Coherent potential approximation for d-wave superconductivity in disordered systems
A. M. Martin
De
´
partement de Physique The
´
orique, Universite
´
de Gene
`
ve, 1211 Gene
`
ve 4, Switzerland
G. Litak
Department of Mechanics, Technical University of Lublin, Nadbystrzycka 36, PL-20-618, Lublin, Poland
B. L. Gyo
¨
rffy and J. F. Annett
H. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
K. I. Wysokin
´
ski
Institute of Physics, Maria Curie-Skłodowska University, Radziszewskiego 10a, PL-20-031, Lublin, Poland
共Received 21 December 1998; revised manuscript received 19 March 1999兲
A coherent-potential approximation CPA is developed for s-wave and d-wave superconductivity in disor-
dered systems. We show that the CPA formalism reproduces the standard pair breaking formula, the self-
consistent Born approximation and the self-consistent T-matrix approximation in the appropriate limits. We
implement the theory and compute T
c
for s-wave and d-wave pairing using an attractive nearest-neighbor
Hubbard model featuring both binary-alloy disorder and a uniform distribution of scattering site potentials. We
determine the density of states and examine its consequences for low-temperature heat capacity.
关S0163-1829共99兲02034-2兴
I. INTRODUCTION
A treatment of disorder is an essential part of the theory
of superconductivity. After all, one must explain why impu-
rity scattering does not cause resistance. Thus it is natural
that as evidence for novel superconducting states multiplies
the foundations of the subject, due mainly to Anderson
1
and
Abrikosov and Gorkov,
2,3
are being reexamined. The experi-
ments which stimulate most strongly the current revival of
interest in the problem are those on the high-temperature
superconductors,
4
which are now universally regarded as
‘‘d-wave superconductors,’’
5
and those involving some of
the heavy fermion systems which display signs of ‘‘p-wave’’
pairing.
6
In what follows, we wish to contribute to the theo-
retical discussion
7–14
of the issues raised by these very inter-
esting developments.
The case of classic, ‘‘s-wave,’’ superconductors is by
now well understood. If the perturbation does not break
time-reversal symmetry and the coherence length is suffi-
ciently long, so that the pairing potential ⌬ does not fluctu-
ate, the Anderson theorem
1
guarantees that there is an abso-
lute gap in the quasiparticle spectrum and the main effect of
disorder is that the density of normal states in the gap equa-
tion is replaced by its average over configurations.
15
On the
other hand, if the perturbation breaks time-reversal invari-
ance, as is the case with paramagnetic impurities, the effect
is more dramatic. For instance, the transition temperature T
c
is reduced from its clean limit value T
c0
, according to the
well-known pair breaking formula
ln
冉
T
c
T
c0
冊
⫽
冉
1
2
冊
⫺
冉
1
2
⫹
c
冊
, 共1兲
where
(x) is the digamma function and
c
⫽ (2
T
c
)
⫺ 1
is
a measure of the strength of the scattering and
⫺ 1
is the
scattering rate.
2,7
By contrast in the case of superconductors whose Cooper
pairs are of exotic p-wave or d-wave character even simple
potential scattering, which does not break time-reversal sym-
metry, causes pair breaking.
7
This fact was noted already in
the early contributions to the field,
16
but has become a sub-
ject of intense scrutiny only recently.
7–14
Of particular inter-
est are two-dimensional models featuring d-wave pairing as
these may be relevant to experiments on high-T
c
supercon-
ductors. Notably, for cuprates many experiments have ex-
plored the variation of T
c
, the density of states and other
properties as a function of Ni and Zn substitutions on the
copper sites
17–24
or irradiation damage.
25–27
Although a wide
variety of theoretical ideas
28–32
and methods
33–38
have been
applied to interpret the experiments, a comprehensive picture
of the role of disorder is far from complete. On a more for-
mal level, an intriguing problem arises from the observation
of Gorkov and Kalugin
10
that the scattering in models where
the order parameter has a line of zeros on the Fermi surface
is highly singular and this may be a manifestation of inter-
esting new physics. Indeed in two dimensions, Nersesyan
and co-workers
14
predict that the quasiparticle density of
states N(E) approaches zero, even in the disordered state, as
power law ⬃
兩
E
兩
␣
, with positive exponent
␣
, instead of go-
ing to a finite value as was found by Gorkov and Kalugin.
