Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing
In n—p bilayer systems an exotic phase-coherent state emerges due to Coulomb pairing of n-layer electrons with p-layer holes. Unlike Josephson junctions, the order parameter phase may be locked by matrix elements of interlayer tunneling in n—p bilayers. Here we show how the phase locking phenomen...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2004
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| Цитувати: | Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing / A.I. Bezuglyj, S.I. Shevchenko // Физика низких температур. — 2004. — Т. 30, № 3. — С. 282-287. — Бібліогр.: 21 назв. — англ. |
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irk-123456789-1194752017-06-08T03:06:06Z Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing Bezuglyj, A.I. Shevchenko, S.I. Свеpхпpоводимость, в том числе высокотемпеpатуpная In n—p bilayer systems an exotic phase-coherent state emerges due to Coulomb pairing of n-layer electrons with p-layer holes. Unlike Josephson junctions, the order parameter phase may be locked by matrix elements of interlayer tunneling in n—p bilayers. Here we show how the phase locking phenomenon specifies the response of the electron—hole condensate to interlayer voltages. In the absence of an applied magnetic field, the phase is steady-state (locked) at low interlayer voltages, V < Vc, but the phase increases monotonically with time (is unlocked) at V > Vc. The change in the system dynamics at V = Vc gives rise to a peak in the differential tunneling conductance. The peak width Vc is proportional to the absolute value of the tunneling matrix element |T₁₂|, but its height does not depend on |T12|; thus the peak is sharp for small |T₁₂|. An in-plane magnetic field reduces the peak height considerably. The present results are in qualitative agreement with the zero-bias peak behavior that has recently been observed in bilayer quantum Hall pseudoferromagnets with spontaneous interlayer phase coherence. 2004 Article Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing / A.I. Bezuglyj, S.I. Shevchenko // Физика низких температур. — 2004. — Т. 30, № 3. — С. 282-287. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 73.43.–f, 73.43.Jn http://dspace.nbuv.gov.ua/handle/123456789/119475 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| spellingShingle |
Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Bezuglyj, A.I. Shevchenko, S.I. Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing Физика низких температур |
| description |
In n—p bilayer systems an exotic phase-coherent state emerges due to Coulomb pairing of
n-layer electrons with p-layer holes. Unlike Josephson junctions, the order parameter phase may be
locked by matrix elements of interlayer tunneling in n—p bilayers. Here we show how the phase
locking phenomenon specifies the response of the electron—hole condensate to interlayer voltages.
In the absence of an applied magnetic field, the phase is steady-state (locked) at low interlayer
voltages, V < Vc, but the phase increases monotonically with time (is unlocked) at V > Vc. The
change in the system dynamics at V = Vc gives rise to a peak in the differential tunneling conductance.
The peak width Vc is proportional to the absolute value of the tunneling matrix element
|T₁₂|, but its height does not depend on |T12|; thus the peak is sharp for small |T₁₂|. An in-plane magnetic
field reduces the peak height considerably. The present results are in qualitative agreement
with the zero-bias peak behavior that has recently been observed in bilayer quantum Hall
pseudoferromagnets with spontaneous interlayer phase coherence. |
| format |
Article |
| author |
Bezuglyj, A.I. Shevchenko, S.I. |
| author_facet |
Bezuglyj, A.I. Shevchenko, S.I. |
| author_sort |
Bezuglyj, A.I. |
| title |
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing |
| title_short |
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing |
| title_full |
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing |
| title_fullStr |
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing |
| title_full_unstemmed |
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing |
| title_sort |
order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119475 |
| citation_txt |
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance of bilayers with electron-hole pairing / A.I. Bezuglyj, S.I. Shevchenko // Физика низких температур. — 2004. — Т. 30, № 3. — С. 282-287. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT bezuglyjai orderparameterphaselockingasacauseofazerobiaspeakinthedifferentialtunnelingconductanceofbilayerswithelectronholepairing AT shevchenkosi orderparameterphaselockingasacauseofazerobiaspeakinthedifferentialtunnelingconductanceofbilayerswithelectronholepairing |
| first_indexed |
2025-07-08T15:56:27Z |
| last_indexed |
2025-07-08T15:56:27Z |
| _version_ |
1837094865526063104 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3, p. 282–287
Order parameter phase locking as a cause of a zero bias
peak in the differential tunneling conductance of bilayers
with electron–hole pairing
A.I. Bezuglyj1 and S.I. Shevchenko2
1 National Science Center «Kharkov Institute of Physics and Technology»
1 Academicheskaya Str., Kharkov 61108, Ukraine
E-mail:bezugly@ic.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 June 13, 2003, revised October 2, 2003
In n—p bilayer systems an exotic phase-coherent state emerges due to Coulomb pairing of
n-layer electrons with p-layer holes. Unlike Josephson junctions, the order parameter phase may be
locked by matrix elements of interlayer tunneling in n—p bilayers. Here we show how the phase
locking phenomenon specifies the response of the electron—hole condensate to interlayer voltages.
