Conductance Distribution in 1D Systems: Dependence on the Fermi Level and the Ideal Leads

Abstract

The correct definition of the conductance of finite systems implies a connection to the system of the massive ideal leads. Influence of the latter on the properties of the system appears to be rather essential and is studied below on the simplest example of the 1D case. In the log-normal regime, this influence is reduced to the change of the absolute scale of conductance, but generally changes the whole distribution function. Under the change of the system length L, its resistance may undergo the periodic or aperiodic oscillations. Variation of the Fermi level induces qualitative changes in the conductance distribution, resembling the smoothed Anderson transition.

Appendices

EVOLUTION OF MOMENTS FOR THE COORDINATE TRANSFER MATRIX

For the coordinate transfer matrix the following evolution equations are valid

$${{z}_{{n + 1}}} = (E - {{V}_{n}}){{z}_{n}} + {{y}_{n}},\quad {{y}_{{n + 1}}} = - {{z}_{n}},$$

((A.1))

where z_n = $\tau _{{12}}^{{(n - 1)}}$, y_n = $\tau _{{11}}^{{(n - 1)}}$ with the initial conditions z₁ = 0, y₁ = 1, or z_n = $\tau _{{22}}^{{(n - 1)}}$, y_n = $\tau _{{21}}^{{(n - 1)}}$ with the initial conditions z₁ = 1, y₁ = 0. For the second moments one has

$$\left( \begin{gathered} \overline {z_{{n + 1}}^{2}} \\ \overline {{{z}_{{n + 1}}}{{y}_{{n + 1}}}} \\ \overline {y_{{n + 1}}^{2}} \\ \end{gathered} \right) = \left( {\begin{array}{*{20}{c}} {{{E}^{2}} + {{W}^{2}}}&{2E}&1 \\ { - E}&{ - 1}&0 \\ 1&0&0 \end{array}} \right)\left( \begin{gathered} \overline {z_{n}^{2}} \\ \overline {{{z}_{n}}{{y}_{n}}} \\ \overline {y_{n}^{2}} \\ \end{gathered} \right),$$

((A.2))

and suggesting their exponential behavior λⁿ with λ = 1 + x, obtains the equation for x

$${{x}^{3}} - {{x}^{2}}({{E}^{2}} - 4) - x({{E}^{2}} - 4) = {{W}^{2}}(2 + 3x + {{x}^{2}}).$$

((A.3))

Setting E² – 4 = 4δ², W² = 4${{\epsilon }^{2}}$δ² and taking the limit δ → 0, $\epsilon$ → 0, δ/${{\epsilon }^{2}}$ = const, one can verify that (A.3) coincides with (44).

Analogously, for the fourth moments one has a system of equations

$$\left( \begin{gathered} \overline {z_{{n + 1}}^{4}} \\ \overline {z_{{n + 1}}^{3}{{y}_{{n + 1}}}} \\ \overline {z_{{n + 1}}^{2}y_{{n + 1}}^{2}} \\ \overline {{{z}_{{n + 1}}}y_{{n + 1}}^{3}} \\ \overline {y_{{n + 1}}^{4}} \\ \end{gathered} \right) = \left( {\begin{array}{*{20}{c}} {{{E}^{4}} + 6{{E}^{2}}{{W}^{2}}}&{4{{E}^{3}} + 12E{{W}^{2}}}&{6{{E}^{2}} + 6{{W}^{2}}}&{4E}&1 \\ { - {{E}^{3}} - 3E{{W}^{2}}}&{ - 3{{E}^{2}} - 3{{W}^{2}}}&{ - 3E}&{ - 1}&0 \\ {{{E}^{2}} + {{W}^{2}}}&{2E}&1&0&0 \\ { - E}&{ - 1}&0&0&0 \\ 1&0&0&0&0 \end{array}} \right)\left( \begin{gathered} \overline {z_{n}^{4}} \\ \overline {z_{n}^{3}{{y}_{n}}} \\ \overline {z_{n}^{2}y_{n}^{2}} \\ \overline {{{z}_{n}}y_{n}^{3}} \\ \overline {y_{n}^{4}} \\ \end{gathered} \right).$$