10
Another interesting and controversial issue is the relative im-
portance of the self-consistent Born approximation 共SCBA兲
and resonant scattering in the unitarity limit.
11,12
Our aim
here is to explore the subject systematically on the basis of
explicit calculations, albeit for a simple, extended Hubbard
model with attractive interactions and site diagonal random-
ness only.
PHYSICAL REVIEW B 1 SEPTEMBER 1999-IIVOLUME 60, NUMBER 10
PRB 60
0163-1829/99/60共10兲/7523共13兲/$15.00 7523 ©1999 The American Physical Society
In short, we will examine the problem of disordered un-
conventional superconductors making use of the coherent-
potential approximation 共CPA兲. The CPA is the most reliable
approximation developed for the theory of electronic struc-
ture of random metallic alloys in the normal state.
39,40
Nota-
bly, it has been shown to be exact in both the weak and the
strong scattering limits, and applicable to systems with low
as well as high concentration of impurities. Significantly, the
CPA reduces to the self-consistent Born approximation
共SCBA兲 for weak scattering impurities, and agrees with the
self-consistent T-matrix approximation 共SCTA兲 results for
strongly scattering impurities of low concentrations. Indeed,
it remains a good approximation in the unitarity limit of
resonant scattering.
11,12
Finally, on account of the fact that it
becomes exact as the number of nearest neighbors goes to
infinity, the CPA is often referred to as a mean-field theory
of disorder.
41
Given these desirable features, it is clearly worthwhile to
explore the consequences of the CPA for disordered super-
conductors. For the case of conventional s-wave pairing this
has already been done, generating many useful results.
42,43
The case of superconductors with Cooper pairs of d symme-
try will be treated here within CPA, developing in detail the
method introduced in our earlier paper
44
and the limited dis-
cussion in Ref. 45.
We will demonstrate that in various limits our formalism
reproduces many of the well-known results for disordered
superconductors, and examine in detail the phase diagram of
the local and nonlocal attractive two-dimensional Hubbard
models. In particular, we study the variations of T
c
with
impurity scattering strength and with impurity concentration
for the case of local s-wave pairing as well as nonlocal 共ex-
tended兲 s-wave and d-wave pairing. We also contrast the
cases for a binary alloy, A-B type, disorder with the case of
uniformly distributed scattering potentials on each site. Fi-
nally, we investigate the density of states 共DOS兲, N(E), at
low energies and its consequences for measurements of the
specific heat.
II. INCORPORATING CPA
INTO BOGOLIUBOV-DE GENNES EQUATION
Our starting point is the single band Hubbard model with
an attractive extended interaction which is described by the
following Hamiltonian:
H⫽
兺
ij
t
ij
c
i
†
c
j
⫹
1
2
兺
ij
U
ij
n
ˆ
i
n
ˆ
j
⫺
兺
i
共
⫺
i
兲
n
ˆ
i
, 共2兲
where c
i
†
and c
i
are, respectively, the usual creation and
annihilation operators for electrons on site i with spin
, and
the local charge operator is n
ˆ
i
⫽ n
ˆ
i↑
⫹ n
ˆ
i↓
with n
ˆ
i
⫽ c
i
†
c
i
.
The chemical potential is
,t
ij
are the hopping integrals 共for
i⫽ j) and
⑀
i
is the local site energy. The interaction term U
ij
can be either a local attractive interaction (U
ii
⬍ 0) giving
rise to s-wave pairing, or a nonlocal attractive interaction
(U
ij
⬍ 0 for i⫽ j) giving rise to d-wave or extended s-wave
pairing. Disorder is introduced into the problem by allowing
the site energies
i
to vary randomly from site to site.
