Transmission resonances anomaly in one-dimensional disordered quantum systems

Connections between the electronic eigenstates and conductivity of one-dimensional (1D) disordered systems is studied in the framework of the tight-binding model. We show that for weak disorder only part of the states exhibit resonant transmission and contribute to the conductivity. The rest of the eigenvalues are not associated with peaks in transmission and the amplitudes of their wave functions do not exhibit a significant maxima within the sample. Moreover, unlike ordinary states, the lifetimes of these “hidden” modes either remain constant or even decrease (depending on the coupling with the leads) as the disorder becomes stronger. In a wide range of the disorder strengths, the averaged ratio of the number of transmission peaks to the total number of the eigenstates is independent of the degree of disorder and is close to the value $\sqrt{2/5}$, which was derived analytically in the weak-scattering approximation. These results are in perfect analogy to the spectral and transport properties of light in one-dimensional randomly inhomogeneous media [Y. P. Bliokh et al., New J. Phys. 17, 113009 (2015)], which provides strong grounds to believe that the existence of hidden, nonconducting modes is a general phenomenon inherent to 1D open random systems, and their fraction of the total density of states is the same for quantum particles and classical waves.

Figure 1

The number of the transmission maxima $N_{\text{TR}}$ (magenta squares) and the number of quasinormal modes $N_{\text{QNS}}$ calculated by integration over the density of states (cyan triangles) for a disordered 1D wire as a function of the length $L$. The cases of: low disorder $\xi \approx 10000$ (left); medium disorder $\xi \approx 100$ (middle); and strong disorder $\xi \approx 25$ (right) are presented. In the low disorder case the number of transmission peaks fits to $N_{\text{TR}}=\sqrt{2/5}L$ (lower black dashed lines), while the integrated density of states follows $L$ (upper black dashed lines). At higher disorders more transmission resonances are seen (i.e., $N_{\text{TR}}>\sqrt{2/5}L)$ due to localization. The missing data of $N_{\text{QNS}}$ at the higher disorder levels are due to numerical inaccuracies in integration over the exponentially high and narrow peaks of $\mathcal{N}\left(E\right)$.

Figure 2

Upper frame: Typical density of states $\mathcal{N}\left(E\right)$ (top red line) and transmission $T_{lr}\left(E\right)$ (bottom blue line) spectra of a particular realization of disorder $\left(L=500,W=1,t_{l/r}=1\right)$. The positions of the isolated Hamiltonian eigenvalues ${\epsilon }_i$ are indicated by the vertical dashed lines. Middle frame: The squared eigenvectors $|{\psi }_i{\left(r\right)|}^2$ of the isolated Hamiltonian as a function of the position along the wire $j$. The 216th eigenstate is located close to the system edge and therefore its transmission resonance is washed out (see upper frame) when the wire is coupled to the leads. Lower frame: The local density of states integrated in the vicinity of the $i\mathrm{th}$ disconnected eigenvalue ${\epsilon }_i,{\rho }_i\left(r\right)={\int }_{{\epsilon }_i-\mathrm{\Delta }/4}^{{\epsilon }_i+\mathrm{\Delta }/4}\rho \left(r,E\right)dE$. For most states ${\rho }_i\left(r\right)\sim {\left|{\psi }_i\left(r\right)\right|}^2$, except for the hidden mode (the 216th eigenstate) for which the local density close to the leads is strongly suppressed.

Figure 3

A color map of the local density of states $\rho (j,E)$ of the system described in Fig. 2. The “ordinary” modes (at $E\sim 1.544,1.552,1.575,1.59,1.604)$ show relatively narrow energy distribution, while the hidden mode originally located at $E\sim 1.58$ (marked with yellow circle on the left) is significantly broadened due to the coupling to the left lead. Similar hidden mode's tail can be noticed at the right end, related to a state hidden at higher energy (long yellow circle).

Figure 4

The ratio of the number of observed transmission peaks to the length of the wire $N_{\text{TR}}/N_{\text{QNS}}$ for various disorder strength $W$ and wire length $L$. Upper inset: Systems with length $L=100$ and $L=200$ for various disorder values. Lower inset: Systems with disorder strength $W=1$ and $W=5$ for various lengths. Main panel: The ratio $N_{\text{TR}}/N_{\text{QNS}}$ as a function of the scaling parameter $L/\xi$ for the results presented in the insets. All curves of $N_{\text{TR}}/N_{\text{QNS}}$ fall on top of each other. For $L/\xi <1,N_{\text{TR}}/N_{\text{QNS}}\sim \sqrt{2/5}$ remains. Once $L/\xi >1$, the ratio increases until $N_{\text{TR}}/N_{\text{QNS}}\to 1$ for large values of $L/\xi$, i.e., for strong localization all modes have transmission resonances. The black dashed line represents the dependence of the $N_{\text{TR}}/N_{\text{QNS}}$ on $L/\xi$ according to Eqs. (28) and (29) with $b=1/4$.

Figure 5

Upper frame: The transmission $T_{lr}\left(E\right)$ for a given realization of disorder at different strengths $W$ from 0.7 (top black line) to 1.3 (bottom red line), for a $L=500$ sample with $t_{l/r}=1$. The eigenenergies of the corresponding isolated wires are marked by circles. Two modes are hidden at low $W$, and become visible only at higher disorder level (marked by arrows). Lower frame: The modulus square of the isolated eigenvectors related the two above hidden states. As the disorder increased, the width of the modes becomes smaller and eventually they disassociate from the states of the wire.

Figure 6

Density of states $\mathcal{N}\left(E\right)$ of the system depicted in Fig. 5 at different energies, as disorder increases. The ordinary states become narrower at larger fluctuations, while the hidden mode (marked with blue arrows) widens. Fit to Lorenzian broadening in accordance with Eq. (30) (blue patterned areas) results in ${\gamma }_i^{0.8}=0.00165,{\gamma }_i^{0.9}=0.00168,{\gamma }_i^{1.0}=0.00172,{\gamma }_i^{1.1}=0.00201$, and ${\gamma }_i^{1.2}=0.00249$, i.e., shorter lifetime at the higher disorder level (see text).

Figure 7

The ratio of the number of observed transmission peaks to the number of QNS, $N_{\text{TR}}/N_{\text{QNS}}$ versus lead-system coupling strength $t_{l/r}$. Top panel: $W=0.1$ for different system length $L$. Bottom panel: $L=100$ for different disorder strength $W$. For both cases as $t_{l/r}\to 0,N_{\text{TR}}/N_{\text{QNS}}\sim 1$, while for $t_{l/r}\to 1,N_{\text{TR}}/N_{\text{QNS}}\sim \sqrt{2/5}$.

Figure 8

The ratio between the number of conductance peaks to the number of QNS, $N_{\text{cp}}/N_{\text{QNS}}$, for different values of temperatures $k_B\mathcal{T}$ and voltages $V$, for a system with $L=500$ and small disorder $W=0.1$ (i.e., $\xi \gg L)$. $N_{\text{cp}}$ is calculated by counting the maxima in the current with changing the chemical potential in the leads $\mu$. Both quantities are scaled by $\mathrm{\Delta }=4t/L$, the mean level spacing of the disconnected wire. The transition from zero temperature behavior and infinitesimal bias $N_{\text{cp}}/N_{\text{QNS}}\sim \sqrt{2/5}$ to the high temperature/voltage behavior $N_{\text{cp}}/N_{\text{QNS}}\to 0$ occurs around $k_B\mathcal{T}/\mathrm{\Delta }=1$, while even for quite large source-drain bias $V/\mathrm{\Delta }=10$ a finite number of modes is still observed in the conductance at low temperatures.