Coupled electron and phonon transport in one-dimensional atomic junctions
J. T. Lü∗ and Jian-Sheng Wang†
Center for Computational Science and Engineering and Department of Physics,
National University of Singapore, Singapore 117542, Republic of Singapore
(Dated: August 5, 2021)
Employing the nonequilibrium Green’s function method, we develop a fully quantum mechanical
model to study the coupled electron-phonon transport in one-dimensional atomic junctions in the
presence of a weak electron-phonon interaction. This model enables us to study the electronic and
phononic transport on an equal footing. We derive the electrical and energy currents of the coupled
electron-phonon system and the energy exchange between them. As an application, we study the
heat dissipation in current carrying atomic junctions within the self-consistent Born approximation,
which guarantees energy current conservation. We find that the inclusion of phonon transport is
important in determining the heat dissipation and temperature change of the atomic junctions.
PACS numbers: 71.38.-k,63.20.Kr,72.10.Bg
I. INTRODUCTION
The electronic transport and phononic transport in
meso- and nano-structures have attracted a great deal
of interest in the past two decades, although their devel-
opment is not so parallel sometimes. These structures
display important quantum effects due to the confine-
ment in one or more directions1. The quantized electrical
conductance2 was observed much earlier than that of the
thermal conductance3 mainly due to the difficulty in mea-
suring the thermal transport properties. Electrons and
phonons are not two isolated systems. Their interactions
are important for both electronic and phononic trans-
port. With the development of both fields there arises
the requirement to study the coupled electron-phonon
transport from time to time. When studying electronic
transport problems, one usually assumes that electrons
interact with some phonon bath where the phonons are in
their thermal equilibrium state characterized by the Bose
distribution. This simple assumption is not able to give
satisfactory results in some cases where the phonons are
driven out of equilibrium by the electrons. This is espe-
cially true in places where the thermal conductance is low
or the phonon relaxation is slow4,5. To take into account
the nonequilibrium phonon effect, one usually introduces
into the electronic transport formalism some phenomeno-
logical parameters that describe the phonon relaxation
process. In engineering applications, as the size of the
electronic devices decreases to nanoscale, the heat dissi-
pation and conduction in these structures become crit-
ical issues, which may influence the electronic proper-
ties dramatically6. Only studying the electronic trans-
port is not enough in these cases. On the other hand,
heat transport in one-dimensional (1D) structures has re-
ceived considerable attention recently6,7,8. Fourier’s law
of heat conduction is no longer valid in many 1D systems.
The microscopic origins of the macroscopic Fourier’s law
remain one of the most frustrating problems in nonequi-
librium statistical mechanics. Since the electrons and
phonons both contribute to the heat conduction, their
relative roles in many nanostructures are still not clear.
Especially in semiconductors, which one carries the ma-
jority of the thermal current is not a trivial problem. To
answer these questions, we need some general models,
which take into account the electron, phonon transport,
and their mutual interactions.
Theoretically, although the development of electronic
transport in 1D structures has been very striking, that
of the phononic transport is relatively slow. Classical
molecular dynamics (MD) and the Boltzmann-Peierls
equation are the widely used methods in phononic trans-
port. MD method is not accurate below the Debye
temperature, while the Boltzmann-Peierls equation can
not be used in nanostructures without translational in-
variance. In both cases, the quantum effect becomes
important1. Only recently, the nonequilibrium Green’s
function method9,10,11,12, which has been widely used
to study the electronic transport, has been applied to
study the quantum phononic transport13,14,15,16,17. As
far as we know, the study of the coupled electronic and
phononic transport in nanostructures is rare18,19,20,21.
In Ref. 19, the authors considered the nonequilibrium
phonons in molecular transport junctions. Galperin and
co-authors analyzed the heat generation and conduction
in molecular systems18. In this paper, using the nonequi-
librium Green’s function method, we study the coupled
electronic and phononic transport in 1D atomic junc-
tions. The formalism is similar to that of Ref. 18. In our
model the electron subsystem is described by a single-
orbital tight-binding Hamiltonian, and the phonon sub-
system is described in a harmonic approximation. We
assume that the electron-phonon interaction is weak so
that the perturbative treatment is valid. The strong-
interaction case is the scope of future work.
The rest of the paper is organized as follows. In Sec. II,
we introduce the 1D model system, and derive expres-
sions for the electrical, energy current of the coupled
electron-phonon system. In Sec. III we show the heat
generation in one- and two-atom structures under differ-
ent model parameters. Sec. IV is the conclusion. In Ap-
pendix A-C we give some technical details of our deriva-
tion.
http://arxiv.org/abs/0704.0723v1
II. COUPLED ELECTRONIC AND PHONONIC
TRANSPORT
A. The Hamitonian
Our model system is an infinite 1D atomic chain as
shown in Fig. 1. The electrons and atoms are only al-
lowed to move in the longitudinal direction. We treat
the atoms as coupled harmonic oscillators, and take into
account their nearest neighbour interactions up to the
second order. We assume that there is only one single
electronic state for each atom and take into account hop-
ping transitions between the nearest states. This corre-
sponds to a single-orbital tight-binding model. Also, we
assume that there is only one spin state for each orbital.
Following Caroli22, we divide the whole system into one
central region and two semi-infinite leads, which act as
electrical and thermal baths (Fig. 1). The Hamiltonian
of the whole system is
α=L,C,R;β=e,ph
α=L,R;β=e,ph
HαCβ +H
+Heph. (1)
The electron-phonon interaction Hamiltonian Heph is
non-zero only in the central region. The electron Hamil-
tonian reads
Hαe =
εαi c
|i−j|=1
tαijc
j , (2)
where c
i and c
i are the electron creation and annihila-
tion operators. εαi is the electron onsite energy, and t
ij is
the hopping energy between adjacent states. i and j run
over the sites in the α region. The coupling Hamiltonian
with the leads is
HLCe =
tLCij c
j , (3)
HCRe =
tCRij c
j . (4)
HCLe and H
e have similar expressions. We also have
tαC = tCα
, α = L,R. For our 1D tight-binding model,
tαC has only one non-zero element. If we label the central
atoms with indices 1 to n as shown in Fig. 1, the non-zero
elements will be tLC01 , t
10 , t
n+1,n, and t
n,n+1.
The phonon Hamiltonian is
Hαph =
u̇αi u̇
|i−j|=0,1
uαi K
j . (5)
uαi and u̇
i are the mass-renormalized atom displacement
and momentum operator. Kαii = 2K
i , and K
εCm+1 ε
εRn+1 ε
KL0,−1
tL0,−1 t
KCL10
tCm+1,m
KCm+1,m
tRCn+1,n
KRCn+1,n
tRn+2,n+1
KRn+2,n+1
Heph Heph
FIG. 1: Shematic diagram of the 1D coupled electron-phonon
system and the parameters used in the model. The big dots
in the bottom line represent atoms, while the small dots in
the upper line represent electron states. They are coupled via
the electron-phonon interaction.
−Kα0 /
mαi m
j (i 6= j). Here Kα0 is the spring constant,
and mαi is the mass of the ith atom in the α region. Like
the electrons, the coupling Hamiltonian with the leads is
HLCph =
uLi K
j , (6)
HCRph =
uCi K
j . (7)
We also have KCα = KαC
. The non-zero elements are
KLC01 , K
10 , K
n+1,n, and K
n,n+1.
The electron-phonon interaction is included within the
adiabatic Born-Oppenheimer approximation. First, the
electron subsystem is solved with all the atoms in their
equilibrium positions. Then, the isolated phonon sub-
system is considered. After that, the electron-phonon in-
teraction is turned on by allowing the atoms to oscillate
around their equilibrium positions. Within this picture,
the electron-phonon interaction is11
Heph =
i,j,k
Mkijc
icjuk. (8)
The interaction matrix element is Mkij =
. All
the operators in Eq. (8) are in the central region, so we
omitted the superscript C. In our model, the electron
operators are in the second quantization, while that of
the phonons are in the first quantization.
B. Green’s functions
The nonequilibrium Green’s function method for the
electronic transport is discussed in Refs. 9,10,11,12, and
that for the phononic transport in Refs. 13,14,15,16,17.
Here we concentrate on the electron-phonon interactions.
The definition of the electron contour-ordered Green’s
function is Gjk(τ, τ
′) = −i〈T {cj(τ)c†k(τ ′)}〉, and the
phonon counterpart is Djk(τ, τ
′) = −i〈T {uj(τ)uk(τ ′)}〉.
Here τ is time on the Keldysh contour, and T {· · ·} is
the contour-ordered operator. We set h̄ = 1 through-
out the formulas. Without the electron-phonon inter-
action, the isolated electron and phonon problem can
be solved exactly. We denote these Green’s functions
as G0(τ, τ
′) and D0(τ, τ
′), respectively. In our case,
it is convenient to write the Hamiltonians as matri-
ces and work in the energy space. The electron re-
tarded and advanced Green’s functions are Gr0(ε) =
†(ε) =
(ε+ iη)I −HCe − ΣrL(ε)− ΣrR(ε)
. I is
an identity matrix, and η → 0+. The retarded self-
energy Σrα = t
Cαgrαt
αC is due to the interactions with
the lead α. The retarded Green’s function of the semi-
infinite lead grα can be obtained analytically (Appendix
A). The “less than” Green’s function is given by G<0 =
Gr0(Σ
L + Σ
0 , where Σ
α = −f eα(Σrα − Σaα). f eα is
the Fermi-Dirac distribution. The phonon retarded and
advanced Green’s functions are23 Dr0(ω) = D
†(ω) =
(ω + iη)2I −KC −ΠrL(ω)−ΠrR(ω)
. The lead re-
tarded self-energy is Πrα(ω) = K
Cαdrα(ω)K
αC . drα also
has analytical expression (Appendix A). The phonon
“less than” Green’s function is D<0 = D
)Da0 ,
where Π<α = f
α −Πaα). fphα is the Bose distribution
function.
