PHYSICAL REVIEW B
VOLUME 57, NUMBER 18
1 MAY 1998-II
From BCS theory for isotropic homogeneous systems to the complete Ginzburg-Landau
equations for anisotropic inhomogeneous systems
Longdao Xu
Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China
Zhenghuang Shu
Jiangsu TV University, Nanjing 210013, People’s Republic of China
Sihui Wang
Basic Science Department, Nanjing University of Chemical Technology, Nanjing 210009, People’s Republic of China
~Received 3 October 1997; revised manuscript received 29 December 1997!
On the basis of the BCS theory of superconductivity, this paper gives the complete expression for the
free-energy density F s as a function of the superconducting energy gap D(T) and temperature T, valid for the
entire superconducting temperature range. Then under the condition D(T)/kT,1, the complete expansion of
the phenomenological Ginzburg-Landau free-energy density is derived from the BCS free-energy density. For
isotropic inhomogeneous systems and systems in magnetic fields, the complete expansions of these two kinds
of free energy and the relations between the microscopic and phenomenological coefficients have been determined. Complete forms of the Ginzburg-Landau equations are obtained for both isotropic and anisotropic
systems. Finally, we have found the nonlinear equation of the energy gap and the expression for the current
density for anisotropic inhomogeneous systems within the framework of BCS theory. The range of validity of
these equations is extended to D(T)/ p k,T<T c rather than the usual T→T c . @S0163-1829~98!04718-3#
I. INTRODUCTION
To study the thermodynamic properties of superconductors, it is important to give a concrete expression for the free
energy. For a uniform system, at constant volume and pressure, and in the absence of an external magnetic field, the
free-energy density of BCS theory takes the form1
F s 2F n 52
1
D~ 0 !
N ~ 0 ! D 2 ~ T ! 2N ~ 0 ! D 2 ~ T ! ln
2
D~ T !
24N ~ 0 ! kT
1
E
\vD
0
ln~ 11e 2 b E ! d j
1 2
p N ~ 0 !~ kT ! 2 ,
3
~1!
where F s and F n are the free-energy densities of the superconducting state and normal state, respectively; D(T) and
D~0! are superconducting energy gaps at temperature T and
at 0 K, respectively; N(0) is the density of states at T
50 K; and j k 5 e k 2 m is the electron’s kinetic energy measured from the Fermi surface, m the chemical potential, and
E k 5 @ j 2k 1D 2 (T) # 1/2; \ is the Plank constant divided by 2p,
v D the Debye circular frequency, k the Boltzmann constant,
and b 51/kT. In the past, because of the difficulties in cal\v
culating the integral * 0 D ln(11e2bE)dj, thermodynamic
properties of conventional superconductors could only be
studied in the limits T→0 K and T→T c with the help of
approximate calculations. We will solve this problem and
give the complete expression for the free-energy density F s
0163-1829/98/57~18!/11654~7!/$15.00
57
as a function of superconducting energy gap D(T) and temperature T, which is valid for the entire superconducting
temperature range. The complete expansions for the freeenergy density will be obtained for D(T)/kT.1 and
D(T)/kT,1. On the other hand, the relation between the
phenomenological Ginzburg-Landau ~GL! equations and
BCS theory is well known.2,3 Our improvement on BCS’s
result surely will lead to modifications to the usual GL equations. Therefore, our next step is to establish the complete
GL equations and to define all the phenomenological coefficients in microscopic terms. The validity of the GL equations
is extended from the usual T→T c to D(T,H)/ p k,T<T c .
The establishment of the complete GL equations is not
merely transforming the microscopic formulas into macroscopic ones and adding new terms of higher orders into the
old equations. More importantly, it allows broad extensions
to more complicated systems such as inhomogeneous or anisotropic systems. We will finally generalize the equations
obtained from isotropic homogeneous systems to inhomogeneous systems and systems in magnetic fields and furthermore to anisotropic systems. A series of important equations
and results will be found.
II. COMPLETE EXPRESSION OF F s FOR ISOTROPIC
HOMOGENEOUS SYSTEMS
Let the integral term on the right side of Eq. ~1! be F 1 ,
and change the integral over j into an integral over E. Because 0<e 2 b E ,1 when 0<T,T c , the integrand, which is
a logarithmic function, can be expanded into a power series,
and the integral upper limit can be extended to `; therefore,
11 654
© 1998 The American Physical Society
57
FROM BCS THEORY FOR ISOTROPIC HOMOGENEOUS . . .
