The time-dependet quantum transport in nanosystems
We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green’...
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2010
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| Назва видання: | Моделювання та інформаційні технології |
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| Цитувати: | The time-dependet quantum transport in nanosystems / A. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є.Пухова НАН України, 2010. — Вип. 58. — С. 64-70. — Бібліогр.: 7 назв. — aнгл. |
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irk-123456789-219562011-06-21T12:06:06Z The time-dependet quantum transport in nanosystems Korostil, A. We propose a time-dependent many-body approach to study the short-time dynamics of correlated electrons in quantum transport through nanoscale systems contacted to metallic leads. This approach is based on the time propagation of the Kadanoff-Baym equations for the nonequilibrium many-body Green’s function of open and interacting systems out of equilibrium. 2010 Article The time-dependet quantum transport in nanosystems / A. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є.Пухова НАН України, 2010. — Вип. 58. — С. 64-70. — Бібліогр.: 7 назв. — aнгл. XXXX-0068 http://dspace.nbuv.gov.ua/handle/123456789/21956 579 en Моделювання та інформаційні технології Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
We propose a time-dependent many-body approach to study the short-time
dynamics of correlated electrons in quantum transport through nanoscale systems
contacted to metallic leads. This approach is based on the time propagation of the
Kadanoff-Baym equations for the nonequilibrium many-body Green’s function of
open and interacting systems out of equilibrium. |
| format |
Article |
| author |
Korostil, A. |
| spellingShingle |
Korostil, A. The time-dependet quantum transport in nanosystems Моделювання та інформаційні технології |
| author_facet |
Korostil, A. |
| author_sort |
Korostil, A. |
| title |
The time-dependet quantum transport in nanosystems |
| title_short |
The time-dependet quantum transport in nanosystems |
| title_full |
The time-dependet quantum transport in nanosystems |
| title_fullStr |
The time-dependet quantum transport in nanosystems |
| title_full_unstemmed |
The time-dependet quantum transport in nanosystems |
| title_sort |
time-dependet quantum transport in nanosystems |
| publisher |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/21956 |
| citation_txt |
The time-dependet quantum transport in nanosystems / A. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є.Пухова НАН України, 2010. — Вип. 58. — С. 64-70. — Бібліогр.: 7 назв. — aнгл. |
| series |
Моделювання та інформаційні технології |
| work_keys_str_mv |
AT korostila thetimedependetquantumtransportinnanosystems AT korostila timedependetquantumtransportinnanosystems |
| first_indexed |
2025-07-02T21:59:17Z |
| last_indexed |
2025-07-02T21:59:17Z |
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1836574109158342656 |
| fulltext |
64 © �. Korostil
��� 579
�. Korostil, Kyiv
THE TIME-DEPENDET QUANTUM TRANSPORT IN NANOSYSTEMS
We propose a time-dependent many-body approach to study the short-time
dynamics of correlated electrons in quantum transport through nanoscale systems
contacted to metallic leads. This approach is based on the time propagation of the
Kadanoff-Baym equations for the nonequilibrium many-body Green’s function of
open and interacting systems out of equilibrium.
1. Introduction
The description of electron transport through nanoscale systems contacted to
metallic leads is currently under intensive investigation especially due to the
possibility of miniaturizing integrated devices in electrical circuits.Several
theoretical methods have been proposed to address the steadystate properties of
these systems.
Ab initio formulations based on time-dependent (TD) density functional theory
(see [1]) (DFT) and current density functional theory [2-4] provide an exact
framework to account for correlation effects both in the leads and the device but lack
a systematic route to improve the level of the approximations. Ad-hoc
approximations have been successfully implemented to describe qualitative features
of the ( )I V characteristic of molecular junctions in the Coulomb blockade regime.
The possibility of including relevant physical processes through an insightful
selection of Feynman diagrams is the main advantage of many-body perturbation
theory (MBPT) over one-particle schemes. Even though computationally more
expensive MBPT offers an invaluable tool to quantify the effects of electron
correlations by analyzing, e.g., the quasiparticle spectra, lifetimes, screened
interactions, etc. One of the most remarkable advances in the MBPT formulation of
electron transport was given by Meir and Wingreen who provided an equation for
the steady-state current through a correlated device region thus generalizing the
Landauer formula.(see [3]). The Meir-Wingreen formula is cast terms of the
interacting Green’s function and self-energy in the device region and can be
approximated using standard diagrammatic techniques. Exploiting Wick’s theorem
a general diagram for the self-energy can be written in terms of bare Green’s
functions and interaction lines. Any approximation to the self-energy which
contains a finite number of such diagrams does, however, violate many
conservation laws.