Knowing the bare electron and phonon Green’s func-
tions G0 and D0, we can include their interaction as per-
turbation. Following the standard procedure of nonequi-
librium Green’s function method, we can express this
interaction as self-energies. The full Green’s functions
are obtained from the Dyson equation, e.g., for elec-
trons Gr,a = G
0 + G
r,a, and G< = GrΣ<t G
Σ<t = Σ
eph + Σ
L + Σ
R is the total self-energy. Keeping
the lowest non-zero order (the second order) of the self-
energies, we have two (Hartree- and Fock-like) terms for
the electrons, and one polarization term for the phonons.
This is the so-called Born approximation (BA)11. The
Fock self-energies are
ΣF,<mn (ε) = iM
G<0 ij(ε− ω)D
0 kl(ω)
M ljn, (9)
ΣF,rmn(ε) = iM
Gr0ij(ε− ω)D
0 kl(ω)
+G<0 ij(ε− ω)D
0kl(ω)
+Gr0ij(ε− ω)D
0kl(ω)
M ljn. (10)
The “less than” Hartree self-energy is zero, and the re-
tarded one is
ΣH,rmn = −iM imnDr0ij(ω
′ = 0)M
G<0 lk(ε)
. (11)
This term is a constant for all energies, which represents
a static potential due to the presence of phonons. The
self-energies for the phonons are
Π<mn(ω) = −iMmlk
G<0 ki(ε)G
0 jl(ε− ω)M
ij , (12)
Πrmn(ω) = −iMmlk
Gr0ki(ε)G
0 jl(ε− ω)
+G<0 ki(ε)G
0jl(ε− ω)
Mnij . (13)
In Eqs. (9-13), sum over internal indices is assumed. The
self-consistent Born approximation (SCBA) is obtained
by replacing all the bare Green’s functions G0 and D0 in
Eqs. (9-13) with the full G and D11. In Appendix B,
we show that the SCBA fulfills the electrical and energy
current conservation, while BA fails.
C. The electrical and energy current
The electrical and energy current can be expressed by
the Green’s functions. The electrical current out of the
lead α is11,24
Jα = e
Tr{G>(ε)Σ<α (ε)−G<(ε)Σ>α (ε)}. (14)
The electron energy current is
JE,eα =
ε Tr{G>(ε)Σ<α (ε)−G<(ε)Σ>α (ε)}. (15)
The electron heat current is obtained from Eqs. (14-
15) as Jh,eα = J
α − µαJα/e. µα is the lead chemical
potential. The derivation of the phonon energy current
runs parallel with that of the electrons17
JE,phα = −
ω Tr{D>(ω)Π<α (ω)−D<(ω)Π>α (ω)}.
For phonons the energy current is the same as the heat
current. When there is no electron-phonon interaction,
the electron energy current is conserved throughout the
structure. So is the phonon energy current. In the pres-
ence of such an interaction, only the total energy current
is conserved due to the energy exchange between them.
The phonons do not carry charges, so in both cases the
electrical current is conserved. Since we can’t get the
exact self-energies in most cases, we need some approxi-
mations. Properly defined self-energies should fulfill the
electrical and energy current conservation
Jα = 0, (17)
(JE,eα + J
α ) = 0, (18)
where α runs over all the leads. We justify that the SCBA
fulfills these conservation laws, while the BA fails to con-
serve the energy current (Appendix B). Provided we sat-
isfy these conservation laws, we can write the electrical
and energy current in symmetric forms. The electrical
current is
J = e
T̃ e(ε) [f eL(ε)− f eR(ε)] . (19)
The transmission coefficient reads
T̃ e = Tr
Gr(ΓL +
Γeph − Se)GaΓR
+GrΓLG
a(ΓR +
Γeph + S
, (20)
where Se is
(f eR + f
L)Γeph + iΣ
f eL − f eR
. (21)
Γα = i(Σ
α − Σaα), α = L,R is the electron level-width
function. Γeph = i(Σ
eph − Σaeph) is due to the electron-
phonon interaction. The total energy current is
T̃ e(ε) [f eL(ε)− f eR(ε)]
T̃ ph(ε)
L (ε)− f
R (ε)
. (22)
The phonon transmission coefficient is
T̃ ph = Tr
Dr(ΛL +
Λeph − Sph)DaΛR
+DrΛLD
a(ΛR +
Λeph + S
, (23)
where Sph is
Sph =
R + f
L )Λeph − iΠ
L − f
. (24)
Λα = i(Π
α − Πaα) is the phonon level-width function.
Λeph = i(Π
eph−Πaeph) is due to the electron-phonon inter-
action. Eqs. (19-24) are the generalization of the Caroli
formula22 to include the electron-phonon interaction.
III. HEAT GENERATION IN CURRENT
CARRYING 1D ATOMIC JUNCTIONS
As an application of the formalism in Sec. II, we
study the heat dissipation in current-carrying 1D atomic
junctions18,25,26,27,28,29,30,31. In the presence of potential
difference between the two leads, there will be electrical
current flowing between them. When the electrons pass
the central region, there is energy exchange between the
electron and phonon systems. The energy dissipated into
the phonon system makes the atom temperature higher
than that of the leads if it is not efficiently conducted to
the leads. If the electron-phonon interaction is weak, the
energy dissipated into the phonon system is only a small
fraction of the electron energy current. But this small
fraction still influences the transport properties of the
atomic junction and even leads to junction breakup32,33,
especially when the thermal conductance is low. Dif-
ferent models have been used to study the local heating
effect. Some simply assume that the phonons are in their
thermal equilibrium states31. Some take into account the
phonon transport by using the rate equations30 or other
semi-classical models26,27,32. Few of them take into ac-
count the quantum effect in heat transport18,34. Our
model treats the electron and phonon transport on an
equal quantum-mechanical footing, and includes their in-
teractions self-consistently. The heat generation is given
by (Eq. (B9))
Q = i
G>nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
. (25)
At zero temperature, we can get an analytical expression
Eq. (C1) for a single-atom structure by using the bare
Green’s functions G0 and D0 in Eq. (25) (Appendix C).
Equation (C1) can reproduce most qualitative features of
heat generation in a single atom, except that it does not
take into account heat conduction in the phonon system.
We first study the case where the lead energy band is
wide compared to the voltage applied to the structure.
For most metallic leads, this condition should hold. In
the weak electron-phonon coupling regime, the Born ap-
proximation should give acceptable results for the heat
generation, although physically it is not a good approx-
imation. Figure 2 shows the heat generation of a single
atom (n = 1 in Fig. 1) computed using Eqs. (B5) and
(B8) under BA and SCBA, respectively. The parameters
used in the calculation are stated in the figure caption.
With these parameters, the electron energy band is in
the range −1 ≤ ε ≤ 1 eV. The chemical potential of each
lead is zero in equilibrium. The phonon energy is ap-
proximately ω = 0.05 eV. In all the results presented in
this section, the temperature is T = 4.2 K, the electron-
phonon coupling matrix M = 0.08 eV/(Å·amu 12 ). The
cut-off energy of the electron system is 2.1 eV, and the
phonon system is 0.2 eV. The energy spacing is dis-
cretized into grids of 1 meV. Equation (B5) gives the
energy decrease of the electron system, while Eq. (B8)
gives the energy increase of the phonon system. Numeri-
cal results from Eq. (B5) and Eq. (B8) under SCBA have
some slight discrepancy. This is due to numerical inac-
curacies. But most of the discrepancy under BA comes
from the difference between the bare and the full Green’s
functions, which may become even larger for some pa-
rameters. So BA should be used with care in the study
the energy exchange between the electron and phonon
system. We also note that although Eqs. (B5) and (B7)
are equivalent, numerical result from Eq. (B5) is unsta-
ble in many cases. The reason is that the energy ex-
change between the electron and the phonon system is
only a small fraction of the total electrical energy cur-
rent. Equation (B5) is the difference between two large
numbers, so our numerical integration has to be accurate
enough to get a reasonable result30. On the contrary,
Eq. (25) is much more stable since we have got the dif-
ference analytically. All the results presented below use
this equation.
