Ferromagnetic-electrodes-induced Hall effect in topological Dirac semimetals

We propose an unconventional type of Hall effect in a topological Dirac semimetal with ferromagnetic electrodes. The topological Dirac semimetal itself has time-reversal symmetry, whereas attached ferromagnetic electrodes break it, causing the large Hall response. This induced Hall effect is a characteristic of the helical surface/edge states that arise in topological materials such as topological Dirac semimetals or quantum spin Hall insulators. We compute the Hall conductance/resistance and the Hall angle by using a lattice model with four-terminal geometry. For topological Dirac semimetals with four electrodes, the induced Hall effect occurs whether the current electrodes or the voltage electrodes are ferromagnetic. When the spins in electrodes are almost fully polarized, the Hall angle becomes as large as that of quantum Hall states or ideal magnetic Weyl semimetals. We show the robustness of the induced Hall effect against impurities and also discuss the spin injection and spin decay problems. This Hall response can be used to detect whether the magnetizations of the two ferromagnetic electrodes are parallel or antiparallel.

Figure 1

Schematic figures of the four-terminal geometries: (a) the current electrodes are ferromagnetic while the voltage electrodes are paramagnetic (I-FM), and (b) the voltage electrodes are ferromagnetic while the current electrodes are paramagnetic (V-FM). The aspect ratio of the system is set to be $L_x\times L_y\times L_z=6L\times L\times L$. The voltage electrodes of width 1 are attached at the center of the top/bottom sides. The blue and red allows on the surface of TDS represent the current carrying states with up and down spins: the helical surface states.

Figure 2

[(a), (b)] Hall resistance $R_{\text{H}}$, [(c), (d)] Hall conductance $G_{\text{H}}$, and [(e), (f)] source-drain conductance $G_{\text{SD}}$ as functions of the polarization of the FM electrodes $P=\frac{n_{\uparrow }-n_{\downarrow }}{n_{\uparrow }+n_{\downarrow }}$. The system size is $L=20$ with the electrode coupling strength $t^{\prime }=t/2$. [(g), (h)] Schematic figure of the current flow in TDSs with $P=1$, i.e., half-metallic electrodes. Blue/red arrows represent the flow of spin-up/down current in the helical surface states. The left panels corresponds to the I-FM and the right panels to the V-FM TDSs.

Figure 3

Hall angle $R_{\text{H}}/R_{\text{SD}}$ as a function of the polarization of the current electrodes $P$ for the (a) I-FM and (b) V-FM TDSs with $L=20$. The Hall angle for the I-FM TDS depends on the coupling strength of the electrodes $t^{\prime }=t/4$ (dotted), $t/2$ (solid), and $t$ (dashed), while that for the V-FM TDS does not.

Figure 4

Hall angle $R_{\text{H}}/R_{\text{SD}}$ as a function [(a), (b)] of the coupling between the sample and electrodes $t^{\prime }$ with $L=20$, and [(c), (d)] of the system size $L$ with $t^{\prime }=t/2$, for $P=1$ (solid), 0.8 (dashed), and $P=0.5$ (dotted), in the [(a), (c)] I-FM and [(b), (d)] V-FM TDSs. Schematic image of spin injection into the helical surface states for (e) a weak $t^{\prime }$ and a small $L$, and for (f) a sufficiently strong $t^{\prime }$ or a large $L$.

Figure 5

Schematic figure of the spin-dependent current flow in (a) I-FM and (b) V-FM TDSs with antiparallelly magnetized FM electrodes. Blue and red electrodes represent the polarization $P=1$ and $P=-1$, respectively. Blue/red arrows represent the flow of spin-up/down current in the helical surface states.

Figure 6

Color map of the Hall angle $\frac{R_{\text{H}}(P_1,P_2)}{R_{\text{SD}}(P_1,P_2)}$ with [(a), (b)] a large electrodes coupling $t^{\prime }=t$ and [(c), (d)] a small electrodes coupling $t^{\prime }=t/4$. Left/right panels are for the clean I-FM/V-FM TDS. The horizontal and vertical axes correspond to the polarization of the pair of FM electrodes. The system size is $L=20$.

Figure 7

Hall angle as a function of Fermi energy $E/t$ in (a) I-FM and (b) V-FM TDSs. The electrodes are half-metallic $P=1$ with coupling $t^{\prime }=t/4$, and the size $L=20$.

Figure 8

[(a), (b)] Averaged Hall conductance $\langle G_{\text{H}}\rangle$ and [(c), (d)] Hall angle $\langle R_{\text{H}}\rangle /\langle R_{\text{SD}}\rangle$ in normal metals (NMs). Left and right panels are for the I-FM and V-FM NM geometries, respectively. The horizontal axis $P$ is the polarization of FM electrodes. Spin scatterers of $W_s=0.5$ and $\rho =5\%$ are induced, and the system size $L=20$ and coupling $t^{\prime }=t/2$. Averaged over 20 000 impurity realizations and the error bars are less than the width of the lines.

Figure 9

Averaged Hall conductance $\langle G_{\text{H}}\rangle$ as a function of the position of the voltage electrodes $l$ for (a) I-FM NM and (b) I-FM TDS with spin scatterers of $W_s=0.5$ and $\rho =5\%$. Only the source electrode is polarized ($P_1=1$ and $P_2=0$). The system size $L_x\times L_y\times L_z$ is (a) $72\times 12\times 12$ and (b) $2000\times 20\times 20$. The electrodes coupling is (a) $t^{\prime }=t$ and (b) $t^{\prime }=t/4$. The solid lines are the functions $cexp(-l/\xi )$ with (a) $c=0.75$, $\xi =20$ and (b) $c=0.2$, $\xi =640$. Averaged over (a) 100 000 and (b) 800 impurity realizations, and the error bars are less than the size of the symbols.

Figure 10

Averaged Hall angle $\langle R_{\text{H}}\rangle /\langle R_{\text{SD}}\rangle$ for disordered (a) I-FM TDS, (b) V-FM TDS, (c) I-FM NM, and (d) V-FM NM as functions of the scattering amplitude $W_s$, with $\rho =5\%$. The system size $L=20$, polarization $P_1=P_2=1$, and coupling $t^{\prime }=t/2$. Averaged over [(a), (b)] 1000 and [(c), (d)] 20 000 impurity realizations so that the error bars are less than the width of the lines.

Figure 11

Hall angle $R_{\text{H}}/R_{\text{SD}}$ for 2D QSH insulators as a function of $P$ for the (a) I-FM and (b) V-FM geometries. The Hall angle in 2D (solid, $m_0/m_2=1$, $L_z=1$) is slightly larger than in 3D (dashed, $m_0/m_2=2$, $L_z=20$). The energy dependence of the Hall angle for the (c) I-FM and (d) V-FM geometries. The size $L_x=120,L_y=20$ and the coupling $t^{\prime }=t/4$.

Figure 12

Band structure of the nanowire (left panels) and V-FM Hall angle (right panels) as a function of energy in (a) ${\mathrm{Na}}_3\mathrm{Bi}$ and (b) ${\mathrm{Cd}}_3{\mathrm{As}}_2$. The system size $L=20$, and the electrodes are half-metallic $P=1$ with coupling $t^{\prime }=t$.