Anderson localization and delocalization of massless two-dimensional Dirac electrons in random one-dimensional scalar and vector potentials

We study Anderson localization of massless two-dimensional Dirac electrons in random one-dimensional scalar and vector potentials theoretically for two different cases, in which the scalar and vector potentials are either uncorrelated or correlated. From the Dirac equation, we deduce the effective wave impedance, in which we derive the condition for total transmission and those for delocalization in our random models analytically. Based on the invariant imbedding theory, we also develop a numerical method to calculate the localization length exactly for arbitrary strengths of disorder. In addition, we derive analytical expressions for the localization length, which are extremely accurate in the weak and strong disorder limits. In the presence of both scalar and vector potentials, the conditions for total transmission and complete delocalization are generalized from the usual Klein tunneling case. We find that the incident angles at which electron waves are either completely transmitted or delocalized can be tuned to arbitrary values. When the strength of scalar potential disorder increases to infinity, the localization length also increases to infinity, both in uncorrelated and correlated cases. The detailed dependencies of the localization length on incident angle, disorder strength, and energy are elucidated and the discrepancies with previous studies and some new results are discussed. All the results are explained intuitively using the concept of wave impedance.

Figure 1

Transmittance $T$ plotted versus incident angle $\theta$, when massless Dirac particles are incident on an inhomogeneous strip of thickness $L$ with the normalized scalar potential $u=2-2x/L$, and the normalized vector potential $a=a_m(1-x/L)$ ($a_m=1.732$, 1, 0, $-1,-1.732$), in the region $0\le x\le L$. In the transmitted region ($x<0$), the potentials are $u=2$ and $a=a_m$. The strip thickness is given by (a) $kL=100$ and (b) $kL=1000$.

Figure 2

Illustration of the comparison between the numerical results obtained using the IIM and the analytical formulas for the localization length presented in Sec. 5. (a) Normalized localization length $k\xi$ plotted versus incident angle $\theta$ for electron waves in model I, when $g_a=0$ and $u_0=0$. The left curve is for $g_u=0.01$ and $a_0=0.5$ and is compared with Eq. (39), while the right curve is for $g_u=100$ and $a_0=-0.3$ and is compared with Eq. (48). (b) Normalized localization length versus incident angle in model I, when $g_a=0.01$, $g_u=0$, $u_0=0.5$, and $a_0=-0.6$. The IIM result is compared with Eq. (39). (c) Normalized localization length versus incident angle in model II, when $f=0.6$ and $u_0=0.5$. The left curve is for $g_u=0.01$ and $a_0=0.2$ and is compared with Eq. (43), while the right curve is for $g_u=100$ and $a_0=-1.2$ and is compared with Eq. (51).

Figure 3

Normalized localization length $k\xi$ plotted versus incident angle $\theta$ for electron waves in model I, when $g_a=0$, $a_0=0$, $\pm 0.5$, $\pm 0.9$, and (a) $g_u=0.01$, $u_0=0$, (b) $g_u=0.01$, $u_0=0.5$, (c) $g_u=0.01$, $u_0=1$, and (d) $g_u=100$. There is no noticeable dependence on $u_0$ when $g_u=100$.

Figure 4

Normalized localization length $k\xi$ plotted versus $|\theta -{\theta }_R|$ in a log-log plot, for electron waves in model I when only the scalar potential is random ($g_a=0$). The four curves are obtained for different values of the parameters $u_0$, $a_0$, and $g_u$, which are shown in the figure. All curves show the divergent behavior $k\xi \propto |\theta -{\theta }_R{|}^{-2}$ near ${\theta }_R$.

Figure 5

Normalized localization length $k\xi$ plotted versus incident angle $\theta$ for electron waves in model I, when $g_a=0.01$, $g_u=0$, $a_0=0$, $\pm 0.5$, $\pm 0.9$ and (a) $u_0=0.5$ and (b) $u_0=1$.

Figure 6

Normalized localization length $k\xi$ plotted versus incident angle $\theta$ for electron waves in model II, when $u_0=0.5$. The values of $a_0$ are designated on the curves. The other parameters are (a) $f=0.6$, $g_u=0.01$, (b) $f=0.6$, $g_u=100$, and (c) $f=1.5$, $g_u=0.01$.

Figure 7

Normalized localization length plotted versus the strength of scalar potential disorder $g_u$ for electron waves in model I, when $g_a=0$, ${\beta }_0=0.5$, and ${\epsilon }_0=0.1$, 0.48, 0.5, and 0.6.

Figure 8

Normalized localization length plotted versus the strength of vector potential disorder $g_a$ for electron waves in model I, when $g_u=0$, ${\beta }_0=0.5$, and ${\epsilon }_0=0.1$, 0.48, 0.5, and 0.6.

Figure 9

Normalized localization length plotted versus the strength of scalar potential disorder $g_u$ for electron waves in model II, when ${\beta }_0=0.5$, ${\epsilon }_0=0.1$, 0.4, 0.48, 0.5, and 0.6 and (a) $f=0.5$ and (b) $f=1.5$.

Figure 10

Energy dependence of the localization length in model I when there is only a random scalar potential. $\xi$ is normalized by the wave number associated with disorder $k_u$ [see Eq. (53)] and $k/k_u$ $[=E/\left(\hslash v_F{k}_u\right)={g_u}^{-1}]$ is the normalized energy variable. In (a), the parameter ${\tilde{u}}_0$ ($=u_{0}k/k_u$) is zero and $\theta ={30}^{\circ }$. The IIM result is compared with the analytical results in the high and low energy limits. (b) ${\tilde{u}}_0=1$, 3, 5 and $\theta ={30}^{\circ }$.

Figure 11

Energy dependence of the localization length in model II with zero average values of the scalar and vector potentials when $\theta ={30}^{\circ }$. The IIM result is compared with the analytical results in the high and low energy limits.

Figure 12

Disorder-averaged transmittance $\langle T\rangle$ versus incident angle for model II with $u_0=0.5$, $a_0=-0.25$, $g_u=0.01$, and $kL=1$ and for $f=-0.5$, 0, 0.5.