Galvano- and thermomagnetic-phenomena in high temperature superconductors, based on kinetic coefficients, are discussed, along with a connection between the electric field and the heat flow in superconductor mixed state. The relationship that determines the transport coefficients of high temperature superconductors in the mixed state based on Seebeck and Nernst effects is developed. It is shown that this relationship is true for a whole transition region of the resistive mixed state of a superconductor. Peltier, Ettingshausen and Righi-Leduc effects associated with heat conductivity as related to high temperature superconductors are also addressed

Meilikhov, E. Z.

English

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THERMOMAGNETIC PHENOMENA IN THE MIXED STATE
OF HIGH TEMPERATURE SUPERCONDUCTORS
E.Z.Meilikhov, RSC '°Kurchatov Institute", 123182 Moscow, Russia
i. Introduction
Galvano- and thermomagnetic phenomena in conductors are
usually described with the help of kinetic coefficients which are
determined with the following expressions for electric field E
and the electron fraction of the heat flow q [I]:
E=PxxJ+Pxy[k×j]+_VT+QB[k×VT ], (1)
q =nj+E_B[ kxj]-_VT+LB[kxVT ]. (2)
Here P×x and Pxy are components of the resistivity tensor; j is a
current density; k=B/B is a unit vector parallel to magnetic
field B; _, Q, H, E, _, and L are Seebeck, Nernst, Peltier,
Ettingshausen, heat conductivity and Righi-Leduc coefficients,
respectively. In accordance with the Onsager principle, some of
these coefficients are interconnected:
H:aT, E_:QT. (3)
On the other hand, there is also a connection between the
electric field and the heat flow in the superconductor mixed
state. If one assumes that both the electric field and the heat
flow arise only under the fluxoid motion then we have Josephson
formula for the electric field:
E=-_[ vfxB ] (4)
and Huebener relationship for the heat flow [2]:
q=S_Tnfvf , (5)
where vf, nf=B/¢0 are an average motion velocity and a fluxoids
density _@0=hc/2e is a magnetic flux quantum), and S_ is an
entropy of a unit section of a fluxoid. It follows from Eqs.(4)
and (5):
q=T (cS4/¢0)[Exk]. (6)
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For given values of the magnetic field and temperature the
velocity of the fluxoid motion vf depends only on the current
density j and the temperature gradient VT. For a general case the
dependence of vf on j and VT has the form:
vf=alJ + a2[jxk ] . a3VT + a4[VT×k ] (7)
(a specific form of coefficients al, a2, a3, and a4 depends on
the choice of the fluxoid motion model, see below). Substituting
Eq.(7) into (4) and (5), we obtain
E=(a2B/c) j + (alB/C)[kxj] + a4BVT - a3B[kxVT], (8)
q=(nfTS4al) j - (nfTS4a2)[kxj] + a3VT- a4[k×VT ]. (9)
The comparison of these relations with general expressions (i)
and (2) for E and q and the use of the Onsager equation (3) lead
us to
s s
as=(CS_/¢0)P×y , QsB=-(cS_/¢0)P×x , (I0)
where index s marks kinetic coefficients relating to a
superconductor in the mixed state. Eqs.(10) which are a
consequence of the connection (6) between q and E in the mixed
state of a superconductor forecast a certain correlation between
s s
"heat" and "current" coefficients (es' Qs and Px×' Pxy) and give
a possibility of checking the correctness of the conception
involved.
The most detailed experimental researches of kinetic
coefficients in the superconductor mixed state have been carried
out for high temperature superconductors [3,4] where their value
is essentially higher and the temperature range accessible for
measurements is noticeably wider than those for traditional
superconductors. The researches revealed the real connections
between kinetic coefficients to be strongly different from those
predicted by relations (i0). Thus, for instance, Hall resistance
Pxy in the mixed state is often a nonmonotonic and even
sign-reversing function of temperature and magnetic field.
