Electron transport and electron density inside quasi-one-dimensional disordered conductors

TL;DR

This paper investigates the statistical behavior of electron density inside quasi-one-dimensional disordered conductors, analyzing different transport regimes through analytical and simulation methods, and proposes a measurement setup.

Contribution

It extends existing techniques to analyze electron density within the sample using scattering properties, covering ballistic, diffusive, and localized regimes.

Findings

Good agreement between analytical and simulation results

Logarithm of electron density exhibits self-averaging behavior

Techniques applicable to other physics branches like electrodynamics and elasticity

Abstract

We consider the problem of electron transport across a quasi-one-dimensional disordered multiply-scattering medium, and study the statistical properties of the electron density inside the system. In the physical setup that we contemplate, electrons of a given energy feed the disordered conductor from one end. The physical quantity that is mainly considered is the logarithm of the electron density, $\ln {\cal W}(x)$, since its statistical properties exhibit a self-averaging behavior. We also describe a {\em gedanken} experiment, as a possible setup to measure the electron density. We study analytically and through computer simulations the ballistic, diffusive and localized regimes. We generally find a good agreement between the two approaches. The extension of the techniques that were used in the past to find information outside the sample is done in terms of the scattering properties of the two segments that form the entire conductor on each side of the observation point. The problem is of interest in various other branches of physics, as electrodynamics and elasticity.

Equations

$$\alpha =\left(u_2\sqrt{1+{\lambda }_2}{v}_2\right)\left(u_1\sqrt{1+{\lambda }_1}{v}_1\right)+\left(u_2\sqrt{{\lambda }_2}{v}_2^{\ast }\right)\left(u_1^{\ast }\sqrt{{\lambda }_1}{v}_1\right).$$

$$t=u_2{v}_2\left[I_N-\frac{1}{2}{u}_1{\lambda }_1{u}_1^{†}-\frac{1}{2}{v}_2^{†}{\lambda }_2{v}_2-(u_1\sqrt{{\lambda }_1}{u}_1^T)(v_2^T\sqrt{{\lambda }_2}{v}_2)+O({\lambda }_i^2)\right]u_1{v}_1.$$

$${\alpha }_2^{†}t$$

$${\beta }_2^{†}t$$

$$\displaystyle{\cal W}(x)=\sum_{n}\frac{\widetilde{k}}{k_{n}}\Bigg{\{}1-2{\rm Re}\left[e^{-2ik_{n}x}(v_{2}^{T}\sqrt{\lambda_{2}}v_{2})\right]_{nn}-(u_{1}\lambda_{1}u_{1}^{\dagger})_{nn}+(v_{2}^{\dagger}\lambda_{2}v_{2})_{nn}$$

$$\displaystyle-2{\rm Re}\left[(u_{1}\sqrt{\lambda_{1}}u_{1}^{T})(v_{2}^{T}\sqrt{\lambda_{2}}v_{2})\right]_{nn}+O(\lambda_{i}^{3/2})\Bigg{\}}$$

$${⟨\ln \frac{\mathcal{W}(x)}{N}⟩}_{s\ll 1,EC}=$$

$$\mathrm{Tr}{⟨v_2^T\sqrt{{\lambda }_2}{v}_2⟩}_{s_2}=\sum _{ab}{⟨\sqrt{({\lambda }_2{)}_b}⟩}_{s_2}{⟨(v_2{)}_{ba}(v_2{)}_{ba}⟩}_0=0,$$

$$-\frac{1}{N}\mathrm{Tr}{⟨{\lambda }_1⟩}_{s_1}=-\frac{1}{N}N\frac{L_1}{\ell }=-\frac{L_1}{\ell }=-\frac{x}{\ell }$$

$$\frac{1}{N}\mathrm{Tr}{⟨{\lambda }_2⟩}_{s_2}=\frac{1}{N}N\frac{L_2}{\ell }=\frac{L_2}{\ell }=\frac{L-x}{\ell }$$

$${⟨u_1\sqrt{{\lambda }_1}{u}_1^T⟩}_{s_1}{⟨v_2^T\sqrt{{\lambda }_2}{v}_2⟩}_{s_2}=0$$

