Quantum transport in topological matters under magnetic fields

Qiang Xiao-Bin; Lu Hai-Zhou

doi:10.7498/aps.70.20200914

Abstract

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.

Keywords:

Authors and contacts

1.
Shenzhen Institute for Quantum Science and Engineering, Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China
2.
Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China

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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Cited By

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].

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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].

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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].

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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].

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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].

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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} $ 和 $ \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].

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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 $ 和$ 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].

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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].

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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} $到$ -{ k} $ 的相干背散射; (c)在长程杂质势作用下, 沿极向的$ \delta { k} $到$ -\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].

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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].

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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].

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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].

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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 $的曲线. 曲线断裂是因为在拍频模式出现时, F和$ \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].

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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].

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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].

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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].

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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].

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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Pb_1–xSn_xSe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 Pb_1–xSn_xSe in experiment^[226]; (d) the theoretical calculated magnetoresistance. Reproduced with permission from Ref. [81].

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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]

Figure 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].

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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]

Figure 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].

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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 $和$ c $是晶轴. 转载自文献[86]

Figure 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].

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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]

Figure 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].

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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	辛	幺正
时间反演	√	√	×
自旋旋转	√	×	×
WL/WAL	WL	WAL	×

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表 2 对于具有不同色散和维度的系统, (48)式中的相移$\phi$. $B_z$和$B_{/\!/}$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	$-5/8(\gamma), 1/8(\delta)$	$5/8(\gamma), -1/8(\delta)$

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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

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表 4 节线半金属的相移$\phi$. $\alpha, \beta, \gamma, \delta$ 是图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

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Vol. 70, Issue 2 - 2021-01-20

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