`
F 1 524N ~ 0 ! kT
3e 2n b E dE
(
n51
J
H
E
~ 21 ! n21
n
`
D~ T !
E
@ E 2 2D 2 ~ T !# 1/2
c s '2N ~ 0 ! D ~ 0 ! k ~ 2 p ! 1/2
~2!
with
`
( ~ 21 ! n21
n51
K 1 ~ nx !
,
n
x5 b D ~ T ! ,
~3!
where K 1 (nx) is the first-order Basset function. A substitution of Eq. ~2! into Eq. ~1! gives the dependence of F s on
D(T) and T, which is valid for the entire superconducting
temperature range:
F s 5F n 2
1
24N ~ 0 ! kTD ~ T ! f „b D ~ T ! …1 p 2 N ~ 0 !~ kT ! 2 . ~4!
3
F( S
m21
k50
1
2z
D
G ~ k1 23 !
k
Because n can be very large in relation ~3!, we cannot
assume nx,1 and directly expand K 1 (nx) into a power series with respect to nx. Using the recurrence formula and the
differential relation
K 2~ z ! 5
2K 1 ~ z !
1K 0 ~ z ! ,
z
x
G ~ 2k1 23 !
G
1
D~ 0 !
F s 5F n 2 N ~ 0 ! D 2 ~ T ! 2N ~ 0 ! D 2 ~ T ! ln
2
D~ T !
S
p kTD ~ T !
1 2
p N ~ 0 !~ kT ! 2 24N ~ 0 ! kT
3
2
3e
2D ~ T ! /kT
S
D
dK 1 ~ z ! 1
5 K 1 ~ z ! 2K 2 ~ z ! ,
dz
z
~9!
d f ~x!
1 f ~ x ! 5xg ~ x ! ,
dx
~10!
with4
g~ x !5
(
n51
~ 21 ! n K 0 ~ nx !
S
1
x
g 1ln
2
4p
5
1O ~ z 2m ! ,
`
~5!
where G(t) is a half-integer order gamma function, and z
5nx. When b D(T)@1, i.e., T!T c , expansion ~5! is retained to the term containing the factor @ b D(T) # 21 , so that
1
D
3kT
.
8D ~ 0 !
~8!
`
For the moment K 1 (n b D) in Eq. ~3! can be expressed
approximately by an asymptotic expansion
A
e 2D ~ 0 ! /kT 11
B. Case b D„T…<1
A. Case b D„T…>1
K 1~ z ! '
S
3/2
where z5nx, combining the above two relations, and summing it over n, we get
1
D~ 0 !
N ~ 0 ! D 2 ~ T ! 2N ~ 0 ! D 2 ~ T ! ln
2
D~ T !
p 2z
e
2z
S D
D~ 0 !
kT
The second term on the right of Eq. ~8! is also a correction to
BCS’s approximate result.
524N ~ 0 ! kTD ~ T ! f ~ x ! ,
f ~ x !5
11 655
1p
(
l51
H
~11!
where g 50.5772 is Euler’s constant. Putting
u ~ x ! 5x f ~ x ! ,
~12!
D
1/2
du ~ x !
5xg ~ x ! .
dx
~13!
An integration of Eq. ~13! gives
~6!
Minimizing F s with respect to D(T) gives the temperature
dependence of D(T). Substituting ln@D(T)/D(0)#'2@1
2D(T)/D(0)# into the relation between D(T) and T and neglecting its secondary terms, we approximately have
S
J
1
1
,
2
2 1/2 1
x
1
2l21
p
2l
p
! # %
@~
$
Eq. ~10! is reduced to
3kT
11
.
8D ~ T !
D ~ T ! 5D ~ 0 ! 2 @ 2 p kTD ~ 0 !# 1/2e 2D ~ 0 ! /kT 11
D
D
3kT
.
8D ~ 0 !
~7!
The last term on the right of Eq. ~7! is a correction to BCS’s
approximate result.
If we neglect the exponential terms in Eqs. ~6! and ~7! and
substitute D(T)5D(0) into Eq. ~6!, relation ~6! will be the
same as BCS’s result at the limit T→0 K and so will be the
thermodynamic critical magnetic field obtained from it. The
specific heat of superconducting electrons given by Eq. ~6! is
approximately
u~ x !5
E
~14!
xg ~ x ! dx1C,
where C is an integral constant which is determined by F s
5F n at T5T c .