From Fig. 2, we can see two threshold values in heat
generation. The first one corresponds to the onset of
phonon emission. Under low temperatures, the equilib-
rium phonon occupation is very small, so the phonon
absorption process seldom takes place. If the applied
bias is smaller than the phonon energy, electrons don’t
 0  0.1  0.2  0.3  0.4  0.5
Voltage (V)
SCBA, B8
SCBA,B5
BA, B8
BA,B5
 0  0.05  0.1
FIG. 2: Comparison of different methods to compute the heat
generation in a single atom structure. The four curves cor-
respond to results from Eqs. (B5) and (B8) under BA and
SCBA, respectively. If we label this single atom as index 1,
its electronic onsite energy is written as εC1 = 0.1 eV. The
onsite energy of the leads is εL = εR = 0 eV. The hopping
energy is tLij = t
ij = 0.5 eV. The non-zero electronic cou-
pling with the lead is tCL10 = t
21 = 0.1 eV. The matrix ele-
ment of the single atom is KC11 = 0.654 eV/(Å
· amu). The
spring constant between the lead atoms is KLij = K
ij = 0.654
eV/(Å2·amu). The non-zero atomic coupling with the leads
is KCL10 = K
21 = 0.127 eV/(Å
·amu).
have enough energy to emit one phonon. So the heat
generation is zero. Once the applied voltage is larger
than the phonon energy, phonon emission turns on. The
heat generation increases almost linearly with the applied
bias (inset of Fig. 2). This is different from the electrical
current, which increases smoothly in this regime. The
second threshold value corresponds to the alignment of
the left lead chemical potential with the electron onsite
energy eV = 2ε0 (positive bias µl > µr). The electron
transmission is nearly unity above the onsite energy. The
larger the transmission, the larger the current and the
heat generation provided that the other parameters re-
main unchanged. These two threshold behaviours may
become less obvious when the coupling with the leads
get stronger. As a result of coupling, the discrete elec-
tron and phonon density of states (DoS) extends to a
small energy region around their discrete values. The
continuous phonon DoS leads to the broadening of the
first threshold behaviour, while the continuous electron
DoS is responsible for that of the second. It is smoothed
out when the coupling is large enough (Fig. 3). Only
electrons whose energies are within the broadened energy
spectrum can tunnel across the central atom. The heat
generation reaches maximum when the electron states in
one lead are all occupied in this energy range, while those
in the other are all empty.
The electron-lead coupling not only leads to the elec-
tron level broadening, but it also influences the elec-
tron tunneling time. The larger this coupling, the less
time electrons spend in the central region. In Fig. 3, we
show the heat generation and the atom temperature for a
single-atom structure under different electronic coupling
strengths. The definition of temperature is ambiguous
in nanostructures6. Here we use the method proposed
in Ref. 18. We can only see one threshold behaviour at
about 0.2 V, which is smoothed out when the coupling is
larger than 0.2 eV. The temperature and the heat gener-
ation show similar trends. The saturate voltage of heat
generation increases with the strengthen of the electron-
lead coupling. This is due to the coupling induced atomic
level broadening. The decrease of the heat generation
and temperature with increasing electron-leads coupling
can be easily understood. The larger this coupling, the
less time electrons spend at the central atom. Since the
electron-phonon interaction takes place there, the heat
generation decreases. We also show the heat generation
as a function of electron-lead coupling in the inset of the
lower panel. The applied voltage is 0.3 V. On one side,
when the coupling is too small, few electrons can tun-
neling through the atom. The heat generation is small.
On the other, when the coupling is very large, the elec-
tron tunneling process is too quick for the phonons to
interact with the electrons. The heat generation is also
small. It has a maximum value at some moderate cou-
pling strength. This is different from the electrical cur-
rent, which increases monotonously with the increase of
coupling strength.
 0  0.1  0.2  0.3  0.4  0.5
Voltage (V)
0.2 0.4
FIG. 3: Heat generation Q and the atom temperature T under
different electron coupling strength tCL10 = t
21 = 0.05, 0.1,
0.2, and 0.3 eV, respectively. Other parameters are the same
with Fig. 2. The inset shows the heat generation as a function
of electron coupling strength at an applied bias V = 0.3 V.
The atom-lead coupling determines how well the gen-
erated heat can be conducted into the surrounding leads.
One of the important reasons why we are interested in
the heat generation in nanostructures is that it may leads
to temperature increase and even structure breakup. To
study the temperature change, we need to take into ac-
count not only the heat generation, but also the heat con-
duction into the leads. In the simplest one-atom struc-
ture, the heat conductance is mainly determined by the
atom-lead coupling. Our model includes this intrinsically.
Figure 4 shows the heat generation and the atom temper-
ature as a function of atom-lead coupling. For the heat
generation, the BA and SCBA results show large differ-
ence around the resonant position, which corresponds to
a perfect atomic junction. For the atom temperature, BA
and SCBA give almost the same results. In the case of
a perfect junction, the heat generation reaches its max-
imum value, while the atom temperature is the lowest.
The reason is that the perfect junction has the best heat
conductance. When the atom-lead coupling is weak, the
heat generation is small. But the poor heat conductance
can still result in a much higher temperature than the
surrounding leads. We also show the heat conductance
as a function of atom-lead coupling in the inset of the
upper panel, which shows a sharp peak at resonance.
 0  0.2  0.4  0.6  0.8  1
K (eV/A2 amu)
0.80.60.40.2
FIG. 4: Heat generation Q and the atom temperature T as
a function of the atom-lead coupling KCL10 = K
21 = K.
Dashed and dotted lines correspond to results of SCBA and
BA, respectively. Other parameters are the same with Fig. 2.
The inset shows the thermal conductance κ as a function of
K. The unit is 1× 10−12 W/K.
In Fig. 5, we show the heat generation as a function
of electron onsite energy at different biases. We assume
that we can tune the the onsite energy via a gate voltage.
When the applied bias is less than the phonon energy,
there will be no heat generation. When the bias energy is
slightly larger than the phonon energy and less than 2ω,
there are two energy positions where the heat generation
is the largest. These two peaks are approximately at
−0.5eV +ω and 0.5eV −ω. They merge into a single one
at a bias of eV = 2ω until it reaches saturation. After
that, this peak broadens, and becomes ladders. All these
behaviour can also be explained by the analytical result
of Eq. (C1).
In Fig. 6 we show the heat generation of a two-atom
structure (n = 2 in Fig. 1). The central region has two
identical atoms. Interaction between them leads to two
discrete energy levels. One is at 0 eV, and the other at
0.4 eV. When the electrical coupling between the leads
and the central region is small (0.1 eV), additional to the
threshold behaviour at eV = ω, there are two ladders
corresponding to the phonon assisted resonant tunneling
across the two electrical levels. If the electrical coupling
0.20.10-0.1-0.2
ε(eV)
FIG. 5: Heat generation Q as a function of electrical onsite
energy εC1 under different biases. From the inner to the outer
side, the applied biases are V = 0.05, 0.10, 0.15, 0.20, 0.25,
0.30, 0.35, 0.40, 0.45, and 0.50 V, respectively.
gets larger (0.2 eV), the two ladders broaden out. Again
this is attributed to the coupling induced level broad-
ening. The heat generation for the two-atom structure
is much larger than that of a single-atom structure. The
more the electrical levels, the larger the electrical current
and heat generation. It is worth noting that for multi-
atom structures the distribution of the electrostatic po-
tential may influence the results significantly35. In the
above calculation, we assume that the two electrical lev-
els don’t change with the applied bias, and that we can
tune their positions via a gate voltage.
 0  0.2  0.4  0.6  0.8  1
Voltage (V)
FIG. 6: Heat generation Q as a function of applied voltage for
a two-atom structure. The two-atom onsite energy is εC = 0.2
eV, the hopping energy is tC = 0.1 eV, and the spring con-
stant is KC = 0.654 eV/(Å2·amu). The two leads are iden-
tical. The electron onsite energy is εL = εR = 0, and the
hopping energy is tL = tR = 0.5 eV. Their spring constants
are the same as the central region. The non-zero coupling cou-
plings with the leads are KCL10 = K
32 = 0.327 eV/(Å
·amu),
and tCL10 = t
32 = 0.1 (solid), 0.2 eV (dashed), respectively.
If one of the metallic leads is replaced by a semicon-
ductor, there will be some new features in the electrical
current and the heat generation. In our simple model,
we can alternate the electron onsite energies between two
values to mimic a simple semiconductor (Appendix A).
In Fig. 7 we show the heat generation and the electri-
cal current for such kind of structure. The alternating
onsite energies of the left lead are −0.1 and −0.2 eV, re-
spectively. This produces an energy band-gap of 0.1 eV.
Other parameters are given in the figure caption. We can
see that there appears negative differential conductivity
in the current-voltage characteristics due to the semi-
conductor band-gap. This qualitatively agrees with the
experimental36 and first-principle37 studies. The heat
generation curve is slightly different. Additional to its
threshold behaviour, the peak and valley positions are
also different. The electrical current has a peak when
the chemical potential of the lead is aligned with the
central electrical level, while the peak of the heat gen-
eration shifts to the right by one phonon energy. This
corresponds to the phonon-assisted resonant tunneling.
The current and heat generation decrease when the sin-
gle electrical level is within the band-gap of the left lead.
The peak-to-valley ratio depends on the coupling with
the semiconductor lead. In the limit of small band-gap
and large coupling, we recover the metallic lead results.
