PHYSICAL REVIEW B
VOLUME 57, NUMBER 13
1 APRIL 1998-I
Quasiparticle transport below T c in the cuprate superconductors
F. Gollnik and R. P. Huebener
Physikalisches Institut, Lehrstuhl Experimentalphysik II, University of Tuebingen, Morgenstelle 14, D-72076 Tuebingen, Germany
~Received 10 September 1997!
The interpretation of experimental data on the Nernst and Seebeck effects in the mixed state of a type-II
superconductor requires a careful treatment of the temperature-dependent electrothermal conductivity
S n (T)/ r n (T) below T c ~S n 5Seebeck coefficient, r n 5resistivity!. For the cuprate superconductors such as
YBa2Cu3O72d, a simple extrapolation from the normal-state behavior above T c is inadequate and can lead to
erroneous results, due to the opening of the superconducting energy gap and its effect on the temperature
dependence of the quasiparticle scattering rate. In particular, the value of the thermal Hall angle of vortex
motion depends sensitively on the correct treatment of the electrothermal conductivity below T c . Taking
experimental data for YBa2Cu3O72d films as an example, we show that the thermal Hall angle of vortex motion
obtained from the data attains the expected small value only if the opening of the energy gap below T c is taken
into account. @S0163-1829~98!07013-1#
II. THERMAL DIFFUSION OF QUASIPARTICLES
AND MAGNETIC FLUX QUANTA
I. INTRODUCTION
In the mixed state of a type-II superconductor, quasiparticles and magnetic flux quanta respond to a temperature
gradient by thermal diffusion, in this way generating the Seebeck and Nernst effects, respectively.1–3 Since there exists a
Hall contribution both to the thermal diffusion of quasiparticles and of magnetic flux quanta, two Hall angles appear in
this vortex dynamics. From an analysis of our measurements
of the Seebeck and Nernst effects in the mixed state of epitaxial c-axis-oriented YBa2Cu3O72d films, we have concluded recently that the Hall angle for the thermal diffusion
of vortices appears to be 30–50 times larger than expected.2,3
In this paper we argue that this discrepancy is due to the
opening of the superconducting energy gap and its effect on
the quasiparticle transport below T c , and that the thermal
Hall angle is consistent with the expected value.
A key ingredient in the previous analysis2,3 has been the
assumption that the electrothermal conductivity S n (T)/ r n (T)
below T c can simply be obtained by extrapolation from its
normal-state value above T c . Here S n and r n are the Seebeck coefficient and the electric resistivity, respectively.
However, various experiments have shown recently that in
the cuprate superconductors below T c the quasiparticle transport is sensitively affected because of the opening of the
superconducting energy gap. This results mainly from the
fact that electron-electron scattering is dominating in these
materials. In the following we show that this modification of
the quasiparticle transport below T c can account for the measured Seebeck and Nernst effects in the mixed state of
YBa2Cu3O72d and that the discrepancy in the thermal Hall
angle is removed.
In Sec. II we summarize the theoretical description of the
vortex motion in the mixed state, leading to the Seebeck and
Nernst effects. The role of the electrothermal conductivity
below T c is emphasized. In Sec. III we discuss the implication of the opening of the superconducting energy gap below
T c for the electrothermal conductivity and how a large thermal Hall angle can appear instead as an artifact, if this implication is ignored.
0163-1829/98/57~13!/7495~4!/$15.00
57
In a temperature gradient the quasiparticles and the vortices present in the mixed state of a type-II superconductor
both experience the thermal force
f th52S * •“ x T.
~1!
Here and in the following, we assume the temperature gradient along the x direction and the magnetic field in the z
direction. In Eq. ~1!, S * denotes the transport entropy of the
species being considered. In the case of charge carriers, S * is
related to the Seebeck coefficient S via the relation
S * 5S•e,
~2!
where e is the elementary charge. ~Note that e is negative for
electrons and positive for holes.! In the case of vortices, the
thermal force ~1! and the transport entropy are usually given
per unit length of vortex line. In the following we denote the
transport entropy per unit length of vortex line by S w* . As
discussed in detail elsewhere,2 the longitudinal ~E x , Seebeck
effect! and transverse ~E y , Nernst effect! electric fields in
the presence of the temperature gradient “ x T are given by
the following expressions:
S w* r m
Sn
Ex
5rm
a,
~ 11 ab ! 1
“ xT
rn
w0
~3!