By substituting Eq. ~11! into Eq. ~14! and considering
relation ~12!, we have the concrete form of f (x) in Eq. ~3!,
which is independent of n,
f ~ x !5
S
D
x
p
1
1
x g 2 1ln
1
4
2
4p
x
H
F S
3 ~ 2l21 ! p 11
`
(
l51
x
~ 2l21 ! p
DG
2 1/2
2
J
x2
C
1 .
4l p
x
~15!
Expanding the terms in the square brackets of Eq. ~15! into a
power series with respect to x 2 / @ (2l21) p # 2 , using D(0)
11 656
LONGDAO XU, ZHENGHUANG SHU, AND SIHUI WANG
5 p e 2 g kT c given by BCS theory,1 and substituting g
5ln@pkTc /D(0)# into Eq. ~15!, a combination of Eqs. ~15!
and ~2! gives
B n ~ T ! 5 ~ 21 ! n 4N ~ 0 !
S
H
3 12
D 2~ T !
F 1 524N ~ 0 ! ~ kT ! 2 C2
4
3
`
1
1 ~ p kT ! 2
~ 2l21 !
l ~ 2l21 !
l51
(
(
l51
F
`
1
(
n52
S(
`
3
~ 2n23 ! !!
~ 2n ! !!
~ 21 ! n21
D
1
D 2n ~ T !
~ 2l21 ! 2n21 ~ p kT ! 2n22
GJ
.
~16!
1
1
2n21 5 12 2n21 z ~ 2n21 ! ,
2
~ 2l21 !
~17!
l51
Using the summation formulas
`
1
52 ln 2,
l ~ 2l21 !
(
l51
`
(
l51
S
D
where z (2n21) is the Riemann zeta function, and substituting Eq. ~17! into Eq. ~16!, we can get the integral term in
relation ~1!. Therefore Eq. ~1! can be rewritten as
S D
T
F s 5F n 1N ~ 0 ! ln
D 2~ T !
Tc
F S(
`
24N ~ 0 !~ kT ! 2 C1 p 2
`
14N ~ 0 !
(
n52
~ 21 ! n
l51
~ 2l21 ! 2
S
1
12
DG
1
~ 2n23 ! !!
12 2n21
2
~ 2n ! !!
D ~T!
3 z ~ 2n21 !
.
~ p kT ! 2n22
F
D
G
(
n52
1
B ~ T ! D 2n ~ T ! ,
n n
T
B 1 ~ T ! 5N ~ 0 ! ln ,
Tc
S DF S DG
7N ~ 0 ! z ~ 3 !
,
8 ~ p kT ! 2
T
Tc
11
B 3 ~ T ! 52
1
T
12
2
Tc
~23!
,
33 ~ 31! z ~ 5 ! N ~ 0 !
.
2 7 ~ p kT ! 4
~24!
~25!
The solution to the above equation is
D 2~ T ! 5
S D
H F
8 ~ p kT ! 2
T
12
7z~ 3 !
Tc
1
33 ~ 31! z ~ 5 !
11
2
@ 7 z ~ 3 !# 2
GS D J
12
T
Tc
. ~26!
The last term on the right of Eq. ~26! is a correction to BCS’s
approximate result. The jump in specific heat at T c is also the
same as BCS’s result. The thermodynamic critical magnetic
field H c (T) near T c is given by
We finally get
`
1
1
B ~ T ! D 4~ T ! 1 B 3~ T ! D 6~ T ! ,
2 2
3
~22!
B 1 ~ T ! '2N ~ 0 ! 12
3 11
1
2
C5 p
~ 2l21 ! .
12 l51
F s 5F n 1B 1 ~ T ! D 2 ~ T ! 1
F s 5F n 1B 1 ~ T ! D 2 ~ T ! 1
B 3 ~ D 2 ! 2 1B 2 D 2 1B 1 50.
~18!
(
~21!
The minimum condition of F s with respect to D 2 (T), that is,
@ dF s /dD 2 (T) # 50, gives the equation
The integral constant C is fixed by the conditions D(T c )
50 and F s 5F n at T5T c ,
2
D
z ~ 2n21 !
