Nonlinear transport theory in the metal with tunnel barrier

TL;DR

This paper develops a nonlinear transport theory for electrons in metals with tunnel barriers using scattering matrix formalism, providing a simplified way to calculate nonlinear conductivity contributions in hybrid structures.

Contribution

It introduces a general expression for mean electrical current that simplifies nonlinear conductivity calculations in metal-tunnel barrier systems.

Findings

Linear approximation matches other theories.

Ballistic transport occurs at effective barrier transmission of 1/3.

Landauer conductivity equals 2e^2/h at this transmission.

Abstract

Within the framework of the scattering matrix formalism the nonlinear Kubo theory for electron transport in the metal with a tunnel barrier has been considered. A general expression for the mean electrical current was obtained. It significantly simplifies the calculation of nonlinear contributions to the conductivity of various hybrid structures. In the model of the tunnel Hamiltonian all linear and nonlinear contributions to a mean electrical current are evaluated. The linear approximation agrees with results of other theories. For effective barrier transmission $\tilde T = 1/3$ the ballistic transport is realized with a value of the Landauer conductivity equal to $G = \frac{{2e^2 }}{h}$.

Equations

$$i\frac{\partial \rho (t)}{\partial t}=\left[H_T(t),\rho (t)\right],$$

$${\rho }_I(t)=\tilde{U}(t){\rho }_0{\tilde{U}}^{+}(t),$$

$$\tilde{U}(t)=S(t,-\infty )=T_t\exp \left[-i\underset{-\infty }{\overset{t}{\int }}{H}_T(t)dt\right],$$

$$\displaystyle\begin{array}[]{l}<I>=Tr(\rho I)=\frac{1}{{\sum\limits_{n}{\left\langle n\right|U^{+}(t)}\rho_{0}U(t)\left|n\right\rangle}}\sum\limits_{n}{\left\langle n\right|U^{+}(t)IU(t)}U^{+}(t)\rho_{0}U(t)\left|n\right\rangle=\\ \quad\quad\frac{1}{{Tr(\tilde{U}(t)\rho_{0}\tilde{U}^{+}(t))}}Tr\left({e^{i\hat{H}_{0}t}Ie^{-i\hat{H}_{0}t}\tilde{U}(t)\rho_{0}\tilde{U}^{+}(t)}\right),\\ \end{array}$$

$$<I>=\frac{1}{Sp({\rho }_I(t))}Sp(I(t){\rho }_I(t))$$

$$e^{A}B{e}^{-A}=\sum _{j=0}^{\infty }\frac{1}{j!}{\left[A,B\right]}_{(j)},$$

$${\left[A,B\right]}_{(j+1)}=\left[A,{\left[A,B\right]}_{(j)}\right]$$

$$\displaystyle\begin{array}[]{l}\rho_{I}(t)=\\ \sum\limits_{j=0}^{\infty}{\frac{{(-i)^{j}}}{{j!}}}\int\limits_{-\infty}^{t}{dt_{j}dt_{j-1}...dt_{1}T_{t}\left\{{\left[{H_{T}(t_{j}),\left[{H_{T}(t_{j-1}),\left[{...}\right[H_{T}(t_{2}),\left[{H_{T}(t_{1}),\rho_{0}}\right]}\right]...}\right]}\right\}}\end{array}$$

$$\left[H_T(t_j),{\rho }_0\right]=H_T(t_j){\rho }_0-{\rho }_0{H}_T(t_j){\rho }_0^{-1}{\rho }_0=A(t_j){\rho }_0,$$

$$A(t_j)=H_T(t_j)-{\rho }_0{H}_T(t_j){\rho }_0^{-1}$$

$$\displaystyle\begin{array}[]{l}{\rm}\left[{H_{T}(t_{2})\left[{H_{T}(t_{1}),\rho_{0}}\right]}\right]=\\ A(t_{1})\left[{H_{T}(t_{2}),\rho_{0}}\right]+\left[{H_{T}(t_{2}),A(t_{1})}\right]\rho_{0}=A(t_{1})A(t_{2})\rho_{0}+\left[{H_{T}(t_{2}),A(t_{1})}\right]\rho_{0}\end{array}$$

