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On the upper critical field of anisotropic superconductors
PhysicaC 173 (1991) North-Holland
347-356
On the upper critical field of anisotropic superconductors Langmann Institut ftir Theoretische Physik, Technische Universitiit, A-8010 Graz, Austria Received 29 May 1990 Revised manuscript received
12 November
1990
We study in general the upper critical field (B,,(T)) for models describing clean, weak-coupling superconductors which are isotropic perpendicular to the magnetic field. Our considerations are based on generalized WHH-equations explicitly taking into account a nontrivial spatial behaviour of the electrons’ pairing interaction and an energy dependence of the electronic density of states. A simple formula for the slope d&( T)/dTI 7=TCis derived. Moreover it is shown LJJJanalytical means that the generalized WHH-&( T)-equations can be simplified in a very good approximation to an “universal” equation for the reduced upper critical field b:, (t) (t= T/T,) depending solely on the shape of the Fermi surface, and the error in this approximation is estimated. We calculate explicit formulas for the series expansions of bz2 (t) in ( I- t) 2 0 and t t 0 up to order 4 and 2, respectively. Finally we apply our results to a model with an “hour-glass” shaped Fermi surface describing layered superconductors with the magnetic field perpendicular to the layers.
1. Introduction The standard BCS-theory [ 1 ] of clean, weak-coupling superconductors presupposes a one-band model with a band relation e(k) and a pseudo potential p(k) representing the electrons and their effective interaction (k is the electrons’ pseudo momentum). Based on such a model, equations are derived relating experimental data to each other and (or) to e(k) and p(k). This scheme (and its various extensions taking into account strong-coupling effects, impurities etc. - cf. e.g. [ 13 ] ) is very successful for “standard” superconductors. The possibility of describing high temperature superconductors (HTSC) by BCS-like models was besides others - discussed by Pint et al. [ 71. They based their considerations on a band energy of the Lawrence-Doniach form E(k)=
&
+qsin*(k,,c/2),
O_
”
c
(1)
and a pseudo potential Qk)=P(&(k))
(2)
describing a layered system with an “hour-glass” shaped Fermi surface (m* is the effective mass of a 092 l-4534/9
l/$03.50
0 1991 - Elsevier Science Publishers
quasi-2D electron gas within each of the layers, q is a coupling parameter and c the distance of two adjacent layers; cf. ibid.), and demonstrated that T,values of about 90 K were possible with reasonable model parameters. Werthamer, Helfand and Hohenberg (WHH) [ 3,4,18] were the first to derive equations for the upper critical field ( IQ1 ( T) ) of isotropic, weak-coupling systems within the framework of BCS-theory. Later on these equations were generalized to incorporate an energy dependence of the electronic density of states [ 141, strong-coupling effects (cf. e.g. [ 111 and [ 15]), anisotropy effects (cf. e.g. [ 21 and [ 10 ] ), and they have been successfully used for an interpretation of experimental data by many groups (see [ 151 and the references cited there). Contributions to an extension of the WHH-theory to layered superconductors were made (besides others; cf. [ 9 ] and references therein ) by Klemm et al. [ 5 ] and - only recently - by Pint [ 81 and Rieck and Scharnberg [ 12 1. The investigations in [ 5 ] were based on the band relation (1) and concentrated on the influence of impurities and Pauli limiting on the upper critical field B& ( T) parallel to the layers. Pint [ 8 ] considered essentially the same model for ‘I= 0 (cor-
B.V. (North-Holland)
348
E. Langmann / On the upper critical field of anisotropic superconductors
responding to a cylindersymmetric Fermi surface). He studied the impurities’ effect on B,‘2( 7’) perpendicular to the layers and discussed the relevance of his results for HTSC. Rieck and Scharnberg [ 12 ] reported on a rather general approach to studying the influence of anisotropic Fermi surfaces on Bc2( T) and presented results obtained numerically for tightbinding models describing layered systems. In the WHH-approach usually the following simplification is adopted: The pseudo potential P(k) is replaced by its k-independent Fermi surface average, and an energy cutoff o, is introduced in order to make this procedure consistent (cf. e.g. eq. (26)fin [ 15 ] ). Recently general Bc2 ( T)-equations were derived without this “local approximation” [ 6 1. These generalized WHH-Bc2 ( T)-equations automatically take into account the energy dependence of the electronic density of states (EDOS) and can be used also for anisotropic models as no restrictions are imposed on the band energy and the pseudo potential of the underlying model. In the limiting case of an energy independent EDOS these equations can be shown [ 201 to be equivalent to the ones studied in [ 121. (We note that in introducing the pseudopotential an approximation is used which amounts to neglect the k-dependence of the order parameter which can be justified only under certain assumptions on the underlying model - see appendix B in [ 6 ] ) . In this paper these equations are studied in detail for models given by a band energy .s( k) and a pseudo potential P(k) with
The plan of the paper is as follows: In section 2 we derive the general Bc2( T) equations and study it by analytical means. We derive a simple - though generally valid - formula for the slope dBcz( T) / dTl T=T,. Furthermore we show that in general the Bc2( T)-equations can be simplified to an universal equation for the reduced upper critical field
b;(t) =
&2
~(k)=@lk,
2. General results (a) The generalized strong-coupling models isotropic perpendicular ref. [ 6 1, eqs. (40-42)
(3)
(k, and k,, are the components of k perpendicular and parallel to the magnetic field, respectively) which describe clean, weak-coupling superconductors isotropic perpendicular to the magnetic field. Our results apply to layered superconductors with the magnetic field perpendicular to the layers. However, the primary concern of this paper is not to explain experiments, but rather to investigate the mathematical structure of the Bc2( T)-theory. We hope that our results provide a complement to numerical work on more realistic models (taking into account impurities, etc. ) such as [ 12 1, and shed some light on the approximations underlying WHH-theory.
j(io, =-
WHH-B,* ( T)-equations for describing systems which are to the magnetic field are (cf. )
1 C j(iw,(iv,)F(iw,-iv,), P meZ
I,k,,), I,k,,)
(4)
’
depending solely on the shape of the Fermi surface, and we discuss the validity of this approximation. Moreover explicit formulas for the power series expansions of bT2(t) in (l-t)20 and t20 are calculated up to order 4 and 2, respectively. In section 3 the special case of an isotropic system is discussed at first. Then - as a nontrivial example - we consider the model given by ( 1) and (2 ) which allows us to demonstrate the strong influence of the shape of an anisotropic Fermisurface on bT2(t). Finally our results are discussed in section 4, and some details of our calculations are given in three appendices.
F(iw,)= c(k)=c(lkl
( T)
-T,~d&,(T)/dTIT=,
(5)
I iv,) I
d3Z
s
xg,(iw,;Z+
d3p exp( - +s2Z: )g,(iw,;
Z-
c)Uiv,;P) ,
g,(iwh X) =g,(iw,; IX, I, XI), Uiv,;p)=Wiv,;
IpI l,h)
!)
