Spin-polarized electron tunneling between charge-density-wave metals
For junctions between metals partially gapped by charge density waves (CDWs), the quasiparticle tunnel currents J(V) and conductances G(V) in external magnetic fields H are calculated as functions of H, the bias voltage V, temperature T, the dielectric gaps ∑, and the gapped portions μ of the Fermi...
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irk-123456789-1207772017-06-13T03:04:00Z Spin-polarized electron tunneling between charge-density-wave metals Ekino, T. Gabovich, A.M. Voitenko, A.I. Электpонные свойства металлов и сплавов For junctions between metals partially gapped by charge density waves (CDWs), the quasiparticle tunnel currents J(V) and conductances G(V) in external magnetic fields H are calculated as functions of H, the bias voltage V, temperature T, the dielectric gaps ∑, and the gapped portions μ of the Fermi surface (FS). The paramagnetic effect of H is taken into account, whereas orbital effects are neglected. General expressions are obtained for different CDW metal electrodes. Analytical formulas are obtained for T = 0. Explicit numerical calculations are carried out for symmetrical junctions. The results are substantially unlike those for junctions between superconductors. It is shown that due to the interplay between quasiparticles from nested and non-nested FS sections the junction properties involve features appropriate to both symmetrical and asymmetrical setups. In particular, for H = 0 discontinuities at eV = ±2∑ and square-root singularities at eV = ±∑ should coexist. Here e is the elementary charge. For H ≠ 0 the former remain intact, while the latter split. It is suggested to use the splitting as a verification of the CDW nature of the pseudogap in high-Tc superconducting oxides. 2005 Article Spin-polarized electron tunneling between charge-density-wave metals / T. Ekino, A.M. Gabovich, and A.I. Voitenko // Физика низких температур. — 2005. — Т. 31, № 1. — С. 77-93. — Бібліогр.: 93 назв. — англ. 0132-6414 PACS: 71.45.Lr, 73.40.Gk, 75.20.En, 75.47.Np http://dspace.nbuv.gov.ua/handle/123456789/120777 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов |
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Электpонные свойства металлов и сплавов Электpонные свойства металлов и сплавов Ekino, T. Gabovich, A.M. Voitenko, A.I. Spin-polarized electron tunneling between charge-density-wave metals Физика низких температур |
| description |
For junctions between metals partially gapped by charge density waves (CDWs), the quasiparticle tunnel currents J(V) and conductances G(V) in external magnetic fields H are calculated as functions of H, the bias voltage V, temperature T, the dielectric gaps ∑, and the gapped portions
μ of the Fermi surface (FS). The paramagnetic effect of H is taken into account, whereas orbital
effects are neglected. General expressions are obtained for different CDW metal electrodes.
Analytical formulas are obtained for T = 0. Explicit numerical calculations are carried out for symmetrical
junctions. The results are substantially unlike those for junctions between superconductors.
It is shown that due to the interplay between quasiparticles from nested and non-nested FS
sections the junction properties involve features appropriate to both symmetrical and asymmetrical
setups. In particular, for H = 0 discontinuities at eV = ±2∑ and square-root singularities at eV = ±∑
should coexist. Here e is the elementary charge. For H ≠ 0 the former remain intact, while the latter
split. It is suggested to use the splitting as a verification of the CDW nature of the pseudogap
in high-Tc superconducting oxides. |
| format |
Article |
| author |
Ekino, T. Gabovich, A.M. Voitenko, A.I. |
| author_facet |
Ekino, T. Gabovich, A.M. Voitenko, A.I. |
| author_sort |
Ekino, T. |
| title |
Spin-polarized electron tunneling between charge-density-wave metals |
| title_short |
Spin-polarized electron tunneling between charge-density-wave metals |
| title_full |
Spin-polarized electron tunneling between charge-density-wave metals |
| title_fullStr |
Spin-polarized electron tunneling between charge-density-wave metals |
| title_full_unstemmed |
Spin-polarized electron tunneling between charge-density-wave metals |
| title_sort |
spin-polarized electron tunneling between charge-density-wave metals |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2005 |
| topic_facet |
Электpонные свойства металлов и сплавов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120777 |
| citation_txt |
Spin-polarized electron tunneling between charge-density-wave metals / T. Ekino, A.M. Gabovich, and A.I. Voitenko // Физика низких температур. — 2005. — Т. 31, № 1. — С. 77-93. — Бібліогр.: 93 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT ekinot spinpolarizedelectrontunnelingbetweenchargedensitywavemetals AT gabovicham spinpolarizedelectrontunnelingbetweenchargedensitywavemetals AT voitenkoai spinpolarizedelectrontunnelingbetweenchargedensitywavemetals |
| first_indexed |
2025-07-08T18:33:45Z |
| last_indexed |
2025-07-08T18:33:45Z |
| _version_ |
1837104762015711232 |
| fulltext |
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1, p. 77–93
Spin-polarized electron tunneling between
charge-density-wave metals
T. Ekino1, A.M. Gabovich2, and A.I. Voitenko2
1Hiroshima University, Faculty of Integrated Arts and Sciences, 1-7-1 Kagamiyama,
Higashi-Hiroshima, 739-8521, Japan
2Crystal Physics Department, Institute of Physics, National Academy of Sciences
46 Prospekt Nauki, Kiev, 03028 Ukraine
E-mail: collphen@iop.kiev.ua
Received April 16, 2004
For junctions between metals partially gapped by charge density waves (CDWs), the quasi-
particle tunnel currents J(V) and conductances G(V) in external magnetic fields H are calculated
as functions of H, the bias voltage V, temperature T, the dielectric gaps �, and the gapped por-
tions � of the Fermi surface (FS). The paramagnetic effect of H is taken into account, whereas or-
bital effects are neglected. General expressions are obtained for different CDW metal electrodes.
Analytical formulas are obtained for T = 0. Explicit numerical calculations are carried out for sym-
metrical junctions. The results are substantially unlike those for junctions between superconduc-
tors. It is shown that due to the interplay between quasiparticles from nested and non-nested FS
sections the junction properties involve features appropriate to both symmetrical and asymmetrical
setups. In particular, for H = 0 discontinuities at eV � �2� and square-root singularities at eV � ��
should coexist. Here e is the elementary charge. For H � 0 the former remain intact, while the lat-
ter split. It is suggested to use the splitting as a verification of the CDW nature of the pseudogap
in high-Tc superconducting oxides.
PACS: 71.45.Lr, 73.40.Gk, 75.20.En, 75.47.Np
1. Introduction
Instabilities of the parent metallic electron spectrum
leading to a formation of charge density waves (CDWs)
[1–3] are in some sense similar to the superconducting
Cooper pairing phenomenon [4]. Namely, although co-
herent properties of the reconstructed low-temperature,
low-T, phases are quite different, the resulting gapping
of the Fermi surface (FS) due to many-body correlations
is described by the same equations, at least in the
weak-coupling limit. Therefore, the so-called semicon-
ducting aspects of both superconductors and excitonic
[1,2] or Peierls [3] insulators are analogous. Neverthe-
less, as has been demonstrated previously [5], they are by
no means identical. It is worthwhile noting that the
quasiparticle electron density of states (DOS) of conven-
tional nondegenerate semiconductors, adequately de-
scribed by the one-body band theory, is nonsingular [6],
contrary to what happens in the models both for super-
conductors [4] and many-body insulators [1–3].
In this publication we want to call attention
once more to the quasiparticle tunneling between
metals partially gapped by CDWs (CDWMs). The
expressions for the tunnel current–voltage (I–V)
characteristics J(V) in the general case of different
CDWM electrodes are obtained, and a number of
practically important particular cases are consid-
ered in more detail. Their analysis shows that due
to the coexistence of gapped and nongapped FS sec-
tions, the I–V characteristics of tunnel junctions
with CDWMs on both sides of the potential barrier
possess some unconventional features. They are
analogous to those observed in the setup where one
of the electrodes is a normal metal without any
electron spectrum distortion and the other one is a
partially-gapped CDW conductor [5,7].
If an external magnetic field H is applied, the dual
nature of the partially-gapped CDWM should result
in the Zeeman (spin) splitting of the peaks in the con-
ductance–voltage (G–V) characteristics G(V),
© T. Ekino, A.M. Gabovich, and A.I. Voitenko, 2005
which, e.g., in the case of superconductivity are ap-
propriate to S–I–N junctions rather than to the S–I–S
ones (compare with Refs. 4,8–10). In this article we
obtain the corresponding expressions for G(V) in a
symmetrical CDWM–I–CDWM structure and demo-
nstrate the existence of the peak splitting. On this ba-
sis, relevant inferences are drawn for recognized CDW
materials and high-Tc cuprates, highly suspected to
belong to this class [11,12].
2. Theory
As has long been understood (see review [13]), the
C–V characteristics for tunneling between two super-
conductors in an external magnetic field H, which in-
duces the Zeeman splitting of the electronic DOS due
to the Pauli paramagnetism of electrons, nevertheless,
does not exhibit splitting of the gap-related peaks.
