Wave transport in one-dimensional disordered systems with finite-size scatterers

We study the problem of wave transport in a one-dimensional disordered system, where the scatterers of the chain are $n$ barriers and wells with statistically independent intensities and with a spatial extension $l_c$ which may contain an arbitrary number $\delta /2\pi$ of wavelengths, where $\delta =k{l}_c$. We analyze the average Landauer resistance and transmission coefficient of the chain as a function of $n$ and the phase parameter $\delta$. For weak scatterers, we find: (i) a regime, to be called I, associated with an exponential behavior of the resistance with $n$; (ii) a regime, to be called II, for $\delta$ in the vicinity of $\pi$, where the system is almost transparent and less localized; and (iii) right in the middle of regime II, for $\delta$ very close to $\pi$, the formation of a band gap, which becomes ever more conspicuous as $n$ increases. In regime II, both the average Landauer resistance and the transmission coefficient show an oscillatory behavior with $n$ and $\delta$. These characteristics of the system are found analytically, some of them exactly and some others approximately. The agreement between theory and simulations is excellent, which suggests a strong motivation for the experimental study of these systems. We also present a qualitative discussion of the results.

Figure 1

Schematic representation of an array of $n$ steps of random height $V_r$ ($r=1,\cdots ,n$) possessing a fixed spatial width $l_c$. The incident energy $E$ is taken larger than all the $|V_r|$'s. Also indicated is the “initial system” with $n$ scatterers and the addition of the building block (BB) consisting of the $(n+1)$-st scatterer.

Figure 2

Schematic representation of the $r$-th scatterer of the chain for the case of a barrier. It has a fixed spatial width $l_c$ and height $V_r$.

Figure 3

Theory and computer simulations for $\langle {\left(R/T\right)}^{\left(n\right)}\rangle$ as a function of $n$ for the 1D system described in the text. For the numerical simulation, an ensemble of ${10}^4$ realizations was used. Results are shown in regime I for a number of $\delta$'s: (a) $1<\delta <2$ and (b) $2.3<\delta <2.9$; $\delta =\pi$, from regime II, is also shown. We can observe the localization properties described in the text. We chose the parameter $y_0=$ 0.09. The error bar due to the finite sample size is very small and is not indicated in the figure: e.g., for $\delta =\pi /2$ and $n=5000$, the error is $\sim {10}^{-2}$.

Figure 4

Perturbation theory and computer simulations for $\langle {(R/T)}^{\left(n\right)}\rangle$ as a function of $n$ for $\delta =3.1200,3.1280,3.1300,3.1380$ and $y_0=0.09$. Perturbation theory was carried out up to second order in $\mathrm{\Delta }\mathrm{\Omega }$ in the eigenvalues and the eigenvectors. The description is reasonable, especially in the first three cases [(a), (b), and (c)]; in the fourth case (d) the agreement deteriorates, as $\delta$ is too close to $\pi$. Notice the oscillatory behavior as a function of $n$. The insets in (a), (b), and (c) are a zoom of the results for the first few oscillations. The estimate of the period from Eq. (3.15) is indicated in each panel and is consistent with the numerical data.

Figure 5

Numerical simulations and analytical solution for $\langle {(R/T)}^{\left(n\right)}\rangle$ vs $n$, obtained diagonalizing numerically the matrix ${\mathrm{\Omega }}_{y_0}\left(\delta \right)$, for $y_0=0.09$ and for four values of $\delta \approx \pi$: [(a)–(c)] $\delta =3.1380,3.1400,3.1405$; (d) $\delta =\pi$. The analytical solution is essentially exact. The results are symmetric around $\delta =\pi$ in the vicinity of this value. As a verification of the theoretical results, also shown are computer simulations using an ensemble of ${10}^4$ realizations. For $\delta =\pi$, the statistical error bar is smaller than ${10}^{-5}$ and is not indicated. The estimate from Eq. (3.15) of the period ${\tau }_n$ of the oscillations is indicated in each panel and is consistent with the numerical simulations and the analytical results.

