Non-local signatures of the chiral magnetic effect in Dirac semimetal Bi$_{0.97}$Sb$_{0.03}$

TL;DR

This study investigates the chiral magnetic effect in Dirac semimetal Bi$_{0.97}$Sb$_{0.03}$ using local and non-local measurements, revealing a long-lived chiral charge polarization indicative of topological properties.

Contribution

It provides experimental evidence of non-local signatures of the chiral magnetic effect in a Dirac semimetal, highlighting the extended lifetime of chiral charge polarization.

Findings

Chiral charge relaxation time exceeds Drude lifetime by over tenfold.

Non-local measurements confirm the topological origin of the CME.

Bi$_{0.97}$Sb$_{0.03}$ exhibits robust chiral magnetic signatures.

Abstract

The field of topological materials science has recently been focussing on three-dimensional Dirac semimetals, which exhibit robust Dirac phases in the bulk. However, the absence of characteristic surface states in accidental Dirac semimetals (DSM) makes it difficult to experimentally verify claims about the topological nature using commonly used surface-sensitive techniques. The chiral magnetic effect (CME), which originates from the Weyl nodes, causes an $\textbf{E}\cdot\textbf{B}$-dependent chiral charge polarization, which manifests itself as negative magnetoresistance. We exploit the extended lifetime of the chirally polarized charge and study the CME through both local and non-local measurements in Hall bar structures fabricated from single crystalline flakes of the DSM Bi$_{0.97}$Sb$_{0.03}$. From the non-local measurement results we find a chiral charge relaxation time which is over one order of magnitude larger than the Drude transport lifetime, underlining the topological nature of Bi$_{0.97}$Sb$_{0.03}$.

Equations

$${\chi }_{\pm }=\frac{1}{2\pi }{\oint }_{FS}({\nabla }_{\text{k}}\times A_{\text{k}\pm })\cdot d{S}_{\text{k}},$$

$$\frac{\partial {\rho }_{\pm }}{\partial t}=f({\Omega }_{\text{k}})\frac{e^3}{4{\pi }^2{\hslash }^2}\text{E}\cdot \text{B},$$

$$V_{\text{NL}}(x)=-{\left(\frac{B}{\gamma +B}\right)}^2{e}^{-|x|/L_{\text{CME}}}+V_{\text{Ohmic}}.$$

$$H_{\text{eff}}=v_{\parallel }(k_x{\sigma }_x{s}_z+k_y{\sigma }_y{s}_0)\pm v_z{k}_z{\sigma }_z{s}_0,$$

$$H_{\text{eff}}=\hslash v_F\left[\begin{array}{cccc}\pm \cos \phi & \sin \phi e^{i\theta }& 0& 0\\ \sin \phi e^{-i\theta }& \mp \cos \phi & 0& 0\\ 0& 0& \pm \cos \phi & -\sin \phi e^{-i\theta }\\ 0& 0& -\sin \phi e^{i\theta }& \mp \cos \phi \end{array}\right],$$

$$\displaystyle\begin{split}\psi_{+}^{1}(\textbf{{k}})&=\big{(}e^{i\theta}\cos{\varphi/2},\,\sin{\varphi/2},\,0,\,0\big{)}^{T}\\ \psi_{-}^{1}(\textbf{{k}})&=\big{(}\,0,\,0,\,-e^{-i\theta}\cos{\varphi/2},\,\sin{\varphi/2}\big{)}^{T}.\end{split}$$

$$\displaystyle\begin{split}\psi_{+}^{2}(\textbf{{k}})&=\big{(}\sin{\varphi/2}\,e^{i\theta},\,\cos{\varphi/2},\,0,\,0\big{)}^{T}\\ \psi_{-}^{2}(\textbf{{k}})&=\big{(}\,0,\,0,\,-\sin{\varphi/2}\,e^{i\theta},\,\cos{\varphi/2}\big{)}^{T}.\end{split}$$

$$A_{+}^1=-\frac{{\cos }^2\phi /2}{k\sin \phi }\hat{\theta }.$$

$${\Omega }_{+}^1=\nabla \times A_{+}^1=\frac{1}{k\sin \phi }\frac{\partial }{\partial \phi }(A_{+}^1\sin \phi )\hat{k}=\frac{1}{2k^2}\hat{k}.$$

$${\chi }_{+}^1=\frac{1}{2\pi }{\int }_0^{2\pi }{\int }_0^{\pi }{\Omega }_{+}^1{k}^2\sin \phi d\phi d\theta =1.$$

$$\begin{array}{rl}H& =\pm \hslash \omega ({\sigma }_{+}a+{\sigma }_{-}{a}^{†})\pm \hslash v_F{\sigma }_z{k}_z\\ H& =\pm \left[\begin{array}{cc}\hslash v_F{k}_z& \hslash \omega a\\ \hslash \omega a^{†}& -\hslash v_F{k}_z\end{array}\right],\end{array}$$

$$H=\left[\begin{array}{cc}\hslash v_F{k}_z& \hslash \omega \sqrt{n}\\ \hslash \omega \sqrt{n}& -\hslash v_F{k}_z\end{array}\right]$$

$$H_{\text{Liu}}=\hslash v_F{\sigma }_x(s_x{k}_y-s_y{k}_x)+\hslash v_F{\sigma }_y{s}_0{k}_z.$$

$$H_{\text{TFK}}=\hslash v_F{\sigma }_x(s_x{k}_x+s_z{k}_y)+\hslash v_F{\sigma }_y{s}_0{k}_z.$$