(6) ff=o, 3 ) (7)
with s=IeB,2(T))“2,p=l/kgT,
(8)
(k, is the Boltzmann constant and e the elementary charge), o,,= (2n+ l)x/Band v,=2mrr/Pthe usual
349
E. Langmann / On the upper criticaljield of anisotropic superconductors
fermionic and bosonic Matsubara frequencies, go, g, the “normal” Green’s functions (cf. below; for a more detailed discussion of the meaning of the symbols in the equations above see ref. [ 6 ] ). It is straightforward to prove that eq. (6) is equivalent to ,?(io,
liv,)
X2,
d&2
( T)
dT
T= T,
with Z=Z(j?, S) ( 13). There is no difficulty in evaluating the expressions entering in eq. ( 14):
iw,; k+ y
P(iv,;
k) ,
(9)
>
with f(k)
= j d3xei”X(x)
denoting the Fourier transform (X=g,, g3 and I’). A “standard” model for a clean, weak coupling superconductor is given by p’(iv,;
k) = P(k) independent
of v,,, ,
_- 2$p(k)(u,(k))2+
1 go,3(iw,; k) =
lo,+
(E(k)-p)
[jdy($T) R
(10)
’
with e(k) =E( -k) the band energy of the electrons in the extended zone scheme and fl the Fermi energy (cf. e.g. eq. ( 50)ffin [ 61). It is easily seen that in this case eq. ( 5) reduces to
(16) with
(11)
l=Z(P,s)
u,(k)=
with Z(/.I,s)=
jj 1 X(io,]iv,). nczz
A straightforward
(12)
calculation
X6( f [&(k+SXI/2)
+&(k--sx,/2)]
(17)
the electrons’ velocity perpendicular field. As the terms
to the magnetic
I ~d(&(k)-i)~(k)(...)(k),
yields
j. d*x,e-':" RZ
Wk) a,k,,
n=O, 1)
j$$
P(k)
-/l--E)
cosh’( /k/2) Xcosh2(j3c/2)+sinh2(/3/4)[e(k+s.rL/2)-e(k-sx,/2)]
(E= ZL+2y/B) are usually rather weakly varying functions of E in the vicinity of E= ,u and /3~>> 1, it is in general a very good approximation to neglect the 2ylpdependence of the integrands in eqs. ( 15) and ( 16 ). This yields
az (13)
(cf. appendix A). Equations ( 11) and ( 13) are our general Bc2 ( T)-equations. From ( 11) and ( 8) one readily obtains (b)
A(P)
(18)
ap,=,=P’ I az ’ 2~ ass=o= 4
-_
10
2
y,~(pL)<~,(k))2>,s
(19)
350
E. Langmann / On the upper critical field ofanisotropic superconductors
(26)
v=L(x,,k,)
with
with the terms (...) - 0(s2/2m*p)omitted. From the T,-equation
~(P,O)lp=,,!fB,=1
(27)
and ( 18) we obtain
-P)P(k)(...)(k)
(21)
denoting the Fermi surface average of a function (...) (k), and y, the numerical constant -
-
log(Z%T,)+l
>
(28)
hence with Z(p, S) =Z(j?, 0) + (Z(B, S) -Z(B,
0))
(22)
Z(P.i)=n(~)log(~k~T=)+l-i-~tanh(y) R
for the slope of B,,( T) at T= T,
p(k)
=0.852557...
Hence we obtain
Z(P>O)=A@)
.
d&(T) dT
7-z7-C
x
16 (kg)* =-- -~ Yl
e
TC
((~i(k))2>Fs ’
sinh2((P/4)vl (kbx) (k)sx) +cosh*y
’
smh2( (jI/4)v,
(23)
or equivalently in units K for T, lo6 m/s for vI (k) and T for B,,(T)
(29)
By arguing similarily as above, one can neglect the 2y/B-dependence of the integrand in this expression. With eqs. (8)) (4) and (23 ), this reduces the B,, ( T) equation ( 11) to
d&(T) dT
7.=7-C
log( l/t)=
= -2.1171...x
1o-4
TC ( (v,(k) )2>Fs .
(24)
Remarkably this formula is very simple and depends only on the critical temperature T, and the mean square ( ( vI (k) ) 2) Fs of the Fermi velocity perpendicular to the magnetic field. (c) The main contributions to the 1xl (-integral in (13)stemfrom Ix,II-l.As
~s2 - le.B,,(T) 2m* m*
f(k) =
$$-)
(l(k))*))
v,(k) ( (u,
(520)
@(<)=j$tanh(y) R
-i
sinh’(x,,h%) sinh’(xfi
(25) the electrons’ effective mass) is usually much smaller than the Fermi energy I*, it is a reasonable approximation to evaluate (13) by using the expansion
SIXL I
=c(k)S.ul(k)Zcosq+(...),
(30)
(k))‘>:/s’
with the function
I
(with
> FS
) + cosh2 y ’
(31)
Equation (30) is rather simple and shows explicitly that the reduced upper critical fields is “universal” and depends solely on the shape of the Fermi-surface c(k) =p.
(d) With series expansion
of the function
0( t;)
(31)> @(r)=<-K2<2+K3<3-K4c4+...) K,=1.36203
....
K3=3.22431...,
K4=10.7248 ... . (32)
E. Langmann / On the upper criticaljield of anisotropic superconductors
it is straightforward derivatives b
~ ”
to evaluate
(- 1” d”&(t) dt”
v!
corresponding (l-t)2O:
to
(30)
v=2, 3,4
1=1’ an
from
expansion
-t)3+b,(
(33)
of
1 -t)4+
the
brz( t )
(...) .
in
351
Butler [ 21 for weakly anisotropic systems in the clean, weak coupling case are essentially the same as the ones obtained above (there are deviations in some of the numerical factors which we assume to be due to print errors in [ 21). In appendix B we discuss in some detail that (e) the approximations leading from (13) to the formula (23) and the eq. (3) are excellent as long as the following conditions are fulfilled:
(34) max I&l< -6kBTC
We obtain bz=wz-312,
x
l(d3k/(2n)3)&&(k) -p--E) p’(k)(v, (k))“’ J-(d3iV(2rr)3)J(e(k)
b3=-w3+2(w2)2-w2+1/3, b,=w,+5(w2)3-5wwzw3-(w2)2+
-1
w,/2-w,/12+1/12
-p)
(‘max (... )’ is the maximum 1, 2, .... and n=2,3,4.
‘%~=G(Cf(k))~“>rs,
(36)
Similarily the behaviour of bz2 (t) in the vicinity t=O can be deduced from the following series:
value as usual) for n = 0,
Ie~Tcd&2(T)/dTI.=,, 1 <<
of
1
(41)
>
(35)
with
p(k) (v, (k) )2n
4m*
1
w
(42)
with
@(<)=J log $ 0
+ 7
+(...)