In the absence of an applied magnetic field, the phase is steady-state (locked) at low interlayer
voltages, V Vc� , but the phase increases monotonically with time (is unlocked) at V Vc� . The
change in the system dynamics at V Vc� gives rise to a peak in the differential tunneling conduc-
tance. The peak width Vc is proportional to the absolute value of the tunneling matrix element
| |T12 , but its height does not depend on | |T12 ; thus the peak is sharp for small | |T12 . An in-plane mag-
netic field reduces the peak height considerably. The present results are in qualitative agreement
with the zero-bias peak behavior that has recently been observed in bilayer quantum Hall
pseudoferromagnets with spontaneous interlayer phase coherence.
PACS: 73.43.–f, 73.43.Jn
The idea that in bilayer n—p structures consisting
of an electronconductivity layer (n-layer) and a
hole-conductivity layer (p-layer) the Coulomb attrac-
tion of electrons and holes may lead a formation of
electron—hole pairs with spatially separated compo-
nents was put forward rather long ago [1,2]. As a re-
sult of Bose—Einstein condensation of these pairs,
there arises a peculiar superfluid (phase-coherent)
state, in which a dissipativeless motion of pairs gives
rise to equal-in-magnitude and oppositely directed
supercurrents in the n- and p-layers. At present, two
variants of the systems have been realized experimen-
tally, where an excitonic condensate with spatially
separated components is formed. In both cases, these
are closely lying GaAs/AlGaAs double quantum
wells, where either interwell excitons are excited by a
laser pulse [3,4], or two-dimensional electron layers
are formed due to doping. In the latter case, the elec-
tron layers must be placed in a strong magnetic field,
normal to the layers, such that the total filling factor
should be � � �T � � �1 2 1 [5]. Since all these systems
have one and the same exciton mechanism for the
interlayer phase coherence [6], the physical properties
of these systems in the coherent state must be qualita-
tively similar.
The present paper has mainly been stimulated by
recent impressive experiments of Spielman et al. [7,8]
who have found that if a bilayer electron system tran-
sitions into a phase-coherent state (in which the quan-
tum Hall effect is observed at �T � 1), then this tran-
sition is accompanied by a sharp rise in the differential
tunneling conductance GT at low interlayer voltages
V. As the temperature is lowered, this peak of tunnel-
ing conductance remains of a finite height and width
in contrast to the tunneling conductance peak of the
Josephson junction. In a magnetic field H parallel to
© A.I. Bezuglyj and S.I. Shevchenko, 2004
the layers the peak height decreases the more drasti-
cally the higher is the field, and at H � 0 6. T the peak
becomes practically indistinguishable.
A number of papers have been devoted to theoreti-
cal interpretation of the experimental results obtained
by Spielman et al. For example, Fogler and Wilczek
[9] have treated the tunneling conductance peak as
a consequence of the Josephson effect in a long in-
homogeneous junction. In Refs. 10,11, the interpreta-
tion of the peak has been based on the notion of a fi-
nite time of phase coherence. Joglekar and
MacDonald [12] have performed both
phenomenological and microscopic calculations of the
tunneling conductance GT value at V � 0. In Ref. 13,
G V HT ( , ) was calculated using the phenomenological
equation similar to the Landau—Lifshitz equation for
the magnetic moment. Such a diversity of theoretical
approaches in the interpretation of experiment [7,8]
gives impetus to a consistent microscopic consider-
ation of the dynamics of phase-coherent bilayer sys-
tems, this being just the subject of the present paper.