$$\displaystyle\begin{split}\psi_{+}(\textbf{{k}})&=\frac{1}{\sqrt{2}}\big{(}\cos{\varphi/2},-e^{i\theta}\sin{\varphi/2},e^{i\theta}\sin{\varphi/2},\cos{\varphi/2}\big{)}^{T}\\ \psi_{-}(\textbf{{k}})&=\frac{1}{\sqrt{2}}\big{(}\sin{\varphi/2},e^{i\theta}\cos{\varphi/2},e^{i\theta}\cos{\varphi/2},-\sin{\varphi/2}\big{)}^{T}.\end{split}$$

$$\begin{array}{rl}{H}_{\text{Liu}}^{\prime }& =\hslash v_F{\sigma }_x(s_x{k}_x+s_y{k}_y)+\hslash v_F{\sigma }_y{s}_0{k}_z\\ & =\hslash v_F\left[\begin{array}{cccc}0& 0& -i{k}_z& k_x-i{k}_y\\ 0& 0& k_x+i{k}_y& -i{k}_z\\ i{k}_z& k_x-i{k}_y& 0& 0\\ k_x+i{k}_y& i{k}_z& 0& 0\end{array}\right].\end{array}$$

$$H_{\text{Liu}}^{\prime }=\left[\begin{array}{cccc}0& 0& -i\hslash v_F{k}_z& \hslash \omega a\\ 0& 0& \hslash \omega a^{†}& -i\hslash v_F{k}_z\\ i\hslash v_F{k}_z& \hslash \omega a& 0& 0\\ \hslash \omega a^{†}& i\hslash v_F{k}_z& 0& 0\end{array}\right],$$

$$\det \left[\begin{array}{cccc}-E_n& 0& -i\hslash v_F{k}_z& \hslash \omega \sqrt{n}\\ 0& -E_n& \hslash \omega \sqrt{n}& -i\hslash v_F{k}_z\\ i\hslash v_F{k}_z& \hslash \omega \sqrt{n}& -E_n& 0\\ \hslash \omega \sqrt{n}& i\hslash v_F{k}_z& 0& -E_n\end{array}\right]=0.$$

$$\begin{array}{rl}{H}_{\text{Liu}}^{\prime \prime }& =U_{\text{R}}{H}_{\text{Liu}}^{\prime }{U}_{\text{R}}^{†}\\ & =\hslash v_F{\sigma }_x(s_y{k}_x+s_z{k}_y)+\hslash v_F{\sigma }_y{s}_0{k}_z\\ & =\hslash v_F\left[\begin{array}{cccc}0& 0& ky-i{k}_z& -i{k}_x\\ 0& 0& i{k}_x& -(ky+i{k}_z)\\ ky+i{k}_z& -i{k}_x& 0& 0\\ i{k}_x& -(ky-i{k}_z)& 0& 0\end{array}\right],\end{array}$$

$$H_{Liu}^{\prime \prime }=\left[\begin{array}{cccc}0& 0& \hslash \omega a& -i\hslash v_F{k}_x\\ 0& 0& i\hslash v_F{k}_x& -\hslash \omega a^{†}\\ \hslash \omega a^{†}& -i\hslash v_F{k}_x& 0& 0\\ i\hslash v_F{k}_x& -\hslash \omega a& 0& 0\end{array}\right].$$

$$\det \left[\begin{array}{cccc}-E_n& 0& \hslash \omega \sqrt{n}& -i\hslash v_F{k}_x\\ 0& -E_n& i\hslash v_F{k}_x& -\hslash \omega \sqrt{n}\\ \hslash \omega \sqrt{n}& -i\hslash v_F{k}_x& -E_n& 0\\ i\hslash v_F{k}_x& -\hslash \omega \sqrt{n}& 0& -E_n\end{array}\right]=0,$$

$$\widehat{{\sigma }_{\perp }}={\sigma }_0\left(\begin{array}{ccc}\frac{1}{1+(\mu B{)}^2}& \frac{\mu B}{1+(\mu B{)}^2}& 0\\ -\frac{\mu B}{1+(\mu B{)}^2}& \frac{1}{1+(\mu B{)}^2}& 0\\ 0& 0& 1\end{array}\right),$$

$$\widehat{{\sigma }_{\parallel }}={\sigma }_0\left(\begin{array}{ccc}1& 0& 0\\ 0& \frac{1}{1+(\mu B{)}^2}& \frac{\mu B}{1+(\mu B{)}^2}\\ 0& -\frac{\mu B}{1+(\mu B{)}^2}& \frac{1}{1+(\mu B{)}^2}\end{array}\right).$$

Full text

I I. Introduction

The electronic structure of bismuth-antimony alloys has been thoroughly studied in the 1960s and 70s and later regained attention when Bi1-xSbx was one of the first topological materials to be discovered Hsieh2008 . It was proposed that at the topological transition point, which is at $x\sim 0.03$, an accidental band touching in the bulk of Bi1-xSbx makes this material a Dirac semimetal (DSM) Kim2013 ; Chuan2018 ; Nagaosa2014 . A popular method to measure the topological nature of DSMs is through the detection of the chiral magnetic effect (CME) in electronic transport measurements. However, these signals are often obscured by the presence of parallel, non-topological conduction channels and are, more importantly, difficult to distinguish from other effects that may cause negative magnetoresistance, such as current jetting Liang2018 . Parameswaran et al. Parameswaran2014 proposed to measure the CME non-locally, using the extended lifetime of chirally polarized charge. For the Dirac semimetal Cd3As2, where two sets of Dirac cones with opposite chirality are separated in momentum space and protected by inversion symmetry, this measurement technique has been succesfully used to measure the CME Zhang2017 . In this work, we present evidence for the presence of the chiral magnetic effect in the accidental Dirac semimetal Bi0.97Sb0.03, based on local and non-local magnetotransport results on crystalline, exfoliated flakes.