((A.4))

Suggesting that the moments behave as λ_n with λ = 1 + x, we have the equation for x

$$\begin{gathered} - {{x}^{5}} + {{x}^{4}}({{E}^{2}}\, + \,1)({{E}^{2}} - 4) \\ + \,{{x}^{3}}({{E}^{2}}\, - \,4)( - {{E}^{4}} + 5{{E}^{2}} + 1) \\ \, - {{x}^{2}}2{{E}^{2}}{{({{E}^{2}} - 4)}^{2}} - x{{E}^{2}}{{({{E}^{2}} - 4)}^{2}} \\ \, + {{W}^{2}}[ - 6{{E}^{2}}({{E}^{2}} - 4) - 15x{{E}^{2}}({{E}^{2}} - 4) \\ \, + {{x}^{2}}( - 12{{E}^{4}} + 60{{E}^{2}} - 6) \\ \, + {{x}^{3}}( - 3{{E}^{4}} + 30{{E}^{2}} - 9) + {{x}^{4}}(6{{E}^{2}} - 3)] = 0, \\ \end{gathered} $$

((A.5))

where terms of the higher order in W² are omitted. Setting E² – 4 = 4δ², W² = 4${{\epsilon }^{2}}$δ² and retaining the main terms at the indicated limiting transition, we come to Eq. (48).

THE ASYMPTOTIC FORMS OF THE INTEGRAL (70)

Using the evenness of the integrand of (70) in variables $\tilde {\varphi }$ = φ – π/2, $\tilde {\theta }$ = θ – π/2 and setting x = sinφ, y = sinθ, we have

$$\begin{gathered} P(\rho ) = \frac{1}{{{{\pi }^{2}}t}}\int\limits_{ - 1}^1 {\frac{{dx}}{{\sqrt {1 - {{x}^{2}}} }}\int\limits_{ - 1}^1 {\frac{{dy}}{{\sqrt {1 - {{y}^{2}}} }}\exp \left\{ { - \frac{{S(x,y)}}{t}} \right\},} } \\ S(x,y) = {{({{\Delta }_{1}}\sqrt {1 + \rho } x + {{\Delta }_{2}}\sqrt \rho y)}^{2}} + \rho (1 - {{y}^{2}}). \\ \end{gathered} $$

((B.1))

Configuration of saddle points is essentially different for ρ < ρ_c and ρ > ρ_c, where ρ_c = $\Delta _{1}^{2}$. In the first case the maximum of the exponent is reached at x = x_c, y = –1 or x = –x_c, y = 1, where x_c = Δ₂$\sqrt \rho$/Δ₁$\sqrt {1 + \rho }$, while in the second case at x = 1, y = –1 or x = –1, y = 1. Let δx and δy are deviations of x and y from the extremum point. For ρ < ρ_c we retain in S(x, y) the quadratic term in δx and the linear term in δy, setting x = x_c, 1 – y² = 2δy in the pre-exponential; then

$$P(\rho ) = \frac{1}{\pi }\sqrt {\frac{1}{{\rho ({{\rho }_{c}} - \rho )}}} ,$$

((B.2))

which is the limiting form of the distribution for t → 0.

For ρ > ρ_c we have

$$S(x,y)\, = \,{{S}_{c}}\, + \,A\delta x\, + \,B\delta y\, + \,C{{(\delta x\, - \,{{x}_{c}}\delta y)}^{2}}\, - \,\rho {{(\delta y)}^{2}},$$

((B.3))

where S_c, A, B are defined in Eq. (73) and C = $\Delta _{1}^{2}$(1 + ρ). For large ρ the linear terms in δx, δy are sufficient in S(x, y) and in radicals; then

$$P(\rho ) = \frac{1}{\pi }\sqrt {\frac{1}{{AB}}} \exp \left( { - \frac{{{{S}_{c}}}}{t}} \right),$$

((B.4))

which is the last asymptotics in (72), (74)–(77).