S w* r m
Ey
Sn
5
1rm
~ b2a !.
“ xT
w0
rn
~4!
Here w 0 is the magnetic flux quantum and r m the electric
resistivity in the mixed state. a 5tan uv and b 5tan uqp ,
where u v and u qp are the Hall angles associated with the
motion of vortices and quasiparticles, respectively. The expressions in Eqs. ~3! and ~4! were derived earlier by Maki4
from the time-dependent Ginzburg-Landau theory. Similar
results were also obtained by Samoilov et al.5 We see that
the longitudinal electric field E x contains a first-order contri7495
© 1998 The American Physical Society
7496
BRIEF REPORTS
bution from the Hall angle a of the thermal diffusion of the
vortices, whereas the transverse electric field E y only contains higher-order contributions from the Hall angle of the
thermal diffusion of the quasiparticles. On the right-hand
side ~RHS! of Eq. ~4!, the difference b 2 a appears, since the
Hall angles of the quasiparticles and of the vortices act in
opposite directions and partly cancel each other. Within the
Bardeen-Stephen model6 we have a 5 b , and the second
term in Eq. ~4! vanishes.
The first term on the RHS in Eq. ~3! indicates that in the
mixed state the Seebeck coefficient also shows the broadening of the transition to the superconducting state typical for
the resistivity r m in the mixed state of the p-doped cuprates.
For epitaxial c-axis-oriented films of YBa2Cu3O72d and
Bi2Sr2CaCu2O81x , it has been found that the first term on
the RHS of Eq. ~3! with ab !1 approximately accounts for
the observed Seebeck coefficient in the mixed state.3,7 In
Ref. 2 a detailed analysis of experimental results for epitaxial
c-axis-oriented YBa2Cu3O72d films has been performed. By
extrapolating the temperature-dependent factor S n (T)/ r n (T)
from the normal state above T c , the difference
Ex
S n~ T !
2 r m~ T !
DS5
“ xT
r n~ T !
S w* r m
w0
a,
r n (T) in the denominator of the second term on the RHS
will turn out to be too large from such a simple extrapolation. Hence, this second term will be too small and the difference DS and the thermal Hall angle too large. In the following we examine this situation in more detail, using only
highly simplified arguments.
We write the electrothermal conductivity S n (T)/ r n (T) in
the form14
E
S n~ T !
1
52
r n~ T !
Te
`
0
~6!
and the thermal Hall angle a has been determined from DS.
In this way a thermal Hall angle in the range a 50.3– 0.5 has
been found. However, this range is much larger than the
value a '0.01 obtained from electric transport measurements. We emphasize that this analysis had been performed
with the assumption that below T c the temperaturedependent electrothermal conductivity S n (T)/ r n (T) can be
obtained simply by extrapolation from its normal-state value
above T c . In the following we critically reevaluate the electrothermal conductivity below T c . In particular, we show
that such a simple extrapolation becomes unjustified because
of the opening of the superconducting energy gap.
III. ELECTROTHERMAL CONDUCTIVITY S n „T…/ r n „T…
BELOW T c
One of the special features of the p-doped cuprate superconductors is the strong indication for the dominance of
electron-electron scattering instead of electron-phonon scattering in the normal state.8,9 Therefore, the opening of the
superconducting energy gap below T c strongly affects the
quasiparticle transport properties. For example, measurements of the microwave surface resistance of YBa2Cu3O72d
have shown a strong decrease of the inelastic quasiparticle
scattering rate with decreasing temperature and an abrupt
change of its temperature dependence near T c . 10,11 Furthermore, below T c a particularly strong decrease of the quasiparticle scattering rate has also been found in measurements
of the thermal conductivity12 and of the thermal Hall
conductivity13 of YBa2Cu3O72d. Hence, below T c , the quasiparticle transport cannot be obtained simply by extrapolation from the normal state above T c . In Eq. ~5! the resistivity
S D
d« ~ «2 m ! 2
]f
s~ « !.
]«
~7!
Here « is the energy, m the chemical potential, and f the
Fermi function. s~«! is the contribution to the conductivity s
from electrons in the interval d« around «,
s5
E S ]] D s
`
0
f
«
d« 2
~ « !.
~8!
