Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article)
The state of the art concerning tunnel measurements of energy gaps in cuprate oxides has been analyzed. A detailed review of the relevant literature is made, and original results calculated for the quasiparticle tunnel current J(V) between a metallic tip and a disordered d-wave superconductor partia...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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| Cite this: | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) / Alexander M. Gabovich Alexander I. Voitenko // Физика низких температур. — 2016. — Т. 42, № 10. — С. 1103-1114. — Бібліогр.: 164 назв. — англ. |
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| citation_txt | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) / Alexander M. Gabovich Alexander I. Voitenko // Физика низких температур. — 2016. — Т. 42, № 10. — С. 1103-1114. — Бібліогр.: 164 назв. — англ. |
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| description | The state of the art concerning tunnel measurements of energy gaps in cuprate oxides has been analyzed. A detailed review of the relevant literature is made, and original results calculated for the quasiparticle tunnel current J(V) between a metallic tip and a disordered d-wave superconductor partially gapped by charge density waves (CDWs) are reported, because it is this model of high-temperature superconductors that becomes popular owing to recent experiments in which CDWs were observed directly. The current was calculated suggesting the scatter of both the superconducting and CDW order parameters due to the samples' intrinsic inhomogeneity. It was shown that peculiarities in the current-voltage characteristics inherent to the case of homogeneous superconducting material are severely smeared, and the CDW-related features transform into experimentally observed peak-dip-hump structures. Theoretical results were used to fit data measured for YBa₂Cu₃O₇–δ and Bi₂Sr₂CaCu₂O₈₊δ. The fitting demonstrated a good qualitative agreement between the experiment and model calculations. The analysis of the energy gaps in high- Tc superconductors is important both per se and as a tool to uncover the nature of superconductivity in cuprates not elucidated so far despite of much theoretical effort and experimental progress.
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| first_indexed | 2025-12-07T17:53:15Z |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10, pp. 1103–1114
Spatial distribution of superconducting
and charge-density-wave order parameters in cuprates
and its influence on the quasiparticle tunnel current
(Review Article)
Alexander M. Gabovich and Alexander I. Voitenko
Institute of Physics, National Academy of Sciences of Ukraine, 46 Nauka Ave., Kyiv 03680, Ukraine
E-mail: gabovich@iop.kiev.ua,
voitenko@iop.kiev.ua
Received April 1, 2016, published online August 29, 2016
The state of the art concerning tunnel measurements of energy gaps in cuprate oxides has been analyzed.
A detailed review of the relevant literature is made, and original results calculated for the quasiparticle tunnel
current J(V) between a metallic tip and a disordered d-wave superconductor partially gapped by charge density
waves (CDWs) are reported, because it is this model of high-temperature superconductors that becomes popular
owing to recent experiments in which CDWs were observed directly. The current was calculated suggesting
the scatter of both the superconducting and CDW order parameters due to the samples’ intrinsic inhomogeneity.
It was shown that peculiarities in the current-voltage characteristics inherent to the case of homogeneous super-
conducting material are severely smeared, and the CDW-related features transform into experimentally observed
peak-dip-hump structures. Theoretical results were used to fit data measured for YBa2Cu3O7–δ and
Bi2Sr2CaCu2O8+δ. The fitting demonstrated a good qualitative agreement between the experiment and model
calculations. The analysis of the energy gaps in high-Tc superconductors is important both per se and as a tool to
uncover the nature of superconductivity in cuprates not elucidated so far despite of much theoretical effort and
experimental progress.
PACS: 71.45.Lr Charge-density-wave systems;
74.55.+v Tunneling phenomena: single particle tunneling and STM;
74.81.–g Inhomogeneous superconductors and superconducting systems, including electronic
inhomogeneities.
Keywords: d-wave superconductivity, charge-density wave, quasiparticle tunnel spectrum, peak-dip-hump struc-
ture, pseudogap, high-Tc superconductor.
Contents
1. High-Tc superconductors and charge-density waves .......................................................................... 1103
2. Theoretical basis................................................................................................................................. 1106
3. Quasiparticle current .......................................................................................................................... 1107
4. Account of inhomogeneity ................................................................................................................. 1108
5. Conductance-voltage characteristics. Background problem ............................................................... 1109
6. Results of calculations and discussion ............................................................................................... 1110
7. Conclusions ........................................................................................................................................ 1111
References .............................................................................................................................................. 1111
1. High-Tc superconductors and charge-density waves
Superconductivity of high- cT oxides is a vast area of
materials science [1]. Nevertheless, although about thirty
years passed since this phenomenon had been discovered,
it remains unexplained so far from a microscopic point
of view [2–6]. The lack of explanation means that we still
do not know for sure what boson (bosons) is responsible
for the Cooper pairing, although the Bardeen–Cooper–
Schrieffer (BCS) character of superconductivity (in the
© Alexander M. Gabovich and Alexander I. Voitenko, 2016
Alexander M. Gabovich and Alexander I. Voitenko
broad sense of this term [7]) is beyond question. Anyway,
the absence of the knowledge on the “glue” between paired
electrons does not allow one to predict and precisely cal-
culate the critical temperature of superconducting transi-
tion, cT , or any other important critical parameter [8]. Only
rare accidental success happens sometimes [9,10], which
might be simply a stroke of luck. One of the problems im-
peding the exact solution of the basic Eliashberg equations
(for the electron-phonon or other mechanisms of supercon-
ductivity [11]) is the ubiquitous and ambiguous Coulomb
pseudopotential [11].
