Conductance characteristics of current-carrying d-wave weak links
The local quasiparticle density of states in the current-carrying d-wave superconducting structures was studied theoretically. The density of states can be accessed through the conductance of the scanning tunnelling microscope. Two particular situations were considered: the current state of the homo...
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irk-123456789-1175712017-05-25T03:03:31Z Conductance characteristics of current-carrying d-wave weak links Shevchenko, S.N. Свеpхпpоводимость, в том числе высокотемпеpатуpная The local quasiparticle density of states in the current-carrying d-wave superconducting structures was studied theoretically. The density of states can be accessed through the conductance of the scanning tunnelling microscope. Two particular situations were considered: the current state of the homogeneous film and the weak link between two current-carrying d-wave superconductors. 2009 Article Conductance characteristics of current-carrying d-wave weak links / S.N. Shevchenko // Физика низких температур. — 2009. — Т. 35, № 11. — С. 1096-1100. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 74.50.+r, 74.78.–w, 74.78.Bz, 85.25.Cp http://dspace.nbuv.gov.ua/handle/123456789/117571 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Shevchenko, S.N. Conductance characteristics of current-carrying d-wave weak links Физика низких температур |
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The local quasiparticle density of states in the current-carrying d-wave superconducting structures was studied theoretically. The density of states can be accessed through the conductance of the scanning tunnelling microscope. Two particular situations were considered: the current state of the homogeneous film and the weak link between two current-carrying d-wave superconductors. |
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Shevchenko, S.N. |
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Shevchenko, S.N. |
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Conductance characteristics of current-carrying d-wave weak links |
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Conductance characteristics of current-carrying d-wave weak links |
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Conductance characteristics of current-carrying d-wave weak links |
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Conductance characteristics of current-carrying d-wave weak links |
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Conductance characteristics of current-carrying d-wave weak links |
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conductance characteristics of current-carrying d-wave weak links |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Conductance characteristics of current-carrying d-wave weak links / S.N. Shevchenko // Физика низких температур. — 2009. — Т. 35, № 11. — С. 1096-1100. — Бібліогр.: 21 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT shevchenkosn conductancecharacteristicsofcurrentcarryingdwaveweaklinks |
| first_indexed |
2025-07-08T12:29:02Z |
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1837081816064851968 |
| fulltext |
Fizika Nizkikh Temperatur, 2009, v. 35, No. 11, p. 1096–1100
Conductance characteristics of current-carrying d-wave
weak links
S.N. Shevchenko
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 27, 2009
The local quasiparticle density of states in the current-carrying d-wave superconducting structures was
studied theoretically. The density of states can be accessed through the conductance of the scanning tunnel-
ling microscope. Two particular situations were considered: the current state of the homogeneous film and
the weak link between two current-carrying d-wave superconductors.
PACS: 74.50.+r Tunneling phenomena; point contacts, weak links, Josephson effects;
74.78.–w Superconducting films and low-dimensional structures;
74.78.Bz High-Tc films;
85.25.Cp Josephson devices.
Keywords: quasiparticle density, d-wave, superconducting structures.
1. Introduction
Unconventional superconductors exhibit different fea-
tures interesting both from the fundamental point of view
and for possible applications [1]. In particular, double de-
generated state can be realized in d-wave Josephson junc-
tions [2]. If the misorientation angle between the banks of
the junction � is taken � / 4, the energy minima of the sys-
tem appear at the order parameter phase difference
� � � � / 2. These degenerate states correspond to the
counter flowing currents along the junction boundary.
Such characteristics make d-wave Josephson junctions
interesting for applications, such as qubits [3]. Our prop-
osition was to make these qubits controllable with the ex-
ternally injected along the boundary transport current [4].
It was shown that the transport current and the spontane-
ous one do not add up — more complicated interference
of the condensate wave functions takes place. This is re-
lated to the phenomena, known as the paramagnetic
Meissner effect [1].
