- Email: [email protected]
Tunneling density of states for the two-band model of superconductivity
Solid State Communications, Vol. 22, pp. 37 1—374, 1977.
Pergamon Press.
Printed in Great Britain
TUNNELING DENSITY OF STATES FOR THE TWO-BAND MODEL OF SUPERCONDUCTIVITY N. Schopohi and K. Scharnberg Abteilung für Theoretische Festkorperphysik, Universität Hamburg, Hamburg, West Germany (Received 28 January 1977 by B. Muhlschlegel) In the presence of interband scattering of quasiparticles by impurities only a single gap in the excitation spectrum is found. The dependence of this gap and the two order parameters on interband coupling and impurity content is discussed. Graphical representations of the density of states for several sets of parameters are given. For certain ranges of parameters our calculated density of states is qualitatively similar to the experimental results. THE TWO-BAND MODEL [1] is often viewed with suspicion because of the large number of parameters it contains. It is therefore desirable to find arguments which can be used to restrict the variation of the model parameters. Such arguments will be presented here, Furthermore, we shall show that some physically measurable quantities are rather insensitive to the choice of parameters. Before one can hope to successfully apply the twoband model to transition metals one has to generalize it to include interband scattering. When this is done, only one gap in the excitation spectrum is found [2, 3]. In the ensuing numerical calculation of the temperature dependent order parameter [2] specific heat [4] and thermal conductivity [5] it is then argued that the ratio of the density of states at the Fermi level ,
=
,
N~(0)/N
2 (0) (1) is very large, and in fact the limit l/~= 0 is taken. In this case a second gap identical to the larger order parameter reappears [2]. Hence, all attempts to interpret experiments on the basis of the two-band model assume the existence of two energy gaps. In view of the actual band structure of transition metals the assumption ~~ 1 seems to be unjustified [6, 7]. The Fermi surface of niobium, for example, consists of three separate sheets. The ratio of the density of states on each of these sheets is given by [61 NJG
:NELL :NOCT
=
4.42:4.10: 1.37.
Hence, if we again approximate the Fermi surface by two spherical sheets [2—5] it seems to be more appropriate to take ~ = 1.2 or ~ = 10, depending on how the three sheets are lumped together to form a two-band model. Since these sheets are not clearly related to distinct s- and d-bands, we prefer to speak of zones rather than bands and use indices 1 and 2. For those values of ~ the theoretical predictions
must be drastically changed when it is recognized that only one energy gap exists. An analytic calculation of the resulting density of states as a function of energy and impurity content, valid for very small and very large impurity concentration, has been given by Moskalenko eta!. [3] under the assumption that the order parameters are known. The impurity dependence of the order parameters has not been calculated by these authors, whose work seems to have escaped general notice. Here, we shall discuss the density of states as thoroughly as is possible on the basis of numerical computations. Before presenting our results we briefly review the derivation of the relevant formulae. The calculation of the electronic temperature Green function of the two-band model [1] supplemented by the electron—impurity interaction Hjmp
exp i(q
=
—
k)R, v~(k,q)c~ocqxa
I~,q~
G
(2)
with q)
~
=
J
dr ~,<(r) Vjmp(r)~iqx(r)
(3)
.
is standard [2 4, 5] Neglectmg all those contributions to the matrix self-energy ~ which contain the impurity * potential in the form vg~(k,q)v (k, q) with K ~zor X ~ i’, following the argument put forward by Rainer [8] the 4 x 4 matrix Green function becomes diagonal with respect to the zone indices [2]: -
.
.
.
.
.
.
,
G~’ =
,
=
—
ea,~r3
iW~To — ~ .-~
—
+
Z~Ti
iW~To+&~1~i.
