Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor
We present a theory which is able to explain enhanced magnetic quantum-oscillation amplitudes in the superconducting state of a layered organic metal with incoherent electronic transport across the layers. The incoherence acts through the deformation of the layer-stacking factor which becomes com...
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irk-123456789-1201232017-06-12T03:03:50Z Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor Gvozdikov, V.M. Wosnitza, J. Свеpхпpоводимость, в том числе высокотемпеpатуpная We present a theory which is able to explain enhanced magnetic quantum-oscillation amplitudes in the superconducting state of a layered organic metal with incoherent electronic transport across the layers. The incoherence acts through the deformation of the layer-stacking factor which becomes complex and decreases the total scattering rate in the mixed state. This novel mechanism restores the coherence by establishing a long-range order across the layers and can compensate the usual decrease of the Dingle factor below the upper critical magnetic field caused by the intralayer scattering. 2006 Article Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor / V.M. Gvozdikov, J. Wosnitza // Физика низких температур. — 2006. — Т. 32, № 2. — С. 152-157. — Бібліогр.: 26 назв. — англ. 0132-6414 PACS: 71.18.+y, 72.15.Gd, 74.70.Kn http://dspace.nbuv.gov.ua/handle/123456789/120123 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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English |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Gvozdikov, V.M. Wosnitza, J. Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor Физика низких температур |
| description |
We present a theory which is able to explain enhanced magnetic quantum-oscillation
amplitudes in the superconducting state of a layered organic metal with incoherent electronic
transport across the layers. The incoherence acts through the deformation of the layer-stacking
factor which becomes complex and decreases the total scattering rate in the mixed state. This novel
mechanism restores the coherence by establishing a long-range order across the layers and can
compensate the usual decrease of the Dingle factor below the upper critical magnetic field caused
by the intralayer scattering. |
| format |
Article |
| author |
Gvozdikov, V.M. Wosnitza, J. |
| author_facet |
Gvozdikov, V.M. Wosnitza, J. |
| author_sort |
Gvozdikov, V.M. |
| title |
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor |
| title_short |
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor |
| title_full |
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor |
| title_fullStr |
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor |
| title_full_unstemmed |
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor |
| title_sort |
incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2006 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120123 |
| citation_txt |
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations in the mixed state of a layered organic superconductor / V.M. Gvozdikov, J. Wosnitza // Физика низких температур. — 2006. — Т. 32, № 2. — С. 152-157. — Бібліогр.: 26 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT gvozdikovvm incoherentinterlayerelectronhoppingasapossiblereasonforenhancedmagneticquantumoscillationsinthemixedstateofalayeredorganicsuperconductor AT wosnitzaj incoherentinterlayerelectronhoppingasapossiblereasonforenhancedmagneticquantumoscillationsinthemixedstateofalayeredorganicsuperconductor |
| first_indexed |
2025-07-08T17:17:13Z |
| last_indexed |
2025-07-08T17:17:13Z |
| _version_ |
1837099945570598912 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, No. 2, p. 152–157
Incoherent interlayer electron hopping as a possible
reason for enhanced magnetic quantum oscillations
in the mixed state of a layered organic superconductor
V.M. Gvozdikov1,2 and J. Wosnitza3,4
1Kharkov National University, 61077, Kharkov, Ukraine
2Max-Planck-Institut f�r Physik Komplexer Systeme, D-01187 Dresden, Germany
3Institut f�r Festk�rperphysik, Technische Universit�t Dresden, D-01062 Dresden, Germany
4Forschungszentrum Rossendorf, Hochfeld-Magnetolabor Dresden (HLD), D-01314 Dresden, Germany
E-mail: Vladimir.M.Gvozdikov@univer.kharkov.ua
Received July 11, 2005
We present a theory which is able to explain enhanced magnetic quantum-oscillation
amplitudes in the superconducting state of a layered organic metal with incoherent electronic
transport across the layers. The incoherence acts through the deformation of the layer-stacking
factor which becomes complex and decreases the total scattering rate in the mixed state. This novel
mechanism restores the coherence by establishing a long-range order across the layers and can
compensate the usual decrease of the Dingle factor below the upper critical magnetic field caused
by the intralayer scattering.