Expanding s~«! around the Fermi energy « F ' m ,
s~ « !5s~ «F!1
~5!
was calculated ~Fig. 3 of Ref. 2!. From Eq. ~3! we have
DS5
57
]s
]«
U
~ «2 m ! ,
~9!
«F
we see that it is the second term on the RHS, describing the
nonsymmetry between electrons and holes, which yields a
nonvanishing contribution to the electrothermal conductivity.
Next, we consider the changes in the electrothermal conductivity due to the opening of the superconducting energy
gap D. In a highly simplified way, we take the quasiparticle
condensation into Cooper pairs into account, by leaving out
the energy interval between m 2D and m 1D in the integrals
~7! and ~8!. Ignoring for the moment any influence of D on
the quasiparticle scattering rate, we obtain
F E
s DÞ0 5 s D50 12
1D/k B T
2D/k B T
dz
1
~ 11e z !~ 11e 2z !
G
~10!
and
S n~ T !
r n~ T !
U
5
DÞ0
S n~ T !
r n~ T !
3
E
U F
12
D50
1D/k B T
2D/k B T
dz
3
p2
G
z2
.
~ 11e z !~ 11e 2z !
~11!
In adopting such a procedure, we have not included the increase in the quasiparticle density of states above m 1D and
below m 2D and the corresponding redistribution of the quasiparticle energy levels. However, near the band edge the
quasiparticle group velocity will be reduced in a similar way
as the density of states is enhanced. Hence the contribution
of these quasiparticles to the current will not be strongly
affected.
It is instructive to compare the two integrands I (0) (z)
5 @ (11e z )(11e 2z ) # 21 and I (2) (z)5(3/p 2 )z 2 I (0) (z) from
Eqs. ~10! and ~11!, respectively. These two functions
I (0) (z) and I (2) (z) are displayed in Fig. 1. From this figure
we see that the opening of the superconducting energy gap
immediately affects the quantity s DÞ0 , whereas the electrothermal conductivity S n (T)/ r n (T) u DÞ0 at first remains
nearly unchanged. For illustrating this point, in Fig. 2 we
BRIEF REPORTS
57
FIG. 1. The functions I ( 0 ) (z)5 @ (11e z )(11e 2z ) # 21 ~solid
line! and I ( 2 ) (z)5(3/p 2 )z 2 I ( 0 ) (z) ~dashed line! plotted vs z
5(«2 m )/k B T.
show the square brackets of Eqs. ~10! and ~11! plotted versus
D/k B T. Taking for D(T) the function D(T)5D(0)
31.74(12T/T c ) 1/2 near T c with D(0)51.76k B T c for a classical BCS superconductor, we obtain the plot of the normalized quantity
S n~ T !
r n~ T !
U Y U
DÞ0
S n~ T !
r n~ T !
,
D50
i.e., of the square bracket of Eq. ~11!, versus T/T c shown by
the solid line in Fig. 3.
Up to now we have ignored the detailed influence of D on
the quasiparticle states close to the energy gap. @One expects
modifications of Eq. ~9! if coherence factors are taken into
account.# Basically, the preceding discussion is a justification for such a simple treatment of the electrothermal conductivity close to T c , since in this temperature regime the
diffusion current due to a temperature gradient is mainly carried by quasiparticles at energies appreciably beyond the
gap. ~Note the position of the peaks of the dashed curve of
Fig. 1.! From Fig. 3 we note that at T/T c 50.8 according to
Eq. ~11! the opening of the energy gap reduces the electrothermal conductivity only by less than 20%. For providing
7497
FIG. 3. Effect of the opening of the superconducting energy gap
on the electrothermal conductivity below T c . The ratio
@ S n (T)/ r n (T) # u DÞ0 / @ S n (T)/ r n (T) # u D50 obtained from Eq. ~11!
plotted vs T/T c ~solid line!. An additional factor of ( j / Aj 2 2D 2 ) in
the corresponding integral for the electrothermal conductivity ~as
discussed in the text! leads to the dotted line, whereas multiplication with the factor (T c /T) 4 results in the dashed curve. The open
circles show the extrapolated values needed to yield the expected
thermal Hall angle a '0.01 based on Eqs. ~5! and ~6!.
additional insight we compare this result with the following
modified calculations. We assume a superconductor with a
geometrically limited mean free path l 0 ~e.g., due to impurities!. Here the reduction of the group velocity v qp
5 v F ( Aj 2 2D 2 / j ) is compensated by an increase of the scattering time t qp5 t F ( j / Aj 2 2D 2 ), where v F is the Fermi velocity, t F 5l 0 / v F , and j 5«2 m . 15 With the quasiparticle
density of states N( j )5N 0 ( j / Aj 2 2D 2 ), this introduces an
additional factor of ( j / Aj 2 2D 2 ) in the corresponding integral for the electrothermal conductivity. However, this divergent factor only leads to an enhancement of
S n~ T !
r n~ T !