In a general Dirac semimetal, 2 Weyl cones of opposite chirality (often labeled as isospin degree of freedom Parameswaran2014 ) are superposed in momentum space as depicted in figure 1(a). Their chiralities are defined through an integral of the Berry connection, $A_{\textbf{k}\pm}=i\langle\psi_{\pm}|\nabla_{\textbf{k}}\psi_{\pm}\rangle$, over the Fermi surface (FS):

[TABLE]

where $\bm{k}$ is the wave vector. Using a basic representation of the Weyl nodes, $\psi_{\pm}=(e^{-i\theta}\sin(\phi/2),-\cos(\phi/2))^{T}$ (where $\theta$ is the in-plane angle in momentum space, with $\theta=0$ along the $k_{x}$-axis, and $\phi$ is the polar angle with respect to $k_{z}$), one finds $\chi_{\pm}=\pm 1$. These nodes with opposite chiralities always come in pairs and, when the degeneracy of the Weyl cones is lifted, they are connected in momentum space by a surface state known as a Fermi arc. In mirror symmetry-protected Dirac systems, such as Cd3As2, different pairs of Weyl nodes can also be connected by Fermi arcs, which has been experimentally observed Moll2016 . Bi0.97Sb0.03 contains accidental Dirac points at the 3 L-points Kim2013 . The crossings at these Dirac points are not protected by any symmetry, and should not be connected by Fermi arcs. However, upon breaking time reversal symmetry with an external magnetic field, the Dirac cones split into two Weyl cones of opposite chirality, in which case Bi0.97Sb0.03 behaves very similar to Cd3As2. A more in-depth discussion on the topological properties of Bi0.97Sb0.03 compared to those of Cd3As2 can be found in appendix A.

The different chiral nodes correspond to a source and drain of Berry curvature ($\Omega_{\textbf{k}\pm}=\nabla_{\textbf{k}}\times A_{\textbf{k}\pm}$), which in turn behaves as a magnetic field in momentum space. Taking this momentum space analogue of the magnetic field into account in the equations of motion, one ends up with an expression for chiral charge pumping $\partial\rho_{\pm}/\partial t$ in external parallel electric and magnetic fields Nielsen1983 ; SonSpivak2013 :

[TABLE]

where $\bm{E}$ and $\bm{B}$ are the electric and magnetic fields respectively, and $f(\Omega_{\textbf{k}})$ represents the chirality-dependence. The result from this semi-classical argument can also been obtained in the quantum limit Zyuzin2012 . This chiral charge pumping creates a difference in chemical potential in the two Weyl cones, causing a net imbalance in chirality, which eventually relaxes by means of impurity scattering. However, the orthogonality of the two degenerate Weyl cones with different isospin and the large momentum difference between different valleys suppress these relaxation processes. As a consequence, chiral charge has an increased lifetime compared to the Drude transport lifetime, which is shortened by many low-energy scattering events. Through the continuity equations, this chiral charge imbalance contributes to the longitudinal conductivity and can be observed in magnetotransport measurements Kim2013 ; QLi2016 .

In Cd3As2, there are 2 scattering events that relax the chiral charge polarization: scattering between degenerate cones of different isospin, and intervalley scattering. Of these two, intervalley scattering should be expected to be the dominant factor as the momentum difference between the valleys is relatively small Zhang2017 . In Bi0.97Sb0.03, where the Dirac points reside at the 3 L-points, intervalley scattering requires a momentum transfer of the order $2\pi/a$, with $a$ being the lattice constant. Because of this required large momentum transfer, we argue that isospin-flip scattering through multiple scattering events is the likely dominant chiral relaxation process in Bi0.97Sb0.03.

II II. Methods and characterization

To characterize the Bi0.97Sb0.03 crystals (which are grown as described by Li et al. Chuan2018 ), several devices with contacts in a Hall bar configuration were fabricated. For all devices in this work, flakes of Bi0.97Sb0.03 were exfoliated from single crystals onto SiO2/Si*++* substrates. Contact leads were defined using standard e-beam lithography, followed by sputter deposition of 120 nm Nb with a few nm of Pd as capping layer, and lift-off. Then, the flakes themselves were structured using another e-beam lithography step, now followed by Ar+ milling. Magnetotransport measurements were conducted at 10 K in He-4 cryostats.

The inset of figure 2(a) shows a schematic overview of the measurement setup as used for local magnetotransport measurements. A current is sourced through the outer contacts and voltages are measured at the contacts in between. As observed earlier by Kim et al. Kim2013 , the magnetoresistance of Bi0.97Sb0.03, shown in figure 2(a), exhibits negative magnetoresistance for parallel electric and magnetic fields. This negative magnetoresistance is considered to be an indication of the CME Kim2013 ; HuiLi2013 ; Caizhen2015 ; QLi2016 . While the CME in Bi0.97Sb0.03 originates from the bulk electrons, the magnetoresistance data shows no Shubnikov-de Haas (SdH) oscillations corresponding to the bulk electron pockets, despite the low effective mass and high mobility of these electrons LiuAllen1995 . We will comment on this later. However, in accordance with previous measurements on flakes of Bi0.97Sb0.03, we do observe SdH oscillations originating from the bulk hole pocket in different samples made of the same single crystal (see appendix B).

