a)), 速度与$ B $的线性相关性导致电导率的$ B^2 $相关性. 在2.1节, 已经证明贝里曲率与$ 1/k^2 $成正比. 考虑到电导率公式的3D积分中存在$ \varOmega^2 $以及$ k^2 $, 则反常电导率部分应与费米波矢成反比, 与$ B^2 $成正比, 即

      $ \delta \sigma (B) \propto {B^2}/{k_{\rm F}^2}, $

      (41)式与文献[28,29]中得到的公式是一致的. 电导率随着$ B^2 $的增加而增加, 产生负磁阻. 由于非平庸的贝里曲率在外尔点处发散, 电导率随费米波矢和载流子密度的减小而增大. 3D时, 载流子密度$ n $$ k_{\rm F}^3 $成正比, 即

      $ \delta \sigma (B) \propto {B^2}/{n^{2/3}}. $

      因此, 为了验证非平庸贝里曲率的负磁阻特性, 有必要对三个性质进行检验, 即角度依赖性、$ B^2 $的磁场依赖性以及$ n^{-2/3} $的载流子密度依赖性. 在实验[61]中, 通过比较不同样品的结果检验了磁阻对载流子密度的依赖性.

      在半经典区域, 电流密度矢量为[95]

      $ {{J}} = -e\int \frac{{\rm{d}}{{k}}}{(2\pi)^3}D_k^{-1}\dot{{r}}f_k, $

      其中$ \dot{{r}} $为半经典波包的群速度, $ f_k $是电子占据$ {{k}} $态的非平衡分布函数, 载流子的输运性质主要由$ \dot{{r}} $$ f_k({{r}}) $所决定. (40)式第一个方程括号内的第三项是与磁场有关的反常速度, 一般来说, 它可以分解成与电场平行和垂直的两个分量. 反常速度项包含$ {{\varOmega}}_{\bf{k}}\cdot{{v}} $, 这表明对于一个外尔锥, 它是沿着磁场方向的, 而对于手性相反的锥, 则相反.

      利用弛豫时间近似$\dot{{k}}\cdot \partial_{{{k}}}f_k({{r}}) \!=\! \dfrac{ f_{\rm{eq}}({{r}}) \!-\! f_k({{r}})}{\tau({{k}})}$, 其中$ f_{\rm{eq}} $为平衡态分布函数, 并由玻尔兹曼方程得到非平衡分布函数为[217]

      $ f_k = f_{\rm{eq}}+\left[eD\tau {{E}}\cdot {{v}}+\frac{e^2}{\hbar}D\tau({{B}}\cdot {{E}})({{v}}\cdot{{\varOmega}})\right]\frac{\partial f_{\rm{eq}}}{\partial \epsilon}, $

      括号内第一项表示电场$ {{E}} $沿着$ {{v}} $的分量引起的载流子分布偏离, 第二项表示手征化学势导致的偏离.

      对于由最小有效模型(7)描述的外尔半金属, 由于手征化学势的影响, 电子通过电场和磁场的共同作用从一个外尔锥转移到另一个手性相反的锥. 如(44)式所示, 偏离平衡分布的这部分与$ {{B}}\cdot {{E}} $成正比, 即当$ {{B}} $$ {{E}} $不垂直时, 与磁场的大小成正比. 在没有倾斜的情况下, 外尔锥在空间上是均匀的, 并且速度$ {{v}} $的大小在费米面上没有变化, 因此$ {{v}} $在单个外尔锥的贡献为零. 而且由于电子分布的偏离, 与磁场$ {{B}} $ 成正比的反常速度项 $ ({{\varOmega}}_{{k}}\cdot{{v}}){{B}} $对两个外尔锥的影响不能相互抵消. 手征化学势和反常速度的共同作用导致了$ {{B}}^2 $依赖的电流密度矢量(43)式, 其纵向分量导致了磁场平方依赖的纵向磁导率$ \sigma_{xx} $, 即手征反常. 它的横向分量产生了磁场平方依赖的霍尔电导率$ \sigma_{yx} $, 即平面霍尔效应.

      对于倾斜的外尔半金属, 其中一个外尔点上的最小有效模型可以写为[217]

      $ H = v{{k}}\cdot{{\sigma}}+{{t}}\cdot {{k}}, $

      其中参数$ {{t}} $为描述倾斜程度的矢量, 当$ |{{t}}| < v $时该模型描述Ⅰ型外尔半金属, 以下讨论主要基于此类体系.

      引入倾斜项后两个具有相反手性的外尔锥沿着相反的方向发生倾斜. 假设$ {{E}} $沿着倾斜的方向, (44)式括号中第一项显示了由电场$ {{E}} $导致的分布偏离. 因此, 两个外尔锥中占据态的反常速度的贡献不会抵消. 但这一项不依赖于磁场$ {{B}} $, 只有反常速度项$ ({{\varOmega}}_{{k}}\cdot{{v}}){{B}} $ 线性依赖于磁场$ {{B}} $, 由此导致的纵向和横向电流分别产生了线性依赖于磁场的$ \sigma_{xx} $$ \sigma_{yx} $. 考虑到(44)式括号中第二项手征化学势的影响后, 不同于普通外尔半金属, 此时$ {{v}} $的大小在费米面上是变化的, 即$ {{v}} $在单个外尔锥上的贡献不会相互抵消. 只要$ {{B}} $$ {{E}} $不垂直, 则手征化学势导致的载流子分布偏离正比于磁场的大小. 导致除了由反常速度引起的2次依赖于磁场的项外进一步产生了线性依赖于磁场的$ \sigma_{xx} $$ \sigma_{yx} $.