 0  0.2  0.4  0.6  0.8  1
Voltage (V)
FIG. 7: Heat generation Q and electrical current J as a
function of applied voltage for a single-atom structure. The
left lead is a semiconductor. Its alternating onsite energies
are −0.2 and −0.1 eV, respectively. The chemical potential
is µL = 0.05 eV higher than the conduction band bottom,
which corresponds to n-type doping. Other parameters are
10 = K
21 = 0.4 eV/(Å
·amu), tCL10 = t
21 = 0.1 eV,
1 = 0.1 eV, K
11 = 0.654 eV/(Å
·amu), εR = −0.05 eV,
L = tR = 0.5 eV, and KL = KR = 0.654 eV/(Å2·amu).
IV. CONCLUSION
We studied the coupled electron and phonon transport
in 1D atomic junctions in the weak electron-phonon inter-
action regime. Base on the nonequilibrium Green’s func-
tion method, we derived the electrical, energy current of
the coupled electron-phonon system, and the energy ex-
change between them. We showed that the SCBA con-
serves the energy current. Using this formalism, we stud-
ied the heat generation in one- and two-atom structures
coupling with different leads under a broad range of pa-
rameters. Especially, we studied the influence of the ther-
mal transport properties on the heat generation and atom
temperature of the central region. The results on semi-
conductor leads agree qualitatively with the experimental
and first-principle studies. This model can be easily ex-
tended to study more realistic structures such as molec-
ular transport junctions and metallic nanowires. The
electron, phonon Hamiltonian, their interaction and lead-
coupling matrices can all be obtained from first-principle
calculations17,38,39. The surface Green’s functions for
bulk leads can be computed by recursive method17,38. It
is also possible to include the electron-electron and the
phonon-phonon interactions14,17.
Acknowledgments
We thank Baowen Li, Sai Kong Chin, Jian Wang, and
Nan Zeng for discussions. This work is supported in part
by a Faculty Research Grant of the National University
of Singapore.
APPENDIX A: SURFACE GREEN’S FUNCTIONS
OF THE 1D LEAD
In this Appendix, we show that for the 1D tight-
binding model the lead self-energies can be expressed
analytically17. The electron and phonon self-energies are
similar in their form. Here we take electrons as an exam-
ple, and give the phonon results directly. We assume that
the onsite energies of the electrons alternate between ε1
and ε2. The hopping energy is t
ij = t0. If ε1 = ε2, we get
a continuum band. This corresponds to a metallic lead.
If they are not equal, we get two bands with a band gap.
We can take the lower as the valence band (VB), and the
upper as the conduction band (CB). We use this method
to mimic a semiconductor lead. In this case, the semi-
infinite lead has two electron states in each period. In the
tight-binding model, only the left- (right-) most state of
the central region is coupled to the left (right) lead. So
we only need to know the surface Green’s function, e.g.,
for the left lead it is g0 = g
00. We assume the retarded
Green’s function is
grij =
i−j state 1,
i−j state 2.
Putting it into the definition of the retarded Green’s func-
tions [(ε+ iη)I −H ]gr = I, we have
− t0c1 + (ε+ iη − ε2)c2 − t0c1λ = 0, (A2)
− t0c2 + (ε+ iη − ε1)c1λ− t0c2λ = 0. (A3)
From Eqs. (A2-A3), we get an equation for λ
2− (ε+ iη − ε1)(ε+ iη − ε2)
λ+ 1 = 0. (A4)
The condition that Eq. (A4) has travelling wave solutions
gives the dispersion relation
(ε1+ε2)−
(ε1−ε2)2+16t
≤ ε ≤ ε1 (VB),
ε2 ≤ ε ≤
(ε1+ε2)+
(ε1−ε2)2+16t
(CB).
We assume ε1 ≤ ε2 without loss of generality. The energy
band-gap is ε2 − ε1. If they are equal, the two bands
merge into one, which corresponds to a metallic lead.
For the surface Green’s function of the left lead, we
also have
(ε+ iη − ε1)c1 − t0c2 = 1. (A6)
From Eqs. (A2,A6), we get
ε+iη−ε2
(1+λ)t2
(VB),
ε+iη−ε1
(1+λ)t2
(CB).
|λ| ≥ 1 is one of the roots of Eq. (A4). The surface
Green’s function of the right lead is identical.
We can also alternate the atom masses to generate a
phonon band-gap. In our model the mass change will
modify the renormalized spring constants. The diagonal
elements of the dynamical matrix will be two alternating
values Kαii = 2k1 or 2k2, while the off-diagonal elements
will be a single value Kαij = −
k1k2, where |i − j| = 1.
If we assume that k2 ≥ k1, the acoustic band (AB) is
0 < ω2 < 2k1, and the optical band (OB) 2k2 < ω
2(k1 + k2). The surface Green’s function is
(1+λ)k1k2
(AB),
(1+λ)k1k2
(OB),
where Ωn = (ω + iη)
2 − 2kn. |λ| ≥ 1 is one of the roots
2− Ω1Ω2
λ+ 1 = 0. (A9)
In all the simulation results of present paper, the two
spring constants are equal (k1 = k2), which correspond
a single continuum phonon band. The electron onsite
energies are also equal (ε1 = ε2) except in Fig. 7, where
we set ε1 = −0.2 eV and ε2 = −0.1 eV to mimic a
semiconductor lead.
APPENDIX B: ENERGY CURRENT
CONSERVATION
In this Appendix, we justify that the SCBA satisfies
the energy current conservation. The justification of the
electrical current conservation is given in the Refs.40,41.
What we need to prove is that
(JE,eα + J
α ) = 0. (B1)
The electron part is
JE,eα =
ε Tr{G>(ε)Σ<α (ε)−G<(ε)Σ>α (ε)}.
Using the important relation39,40
G>Σ<t −G<Σ>t
= 0, (B3)
we get
JE,eα = −
ε Tr{G>(ε)Σ<eph(ε)−G
<(ε)Σ>eph(ε)}.
The Hartree term does not contribute to the current di-
rectly. It’s just like a static potential which only modifies
the Green’s function. Putting the Fock self-energy into
Eq. (B4), we have
JE,eα
G>nm(ε)M
kl(ω)G
ij(ε− ω)M
−G<nm(ε)MkmiD>kl(ω)G
ij(ε− ω)M ljn
. (B5)
Sum over all the indices is assumed. The heat generation
Q is the energy decrease of the electron system, which
should also be the energy increase of the phonon system.
Replacing ω by −ω, using the symmetric properties of
the phonon Green’s functions17, replacing ε by ε − ω,
and finally changing dummy variables, we get
G<nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
G<nm(ε)M
kl(−ω)G
ij(ε+ ω)M
(ε− ω)
G<nm(ε− ω)MkmiD<lk(ω)G
ij(ε)M
(ε− ω)
G>nm(ε)M
kl(ω)G
ij(ε− ω)M
Putting Eq. (B6) back into Eq. (B5), we get
JE,eα
G>nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
6= 0. (B7)
For the phonon energy current we have
JE,phα
D>nm(ω)M
ki(ε)G
jl(ε− ω)M
−D<nm(ω)MmlkG>ki(ε)G
jl(ε− ω)M
. (B8)
Following the same procedure as electrons, finally we get
JE,phα
G>nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
6= 0. (B9)
So we still have
JE,eα + J
= 0. (B10)
Eqs. (B7, B9) give the energy exchange between the
electron and the phonon system, which is also the heat
generation of the atomic junction. Replacing D<, G<
by D<0 , G
0 in Eq. (B7), and G
>, G< by G>0 , G
Eq. (B9), we get the results under BA. We can find that
the energy increase of the phonons does not equal to the
energy decrease of the electrons under BA.
APPENDIX C: ANALYTICAL RESULT AT ZERO
TEMPERATURE
At zero temperature, we can get an analytical expres-
sion for heat generation in a single-atom structure by
using the bare Green’s functions in Eq. (B7, B9). We
only take into account the imaginary part of the lead self-
energies and ignore their energy dependence (the wide-
band limit)24. Finally, we assume that the phonons are
in their equilibrium states. Under these approximations,
the heat generation is (assuming eV ≥ ω0)
Q ≈ 1
M2ΓLΓR
[(ε− ε0)2 + Γ2/4] [(ε− ε0 − ω0)2 + Γ2/4]
M2ΓLΓR
4π(ω20 + Γ
(eV/2− ε0)2 + Γ2/4
(eV/2− ε0 − ω0)2 + Γ2/4
(−eV/2− ε0 + ω0)2 + Γ2/4
(−eV/2− ε0)2 + Γ2/4
arctan
eV/2− ε0
+ arctan
eV/2− ε0 − ω0
−arctan
−eV/2− ε0
−arctan
−eV/2− ε0 + ω0
. (C1)
ω0 is the phonon energy, ε0 is the electron onsite energy,
V is the applied bias, and Γ = ΓL+ΓR. The heat genera-
tion is zero when eV ≤ ω0. Equation (C1) can reproduce
most the qualitative features of heat generation in a sin-
gle atom, except that it does not take into account heat
conduction in the phonon system.
∗ Electronic address: tower.lu@gmail.com
http://staff.science.nus.edu.sg/~phywjs/
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  Employing the nonequilibrium Green's function method, we develop a fully
quantum mechanical model to study the coupled electron-phonon transport in
one-dimensional atomic junctions in the presence of a weak electron-phonon
interaction. This model enables us to study the electronic and phononic
transport on an equal footing. We derive the electrical and energy currents of
the coupled electron-phonon system and the energy exchange between them. As an
application, we study the heat dissipation in current carrying atomic junctions
within the self-consistent Born approximation, which guarantees energy current
conservation. We find that the inclusion of phonon transport is important in
determining the heat dissipation and temperature change of the atomic
junctions.