Starting from Eq. 共2兲 we apply the
Hartree-Fock-Gorkov
46,47
approximation, which results in
the following Bogoliubov-de Gennes equation:
兺
l
冉
共
ı
n
⫺
i
⫹
兲
␦
il
⫹ t
il
⌬
ij
⌬
ij
*
共
ı
n
⫹
i
⫺
兲
␦
il
⫺ t
il
冊
⫻
冉
G
11
共
l,j;ı
n
兲
G
12
共
l,j;ı
n
兲
G
21
共
l,j;ı
n
兲
G
22
共
l,j;ı
n
兲
冊
⫽
␦
ij
冉
10
01
冊
, 共3兲
for the Green’s-function matrix G(i,j;ı
n
) at the Mastus-
bara frequency
n
⫽ (2n⫹ 1)
k
B
T, in units where ប⫽ 1. For
computational convenience we shall take the hopping inte-
gral t
ij
to be nonzero only when the sites i and j are nearest
neighbors. The mean-field pairing potential ⌬
ij
can either be
local (i⫽ j) or nearest-neighbor nonlocal. Of course, the
above equations are completed by the self-consistency con-
dition that
⌬
ij
⫽
兩
U
ij
兩
1

兺
n
e
ı
n
G
12
共
i,j;ı
n
兲
, 共4兲
where
is a positive infinitesimal. To simplify matters we
have assumed that the normal Hartree and exchange terms
can be absorbed into the definitions of the chemical potential
or the hopping integrals t
ij
. As usual, Eqs. 共3兲 and 共4兲 are
to be solved subject to the requirement on the chemical po-
tential that
n
i
⫽
2

兺
n
e
ı
n
G
11
共
i,i;ı
n
兲
, 共5兲
where n
i
is the number of electrons at site i. Clearly, the
Green’s-function matrix G(i,j;ı
n
) determined by the above
equations depends on the set of site energies
兵
i
其
. Our task is
to find the configurationally averaged Green’s-function ma-
trix
具
G(i,j;ı
n
)
典
. Evidently, this is made much easier if we
assume that the pairing potential does not fluctuate from con-
figuration to configuration. As was argued by Gyo
¨
rffy, Litak,
and Wysokin
´
ski
15
for s-wave superconductors this is a good
approximation when the T⫽ 0 coherence length
0
is large.
Thus our specific results will have to be treated with appro-
priate care when applied to superconductors with d-wave
symmetry or short coherence length such as superconducting
cuprates.
Let us now proceed to deploy the CPA strategy for cal-
culating the averaged Green’s function matrix
具
G(i,j;ı
n
)
典
subject to the self-consistency conditions:
⌬
¯
ij
⫽
兩
U
ij
兩
1

兺
n
e
ı
n
具
G
12
共
i,j;ı
n
兲
典
, 共6兲
n
¯
⫽
2

兺
n
e
ı
n
具
G
11
共
i,i;ı
n
兲
典
. 共7兲
The first move in deriving the fundamental equations of
the coherent potential approximation is to define a coherent
medium Green’s-function matrix G
c
(i,j;ı
n
)by
兺
l
冉
关
ı
n
⫹
⫺ ⌺
11
共
ı
n
兲
兴
␦
il
⫹ t
il
⌬
¯
il
⌬
¯
il
*
关
ı
n
⫺
⫺ ⌺
22
共
ı
n
兲
兴
␦
il
⫺ t
il
冊
G
c
共
l,j;ı
n
兲
⫽
␦
ij
冉
10
01
冊
. 共8兲
7524 PRB 60A. M. MARTIN et al.
As will be clear later, G
c
(i,j;ı
n
)⫽
具
G(i,j;ı
n
)
典
and
hence ⌺
11
(ı
n
) and ⌺
22
(ı
n
) are the diagonal components
of the usual self-energy. Note that we did not introduce any
off-diagonal self-energies such as ⌺
12
(ı
n
) and ⌺
21
(ı
n
),
because for the single site perturbations of our model they
are zero. The next step is to consider the scattering of the
quasiparticles, propagating according to G
c
(i,j;ı
n
) by the
defects described by the potentials:
V
l
共
ı
n
兲
⫽
冉
l
0
0 ⫺
l
冊
⫺
冉
⌺
11
共
ı
n
兲
0
0 ⌺
22
共
ı
n
兲
冊
, 共9兲
where l labels one of the m different site energies we wish to
consider.