Conserving approximations [5] require the resummation of an infinite number
of diagrams and are of paramount importance in nonequilibrium problems as they
guarantee satisfaction of fundamental conservation laws such as charge
conservation. Examples of conserving approximations are the Hartree-Fock (HF),
second Born (2B), GW, T-matrix, and fluctuation exchange (FLEX)
65
approximations [6]. The success of the GW approximation in describing spectral
features of atoms and molecules as well as of interacting model clusters prompted
efforts to implement the Meir-Wingreen formula at the GW level in simple
molecular junctions and tight-binding models.
The advantage of using molecular devices in future nanoelectronics is not
only the miniaturization of integrated circuits. Nanodevices can work at the
terahertz regime and hence perform operations in a few picoseconds or even faster.
Space and time can both be considerably reduced. Nevertheless, at the
subpicosecond time-scale stationary steady-state approaches are inadequate to
extract crucial quantities such as, e.g., the switching or charging time of a
molecular diode, and consequently to understand how to optimize the device
performance.
Recently several practical schemes have been proposed to tackle TD quantum
transport problems of noninteracting electrons. In some of these schemes the
electronelectron interaction can be included within a TDDFT framework [7] and
few calculations on the transient electron dynamics of molecular junctions have
been performed at the level of the adiabatic local density approximation. So far,
however, no one has extended the diagrammatic MBPT formulation of Meir and
Wingreen to the time domain. As in the steady-state case the MBPT formulation
allows for including relevant scattering mechanisms via a proper selection of
physically meaningful Feynman diagrams. The appealing nature of diagrammatic
expansions renders MBPT an attractive alternative to investigate out-of
equilibrium systems.
In this paper we consider a time-dependent MBPT formulation of quantum
transport which is based on the realtime propagation of the Kadanoff-Baym (KB)
equations for open and interacting systems. The KB equations are equations of
motion for the nonequilibrium Green’s function from which basic properties of the
system can be calculated. It is the purpose of this paper to give a detailed account
of the theoretical derivation and to extend the numerical analysis to quantum wires
connected to two-dimensional leads. For practical calculations we have
implemented the fully self-consistent HF, 2B, and GW conserving approximations.
Our results reduce to those of steady-state MBPT implementations in the
long-time limit. However, having full access to the transient dynamics we are also
able to extract novel information such as the switching and charging times, the
time-dependent renormalization of the electronic levels, the role of initial
correlations, the time-dependent dipole moments etc.
2. The model of quantum system
We consider a class of quantum correlated open systems (Fig. 1, the left part)
(which we call central regions), coupled to noninteracting reservoirs, leads) with
the Hamiltonian
,
( ) ( ) ( )C T
L R
H t H t H t H N�
�
�
�
� � � �� , (1)
66
where CH , H� and TH are the central region, the lead ,L R� � and the tunneling
Hamiltonians, respectively, and N is the particle number operator coupled to
chemical potential � .
Fig.1. Sketch of the transport setup. The correlated central region (C) is coupled to semi-
infinite left (L) and right (R) tight-binding leads via tunneling Hamiltonians H_C and HC
( ,L R� � ).
We assume that there is no direct coupling between the leads. The explicit
expressions for these Hamiltonians have the form
' '
, , , , ,
, '
( ) ( )C ij i j ijkvl i i k l
i j i j k l
H t h t d d v d d d d� � � � � �
�
� �
� � �� �� � , (2)
where ,i j label a complete set of one-particle states in the central region, � , '�
are the spin indices, d � , d are the creation and annihilation operators,
respectively. The one-body part of the Hamiltonian ( )ijh t may have an arbitrary
time dependence, describing, e.g., a gate voltage or pumping fields. The two-body
part accounts for interactions between the electrons, where ijklv are, for example in
the case of a molecule, the standard two-electron integrals of the Coulomb
interaction. The lead Hamiltonians have the form
, ,
( ) ( ) ij i j
i j
H t U t N h c c�
� � � �� ��
�
�� � � , (3)
where the creation and annihilation operators for the leads are denoted by c� and
c . Here i ii
N c c�� ���
�� � is the operator describing the number of particles in
lead� . The one-body part of the Hamiltonian ijh describes metallic leads and can
be calculated using a tight-binding representation, or a real space grid or any other
convenient basis set. We are interested in exposing the leads to an external electric
field which varies on a time-scale much longer than the typical plasmon time-scale.
