Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems
A possibility of nondissipative transmission of electrical current from the source to the load using superfluid electron-hole pairs in bilayers is studied. The problem is considered with reference to quantum Hall bilayers with the total filling factor νT = 1. At nonzero interlayer tunneling the cu...
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Fil, D.V. Shevchenko, S.I. 2017-06-13T10:34:17Z 2017-06-13T10:34:17Z 2007 Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems / D.V. Fil, S.I. Shevchenko // Физика низких температур. — 2007. — Т. 33, № 9. — С. 1023–1026. — Бібліогр.: 17 назв. — англ. 0132-6414 PACS: 73.43.Jn, 74.90.+n https://nasplib.isofts.kiev.ua/handle/123456789/120927 A possibility of nondissipative transmission of electrical current from the source to the load using superfluid electron-hole pairs in bilayers is studied. The problem is considered with reference to quantum Hall bilayers with the total filling factor νT = 1. At nonzero interlayer tunneling the current pattern looks as a sum of uniform planar counterflow currents and Josephson vortices. The difference of electrochemical potentials of the layers (that is required to support the current in the load circuit) causes the motion of the Josephson vortices. In such a situation the second superfluid viscosity comes into play and results in dissipation of energy. It is found that the power of losses is proportional to the square of the matrix element of the interlayer tunneling and depends nonlinearly on the load resistance. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems Article published earlier |
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Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems |
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Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems Fil, D.V. Shevchenko, S.I. International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| title_short |
Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems |
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Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems |
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Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems |
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Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems |
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interlayer tunneling and the problem of superfluidity in bilayer quantum hall systems |
| author |
Fil, D.V. Shevchenko, S.I. |
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Fil, D.V. Shevchenko, S.I. |
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International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
| topic_facet |
International Conference "Statistical Physics 2006. Condensed Matter: Theory and Application" |
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2007 |
| language |
English |
| container_title |
Физика низких температур |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Article |
| description |
A possibility of nondissipative transmission of electrical current from the source to the load using
superfluid electron-hole pairs in bilayers is studied. The problem is considered with reference to quantum
Hall bilayers with the total filling factor νT = 1. At nonzero interlayer tunneling the current pattern looks as a
sum of uniform planar counterflow currents and Josephson vortices. The difference of electrochemical potentials
of the layers (that is required to support the current in the load circuit) causes the motion of the
Josephson vortices. In such a situation the second superfluid viscosity comes into play and results in dissipation
of energy. It is found that the power of losses is proportional to the square of the matrix element of the
interlayer tunneling and depends nonlinearly on the load resistance.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120927 |
| citation_txt |
Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems / D.V. Fil, S.I. Shevchenko // Физика низких температур. — 2007. — Т. 33, № 9. — С. 1023–1026. — Бібліогр.: 17 назв. — англ. |
| work_keys_str_mv |
AT fildv interlayertunnelingandtheproblemofsuperfluidityinbilayerquantumhallsystems AT shevchenkosi interlayertunnelingandtheproblemofsuperfluidityinbilayerquantumhallsystems |
| first_indexed |
2025-11-24T20:20:44Z |
| last_indexed |
2025-11-24T20:20:44Z |
| _version_ |
1850495194283114496 |
| fulltext |
Fizika Nizkikh Temperatur, 2007, v. 33, No. 9, p. 1023–1026
Interlayer tunneling and the problem of superfluidity
in bilayer quantum Hall systems
D.V. Fil1 and S.I. Shevchenko 2
1
Institute for Single Crystal National Academy of Sciences of Ukraine
60 Lenin Ave., Kharkov 61001, Ukraine
E-mail: fil@isc.kharkov.ua
2
B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkov 61103, Ukraine
E-mail: shevchenko@ilt.kharkov.ua
Received January 18, 2007
A possibility of nondissipative transmission of electrical current from the source to the load using
superfluid electron-hole pairs in bilayers is studied. The problem is considered with reference to quantum
Hall bilayers with the total filling factor �T �1. At nonzero interlayer tunneling the current pattern looks as a
sum of uniform planar counterflow currents and Josephson vortices. The difference of electrochemical po-
tentials of the layers (that is required to support the current in the load circuit) causes the motion of the
Josephson vortices. In such a situation the second superfluid viscosity comes into play and results in dissipa-
tion of energy. It is found that the power of losses is proportional to the square of the matrix element of the
interlayer tunneling and depends nonlinearly on the load resistance.