1
,
2 2n21 ~ p kT ! 2n22
By fixing the constant C, the divergent terms in the sum over
l and the terms independent of D(T) in Eq. ~18! are eliminated, so that F s in relation ~19! is purely an accurate and
complete expansion with respect to D(T).
From Eqs. ~19!–~21! we can see that ~1! the system returns to the normal state at T5T c ; ~2! the range of validity
is extended from @ D(T)/kT # ,1 to @ D(T)/kT # , p ; ~3! the
expansion for the free-energy density only includes even order powers, which is similar to the expansion of the freeenergy density in a power series of the order parameter u C 2 u
in the Ginzburg-Landau phenomenological theory of
superconductivity;5 and ~4! all the microscopic coefficients
of this expansion can be given and they all depend on N(0)
and T. Each term of the expansion changes its negative or
positive sign alternately.
For the moment the relation between D and T can be
found in the vicinity of T c . When relation ~19! is retained to
the term containing D 6 (T) and ln(T/Tc) is retained to the
term containing (12T/T c ) 2 , we have
B 2~ T ! 5
2n
`
n ~ 2n23 ! !!
~ 2n ! !!
n52,3,4, . . . .
1
T 1
3ln 2 D 2 ~ T ! ln 21 D 2 ~ T !
Tc 2
4
`
57
~19!
~20!
S DF
H c ~ T ! 51.74H c ~ 0 ! 12
T
Tc
S DG
110.727 12
T
Tc
.
~27!
The last term on the right of the above equation is a correction to BCS’s approximate result.
57
FROM BCS THEORY FOR ISOTROPIC HOMOGENEOUS . . .
From Eq. ~25! we can roughly estimate the range of validity for temperature in expansion ~22!. From Eq. ~25!,
D 2 (T) can be solved:
D ~ T !5
2
2B 2 @ 12 ~ 124B 1 B 3 /B 22 ! 1/2#
2B 3
.
~28!
D 2 (T) has a real root only for 4B 1 B 2 /B 22 ,1; therefore, the
temperature interval is
T c >T.0.683T c '0.7T c .
~29!
The thermodynamic quantities given by Eq. ~22! are meaningful only in the temperature range given by Eq. ~29!. It is
certain that as T→T c , the results are more and more accurate.
S DH S D J
C 2 ~ T ! 5n s* ~ T ! 5n s* ~ 0 ! 12
Under the condition D(T)/ p kT,1, the complete expansion of the BCS free-energy density for isotropic homogeneous systems in zero magnetic field and its coefficients are
expressed in Eqs. ~19!–~21!. In Ginzburg-Landau theory, F s
is usually expanded to the order of C 4 as T→T c . 5 Now we
extend the condition of temperature, assuming that F s has
the form
`
F s 5F n 1 a ~ T ! C 2 ~ T ! 1
(
N52
1
b ~ T ! C 2n ~ T ! ,
n n
~30!
where a and b n are to be determined. The free-energy densities in Eqs. ~19! and ~30! should be equal to each other.
Comparing terms of the same order in Eqs. ~19! and ~30!, we
have
B 1~ T ! D ~ T ! 5 a ~ T ! C ~ T ! ,
2
B n ~ T ! D 2n ~ T ! 5 b n ~ T ! C 2n ~ T ! ,
2
n52,3,4, . . . , ~31!
where B 1 and B n are defined in Eqs. ~20! and ~21!. The
energy gap can be generalized from Eq. ~26! as
D 2~ T ! 5
S D
G
H F
S D S D J
8 ~ p kT ! 2
T
12
7z~ 3 !
Tc
a~ T !5
T
T
3 12
1O 2 12
Tc
Tc
2
1¯ .
~32!
Similar to Eq. ~26!, we will try to expand C 2 (T) in Eq.
~30! into a power series of (T2T c ). Note that C 2 (T)
5n s* (T), where n s* is the superconducting electron ~actually
Cooper pairs! density. Through average approximation, i.e.,
@ dn s* ~ T ! /dT # u T c '2n s* ~ 0 ! /T c ,
and because n s* (T c )50, we have
11O 1 12
8 ~ p kT ! 2 N ~ 0 !
ln
7 z ~ 3 ! n s* ~ 0 !
T
1¯ .
Tc
~33!
T
.
Tc
~34!