In a straightforward application of the CPA principles,
40
⌺(ı
n
) and therefore G
c
(i,j;ı
n
) is determined by the con-
dition that these defects do not scatter on the average, i.e.,
兺
l⫽ 1
m
c
l
T
l
共
ı
n
兲
⫽ 0, 共10兲
where
T
l
共
ı
n
兲
⫽ V
l
共
ı
n
兲
关
1⫺ G
c
共
i,i;ı
n
兲
V
l
兴
⫺ 1
共11兲
and the concentration of sites of energy
l
is c
l
, obeying
兺
l⫽ 1
m
c
l
⫽ 1. 共12兲
From Eqs. 共10兲 and 共11兲 it is now possible, in conjunction
with Eqs. 共6兲–共8兲, to calculate ⌺(ı
n
) and G
c
(i,j;ı
n
). The
numerical methodology for calculating G
c
(i,j;ı
n
) and
⌺(ı
n
) closely follows that in Ref. 48.
A number of recent studies of superconductors with un-
conventional pairing suggest that the consequences of disor-
der depend sensitively on the models used to describe the
randomness.
49,50
Thus we are going to investigate two differ-
ent models. The first corresponds to binary-alloy disorder,
where m⫽ 2. Namely, we consider two types of sites with
site energies
1
and
2
and concentrations of c and 1⫺ c,
respectively. The second model is described by a uniform
distribution of site energies. Here we shall have in mind the
limit where m→⬁ with
l
苸
关
⫺
␦
/2,
␦
/2
兴
. Consequently,
in Eq. 共10兲 the sum
兺
l
becomes the integral (1/
␦
)
兰
d
l
.
In the bimodal case, where m⫽ 2, we can simplify Eq.
共10兲 to find
⌺
11
共
ı
n
兲
⫽
共
2c⫺ 1
兲
␦
2
⫺
冉
␦
2
⫺ ⌺
11
共
ı
n
兲
冊
⫻ G
11
c
共
ı
n
兲
冉
⫺
␦
2
⫺ ⌺
11
共
ı
n
兲
冊
, 共13兲
where
兩
1
⫺
2
兩
⫽
␦
, while for uniform distribution one gets
⌺
11
共
ı
n
兲
⫽⫺
1
G
11
c
共
ı
n
兲
⫹
␦
2
1
tanh
冉
␦
G
11
c
共
ı
n
兲
2
冊
. 共14兲
Thus our CPA calculations will consist of solving numeri-
cally either Eq. 共13兲 for the bimodal distribution of the site
energies, or Eq. 共14兲 for the case of uniform distribution, to
determine the self-energies ⌺
11
(ı
n
) and ⌺
22
(ı
n
).
III. PAIR BREAKING FORMULA IN CPA
We now relate the CPA formulas derived above to the
usual results of disordered superconductors, corresponding to
the well-known pair breaking formula Eq. 共1兲. As is well
known,
7
the pair breaking formula was first derived for mag-
netic impurities in s-wave superconductors
2
but it also ap-
plies in many other interesting circumstances such as our
present concern, namely, the case of nonmagnetic impurities
in d-wave superconductors.
16
To derive it within the CPA let us start with the gap
equation
⌬
k
ជ
⫽
1
N
兺
q
ជ
U
k
ជ
⫺ q
ជ
1

兺
n
G
12
c
共
q
ជ
;ı
n
兲
e
ı
n
␦
. 共15兲
As a motivation for our argument we recall the method of
Abrikosov and Gorkov
2
for solving the gap equation at T
c
for a clean superconductor. In that case, to find T
c
we lin-
earize the analog of Eq. 共15兲 by approximating the off-
diagonal Green’s function G
12
c
as follows:
G
12
c
共
q
ជ
;ı
n
兲
⬵
⫺ ⌬
q
ជ
共
ı
n
⫺
q
ជ
兲
共
ı
n
⫹
q
ជ
兲
, 共16兲
where
q
ជ
⫽
q
ជ
⫺
, and for our tight-binding model with a
square lattice
q
ជ
⫽⫺2t
关
cos(q
x
)⫹cos(q
y
)
兴
. Then, we note that
the kernel of the linear integral equation for ⌬
k
ជ
is a four-term
degenerate kernel:
U
共
k
ជ
⫺ q
ជ
兲
⫽
兩
U
兩
冉
k
ជ
q
ជ
⫹
␥
k
ជ
␥
q
ជ
4
⫹ 2 sin k
x
sin q
x
⫹ 2 sin k
y
sin q
y
冊
, 共17兲
where
k
ជ
⫽ 2
关
cos(k
x
)⫺cos(k
y
)
兴
and
␥
k
ជ
⫽ 2
关
cos(k
x
)⫹cos(k
y
)
兴
.