$${\rho }_I(t)=\sum _{j=0}^{\infty }\frac{(-i{)}^j}{j!}\underset{-\infty }{\overset{t}{\int }}d{t}_{j}d{t}_{j-1}...d{t}_1{T}_t\left\{A(t_1)A(t_2)...A(t_j){\rho }_0\right\}$$

$$<I>=\frac{1}{Tr({\rho }_I(t))}\sum _{j=0}^{\infty }\frac{(-i{)}^j}{j!}\underset{-\infty }{\overset{t}{\int }}d{t}_{j}d{t}_{j-1}...d{t}_{1}Tr(T_t\left\{I(t)A(t_1)A(t_2)...A(t_j){\rho }_0\right\})$$

$$\frac{1}{Tr({\rho }_I(t))}Tr(T_t\left\{I(t)A(t_1)A(t_2)...A(t_j){\rho }_0\right\})=<T_t\left\{I(t)A(t_1)A(t_2)...A(t_j){\rho }_0\right\}{>}_{0con.}$$

$$<I>=\sum _{j=0}^{\infty }(-i{)}^j\underset{-\infty }{\overset{t}{\int }}d{t}_{j}d{t}_{j-1}...d{t}_1<T_t\left\{I(t)A(t_1)A(t_2)...A(t_j){\rho }_0\right\}{>}_{0con.}$$

$$\hat{H}={\hat{H}}_0+H_T,$$

$$H_T=\sum _{\mathbf{kp}}\left(T_{\mathbf{kp}}{a}_{\mathbf{k}}^{+}{a}_{\mathbf{p}}+T_{\mathbf{kp}}^{\ast }{a}_{\mathbf{p}}^{+}{a}_{\mathbf{k}}\right)$$

$$e^{-\beta {\hat{H}}_L}=e^{-\beta \sum _{\mathbf{k}}{\xi }_{\mathbf{k}}{n}_{\mathbf{k}}}=\prod _{\mathbf{k}}\left\{E_{\mathbf{k}}+\left(e^{-\beta {\epsilon }_{\mathbf{k}}}-1\right)n_{\mathbf{k}}\right\},$$

$$Tr(e^{-\beta {\hat{H}}_L})=\prod _{\mathbf{k}}\left\{1+e^{-\beta {\epsilon }_{\mathbf{k}}}\right\}$$

$$\left[{\hat{H}}_0,H_T\right]=\sum _{\mathbf{kp}}({\xi }_{\mathbf{k}}-{\xi }_{\mathbf{p}})\left\{T_{\mathbf{kp}}{a}_{\mathbf{k}}^{+}{a}_{\mathbf{p}}-T_{\mathbf{kp}}^{\ast }{a}_{\mathbf{p}}^{+}{a}_{\mathbf{k}}\right\}$$

$$\left[{\hat{H}}_0\left[{\hat{H}}_0,H_T\right]\right]=\sum _{\mathbf{kp}}({\xi }_{\mathbf{k}}-{\xi }_{\mathbf{p}}{)}^2\left\{T_{\mathbf{kp}}{a}_{\mathbf{k}}^{+}{a}_{\mathbf{p}}+T_{\mathbf{kp}}^{\ast }{a}_{\mathbf{p}}^{+}{a}_{\mathbf{k}}\right\}$$

$$\displaystyle\begin{array}[]{l}A(t_{j})=\\ \quad\quad\sum\limits_{\bm{kp}}{\left\{{\left({1-e^{\beta(\xi_{\bm p}-\xi_{\bm k})}}\right)T_{\bm{kp}}a_{\bm k}^{+}(t_{j})a_{\bm p}(t_{j})+\left({1-e^{-\beta(\xi_{\bm p}-\xi_{\bm k})}}\right)T_{\bm{kp}}^{*}a_{\bm p}^{+}(t_{j})a_{\bm k}(t_{j})}\right\}}\\ \end{array}$$