(4)
The last equation defines the quantities and L~i,<. Inserting this Green function into the approximate expression for ~ gives a system of coupled equations for i~ and which can be simplified by considering the quantities ~,
371
TWO-BAND MODEL OF SUPERCONDUCTIVITY
372 0
0.5
1.0
2.0
/ 1.8
_~_
0
0.5
10
2.0
// /
Vol. 22, No.6
//
V
2.2
V
_7
—
18 1.7654
/_-V
0.0
—
— — —
_——~
—
_____
_____
_______
1.7654 1.6
0.0 \
2.2 0.5
14 1.2
0.88 0.44
/
~1
1.0
1 2
0.9
2.2044
—
— ——
/ / 0
/
7 V
// ~________ Q ___________ 176 / / / ~2
/ / /
V
/
1.0
V
-V
/ I
/00
/
V
0.8
1.0 1
I-
175
.—
.-,_~.0
/
//
/
0~6,~U
.I1~1~
as 1.0 Fig. 1. Normalized energy gap (full curves) and order parameters (dashed curves) as functions of interband coupling ~ = Ui2/U~x.The parameter varied is p/T~°. Order parameters are shown for p = 0 and p = only. For p = 0, Wg = mm (Lxi ‘~2)• The inset shows the dependence of T~on i~and p for the two values of Dashed curves represent 1~(18),full curves T~(p= 0.44T~~). 2/utTux
00
,
~-.
(5)
u~(iw~) = After analytic continuation one obtains uK~ = w +
(U~—
2r~~
[21:
0
0.2
12/u~ax ~1.0
0.4
Fig. 2. Same as Fig. 1, but with ~ = 1 .2. In this case L~ii/T~ differs more strongly from the BCS-value, the deviation being maximal for p ~ 2.2T~,°. The inset shows the rounding off of the maximum of wg/Tc at ~ = 0.99586 (17) with increasing p. To determine the gap we eliminate ux from (6) and consider the resulting equations as defining two implicit functions wi(2)(u) [10]. If ~2 is the lower order parameter, then the function
u~)(u~ l)h/2 —
with
=
U
+ 1 +hi2 (~i
—
~2)
}
(9)
(6) K~X
~,>0.
h12
These equations contain only the interband scattering times 19]
i
=
/u—1i’(
~ ~ 2r21 + i~) 2. (7) its concave for 0
Vol. 22, No.6
TWO-BAND MODEL OF SUPERCONDUCTIVITY 5.0-
2.0-
373
4.0-
3.0 N(w)
-
N(0) N(w)
2.0
1.0—
-
1.00.0—
I
0.5
I 1.5
________________
1.0
2.0
2.5
3.0
Fig. 3. Partial and total density of states at T = 0 for U12 = 0, ~ = 1.2 and p = 0.66T. With these parameters one obtains TC/TC° = 0.8595 and wg/T~= 1.273. Near the transition temperature T~we can expand u~(6) in terms of z~.To first order we obtain -i
U~
Ii
~
=+y~
1 ~ I ~ + ~pj 2PTKX —
~
w>0 (11)
K~rX
with p
=
1/2r12 + l/2r21
=
(1 + ~)/2r12.
(12)
Inserting this into (10) gives: 0
=~
X
~
1 r0—U (roJ(T~~o~~ )+—~ ~+
)
/
10
—
~
~
)
1\
0.00.5
21
32
H
If
1.0
U
=
(UKX)
=
I
=
in
{~
~—
~
+~
The choice of indices is such that for U12 = 0 we have U11 = = 0.2835, U22 = = 0.2370 and con-
is limited to the range 0~U12 XmaxXmin
1.13w0 exp (
U12
_f 0.0213 —
for~=1.2
0.0074 for ~ = 10.
responding purity of the niobium it might be helpful to relate p to an effective mean free path through the dimensionless parameter p/Ti?. To visualize the cor-
-
(15)
1/Xmax).