The nonexistance of the splitting in S�–I–S junctions
is explained by equal shifts in energy of electron
subbands possessing the same spin projection value on
both sides of the barrier and the conservation of spin
direction while tunneling in the absence of the spin–or-
bital effect [14]. On the other hand, tunneling across
S–I–N junctions reveals such a splitting, because in
this situation G(V) is proportional to the supercon-
ducting and normal electron DOSs shifted with re-
spect to each other in the magnetic field [4,8–10].
As concerns the paramagnetic properties, a CDWM
described either by the excitonic [1,2] or Peierls
[3] models is quite similar [15–18] to an s-wave
Bardeen—Cooper—Schrieffer (BCS) superconductor
[13,14]. This means that for H not exceeding a certain
value, mathematically analogous to the Clogs-
ton—Chandrasekhar paramagnetic limit [4], the
CDW gap �( )T may be considered as a BCS-like one,
not dependent on H. Such an expectation is supported
by experiment. For example, the destructive influence
of H on the critical temperature, Td , of the structural
phase transition was observed to be extremely small at
low fields for such different substances with CDWs as
the A15 compound V3Si [19] (Td = 20.15 K at H = 0
and is reduced by –0.6 K at H = 156 kOe) and the
quasi-one-dimensional organic metal Per2 [Au(mnt)2]
[20], and could not be detected for any other CDW
compounds.
On the other hand, in the following analysis we are
going to completely ignore the diamagnetic response
of the CDWM. The experimental reason for this ne-
glect was cited above. From the theoretical point of
view, it may be justified as follows. Due to the differ-
ent type of long-range order in comparison to that for
superconductors, the Meissner effect is absent in
excitonic or Peierls insulators [1,2,11,12], although
other interesting coherent phenomena may occur
[21–24]. The more conventional orbital effects of the
magnetic field should exist, but their influence on the
CDW phase is not destructive. On the contrary, ac-
cording to Refs. 25, 26, the inevitable paramagnetic ef-
fects are augmented by diamagnetic ones, favorable to
CDWs due to the reduction of the electron spectrum
dimensionality for large H [4]. These considerations are
supported by recent experiments for the organic metal
�-(BEDT-TTF)2KHg(SCN)4 [27], where a series of
phase transitions between subphases with different va-
lues of the nesting vector Q was observed. A stabiliza-
tion of CDWs by the restricted orbital motion in the
magnetic field is analogous to the emergence of field-in-
duced spin-density waves (SDWs) in (TMTSF)2X or-
ganic salts [28]. Therefore, this phenomenon, which
preserves CDWs, would be helpful for the spin split-
ting of CDW-driven peaks in G(V), although it might
make the interpretation of the spectra more ambiguous.
2.1. CDW metal
2.1.1. Zero magnetic field
The starting point of our approach is the mean-field
Hamiltonian of the partially-gapped superconducting
CDW metal proposed by Bilbro and McMillan [29].
For our current purposes we need a simpler case of a
normal CDW metal, which can be obtained from the
original model when the superconducting gap is iden-
tically zero [5]. According to this model, the FS of the
CDWM is split into degenerate (nested, i = 1, 2) and
nondegenerate (non-nested, i = 3) sections. For the for-
mer, the bare quasiparticle spectrum branches reckoned
from a common Fermi level are linked by the relation
� �1 2( ) – ( )p p Q� + , (1)
where Q is the CDW vector. Due to the interaction of
quasiparticles from different (i = 1, 2) nested FS sec-
tions, a many-body correlation (leading to a pairing,
which is a close analog of Cooper pairing) appears be-
tween them. The CDW pairing can be described by a
dielectric order parameter ~,� and a relevant dielectric
gap � emerges at both nested sections. If this interac-
tion is mainly of a Coulomb origin [1,2], and the
branches �12, ( )p represent the electron and hole
bands, respectively, the CDW gapping corresponds to
the excitonic insulator. Another possibility appears if
the degenerate spectrum �12, ( )p is quasi-one-dimen-
sional and the quasiparticle interaction is mediated by
phonons. Then CDW gapping results in an emergence
of the Peierls insulator state [3]. In both CDW cases,
the coupling occurs between quasiparticles with op-
positely directed spins (singlet pairing). Those alter-
natives can be considered in the framework of the
same approach. The rest of the FS remains undis-
78 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
torted by CDWs and is described by the nondegene-
rate electron spectrum branch �3( )p . The portion of
the FS gapped by the CDW instability (partial gap-
ping) is determined by the dielectric-gapping (dielec-
trization) parameter
� � N /Nd0 00 0( ) ( ), (2)
where N N Nn d0 0 00 0 0( ) ( ) ( )� is the total initial
(above Td) electronic DOS on the FS, and Nd0 0( )
and Nn0 0( ) are the relevant DOSs on the degenerate
(d) and nondegenerate (n) FS sections, respectively.
In principle, CDWs may be commensurate or in-
commensurate with the background crystal lattice. In
the excitonic insulator model, the Coulomb-induced
distortion below the transition temperature, Td , is
commensurate. Moreover, the phase of the order pa-
rameter in excitonic insulators is always pinned
[30,31] and ~� is an either positive or negative quantity
[2,11,12,32,33]. On the other hand, in Peierls insula-
tors, incommensurate CDWs with the order parame-
ters ~� �� e i
may exhibit a rich dynamics, although
in the direct current measurements they are usually
pinned with arbitrarily frozen phases
[3,23].
In the framework of the approach adopted, the
partially-gapped nonsuperconducting CDW metal
(CDWM) in an absence of the external magnetic field
H is described by the following temporal Green’s
functionsGij ( )� , where i,j = 1, 2, 3 are the subscripts
labeling the FS sections (see above):
G G Gd11 22� � , (3)
G G Gc12 21� � , (4)
G Gn33 � . (5)
For all the other ij combinations, Gij � 0. The func-
tion Gc describes the electron–hole pairing. It is
«normal» in a conventional sense [10], since it is not
a product of either creation or annihilation operators
only, but is, nevertheless, «anomalous» in analogy
with the Gor’kov Green’s function, because it is pro-
portional to the CDW order parameter ~�. All techni-
cal details of the calculations and explicit expressions
for the functions Gd , Gc, and Gn can be found in our
previous publications [5,11,12].
Thus, one sees that in the Bilbro–McMillan model
[29] adopted by us here and in accord with the divi-
sion of the FS into d and n sections, the electron states
are of two different kinds, dubbed from here on as n
and d states. Nevertheless, it is important to compre-
hend that whatever the distinctions of the electron
spectrum between quasiparticle branches, the whole
system has a common chemical potential pinned to
that of the metallic n component and disposed inside
the dielectric gap � inherent to d states. On the other
hand, in the model of the doped excitonic insulator,
the Fermi level is supposed to be located above or be-
low the gap edge in its nearest neighborhood. Decades
ago, a significant enhancement of the superconducting
Tc due to the DOS increase in the indicated energy
range was expected to happen if such a situation
would have been realized [2,34]. Unfortunately, these
hopes turned out to be vain and in all compounds in
which superconductivity and CDW gapping have been
proved to coexist, the latter is detrimental to the for-
mer [11,12].
So, in the reconstructed CDW phase below Td , the
density of the n states, Nn ( )� , may be considered as
that in the absence of CDW gapping. Hereafter, the
energy distance from the Fermi level will be denoted
as �. Since the phenomena investigated in this publi-
cation are determined only by the states in a narrow
shell near the FS, the energy dependence of Nn ( )� can
be neglected, i.e. N Nn n( ) ( )� � 0 0 . At the same time,
the energy spectrum of the d states involves a dielec-
tric gap below Td , so that its DOS takes on a super-
conducting-like appearance [2]
N Nd d( ) ( )
( )
�
�
�
�
�
�
�
0
2 2
0
�
�
. (6)
As was shown by Frenkel [35,36] (see also extensive
accounts in Refs. 9, 37), the tunnel current J across the
biased barrier between metal electrodes is given by an
algebraic sum of the forward and backward compo-
nents. The voltage dependence of J is exponential for
large and Ohmic for small V [37]. We shall not extend
the subsequent analysis beyond the Ohmic regime,
since experimentally relevant dielectric gap energies
fall within the range of 0.5 meV < � < 30 meV,
whereas the deviations from Ohm’s law, indicating a
changeover to the Fowler–Nordheim tunneling, emerge
when the electron energy gain eV becomes comparable
to the conduction band width W for either of elec-
trodes. (Hereafter, e > 0 is the elementary charge).
Really, in the majority of metals the energy W ex-
ceeds 1 eV, so that the existing power-law corrections
to conductances G V dJ/dV( ) � of the tunnel junc-
tions involving such metals [9] are not important for
our purposes. One can imagine, however, a hypotheti-
cal situation when more than one conduction band for
each metal take part in tunneling, which is plausible
for narrow-band metals. Then, additional features in
G(V) may appear [38].
