Spectral Analysis of Universal Conductance Fluctuations

Abstract

Universal conductance fluctuations are usually observed in the form of aperiodic oscillations in the magnetoresistance of thin wires under variation of the magnetic field B. A Fourier analysis of aperiodic oscillations observed in the classical experiments by Webb and Washburn reveals a practically discrete spectrum in agreement with the scenario based on the analogy with one-dimensional systems, according to which conductance fluctuations are due to the superposition of incommensurate harmonics. A more detailed analysis reveals the existence of a continuous component, whose smallness is explained theoretically. A lot of qualitative results are obtained that confirm the presented picture: the distribution of phases, frequency differences, and the growth exponents is consistent with theoretical predictions; discrete frequencies weakly depend on the processing procedure; and the discovered shift oscillations confirm the analogy with one-dimensional systems. Microscopic estimates show that the results obtained are consistent with the geometrical dimensions of the sample.

Appendices

APPENDIX A

1.1 Solution of Equation (19)

We restrict the analysis to the metallic region when typical values of ρ are small. Considering the eigenvalue problem for the operator on the right-hand side of Eq. (19) and restricting ourselves to the lowest order in ρ, we have the equation

$$ - \lambda P = P_{\rho }^{'} + \rho P_{{\rho \rho }}^{{''}}.$$

(A.1)

Assuming that ρ varies in the range from zero to R, we impose the finiteness condition at ρ = 0 and the zero boundary condition at ρ = R. Then the eigenvalues and eigenfunctions have the form

$${{\lambda }_{s}} = \mu _{s}^{2}{\text{/}}4R,\quad {{e}_{s}}(\rho ) = {{J}_{0}}({{\mu }_{s}}\sqrt {\rho {\text{/}}R} ),$$

(A.2)

where μ_s are the roots of the Bessel function J₀(x). Solving Eq. (19) with the initial condition (22) by expanding P(ρ) in terms of eigenfunctions (A.2), we have

$$P(\rho ,t) = \int\limits_0^\infty {2\mu d\mu {{{\text{e}}}^{{ - {{\mu }^{2}}t}}}{{J}_{0}}(2\mu \sqrt {{{\rho }_{0}}} ){{J}_{0}}(2\mu \sqrt \rho ),} $$

(A.3)

where t = α(L – L₀) and it is taken into account that, for large R, the spectrum of the eigenvalues μ_s becomes quasicontinuous and the summation over s can be replaced by integration with respect to μ using the asymptotics μ_s = πs + const for large s. Calculating the integral in (A.3), we obtain

$$P(\rho ,t) = \frac{1}{t}\exp \left\{ { - \frac{{\rho + {{\rho }_{0}}}}{t}} \right\}{{I}_{0}}\left( {\frac{{2\sqrt {\rho {{\rho }_{0}}} }}{t}} \right),$$

(A.4)

where I₀(x) = J₀(ix). For ρ $\lesssim$ t and t ≫ ρ₀, distribution (A.4) transforms into (21), and, for ρ ≈ ρ₀ and t ≪ ρ₀, it becomes Gaussian,

$$P(\rho ,t) = {{\left( {\frac{1}{{4\pi {{\rho }_{0}}t}}} \right)}^{{1/2}}}\exp \left\{ { - \frac{{{{{(\rho - {{\rho }_{0}})}}^{2}}}}{{4{{\rho }_{0}}t}}} \right\}.$$

(A.5)

The closeness of (A.4) to the Gaussian distribution allows us to characterize it by the first two moments. Multiplying (19) by ρⁿ and integrating with respect to ρ, we obtain evolution equations for the moments of the distribution P(ρ); their solution for the initial condition (22) has the form

$$\langle \rho \rangle = - \frac{1}{2} + \frac{{1 + 2{{\rho }_{0}}}}{2}{{{\text{e}}}^{{2t}}},$$

(A.6)

$$\langle {{\rho }^{2}}\rangle = \frac{1}{3} - \frac{{1 + 2{{\rho }_{0}}}}{2}{{{\text{e}}}^{{2t}}} + \left[ {\rho _{0}^{2} + \frac{{1 + 2{{\rho }_{0}}}}{2} - \frac{1}{3}} \right]{{{\text{e}}}^{{6t}}}.$$

For small ρ, the expressions are simplified,

$$\begin{gathered} \langle \rho \rangle = {{\rho }_{0}} + t, \\ \langle {{\rho }^{2}}\rangle = \rho _{0}^{2} + 4{{\rho }_{0}}t + 2{{t}^{2}}, \\ {{\sigma }^{2}} = 2{{\rho }_{0}}t + {{t}^{2}}, \\ \end{gathered} $$

(A.7)

and correspond to distribution (A.4). These results are valid for L > L₀. The description of evolution in the range 0 < L < L₀ is complicated by the need to satisfy two conditions (20) and (22), which is possible only under certain restrictions on the realization of the random potential. These restrictions are imposed on the interval (0, L₀) as a whole and are not significant for small L and L close to L₀. In the first case, we have 〈ρ〉 = σ in accordance with distribution (21). In the second case, the situation is determined by the fact that the broadening of the distribution occurs symmetrically when L deviates to the right or left of L₀.¹¹ For the quantity 〈ρ〉, conditions (20) and (22) are automatically satisfied if ρ₀ is chosen equal to the mean value of distribution (21) for L = L₀. As a result of the aforesaid, we arrive at the picture shown in Fig. 8.

