Energy pumping in a quantum nanoelectromechanical system
Fully quantized mechanical motion of a single-level quantum coupled to two voltage biased electronic leads is studied. It is found that there are two different regimes depending on the applied voltage. If the bias voltage is below a certain threshold (which depends on the energy of the vibrationa...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2005
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
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irk-123456789-1216632017-06-16T03:02:28Z Energy pumping in a quantum nanoelectromechanical system Nord, T. Gorelik, L.Y. Низкоразмерные и неупорядоченные системы Fully quantized mechanical motion of a single-level quantum coupled to two voltage biased electronic leads is studied. It is found that there are two different regimes depending on the applied voltage. If the bias voltage is below a certain threshold (which depends on the energy of the vibrational quanta) the mechanical subsystem is characterized by a low level of excitation. Above a threshold the energy accumulated in the mechanical degree of freedom dramatically increases. The distribution function for the energy level population and the current through the system in this regime is calculated. 2005 Article Energy pumping in a quantum nanoelectromechanical system / T. Nord, L.Y. Gorelik // Физика низких температур. — 2005. — Т. 31, № 6. — С. 703-707. — Бібліогр.: 20 назв. — англ. 0132-6414 PACS: 73.23.–b, 73.63.–b, 85.85.+j http://dspace.nbuv.gov.ua/handle/123456789/121663 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
| spellingShingle |
Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Nord, T. Gorelik, L.Y. Energy pumping in a quantum nanoelectromechanical system Физика низких температур |
| description |
Fully quantized mechanical motion of a single-level quantum coupled to two voltage biased
electronic leads is studied. It is found that there are two different regimes depending on the applied
voltage. If the bias voltage is below a certain threshold (which depends on the energy of the
vibrational quanta) the mechanical subsystem is characterized by a low level of excitation. Above
a threshold the energy accumulated in the mechanical degree of freedom dramatically increases.
The distribution function for the energy level population and the current through the system in
this regime is calculated. |
| format |
Article |
| author |
Nord, T. Gorelik, L.Y. |
| author_facet |
Nord, T. Gorelik, L.Y. |
| author_sort |
Nord, T. |
| title |
Energy pumping in a quantum nanoelectromechanical system |
| title_short |
Energy pumping in a quantum nanoelectromechanical system |
| title_full |
Energy pumping in a quantum nanoelectromechanical system |
| title_fullStr |
Energy pumping in a quantum nanoelectromechanical system |
| title_full_unstemmed |
Energy pumping in a quantum nanoelectromechanical system |
| title_sort |
energy pumping in a quantum nanoelectromechanical system |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2005 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/121663 |
| citation_txt |
Energy pumping in a quantum nanoelectromechanical
system / T. Nord, L.Y. Gorelik // Физика низких температур. — 2005. — Т. 31, № 6. — С. 703-707. — Бібліогр.: 20 назв. — англ. |
| series |
Физика низких температур |
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AT nordt energypumpinginaquantumnanoelectromechanicalsystem AT gorelikly energypumpinginaquantumnanoelectromechanicalsystem |
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2025-07-08T20:18:14Z |
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2025-07-08T20:18:14Z |
| _version_ |
1837111333536923648 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, No. 6, p. 703–707
Energy pumping in a quantum nanoelectromechanical
system
T. Nord and L.Y. Gorelik
Department of Applied Physics, Chalmers University of Technology and Göteborg University
SE-412 96 Göteborg, Sweden
E-mail: nord@fy.chalmers.se
Received September 14, 2004
Fully quantized mechanical motion of a single-level quantum coupled to two voltage biased
electronic leads is studied. It is found that there are two different regimes depending on the ap-
plied voltage. If the bias voltage is below a certain threshold (which depends on the energy of the
vibrational quanta) the mechanical subsystem is characterized by a low level of excitation. Above
a threshold the energy accumulated in the mechanical degree of freedom dramatically increases.
The distribution function for the energy level population and the current through the system in
this regime is calculated.
