How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors?
The bias-voltage, V, dependences of the differential tunnel conductance G(V) = dJ/DV were calculated for the quasiparticle current J flowing in the ab plane across the break junction made of d-wave superconductors. The tunnel directionality effect was taken into account by introducing an effective t...
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nasplib_isofts_kiev_ua-123456789-1752192025-02-23T17:29:35Z How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? Gabovich, A.M. Voitenko, A.I. Свеpхпpоводимость, в том числе высокотемпеpатуpная The bias-voltage, V, dependences of the differential tunnel conductance G(V) = dJ/DV were calculated for the quasiparticle current J flowing in the ab plane across the break junction made of d-wave superconductors. The tunnel directionality effect was taken into account by introducing an effective tunneling cone described by the angle 2θ₀. It was shown that G(V) looks like predominantly d-wave or isotropic s-wave ones, depending on the magnitude of θ₀ and the rotation angles of the crystal lattices of electrodes with respect to the junction plane. In certain configurations, the G(V) dependences of nominally symmetric S-I-S junctions may turn out similar to those for non-symmetric S–I–N junctions (here, S, I, and N denote superconductors, insulators, and normal metals, respectively) and provide misleading information about the actual energy gap. At finite temperatures, sub-gap structures appear, which possess features appropriate to both d- and s-wave superconductors and are dependent on the problem parameters. The work was partially supported by the Project No. 24 of the 2015–2017 Scientific Cooperation Agreement between Poland and Ukraine. 2017 Article How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? / A.M. Gabovich, A.I. Voitenko // Физика низких температур. — 2017. — Т. 43, № 10. — С. 1471-1481. — Бібліогр.: 119 назв. — англ. 0132-6414 PACS: 74.20.Rp, 74.55.+v, 74.72.–h https://nasplib.isofts.kiev.ua/handle/123456789/175219 en Физика низких температур application/pdf 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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The bias-voltage, V, dependences of the differential tunnel conductance G(V) = dJ/DV were calculated for the quasiparticle current J flowing in the ab plane across the break junction made of d-wave superconductors. The tunnel directionality effect was taken into account by introducing an effective tunneling cone described by the angle 2θ₀. It was shown that G(V) looks like predominantly d-wave or isotropic s-wave ones, depending on the magnitude of θ₀ and the rotation angles of the crystal lattices of electrodes with respect to the junction plane. In certain configurations, the G(V) dependences of nominally symmetric S-I-S junctions may turn out similar to those for non-symmetric S–I–N junctions (here, S, I, and N denote superconductors, insulators, and normal metals, respectively) and provide misleading information about the actual energy gap. At finite temperatures, sub-gap structures appear, which possess features appropriate to both d- and s-wave superconductors and are dependent on the problem parameters. |
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Gabovich, A.M. Voitenko, A.I. |
| author_facet |
Gabovich, A.M. Voitenko, A.I. |
| author_sort |
Gabovich, A.M. |
| title |
How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? |
| title_short |
How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? |
| title_full |
How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? |
| title_fullStr |
How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? |
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How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? |
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how does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? |
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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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https://nasplib.isofts.kiev.ua/handle/123456789/175219 |
| citation_txt |
How does the break-junction quasiparticle tunnel conductance look like for d-wave superconductors? / A.M. Gabovich, A.I. Voitenko // Физика низких температур. — 2017. — Т. 43, № 10. — С. 1471-1481. — Бібліогр.: 119 назв. — англ. |
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Физика низких температур |
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AT gabovicham howdoesthebreakjunctionquasiparticletunnelconductancelooklikefordwavesuperconductors AT voitenkoai howdoesthebreakjunctionquasiparticletunnelconductancelooklikefordwavesuperconductors |
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2025-11-24T02:52:41Z |
| last_indexed |
2025-11-24T02:52:41Z |
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1849638531029270528 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10, pp. 1471–1481
How does the break-junction quasiparticle tunnel
conductance look like for d-wave superconductors?
Alexander M. Gabovich and Alexander I. Voitenko
Institute of Physics, National Academy of Sciences of Ukraine, 46 Nauki Ave., Kyiv 03680, Ukraine
E-mail: gabovich@iop.kiev.ua, voitenko@iop.kiev.ua
Received February 28, 2017, published online August 27, 2017
The bias-voltage, V, dependences of the differential tunnel conductance G(V) = dJ/DV were calculated for the
quasiparticle current J flowing in the ab plane across the break junction made of d-wave superconductors. The
tunnel directionality effect was taken into account by introducing an effective tunneling cone described by the
angle 2θ0. It was shown that G(V) looks like predominantly d-wave or isotropic s-wave ones, depending on the
magnitude of θ0 and the rotation angles of the crystal lattices of electrodes with respect to the junction plane. In
certain configurations, the G(V) dependences of nominally symmetric S–I–S junctions may turn out similar to
those for non-symmetric S–I–N junctions (here, S, I, and N denote superconductors, insulators, and normal met-
als, respectively) and provide misleading information about the actual energy gap. At finite temperatures, sub-
gap structures appear, which possess features appropriate to both d- and s-wave superconductors and are de-
pendent on the problem parameters.