APPENDIX B

1.1 Evolution of 〈ρ〉 for the Initial Condition (22)

The calculation of 〈ρ〉 technically reduces to studying the evolution of the second moments for the transfer matrix with complex elements T_ij [8]:

$$\begin{gathered} z_{1}^{{(l)}} = \langle {\text{|}}T_{{11}}^{{(l)}}{{{\text{|}}}^{2}}\rangle ,\quad z_{2}^{{(l)}} = \langle T_{{11}}^{{(l)}}T_{{12}}^{{(l)*}}\rangle , \\ z_{3}^{{(l)}} = \langle T_{{11}}^{{(l)*}}T_{{12}}^{{(l)}}\rangle ,\quad z_{4}^{{(l)}} = \langle {\text{|}}T_{{12}}^{{(l)}}{{{\text{|}}}^{2}}\rangle . \\ \end{gathered} $$

(B.1)

They satisfy the system of difference equations, whose general solution has the form [8]

$$\left( \begin{gathered} z_{1}^{{(l)}} \\ z_{2}^{{(l)}} \\ z_{3}^{{(l)}} \\ z_{4}^{{(l)}} \\ \end{gathered} \right) = {{C}_{0}}\left( \begin{gathered} - 1 \\ 0 \\ 0 \\ 1 \\ \end{gathered} \right) + \sum\limits_{i = 1}^3 {{{C}_{i}}} \left( \begin{gathered} 1 \\ {{e}_{2}}({{x}_{i}}) \\ {{e}_{3}}({{x}_{i}}) \\ 1 \\ \end{gathered} \right)\exp ({{x}_{i}}l),$$

(B.2)

where x₁, x₂, and x₃ are the roots of the first equation in (37) and

$${{e}_{2}}(x) = \frac{{\mathcal{A}x + \mathcal{B}}}{{p(x)}},\quad {{e}_{3}}(x) = \frac{{\mathcal{A}{\text{*}}x + \mathcal{B}{\text{*}}}}{{p(x)}},$$

$$\mathcal{A} = 2{{\epsilon }^{2}} - 2i\Delta ,\quad \mathcal{B} = 4\alpha \Delta + 4i{{\epsilon }^{2}}(\alpha - \Delta ),$$

(B.3)

$$p(x) = {{x}^{2}} + 2{{\epsilon }^{2}}x + 4{{\alpha }^{2}}.$$

Here α = –Δ₂δ, Δ = Δ₁δ, and δ and ${{\epsilon }^{2}}$ were defined after (23),

$${{\Delta }_{1}} = \frac{1}{2}\left( {\frac{k}{{\bar {k}}} - \frac{{\bar {k}}}{k}} \right),\quad {{\Delta }_{2}} = \frac{1}{2}\left( {\frac{k}{{\bar {k}}} + \frac{{\bar {k}}}{k}} \right),$$

(B.4)

where $\bar {k}$ and k are the Fermi momenta in the system under study and the ideal leads connected to it. In contrast to [8], as the initial condition, we take the general transfer matrix

$$T = \left( {\begin{array}{*{20}{c}} {\sqrt {\rho + 1} {{e}^{{i\varphi }}}}&{\sqrt \rho {{e}^{{i\theta }}}} \\ {\sqrt \rho {{e}^{{ - i\theta }}}}&{\sqrt {\rho + 1} {{e}^{{ - i\varphi }}}} \end{array}} \right),$$

(B.5)

with ρ = ρ₀ and ψ = θ – φ, rather than the identity matrix. The quantity $z_{4}^{{(n)}}$ immediately determines 〈ρ〉, giving a general expression for the latter:

$$\langle \rho \rangle = {{C}_{0}} + {{C}_{1}}{{{\text{e}}}^{{{{x}_{1}}l}}} + {{C}_{2}}{{{\text{e}}}^{{{{x}_{2}}l}}} + {{C}_{3}}{{{\text{e}}}^{{{{x}_{3}}l}}},$$

(B.6)

where

$$\begin{gathered} {{C}_{0}} = - \frac{1}{2},\quad {{C}_{i}} = {{( - 1)}^{{i + 1}}}\left[ {(1 + 2{{\rho }_{0}})\frac{{{{Q}_{i}}}}{{2Q}}} \right. \\ \left. {\, + {{K}_{1}}\frac{{{{R}_{i}}}}{Q} + {{K}_{2}}\frac{{{{S}_{i}}}}{Q}} \right],\quad i = 1,2,3 \\ \end{gathered} $$