Coupled electron and phonon transport in one-dimensional atomic junctions
J. T. Lü∗ and Jian-Sheng Wang†
Center for Computational Science and Engineering and Department of Physics,
National University of Singapore, Singapore 117542, Republic of Singapore
(Dated: August 5, 2021)
Employing the nonequilibrium Green’s function method, we develop a fully quantum mechanical
model to study the coupled electron-phonon transport in one-dimensional atomic junctions in the
presence of a weak electron-phonon interaction. This model enables us to study the electronic and
phononic transport on an equal footing. We derive the electrical and energy currents of the coupled
electron-phonon system and the energy exchange between them. As an application, we study the
heat dissipation in current carrying atomic junctions within the self-consistent Born approximation,
which guarantees energy current conservation. We find that the inclusion of phonon transport is
important in determining the heat dissipation and temperature change of the atomic junctions.
PACS numbers: 71.38.-k,63.20.Kr,72.10.Bg
I. INTRODUCTION
The electronic transport and phononic transport in
meso- and nano-structures have attracted a great deal
of interest in the past two decades, although their devel-
opment is not so parallel sometimes. These structures
display important quantum effects due to the confine-
ment in one or more directions1. The quantized electrical
conductance2 was observed much earlier than that of the
thermal conductance3 mainly due to the difficulty in mea-
suring the thermal transport properties. Electrons and
phonons are not two isolated systems. Their interactions
are important for both electronic and phononic trans-
port. With the development of both fields there arises
the requirement to study the coupled electron-phonon
transport from time to time. When studying electronic
transport problems, one usually assumes that electrons
interact with some phonon bath where the phonons are in
their thermal equilibrium state characterized by the Bose
distribution. This simple assumption is not able to give
satisfactory results in some cases where the phonons are
driven out of equilibrium by the electrons. This is espe-
cially true in places where the thermal conductance is low
or the phonon relaxation is slow4,5. To take into account
the nonequilibrium phonon effect, one usually introduces
into the electronic transport formalism some phenomeno-
logical parameters that describe the phonon relaxation
process. In engineering applications, as the size of the
electronic devices decreases to nanoscale, the heat dissi-
pation and conduction in these structures become crit-
ical issues, which may influence the electronic proper-
ties dramatically6. Only studying the electronic trans-
port is not enough in these cases. On the other hand,
heat transport in one-dimensional (1D) structures has re-
ceived considerable attention recently6,7,8. Fourier’s law
of heat conduction is no longer valid in many 1D systems.
The microscopic origins of the macroscopic Fourier’s law
remain one of the most frustrating problems in nonequi-
librium statistical mechanics. Since the electrons and
phonons both contribute to the heat conduction, their
relative roles in many nanostructures are still not clear.
Especially in semiconductors, which one carries the ma-
jority of the thermal current is not a trivial problem. To
answer these questions, we need some general models,
which take into account the electron, phonon transport,
and their mutual interactions.
Theoretically, although the development of electronic
transport in 1D structures has been very striking, that
of the phononic transport is relatively slow. Classical
molecular dynamics (MD) and the Boltzmann-Peierls
equation are the widely used methods in phononic trans-
port. MD method is not accurate below the Debye
temperature, while the Boltzmann-Peierls equation can
not be used in nanostructures without translational in-
variance. In both cases, the quantum effect becomes
important1. Only recently, the nonequilibrium Green’s
function method9,10,11,12, which has been widely used
to study the electronic transport, has been applied to
study the quantum phononic transport13,14,15,16,17. As
far as we know, the study of the coupled electronic and
phononic transport in nanostructures is rare18,19,20,21.
In Ref. 19, the authors considered the nonequilibrium
phonons in molecular transport junctions. Galperin and
co-authors analyzed the heat generation and conduction
in molecular systems18. In this paper, using the nonequi-
librium Green’s function method, we study the coupled
electronic and phononic transport in 1D atomic junc-
tions. The formalism is similar to that of Ref. 18. In our
model the electron subsystem is described by a single-
orbital tight-binding Hamiltonian, and the phonon sub-
system is described in a harmonic approximation. We
assume that the electron-phonon interaction is weak so
that the perturbative treatment is valid. The strong-
interaction case is the scope of future work.
The rest of the paper is organized as follows. In Sec. II,
we introduce the 1D model system, and derive expres-
sions for the electrical, energy current of the coupled
electron-phonon system. In Sec. III we show the heat
generation in one- and two-atom structures under differ-
ent model parameters. Sec. IV is the conclusion. In Ap-
pendix A-C we give some technical details of our deriva-
tion.
http://arxiv.org/abs/0704.0723v1
II. COUPLED ELECTRONIC AND PHONONIC
TRANSPORT
A. The Hamitonian
Our model system is an infinite 1D atomic chain as
shown in Fig. 1. The electrons and atoms are only al-
lowed to move in the longitudinal direction. We treat
the atoms as coupled harmonic oscillators, and take into
account their nearest neighbour interactions up to the
second order. We assume that there is only one single
electronic state for each atom and take into account hop-
ping transitions between the nearest states. This corre-
sponds to a single-orbital tight-binding model. Also, we
assume that there is only one spin state for each orbital.
Following Caroli22, we divide the whole system into one
central region and two semi-infinite leads, which act as
electrical and thermal baths (Fig. 1). The Hamiltonian
of the whole system is
α=L,C,R;β=e,ph
α=L,R;β=e,ph
HαCβ +H
+Heph. (1)
The electron-phonon interaction Hamiltonian Heph is
non-zero only in the central region. The electron Hamil-
tonian reads
Hαe =
εαi c
|i−j|=1
tαijc
j , (2)
where c
i and c
i are the electron creation and annihila-
tion operators. εαi is the electron onsite energy, and t
ij is
the hopping energy between adjacent states. i and j run
over the sites in the α region. The coupling Hamiltonian
with the leads is
HLCe =
tLCij c
j , (3)
HCRe =
tCRij c
j . (4)
HCLe and H
e have similar expressions. We also have
tαC = tCα
, α = L,R. For our 1D tight-binding model,
tαC has only one non-zero element. If we label the central
atoms with indices 1 to n as shown in Fig. 1, the non-zero
elements will be tLC01 , t
10 , t
n+1,n, and t
n,n+1.
The phonon Hamiltonian is
Hαph =
u̇αi u̇
|i−j|=0,1
uαi K
j . (5)
uαi and u̇
i are the mass-renormalized atom displacement
and momentum operator. Kαii = 2K
i , and K
εCm+1 ε
εRn+1 ε
KL0,−1
tL0,−1 t
KCL10
tCm+1,m
KCm+1,m
tRCn+1,n
KRCn+1,n
tRn+2,n+1
KRn+2,n+1
Heph Heph
FIG. 1: Shematic diagram of the 1D coupled electron-phonon
system and the parameters used in the model. The big dots
in the bottom line represent atoms, while the small dots in
the upper line represent electron states. They are coupled via
the electron-phonon interaction.
−Kα0 /
mαi m
j (i 6= j). Here Kα0 is the spring constant,
and mαi is the mass of the ith atom in the α region. Like
the electrons, the coupling Hamiltonian with the leads is
HLCph =
uLi K
j , (6)
HCRph =
uCi K
j . (7)
We also have KCα = KαC
. The non-zero elements are
KLC01 , K
10 , K
n+1,n, and K
n,n+1.
The electron-phonon interaction is included within the
adiabatic Born-Oppenheimer approximation. First, the
electron subsystem is solved with all the atoms in their
equilibrium positions. Then, the isolated phonon sub-
system is considered. After that, the electron-phonon in-
teraction is turned on by allowing the atoms to oscillate
around their equilibrium positions. Within this picture,
the electron-phonon interaction is11
Heph =
i,j,k
Mkijc
icjuk. (8)
The interaction matrix element is Mkij =
. All
the operators in Eq. (8) are in the central region, so we
omitted the superscript C. In our model, the electron
operators are in the second quantization, while that of
the phonons are in the first quantization.
B. Green’s functions
The nonequilibrium Green’s function method for the
electronic transport is discussed in Refs. 9,10,11,12, and
that for the phononic transport in Refs. 13,14,15,16,17.
Here we concentrate on the electron-phonon interactions.
The definition of the electron contour-ordered Green’s
function is Gjk(τ, τ
′) = −i〈T {cj(τ)c†k(τ ′)}〉, and the
phonon counterpart is Djk(τ, τ
′) = −i〈T {uj(τ)uk(τ ′)}〉.