Consequently, the general ⌬
k
ជ
will be a linear superposition
of
k
ជ
,
␥
k
ជ
, sin k
x
, and sink
y
. However, when the internal
symmetry of the singlet Cooper pair is pure d wave we may
take ⌬
k
ជ
to be of the form
⌬
k
ជ
⫽ ⌬
k
ជ
. 共18兲
Then the condition for nonzero order parameter becomes
1⫽
兩
U
兩
N
兺
q
ជ
q
ជ
2
4
T
c0
兺
n
1
n
2
⫹
q
ជ
2
. 共19兲
Let us now define a d-wave weighted density of states:
N
d
共
E
兲
⫽
1
N
兺
q
ជ
q
ជ
2
4
␦
共
E⫺
q
ជ
兲
共20兲
and write the above condition, which determines the transi-
tion temperature T
c0
,as
1⫽
兩
U
兩
冕
⫺ ⬁
⬁
dEN
d
共
E
兲
T
c0
兺
n
⬎ 0
2
n
2
⫹ E
2
, 共21兲
PRB 60 7525COHERENT POTENTIAL APPROXIMATION FOR d-WAVE . . .
where
n
⫽
T
c0
(2n⫹ 1). In the above equation the integral
and the sum are divergent, so we need to introduce a cutoff,
n
c
. In the usual way we assume the density of states N
d
(E)
is slowly varying up to the cut-off energy, so we will make
the approximation N
d
(E)⫽ N
d
(0). Then, considering that
N
d
共
0
兲
冕
⫺ ⬁
⬁
dE
1
n
2
⫹ E
2
⫽
N
d
共
0
兲
1
n
, 共22兲
we can write
1⫽
兩
U
兩
N
d
共
0
兲
2
T
c0
兺
n
⬎ 0
n
c
1
n
共23兲
and hence rewrite Eq. 共21兲 as
1
兩
U
兩
N
d
共
0
兲
⫽
冉
1
2
⫹
n
c
2
T
c0
冊
⫺
冉
1
2
冊
⬇ln
冉
␥
n
c
2
T
c0
冊
.
共24兲
This is the BCS result for the superconducting transition
temperature in the case of d-wave pairing.
46
It differs from
the conventional result only in that the d-projected density of
states N
d
(0) has replaced the usual full density of states
N(0).
Let us now return to disordered superconductors and ex-
amine how the above well-known argument is modified
when the randomness is dealt with within the CPA. Using
Eq. 共8兲 it can be easily seen that instead of Eq. 共16兲 we
should use
G
12
c
共
q
ជ
;ı
n
兲
⫽
⫺ ⌬
¯
q
ជ
共
ı
n
⫺
q
ជ
⫺ ⌺
11
共
ı
n
兲兲
关
ı
n
⫹
q
ជ
⫺ ⌺
22
共
ı
n
兲
兴
共25兲
to linearize Eq. 共15兲 at T
c
. Thus noting that
⌺
22
共
ı
n
兲
⫽⫺⌺
11
共
⫺ ı
n
兲
, 共26兲
the condition which determines T
c
can be written as
1⫽⫺
兩
U
兩
冕
⫺ ⬁
⬁
dEN
d
共
E
兲
T
c
⫻
兺
n
⬎ 0
2
关
ı
n
⫺ E⫺ ⌺
11
共
ı
n
兲
兴关
ı
n
⫹ E⫹ ⌺
11
共
⫺ ı
n
兲
兴
.
共27兲
Now, at this point we need to know the form of ⌺
11
(ı
n
)to
progress any further. As a first approximation we assume
that the most important component to the self-energy is the
component at the Fermi energy E⫽E
F
⫽
. Later on we will
test the accuracy of this approximation by examining our
numerical results for ⌺
11
(ı
n
). For now, however, let us
proceed by taking
⌺
11
共
ı
n
兲
⫽ ı
兩
⌺
0
兩
sgn
共
n
兲
. 共28兲
Evidently this leads to
1⫽⫺
兩
U
兩
冕
⫺ ⬁
⬁
dEN
d
共
E
兲
T
c
⫻
兺
n
⬎ 0
2
共
ı
n
⫺ E⫹ ı
兩
⌺
0
兩
兲
共
ı
n
⫹ E⫹ ı
兩
⌺
0
兩
兲
. 共29兲
Again taking N
d
(E) outside of the integration as N
d
(0) and
performing the integration over E,wefind
1⫽
兩
U
兩
N
d
共
0
兲
2
T
c
兺
n
⬎ 0
n
c
1
n
⫹
兩
⌺
0
兩
, 共30兲
where again the sum is cut off, as in the clean limit, by
n
c
.