=
max
)
2ir 3cP I ~,(i + —1 I
model parameters. In view of the Fermi surface of Nb it seems reasonable to assume that both zones contribute to superconductivity, i.e. U(l4) must be positive definite. For U12 = 0 and p = 0 both zones could then have positive transition temperatures T~,° > T1~<>0. The larger of the two eigenvalues of Uis determined from experimental values of T~° (Nb: 9.2 K) according to =
—
Assuming identical cut-offs for the three interactions V,<~we have chosen the value w~/T~° = 30, appropriate forNb. The amount of interband scattering is controlled by
The purity dependent T~is then determined from the secular equation corresponding to (13). Before we can solve (6) and (10) we have to choose
C
3.0
(14)
tanh~w/2 W + 1P —
2.5
More or less arbitrarily we choose T~<= T~°/2. This determines Xmjn. With U2~= U12~as variable parameter we can calculate the diagonal elements from the identity 2—UTrU+r [12] U 0detU= 0.
2~/~
)
2.0
sequently ~i > ~ According to Leggett, the phase of the Bloch[13]. functions be chosen the suchU,~ thatareU12 is U12 positive Since,can furthermore, real, (13)
(V,,~ANx(0))
J(T~,p,~) = Rejdw
1.5
Fig. 4. Total density of states at T = 0 for U12 = 0 and ~ = 1.2 for three values of p: (1) p = 0.165T~°{T~/T~° = 0.948, wg/Tc = 1 .0l5}; (2) p = 2.2T~{T~/T~ = 0.793, wg/Tc = 1 .554}; (3) p = 6.6T~°{T~/T~° = 0.767, oJg/T~ = 1 .694}. The arrows mark the positions of ~j/~g and ~2 /wg, respectively.
where the matrix Uis given by and
1
w
(16)
1
.~_±_c
3500 A. VF — 1 + ~ 2p~ ~ p From (6) and (10) one can easily see that for U =
=
11 + U12 U22 + U21, which can be fulfilled for ~~ 1 and is then
equivalent to 2V’f U~2/U!r= 1
(17)
the two order parameters become equal, i.e. the twoband model reduces to the one-band BCS-modei with impurity independent T~.This can be seen in Figs. I and 2, where order parameters and energy gaps are shown as
374
TWO-BAND MODEL OF SUPERCONDUCTIVITY
Vol. 22, No. 6
LI®
function of ~ for various values of p/Tx. We note that for i~> 2’~/~/(1 + ~) we have ~2 > ~i even though
8.0 -j
U
11> U22 and Ni(0)>N2(0). Another important result evident from these figures is that one always has wg/Tc ~ L~BCS/T~CS= 1.7654. With increasing p the gap wg/T~approaches monotonically the BCS-value 1.7654, independent of~,whereas the normalized order parameters, although closer to the BCS-value, remain to be functions of~evenfor i/p = 0. The limiting temperature ~ = ~ T~(p)shown in the inset of Fig. 1 has been calculated from equation
T~= 1.13w0 exp
—
1/
Xmin
+
N(O) 60
-
40
-
2.0
1+c11
with
(Xma,~
00
Xmin)
-
-
(18) I~(~ ii)
=
~
1 +~
1 +~
For the density of states (Figs. 3,4 and 5) shows a gradual transition from the two square root singularities of the pure two-band superconductor to BCS-behaviour when p is increased. For ~ = 10 and r~= 0 the first peak due to N2(w) is strongly reduced, so that even for fairly pure materials (p/T~° > 0.1) the main contribution to N(w) for w < z~icomes from N1 (w) (Fig. 3). For = 10 and i~> 2~,/~/(l + ~) (Fig. 5) the structure of
______________________________
1.0
1.5
-~
2.0
Fig.5.SameasFig.4,butwith~=lOandU12= (l)p =0.033Tc°{Tc/Tc°=0.995,wg/Tc=1.502, U~: = 0.9994}; (2) p = 0.33Tc°{Tc/Te° = 0.960 wg/Tc = 1.565, L~1I Wg = 0.9965}.
is qualitatively similar to what is found experimentally. The fact that band structure effects can produce such structure might help to remove the difficulties encountered when strong-coupling theory is applied to Nb [15]. N(w)
REFERENCES 1. 2.