PACS: 73.43.Jn Tunneling;
74.90.+n Other topics in superconductivity.
Keywords: electron-hole superfluidity, Josephson vortices.
Thirty years ago in the papers [1,2] the idea of unusual
mechanism of superconductivity that can be realized in
electron-hole bilayers was proposed. The idea is based on
the fact that in such systems an electron and a hole may
form a bound state that can be considered as a composite
boson. Superfluid motion of such pairs is equivalent to
two nondissipative electrical currents (of opposite direc-
tion) in the layers. Using the contacts that allows one
to access the layers separately one can realize the counter-
flow setup and use the bilayer for the transmission of the
current from the source to the load. One can expect that
such a bilayer will work as a double-wire superconduct-
ing transmission line.
The specifics of the electron-hole superfluidity is that
the layers should be situated rather close to each other and
the interlayer tunneling cannot be neglected. In the pres-
ence of tunneling the bilayer system cannot support the
uniform planar supercurrent, and the current state be-
comes a soliton-like [3–5]. This state is similar to the
Josephson vortex state in long contacts between two su-
perconductors. The aim of this study is to clarify whether
the soliton current state in bilayers remains nondissi-
pative. Our consideration is based on the statement that in
the counterflow setup the soliton-like current state be-
comes nonstationary one. The reason is the following.
Nonzero current in the load circuit can appear only if the
electrochemical potentials of the layers differ from each
other, and in the presence of the electrochemical potential
difference the solitons (Josephson vortices) begin to
move. We find that such motion results in dissipation of
energy.
In this paper the electron-hole superfluidity is studied
with reference to a bilayer quantum Hall system. As was
pointed out in [6,7] in bilayer systems with the same type
of conductivity of the layer subjected by a quantizing
magnetic field one can realize the electron-hole super-
fluidity. The condition is that the sum of filling factors of
the layers is equal to unity. The idea [6,7] greatly in-
creases the interest to the experimental and theoretical
study of the electron-hole superfluidity. The results of re-
cent experimental investigations of quantum Hall bilay-
ers [8–14] support the theoretical prediction on Bose-Ein-
© D.V. Fil and S.I. Shevchenko, 2007
s te in condensa t ion (BEC) and super f lu id i ty of
electron-hole pairs. In particular, in the counterflow ex-
periments a huge increase of longitudinal conductivity
was observed [13,14]. Other important observations are a
strong low bias tunnel conductivity [8,12], a Godstone
collective mode [9], and a quantized Hall drag between
the layers [10,11]. Nevertheless, genuine nondissipative
counterflow state was not achieved in the experiments.
The BEC of electron-hole pairs in quantum Hall bi-
layers can be interpreted as the development of the spon-
taneous phase coherence between electrons belonging to
adjacent layers. The coherent phase � is the phase of the
order parameter that describes the electron-hole pairing.
The superflow of the pairs is associated with nonzero gra-
dient of the phase. In the continuous approximation the
energy of the system can be written as:
E E d r
d
dx
t
s� � �
�
�
�
� �
�
�
�
�
�
�
��0
2
2
2
1
2 2
1�
�
�
�
~
(cos )
�
, (1)
where we assume that the system is uniform in y direction
(direction perpendicular to the current) and neglect the
variations of the modulus of the order parameter. In (1) E0
is the energy of the ground state,
~
t t� 2 0� , the tunneling
energy (t is the matrix element of the interlayer tunnel-
ing),
�
��
�
s
e d d d
�
�
�
�
�
�
�
�
�
�
�
�
� 0
2
2 2
2
2
24 2 2 2
1
� � � �
exp erfc
�
�
�
�
�
�
�
�
�
�
�
�
d
�
,
(2)
the superfluid stiffness, � 0 1� �� �( ), the modulus of
the order parameter (the filling factors of the layers are
� �1 � and � �2 1� � ), d is the interlayer distance, � is the
magnetic length. Eq. (1) is obtained from the equation
E H� � �� �| | , where H is the Hamiltonian of the
bilayer system in the lowest Landau level approximation,
| ( )|�� � � �� � �u v e h vacX
X
X X X2 1
is the BCS-like many particle wave function, X is the
guiding center of the orbit, eiX
� (hiX
� ), the creation opera-
tors for the electron (hole) in the ith layer, the vacuum
state | vac� is defined as the state with empty layer 2 and
completely filled (�1 1� ) layer 1.