One can see that both Eqs. ~20! and ~34! have the correspondent factor ln(T/Tc). Substituting a (T) in the first equation of
Eqs. ~31! with Eq. ~34!, we have
C 2~ T ! 5
7 z ~ 3 ! n s* ~ 0 !
8 ~ p kT ! 2
D 2~ T ! .
~35!
Equation ~35! will become Gorkov’s result, if T in the
denominator is replaced by T c . 2 However, Eq. ~35! is not
restricted to T→T c . As indicated by the condition
D(T)/ p kT,1, the applicable temperature range has been
extended. A combination of Eqs. ~21! and ~35! and the second equation of Eqs. ~31! gives
b n 5B n
S
D 2~ T !
C 2~ T !
5 ~ 21 ! n
3
D
n
2 3n12 n ~ 2n23 ! !!
~ 2n ! !!
N ~ 0 ! z ~ 2n21 !
@ 7 z ~ 3 ! n s* ~ 0 !#
n
S
12
1
2
2n21
D
~ p kT ! 2 .
~36!
Equation ~30! together with Eqs. ~34! and ~36! gives the
complete and concrete expansion of F s in a power series of
C 2 . Every parametric coefficient in the expansion is an explicit function of T, and every one is expressed in terms of
microscopic parameters. As a further verification of these
results, we now expand Eq. ~30! to the order of C 6 . By
minimizing F s with respect to C 2 , the dependence of C 2 on
T will be found, which can be compared with Eq. ~32!. Now
we write
F s 5F n 1 a C 2 1
1
33 ~ 31! z ~ 5 !
3 11
11
2
@ 7 z ~ 3 !# 2
T
Tc
Comparing Eq. ~19! with Eq. ~30!, the term of the order
D 2 (T) ~or D 2n ! corresponds to that of C 2 (T) ~or C 2n !. It
may be assumed that the expansions in the braces $¯% of
Eqs. ~32! and ~33! are equal @see Eq. ~40! in this paper#. A
combination of Eqs. ~20!, ~32!, and ~33! and the first equation of Eqs. ~31! gives
III. COMPLETE EXPANSION OF THE GL FREE-ENERGY
DENSITY FOR ISOTROPIC SYSTEMS
A. Isotropic homogeneous systems in zero magnetic field
11 657
1
1
b C 41 b 3C 6,
2 2
3
~37!
where a is defined in Eq. ~34!; ln(T/Tc) in it should be expanded to the order of (12T/T c ) 2 , and b 2 and b 3 are
b 25
b 3 52
8 ~ p kT ! 2 N ~ 0 !
7 z ~ 3 ! n s* 2 ~ 0 !
4333 ~ 31! z ~ 5 ! N ~ 0 !
@ 7 z ~ 3 ! n s* ~ 0 !# 3
~38!
,
~ p kT ! 2 .
~39!
To minimize F s in Eq. ~37! with respect to C 2 , the condition
] F s / ] C 2 50 gives
b 3 ~ C 2 ! 2 1 b 2 C 2 1 a 50.
11 658
LONGDAO XU, ZHENGHUANG SHU, AND SIHUI WANG
The approximate solution to this equation with accuracy of
the order of (12T/T c ) 2 is
C 2 52
H F
b2
4 ab 3
12 12
2b3
b 22
S DH F
5n s ~ 0 ! 12
T
Tc
11
GJ
1/2
1
33 ~ 31! z ~ 5 !
11
2
@ 7 z ~ 3 !# 2
GS D J
12
T
Tc
B 1 and B n are still defined in Eqs. ~20! and ~21!. D is an
unknown parameter that will be determined later in Eq. ~46!.
In this case, the order parameter C is also a complex
variable of rW . According to Ginzburg-Landau theory,5,6 Eq.
~30! can now be generalized as
`
F s 5F n 1 a u C ~ rW ! u 2 1
.
~40!
A comparison of Eq. ~40! with Eq. ~32! shows that the expansions in these two pairs of braces $¯% are equal to each
other to the order of (12T/T c ). It indirectly shows that the
assumption we made above is reliable. Therefore, the consequent expressions from Eqs. ~34!–~36! are also correct. The
last term on the right of Eq. ~40! is a corrective term near T c
to the C 2 ;T relation in GL theory.
As in Eq. ~29!, Eqs. ~37! and ~40! are valid for T c >T
.(0.7)T c .