In studying tunnel currents between CDW metals,
we shall use a fruitful analogy between the latter and
BCS superconductors. The standard way of handling
tunneling between superconductors is the tunnel
Hamiltonian method [4,39,40]. Then, J(V) constitutes
an integral of electronic DOSs and the difference be-
Spin-polarized electron tunneling between CDW metals
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 79
tween the Fermi distribution functions for both elec-
trodes [4,9,41]. Insofar as the conductivity in the super-
conducting state is Ohmic, one can introduce a unique
parameter R representing the junction resistance in the
normal state. The quantity R is inversely proportional to
the averaged square of the tunnel matrix elements
[4,42]. The same theoretical approach has been demon-
strated to be applicable for tunneling between normal
CDW metals and superconductors [5,11,12].
2.1.2. Nonzero magnetic field
If the external magnetic field H (the spatial z axis
is chosen to be aligned with H) is switched on, the d
and n states exhibit quite different paramagnetic pro-
perties. To describe them properly, it is convenient at
first to restrict the respective reasoning to the limiting
situation T = 0. For nonzero H (hereafter we consider
values of H less than the paramagnetic limit Hp for
CDWM, see below), electrons with the spin projec-
tion s /z � 1 2 onto H increase their energies by �BH� ,
while the electrons with the opposite spin direction,
s /z � �1 2, reduce their energies by the same amount
[43]. Here �B e / m c� �� � 2 is the effective Bohr mag-
neton, � is Planck’s constant, c is the velocity of light,
and m� is the effective mass of the current carriers.
Henceforth, quasiparticles with either spin direction
will be labeled by «+» or «–». The quasiparticle level
scheme is shown in Fig. 1.
Quasiparticles belonging to the n section, for which
the Fermi level segregates occupied and empty states,
behave in a conventional manner inherent to normal
metals [43,44]. Namely, the states from the «+» spin
subband, for H = 0 coinciding in energy with its «–»
counterpart and, therefore, equally populated, shift
upwards in energy. As a consequence of the quasi-
particle transfer from the «+» to the «–» subband, the
former becomes more and more depleted as H in-
creases, whereas the number of occupied states in the
«–» subband rises simultaneously by the same amount.
This field-induced spin-polarization results in a change
of the chemical potential ~�, the latter coinciding with
the Fermi energy EF of n electrons at T = 0. For small
H, the relative corrections to ~� are of the order of
( )�B FH/E� 2. Since we are interested in the effects
when �BH� is, at least, smaller than �, the inequality
( )�B FH/E� ��2 1 is valid and we may neglect the
changes to ~� altogether. It is necessary to remind that
in itinerant Stoner ferromagnets this is not the case
and ~� is altered conspicuously by the respective spin
polarizations [44] (see also an account of concomitant
phenomena in Refs. 45, 46).
Thus, the electronic DOS and the Fermi distribu-
tion function in the phenomenological expressions for
the tunnel current are not affected, in a first approxi-
mation, by magnetic fields, which are of the order of
the energy parameters reflecting many-body gapping
of the parent electron spectrum [4].
The paramagnetic splitting of quasiparticle states
from the gapped FS sections can be examined analo-
gously to that for superconductors [13,14]. The reason
of the similarity is due to the fact that both CDW
(electron–hole) and Cooper pairs are spin-singlet and
therefore are prone to the destructive action of the
Zeeman splitting [15–18,47,48]. As a result, the qua-
siparticles of the gapped «+» and «–» subbands shift
in opposite directions in energy for H � 0. All spin-flip
processes leading to the smearing of the ideal splitting
are ignored hereafter, because we are interested in a
qualitative picture only.
Once formed from the praphase, the electron sys-
tem of the partially-gapped CDWM is stable against
the influence of magnetic fields in the range defined
below, the chemical potential ~� being pinned at the
original Fermi level.
When T is finite, the Fermi distribution factors are
no longer step-like functions and the thermally ex-
cited electron-like and hole-like quasiparticles appear
above and below the gap �, respectively. At the same
80 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
d d nn
CDWM� CDWM
�
�
�
�
�
�
�
�Fermi level Fermi level
� �+ H�B�� �+ H�
B
� �– H�B
–� �+ H�B
–� �– H�B
�� �– H�
B
–�� �+ H�B
–�� �– H�B
Fig. 1. The energy level scheme in a tunnel junction be-
tween partially gapped metals with charge-density waves
(CDWs) affected by an external magnetic field H at zero
bias voltage. Quasiparticle energies for non-nested (n) and
nested (d) Fermi surface (FS) sections are depicted sepa-
rately on both sides of the insulating barrier. �� and � are
the CDW gaps on the left and on the right, respectively.
�B
� is the Bohr magneton. «+» and «–» denote spin sub-
bands with projections along H and in the opposite direc-
tion. Dashed and solid lines correspond to the possible
tunnel transitions without spin flipping, which make con-
tribution to the current components that do not depend or
depend on H, respectively.
time, the chemical potential ~� decreases with T, the
relative correction being of the order ( )k T/EB F
2 [4].
Here kB is the Boltzmann constant.
In what follows, we shall describe tunneling in a
magnetic field H � 0 with the help of the Green’s
functions Gd, Gc, and Gn mentioned above. The only
modification, in comparison to the expressions of
Refs. 5, 11, 12, is that now the number of Green’s
functions is doubled: six relevant H-dependent func-
tions Gds, Gcs, and Gns are denoted by an extra sub-
script s � �. They are functions of the relevant vari-
ables � �� BH� , the signs beeng inverse to those of s.
2.1.3. Paramagnetic limit for CDWMs
As has been indicated above, there exists Pauli li-
mitation to CDW pairing similar to the Clog-
ston—Chandrasekhar limit [47,48] for superconduc-
tors. Since the Meissner orbital effect is absent in
excitonic or Peierls insulators [1,2,12], the paramag-
netic effect manifests itself here just as it is. The eval-
uation of the paramagnetic limit Hp for CDWMs is
methodologically the same as in the case of BCS su-
perconductors [4]. Specifically, one should compare
the free energy of the partially-gapped phase �FCDWM
with that of the paramagnetic state in the presence of
the magnetic field �Fp . Moreover, we should take into
account that the paramagnetic CDWM phase (the
analog of the Sarma state in superconductors) is ener-
getically unfavorable [49].
Since the gap �( )T appears only on the d (nested)
FS sections, we obtain
( )�
�
B pH� �2
0
2
2
� , (7)
where � �0 0� �( )T , so that
Hp
B
�
�
�0
2�
�
. (8)
The reduction of the actual Hp in comparison to the
limiting value of the complete gapping (� � 1) should be
allowed for when comparatively analyzing orbital and
spin effects in CDW substances. In particular, one
should mention organic substances�-(ET)2MHg(SCN)4
(M = K, Tl, Rb, etc.) [18,50–52].
2.2. Current–voltage characterictics
General expressions for quasiparticle currents across
tunnel junctions between dissimilar CDWMs (a
CDWM�–I–CDWM junction) are given in Appendix A.
But the main features of the investigated phenomena
are appropriate also to a simpler case of symmetrical
junctions with identical CDWM electrodes. In the
Bilbro—McMillan model [29] this means equality of
the parameters ~� and �. At the same time, the I–V and
G–V characteristics become much less cumbersome.
Indeed, the singularity positions, depending on the
CDWM� and CDWM gap magnitudes, merge in the
symmetrical case, and certain pre-integral factors be-
come equal.
But in making use of the emerging simplifying sym-
metry one should be very careful. For example, con-
sider the pair Jcn and Jnc�. It is easy to ascertain from
generic equations (A.3) or by analyzing the transla-
tion-containing symmetry properties (A.17) and
(A.18) of the current components, that J eVcn ( )
��J eVnc ( ) 0. If we calculate the overall charge
transfer regardless of the spin projection, the pairs of
components like Jcn and Jnc� may be neglected from
the outset. However, it is a spin-splitting analysis. A
premature mutual cancellation of the terms Jcn and
Jnc� would result in misleading results for each of the
«+» and «–» components. Therefore, the best way to
automatically avoid such traps is to add up the compo-
nents with a certain s separately before making the fi-
nal comparison between J V ( ) and J V�( ).
Nevertheless, for symmetrical CDWM–I–CDWM
junctions, we can exclude the cd � and dc � compo-
nents from consideration, since J eV J eVcds dcs( ) ( )� �
for each s.