      文献[217]进一步分析了磁场的角度和相空间体积的修正对电子输运的影响, 并指出该半经典理论可以推广到Ⅱ型外尔半金属, 为未来可能的实验验证提供指导.

    • 拓扑半金属中的手征反常[26,210,211]被广泛认为是产生负磁阻的原因[41-43,46,55,56,60,61,63,65,66,68,224]. 然而, 在无法定义手征的拓扑绝缘体中, 也可以观察到负磁阻现象, 这导致了对负磁阻进行理论解释的巨大混乱[152-155,225-228]. 在文献[80]中, 我们使用具有贝里曲率和轨道磁矩修正后的半经典玻尔兹曼方程来解释拓扑绝缘体中的负磁阻, 并与实验结果进行了定量的比较(图12).

      图  12  理论计算的负磁阻与实验[152-154]的比较. 转载自文献[80]

      Figure 12.  The comparison between the theoretical negative magnetoresistance and the experiments[152-154]. Reproduced with permission from Ref. [80].

      实验中, 负磁阻存在于$ T = 100 $ K[154]的温度以上, 因此可以排除量子干涉机制. 此外, 由于拓扑绝缘体Bi2Te3和Bi2Se3[229]的迁移率较差, 当磁场达到6 T时, 朗道能级不能很好地形成. 半经典运动方程(39)的速度项为

      $ \dot{{r}} = \frac{1}{\hbar}\nabla_{{{k}}}\widetilde{\varepsilon}_{{{k}}} -\dot{{{k}}}\times {{\varOmega}}_{{{k}}}, $

      不同于(39)式, 此处的$ \widetilde{\varepsilon}_{{{k}}} = \varepsilon_{{{k}}}-{{m}} \cdot {{B}} $是经过轨道磁矩$ {{m}} $修正后的能量, $ \varepsilon_{{{k}}} $是能带色散. (41)式和(42)式表明贝里曲率可以导致一个随磁场$ B^2 $ 增长的电导率修正. 这种机制在拓扑绝缘体中非常显著[80], 在几个特斯拉的平行磁场中, 相对磁阻可以超过–1%[152-155,225-227].

      在玻尔兹曼理论中, 纵向电导率$ \sigma^{\mu\mu} $由穿过费米能级的所有能带决定, 对于能带$ n $[223],

      $ \sigma^{\mu\mu} = \int\frac{{\rm d}^{3}{{k}}}{(2\pi)^{3}} \frac{e^{2}\tau}{D_{{k}}} \left(\widetilde{v}^{\mu}_{{k}} +\frac{e}{\hbar}B^{\mu}\widetilde{v}^{\nu}_{{k}}\varOmega^{\nu}_{{k}}\right)^2 \left(-\frac{\partial f_{\rm {eq}}}{\partial\widetilde{\varepsilon}}\right), $

      其中$ f_{\rm{eq}} $是平衡态的费米分布, 弛豫时间$ \tau $假定为常数[29].

      图12表明了负磁阻的实验和数值计算结果之间的一致性. 在实验[152]中, 温度为1.8 K, 因此原始数据在零场附近具有正磁阻, 这是由弱反局域化导致的[206,208,230-233]. 在实验[153,154]中负磁阻随温度变化不大这与理论保持一致, 显示了负磁阻的半经典性质. 在大多数使用半经典输运理论研究磁阻的文献中, 轨道磁矩均被忽略[28,223]. 如文献[234,235], 轨道磁矩对于外尔半金属中磁阻的各向异性是必不可少的. 文献[80]表明, $ {{m}}\cdot {{B}} $$ \varepsilon({{k}}) $的修正可以有效地增加负磁阻.

      其他的一些工作工作[236,237]发展了一般3D非磁性金属的具有二阶精度的半经典理论. 这些工作给出了几个令人惊讶的结果. 首先, 由于$ \delta \sigma^{\rm{int}}/\sigma_0 $与弛豫时间无关, 存在本征磁导$ \delta \sigma^{\rm{int}} $. 第二, 显著的$ \delta \sigma^{\rm int} $ 项可能导致对科勒规则的偏离. 第三, $ \delta \sigma^{\rm{int}} $可能导致正的纵向磁导. 在仅占据最低朗道能带的量子极限下, 磁阻微妙地依赖于散射机制[74,76,238], 而不是贝里曲率和轨道磁矩. 最近的理论工作[227,239]为理解负磁阻提供了新的观点. 值得注意的是文献[240]利用格林函数方法系统地计算了有质量狄拉克费米子在磁场下的输运行为, 该理论给出了负的且正比于$ B^2 $ 的纵向磁阻, 这一结论有望澄清长久以来对拓扑物态中负磁阻解释的困难.

