Recent progress of transport theory in Dirac quantum materials

Wang Huan-Wen; Fu Bo; Shen Shun-Qing

doi:10.7498/aps.72.20230672

Abstract

Dirac quantum materials comprise a broad category of condensed matter systems characterized by low-energy excitations described by the Dirac equation. These excitations, which can manifest as either collective states or band structure effects, have been identified in a wide range of systems, from exotic quantum fluids to crystalline materials. Over the past several decades, they have sparked extensive experimental and theoretical investigations in various materials, such as topological insulators and topological semimetals. The study of Dirac quantum materials has also opened up new possibilities for topological quantum computing, giving rise to a burgeoning field of physics and offering a novel platform for realizing rich topological phases, including various quantum Hall effects and topological superconducting phases. Furthermore, the topologically non-trivial band structures of Dirac quantum materials give rise to plentiful intriguing transport phenomena, including longitudinal negative magnetoresistance, quantum interference effects, helical magnetic effects, and others. Currently, numerous transport phenomena in Dirac quantum materials remain poorly understood from a theoretical standpoint, such as linear magnetoresistance in weak fields, anomalous Hall effects in nonmagnetic materials, and three-dimensional quantum Hall effects. Studying these transport properties will not only deepen our understanding of Dirac quantum materials, but also provide important insights for their potential applications in spintronics and quantum computing. In this paper, quantum transport theory and quantum anomaly effects related to the Dirac equation are summarized, with emphasis on massive Dirac fermions and quantum anomalous semimetals. Additionally, the realization of parity anomaly and half-quantized quantum Hall effects in semi-magnetic topological insulators are also put forward. Finally, the key scientific issues of interest in the field of quantum transport theory are reviewed and discussed.

Keywords:

Authors and contacts

1.
School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China
2.
Department of Physics, The University of Hong Kong, Hong Kong 999077, China

Corresponding author: Wang Huan-Wen, wanghw@uestc.edu.cn ; Fu Bo, fubo@gbu.edu.cn ; Shen Shun-Qing, sshen@hku.hk

Funds: Project supported by the National Key R&D Program of China (Grant No. 2019YFA0308603), the Research Grants Council of the Hong Kong Government, University Grants Committee, China (Grant Nos. C7012-21G, 17301220), the Scientific Research Starting Foundation of University of Electronic Science and Technology of China (Grant No. Y030232059002011), and the International Postdoctoral Exchange Fellowship Program, China (Grant No. YJ20220059)

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图 1 有质量狄拉克费米子的内禀磁阻　(a)有质量狄拉克费米子的横向磁阻和纵向磁阻, 其中纵向磁阻为负, 横向磁阻为正; (b)无量纲系数随能带展宽的变化关系, $c_\alpha$在弱散射下趋于一个常数. 转载自文献[48]

Figure 1. Intrinsic Magnetoresistivity in massive Dirac fermion: (a) Transversal and longitudinal magnetoresistivity, where the longitudinal one is negative and transversal one is positive; (b) dimensionless parameter $c_\alpha$ as functions of band broadening, here $c_\alpha$ tends to a constant in weak scattering. Reproduced with permission from Ref. [48]

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图 2 Mn掺杂的拓扑绝缘体薄膜中磁导在不同磁场强度下的温度依赖行为, 其中(a) $x_{\mathrm{Mn}} = 0{\text{%}}$和(b) $x_{\mathrm{Mn}} = 8{\text{%}}$. 图中的空心方块是从文献[63]中获得的实验数据, 实线是(25)式在不同磁场下的拟合结果. 转载自文献[72]

Figure 2. Magnetoconductivity as a function of temperature at different magnetic field strength for two Mn-doped topological insulator thin films of (a) $x_{{\rm{Mn}}} = 0{\text{%}}$ and (b) $x_{{\rm{Mn}}} = 8{\text{%}}$. The open squares are the experimental data extract from Ref. [63]. The solid red lines are the fitting results at different magnetic filed B by using the formula in Eq. (25). Reproduced with permission from Ref. [72]

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图 3 库珀子能隙和权重因子关于能量的函数关系, 其中(a) 拓扑绝缘体$(mb>0)$, (b) 平庸绝缘体$(mb<0)$, (c)半金属($mb=0$). $F_\mathrm{tot}=\displaystyle\sum\nolimits_i F_i$是总的权重因子. 转载自文献[75]

Figure 3. Dimensionless Cooperon gap $\ell_e^2/\ell_i^2$ and weighting factor $F_i$ as a function of the Fermi energy $\mu$ for (a) topological insulator ($mb>0$), (b) trivial insulator ($mb<0$), and (c) Dirac semimetal ($mb=0$). $F_\mathrm{tot}$ is the total weighting factors defined as $F_\mathrm{tot}=\displaystyle\sum\nolimits_i F_i$. Reproduced with permission from Ref. [75]