Another problem arising while making attempts to carry
out reliable calculations of superconducting properties is
the proper account of strong electron-electron correlations,
which, in principle, may even dominate in the normal-state
and superconducting behavior of perspective materials
[12,13]. Many-body correlations can lead to other compet-
ing electron-spectrum instabilities different from Cooper
pairing. In particular, these are charge density waves (CDWs),
which distort the nested Fermi surface (FS) sections by the
emergence of dielectric energy gaps [14,15]. Experiments
show that CDWs do appear in cuprates [16–24] and the in-
tertwining between the CDW and superconducting gapp-
ings is a hot issue of condensed matter physics [25–30]. It
has to be indicated that cuprates also host the so-called
pseudogaps [31,32], which we identify as CDW related
gaps. Nevertheless, it might also happen that the both phe-
nomena are interrelated, but not identical.
The existence of intrinsic disorder in high- cT oxides is
another factor, which makes the identification of CDW
gaps more complicated than that of their superconducting
counterparts [33–38] as comes about from the scanning-
tunnel-microscopy (STM) measurements. The spread of
the latter over the sample surface is also quite wide, alt-
hough the distribution histogram for the superconducting
gaps is narrower than that for the CDW ones in the cases
when the both kinds can be distinguished [39]. Examples
of such histograms taken from Refs. 40–49 are shown in
Fig. 1.
We would like to attract attention to the variety of gap
distributions. In particular, the superconductivity- and
CDW-driven gap histograms can be either strongly over-
lapped or well separated. Concerning the latter case (see
Fig. 1, panels (c) and (j)–(l)), it should be emphasized that,
in the framework of our model (see Sec. 2), the smaller-
gap clusters are associated with the distributions of a pure
d -wave superconducting gap, whereas the larger-gap ones
with those of a combined (superconducting + CDW) gap.
However, near cT (see panels (d) and (e)), the distribution
is governed predominately (below cT ) or exclusively
(above cT ) by the spread of CDW strength.
The reasons of the electronic inhomogeneity leading to
the inhomogeneous electron-spectrum gapping may be
different, with the oxygen non-stoichiometry [50] being an
indefeasible factor for cuprates [51]. Both CDWs and the
electronic nonhomogeneity should be crucial for tunnel
spectroscopy, which studies voltage, V , dependences of
the quasiparticle current J and the corresponding conduct-
ance ( ) = /G V dJ dV [52–56]. Some time ago we suggest-
ed that CDWs have already been detected in the quasi-
particle tunnel current-voltage characteristics [30,57–60].
Specifically, ( )G V of junctions (either symmetric or non-
symmetric ones) involving high- cT oxides reveal peak-dip-
hump structures at energies eV much larger than the su-
perconducting gap-edge positions ∆ [25,31,55,61,62].
Here, > 0e is the elementary electric charge.
There are other scenarios, alternative to our approach
(see references in our publications [30,60]), which attempt
to explain the existence of those structures. Their main
difficulty consists in inevitable conspicuous CDW traces in
tunnel junctions with cuprate electrodes; such traces were
earlier observed in a number of partially CDW-gapped
normal metals and superconductors [63–65]. This phenom-
enon is a simple consequence of the dielectric gap emer-
gence on FS sections, which is similar to the superconduct-
ing gapping, because the coherence factors disappear from
the expression for ( )J V [66]. That is why the basic model
of the quasiparticle tunneling between superconductors is
called a “semiconductor” one. Hence, the CDW manifesta-
tions should be at least as strong as superconducting ones.
One can see that this is the case with the temperature above
cT , when the pseudogap-induced depletion in the density of
states is easily detected [31,55,67,68]. The evidence is not
so clear for temperatures below cT , since one observes
either weak peak-dip-hump structures mentioned above
(and discussed in more detail in following sections) or co-
existing distinct CDW- and superconductivity-driven gap-
edge peaks [69,70]. The latter interpretation was however
questioned [71], because heating effects might spoil
the picture (see also the earlier work [72] dealing with
BaPb1–xBixO3 ceramics, the predecessor of high- cT oxides).
The observed CDW structures in cuprates are not so
pronounced as, say, their counterparts in tri- [63,65,73] or
dichalcogenides [63,74–76]. The indicated disorder leading
to the spatial averaging of CDW manifestations is one of
the reasons. An underdevelopment of superlattice struc-
ture, i.e., its short-range or fluctuating CDW character
[17,77–79] in high- cT oxides may be another reason of its
apparent suppression in tunnel measurements. The weak-
ness of CDW distortions, accompanied by fluctuations, and
the smallness of CDW domains may be intimately con-
nected to disorder [80]. This fact makes the elaboration of
adequate microscopic theories rather involved.
Moreover, one should point at the commonly occurring
four-fold 4C symmetry loss of the reconstructed cuprate
two-dimensional electron spectrum. As a result, there
emerges the 2C (nematic or smectic) charge order, which
has already been found not only in high- cT oxides but also
in iron-based superconductors [22,81–86]. The relevant
experiments for cuprates demonstrate that the symmetry
1104 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10
Spatial distribution of superconducting and charge-density-wave order parameters in cuprates
violation is not mandatory. Therefore, both bidirectional
(checkerboard) CDWs, which preserve the 4C symmetry,
and unidirectional (stripe) patterns corresponding to ne-
matic or smectic states are possible. Hence, they should be
considered while studying tunnel currents in junctions
made of CDW superconductors.