Using the thermodynamic quantities that may be obtained
from the phenomenological expansion ~30!, and considering
the equations from Eq. ~34! to ~36!, the corresponding thermodynamic quantities in BCS theory can be rewritten microscopically. For instance, as T→T c , lnT/Tc'2(12T/Tc), Eq.
~30! is expanded to C 4 . According to Ginzburg-Landau
theory, the thermodynamic critical field H c (T) satisfies
1
a ~T!
H 2c ~ T ! 52
.
8p
2b2
2
F s 2F n 52
Substituting Eqs. ~34! and ~38! into this equation, we have
H c ~ T ! 54
S
S D
S DS D
2pN~ 0 !
7z~ 3 !
5H c ~ 0 ! 2e g
D
1/2
p kT c 12
2
7z~ 3 !
1/2
T
Tc
T
12
,
Tc
~41!
in which BCS’s result has been used, i.e., H c (0)
52 @ p N(0) # 1/2D(0), D(0)5 p e 2 g kT c , where g 50.5772 is
Euler’s constant. One can see that Eq. ~41! is the same as
BCS’s result.1 It can also be shown that when Eq. ~30! is
expanded to the order of C 6 , H c (T) will be the same as that
obtained from Eq. ~26!, i.e., Eq. ~27!.
B. Isotropic inhomogeneous systems and systems
in magnetic fields
For isotropic inhomogeneous systems and systems in
magnetic fields, D is a complex variable of Wr . Equation ~1!
can be generalized as3
`
F s 5F n 1B 1 u D ~ rW ! u 2 1
US
(
n52
D U
\2
2m *
US
2i¹2
D U
2
e*
h2
AW C ~ rW ! 1
,
\c
8p
~43!
where m * is the mass of the Cooper electron pairs. After
introducing the transformation formulas
C ~ rW ! 5
a5
\2
B ,
2m * D 1
S
b n5
2m * D
\
S
D
\2
2m * D
1/2
D ~ rW ! ,
D
~44!
n
Bn ,
n52,3,4, . . . ,
~45!
a comparison of Eqs. ~35! and ~44! gives
\ 2 7 z ~ 3 ! n s* ~ 0 !
D5
.
2m * 8 ~ p kT ! 2
~46!
Equation ~45! together with Eqs. ~20!, ~21!, and ~46! gives
the explicit expressions of a and b n , which are equal to
those given in Eqs. ~34! and ~36!. Therefore, the transformation formulas ~44!–~46! act as a bridge between Eqs. ~42!
and ~43!. A substitution of Eq. ~46! into Eq. ~42! gives the
complete expression of F s under the condition @ D(r)/ p kT #
,1 for isotropic inhomogeneous systems in magnetic fields.
IV. COMPLETE GL EQUATIONS FOR ISOTROPIC
SYSTEMS
As usually done in GL theory, the free energy of a superconductor is given by integrating F s in Eq. ~43! over the total
volume of the superconductor, * F s dV. The condition of
thermodynamic stability requires that * F s dV be minimum
W . This
with respect to an arbitrary variation of C * , C, and A
condition gives the complete Ginzburg-Landau equations
S
e*
1
W
A
2i\¹2
*
2m
c
Wj 52
D
`
2
1 a C1
(
n52
b n u C u 2n22 C50,
~47!
i\e *
c
~ e*!2
W5
uCu2A
¹3hW
~ C * ¹C2C¹C * ! 2
2m *
m *c
4p
~48!
S
nW • 2i\¹2
2
e*
h
AW D ~ rW ! 1
1D 2i¹2
,
\c
8p
1
1
b u C ~ rW ! u 2n
n n
(
n52
and the boundary condition
Bn
u D ~ rW ! u 2n
n
2
57
~42!
where e * is the charge of the Cooper pairs, \ is Plank conW is the vector
stant divided by 2p, c is the speed of light, A
potential, and hW 5“3AW is the microscopic magnetic field.
D
e*
W C50,
A
c
~49!
which applies if the superconductor is in contact with insulators. Here Wj is the supercurrent density, a and b n have been
defined in Eqs. ~34! and ~36! in terms of the microscopic
quantity N(0), both a and b n depend on T, and nW is the
normal unit vector of the boundary surface. By retaining
57
FROM BCS THEORY FOR ISOTROPIC HOMOGENEOUS . . .
only the term of b 2 after the summation sign, Eq. ~47! will
reduce to the usual Ginzburg-Landau equation near T c .
