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Effects of lattice distortion on the energy gap of a BCS superconductor
Physica 84A (1976) 371-391 © North-Holland Publishing Co.
EFFECTS OF LATTICE DISTORTION ON THE ENERGY GAP O F A BCS S U P E R C O N D U C T O R J.G. BRANKOV and N.S. TONCHEV Joint Institute for Nuclear Research, Dubna, USSR
Received 6 January 1976
The model of Mattis and Langer for superconductors with a structural distortion associated with doubling of the lattice periodicity is studied. New results for the zero-temperature superconducting gap are foufid in the case of more than a half-filled band. The modification of both the electron density of states and the reduced BCS interaction is taken into account. A comparison with the results found for the additive type model is given.
I. Introduction
Recently, for some superconductors with high transition temperature Ts, a p r o n o u n c e d correlation between Ts and the temperature Td at which they undergo a structural distortion has been observed1). This fact inspires an increasing n u m ber of works on model systems* which display b o t h a structural and a superconducting phase transition a - la). In the case o f three-dimensional systems, the possibility for enhancement o f Ts due to a structural distortion has been pointed out in refs. 3 and 4. Further this problem has been extensively investigated s - l a ) and different approaches have been p r o p o s e d 9' 1o). One of the expected mechanisms for influence o f structural phase transitions on superconductivity is based on the change in the electron density o f states4). A particular model for the lattice distortion has been exploited and the effect o f the corresponding modification in the electron density o f states on Ts has been c o m p u t e d in the weak coupling limit 6' 1 l - 13). A n o t h e r a p p r o a c h has been p r o p o s e d in1°), which circumvents the p r o b l e m o f identifying the precise model for the structural transition and attempts to c o n * Here we confine ourselves to the three-dimensional case. For discussion of one-dimensional models see2) and references therein. 371
372
J.G. BRANKOV AND N.S. TONCHEV
struct instead a direct relation between the amount of phonon softening and the enhancement of the electron-phonon coupling parameter. The model we investigate in this paper has been suggested by Mattis and Langera). It allows one to study in a rather simple way the effects due to the modification in both the effective Bardeen-Cooper-Schrieffer (BCS) coupling and the electron density of states, associated with the lattice distortion. The model system under consideration describes a single-band metal with electron dispersion ek having the symmetry property
ek = - e k + a ,
Q = = (__1, __1, + 1 ) ,
(1)
(d
where a is the lattice spacing (2Q is a vector in the reciprocal lattice). However, the results obtained here apply equally to the case of a two-band semimetal with band extrema separated by a vector Q in momentum space (see e.g. ref. 11). It has been found 3,6) that the crystal structure of a half-filled single-band metal with an electron spectrum ek, satisfying relation (1), is unstable under infinitesimally small electron-phonon coupling. At a certain temperature To such a system undergoes a metal-insulator phase transition. The lattice has been found unstable also when the deviation of the chemical potential ~o from the half-filled band case (/~o = 0) is sufficiently small in comparison with the interaction parameter Ao11.13)
I~ol ~ ½&.
(2)
Here the parameter J o corresponding to the value of the insulating gap for a half-filled band is related to the band width 2w, the electron-phonon coupling strength g(Q) and the phonon energy e)(Q) as follows 3'6) .d o = w sinh -~ - w -, g~
gd = -2g -2 (Q) c'~(Q)
(3)
The phase transition in this case is of the type of metal-degenerate semiconductor3). In general, the essential condition for the lattice instability to exist is the presence of fiat regions of the Fermi surface which coincide (or "almost" coincide3'6)) under translation by some fixed wave vector14). This three-dimensional instability is analogous to the well-known Peierls instability of one-dimensional metals*. * Notice that the one-dimensional instability appears for Q = 2kv, kv being the Fermi wave vector, whereas in the case of three-dimensional systems the corresponding instability wavevector is always Q = ~a- 1 (+ l, + 1, + 1) and it coincides with 2kv only if the electron band is half-filled.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
373
The idea proposed by Mattis and Langer in 3) consists in deriving the effective electron-electron pairing interaction from the Frohlich hamiltonian in the spirit of the usual BCS theory but with the appearance of a macroscopic lattice distortion taken into account. It should be emphasized that when performing the conventional procedure 15) for eliminating (in second order perturbation theory) the linear electron-phonon coupling from the hamiltonian, one does not presuppose a macroscopic filling of any phonon mode to take place. However, a lattice distortion changes both the electron and phonon dispersions and, in particular, causes a shift in the Fermi level. These effects, if taken into account, may lead to an explicit dependence of the BCS-type interaction on the order parameter of the distorted phase A. The aim of this paper is to provide a more detailed analysis of the possible effects of a structural phase transition on superconductivity, based on the explicit solutions for the zero-temperature superconducting gap found here for a relevant model problem.