0
a, =0.3273...
b. = 0.5896...,
(37) &l(p++-eeB/4m*)
We obtain
a&
b,*(O) =bo exp( -rs) d&(t) dt
(38)
=o I=0
d2&z(t) dt2
,
= -2o!,
(39)
I=0
and, by neglecting
the terms ‘( ...)’ in eq. (37),
6 is an energy characterizing the electrons’ interaction in the system. We note that for reasonably wellbehaved band energies c(k) it can be expected that the validity of (41) for n=O implies its validity for n>o. Assuming e.g. a “Debye-potential” P(k) = Qb(E(k) -P), Fb(E) =&T@(w, - I&l )
Usually the expressions (34) (with the terms ‘(...)’ omitted) and eq. (40) can be expected to give reasonable approximations (with an error < - 5 x 10 -“) to the eq. (30) for -0.9-ctI 1 and OIt< -0.15, respectively. We note that the formulas for bZ2 (0) and the power series of Bc2 in ( 1 - t) 2 0 up to order 2 presented by
(43)
(wb is the Debye energy) the condition reads N(P+E) max I&l<-bk~T, N(P)
- 1 < 1
(44) (4 1) for n = 0
(45)
with N(P+E) the electronic density of states as usual. Equation (42) can be shown to hold for l/c3 (43) if it holds for 1/O= 1/ob and
352
-=----
E. Langmann / On the upper criticaljield of anisotropic superconductors
N(~+~D)-N(~-wD) N(P)
I.
(46)
for the r.h.s. of eq. Values less than N 10-4-10-3 (42) are typical for “standard” superconductors, and kBTCis usually a factor N 102- 1O3 smaller than the energy scale where a significant change of the electronic density of states can be expected. Hence in general it can be assumed that the conditions (4143) are certainly fulfilled (possible exceptions are mentioned in section 4).
3. Models It follows the reduced (30) which model only
from the analysis of the last section that upper critical field b$ ( t ) is given by eq. depends on the characteristics of a given via the expression
(F(Cf(k))2)>Fs,
f(k)=v.(k)l((v,(k))2)~~2
for the Fermi surface average of a certain universal function F( . ) - at least for models with the conditions (4 l-43 ) fulfilled. In the following we discuss bz2( t) for some specific models.
One can easily see that for models s(k) =E( ]k( ) and P(k) = P( ]k] ) one obtains
with
The band energy ( 1) with the pseudo potential (2) provides a rather simple model displaying the influence of the shape of the Fermi surface on bT2(t). It is easy to see that the Fermi surface e(k) =p looks similar to an hour-glass with a “waist’‘-factor xw_
Ik,I(k,,=~/c) z&q; Ik, I(4 =O)
+;
(51)
(cf. [7]). From eqs. (1) and (2) we obtain pendix C)
(cf(k))*“>Fs=
Q.
I
(47) eXP( -
0
< cf(k))2”>Fs=2*6”
3.2. Layered systems
(cf. ap-
hence
3.1. Isotropic systems
Simple calculations
formula (40) is not applicable to isotropic systems. The reduced upper critical field evaluated from eq. (30) with eq. (47) coincides essentially with the standard clean-limit WHH-result (there is a deviation ~0.4% which we expect to be due to the additional approximation used by WHH for evaluating the Fourier transform of the “normal” Green’s functions &s (10) (see [ 191, eq. (54)); the value for b:,(O) given in ref. [4] is 0.7273).
n!(2v)! (n- v)!(v!)3
* ( > ”
- :
(=N)
= (1+2)+JI_rj (neN)
>
(48)
ew( -Fs) =e*/6,
>
(53)
<10gcf(k))2>Fs) 2(1-q/2)
yield n!(n+l)! (2n+2)!
1 (l_r1,2)”
(54)
and
(55)
and
From eq. (35) we obtain the numbers b, = 0.1344, bj= -0.9330 and b4=3.003 characterizing bE2(t) in (22) gives the vicinity of t= 1, and eq. bz2 (0) ~0.7261. Equation (50) suggests that the
In fig. 1 br2 (0) (cf. eq. (38) ) is plotted as a function of the waist-factor xw. The values of b, ( u= 2, 3, 4 ), b:;(O), and (Y (cf. eqs. (34), (35), (38), (39) and (5 1) ) obtained with these formulas are listed in table 1 for five different values of xw, and the corresponding br2(t)-curves obtained numerically from eqs. (30) and (52) are plotted in fig. 2.
E. Langmann / On the upper critical field
of anisotropic superconductors
353
00 00
0.2
0.4
Fig. 1. bri (0) for the hour-glass shaped Fermi surface as a function of the waist-factor xw (cf. eqs. (54), (38) ). x,= 1 corresponds to a cylinder symmetric Fermi surface. Table 1 b$( t) for the hour-glass shaped Fermi surface: parameters the series expansions b:2(t)=(1-t)+b,(l-t)2+b,(l-t)3+ b4( 1 - t)4 and b$( t) = bzz (0) --at* for different waist-factors 0.7
XW
1.0
bz
-0.1380 -0.5428 1.1273 0.5896 0.3273
b, b, & (0) (Y
-0.0582 -0.7418 1.9866 0.6080 0.3483
It is worth mentioning
d*&(t) dt2
r=1
0.5 0.1072 - 1.0732 3.5332 0.6551 0.4091
06
06
1.0
t
%l
x,.
0.1
0.3 0.3367 - 1.3518 5.3706 0.7605 0.5946
for
0.5163 - 1.4229 6.7694 0.9843 1.6528
that the curvature
Fig. 2. b:,(t) for the hour-glass ent waist-factors x,.
shaped Fermi surface for differ-
a spherical symmetric system, and the b:*( t)-curves for these cases are essentially the same close to t = 1; remarkably these two curves deviate significantly from each other in the vicinity of t=O (we obtain b:, (0) -0.665 and -0.726, respectively). This example shows that b$( t)-curves depend rather sensitively on the shape of the Fermi surface, and the theoretical understanding of the former requires a rather detailed knowledge of the latter (e.g. from band structure calculations ). For Q=O the Fermi surface has the shape of a cylinder; in this case we recover the clean-limit result of Pint [ 81. We note that due to eq. (30) this b:;(t)-curve is valid for all cylinder symmetric modelswith e(k)=~(]k~]) and P(k)=P(]k,]).
= 2bz = 0.2716ij2-0.2760( (l-Q-/2)2
4. Discussion
1 -ii) .