$$\frac{\left(E_{\mathbf{k}}+\left(e^{-\beta {\xi }_{\mathbf{k}}}-1\right)n_{\mathbf{k}}\right)\left(E_{\mathbf{p}}+\left(e^{-\beta {\xi }_{\mathbf{p}}}-1\right)n_{\mathbf{p}}\right)}{Tr\left\{\left(E_{\mathbf{k}}+\left(e^{-\beta {\xi }_{\mathbf{k}}}-1\right)n_{\mathbf{k}}\right)\left(E_{\mathbf{p}}+\left(e^{-\beta {\xi }_{\mathbf{p}}}-1\right)n_{\mathbf{p}}\right)\right\}}=$$

$$\frac{\left(E_{\mathbf{k}}+\left(e^{-\beta {\xi }_{\mathbf{k}}}-1\right)n_{\mathbf{k}}\right)\left(E_{\mathbf{p}}+\left(e^{-\beta {\xi }_{\mathbf{p}}}-1\right)n_{\mathbf{p}}\right)}{\left(1+e^{-\beta {\xi }_{\mathbf{k}}}\right)\left(1+e^{-\beta {\xi }_{\mathbf{p}}}\right)}$$

$$\tilde{A}(t_j)=\sum _{\mathbf{kp}}\left[f({\xi }_{\mathbf{p}})-f({\xi }_{\mathbf{k}})\right]\left\{T_{\mathbf{kp}}{a}_{\mathbf{k}}^{+}(t_j)a_{\mathbf{p}}(t_j)-T_{\mathbf{kp}}^{\ast }{a}_{\mathbf{p}}^{+}(t_j)a_{\mathbf{k}}(t_j)\right\},$$

$${\hat{H}}^{\prime }={\hat{H}}_0^{\prime }+H_T,$$

$${\hat{H}}_0^{\prime }=\sum _{\mathbf{k}}{\epsilon }_{\mathbf{k}}{n}_{\mathbf{k}}+\sum _{\mathbf{p}}{\epsilon }_{\mathbf{p}}{n}_{\mathbf{p}}=\sum _{\mathbf{k}}{\xi }_{\mathbf{k}}{n}_{\mathbf{k}}+\sum _{\mathbf{p}}{\xi }_{\mathbf{p}}{n}_{\mathbf{p}}+{\mu }_L{N}_L+{\mu }_R{N}_R$$

$$\displaystyle\begin{array}[]{l}\tilde{a}_{\bm k}(t_{j})=e^{-it_{j}\mu_{L}}a_{\bm k}(t_{j})\\ \tilde{a}_{\bm k}^{+}(t_{j})=e^{it_{j}\mu_{L}}a_{\bm k}^{+}(t_{j})\\ \end{array}$$

$$H_T(t_j)=\sum _{\mathbf{kp}}\left\{e^{ieV{t}_j}{T}_{\mathbf{kp}}{a}_{\mathbf{k}}^{+}(t_j)a_{\mathbf{p}}(t_j)+e^{-ieV{t}_j}{T}_{\mathbf{kp}}^{\ast }{a}_{\mathbf{p}}^{+}(t_j)a_{\mathbf{k}}(t_j)\right\}$$

$$\tilde{A}(t_j)=\sum _{\mathbf{kp}}\left\{{\tilde{T}}_{1\mathbf{kp}}{e}^{ieV{t}_j}{a}_{\mathbf{k}}^{+}(t_j)a_{\mathbf{p}}(t_j)+{\tilde{T}}_{2\mathbf{kp}}{e}^{-ieV{t}_j}{a}_{\mathbf{p}}^{+}(t_j)a_{\mathbf{k}}(t_j)\right\},$$

Full text

1 Introduction

At present the method for solving a problem of electron transport in the hybrid structures is one of the key tasks from the theoretical as well as the practical point of view. On the one hand, a derivation of the theory in the framework of modern diagrammatic methods based on the scattering matrix formalism makes it possible to better understand the nature of electronic dynamics in nonequilibrium processes and also to study nonlinear effects in condensed matter. On the other hand, the design of the electron devices demands the sufficiently detailed information on processes of the electron scattering on contacts, barriers and impurities to be caused by technology of industry. Also, in the miniaturization of working medium, the problem of accounting for quantum properties in nanocontacts and nanowires somehow arises.