Meanwhile Seebeck coefficient (thermal power) usually rises
monotonically with the T or B increase and its behavior, as a
rule, resembles that of the longitudinal resistance'e a Pxx"
Besides, according to Eq.(10), the Seebeck coefficient goes to a
zero on the normal state transition (since in this case entropy
S_ goes to a zero), although this coefficient obviously should
take the value peculiar for the normal state. All this testifies
to the fact that Eq.(10) is based on incorrect premises. It is
clear that these are only Eqs. (4) and (5) (ensuing from an
assumption that the electric field generation and the heat
319
transfer and, consequently, all the kinetic phenomena in the
superconductor mixed state are related solely to the fluxoid
motion) may be erroneous.
This approach is applicable in full measure only far from the
superconducting transition, where fraction N =n /N of normaln n
quasiparticles is small (N and nn are normal quasiparticles
concentrations at T>Tc and T<Tc, accordingly). In conformity with
the weak coupling BCS theory such particles originate only due to
their excitation via energy gap _, and hence in low temperature
superconductors (with a weak electron-phonon interaction) normal
quasiparticles appear in a noticeable quantity only in a direct
proximity to T when the width of superconducting gap _ becomesc
close to (or smaller than) thermal energy kT. Thus, this approach
is valid almost everywhere but for a narrow temperature range
near the transition.
A different situation occurs in high-temperature super-
conductors with a marked tendency towards a strong
electron-phonon interaction resulting in the finite density of
states within the energy gap [5]. In this case normal
quasiparticles arise mainly owing to filling these states and
their concentration is determined by a power-law dependence
(e.g , by law Nn_(T/Tc)4• , known for a Gorter-Casimir two- liquid
model [6]) rather than by temperature exponential dependence
N (T) a exp(-A/kT). The temperature range where fraction ofN w
normal quasiparticles is large, grows much broader and there is
no groun_ to neglect their contribution into different kinetic
phenomena In Ref.[5] numerous references are made to the works
which point out that the temperature dependences of thermal
properties of high temperature superconductors near T are much
c
better described by the Gorter-Casimir phenomenological model
than by weak coupling BCS theory. In this connection the
description of kinetic properties of the mixed state of high
temperature superconductors requires a simultaneous and
self-consisted account for the motion of fluxoids and normal
quasiparticles.
The most straightforward way here is to proceed from an
assumption that corresponding contributions into kinetic
coefficients are independent and additive. And it exactly this
way which is usually taken (see, e.g., Refs.[7 and 8]). This
approach presupposes, in particular, that the presence of normal
quasiparticles effects in no way the fluxoid motion (and,
consequently, the electric field in a superconductor). This key
1
Here by the term "normal quasiparticles" we understand not the
particles which are "bound" in the fluxoid cores (with the
concentration of N) but those which are outside the fluxoid cores
and are "free" (their concentration is n <N).
n
320
assumption is corroborated with no due arguments and is simply
postulated although it does not look absolutely obvious• So one
needs to get rid of this limitation and to construct a
self-consisted theory of galvano- and thermomagnetic phenomena in
the superconductor mixed state within a frame work of a single
model. Since at present the fluxoid motion is described
approximately even in neglect for normal quasiparticles [9] a
phenomenological approach analogous to that used in the classical
work by Gorter-Casimir [6] might be used.
2. Seebeck and Nernst Effects
Average density Jn of the normal quasiparticle current is
determined by the standard relationship:
N E+aN [Exk]__NIVT__N2[VTxk ] (Ii)Jn=Oxx xy
N and oN
which takes into account that conductivity (axx xY) of a
system of normal quasiparticles (metal) is proportional to their
concentration Nn while Seebeck (aN) and Nernst (QN) coefficients
• N N
are independent on it2 Here _N]=aNaxx+QNBaxy and
N N
_N2=_Naxy-Q_Baxx. The electric field in a conductor with finite
conductance is determined by well-known relations, namely, the
Ohm's law
N . N
E=Pxx]n+Pxy[kxJn]+aNVT+QNB[kxVT ] (12)
for a normal metal and Josephson's formula (4) for a
superconductor in the mixed state without normal quasiparticles.