Varying the energy (1) with respect to the phase and
equating the result to zero one obtains the stationary con-
tinuity equation
� � �
e d
dx
e ts� �
�
�
� �
2
2 22
0
~
sin ,
�
(3)
where the first term is the divergence of the planar super-
current density
j j
e d
dx
s s s
1 2
( ) ( )� � � �
� �
�
, (4)
and the second term is the density of interlayer super-
current, flowing from the layer 1 to the layer 2.
I
e tT
1 2 22
� �( )
~
sin .
� �
�
�
(5)
Equation (3) coincides in form with the equation of mo-
tion of nonlinear pendulum. At given input current (cur-
rent from the source) the solution of (3) should satisfy the
boundary condition j j
s
1
0
( )
( ) � in , and it should corre-
spond the conditional minimum of energy (1). Note that
in difference with the pendulum problem, energy (1) does
not coincide with the integral of motion of (3). Using the
known solutions for the pendulum problem we find that
there is a critical input current j e /c s� 2 � �� (where
� ��� � 2 s /t
~
is the Josephson length) that separates two
regimes.
1 . A t j j cin � t h e p l a n a r c u r r e n t j
s
1
( ) �
� �j x x /csech(( ) )0 � ( x j /jc0
1� �� cosh ( )in ) decreases
exponentially with distance from the source edge, and for
long systems Lx �� � the current at the load edge will be
exponentially small. Therefore, at small input current the
bilayer system cannot work as the transmission line.
2. At j j cin � the planar current j j /
s
c1
( )
( )� �� dn
�( / , )x � � � (� � j /jc
2 2
in , dn( , )x k is the Jacobi elliptic
function) contains the constant part and spatially oscillat-
ing part. The total current is a sum of the uniform current
and the current of Josephson vortices. The planar current
reaches the load edge and the bilayer system works as the
transmission line.
In the counterflow setup the current withdrawn from
one layer should be redirected into the other layer through
the load circuit. To support the current in the load circuit
the difference of electrochemical potential of the layer is
required. At such difference the phase of the order para-
meter changes in time and the current state becomes non-
stationary.
We describe the nonstationary state by the set of two
equations for the phase and for the local difference of the
electrochemical potentials V . The first equation is the
nonstationary continuity equation
e
n
t
j
x
I
s
�
� ��
~ ( )
1
1 2 0, (6)
where ~n is the excess local density of the electron-hole
pairs that corresponds to the excess local charge densities
in the layers. It is connected with V by the capacitor equa-
tion en CV~ � with
C
d d
d d
� � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
4
1
2
1
2 2
2
2
�
� �
exp erfc
�
!
"
#
$#
%
&
#
'#
�1
,
1024 Fizika Nizkikh Temperatur, 2007, v. 33, No. 9
D.V. Fil and S.I. Shevchenko
the effective capacity of the bilayer system per unit area.
In Eq. (6) and below we neglect the normal component of
the gas of electron-hole pairs.
The second equation is the generalized Josephson
equation in which dissipative terms are taken into ac-
count. Without derivation this equation was given in [15].
One can justify this equation basing on general argu-
ments. In the superfluid hydrodynamics [16] the equation
for the superfluid velocity (in linear approximation) with
dissipative terms has the following form:
( )
� �* � * * +
v
v
s
m s s
t
n, - 3 , (7)
where,m is the chemical potential per unit mass , ,m /m�
(m is the mass of the Bose particle), ns is the superfluid
density, - 3 is the second viscosity for the superfluid com-
ponent.