Though we consider the n—p system in the absence of
a perpendicular magnetic field, the exciton nature of
the collective state in all the above-mentioned systems
encourages us to believe that the present results pro-
vide a qualitative description of experiments by
Spielman et al. [7,8].
An important but still not completely resolved
problem for the systems with electron—hole pairing is
the problem of phase locking by interband transitions*
[14], which coincide with interlayer tunneling transi-
tions in the systems under consideration. The tunnel-
ing transitions lift the degeneracy in the phase of the
order parameter, thereby locking the phase and mak-
ing it equal to the phase of tunneling matrix elements.
The last statement is valid in the absence of a mag-
netic field parallel to the layers. Kulik and one of the
present authors [15] have shown that in the magnetic
field parallel to the layers the phase is locked only in
the fields H Hc� 1. (The critical field H Tc
/
1 12
1 2� | | ,
where T T i
12 12� | | e � is the matrix element of
interlayer tunneling.) At H Hc� 1 the phase locking is
lifted and the phase changes monotonically in the di-
rection normal to the field, this giving rise to spatial
oscillations of the tunneling current (vortex state).
The phase locking phenomenon appears to exert an es-
sential effect not only on the thermodynamic proper-
ties of n—p systems but also on their kinetics.
Relying on the microscopic approach, the present
paper deals with the response of a phase-coherent n—p
system to the interlayer voltage V. We demonstrate
that similarly to the existence of the critical field Hc1,
in the case under consideration there exists a threshold
voltage V Tc( | | )� 12 that quantitatively characterizes
the degree of phase locking in the n—p system. At low
voltages, V Vc� , the order parameter phase is locked
(steady-state), and the direct tunneling current is pro-
portional toV. The Ohmic character of a spatially uni-
form tunneling current at V Vc� means that in the
phase-coherent n—p system there is no dc Josephson
effect [16]. (The absence of the dc Josephson effect in
the two-layer electron system has been established by
Joglekar and MacDonald [12].) At voltages V Vc� ,
the phase changes monotonically with time, and this
results in tunneling current oscillations with fre-
quency � � �e V Vc
2 2 (here e is the elementary
charge, and � � 1). So, at V Vc� the n—p system re-
tains the essential feature of the ac Josephson effect in
superconductors, namely, tunneling current oscilla-
tions at a constant applied voltage. At the same time,
the dissipative character of the oscillating tunneling
current (see below), the nonuniversality of the volt-
age dependence of �, and the presence of a threshold
voltage Vc are specific to phase-coherent bilayer n—p
systems.
Further on, we show that the above-described «lib-
eration» of the order parameter phase at V Vc� re-
sults in a sharp peak of G VT ( ), the height of which is
independent of | |T12 , and its width is equal to 2Vc, i.e.,
for small | |T12 the peak will be high and sharp. Thus,
in our opinion, the nature of the tunneling conduc-
tance peak observed in the experiments of Spielman et
al. is closely connected with the phenomenon of order
parameter phase locking by tunneling transitions. The
experimentally observed suppression of G VT ( ) peak
with an increasing parallel magnetic field [8] also
lends support in favor of this interpretation, because,
as indicated above, a sufficiently strong in-plane mag-
netic field eliminates the phase locking.
We are now coming to the analysis of the dynamics
of a phase-coherent n—p system in the limit of a high
pair density, when the average distance between the
electron—hole pairs is small compared to the charac-
teristic pair size. The advantage of the high-density
limit lies in the possibility of considering the phase-co-
herent system dynamics in the gapless state, when the
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 283
* R.R. Gusejnov and L.V. Keldysh have first demonstrated that interband transitions lift the phase degeneracy of the wave
function of the electron—hole condensate in the exciton dielectric.
gap in the excitation spectrum becomes zero under the
action of strong depairing, and the order parameter
is reduced but remains nonzero* [17]. For the n—p
bilayer the order parameter is proportional to the av-
erage
1 2( , ) ( , )r rt t� , where
i i
� ( ) is the electron
creation (annihilation) operator in the layer i. An es-
sential simplification consists in the fact that the ab-
sence of the gap makes it possible to describe the dy-
namics of the phase-coherent system only in terms of
the complex order parameter ( | | ) � e i� without in-
volving the dynamics of the quasiparticle distribution
function.