It should be noted that the researchers have accumulated a considerable experience in solving these problems [1, 2, 3, 4, 5] for both normal metals and superconductors. Nevertheless, there are still a number of unresolved questions related to the effects of nonlinear contributions to the current-voltage characteristics of various solid-state structures. One of the problems is related to the fact that in the calculation of a mean current, it is necessary to take into account the time-dependent evolution of a wave function and a dependence of current operator on the real time $t$. Unfortunately, the developed method of Matsubara [6] for evaluating a mean value of the operators at T$\neq 0$ implies a transition to the imaginary time. Indeed, in this case the density matrix is used which satisfies the Bloch equation. As a result, a formula for calculating the S-matrix contains integrals over the imaginary time $\tau$ = it. This discrepancy raises a problem of the passage to the limiting case of zero temperatures. Therefore, we will study this question in more detail, which will allow us to generalize the Kubo theory for transport in the nonlinear limit using the S-matrix. In [7], in studying the statistical mechanics of irreversible processes R. Kubo presented a general expression for evolution of the density matrix in the quantum-mechanical case, which was expressed in terms of commutators of the perturbation operators in the interaction representation with unperturbed density matrix. However, the obtained formula is technically very difficult to use because of the presence of complicated commutators. Therefore, in the majority of cases [8, 9] only a linear response is considered that does not give a complete picture of the influence of nonlinear contributions to the transport.

Taking into account the above, the structure of the work is as follows. Section 2 develops the Kubo theory for the chronological products of perturbation operators. This allows us to do away with complex commutators of the time-dependent operators and also to obtain a simplified Kubo formula for the mean current. Section 3 discusses the specific application of the resulting formula for the mean current in a hybrid metal structure with a tunnel barrier in the linear approximation. The results obtained are consistent with the already known data. In Section 4, the theory is further generalized to the contribution to the mean current in any order of time perturbation theory. A quantum-statistical correction is obtained to the already known Landauer formula for conductance, It is also shown that the tunneling Hamiltonian correctly describes the transport properties of hybrid structures.

2 Scattering matrix formalism in the nonlinear Kubo theory for electron transport

In this section, we consider the effect of the barrier and the applied voltage on the electron transport in a metal at the nonzero temperature T. In the framework of perturbation theory, the evolution of the wave function in the interaction representation is determined by the perturbation only. It simplifies the calculations considerably. Taking into account the above, we start with the Liouville equation for the evolution of the density matrix in real time ( $\hbar=1$):

[TABLE]

where $\rho(t)=e^{-i\hat{H}_{0}t}\rho_{I}(t)e^{i\hat{H}_{0}t}$, $\rho_{I}(t)$ and $H_{T}(t)$ are the density matrix $\rho(t)$ and perturbation Hamiltonian $H_{T}$ in the interaction representation. Also, it is easy to show that

[TABLE]

where the operator $\tilde{U}(t)=e^{i\hat{H}_{0}t}e^{-i\hat{H}t}$. The operator $\rho_{0}=\frac{{e^{-\beta\hat{H}_{0}}}}{{Tr(e^{-\beta\hat{H}_{0}})}}$ is a standard expression for unperturbed density matrix at ${\textit{t}=-}\infty$ and ${\rm 1/}\beta{\textit{= T}}$ is the temperature. Here, it is supposed an infinite slow switching on a perturbation $H_{T}$ to the unperturbed part $\hat{H}_{0}$ up to full Hamiltonian $\hat{H}$. The Schr${\rm\ddot{o}}$dinger evolution operator $U(t)=e^{-i\hat{H}_{0}t}\tilde{U}(t)$ of the wave function, where $\tilde{U}(t){\rm}$ is expressed in terms of scattering matrix ${\textit{S(t,}-\infty)}$ by the next manner

[TABLE]

Here, T${}_{\textit{t}}$ is the time-ordered operator. Apparently, that the Hermitian conjugate operator $U^{+}(t)=\tilde{U}^{+}(t)e^{i\hat{H}_{0}t}$. Then the mean current is determined by expression