N N
are components of resistivity tensor of a normalHere Pxx and Pxy
metal. How could one generalize Eqs.(4), (12) corresponding to
two limiting situations (N =0 and N =I) for a common case whenn n
0<N <i? In the absence of an exact answer to this question one
n
may try the interpolation formula
S:-f {_[ vfxS ]}+fn{p: " N }3n+Pxy[kxj ]+eNVT+QNB[kxVT ] (13)s x n
where weight factors fs and fn have to satisfy conditions
2In normal metal _, Q _ kT/EF [i] where Fermi energy aF is determ
the total concentration of charge carriers and is independent on Nn.
321
fs(Nn=0) =i, fs(Nn=0)=0, f (Nn=l) =I which ensure the transfer ofn
Eq.(13) into "correct" expressions (4) and (12) in the above
cited limiting cases. The expression for an electric field which
is obtained after substitution of Eq.(ll) into (13) contains no
terms proportional to VT (it is required for obvious equality E=0
to take place at vf=0) and has the form
f
E s
As it can be seen from Eq.(14), in order to determine an electric
field it is necessary to find the velocity of fluxoid motion vf
for which purpose let us use the equation of motion suggested in
Ref [I0] (when N =0, VT=0) :• n
_m[VsXk] - Dm[Vfxk] - _±vs - _,vf = 0, (15)
where um=nse_0/c, ns is a concentration of superconducting
carriers; vs is their (average) current velocity; and _i and _ll
are viscosity coefficients. The existence of normal current Jn
(for N /0) should have brought about the appearance of an
v -dependent function A(vn) in Eq (17) Since at N _ 1 all the• " nn
terms occurring in Eq.(17) go to a zero, the equation of fluxoid
motion in this case come to relation A(Vn)=0 coinciding with the
equation of normal quasiparticles motion, i.e., to the Ohm's law
<I variables v and v in the mo-(12). We may assume that at Nn s n
tion equation are also "divided", in other words, the equation is
"split" into two expressions: A(Vn)=0 (coincides with Eq.(ll) )
and Eq.(15).
Now let us discuss the way of modification of the fluxoid
motion equation (15) with the presence of the temperature
gradient. Firstly, "thermodynamic" force FT! = -S_VT must
manifest itself in it. Here S_=S_(T) is a temperature dependent
entropy of a fluxoid section of a unit length. This force is an
analogy [2] for Lorenz force _m[Vs×k] and can be derive from the
latter by substituting Vs4 v*=(S_/_m)[VTxk]. Simultaneously force
FT2 .analogous to "viscosity" force -U±v s from Eq.(15) should be
introduced; it is derived from the latter by means of the same
v*: = -S_(_/_m) [VTxk]-substitution vs FT2
Another reason for appearance of "thermal" forces in the
fluxoid motion equation is the origination of counterflow
supercurrent in-between their normal cores Ill]. If there is a
temperature gradient inside the fluxoid core (that is in the
322
region where Nn=l) there appears additional normal current with
the density, according to Eq. (ii), equal to JT = -_NIVT-
-_N2[VTxk]. This current is balanced with externally flowing
counterf low current with an average density jb=Jbs+Jbn= -JT'
which is a sum of supercurrent Jbs and normal current Jbn" The
superconducting component of this current equals Jbs = NsJb =
V it is a
-NsJT. Similarly to the transport supercurrent Js=ens s
source of forces effecting fluxoids [12]. Hence two more forces
FT3=- (_m/Ne)(_NI[VTxk]-_N2VT) ' (16)
FT4=(n±/Ne) (_NIVT+_N2[VTxk]
should be introduced into fluxoid motion equation (15) related to
the case when VT=0. These forces are derived from Lorenz force
_m[VsXk] and viscous force -_±vs by means of substitution v3 s
-v**, where v**= Jbs/ens = - (I/eN)JT
Thus, in the presence of VT the fluxoid motion equation
acquires the form:
_m [VsXk]- __Vs - [S_ [_] + [_] (_Tm/_NI-_7_L_N2)] [VTxk] -
Using expression Js=ensVs=j-jn (where the normal current follows
Eq. (Ii) ) and substituting relationship Vs=C[ExB][ (l-Nnfn)/fsI/B2
ensuing from Eq.(14) into Eq.(17) we obtain the equation for the
electric field. The solution of that equation in respect to E
gives the relationship determining the transport coefficients of
superconductors in the mixed state : a Seebeck coefficient
(thermal power)
3The negative sign appeared in the v _ -v** substitution is as
result of the reverse (in respect to the normal current )
direction of supercurrent Jbs" The nondiscrepancy of the fluxoid
motion equation (17) thus obtained is confirmed by the
fulfillment of the Onsager relationship for calculated kinetic
coefficients (see below).