PACS: 71.18.+y, 72.15.Gd, 74.70.Kn
Keywords: magnetic quantum oscillations, incoherence, layered organic superconductor
1. Introduction
It is well known that magnetic field and super-
conductivity are antagonists in the sense that a super-
conductor tend to expel magnetic field from its
interior: completely in the Meissner state and parti-
ally in the Abrikosov vortex state. Because of that,
a foundation in 1976 the coexistence of magnetic
quantum oscillations and superconductivity in layered
dichalcogenides of transient metals [1] was at first
considered just as a strange experimental fact. Only a
decade and a half later was it recognized that the
magnetic quantum oscillations in layered super-
conductors is not an artefact but a general and
complex phenomenon waiting for systematic studies.
A history of these studies and a survey of the related
theoretical ideas can be found in the review paper [2].
Since the first observation of the de Haas—van
Alphen (dHvA) oscillations in the mixed state of
layered superconductors it has been firmly established
in numerous experiments that these oscillations are
always damped below the upper critical field Bc2. This
damping was explained by several mechanisms re-
viewed in [2]. The recent finding that both the dHvA
and the Shubnikov—de Haas (SdH) amplitudes are
enhanced in the mixed state of the layered organic
superconductor ��� -(BEDT-TTF)2SF5CH2CF2SO3 ( ���
salt in the following) [3] was a real surprise because it
is in sharp contrast to all experiments and theories
known so far [2]. Up to now it has been assumed
that the quasiparticle scattering by the «vortex
matter» is the main mechanism of the damping which
acts through additional Dingle-like factor Rs �
� �exp ( )2� �/� sc . Here � is the cyclotron frequency,
� sc is the scattering time. The Landau-level broaden-
ing, �/� sc, is a sum of the two terms: �/� sc1 �
� � �2 1 2( )� / /
� , which is due to the intralayer
scattering at vortices [4–6], and �/ /� � �sc2
2
0� � � ,
which takes account of the additional cooperative
scattering effect by impurities and vortices within the
layers [7] ( is the chemical potential, � is the
© V.M. Gvozdikov and J. Wosnitza, 2006
superconducting order parameter which just below the
Bc2 is less than the Landau-level separation � �
� ,
and � � 1). It is important to note that all theories so
far were developed for the 2D superconductors. On
the other hand, it was shown yet in 1984 [8] that
electron hopping across the layers strongly affects
oscillations through a special layer-stacking factor
which together with the standard Dingle, spin, and
temperature factors modulates the dHvA amplitudes.
The case of the SdH oscillations is more complex
because of the incoherence and localization associated
with the electron hopping between the layers [9].
In the 2D superconductors the additional Dingle-
like factor Rs is due to the intralayer scattering only.
It describes the extra damping of the dHvA and SdH
amplitudes so that the enhancement of the oscillations
in the mixed state of the ��� is absolutely unclear. To
understand the anomalous enhancement of the mag-
netic quantum oscillations in the ��� one can first rise a
question what is unusual in this superconductor
compared to conventional layered superconductors.
The answer is that the ��� is the only known super-
conductor which (i) displays enhanced magnetic
quantum-oscillation amplitudes in the mixed state
both in the dHvA and the SdH signals and (ii)
exhibits an incoherent electronic transport across the
layers [10]. The incoherence means that the electronic
properties of layered quasi-2D metals can not be
described within the usual fundamental concept of
an anisotropic 3D Fermi surface (FS) [11]. The ���
salt belongs to organic conductors of the type
(BEDT-TTF)2X, where BEDT-TTF stands for bi-
sethylenedithio-tetra-thiafulvalene and X for a mono-
valent anion. This class of materials displays a number
of unique properties such as an unconventional
electronic interlayer transport, clear deviations of
their magnetic quantum oscillations [12] from the
standard three-dimensional (3D) Lifshitz—Kosevich
theory [13], and puzzling enhancement of the mag-
netic quantum oscillations in the superconducting
mixed state. This enhancement cannot be explained by
the 2D theories.