[TABLE]

where I is the current operator and the symbol $Tr=\sum\limits_{n}{\left\langle n\right|}...\left|n\right\rangle$ denotes trace with summation over complete set of states. By using both the cyclic property of the trace and the definition of the current operator in the interaction representation, we write the expression for a mean current in the form:

[TABLE]

It is easy to present the expression  (2) for $\rho_{I}(t)$ as a series using the Baker-Campbell-Hausdorff formula [10]:

[TABLE]

where the symbol $\left[{A,B}\right]_{(j)}$ denotes the k-order commutator determined in accordance with recurrence relation

[TABLE]

At that $\left[{A,B}\right]_{(0)}=B$ and $\left[{A,B}\right]_{(1)}=\left[{A,B}\right]=AB-BA$ is the ordinary commutator. Assuming that the perturbation ${H_{T}}$ is a Hermitian operator, we write the final expression for $\rho_{I}(t)$ in any order of perturbation theory:

[TABLE]

In the first order in from Eq. (2) it is easy to obtain the linear response in the framework of the Kubo theory for charge transport. A generalization to higher orders of perturbation theory causes certain difficulties due to the presence of a large number of commutators under the signs of integrals. It considerably complicates their calculation. Therefore, for convenience, we transform the integrand  (12). It is obvious that for any tj the identity holds

[TABLE]

where

[TABLE]

Let consider the commutator in the second order of the expansion  (12):

[TABLE]

The last term in Eq. (17) under the signs of the integrals and Tt equals identically zero, since the operator ${H_{T}}(t_{2})$ is a Bose type. Thus, the sign of the commutator term with minus sign after time ordering is not changed. Obviously, the commutators of higher orders are simplified in a similar way. Then Eq. (12) can be written in the form

[TABLE]

Substituting Eq. (18) into Eq. (7) we obtain the formula for mean current

[TABLE]

This expression is essentially simplified by using the linked-diagram theorem [11], which excludes from consideration the diagrams containing unlinked blocks. This theorem is typical in quantum field methods of statistical physics. Let us introduce the notation

[TABLE]

for the contributions from linked diagrams. Then Eq. (19) takes the most simple form

[TABLE]

Expression  (20) is the main when we consider an influence of the nonlinear contributions of perturbation theory on transport in the electron systems.

3 Tunnel barrier in the hybrid structure : normal metal -barrier- normal metal

Let us consider a simple task on electron transport in the system to be consisted from two layers of normal metals which are separated by oxide film creating the tunnel barrier. In Fig.1 the energy structure of a given model is figured and one can be presented in the form of Hamiltonian

[TABLE]

where $\hat{H}_{0}=\hat{H}_{L}+\hat{H}_{R}$. The unperturbed parts $\hat{H}_{L}=\sum\limits_{\bm k}{(\varepsilon_{\bm k}-\mu_{L})n_{\bm k}}=\sum\limits_{\bm p}{\xi_{\bm k}n_{\bm k}}$ and $\hat{H}_{R}=\sum\limits_{\bm p}{(\varepsilon_{\bm p}-\mu_{R})n_{\bm p}}=\sum\limits_{\bm p}{\xi_{\bm p}n_{\bm p}}$. Here, ${\varepsilon_{\bm k}}$ and ${\varepsilon_{\bm p}}$ are electron energies. The number operators $n_{\bm k}=a_{\bm k}^{+}a_{\bm k}$ and $n_{\bm p}=a_{\bm p}^{+}a_{\bm p}$ for electron states ${\bm k}$ and ${\bm p}$ and also the chemical potentials ${\mu_{L}}$ and ${\mu_{R}}$ are designated for left- and right-hand sides of hybrid structure, respectively.