If we now add and subtract the terms corresponding to ⌺
0
⫽ 0 共the clean case兲 we find
1
兩
U
兩
N
d
共
0
兲
⫽ 2
T
c
兺
n
⬎ 0
n
c
1
n
⫹ 2
T
c
兺
n
⬎ 0
n
c
冉
1
n
⫹
兩
⌺
0
兩
⫺
1
n
冊
.
共31兲
Clearly the term 1/
兩
U
兩
N
d
(0) on the left-hand side of Eq. 共31兲
can be replaced by ln
关
␥
(
n
c
/2
T
c0
)
兴
on account of Eq. 共24兲.
With the same accuracy, the first sum on the RHS of Eq. 共31兲
equals ln
关
␥
(
n
c
/2
T
c
)
兴
and the second sum is convergent.
Hence the cutoff
n
c
can be extended to infinity. As has been
noted frequently before, this infinite sum can be readily
performed
2
and we find
ln
冉
T
c
T
c0
冊
⫽
冉
1
2
冊
⫺
冉
1
2
⫹
c
冊
, 共32兲
where
c
⫽
兩
⌺
0
兩
2
T
c
. 共33兲
Equations 共32兲 and 共33兲 are the central results of this sec-
tion. Reassuringly, while Eq. 共32兲 is the standard pair break-
ing formula,
7
Eq. 共33兲 is a very natural CPA expression for
the pair breaking parameter
c
. Recall that our derivation of
the above result from CPA involved the approximation
⌺
11
(ı
n
)⬵ı⌺
0
. To test the validity of this approximation we
wish to compare exact CPA numerical results with the pre-
dictions of the analytical expression: Eqs. 共32兲 and 共33兲. Us-
ing numerical solutions of the CPA equation, to be discussed
latter, Fig. 1 plots the pair breaking strength
c
vs
␦
/t, the
disorder strength for the binary alloy-type disorder. To find
pair breaking parameter
c
we calculated T
c
for each disor-
der strength
␦
/t and inverted Eq. 共32兲 to obtain an effective
c
. The exact CPA
c
can then be compared to the solid line
in Fig. 1 where we have taken our numerically calculated
values for ⌺
0
and directly calculated
c
, via Eq. 共33兲. Fi-
nally, the dashed line in Fig. 1 corresponds to
c
obtained
using the self-consistent Born approximation 共SCBA兲. Evi-
dently, the self-energy at the Fermi energy, E⫺
⫽ 0, ⌺
0
,
gives a good description of the pair breaking parameter
c
via Eq. 共32兲. Also it is clear that, as expected, the self-
consistent Born approximation ⌺
0
⬅ប/
⫽
␦
2
N(0) only
works well in the weak scattering limit.
7526 PRB 60A. M. MARTIN et al.
IV. ANALYTICAL FEATURES AND PREDICTIONS
OF CPA EQUATIONS
In this section we examine various analytically accessible
limits of the CPA formalism described above. First, we dem-
onstrate that Anderson’s theorem is obeyed for s-wave su-
perconductors and the CPA equations are consistent with the
results of Abrikosov and Gorkov.
2
Second, we show that for
d-wave superconductors the quasiparticle density of states at
the Fermi energy N(0) is nonzero in the presence of non-
magnetic disorder scattering and is consistent with the results
of Gorkov and Kalugin.
