Non-Equilibrium Green Functions of Electrons in Single-Level Quantum Dots at Finite Temperature

In the quantum field theory with the vacuum being the ground state the Green functions are the vacuum expectation values of the chronological, retarded or advanced products of the field operators (Bjoken & Drell, 1964; Itzykson & Zuber, 1985; Peskin & Schroeder, 1995). They are the generalized functions of the real time variables ti (and also other spatial coordinates). For the application of the Green function technique to the study of the timeindependent phenomena in equilibrium many-body systems at a finite temperature, the Matsubara imaginary time Green functions were introduced and widely used (Abrikosov et al., 1975; Bruuns & Flensberg, 2004; Haken, 1976). They are the mean values over a statistical ensemble at a finite temperature of the chronological products of the imaginary timedependent operators. Both these types of Green functions are inadequate for the application to the study of the time-dependent phenomena in the many-body systems with a finite density and at a finite temperature, in particular the non-equilibrium systems. For the application to the study of the time-dependent dynamical processes in non-equilibrium many-body systems Keldysh (Keldysh, 1965) has introduced a more general class of timedependent Green functions at finite temperature and density. They are the mean values of the time-ordered products of quantum operators in the Heisenberg picture over statistical ensembles of many-body systems with finite densities and at finite temperatures (which may be non-vanishing). The simplest example is the two-point Green function