PACS: 73.23.–b, 73.63.–b, 85.85.+j
During the past few years experimental methods of
physics has seen an advancing capability to manufac-
ture smaller and smaller structures and devices. This
has lead to many new interesting investigations of
nanoscale physics. Examples include, for instance, ob-
servation of the Kondo effect in single-atom junctions
[1], manufacturing of single-molecular transistors [2],
and so on. There has also been a great interest in the
promising field of molecular electronics [3]. One of
the main features of the conducting nanoscale compos-
ite systems is its susceptibility to significant mechani-
cal deformations. This results from the fact that on the
nanoscale level the mechanical forces controlling the
structure of the system are of the same order of magni-
tude as the capacitive electrostatic forces governed by
charge distributions. This circumstance is of the ut-
most importance in the so called electromechanical
single-electron transistor (EM–SET), which has been
in focus of recent research. The EM–SET is basically a
double junction system where the additional (mechan-
ical) degree of freedom, describing the relative posi-
tion of the central island, significantly influences the
electronic transport. Experimental work in relation to
EM–SET structures range from the macroscopic [5] to
the micrometer scale [6–8] and down to the
nanometer scale [9]. Various aspects of electronic
transport in such systems have been theoretically in-
vestigated in a series of articles [11–20].
In Ref. 4 and Ref. 15 it was, among other things,
shown that coupling the mechanical degree of freedom
of an EM–SET to the nonequilibrium bath of elec-
trons constituted by the biased leads, can lead to dy-
namical self excitations of the mechanical subsystem
and as a result bring the EM–SET to the shuttle re-
gime of charge transfer. This phenomena is usually re-
ferred to as a shuttle instability. In these papers the
grain dynamics are treated classically and the key is-
sue is that the charge of the grain, q t( ), is correlated
with its velocity, �( )x t , in a way so that the time aver-
age, q t x( ) � � 0.
Decreasing the size of the central island in the
EM–SET structure to the nanoscale level results in
the quantization of its mechanical motion. Charge
transfer in this regime was studied theoretically in
Ref. 10. However, the strong additional dissipation in
the mechanical subsystem suggested in this paper
keeps the mechanical subsystem in the vicinity of its
ground state and prevents the developing of the me-
chanical instability. The aim of our paper is to investi-
gate the behavior of the EM–SET system in the quan-
tum regime when its interaction with the external
thermodynamic environment generating additional
dissipation processes can be partly ignored in such a
manner that the mechanical instability becomes possi-
ble. We will show that in this case at relatively low
bias voltages, intrinsic dissipation processes bring the
© T. Nord and L.Y. Gorelik, 2005
mechanical subsystem to the vicinity of the ground
state. But if the bias voltage exceeds some threshold
value, the energy of the mechanical subsystem, ini-
tially located in the vicinity of the ground state, starts
to increase exponentially. We have found that intrin-
sic processes alone saturate this energy growth at some
level of excitation. The distribution function for the
energy level population and the current through the
system in this regime is calculated.
We will consider a model EM–SET structure con-
sisting of a one level quantum situated between two
leads (see Fig. 1). To describe such a system we use
the Hamiltonian
H E a a E D
X
x
c c
m
Pk
k
k k� � � � �� ,
,
, , (
�
) �
�
�
� �
† †
0
0
21
2
� � �
1
2 0
2 2m X T X a c c ak k k� � � �
� ( �)[ ], , ,
† † . (1)
The first term describes electronic states with energies
Ek,� and where ak,�
† (ak,�) are creation (annihila-
tion) operators for these noninteracting electrons
with momentum k in the left (� � L) or right (� � R)
lead. The second term describes the interaction of the
electronic level on the with the electric field so that
c† (c) is the creation (annihilation) operator for the
level electrons and E0 is the energy level. The scalar
D represents the strength of the Coulomb force acting
on a charged grain, �X is the position operator, and
x /m0 0� � � is the harmonic oscillator length scale
for an oscillator with mass m and angular frequency
�0. The third and fourth terms describe the center of
mass movement of the in a harmonic oscillator poten-
tial so that �P is the center of mass momentum opera-
tor. The last term is the tunneling interaction be-
tween the lead states and the level and T Xk, ( �)� is the
tunneling coupling strength. We will consider the
case when the tunneling coupling depends exponen-
tially on the position operator �X, i.e. T Xk R, ( �) �
� T X/R exp { � � and T X T X/k L L, ( �) exp { � }� � � , whe-
re TR and TL are constants and � is the tunneling
length.