D/k B T.
line! and
plotted vs
DÞ0
S n~ T !
r n~ T !
D50
of about 40% at T/T c 50.8, as is shown by the dotted line in
Fig. 3. We conclude that a substantial increase of the electrothermal conductivity below T c has to involve a considerable reduction of the scattering rate for quasiparticles at
higher energies. Since electron-electron scattering appears to
be the dominating process in high-T c materials, the scattering rate is expected to be proportional to the quasiparticle
concentration n qp . Not too far below T c , n qp of a superconductor with an isotropic gap is well approximated by the
corresponding expression of the two-fluid model,16 yielding
n qp;(T/T c ) 4 . Hence the opening of the energy gap at temperatures below T c is expected to introduce the additional
factor (T c /T) 4 into the normalized quantity
S n~ T !
r n~ T !
1D/k T
FIG. 2. $ 12 * 2D/k B T dz @ 1/(11e z )(11e 2z ) # % ~solid
B
k
1D T
$ 123/p 2 * 2D/kB T dz @ z 2 /(11e z )(11e 2 ) # % ~dashed line!
B
U Y U
U Y U
DÞ0
S n~ T !
r n~ T !
,
D50
derived from Eq. ~11!, because of the reduced quasiparticle
scattering. This curve is shown in Fig. 3 by the dashed line.
We emphasize that it is mainly this effect of the opening
of the superconducting energy gap on the electrothermal
conductivity below T c which is not taken into account by
7498
BRIEF REPORTS
57
simply extrapolating this quantity from above T c . Numerical
calculations on the electron-electron scattering in the cuprates below T c have been reported by Rieck et al.17
In our discussion we have not yet included the vortex
structure present in the mixed state. The experiments mentioned in Refs. 10–12 also deal only with the case B50.
Turning now to the case BÞ0 ~with B oriented parallel to
the c axis!, we note that in YBa2Cu3O7 the vortex core is
close to the superclean limit.18,19 In this limit the electronic
structure of the vortex core is given by the discreet levels of
the Andreev bound states, with the level spacing D«
5D 2 /« F being larger than the level smearing d «5\/ t due
to scattering ~t 5quasiparticle scattering time!. In
YBa2Cu3O7 it appears that only a single Andreev bound state
exists in the vortex core.20,21 Therefore, the quasiparticle
scattering is not much affected by the vortex cores, and our
arguments given above for the case B50 can be extended
also to the case BÞ0 without introducing a large error. However, this approach will be restricted to the low-field limit
B!B c2 .
The (T/T c ) 4 factor reduces the quasiparticle scattering
rate at 0.8T c by a factor of 2.5 compared to its value at T c .
This appears consistent with the other experimental data for
YBa2Cu3O72d. 10–13 The temperature-dependent electrother-
mal conductivity displayed by the dashed curve in Fig. 3 also
removes the discrepancy of the thermal Hall angle a discussed above. In Fig. 3 we show for six temperatures below
T c ~open circles! the factor by which the quantity
S n (T)/ r n (T) must be enhanced in order to obtain the expected value a '0.01 using Eqs. ~5! and ~6!. These values
were calculated from the experimental data presented in Fig.
3~b! of Ref. 2, and they are close to the dashed line in Fig. 3.
We conclude that our simplified treatment leads to a reasonable understanding of the Seebeck and Nernst effect data in
the mixed state of YBa2Cu3O72d, without envoking an unusually large value of the thermal Hall angle.
Finally, we emphasize that our discussion given above
has been highly simplified, only pointing out the main aspects of the underlying physics. In particular, we have ignored implications of the possible d-wave symmetry of the
order parameter in the cuprates. A detailed theoretical analysis of these questions needs to be done and remains an interesting task.
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ACKNOWLEDGMENT
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