In this paper we consider a new mechanism for the
SdH and dHvA amplitude modulation in the mixed
state. This mechanism goes beyond the 2D consider-
ation and takes account of the incoherent interlayer
hopping. The physical picture behind this mechanism
is as follows. The incoherence, or disorder in direction
perpendicular to the layers, hampers the electron
hopping between neighboring layers. This enhances
the scattering at impurities within the layers since
electrons on the Landau orbit interact with the same
impurities many times before a hopping to the neigh-
boring layer. In the superconducting state a long-
range order establishes across the layers which allows
quasiparticles to escape the intralayer multiple
scattering by the Josephson tunnelling between the
layers. This reduces the scattering rate by impurities
and enhances the Dingle factor in the superconducting
state. For the ��� salt this effect most likely plays a
dominant role. In the coherent case, there is no
interlayer scattering and electrons (quasiparticles)
can move freely across the layers with the fixed
momentum both in the normal and superconducting
states which render the above mechanism much less
effective. Numerically, this effect is described by the
layer-stacking factor [Eq. (3)] which itself contains
a Dingle-like exponent in the case of incoherent
interlayer hopping [8]. Superconductivity restores the
coherence across the layers by renormalizing the hop-
ping integrals [14]. This reduces the interlayer scat-
tering and enhances the oscillation amplitudes.
Anomalous dHvA oscillations have also been
observed in YNi B C2 2 [15,16]. These oscillations per-
sist down to the surprisingly low field 0.2Bc2 [16].
A Landau-quantization scheme for fields well below
Bc2 in the periodic vortex-lattice state was developed
in [17] and for a model with an exponential decrease
of the pairing matrix elements in [7]. The observed
recovery of the dHvA amplitudes for B Bc
2 in the
borocarbide YNi B C2 2 was explained by the enhance-
ment of the special vortex-lattice factor depending on
the Landau bands which become narrower when the
vortex lattice grows thinner [7].
The anomalous magnetic quantum oscillations in
the mixed state of the ��� salt is much more mysterious
and poses the question on the peculiarity of this
material.
2. The basic equations and experiment
In this section we consider a new mechanism for the
quasiparticle scattering that goes beyond the usual 2D
consideration of intralayer scattering by taking into
account the interlayer-hopping contribution to the
total scattering rate. This can explain an oscillation-
amplitude enhancement in the superconducting state
for layered conductors with incoherent hopping across
the layers.
In case the momentum across the layers is not
preserved, the electron interlayer hopping maybe de-
scribed in terms of an energy � that is distributed with
the density of states (DOS) g( )� . The energy spectrum
of a layered conductor in a perpendicular magnetic
field is, therefore, given by E n /n ( ) ( )� �� � ��� 1 2
[8]. The total DOS then follows from the standard
Green-function definition
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations
Fizika Nizkikh Temperatur, 2006, v. 32, No. 2 153
N E
l
d
g
E E E
n n n
( ) Im
( )
( )
�
� � ���
�
�
1
2 2 2
0�
�
�
�
, (1)
where
n E( ) is the average self energy corresponding
to the nth Landau level and l e/cB� ( ) /
�
1 2 is the
magnetic length. For large energies and large
n E/� ���� 1 (relevant for the magnetic quantum
oscillations here) the self energy is independent on
the index n and the summation in Eq. (1) by use of
Poisson’s formula yields
N E
N
R p E I
ipEp
p
D p
( )
( )
Re ( ) ( , ) exp ,
0
1 2 1
2
1
� � � �
�
�
�
�
�
�
�
� �
��
(2)
where N( )0 is the 2D electron-gas DOS and the
function R p E p E /D( , ) exp ( | Im ( )| )� �2�
�� gene-
ralizes the Dingle factor to the case of the ener-
gy-dependent self energy
( )E . The layer-stacking
factor in Eq. (2),
I d g
ip
p � �
�
�
�
�
�� � �
� �
( ) exp
2
��
, (3)
is an important factor in the theory of magnetic
quantum oscillations in normal and superconducting
layered systems [7,18]. It describes contributions to
the oscillations coming from the interlayer hopping.