PACS: 74.20.Rp Pairing symmetries (other than s-wave);
74.55.+v Tunneling phenomena: single particle tunneling and STM;
74.72.–h Cuprate superconductors.
Keywords: d-wave superconductivity, quasiparticle tunnel conductivity, tunnel directionality, break junctions,
high-Tc superconductors.
1. Introduction
Although high- cT cuprate superconductors were dis-
covered more than three decades ago [1,2], two major
problems in their physics still remain unresolved. These
are the mechanism of Cooper pairing, i.e., the nature of
mediating boson [3–8], and the superconducting order pa-
rameter symmetry [9,10]. The both issues are interlinked to
a certain extent [11,12]. One should bear in mind that, in
principle, superconductivity in high- cT oxides might be
induced by bipolaron condensation [13] or another, even
more exotic phenomenon [14]. Nevertheless, the whole
body of experimental data testifies that the old good Bar-
deen–Cooper–Schrieffer (BCS) scenario of superconduc-
tivity [15,16] is valid for those materials if the correspond-
ing strong-coupling modifications are made [8,17].
Therefore, in this work, we will restrict the interpretation
of superconductivity in oxides to the BCS scheme, what-
ever the specific pairing mechanism.
Nevertheless, even if the BCS picture is taken for
granted, the gluing agent cannot be unambiguously eluci-
dated. Indeed, phonons always exist in solids, whereas spin
fluctuations, which are revealed in direct experiments for
cuprate samples in certain oxygen doping ranges [18–26],
might be relevant to superconductivity or not, but the con-
clusion cannot be made solely on the basis of their persis-
tence. On the other hand, the order parameter symmetry
can be checked directly in phase-sensitive studies dealing
with the Josephson tunnel current [10,27–35]. The vast
majority of the scientists working in this field think that the
d -wave symmetry of the superconducting order parameter
in cuprates has been unambiguously proved by the experi-
ments indicated above. However, there are other, although
less numerous, phase-sensitive measurements, which testi-
fy that the s-wave (isotropic) contribution to the overall
order parameter can be at least very significant [9,35–42].
Hence, the problem still remains to be solved.
At the same time, quasiparticle tunnel measurements
can also probe the pairing symmetry, although indirectly. It
is so, because the form of the resulting dependences of the
junction tunnel conductance G on the bias voltage V ap-
plied across the tunnel junction ( ( )G V , the conductance-
voltage characteristic, CVC) is associated with the struc-
ture of the superconducting order parameter in the momen-
tum space. In particular, since the isotropic s-wave super-
© Alexander M. Gabovich and Alexander I. Voitenko, 2017
Alexander M. Gabovich and Alexander I. Voitenko
conductor (Si) has no gap nodes on its Fermi surface (FS),
the zero-temperature ( = 0T ) conductance equals zero
( = 0)G in the zero-voltage vicinity <V ∆ for the Si–
insulator–normal metal (Si–I–N) junction and < 'V ∆ + ∆
for the S –I–Si i′ ones [43–45], where ∆ and ′∆ are the gaps
in the corresponding s-wave electrodes. Tunnel junctions
involving the conventional weak-coupling isotropic super-
conductors, e.g., tin or aluminum, reproduce the behavior
of quasiparticle current theoretically predicted pretty well
making no allowance for the directional tunneling [46–48],
because this phenomenon does not reveal itself for iso-
tropic superconductors.
On the other hand, since the d -wave superconductor
(S) has a cosine-like order parameter in the momentum
space ( ( ) cos 2∆ θ ∝ θ; hereafter, the angle θ is reckoned
from the xk -axis direction) and thus includes nodes
( = 0∆ ) on the FS, the zero-temperature CVCs for tunnel
junctions including d -wave electrodes are supposed to be
quite different. In particular [49,50] (see Fig. 1), the de-
pendence ( )G V has a V-shape form near the bias-voltage
origin = 0V for the S–I–N junctions (panel a), and an U-like
form for the S–I–S ones (panel b). We emphasize that the
demonstrated CVC profiles [49,50], as well as the CVCs
calculated in the framework of the modified Won–Maki
approach [51], were obtained both assuming the non-
coherent quasiparticle tunneling through the junction and
neglecting the tunnel directionality. This issue will be dis-
cussed below.