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图 4 (a) (27)式对实验中Cd₂As₃样品^[64]相对纵向磁阻的理论拟合; (b) 拟合得到的相干长度关于温度的函数, 可以被$\ell_{\phi}\propto T^{-0.75}$很好地拟合; (c)不同磁场强度下的相对磁阻关于温度的函数. 转载自文献[75]

Figure 4. Theoretical fitting to the relative longitudinal magnetoresistance in a $\mathrm{Cd}_{2}\mathrm{As}_{3}$ sample^[64]; (b) fitted phase coherence length $\ell_\phi$ as a function of temperatures (open squares), which can be well-fitted by $\ell_{\phi}\propto T^{-0.75}$; (c) measured relative magnetoresistance as a function of temperatures at $B=1, 2, 3\;\mathrm{T}$. Reproduced with permission from Ref. [75]

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图 5 ZrTe₅和HfTe₅电阻反常效应　(a)关于ZrTe₅温度依赖的能谱在实验 (根据文献中ARPES测量得到) 和理论 (实线) 上的比较, 随着温度升高, 化学势由导带变化至价带; (b)—(d) 分别为不同载流子浓度下计算得到的电阻反常行为、霍尔系数和塞贝克系数. 转载自文献[121]

Figure 5. Resistivity anomaly in ZrTe₅ and HfTe₅: (a) Comparison of experimental (according to the ARPES measurements in literature) and theoretical (solid lines) temperature-dependent energy spectrum. The chemical potential varies from valence band to conduction band with the increasing of temperature. (b)–(d) The resistivity anomaly, Hall coefficients, and Seebeck coefficient for several different carrier concentrations. Reproduced with permission from Ref. [121]

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图 6 不同温度下的(a)横向电阻, (b)霍尔电阻, (c)塞贝克系数和(d)能斯特系数的磁场依赖. 转载自文献[121]

Figure 6. Magnetic field dependence of (a) the transverse magnetoresistance $\rho_xx$, (b) the Hall resistivity $\rho_{xy}$, (c) the Seebeck coefficient and (d) the Nernst coefficient for different temperatures. Reproduced with permission from Ref. [121]

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图 7 连续性方程(32)和(34)中系数$C_{\rm{h}}{\rm{}}$和$C_5$的比较. 转载自文献[50]

Figure 7. Comparison of the coefficients $C_{{{{\rm{h}}}}/5}$ in the equations for the divergence of the helical current and axial vector currents in Eqs. (32) and (34). Reproduced with permission from Ref. [50]

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图 8 宇称反常半金属示意图　(a) Haldane模型: 无质量和有质量的狄拉克锥在动量空间分开; (b)三维半磁性拓扑绝缘体: 无质量和有质量的狄拉克锥在实空间分开; (c)宇称反常半金属中低能电子态的分布以及幂律衰减的边界流. 转载自文献[123]

Figure 8. Illustration of parity anomaly semimetals: (a) Haldane model where massive and massless Dirac cone separated in momentum space; (b) semi-magnetic 3D topological insulator in which a massive and a massless Dirac cone separated in position space; (c) distribution of a set of low energy states and the power law decay edge current in the parity anomalous semimetal for open boundary condition. Reproduced with permission from Ref. [123]

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图 9 (a)关于磁化强度和体拓扑磁电效应关系的示意图, 外加电场产生了表面电流和磁化强度; (b)沿着x方向的局域电流密度关于位置z的函数; (c)轴子角$\theta$关于位置z的函数关系. 这里电场沿着y方向. 转载自文献[125]

Figure 9. (a) Schematic diagram of the relation between magnetization and bulk topological magnetoelectric effect. A surface current is produced by an electric field due to the magnetization. (b) Local current density along the x–direction as a function of slab position z. (c) Spatial dependent $\theta$ along the z direction. The electric field is applied along the y-direction. Reproduced with permission from Ref. [125]

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图 10 ${\cal{F}}(x)$关于x的函数关系, 其中蓝色虚线表示$ \mathcal{F}(x)= 1 $的位置. 插图是$x\leqslant3$的函数曲线, 绿色虚线是$x\to0$下的线性拟合${\cal{F}}(x)={x}/{2}$

Figure 10. Function relation between ${\cal{F}}(x)$ and x, the dashed blue line indicates the position of $ \mathcal{F}(x)=1 $. Insert is the function curve for $x\leqslant3$, the dashed green line is the linear fitting with ${\cal{F}}(x)={x}/{2}$ for $x\to0$