It was demonstrated both experimentally [5] and theo-
retically [6,7] that at the boundary of some high-Tc super-
conductors placed in external magnetic field the current
flows in the direction opposite to the diamagnetic Meiss-
ner supercurrent which screens the external magnetic
field. This countercurrent is carried by the surface-in-
duced quasiparticle states. These nonthermal quasipar-
ticles appear because of the sign change of the order pa-
rameter along the reflected quasiparticle trajectory. Such
a depairing mechanism is absent in the homogeneous si-
tuation. Note that in a homogeneous conventional super-
conductor at zero temperature the quasiparticles ap-
pear only when the Landau criterion is violated, at
v s Fp� �0 / . Here v s is the superfluid velocity which
parameterizes the current-carrying state, �0 stands for the
bulk order parameter, and pF is the Fermi momentum.
The appearance of the countercurrent can be understood
as the response of the weak link with negative self-induc-
tance to the externally injected transport supercurrent.
The state of the junction in the absence of the transport
supercurrent at zero temperature is unstable at � � � from
the point of view that small deviations �� � � 0change the
Josephson current from 0to its maximal value [8]. The re-
sponse of the Josephson junction to small transport super-
current at � � �produces the countercurrent [9]. It is simi-
lar to the equilibrium state with the persistent current in
1D normal metal ring with strong spin-orbit interaction:
there is degeneracy at zero temperature and � � �, and the
response of the ring is different at �� 0or B 0, where B
is the effective magnetic field which enters in the Hamil-
tonian through the Zeeman term (which breaks time-re-
versal symmetry) [10]. The degeneracy is lifted by small
effective magnetic field so that the persistent current rap-
idly changes from 0 to its maximum value. In the case of
© S.N. Shevchenko, 2009
the weak link between two superconductors in the ab-
sence of the transport supercurrent there is degeneracy
between
p y and �p y zero-energy states; both the time-
reversal symmetry breaking by the surface (interface)
order-parameter and the Doppler shift (due to the trans-
port supercurrent or magnetic field) lift the degeneracy
and result in the surface (interface) current [5].
In recent years mesoscopic superconducting structures
continue to attract attention because of the possible appli-
cation as qubits, quantum detectors etc. (e.g. [3,11]). In
particular, such structures can be controlled by the trans-
port supercurrent and the magnetic flux (through the
phase difference on Josephson contact). This was in the
focus of many recent publications, e.g. [4,12–16]. Here
we continue to study the mesoscopic current-carrying
d-wave structures. Particularly, we study the impact of
the transport supercurrent on the density of states in both
homogeneous film and in the film which contains a weak
link.
2. Model and basic equations
We consider a perfect contact between two clean sin-
glet superconductors. The external order parameter phase
difference � is assumed to drop at the contact plane at
x � 0. The homogeneous supercurrent flows in the banks
of the contact along the y-axis, parallel to the boundary.
The sample is assumed to be smaller than the London pen-
etration depth so that the externally injected transport
supercurrent can indeed be treated as homogeneous far
from the weak link. The size of the weak link is assumed
to be smaller than the coherence length. Such a system
can be quantitatively described by the Eilenberger equa-
tion [8]. Taking transport supercurrent into account leads
to the Doppler shift of the energy variable by p vF s� . The
standard procedure of matching the solutions of the bulk
Eilenberger equations at the boundary gives the Matsu-
bara Green’s function � ( )G
0 at the contact at x � 0 [4].
Then for the component G g
11 � ( , )r of �G
, which de-
fines both the current density and the density of states
(see below), we obtain in the left (L) and right (R) banks of
the junction:
g g g gL R L R L R
L R
x
, , ,
,
( ) ( ) [ ( ) ( )] exp
| |
| |
r
r
� �
� � �
�
�
�
�0
2 �
v
�
�
�
� ,
(1)
g L R
L R
,
,
( )
~
� �
�
, (2)
g
iL R x L R
L R L R
( )
~( ) ( ) sin
~ cos
0
2
�
� �
�
� � � �
� � � �
sgn v
. (3)
Here
��
T n( )2 1 are Matsubara frequencies, �L R,
stands for the order parameter in the left (right) bank, and
~
�
�i F sp v , � �L R L R, ,
~�
2 2 . (4)
The direction-dependent Doppler shift p vF s� results in
the modification of current-phase dependencies and in the
appearance of the countercurrent along the boundary.