    • 2.1节可知, 在$ z $方向的磁场中能谱演化为一系列1D的朗道能带[74,76] (图2), 这导致了电阻的SdH振荡. 一般用栗弗席兹-科塞维奇(Lifshitz–Kosevich)公式[241]来描述:

      $ \rho \sim \cos[2\pi(F/B+\phi)], $

      式中, $ \phi $为相移, $ F $为振荡频率, $ B $为磁场大小. 每个频率分量的相移为$ -{1}/{2}+{\phi_{\rm B}}/{2\pi}+\phi_{\rm{3 D}} $, 其中$ \phi_{\rm{3 D}} = \mp1/8 $为仅针对3D的修正, $ \phi_{\rm B} $为贝里相位[95,242]. 费米面沿磁场方向的曲率决定了$ \phi_{\rm{3 D}} $的符号[243,244]. 拓扑物态为探索非平庸贝里相位提供了一个新的平台[50-52,54,67,245-255]. 拓扑半金属中具有额外大小为$ \pi $的贝里相位[122,242], 因此在2D和3D体系中相移分别为$ 0 $[256]$ \pm1/8 $[257]. 在文献[77]中, 我们发现在栗弗席兹点附近, 量子振荡的相移可以超过一般的$ \pm1/8 $$ \pm5/8 $, 并且非单调地向更宽范围$ \pm7/8 $$ \pm9/8 $之间移动. 到目前为止, 在HfSiS[258-264], ZrSi(Se/Te)[265] 和ZrGe(S/Se/Te)[266]中对量子振荡都进行了实验研究, 但它们的相移并不相同. 表2 总结了具有线性和抛物线色散的 2 维和 3 维能带的相移.

      系统 电子载流子 空穴载流子
      2D抛物线 –1/2 1/2
      3D抛物线 –5/8 5/8
      2D线性 0 0
      3D线性 –1/8 1/8
      磁场$B_z$中的节线 $-5/8(\alpha), 5/8(\beta)$ $5/8(\alpha), -5/8(\beta)$
      磁场$B_{/\!/}$中的节线 $-5/8(\gamma), 1/8(\delta)$ $5/8(\gamma), -1/8(\delta)$

      表 2  对于具有不同色散和维度的系统, (48)式中的相移$\phi$. $B_z$$B_{/\!/}$是节线平面内外的磁场. $\alpha$, $\beta$, $\gamma$, δ 对应于图14中费米面的截面. 转载自文献[79]

      Table 2.  For systems with different dispersion and dimensions, the phase shift ϕ in Eq. (48). $B_z$ and $B_{/\!/}$ are magnetic fields outside and inside the nodal-line plane. $\alpha$, $\beta$, $\gamma$, and $\delta$ correspond to the cross sections of Fermi surface in Fig. 14. Reproduced with permission from Ref. [79].

      实验中, 在$ B $轴上的峰值位置或谷值位置给出整数朗道指数$ n $, 进而从$ n $$ 1/B $的图中得到相移和频率$ F $. 尽管如此, 对于是峰值[50-52,66,67,246,257] 还是谷值[54,248,251,267]应该被赋予朗道指数尚不明晰. 我们的计算结果表明, 在朗道能带边缘出现了与整数朗道指数对应的电阻率$ \rho_{zz} $$ \rho_{xx} $的峰值. 利用线性响应理论[268-271]估算电阻率分量[272,273]. 对于纵向情况, 电阻率$ \rho_{zz} $ = $ 1/\sigma_{zz} $. 在带边附近, 电导率$ \sigma_{zz} $呈现谷值, 因此$ \rho_{zz} $ 呈现峰值. 对于横向情况, $ \rho_{xx} = \sigma_{yy}/(\sigma_{yy}^2+\sigma_{xy}^2) $, 纵向电导率和场致霍尔电导率分别为

      $ \begin{split} &\sigma_{yy} = \frac{\sigma_0(1+\delta)}{1+(\mu B)^2}, \\ &\sigma_{yx} = \frac{ \mu B \sigma_0 }{1+(\mu B)^2}\left[1-\frac{\delta}{(\mu B)^2} \right], \end{split} $

      式中$ \delta\ll 1 $为振荡部分, $ \sigma_0 $为零场电导率. 通过详细分析[74,269,274]可知, 峰的位置遵循关系式$\rho_{zz} \sim \sigma_{zz}^{-1} \sim \sigma_{yy} \sim \rho_{xx}$, 因此$ \rho_{zz} $$ \rho_{xx} $在朗道能带边缘出现的峰及其相移是相同的.

      图13可见, 数值结果与理论分析预测高度一致. 定义$ E_A = A k_w $$ E_M = Mk_w^2 $, 对于$ E_M\neq E_A $, 随着拍频模式的出现, $ \phi {\text{-}} E_{\rm F} $曲线开始出现断裂. 在图13(c)中, 对于$ E_A < E_M $相移降到$ -5/8 $以下, 而不是在栗弗席兹点周围从–1/8单调地移到–5/8(即$ E_{\rm F} = E_M $), 这是因为没有简单的$ k_z^2 $依赖关系[77]. 在栗弗席兹点, 可以解析地证明相移为$ -9/8 $, 这与图13(c)中的相移一致. 它相当于$ -1/8 $, 通常认为它起源于大小为$ \pi $的贝里相位.

      图  13  对于(1)式描述的外尔半金属, 数值(散点)和解析(实线)得到的频率$ F $的曲线 (a)固定$ M $对应不同的$ A $; (b)固定$ A $对应不同的$ M $. (c)固定$ E_M $ 不同的$ E_A $ 对应的相移$ \phi $的曲线. 曲线断裂是因为在拍频模式出现时, F$ \phi $无法拟合. 垂直虚线表示栗弗席兹点. 转载自文献[77]

      Figure 13.  For the Weyl semimetal described in Eq. (1), the frequency $ F $ obtained by numerical (scatters) and analytical (solid curves): (a) Fixed $ M $ corresponds to different $ A $; (b) fixed $ A $ corresponds to different $ M $. (c) Fixed $ E_M $, for different $ E_A $ corresponds to the curve of phase shift $ \phi $. The curve breaks because $ F $ and $ \phi $ can not fit when beating patterns occur. The vertical dashed lines represents the Lifshitz point. Reproduced with permission from Ref.[77].