The dynamical equation for the order parameter of
the n—p system was derived by the Green function
technique in our paper [18] and has the following
form:
� � �
� � �
�
�
� ��
�
�
��
�
�
�
��
�
�
�
( � )
| | ( )
ieV
A B D
ie
c
2
1 2
2
r
A A
��
� �
T12 0
��
.
(1)
The equation obtained is in a perfect agreement with
the general theory of a relaxation of an order parame-
ter near the point of a second-order phase transition
(see, for instance [19]). In accordance with this the-
ory a state of a physical system under a second-order
phase transition can be described by an order parame-
ter, that is nonzero below the transition point and
equal to zero above this point. An equilibrium value
of the order parameter can be found from the condi-
tion that the variation of the corresponding thermo-
dynamic potential is equal to zero. In the absence of
interband hybridization the thermodynamic potential
for a condensate of electron—hole pairs with spa-
tially separated components can be presented in the
form
F D i
e
c
A B� �
�
�
� ��
�
�
��
� �
�
�
�
��
�
�
�
��
r
A A( ) | | | |1 2
2
2 41
2
� dr .
(2)
Expression (2) is similar to the thermodynamic po-
tential for Cooper pairs in the Ginzburg—Landau
theory, but here the term 2eA is replaced by the term
e( )A A1 2� . Such a modification is quite natural. In-
deed, for the case of electron—hole pairs with spa-
tially separated components an electron in the layer 1
«sees» the vector potential A1, while a hole in the
layer 2 «sees» the vector potential A2. Since the signs
of the electron and hole charges are different the vec-
tor potentials A1 and A2 are subtracted from each
other in Eq. (2). In equilibrium the order parameter
( )r is found from the condition � �F/ � ( )r � 0. For
a small deviation from equilibrium, when the deriva-
tive � �F/ � ( )r is nonzero but small, the order param-
eter relaxation rate (the derivative � � / t) is also
small. In the mean field approximation these two de-
rivatives should be proportional to each other. But it
is necessary to take into account that due to the gauge
invariance of the theory the derivative � �/ t can enter
into the equation in a combination with the term
ie V V( )1 2� , where V1 and V2 are the electrostatic po-
tentials in layers 1 and 2, respectively. As a result, in
the absence of interband hybridization one arrives at
Eq. (1), where T12 0� .
In the presence of interband hybridization the
Hamiltonian of the system contains terms linear in the
order parameter and in the matrix elements T12 and
the corresponding conjugate terms (and that means
that the thermodynamic potential contains the same
terms). These terms play the role of a source of elec-
tron—hole pairs. They are analogous to the terms that
appear in the Hamiltonian of a ferromagnet in an ex-
ternal magnetic field. For the case of a magnet it re-
sults in the appearance of a term linear in the magnetic
field in the equation for the order parameter. Since for
the system considered the matrix element T12 is analo-
gous to the magnetic field, a term linear in T12 should
appear in the equation for the order parameter in the
presence of interband hybridization. We see that Eq.
(1) does in fact contain this term. The microscopic
analysis shows that although of the phenomenological
arguments presented look quite general, in reality
Eq. (1) is valid only in a rather narrow interval of im-
purity concentration, in similarity with the Gor’-
kov—Eliashberg equation for superconductors with
paramagnetic impurities [20].
In the gapless situation under consideration, the co-
efficients of the dynamic Ginzburg—Landau equa-
tion (1) have the forms A T / T Tc( ) ( ) ( )� �2 32 2 2� � ,
B m / M� 4 3� , D p /M� 0
2 2� [18]. Here � is the elec-
tron elastic scattering time (for simplicity, it is
considered equal to the hole elastic scattering time), T
is the temperature (kB � 1), Tc is the critical tempera-
ture, M m m� �1 2 is the pair mass, m m m /M� 1 2 is
the reduced mass of pair, p0 is the Fermi momentum of
electrons and holes, and � is the dimensionless con-
stant of the Coulomb interaction [2]. It should be
284 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
A.I. Bezuglyj and S.I. Shevchenko
* Similarly to magnetic impurities in superconductors, the usual (nonmagnetic) impurities and crystal lattice distortions in
the n—p system suppress the order parameter and lead to the transition of the system to the gapless state in the narrow
range of defect concentrations in the vicinity of the critical concentration.
noted that Eq. (1) is derived by expanding of the
anomalous Green function as a power series in ( ) /Tc
[18]. Since a term linear in the matrix element T12 ap-
pears in the expression for the order parameter, it is
necessary that | |T Tc12 �� for the validity of Eq. (1).