(56)
of bzz( t) at t= 1 is negative for small ij-values; for fl larger than 0.6208 (or equivalently: for waist-factors less than x,=0.6158) the b:z(t)-curve has an upward curvature at t= 1. For q-O.775 (x,-0.475) we have the same curvature of b:?( t) at t= 1 as for
The great success of BCS-theory is partly due to the fact that it is universal, i.e. the quantities which are compared with experimental data are usually in a very good approximation independent of the details of the underlying model. For example, the BCST,-equation
354
E. Langmann /On the upper criticaljieldof anisotropicsuperconductors
kaT,=1.13wDe-(‘lgN(~))
(57)
evaluated usually for the special (and rather idealized) model with the Sommerfeld band energy
of ref. [ 61 general-
netic field and k,, the equations izing ( 11) are
FN=
2
~(N,,)(P,s)FN-,
(N=O, 1,2 )... ). (59)
j=--m
E(k)=
&
and the “Debye potential” (44) is in fact generally valid for any model with a band energy e(k) and a pseudo potential p(k) if 1.13~~ is replaced by some energy 52 (characteristic for the model) and gN(p) byA (20) (thisfollowsfrom (11) for.s=Oand ( 15 ); see also [ 71). In this paper we showed by analytical means that this universality does hold also for the upper critical field: The formula (23 ) for dBcz ( T) /dT] T= =, is universal, and the clean-limit result of WHH [ 41 for bz2 ( t) evaluated with eq. (58 ) and the local approximation (i.e. by neglecting the k-dependence of the pseudo-potential P(k) ) is valid for all isotropic models. Hence one can consider the approximation leading from eq. ( 13 ) to the universal bz! ( t )-equation (30) as a rigorous version (hence justification) of the “local approximation” of the WHH-approach. Due to the discussion in section 3, this universality cannot be expected for systems where the electronic density of states N(~+E) varies appreciably in the vicinity ( 1E1< - 6k,T,) of the Fermi surface e=O. In such cases, the general eq. ( 13) has to be used, and the predictive power of the BCS-approach is lost as the B,, ( T)-curves depend on details of&(k) and P(k). For example, this is to be expected for the A 15compounds [ 141. We hope that our results contribute to the discussion whether or not BCS-like models are adequate for a description of HTSC. In ref. [ 71 it is argued that experimental data for normal state properties and the high-T,-values of La-Sr-Cu-0 and Y-BaCu-0 are consistent with BCS-theory based on a model given by the band relation ( 1). Our results can provide supplements to such considerations. In this paper we studied models for layered superconductors given by eqs. ( 1) and (2 ), and restricted ourselves to the assumption that the magnetic field is parallel to k,, (i.e. the axes of rotational symmetry of the system) because this case is exceptionally simple and allows explicit results. In the general case of an arbitrary angle 8 between the mag-
with N a Landau quantum number and I,,, (/3, s) given by expressions similar to eq. (13); B,,(T) = B,, ( T, @) has to be determined from the maximum eigenvalue s=s(/3) of this infinite system of linear equations (cf. eq. ( 8 ) ) . For @= 0 the equations for the different Landau quantum numbers N decouple (i.e., I(/?, S)NjKSj,,), and they can be restricted to N= 0 leading to ( 11). It is worth mentioning that eq. (59) can be solved analytically for the slope of Bc2( T, 8) at T= T,. The result is [ 20 ] :
16 (ke)’
d&2( T @) dT
T= T,
=-y,---
e
TC (
(UL
(k))2)FS
1 XJcos2@+(2<(~l,(k))2),)/<~l(k))2)Fs)sin2@’
(60) (u,, (k) = a&(k) /ak,, ) which predicts the same angular dependence of the upper critical field of layered systems as the GL-equations when
is identified
with the GL anisotropic
mass ratio m/
A4 (cf. e.g. [5], eq. (1)).
As discussed above, considerations based on formulas such as eq. (57) and (23) implicitly assume the universality of the underlying BCS-model. As the band-width in reasonable BCS-like models for HTSC have to be assumed rather small and kBTc is large, this assumption might be false.
Acknowledgement I would like to thank Dr. E. Schachinger for critical comments and Dr. W. Pint for discussion.
Appendix A Derivation
2) -p]
of eq.
we obtain
(13). With from eq. ( 10)
ct = [e(k+sxL/
E. Langmann / On the upper critical field of anisotropic superconductors
=
jjzz &) (&
=
(E+L_)
- -)= io, + E+
(E+
(...I
sinh(/?(C++L)/2) cosh(/?(E;+E_)/2)+sinh(/I(E+-E-)/2)
this term to the first integral of the r.h.s. of ( 15) can be estimated by eqs. (42) and (43). (b) The approximations leading to eq. (30). Similarily as above it can be seen that the 2y/&dependence in eq. (29 ) is negligible if eq. (41) holds for n=O, 1, 2, .... In deriving eq. (29) from ( 13 ), we neglected the term lx1 12s2/8m* in the argument of the Gfunction. As in general
s*
Ie&2(T)I
8m*
8m*
> -=
=(...)=j$tanh(@/2)J(
[E; +~_]/2-E)
<
T
B
(64)
c’
this is also justified by (41) - at least for the third term on the r.h.s of eq. (29 ). The term that remains to be considered is (cf. (20) and (13) for ~0)
R cosh’( P&/2) ’ cosh*(/k/2)+sinh*(/?(E+
355
-e-)/4)
leading to eq. (13).
+&-&xg11(8,0)+A(p).A. (65) The approximations leading to
(a)
the approximation the r.h.s. (14) is
tegral
icosh ( 0
2
~(P+%T~Y)
(Y)
‘4(P)
_ 1 >.
(23). The
18) to
s
+%(,%I
12 and performing
a partial in-
(61) 1
As LI(p + E) is bounded in “reasonable” models, this error usually can be estimated by the 1.h.s. of eq. (41) for n=O. Similarly the estimate eq. (41) for n= 1 is obtained for the relative error of the approximation ( 19) to the first integral of the r.h.s. of eq. ( 16). For simplicity we assume in the following a band relation of the form
E(k)=
By introducing r=r: tegration we obtain
(62)
aA(p++-@*/4m*)
a.5
(66)
In deriving eq. (29) this term was neglected. This is a good approximation if eq. (42) holds with 1/c.G (43).
Appendix C
Derivation of eq. (52). With eqs. ( 1), (2) and (2 1) we obtain
with some reasonable function E,,( - ). Then the absolute value of the second integral on the r.h.s. of eq. ( 15 ) can be written as
(63) +f,r sin2($)-~)G(~) It follows from eqs. (19) and (23) that the ratio of
.
E. Langmann / On the upper criticalfield of anisotropic superconductors
356
2rr
m
0
0
d(x2)6(x2+qsin2X-
l)G
(67) (with kF=qp, x= Ik, I /kF), hence
fj=~l,u~l,
x=k,,c/2,
and
and <(u.(k))2)Fs=(k,lm*)2(l-9/2). Eqs. (67-69)
(69)
lead to (52).
References [ 1] E. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [2] W.H. Butler, Phys. Rev. Lett. 44 (1980)
1516.