      由(11)式描述的狄拉克半金属具有时间反演对称性[105,275]]. 对于狄拉克半金属, 总相移可以取两个值, $ \alpha\in [0, 1/4] $$ [3/4, 1] $时为$ -1/8 $, $\alpha\in [1/4, 3/4]$$ -5/8 $. 在栗弗席兹点附近, 总相移可能在两个值之间变化.

      一般情况下电子载流子应该产生负相移, 空穴载流子应该产生正相移[257]. 然而, 在狄拉克半金属Cd3As2的实验中, 电子载流子的相移为正值[50,51,54]. 一种可能的解释是, 对于实验中的相移1/8—3/8, 由于$ 2\pi $的周期性它们的实际值应为–7/8—–5/8. 表3列出了相移实验值的对应项. 在文献[77]中, 除了轨道量子干涉[276]、塞曼劈裂[52,247,277]和费米面嵌套[51]等机制, 我们还证明了由于能带反转而出现的拍频模式.

      文献 $\phi_{\rm{exp}}$ $\phi_{\rm{Weyl}}$ $\phi_{\rm{Dirac}}$
      [50] 0.06 — 0.08 –0.94 — –0.92 –5/8
      [51] 0.11 — 0.38 –0.89 — –0.62 –5/8
      [54] 0.04 –0.96 –5/8

      表 3  从Cd3As2的实验中得到的相移$\phi_{\exp}$. 转载自文献[77]

      Table 3.  The phase shift $\phi_{\exp}$ obtained from the experiment of Cd3As2. Reproduced with permission from Ref. [77].

    • 对于节线半金属无论是面内还是面外, 轮胎状费米面都具有最大截面和最小截面. 其次, 贝里相位沿与节线平行的圆为0, 沿包围节线的圆为$ \pi $. 根据昂萨格关系得到$ F = (\hbar/2\pi e) A $, 其中$ A $是费米面上垂直于磁场的极值截面面积. 当节线平面垂直于磁场, 即磁场取$ B_z $时, $ k_z = 0 $平面上有两个极值截面(图14(b)). 通过理论分析发现, 外圆的高频项为$F_\alpha = m(u+E_{\rm F})/(\hbar e)$, 内圆的低频项为$F_\beta = m(u- E_{\rm F})/(\hbar e)$. 这两个频率可能导致拍频模式的出现[278]. 在文献[79]中, 通过分析电阻率的计算结果得到了节线半金属的相移和频率. 需要指出的是, 当保护节线的对称性破缺时, 出现大小为$ \varDelta $的能隙将导带和价带分开, 此时贝里相位为$ \phi_{\rm B} = \pm\pi\big[1-\delta/(2 E_{\rm F})\big] $. 表4进一步总结了随机情况下相移的一般规则, 这些一般的规则可以帮助分析一些其他的材料, 例如ZrSiS和Cu3PdN[259-264].

      贝里相位 最大/
      最小
      电子 空穴
      $\alpha$ 0 最大 $ -1/2+0-1/8 = -5/8 $ +5/8
      $\beta$ 0 最小 $-1/2+0+1/8 = -3/8 \leftrightarrow 5/8$ –5/8
      $\gamma$ 0 最大 $ -1/2+0 - 1/8 = - 5/8 $ +5/8
      δ π 最小 $-1/2+\pi/2\pi+1/8 = 1/8$ –1/8

      表 4  节线半金属的相移$\phi$. $\alpha, \beta, \gamma, \delta$图14中的极值截面. 转载自文献[79]

      Table 4.  The phase shift $\phi$ of the nodal-line semimetal. $\alpha, \beta, \gamma, \delta$ are the extremal cross sections in Fig. 14. Reproduced with permission from Ref. [79].

      图  14  (a) (17)式中节线半金属的节线(虚线环), 轮胎状和鼓形费米面, $ E_{\rm F} $是费米能, $ u $ 是模型参数; (b)轮胎状费米面在节线平面内的最大($ \alpha $)和最小($ \beta $)截面; (c)轮胎状费米面在节线平面外的最大($ \gamma $)和最小($ \delta $)截面. 转载自文献[79]

      Figure 14.  (a) In the model of nodal-line semimetal Eq.(17), the nodal line (dashed ring), torus and drum Fermi surface, $ E_{\rm F} $ is Fermi energy, $ u $ is model parameter; (b) the maximum ($ \alpha $) and minimum ($ \beta $) cross sections of the torus Fermi surface; (c) the maximum ($ \gamma $) and minimum ($ \delta $) cross sections of the Fermi surface outside the nodal-line plane. Reproduced with permission from Ref. [79].

      对于节线半金属, 大多数量子振荡实验都是针对ZrSiS族材料[258-266], 其中费米能处既有电子型也有空穴型口袋[140,141,259,264]. 文献[79]进一步分析了Cu3PdN[120,121]的相移.

    • 2D量子霍尔效应的发现打开了拓扑物态领域的大门[157,279]. 在3D电子气中, 由于沿磁场方向的波矢$ k $是好量子数, 量子霍尔效应通常只能在2D系统中观察到[157,256,280-282]. 在文献[78,283]中, 我们描述了拓扑半金属中的3D量子霍尔效应. 拓扑半金属在平行于外尔点方向的表面处有拓扑保护的表面态, 即费米弧[284,285-288](图15(a)图15(b)).