Thus, a complete set of components of the qua-
siparticle tunnel current through a symmetrical
CDWM-I-CDWM junction is as follows:
J
eR
d K f eV f eVdd �
��
�
� � ��
�
� � � � �
2
4
| | ( , )| | ( , )� � � �� �
(9)
J
eR
d K f eVcc�
��
�
� � ��
( )
( ) ( , ) ( )
�
� � � �
�
�
2
4
sgn sgn� � �
� �f eV( , )�� � , (10)
J
V
Rnn� �
�( )1
2
2�
, (11)
J
eR
d K fdn�
��
�
�
�
�
� �
� � �
( )
| | ( , ),
1
4 � � � (12)
J
eR
d K eV f eVnd �
��
�
�
�
� ��
� �
� � �
( )
| | ( , ),
1
4 � � �
(13)
J
eR
d K fcn�
��
�
�
�
�
� �
� � �
( )
( ) ( , )
1
4
�
�sgn � � , S(14)
J
eR
d K eV f eVnc�
��
�
�
�
� ��
� �
� � �
( )
( ) ( , ),
1
4
�
�sgn � �
(15)
Spin-polarized electron tunneling between CDW metals
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 81
where the factor K K eV� ( , )� , generated by the
Fermi distributions of current carriers in both elec-
trodes, and the tunnel resistance R are determined by
Eqs. (A.5) and (A.4), respectively.
For T = 0, all the current components, except the
trivial Ohmic term Jnn� , can be expressed in terms of
elliptic integrals. The relevant expressions are given
in Appendix B. For T � 0, numerical calculations were
made (see the next Section).
3. Calculations
The representative quantities of the setup under in-
vestigation (a symmetrical CDWM–I–CDWM junc-
tion in a magnetic field) are as follows: the critical
temperature of the CDW phase transition Td or,
equivalently, the zero-temperature dielectric gap
�0 � ( )� �� Td , and the gapping parameter � [Eq. (2)]
in either electrode, the junction resistance R [Eq.
(A.4)], the temperature T, and the external magnetic
field H. Here � � 178. ... is the Euler constant. Hereaf-
ter, we use the dimensionless parameters: t k T/B� �0
and h H/B� �� �0.
3.1. Conductance—voltage characteristics
It is well-known that the differential tunnel G–V
characteristics dJ(V)/dV are much more informative
than the original I–V characteristics J(V), with the
former acting as an amplifier of the gap-driven pecu-
liarities [9]. In particular, the G–V characteristics
give direct information about an energy dependence of
the electronic DOS, renormalized due to Cooper
[4,8,9,53] or CDW [5,54] pairings for junctions be-
tween «normal» electrodes and those having a «gap-
ped» electron spectrum. Thus, for brevity, we shall
confine ourselves below to the analysis of tunnel G–V
characteristics and introduce dimensionless spin-de-
pendent conductance components
g RdJ /dVijs ijs� . (16)
The G–V characteristics for a symmetrical
CDWM–I–CDWM tunnel junction are shown in
Fig. 2,a for the cases where the external magnetic
field H is absent or present. For the sake of definite-
ness, we shall restrict the numerical calculations in
this Section to the case
� 0. The main properties of
the overall conductance versus voltage dependence
and its splitting in the magnetic field survive for arbi-
trary
(see a discussion in Appendix C). One can
readily see that each square-root singularity from the
positive or negative voltage branch is split into two
peaks, whereas the step-like peculiarities remain
unsplit. Moreover, the conductance in each split peak
has a predominant (not unique) polarization indicated
82 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
� = 0.5
� �
g
=
R
d
J/
d
V
3
2
1
0
t = 0.01
a
eV/� 0
–3 –2 –1 0 1 2 3
h = 0.2
h = 0
g
=
R
d
J/
d
V
2
1
0
eV/� 0
–3 –2 –1 0 1 2 3
g
g
g + g
–
+
–+
b
Fig. 2. The dimensionless conductance g = RdJ/dV of the
symmetrical tunnel junction between similar CDW metals is
shown as a function of the dimensionless bias voltage
eV/�0. Here R is a tunnel junction resistance in the undis-
torted state above the critical temperature Td of the CDW
transition, �0 � � �T /d is the CDW gap at T = 0, J is the
quasiparticle current, e is the elementary charge, � = 1.78...
is the Euler constant, T is temperature, t k T/B� �0 is the
dimensionless temperature, kB is the Boltzmann constant,
h H/B� �� �0, � � N /Nd0 00 0( ) ( ), N0(0) = Nn0(0) + Nd0(0),
Nd0(0) and Nn0(0) are the initial (above Td) electronic den-
sities of states on the d and n FS sections, respectively. The
signs + and – indicate a predominant spin polarization of the
peaks. (b) A decomposition of the total conductance g, dis-
played in panel (a), into two summands gs, each comprising
contributions of current carriers with a corresponding spin
polarization s � � .
by a or � sign. Figure 2,b illustrates a decomposition
of the resulting G–V characteristic for H � 0 into two
components with different spin polarization of current
carriers. One also sees a novel remarkable feature,
namely, the peaks of the «+» component move apart
in the magnetic field, whereas their «–» counterparts
converge. Such a behavior differs drastically from that
appropriate to S–I–N junctions, for which the g V ( )
and g V�( ) peaks move in opposite directions, irrespec-
tive of the voltage polarity. As to the tunneling
through a S�–I–S junction, the peak-to-peak separa-
tion does not depend on H and is the same for either
sign of s [13,14]. To explain the distinction between
the superconducting and CDW cases, we should con-
sider in detail each current constituent involved (re-
call that the cd � and dc � components were excluded
from analysis for the symmetrical CDWM–I–CDWM
configuration due to their mutual compensation).
First, as stems from the speculations in Appen-
dix A, the tunneling between the gapped FS sections,
as well as the transfer of the electron–hole pairs across
the barrier, do not induce any peak splitting in the as-
sumed absence of spin flips. This concerns the contri-
butions dd� , cc � , cd� , and dc � to both J(V) and
G(V), and is similar to what happens in S�–I–S junc-
tions. At the same time, the «normal» nn � compo-
nents should reveal no noticeable magnetoresistance
under the action of relatively small magnetic fields
H H /B p� � �0 2 2 80. .� � �, which are, e.g., far below
the fields experimentally found necessary for orbital
quantization in the organic material �-(BEDT-TTF)2
KHg(SCN)4 [27,51]. Here, relationship (8) was taken
into account. Had it not been for other components,
the G–V characteristics of CDWM–I–CDWM junc-
tions would have possessed only feature points at
eV � �2�, as it is the case for S–I–S tunneling
[4,8,9].
Now let us pass on to the tunnel processes that con-
nect the n FS section of one electrode and the d FS
section of the another one (Fig. 3). For the sake of
definiteness we start our analysis with the «+»-polar-
ization conductance. The components dn+ and nd+ are
contributions to the G–V characteristics of the type
known from the theory of superconducting splitting
[13,14]. Using this analogy, it can be shown that the
dimensionless conductances gdn+, nd+ = RdJdn+, nd+ /dV
can be represented as
g eV H d K eV F Hdn
��
�
�� ��( , ) ( ) ( , ),� � �0 (17)
g eV H d K F eV Hnd
��
�
�� ��( , ) ( ) ( , )� � �0 . (18)
Here, F H N Hd
�
��( ) ( )( , ) ( , )� � , i.e., the densities of
the gapped « »-states for the relevant electrodes
N H N
H H
d d
B B
� �
� � �
�
� � �
�
( ) ( )
( )
( , ) ( )
| | (| | )
(
�
� �
� �
� �
0 0
2
�
BH� ��) ( )2 2�
(19)
[cf. Eq. (6)], and the kernels K0
�( ) are the derivatives
of the kernel K eV( , )� [Eq. (A.5)] in the integrand of
Eq. (A.3). The quantities K0
�( ) are �-like functions
with the maxima at the Fermi levels of the unprimed or
primed electrodes, respectively, in the degenerate case
considered. In the limiting case T = 0, the V depend-
ence of the conductance g eVdn ( ) coincides with that
Spin-polarized electron tunneling between CDW metals
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 83
CDWM�
CDWM
g (eV)
–V +V
ar
b
.u
n
its
ar
b
.u
n
its
ar
b
. u
n
its
eV
�
a
b
cg
dn+ gnd+
1�H
1�0 2
2
�0
�H
K ( )��0
N ( )��
d+
N ( )�d+
K ( )�0
1 1 2 20 H 0 H
dn+, nd+
1H2H
2�H1�H
10 2�0+1�0 20+
0 � �+ H�B� �– H�B–� �+ H�B–� �– H�B –� �
Fig. 3. The scheme of the peak spin-splitting for the con-
duction components gdn+ and gnd+ in the magnetic field
H. (a) The energy dependence of the gapped DOSs
Nd
� ( )� for the primed electrode at H = 0 (dotted curves,
subscript 0) and H � 0 (solid curves, subscript H). The
function K0
�( )� is a T-dependent kernel [Eq. (18)]
originating from the Fermi distribution of the n electrons.
(b) The same as in panel (a), with an accuracy up to no-
tation, for the unprimed electrode. All elements of this
panel are shifted with respect to those in panel (a) by the
value of eV if a bias voltage V is applied. (c) The result-
ing contributions gdn+ and gnd+ to the G–V characteristic.