Introduction
In the quantum field theory with the vacuum being the ground state the Green functions are the vacuum expectation values of the chronological, retarded or advanced products of the field operators (Bjoken & Drell, 1964;Itzykson & Zuber, 1985;Peskin & Schroeder, 1995). They are the generalized functions of the real time variables t i (and also other spatial coordinates). For the application of the Green function technique to the study of the timeindependent phenomena in equilibrium many-body systems at a finite temperature, the Matsubara imaginary time Green functions were introduced and widely used (Abrikosov et al., 1975;Bruuns & Flensberg, 2004;Haken, 1976). They are the mean values over a statistical ensemble at a finite temperature of the chronological products of the imaginary timedependent operators. Both these types of Green functions are inadequate for the application to the study of the time-dependent phenomena in the many-body systems with a finite density and at a finite temperature, in particular the non-equilibrium systems. For the application to the study of the time-dependent dynamical processes in non-equilibrium many-body systems Keldysh (Keldysh, 1965) has introduced a more general class of timedependent Green functions at finite temperature and density. They are the mean values of the time-ordered products of quantum operators in the Heisenberg picture over statistical ensembles of many-body systems with finite densities and at finite temperatures (which may be non-vanishing). The simplest example is the two-point Green function as complex times. For the definition of the ordering of the complex variables z, z' it was proposed to use some contour C in above-mentioned stripe with some initial point t 0 on the real axis and the final point 0 ti   such that all the complex numbers z, z'… belong to this contour. Then the "chronological" ordering T C of the complex times z, z' … is defined as the ordering along the contour C. The complex time-dependent operators a(z), b(z) and () , () az bz , for example, are defined in the analogy with the operators in the Heisenberg picture T C denoting the "chronological" ordering along the contour C. The Green functions of the form (1.3), usually called the Keldysh complex time-dependent Green functions at finite density and temperature, some time also simply called non-equilibrium Green functions, are widely used in quantum statistical physics and many-body theories (Chou et al., 1985;Kapusta, 1989;Le Bellac, 1996).
In practice we need to know the Green functions at the real values of the time variables. For the convenience we chose the contour C to consists of four parts 123 , CC C C C    C 1 being the part of the straight line over and infinitely close to the real axis from some point 0 ti o  to infinity io  , C 2 being the part of the straight line under and infinitely close to the real axis from infinity io  to the point 0 ti o  , C 3 and C  being the segments 00 [, ] tt i  and [, ] io io      parallel to the axis Oy (figure 1).
The contributions of the segment [, ] io io   to all physical observables are negligibly small, because of its vanishing length. Therefore this segment plays no role, and the contour C can be considered to consist of only three parts C 1 , C 2 and C 3 . Then the function () ab C Gzz   with the complex time variables z and z' on the contour C effectively becomes a set of nine functions of two variables, each of which has the values on one among three segments C 1 , C 2 and C 3 . When both variables z and z' belong to the line C 1 , the function (1.3) is the quantum statistical average of the usual chronological product of two quantum operators a(t) and b(t') in the Heisenberg picture over a statistical ensemble of a many-body system at finite density and temperature, and can be denoted by In the study of stationary physical processes one often uses the complex time Green functions of the form (1.3) in the limit 0 t . Because the interaction must satisfy the "adiabatic hypothesis" and therefore vanishes at this limit, the segment C 3 also gives no contribution to the stationary physical processes. In this case the contour C can be considered to consist of only two segments C 1 and C 2 , and the complex time The electrons transport through a single-level quantum dot (QD) connected with two conducting leads has been the subject for theoretical and experimental studies in many works since the early days of nanophysics (Choi et al., 2004;Costi et al., 1994;Craco & Kang, 1999;Fujii & Ueda, 2003;Hershfield et al., 1991;Inoshita et al., 1993;Izumida et al., 1997Izumida et al., , 1998Izumida et al., , 2001Konig & Gefen, 2005;Meir et al., 1991Meir et al., , 1993Ng, 1993;Nguyen Van Hieu & Nguyen Bich Ha, 2005Nguyen Van Hieu et al., 2006a, 2006bPustilnik & Glasman, 2004;Sakai et al., 1999;Swirkowicz et al., 2003Swirkowicz et al., , 2006Takagi & Saso, 1999a, 1999bTorio et al., 2002;Wingreen & Meir, 1994;Yeyati et al., 1993). Two observable physical quantities, which can be measured in experiments on electron transport, are the electron current through the QD and the time-averaged value of the electron number in the QD. Both can be expressed in terms of the single-electron Green functions. In the pioneering theoretical works (Meir et al., 1991(Meir et al., , 1993 on the electron transport through a single-level QD, the differential equations for the non-equilibrium Green functions were derived with the use of the Heisenberg equations of motion for the electron destruction and creation operators. Due to the presence of the strong Coulomb interaction between electrons in the QD, the differential equations for the single-electron Green functions contain multi-electron Green functions, and all the coupled equations for these Green functions form an infinite system of differential equations. In order to have a finite closed system of equations, one can assume some approximation to decouple the infinite system of equations. Moreover, since the electron transport is a non-equilibrium process, one should work with the Keldysh formalism of non-equilibrium complex time Green functions.
As the simplest explanation of the calculation methods for establishing the differential equations of non-equilibrium Green functions and deriving their exact solutions, in Section 2 we present the theory of non-equilibrium Green functions of free electron in a single-level quantum system. In Section 3 we study non-equilibrium Green functions of interacting electron in an isolated single-level QD. The elaborated calculation methods are then applied in Section 4 to the study of non-equilibrium Green functions of electrons in a single-level QD t 0 +io www.intechopen.com connected with two conducting leads. Due to the electron tunneling between QD and conducting leads there does not exist a closed finite system of differential equations for some finite number of Green functions. In order to truncate the infinite system of differential equations for the infinite number of Green functions we can apply some suitable approximation. In Section 5 the mean-field approximation was used to truncate the infinite system of differential equations for the Green functions. As the result we establish a closed system of Dyson equations for a finite number of Green functions. This system of differential equations can be exactly solved. The asymptotic analytical expressions of these Green functions at the resonances, Kondo and Fano resonances, are derived in Section 6. Section 7 is the Conclusion.

Non-equilibrium Green function of free electrons in a single-level quantum system
For the demonstration of the calculation methods to derive the differential equations and the expressions of the non-equilibrium Green functions let us consider a simplest quantum system -that of free electrons at a single energy level E. Denote by c  and c   the destruction and creation operators of the electron with the spin projection ,   in the Schrödinger picture and by H 0 the Hamiltonian of this system. We have 0 HEc c The non-equilibrium Green function of electron system with Hamiltonian (2.1) is defined as follows: First consider three cases when both variables z and z' belong to one and the same segment T C is the usual chronological ordering T of the real times t and t':

Tc tc t t t c tc t t tc t c t
we rewrite equations (2.8.1), (2.8.2), (2.17) and (2.18) in the unified form (2.20) From above presented reasonnings and relations determining nine functions ()

E ij
Sz z   , and formula (2.1) for total Hamiltonian, it is straightforward to derive explicit expressions of these functions. They depend on the average electron number with a definite spin projection We obtain following results: For concluding this Section we consider the Fourier transformation of the functions () The explicit expressions of Green functions of free electrons presented in this Section are often used in the theoretical studies of non-equilibrium processes by means of the perturbation theory.