Here τ is time on the Keldysh contour, and T {· · ·} is
the contour-ordered operator. We set h̄ = 1 through-
out the formulas. Without the electron-phonon inter-
action, the isolated electron and phonon problem can
be solved exactly. We denote these Green’s functions
as G0(τ, τ
′) and D0(τ, τ
′), respectively. In our case,
it is convenient to write the Hamiltonians as matri-
ces and work in the energy space. The electron re-
tarded and advanced Green’s functions are Gr0(ε) =
†(ε) =
(ε+ iη)I −HCe − ΣrL(ε)− ΣrR(ε)
. I is
an identity matrix, and η → 0+. The retarded self-
energy Σrα = t
Cαgrαt
αC is due to the interactions with
the lead α. The retarded Green’s function of the semi-
infinite lead grα can be obtained analytically (Appendix
A). The “less than” Green’s function is given by G<0 =
Gr0(Σ
L + Σ
0 , where Σ
α = −f eα(Σrα − Σaα). f eα is
the Fermi-Dirac distribution. The phonon retarded and
advanced Green’s functions are23 Dr0(ω) = D
†(ω) =
(ω + iη)2I −KC −ΠrL(ω)−ΠrR(ω)
. The lead re-
tarded self-energy is Πrα(ω) = K
Cαdrα(ω)K
αC . drα also
has analytical expression (Appendix A). The phonon
“less than” Green’s function is D<0 = D
)Da0 ,
where Π<α = f
α −Πaα). fphα is the Bose distribution
function.
Knowing the bare electron and phonon Green’s func-
tions G0 and D0, we can include their interaction as per-
turbation. Following the standard procedure of nonequi-
librium Green’s function method, we can express this
interaction as self-energies. The full Green’s functions
are obtained from the Dyson equation, e.g., for elec-
trons Gr,a = G
0 + G
r,a, and G< = GrΣ<t G
Σ<t = Σ
eph + Σ
L + Σ
R is the total self-energy. Keeping
the lowest non-zero order (the second order) of the self-
energies, we have two (Hartree- and Fock-like) terms for
the electrons, and one polarization term for the phonons.
This is the so-called Born approximation (BA)11. The
Fock self-energies are
ΣF,<mn (ε) = iM
G<0 ij(ε− ω)D
0 kl(ω)
M ljn, (9)
ΣF,rmn(ε) = iM
Gr0ij(ε− ω)D
0 kl(ω)
+G<0 ij(ε− ω)D
0kl(ω)
+Gr0ij(ε− ω)D
0kl(ω)
M ljn. (10)
The “less than” Hartree self-energy is zero, and the re-
tarded one is
ΣH,rmn = −iM imnDr0ij(ω
′ = 0)M
G<0 lk(ε)
. (11)
This term is a constant for all energies, which represents
a static potential due to the presence of phonons. The
self-energies for the phonons are
Π<mn(ω) = −iMmlk
G<0 ki(ε)G
0 jl(ε− ω)M
ij , (12)
Πrmn(ω) = −iMmlk
Gr0ki(ε)G
0 jl(ε− ω)
+G<0 ki(ε)G
0jl(ε− ω)
Mnij . (13)
In Eqs. (9-13), sum over internal indices is assumed. The
self-consistent Born approximation (SCBA) is obtained
by replacing all the bare Green’s functions G0 and D0 in
Eqs. (9-13) with the full G and D11. In Appendix B,
we show that the SCBA fulfills the electrical and energy
current conservation, while BA fails.
C. The electrical and energy current
The electrical and energy current can be expressed by
the Green’s functions. The electrical current out of the
lead α is11,24
Jα = e
Tr{G>(ε)Σ<α (ε)−G<(ε)Σ>α (ε)}. (14)
The electron energy current is
JE,eα =
ε Tr{G>(ε)Σ<α (ε)−G<(ε)Σ>α (ε)}. (15)
The electron heat current is obtained from Eqs. (14-
15) as Jh,eα = J
α − µαJα/e. µα is the lead chemical
potential. The derivation of the phonon energy current
runs parallel with that of the electrons17
JE,phα = −
ω Tr{D>(ω)Π<α (ω)−D<(ω)Π>α (ω)}.
For phonons the energy current is the same as the heat
current. When there is no electron-phonon interaction,
the electron energy current is conserved throughout the
structure. So is the phonon energy current. In the pres-
ence of such an interaction, only the total energy current
is conserved due to the energy exchange between them.
The phonons do not carry charges, so in both cases the
electrical current is conserved. Since we can’t get the
exact self-energies in most cases, we need some approxi-
mations. Properly defined self-energies should fulfill the
electrical and energy current conservation
Jα = 0, (17)
(JE,eα + J
α ) = 0, (18)
where α runs over all the leads. We justify that the SCBA
fulfills these conservation laws, while the BA fails to con-
serve the energy current (Appendix B). Provided we sat-
isfy these conservation laws, we can write the electrical
and energy current in symmetric forms. The electrical
current is
J = e
T̃ e(ε) [f eL(ε)− f eR(ε)] . (19)
The transmission coefficient reads
T̃ e = Tr
Gr(ΓL +
Γeph − Se)GaΓR
+GrΓLG
a(ΓR +
Γeph + S
, (20)
where Se is
(f eR + f
L)Γeph + iΣ
f eL − f eR
. (21)
Γα = i(Σ
α − Σaα), α = L,R is the electron level-width
function. Γeph = i(Σ
eph − Σaeph) is due to the electron-
phonon interaction. The total energy current is
T̃ e(ε) [f eL(ε)− f eR(ε)]
T̃ ph(ε)
L (ε)− f
R (ε)
. (22)
The phonon transmission coefficient is
T̃ ph = Tr
Dr(ΛL +
Λeph − Sph)DaΛR
+DrΛLD
a(ΛR +
Λeph + S
, (23)
where Sph is
Sph =
R + f
L )Λeph − iΠ
L − f
. (24)
Λα = i(Π
α − Πaα) is the phonon level-width function.
Λeph = i(Π
eph−Πaeph) is due to the electron-phonon inter-
action. Eqs. (19-24) are the generalization of the Caroli
formula22 to include the electron-phonon interaction.
III. HEAT GENERATION IN CURRENT
CARRYING 1D ATOMIC JUNCTIONS
As an application of the formalism in Sec. II, we
study the heat dissipation in current-carrying 1D atomic
junctions18,25,26,27,28,29,30,31. In the presence of potential
difference between the two leads, there will be electrical
current flowing between them. When the electrons pass
the central region, there is energy exchange between the
electron and phonon systems. The energy dissipated into
the phonon system makes the atom temperature higher
than that of the leads if it is not efficiently conducted to
the leads. If the electron-phonon interaction is weak, the
energy dissipated into the phonon system is only a small
fraction of the electron energy current. But this small
fraction still influences the transport properties of the
atomic junction and even leads to junction breakup32,33,
especially when the thermal conductance is low. Dif-
ferent models have been used to study the local heating
effect. Some simply assume that the phonons are in their
thermal equilibrium states31. Some take into account the
phonon transport by using the rate equations30 or other
semi-classical models26,27,32. Few of them take into ac-
count the quantum effect in heat transport18,34. Our
model treats the electron and phonon transport on an
equal quantum-mechanical footing, and includes their in-
teractions self-consistently. The heat generation is given
by (Eq. (B9))
Q = i
G>nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
. (25)
At zero temperature, we can get an analytical expression
Eq. (C1) for a single-atom structure by using the bare
Green’s functions G0 and D0 in Eq. (25) (Appendix C).
Equation (C1) can reproduce most qualitative features of
heat generation in a single atom, except that it does not
take into account heat conduction in the phonon system.
We first study the case where the lead energy band is
wide compared to the voltage applied to the structure.
For most metallic leads, this condition should hold. In
the weak electron-phonon coupling regime, the Born ap-
proximation should give acceptable results for the heat
generation, although physically it is not a good approx-
imation. Figure 2 shows the heat generation of a single
atom (n = 1 in Fig. 1) computed using Eqs. (B5) and
(B8) under BA and SCBA, respectively. The parameters
used in the calculation are stated in the figure caption.
With these parameters, the electron energy band is in
the range −1 ≤ ε ≤ 1 eV. The chemical potential of each
lead is zero in equilibrium. The phonon energy is ap-
proximately ω = 0.05 eV. In all the results presented in
this section, the temperature is T = 4.2 K, the electron-
phonon coupling matrix M = 0.08 eV/(Å·amu 12 ). The
cut-off energy of the electron system is 2.1 eV, and the
phonon system is 0.2 eV. The energy spacing is dis-
cretized into grids of 1 meV. Equation (B5) gives the
energy decrease of the electron system, while Eq. (B8)
gives the energy increase of the phonon system. Numeri-
cal results from Eq. (B5) and Eq. (B8) under SCBA have
some slight discrepancy. This is due to numerical inac-
curacies. But most of the discrepancy under BA comes
from the difference between the bare and the full Green’s
functions, which may become even larger for some pa-
rameters. So BA should be used with care in the study
the energy exchange between the electron and phonon
system. We also note that although Eqs. (B5) and (B7)
are equivalent, numerical result from Eq. (B5) is unsta-
ble in many cases. The reason is that the energy ex-
change between the electron and the phonon system is
only a small fraction of the total electrical energy cur-
rent. Equation (B5) is the difference between two large
numbers, so our numerical integration has to be accurate
enough to get a reasonable result30. On the contrary,
Eq. (25) is much more stable since we have got the dif-
ference analytically. All the results presented below use
this equation.