Figure 2(b) shows the results of a Hall-type measurement. By tensor inversion of the measured longitudinal and Hall resistances, the longitudinal and transverse conductances were obtained. In figure 2(c), the conductances are fitted using a multi-band model, which takes two surface and two bulk conduction channels into account Chuan2018 . For the bulk electrons, we obtain a bulk electron density of $n_{B}^{e}=$ 3.0 $\cdot 10^{22}$ m*-3*, where we have used a flake thickness of 200 nm. For anisotropic Fermi velocities of $v_{1}=$ 0.8$\cdot 10^{5}$ m/s and $v_{2}=$ 10$\cdot 10^{5}$ m/s Hsieh2008 , this would indicate that the Fermi energy lies only $E_{F}=\hbar(\pi^{2}n_{B}^{e}ev_{1}v_{2}/3)^{1/3}=$ 13 meV above the Dirac point. The bulk charge carrier mobilities as obtained from the multi-band fit are lower than those found in unstructured devices Chuan2018 . This is in line with the absence of SdH oscillations in this measurement, which can be attributed to the device dimensions being of the same order as the cyclotron radius, and to disorder due to etching at the device edges. The consequential broadening of the Landau levels does not hamper the presence of the CME Zhang2017 . For a conservative effective mass of $m_{e,h}=0.05\,m_{0}$ Chuan2018 , the bulk electron and hole mobilities of $\mu=$0.65 m2/Vs give us an estimate of the momentum relaxation time: $\tau_{c}=\mu m/e\approx 1.8\cdot 10^{-13}$ s. This is in line with the absence of SdH oscillations in this measurement, which can be attributed to the device dimensions being of the same order as the cyclotron radius.

The non-local measurement setup is designed such that we can measure the coupling between the polarization of chirality and an external magnetic field at different distances from the polarization source, and is shown in figure 3. A current is sourced from contact 8 to 1, as indicated by the blue arrow. By applying a magnetic field parallel to the current, a chiral charge imbalance is induced. As the charge diffuses away from the polarizing source, the polarization becomes weaker and so does the measurable voltage of the polarized charge in the external magnetic field. We measure the voltages locally ($V_{L}\equiv V_{81}=V_{8}-V_{1}$) and non-locally ($V_{1}\equiv V_{72}$, $V_{2}\equiv V_{63}$ and $V_{3}\equiv V_{54}$). To be able to distinguish the Ohmic (i.e. normal diffusion) and CME signals, the voltage terminals are located at distances similar to both the expected Ohmic and chiral relaxation lengths.

When studying the CME, in the ideal case one measures the non-local response of the chiral anomaly as a function of the applied magnetic field only, i.e. keeping the applied electric field at the source contacts constant. However, due to the low resistance of these samples, we cannot voltage bias our sample and must resort to a current source, thereby causing the current to be constant as a function of the applied magnetic field. When measuring the local electric field, we find that this field is dependent on the magnetic field as well.

In order to obtain a data set with a constant electric field at the source contacts, measurements were performed by sweeping the source current from 0 to 10 $\mu$A for every magnetic field point, resulting in local and non-local magnetoresistance curves for a range of source currents $V_{\text{L,1,2,3}}(I_{s},B)$. An example of such a set of magnetoresistance curves for a fixed source current is shown in figure 4(a). The local voltage, $V_{\text{L}}(I_{s},B)$, is shown in the inset and should be kept constant, which is achieved by varying the source current $I_{s}(V_{\text{L}}=\text{cst.\,},B)$. In figure 4(b), we present the resulting voltages $V_{\text{NL}}(I_{s}(B),B)$, when the local electric field is kept constant in this way. Note that the local voltage, shown in the inset of figure 4(b), is now constant. Using this method, the magnetic field dependence of the non-local signals can be studied without side effects from the local magnetoresistance. For more information on this procedure, see appendix C.

III III. Results

The measured local voltage $V_{\text{L}}$, presented in the inset of figure 4(a), is in good agreement with the expected resistance based on the 4-point resistance, taking the size of the channels into account. This indicates that the effects of contact resistances are negligible. Figure 4(b) shows the non-local voltages measured at different distances as a function of the applied magnetic field. Here, the electric field at the source side is kept constant. At zero magnetic field, the measured voltages drop with increasing distance from the source. Furthermore, at all distances we observe a decreasing voltage with increasing magnetic field, which we attribute to the CME. The CME is not dominant for the entire magnetic field range as both at low and high fields, the voltage increases slightly with magnetic field. Kim et al. attribute the low field MR to weak anti-localization Kim2013 . High field deviations from the CME signal may originate from higher order terms, which are not taken into account in this work.

We have identified three possible causes of the small asymmetry of the data. First and foremost is the device asymmetry with respect to the source channel, but variations in sample thickness and imperfections in the structuring process may also be contributing factors. Figure 4(c) shows the symmetrized non-local voltages. We fitted the intermediate field data between 2 T and 5.5 T with a model that subtracts the constant Ohmic contribution, and extracts the diffusion of the chiral charge as given by Parameswaran et al. Parameswaran2014 :

[TABLE]

Here $x$ is the distance between the source and the non-local probes, $\gamma$ is proportional to the conductance at the metal contact and $L_{\text{\text{CME}}}$ is the diffusion length of the chiral charge polarization. The fit agrees well with the data for intermediate magnetic fields and it gives a diffusion length of 1.07 $\mu$m. Using $L_{\text{\text{CME}}}=\sqrt{D\,\tau_{\text{\text{CME}}}}$ and $D=(1/3)\,v_{F}\,l_{m}$, with $l_{m}=v_{F}\,\tau_{\text{\text{CME}}}$, we find a chiral polarization lifetime of $\tau_{\text{\text{CME}}}\approx 3.7\cdot 10^{-12}$ s, which is over one order of magnitude longer than the Drude transport lifetime $\tau_{c}\approx 1.4\cdot 10^{-13}$ s.

The dependence of the normalized Ohmic and CME contributions to the measured voltages is presented in figure 5, along with an atomic force microscopy (AFM) image of the device. Here, the amplitudes of the best fits are used to represent the CME strength. The measured Ohmic (zero-field) contribution at all voltage terminals is shown for comparison. The Ohmic contribution of the device is also modeled numerically, where the shown solid curve is a line cut along the horizontal part of the device (see appendix Dl). The simulated Ohmic contributions fit very well to the measured data, emphasizing the good homogeneity of the flake. The most notable feature of figure 5 is that the chiral polarization of the charge carriers has a significantly longer relaxation length than the Drude transport lifetime.