Non-equilibrium Green functions of electrons in isolated single-level quantum dot
The calculation methods and reasonnings presented in the preceding Section are now applied to the study of the Keldysh non-equilibrium Green functions of interacting electrons in the simplest nanosystem -the isolated single-level quantum dot (QD) with total Hamiltonian , where U is the value of a potential energy, ,    denotes the spin projection (if  then   and vice versa) and is the number of electrons with the spin projection σ. The second term in Hamiltonian (3.1) is the potential energy of the Coulomb electron-electron interaction (two electrons with different spin projections in one and the same energy level). The interacting nanosystem with total Hamiltonian (3.1) is an exactly solvable model. There are four exactly determined eigenstates and eigenvalues of H: the vacuum with vanishing energy, two degenerate single-electron states with two different spin projections and the same energy E, and a twoelectron state with total energy 2E+U. The Keldysh complex time-dependent two-point Green function of two operators () cz and () cz  is defined as follows with total Hamiltonian (3.1). They have the form As in the preceding Section, we choose the contour C to consist of three segments C 1 , C 2 and C 3 .
The calculations of these functions are straightforward, as they have been done in the preceding Section for free electrons at a single energy level. We obtain following results: In the study of non-equlibrium dynamical processes by means of the perturbation theory one often needs to use the Fourier transformation of four functions ( We have following exact expressions of their Fourier transforms: Now we derive the system of differential equations for two-point Green functions . Consider first the function with i = j = 1: From the Heisenberg quantum equation of motion Substituting this expression of () dc t i dt  into the r.h.s. of equation (3.9), we obtain . In order to derive the differential equation for this new Green function it is necessary to calculate the time derivatives of both sides of equation (3.13). Note that Nσ commutes with H and therefore does not depend on t. Moreover, it has following property 2 .

NN    
Multiplying both sides of relation (3.11) with N   and using these two above-mentioned properties of N  , we obtain (3.14) www.intechopen.com Differentiating both sides of equation (3.13) and using relation (3.14), we derive following differential equation In preceding Section we have shown that Equations (3.18)  and () C Gz z   satisfies differential equation

Non-equilibrium Green functions of electrons in single-level quantum dot connected with two conducting leads
Consider the single-electron transistor (SET) consisting of a single-level quantum dot (QD) connected with two conducting leads through two potential barriers. The electron transport through this SET was investigated experimentally and studied theoretically in many works (Choi et al., 2004;Costi et al., 1994;Craco & Kang, 1999;Fujii & Ueda, 2003;Hershfield et al., 1991;Inoshita et al., 1993;Izumida et al., 1997Izumida et al., , 1998Izumida et al., , 2001Meir et al., 1991Meir et al., , 1993Ng, 1993;Pustilnik & Glasman, 2004;Sakai et al., 1999;Swirkowicz et al., 2003Swirkowicz et al., , 2006Takagi & Saso, 1999a, 1999bTorio et al., 2002;Wingreen & Meir, 1994;Yeyati et al., 1993). It was assumed that the electron system in this SET has following total Hamiltonian In order to define the complex time-dependent Green functions we introduce the complex time-dependent quantum operators Because there is no magnetic interaction, all Green functions (4.3)-(4.10) and other ones are proportional to    . From Heisenberg quantum equations of motion and equal-time canonical anti-commutation relations for the electron destruction and creation operators it follows the differential equations for these operators: and similarly for (;) bz  k and (;) bz  k .
By using differential equation (4.11) and the equal-time canonical anti-commutation relation between () cz  and () cz   , it is easy to derive the differential equation for the Green function () . These new functions must satisfy following differential equations which can be also derived by using differential equations (4.11)-(4.14): .
The presented calculations for deriving differential equations of Green function showed that there does not exist a closed system of a finite number of differential equations for a finite number of Green functions. Some approximation should be used for truncating the infinite system of all differential equations at some step. The mean-field approximation is the most appropriate one. In order to apply this approximation we rewrite equations (4.23)-(4.25) in the form of integral equations:

Dyson equations for non-equilibrium Green functions of electrons in single-level quantum dot connected with two conducting leads and their solutions
The r.h.s. of equation ( . Applying the mean-field approximation to each of others above-mentioned multi-electron Green functions in any manner, we always obtain the vanishing mean value in the lowest order the perturbation theory with respect to the effective tunnelling coupling constants   , ab V k . Note that these functions enter the r. h. s. of the equation (29) with the coefficients of the second order with respect to the effective tunnelling coupling constants. This means that in this second order they do not give contributions. Thus in the second order approximation the equation ( (5.17) The system of Dyson equations (4.21) and (5.5) is the mathematical tool for the study of the electron transport through a single-level QD. Since this is a stationary process one can apply the Keldysh non-equilibrium Green function formalism in the limit 0 t . Because the interaction vanishes at this limit, the contour C can be considered to consist of only two segment 1 [,] Ci (2) (2) (2) 11 22  at the resonances. This will be done in the subsequent Section.

Kondo and Fano resonances in electron transport through single-level quantum dot
In this Section we study the appearance of the resonances in the expressions of the  (1) (0) (0) 11 (1)  When Y() vanishes due to the cancellation between the finite terms and those containing divergent integrals, there appear the resonances. Therefore in order to study the resonances it is necessary to investigate the divergence of the integrals in the r.h.s. of the formulae (6.5)-(6.7).

Conclusion
The present Chapter is an introductory review of the Keldysh non-equilibrium Green functions of electrons in simplest nanosystems: isolated single-level QD and single-level QD connected with two conducting leads. In the case of an isolated single-level QD the closed system of a finite number of differential equations for a finite number of Green functions was established by using the Heisenberg quantum equations of motion for the electron destruction and creation operators. The exact expressions of the Green functions were derived. In the case of the nanosystem consisting of a single-level QD connected with two conducting leads there does not exist a finite closed system of differential equations for some finite number of Green functions. In the differential equations for n-point Green functions there appear the contributions from (n+2)-point Green functions. Therefore, the exact system of differential equations contains an infinite number of equations for an infinite number of Green functions. In order to truncate this infinite system of differential equations we have applied the mean-field approximation to the products of four electron quantum operators and limited at the terms of the second order with respect to the effective tunnelling coupling constants. As the result we have derived a closed system of Dyson equations for two types of 2-point Green functions. All the crossing terms are included into the equations. The exact solution of the system of Dyson equations may have the resonances of four types in the dependence on the physical parameters of the system: the Kondo resonances at the Fermi surface, whose origin is similar to that of the Kondo effect in the scattering of electrons on magnetic impurities, the Fano resonances due to the presence of the electron quasi-bound state at the lower edge of the energy band of the conducting electrons, the Kondo resonances in the crossing terms and the Fano resonances in the crossing terms. The analytical asymptotic expressions of the single-electron Green function at these resonances were derived. These results agree well with the numerical calculations in references on the electron Green functions in QD (Yeyati et al., 1993;Costi et al., 1994;Izumida et al., 1997Izumida et al., , 1998Izumida et al., , 2001Sakai et al., 1999;Torio et al., 2002).
The theoretical study of the non-equilibrium Green functions of electrons in QDs would signify the beginning of the development of the quantum dynamics of physical processes in QD-based nanodevices. The next step would be the elaboration of the theory of non-equilibrium Green functions of phonons in QDs as well as of electrons and phonons of interacting electron-phonon systems in QDs. The quantum dynamical theory of QD-based optoelectronic and photonic nanodevices necessitates also the study of non-equilibrium Green functions of electrons and phonons confined in QDs in the presence of the electronphonon interactions as well as the interaction of photons with confined electron-phonon systems. The methods and reasonnings presented in this Chapter could be generalized for the application to the study of all above-mentioned non-equilibrium Green functions.

Acknowledgment
I would like to express the gratitude to Institute of Materials Science and Vietnam Academy of Science and Technology for the support to my work on the subject of this review during many years. I thank also Academician Nguyen Van Hieu for suggesting the main ideas of the series of publications on this subject. The book "Fingerprints in the optical and transport properties of quantum dots" provides novel and efficient methods for the calculation and investigating of the optical and transport properties of quantum dot systems. This book is divided into two sections. In section 1 includes ten chapters where novel optical properties are discussed. In section 2 involve eight chapters that investigate and model the most important effects of transport and electronics properties of quantum dot systems This is a collaborative book sharing and providing fundamental research such as the one conducted in Physics, Chemistry, Material Science, with a base text that could serve as a reference in research by presenting up-to-date research work on the field of quantum dot systems.