The function g ( , )
r defines the current density, as fol-
lowing:
j v v� � �
�
�4 0
0
�
eN T gF
n
v � �Im . (5)
Here N 0 is the density of states at the Fermi level, � �� �v
denotes averaging over the directions of Fermi velocity
vF , � /v v� F Fv is the unit vector in the direction of vF .
Analytic continuation of g ( )
, i.e.,
g g i( ) ( )�
� �� � �
, (6)
gives the retarded Green’s function, which defines the
density of states:
N g( , ) ( , )� �r r� Re . (7)
Here � is the relaxation rate in the excitation spectrum of
the superconductor.
The local density of states can be probed with the
method of the tunnelling spectroscopy by measuring the
tunnelling conductance G dI dV� / of the contact between
our superconducting structure and the normal metal scan-
ning tunnelling microscope’s (STM) tip. At low tempera-
ture the dependence of the conductance on the bias volt-
age V is given by the following relation [17]:
G eV G D N eVN F F( ) ( ) ( , )� � �p p , (8)
where GN is the conductance in the normal state; D F( )p
is the angle-dependent superconductor-insulator-normal
metal barrier transmission probability. The barrier can be
modelled, e.g., as in Ref. 7 with the uniform probability
within the acceptance cone � �� �� c , where � is the polar
angle and the small value of � c describes the thick tun-
nelling barrier:
D
c
c( ) ( )� �
�
� ��
1
2
2 2 , (9)
where (... ) is the theta function.
3. Conductance characteristics of the homogeneous
current-carrying film
Before studying the current-carrying weak link we
consider the homogeneous situation. We will consider the
d-wave film as shown in the left inset in Fig. 1. The moti-
vation behind this study is twofold: first, to demonstrate
the application of the theory presented above, and second,
to describe recent experimental results [14].
Conductance characteristics of current-carrying d-wave weak links
Fizika Nizkikh Temperatur, 2009, v. 35, No. 11 1097
The system considered consists of the d-wave film, in
which the current is injected along the y-axis, and the
STM normal metal tip (another STM contact is not shown
in the scheme for simplicity; for details see [14]). Follow-
ing the experimental work [14], we consider the c-axis
along the x-axis and the misorientation angle between
a-axis and the direction of current ( y-axis) to be � / 4.
Such problem can be described with the equations pre-
sented in the previous section as following [7,15].
Consider the specular reflection at the border, when
the boundary between the current-carrying d-wave su-
perconductor and the insulator can be modelled as
the contact between two superconductors with the order
parameters given by � � �L � � � � �( ) cos ( )0 2 � and
� � �R � �� �( ) and with � � 0. Then from Eq. (3) we
have the following:
g ( )
~( )
~
�
� �
�� ��2
, (10)
where � ��
~
2 2 and � ��
~
2 2 . This expression
is valid for any relative angle � between the a-axis and the
normal to the boundary; in particular,
g ( )
~
�
�
, � � 0 ( ( ) cos )� �� � �0 2 , (11)
g ( ) ~
�
�
, �
�
�
4
( ( ) sin )� �� � �0 2 . (12)
The accurate dependence of the gap function � �0 0� ( , )v s �
can be obtained from Ref. 4 with introducing � as follow-
ing: p v p vF s F s i� � � � � (which is analogous to Eq. (6)).
The energy values in this paper are made dimension-
less with the zero-temperature gap at zero current:
� �00 0 0� �( )v s .