Fig. 1. (Color online) Histograms of energy-gap distributions in various cuprates revealed by scanning tunnel microscopy (STM) exper-
iments: (a) overdoped Bi2Sr2CaCu2O8+δ (BSCCO), the critical temperature = 84 KcT [40]; (b) overdoped BSCCO, = 86–87 KcT [41];
(c) overdoped BSCCO, = 65 KcT , measured at various temperatures T’s [42]; (d) underdoped BSCCO, = 87 KcT , measured at
= 82 KT [43]; (e) underdoped BSCCO, = 87 KcT , measured at = 92 KT [43]; (f) overdoped BSCCO, = 68 KcT [44]; (g) optimally
doped BSCCO, = 93 KcT [44]; (h) overdoped BSCCO, = 68 KcT [45]; (i) overdoped Bi2Sr2–xLaxCuO6+δ, (bottom) = 0.1x ,
= 14 KcT ; (top) = 0.2x , = 31 KcT [46]; (j) (Cu,C)Ba2Ca3Cu4O12+δ, = 117 KcT [47]; (k) Ba2Ca4Cu5O10(O0.17F0.83)2, = 70 KcT [48];
(l) TlBa2Ca2Cu2O10–δ, = 91 KcT [49].
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 1105
Alexander M. Gabovich and Alexander I. Voitenko
2. Theoretical basis
The idea that the electron spectrum peculiarities of
high- cT oxides can be associated with the existence of
CDWs was put forward by us not long ago after the
cuprates had been discovered [87,88]. For d -wave super-
conductors with s-wave CDWs, we use a theoretical
framework suggested earlier [25,27,60,89]. This treatment,
in agreement with the experimental data for cuprates, as-
sumes a 2 2x yd − -wave BCS-like superconducting order
parameter and an s-wave CDW order parameter attributed
to certain (nested) FS sections at temperatures below the
critical one, sT , which is usually substantially higher than
cT . This includes = 4N (bidirectional CDWs) or 2 (unidi-
rectional CDWs) sectors.
The mean-field Hamiltonian has the form
kin= ,BCS CDWH H H H+ + (1)
where
†
kin ,,, = ,
= ,
= ( ) ,i ii
i d nd
H a a σσσ ↑ ↓ ξ∑ k,k,k k (2)
† †
, ,
= ,
= ( ) c.c.,BCS i i
i d nd
H a a
↑ − ↓
∆ +∑ ∑ k, k,
k
k (3)
†
,,
, = , =
= ( ) c.c.CDW ii
i d
H a a + σσ
σ ↑ ↓
Σ +∑ ∑ k Q,k,
k
k (4)
Here, ( )iξ k is the initial quasiparticle spectrum; the d-wave
superconducting momentum-dependent order parameter,
( )∆ k , and the s-wave dielectric one, ( )Σ k , are described
below; †a and a are the creation and annihilation opera-
tors, respectively; σ is the quasiparticle spin projection, Q
is the CDW vector, and the notations “d” and “nd” indicate
the CDW-gapped or in other words dielectrized (to distin-
guish this pairing mechanism from the superconducting
one, we will use hereafter the term “dielectrized”) and non-
dielectrized FS sections, respectively.
In the parent non-superconducting CDW state, the s-wave-
like CDW (dielectric) complex order parameter 0 ( ) eiT ϕΣ
(here, ϕ is the CDW phase) is constant within any of N
dielectrized sectors directed along the xk and yk axes
(see Fig. 2). Each of the sectors spans the angle 2 <α Ω
( = 90Ω ° for = 4N or 180° for = 2N ); this is the model of
partial FS dielectrization. In the adopted s-wave theory
[14,90,91], the zero-temperature magnitude of CDW order
parameter equals 0 0( = 0) = ( / ) sT TΣ π γ , where = 1.78γ
is the Euler constant, the Boltzmann constant = 1Bk , and
0 =s sT T since there is no influence of ∆ in the normal state.
The CDW order parameter itself has the conventional s-wave
dependence on T,
0 0 0( ) = (0)Mu ( / )s sT T TΣ Σ , (5)
where Mu ( )s x is the reduced [Mu (0) = 1s ] s-Mühlschlegel
function. The angle θ in the two-dimensional k -plane is
reckoned from the xk axis. The profile 0 ( , )TΣ k , or
0 ( , )TΣ θ , over the whole FS contains the factor ( )fΣ θ ,
which is equal to 1 inside each dielectrized sector and 0
otherwise (see Fig. 2), so that
0 0( , ) = ( ) ( ).T T fΣΣ θ Σ θ (6)
The parent non-dielectrized BCS 2 2x y
d
−
-wave super-
conductor (dBCS) [92] is characterized by the supercon-
ducting order parameter 0 (0)∆ at = 0T and
0 0= ( e / 2 ) (0)cT γ π ∆ , where e is the base of natural loga-
rithm. The order parameter lobes are oriented in the xk and
yk directions, i.e., in the same (antinodal) directions as
the bisectrices of CDW sectors. The profile 0 ( , )T∆ θ in the
k -space spans the whole FS
0 0( , ) = ( ) ( ),T T f∆∆ θ ∆ θ (7)
where (see Fig. 2)
( ) = cos 2f∆ θ θ (8)
and
0 0 0( ) = (0)Mu ( / ).d cT T T∆ ∆ (9)
Here, Mu ( )d x is the conventional d -wave superconduct-
ing order parameter dependence.