To apply Eq. (7) for a description of the bilayer elec-
tron-hole superfluidity we should replace the 3D diver-
gence of the density of the flow with the 2D divergence
of the planar supercurrent plus the term accounted for
the leakage caused by the the tunnel supercurrent:
* . � � �( ) ( )( )
( )
,n /e j / x Is s
s
Tv 1
1 1 2 . The chemical po-
tential should be replaced with the difference of electro-
chemical potentials ,. �eV . Using Eqs. (4), (5) and the
definition of the superfluid velocity v s /m� *� � (m is the
mass of the electron-hole pairs) we obtain from (7) the
following equation
�
� �
�
�
�
�
�
�
�
/ � �
�
��
t
eV
x
ts2 2
2
2
�
~
sin , (8)
where / � is dimensional less parameter proportional to
the second viscosity. It is just the equation given in [15].
Equation for the phase that coincides in form with (8) was
also derived in [17] for the electron-hole bilayers in zero
magnetic field.
Note that nonzero second viscosity does not yield au-
tomatically the dissipation. In particular, one can see from
Eq. (8) that the viscosity term is equal to zero if the sta-
tionary continuity equation (3) is fulfilled. We will show
that under motion of Josephson vortices the second vis-
cosity results in dissipation.
We consider the case of small dissipation when the
power of losses is much smaller than the input power. We
restrict our consideration to the case of large input current
(much larger than j c). At large input current the approxi-
mate solution of Eqs. (6), (8) can be presented in the form
� 0 0 0� �( , ) sin ( ) cos ( )x t t kx A t kx B t kx� � � � � � ,
(9)
V x t V A t kx B t kxV V( , ) [ sin ( ) cos ( )]� � � � �0 1 0 0 ,
(10)
where V0 is the average difference of the electrochemical
potentials of the layers, 0 � eV /0 �, and k is wave number
connected with the average planar supercurrent by the re-
lation j e k/s0 � � �. The presence of oscillating terms in
(9), (10) is caused by the interlayer tunneling: all the coef-
ficients A� , B� , AV , BV are proportional to
~
t . The explicit
form of these coefficient can be obtained by substituting
of (9), (10) into (6), (8) and taking into account the term
linear in
~
t . The solution (9), (10) corresponds the vortex
structure that moves with the velocity v /kv � �0
� e R L /s l y
2 2� � (V0 and j jout 1 0 satisfy the Ohm’s law
V j L Ry l0 � out , where Rl is the load resistance).
The power of losses for the transmission line is deter-
mined as the difference between the input and the output
power P j j L Vl y� �( )in out 0. The difference j jin out�
emerges if the average interlayer current IT becomes non-
zero: j j I LT xin out� � . Thus, the power of losses per unit
area is proportional to the average interlayer current:
p P L L I Ll l x y T x� �/ (for the case of small dissipation).
One can see from Eq. (4) that the average interlayer cur-
rent is proportional to B� : I et B /T � ~
� �4 2
� �. The explicit
expression for B� reads as
B
t
k
V
V V /Vs C C
� �
�� �
/
2
2 /
�
� �
~
( )
,
2 12 2
0
2
2 2 2
0
2 2
�
(11)
where 2 �2
0
2 2�CV / ks and V e/ CC � 2 2�� . As follows
from (11) B� 3 0 and pl 3 0 due to nonzero second vis-
cosity / -� 4 3. We emphasize that our analysis is not
valid for 2 that corresponds R R / eL Cl c y s� � � ( )� . At
such Rl the dissipation is large. At Rl not very close to Rc
the power of losses is given by the equation
p
t
R
R
R
R
l
l
c
l
c
�
�
�
��
�
�
�
�
��
�
�
�
�
�
�
/
�
�
~ 2
2
4
2
24
1
� �
. (12)
Thus, the power of losses is proportional to the square of
the tunneling amplitude and depends nonlinearly on the
load resistance.
In conclusion, we have shown that in bilayer systems
with interlayer phase coherence the dissipation may ap-
pear under the transmission of the current from the source
to the load in the counterflow regime. The dissipation is
caused by joint action of two factors. The first factor is
nonzero interlayer tunneling. The tunneling results in a
formation of the vortex current state. The second factor is
the motion of vortices. They begin to move due to the dif-
ference of electrochemical potentials between the layers.
The dissipation emerges due to second viscosity that co-
mes into play only for moving vortices. Since one of the
factors is the motion of vortices, one can hope that the
dissipation can be eliminated by vortex pinning.
Interlayer tunneling and the problem of superfluidity in bilayer quantum Hall systems
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D.V. Fil and S.I. Shevchenko
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