If the stacking is irregular Ip becomes complex and
contains a Dingle-like exponent [8].
The inverse scattering time 1/ E E�( ) | Im ( )|/�
�
in the self-consistent Born approximation (SCBA)
was found to be proportional to the total DOS
[19–21]. Accordingly, the relation N E /N( ) ( )0 �
� � �0/ E( ) holds, where �0 is the intralayer scattering
time [22,23]. Substituting this into Eq. (2) leads to an
equation for �( )E showing that it oscillates as a
function of 1/B.
In case the highly anisotropic electronic system has
a 3D Fermi surface the DOS related to the interlayer
hopping is symmetric, g g( ) ( )� �� � , and becomes for
nearest-neighbor hopping g t( ) ( ) /� � �� �� �1 2 2 1 24 .
The corresponding layer-stacking factor then is given
by I J tp/p � 0 4( )� �� [8]. This Bessel function oscil-
lates as a function of 1/B which is just another way to
describe the well-known bottle-neck and belly oscilla-
tions of a corrugated 3D FS. Oscillating corrections to
the Ginzburg—Landau expansion coefficients caused
by the factor J tp/0 4( )� �� were also calculated in
[24].
For the incoherent case, on the other hand, the
translation invariance across the layers is lost. The
irregular interlayer hopping means that the DOS
deviates from the function g t /( ) ( )� � �� �� �1 2 2 1 24
and loses the symmetry g g( ) ( )� �� � which implies
that Im Ip � 0. As will be shown below, this results
in a special contribution to the electron scattering
time which can be negative below Bc2 and acts against
the known intralayer damping mechanisms of the
quantum oscillations in the mixed state [2]. However,
this contribution vanishes if the hopping between the
layers preserves the interlayer momentum leading to a
corrugated 3D FS cylinder. This is an important point
in our consideration.
Using Eq. (2) and N E /N / E( ) ( ) ( )0 0� � � we
write �( )E �1 as a sum of the coherent (symmetric)
and incoherent (asymmetric) terms:
� � �( ) ( ) ( )E E Es a
� � �� �1 1 1, (4)
�
�
�0
1
1 2 1
2
s
p
p
D pR p E I
pE
� � � �
�
�
�
�
�
�
�
�( ) ( , )Re cos
��
, (5)
�
�
�0
1
2 1
2
a
p
p
D pR p E I
pE
� � �
�
�
�
�
�
�
�
�( ) ( , ) Im sin
��
. (6)
With the help of the summation rule
S pp
p
p( , ) ( ) cos
cos
| |� � �
�
� �
�� � �
�
���
�
�� 1 e
sinh
cosh
(7)
one can rewrite Eqs. (5) and (6) in the integral form
1 1
0� �
� � �� � �
s a
s ad g S E
( )
( )( ) [ ( , )]� � . (8)
Here g gs s( ) ( )� �� � is the symmetric and
g ga a( ) ( )� �� � � is the antisymmetric part of the DOS
g( )� , � � �( )E /� 2 � , and � � � �( , ) ( )E E /� �2 ��. The
SCBA, as well as Eqs. (4)–(8), are valid not only for
point-like impurities but also for a smooth random
potential provided its correlation radius is less than
the Larmor radius, which holds for large n [21]. One
can see from Eqs. (4)–(8) that, in general, the
incoherent contribution, � ��( )E a
1, to the total
scattering rate, �( )E �1, is essential. The integral
equation for �( )E �1 is very complex and can be
solved only perturbatively in the case � �� �. In the
limit � � �, when S( )�� � � 1, we have � �s
� ��1
0
1 and
�a
� �1 0. For finite, but large � the parameter
R p ED
p( , ) �
�e � 1. Even if �� � 1, the quantity
e �
� 1 and Eqs. (5) and (6) are just a series
expansion in powers of the small parameter e ��.