By substituting Eqs. ~44!–~46! and ~20!, ~21! into Eqs.
~47! and ~48!, one can find the equation of the energy gap
and the expression for the current density within the framework of BCS theory @cf. the equations for anisotropic systems, Eqs. ~58! and ~59!#.
condition @ D(rW )/ p kT # ,1, in which B 1 , B n , a, and b n are
defined in Eqs. ~20!, ~21!, ~34!, and ~36!.
B. Complete GL equations
As has been done in Sec. IV, the complete anisotropic GL
equations may be derived as follows:
3
In Ginzburg-Landau theory, the order parameter C of an
anisotropic superconductor is still a scalar variable. The anisotropic properties are described through three different effective masses at three principle axes of superconducting
electrons ~actually Cooper pairs!.6–8 The complete GinzburgLandau free-energy density then has the form
3
1
US
\2
(
2m m*
m 51
1
b u C u 2n
n n
(
n52
2i¹ m 2
DU
2
e*
h2
Am C 1
, ~50!
\c
8p
where m m* ( m 51,2,3) is the effective mass component at the
principle m axis, and a and b n are still scalar parameters. In
analogy with the isotropic cases, we will give the microscopic free-energy density by introducing a scalar effective
energy gap in place of the above isotropic energy gap. The
anisotropic properties are also described through effective
masses m m* . Then the anisotropic BCS free-energy density,
which is similar to Eq. ~42! in form, may be written as
`
F s 5F n 1B 1 u D u 2 1
US
3
1
(
m 51
Dm
DU
2
e*
h2
Am D 1
2i¹ m 2
.
\c
8p
C ~ rW ! 5
a5
S
\
\2
2m m* D m
B1 ,
D
~51!
~ m 51,2,3 ! ,
b n5
S
\2
2m m* D m
D
~52!
n
Bn .
~53!
As we have done before @see Eq. ~46!#, we have
D m5
\ 2 7 z ~ 3 ! n s* ~ 0 !
.
2m m* 8 ~ p kT ! 2
~54!
Therefore,
2i\¹ m 2
e*
A
c m
D
2
C1 a C
2m m*
b n u C u 2n22 C50,
~56!
~ C * ¹ m C2C¹ m C * ! 2
c
~ ¹3hW ! m
4p
~ e*!2
m m* c
uCu2A m
~ m 51,2,3 ! .
~57!
By retaining only the term of b 2 after the summation sign,
Eq. ~56! will reduce to the anisotropic GL equation near T c .
Actually Eqs. ~56! and ~57! are simultaneous GL equations,
and so are Eqs. ~47! and ~48!.
By substituting Eqs. ~52!–~54! into Eqs. ~56! and ~57!, we
find
3
(
m 51
1
2m m*
1
S
D
S D
2i\¹ m 2
8 ~ p kT ! 2 N ~ 0 !
7 z ~ 3 ! n s* ~ 0 !
e*
A
c m
ln
2
D ~ rW !
`
T
D ~ rW ! 1
~ 21 ! n
Tc
n52
(
1
2 5 n ~ 2n23 ! !! z ~ 2n21 ! N ~ 0 !
~ 2n ! !!
7 z ~ 3 ! n s* ~ 0 ! ~ p kT ! 2n24
S
D
1
u D ~ rW ! u 2n22 D ~ rW ! 50,
2 2n21
H
~58!
7 z ~ 3 ! n s* ~ 0 ! i\e *
@ D * ~ rW ! ¹ m D ~ rW !
8 ~ p kT ! 2
2m m*
2D ~ rW ! ¹ m D * ~ rW !# 1
e *2
m m* c
U U J
D ~ rW ! 2 A m ,
~59!
which are the complete nonlinear equation of the energy gap
and the expression for the current density for anisotropic
inhomogeneous systems in magnetic fields within the framework of BCS theory. For m *
1 5m *
2 5m *
3 , Eqs. ~58! and ~59!
reduce to the case of isotropic superconductors. Replacing T
on the right side in ( p kT) 2 of Eq. ~59! by T c , and m *
52m, e * 52e, Eq. ~59! gives the same result as obtained by
using the Green’s functions method.1,2
VI. CONCLUSION
\ 7 z ~ 3 ! n s* ~ 0 !