$$⟨{\left[\mathrm{Tr}\left(v_2^T\sqrt{{\lambda }_2}{v}_2\right)\right]}^2⟩=0$$

$$⟨{\left[\mathrm{Tr}\left(v_2^{†}\sqrt{{\lambda }_2}{v}_2^{\ast }\right)\right]}^2⟩=0$$

$$⟨\mathrm{Tr}\left(v_2^T\sqrt{{\lambda }_2}{v}_2\right)\mathrm{Tr}\left(v_2^{†}\sqrt{{\lambda }_2}{v}_2^{\ast }\right)⟩=\frac{2}{N+1}N\frac{L_2}{\ell },$$

$$(\text{ref}e)=-\frac{2}{N(N+1)}\frac{L_2}{\ell }.$$

$$\sum _{a,n=1}^N{v}_{an}{v}_{an}$$

$$\equiv$$

$$\mathrm{with}:s{s}^{†}$$

$$p({\theta }_1,\cdots ,{\theta }_N)=C_N\prod _{1=a<b}^N\left|{\mathrm{e}}^{i{\theta }_a}-{\mathrm{e}}^{i{\theta }_b}\right|.$$

$${⟨\ln \frac{\mathcal{W}(0)}{N}⟩}_{s\gg N,EC}$$

$${⟨\ln \mathcal{W}(0)⟩}_{s\gg 1}={\int }_0^{2\pi }\ln (1-\cos \theta )\frac{d\theta }{2\pi }+\ln 2=-\ln 2+\ln 2=0,$$

$$p({\theta }_1,{\theta }_2)=\frac{1}{8\pi \sqrt{2}}\sqrt{1-\cos ({\theta }_1-{\theta }_2)}.$$

$${⟨\ln \left(1-\frac{\cos {\theta }_1+\cos {\theta }_2}{2}\right)⟩}_0$$

$$={\int }_0^{2\pi }d{\theta }_1{\int }_0^{2\pi }d{\theta }_2\ln \left(1-\frac{\cos {\theta }_1+\cos {\theta }_2}{2}\right)p({\theta }_1,{\theta }_2)$$

$$=-0.122351,$$

Full text

I Calculation of the intensity in the ballistic regime

The submatrix $\alpha$ of the transfer matrix (Eq. (A2) of the main text) for the full system is given in terms of the polar parameters of the two individual sections of the wire as

[TABLE]

The transmission matrix $t$, written in Eq. (A4b) of the main text as $t=1/\alpha^{\dagger}$, is given, in the ballistic regime, up to terms $\lambda_{i}^{3/2}$, by

[TABLE]

The terms $\alpha_{2}^{\dagger}t$ and $\beta_{2}^{\dagger}t$, appearing in Eq. (2.9a) of the main text, then take the form

[TABLE]

Substituting expressions (3a) and (3b) in Eq. (2.9a) of the text, we find

[TABLE]

In an equivalent-channel (EC) approximation, $k_{n}^{EC}\equiv k_{0}\;\;\forall n$, an ensemble average of the logarithm of the intensity can be written, upon expanding the logarithm, as

[TABLE]

In Eq. (5) we have

[TABLE]

Putting the various terms together, we find the result (3.3) of the text.

II The localized regime

In order to compute the expectation value appearing in Eq. (3.21) of the main text, we first define the unitary, symmetric matrix $s=vv^{T}$ and its eigenvalues ${\rm e}^{i\theta_{a}}$. We can write

[TABLE]

If $v$ is distributed according to the invariant measure of the group $U(N)$, $s$ is distributed according to the invariant measure of unitary, symmetric $N$-dimensional matrices, and its eigenvalues according to the distribution [dyson, , mehta, ]

[TABLE]

The expectation value in Eq. (3.21) of the main text can thus be written as

[TABLE]

Here, the index [math] indicates that the average over $v$ is computed according to the invariant measure of the unitary group $U(N)$.

As an example, Eq. (9) gives, for $N=1$,

[TABLE]

verifying the result of Ref. [cheng-yepez-mello-genack, ].

For $N=2$, Eq. (2.2) gives

[TABLE]

The integral needed in Eq. (9) was evaluated numerically, with the result

[TABLE]

as quoted in Eq. (3.22) of the main text.