The results (B.2), (B.4) are not applicable for ρ close to ρ_c, since the coefficient A turns to zero at ρ = ρ_c. Setting δ$\tilde {x}$ = δx – x_cδy and omitting the last term in (B.3), we have

$$S(x,y) = {{S}_{c}} + A\delta \tilde {x} + C{{(\tilde {\delta }\tilde {x})}^{2}} + 2\rho \delta y.$$

((B.5))

For A² ≪ Ct fluctuations of δ$\tilde {x}$ are determined by the quadratic term, and Aδ$\tilde {x}$ can be omitted; then in the small vicinity of ρ_c

$$P(\rho ) = \frac{1}{{2\pi }}\sqrt {\frac{1}{{2\pi \rho }}} \frac{{\Gamma (1{\text{/}}4)}}{{{{{(Ct)}}^{{1/4}}}}},\quad {\text{|}}\rho - {{\rho }_{c}}{\text{|}} \lesssim {{(Ct)}^{{1/2}}},$$

((B.6))

so divergency at ρ → ρ_c is eliminated and the third asymptotics in (72) and (75) is recovered. Under the indicated condition, this result remains valid for ρ < ρ_c, when A is negative, but small in modulus. In fact, Eq. (B.6) takes place in the cases t ≪ $\Delta _{1}^{2}$ ≪ 1 and t ≪ 1 ≪ $\Delta _{1}^{2}$, when the fluctuation δy ~ t/ρ is small in comparison with δ$\tilde {x}$ ~ $\sqrt {t{\text{/}}C}$, so δ$\tilde {x}$ ≈ δx and one can set 1 – x² = 2δ$\tilde {x}$, 1 – y² = 2δy in the pre-exponential. The inverse situation is realized in the case $\Delta _{1}^{2}$ ≫ 1, t ≫ 1, when δy ≫ δ$\tilde {x}$ and 1 – x² ≈ 2δx = 2δ$\tilde {x}$ + 2x_cδy ≈ 2δy in the preexponential; then we come to the result

$$\begin{gathered} P(\rho ) = \frac{1}{{2{{\pi }^{2}}}}\sqrt {\frac{\pi }{{\rho \Delta _{1}^{2}t}}} \ln \frac{{\Delta _{1}^{2}t}}{\rho }\exp \left( { - \frac{{{{S}_{c}}}}{t}} \right), \\ {{\rho }_{c}}{\text{/}}t \lesssim {{\rho }_{c}} \lesssim {{\rho }_{c}}t, \\ \end{gathered} $$

((B.7))

determining the penultimate asymptotics in (76), (77). The condition A² ≪ Ct corresponds to (ρ – ρ_c)² ≪ $\Delta _{1}^{2}$ρt, which for t ≫ 1 is reduced to the given in (B.7). If (B.6) is valid in the small vicinity of ρ_c, the wide range of validity arises for (B.7).

For small ρ the saddle point approximation is not applicable, and the initial form of (70) is convenient. If t ≪ $\Delta _{1}^{2}$, then saddle point integration over φ is still possible and leads to expression

$$\begin{gathered} P(\rho ) = \frac{1}{{{{\pi }^{2}}t}}\int\limits_{ - \pi /2}^{\pi /2} {d\theta \sqrt {\frac{{\pi t}}{{\Delta _{1}^{2} - \rho + \Delta _{2}^{2}\rho {{{\sin }}^{2}}\theta }}} } \\ \times \exp \left\{ { - \frac{\rho }{t}{{{\sin }}^{2}}\theta } \right\}. \\ \end{gathered} $$

((B.8))

For $\Delta _{1}^{2}$ ≪ 1 one has Δ₂ ≈ 1; if ρ ≪ t, then only $\Delta _{1}^{2}$ can be retained in the denominator, while integration over θ can be produced by expansion of the exponent

$$P(\rho ) = \frac{1}{\pi }\sqrt {\frac{\pi }{{\Delta _{1}^{2}t}}} \exp \left( { - \frac{\rho }{{2t}}} \right),\quad \rho \ll t,$$

((B.9))

which is the first result in (72), (75), (76). For ρ $\gtrsim$t one returns to the saddle point results (B.2), (B.4), (B.5).