When CDWs and superconductivity coexist (we call
this state SCDW), the order parameter dependences ( )TΣ
and ( )T∆ differ from those of the pure phases, i.e., 0 ( )TΣ
and 0 ( )T∆ , respectively [26,27]. The resulting self-
consistent set of gap equations, which determines ( )TΣ and
( )T∆ for the given set of the input model parameters
0( (0)∆ , 0 (0)Σ — for brevity, they are denoted below as 0∆
and 0Σ , respectively — α, and N ) has the following form
obtained earlier [25–27,30]:
2 2 2
0( 2 ), , ) = 0,cosMI T d
α
−α
Σ + ∆ θ Σ θ∫ (10)
Fig. 2. (Color online) Angular factors of order parameters for the
partially gapped CDW d-wave superconductor.
1106 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10
Spatial distribution of superconducting and charge-density-wave order parameters in cuprates
2 2 2 2
0( 2 , , cos 2 ) 2cos cosMI T d
α
−α
Σ + ∆ θ ∆ θ θ θ+∫
2
0( cos 2 , , cos 2 ) 2 = 0,cosMI T d
Ω−α
α
+ ∆ θ ∆ θ θ θ∫ (11)
here
2 2
0 2 2 2 2
0 0
1 1( , , ) = tanh
2ξ
MI T d
T
∞ ξ + ∆ ∆ ∆ − ξ
+ ∆ ξ + ∆
∫
(12)
is the Mühlschlegel integral of the BCS theory. Due to the
order parameter interplay, the lowest of the initial tempera-
tures 0cT or 0sT is suppressed due to the competition, so
that the actual critical temperatures become 0<c cT T or
0<s sT T . For existing CDW superconductors, it was found
that <c sT T [14,25,65].
The overall gap on the whole FS (the gap rose) can be
written in the form
2 2( , ) = ( , ) ( , ),D T T Tθ Σ θ + ∆ θ (13)
where
( , ) = ( ) ( ),T T f∆∆ θ ∆ θ (14)
( , ) = ( ) ( ).T T fΣΣ θ Σ θ (15)
In other words, in the mixed phase, a combined gap
2 2( , ) = ( ) ( , )D T T Tθ Σ + ∆ θ
determined by both order parameters, appears on the d FS
sections, and the gap ( , )T∆ θ depending only on the super-
conducting order parameter exists on the nd ones.
3. Quasiparticle current
In the framework of general approach, we consider
quasiparticle tunneling along the c-axes of both SCDW
electrodes, i.e., between the superconducting planes of
the same crystal or between two cuprate crystals, which are
suggested to be partially CDW-gapped d -wave SCDWs
with their superconducting planes (a–b facets) oriented
parallel to the junction interface. Such a configuration can
be realized in mesas [93,94], twist-crystal structures made
of bicrystals [95], the artificial cross-whiskers [96], or the
natural cross-whiskers [97]. We assume the strongly inco-
herent tunneling in the c-direction between electrodes sup-
ported by the experimental evidence for the Josephson
current [98]. However, the opposite case of the c-axis co-
herent tunneling would give similar results.
The formulas for the current ( )J V through the junction
concerned were obtained in the tunnel-Hamiltonian ap-
proach [14,57,99–103]
( )2
1( ) =
2 2
J V d d d
eR
π π ∞
−π −π −∞
′θ θ ω×
π
∫ ∫ ∫
( , , ) ( , ) ( , ),K V T P P eV′ ′× ω ω θ ω− θ (16)
where
( , , ) = tanh tanh ;
2 2
eVK V T
T T
ω ω−
ω − (17)
and the P -factors describe SCDW electrodes. All the tun-
nel barrier properties were incorporated into the single
constant R describing the normal-state resistance. The prim-
ed quantities are associated with the electrode that the po-
tential V is applied to (the V -electrode); its counter elec-
trode will be referred to as 0-electrode. In particular, for
the 0-electrode,
( )
2 2
( , )
( , ) =
( , )
D T
P
D T
Θ ω − θ
ω θ ×
ω − θ
sign cos ( , ) ,T × ω + ω ϕΣ θ (18)
where ( )xΘ is the Heaviside step-function, the CDW
phase ϕ is usually pinned by the junction interface and
acquires the values 0 or π (see discussion in Refs. 14, 25).
For ( , )' 'P eVω− θ , one has eVω− rather than ω in formu-
la (18), whereas all other parameters have to be primed,
i.e., associated with the V -electrode. The peculiar term in
the brackets of Eq. (18) is generated by the electron-hole-
pairing Green’s function ibG , which is dubbed “normal”
because it is proportional to the product †
rlc c [104,105].
However, this term can also be called “anomalous”, since
it contains the CDW order parameter as a factor, in the same
way as the Gor’kov Green’s function F is proportional to
the factor ∆ [99]. Here, the subscripts l and r correspond
to two different nested FS sections.
