Universal Conductance Fluctuations in Dirac Materials in the Presence of Long-range Disorder

We study quantum transport in Dirac materials with a single fermionic Dirac cone (strong topological insulators and graphene in the absence of intervalley coupling) in the presence of non-Gaussian long-range disorder. We show, by directly calculating numerically the conductance fluctuations, that in the limit of very large system size and disorder strength, quantum transport becomes universal. However, a systematic deviation away from universality is obtained for realistic system parameters. By comparing our results to existing experimental data on $1/f$ noise, we suggest that many of the graphene samples studied to date are in a nonuniversal crossover regime of conductance fluctuations.

Figure 1

(a) $⟨\sigma ⟩$ as a function of $n$ for $n_{\mathrm{imp}}=(3,5,9)\times {10}^{12}{\mathrm{cm}}^{-2}$ from top to bottom. (b) $⟨\sigma ⟩$ at the DP as a function of $n_{\mathrm{imp}}$ for the case of Coulomb disorder. The inset shows $⟨\sigma ⟩$ as a function of sample length $L\equiv x$ for $W=160\mathrm{nm}$, and $n=(3,2,1)\times {10}^{12}{\mathrm{cm}}^{-2}$ from top to bottom. The dashed lines show $d\sigma /d\ln L=4e^2/\pi h$.

Figure 2

(a) $⟨(\delta G{)}^2⟩$ as function of $n$ for the case of long-range disorder ($L=160\mathrm{nm}$) and different values of $n_{\mathrm{imp}}$; from top to bottom (for $n=2\times {10}^{12}{\mathrm{cm}}^{-2}$): $n_{\mathrm{imp}}=(0.682,1,3,5,9,16)\times {10}^{12}{\mathrm{cm}}^{-2}$. (b) $⟨(\delta G{)}^2⟩$ as a function of $\mu$ for different types of disorder potential (see text). (c) $⟨(\delta G{)}^2⟩$ as a function of $k_F\xi$ for Gaussian disorder. (d) $⟨(\delta G{)}^2⟩$ at the DP as a function of $n_{\mathrm{imp}}$ for the case of long-range disorder ($L=160\mathrm{nm}$). In the inset $⟨(\delta G{)}^2⟩$ as a function $L/W$; the top line shows the results for $n_{\mathrm{imp}}=5\times {10}^{12}{\mathrm{cm}}^{-2}$ and $n=3\times {10}^{12}{\mathrm{cm}}^{-2}$; the solid line below, and the bottom one, show the results for the same $n_{\mathrm{imp}}$ and $n={10}^{12}{\mathrm{cm}}^{-2}$ and $n=0$, respectively. Between these two lines there is the line showing the results for the case of Gaussian disorder with $K_0=1$ and $k_F\xi =0.8$. The dashed lines in all the panels show the universal value of Eq. (1).

Figure 3

(a) $⟨(\delta G{)}^2⟩/G^2$ as function of $n$ for the case of long-range disorder ($L=W=160\mathrm{nm}$) and different values of $n_{\mathrm{imp}}$; from top to bottom (for $n=2\times {10}^{12}{\mathrm{cm}}^{-2}$): $n_{\mathrm{imp}}=(5,3,1,0.682)\times {10}^{12}{\mathrm{cm}}^{-2}$. (b) $⟨(\delta G{)}^2⟩/G^2$ as a function of $k_F\xi$ for the case of Gaussian disorder for $K_0=4$, top, and $K_0=1$, bottom. $L=W=75\xi$.