To study the temperature dependence of the CME in Bi0.97Sb0.03, another device was measured at higher temperatures. This device has larger channel widths and spacing as can be seen in the AFM image of the device in figure 6(a). Figure 6(b) shows the measured voltages at the non-local contacts closest to the source, which present the most striking features. It is apparent that for all temperatures displayed in this figure, the magnetoresistance is strongly negative and that this non-local voltage decreases as temperature increases. Through the same fitting procedure described above, the chiral charge polarizarion diffusion length is extracted for each temperature and shown in figure 6(c) (more information on this procedure can be found in appendix F). In contrast to what has been found for Cd3As2 Zhang2017 , the chiral diffusion length in Bi0.97Sb0.03 does not seem to be constant with increasing temperature, but rather decreases linearly. The increasing relaxation rate $\tau_{\text{\text{CME}}}^{-1}$ with increasing temperature, indicates that inelastic processes are responsible for the relaxation of the chiral polarization in Bi0.97Sb0.03.

IV IV. Conclusions

In summary, we studied the chiral magnetic effect in Bi0.97Sb0.03 through transport measurements in local and non-local configurations. First, we characterized the exfoliated crystalline Bi0.97Sb0.03 flakes using a Hall-type measurement. Here, we identified contributions from two bulk bands, one of them corresponding to the electron pockets with a linear dispersion and a Fermi level close to the Dirac point. When subjected to parallel electric and magnetic fields, local measurements on our Bi0.97Sb0.03 devices show a pronounced negative magnetoresistance, an indication of the chiral magnetic effect.

In a non-local configuration, we measured voltages that strongly decrease with increasing magnetic field, which we attribute to the chiral magnetic effect. As voltage contacts are located further away from the polarization source, the measured chiral magnetic effect weakens. This weakening occurs at a much lower rate than the decay of the Ohmic signal, which is a consequence of the long lifetime of the chiral polarization $\tau_{\text{CME}}$. Furthermore, measurements at different temperatures show that the chiral charge diffusion length decreases with increasing temperature, emphasizing the role of inelastic scattering in the chiral charge relaxation process in Bi0.97Sb0.03. Both local and non-local measurements provide strong evidence of the presence of the chiral magnetic effect in the three-dimensional Dirac semimetal Bi0.97Sb0.03.

Acknowledgements.

V Acknowledgements

The authors would like to thank T. Hashimoto for fruitful discussions and acknowledge financial support from the European Research Council (ERC) through a Consolidator Grant and from the Netherlands Organization for Scientific Research (NWO) through a Vici Grant.

Appendix A Appendix A: Effective Hamiltonian and the chiral magnetic effect

A.1 A1. Cd3As2

Cd3As2 is a Dirac semimetal belonging to the point group $D_{\text{\text{4h}}}$. The 2-fold inversion symmetry in this system ensures two topologically protected Dirac points along the $k_{z}$-axis. To explore the electronic structure of Cd3As2 we utilize a linearized version of the $k\cdot p$ model Hamiltonian as defined by B.J. Yang and Nagaosa SNagaosa2014 :

[TABLE]

where $\pm$ is used to switch between the nodes residing at $k_{z}^{\prime}=\pm k_{D}$ and $k_{z}$ is measured relative to the center of the node. The total angular momentum and orbital degrees of freedom are indicated by $s$ and $\sigma$, respectively. We can see that Cd3As2 exhibits orbital-momentum locking and is of the form $(\bm{k}\cdot\bm{\sigma})s_{0}$. Assuming isotropic spin-orbit coupling strength ($v_{\parallel}=v_{z}=v_{F}$) and switching to spherical coordinates with $\theta$ the angle in the $xy$-plane and $\varphi$ the polar angle measured from $+k_{z}$, we get

[TABLE]

where the basis is taken as $\{|s\uparrow\rangle,|p_{x}+ip_{y}\uparrow\rangle,|s\downarrow\rangle,|p_{x}-ip_{y}\downarrow\rangle\}$. In this case, the eigenenergies are simply $E_{\pm}=\pm v_{F}\textbf{{k}}$. For a system doped to the n-type regime so that $E>0$, we can find the normalized spinors at the $k_{z}=-k_{D}$ Dirac point, which we will refer to as node 1:

[TABLE]

For node 2, located at $k_{z}=+k_{D}$, we have

[TABLE]

To find the topological properties of node 1, we first need to find the Berry connection $A_{+}^{1}=i\langle\psi_{+}^{1}(\textbf{{k}})|\nabla_{k}\psi_{+}^{1}(\textbf{{k}})\rangle$. Because in spherical coordinates $\nabla_{k}=\frac{\partial}{\partial k}\hat{k}+\frac{1}{k}\frac{\partial}{\partial\varphi}\hat{\varphi}+\frac{1}{k\sin{\varphi}}\frac{\partial}{\partial\theta}\hat{\theta}$, we get

[TABLE]

From this, we find that the Berry curvature

[TABLE]

This Berry curvature can be seen as a magnetic field in $k$-space and indicates the monopole character of the node. The flux coming from each node gives us, once normalized by $2\pi$, the “chirality” or “Chern number” of the node:

[TABLE]

Similarly, we find $\chi_{-}^{1}=-1$ and for node 2 we find $\chi_{+}^{2}=-1$ and $\chi_{-}^{2}=+1$. First of all, this shows us that the Dirac points in this material have Chern number $\pm 1$, which makes Cd3As2 topological. Furthermore, it seems that the degenerate Dirac cones have opposite chirality, and that the nodes at $k_{z}\rightarrow\mp k_{D}$ also have opposite chiralities (figure S1(a)), reflecting the 2-fold inversion symmetry of the system.