Equation (7) shows the convergence of this series for
any � � 0 allowing a perturbative solution. The
perturbative terms oscillate as a function of E and can
be written as the series � �� �� � � �1
0
1
1 21( ) [E X X
� �O( )]e 3 0� , with X1 0� �e � , X2
2 0� �e � , and
� � �0 02� /� . The first nonzero correction averaged
154 Fizika Nizkikh Temperatur, 2006, v. 32, No. 2
V.M. Gvozdikov and J. Wosnitza
over an oscillation period is proportional to
X X I1 2 0
2
1
22 0� � � �� �e | | and yields
1 1
1
4
0 0
0 2
1
2
1
2
� �
�
�
� � �
�
�
!
"
#�
( ) (Re Im )R I ID . (9)
The Dingle factor R /D
0
02� �exp ( )� �� is a small
parameter in our perturbative solution. The term
Im I1
2 in Eq. (9) appears due to the incoherence.
It was established that in the ��� salt electron
hopping across the layers is most probably incoherent,
i.e., the momentum perpendicular to the layers is not
preserved and there is no 3D Fermi surface [10]. The
reason for this remarkable feature is unknown so far.
It might be that some kind of disorder, such as dif-
ferent spatial configurations in the extraordinary large
and complex anion-molecule layer may induce random
hopping integrals, in analogy with intercalated layer-
ed compounds [8]. Furthermore, the ��� salt is the
only material so far studied (not only among the
BEDT-TTF salts) which displays an enhancement of
the magnetic quantum-oscillation amplitude in the
superconducting state [3].
This important result is summarized in Fig. 1. In
this Dingle plot for a 2D metal the amplitudes A1 of
the fundamental dHvA frequency (F � 198 T) and A2
of the second harmonic (2F) are normalized by
B X Tsinh ( ) �1 and plotted on a logarithmic scale as a
function of 1/B, with X k m T/e BB c� 2 2� � and mc the
effective cyclotron mass (see [3] for more details).
Upon entering the superconducting state the
oscillation amplitude A1 is enhanced compared to the
normal-state dependence (solid lines). For the second
harmonic A2 neither an additional damping nor an
enhancement is observed.
In the superconducting state long-range order
across the layers evolves through the renormalization
of the hopping integrals [14]. This can be understood
as follows. The quasiparticle hopping between the
layers is an independent degree of freedom with
respect to the in-plane Landau quantization. In the
normal state the Green-function equation related to
the interlayer hopping is given by
[( ) ] ( )
m
i im im ij ijt G� � � �� � � � �0 , (10)
where the electron energy in the layer � i and the
hopping integrals t tim i i m i m� �� �( ), ,� �1 1 are as-
sumed to depend on the layer indices for the sake of
generality. In the superconducting state the order
parameters in the layers, � i , become nonzero and the
Gor’kov equation for the Green functions Gij can be
written as
[( ) � ] ( )
m
i im im ij ijt G� � � �� � � � � , (11)
� ( ) *t t Gim im i im m� � �� �0 � . (12)
The star means the complex conjugate. When com-
paring Eq. (10) with Eq. (11) it is seen that in the
superconducting state the effective hopping integrals,
given by Eq. (12), become nonzero not only for
next-nearest-neighbor hopping. In that case the
nonvanishing retarded Green-function components
Gim
0 ( )�� result in nonzero �tim for electron hopping
between arbitrary sites i and m. In fact, Eq. (12)
simply reflects how the superconducting long-range
order is established across the layers. In general
the complex order parameters in the layers � i �
� | | exp ( )� i i$ appear and result in interlayer (in-
trinsic) Josephson coupling [14]. In the absence of
Josephson currents the order parameter can be chosen
to be real and independent of the layer index
� � �i i� �| | . The correction to the DOS due to this
mechanism is
� �
�
�g
g
t
G
ijij
ij( )
( )
( )� �
%
%
���2 0 . (13)
The second effect we have to take into account is
caused by the vortex matter in the mixed state. Here,
the vortices are disordered for fields slightly below
Bc2. They convert the degenerate Landau levels
into asymmetric Landau bands as was shown in
Refs. 25, 26. Thus, in the mixed state the quasiparticle
tunneling between the layers implies the quantum
transition between these Landau-band states which
results in the additional contribution to the simple
nearest-neighbor DOS
� �
�
g
g
�
�
�
�
( )
( )
�
%
%
�
�
��
�
�
��
�
2
2
0
. (14)
Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations
Fizika Nizkikh Temperatur, 2006, v. 32, No. 2 155
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
1/B, T –1
10
0
10
1
10
2
10
3
10
4
A
iB
si
n
h
X
/
T
,a
rb
.u
n
its A2
A1
T, K
0.03
0.12
0.35
0.54
� -( BEDT-TTF ) SF CH CF SO2 5 2 2 3
Fig. 1. Dingle plot of the fundamental (A1) and the se-
cond harmonic (A2) dHvA amplitudes of the ��� salt ex-
tracted from modulation-field data for different tempe-
ratures. The solid lines are fits to the data above 6 T.
The total correction to the DOS in the mixed state
can be written as � � �g Gtot ( ) ( )� �2 , where G( )� is
directly defined by Eqs. (13) and (14). Inserting
� �gtot ( ) into Eq. (8) and averaging over a period in E
results in
�
� �
� � �� � �
�
&
1 2
0
2
0
�
�
�
�
�
� � � ��
� �
d G S E( ) ( ( , )) . (15)
Since the DOS is normalized (� �d g� �( ) 1) the
function G( )� satisfies the condition � �d G� �( ) 0.
This means that it is alternating in sign and & might
be negative because S E( , ( , ))� � �� � 0. The studied
system is too complex to calculate & in general. In
the limit � � � this coefficient vanishes since
S E( , ( , ))� � �� � 1. It is instructive to consider a cor-
rection to the scattering rate in Eq. (9) in the mixed
state. The variation of the layer-stacking factor �I1 is
given by Eq. (3) with the DOS replaced by � �gtot ( ).
The broadening of the Landau levels, caused by
� �gtot ( ), is of the order of the width of this
function and much less then �� in order to observe
the oscillations. Therefore, in first approximation
Re �I1 0� and
Im ( )sin� � �
�� � �
I d G1
2 22 2
� �
�
�
�
�
� �
' (�
�
�
�
�
���
�
�
�� �
,
where ' ( � �� � � �d G( ) . (For G G( ) ( )� �� � � , Re �I1 is
zero exactly.) Thus, the correction to the scattering
rate in the mixed state near Bc2, caused by the
interlayer-hopping mechanism, is given by
�
� �
�
�
� �1 2 4 22 1
0 0
0 2�
�
�
�
�
� � �
�
�
��
�
�
��
' (�
�
�
��
� �
Im
( )
I
RD
� �
�. (16)
For ' ( �� Im I1 0, this gives a decrease in the scat-
tering rate. Note that the latter is nonzero only if the
system is incoherent in the normal state and
Im I1 0� . This strongly supports the relevance of this
mechanism for the ��� salt, since only this organic
metal displays both incoherence in the normal state
and an enhancement of the SdH and dHvA
amplitudes in the mixed state.
Thus, the overall effect superconductivity has on
Rs is determined by the balance between positive and
negative contributions to the scattering rate. The
already mentioned positive contribution from the in-
tralayer scattering at vortices and defects is
�
��
�
�
�sc
�
�
�
��
�
�
�� �
�
�
!
"
#
#
�
� �
2
1 2
0
/
. (17)
The new additional interlayer mechanism we discuss
here results in a negative contribution given by
� � � &( ) ( )1 2
0/ /� � [Eq. (15)]. Since little is known
about the DOS of the studied system, even in the
normal state, we cannot calculate the coefficient &
quantitatively. However, contrary to Eq. (17), for
this term the small factor 1/ is absent, so that the
overall correction to the scattering rate might be
negative. The experimental facts [3] give us
confidence that this is the case at least for the ��� salt.