Any peak in a certain conductance component appears if a
DOS singularity in one panel is overlapped by the kernel
function in the other one [those three relevant elements
(the peak, the DOS, and the kernel) are drawn using the
same style of the curve]. The numbering of the peaks cor-
responds to the «parent» DOS singularities in panels (a)
and (b). See further explanations in the text.
of N eV Hd
� ( , ) [see Eq. (17)], while g eVnd ( ) be-
comes proportional to N eV Hd �( , ) [see Eq. (18)].
When H = 0, the electrode Fermi levels are located
at the center of the relevant CDW gaps and coincide
in the absence of the bias voltage. The bias magni-
tudes, needed either to shift the Fermi level of the
unprimed electrode (see Fig. 3,b, K0) downwards in
reference to the lower edge of the CDW gap of the
other electrode (Fig. 3,a, 10
� , dotted) or to step it up
with respect to the upper edge of the same gap (Fig.
3,a, 20
� , dotted) are equal. Hence, the positions of rel-
evant singularities in the term g eVdn ( ) are equidis-
tant from V = 0 (Fig. 3,c, component 10
� of the com-
bined peak 1 20 0
� and component 20
� of the combined
peak1 20 0 � , dotted). The same is valid for the contri-
bution g eVnd ( ) [Eq. (18)] which contains peak 10,
positioned at the same bias as the 20
� one, and peak 20,
disposed at the same bias as the 10
� one. Their ampli-
tudes are also pairwise equal, since for the symmetri-
cal junction K K0 0( ) ( )� �� � and N Nd d( ) ( )� �� � . Ev-
idently, one cannot distinguish between contributions
of the pair elements to the corresponding features of
the G–V characteristic, the latter therefore being
unsplit (Fig. 3,c, dotted peaks 1 20 0
� and 1 20 0 � ).
When the magnetic field is switched on but the
junction is not biased, the «+» subsystem in each elec-
trode shifts upwards in energy by ��BH relative to
their common Fermi level, which remains fixed (see
Fig. 3,a, solid curves, and Fig. 3,b, dashed curves).
Thus, different bias voltages with different H-driven
offsets should be applied now to obtain peaks in either
of conductance terms. In particular, the singularities
in the g eVdn ( ) component shift to � �� �BH posi-
tions (Fig. 3,c, solid peaks 1H
� and 2H
� ) and the singu-
larities in the g eVnd ( ) component shift to � � �� �BH
positions (Fig. 3,c, dashed peaks 1H and 2H). The no-
menclature of peaks in Fig. 3,c coincides with that in
Figs.3,a or 3,b, explicitly indicating the gap edge re-
sponsible for each feature. Thus, either of the G–V
characteristic peaks, being combined in a zero mag-
netic field, splits into two smaller ones for H � 0. The
peaks belonging to the gdn or gnd components shift
symmetrically in opposite directions of the V axis,
which can be deduced directly from Eqs. (17) and
(18). On the other hand, the apparent motion of the
peak pairs (1 1H H
� � and 2 2H H
� � ), each originating
from both gdn and gnd terms, is directed inwards
and outwards, respectively. But even this rather com-
plicated picture does not signify the end of the story.
The involvement of the gcn nc , components, di-
rectly descending from the electron–hole pairing,
changes the situation radically and makes the resul-
ting G–V characteristics highly unconventional. In
particular, the component gcn is also of the form (17)
but with another function F H N Hd
�
��( ) ( )( , ) ( , )� � .
The term g Vcn ( ) has the same functional depen-
dences and amplitude of singularity at the same volt-
age values as the component gdn does, but, contrary
to the latter and due to the non-trivial properties of
the Green’s function Gc( )� , it is antisymmetric with
respect to �. As a consequence, the gcn singularity
enhances its counterpart of gdn on the positive volt-
age branch and almost compensates the singularity of
gdn on the negative-V branch, transforming it into a
cusp. This is illustrated in Fig. 4. The resulting pattern
moves as a whole along the V axis towards larger posi-
tive V if an external field H is applied. At the same
time, the V dependence of the sum g gnd nc consti-
tutes a specular reflection of the curve g Vdn ( )
g Vcn ( ) relative to the g axis. Therefore, the overall
conductance g V ( ), which is a sum of all the four re-
levant contributions discussed above, has two peaks
symmetrically moving apart and two cusps symmetri-
cally crowding together along the V axis as the mag-
netic field H grows.
The conductance behavior obtained for the
CDWM–I–CDWM sandwich is due to the fact that
the FSs have both n and d sections. This means that
the whole configuration can be viewed as a combina-
tion of two asymmetrical junctions. Hence, there are
two unequal current components Jnd and Jdn con-
84 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
g
dn+g
=
R
d
J/
d
V
1
0
eV/� 0
–3 –2 –1 0 1 2 3
g
dn+
g
cn+
g
cn++
Fig. 4. An illustration of the component compensation:
the selected contributions gdn+, gcn+ to the overall con-
ductance g and their sum. An almost complete mutual
compensation of the logarithmic singularities for negative
V and their amplification for positive V is demonstrated.
The relevant parameters are the same as in Fig. 2.
necting the gapped and non-gapped quasiparticle sub-
systems. Moreover, the terms Jcn and Jnc , related
to the CDW pairing, result in the appearance of
antisymmetrical conductance peaks. The interplay of
all constituents leads to the effect described above.
An analysis of the sum g g g gdn cn nd nc� � � �
is performed in the same way. The resulting two peaks
converge symmetrically, whereas the two cusps sym-
metrically move apart, with increasing H. Since the
cusps of one polarization superpose on the singulari-
ties of the other polarization and there is no spin filter
in the circuit, the cusps may be inconspicuous against
the background of the singularities (see Fig. 2,b).
Taking into account the other components, not exhib-
iting a peak splitting of any nature, we obtain the
G–V characteristic shown in Fig. 2, which reveals
four H-dependent polarized peaks at eV HB� � � �� �
and two fixed jumps at eV � �2�.
The pattern obtained is a consequence of the choice
� 0 made above. If one assumes another realistic sit-
uation with the order parameter phase averaged out
(see discussion in Sec. 3.2), components (14) and (15)
should disappear, so that the G–V characteristics will
change substantially. To easily embrace all possible
cases with varying
, it is convenient to analyze all
conductance components for T = 0. The results are
summarized in Appendix C.
3.2. Influence of different factors on the G–V
characteristics
Figure 2 distinctly reveals the main peculiarities
of the G–V characteristics for a symmetrical
CDWM–I–CDWM junction. Namely, there exist dis-
continuities at H-independent locations eV � �2�, de-
termined merely by a dielectric gap value. Besides,
there are H-driven square-root singularities shifted by
� ��BH from the basic eV � �� bias values. It is obvi-
ous that the larger is the magnetic field H, the stron-
ger are the inward and outward displacements of the
singular conductance peaks. As was clearly demon-
strated above, the apparent splitting has a dual nature
reflecting both the intrinsic configurational asymmetry
of the junction concerned and the Zeeman effect.
Nonzero temperatures smear the overall curves
g(V) and especially the singular peaks. It is shown in
Fig. 5,a that these dependences are highly sensitive to
the dimensionless parameter t. Therefore, to observe
the predicted splitting in the magnetic field one
should either heavily reduce T or use CDWMs with
large Td ’s and hence CDW gaps �0.
The influence of the control gapping parameter � on
the G–V characteristics is demonstrated in Fig. 5,b. It
is readily seen that the increase of � reduces the mi-
nimal value of g(V), determined by the gnn� contri-
butions, and enhances the jump amplitude at eV � �2�.
As stems from Fig. 5,b and Eqs. (12)–(15), to improve
the observability of the predicted splitting effect one
should maximize the factor � �( )1 � , i.e., those sub-
stances with� close to 0.5 appear to be more promising.
4. Discussion and conclusions
The predicted splitting of the G–V characteristic
induced by the paramagnetic action of the magnetic
field H can be observed, in principle, for any CDW
metal, i.e., the electron spectrum gapping should be
incomplete, which is usually the case for a large num-
ber of low-dimensional Peierls metals with incommen-
surate CDWs. The specific requirement is to maintain
a balance between d and n portions of the FS (� � 0 5. ,
see Sec. 3.2). This means that an external control of
Spin-polarized electron tunneling between CDW metals
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 85
� = 0.5
t = 0.03
t = 0.2
t = 0.1
h = 0.2
g
=
R
d
J/
d
V
0.5
0
eV/� 0
–3 –2 –1 0 1 2 3
1.0
1.5
t = 0.05
h = 0.2
g
=
R
d
J/
d
V
0.5
0
eV/� 0
–3 –2 –1 0 1 2 3
1.0
1.5
a
b
� = 0.5
� = 0.7
� = 0.1
Fig. 5. The dependences g eV/( )�0 for different t (a) and
different � (b).
the parameter � (e.g., by the applied pressure) may be
crucial for the success of the experiment. The magni-
tude of the CDW gap � defines the natural scale for
the field H. Nevertheless, large H and � do not mean
that the investigations can be carried out at high T.