      图  15  (a)外尔半金属中的费米弧和体态的色散, $ k_{/\!/} $表示$ (k_x, k_y) $; (b)在$k_z\text-k_x$平面上$ y = L/2 $, $ E_{\rm F} = E_w $处的费米弧; (c)宽度为$ W $, 厚度为$ L $的外尔半金属板; (d)在$E_{\rm F} = E_w$处的费米弧(实线); (e)−(g)波函数在$ k_z = 0 $处沿$ y $轴的分布; (h) 3D量子霍尔效应中的朗道能级和边缘态; (i)单一表面的电子无法被y方向的磁场$ B $驱动完成一个完整的回旋运动. 转载自文献[78,283]

      Figure 15.  (a) The energy dispersions of the Fermi arc and bulk states in a Weyl semimetal, $ k_{/\!/} $ stands for $ (k_x, k_y) $; (b) the Fermi arc at $ y = L/2 $ and $ E_{\rm F} = E_w $ in the $ k_z\text-k_x $ plane; (c) a Weyl semimetal slab with width $ W $ and thickness $ L $; (d) Fermi arc (solid) at $ E_{\rm F} = E_w $; (e)−(g) the distribution of wave function along $ y $-axis at $ k_z = 0 $; (h) the Landau levels and edge states in the 3D quantum Hall effect; (i) an electron in single surface could not be driven in a y-direction magnetic field $ B $ to perform a complete cyclotron motion. Reproduced with permission from Refs. [78,283].

      单一表面上, 费米弧无法形成一个闭合的费米环, 而这一点对量子霍尔效应至关重要. 然而, 在外尔半金属中, 来自相反表面的费米弧(图15(c))可以形成所需的闭合费米环(图15(d)). 电子通过外尔点在相对表面的费米弧之间实现隧穿(图15(e)图15(g)), 通过这一机制电子可以完成完整的回旋运动, 从而实现3D量子霍尔效应. 在实际材料中, 隧穿距离受平均自由程的限制, 在高迁移率的拓扑半金属[60,103]中, 平均自由程大约为$ 100\; {\rm {nm}} $, 在一些材料中甚至可以达到$ 1$ µm[289], 在我们的计算中块体厚度为100 nm. 为使量子霍尔效应仅来自费米弧的贡献需将费米能级调到外尔点上来耗尽体态载流子[290]. 外尔半金属TaAs家族[291-295]和狄拉克半金属Cd3As2和Na3Bi具有量子霍尔效应所要求的极高迁移率[46,48,50,51,54]、低载流子密度[224]等特性. 通过费米弧的3D量子霍尔效应预计可以存在于Cd3As2 [224,296-298], Na3Bi以及TaAs家族的材料中[61,290].

      根据久保(Kubo)公式, 我们计算了霍尔电导[78,299-303]. 当费米能远离外尔点时, 霍尔电导服从通常的$ 1/B $依赖关系. 当费米能向外尔点移动时, 开始出现$ \sigma_{\rm H}^s $ 的量子化平台.

      图15(c)可见, 费米弧的边界态具有独特的3D空间分布. 具体而言, 上边界态向左传播(绿色箭头), 下边界态向右传播(橙色箭头). 这种费米弧边缘态的独特3D分布可以通过扫描隧道显微镜[304]或微波阻抗显微镜[305]来探测. 与拓扑绝缘体[281,282] 不同, 基于费米弧的量子霍尔效应需要两个表面的共同作用. 最近的工作[306]系统地探讨了外尔半金属中3D量子霍尔效应的边缘态图像. 外尔半金属中不同手性外尔点处的速度相反, 强磁场使体态进入手征朗道能带, 这一点显著地影响了边缘态的载流子输运. 除了上下表面的费米弧通过隧穿形成一个闭环[78]外, 在磁场作用下拓扑半金属中还观察到其他奇异的量子霍尔现象[296,307,308]. 在垂直磁场$ {{B}} $中, 基于半经典运动方程从理论上给出了边缘态的闭合3D轨道. 在$ x $方向, 由于手征朗道能带的影响, 上(下)表面费米弧态的半圆轨道形成跳跃边缘态. 在$ z $方向, 手征朗道能带和侧面的费米弧态共同构成闭合轨道. 此外, 由于手征朗道能带沿磁场方向色散, 边缘态的量子通道可以由倾斜的磁场控制, 导致边缘态的分布和霍尔电导的行为发生了显著的变化. 当调整$ {{B}}_z $并保持$ {{B}}_y $不变时, 边缘态的位置会从一侧变化到另一侧, 霍尔电导也随之改变符号. 文献[306]利用数值方法进一步分析了局域态密度分布, 以及霍尔电导随磁场大小和方向的变化.

      3D量子霍尔效应也可以由电荷密度波(CDW)机制[309]实现, 最近在ZrTe5[310]的实验中观察到了这一机制, 并看到了清晰的量子化平台. 在文献[87]中我们详细地探讨了这一机制, 并给出了一个完备的理论可以解释实验的主要特征. 我们发现电荷密度波可以在1D的朗道能带上产生, 而且强烈依赖于磁场的大小. 磁场诱导了3D电子气向电荷密度波相的二阶相变, 导致朗道能带打开一个能隙从而使体态绝缘, 即3D电子气变为一系列的2D量子霍尔态, 从而实现3D量子霍尔效应(图16).

      图  16  左图: 2D电子气在磁场中形成量子霍尔态. 中间图: 3D时朗道能级变为一系列2D的朗道能带. 右图:电荷密度波使朗道能带打开能隙, 使体态绝缘, 可以观察到3D量子霍尔效应. 转载自文献[87]

      Figure 16.  Left: the quantum Hall state in 2D electron gas under magnetic field. Center: in 3D, the Landau levels turn to one dimensional Landau bands. Right: the charge density wave gap the Landau band, so that the bulk is insulating and the 3D quantum Hall effect can be observed. Reproduced with permission from Ref. [87].