If one of the electrodes is a normal metal we arrive at
a specific junction variant, which is typical of STM studies
of the quasiparticle tunnel current between the tip and the
superconducting cuprate plane (the STM configuration). In
this case, the ( , )P ω θ function corresponding to the normal
metal becomes identical to unity and drops out of consid-
eration. Below, we will make specific calculations just for
this case (although, in principle, the symmetric one is not
more difficult for studying). In the case of non-symmetric
configuration, the CVC is non-symmetric [106] in accord-
ance with ( )G V ’s observed for high- cT superconductors
[55,56,107].
We emphasize that the appearance of the dielectric gap Σ
in expression (16) for the tunnel current is justified by the
same arguments as the now-conventional emergence of its
superconducting counterpart ∆. The problem of the collec-
tive-state-gap manifestations started with the Harrison’s
analysis [108] carried out in the framework of the independ-
ent-particle picture and using quite a reasonable Wentzel–
Kramers–Brillouin approximation [109]. This author show-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 1107
Alexander M. Gabovich and Alexander I. Voitenko
ed that in the one-dimensional case the density-of-states
factor should be canceled out from the final expression for
the tunnel current due to the dependence of the tunneling
matrix element 12T on this factor [108]. As was demon-
strated experimentally by Giaever [110,111], this approach
fails in the case of quasiparticle tunneling between super-
conductors or between a normal metal and a superconduc-
tor, because many-body correlations enter into the game.
Thus, the tunnel current becomes a functional of the so-
called superconducting densities of states, which are propor-
tional to 2 2 1/2( )−ω − ∆ . Bardeen explained this situation
from the theoretical point of view by suggesting his tunnel-
Hamiltonian method and speculating that the Cooper-
pairing correlations die out inside the barrier, whereas the
single-particle densities of states are renormalized by su-
perconductivity [112]. Bardeen’s insight proved to be qual-
itatively valid and even quantitatively correct for quasipart-
icle currents. At the same time, it failed when Bardeen
objected [113] to the possibility of the Josephson coherent
tunneling between superconductors [114,115]. Hence, su-
perconducting correlations should preserve, although being
weakened, inside the hostile insulating or normal-metal
environment.
Peierls [116] and excitonic [117] insulators, as well as
any other completely or partially CDW-gapped systems,
are a consequence of the electron-hole many-body correla-
tions. The latter, physically and mathematically, are very
similar (although not identical!) to superconducting ones.
Hence, their manifestations in the quasiparticle currents
should also be similar. In this sense, the CDW gaps Σ be-
have differently from the band gaps GE in Esaki diodes.
They appear in the ( )J V -dependence and its derivatives as
conspicuous features [63–65].
4. Account of inhomogeneity
It is well-known that the dependence of the quasi-
particle tunnel current J on the bias voltage V across the
junction is much less informative than the same depend-
ence of the tunnel conductance = /G dJ dV , the CVC.
Technically, the latter is determined as the ratio /J V∆ ∆ ,
where J∆ is the increment of the current between two
voltage values separated by V∆ . Plenty of experiments for
various solids testify that, if V∆ is small enough, the ratio
/J V∆ ∆ calculated from the data adequately reproduces
the theoretically expected /dJ dV even if the latter contains
such peculiarities as jumps, cusps, and so on [99,115].
Nevertheless, if we tried to use the formulas of Section 3
(see relevant calculations in Ref. 30) for the comparison
with ( )G V -dependences actually measured for symmetric
or non-symmetric junctions involving high- cT oxides, the
result would be discouraging.
The key origin of this discrepancy consists in that high-
cT oxides, even single crystal samples, are intrinsically
inhomogeneous objects [51]. The most compelling origin
of this inhomogeneity is the non-uniform distribution of
oxygen atoms in those non-stoichiometric materials. How-
ever, the inhomogeneity may be traced to other causes of
the nanoscale phase separation observed not only in super-
conducting and magnetic oxides [33–37,118,119] but also
in other magnets [120,121], non-equilibrium alloys [122],
porous systems [123,124], and colloids [125,126].
In the case of STM spectroscopy, the tunnel current is
apparently harvested from a small area of the atomic size
on the substrate [127]. If the substrate metal is in the nor-
mal state, this is really the case. However, if the substrate
is a superconductor, the electron properties of the patch
under the tip are formed by many-body correlations ex-
tending over a linear distance of about the Pippard-BCS
coherence length ξ0 [128]. If electron-hole (CDW) pair-
ing [129,130] is also present, the corresponding correlation
length is of importance as well. The both lengths, although
reduced owing to defects, may be much larger than
the lattice constants [131]. In other words, the measured
quasiparticle tunnel current is actually averaged over rather
a large region of the electrode surface. Nevertheless, while
scanning along the sample surfaces of various cuprates, the
measured CVCs change appreciably [34,39–49,55], which
brings us to a conclusion that SCDWs are really electroni-
cally inhomogeneous objects characterized by certain dis-
tributions of their parameters. As a result, the calculation
of the tunnel current should include averaging over those
distributions. Since we calculate CVCs for a fixed tip posi-
tion over the oxide surface, the chosen distribution is con-
sidered as a characteristics of this position. Another loca-
tion of the tip may reveal a different distribution, in ac-
cordance with the experiment.