First, the spin splitting will be smeared and not re-
solved even for moderate T/� (see Fig. 5). Second, in
the close neighborhood of the paramagnetic limit it is
possible to enter the region, where, in analogy with
the case of superconductors, fluctuations [55] and the
influence of the magnetic and spin–orbital scatterings
[56], not covered by the present theory, may become
important.
To make the predicted effects observable, it is cru-
cial (at least in the symmetrical setup) for the compo-
nents Jdn and Jnd , describing the tunnel current
linking d and n FS sections, to survive. In our
phenomenological approach, when all matrix elements
of the tunnel Hamiltonian [8] are considered equal
(tunneling is not considered directional), it is the
case. In the other extreme limit of the complete tun-
neling directionality [57], the components Jdn and
Jnd may not exist and the spin splitting may disap-
pear. In principle, any degree of directionality is pos-
sible. The actual realization of the intermediate
situation stems from the analysis carried out for tun-
nel spectroscopic studies of high-Tc oxides [58–63].
Therefore, one should consider the limit of no direc-
tionality and equal probability of all processes con-
necting different FS sections (a unique quantity R) as
an idealized picture, so that for some junctions the
feature points at eV � �� might be weakened in com-
parison to those at eV � �2�. As a consequence, the
spin splitting might be also partially suppressed.
The appearance of superconductivity for smaller
T T Tc d� � in any specific CDW substance (see re-
views [11,12]) may serve as a clear indication that this
material is a metal rather than an insulator, and,
therefore, of its ability to demonstrate the Zeeman
spin splitting. Hence, the low-dimensional metals ex-
hibiting CDW instabilities, such as NbSe3, Nb3Te4,
Li0.9Mo6O17, Tl2Mo6Se6, layered dichalcogenides, al-
loys with the A15 and C15 structures, Lu5Ir4Si10,
P4W14O50, tungsten bronzes doped by alkali metals, and
solid solutions BaPb1–xBixO3 may serve as good candi-
dates. Other monophosphate bronzes (PO2)4(WO3)2m,
doped and undoped by alkalis, are also suitable par-
tially gapped CDWMs [64,65].
An important conjecture should be made concern-
ing the magnitude of the CDW gaps. For supercon-
ductors, the ratio 2 0�( )/k TB c is usually of the order
or somewhat larger than the BCS value 2� �/ � 3.52
[4]. The only exception is MgB2. In that case, the very
character of the superconductivity is as yet ambiguous
and an intrinsic two-gap scenario is often accepted
(see the relevant critical discussion of this concept in
Refs. 66, 67). On the other hand, the observed depen-
dence �( )T in CDWMs and CDW insulators has a
generalized BCS-like form. Namely, in the coordinates
� �( ) ( )T / T � 0 vs T/Td , the data follow the Mühlsch-
legel curve, whereas the ratio 2 0�( )/k TB d essentially
exceeds the BCS weak-coupling limit (such a behavior
is described by the phenomenological scheme [68]).
For example, this quantity is about 13 in the insula-
ting La1.67Sr0.33NiO4 [69]. Layered dichalcogenides
2H–TaSe2, 2H–TaS2 and 2H–NbSe2 are marked by
extremely large values 2 0�( )/k TB d=15.2, 15.4, and
23.9, respectively [70]. In NbSe3, with its two CDW
transitions at Td
low K� 59 and Td
high K� 145 [11,12],
the respective ratios, as was shown by direct tunneling
studies [71], fall into the ranges 2 0� low low( )/k TB d �
� 11.8–14.3 and 2 0� high high( )/k TB d � 11.4–14.4.
Taking the observed Gaussian spread of the CDW
gaps into account gives somewhat lower values
2 0� low low( )/k TB d � 9.2 and 2 0� high high( )/k TB d � 8.2
[72]. Larger gap-to-Td ratios are favorable for our pur-
poses, since to clearly observe the splitting, one
should avoid high temperatures during the experi-
ment, although large gaps are convenient.
Superconducting cuprates can be suggested as an-
other class of substances in which the CDW-triggered
spin-splitting in the magnetic field can be observed.
Two kinds of features testify that CDWs exist in a
number of high-Tc oxides. The first one is a dip–hump
structure of G(V) for voltages exceeding the positions
of the superconducting gap maxima [72,73], while the
other one is the so-called pseudogap, �� , persisting
both above and below Tc [11,12,74–78]. We think
that the CDW origin of those peculiarities is quite
plausible, whereas the most popular interpretation
based on the precursor Cooper pairing (see, e.g., re-
view [79] and references therein) should be rejected,
at least because the applied magnetic field influences
true gaps and the �� ’s in a different way [80–82].
Moreover, the predominantly paramagnetic character
of the magnetic field influence on the �� ’s is attested
by the existence of Zeeman scaling (proportionality)
between the pseudogap-closing field Hpg and the
characteristic pseudogap temperature T� resistively
determined in Bi2Sr2CaCu2O8—y [83,84]. It means
that the huge orbital Meissner effect is absent for �� ,
so that at least in Bi2Sr2CaCu2O8—y it may be identi-
fied with �. We recommend recent comprehensive re-
views [11,12,74–78,85] to compare the arguments be-
longing to various analysts in this field.
Low T of measurements may turn out to be a neces-
sary condition for resolving the spin splitting of CDW
gaps (pseudogaps). Since one can make more definite
86 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
conclusions for pure CDW phase above Tc, it is advan-
tageous to carry out investigations in magnetic field
for substances with relatively low Tc much smaller
than both Td and �( )T . That is why we consider the
experiments of Ref. 86 very important. Specifically, a
well-resolved superconducting gap � and a pseudogap
� were found by the authors of Ref. 86 for
Bi2(Sr2–xLax)CuO6+� mesas with 10 K � Tc � 32 K. In
a search for the predicted effect in cuprates, one
should look through substances in which the follow-
ing conditions are satisfied: (i) a clear-cut resolution
between � and � [73,80–82,86–88], (ii) all the four co-
existent features, positioned at �� and �2�, should be
manifested, and (iii) the temperature of observation
must be as low as possible. The last requirement might
not be so severe as it seems at the first glance, because
the same H that drives the Zeeman peak splitting would
suppress superconductivity, making the CDW gap itself
open for probing.
Acknowledgements
A.M.G. is grateful to the Japan Society for the Pro-
motion of Science for support of his visit to the Hiro-
shima University (Grant ID No. S-03204) and to the
Mianowski Foundation for support of his visit to War-
saw University. The research has been partly sup-
ported by the NATO grant PST.CLG.979446 and the
grants COE (No. 13CE2002) and Scientific Research
(No. 15540346) of the Ministry of Education, Cul-
ture, Sports, Science and Technology of Japan. The
authors are also grateful to Jun Akimitsu (Tokyo),
Serguei Brazovskii (Kyoto), Kenji Ishida (Kyoto),
Yoshiteru Maeno (Kyoto), and Mai Suan Li (War-
saw) for fruitful discussions.
Appendix A: Tunnel current components
Let us consider the general case of two different
CDWMs with relevant parameters (~ �� , �� ) and (~, )� �
at both sides of the potential barrier created by an in-
sulating interlayer. The current–voltage (I–V) char-
acteristics J(V) for the quasiparticle tunnel current in
this junction are calculated by the tunnel Hamiltonian
method [4,10,40,42] in the first order of the perturba-
tion scheme [32,89]. Under the assumption that there
is no spin flipping while tunneling, the overall tunnel
current J can be described as consisting of the follow-
ing 18 terms:
J dij �
�
��
�
� ��Re �
�
� �
��
� � � � �
�d
G H G H
eV i
is B js B
�
� � � �
� �
Im ( ) ( )� �
0
, (A.1)
which correspond to various combinations of Green’s
functions and spin projections (s � �) for the two
electrodes. Here the quantities related to different
electrodes are primed or unprimed, the subscripts i
and j of the Green’s functions G( )� ’s equal to d, n, or
c [see Eqs. (3)–(5) for notations], and the signs in
the Green’s functions’ arguments are opposite to
those of the spin projection. Hereafter, the potential
of the primed electrode is taken as zero, so that the
bias voltage V comprises the electrostatic potential of
the unprimed electrode. After standard calculations
following the pattern of Refs. 5, 11, 12, we obtain the
expression
J Jijs
i j d n c
s
�
�
� �
�
, , ,
,
, (A.2)
with each component and the overall current depend-
ing on the bias voltage V, temperature T, and H. All
current components Jijs have the general form
J
eR
d K Z Hij i B�
��
�
�� � ��
1
4
� � �( ,~ )� �
� ��Z H eVj B( ,~)� �� � . (A.3)
The quantity
R e N N T
FS
� ��1 2
0 0
2
4 0 0� ( ) ( ) pq (A.4)
is the conductance (inverse resistance) of the junction
above Td, where both electrodes are in the non-gapped
state. The square of the modulus of the tunnel matrix
element Tpq is averaged over the FS, i.e., all matrix
elements of the tunnel Hamiltonian are taken equal.