The applied electrical voltage V shifts the chemical potentials relatively one another in a such manner that the relation $\mu_{L}-\mu_{R}=eV$ is fulfilled, where e is a modulus of the electron charge. The perturbation Hamiltonian ${H_{T}}$ describing the electron tunneling from left- to right-hand side of hybrid structure has a form

[TABLE]

To evaluate a mean electrical current it is necessary to find the expression for operators ${A(t_{j})}$ entering in Eq. (20). From the whole set of states with wave vectors ${\bm k}$ we will fix one ${\bm k}$. Then one can consider subspace of two wave functions $\left|{1_{\bm k}}\right\rangle$ and $\left|{0_{\bm k}}\right\rangle$ with and without electron in ${\bm k}$ -state, respectively. It is obvious that space of all states of the wave vectors is the direct product of these subspaces for each vector ${\bm k}$. For example, the operator exponent for the left-hand side of the metal, which determines the unperturbed density matrix, can be represented as a direct product:

[TABLE]

where E_{\bm k}=\left({\begin{array}[]{*{20}c}1&0\\ 0&1\\ \end{array}}\right) is the unit matrix and n_{\bm k}=\left({\begin{array}[]{*{20}c}1&0\\ 0&0\\ \end{array}}\right). Obviously, that

[TABLE]

One can simplify the expression (10) using Eq. (8), i.e. the task is reduced to calculation of the commutators of ${\hat{H}_{0}}$ and ${H_{T}}$. Indeed, it is not difficult to find a commutator in the first order of expansion  (8), which is expressed as follows:

[TABLE]

It can be shown that the commutator of the next order

[TABLE]

The structure of the commutators of higher order is evident. Therefore, we immediately write the operator ${A(t_{j})}$ from Eq. (14):

[TABLE]

This operator can not be substituted in Eq. (20) for calculation $<I>$ , because in the correlators it is necessary to carry out convolutions with respect to the indices ${\bm k}$ and ${\bm p}$. On the other hand, these same indices are also included in ${\rho_{0}}$ of expression  (20). Therefore, we select the factors with these indices from ${\rho_{0}}$ and substitute ones in Eq. (27) that leads to a replacement ${A(t_{j})}$ in Eq. (20) by ${\tilde{A}(t_{j})}$ . This is easy to do if we take into account that the factors in $\rho_{0}$ with the indices ${\bm k}$ and ${\bm p}$ have the form

[TABLE]

[TABLE]

As a result of this substitution we obtain the expression for ${\tilde{A}(t_{j})}$ :

[TABLE]

which one is substituted in Eq. (20) instead of ${A(t_{j})}$ . Now in the density matrix ${\rho_{0}}$ the factors with indices ${\bm k}$ and ${\bm p}$ are absent, i.e. ${\rho_{0}}$ is replaced by ${\tilde{\rho}_{0}}$ .

It should be noted that the theory presented in Section 3 described the evolution of the density matrix for a system with a fixed number of particles. In reality, there are two subsystems with chemical potentials $\mu_{L}$ and $\mu_{R}$. Due to the current, the number of particles at time t in the left- and right-hand sides is not fixed. Therefore, evolution in a system with current must be described by the complete Hamiltonian [8]:

[TABLE]

where

[TABLE]

${N_{L}=\sum\limits_{\bm k}{n_{\bm k}}}$ and ${N_{R}=\sum\limits_{\bm p}{n_{\bm p}}}$ is the total number of electrons in the left- and right-hand side of the given hybrid structure, respectively. Taking into account the above, the creation and annihilation operators in the expressions for ${H_{T}(t_{j})}$ and ${\tilde{A}(t_{j})}$ take the form

[TABLE]

and likewise for the right-hand side of the metal. Thus, we have

[TABLE]

[TABLE]

where

[TABLE]

Since ${\frac{{d(N_{L}+N_{R})}}{{dt}}=0}$ and ${\frac{{dN_{L}(t)}}{{dt}}=i\left[{H,N_{L}}\right]=-i\left[{N_{L},H_{T}}\right]}$, then the current operator is determined as

[TABLE]

and

[TABLE]

The initial density matrix ${\rho_{0}}$ is determined by the unperturbed Hamiltonian $\hat{H}_{0}$ since in each subsystems in absence current the particle numbers are fixed.