In our previous calculations [60], we showed that taking
the spread of only one of SCDW parameters into account
(in the case concerned, it was the parameter 0Σ ) was not
sufficient to reproduce the quasiparticle CVCs for non-
symmetric junctions with SCDWs. In this work, we let
all SCDW initial parameters ( )0 0= , ,X ∆ Σ α vary. We
assumed that the distribution for each of them is de-
scribed by the bell-shaped function ( )w X proportional to
4 2 2 2
00 0
(( ) )P PX X−δ − − δ within the corresponding inter-
vals 0 < 00
,XX X − δ and 0 0 > 0
, XX X + δ , equal to zero
outside them, and with the unity normalization in the inter-
val 0 0 < 0 >0 0
( ) = ,X XZ X X X − δ + δ . Below, for brief,
we mark this distribution as ( ) > 0
0 < 0
P
P
X
+δ
−δ .
The procedure of finding G at the bias voltage V was as
follows. First, we calculated the normalized average cur-
rents j at the voltages V V− ∆ and V V+ ∆ . The voltage
increment V∆ was selected small enough for the final re-
sult not to depend on it. Each averaged current
( )j V V± ∆ was calculated as the weighted integral
1108 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10
Spatial distribution of superconducting and charge-density-wave order parameters in cuprates
0 0( ) = ( , ) ( ) ( ) ( ) ,j V j X V w w w dX∆ Σ α∫ (19)
where ( , )j X V is the current ( )J V calculated for the pa-
rameter set ( )0 0= , ,X ∆ Σ α using formula (16) but without
the algebraic multiplier before the integral. It is easy to
verify that, with this normalization, the dependence ( )j V
has the Ohmic asymptotics ( )j V V→ ±∞ → . Integr-
ation was carried out in the parameter region
0 0( ) ( ) ( )Z Z Z∆ × Σ × α using the Monte Carlo method. Af-
ter reaching the required accuracy, the normalized tunnel
conductance was found according to the formula
( ) ( )
( ) .
2
j V V j V V
g V
V
+ ∆ − − ∆
≈
∆
(20)
The results of calculations (see Sec. 6) showed that
the relevant experimental CVCs better correspond to small
α-values. In this case, the variation of α (also in narrow
limits!) gave rise to small CVC changes. Therefore, we se-
lected α to be a fixed parameter. This choice corresponds
to putting 0( ) = ( )w α δ α −α in Eq. (19), and the averaging
integration was carried out in the parameter region
0 0( ) ( )Z Z∆ × Σ .
5. Conductance-voltage characteristics.
Background problem
The electric conductance in tunnel structures is a com-
plex phenomenon including many-body (dynamic electron-
plasmon) interactions [132–134]. In the entire voltage
range between the Ohmic and Fowler–Nordheim limits, it
demonstrates both universal features [135–138] and mate-
rial-dependent peculiarities [139,140]. In the case of high-
cT oxides, the CVC behavior is even more involved. For
instance, the CVCs often include a background, linear and
even asymmetric, which extends over hundreds of milli-
volts [141,142]. The origin of this phenomenon has not
been clearly identified, so that the exact form of this con-
tribution remains still unknown for both symmetric and
non-symmetric junctions with high- cT oxide SCDWs
[141–145]. Moreover, the background component depends
not only on the measurement set-up, but also on the studied
specimen (see, e.g., measurements [146–149]). Relevant
information is too scarce for definite conclusions and as-
sumptions to be made. Let us consider this problem in brief
using, in particular, the data of publications [146–149].
The authors of Ref. 149, while studying under- and
overdoped Bi2212 oxides, assumed that the measured tun-
nel current consisted of a true gap-dependent component
and a background, the latter being provided by the speci-
men itself and the measuring equipment. This background
was found to vary from one specimen to another, which is
quite reasonable, but it was supposed to be independent of
the temperature for the fixed combination “specimen +
+ measurement unit”. Therefore, by assuming that the
specimen is a normal metal at > cT T , the background cur-
rent (more precisely, the background CVC) was measured
at those temperatures. Then, the “genuine” CVC was cal-
culated by subtracting this reference CVC from the meas-
ured one. However, we know that, at temperatures above
cT , CVCs for high- cT oxides demonstrate a pseudogap
behavior, which survives up to temperatures much higher
than cT [31]. At the same time, the presented CVCs, in-
cluding the reference one, also include a background con-
tribution. It is so because, in the absence of background
signal, (i) the measured CVCs have to approach the nor-
malized Ohmic asymptotic Ohmic ( ) = 1g eV at high, by
magnitude, bias voltages, and (ii) they must satisfy the
“sum rule”: the summed up areas between the plots of a
specific CVC and its Ohmic asymptotic to the both sides
from the latter must compensate each other. The “sum
rule” is known to be dictated by the conservation law for
the number of quasiparticle states in the semiconducting
model of BCS superconductors [66]. It has to be strictly
obeyed for the exact junction conductance. The experi-
mental determination of the latter would require current
measurements at two infinitesimally close bias voltages,
which is, of course, practically impossible. Nevertheless,
we believe that, if the tunnel conductance is calculated as a
finite difference across a sufficiently short bias voltage
interval, the “sum rule” remains a good criterion for the
absence of any background signal in the CVC.