As the $k$-space analogue, the Berry curvature also has to be taken into account in the equations of motion that describe the equilibrium state of the system. Since the chiralities tell us that some Dirac points act as sources of Berry curvature and others as drains, the signs of the forces due to the Berry curvature are also opposite. This results in a net imbalance between cones of opposite chirality, a so called “chiral charge imbalance”. In this section, we will derive the resulting charge pumping by studying the Landau levels.

As Eqn. (S2) consists of 2 decoupled Weyl cones of the form $H=\pm v_{F}\bm{\sigma}\cdot\bm{p}$, where $\bm{\sigma}$ denotes the orbital degree of freedom, one can easily find the Landau level dispersion relations as described by Zyuzin et al. SZyuzin2012 . In short, one includes the vector potential correction to the momentum as $\bm{\pi}=\bm{p}+e\bm{A}$ and rewrites the Hamiltonian in terms of $a=(\pi_{x}-i\pi_{y})/\sqrt{2\hbar eB}$ and $a^{\dagger}=(\pi_{x}+i\pi_{y})/\sqrt{2\hbar eB}$ (for a magnetic field $\bm{B}=B_{z}$):

[TABLE]

where $\sigma_{\pm}=(\sigma_{x}\pm i\sigma_{y})/2$. Assuming a wavefunction of the form $\psi_{n}=(u_{n}|n-1\rangle,v_{n}|n\rangle)^{T}$, the creation and annihilation operators can be replaced according to $a|n\rangle=\sqrt{n}|n-1\rangle$ and $a^{\dagger}|n-1\rangle=\sqrt{n}|n\rangle$. To find the dispersion relation one can solve $\det{(H-\sigma_{0}E_{n})}=0$ with

[TABLE]

to find $E_{n}=\pm\hbar\omega\sqrt{n+(v_{F}k_{z}/\omega)^{2}}$. Here, $\pm$ corresponds to the chirality of the accompanying wavefunction, as shown in Eqn. S7. For $n=0$, this gives us the dispersion of the zeroth Landau level: $E_{0}=\pm\hbar v_{F}k_{z}$, which describes a linear dispersion with a Fermi velocity parallel or anti-parallel to $\bm{B}=B_{z}$, depending on the chirality of the Weyl node. It can be shown in a rather easy way that a dispersion relation of this form leads to a chiral charge imbalance SZyuzin2012 . The resulting charge pumping between Dirac cones of opposite chirality is balanced by relaxation processes (see figure S1(b)). Orthogonality of isospin and the large $\Delta\textbf{{k}}$ involved with intervalley scattering significantly increase the relaxation times ($\tau_{i}$ and $\tau_{v}$ respectively), so that the chiral charge imbalance becomes a relevant quantity in transport measurements.

A.2 A2. Bi1-xSbx

Bi1-xSbx belongs to the point group $D_{\text{\text{3D}}}$ and has 3-fold rotation symmetry. There are no symmetries in this system that ensure the presence of Dirac cones, and the Dirac cones that do reside at the L-points are therefore labelled as accidental band touchings. Due to the 3-fold rotation symmetry, Bi1-xSbx has three accidental Dirac cones, separated by 120∘. Topological Bi compounds can generally be described using the model Hamiltonian for topological insulators as developed by Liu et al. SLiu2010 . Assuming isotropic Fermi velocities and taking $m\rightarrow 0$, the linearized Hamiltonian can be written as

[TABLE]

Teo et al. STeo2008 described the Dirac physics around a single L-point in Bi1-xSbx in great detail using a modified form of this Hamiltonian, which can be obtained as $H_{\text{\text{TFK}}}=U_{\text{\text{R}}}^{\dagger}H_{\text{\text{Liu}}}U_{\text{\text{R}}}$ with $U_{\text{\text{R}}}=(s_{0}-is_{x}-is_{y}-is_{z})/2$ (a $-2\pi/3$ rotation along the [111]-axis in spin space) and taking $v_{x}\rightarrow-v_{x}$:

[TABLE]

The spinor part of the two wavefunctions for the conduction band side of the cone can be written as

[TABLE]

In the same manner as for Cd3As2, we can use these spinors to find the Berry curvature $\Omega_{+}^{1}=\nabla\times A_{\pm}=\pm\frac{1}{2k^{2}}\hat{k}$ and the chirality $\chi_{\pm}=\pm 1$. The unitary transformation $U_{\text{\text{R}}}$ can be used to transform $\psi_{\text{\text{TFK}}}$ into $\psi_{\text{\text{Liu}}}$, so that the topological properties of $H_{\text{\text{Liu}}}$ are the same as those found for $H_{\text{\text{TFK}}}$. To find out of this leads to the chiral magnetic effect in Bi1-xSbx, we consider the full 4x4 Hamiltonian.

For a magnetic field along the $c$-axis we use $H_{\text{\text{Liu}}}$, with the spin space rotated by 90∘ along the $s_{z}$-axis so that, in the basis $\{|\sigma_{1}\uparrow\rangle,|\sigma_{1}\downarrow\rangle,|\sigma_{2}\uparrow\rangle,|\sigma_{2}\downarrow\rangle\}$, the effective Hamiltonian takes the form:

[TABLE]

The orbital shift due to the vector potential is included as $\bm{\pi}=\bm{p}+e\bm{A}$, which we write again as creation and annihilation operators: $a=(\pi_{x}-i\pi_{y})/\sqrt{2\hbar eB}$ and $a^{\dagger}=(\pi_{x}+i\pi_{y})/\sqrt{2\hbar eB}$. The corresponding raising and lowering matrices are $s_{\pm}=\sigma_{x}(s_{x}\pm is_{y})$. Then

[TABLE]

where $\hbar\omega=v\sqrt{2\hbar eB}$. As a trial wavefunction, we double the basis used for the 2x2 case: $\psi_{n}=(u_{n}^{1}|n-1\rangle,v_{n}^{1}|n\rangle,u_{n}^{2}|n-1\rangle,v_{n}^{2}|n\rangle)^{T}$. To find the dispersion relations for this wavefunction, we solve

[TABLE]

This gives us $E_{n}=\pm\hbar\omega\sqrt{n+(v_{F}k_{z}/\omega)^{2}}$, exactly the same result as we found for Cd3As2, including the linear zeroth Landau levels.