To introduce a connection to the quantified vibra-
tional states of the oscillator we perform a unitary
transformation of the Hamiltonian (1) so thatUHU † �
� ~H, where U iPd c c/� exp ( � )0
†
� . In this paper we
consider the situation when ~H has the most symmetric
form:
~
,
,
, ,
~
H E a a E c c b bk
k
k k� � � �
�
�
�
� �� �
�
� � �† † †
0 0
1
2
�
� � �� � �T a c x b x b
k
k R0 [ exp ( ),
† †
� � � �� �a c x b x bk L, exp ( )]† † h.c. (2)
Here b† (b) is a bosonic creation (annihilation) opera-
tor for the vibronic degree of freedom, and the
dimensionless parameters x / x / d /x� � �1 2 0 0 0( )�
(where d D/ x m0 0 0
2� ( )� ) characterize the strength of
the electromechanical coupling. Furthermore, T0 is
the renormalized tunneling coupling constant, and
~ ( )E E Dd / x m d /0 0 0 0 0
2
0
22 2� � � � is the shifted dot
level. For simplicity, but without loss of generality,
we choose ~~E0 equal to the chemical potential of the
leads at zero bias voltage.
First let us study the situation when the mechani-
cal subsystem is characterized by a low level of excita-
tion. We will consider the case of small electrome-
chanical coupling. This means that the dimensionless
parameters x� �� 1 and that only elastic electronic
704 Fizika Nizkikh Temperatur, 2005, v. 31, No. 6
T. Nord and L.Y. Gorelik
eV
eV
2
2
h�0
h�0
a
b
Fig. 1. Model system consisting of a one level quantum
dot placed between two leads. The level of the dot equals
the chemical potential of the leads and a bias voltage of V
is applied between the leads. The center of mass move-
ment of the dot is in a harmonic oscillator potential with
the vibrational quanta ��0. The applied bias voltage is
such that eV/2 0� �� (a). Same as (a) but with the ap-
plied bias voltage larger than 2 0�� /e (b).
transitions and transitions accompanied by emission or
absorption of a single vibronic quantum (sin-
gle-vibronic processes) are important. If the applied
voltage is smaller than 2 0�� /e and the temperature is
equal to zero, the six allowed transitions of this type
are the ones described in Fig. 1,a. Here we see that
only elastic tunneling processes and tunneling pro-
cesses in which the vibronic degree of freedom absorbs
one vibronic quantum are allowed and as a result the
rate equation for the distribution function of the en-
ergy level population P n t( , ) has the form:
��
� �� � � � � � �1 2 2 1 1tP n t P n t x x n P n t( , ) ( , ) ( )( ) ( , )
� � �� �( ( )) ( , )1 2 2n x x P n t ,
where � � 2 0
2� �T /� and � is the density of states in
the leads.
It is straightforward to solve these equations and
find that the solution exponentially fast approaches
the stable solution P( )0 1� and P n( ) � 0 for all n � 0.
As a result, the dimensionless average extra energy ex-
cited in the vibronic subsystem,
E t nP n t
n
( ) ( , )�
�
�
�
0
, (3)
goes to 0.
If the applied bias voltage is increased above the
threshold value V /ec � 2 0�� we instead get the al-
lowed transitions described in Fig. 1,b, i.e. two ab-
sorption processes has changed into emission processes
where the energy quantum ��0 is transferred to the
vibronic degree of freedom. These transitions lead to
the following equation for P n( ):
��
� �� � � � � � �1 2 21 1tP n t x n x P n t( , ) ( ( ) ) ( , )
� � � � � �� �x n P n t P n t x nP n t2 21 1 1( ) ( , ) ( , ) ( , ). (4)
One can find from this equation that the time evo-
lution of the exited energy is given by the formula:
E t
x
x x
x x t( ) [exp ( ( ) ) ]�
�
� ��
� �
� �
2
2 2
2 2 1� , (5)
i.e. energy is continuously pumped into the mechani-
cal subsystem, which is strong evidence that the low
exited regime is unstable if the bias voltage exceeds
the critical value Vc. Furthermore, it is necessary to
remark here that for this case we thus have a linear
increase in the energy as a function of time even when
x�
2 approaches x�
2.
As the excitation of the vibronic subsystem in-
creases multi-vibronic processes become important.
They give rise to an additional dissipation which satu-
rates the energy growth induced by the single-vibronic
processes. As a result the system comes to a stationary
regime which is characterized by a significant level of
excitation of the vibronic subsystem. To demonstrate
this we will now expand our analysis by taking into
account electronic transitions accompanied by the
emission or absorption of two vibronic quanta
(two-vibronic processes). To describe such transitions
one has to take into account second order terms in b†
and b in the tunneling part of the Hamiltonian (2). As
illustrated in Fig. 2 these terms will generate four pro-
cesses in which two vibrational quanta are absorbed
by the electron during the tunneling event. There is
also a renormalization of the elastic channel coming
from the inclusion of these terms. Now the equation
for the distribution function of the energy level popu-
lation has the form:
�� � � � � � � � �1 21 1 1tP n t nP n t n n( , ) ( , ) [ ( ) ]� � � �
� � � � �P n t n P n t( , ) ( ) ( , )� 1 1
� � � � ��( )( ) ( , )n n P n t1 2 2 0, (6)
where we have introduced the constants
� � �� � �( ( ))x x / x4 4 24 and � � � �x /x2 2 .