4 Linear response in the metal with a tunnel barrier

Here, we will consider the use of Eq. (20) to find electrical current in a linear approximation. The expression  (20) takes the form

[TABLE]

Substituting in Eq. (39) the expressions  (34) and  (38) we obtain the linear current

[TABLE]

Factor 2 arises from summing up over electron spins. Keeping in Eq. (43) nonzero correlators only and taking into account for definitions  (37) we have

[TABLE]

The correlators obtained in Eq. (47) are decoupled by Wick’s theorem which allows us to write down

[TABLE]

In Eq. (51) the operator ${\tilde{\rho}_{0}}$ disappears since it has not multipliers with indices ${\bm k}$ and ${\bm p}$. Also, in Eq. (51) there are unperturbed Green’s functions which one can define as follows:

[TABLE]

since ${Tr(E_{\bm k}-n_{\bm k})=Tr(n_{\bm k})=1}$. The Fourier transform for ${G_{\bm k}(t-t_{1})}$ has a form:

[TABLE]

where the infinitesimal imaginary corrections are due to adiabatic switching on and off of the perturbation. The Fourier transform of the Green’s functions in Eq. (51) gives

[TABLE]

As a result of integration over variable $t_{1}$ we obtain

[TABLE]

After substituting the expressions for Fourier transforms of the Green functions in Eq. (60), it is easy to integrate with respect to the variables $\omega_{1}$ and $\omega_{2}$. The contours of integration are closed in the upper or lower half-planes of complex variables, depending on the presence of the minimal number of poles. Thus, we have

[TABLE]

In order to perform the summation in Eq. (63) over the wave vectors let us assume $\left|{T_{kp}}\right|^{2}=\left|T\right|^{2}$ and put for simplicity the constant electron densities of the states $D_{L}$ and $D_{R}$ and denote the integration variables $\xi_{L}$ and $\xi_{R}$ for the left-hand and right-hand sides of the metal, respectively. In accordance with the Landau rule for bypass the poles one can write expression (34) in the form:

[TABLE]

where $\delta(x)$ is the Dirac delta function. Let us consider the case of temperatures        $T<<E_{F}$ where $E_{F}$ is the Fermi energy. Then the Fermi distribution function can be replaced by the Heaviside step function $\theta(-x)$. As a result, we have

[TABLE]

This expression reflects the Ohm’s law and coincides with a similar formula presented in [8] within the framework of the linear Kubo theory for electron transport.

5 Nonlinear contribution in the electron transport

Using Eq. (20) it is not difficult to generalize an expression for the current in any order of perturbation theory. Obviously, the next nonzero contribution appears only in the third order since the second order of series expansion  (20) includes an odd number of creation and annihilation operators for each metal subsystems. One can write the contribution of the third order from expression  (20):

[TABLE]

Substituting in Eq. (71) the expression for ${\tilde{A}(t_{j})}$ from Eq. (27) we have

[TABLE]

An additional factor of 4 arises from summing up over spin indices. Carrying out the decoupling of the correlators entering in Eq. (76), we obtain

[TABLE]

where $\varphi_{\bm{kpk_{1}p_{1}}}=\left|T\right|^{4}\left[{f(\xi_{\bm p})-f(\xi_{{\bm k}_{1}})}\right]\left[{f(\xi_{{\bm p}_{1}})-f(\xi_{{\bm k}_{1}})}\right]\left[{f(\xi_{{\bm p}_{1}})-f(\xi_{\bm k})}\right]$. As in     Section 3, the Fourier transform of the Green functions and the following integration over $t_{1}$, $t_{2}$ and $t_{3}$ allows to write

[TABLE]

where

[TABLE]

From Eq. (83) it follows that $<I>_{3}$ as $<I>_{1}$ are determined by difference of the complex conjugate values, i.e. $<I>_{3}$ is real and equal to double imagine part of one of the integrals. Obviously, this is due to the structure of the current operator, which describes the electron hopping through the barrier in one as well as in the opposite directions independently on the order of the perturbation theory. The integrals in Eq. (83) are calculated using the theory of residues. Denoting the fourfold integral in Eq. (83) through $J_{\bm{k,p,k_{1},p_{1}}}(\delta)$ , we can write its expression in the form:

[TABLE]

where

[TABLE]

The factor 2 in the last term of Eq. (89) appears due to the fact that in Eq. (84) there should be 5 terms, but 2 of them are the same. It is seen from Eq. (89) that the poles are located both in the upper and lower complex planes. For calculation of the current in a limit $\delta\rightarrow 0$ the numerical factors in front of $\delta$ does not play any role. Also, in Eq. (84) only the differences of complex conjugate quantities enter. Thus, in the expression for $<I>_{3}$ the integrals in the sense of a principal value are absent. By analogy with Section 3, taking into account the Landau rule for bypassing poles we have