The simplified Cd3As2 Hamiltonian we employed earlier has isotropic orbital-momentum locking $\hbar v_{F}\bm{\sigma}\cdot\bm{k}$, so that the Landau level formation is the same for magnetic fields in all directions. $H_{\text{\text{Liu}}}^{\prime}=\hbar v_{F}\sigma_{x}(s_{x}k_{x}+s_{y}k_{y})+\hbar v_{F}\sigma_{y}s_{0}\,k_{z}$ has a clear difference between the spin-momentum locking in the in-plane $k_{x}$ and $k_{y}$ directions, and the $k_{z}$ direction, so a different response to magnetic field from different directions can be expected. To study the effect of a magnetic field along the $k_{x}$-axis ($B_{x}$), we first perform a rotation along the [111]-axis in spin space ($U_{\text{\text{R}}}=(s_{0}-is_{x}-is_{y}-is_{z})/2$) to get:

[TABLE]

which makes the following operations easier. With the modified raising and lowering matrices $s_{\pm}=\sigma_{x}s_{z}\pm i\sigma_{y}s_{0})$ and operators $a=(\pi_{y}-i\pi_{z})/\sqrt{2\hbar eB}$ and $a^{\dagger}=(\pi_{y}+i\pi_{z})/\sqrt{2\hbar eB}$, we can rewrite $H_{\text{\text{Liu}}}^{\prime\prime}$ into

[TABLE]

With $\psi_{n}=(u_{n}^{1}|n-1\rangle,v_{n}^{1}|n\rangle,v_{n}^{2}|n\rangle,u_{n}^{2}|n-1\rangle)^{T}$, we find from

[TABLE]

that the zeroth Landau level disperses as $E_{0}=\pm\hbar v_{F}k_{x}$, which is same linear dispersion as for the $B_{z}$ field.

For a magnetic field in the $k_{y}$-direction ($B_{y}$), we can rotate $H_{\text{\text{Liu}}}$ around the [111]-axis in spin space in the different direction to get $H_{\text{\text{Liu}}}^{\prime\prime\prime}=U_{R}^{\dagger}H_{Liu}^{\prime}U_{R}=\hbar v_{F}\sigma_{x}(s_{z}k_{x}+s_{x}k_{y})+\hbar v_{F}\sigma_{y}s_{0}\,k_{z}$, which gives a result analogous to the $B_{x}$ case: $E_{0}=\pm\hbar v_{F}k_{y}$. This shows that the chiral zeroth Landau levels, and therefore also the CME, are expected to occur in every direction in Bi1-xSbx.

Appendix B Appendix B: Local transport

Hall bar samples have been fabricated to characterize the Bi0.97Sb0.03 flakes used for this work. The Hall bars were fabricated by using standard e-beam lithography, followed by sputter deposition of Nb with a capping layer of a few nm of Pd. Measurements were performed at $T=$10 K so as to not induce superconductivity in the Bi0.97Sb0.03 flake.

The longitudinal magnetoresistance, as shown in figure S2(a), exhibits a rather sharp angle dependence, with a negative magnetoresistance of about 12% for parallel electric and magnetic fields. Although subtle, Shubnikov-de Haas (SdH) oscillations can be observed. Figure S2(b) shows the SdH oscillations for perpendicular electric and magnetic fields, extracted by subtracting a simple smooth function from the measured data. The inset shows the obtained FFT spectrum, which reveals a single oscillation frequency $F\approx$17 T. The oscillation frequency shifts upon increasing the angle between the magnetic field and the $c$-axis of the crystal. Figure S2(c) shows the angle-dependence of this frequency, fitted with an ellipsoidal Fermi surface with an anisotropy of $k_{z}=3.3\cdot k_{x,y}$. This indicates that the oscillations originate from the bulk hole pocket SChuan2018 .

Appendix C Appendix C: Constant local electric field

When studying the CME, in the ideal case one measures the non-local response of the chiral anomaly as a function of the applied magnetic field only, i.e. keeping the applied electric field at the source contacts constant. However, due to the low resistance of our sample, we can not voltage bias our sample and must resort to a current source, thereby causing the current to be constant as a function of the applied magnetic field. When measuring the local electric field, we find that this field is dependent on the magnetic field as well. Hence, sweeping the magnetic field changes both the strength of E and B, as follows directly from the chiral magnetic effect SQLi2016 .

In order to obtain a data set with a constant electric field at the source contacts, we measured the local and non-local voltages as a function of both the applied current and magnetic field. The dependence of the local voltage on both parameters can be seen in figure S3(a). From this map, we can find a line for which $V_{\text{L}}(I_{s},B)$ is constant. This line is plotted on top of the map. By retracing the same line on the non-local voltage maps, we can extract the non-local voltages that correspond to the same constant local electric field and study the magnetic field dependence. In other words, we measure the local voltage $V_{\text{L}}(I_{s},B)$, extract the source current $I_{s}(V_{\text{L}}=\text{cst.\,},B)$, and use this to find the non-local voltages for a constant local electric field: $V_{\text{NL}}(I_{s}(B),B)$.