To find the stationary solution of this equation we
introduce the generating function:
P ( ) ( )z z P nn
n
�
�
�
�
0
,
where z is a complex number inside the unit circle.
Rewriting Eq. 6 we find the equation for P ( )z
Energy pumping in a quantum nanoelectromechanical system
Fizika Nizkikh Temperatur, 2005, v. 31, No. 6 705
eV
2
h�0
Fig. 2. Illustration of the second order case where elastic
tunneling and inelastic tunneling exchanging two or less
vibrational quanta are included. The level of the dot is
equal to the chemical potential and the bias voltage is set
so that 2 40 0� �� �/e V /e� � .
� �( ) ( ) ( ) ( ) ( )z z z z zz z� � � � � � �1 02P P P .
The solution to this equation is
P ( ) exp
( )
z dz
z
z
z
� � �
� � �
��
�
�
��
�
�
��
��
1
1
� �
�
� � ��
� � ��
�� �
�
�
��
�
�
�� �� �
�
dz dz
z
z
C
z
z
z z
0 1
1
1
exp
( ) (
� �
� � �
�
�
�
��
!
�
"�
1)
,
where C1 and z0 are constants. Since the probabilities
P n( ) are positive and normalized, the sum
� � � � ��
�
n
n P n z0 1 1( ) ( ) ( )P converges absolutely.
This is true only for z0 1� � . The second constant C1
can be determined from the normalization condition
P ( )z � �1 1 to be C / dx x /1
1
1
2 1� � �
�
�� �# exp (( ) )
� � �( )x 1 1# , where we have introduced the constant
# � � �� � �( )1 / . Therefore the final expression for
P ( )z is
P ( )
exp
( )
exp (z
C
z
z
dz
z
z
z
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�1
1
1
1
1
�
�
�#
�� �1 1)# .
(7)
We can now calculate the average energy excited in
the harmonic oscillator, which is just � z zP ( ) calcu-
lated at z � 1,
E C� � �
1
2
2 1�
�#( ). (8)
One can show that C1( )� decays exponentially as
exp ( )�const/� when � $ 0 so for small � we get
E / O� � �( ) ( )1 2 1� � .
To see how the energy pumped into the harmonic
oscillator affects the charge transport we calculate the
current I through the system in units of e�. For volt-
ages belowVc the current is only mediated by the elas-
tic channel and is thus I /� 1 2.
For voltages in the range 2 40 0� �� �/e V /e� �
the current can be calculated to leading order in � as
I x x x x z� � �
�
�
�
� � �� � � � ��1
4
14 4 2 2( ) ( )P
� � � �� � �( ) ( )x x z2 1 1P . (9)
In Fig. 3 we have chosen a set of numerical values
and plotted (solid line) the calculated current as a
function of the bias voltage. For comparison we have
also plotted (dashed line) the current as given in the
high dissipation limit where the harmonic oscillator
goes to the ground state between tunneling events. It
is clear that the current in the regime characterized by
a high level of excitation is much greater than the one
in the regime of low level of excitation.
In conclusion, we have studied fully quantized me-
chanical motion of a single-level quantum coupled to
two voltage biased electronic leads. We have shown
that above a certain threshold voltage the energy ac-
cumulated in the mechanical subsystem increases dra-
matically. We have also shown that second order in-
elastic tunneling events are enough to stabilize this
pumping of energy. Finally the current through the
system was calculated and it was found that the devel-
opment of the mechanical instability is accompanied
by a dramatic increase in the current.
The authors would like to thank Robert Shekhter,
Jari Kinaret, and Anatoli Kadigrobov for valuable dis-
cussions related to this manuscript.
This work was supported in part by the European
Commission throuh project FP-003673 CANEL of the
IST Priority. The views expressed in this publication
are those of the authors and do not necessarily reflect
the official European Commission’s view on the sub-
ject. Financial support from Swedish SSF and Swed-
ish VR is also gratefully acknowledged.
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Fizika Nizkikh Temperatur, 2005, v. 31, No. 6 707
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