At low fields and currents, the modulus of the or-
der parameter varies only slightly in space and time.
Assuming | | to be constant equal to 0, the imaginary
part of Eq. (1) can be written as follows:
� [ ] sin .� �
�
�
��
�
�
�
!!
"
#
$$ � � � �D
d
eV eVcr r
Hn
2
0
0
�
%
(3)
Here, the gradient-invariant phase � � � �& '
� ( )2 0�d/ Az% is introduced, d is the interlayer dis-
tance, and %0 � hc/e is the magnetic flux. The unit
vector n � ( , , )0 01 is normal to the layers and is di-
rected from layer 1 (electron layer) to layer 2 (hole
layer). The threshold voltage V T ec �| |/( )12 0�� .
It is readily seen that in the uniform case Eq. (3)
for the phase � is different from the equation �� � eV
that appears in a number of papers and is treated as
the Josephson relation for phase-coherent bilayer sys-
tems. The occurrence of the term proportional to | |T12
in the dynamical equation for the phase radically
changes the solutions of this equation. Thus, in the ab-
sence of external fields the stable steady-state and uni-
form solution of Eq. (3) is � � 0, i.e., & '� , and this
means that the interlayer tunneling transitions hold
the order parameter phase locked. Below, we consider
in detail how the phase locking phenomenon influ-
ences the dynamic properties of n—p systems.
We start from the analysis of dynamics of the n—p
system in the phase-coherent state for the spatially
uniform case in the absence of magnetic field. Let the
n—p tunneling junction be incorporated into an elec-
trical circuit having resistance R and a voltage source
�. The resulting voltage V across the n—p tunnel
junction determines the difference of electrochemical
potentials of the layers and thereby dictates the carrier
density in the n and p layers. If �n is the deviation of
the electron density from equilibrium one, then the
equality eV n/N� � � *( )0 is valid, where the
renormalized density of states on the Fermi surface is
N N
e m
C*( ) ( )0 0 1
2 1
� �
�
!
!
"
#
$
$
�
�
(C is the capacitance of the bilayer system per unit
area; N m/( )0 � �).
In the approximation linear in T12 the density of
tunneling current from layer 1 to layer 2 is equal to
J Jc� �sin , where J eN T /c � 4 0 12 0( )| | � [18]. The
charge balance equation for the layer 1 can be written
as
eS n
V
R
Ic� � sin ,�
�
� � �
�
0 (4)
where S is the area of the n—p junction, and
I SJc c� . Though below we assume � � const, it
should be noted that Eq. (4) also holds for a time-de-
pendent voltage source.
Making use of the relationship between �n and V,
from Eqs. (3) and (4) one can derive the equation for
the phase �. In terms of dimensionless variables, this
equation takes on the following form, well known in
the theory of Josephson junctions:
�� ( cos )� sin� � � � � � � �
1
1
(
) * . (5)
Here the following dimensionless parameters are in-
troduced: ( � e tc� 0, ) � eV tc 0, * � � �/ c, where
�c c cV I R� � and t e N RS0
2 0� *( ) . The time is mea-
sured in the 1 0/� units, where �0 0
1 2� ( ) /e /tc� .
Despite the coincidence of Eq. (5) with the dyna-
mical equation for the phase difference across the
Josephson junction, the different meaning of the pa-
rameter * entering into these equations leads (as will
be seen from what follows) to a substantially different
behavior of n—p systems and Josephson junctions.