The quasiparticle current in structures involving layered
cuprates may be directed perpendicularly to the layers, i.e.
along the crystal c axis. Such a situation is realized, e.g., in
mesas [52–57]. In this case, tunneling is predominantly inco-
herent [9], and it was considered by us earlier [58–61] with
the special emphasis on the possible influence of the charge
density wave (CDW) manifestations. It might also happen,
that in the specific case of Bi2Sr2CaCu2O8+δ, the incoherent
tunneling along the c axis, which is inherent to underdoped
oxide compositions, becomes coherent for overdoped ones
due to the “metallization” of the Bi–O layers [56].
There is, however, another effective method to probe
the gapped energy spectrum of superconductors, namely
the break-junction technique [62–70]. It should be noted
that break junctions are most often fabricated from single
crystals in such a manner that the measured Josephson or
quasiparticle tunnel currents should flow along the CuO2
planes [71–73]. However, the crystals can be easily
cleaved along the ab planes during the junction fabrication,
and there is a large probability that a configuration is
formed in which tunneling occurs along the c axis [74].
Break junctions with high- cT oxides can also be intention-
ally produced to ensure tunneling along the c axis with
clear-cut gap-edge features [75]. Actually, quasiparticle
current conductances of break junction samples, descend-
ing from the same crystal batch, can demonstrate a certain
variety of different tunnel directions with respect to the
crystal axes and some kind of gap-averaging in the lateral
junction plane [63]. It seems reasonable that broken poly-
crystalline samples should exhibit a mixture of c-axis and
in-plane properties of ( )G V with peculiarities located at
the same gap positions, as was shown, e.g., for
Bi2Sr2CaCu2O8+δ [75].
In this article, we calculate the tunnel conductance for
break junctions between d-wave superconductors. This for-
mulation allows us to describe not only cuprates but also
other layered superconductors [42], where such an order
parameter symmetry is realized. To reduce the number of
problem parameters, we leave complications connected
with the possible c axis tunnel current leakage beyond the
Fig. 1. (Color online) Dimensionless conductance-voltage charac-
teristics (CVCs) ( )g v for the quasiparticle current J through non-
symmetric (S–I–N, panel (a)) and symmetric (S–I–S, panel (b))
tunnel junctions with d-wave superconductors at various reduced
temperatures / cT T making no allowance for the directionality
(the Won–Maki model [49]). S, I, and N mean d-wave supercon-
ductors, insulators, and normal metals, respectively; = / ;g R dJ dV
0= / ;eV ∆v R is the normal-state junction resistance; V is the
bias voltage; 0∆ is the superconducting order parameter ampli-
tude at = 0T ; cT is the critical temperature.
3
2
1
0
0 2–2
v = /eV ∆0
T T/ =c
0
0.5
0.7
0.9
g
=
R
dJ
d V/
2
1
0
0 2–2
v = /eV ∆0
g
=
R
dJ
dV/
(a)
(b)
1472 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
How does the break-junction quasiparticle tunnel current conductance look like for d-wave superconductors?
scope of consideration and analyze the coherent break-
junction quasiparticle tunneling directed strictly along the
ab plane. At the same time, the two broken pieces (the
junction electrodes) of the original d -wave superconduct-
ing sample can be arbitrarily rotated (in this plane) with
respect to the junction face. The theoretical picture is in-
tentionally somewhat idealized to trace only the gross fea-
tures of the superconductivity itself; in particular, without
the inevitable (for high- cT oxides) CDWs [26,60,76–78],
although the additional energy gaps induced by the latter
are well known to severely distort the actual conductance
( )G V dependences of high- cT oxides [58–61].
The quasiparticle tunneling is assumed coherent (see
below), because the coherent Josephson currents are usual-
ly observed in such geometrical configurations [79–81]. In
the coherent tunneling approximation, the directionality
effects [82–86] are very important. They are treated here in
the phenomenological manner, because the details are not
crucial. The results obtained show how possible electrode
rotations, when combined with the tunnel directionality,
can drastically change the ( )G V patterns in break-junction
experiments. For instance, the behavior of ( )G V for the
intrinsically d-wave superconductors can resemble that for
the isotropic s-wave ones. We show also that sometimes
the nominally S–I–S configuration, with break junctions
being a subset, may demonstrate an apparent S–I–N behav-
ior. Finally, we attract attention to the insensitivity of the
quasiparticle tunnel current to the phase of the anisotropic
order parameter. Therefore, contrary to the case of the Jo-
sephson current [28,31,79–81,87–91], our results com-
pletely coincide with those for the so-called extended s-
wave superconducting order parameter [92–97] propor-
tional to cos 2θ in the k space.