And now with Eqs. (12) and (6)–(8) we plot the STM
conductance for the current-carrying d-wave film in
Fig. 1. We obtain the suppression of the zero-bias conduc-
tance peak by the transport supercurrent, as was studied
in much detail in Ref. 14. Our results are in agreement
with their Fig. 1. Also the authors of Ref. 14 developed
the model based on phase fluctuations in the Blon-
der–Tinkham–Klapwijk (BTK) formalism to explain the
suppression of the zero-bias conductance peak. However,
their theoretical result, Fig. 2, describes the experimental
one only qualitatively, leaving several distinctions.
They are the following: (i) position of the minima
( / ~ .eV �00 0 5and 1for the experiment and the theory re-
spectively); (ii) height of the zero-bias peak at zero trans-
port current (~ 2.5 and 4 respectively); (iii) height of the
peak at maximal transport current (~ 1.3 and 2.5 respec-
tively); (iv) presence/absence of the minima for all
curves. Our calculations, Fig. 1, demonstrate agreement
with the experiment in all these features. The agreement
we obtained with two fitting parameters, � and � c .
To further demonstrate the impact of the two fitting pa-
rameters of our model, � and� c , in Fig. 2 we plot the nor-
malized conductance fixing one of them and changing an-
other. The figure clearly demonstrates how they change
the shape of the curves: the position of the minima, split-
ting of the zero-bias peak etc. Note that the splitting is
suppressed at small � c and high �. This absence of the
splitting was observed in the experiment [14] and studied
in several articles, e.g. [18].
4. Conductance characteristics
of the current-carrying weak link
Consider now the weak link between two d-wave cur-
rent-carrying banks. For studying the effect of both the
1098 Fizika Nizkikh Temperatur, 2009, v. 35, No. 11
S.N. Shevchenko
–4 –3 –2 –1 0 1 2 3 4
0.5
1.0
1.5
2.0
2.5
STM tipz
y
x
d
I/
d
V
eV 00/�
p vF s 00= 0, 0.4, 0.5/�
Fig. 1. Normalized (divided by GN ) conductance dI dV/ for
the homogeneous current-carrying state in the d-wave film for
different values of the transport current. The curves are plotted
with � / .�00 015� and � �c 01. � [� �00 0 0� �( )vs ]. Left and
right insets show the schemes for probing the density of states
in the current-carrying d-wave film and in the weak link (see
text for details).
–2 –1 0 1 2
0 5.
1 0.
1 5.
2 0. �c/ =
0.05
0.15
0.5
� = 0.15
d
I/
d
V
d
I/
d
V
eV 00/� eV 00/�
–2 –1 0 1 2
0 5.
1 0.
1 5.
2 0.
2 5. � =
0.05
0.1
0.3
c= 0.1�
Fig. 2. Normalized conductance dI dV/ for the homogeneous
current-carrying state in the d-wave film for different values of
� and �c at pF sv / �00 � 0.5.
transport current and the phase difference on the density
of states in the contact, we propose the scheme, presented
in the right inset in Fig. 1. The supercurrent is injected
along y-axis in the superconducting film, as it was dis-
cussed in the previous section. Besides, the weak link is
created by the impenetrable for electrons partition at x � 0.
The small break in this partition (a � !0 ) plays the role of
the weak link in the form of the pinhole model [8,15,19].
The STM tip in the scheme is positioned above the weak
link to probe the density of states in it. Two more contacts
along the x -axis provide the order parameter phase differ-
ence � along the weak link. This can be done, for exam-
ple, by connecting the contacts with the inductance, as
shown in the scheme, and applying magnetic flux "e to
this inductance. Then one obtains the phase control of the
contact with the relation: � � " "e / 0 .