A detailed analysis of the dynamic states of the sys-
tem described by Eq. (5) was performed by Belykh et
al. [21]. Without going into the details of that analy-
sis, we shall mention its main results. For each value
of the parameter ) one can find the corresponding
number (1. At ( (� 1 (large resistances R), the range
of * values is split into three adjacent intervals:
0 � �* *c, * *c � � 1 , * � 1 (* ( )c( , ) is the bifurcation
value of the parameter * [21]). In the first interval,
there is only one stable solution, � � arcsin *; in the
third interval the only stable state is the limit cycle
embracing the phase cylinder. In the intermediate
(second) interval both solutions, � � arcsin * and
the limit cycle, are stable. This nonuniqueness of
the solution of Eq. (5) results in hysteresis of
current—voltage characteristic (CVC) at ( (� 1. For
( (� 1 (low resistances R) the stable solutions will be
� � arcsin * at 0 1� �* and the limit cycle at * � 1,
while the interval of * with two stable states drops
out. Correspondingly, at( (� 1 the CVCs have no hys-
teresis.
Further on, we find the CVC and the differential
tunneling conductance of the n—p system in the sim-
ple, but physically rather demonstrative, case R � 0.
In this limit, no distinction may be made between V
and� , Vc and�c, and the dynamics of the system may
be analyzed relying on Eq. (3) (without spatial deriv-
atives). Since in the case considered we have ( (� 1,
then the hysteresis of the CVC is absent.
Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 285
If the system is spatially uniform and the voltageV
does not depend on the time, Eq. (3) can be inte-
grated. One can see that for V Vc� Eq. (3) has the
time-independent solution � �0 arcsinV/Vc. In such a
case the tunnel current, also independent of time, is
equal to I I V/V V/Rc c c csin � � +0 . This current is
proportional to the applied voltage V and it is a usual
dissipative current.
The corresponding tunneling conductance is given
by
G
dI
dV
R e N ST c� � ��1 2
0
24 0( )� . (6)
Note that at V Vc� the tunneling conductance is con-
stant and is independent of the value of the tunneling
matrix element | |T12 . This independence of the tunnel-
ing conductance from | |T12 and also its proportionality
to 0
2 ( )T are in agreement with the result of Joglekar
and MacDonald [12] for GT at V � 0.
In case of V Vc� the integration yields the tunnel
current, equal to
I t I
t
tc( )
tan
( )
tan
( )
�
�
�
�
2 2
1
2
2
(7)
where
tan
( )
( )
tan ( ) ( )
�
� � � ,
, � ��
�
t
V /V V /V
e
V V t t
c c
c
/
2
1
2
2
2 2 1 2
0
�
��
. (8)
One can see that the interlayer current oscillates with
the frequency � � �e V Vc( ) /2 2 1 2 and that this cur-
rent is not sinusoidal. Due to the nonsinusoidal char-
acter of the oscillations the average value of the tun-
nel current is nonzero. The average current is a
function of the voltage V:
I I /V V V Vc c c� � �( )( ).2 2 (9)
The behavior of the system considered is similar to
the behavior of a Josephson junction between two su-
perconductors in a circuit in which the junction is con-
nected in series with a resistor and a voltage genera-
tor. But in the former case the essential difference is
that the resistor (with R V /Ic c c� ) is embedded in
the junction and it cannot be deleted from the circuit.
Thus, there is not any transverse superconductivity in
the systems considered.
Since according to Eq. (9) the tunneling current
decreases with increasing voltage, the differential tun-
neling conductance at V Vc� is negative:
G V I /V V V VT c c c
/( ) ( )[ ( ) ]� � � ��2 2 1 2 1 . (10)
The conductance G VT ( ) has its maximum (con-
stant) value at | |V Vc� and the discontinuity points at
V Vc� - . At | |V Vc� , as | |V increases, the tunneling
conductance monotonically tends to zero, remaining
negative. If we take into account the fluctuation
smoothing of the CVC, then the dependence of GT on
V will look like a smooth curve with the maximum at
V � 0 (approximately 2Vc in width) and two minima
atV Vc. - . It is just this behavior of theG VT ( ) curve
that was observed in experiment [8] in the absence of
the magnetic field parallel to the layers.
It should be noted that both at V Vc� and V Vc�
the spatially uniform tunneling current is dissipative.
The reason for the dissipation lies in the fact that the
uniform interlayer current causes the order parameter
phase to deviate from its equilibrium value, and a con-
tinuous input of energy is required to maintain this
nonequilibrium state.