Thus, we assume a unique tunnel resistance for every
current component. From the physical point of view
this means, in particular, no tunnel directionality,
which is possible, in principle [58–63]. The factor
K K V T
T
eV
T
� � �
�
( , , ) tanh�
� �
2 2
tanh (A.5)
stems from the Fermi distribution functions of the
two electrodes.
Each Z function in the integrand of Eq. (A.3) con-
stitutes a product
Z z fi i i( ,~) ( ,~) ( ,~)� � �� � �� (A.6)
of one of the characteristic functions
zd ( ,~) | |� � �� � , (A.7)
zc( ,~) ( ) ~� � �� �� sgn , (A.8)
zn ( ,~) ( )� � �� � �1 , (A.9)
and the factor
Spin-polarized electron tunneling between CDW metals
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 87
fi ( ,~)
( )
�
�
�
�
�
�
�
�
�2 2
, (A.10)
which describes the presence (~� � 0 for i d� and c) or
absence (~� � 0 for i n� ) of gapping;
( )x is the
Heaviside step function. For i n� , the Zn function re-
duces to
Zn ( ,~)� �� � �1 . (A.11)
When the gapping is absent in both electrodes
( )� �� � � 0 , the total current consists of Jnn� compo-
nents only and has a conventional Ohm’s form V/R
in the whole voltage range, independent of T and H.
Of course, in the general case, Ohm’s law is restored
for large enough voltages substantially exceeding ��
and �.
Tunnel currents between CDWMs across the barri-
ers of different transparencies making allowance for
the dependences of J on
and �
were studied in a
number of papers [21,90–93]. Contrary to some of
them [91–93], we shall consider the most general
setup, when
and �
are independent of each other.
The situation when
= �
will stem from the general
equations as a particular case.
Let us analyze the symmetry properties of different
current components. It is convenient to start from the
case H = 0. Such an analysis has been done earlier [5],
but to investigate below a more involved situation
with nonzero H, it is necessary to carry out an addi-
tional examination. We shall use the notation Jij0 �
� �J Hijs ( )0 , since in this case � �� � in the inte-
grands of (A.3), and therefore J Hij � �( )0
� ��J Hij ( )0 .
The components can be divided into symmetrical
and asymmetrical ones with respect to the voltage V.
The symmetrical components, for which an unusual
relation [5,54]
J V J Vij ij0 0( ) ( )� � (A.12)
holds, are those that contain the «anomalous»
Green’s function Gc once in the integrand [Eq.
(A.1)]. These are Jdc0, Jcd0, Jnc0, and Jcn0. Other
components, namely, Jdd0, Jcc0, Jnn0, Jdn0, and Jnd0,
satisfy the conventional equation [9]
J V J Vij ij0 0( ) ( )� � � . (A.13)
It is worthwhile mentioning that the integrand of Jcc0
includes a product of two Green’s functions Gc, the
anomalous symmetry properties of which compensate
each other.
The differential conductance G V dJ V /dV( ) ( )� is
a quantity of primary interest to experimentalists. The
same is true for its symmetry properties. The depen-
dence G(V) and similar V dependences of its compo-
nents Gij will be henceforth called the conduc-
tance–voltage (G–V) characteristics. The symmetry
relationships for G Vij0( ) are easily deduced from
those for J Vij0( ). Namely, for the dd0, cc0, nn0, dn0,
and nd0 components, they have the standard form [9]
G V G Vij ij0 0( ) ( )� � , (A.14)
whereas for the dc0, cd0, nc0, and cn0 terms, the
symmetry properties are anomalous:
G V G Vij ij0 0( ) ( )� � � . (A.15)
Going to the case H � 0, we should consider the
changes of the CDWM electron spectrum on both
sides of the junction under the influence of the exter-
nal magnetic field (see Fig. 1). As was indicated in
Sec. 2.1.2, the Fermi levels (chemical potentials) in
both electrodes, differing from each other by eV, re-
main practically intact when H is switched on. Hence,
the electron spectrum on the n FS sections is also un-
changed. On the other hand, the quasiparticle energy
subbands with the spin projection s = + on the d FS
sections in both electrodes shift upwards by the value
�BH� , while the subbands with the projection s = –
shift downwards by the same value.
Since tunneling is assumed to preserve spin values,
it is clear that a current component depends on H only
in the case where the subbands involved (one from the
primed electrode and the other from the unprimed
one) change their energy difference with increasing
magnetic field. Relevant links are shown in Fig. 1 by
dashed lines. Therefore, all 18 components can be di-
vided into three groups. The first one contains those
terms which do not depend on H. They are Jdds, Jccs,
Jdcs, Jcds, and Jnns, and for them
J eV H J eVijs ijs( , ) ( , )� 0 . (A.16)
The next group of current components, Jdn+, Jcn+ ,
Jnd–, and Jnc–, is shifted towards larger voltages for
H � 0, i.e.,
J eV H H J eVijs B ijs( , ) ( , ).� ��� 0 (A.17)
The remaining terms Jdn–, Jcn–, Jnd+, and Jnc+ move
in the opposite direction of the V axis, i.e.,
J eV H H J eVijs B ijs( , ) ( , ). ��� 0 (A.18)
Various links describing such H-dependent current
components are specified by solid lines in Fig. 1.
Those components play a crucial role, because they
lead to a new phenomenon revealed in junctions be-
tween CDWMs exposed to a magnetic field. Spe-
cifically, the overall tunnel current and peaks in
G(V), originating from the CDW gapping, are
spin-split even in the symmetrical setup. This is unlike
the total absence of splitting when both electrodes are
88 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
superconducting, whatever the relative magnitudes of
the superconducting gaps [13].
It is worthwhile to discuss one peculiarity, con-
cerning the shifts (A.17) and (A.18) of the I–V (and,
by implication, the G–V) characteristics along the V
axis. It happens that J eV Hij � � � �( , )0 0 0 for those
current components, although each component in the
sum (A.2) becomes zero in the absence of H and in the
absence of voltage: J eVij0 0 0( )� � . But this does not
signify any violation of the laws of quasi-stationary
electrodynamics, since only the total current (A.2),
for which J eV H( , )� � �0 0 0 irrespective of the mag-
nitude of H, has physical meaning.
Appendix B: Analytical expressions for tunnel
current components in asymmetrical and
symmetrical junctions at T = 0 and H = 0
Here, the analytical expressions for tunnel current
components across the insulating barrier between dis-
similar CDWM electrodes at T = 0 and in the absence
of the magnetic field are calculated. The derivation of
the corresponding expressions is straightforward al-
though cumbersome. The final results are displayed
below. Only the branches V > 0 of the components of
the I–V characteristic are explicitly shown, because
their negative V counterparts can be easily obtained
using the symmetry properties (A.12) and (A.13).
J V
eR
Ndd ( ) [ ( )� �
�
�
�
!
��0 2
� �
�� �]E
�
"
#
$
�
� �
�
�4
2
� � � �
� �
� �
[ ]
[ ]
( ) [ ( )]
N
N
eVK �
;
(B.1)
J Vcc( )� �0
� �
� � � � �
�
4
2
� �
�
% &� � � �
� �
cos cos ( ) ( )
[ ]
K eV
eR N
;
(B.2)
J V
eR
eV eVdn ( ) { ( ) ( ) ( ) }� � � � � � � �0
1
1 2 2� �
� � ;
(B.3)
J V
eR
eV eVnd ( ) { ( ) ( ) ( ) }� � � � � �0
1
1 2 2� �
� � .
(B.4)
J V
eV
eRNcd ( )
( )]
cos� �
� � � �
� �0
2� �
%
� � � �
�
� � �
� �
'
(
)
*
+
, �
�
�
!
"
#
$
2
2
Ï K
�
, , ( )
eV
eV
k k
� �
� �
; (B.5)
J V
eV
eRNdc( )
( )
cos� �
� � � �
�0
2� �
% &
� � � �
� �
� � �
� �
'
(
)
*
+
,
�
�
!
"
#
$
2
2
Ï K
�
, , ( )
eV
eV
k k
� �
� �
; (B.6)
J V
eR
eVcn ( ) cos ( )� � � � � � � �0
1
�
� �
�
� �
�
ln
( )eV eV 2 2�
�
; (B.7)
J V
eR
eVnc( ) cos ( )� � � �0
1
�
� �
�
�
ln
( )eV eV 2 2�
�
. (B.8)
Expression (11) for the current between n FS sections
conserves its form. Here K(x), E(x) and Ï(�/2,x,y)
are the complete elliptic integrals of the first, second,
and third kind, respectively. Their arguments are
k
eV
N
�
� � ( ) ( )2 2� �
(B.9)
and
� �
� �
�
N
N
2
2
� �
� �
, (B.10)
whereas the quantity N is equal to
N eV� � � �( ) ( )2 2� � . (10)
One can readily see that the analytical expressions
for the tunnel current between dissimilar CDWMs
differ substantially from their well-known counter-
parts in the case of the quasiparticle current between
different superconductors [8]. Namely, there is a sin-
gle term J Vsc( ) for the superconducting junction co-
inciding with our component J Vdd ( ) with an accu-
racy up to substitution of the superconducting gaps ��
and � for the CDW ones �� and �, while setting
� � �� � 1. Extra terms originate from the pairwise
combinations of the Green’s functions (3), (4) and
(5) appropriate to the currently investigated case of
the junction involving CDWMs.