From Fig. 2, we can see two threshold values in heat
generation. The first one corresponds to the onset of
phonon emission. Under low temperatures, the equilib-
rium phonon occupation is very small, so the phonon
absorption process seldom takes place. If the applied
bias is smaller than the phonon energy, electrons don’t
 0  0.1  0.2  0.3  0.4  0.5
Voltage (V)
SCBA, B8
SCBA,B5
BA, B8
BA,B5
 0  0.05  0.1
FIG. 2: Comparison of different methods to compute the heat
generation in a single atom structure. The four curves cor-
respond to results from Eqs. (B5) and (B8) under BA and
SCBA, respectively. If we label this single atom as index 1,
its electronic onsite energy is written as εC1 = 0.1 eV. The
onsite energy of the leads is εL = εR = 0 eV. The hopping
energy is tLij = t
ij = 0.5 eV. The non-zero electronic cou-
pling with the lead is tCL10 = t
21 = 0.1 eV. The matrix ele-
ment of the single atom is KC11 = 0.654 eV/(Å
· amu). The
spring constant between the lead atoms is KLij = K
ij = 0.654
eV/(Å2·amu). The non-zero atomic coupling with the leads
is KCL10 = K
21 = 0.127 eV/(Å
·amu).
have enough energy to emit one phonon. So the heat
generation is zero. Once the applied voltage is larger
than the phonon energy, phonon emission turns on. The
heat generation increases almost linearly with the applied
bias (inset of Fig. 2). This is different from the electrical
current, which increases smoothly in this regime. The
second threshold value corresponds to the alignment of
the left lead chemical potential with the electron onsite
energy eV = 2ε0 (positive bias µl > µr). The electron
transmission is nearly unity above the onsite energy. The
larger the transmission, the larger the current and the
heat generation provided that the other parameters re-
main unchanged. These two threshold behaviours may
become less obvious when the coupling with the leads
get stronger. As a result of coupling, the discrete elec-
tron and phonon density of states (DoS) extends to a
small energy region around their discrete values. The
continuous phonon DoS leads to the broadening of the
first threshold behaviour, while the continuous electron
DoS is responsible for that of the second. It is smoothed
out when the coupling is large enough (Fig. 3). Only
electrons whose energies are within the broadened energy
spectrum can tunnel across the central atom. The heat
generation reaches maximum when the electron states in
one lead are all occupied in this energy range, while those
in the other are all empty.
The electron-lead coupling not only leads to the elec-
tron level broadening, but it also influences the elec-
tron tunneling time. The larger this coupling, the less
time electrons spend in the central region. In Fig. 3, we
show the heat generation and the atom temperature for a
single-atom structure under different electronic coupling
strengths. The definition of temperature is ambiguous
in nanostructures6. Here we use the method proposed
in Ref. 18. We can only see one threshold behaviour at
about 0.2 V, which is smoothed out when the coupling is
larger than 0.2 eV. The temperature and the heat gener-
ation show similar trends. The saturate voltage of heat
generation increases with the strengthen of the electron-
lead coupling. This is due to the coupling induced atomic
level broadening. The decrease of the heat generation
and temperature with increasing electron-leads coupling
can be easily understood. The larger this coupling, the
less time electrons spend at the central atom. Since the
electron-phonon interaction takes place there, the heat
generation decreases. We also show the heat generation
as a function of electron-lead coupling in the inset of the
lower panel. The applied voltage is 0.3 V. On one side,
when the coupling is too small, few electrons can tun-
neling through the atom. The heat generation is small.
On the other, when the coupling is very large, the elec-
tron tunneling process is too quick for the phonons to
interact with the electrons. The heat generation is also
small. It has a maximum value at some moderate cou-
pling strength. This is different from the electrical cur-
rent, which increases monotonously with the increase of
coupling strength.
 0  0.1  0.2  0.3  0.4  0.5
Voltage (V)
0.2 0.4
FIG. 3: Heat generation Q and the atom temperature T under
different electron coupling strength tCL10 = t
21 = 0.05, 0.1,
0.2, and 0.3 eV, respectively. Other parameters are the same
with Fig. 2. The inset shows the heat generation as a function
of electron coupling strength at an applied bias V = 0.3 V.
The atom-lead coupling determines how well the gen-
erated heat can be conducted into the surrounding leads.
One of the important reasons why we are interested in
the heat generation in nanostructures is that it may leads
to temperature increase and even structure breakup. To
study the temperature change, we need to take into ac-
count not only the heat generation, but also the heat con-
duction into the leads. In the simplest one-atom struc-
ture, the heat conductance is mainly determined by the
atom-lead coupling. Our model includes this intrinsically.
Figure 4 shows the heat generation and the atom temper-
ature as a function of atom-lead coupling. For the heat
generation, the BA and SCBA results show large differ-
ence around the resonant position, which corresponds to
a perfect atomic junction. For the atom temperature, BA
and SCBA give almost the same results. In the case of
a perfect junction, the heat generation reaches its max-
imum value, while the atom temperature is the lowest.
The reason is that the perfect junction has the best heat
conductance. When the atom-lead coupling is weak, the
heat generation is small. But the poor heat conductance
can still result in a much higher temperature than the
surrounding leads. We also show the heat conductance
as a function of atom-lead coupling in the inset of the
upper panel, which shows a sharp peak at resonance.
 0  0.2  0.4  0.6  0.8  1
K (eV/A2 amu)
0.80.60.40.2
FIG. 4: Heat generation Q and the atom temperature T as
a function of the atom-lead coupling KCL10 = K
21 = K.
Dashed and dotted lines correspond to results of SCBA and
BA, respectively. Other parameters are the same with Fig. 2.
The inset shows the thermal conductance κ as a function of
K. The unit is 1× 10−12 W/K.
In Fig. 5, we show the heat generation as a function
of electron onsite energy at different biases. We assume
that we can tune the the onsite energy via a gate voltage.
When the applied bias is less than the phonon energy,
there will be no heat generation. When the bias energy is
slightly larger than the phonon energy and less than 2ω,
there are two energy positions where the heat generation
is the largest. These two peaks are approximately at
−0.5eV +ω and 0.5eV −ω. They merge into a single one
at a bias of eV = 2ω until it reaches saturation. After
that, this peak broadens, and becomes ladders. All these
behaviour can also be explained by the analytical result
of Eq. (C1).
In Fig. 6 we show the heat generation of a two-atom
structure (n = 2 in Fig. 1). The central region has two
identical atoms. Interaction between them leads to two
discrete energy levels. One is at 0 eV, and the other at
0.4 eV. When the electrical coupling between the leads
and the central region is small (0.1 eV), additional to the
threshold behaviour at eV = ω, there are two ladders
corresponding to the phonon assisted resonant tunneling
across the two electrical levels. If the electrical coupling
0.20.10-0.1-0.2
ε(eV)
FIG. 5: Heat generation Q as a function of electrical onsite
energy εC1 under different biases. From the inner to the outer
side, the applied biases are V = 0.05, 0.10, 0.15, 0.20, 0.25,
0.30, 0.35, 0.40, 0.45, and 0.50 V, respectively.
gets larger (0.2 eV), the two ladders broaden out. Again
this is attributed to the coupling induced level broad-
ening. The heat generation for the two-atom structure
is much larger than that of a single-atom structure. The
more the electrical levels, the larger the electrical current
and heat generation. It is worth noting that for multi-
atom structures the distribution of the electrostatic po-
tential may influence the results significantly35. In the
above calculation, we assume that the two electrical lev-
els don’t change with the applied bias, and that we can
tune their positions via a gate voltage.
 0  0.2  0.4  0.6  0.8  1
Voltage (V)
FIG. 6: Heat generation Q as a function of applied voltage for
a two-atom structure. The two-atom onsite energy is εC = 0.2
eV, the hopping energy is tC = 0.1 eV, and the spring con-
stant is KC = 0.654 eV/(Å2·amu). The two leads are iden-
tical. The electron onsite energy is εL = εR = 0, and the
hopping energy is tL = tR = 0.5 eV. Their spring constants
are the same as the central region. The non-zero coupling cou-
plings with the leads are KCL10 = K
32 = 0.327 eV/(Å
·amu),
and tCL10 = t
32 = 0.1 (solid), 0.2 eV (dashed), respectively.
If one of the metallic leads is replaced by a semicon-
ductor, there will be some new features in the electrical
current and the heat generation. In our simple model,
we can alternate the electron onsite energies between two
values to mimic a simple semiconductor (Appendix A).
In Fig. 7 we show the heat generation and the electri-
cal current for such kind of structure. The alternating
onsite energies of the left lead are −0.1 and −0.2 eV, re-
spectively. This produces an energy band-gap of 0.1 eV.
Other parameters are given in the figure caption. We can
see that there appears negative differential conductivity
in the current-voltage characteristics due to the semi-
conductor band-gap. This qualitatively agrees with the
experimental36 and first-principle37 studies. The heat
generation curve is slightly different. Additional to its
threshold behaviour, the peak and valley positions are
also different. The electrical current has a peak when
the chemical potential of the lead is aligned with the
central electrical level, while the peak of the heat gen-
eration shifts to the right by one phonon energy. This
corresponds to the phonon-assisted resonant tunneling.
The current and heat generation decrease when the sin-
gle electrical level is within the band-gap of the left lead.
The peak-to-valley ratio depends on the coupling with
the semiconductor lead. In the limit of small band-gap
and large coupling, we recover the metallic lead results.