Let now the bilayer n—p structure be placed in
a magnetic field H parallel to the layers and direc-
ted along the x axis. If H H / dc� � ,1 0
22( )% �
, ( )J M/enc s
/1 2 (the two-dimensional density of
pairs n p N /Ms � 4 00
2
0
2( )( )� ), then the magnetic
field between the layers has a nonuniform (vortex)
component. We shall show that the CVCs of the n—p
system in the magnetic field strongly differ from
the CVCs in zero field and are substantially different
at both low and high resistances R. In the limiting
case R � 0 (and H Hc�� 1), the solution of Eq. (3) can
be derived using perturbation theory. Putting
� � � � �0 1, where � � �0 ky t� (k dH/� 2 0� % ,
� � eV) and taking into account the correction term �1
(proportional to a small T12 value) as a perturbation,
we obtain the following expression for the average
tunneling current density:
J J
eV
Dk
c
c�
�2 2 2 2
�
�( )
. (11)
So, for R � 0 the CVC has a wide diffusion maximum
at � � Dk2.
At high R values, the charge transport from one
layer to the other over the electrical circuit is insignif-
icant, and the electron density dynamics in layer 1 is
determined by the continuity equation
e n Jc� � sin ,� � �div2j (12)
where div2j denotes the two-dimensional divergence
of the intralayer current
j
r
Hn� �
��
�
�
�
!!
"
#
$$
en
M
ds 2
0
�
%
[ ] .
286 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
A.I. Bezuglyj and S.I. Shevchenko
In the assumption that eV �� � 0
2 , the above-de-
scribed perturbation-theory procedure yields the fol-
lowing equation for �1:
��
�
( )
sin ,
*
� �
� �
�
�
� �
�
� � �1
2
1
2 0
2
2
1
2 00
D u
J
eN
c
r r
(13)
where u n /MNs0
1 20� ( ( ))*
/ . Unlike the R � 0 case,
the left-hand side of Eq. (13) has a wave character
rather than a diffusion character. Correspondingly,
the expression for the average tunneling current den-
sity
J J
k
/u k k
c
J
�
� �
1
2 2
2
2
0
2 2 2 2 2/
�0
� �0( ) ( )
(14)
has a resonance at � � u k0 , the width of which is de-
termined by the attenuation 0 � D/u0
2. This reso-
nance results from the coincidence between the
plasmon velocity u0 in the bilayer structure and the
velocity of the magnetic-field vortices. The parameter
/ J equals to ( ) /en /MJs c
1 2.
From relations (11) and (14) it follows that at
H Hc�� 1 the GT ( )0 value is proportional to a small
| |T12
2 value, i.e., the differential tunneling conduc-
tance peak (occurring at H � 0) is strongly sup-
pressed. The reason for this suppression lies in the fact
that at H Hc� 1 the phase � varies monotonically with
the coordinate, and in this case Eq. (3) has no station-
ary solution at finite voltage, i.e., no phase locking
arises.
Thus, the present work has demonstrated in the
framework of a consistent microscopic approach that
in phase-coherent bilayer n—p systems the known
phenomenon of order-parameter phase locking by tun-
neling matrix elements T12 leads to a sharp peak in dif-
ferential tunneling conductance G VT ( ) at V � 0. The
peak height is independent of | |T12 and its width is
proportional | |T12 , i.e., at weak tunneling the peak is
high and sharp. These results are in qualitative agree-
ment with the peculiarities ofG VT ( ) observed in elec-
tron bilayer systems in the regime of an integral quan-
tum Hall effect at the total filling factor �T � 1. We
stress once again that though the theory developed
here describes the n—p system without a transverse
magnetic field, the present results are in qualitative
agreement with the data from experiments on electron
bilayer systems in a strong transverse magnetic field.
This agreement does not seem to be accidental. The
reason is that most likely the strong magnetic field
does not affect the structure of the equation that de-
fines the dynamics of the order parameter, but only
changes the values of the coefficients entering into
this equation.
This work was supported by the INTAS program,
grant No 01-2344.
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Order parameter phase locking as a cause of a zero bias peak in the differential tunneling conductance
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 287
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