Unfortunately, the majority of experimental CVCs, in-
cluding those in measurements [146–149], do not satisfy
any of those requirements. Attempts to fulfill condition (i)
by taking a simple smooth curve with its high-voltage
asymptotics repeating those of the specific CVC resulted in
the violation of condition (ii). Therefore, the true back-
ground is a more complicated function. Now, we have no
information on what model should be used in fitting.
Moreover, the change of the sought background voltage
dependence can strongly affect the resulting “genuine”
CVC, which makes the whole problem of the background
current determination a non-trivial task.
Taking all the aforesaid into account and bearing in
mind the qualitative character of the current analysis, we
selected a few CVCs presented in the literature, the back-
ground of which can be modeled by a single smooth curve
bk ( )g v satisfying conditions (i) and (ii). This curve had
prescribed high-v linear asymptotics with a smooth transi-
tion between them:
bk
1( ) = ( , ) 1 tanh ( , ) 1 tanh ,
2
g A A
X X
−∞ − + +∞ +
v vv v v
(21)
where
( , ) = s sA s B v C+v (22)
is the linear asymptotic at s→v , the parameter =s ±∞
identifies the corresponding asymptotic, and the parameter
> 0X determines the transition interval between two
Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10 1109
Alexander M. Gabovich and Alexander I. Voitenko
asymptotics. The procedure of finding bk ( )g v was nothing
else but an attempt to approximate the given CVC by de-
pendence (21). At every fitting routine, the fixed asymptot-
ic parameters sB and sC were selected on the basis of
some reasonable considerations concerning the behavior of
analyzed CVC. The fitting procedure was performed using
the software package Origin 2016. For plenty of examined
CVCs, the resulting X-values turned out too small. This
result meant that either the transition interval was exclu-
sively narrow or, more probably, the selected function
bk ( )g v with prescribed asymptotics did not satisfy condi-
tion (ii). In both cases, such CVCs were rejected from fur-
ther consideration.
6. Results of calculations and discussion
Guided by the considerations outlined in Sec. 5, we
made attempts to fit ( )G V for high- cT oxides YBa2Cu3O7–δ
with 90 KcT ≈ (Ref. 148) and Bi2Sr2CaCu2O8+δ with
92 KcT ≈ (Ref. 146). In the both cases, the experimental
measurements were carried out at the temperature of
= 4.2 KT , so that cT T . Therefore, our specific calcula-
tions were performed for the temperature = 0 KT . The
relevant experimental data together with model back-
ground (21), the “true” CVCs, and their fitting curves are
shown in Figs. 3 and 4, respectively. As one can see, the
main features of the experimental CVCs are reproduced
very well in the former case. Specifically, these are (i) the
d -wave-like character of the region between the ∆-gap
edges; (ii) the asymmetry of the CVCs with reference
to the voltage; and (iii) the dip-hump structure outside
the d -wave gap region. We should mention that the ap-
pearance of this structure in the negative-voltage CVC
branch corresponds to the selection cos = 1ϕ − or =ϕ π for
the CDW phase (see Sec. 2). One should note that, in the
framework of the suggested theory, a slight asymmetry
between the inner peaks, which is also typical of the over-
whelming majority of experimental CVCs, can also be
explained, in addition to the influence of unknown back-
ground current, by a slight deviation of the CDW phase
from this value (see the relevant discussion in Ref. 106).
At the same time, Fig. 4 demonstrates the importance of
the background choice in accordance with the “sum rule”
for the density of states. An unsatisfactory selection result-
ed in that we did not succeed in reproducing the magni-
tudes of the peaks and humps precisely. We think that the
ambiguity of the background account is to blame for this
shortcoming. It is worth noticing that various kinds of the
quite reasonable normalization intended to exclude the
background influence cannot solve this problem (see, e.g.,
Ref. 39). However, even the CVC shown in Fig. 4 can
qualitatively be understood as a consequence of a competi-
tion between the CDWs and superconductivity.
The results obtained demonstrate that the FS dielectri-
zation degree in cuprates is low. Nevertheless, it is suffi-
cient to make the CVCs for non-symmetric junctions also
non-symmetric and allow a conclusion about the presence
of CDWs in cuprates to be made.
The most frequently expressed alternative viewpoint
[150–153] attributes the dip-hump structures observed in
cuprates [25,55,56] to the extremely strong-coupling ef-
Fig. 3. (Color online) (a) Experimental current-voltage character-
istic (CVC) for YBa2Cu3O7–δ [148] (solid curve) and its assumed
background component (dotted curve). (b) Normalized back-
ground-free CVC (solid curve) and its fitting in the framework of
SCDW model (dashed curve); the corresponding fitting parame-
ters are indicated.
Fig. 4. (Color online) The same as in Fig. 3, but for BSCCO
[146].
1110 Low Temperature Physics/Fizika Nizkikh Temperatur, 2016, v. 42, No. 10
Spatial distribution of superconducting and charge-density-wave order parameters in cuprates
fects, well-known for other superconductors with high cT
[56,154]. However, in order to implement corresponding
models, which suggest an anomalously strong electron
interaction with gluing bosons (these models postulate that
Cooper pairing in high- cT oxides is driven by spin fluctua-
tions [5,155–158]), one should additionally conceive the
important role of the Van Hove singularities in the electron
spectrum [62,159]. Such a reconstruction of the observed
features is, of course, possible. Nevertheless, our interpre-
tation discussed here seems to be more appealing. First, it
invokes CDWs, which are real phenomena intrinsic to
cuprates. Second, CDW effects in tunneling, if not de-
pressed by disorder, are very strong on their own [30], so
one does not need any additional amplification by another
cause. Third, we can easily explain the symmetry breaking
in CVCs as the direct consequence of the CDW existence
due to the role of the normal-anomalous Green’s function
ibG [106].