[TABLE]

Assuming that the electronic densities of states are constant, we replace the sums over wave vectors by integrals with the same integration variables as in Section 3. It is easy to find that

[TABLE]

Apparently, in the fifth order of the perturbation theory we have the contribution in a mean electrical current

[TABLE]

and so on. Summing up the contribution for all orders we obtain the geometric series at condition $2\pi^{2}\left|T\right|^{2}D_{L}D_{R}\left[{f(\xi_{L})-f(\xi_{R})}\right]^{2}<1$. Thus, the common factor of all the contributions to the current is twice the geometric ratio. Then the general expression for the mean current $<I>$ at finite temperatures takes the form:

[TABLE]

In particular, at zero temperature a mean electrical current is presented in form of the renormalized Landauer [12] formula with $\hbar\neq 1$ :

[TABLE]

from which it follows that barrier transmission must be determined as $\tilde{T}=2\pi^{2}\left|T\right|^{2}D_{L}D_{R}$. It is connected with the probability of the barrier tunneling and depends on electron densities of the left- and right-hand sides of metal.

From Eq. (94) it follows that for barrier transmission $\tilde{T}=1$ the current becomes infinity since the scattering centers are absent. In a single electron approximation the presence of the finite Landauer resistance $R_{k}=\frac{h}{{2e^{2}}}$ at full barrier transparency gives rise to contradiction that are explained by many- channel processes in the leads [5, 6]. However, from general Eq. (94) it follows that with the nonlinear contributions the pointed contradiction is removed. The barrier transmission is determined by repeated electron penetrations and reflections. The electron statistic does not allow to separate the penetration and reflection electron processes. Thus, one can say about effective barrier transmission in hybrid structures. Only at $\tilde{T}=1/3$ we obtain the ballistic transport with Landauer conductivity $G=\frac{{2e^{2}}}{h}$. Also, the account for electron quantum statistics leads to the fact that the effective coefficient of electron reflection ${\tilde{R}}$ from the barrier can not be represented in a form $\tilde{R}=1-\tilde{T}$ as for one-electron approximation [13].

It can be seen from Eq. (94) that the value $\left|T\right|$ of the tunnel matrix element that determines the probability of penetration of an electron through a barrier can not exceed a value $\frac{1}{{2\pi\sqrt{D_{L}D_{R}}}}$ . For this maximum value the transport of electrons is equivalent to transport in systems without a barrier and scattering. The form of Eq. (93), apparently, removes the contradiction of use the tunnel Hamiltonian [8], where it was indicated that the exponential growth $\left|T\right|^{2}$ as a function $\xi_{\bm k}$ can exceed the power-law drop of the spectral density. Then the main contribution to transport is caused by the top of the barrier. That is an unphysical result. As seen from Eq. (93), $\left|T\right|^{2}$ enters both in the numerator and the denominator. It significantly complicates the relationship between the spectral density and height of the barrier.

6 Summary

The paper presents a modified nonlinear Kubo theory for electron transport in hybrid structures of normal metals with a tunnel barrier. The results of the work can be generalized to the case of superconducting layers and Josephson junctions. The general expression for a mean current is obtained with account for all contributions of the time-dependent perturbation theory. It is found that the nonlinearity of the theory causes a substantial renormalization of the Landauer formula for conductivity in the ground state. It turns out to be the Landauer formula works in the framework of the one-electron approximation only. The ballistic transport conductivity $G=\frac{{2e^{2}}}{h}$ is realized for barrier transmission $\tilde{T}=1/3$. The validity of the tunneling Hamiltonian for transport phenomena in hybrid structures is also justified.

Acknowledgements

It is a pleasure to acknowledge a number of stimulating discussions with E.M. Rudenko, M.A.Belogolovskii and A. P. Shapovalov.

Funding

The study was carried out within the Fundamental Research Programme funded by the MES of Ukraine (Project No. 0117U002360).