[ 31 E. Helfand and N.R. Werthamer, Phys. Rev. Lett. 13 ( 1964) 686. [4] E. Helfand and N.R. Werthamer, Phys. Rev. 147 (1966) 288. [ 51 R.A. Klemm, A. Luther and M.R. Beasley, Phys. Rev. B12 (1975) 877. [ 61 E. Langmann, Physica C 159 ( 1989) 56 1. [7] W. Pint, E. Langmann and E. Schachinger, Physica C 157 (1989) 415. [8] W. Pint, Physica C 168 (1990) 143. [ 9 ] D.E. Prober, R.E. Schwa11 and M.R. Beasley, Phys. Rev. B2 I (1980) 2717. [lo] M. Prohammerand E. Schachinger, Phys. Rev. B36 ( 1987) 8353. [ I 1 ] D. Rainer and G. Bergmann, J. Low. Temp. Phys. 14 ( 1974) 501. [ 121 CT. Rieck and K. Scharnberg, Physica B 163 ( 1990) 670. [ 131 D.J. Scalapino, J.R. Schriefferand J.W. Wilkins, Phys. Rev. 148 (1966) 263. [ 141 M. Schossmann and E. Schachinger, Phys. Rev. B30 ( 1984) 1349. [ 151 M. Schossmann and E. Schachinger, Phys. Rev. B 33 ( 1986) 6123. [ 161 U. Welp, M. Grimsditch, H. You, W.K. Kwok, M.M. Fang, G.W. Crabtreeand J.Z. Liu, Physica C 161 (1989) 1. [ 171 U. Welp, W.K. Kowk, G.W. Crabtree, K.G. Vandervoort and J.Z. Liu, Phys. Rev. B40 ( 1989) 5263. [ 181 N.R. Werthamer, E. Helfand and P.C. Hohenberg, Phys. Rev. 147 (1966) 295. [ 191 N.R. Werthamer, in: Superconductivity, ed. R.D. Parks (Marcel Dekker, New York, 1969). [ 201 E. Langmann, in preparation.
347-356
On the upper critical field of anisotropic superconductors Langmann Institut ftir Theoretische Physik, Technische Universitiit, A-8010 Graz, Austria Received 29 May 1990 Revised manuscript received
12 November
1990
We study in general the upper critical field (B,,(T)) for models describing clean, weak-coupling superconductors which are isotropic perpendicular to the magnetic field. Our considerations are based on generalized WHH-equations explicitly taking into account a nontrivial spatial behaviour of the electrons’ pairing interaction and an energy dependence of the electronic density of states. A simple formula for the slope d&( T)/dTI 7=TCis derived. Moreover it is shown LJJJanalytical means that the generalized WHH-&( T)-equations can be simplified in a very good approximation to an “universal” equation for the reduced upper critical field b:, (t) (t= T/T,) depending solely on the shape of the Fermi surface, and the error in this approximation is estimated. We calculate explicit formulas for the series expansions of bz2 (t) in ( I- t) 2 0 and t t 0 up to order 4 and 2, respectively. Finally we apply our results to a model with an “hour-glass” shaped Fermi surface describing layered superconductors with the magnetic field perpendicular to the layers.
1. Introduction The standard BCS-theory [ 1 ] of clean, weak-coupling superconductors presupposes a one-band model with a band relation e(k) and a pseudo potential p(k) representing the electrons and their effective interaction (k is the electrons’ pseudo momentum). Based on such a model, equations are derived relating experimental data to each other and (or) to e(k) and p(k). This scheme (and its various extensions taking into account strong-coupling effects, impurities etc. - cf. e.g. [ 13 ] ) is very successful for “standard” superconductors. The possibility of describing high temperature superconductors (HTSC) by BCS-like models was besides others - discussed by Pint et al. [ 71. They based their considerations on a band energy of the Lawrence-Doniach form E(k)=
&
+qsin*(k,,c/2),
O_
”
c
(1)
and a pseudo potential Qk)=P(&(k))
(2)
describing a layered system with an “hour-glass” shaped Fermi surface (m* is the effective mass of a 092 l-4534/9
l/$03.50
0 1991 - Elsevier Science Publishers
quasi-2D electron gas within each of the layers, q is a coupling parameter and c the distance of two adjacent layers; cf. ibid.), and demonstrated that T,values of about 90 K were possible with reasonable model parameters. Werthamer, Helfand and Hohenberg (WHH) [ 3,4,18] were the first to derive equations for the upper critical field ( IQ1 ( T) ) of isotropic, weak-coupling systems within the framework of BCS-theory. Later on these equations were generalized to incorporate an energy dependence of the electronic density of states [ 141, strong-coupling effects (cf. e.g. [ 111 and [ 15]), anisotropy effects (cf. e.g. [ 21 and [ 10 ] ), and they have been successfully used for an interpretation of experimental data by many groups (see [ 151 and the references cited there). Contributions to an extension of the WHH-theory to layered superconductors were made (besides others; cf. [ 9 ] and references therein ) by Klemm et al. [ 5 ] and - only recently - by Pint [ 81 and Rieck and Scharnberg [ 12 1. The investigations in [ 5 ] were based on the band relation (1) and concentrated on the influence of impurities and Pauli limiting on the upper critical field B& ( T) parallel to the layers. Pint [ 8 ] considered essentially the same model for ‘I= 0 (cor-
B.V. (North-Holland)
348
E. Langmann / On the upper critical field of anisotropic superconductors
responding to a cylindersymmetric Fermi surface). He studied the impurities’ effect on B,‘2( 7’) perpendicular to the layers and discussed the relevance of his results for HTSC. Rieck and Scharnberg [ 12 ] reported on a rather general approach to studying the influence of anisotropic Fermi surfaces on Bc2( T) and presented results obtained numerically for tightbinding models describing layered systems. In the WHH-approach usually the following simplification is adopted: The pseudo potential P(k) is replaced by its k-independent Fermi surface average, and an energy cutoff o, is introduced in order to make this procedure consistent (cf. e.g. eq. (26)fin [ 15 ] ). Recently general Bc2 ( T)-equations were derived without this “local approximation” [ 6 1. These generalized WHH-Bc2 ( T)-equations automatically take into account the energy dependence of the electronic density of states (EDOS) and can be used also for anisotropic models as no restrictions are imposed on the band energy and the pseudo potential of the underlying model. In the limiting case of an energy independent EDOS these equations can be shown [ 201 to be equivalent to the ones studied in [ 121. (We note that in introducing the pseudopotential an approximation is used which amounts to neglect the k-dependence of the order parameter which can be justified only under certain assumptions on the underlying model - see appendix B in [ 6 ] ) . In this paper these equations are studied in detail for models given by a band energy .s( k) and a pseudo potential P(k) with
The plan of the paper is as follows: In section 2 we derive the general Bc2( T) equations and study it by analytical means. We derive a simple - though generally valid - formula for the slope dBcz( T) / dTl T=T,. Furthermore we show that in general the Bc2( T)-equations can be simplified to an universal equation for the reduced upper critical field
b;(t) =
&2
~(k)=@lk,
2. General results (a) The generalized strong-coupling models isotropic perpendicular ref. [ 6 1, eqs. (40-42)
(3)
(k, and k,, are the components of k perpendicular and parallel to the magnetic field, respectively) which describe clean, weak-coupling superconductors isotropic perpendicular to the magnetic field. Our results apply to layered superconductors with the magnetic field perpendicular to the layers. However, the primary concern of this paper is not to explain experiments, but rather to investigate the mathematical structure of the Bc2( T)-theory. We hope that our results provide a complement to numerical work on more realistic models (taking into account impurities, etc. ) such as [ 12 1, and shed some light on the approximations underlying WHH-theory.
j(io, =-
WHH-B,* ( T)-equations for describing systems which are to the magnetic field are (cf. )
1 C j(iw,(iv,)F(iw,-iv,), P meZ
I,k,,), I,k,,)
(4)
’
depending solely on the shape of the Fermi surface, and we discuss the validity of this approximation. Moreover explicit formulas for the power series expansions of bT2(t) in (l-t)20 and t20 are calculated up to order 4 and 2, respectively. In section 3 the special case of an isotropic system is discussed at first. Then - as a nontrivial example - we consider the model given by ( 1) and (2 ) which allows us to demonstrate the strong influence of the shape of an anisotropic Fermisurface on bT2(t). Finally our results are discussed in section 4, and some details of our calculations are given in three appendices.