323
N N +Q.B( a - o ) + (cS_/_0)pxy (18)a=N n aN(Pxx°xx+Pxy°xy ) Pxx xy Pxy xx
and Nernst coefficient
N _ (aN/B) (PxxoN_ _ ]Q=Nn °xx+Pxy°xy)- xy Pxy°xx) -(cS_/B_0)Pxx" (19)
It is easy to see that for normal metal (Nn=I, S_ =0) Eqs. (18)
and (19) give _=aN and Q=QN' and in the superconducting state (no
fluxoid motion, P<Pc and Pxx = Pxy = 0), as it might be expected,
a=Q=0. Thus, Eqs. (18) and (19) are true for a whole transition
region of the resistive mixed state of a superconductor.
3. Heat conductivity.
Peltier, Ettingshausen and Righi-Leduc Effects
Heat flow qs' related to the entropy transfer with moving
fluxoids, equals qs=S_TnefVf, where nef is effective density of
moving fluxoids which is different from their real density
nf=B/_0. Nevertheless, rewriting Eq. (14) in the form E=-(1/c) [vf
x B](nef/nf) we come to the relation
qs=T (cS_/_0)[E×k],
which is an analogy to Eq.(6). As to the heat flow due to the
normal quasiparticle motion, it is defined by the relationship
qn=nNJn+EN_NB[kxj,]+Nn [-_N_T+LNB[kxVT ]], (20)
which accounts for _N and LN being proportional to current
carrier concentration Nn while HN and the product EN_N are
independent on it.
Thus, the electron component of the heat flow in a
superconductor equals
.cs .
q:qs+qn:T [ ¢0][E×k ]+HNJn+E_B[ kxjn]+Nn[-_NVT+LNB[kxVT ]]. (21)
To deduce corresponding kinetic coefficients is possible by means
of expressing q in terms of total transport current j. First let
us find the connection between normal and total currents
substituting Eq.(1) into Eq.(ll) :
[Jn=Nn J(Pxx°_x+Pxy°_y)+[kxj](Pxy°xx-Pxxax
324
-(_NI-Nn_I)vT - (_N2-Nn_2)[VTxk] (22)
where _I=(°N _+°_yQB)xx , _2=(o_ya-O_xQB) . Next, substituting
Eqs.(1) and (22) into (21) we come to expression (2) where
Peltier coefficient
[ N N N N ]_=N _N (Pxxon xx+Pxyaxy)+EN_NB(Pxx°xy-Pxy°xx) + T(cS_/_0) Pxy (23)
Ettingshausen coefficient
(E_B)=Nn N_NB(PxxOxx+PxyOxy)-gN(OxxOxy-PxyOxx ) -T(cS¢/¢0)Pxx, (24)
heat conductivity coefficient
_=Nn_N+nN (_NI-Nn_I)+EN_NB(_N2-Nn_2)-T(cS_/@o) QB, (25)
and Righi-Leduc coefficient
LB=NnLNB+HN (_N2-Nn_2)-EN_NB (_NI-Nn_])-T (cS_/¢0)a. (26)
Proceeding from the Onsager equation for a normal metal
(HN=aNT, EN_N=QNT) we make sure that Onsager equations (3) are
also valid for above given Seebeck and Peltier coefficients
(compare Eqs.(18) and (23)) as well as for Nernst and
Ettingshausen coefficients (compare Eqs.(19) and (24)) for a
superconductor in the mixed state. This is an evidence of
inherent nondiscrepancy of the model suggested and, in
particular, of a correct form of the fluxoid motion equation
(17). The latter might be obtained without defining concretely
expressions for "thermal" forces FTI-FT4. It would be sufficient
first to obtain relations (18) and (19) for a and Q, proceeding
from Onsager relations and Eqs. (23) and (24) for H and E
coefficients, and then to "restore" the fluxoid motion equation
(17).