(B.7)

and

$${{Q}_{1}} = ({{x}_{2}} - {{x}_{3}})p({{x}_{1}}),\quad {{Q}_{2}} = ({{x}_{1}} - {{x}_{3}})p({{x}_{2}}),$$

$${{Q}_{3}} = ({{x}_{1}} - {{x}_{2}})p({{x}_{3}}),$$

$$\begin{gathered} Q = {{Q}_{1}} - {{Q}_{2}} + {{Q}_{3}} \\ = x_{1}^{2}({{x}_{2}} - {{x}_{3}}) - x_{2}^{2}({{x}_{1}} - {{x}_{3}}) + x_{3}^{2}({{x}_{1}} - {{x}_{2}}). \\ \end{gathered} $$

$${{R}_{1}} = [{{x}_{2}}p({{x}_{3}}) - {{x}_{3}}p({{x}_{2}})]p({{x}_{1}}),$$

$${{R}_{2}} = [{{x}_{1}}p({{x}_{3}}) - {{x}_{3}}p({{x}_{1}})]p({{x}_{2}}),$$

$${{R}_{3}} = [{{x}_{1}}p({{x}_{2}}) - {{x}_{2}}p({{x}_{1}})]p({{x}_{3}}),$$

$${{S}_{1}} = [p({{x}_{3}}) - p({{x}_{2}})]p({{x}_{1}}),\quad {{S}_{2}} = [p({{x}_{3}}) - p({{x}_{1}})]p({{x}_{2}}),$$

$${{S}_{3}} = [p({{x}_{2}}) - p({{x}_{1}})]p({{x}_{3}}),$$

$$\begin{gathered} {{K}_{1}} = \frac{{{{\epsilon }^{2}}\sin \psi - \Delta \cos \psi }}{{4[\alpha {{\Delta }^{2}} + {{\epsilon }^{4}}(\alpha - \Delta )]}}\sqrt {{{\rho }_{0}}(1 + {{\rho }_{0}})} , \\ {{K}_{2}} = \frac{{{{\epsilon }^{2}}(\alpha - \Delta )\cos \psi + \alpha \Delta \sin \psi }}{{2[\alpha {{\Delta }^{2}} + {{\epsilon }^{4}}(\alpha - \Delta )]}}\sqrt {{{\rho }_{0}}(1 + {{\rho }_{0}})} . \\ \end{gathered} $$

(B.8)

Using the asymptotics for the roots x₁, x₂, and x₃ in the metallic regime [8], we arrive at the results (23) and (25). The first of them is valid for “natural” ideal leads,¹² which differ from the studied system only by the absence of the random potential in them; in this case, k = $\bar {k}$, Δ₁ = 0, and the oscillations appear in the first order in the small parameter ${{\epsilon }^{2}}$/δ. The result (25) is valid for foreign leads when Δ₁ ≠ 0 and oscillations occur in the zero order in ${{\epsilon }^{2}}$/δ. Using the asymptotics of x₁, x₂, and x₃ in the “critical” region [8] leads to the result (24), which is valid near the edge of the initial band; it is given for “natural” leads, since the situation for foreign leads is sufficiently characterized by the formulas given in [8].

APPENDIX C

1.1 On the Choice of the Natural Origin

According to the Onsager relations, conductance is an even function of the magnetic field B, and, when x = B, the function f(x) in (3) is even. Let us choose a smoothing function in the form G(x – a) + G(x + a) with even G(x), i.e., in a form symmetrized with respect to x. Then the Fourier transform is real and coincides, up to sign, with its modulus; therefore, it does not contain shift oscillations. Now we eliminate the function G(x + a). The resulting Fourier transform F(ω) of the function f(x)G(x – a) is complex; its real part is half the previous one and does not contain shift oscillations.¹³ The latter are also absent in ImF(ω), since they equally affect the real and imaginary parts. After the shift x → x + a, the Fourier transform takes the form

$$F(\omega ) = {{{\text{e}}}^{{i\omega a}}}\int {f(x + a)G(x){{{\text{e}}}^{{i\omega x}}}dx} $$

and the resulting integral for a = μ₀ corresponds to that considered in Section 2, while the factor e^iωa corresponds to the shift oscillations. However, the sign of a obtained in this case does not correspond to that found empirically.

The reason for the contradiction is that the Onsager symmetry distinguishes not only the value B = 0, but also B = ∞; it is the latter value that corresponds to the empirical situation. When choosing x = 1/B, we can repeat all the previous arguments; however, the increments of B and x have opposite signs, and, for a qualitative agreement with Section 2, we should change the sign of ω. This will give rise to a factor e^iωa with the correct sign of a.