10
A. Anderson’s theorem in coherent-potential approximation
Formally, the CPA Eqs. 共10兲 and 共11兲 can be written in
terms of renormalized Matsubara frequencies
˜
n
, pairing
parameter ⌬
˜
k
ជ
, and particle energies
˜
k
ជ
. These quantities are
defined as follows:
˜
n
⫽
n
冉
1⫺
Im ⌺
11
共
ı
n
兲
n
冊
, 共34兲
⌬
˜
k
ជ
⫽ ⌬
¯
k
ជ
, 共35兲
˜
k
ជ
⫽
k
ជ
⫺
⫹ Re ⌺
11
共
ı
n
兲
, 共36兲
consequently
G
11
共
ı
n
兲
⫽
1
N
兺
k
ជ
ı
˜
n
⫹
˜
k
ជ
共
ı
˜
n
兲
2
⫺
˜
k
ជ
2
⫺ ⌬
˜
k
ជ
2
共37兲
and for the alloy-type disorder with c⫽ 0.5 and
l
⫽⫾
␦
/2,
the self-energy ⌺
11
(ı
n
) which renormalizes
n
,⌬
k
ជ
, and
k
ជ
, and is defined by Eq. 共13兲, can be written as
⌺
11
共
ı
n
兲
⫽
共
␦
2
/4
兲
G
11
c
共
ı
n
兲
1⫹ G
11
c
共
ı
n
兲
⌺
11
共
ı
n
兲
. 共38兲
The alternative expression for ⌺
11
(ı
n
) in the case of a uni-
form distribution of local potentials, ⫺
␦
/2⬍
l
⬍
␦
/2, is
given in Eq. 共14兲.
Note that in the case of a nonisotropic, d-wave gap, ⌬
k
ជ
is
not renormalized if the disorder is diagonal both in site and
Nambu space. This is different from the case of local s-wave
pairing where, as can be readily shown, ⌬
k
ជ
is renormalized
by the same factor as
˜
n
in Eq. 共34兲. Thus for conventional
superconductors, in contradiction to Eqs. 共34兲–共37兲,wefind
that the CPA yields
˜
n
n
⫽
⌬
˜
⌬
, 共39兲
in agreement with Born approximation or Abrikosov-Gorkov
theory.
2
As is widely appreciated,
2,15
the above equation im-
plies the Anderson’s theorem in s-wave superconductors. By
contrast in the d-wave case represented by Eqs. 共34兲, Eq.
共39兲 does not hold and hence there is no Anderson theorem.
Finally, in concluding this section, we would like to stress
that Eqs. 共34兲–共38兲 represent strictly a pure d-wave result.
Even if we stick to the singlet case, a more general solution
of the CPA equation will imply a renormalization of ⌬
¯
k
ជ
to
⌬
˜
k
ជ
. A good example of such a situation is a case where the
symmetry of ⌬
¯
k
ជ
is of extended s-wave symmetry s
*
⬀
关
cos(k
x
)⫹cos(k
y
)
兴
type. We shall encounter this interesting
circumstance later on in this paper.
B. Density of states N„0… in d-wave superconductors
Moving on and returning to the d-wave case, we observe
that the form of Eqs. 共34兲–共38兲 are the same as were found
by Larkin.
16
Thus again the CPA reproduces the expected
general form of the gap and frequency renormalizations, but
with an improved description of the disorder. The most
prominent feature of a conventional superconductor is van-
ishing of the quasiparticle density of states N(E) for energies
E measured from the Fermi energy E
F
, less than ⌬. In the
case of clean, d-wave superconductors, the line of zeros of
⌬
k
ជ
on the Fermi surface leads to finite N(E) for all E except
E⫽ 0. In fact, as is well known,
7
N(E) approaches zero lin-
early in E. In the present section we shall investigate what
happens to N(E) in the presence of disorder.
As it turns out for a given gap parameter ⌬
¯
k
ជ
⫽ ⌬
k
ជ
and in
the limit of small disorder
␦
→0 the CPA equations can be
solved analytically. To affect the solution note that in Eq.
FIG. 1. The effective pair breaker
c
as calculated 共i兲 by nu-
merically finding T
c
/T
c0
and inverting Eq. 共32兲共squares兲, 共ii兲 nu-
merically finding
兩
⌺
0
兩
and using this in Eq. 共33兲共solid line兲, and
共iii兲 using the self-consistent Born approximation to find
兩
⌺
0
兩
and
then evaluating Eq. 共33兲共dashed line兲.
FIG. 2. A comparison of the density of states at the chemical
potential vs different strengths of alloyed disorder
⑀
i
⫽⫾
␦
/2. The
solid line is the analytical form derived in Eq. 共43兲 and the dashed
one represents our self-consistent numerical calculations.
PRB 60
7527COHERENT POTENTIAL APPROXIMATION FOR d-WAVE . . .