2. Formulation of the problem
We consider the quasiparticle tunneling in the ab plane
between two pieces of a layered 2 2x
d y− - superconductor
formed as a result of break-junction fabrication [62–70]. The
phenomenological tunnel-Hamiltonian approach is used
[44,98]. As was indicated above, we assume the coherent
character of tunneling, i.e., when a quasiparticle preserves its
momentum (and spin projection) while passing through the
junction barrier. Bearing in mind the electron structure of
high- cT oxides, we confine the consideration to the strictly
two-dimensional case (see Fig. 2), i.e. we neglect possible
current deviations into c-axis-directed trajectories [74]. Then,
the expression for the tunnel current between the unbiased
(0-) and V-biased (V-) electrodes looks like
/2
/2
( ) cos ( , )J V d d Y
π ∞
−π −∞
θ θ ω ω θ ×∫ ∫
( , , ) ( , ) ( , ),K V T N N eV′× ω ω θ ω− θ (1)
where e is the elementary charge, and the other notations
are as follows.
The unprimed and primed variables correspond to the
0- and V-electrodes, respectively (Fig. 2). The integration
variable θ is the angle in the ab plane at which the
quasiparticle crosses the junction (the angle between the
quasiparticle momentum and the normal vector to the junc-
tion, n). The integration interval [ ]/2, /2−π π for θ means
that only quasiparticles in the 0-electrode with “positively
directed” projections of their momenta on the normal to the
junction give a contribution to the total current. The inverse
current through the barrier, taken into account by Eq. (1)
along with the direct-current counterpart, is provided by
quasiparticles in the V-electrode with “negatively directed”
projections of their momenta on the normal to the junction.
However, since the order parameters in both electrodes are
symmetric with respect to their rotation by an angle of π,
the integration interval over the angle ′θ within the interval
[ ]/2, 3 /2π π is equivalent to the integration over the interval
[ ]/2, /2−π π . The functions ( , )N ω θ and ( , )N eV′ ω− θ are
the partial quasiparticle densities of states in the 0- and V-
electrodes, respectively. The energy variable ω is reckoned
from the 0-Fermi level, and, in the anisotropic BCS ap-
proximation,
2 2
( , ) = Re ,
( , )
FN N
T
ω
ω θ
ω −∆ θ
(2)
where FN is the normal-state electron density of states at
the Fermi level. The function ( , )T∆ θ is the d-wave (or
extended s-wave) superconducting gap profile in the mo-
mentum space at the temperature T (the Boltzmann con-
stant = 1Bk ).
The coherent character of tunneling means that the tran-
sition takes place only between those quasiparticle states
on both sides of the barrier that are characterized by the
same angle θ. The appearance of the Fermi-distribution-
driven function
( , , ) = tanh tanh
2 2
eVK V T
T T
ω ω−
ω − (3)
Fig. 2. (Color online) Schematic diagram illustrating the tunnel
break-junction configuration and its main parameters: γ and ′γ
are the rotation angles of the 0- and V-electrodes with respect to
the normal n to the junction plane, and 02θ is the directionality
cone. See further explanations in the text.
2θ0
n
S I S
0-electrode V-electrode
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10 1473
Alexander M. Gabovich and Alexander I. Voitenko
is associated with the temperature-dependent evolution of
the quasiparticle states in both electrodes.
The quantity ( , )Y ω θ is the barrier penetration factor
[83,99,100] for quasiparticles with the energy ω that move
in the direction comprising the angle θ with the junction
normal. Note that, strictly speaking, for quasiparticles
moving in the backward direction, the barrier transparency
factor is ( , )Y eVω− θ rather than ( , )Y ω θ . Anyway, the
actual tunnel barriers in break-junction experiments are
rather high, so that the main contribution to the current is
made by quasiparticles located in the vicinity of the Fermi
level. Moreover, relevant voltages are of the order
( , )/V T e≤ ∆ θ . Hence, we may put ( , )Y eVω− θ to be ener-
gy-independent, so that
( , ) ( , ) ( ),Y eV Y AYω− θ ≈ ω θ → θ (4)
where the coefficient A is identical for all relevant .ω As a
result, the factor A , as well as the electron densities of
states FN and FN ′ , can be factorized out of the integrals to
form, together with other pre-integral multipliers, the junc-
tion resistance in the normal state, R . The standard factor
cosθ makes allowance only for the normal projection of
the quasiparticle motion with respect to the tunnel junction
plane while calculating the total tunnel current through the
junction [100–102]. This factor, together with the function
( )Y θ [formula (4)], makes the contribution of a tunneling
quasiparticle into the total current dependent on the angle
at which the quasiparticle transverses the junction. This is
the so-called tunnel directionality [55,82–86,103].
The importance of the tunnel directionality, which was
recognized long ago, can strongly manifest itself when
interpreting the quasiparticle currents across the break-
junctions. Indeed, the voltage dependences of tunnel con-
ductance ( )G V , i.e. the differential quasiparticle current-
voltage characteristics, for symmetric junctions are rou-
tinely used to determine the superconducting energy gap in
the junction electrodes. Namely, the energy distance be-
tween the gap edges (the peak-to-peak separation) is as-
sumed equal to 4∆ [73,104–106]. However, the phenome-
non of tunnel directionality for a break junction fabricated
from a d-wave superconductor may mislead the observer.