SUHL H., MATTHIAS B.T. & WALKER L.R., Phys. Rev. Lett. 3, 552 (1959). SUNG C.C. & WONG V.K., J. Phys. Chem. Solids 28, 1933 (1967).
3.
MOSKALENKO V.A., URSU A.M. & BOTOSHAN N.J., Phys. Lett. 44A, 183 (1973).
4. 5.
CHOW W.S., Phys. Rev. 172,467 (1968); BURKEL R.H. & CHOW W.S., Phys. Rev. B3, 779 (1971). KUMAR P. & GUPTA S.N.,Phys. Rev. B6, 2642 (1972).
6.
MATTHEISS L.F.,Phys. Rev. Bi, 373 (1970).
7. 8.
ENTEL P., Z. Phys. B23, 321 (1976). RAINER D., Forschungslaboratorium der Siemens AG, Erlangen, Aktenvermerk l—2954 vom 6.9.1967 (unpublished). MOSKALENKO V.A. & PALISTRANT M.E., JETP 22, 536 (1966). MAKI K., Superconductivity (Edited by PARKS R.D.), p. 1046. Marcel Dekker, New York (1969).
9. 10. 11. 12.
ABRIKOSOV A.A., GORKOV L.P. & DZYALOSHINSKI I.E., Methods of Quantum Field Theory in Stattistical Physics, p. 337. Prentice-Hall, Englewood Cliffs, New Jersey, (1963). GREUB W.H., Multilinear Algebra, p. 164. Springer, Berlin (1967).
13.
LEGGETT A.J., Progr. Theoret. Phys. 36, 901 (1966).
14.
This expression has been given before by RAINER [8]. Everywhere else in the literature [e.g. reference [9] and KUSAKABE T., Progr. Theoret. Phys. 43,907(1970)] a different formula is obtained from (13) by omitting p from the logarithmic term in (15). The TC°°thus obtained is somewhat smaller than (18).
15.
BOSTOCK J., DIADIUK V., CHEUNG W.N., LU K.H., RUSE R.M. & MacVICAR M.L.A.,Phys. Rev. Lett. 36, 603 (1976).
Pergamon Press.
Printed in Great Britain
TUNNELING DENSITY OF STATES FOR THE TWO-BAND MODEL OF SUPERCONDUCTIVITY N. Schopohi and K. Scharnberg Abteilung für Theoretische Festkorperphysik, Universität Hamburg, Hamburg, West Germany (Received 28 January 1977 by B. Muhlschlegel) In the presence of interband scattering of quasiparticles by impurities only a single gap in the excitation spectrum is found. The dependence of this gap and the two order parameters on interband coupling and impurity content is discussed. Graphical representations of the density of states for several sets of parameters are given. For certain ranges of parameters our calculated density of states is qualitatively similar to the experimental results. THE TWO-BAND MODEL [1] is often viewed with suspicion because of the large number of parameters it contains. It is therefore desirable to find arguments which can be used to restrict the variation of the model parameters. Such arguments will be presented here, Furthermore, we shall show that some physically measurable quantities are rather insensitive to the choice of parameters. Before one can hope to successfully apply the twoband model to transition metals one has to generalize it to include interband scattering. When this is done, only one gap in the excitation spectrum is found [2, 3]. In the ensuing numerical calculation of the temperature dependent order parameter [2] specific heat [4] and thermal conductivity [5] it is then argued that the ratio of the density of states at the Fermi level ,
=
,
N~(0)/N
2 (0) (1) is very large, and in fact the limit l/~= 0 is taken. In this case a second gap identical to the larger order parameter reappears [2]. Hence, all attempts to interpret experiments on the basis of the two-band model assume the existence of two energy gaps. In view of the actual band structure of transition metals the assumption ~~ 1 seems to be unjustified [6, 7]. The Fermi surface of niobium, for example, consists of three separate sheets. The ratio of the density of states on each of these sheets is given by [61 NJG
:NELL :NOCT
=
4.42:4.10: 1.37.