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表 1 狄拉克哈密顿量中利用狄拉克伽马矩阵表示的16个物理量及无序根据时间反演(${\cal{T}}$)、宇称(${\cal{I}}$)以及手性对称性(${\cal{C}}$)的分类. 转载自文献[49]

Table 1. Various types of physical quantities and disorder represented by fermionic bilinears ($i = 1, $$ 2, 3$), their symmetries under time-reversal (${\cal{T}}$), parity (${\cal{I}}$), and continuous chiral rotation (${\cal{C}}$). Reproduced with permission from Ref. [49]

Bilinear ($\hat{\cal{S} }_{\mathtt{A} }\propto\bar{\varPsi}{\boldsymbol{\gamma}}^{\mathtt{A} }\varPsi$)	Physical quantity	${\cal{T}}$	${\cal{I}}$	${\cal{C}}$	Disorder
$\bar{\varPsi}{\boldsymbol{\gamma}}^{0}\varPsi$	Total charge $(J^{0})$	$\checkmark$	$\checkmark$	$\checkmark$	$\varDelta$
$\bar{\varPsi}{\boldsymbol{\gamma}}^{0}{\boldsymbol{\gamma}}^{5}\varPsi$	Axial charge $(J^{a0})$	$\checkmark$	$\times$	$\checkmark$	$\varDelta_{{\rm{a}}}$
$\bar{\varPsi}\varPsi$	Scalar mass $({n}_{\beta})$	$\checkmark$	$\checkmark$	$\times$	$\varDelta_{{\rm{m}}}$
$\bar{\varPsi}{\rm{i}}{\boldsymbol{\gamma}}^{5}\varPsi$	Pseudo-scalar density $({n}_{{\rm{P}}})$	$\times$	$\times$	$\times$	$\varDelta_{{\rm{P}}}$
$\bar{\varPsi}{\boldsymbol{\gamma}}^{i}\varPsi$	Current $(J^{i})$	$\times$	$\times$	$\checkmark$	$\varDelta_{{\rm{c}}}$
$\bar{\varPsi}\gamma^{i}\gamma^{5}\varPsi$	Axial current $(J^{ai})$	$\times$	$\checkmark$	$\checkmark$	$\varDelta_{{\rm{ac}}}$
$\bar{\varPsi}{\rm{i}}{\boldsymbol{\gamma}}^{0}{\boldsymbol{\gamma}}^{i}\varPsi$	Electric polarization $({p}_{i})$	$\checkmark$	$\times$	$\times$	$\varDelta_{{\rm{p}}}$
$\bar{\varPsi }{\boldsymbol{\gamma}}^{5}{\boldsymbol{\gamma} }^{0}{\boldsymbol{\gamma} }^{i}\varPsi$	Magnetization $({m}_{i})$	$\times$	$\checkmark$	$\times$	$\varDelta_{{\rm{M}}}$

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表 2 4个库珀子通道$i = s, t_{0, \pm}$的库珀子能隙(以$ \ell_{e}^{-2} $为单位)和权重因子, 其中$\eta = mv^2/\mu$是狄拉克费米子的自旋极化. 转载自文献[72]

Table 2. Components of four Cooperon channels $i = s, t_{0, \pm}$ in the basis of spin-triplet and singlet $|s, s_{z}\rangle$, the Cooperon gap $\ell_{i}^{-2}$ in unit of the mean free path $\ell_{{\rm{e}}}^{-2}$ and the weighting factors $w_{i}$, where $\eta = mv^2/\mu$ is the orbital polarization of Dirac fermion. Reproduced with permission from Ref. [72]

i	Cooperon in $\|s, s_z\rangle$	$w_i$	$\ell_{\rm{e}}^2/\ell_i^2$
s	$\|0, 0\rangle$	$-\dfrac{(1-\eta^{2})^{2}}{2(1+3\eta^{2})^{2}}$	$\dfrac{(1-\eta^{2})\eta^{2}}{(1+\eta^{2})^{2}}$
$t_{+}$	$\|1, 1\rangle$	$\dfrac{4\eta^{2}(1+\eta^{2})}{(1+3\eta^{2})^{2}}$	$\dfrac{4(1-\eta)^{2}\eta^{2}}{(1+3\eta^{2})(1+\eta)^{2}}$
$t_{0}$	$\|1, 0\rangle$	0	$\infty$
$t_{-}$	$\|1, -1\rangle$	$\dfrac{4\eta^{2}(1+\eta^{2})}{(1+3\eta^{2})^{2}}$	$\dfrac{4(1+\eta)^{2}\eta^{2}}{(1+3\eta^{2})(1-\eta)^{2}}$

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map

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