One-dimensional disordered wires with Pöschl-Teller potentials

We study the electronic properties of a one-dimensional disordered chain made up of Pöschl-Teller potentials. The features of the whole spectrum of the random chain in the thermodynamic limit are analyzed in detail by making use of the functional equation formalism. The disordered system exhibits a fractal distribution of states within certain energy intervals and two types of resonances exist for the uncorrelated case. These extended states are characterized by different values of their critical exponents and different behaviors near the critical energies when the system is finite, but their existence can be defined by a single common condition. The chain is also considered including a natural model of short-range correlated disorder. The results show that the effects of the correlations considered are independent of the potential model.

Figure 1

Pöschl-Teller potential defined in Eq. (1).

Figure 2

Bound states for the Pöschl-Teller well. The solid curves mark the position of the eigenstates given by Eq. (8). The dashed line mark the position of the bottom of the well $(-\sqrt{\mid V\mid })$. The dotted grids highlight the integer and the half-integer states.

Figure 3

Potential of a disordered Pöschl-Teller wire.

Figure 4

(Color online) DOS and Lyapunov exponent (black) for several disordered Pöschl-Teller wires with parameters (a) $V_{\gamma }\left[c_{\gamma }\right]$: $-4\left[0.5\right]$, 3[0.5], (b) $V_{\gamma }\left[c_{\gamma }\right]$: 5[0.5], 7[0.5], (c) $V_{\gamma }\left[c_{\gamma }\right]$: $-1\left[0.5\right]$, $-3\left[0.5\right]$, and (d) $V_{\gamma }\left[c_{\gamma }\right]$: 1[0.2], 6[0.2], $-5\left[0.2\right]$, $-1\left[0.2\right]$, 9[0.2]. Notice the different vertical scales for DOS (left) and $\lambda$ (right). The negative part of the abscissa axis represents negative energies $(-\eta ∕\alpha )$.

Figure 5

DOS for a binary disordered Pöschl-Teller wire with parameters $V_1=1$ and $V_2=-1$ and equal concentrations. $d_f$ indicates the fractal dimension of the portion of the DOS contained between the dashed lines.

Figure 6

(Color online) Comparison of Lyapunov exponent (a) and IPR (b) as functions of the energy, for different binary disordered chains. The parameters for the different chains are $V_{\gamma }\left[c_{\gamma }\right]$: $-1.5\left[0.5\right]$, $-4\left[0.5\right]$ (solid), 1[0.5], 3[0.5] (dash), 5[0.5], 7[0.5] (dash-dot). The IPR is obtained averaging over 100 realizations of a 1000-atom array.

Figure 7

(Color online) Evolution of the Lyapunov exponent vs energy and concentration for disordered binary wires with parameters $V_1=-2.5$, $V_2=5$. Resonances occur at ${ϵ}_c=1.809$, ${ϵ}_c=2.226$. The top mapping shows the Lyapunov exponent in a color scale. Notice how the resonances are present for all concentrations.

Figure 8

(Color online) DOS and Lyapunov exponent (black) for binary chains. (a) Two species with symmetric cutoff: ${\alpha }_1=\alpha$, $V_1=-1$, $d_1=4.5∕{\alpha }_1$ and ${\alpha }_2=1.95\alpha$, $V_2=2$, $d_2=4.1∕{\alpha }_2$. The three first resonances occur at ${ϵ}_c=1.6818$, ${ϵ}_c=2.3986$, and ${ϵ}_c=3.1064$. Notice that the first two resonances are located below the barrier height ${ϵ}_{\max }=1.95\sqrt{2}\simeq 2.76$. (b) Two species with asymmetric cutoff: ${\alpha }_1=\alpha$, $V_1=1$, $d_1^L=5.2∕{\alpha }_1$, $d_1^R=4.2∕{\alpha }_1$ and ${\alpha }_2=1.5\alpha$, $V_2=3$, $d_2^L=6.8∕{\alpha }_2$, $d_2^R=5.3∕{\alpha }_2$. The first four resonances occur at ${ϵ}_c=1.7434$, ${ϵ}_c=2.0618$, ${ϵ}_c=2.3865$, and ${ϵ}_c=2.7143$. Notice that the first three resonances are located below the highest barrier ${ϵ}_{\max }=1.5\sqrt{3}\simeq 2.60$.

Figure 9

(Color online) Lyapunov exponent (solid line) for a binary chain with random parameters ${\alpha }_1=\alpha$, $V_1=-4$, $d_1^L=4∕{\alpha }_1$, $d_1^R=5∕{\alpha }_1$ and ${\alpha }_2=1.8\alpha$, $V_2=1$, $d_2^L=6∕{\alpha }_2$, $d_2^R=4.5∕{\alpha }_2$. The dashed line corresponds to the Euclidean norm of the commutator of the transmission matrices. Shaded zones highlight the common permitted spectrum of both species. The upper box combines density of states and Lyapunov exponent.

Figure 10

(Color online) DOS and Lyapunov exponent (black) for binary chains with parameters (a) $V_1=-4$, $V_2=1.571051$ and (b) $V_1=4$, $V_2=7.217957$ with equal concentrations in both cases. The CBE resonances are located at ${ϵ}_c=0.814459$ and ${ϵ}_c=2.56389$, respectively.

Figure 11

Behavior of DOS and Lyapunov exponent near critical energies for binary wires. Symbols mark numerical results. (a) $V_1=-4$ and $V_2=1.571051$ with equal concentrations. CBE resonance located at ${ϵ}_c=0.814459$. Notice that for the Lyapunov exponent the vertical scales are different before and after the resonance to ensure an optimal visualization. Solid lines correspond to fits according to expression (32) for $\lambda \left(ϵ\right)$ and $g\left(ϵ\right)\sim {\mid ϵ-{ϵ}_c\mid }^{-1∕2}$. (b) $V_1=5$ and $V_2=7$ with different concentrations. Commuting-resonance located at ${ϵ}_c=2.601$. Solid lines correspond to quadratic fits for $\lambda \left(ϵ\right)$ and linear fits for DOS.