F(iw,)= c(k)=c(lkl
( T)
-T,~d&,(T)/dTIT=,
(5)
I iv,) I
d3Z
s
xg,(iw,;Z+
d3p exp( - +s2Z: )g,(iw,;
Z-
c)Uiv,;P) ,
g,(iwh X) =g,(iw,; IX, I, XI), Uiv,;p)=Wiv,;
IpI l,h)
!)
(6) ff=o, 3 ) (7)
with s=IeB,2(T))“2,p=l/kgT,
(8)
(k, is the Boltzmann constant and e the elementary charge), o,,= (2n+ l)x/Band v,=2mrr/Pthe usual
349
E. Langmann / On the upper criticaljield of anisotropic superconductors
fermionic and bosonic Matsubara frequencies, go, g, the “normal” Green’s functions (cf. below; for a more detailed discussion of the meaning of the symbols in the equations above see ref. [ 6 ] ). It is straightforward to prove that eq. (6) is equivalent to ,?(io,
liv,)
X2,
d&2
( T)
dT
T= T,
with Z=Z(j?, S) ( 13). There is no difficulty in evaluating the expressions entering in eq. ( 14):
iw,; k+ y
P(iv,;
k) ,
(9)
>
with f(k)
= j d3xei”X(x)
denoting the Fourier transform (X=g,, g3 and I’). A “standard” model for a clean, weak coupling superconductor is given by p’(iv,;
k) = P(k) independent
of v,,, ,
_- 2$p(k)(u,(k))2+
1 go,3(iw,; k) =
lo,+
(E(k)-p)
[jdy($T) R
(10)
’
with e(k) =E( -k) the band energy of the electrons in the extended zone scheme and fl the Fermi energy (cf. e.g. eq. ( 50)ffin [ 61). It is easily seen that in this case eq. ( 5) reduces to
(16) with
(11)
l=Z(P,s)
u,(k)=
with Z(/.I,s)=
jj 1 X(io,]iv,). nczz
A straightforward
(12)
calculation
X6( f [&(k+SXI/2)
+&(k--sx,/2)]
(17)
the electrons’ velocity perpendicular field. As the terms
to the magnetic
I ~d(&(k)-i)~(k)(...)(k),
yields
j. d*x,e-':" RZ
Wk) a,k,,
n=O, 1)
j$$
P(k)
-/l--E)
cosh’( /k/2) Xcosh2(j3c/2)+sinh2(/3/4)[e(k+s.rL/2)-e(k-sx,/2)]
(E= ZL+2y/B) are usually rather weakly varying functions of E in the vicinity of E= ,u and /3~>> 1, it is in general a very good approximation to neglect the 2ylpdependence of the integrands in eqs. ( 15) and ( 16 ). This yields
az (13)
(cf. appendix A). Equations ( 11) and ( 13) are our general Bc2 ( T)-equations. From ( 11) and ( 8) one readily obtains (b)
A(P)
(18)
ap,=,=P’ I az ’ 2~ ass=o= 4
-_
10
2
y,~(pL)<~,(k))2>,s
(19)
350
E. Langmann / On the upper critical field ofanisotropic superconductors
(26)
v=L(x,,k,)
with
with the terms (...) - 0(s2/2m*p)omitted. From the T,-equation
~(P,O)lp=,,!fB,=1
(27)
and ( 18) we obtain
-P)P(k)(...)(k)
(21)
denoting the Fermi surface average of a function (...) (k), and y, the numerical constant -
-
log(Z%T,)+l
>
(28)
hence with Z(p, S) =Z(j?, 0) + (Z(B, S) -Z(B,
0))
(22)
Z(P.i)=n(~)log(~k~T=)+l-i-~tanh(y) R
for the slope of B,,( T) at T= T,
p(k)
=0.852557...
Hence we obtain
Z(P>O)=A@)
.
d&(T) dT
7-z7-C
x
16 (kg)* =-- -~ Yl
e
TC
((~i(k))2>Fs ’
sinh2((P/4)vl (kbx) (k)sx) +cosh*y
’
smh2( (jI/4)v,
(23)
or equivalently in units K for T, lo6 m/s for vI (k) and T for B,,(T)
(29)
By arguing similarily as above, one can neglect the 2y/B-dependence of the integrand in this expression. With eqs. (8)) (4) and (23 ), this reduces the B,, ( T) equation ( 11) to
d&(T) dT
7.=7-C
log( l/t)=
= -2.1171...x
1o-4
TC ( (v,(k) )2>Fs .
(24)
Remarkably this formula is very simple and depends only on the critical temperature T, and the mean square ( ( vI (k) ) 2) Fs of the Fermi velocity perpendicular to the magnetic field. (c) The main contributions to the 1xl (-integral in (13)stemfrom Ix,II-l.As
~s2 - le.B,,(T) 2m* m*
f(k) =
$$-)
(l(k))*))
v,(k) ( (u,
(520)
@(<)=j$tanh(y) R
-i
sinh’(x,,h%) sinh’(xfi
(25) the electrons’ effective mass) is usually much smaller than the Fermi energy I*, it is a reasonable approximation to evaluate (13) by using the expansion
SIXL I
=c(k)S.ul(k)Zcosq+(...),
(30)
(k))‘>:/s’
with the function
I
(with
> FS
) + cosh2 y ’
(31)
Equation (30) is rather simple and shows explicitly that the reduced upper critical fields is “universal” and depends solely on the shape of the Fermi-surface c(k) =p.
(d) With series expansion
of the function
0( t;)
(31)> @(r)=<-K2<2+K3<3-K4c4+...) K,=1.36203
....
K3=3.22431...,
K4=10.7248 ... . (32)
E. Langmann / On the upper criticaljield of anisotropic superconductors
it is straightforward derivatives b
~ ”
to evaluate
(- 1” d”&(t) dt”
v!
corresponding (l-t)2O:
to
(30)
v=2, 3,4
1=1’ an
from
expansion
-t)3+b,(
(33)
of
1 -t)4+
the
brz( t )
(...) .
in
351
Butler [ 21 for weakly anisotropic systems in the clean, weak coupling case are essentially the same as the ones obtained above (there are deviations in some of the numerical factors which we assume to be due to print errors in [ 21). In appendix B we discuss in some detail that (e) the approximations leading from (13) to the formula (23) and the eq. (3) are excellent as long as the following conditions are fulfilled:
(34) max I&l< -6kBTC
We obtain bz=wz-312,
x
l(d3k/(2n)3)&&(k) -p--E) p’(k)(v, (k))“’ J-(d3iV(2rr)3)J(e(k)
b3=-w3+2(w2)2-w2+1/3, b,=w,+5(w2)3-5wwzw3-(w2)2+
-1
w,/2-w,/12+1/12
-p)
(‘max (... )’ is the maximum 1, 2, .... and n=2,3,4.