To increase the accuracy of the maps - we measured the field dependent data for 51 different values of the excitation current - we linearly interpolated our data as a function of $I_{s}$. Although recent work on Bi1-xSbx suggests that Ohm’s law is violated in this system SShin2017 , our generated electric field is well above the non-linear regime.

Figure S3(b) shows the raw data, i.e. when we omit this procedure and would measure the voltages by applying a constant current. We clearly observe that the local voltage is not constant with respect to the magnetic field. Figure S3(c) shows the data when we follow our procedure. The local voltage is now constant as a function of the magnetic field and the large dips that were present in figure S3(b) are less pronounced in figure S3(c).

Appendix D Appendix D: Modelling the Ohmic contribution

The measured data comprises a combination of a chiral signal and a normal, i.e. Ohmic, contribution. To gain better insight in the Ohmic contribution, we modelled this contribution in COMSOL Multiphysics. The device geometry used for the measurements was replicated in COMSOL. One of the source leads was defined as a current source, sourcing 1 $\mu$A, while the other current lead was set as ground. The other terminals were voltage probes.

For simulations with perpendicular electric and magnetic fields (i.e. $B$ in the $z$-direction and $I$ in the $x$-direction), the three-dimensional conductivity tensor for a single band can be obtained from the Drude model SDatta ;

[TABLE]

where $\sigma_{0}=ne\mu$. For parallel electric and magnetic fields (i.e. $B$ in the $x$-direction), one can show that the conductivity tensor is given by

[TABLE]

Upon changing the tensor in COMSOL we effectively rotate the magnetic field with respect to the sample. Figure S4 shows the results of simulations for different structures (depicted as insets within the figures) and different orientations of the magnetic field with respect to the electric field. For every simulation we used a carrier density of $n=4\cdot 10^{17}$ m*-2* and mobility $\mu=2$ m2/Vs.

Figure S4(a) shows the simulation results for a magnetic field perpendicular to the plane (and thus to the current). The sample is shaped into a non-local structure with a geometry as used in the experiments. We observe decreasing non-local voltages at the nearest contact with increasing perpendicular magnetic field. As the used model only captures the Drude conductivity, the observed decrease cannot be explained by a chiral component. The contacts further away from the source all returned negigibly small voltages.

Figure S4(b) shows the non-local voltages for magnetic fields parallel to the electric source field. The strongly decreasing magnetoresistance can be explained by an effect called current jetting. Current jetting, as follows from the conductivity tensor for parallel fields, is an effect where the magnetic field suppresses current flow in the transverse direction SPippard ; Hirsch; dosReis. This reduces the amount of current that spreads out towards the non-local voltage probes, resulting in a decrease in the voltages that are measured. The fact that current jetting can also introduce negative magnetoresistance, is (among other effects) one of the main reasons that observing negative magnetoresistance itself is not a proof of observing the chiral anomaly. However, the simulated non-local voltages decrease (even vanish) much faster than observed in experiments. From this, we conclude that the negative magnetoresistance in our samples is not caused by current jetting.

In panel (a), we observed a decreasing non-local voltage while the electric and magnetic fields were perpendicular to each other. To further investigate the nature of this decrease, simulations were performed on a rectangular, unstructured flake with 8 contacts, as shown in figure. S4(c). The non-local voltages now show an upturn with increasing parallel magnetic field. By comparing panels (a) and (c), which make use of the same conductivity tensor and only differ in terms of geometry, we conclude that the decreasing non-local voltage in panel (a) is caused by the geometry of the device.

Appendix E Appendix E: Non-local measurements at perpendicular electric and magnetic fields

Measurements have been performed for perpendicular electric and magnetic fields. In figure S5 we present the data for two different samples. Striking is that for panel (a) we observe a decrease of $V_{1}$ as a function of magnetic field, while in the panel (d) we only observe the standard upturn with magnetic field, similar to the data measured for the Hall bar sample.

In the simulations of the previous section, we found that this negative magnetoresistance in perpendicular fields is a geometrical effect. Comparing panels (a) and (b) of figure S5 to panel (a) of the simulations (figure S4), and panels (c) and (d) of figure S5 to panel (c) of the simulations, we see that the experimental data confirms this. That the simulations are much more distinctive on the two different geometries, is because of the simulation taking only a single band into account for the conductivity, while in reality the conductivity is mediated by four channels. Furthermore, as can be seen from panel (c), the flakes are not always uniform in thickness and are not as well defined as the geometries in our simulations.

Appendix F Appendix F: Temperature dependence of the chiral diffusion length

The device used in the main text to study the temperature dependence had one broken voltage probe. As a consequence, only 2 non-local voltages were measured. This device has contacts spaced 1.5 $\mu$m apart, with twice this spacing in the middle, so that the contacts $V_{\text{NL1,2}}$ are located at 3.0 $\mu$m and 7.5 $\mu$m respectively.

The measurements for this device were performed using a single, constant current. Because of the linearity of the measured voltage as a function of the applied current (see section C), we can still extrapolate the data by using the recorded data point and the fact that $V_{L}(I_{s}=0)=0$ V. Then, the procedure as outlined in section C was followed to convert the mapped data into data for which the local electric field is effectively constant. In figure S6 we present some examples of fits to the data to extract the temperature-dependent chiral diffusion length as presented in the main text. As described in the main text, all fits were performed on the intermediate field data (2 T-5.5 T).

Lastly, figure S7 shows the temperature dependence of the non-local voltage of the smallest sample shown in the main text. Here, we also observe a decreasing strength of the anomaly with increasing temperature. The observed peak in the curve drifts to higher magnetic fields with temperature. For temperatures above 100 K, we do not observe negative magnetoresistance.