The main qualitative distinction between supercon-
ductor- and CDWM-based junctions consists in the dif-
ferent form of the feature points. There is one break
point in the superconducting junction at eV � � � �,
where the current steeply changes from zero to the value
�
� �
J
eRsc �
��
2
. (B.12)
Spin-polarized electron tunneling between CDW metals
Fizika Nizkikh Temperatur, 2005, v. 31, No. 1 89
At the same time, from Eqs. (B.1), (B.2), (B.5) and
(B.6) it comes about that
� � �
� �
J eV
eR
( ) ( cos )( cos )� � �
� �
� �
�� �
2
1 1 .
(B.13)
One should note that there is an intrinsic asymme-
try in Eq. (B.13), i.e., �J depends on each of the
phases �
and
separately and in a different way. Both
phase factors cos �
and cos
will change their signs
for the opposite voltage polarity V < 0. A similar
asymmetry takes place in an asymmetrical junction
between CDW conductors [5]. The jump disappears in
the particular case of an excitonic insulator, where
each of the phase angles is either 0 or �, contrary to
what is appropriate to the superconducting tunneling.
The emergence of root singularities in Jdn, Jnd, Jcn
and Jnc components [see Eqs. (B.3), (B.5), (B.7) and
(B.8), respectively] is another important phenomenon
appropriate to asymmetrical junctions involving
CDWMs.
The phase dependences of the current components
Jcd, Jdc, Jcn and Jnc, represented by Eqs. (B.5),
(B.6), (B.7) and (B.8), respectively, were obtained
assuming definite constant values of the phases �
and
for both electrodes. Actually, an averaging of the
currents over the junction cross section, as a conse-
quence of the phase randomness, may wipe out these
terms. To preserve them, the usage of the break-junc-
tion technique confining the contact area would be of
benefit. Nevertheless, the resulting current J(V)
would differ substantially from its superconducting
analog even in this case. In particular, the jump
� � �J eV( )� � , expressed by Eq. (B.13), disappears,
contrary to what stems from the BCS theory and expe-
riments carried out for S�–I–S junctions [9]. Another
very important phenomenon that survives the averag-
ing is the square-root dependence of the components
Jdn and Jnd on the voltage [see Eqs. (B.3) and (B.4)].
The general results (B.1)–(B.8) can be substantially
simplified in the case of identical CDW parameters in
both electrodes (~ ~� �� � and � �� �). In that case the ex-
pression for the tunnel current takes the following form:
J V
eR
eV eV( ) ( ) ( ) ( )� � � � � �
�
�
!
0
1
1 2 12� � �
�
� � � �( ) ( )eV eV2 2 2 2� ��
� �
-
.
/
/
0
1
2
2
"
( ) ( )
( ) ( cos )
eV
eV
eVe
e2
4 1
2
2
�
�% � &
�
E
K
�
�
#
$
.
(B.14)
Here
� e
eV
eV
�
�
2
2
�
�
. (B.15)
For the pinned phase of the commensurate excitonic
insulator (
� 0 or �), the results of our previous
work [5] are reproduced.
One can imagine a plausible situation wherein a tun-
nel current is assembled from a large enough contact
area, so that the CDW phases vary substantially over
the contact plane. Then all
-dependent terms should be
averaged out. This means that all components directly
involving CDW pairing amplitude, i.e., possessing at
least one c-subscript, must vanish in an asymmetrical
configuration. For the CDWM–I–CDWM junction,
this will result in a substitution of 1/2 for cos2
.
Appendix C: Tunnel conductance in a magnetic
field at T = 0
It stems from the basic equations (9)–(15) that
the C–V characteristics for CDWM�–I–CDWM junc-
tions should differ substantially from their coun-
terparts for S�–I–S tunnel structures. Moreover,
CDWM�–I–CDWM junctions exhibit peculiar Zee-
man splitting (see Sec. 3.1) totally absent for currents
between superconducting electrodes. Below, analyti-
cal formulas for G V� ( ) components are represented as
direct illustrative evidence for the predicted pheno-
mena. Likewise in Appendix B only the branches
V > 0 of the components of the C–V characteristics
are explicitly shown.
In particular, the field-independent terms of the to-
tal conductance have the form
G V G V G V G V
R
eVdd cc dd cc � �� � � � � � �( ) ( ) ( ) ( ) (0 0 0 0
2
2
2�
�
�
�
�
�
)
( )
( ) ( cos )
( )
( )
(
�
�
�
�
-
.
/
/
E K�
�e e
eV
eV eV
2 2 22 1
2
4
1
4
2
2 2
�
�
0
1
2
2
cos )
( )
,
eV
eV �
(C.1)
90 Fizika Nizkikh Temperatur, 2005, v. 31, No. 1
T. Ekino, A.M. Gabovich, and A.I. Voitenko
G V G V
Rnn nn �� � � �
�
( ) ( )
( )
0 0
1
2
2�
. (C.2)
The notations here coincide with those of Appendix
B. Other components split in a magnetic field and ap-
pear as
G Vdn� � �( )0
�
� � � �
� �
� �
�
� � �
�
�
( )( ) ( )
( )
1
2 2 2
eV H eV H
R eV H
B B
B
�
�
, (C.3)
G Vnd � � �( )0
�
� �
�
� �
�
� � �
�
�
( )( ) ( )
( )
1
2 2 2
eV H eV H
R eV H
B B
B
� �
�
�
�
, (C.4)
G V
eV H
R eV H
cn
B
B
�
�
�
� � �
� �
�
( )
( ) cos ( )
( )
0
1
2 2 2
� �
�
�
� �
�
�
�
,
(C.5)
G V
eV H
R eV H
nc
B
B
�
�
�
� �
� � �
� �
( )
( ) cos ( )
( )
0
1
2 2 2
� �
�
�
� �
�
.
(C.6)
A sum of components (C.1)–(C.6) gives the overall
conductance G(V), its shape shown in Fig. 2.
It comes about from Eqs. (C.1) and (A.14) that
� �G eV
R
( )� � �2
2
2��
, (C.7)
regardless of the order parameter phase
. It is neces-
sary to underline that the tunneling conductance for
the superconducting symmetrical junction is deter-
mined by quite a different expression [8,10],
G V
eV
eV R
eV
eVsc s( )
( )
( )
( )
( )
� �
�
�
�
-
.
/
/
0
2
2
22 2
�
�
�
�
E
�
0
1
2K( )
( )
,� s
eV
eV
4
2
� �
� (C.8)
where � is the superconducting gap and
� s
eV
eV
�
�
2
2
�
�
. (C.9)
The conductance G Vsc( ) diverges when eV tends to
2� from above:
G V
eV Rsc eV
( )
( )3
4 344444
�2 0 2�
�
�
�
. (C.10)
The distinction between properties (C.7) and (C.10)
is due to the fact that the pertinent tunnel current in
superconductors comprises a direct one-term convolu-
tion of the Fermi-distribution factor and two gapped
DOSs. On the other hand, for CDWMs, the singula-
rities in the terms Gdds(V) at eV � �2� are to a cer-
tain extent compensated by the contributions from the
terms Gccs(V) formed by two «intersection» Green’s
functions Gc [5,11,12].
As for the spin-split peaks at two times smaller
voltages eV � �� in G(V) for CDWM-I-CDWM tun-
nel junctions, they should be observed only for par-
tially gapped metals. Therefore, such substances, as,
e.g., TTF–TCNQ, (TaSe4)2I, 5-TaS3, or K0.33MoO3,
which are insulating below Td [3], can not reveal the
predicted phenomenon of anomalous magnetic-field-in-
duced splitting.
1. B.I. Halperin and T.M. Rice, Solid State Phys. 21,
115 (1968).
2. Yu.V. Kopaev, Trudy Fiz. Inst. Akad. Nauk SSSR
86, 3 (1975).
3. G. Grüner, Density Waves in Solids, Addison-Wesley
Publishing Company, Reading, Massachusetts (1994).
4. A.A. Abrikosov, Fundamentals of the Theory of
Metals, North-Holland, Amsterdam (1987).
5. A.M. Gabovich and A.I. Voitenko, Phys. Rev. B52,
7437 (1995).
6. A.I. Anselm, Introduction into the Semiconductor
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