Another interpretation of the dip-hump structures, which
has been published recently [160], is based on the exist-
ence of the normal layer at the degraded superconductor
surface and the appearance of the bound state in this layer.
However, the corresponding dip-hump features turned out
symmetric with respect to the bias voltage in the non-sym-
metric tunnel junction geometry, contrary to our results
and to the experimental ones.
It is instructive to note that the interpretation of the same
phenomena in high- cT oxides steadily changes with time,
highly depending on the dominating viewpoint in the com-
munity. For instance, the appearance of extra peaks in point-
contact CVCs with YBa2Cu3O7–δ [161] was attributed to
phonon manifestations. Similarly, point-contact and tunnel
studies of YBa2Cu3O7–δ [162] and Bi2Sr2CaCu2O8+δ [163],
which revealed sub-gap peculiarities, were also considered
as the evidence of the phonon-driven Cooper pairing me-
chanism, as well as the s-symmetry of superconducting order
parameter. This scientific group adopts the more or less con-
ventional character of superconductivity in cuprates [164].
7. Conclusions
In this work, we analyzed the issue about the energy
gap scatter in cuprates observed by means of tunnel spec-
troscopy. We showed that the gap spread is due to the dis-
persion of both the genuine superconducting gap ∆ and
the competing CDW gap Σ . The non-symmetricity of the
CVCs and the existence of the peak-dip-hump structures
for one voltage polarity are other important features re-
vealed in the experimental data. Using our model of the
coexistence between d -wave superconductivity and CDWs,
we calculated tunnel CVCs and fitted experimental data
for YBa2Cu3O7–δ and Bi2Sr2CaCu2O8+δ. The results are
satisfactory taking into account the background uncertain-
ty, making the comparison of theoretical and experimental
values for the conductance ( )G V ambiguous.
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. High-Tc superconductors and charge-density waves
2. Theoretical basis
3. Quasiparticle current
4. Account of inhomogeneity
5. Conductance-voltage characteristics. Background problem
6. Results of calculations and discussion
7. Conclusions
|
| id | nasplib_isofts_kiev_ua-123456789-129307 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:53:15Z |
| publishDate | 2016 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Gabovich, A. M. Voitenko, A.I. 2018-01-18T17:58:59Z 2018-01-18T17:58:59Z 2016 Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) / Alexander M. Gabovich Alexander I. Voitenko // Физика низких температур. — 2016. — Т. 42, № 10. — С. 1103-1114. — Бібліогр.: 164 назв. — англ. 0132-6414 PACS: 71.45.Lr, 74.55.+v, 74.81.–g https://nasplib.isofts.kiev.ua/handle/123456789/129307 The state of the art concerning tunnel measurements of energy gaps in cuprate oxides has been analyzed. A detailed review of the relevant literature is made, and original results calculated for the quasiparticle tunnel current J(V) between a metallic tip and a disordered d-wave superconductor partially gapped by charge density waves (CDWs) are reported, because it is this model of high-temperature superconductors that becomes popular owing to recent experiments in which CDWs were observed directly. The current was calculated suggesting the scatter of both the superconducting and CDW order parameters due to the samples' intrinsic inhomogeneity. It was shown that peculiarities in the current-voltage characteristics inherent to the case of homogeneous superconducting material are severely smeared, and the CDW-related features transform into experimentally observed peak-dip-hump structures. Theoretical results were used to fit data measured for YBa₂Cu₃O₇–δ and Bi₂Sr₂CaCu₂O₈₊δ. The fitting demonstrated a good qualitative agreement between the experiment and model calculations. The analysis of the energy gaps in high- Tc superconductors is important both per se and as a tool to uncover the nature of superconductivity in cuprates not elucidated so far despite of much theoretical effort and experimental progress. The work was partially supported by the Project No. 24
 of the 2015–2017 Scientific Cooperation Agreement between
 Poland and Ukraine. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 30-летию открытия высокотемпературной сверхпроводимости Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) Article published earlier |
| spellingShingle | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) Gabovich, A. M. Voitenko, A.I. К 30-летию открытия высокотемпературной сверхпроводимости |
| title | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) |
| title_full | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) |
| title_fullStr | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) |
| title_full_unstemmed | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) |
| title_short | Spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (Review Article) |
| title_sort | spatial distribution of superconducting and charge-density-wave order parameters in cuprates and its influence on the quasiparticle tunnel current (review article) |
| topic | К 30-летию открытия высокотемпературной сверхпроводимости |
| topic_facet | К 30-летию открытия высокотемпературной сверхпроводимости |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129307 |
| work_keys_str_mv | AT gabovicham spatialdistributionofsuperconductingandchargedensitywaveorderparametersincupratesanditsinfluenceonthequasiparticletunnelcurrentreviewarticle AT voitenkoai spatialdistributionofsuperconductingandchargedensitywaveorderparametersincupratesanditsinfluenceonthequasiparticletunnelcurrentreviewarticle |