‘%~=G(Cf(k))~“>rs,
(36)
Similarily the behaviour of bz2 (t) in the vicinity t=O can be deduced from the following series:
value as usual) for n = 0,
Ie~Tcd&2(T)/dTI.=,, 1 <<
of
1
(41)
>
(35)
with
p(k) (v, (k) )2n
4m*
1
w
(42)
with
@(<)=J log $ 0
+ 7
+(...)
0
a, =0.3273...
b. = 0.5896...,
(37) &l(p++-eeB/4m*)
We obtain
a&
b,*(O) =bo exp( -
(38)
=o I=0
d2&z(t) dt2
,
= -2o!,
(39)
I=0
and, by neglecting
the terms ‘( ...)’ in eq. (37),
6 is an energy characterizing the electrons’ interaction in the system. We note that for reasonably wellbehaved band energies c(k) it can be expected that the validity of (41) for n=O implies its validity for n>o. Assuming e.g. a “Debye-potential” P(k) = Qb(E(k) -P), Fb(E) =&T@(w, - I&l )
Usually the expressions (34) (with the terms ‘(...)’ omitted) and eq. (40) can be expected to give reasonable approximations (with an error < - 5 x 10 -“) to the eq. (30) for -0.9-ctI 1 and OIt< -0.15, respectively. We note that the formulas for bZ2 (0) and the power series of Bc2 in ( 1 - t) 2 0 up to order 2 presented by
(43)
(wb is the Debye energy) the condition reads N(P+E) max I&l<-bk~T, N(P)
- 1 < 1
(44) (4 1) for n = 0
(45)
with N(P+E) the electronic density of states as usual. Equation (42) can be shown to hold for l/c3 (43) if it holds for 1/O= 1/ob and
352
-=----
E. Langmann / On the upper criticaljield of anisotropic superconductors
N(~+~D)-N(~-wD) N(P)
I.
(46)
for the r.h.s. of eq. Values less than N 10-4-10-3 (42) are typical for “standard” superconductors, and kBTCis usually a factor N 102- 1O3 smaller than the energy scale where a significant change of the electronic density of states can be expected. Hence in general it can be assumed that the conditions (4143) are certainly fulfilled (possible exceptions are mentioned in section 4).
3. Models It follows the reduced (30) which model only
from the analysis of the last section that upper critical field b$ ( t ) is given by eq. depends on the characteristics of a given via the expression
(F(Cf(k))2)>Fs,
f(k)=v.(k)l((v,(k))2)~~2
for the Fermi surface average of a certain universal function F( . ) - at least for models with the conditions (4 l-43 ) fulfilled. In the following we discuss bz2( t) for some specific models.
One can easily see that for models s(k) =E( ]k( ) and P(k) = P( ]k] ) one obtains
with
The band energy ( 1) with the pseudo potential (2) provides a rather simple model displaying the influence of the shape of the Fermi surface on bT2(t). It is easy to see that the Fermi surface e(k) =p looks similar to an hour-glass with a “waist’‘-factor xw_
Ik,I(k,,=~/c) z&q; Ik, I(4 =O)
+;
(51)
(cf. [7]). From eqs. (1) and (2) we obtain pendix C)
(cf(k))*“>Fs=
Q.
I
(47) eXP( -
0
< cf(k))2”>Fs=2*6”
3.2. Layered systems
(cf. ap-
hence
3.1. Isotropic systems
Simple calculations
formula (40) is not applicable to isotropic systems. The reduced upper critical field evaluated from eq. (30) with eq. (47) coincides essentially with the standard clean-limit WHH-result (there is a deviation ~0.4% which we expect to be due to the additional approximation used by WHH for evaluating the Fourier transform of the “normal” Green’s functions &s (10) (see [ 191, eq. (54)); the value for b:,(O) given in ref. [4] is 0.7273).
n!(2v)! (n- v)!(v!)3
* ( > ”
- :
(=N)
= (1+2)+JI_rj (neN)
>
(48)
ew( -
>
(53)
<10gcf(k))2>Fs) 2(1-q/2)
yield n!(n+l)! (2n+2)!
1 (l_r1,2)”
(54)
and
(55)
and
From eq. (35) we obtain the numbers b, = 0.1344, bj= -0.9330 and b4=3.003 characterizing bE2(t) in (22) gives the vicinity of t= 1, and eq. bz2 (0) ~0.7261. Equation (50) suggests that the
In fig. 1 br2 (0) (cf. eq. (38) ) is plotted as a function of the waist-factor xw. The values of b, ( u= 2, 3, 4 ), b:;(O), and (Y (cf. eqs. (34), (35), (38), (39) and (5 1) ) obtained with these formulas are listed in table 1 for five different values of xw, and the corresponding br2(t)-curves obtained numerically from eqs. (30) and (52) are plotted in fig. 2.
E. Langmann / On the upper critical field
of anisotropic superconductors
353
00 00
0.2
0.4
Fig. 1. bri (0) for the hour-glass shaped Fermi surface as a function of the waist-factor xw (cf. eqs. (54), (38) ). x,= 1 corresponds to a cylinder symmetric Fermi surface. Table 1 b$( t) for the hour-glass shaped Fermi surface: parameters the series expansions b:2(t)=(1-t)+b,(l-t)2+b,(l-t)3+ b4( 1 - t)4 and b$( t) = bzz (0) --at* for different waist-factors 0.7
XW
1.0
bz
-0.1380 -0.5428 1.1273 0.5896 0.3273
b, b, & (0) (Y
-0.0582 -0.7418 1.9866 0.6080 0.3483
It is worth mentioning
d*&(t) dt2
r=1
0.5 0.1072 - 1.0732 3.5332 0.6551 0.4091
06
06
1.0
t
%l
x,.
0.1
0.3 0.3367 - 1.3518 5.3706 0.7605 0.5946
for
0.5163 - 1.4229 6.7694 0.9843 1.6528
that the curvature
Fig. 2. b:,(t) for the hour-glass ent waist-factors x,.
shaped Fermi surface for differ-
a spherical symmetric system, and the b:*( t)-curves for these cases are essentially the same close to t = 1; remarkably these two curves deviate significantly from each other in the vicinity of t=O (we obtain b:, (0) -0.665 and -0.726, respectively). This example shows that b$( t)-curves depend rather sensitively on the shape of the Fermi surface, and the theoretical understanding of the former requires a rather detailed knowledge of the latter (e.g. from band structure calculations ). For Q=O the Fermi surface has the shape of a cylinder; in this case we recover the clean-limit result of Pint [ 81. We note that due to eq. (30) this b:;(t)-curve is valid for all cylinder symmetric modelswith e(k)=~(]k~]) and P(k)=P(]k,]).
= 2bz = 0.2716ij2-0.2760( (l-Q-/2)2
4. Discussion
1 -ii) .