The two half-plains (for x � 0and x � 0) play the role of
the two banks of the contact, which we also call left and
right superconductors. In our scheme the banks carry the
transport current along the boundary, and the Josephson
current along the contact is created due to the phase dif-
ference. The banks we consider to be d-wave supercon-
ductors with c-axis along the z-axis and with the
misorientation angles �L � 0 and � �R � / 4. Now we can
apply the equations presented in Sec. 2 to describe the
conductance characteristics of the contact between cur-
rent-carrying d-wave superconductors. This is done in
Fig. 3, where the normalized conductance is plotted for
two values of the phase difference, for � � � � / 2and with
� / .�00 01� . The two values of the phase difference,
� � � � / 2, are particularly interesting for the application
since they correspond to the double-degenerate states
[3,4]. So, the density of states is the same in the absence
of the transport supercurrent in both panels in Fig. 3
with mid-gap states (at eV � �00 ) which create the
spontaneous current along the boundary. The transport
supercurrent (v s 0) removes the degeneracy by signifi-
cantly changing the mid-gap states (Fig. 3), which explains
different dependencies of the current in the contact on the
applied transport current (i.e., on v s ), studied in [4,9].
5. Distribution of the current in the vicinity
of the contact
To illustrate the spatial distribution of the current den-
sity in the vicinity of the contact we study the case of bal-
listic point contact between d-wave superconductors (see
also [9] and [20]). The position-dependent current den-
sity j( )r is calculated with Eq. (5), where the function g
is given either by g ( )0 , Eq. (3), for transit trajectories
(which pass through the orifice) or by g L R, ( )� , Eq. (2),
for non-transit trajectories (which are reflected from the
partition); see [21] for more details about such calcula-
tions. The spatial distribution of the current is shown in
Fig. 4, where thick lines denote the impenetrable partition
between the superconducting banks.
Although the condition that the contact size is smaller
than the coherence length a � !0 is hardly realizable for
high-Tc superconductors, we consider this model as an il-
lustrative case to show: (a) how the current is distributed
in the ground state of the contact; (b), (c) how the trans-
port supercurrent modifies the current distribution in the
ground state (qualitatively, the resulting current is a sum
of what was in the absence of v s and of the transport cur-
rent); (d) how the appearance of the countercurrent re-
sults in the vortex-like current distribution.
Conductance characteristics of current-carrying d-wave weak links
Fizika Nizkikh Temperatur, 2009, v. 35, No. 11 1099
–2 –1 0 1 2
0.5 0.5
1.0 1.0
1.5
� �= /2 � �= – /2
–2 –1 0 1 2
0
0.2
0.8
d
I/
d
V
d
I/
d
V
eV 00/�eV 00/�
p vF s 00/�
Fig. 3. Normalized conductance dI dV/ at the contact between
two current-carrying d-wave superconductors for different
values of the transport supercurrent (vs) and for two values of
the phase difference � � � � / 2.
a b
c d
Fig. 4. Distribution of the current density in the vicinity of the
contact for vs � 0, � � � / 2 (a); pF sv / .�00 0 2� and � � � / 2 (b);
pF sv / .�00 0 2� and � � �� / 2 (c); pF sv / .�00 0 2� and � � �
(d).
6. Conclusion
We have studied the density of states in the cur-
rent-carrying d-wave structures. Namely, we have consid-
ered, first, the homogeneous situation and, second, the
superconducting film with the weak link. The former case
was related to recent experimental work, while the latter
is the proposition for the new one. The local density of
states was assumed to be probed with the scanning tunnel-
ling microscope. The density of states at the weak link
and the current (i.e., its components through the contact
and along the contact plane) are controlled by the values
of � and v s. The system is interesting because of possi-
ble applications: in the Josephson transistor with control-
ling parameters � and v s governed by external magnetic
flux and the transport supercurrent [11], and in solid-state
qubits, based on a contact of d-wave superconductors [3].
The author is grateful to Yu.A. Kolesnichenko and
A.N. Omelyanchouk for helpful discussions. This work
was supported in part by the Fundamental Researches
State Fund (grant number F28.2/019) and by the program
«Nanosystems, nanomaterials, and nanotechnology» of
the National Academy of Sciences of Ukraine.
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