Namely, for certain electrode configurations, it can effec-
tively “cut off” the contribution of those Fermi surface
sections that fall outside the “tunneling cone” 0( >θ θ , see
Fig. 2), so that the corresponding gap values will not affect
the actual CVCs. The consequences are considered below
in more detail. To make them more illustrative, in this pa-
per, we analyzed the following parabolic barrier-
penetration function:
2 2
0 0if | | ,
( )
0 overwise.
Y
θ − θ θ ≤ θ′θ
(5)
The selected phenomenological model totally excludes
the participation of the Fermi surface sections falling out-
side the tunneling cone 0θ in the CVC formation. It was
done intentionally in order to make the directionality ef-
fect more dramatic and bearing in mind that, unfortunate-
ly, one cannot select the most adequate ( )Y θ dependence
on the basis of experiment. Nevertheless, the very exist-
ence and the character of the tunnel directionality (see,
e.g., Refs. 55, 82–86) are satisfactorily described by the
simple phenomenological function (5).
Furthermore, it is clear that, in the framework of the
model concerned, the effectiveness of certain Fermi sur-
face sections in the CVC formation strongly depends on
the orientation of the break-junction electrode crystal lat-
tices relative to the junction plane. This orientation will be
described by the angle γ for the 0-electrode and ′γ for the
V-one; the both being reckoned from the junction normal
to the corresponding superconducting lobe bisectrix. Since
we consider symmetric junctions between two identical d-
wave BCS superconductors, the common superconducting
order parameter amplitude 0∆ can be chosen as the energy
scale and used to present the results in the normalized di-
mensionless form. As a result, the angular characteristics
0θ , γ , and ′γ (together with the temperature T ) are the only
parameters of the problem.
Taking all the aforesaid into account, formula (1) can
be rewritten in the form
/2
/2
1( ) = cos ( )
4
( , , ) ( , ) ( , ),
J V d Y
R
d K V T P P eV
π
−π
∞
−∞
θ θ θ ×
π
′× ω ω ω θ ω− θ
∫
∫
(6)
where
( )
2 2
( , )
( , ) = ,
( , )
T
P
T
ω Θ ω −∆ θ
ω θ
ω −∆ θ
(7)
and ( )Θ ω is the Heaviside unit step function. If the barrier-
transparency function is normalized to unity,
/2
/2
( ) = 1,d Y
π
−π
θ θ∫ (8)
the pre-integral coefficient ensures the correct (Ohmic)
current asymptotics at large V ’s:
( ) = .lim
V
VJ V
R→±∞
(9)
One should pay attention that the quantity
1 ( , , ) ( , ) ( , )d K V T P P eV
R
∞
−∞
′ω ω ω θ ω− θ∫ (10)
1474 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
How does the break-junction quasiparticle tunnel current conductance look like for d-wave superconductors?
in Eq. (6) is proportional to the tunnel current through a
symmetric junction between two s-wave superconductors
[45]. Hence, formula (6) can be interpreted as the coherent
averaging of quasiparticle current in the momentum space
(over the angle θ) with the weight cos ( )Yθ θ .
It is known [46–48] that the voltage dependence of the
tunnel conductance = /G dJ dV , i.e. ( )G V , is much more
informative than the voltage dependence of the tunnel cur-
rent itself, i.e. ( )J V . By making trivial transformations, the
dependence ( )G V can be reduced to the dimensionless
form ( )g v , with =g RG and 0= /eV ∆v . So, the depend-
ences ( )g v were calculated by simulating the experimental
routine. In particular, a formula that can be reduced to the
expression
( ) ( )( ) =
2
J V V J V Vg R
V
+ δ − − δ
δ
v , (11)
in which the current J was calculated numerically, since
analytical integration was impossible even in the case
= 0T . (For a more detail discussion of this issue, see, e.g.,
Ref. 107.) The increment δv of the dimensionless bias
voltage for the numerical differentiation of the dependence
( )J V was selected to equal 0= / = 0.001e Vδ δ ∆v .
3. Results of calculation and discussion
The general arrangement of the rotated d-wave (or ex-
tended s-wave) superconducting electrodes on the both
sides of the break junction is shown in Fig. 2. In this work
dealing with the combined effect of tunnel directionality
and the anisotropy of the superconducting gap in d-wave
(extended s-wave) superconductors on the quasiparticle
current, we confined the analysis to the cases when either
or both electrodes are oriented at an angle of 0° or 45° with
respect to the junction normal n. The both orientations are
shown in Fig. 2. The figure also demonstrates the “tunnel
cones” 02θ , which “give” a contribution to the tunnel cur-
rent. From the figure, one can easily get an idea how the
rotation of electrodes engages the FS sections that effec-
tively participate in quasiparticle tunneling.