      通过沿$ z $方向的电子-电子或电子-声子相互作用, 可使$ k_{\rm F} $$ -k_{\rm F} $附近的电子之间产生耦合从而打开电荷密度波能隙(由序参量$ \varDelta $描述). 电子-电子相互作用为[311,312]

      $ H_{\rm {e e}} = -\sum\limits_{{{k}}}|\varDelta|\left({\rm{e}}^{{\rm{i}} \phi} \hat{d}_{{{k}}+}^{\dagger} \hat{d}_{{{k}}-}+{\rm{H.c. }}\right)+\frac{2|\varDelta|^{2} V}{U\left(2 k_{\rm F}\right)}, $

      其中, 序参量定义为 $\varDelta = \varDelta_{\rm {e e}} = \left[U\left(2 k_{\rm F}\right) /( 2 V)\right] \cdot \sum\nolimits_{{{k}}}\left\langle\hat{d}_{{{k}}-2 k_{\rm F}}^{\dagger} \hat{d}_{{{k}}}\right\rangle$, $ \quad V \quad $表示体积, $ \varDelta = |\varDelta| {\rm{e}}^{{\rm{i}} \phi} $. 电子-电子相互作用势采用汤川势$U\left(2 k_{\rm F}\right) = e^{2} /\big\{\epsilon_{\rm r} \epsilon_{0}\cdot \big[\left(2 k_{\rm F}\right)^{2}+\kappa^{2}\big]\big\}$, 其中 $ \epsilon_{\rm r}\left(\epsilon_{0}\right) $表示介电常数, $ 1 / \kappa $ 为屏蔽长度. 在随机相位近似下$ \kappa = \sqrt{e^{3} B /\left(4 \pi^{2} \epsilon \hbar^{2} v_{\rm F}\right)} $, 此处$ \epsilon = \epsilon_{0} \epsilon_{\rm r} $. 电声相互作用的哈密顿量为[313,314]

      $ H_{\rm{e{\text{-}}ph}} = \sum\nolimits_{{k}}|\varDelta|\left({\rm{e}}^{{\rm{i}} \phi} \hat{d}_{{{k}}+}^{\dagger} \hat{d}_{{{k}}-}+{\rm{H.c.}}\right), $

      其中$ \varDelta = \varDelta_{{\rm e}{\text{-}}{\rm {ph}}} = \left(\alpha_{{{{q}}}} / V\right)\left(\left\langle\hat{b}_{{q}}\right\rangle+\left\langle\hat{b}_{-{ q}}^{\dagger}\right\rangle\right) $, 电声耦合势$ \alpha_{{ q}} $同样采用汤川势的形式. 在$ \pm k_{\rm F} $附近, $ 0+ $朗道能带的平均场哈密顿量为

      $ { {\cal{H}}}_{k_{z}}^{0+} = \begin{bmatrix} \hbar v_{\rm F}\left(k_{z} \pm k_{\rm F}\right) & \varDelta \\ \varDelta^{*} & -\hbar v_{\rm F}\left(k_{z} \pm k_{\rm F}\right) \end{bmatrix}, $

      其中, 费米速度$\hbar v_{\rm F} \! \equiv\! \big|\partial E_{k_{z}}^{(0+)} / \partial k_{z}\big|_{k_{z} = k_{\rm F}}$, 电荷密度波序参量可以通过由$ \partial E_{\rm g}/\partial|\varDelta| $定义的能隙方程自洽计算得到, 其中基态能量$ E_{\rm g}\equiv\langle H\rangle $. 进一步计算发现[87], 使用电子-电子相互作用计算的序参量仅在大于10 T的阈值磁场时是足够大的, 但比实验测量的值大. 另一方面, 对于具有适当耦合常数的电子-声子相互作用, 由非欧姆的$ I{\text{-}}V $关系确定的阈值磁场可以小于1.5 T, 因此, 电子-声子相互作用可能是ZrTe5实验中[310]导致电荷密度波的主要机制.

      在实验中, 霍尔电阻率的平台覆盖了1.7— 2.1 T的较宽范围, 以$ (e^2/h) $为单位的霍尔电导率由电荷密度波的层数决定. 电荷密度波波长$ \lambda_{\rm{cdw}} $与费米波长相关:

      $ \lambda_{\rm{cdw}} = \lambda_{\rm F}/2 = \pi/k_{\rm F}. $

      在1.7—2.1 T之间观察到的$ \rho_{xx} $平台意味着存在公度的电荷密度波, 即电荷密度波波长与晶格常数$ a $的整数倍相称. 通过比较公度和非公度的电荷密度波在2.1 T附近的基态能量, 发现对于公度和非公度的电荷密度波在磁场分别在$ [1.7, 2.1] $ T以及$ [2.1, 3.0] $ T的区间时具有较低的能量, 因此在两种电荷密度波之间存在交叉. 在$ B\in[1.7, 2.1] $ T范围内固定的$ \lambda_{\rm{cdw}} $表示费米能不变, 即系统属于巨正则系综载流子的数量可以改变. 相比之下, 非公度电荷密度波中的载流子数量不能改变. 因此, 电子能量的变化导致$ B\in[1.7, 2.1] $ T范围内公度的电荷密度波基态能量降低. 进一步将磁场增加到2.1 T以上, 磁场将把费米能推得更低(最终到达带底), 进一步说明由磁场诱导的电荷密度波存在从公度到非公度的交叉.