Hence, if we again approximate the Fermi surface by two spherical sheets [2—5] it seems to be more appropriate to take ~ = 1.2 or ~ = 10, depending on how the three sheets are lumped together to form a two-band model. Since these sheets are not clearly related to distinct s- and d-bands, we prefer to speak of zones rather than bands and use indices 1 and 2. For those values of ~ the theoretical predictions
must be drastically changed when it is recognized that only one energy gap exists. An analytic calculation of the resulting density of states as a function of energy and impurity content, valid for very small and very large impurity concentration, has been given by Moskalenko eta!. [3] under the assumption that the order parameters are known. The impurity dependence of the order parameters has not been calculated by these authors, whose work seems to have escaped general notice. Here, we shall discuss the density of states as thoroughly as is possible on the basis of numerical computations. Before presenting our results we briefly review the derivation of the relevant formulae. The calculation of the electronic temperature Green function of the two-band model [1] supplemented by the electron—impurity interaction Hjmp
exp i(q
=
—
k)R, v~(k,q)c~ocqxa
I~,q~
G
(2)
with q)
~
=
J
dr ~,<(r) Vjmp(r)~iqx(r)
(3)
.
is standard [2 4, 5] Neglectmg all those contributions to the matrix self-energy ~ which contain the impurity * potential in the form vg~(k,q)v (k, q) with K ~zor X ~ i’, following the argument put forward by Rainer [8] the 4 x 4 matrix Green function becomes diagonal with respect to the zone indices [2]: -
.
.
.
.
.
.
,
G~’ =
,
=
—
ea,~r3
iW~To — ~ .-~
—
+
Z~Ti
iW~To+&~1~i.
(4)
The last equation defines the quantities and L~i,<. Inserting this Green function into the approximate expression for ~ gives a system of coupled equations for i~ and which can be simplified by considering the quantities ~,
371
TWO-BAND MODEL OF SUPERCONDUCTIVITY
372 0
0.5
1.0
2.0
/ 1.8
_~_
0
0.5
10
2.0
// /
Vol. 22, No.6
//
V
2.2
V
_7
—
18 1.7654
/_-V
0.0
—
— — —
_——~
—
_____
_____
_______
1.7654 1.6
0.0 \
2.2 0.5
14 1.2
0.88 0.44
/
~1
1.0
1 2
0.9
2.2044
—
— ——
/ / 0
/
7 V
// ~________ Q ___________ 176 / / / ~2
/ / /
V
/
1.0
V
-V
/ I
/00
/
V
0.8
1.0 1
I-
175
.—
.-,_~.0
/
//
/
0~6,~U
.I1~1~
as 1.0 Fig. 1. Normalized energy gap (full curves) and order parameters (dashed curves) as functions of interband coupling ~ = Ui2/U~x.The parameter varied is p/T~°. Order parameters are shown for p = 0 and p = only. For p = 0, Wg = mm (Lxi ‘~2)• The inset shows the dependence of T~on i~and p for the two values of Dashed curves represent 1~(18),full curves T~(p= 0.44T~~). 2/utTux
00
,
~-.
(5)
u~(iw~) = After analytic continuation one obtains uK~ = w +
(U~—
2r~~
[21:
0
0.2
12/u~ax ~1.0
0.4
Fig. 2. Same as Fig. 1, but with ~ = 1 .2. In this case L~ii/T~ differs more strongly from the BCS-value, the deviation being maximal for p ~ 2.2T~,°. The inset shows the rounding off of the maximum of wg/Tc at ~ = 0.99586 (17) with increasing p. To determine the gap we eliminate ux from (6) and consider the resulting equations as defining two implicit functions wi(2)(u) [10]. If ~2 is the lower order parameter, then the function
u~)(u~ l)h/2 —
with
=
U
+ 1 +hi2 (~i
—
~2)
}
(9)
(6) K~X
~,>0.