The Hall conductivity (resistance) is known to be
sufficiently lower than the longitudinal conductivity
(resistance) at any temperature [3,4]. It gives a possibility to
simplify noticeably the expression obtained for kinetic
coefficients:
a=HIT _ NnaN(pxx/pn)+(cS_/¢O)Pxy _ Nn(_N(Pxx/PN), (27)
Q:(E/gT) _ NnQN(Pxx/PN)-(cS_/B_o)Pxx :_- (cS#/B¢0)Pxx, (28)
325
_ Nn_N , (29)
LB_NnLNB-T(cS_/¢O)_-NnaNT(CS_/_O)(Pxx/PN). (30)
The first two of them were derived in earlier works in a not
quite correct way [i, 2, ii, 13].
Let us emphasize that the expressions derived for
coefficients a, Q, H, E, _ and L are universal, i.e., independent
on the choice of coefficients f and f . The latter determine
s n
only the connection between total conductivity of a
superconductor in the mixed state and "partial conductivities",
related to normal quasiparticles and fluxoids [14].
From the expressions obtained for kinetic coefficients e, Q,
H, E, _, and L of a superconductor in the mixed state it is seen
that each of them consists of two terms, one of which is
proportional to concentration N of normal quasiparticles and then
other is to the S_ entropy of fluxoids. These two parts may be
4
conditionally called "normal" and "fluxoid" contributions Their
role in different kinetic coefficients is not identical.
Longitudinal effects (described with coefficients e, H, and _)
are defined basically with the "normal" contribution, while
transverse effects (coefficients Q, E, and L) are governed by the
"fluxoid" one. It is due to the following:
I) Seebeck and Peltier coefficients e, and H : a relatively
minute value of the fluxoid entropy (cS_PN/aN_ 0 _ R << Pxx/Pxy)
and a absolutely small Hall angle (p×x/Pxy >> i); the vortices are
almost immovable along the temperature gradient and transfer
small (in comparison with normal electrons) energy;
2) Nernst and Ettingshausen coefficient: a relatively large
value of the fluxoid entropy (cS_PN/_N_0 _ R >> I) and a
relatively small Nernst coefficient QN in the normal state
(QNB/eN _ 7b << i); fluxoid transfer high (as compared to normal
electrons) energy in the direction perpendicular to the
temperature gradient;
3) Heat conductance and Righi-Leduc coefficients _ and L: all
the above-said conditions and a relatively high electron heat
conductance in the normal state (_NPN/Tc_N >>i) ; the vortices are
4The conventionality of this division can be seen if only from
the fact that the "normal" contribution into kinetic coefficients
is related to the longitudinal resistivity Pxx' which, at least
to a certain extent, is determined by the fluxoid motion.
326
almost immovable along the temperature gradient ant transfer
small (in comparison with normal electrons) energy.
A good agreement with experimental data for HTSC of different
compositions has been achieved for a different selection of
=N_ f =Nu [14]. The valuesinterpolation factors of the form fs s' n n
= u =I for YBa2Cu307_6 and _ = u =3/2 for Bi-2223 were shown to
provide the best fit.. A possible reason for these different
values is the different degree of anisotropy of the two compounds
in question. This problem, however, is to be further analyzed.
REFERENCES
I. A.A.Abrikosov, Fundamentals of the Theory of Metals, Elsevier,
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Temperature Superconductors, Preprint, 1992
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6. C.J.Gorter, H.B.G.Casimir, Zs.Phys., 1934 voi.35, p.963;
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Thermomagnetic phenomena in the mixed state of high temperature superconductors

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