    • 选择随机高斯势来探讨量子极限下无序散射对磁阻的影响

      $ U({{r}}) = \sum\limits_{i}\frac{u_{i}}{(d\sqrt{2\pi})^{3}}{\rm{e}}^{-|{{r}}-{{R}}_{i}|^{2}/(2d^{2})}, $

      式中, $ u_{i} $表示随机分布杂质在$ {{R}}_{i} $处的散射强度, $ d $是确定散射势范围的参数. 杂质势范围$ d $和磁长度$ l_{B} $定义了两个区域, 长程势区域$ d\gg l_{B} $和短程势极限$ d\ll l_{B} $. 在强磁场极限下, 如果$ B > 10 $ T, 磁长度 $ l_{B} $ 小于$ 10 \;{\rm {nm}} $.

      通过考虑散射时间对磁场的依赖性, 我们发现在强场极限($ B\rightarrow\infty $)下, 即对于第0个朗道能带, 费米面上的态间散射的弛豫时间为$\tau \!=\! \dfrac{\hbar^{2}v_{\rm F}^{0}\pi l_{B}^{2}}{V_{\rm{imp}}}$[76], 得到强场极限下的电导率

      $ \sigma_{zz,0}^{\rm {sc}} = \frac{e^{2}}{h}\frac{(\hbar v_{\rm F}^{0})^{2}}{V_{\rm{imp}}}. $

      因此$ \sigma_{zz, 0}^{\rm{sc}} $的磁场依赖性由费米速度$ v_{\rm F}^{0} $给出. 当忽略费米速度的磁场依赖性时, 得到与$ B $无关的电导率, 这与先前工作中速度是常数的结果是一致的[315]. 文献[76]中进一步探讨了费米速度的磁场依赖性可以导致正、负磁导的不同情况.

      当磁场为零时, 磁长度发散, $ d/l_B\rightarrow 0 $, 给出最小电导率[76]

      $ \sigma_{zz}(0) = \frac{e^{2}}{h}\frac{ 4(Mk_w)^{2}}{V_{\rm{imp}} }{\rm e}^{4d^{2}k_w^{2}}. $

      注意外尔点处态密度在零磁场时趋于0, 类似的结果在没有朗道能级的情况下也曾被发现过[316]. 通过进一步的计算得到磁导[76]

      $ \Delta\sigma_{zz}(B) \equiv \dfrac{\sigma_{zz}(B)-\sigma_{zz}(0)}{\sigma_{zz}(0)} = \dfrac{B}{B_{0}}, $

      其中$B_{0} = \hbar/(2 ed^{2})$. 可见, 磁导是由杂质势的范围给出的, 与模型参数无关. 这意味着对外尔半金属有$ \hat{{z}} $方向的正线性磁导. 通过数值计算发现, 对于长程无序系统也具有正的磁导, 尽管对于弱的短程无序, 系统往往趋向于具有负的磁导[40].

      图17给出了不同极限下的纵向和横向磁导. 特别地, 图17(f)中长程势极限下的$ \sigma_{xx}\propto1/B $. 垂直于$ x{\text{-}}y $平面的场中, 还存在霍尔电导率$\sigma_{yx} = {\rm{sgn}}(M)(k_w/\pi)e^{2}/h+en_{0}/B$, 其中第一项是反常霍尔电导率, 第二项是经典电导率. 在弱场中, 经典霍尔效应占主导地位, $ \sigma_{xx} $$ \sigma_{yx} $都与$ 1/B $成正比, 在磁场$ B $中电阻$ \rho_{xx} = \sigma_{xx}/(\sigma_{xx}^{2}+\sigma_{yx}^{2}) $是线性的. 注意, 这里垂直场中的线性磁阻与以前的机制并不相同[274,317].

      图  17  不同的势范围下, 外尔半金属在$ \hat{{z}} $方向磁场B中的纵向电导率$ \sigma_{zz} $和横向电导率$ \sigma_{xx} $. 转载自文献[76]

      Figure 17.  The longitudinal conductivity $ \sigma_{zz} $ and transverse conductivity $ \sigma_{xx} $ of the Weyl semimetal in the $ \hat{{z}} $-direction magnetic field B under the different potential ranges. Reproduced with permission from Ref. [76].

    • 拓扑绝缘体作为拓扑物态中最先被关注的体系, 为探索奇异的拓扑物态[318-325] 提供了诸多线索. 在强磁场中, 2D拓扑绝缘体的最低朗道能级相互交叉, 该特征可以作为量子自旋霍尔相的标志[326,327]. 同样在2D情况下, 干涉效应[328]也可以用来探测量子自旋霍尔相. 然而在3D时, 如何利用最低朗道能带来识别拓扑绝缘体却很少被讨论. 在文献[81]中, 我们研究了3D拓扑绝缘体在强场量子极限下的磁阻(图18(b)). 在临界磁场中, 背散射在量子极限下被完全抑制, 这一效应可以用来识别拓扑绝缘体. 不过这种禁止的背散射在拓扑半金属中是不存在的[74,76,238]. 这一理论与最近的实验展现出惊人的一致性(图18(c)图18(d)). 此外, 这一机制可以很好地应用于那些能隙较小的材料, 例如ZrTe5[329,330]和Ag2Te [331].