(56)
of bzz( t) at t= 1 is negative for small ij-values; for fl larger than 0.6208 (or equivalently: for waist-factors less than x,=0.6158) the b:z(t)-curve has an upward curvature at t= 1. For q-O.775 (x,-0.475) we have the same curvature of b:?( t) at t= 1 as for
The great success of BCS-theory is partly due to the fact that it is universal, i.e. the quantities which are compared with experimental data are usually in a very good approximation independent of the details of the underlying model. For example, the BCST,-equation
354
E. Langmann /On the upper criticaljieldof anisotropicsuperconductors
kaT,=1.13wDe-(‘lgN(~))
(57)
evaluated usually for the special (and rather idealized) model with the Sommerfeld band energy
of ref. [ 61 general-
netic field and k,, the equations izing ( 11) are
FN=
2
~(N,,)(P,s)FN-,
(N=O, 1,2 )... ). (59)
j=--m
E(k)=
&
and the “Debye potential” (44) is in fact generally valid for any model with a band energy e(k) and a pseudo potential p(k) if 1.13~~ is replaced by some energy 52 (characteristic for the model) and gN(p) byA (20) (thisfollowsfrom (11) for.s=Oand ( 15 ); see also [ 71). In this paper we showed by analytical means that this universality does hold also for the upper critical field: The formula (23 ) for dBcz ( T) /dT] T= =, is universal, and the clean-limit result of WHH [ 41 for bz2 ( t) evaluated with eq. (58 ) and the local approximation (i.e. by neglecting the k-dependence of the pseudo-potential P(k) ) is valid for all isotropic models. Hence one can consider the approximation leading from eq. ( 13 ) to the universal bz! ( t )-equation (30) as a rigorous version (hence justification) of the “local approximation” of the WHH-approach. Due to the discussion in section 3, this universality cannot be expected for systems where the electronic density of states N(~+E) varies appreciably in the vicinity ( 1E1< - 6k,T,) of the Fermi surface e=O. In such cases, the general eq. ( 13) has to be used, and the predictive power of the BCS-approach is lost as the B,, ( T)-curves depend on details of&(k) and P(k). For example, this is to be expected for the A 15compounds [ 141. We hope that our results contribute to the discussion whether or not BCS-like models are adequate for a description of HTSC. In ref. [ 71 it is argued that experimental data for normal state properties and the high-T,-values of La-Sr-Cu-0 and Y-BaCu-0 are consistent with BCS-theory based on a model given by the band relation ( 1). Our results can provide supplements to such considerations. In this paper we studied models for layered superconductors given by eqs. ( 1) and (2 ), and restricted ourselves to the assumption that the magnetic field is parallel to k,, (i.e. the axes of rotational symmetry of the system) because this case is exceptionally simple and allows explicit results. In the general case of an arbitrary angle 8 between the mag-
with N a Landau quantum number and I,,, (/3, s) given by expressions similar to eq. (13); B,,(T) = B,, ( T, @) has to be determined from the maximum eigenvalue s=s(/3) of this infinite system of linear equations (cf. eq. ( 8 ) ) . For @= 0 the equations for the different Landau quantum numbers N decouple (i.e., I(/?, S)NjKSj,,), and they can be restricted to N= 0 leading to ( 11). It is worth mentioning that eq. (59) can be solved analytically for the slope of Bc2( T, 8) at T= T,. The result is [ 20 ] :
16 (ke)’
d&2( T @) dT
T= T,
=-y,---
e
TC (
(UL
(k))2)FS
1 XJcos2@+(2<(~l,(k))2),)/<~l(k))2)Fs)sin2@’
(60) (u,, (k) = a&(k) /ak,, ) which predicts the same angular dependence of the upper critical field of layered systems as the GL-equations when
is identified
with the GL anisotropic
mass ratio m/
A4 (cf. e.g. [5], eq. (1)).
As discussed above, considerations based on formulas such as eq. (57) and (23) implicitly assume the universality of the underlying BCS-model. As the band-width in reasonable BCS-like models for HTSC have to be assumed rather small and kBTc is large, this assumption might be false.
Acknowledgement I would like to thank Dr. E. Schachinger for critical comments and Dr. W. Pint for discussion.
Appendix A Derivation
2) -p]
of eq.
we obtain
(13). With from eq. ( 10)
ct = [e(k+sxL/
E. Langmann / On the upper critical field of anisotropic superconductors
=
jjzz &) (&
=
(E+L_)
- -)= io, + E+
(E+
(...I
sinh(/?(C++L)/2) cosh(/?(E;+E_)/2)+sinh(/I(E+-E-)/2)
this term to the first integral of the r.h.s. of ( 15) can be estimated by eqs. (42) and (43). (b) The approximations leading to eq. (30). Similarily as above it can be seen that the 2y/&dependence in eq. (29 ) is negligible if eq. (41) holds for n=O, 1, 2, .... In deriving eq. (29) from ( 13 ), we neglected the term lx1 12s2/8m* in the argument of the Gfunction. As in general
s*
Ie&2(T)I
8m*
8m*
> -=
=(...)=j$tanh(@/2)J(
[E; +~_]/2-E)
<
T
B
(64)
c’
this is also justified by (41) - at least for the third term on the r.h.s of eq. (29 ). The term that remains to be considered is (cf. (20) and (13) for ~0)
R cosh’( P&/2) ’ cosh*(/k/2)+sinh*(/?(E+
355
-e-)/4)
leading to eq. (13).
+&-&xg11(8,0)+A(p).A. (65) The approximations leading to
(a)
the approximation the r.h.s. (14) is
tegral
icosh ( 0
2
~(P+%T~Y)
(Y)
‘4(P)
_ 1 >.
(23). The
18) to
s
+%(,%I
12 and performing
a partial in-
(61) 1
As LI(p + E) is bounded in “reasonable” models, this error usually can be estimated by the 1.h.s. of eq. (41) for n=O. Similarly the estimate eq. (41) for n= 1 is obtained for the relative error of the approximation ( 19) to the first integral of the r.h.s. of eq. ( 16). For simplicity we assume in the following a band relation of the form
E(k)=
By introducing r=r: tegration we obtain
(62)
aA(p++-@*/4m*)
a.5
(66)
In deriving eq. (29) this term was neglected. This is a good approximation if eq. (42) holds with 1/c.G (43).
Appendix C
Derivation of eq. (52). With eqs. ( 1), (2) and (2 1) we obtain
with some reasonable function E,,( - ). Then the absolute value of the second integral on the r.h.s. of eq. ( 15 ) can be written as
(63) +f,r sin2($)-~)G(~) It follows from eqs. (19) and (23) that the ratio of
.
E. Langmann / On the upper criticalfield of anisotropic superconductors
356
2rr
m
0
0
d(x2)6(x2+qsin2X-
l)G
(67) (with kF=qp, x= Ik, I /kF), hence
fj=~l,u~l,
x=k,,c/2,
and
and <(u.(k))2)Fs=(k,lm*)2(l-9/2). Eqs. (67-69)
(69)
lead to (52).
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1516.
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