Symmetries and weak localization and antilocalization of Dirac fermions in HgTe quantum wells

We perform a symmetry analysis of a 2D electron system in HgTe/HgCdTe quantum wells in the situation when the chemical potential is outside of the gap, so the bulk of the quantum well is conducting. In order to investigate quantum transport properties of the system, we explore symmetries of the low-energy Hamiltonian, which is expressed in terms of two flavors of Dirac fermions, and physically important symmetry-breaking mechanisms. This allows us to predict emerging patterns of symmetry breaking that control the weak localization and antilocalization showing up in transverse-field magnetoresistance.

Figure 1

Schematic diagram showing the symmetry patterns in the $L$-$E_F$ plane. Here $L$ is the system size, phase-breaking length $L_{\phi }$, or magnetic length $L_H$, whichever is shorter. For simplicity, the transport scattering length $l_{\mathrm{tr}}$ (vertical dotted line) is assumed to be energy independent (this assumption does not affect the “topology” of the diagram). The “phase boundaries” are shown by solid curves. Boundaries of the 2U regions are determined by $l_m={\left(D{\tau }_m\right)}^{1/2}$ for 2Sp $\to$ 2U crossover, and by $l_A={\left(D{\tau }_A\right)}^{1/2}$ for 2O $\to$ 2U crossover. (Left panel) Inverted band structure (thick quantum well) with no block mixing. Dashed line shows energy for which $m\left(k_F\right)=-\left|M\right|+B{k}_F^2=0.$ (Middle panel) Normal band structure (thin quantum well) with no block mixing. (Right panel) Inverted band structure with block mixing, characterized by $l_{\Delta }={\left(D{\tau }_{\Delta }\right)}^{1/2}$ and $l_{\mathrm{SO}}={\left(D{\tau }_{\mathrm{SO}}\right)}^{1/2}$ (dash-dotted), with $\tau <{\tau }_{\Delta }<{\tau }_{\mathrm{SO}}$. The energies $E_F=E_1...E_5$ (from bottom to top) mark different horizontal cross sections of the “phase-diagram” corresponding to the patterns 2O $\to$ 1Sp, 2O $\to$ 2U $\to$ 1Sp, 2U $\to$ 1Sp, 2Sp $\to$ 2U $\to$ 1Sp, and 2Sp $\to$ 1Sp, respectively, that appear with increasing $L$. The magnetoresistance curves corresponding to these energies are shown schematically in Fig. 2. The diagram for the case of normal band structure with the block mixing is qualitatively similar.

Figure 2

Schematic view of the magnetoresistance curves ${\rho }_{xx}\left(B\right)$ in linear-logarithmic scale. Panels correspond to energies $E_1...E_5$ from the right panel of Fig. 1 (inverted band structure). The numbers indicate the prefactor $\alpha =(\pi h/e^2)\partial \sigma /\partial {\tau }_H$ (here ${\tau }_H=L_H^2/D$) of the logarithmic magnetoresistance in different domains of magnetic field. For the purpose of visualization, the crossover regions (where ${\tau }_H$ is of the order of the corresponding symmetry-breaking time) are shown as cusps.