 0  0.2  0.4  0.6  0.8  1
Voltage (V)
FIG. 7: Heat generation Q and electrical current J as a
function of applied voltage for a single-atom structure. The
left lead is a semiconductor. Its alternating onsite energies
are −0.2 and −0.1 eV, respectively. The chemical potential
is µL = 0.05 eV higher than the conduction band bottom,
which corresponds to n-type doping. Other parameters are
10 = K
21 = 0.4 eV/(Å
·amu), tCL10 = t
21 = 0.1 eV,
1 = 0.1 eV, K
11 = 0.654 eV/(Å
·amu), εR = −0.05 eV,
L = tR = 0.5 eV, and KL = KR = 0.654 eV/(Å2·amu).
IV. CONCLUSION
We studied the coupled electron and phonon transport
in 1D atomic junctions in the weak electron-phonon inter-
action regime. Base on the nonequilibrium Green’s func-
tion method, we derived the electrical, energy current of
the coupled electron-phonon system, and the energy ex-
change between them. We showed that the SCBA con-
serves the energy current. Using this formalism, we stud-
ied the heat generation in one- and two-atom structures
coupling with different leads under a broad range of pa-
rameters. Especially, we studied the influence of the ther-
mal transport properties on the heat generation and atom
temperature of the central region. The results on semi-
conductor leads agree qualitatively with the experimental
and first-principle studies. This model can be easily ex-
tended to study more realistic structures such as molec-
ular transport junctions and metallic nanowires. The
electron, phonon Hamiltonian, their interaction and lead-
coupling matrices can all be obtained from first-principle
calculations17,38,39. The surface Green’s functions for
bulk leads can be computed by recursive method17,38. It
is also possible to include the electron-electron and the
phonon-phonon interactions14,17.
Acknowledgments
We thank Baowen Li, Sai Kong Chin, Jian Wang, and
Nan Zeng for discussions. This work is supported in part
by a Faculty Research Grant of the National University
of Singapore.
APPENDIX A: SURFACE GREEN’S FUNCTIONS
OF THE 1D LEAD
In this Appendix, we show that for the 1D tight-
binding model the lead self-energies can be expressed
analytically17. The electron and phonon self-energies are
similar in their form. Here we take electrons as an exam-
ple, and give the phonon results directly. We assume that
the onsite energies of the electrons alternate between ε1
and ε2. The hopping energy is t
ij = t0. If ε1 = ε2, we get
a continuum band. This corresponds to a metallic lead.
If they are not equal, we get two bands with a band gap.
We can take the lower as the valence band (VB), and the
upper as the conduction band (CB). We use this method
to mimic a semiconductor lead. In this case, the semi-
infinite lead has two electron states in each period. In the
tight-binding model, only the left- (right-) most state of
the central region is coupled to the left (right) lead. So
we only need to know the surface Green’s function, e.g.,
for the left lead it is g0 = g
00. We assume the retarded
Green’s function is
grij =
i−j state 1,
i−j state 2.
Putting it into the definition of the retarded Green’s func-
tions [(ε+ iη)I −H ]gr = I, we have
− t0c1 + (ε+ iη − ε2)c2 − t0c1λ = 0, (A2)
− t0c2 + (ε+ iη − ε1)c1λ− t0c2λ = 0. (A3)
From Eqs. (A2-A3), we get an equation for λ
2− (ε+ iη − ε1)(ε+ iη − ε2)
λ+ 1 = 0. (A4)
The condition that Eq. (A4) has travelling wave solutions
gives the dispersion relation
(ε1+ε2)−
(ε1−ε2)2+16t
≤ ε ≤ ε1 (VB),
ε2 ≤ ε ≤
(ε1+ε2)+
(ε1−ε2)2+16t
(CB).
We assume ε1 ≤ ε2 without loss of generality. The energy
band-gap is ε2 − ε1. If they are equal, the two bands
merge into one, which corresponds to a metallic lead.
For the surface Green’s function of the left lead, we
also have
(ε+ iη − ε1)c1 − t0c2 = 1. (A6)
From Eqs. (A2,A6), we get
ε+iη−ε2
(1+λ)t2
(VB),
ε+iη−ε1
(1+λ)t2
(CB).
|λ| ≥ 1 is one of the roots of Eq. (A4). The surface
Green’s function of the right lead is identical.
We can also alternate the atom masses to generate a
phonon band-gap. In our model the mass change will
modify the renormalized spring constants. The diagonal
elements of the dynamical matrix will be two alternating
values Kαii = 2k1 or 2k2, while the off-diagonal elements
will be a single value Kαij = −
k1k2, where |i − j| = 1.
If we assume that k2 ≥ k1, the acoustic band (AB) is
0 < ω2 < 2k1, and the optical band (OB) 2k2 < ω
2(k1 + k2). The surface Green’s function is
(1+λ)k1k2
(AB),
(1+λ)k1k2
(OB),
where Ωn = (ω + iη)
2 − 2kn. |λ| ≥ 1 is one of the roots
2− Ω1Ω2
λ+ 1 = 0. (A9)
In all the simulation results of present paper, the two
spring constants are equal (k1 = k2), which correspond
a single continuum phonon band. The electron onsite
energies are also equal (ε1 = ε2) except in Fig. 7, where
we set ε1 = −0.2 eV and ε2 = −0.1 eV to mimic a
semiconductor lead.
APPENDIX B: ENERGY CURRENT
CONSERVATION
In this Appendix, we justify that the SCBA satisfies
the energy current conservation. The justification of the
electrical current conservation is given in the Refs.40,41.
What we need to prove is that
(JE,eα + J
α ) = 0. (B1)
The electron part is
JE,eα =
ε Tr{G>(ε)Σ<α (ε)−G<(ε)Σ>α (ε)}.
Using the important relation39,40
G>Σ<t −G<Σ>t
= 0, (B3)
we get
JE,eα = −
ε Tr{G>(ε)Σ<eph(ε)−G
<(ε)Σ>eph(ε)}.
The Hartree term does not contribute to the current di-
rectly. It’s just like a static potential which only modifies
the Green’s function. Putting the Fock self-energy into
Eq. (B4), we have
JE,eα
G>nm(ε)M
kl(ω)G
ij(ε− ω)M
−G<nm(ε)MkmiD>kl(ω)G
ij(ε− ω)M ljn
. (B5)
Sum over all the indices is assumed. The heat generation
Q is the energy decrease of the electron system, which
should also be the energy increase of the phonon system.
Replacing ω by −ω, using the symmetric properties of
the phonon Green’s functions17, replacing ε by ε − ω,
and finally changing dummy variables, we get
G<nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
G<nm(ε)M
kl(−ω)G
ij(ε+ ω)M
(ε− ω)
G<nm(ε− ω)MkmiD<lk(ω)G
ij(ε)M
(ε− ω)
G>nm(ε)M
kl(ω)G
ij(ε− ω)M
Putting Eq. (B6) back into Eq. (B5), we get
JE,eα
G>nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
6= 0. (B7)
For the phonon energy current we have
JE,phα
D>nm(ω)M
ki(ε)G
jl(ε− ω)M
−D<nm(ω)MmlkG>ki(ε)G
jl(ε− ω)M
. (B8)
Following the same procedure as electrons, finally we get
JE,phα
G>nm(ε)M
kl(ω)G
ij(ε− ω)M ljn
6= 0. (B9)
So we still have
JE,eα + J
= 0. (B10)
Eqs. (B7, B9) give the energy exchange between the
electron and the phonon system, which is also the heat
generation of the atomic junction. Replacing D<, G<
by D<0 , G
0 in Eq. (B7), and G
>, G< by G>0 , G
Eq. (B9), we get the results under BA. We can find that
the energy increase of the phonons does not equal to the
energy decrease of the electrons under BA.
APPENDIX C: ANALYTICAL RESULT AT ZERO
TEMPERATURE
At zero temperature, we can get an analytical expres-
sion for heat generation in a single-atom structure by
using the bare Green’s functions in Eq. (B7, B9). We
only take into account the imaginary part of the lead self-
energies and ignore their energy dependence (the wide-
band limit)24. Finally, we assume that the phonons are
in their equilibrium states. Under these approximations,
the heat generation is (assuming eV ≥ ω0)
Q ≈ 1
M2ΓLΓR
[(ε− ε0)2 + Γ2/4] [(ε− ε0 − ω0)2 + Γ2/4]
M2ΓLΓR
4π(ω20 + Γ
(eV/2− ε0)2 + Γ2/4
(eV/2− ε0 − ω0)2 + Γ2/4
(−eV/2− ε0 + ω0)2 + Γ2/4
(−eV/2− ε0)2 + Γ2/4
arctan
eV/2− ε0
+ arctan
eV/2− ε0 − ω0
−arctan
−eV/2− ε0
−arctan
−eV/2− ε0 + ω0
. (C1)
ω0 is the phonon energy, ε0 is the electron onsite energy,
V is the applied bias, and Γ = ΓL+ΓR. The heat genera-
tion is zero when eV ≤ ω0. Equation (C1) can reproduce
most the qualitative features of heat generation in a sin-
gle atom, except that it does not take into account heat
conduction in the phonon system.
∗ Electronic address: tower.lu@gmail.com
http://staff.science.nus.edu.sg/~phywjs/
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