The conductances ( )G V for the symmetric S–I–S junc-
tion obtained by cracking a single piece of d-wave super-
conductor without any rotations in the ab plane ( = = 0 )′γ γ
are shown in Fig. 3 for explicitly indicated tunneling cone
parameters 0θ and various reduced temperatures / cT T .
Here, 0
e
=
2
E
cT γ
∆
π
is the critical temperature for the
Fig. 3. (Color online) CVCs for break junctions with various 0θ ’s and at various / cT T ’s. The electrode orientations are = = 0′γ γ .
–3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3
eV/∆ eV/∆
0.7 0.9
3
2
1
0
R
dJ
dV/
eV/∆eV/∆
3
2
1
0
R
dJ
dV/
0.5
θ0 =
T T/ = 0c
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10 1475
Alexander M. Gabovich and Alexander I. Voitenko
2 2x
d y− -wave (extended s-wave) superconductor,
0.5771Eγ ≈ is the Euler constant, and e ≈ 2.718 is the
base of natural logarithms. One sees that at / = 0cT T , the
CVC behavior near = 0V varies from the s-wave-like
[47] (for small 0θ ’s) to the d-wave-like (for large 0θ ’s)
one [49]. However, the overall behavior and especially
the form of ( )G V near the gap edges conspicuously differ
from the curves for S–I–S junctions [49] depicted in
Fig. 1 even for thin interlayers (at large 0θ ’s). It is no
wonder, because the directionality effects were not taken
into account in Ref. 49.
Specifically, the authors of Ref. 49 calculated ( )G V
both for S–I–S and S–I–N junctions by means of separate
angular (over θ and ′θ ) averaging of the gapped electron
densities of states (2) for each electrode and subsequent
integration over the energy variable ω (in our notations).
Therefore, only one angular integration was performed
while calculating ( )G V for S–I–N junctions and two integra-
tions in the case of the S–I–S ones. In essence, this proce-
dure is the realization of the incoherent tunneling regime.
Moreover, the “directionality factors” cosθ and ( )Y θ were
omitted from consideration, which allowed the cited authors
to obtain analytical expressions. (One should pay attention
that the interpretation of the angle θ, at which a quasiparticle
crosses the barrier, faces difficulties in the case of incoherent
tunneling.) On the contrary, we take the directionality into
account, so that the both “directionality factors” enter the
consideration. Hence, we integrate over the single angle θ,
which is the feature appropriate to the coherent tunneling.
That is why our CVCs for S–I–S junctions are similar to
those for S–I–N junctions found in Ref. 49.
All the aforesaid means that the uncontrollable direc-
tionality determines the apparent form of ( )G V and makes
CVCs neither truly d-wave nor s-wave ones. As a conse-
quence, the tunnel break junction CVC measurements can-
not serve as an unambiguous evidence of the supercon-
ducting order parameter symmetry, contrary to what is
frequently argued. Thus, discrepancies between the results
obtained even for break junctions produced from cuprate
samples taken from the same batch can be interpreted as
caused by different directionality cones 0θ ’s, which can be
associated, e.g., with different interelectrode distances.
The same phenomena might have already been observed in
break-junction experiments with other layered materials
demonstrating a controversy about the order parameter
symmetry, e.g., iron-based superconductors [108,109].
For finite temperatures, the gap amplitude is reduced in
a usual way and decreases the distance between the gap-
related maxima in ( )G V . A new feature, which appears at
= 0V and finite T ’s, is proportional to lnV and rapidly
grows with T . This zero-bias peak is well-known for iso-
tropic s-wave superconductors and reflects the rise of con-
ductance resulting from the thermal filling of quasiparticle
states primarily just below and above the energy gaps in
both electrodes. The effect is especially strong when the
upper and lower gap edges almost coincide ( 0V → ) [44].
Such a behavior persists for the configuration concerned,
although the order parameters are not constant but propor-
tional to cos 2θ for the anisotropic superconductivity. The
low-V peculiarity should be observed in d-wave break
junctions, mimicking the isotropic pairing picture. Note
that this peculiarity is much more pronounced than that
predicted in the Won–Maki model [49,50] (see Fig. 1(b)).