3. Conclusions
We conclude with a qualitative picture of the effect
discussed here. The incoherence means that the
hopping time between the layers � �z t� ���/| | 0 so
that an electron scatters many times within a layer
before leaving it [11]. Here the quantity | |t is some
averaged hopping integral that in the ��� salt may be
assumed to be the smallest parameter in energy.
Indeed, experimentally | |t cannot be resolved in the ���
salt reflecting the fact that the hopping integral is one
of the smallest for all known 2D organic metals so far
[10]. Consequently, even small spatial fluctuations of
the hopping probability within and across the layers
render the electron motion across the layers incohe-
rent. On the other hand, for the evolution of super-
conductivity some interlayer (Josephson) coupling is
vitally important. Long-range order is established
below Bc2, thereby renormalizing the hopping in-
tegral. According to Eq. (12), the renormalized � z in
the superconducting state may be estimated as
� z / t /t� �� | |�2 . For � �� | |t the hopping time re-
duces considerably and becomes � z t� �| |/�2. The
latter means that the quasiparticles spend less time
within the (impurity-containing) layers decreasing
the scattering rate and, consequently, enhancing the
Dingle factor. In the ��� salt this effect is strong
because of the smallness of | |t . Thus, our mechanism
relates the two unusual effects observed in the ��� salt:
the incoherent interlayer hopping transport and the
enhancement in the quantum-oscillation amplitudes in
the mixed state.
One final remark is in order. Qualitatively, in the
normal state the conductivity is proportional to the
scattering time of electrons on impurities. A transition
into the superconducting state means that new quasi-
particles (the Cooper pairs) do not scatter on im-
purities and the scattering time became infinite. In the
incoherent quasi-2D conductors there are two contri-
butions to the Dingle factor. The intralayer scattering
on the Abrikosov vortices decreases the scattering time
and, correspondingly, the Dingle factor. Contrarily,
the interlayer scattering time in the incoherent layered
conductors increases in the superconducting state be-
cause quasiparticles move more freely across the layers
when the long-range-order establishes in this di-
156 Fizika Nizkikh Temperatur, 2006, v. 32, No. 2
V.M. Gvozdikov and J. Wosnitza
rection. The corresponding effect on the Dingle factor
is its enhancement. If this enhancement is larger than
decrease due to the intralayer scattering, the total
effect of the superconductivity on the SdH and dHvA
oscillations would be an increase of their amplitudes
since both SdH and dHvA oscillations depend on the
Dingle factor. Therefore, the long-range-order across
the layers decreases the interlayer scattering rate in
the superconducting state, restores the coherence, and
decreases the Dingle factor. In coherent layered
conductors with the 3D FS the momentum across the
layers is preserved which assumes no scattering in the
interlayer electron hopping. Thus, the simple and
«counterintuitive» physics we discuss here works only
if in the normal state electron hopping is incoherent.
All theories of the quantum magnetic oscillations in
layered superconductors so far have ignored the in-
terlayer hopping and have formed a wrong intuition
that oscillation amplitudes should always be de-
creased in the superconducting state. We have shown
in this paper that this may be not the case in the
incoherent layered conductors like the ��� salt. The
problem is complex and further experiments with
layered organic conductors and/or artificially created
superlattices are necessary to check the ideas of our
paper.
This work was supported in part by INTAS, project
INTAS-01-0791, and the NATO Collaborative Linkage
Grant No. 977292. V.M.G. thanks P. Fulde and
S. Flach for the hospitality at the MPIPKS in Dres-
den and P. Wyder, I. Vagner, T. Maniv, P. Grigoriev,
A. Dyugaev, M. Kartsovnik, and W. Biberacher for
useful discussions.
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Incoherent interlayer electron hopping as a possible reason for enhanced magnetic quantum oscillations
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