Figure 6

Theory and numerical simulations for $\langle {(R/T)}^{\left(n\right)}\rangle$ as a function of $\delta$, for a chain of $n=5000$ scatterers and for $y_0=0.09$, in regimes I and II. The simulations use ${10}^5$ realizations. In regime II, Eqs. (3.6) and (B6) were employed, diagonalizing numerically the matrix $\mathrm{\Omega }$; the computer simulations (with a statistical error bar $\sim {10}^{-5}$ for $\delta =\pi$) constitute a verification of the theoretical results. Well inside regime II, we observe the dramatic enhancement of the average resistance by nearly three orders of magnitude, explained in the text. The inset in panel (a) and panels (b) and (c) show this latter region in greater detail. Notice the oscillatory behavior of the average Landauer resistance as a function of $\delta$ for fixed $n$. The period of the oscillations can be estimated from Eq. (3.16) as ${\tau }_{\delta }\sim 6\times {10}^{-4}$, which is consistent with what we observe in the figure (in spite of $\delta$ being quite close to $\pi$).

Figure 7

Theory and numerical simulations for the average transmission coefficient $\langle T\rangle$ as a function of the number $n$ of scatterers and for various values of the phase parameter $\delta$ in regime I, in the ranges (a) $1<\delta <2$ and (b) $2.3<\delta <2.9$, as in Fig. 3. For the simulation, an ensemble of ${10}^4$ realizations was used. As usual, we chose the parameter $y_0=$ 0.09. The theoretical results, obtained from Eq. (4.7), lie on top of the numerical ones. The error bar due to the finite sample is not indicated in the figure: e.g., for $\delta =\pi /2$ and $n=5000$, the error is $\sim {10}^{-2}$.

Figure 8

The theoretical average transmission coefficient $\langle T\rangle$ vs $n$, obtained from the approximation of Eq. (4.8), for $y_0=0.09$ and for four values of $\delta \approx \pi$: [(a)–(c)] $\delta =3.1380,3.1400,3.1405$ and (d) $\delta =\pi$ (as in Fig. 5), compared with numerical simulations. The agreement is excellent, suggesting that the approximation of Eq. (4.8) is justified. The estimate from Eq. (3.15) of the period ${\tau }_n$ of the oscillations is indicated in each panel and is consistent with the analytical results and the numerical simulations.

Figure 9

Theoretical results (as described in the text) and numerical simulations for $\langle T\rangle$ vs $\delta$ in regimes I and II for a chain of $n=5000$ scatterers and ${10}^5$ realizations. The main figure shows the “gross-structure” behavior and the dip for $\delta \approx \pi$ exhibiting the formation of a band gap or forbidden region: A zoom of the latter is shown in the insets. The agreement between simulation and theory is excellent. Notice the interference fringes in the inset; the period ${\tau }_{\delta }$ of the oscillations was estimated using Eq. (3.16) and agrees well with the data (in spite of $\delta$ being quite close to $\pi$). The statistical error bar for $\delta =\pi$ is $\sim {10}^{-5}$.

Figure 10

Results of perturbation theory, up to second order for eigenvalues and eigenvectors, and computer simulations for the average Landauer resistance as a function of $\delta$ for $n=5000$ scatterers. The matrix elements of $\mathrm{\Delta }\mathrm{\Omega }$ of the perturbation were calculated numerically. The tendency of $\langle R/T\rangle$ to decrease with increasing $\delta$ and subsequently recover as $\delta$ approaches $\pi$ is reproduced by the perturbative approach. Very close to $\delta =\pi$, the approximation clearly fails.

Figure 11

The transmission coefficient for a stretch of 5000 identical scatterers, each consisting of a barrier and a well. Each unit cell has a length $d$. In the vicinity of $kd=\pi , T$ shows a dip, which becomes ever deeper as $n$ increases. One observes interference fringes on each side of the dip.

Figure 12

Results of computer simulations for the average Landauer resistance for the present model and the DWSL model of Ref. [8]. The parameters chosen for each model are indicated in the figure and conform to the criteria explained in the text. The agreement is excellent.