Interplay of Lorentz-Berry forces in position-momentum spaces for valley-dependent impurity scattering in $\alpha -T_3$ lattices

The Berry-phase mediated valley-selected skew scattering in $\alpha -T_3$ lattices is demonstrated. The interplay of Lorentz and Berry forces in position and momentum spaces is revealed and analyzed. Many-body screening of the electron-impurity interaction is taken into account to avoid overestimation of back and skew scattering of electrons in the system. Triplet peak from skew interactions at two valleys is found in near-vertical and near-horizontal forward- and backward-scattering directions for small Berry phases and low magnetic fields. Magnetic-field dependence in both nonequilibrium and thermal-equilibrium currents is also presented for valley-dependent longitudinal and transverse transports mediated by a Berry phase. Mathematically, two Boltzmann moment equations are employed for computing scattering-angle distributions of nonequilibrium skew currents by using microscopic inverse energy- and momentum-relaxation times. Meanwhile, a valley-dependent unbalanced thermal-equilibrium anomalous Hall current induced by the Berry force in momentum space, due to different mobilities for two valleys, is also computed for comparisons.

Figure 1

(a) $\alpha -T_3$ lattice with three atoms ($A, B, C$) per unit cell within the $(x,y)$ plane, where the $\alpha =tan\varphi$ parameter characterizes the ratio of the bonding strengths between $A-C$ and $A-B$ atoms. (b) Illustration for a band structure featuring three bands of $\alpha -T_3$ lattice, where the middle one is flat. (c) Schematic diagram for a scattering angle ${\beta }_s$ of an incident electron with wave vector ${\mathbf{k}}_{\mathrm{in}}$ by different impurities at two valleys characterized by $\tau =\pm 1$ under an applied electric field ${\mathbf{E}}_x$ along the $x$ direction, where an external nonquantizing magnetic field ${\mathbf{B}}_z$, and the internal Berry curvature ${\mathbf{\Omega }}_k$ as well, are along the $z$ direction and the longitudinal (transverse) scattering is labeled by $L$ ($T$), respectively. Here the Berry phases are calculated in Sec. 1 as ${\mathrm{\Phi }}_s^{\tau }\left(\varphi \right)=\tau \pi cos2\varphi$ for $s=\pm 1$ and ${\mathrm{\Phi }}_0^{\tau }\left(\varphi \right)=-2{\mathrm{\Phi }}_s^{\tau }\left(\varphi \right)$. For the sake of clearness, we call the geometry phase $\varphi$ as the “Berry phase” in this paper.

Figure 2

Calculated real part of the polarization function $\mathrm{Re}\left[{\mathcal{Q}}_{\varphi }({\mathbf{q}}_{\parallel },\omega )\right]$ from Eq. (B2) with $\varphi =\pi /4$ (dice, green), $\pi /6$ (black), $\pi /8$ (blue), and 0 (graphene, red) as a function of $q_{\parallel }$ at $\hslash \omega =0$ (a) and $\hslash \omega /E_F=0.5$ (b), as well as a function of $\hslash \omega$ at $q_{\parallel }/k_F=0.3$ (c) and $q_{\parallel }/k_F=0.7$ (d). Here the unit of $(k_F^2/E_F)$ has been used for scaling ${\mathcal{Q}}_{\varphi }({\mathbf{q}}_{\parallel },\omega )$ in Eq. (B2).

Figure 3

Calculated square of the dimensionless form factor $|{\mathcal{F}}_{\tau ,\varphi }(k_{\parallel },{\beta }_s){|}^2$ from Eq. (15) with $\varphi =\pi /6$ and $\pi /8$ as a function of ${\beta }_s$ at $k_{\parallel }/k_F=0.8$ (a) and as a function of $k_{\parallel }$ at ${\beta }_s=\pi /8$ (b) for $\tau =1$ (black) and $\tau =-1$ (red); as well as thermally averaged energy-relaxation time ${\overline{\tau }}_{\varphi }(k_F,\tau )$ calculated from Eq. (14) as a function of $\varphi$ for $\tau =1$ (black) and $\tau =-1$ (green) under both unscreened (c) and screened (d) conditions. Here the unit of $({\pi }^2\hslash /4E_F)$ has been used for scaling ${\overline{\tau }}_{\varphi }(k_F,\tau )$.

Figure 4

Calculated diagonal elements $b_{xx}(k_F,\tau ,\varphi )$ for $\tau =1$ (a) and both $b_{xx}(k_F,\tau ,\varphi )$ and $b_{yy}(k_F,\tau ,\varphi )$ for $\tau =-1$ (b) of the inverse momentum-relaxation-time tensor $\overleftrightarrow{\mathcal{T}}{}_p^{-1}(k_F,\tau ,\varphi )$ in Eq. (17) as functions of $\varphi$, where the difference $\delta b\equiv b_{xx}(k_F,\tau ,\varphi )-b_{yy}(k_F,\tau ,\varphi )$ for $\tau =1$ is also presented in the inset (i1) and the dashed line corresponds to $\delta b=0$ to highlight its sign switching. Here the unit of $1/{\tau }_0=4E_F/{\pi }^2\hslash$ has been used for scaling $b_{xx}(k_F,\tau ,\varphi )$ and $b_{yy}(k_F,\tau ,\varphi )$.