Figure 12

(Color online) IPR (a) and Lyapunov exponent (b) near a CBE resonance for a binary chain with parameters $V_1=2$ and $V_2=-2.6422653344$ with equal concentrations. The critical energy is ${ϵ}_c=0.9270063568$. The horizontal dotted lines mark the inverse of the length of the system in the different cases considered. Inside graphics (a) and (b) the length of the chain increases from top to bottom. For a given length, data of all graphics correspond to the same disordered sequence. Only one realization of the disorder has been considered for each length. Graphics at the bottom show the envelope of the electronic state for the chain with 1000 atoms for three different energies corresponding to transmissions (c.1) $T=3.7\times {10}^{-7}$, (c.2) $T=0.8207$, (c.3) $T=0.9955$.

Figure 13

(Color online) IPR (a) and Lyapunov exponent (b) near the critical energy ${ϵ}_c=1.808977$ for a binary wire with parameters $V_1=-2.5$ and $V_2=5$ with equal concentrations. The horizontal dotted lines mark the inverse of the different lengths considered. Inside graphics (a) and (b) the length of the chain increases from top to bottom. For a given length, data of all graphics correspond to the same disordered sequence. Only one realization of the disorder has been considered for each length. Graphics at the bottom show the envelope of the electronic state for the chain with 1000 atoms for three different energies corresponding to transmissions (c.1) $T=0.1067$, (c.2) $T=0.9999$, (c.3) $T=0.8537$.

Figure 14

Evolution of the envelope of the electronic state at a commuting critical energy as a function of concentration, for binary disordered wires with open boundary conditions. (a) $V_1=-2.5$, $V_2=5$ and ${ϵ}_c=1.808977$, (b) $V_1=5$, $V_2=7$ and ${ϵ}_c=2.600753$. In all cases the system includes 500 atoms and the states are properly normalized to make the comparison possible.

Figure 15

(Color online) Multifractal analysis for a binary wire with parameters $V_1=-2.5$, $V_2=5$ with equal concentrations. The moments considered range from $q=2$ to $q=8$. Dashed lines correspond to equation ${\mu }_q\left(N\right)=N^{-(q-1)}$.

Figure 16

(Color online) DOS and Lyapunov exponent (black) for binary chains. (a) $V_1=-0.5$, $V_2=-3.399309$ with equal concentrations. The CBE extended state occurs at ${ϵ}_c=-0.297183$. (b) ${\alpha }_1=\alpha$, $V_1=-3.1$, $d_1^L=4.5∕{\alpha }_1$, $d_1^R=4∕{\alpha }_1$ and ${\alpha }_2=2\alpha$, $V_2=-2.5$, $d_2^L=5.5∕{\alpha }_2$, $d_2^R=4.5∕{\alpha }_2$ with equal concentrations. The commuting extended bound state is located at ${ϵ}_c=-0.314670$. The negative part of the abscissa axis corresponds to $-\eta ∕\alpha$.

Figure 17

(Color online) DOS and Lyapunov exponent (black) for binary chains including wells in equal concentrations with parameters (a) $b_1=3∕2$, $b_2=5∕2$ with a common eigenstate for $\eta ∕\alpha =0.5$ and (b) $V_1=-2.8125$ and $V_2=-7.3125$ showing common eigenstates at $\eta ∕\alpha =0.25,1.25$. The commuting extended bound states are located at ${ϵ}_c=-0.217142$ and ${ϵ}_c=-1.25$. Notice the change in the vertical scale at $-0.75$ in the (b) example. The negative part of the abscissa axis represents $-\eta ∕\alpha$.

Figure 18

Optimal representation of the correlation space for two species as a function of the concentration: if $c_1⩽0.5$ then $p_{12}$ vs $c_1$, if $c_1>0.5$ then $p_{21}$ vs $c_1$. The dashed line corresponds to the completely random configurations.

Figure 19

DOS for correlated configurations of a binary chain with parameters $V_1=-2.5$ and $V_2=2$ with concentration $c_1=0.4$.

Figure 20

(Color online) Evolution of the Lyapunov exponent (a) and the IPR (b) with the correlations for a binary chain with parameters $V_1=-1$, $V_2=1$, $c_1=0.5$. Notice the presence of a critical energy at ${ϵ}_c=1.095$. For each configuration the IPR has been obtained by averaging 100 sequences of a 1000-atom array.

Figure 21

Transmission probability vs energy for a 1000-atom binary disordered chain in different correlation regimes. The circular point inside the insets marks the configuration on the configurational space. Only one realization of the disorder has been considered for each case. Parameters: $V_1=-2.5$, $V_2=5$, $c_1=0.5$ and correlation from top to bottom $p_{12}=0.5,0.1,0.9$.

Figure 22

(Color online) Transmission efficiency over correlation space for binary chains: (a) $V_1=-2.5$, $V_2=5$ and integration interval [0, 3.7], (b) $V_1=2.5$, $V_2=5$ and integration interval [0, 3].

Figure 23

(Color online) Transmission efficiency vs length for different configurations of a 1000-atom binary chain with parameters $V_1=-1$, $V_2=1$, $c_1=0.4$, integration interval [0, 1.75]. $\left(R\right)$ marks the completely random situation. The inset shows the relative differences $\Delta T_{\mathrm{eff}}=T_{\mathrm{eff}}-T_{\mathrm{eff}}\left(R\right)$.