h12
These equations contain only the interband scattering times 19]
i
=
/u—1i’(
~ ~ 2r21 + i~) 2. (7) its concave for 0
Vol. 22, No.6
TWO-BAND MODEL OF SUPERCONDUCTIVITY 5.0-
2.0-
373
4.0-
3.0 N(w)
-
N(0) N(w)
2.0
1.0—
-
1.00.0—
I
0.5
I 1.5
________________
1.0
2.0
2.5
3.0
Fig. 3. Partial and total density of states at T = 0 for U12 = 0, ~ = 1.2 and p = 0.66T. With these parameters one obtains TC/TC° = 0.8595 and wg/T~= 1.273. Near the transition temperature T~we can expand u~(6) in terms of z~.To first order we obtain -i
U~
Ii
~
=+y~
1 ~ I ~ + ~pj 2PTKX —
~
w>0 (11)
K~rX
with p
=
1/2r12 + l/2r21
=
(1 + ~)/2r12.
(12)
Inserting this into (10) gives: 0
=~
X
~
1 r0—U (roJ(T~~o~~ )+—~ ~+
)
/
10
—
~
~
)
1\
0.00.5
21
32
H
If
1.0
U
=
(UKX)
=
I
=
in
{~
~—
~
+~
The choice of indices is such that for U12 = 0 we have U11 = = 0.2835, U22 = = 0.2370 and con-
is limited to the range 0~U12 XmaxXmin
1.13w0 exp (
U12
_f 0.0213 —
for~=1.2
0.0074 for ~ = 10.
responding purity of the niobium it might be helpful to relate p to an effective mean free path through the dimensionless parameter p/Ti?. To visualize the cor-
-
(15)
1/Xmax).
=
max
)
2ir 3cP I ~,(i + —1 I
model parameters. In view of the Fermi surface of Nb it seems reasonable to assume that both zones contribute to superconductivity, i.e. U(l4) must be positive definite. For U12 = 0 and p = 0 both zones could then have positive transition temperatures T~,° > T1~<>0. The larger of the two eigenvalues of Uis determined from experimental values of T~° (Nb: 9.2 K) according to =
—
Assuming identical cut-offs for the three interactions V,<~we have chosen the value w~/T~° = 30, appropriate forNb. The amount of interband scattering is controlled by
The purity dependent T~is then determined from the secular equation corresponding to (13). Before we can solve (6) and (10) we have to choose
C
3.0
(14)
tanh~w/2 W + 1P —
2.5
More or less arbitrarily we choose T~<= T~°/2. This determines Xmjn. With U2~= U12~as variable parameter we can calculate the diagonal elements from the identity 2—UTrU+r [12] U 0detU= 0.
2~/~
)
2.0
sequently ~i > ~ According to Leggett, the phase of the Bloch[13]. functions be chosen the suchU,~ thatareU12 is U12 positive Since,can furthermore, real, (13)
(V,,~ANx(0))
J(T~,p,~) = Rejdw
1.5
Fig. 4. Total density of states at T = 0 for U12 = 0 and ~ = 1.2 for three values of p: (1) p = 0.165T~°{T~/T~° = 0.948, wg/Tc = 1 .0l5}; (2) p = 2.2T~{T~/T~ = 0.793, wg/Tc = 1 .554}; (3) p = 6.6T~°{T~/T~° = 0.767, oJg/T~ = 1 .694}. The arrows mark the positions of ~j/~g and ~2 /wg, respectively.