      图  18  (a) 3D拓扑绝缘体的零场能谱($ k_x = k_y = 0 $); (b)在强磁场中, 费米能只穿过$ 0+ $朗道能带; (c)实验测得Pb1–xSnxSe的磁阻[226]; (d)理论计算出的磁阻. 转载自文献[81]

      Figure 18.  (a) The zero field energy spectrum of 3D topological insulator ($ k_x = k_y = 0 $); (b) in a strong magnetic field, fermi energy $ E_{\rm F} $ can only crosses the $ 0+ $ Landau energy band; (c) the magnetoresistance of Pb1–xSnxSe in experiment[226]; (d) the theoretical calculated magnetoresistance. Reproduced with permission from Ref. [81].

      在沿$ z $方向的强磁场$ B $中, 能谱量子化为一系列1D的朗道能带(图18(a)图18(b)). 最低的两个朗道能带分别表示为$ 0+ $$ 0- $, 其能量为$E_{0\pm} = C_0+C_zk_z^2+C_\bot/l_B^2\pm\sqrt{m^2+V_n^2 k_z^2}$, 质量项$m = M_0+M_zk_z^2+ M_\bot/l_B^2$. 当费米能仅与 $ 0+ $ 朗道能带相交, 其本征态为

      $ |0,+,k_x,k_z\rangle = \begin{bmatrix} 0\\ -{\rm{i}} \sin (\theta/2)\\ 0\\ \cos(\theta/2) \end{bmatrix} |0,k_x,k_z\rangle, $

      其中$ \cos\theta \equiv -m /\sqrt{m^2 + (V_n k_z)^2} $. 在固体中, 电子输运相对地受背散射的影响, 在1D的朗道能带上尤其显著, 因为费米能上只有两种态, 如图18中的$ k_{\rm F} $$ -k_{\rm F} $所示. 这两种态之间的背散射由它们之间的散射矩阵元刻画. 由(54)式中的旋量本征态可以发现, $ k_{\rm F} $$ -k_{\rm F} $ 之间的散射矩阵元的模方正比于形状因子:

      $ I_{\rm S} = \left.\cos^2\theta\right|_{k_z = k_{\rm F}}. $

      $ m $在由$ M_0+M_zk_{\rm F}^2+ M_\bot eB_{\rm c} /\hbar = 0 $决定的临界磁场$ B_{\rm{c}} $下为0, 其中$ k_z $等于费米能对应的费米波矢$ k_{\rm F} $. 对于拓扑绝缘体, $ M_0 M_z < 0 $ 并且$ M_0 M_\bot < 0 $, $ B_{\rm{c}} $有解, 因此背散射被完全抑制. 这种被禁止的背散射将导致电阻随磁场的变化而下降, 可以作为在实验中区分拓扑绝缘体相的一个显著特征.

    • 当外界强磁场与外尔费米子耦合时其手性消失并获得质量, 而这是外尔费米子最明显的两个特征, 在这个意义上, 外尔费米子被湮灭. 在外尔半金属TaP中我们实现了这种磁耦合[64].

      2.1节给出了沿$ z $方向磁场中的朗道能带. 现在把这种情况推广到垂直于$ y $方向${{B}} = B(\sin\phi, 0, \cos\phi)$的任意场, 其中$ \phi $$ z $方向和磁场之间的夹角. 朗道规范可选为$ {{A}} = (-B_z y, 0, B_x y) $, 在皮尔斯(Pierls)代换下

      $ {{k}} \rightarrow \left(k_x-\frac{y \cos \phi}{l_B^2},-{\rm{i}}\partial_y,k_z+ \frac{y \sin \phi}{l_B}\right), $

      哈密顿量变为

      $ H({{k}})\! \rightarrow \!\left[\!\!\! \begin{array}{cc}{\rm i} {\cal{M}}_k^B & A\!\left(k_x\!-\!\frac{y \cos \phi}{l_B^2} \!-\!\partial_y\!\right)\!\! \\ A\!\left(k_x\!-\!\frac{y \cos \phi}{l_B^2}\! +\!\partial_y\!\right)\! & \!-\!{\cal{M}}_k^B \\ \end{array}\!\!\! \right],\\ $

      其中${\cal{M}}_k^B = M_1[k_{\rm{c}}^2-(k_x-y\cos\phi/l_B^2 )^2+\partial_y^2-(k_z+ y\sin\phi/l_B^2)^2]$. 在$ \phi = 0 $即磁场沿着$ z $方向时, 完全回到2.1节给出的结果. 如果$ \phi\neq 0 $, 能谱为

      $ E^{\nu\pm}_{k} = \pm \sqrt{{\cal{M}}_\nu^2+(A k_{/\!/}\sin\phi)^2}, $

      其中 ${\cal{M}}_\nu \equiv M_1 \Big[k_{\rm c}^2- k_{/\!/}^2 - \dfrac{2}{l_B^2} (\nu + {1}/{2})\Big]$. 通过数值求解得到$x{\text{-}}z$平面上任意方向的能谱.

      图19给出了外尔半金属的朗道能带. 当磁场沿$ z $方向($ \theta = 0 $)时, 最低的朗道能带(红色)穿过费米能(虚线). 随着磁场从$ z $方向旋转到$ x $方向($ \theta = \pi/2 $), 最低朗道能带发生了移动. 当磁场沿$ x $方向时, 朗道能带的能谱是粒子-空穴对称的, 并且由于外尔费米子之间的耦合存在一个能隙. 这就是为什么在外尔半金属TaP[64]的强场量子极限下, 霍尔电阻出现明显的符号反转. 由于能隙的存在, 外尔费米子获得质量并失去其手性. 由于具有手性和无质量是外尔费米子的两个主要特征, 因此霍尔信号表明外尔费米子被湮灭.