Then, the coarse-grained time evolution can be performed assuming a perfect
instantaneous screening in the leads and the homogeneous time-dependent field
( )U t� can be interpreted as the sum of the external and the screening field, i.e., the
applied bias. This effectively means that the leads are treated at a Hartree mean
field level. We finally consider the tunneling Hamiltonian TH ,
67
,
[ ]T ij i j j i
ij
H V d c c c� � �� �� �
��
� �� �� , (4)
which describes the coupling of the leads to the interacting central region. This
completes the full description of the Hamiltonian of the system. In the next section
we study the equations of motion for the corresponding Green’s function.
2. Equations for the Keldysh Green functions.
We assume the system to be contacted and in equilibrium at inverse
temperature before time 0t � and described by Hamiltonian 0H . For times
0t t
the system is driven out of equilibrium by an external bias and we aim to
study the time-evolution of the electron density, current, etc. In order to describe
the electron dynamics in this system we use Keldysh Green’s function theory
which allows us to include many-body effects in a diagrammatic way. The
Keldysh Green’s function is defined as the expectation value of the contour-
ordered product
� �
�
� �
�
1 1
1 1
Sp exp ( ) ( ) ( ')
( , ')
Sp exp ( )
r s
rs
T i dz H z a z a z
G z z i
T i dz H z
��
� �
�
�
�
, (5)
where a� and a are either lead or central region operators and the indices r and s
are collective indices for position and spin. The variable z is a time contour
variable that specifies the location of the operators on the time contour. The
operator T orders the operators along the Keldysh contour displayed in Fig. 2,
consisting of two real-time branches and the imaginary track running from 0� to
0 i� � . In the definition of the Green’s function the trace is taken with respect to
the many-body states of the system.
All time-dependent one-particle properties can be calculated from G. For
instance, the time-dependent density matrix is given as
( ) ( , )rs rsn t iG t t� �� � ,
where the times t_ lie on the lower/upper branch of the contour. The equations of
motion for the Green’s function of the full system can be easily derived from the
definition Eq. 5 as
1 1 1
1 1 1
( ; ') 1 ( , ') ( ) ( , ') ( , ) ( '),
( ; ') 1 ( , ') ( , ') ( ') ( ; ) ( , ').
'
MB
MB
di G z z z z H z G z z dz z z G z z
dz
di G z z z z G z z H z dz G z z z z
dz
�
�
�
�
� � � �
� � � � �
�
�
(6)
where 1( , )MB z z� is the many-body self-energy, ( )H z is the matrix representation
of the one-body part of the full Hamiltonian and the integration is performed over
the Keldysh contour (Fig.1, right part). This equation of motion needs to be solved
with the boundary conditions
68
0 0 0 0( , ') ( , '), ( , ) ( , ),G t z G t i z G z t G z t i � � � � � �
which follow directly from the definition of the Green’s function Eq. 5. Explicitly,
the one-body Hamiltonian H for the case of two leads, left L and right R
connected to a central region C , is || ||, ( , , , )H H L C R� � � � , where
0RL LRH H� � . Here the different block matrices describe the projections of the
one-body part H of the Hamiltonian onto different subregions. They are explicitly
given as
� � � �
� � � �
, '. '
, '. ' . '
( ) [ ( ) ( ) ] ,
( ) [ ( ) ] , ( ) ( ) .
ij ij iji j
CC ij ij C iji j i j
H z h z U z
H z h z H z V z
�� � � �� �
� � � �� � � �
� � � �
� � � �
� � � �
� � �
The dynamical processes occurring in the central region are described by the
Green’s function CCG projected onto region C . The many-body self-energy in Eq.
7 has nonvanishing entries only for indices in region C . This implies that
( )MB
CCG� is a functional of CCG only. From these considerations it follows that in
the one-particle basis the matrix structure of MB� is given as
� � ( )MB MB
CC CCG
�
� � � , ,|| ||MB MB
CC C C� � �� � � .
