Magnetoresistance of compensated semimetals in confined geometries

Two-component conductors – e.g., semimetals and narrow-band semiconductors – often exhibit unusually strong magnetoresistance in a wide temperature range. Suppression of the Hall voltage near charge neutrality in such systems gives rise to a strong quasiparticle drift in the direction perpendicular to the electric current and magnetic field. This drift is responsible for a strong geometrical increase of resistance even in weak magnetic fields. Combining the Boltzmann kinetic equation with sample electrostatics, we develop a microscopic theory of magnetotransport in two and three spatial dimensions. The compensated Hall effect in confined geometry is always accompanied by electron-hole recombination near the sample edges and at large-scale inhomogeneities. As the result, classical edge currents may dominate the resistance in the vicinity of charge compensation. The effect leads to linear magnetoresistance in two dimensions in a broad range of parameters. In three dimensions, the magnetoresistance is normally quadratic in the field, with the linear regime restricted to rectangular samples with magnetic field directed perpendicular to the sample surface.

Figure 1

Typical semiclassical trajectories for oppositely charged quasiparticles in two-component systems at charge neutrality. The two panels illustrate electron-hole asymmetric (a) and symmetric (b) systems. As a manifestation of the compensated Hall effect, both quasiparticle currents are flowing in the same direction in the bulk of the sample. In the symmetric sample (b), the quasiparticle flow, $\mathit{P}={\mathit{j}}_e+{\mathit{j}}_h$, is orthogonal to the electric current. In the asymmetric case (a), the longitudinal component of $\mathit{P}$ is also finite. Such a flow leads to quasiparticle accumulation at the boundaries of the otherwise homogeneous sample. The excess quasiparticle density appears in a boundary region of the width of the order of the electron-hole recombination length. Contributions of the bulk and boundary regions to the sheet resistance exhibit different dependence on the magnetic field. In classically strong fields, the boundary region may dominate leading to linear magnetoresistance.

Figure 2

Lateral profiles of the quasiparticle density $\delta \rho \left(y\right)$, charge density $\delta n\left(y\right)$, quasiparticle flow $P_{x,y}\left(y\right)$, and electric current density $J_x\left(y\right)$ in a 2D two-component system away from charge neutrality, see Fig. 1, panel (a), calculated within the theory presented in Sec. 2. For concreteness, we chose the carrier parameters of a typical topological-insulator film: electron and hole mobilities ${\mu }_e=20{\mu }_h,{\mu }_h=1{\mathrm{m}}^2/\left(\mathrm{V}\mathrm{s}\right)$ and velocities $v_e={10}^6\mathrm{m}/\mathrm{s},v_h=0.5v_e$. The sample is assumed have the width $W=10\mu \mathrm{m}$, with the distance to the gate $d=0.5\mu \mathrm{m}$ and the dielectric constant of the surrounding insulator $\epsilon =5$. The carrier densities were calculated using a generic two-band model with the energy gap $\mathrm{\Delta }=4$ meV at room temperature $T=300$ K. The recombination length is assumed to take the value ${\ell }_R=0.46\mu \mathrm{m}$ at $B=2$ T. All curves are normalized to the maxima of their absolute values.

Figure 3

(a) Sheet resistance $R_{\square }\left(B\right)$ of a 2D symmetric, two-component system at charge neutrality for three different values of the ratio of the sample width to the zero-field recombination length, $W/{\ell }_0=0.01,2,1000$, represented by the solid red, dotted blue, and dashed green lines, respectively (the curves are rescaled for clarity). (b), (c), (d) $R_{\square }\left(B\right)$ of a 2D asymmetric two-component system for three different values $W/{\ell }_0=0.008$ [panel (b)], $W/{\ell }_0=2.1$ [panel (b)], and $W/{\ell }_0=1000$ [panel (c)]. In all three plots, solid lines correspond to the charge neutrality point, while dashed lines show results away from neutrality. The curves were calculated using the theory presented in Sec. 2, with the parameter values corresponding to typical topological-insulator films (see the caption to Fig. 2), with the recombination length ${\ell }_0=4.8\mu \mathrm{m}$.

Figure 4

The magnetic field dependence of the sheet longitudinal (left) and Hall resistance (right) given by Eqs. (79) and (80). The numerical values were obtained using the typical experimental parameters of topological-insulator films: electron and hole mobilities ${\mu }_e=20{\mu }_h,{\mu }_h=1{\mathrm{m}}^2/\left(\mathrm{V}\mathrm{s}\right)$, and velocities $v_e={10}^6\mathrm{m}/\mathrm{s},v_h=0.5v_e$, and the sample parameters are $W=10\mu \mathrm{m},d=0.5\mu \mathrm{m}$, and $\epsilon =5$. The carrier densities were calculated using a generic two-band model with the energy gap $\mathrm{\Delta }=4$ meV at room temperature $T=300$ K. The recombination length in the absence of magnetic field at charge neutrality is ${\ell }_0=0.37\mu \mathrm{m}$. The solid line corresponds to charge neutrality, $\delta n=0$, while the other lines correspond to negative densities $\delta n=0.3,0.5,0.9,1.3,2.1\times {10}^{11}{\mathrm{cm}}^{-2}$. The inset in the left panel shows the magnetoresistance $R_{\square }$ for a symmetric model with ${\mu }_{\alpha }=20{\mathrm{m}}^2/\left(\mathrm{V}\mathrm{s}\right)$.

Figure 5

3D sample with the shape of a slab in an oblique magnetic field. The magnetic field vector lies in the $xy$ plane.

Figure 6

3D two-component system at charge neutrality in an oblique magnetic field. The recombination length in zero magnetic field is taken to be equal to the sheet width: $W/{\ell }_0=1$. The solid red curves represent the calculated values of the magnetoresistance. The field is directed at the angle $\theta =0.5^{\circ }$. The two panels show the same result in the two different ranges of the parameter ${\omega }_c\tau$: the panel (a) shows the onset of the intermediate, nearly linear behavior, while the panel (b) shows the recovery of the quadratic magnetoresistance in strong fields. The dashed blue lines are guides to the eye.

Figure 7

Spatial profiles of the lateral electric field (left) and charge density (right) in the classical Hall effect in a sample with the slab geometry of Fig. 5.