Tailoring weak and metallic phases in a strong topological insulator by strain and disorder: Conductance fluctuations signatures

Transport measurements are readily used to probe different phases in disordered topological insulators (TIs), where determining topological invariants explicitly is challenging. On that note, universal conductance fluctuations (UCF) theory asserts the conductance $G$ for an ensemble has a Gaussian distribution, and that standard deviation $\delta G$ depends solely on the symmetries and dimensions of the system. Using a real-space tight-binding Hamiltonian on a system with Anderson disorder, we explore conductance fluctuations in a thin ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ film and demonstrate the agreement of their behavior with UCF hypotheses. We further show that magnetic field applied out-of-plane breaks the time-reversal symmetry and transforms the system's Wigner-Dyson class from symplectic to unitary, increasing $\delta G$ by $\sqrt{2}$. Finally, we reveal that while ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ is a strong TI, weak TI and metallic phases can be stabilized in presence of strain and disorder, and detected by monitoring the conductance fluctuations.

Figure 1

(a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ crystal structure. (b) A quintuple layer (QL) of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The atoms are stacked in layers in the $z$ direction in an ABCABC sequence. (c) Lattice structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ in the $x\text{−}y$ plane. Each unit cell has two neighboring unit cells in the $z$ direction connected by vector ${\mathbf{n}}_{\mathbf{4}}$, and six neighboring unit cells in the $x\text{−}y$ plane connected by vectors ${\mathbf{n}}_{\mathbf{1},\mathbf{2},\mathbf{3}}$.

Figure 2

(a) The mean conductance $\langle G\rangle$ and (b) the standard deviation of conductance $\delta G$ of disordered ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ as a function of disorder strength $W$, with $N_{\text{QL}}=4$ and the number of in-plane sites $L_x$ = $L_y$ = 10. $E_F$ = 0.05 eV is related to surface states (any $E_F$ < 0.15 eV is), and as shown in (b), conductance fluctuations are negligible for weak disorder, indicating surface states' robustness to disorder.

Figure 3

(a) Conductance fluctuation versus the thickness of the sample $N_{\text{QL}}$. $\delta G$ depends on $N_{\text{QL}}$/$L_x$ and $N_{\text{QL}}$/$L_y$ ratio, and the conductance fluctuations are almost constant, for fixed $L_x$ and $L_y$ of the thin film. (b) ${\mathrm{c}}_d$ values calculated here, compared to those extracted from Ref. [12], as a function of thickness. (c) Conductance fluctuation versus Fermi energy. Considering the symplectic class where SR symmetry does not exist, increasing the Fermi energy does not change the Wigner-Dyson class of the sample. (d) Conductance versus Fermi energy for 200 samples for each $E_F$.

Figure 4

Magnetization of the system leads to TRS collapse, a change from symplectic ($\beta$=4) to unitary ($\beta =2$) class, and a $\sqrt{2}$-fold rise in $\delta G$ in the absence of TRS.

Figure 5

(a) ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ bulk gap (${\mathrm{\Delta }}_{\mathrm{\Gamma }}$) as a function of out-of-plane (${\epsilon }_{\perp }$) and in-plane strain (${\epsilon }_{\parallel }$). (b) The behavior of conductance in terms of out-of-plane strain for $W=1$, 3, 5, and 7 eV. In each case, the considered Fermi energy is located on the first conduction band. Transport calculations may be used to identify the topological phase transition in nontranslationally symmetric systems. When the system is in a strong phase, changes of $\langle G\rangle$ are minimal and, fluctuations in conductance may be noticed during the weak phase.

Figure 6

Phase diagram for strong, weak, and metallic states in 3D TI in terms of out-of-plane strain and strength of disorder. Both strong and weak phases exist for weak disorders, and the metallic phase is detectable after sufficiently strong disorder is present.