In the case $\Delta _{1}^{2}$ ≪ 1 and t ≫ $\Delta _{1}^{2}$ the region of small ρ is determined by the condition ρ(1 + ρ) ≪ t²/$\Delta _{1}^{2}$ and the integral (70) is calculated by expansion over S(φ, θ)/t till the second order, which gives the first result in (74), (77); for the opposite inequality we have A ≈ B ≈ 2Δ₁Δ₂$\sqrt {\rho (1 + \rho )}$ ≫ t and A² ≫ Ct, which is sufficient for validity of (B.4).

In the case $\Delta _{1}^{2}$ ≫ 1 and t ≪ $\Delta _{1}^{2}$Eq. (B.8) remains valid, but its analysis is more complicated. For t ≪ 1 we have the familiar situation: in the interval ρ $\lesssim$t one can retain $\Delta _{1}^{2}$ in the denominator and obtain (B.9), while in the interval ρ $\gtrsim$t the saddle point results (B.2), (B.4), (B.5) are valid. For 1 ≪ t ≪ $\Delta _{1}^{2}$ the result (B.9) is valid only for ρ $\lesssim$ 1. In the interval 1 $\lesssim$ ρ $\lesssim$t the term $\Delta _{2}^{2}$ρsin²θ is dominated in the denominator of (B.8), while the quantity $\Delta _{1}^{2}$ – ρ is necessary only for cutoff of the logarithmic divergency:

$$P(\rho ) = \frac{1}{{{{\pi }^{2}}}}\sqrt {\frac{\pi }{{\rho \Delta _{1}^{2}t}}} \ln \frac{{\Delta _{1}^{2}\rho }}{{{{\rho }_{c}} - \rho }},\quad 1 \lesssim \rho \lesssim t.$$

((B.10))

In the interval t$\lesssim$ ρ $\lesssim$ ρ_c the exponent restricts integration in (B.8) by values θ²$\lesssim$t/ρ, so $\Delta _{1}^{2}$ρ in (B.10) is changed by $\Delta _{1}^{2}$t. In fact, both results are actual only for ρ ≪ ρ_c, since in the interval ρ $\gtrsim$ ρ_c/t divergency at ρ → ρ_c is eliminated due to nonlinear terms in (B.3) and the result (B.7) is valid; so we have ln ρ for 1 $\lesssim$ ρ $\lesssim$t and ln t for t$\lesssim$ ρ $\lesssim$ ρ_c/t, as is reflected in (76).

In the case $\Delta _{1}^{2}$ ≫ 1 and t ≫ $\Delta _{1}^{2}$, expansion over S(φ, θ)/t is possible in the interval ρ $\lesssim$t/$\Delta _{1}^{2}$ and leads to the first result (77). In the interval ρ $\gtrsim$t/$\Delta _{1}^{2}$ expression (B.8) is valid, where |$\Delta _{1}^{2}$ – ρ| ≪ t, $\Delta _{2}^{2}$ ≫ ρ≫ t and the latter term is dominant in the denominator; the logarithmic divergency is removed due to restriction ($\Delta _{2}^{2}$ρ/t)sin²θ $\gtrsim$ 1, which is necessary for the saddle point integration over φ and validity of (B.8). If ρ $\lesssim$t, then the exponent in (B.8) is not essential and the second result (77) holds. If ρ $\gtrsim$t, then we have the saddle point situation and validity of (B.7) and (B.4).