.
2 8 ~ p kT ! 2
2
D 1m *
1 5D 2 m *
2 5D 3 m *
35
5
i\e *
j m 52
1/2
D ~ rW !
j m 52
(
n52
3 12
The parameters in Eqs. ~50! and ~51! satisfy
2m m* D m
1
3
1
B u D u 2n
n n
(
n52
2m m*
S
`
A. Complete free-energy density
F s 5F n 1 a u C u 2 1
1
(
m 51
V. ANISOTROPIC INHOMOGENEOUS SYSTEMS
IN MAGNETIC FIELDS
`
11 659
~55!
A substitution of Eq. ~54! into Eq. ~51! gives the complete
formal expression of the BCS free-energy density under the
Starting from the usual BCS free-energy density ~1! for
isotropic homogeneous systems, we have given the analytical expression of F s in Eq. ~4!, which is valid for the entire
superconducting temperature range. In principle, by using
relations ~3!–~5! and ~19!–~21!, one can make order-by-
11 660
LONGDAO XU, ZHENGHUANG SHU, AND SIHUI WANG
order corrections to the free-energy density and other thermodynamic quantities obtained from it, so that the approximate results usually obtained in BCS theory at the limits T
→0 K and T→T c can be corrected order by order. Under the
condition D(T)/kT,1, we find an expansion of F s in a
power series of D(T) with all coefficients defined explicitly
in Eqs. ~20! and ~21!. Under the condition D(T)/ p kT,1, by
comparing the BCS free-energy density to that of GL theory,
the explicit relations between the macroscopic and microscopic coefficients are determined. Then the results are generalized to inhomogeneous systems and systems in external
magnetic fields. The energy gap decreases with the increase
of H; therefore, the temperature conditions change with the
field, which becomes D(T,H)/ p k,T<T c (H). In this case
D(T,H) is a complex variable related to the order parameter
C in GL theory, so that further studies are made within the
framework of GL theory. The complete GL equations in this
case are Eqs. ~47! and ~48!. The results have also been generalized to anisotropic superconductors, in which case the
effective masses of superconducting electrons are very important. The complete GL equations for anisotropic superconductors are Eqs. ~56! and ~57!. Finally, we return to terms
1
A. L. Fetter and J. D. Walecka, Quantum Theory of Manyparticle Systems ~McGraw-Hill, New York, 1971!, Chap. 13.
2
L. P. Gorkov, Sov. Phys. JETP 9, 1364 ~1959!.
3
P. G. De Gennes, Superconductivity of Metals and Alloys ~Benjamin, New York, 1966!.
4
W. Magnus und F. Oberhettinger, Formeln und Satze fur die
speziellen Funktionen der Mathematischen Physik ~SpringerVerlag, Berlin, 1948!, p. 60.
57
of BCS theory and find the complete equations for energy
gap and current density in Eqs. ~58! and ~59! for anisotropic
inhomogeneous superconducting systems in magnetic fields.
As the usual GL equations are only valid near T c , the
temperature condition for the complete GL equations has
been extended. However, the complete GL equations are
nonlinear equations including expansions of an infinite
power series of C. Presently nonlinear equations like these
cannot be solved exactly. The vortex state for type-II superconductors proposed by Abrikosov9 is only an approximate
solution of Eqs. ~47! and ~48! by retaining only the term n
52. For more terms of higher order, a solution will be more
difficult to find. At present, the phenomenological theory
based on anisotropic GL equations has been proved effective
and is often used to study the macroscopic properties of
high-T c superconductors. But the relations between the phenomenological and microscopic parameters depend on the
microscopic mechanism of superconductivity, as is done in
this paper for BCS’s model. The validity of one microscopic
mechanism can be verified by comparing to experimental
results through these relations.
5
V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064
~1950!.
6
V. L. Ginzburg, Zh. Eksp. Teor. Fiz. 23, 236 ~1952!.
7
L. P. Gorkov and T. K. Melik-Barkhudarov, Sov. Phys. JETP 18,
1031 ~1964!.
8
D. K. Tilly, Proc. Phys. Soc. London 85, 1177 ~1965!; 86, 289
~1965!.
9
A. A. Abrikosov, Zh. Eksp. Teor. Fiz. 32, 1442 ~1957! @Sov.
Phys. JETP 5, 1174 ~1957!#.