The projection of the equation of motion (6) onto regions CC and C�
yields
1 1 1
( ) ( , ') ( , ')1
( , ') ( , ) ( '),
CC CC
MB
C C CC CC
di H z G z z z z
dz
H G z z dz z z G z z� �
� �
�� �� � �� �
� �
� �� �
(7)
for the central region and
( ) ( , ') ( , ')C C CC
di H z G z z H G z z
dz �� � �
� �� �� �
� �
, (8)
for the projection on C . The latter equation can be solved for C� , taking into
account the boundary conditions we obtain
1 1 1( , ') ( , ) ( ')C C CCG z z dz g z z H G z z� �� �
�
� � , (9)
where the integral is along the Keldysh contour. Here we defined g�� as the
solution of
( ) ( , ') ( , ')C C CC
di H z G z z H G z z
dz �� � �
� �� �� �
� �
, (10)
with the above mentioned boundary conditions. The function g�� is the Green’s
function of the isolated and biased � -lead. We wish to stress that a Green’s
function g�� with boundary conditions automatically ensures the correct boundary
conditions for the ( , ')CG z z� in Eq. (9). Any other boundary conditions would not
69
only lead to an unphysical transient behavior but also to different steady-state
results. This is the case for, e.g., initially uncontacted Hamiltonians in which the
equilibrium chemical potential of the leads is replaced by the electrochemical
potential, i.e., the sum of the chemical potential and the bias
Taking into account Eq. (9) the second term on the righthand side of Eq. (7)
becomes
1 1 1( , ') ( , ) ( ')C C em CCH G z z dz z z G z z� �
� �
� �� � , (11)
where we have introduced the embedding self-energy
,( , ') ( , ') ( , ')em C Cz z z z H g z z H� � � �� �
� �
� � � �� � .
which accounts for the tunneling of electrons from the central region to the leads
and vice versa. The embedding selfenergies ,em �� , are independent of the
electronic interactions and hence of CCG , and are therefore completely known once
the lead Hamiltonians H� are specified. Inserting Eq. (11) back to Eq. (9) then
gives the equation of motion
� �1 1 1
( ) ( , ') ( , ')1
( , ') ( , ) ( '). (12)
CC CC
MB
C C CC em CC
di H z G z z z z
dz
H G z z dz z z G z z� �
� �
�� �� � �� �
� �
� � � �� �
Equation (12) is an exact equation for the Green’s function CCG . To solve the
equation of motion Eq. (12)_ we need to find an approximation for the many-body
self-energy [ ]MB
CC CCG� as a functional of the Green’s function CCG . This
approximation can be constructed using a variational derivative method which can
straightforwardly be extended to the case of contourordered Green’s functions. In
our case the perturbative expansion is in powers of the two-body interaction and
the unperturbed system consists of the noninteracting, but contacted and biased
system. We stress, however, that eventually all our expressions are given in terms
of fully dressed Green’s functions leading to fully self-consistent equations for the
Green’s function.
2. The equations for a time-dependent current.
An equation for the time-dependent current flowing into the lead � can be
derived from the time-derivative of the number of particles in lead � using the
equation of motion for the lead’s Green’s function G�� . This yields
� �( ) 2ReSp ( , )C CI t G t t H� � �
�� ,
where '( , ') ( , )G t t G t t
�
�� � . If we insert the adjoint of Eq. (9) and extract the
different components from the resulting equation, the equation for the cdurrent
becomes
70 © �.�. ���
��, �.�.
�������,
.�. ��� �� ��
0 0
1 1 , 1 1 1 , 1( ) 2ReSp ( , ) ( , ) ( , ) ( , )
t t
A R
CC em CC em
t t
I t dt G t t t t dt G t t t t� � �
� �
�
� � � � ���
�
� �
0
| |
1 ,( , ) ( , ) .
t
CC em
t
d G t t�� � �
� � �
� �
!
�
Here |
0( , ) ( , )CCG t G t t i� �
�
�� � and
,
|
, 0( , ) ( , )
em emt t i t
� �� �
�
� � � � . The first two terms
in this equation contain integration over early times from 0t to t and take into
account the nontrivial memory effects arising from the time-nonlocality of the
embedding self-energy and Green’s functions. As anticipated the last term in this
equation explicitly accounts for the effects of initial correlations and initial-state
dependence. If one assumes that both dependences are washed out in the long-time
limit t " # , then the last term vanishes and we can safely take the limit
0t " �# .
1. Evers, F. Weigend, Koentopp M. Phys. Rev. B 69, 235411 (2004).
2. Koentopp M., Burke K., Evers F. Phys. Rev. B 73, 121403R (2006).
3. Meir Y., Wingreen N. S. Phys. Rev. Lett. 68 2512 (1992).
4. Jauho A. P., Wingreen N. S., Meir Y. Phys. Rev. B 50 5528 (1994).
5. Baym G., Kadanoff L.P. Phys. Rev. 124, 287 (1961).
6. Leeuwen R., Dahlen N. E., Stan A. Phys. Rev. B 74 195105 (2006).
7. Kurth S. Kurth,. Stefanucci G, Almbladh C.-O., Rubio A., Gross E. K. U. Phys. Rev. B
72, 035308 (2005).
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