Figure 5

Calculated diagonal elements ${\mu }_{xx}(k_F,\tau ,\varphi )$ (a) and (b) and ${\mu }_{yy}(k_F,\tau ,\varphi )$ (e) and (f), as well as off-diagonal element ${\mu }_{xy}(k_F,\tau ,\varphi )$ in logarithm scale (c) and (d), of the mobility tensor $\overleftrightarrow{\mathbf{\mu }}(k_F,\tau ,\varphi )$ given by Eq. (18) as a function of $B_z$ with $\varphi =\pi /4$ (green), $\varphi =\pi /6$ (blue), $\varphi =\pi /8$ (red), and $\varphi =0$ (black) for $\tau =1$ (a), (c), and (e) and $\tau =-1$ (b), (d), and (f). Here ${\mu }_0=4e/{\pi }^2\hslash k_F^2$ has been used for scaling all elements of $\overleftrightarrow{\mathbf{\mu }}(k_F,\tau ,\varphi )$ and $B_0=\hslash k_F^2/e$.

Figure 6

Calculated integrands of longitudinal $j_L(\tau ,\varphi )$ (a) and (b) and transverse $j_T(\tau ,\varphi )$ (c) and (d) scattering currents from Eq. (13) as a function of ${\beta }_s\in [-\pi ,\pi ]$ with $\varphi =\pi /4,B_z/B_0=0.01$ (blue), $\varphi =\pi /6,B_z/B_0=0.01$ (red), and $\varphi =\pi /6,B_z/B_0=0.005$ (black) for $\tau =1$ (a) and (c) and $\tau =-1$ (b) and (d). Here the unit of $j_0=n_{i}e{v}_F$ has been used for scaling both $j_L(\tau ,\varphi )$ and $j_T(\tau ,\varphi )$ and $B_0$ is given in Fig. 5.

Figure 7

(a)–(d) Back-scattering current-distribution component $C_x(k_F,\tau ,\varphi ,{\beta }_s)$ from Eq. (19) as a function of $B_z$ (a) and (b) with $\varphi =\pi /4,{\beta }_s=\pi /6$ (green), $\varphi =\pi /6,{\beta }_s=\pi /6$ (black), $\varphi =\pi /4,{\beta }_s=\pi /3$ (blue), and $\varphi =\pi /6,{\beta }_s=\pi /3$ (red) for $\tau =1$ (a) and $\tau =-1$ (b), as well as a function of ${\beta }_s$ with $\varphi =\pi /6,B_z/B_0=0.05$ (black), $\varphi =\pi /6,B_z/B_0=0.1$ (red), and $\varphi =\pi /4,B_z/B_0=0.1$ (blue) for $\tau =1$ (c) and $\tau =-1$ (d). 2D contour plots of $C_x(k_F,\tau ,\varphi ,{\beta }_s)$ (e) and (f) as a function of both $\varphi$ and $B_z$ for ${\beta }_s=-5\pi /8$ and $\tau =1$ (e) and for ${\beta }_s=-9\pi /40$ and $\tau =-1$ (f). Here two green circles in (c) and (d) indicate large back-scattering current peaks at ${\beta }_s\approx -5\pi /8$ (${\beta }_s\approx -9\pi /40$) for $\tau =1$ ($\tau =-1$), respectively. In addition, the unit of $C_0=4k_F{v}_F^2/{\pi }^2$ has been used for scaling $C_x(k_F,\tau ,\varphi ,{\beta }_s)$ and $B_0$ is given in Fig. 5.

Figure 8

(a)–(d) Skew-scattering current-distribution component $C_y(k_F,\tau ,\varphi ,{\beta }_s)$ from Eq. (20) as a function of $B_z$ (a) and (b) with $\varphi =\pi /4,{\beta }_s=\pi /6$ (green), $\varphi =\pi /6,{\beta }_s=\pi /6$ (black), $\varphi =\pi /4,{\beta }_s=\pi /3$ (blue), and $\varphi =\pi /6,{\beta }_s=\pi /3$ (red) for $\tau =1$ (a) and $\tau =-1$ (b), as well as a function of ${\beta }_s$ with $\varphi =\pi /6,B_z/B_0=0.05$ (black), $\varphi =\pi /6,B_z/B_0=0.1$ (red), and $\varphi =\pi /4,B_z/B_0=0.1$ (blue) for $\tau =1$ (c) and $\tau =-1$ (d). (e) and (f) 2D contour plots of $C_y(k_F,\tau ,\varphi ,{\beta }_s)$ as a function of both $\varphi$ and $B_z$ for ${\beta }_s=-3\pi /10$ and $\tau =1$ (e) and for ${\beta }_s=-\pi /4$ and $\tau =-1$ (f). Here two green circles in (c) and (d) indicate large skew-current peaks at ${\beta }_s\approx -3\pi /10$ (${\beta }_s=-\pi /4$) for $\tau =1$ ($\tau =-1$), respectively. In addition, $C_0$ and $B_0$ are given in Figs. 7 and 5, respectively.

Figure 9

Calculated nonequilibrium total back-scattering current $j_{1x}(\tau ,\varphi )$ (a) and (b) and total skew-scattering current $j_{1y}(\tau ,\varphi )$ (c) and (d) from Eq. (11) as a function of $B_z$ with $\varphi =\pi /4$ (green), $\varphi =\pi /6$ (blue), $\varphi =\pi /8$ (red), and $\varphi =0$ (black) for $\tau =1$ (a) and (c) and $\tau =-1$ (b) and (d). Here $B_0$ and $j_0$ are given in Figs. 5 and 6, respectively.

Figure 10

Calculated thermal-equilibrium Berry-curvature induced longitudinal current $j_{2x}(\tau ,\varphi )$ (a) and (b) and Hall current $j_{2y}(\tau ,\varphi )$ (c) and (d) from Eq. (12) as a function of $B_z$ with $\varphi =\pi /4$ (green), $\varphi =\pi /6$ (blue), $\varphi =\pi /8$ (red), and $\varphi =0$ (black) for $\tau =1$ (a) and (c) and $\tau =-1$ (b) and (d). Here $B_0$ and $j_0$ are given in Figs. 5 and 6, respectively.