It is well-known that, in principle, the electrode surfac-
es in break junctions are not atomically smooth. Instead,
they are usually very rough. As a result, the junction inter-
layer can be oriented (this orientation is determined by the
orientation of the vector n normal to both electrodes, see
Fig. 2) at an angle 0−γ ≠ ° with respect to the xk axis. This
configuration can be interpreted as both electrodes simul-
taneously rotated by the angle γ with respect to the junc-
tion plane. For simplicity, we consider the limiting config-
uration: a symmetric junction with = = 45′γ γ °. The
corresponding ( )G V is shown in Fig. 4. In such a junction,
whatever the directionality cone 0θ , the conductance ( )G V
has the V-form near the bias-voltage origin, thus being
apparently a d -wave like from this point of view. At the
same time, for large interelectrode distances (small 0θ ’s),
the peak-to-peak separation is much smaller than that esti-
mated from the independently measured superconducting-gap
amplitude 0∆ . Furthermore, the peaks are smeared and low.
Only for extremely large 0θ ’s, the patterns looking like the
“textbook” d -wave CVCs (see Fig. 1) are reproduced with an
accuracy of the angular integration peculiarities (see the dis-
cussion above). At finite temperatures ( 0T ≠ ), the peak-to-
peak separation is reduced and the logarithmic peak appears
in the vicinity of = 0V , being quite similar to its counterpart
in the configuration = = 0′γ γ ° (Fig. 3).
One can see, that the CVCs obtained in both considered
cases ( = = 0′γ γ ° and = = 45′γ γ °) are rather different. In
addition, owing to the break-junction gap roughness, the
interelectrode gap in the “active” junction section can be
oriented at an arbitrary angle −γ within the interval 0–45°
(those angles are not considered here). Furthermore, if we
adopt that quasiparticle tunneling can take place at several
points over the interelectrode gap with different values of
the normal vector n, we must be ready to approximate ex-
perimental CVCs by a linear sum of theoretical CVCs cor-
responding to different n orientations and taken with un-
known weights (different R values). Therefore, one can
hardly find such a set of break-junction measurements that
can be directly compared to CVCs for configurations
= = 0′γ γ ° and = = 45′γ γ °. However, experiments, very
similar to the break-junction ones, were made to study tun-
neling between thin crystals of Bi2Sr2CaCu2O8+δ in a
crossing configuration [110]. Those measurements showed
that the peak-to-peak separation was approximately twice
as much for = = 0′γ γ ° than for = = 45′γ γ °. The observa-
tions agree with our (purely d-wave!) calculations with
1476 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
How does the break-junction quasiparticle tunnel current conductance look like for d-wave superconductors?
quite a reasonable 0 30θ ≈ °. Nevertheless, the forms of the
CVCs differ substantially from the theoretical curves.
Thus, it is impossible to consider those experiments as a
reliable confirmation of the d -wave order parameter sym-
metry in Bi2Sr2CaCu2O8+δ, although the inferred energy
gap is undoubtedly anisotropic.
Finally, let us consider a possible symmetric S–I–S con-
figuration with = 45γ ° and = 0′γ °. It is of special interest,
since in this case the apparent s- and d-like features of
both pieces interfere in the resulting CVCs. The results of
calculation are demonstrated in Fig. 5. One can readily see
that, at / = 0cT T , the conductance curves ( )G V are more or
less of the s-type for all directionality cones. At small
cones, 0 10θ ≤ °, one more interesting property is observed.
Namely, the gap-related peaks are located at 0 /V e≈ ±∆
rather than at 02 /V e≈ ± ∆ , as it should be for S–I–S con-
figurations. Hence, the CVCs look like those inherent to
Si–I–N junctions made of s-wave superconductors. For larg-
er 0θ ’s up to 0 = 90θ °, the ( )G V dependences are distorted,
so that there are deviations of gap edges towards larger V ,
although the locations 02 /e± ∆ , which are characteristic of
S–I–S junctions, are never achieved. Such apparent Si–I–N
conductance-voltage characteristics were observed, e.g., in
nominally S–I–S junctions with YBa2Cu3O7–δ [111] and
La1.85Sr0.15CuO4 [112] electrodes.
Thermal effects in the ( = 45 , = 0 )′γ ° γ ° configuration
are very peculiar. First of all, the CVCs at 0T ≠ include
conspicuous negative blowouts in the gap region. Negative
portions of ( )G V are not forbidden by any general consid-
eration [107], still being somewhat exotic. They were ob-
served, e.g., in Bi2Sr2CaCu2O8+δ break junctions [113].
Their origin was attributed either to CDWs [107] or reso-
nance spin excitations [113], the both assumptions seeming
quite plausible. In our theory, any complications of this
type are absent, so that the effect of negative ( )G V is
caused by the difficulty of tunneling between thermally
filled states above and below the gap edges, when the
quasiparticles tunnel from the node of one electrode into
the antinode FS region of the other one. One should expect
that any disorder effects will substantially smear the nega-
tive spikes.