where the matrix Uis given by and
1
w
(16)
1
.~_±_c
3500 A. VF — 1 + ~ 2p~ ~ p From (6) and (10) one can easily see that for U =
=
11 + U12 U22 + U21, which can be fulfilled for ~~ 1 and is then
equivalent to 2V’f U~2/U!r= 1
(17)
the two order parameters become equal, i.e. the twoband model reduces to the one-band BCS-modei with impurity independent T~.This can be seen in Figs. I and 2, where order parameters and energy gaps are shown as
374
TWO-BAND MODEL OF SUPERCONDUCTIVITY
Vol. 22, No. 6
LI®
function of ~ for various values of p/Tx. We note that for i~> 2’~/~/(1 + ~) we have ~2 > ~i even though
8.0 -j
U
11> U22 and Ni(0)>N2(0). Another important result evident from these figures is that one always has wg/Tc ~ L~BCS/T~CS= 1.7654. With increasing p the gap wg/T~approaches monotonically the BCS-value 1.7654, independent of~,whereas the normalized order parameters, although closer to the BCS-value, remain to be functions of~evenfor i/p = 0. The limiting temperature ~ = ~ T~(p)shown in the inset of Fig. 1 has been calculated from equation
T~= 1.13w0 exp
—
1/
Xmin
+
N(O) 60
-
40
-
2.0
1+c11
with
(Xma,~
00
Xmin)
-
-
(18) I~(~ ii)
=
~
1 +~
1 +~
For the density of states (Figs. 3,4 and 5) shows a gradual transition from the two square root singularities of the pure two-band superconductor to BCS-behaviour when p is increased. For ~ = 10 and r~= 0 the first peak due to N2(w) is strongly reduced, so that even for fairly pure materials (p/T~° > 0.1) the main contribution to N(w) for w < z~icomes from N1 (w) (Fig. 3). For = 10 and i~> 2~,/~/(l + ~) (Fig. 5) the structure of
______________________________
1.0
1.5
-~
2.0
Fig.5.SameasFig.4,butwith~=lOandU12= (l)p =0.033Tc°{Tc/Tc°=0.995,wg/Tc=1.502, U~: = 0.9994}; (2) p = 0.33Tc°{Tc/Te° = 0.960 wg/Tc = 1.565, L~1I Wg = 0.9965}.
is qualitatively similar to what is found experimentally. The fact that band structure effects can produce such structure might help to remove the difficulties encountered when strong-coupling theory is applied to Nb [15]. N(w)
REFERENCES 1. 2.
SUHL H., MATTHIAS B.T. & WALKER L.R., Phys. Rev. Lett. 3, 552 (1959). SUNG C.C. & WONG V.K., J. Phys. Chem. Solids 28, 1933 (1967).
3.
MOSKALENKO V.A., URSU A.M. & BOTOSHAN N.J., Phys. Lett. 44A, 183 (1973).
4. 5.
CHOW W.S., Phys. Rev. 172,467 (1968); BURKEL R.H. & CHOW W.S., Phys. Rev. B3, 779 (1971). KUMAR P. & GUPTA S.N.,Phys. Rev. B6, 2642 (1972).
6.
MATTHEISS L.F.,Phys. Rev. Bi, 373 (1970).
7. 8.
ENTEL P., Z. Phys. B23, 321 (1976). RAINER D., Forschungslaboratorium der Siemens AG, Erlangen, Aktenvermerk l—2954 vom 6.9.1967 (unpublished). MOSKALENKO V.A. & PALISTRANT M.E., JETP 22, 536 (1966). MAKI K., Superconductivity (Edited by PARKS R.D.), p. 1046. Marcel Dekker, New York (1969).
9. 10. 11. 12.
ABRIKOSOV A.A., GORKOV L.P. & DZYALOSHINSKI I.E., Methods of Quantum Field Theory in Stattistical Physics, p. 337. Prentice-Hall, Englewood Cliffs, New Jersey, (1963). GREUB W.H., Multilinear Algebra, p. 164. Springer, Berlin (1967).
13.
LEGGETT A.J., Progr. Theoret. Phys. 36, 901 (1966).
14.
This expression has been given before by RAINER [8]. Everywhere else in the literature [e.g. reference [9] and KUSAKABE T., Progr. Theoret. Phys. 43,907(1970)] a different formula is obtained from (13) by omitting p from the logarithmic term in (15). The TC°°thus obtained is somewhat smaller than (18).
15.
BOSTOCK J., DIADIUK V., CHEUNG W.N., LU K.H., RUSE R.M. & MacVICAR M.L.A.,Phys. Rev. Lett. 36, 603 (1976).