Heating also leads to the appearance of sub-gap struc-
tures in the ( = 45 , = 0 )′γ ° γ ° junctions (Fig. 5). They,
however, differ significantly from their counterparts in the
= = 0′γ γ ° and = = 45′γ γ ° junctions (Figs. 3 and 4, re-
spectively). Specifically, for large 0θ ’s, a broad maximum
develops near = 0V and replaces the sharp logarithmic
singularity intrinsic to = ′γ γ configurations (Figs. 3 and 4).
This phenomenon is associated with the superposition of con-
tributions from different segments of the FSs in the left (0-)
Fig. 4. (Color online) The same as in Fig. 3, but for = = 45′γ γ .
–3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3
eV/∆ eV/∆
0.7 0.9
3
2
1
0
R
dJ
dV/
eV/∆eV/∆
3
2
1
0
R
dJ
dV/
0.5
θ0 =
T T/ = 0c
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10 1477
Alexander M. Gabovich and Alexander I. Voitenko
and right (V-) electrodes. The picture changes drastically
for smaller 0θ ’s. Then, the logarithmic singularity is re-
stored, but it corresponds now to the difference between
the finite temperature-dependent gap value at = 0′γ ° to
the right and a negligibly small gap at = 45γ ° to the left
of the junction. Such a property is well known for iso-
tropic s-wave superconductors [114]. The pair of maxima
at finite V values, which appear when 0T ≠ , become more
pronounced for decreasing 0θ , but they are never as sharp
as the logarithmic singularities near = 0V emerging in the
= = 0′γ γ ° and = = 45′γ γ ° junctions.
For other rotations of crystal lattices with respect to the
break-junction plane, the CVC patterns become more
complex and involve features appropriate to their symmet-
ric counterparts. The sub-gap features that arise due to dif-
ferent relative electrode orientations deserve to be sought
in the measurement results obtained for both cuprates and
other layered superconductors, which are suspected to pos-
sess the d-wave order parameter symmetry (for instance,
the attribution of peculiarities in intrinsic mesa junctions
made of Bi2Sr2CaCu2O8+δ [56] may be revised). The actu-
al ( )G V are of course even more intricate, because addi-
tional pseudogap (i.e. CDW) contributions appear in many
of such materials [26,55,60,73,76–78,107,115–118].
To trace the evolution of CVCs with the interelectrode
distance and rotation angles in detail, it would be useful to
complement break-junction measurements by other exper-
iments with angle-resolved electron tunneling between
layered d-wave superconductors. In particular, the crossing
tunneling [110] may be used, or ring ramp-edge high- cT
oxide samples [10,32,119] should be harnessed with the
counter-electrode also made of the same high- cT oxide.
Then, one should be able to controllably change the junc-
tion configurations with small steps in γ and ′γ .
4. Conclusions
Our study of the tunneling in the ab plane in break junc-
tions composed of layered d-wave superconductors
demonstrated a strong dependence of the corresponding
CVCs on the directionality cone 02θ width, i.e. on the in-
terlayer distance, and the rotation angles γ and ′γ of the
electrode crystal lattices with respect to the junction
plane. It was shown that, although the intrinsic Cooper
pairing symmetry in the electrode material is the d-wave
one, the conductances ( )G V may exhibit predominantly
d-wave, s-wave or mixed forms, depending on 0θ , which
in its turn may be changed by adjusting the interelectrode
Fig. 5. (Color online) The same as in Fig. 3, but for = 45γ and = 0′γ . All plots with 0 > 30θ are very similar to the corresponding
plots with 0 = 30θ and therefore are not included.
–2 –1 0 1 2 –2 –1 0 1 2
eV/∆ eV/∆
0.7 0.9
R
dJ
dV/
eV/∆eV/∆
3
2
1
0
R
dJ
dV/
0.5
θ0 =
T T/ = 0c
–1
4
3
2
1
0
–1
4
1478 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 10
How does the break-junction quasiparticle tunnel current conductance look like for d-wave superconductors?
distance. Moreover, the CVCs for essentially S–I–S junc-
tions may acquire forms similar to those appropriate to the
S–I–N junctions if the angular configuration involves 45°
and = 0′γ °. For finite temperatures, the sub-gap features
demonstrate peculiar forms reflecting the spread of the
electrode energy gap values and the absence of matching
between the left and right gap amplitudes. Our calculations
can be applied not only to break junctions, but also to other
experimental setups, for instance, to those used to study the
Josephson currents flowing in the ab plane of specially
prepared cuprate structures [10,32,110,119]. The results
obtained above were discussed with an emphasis on high-
cT oxides. However, the adopted basic model is common
to any d-wave layered superconductors and layered super-
conductors with the hypothetical extended s-wave sym-
metry of their order parameter.
Acknowledgments
The work was partially supported by the Project No 24
of the 2015–2017 Scientific Cooperation Agreement be-
tween Poland and Ukraine.
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1. Introduction
2. Formulation of the problem
3. Results of calculation and discussion
4. Conclusions
Acknowledgments
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