2. The model hamiltonian
It seems useful to treat first the simplest case of Einstein phonons and momentum-independent electron-phonon coupling parameter3). Thus the hamiltonian we start with is ~/ = Z (~k - ~) ak,~,ak,¢, + + o ) ~ (b+b~ + ½) k, ~
+ gv-
tl
(bq + b+q) 2 E ak+,,~ak,~ + q
(4)
k,o"
Here aj,,+ ~, ak,~ are creation and annihilation operators, respectively, for electrons in states of wave vector k and spin ~ (o = _ 1); b +, bq are the corresponding operators for phonon modes of wave vector q. In eq. (4) V denotes the volume of the system and # is the chemical potential by means of which we adjust the number of electrons. The one-electron energies ek are measured relative to the band center and satisfy eq. (1). Due to the appearance of a macroscopic occupation number of the phonon _x _~ + mode of wave vector Q (below Td), the operators V b+Q and V b+Q can be replaced by a c-number (following Bogoliubov's ideat6))
V-~b*+o --* - ( 2 g ) -1 A,
(5)
where the trial parameter A has to be determined from the absolute minimum principle for the free energy density (see ref. 17). It should be noted that under
374
J.G. BRANKOV AND N.S. TONCHEV
some general conditions the substitution (5) has been rigorously justified in ref. 18. Further we consider instead of hamiltonian (4) the thermodynamically equivalent one (6)
H ( A ) = Ho(A) + H , ,
where Ho(A) = ~ [(eL - ~) a k+, a (l k , cr - ~la+o , oak, o] + ½V(J2/'ga) k,a
H~ = '" Z
(b+bq + ½) + g V -
Z ak+~, + ,ak,~ (bq + b+q).
~ ~
qvaO
qq-O
k, a
Following ref. 3, we diagonalize the term Ho(A) using the canonical transformation ak,a
=
UkCk,a
"1- L l k ( ' k + O , a ~
(7) ak+Q, a ~
- - U k C k , a -q- l l k C k + Q , a ~
with ck = s g n e k ~
u~ = - 1 I 1 + 2)~ ,]2 (e~ +[~:~1 _:1 1 ~ ,
1
(~ +[ek[.J~)~ 1 ~.
The resulting hamiltonian is +
1
k,a
q~Q
~ (bo + b_+O[l(k + q, k) cLo, oc,,o
+ gg-~ y q~-Q
+p(k
2
k, a
+ q,k)
C k++ q , a C k + Q , a ] ~
where ! (k, k') = u~uk, + vkv~,, p (k, k') = ukv~, - vku,,.
(8)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
375
In eq. (8) the dispersion Ek for electrons in the distorted lattice is given as ER = sgn eh (e~ + A2) ~.
(9)
Next, applying the procedure of Bardeen and Pines ~5) for eliminating the linear electron-phonon coupling from the hamiltonian, Mattis and Langer 3) have shown that the effective hamiltonian of the reduced BCS type is
yc(~)
y~ (E~
=
-
~) c k+, ° e ~ . .
+ -1-v(A:lg~)
k, ex
+ 2g2oJV_ 1 ~ 1 + A2/EkEk, + + k,k' (Ek -- Ek') 2 -- o) 2 Ck. l C - k , - l C - k , , - l e k , , 1 .
(10)
It should be mentioned that in eq. (10) effects of renormalization of the phonon frequencies have been neglected: Now, in contrast with ref. 3, we make explicitly the BCS approximation ag) g2(o (E~ -
-g~2 (lEg -- /~l) 2 ([Ek, -- ,ul),
Ek,) 2 - - (,~2
(11)
where lEg - # [ < OgD IEk--#l >o'o.
I1,
2(IEk--,ul)=
0,
(12)
The only difference between the model studied here and the one originally proposed by Mattis and Langer a) is that we introduce an independent interaction parameter gs (of the order of magnitude g2/o)) and a cut-off energy (Oo (of the order of (o) which are in principle phenomenological parameters. The model of ref. 3 corresponds to the particular case g~ = g2/oJ and (oD = o). Thus, with the aid of approximation (11) hamiltonian (10) reduces to the following exactly solvable one ~(A)
y~ ( E k -- ~,)
=
ck,,ck,~ + ½V(Zl2/g.) +
R, ~r
j2 )
-
-
gs v - 1 ~,~,,',,, 2(IEk -- #11 2 (IE,,, -- #1) (1 + - EkEk, +
+
X Ck, a t - k ,
- o C - - k ' , --aCk,,~r.
(13)
The system defined by eq. (13) belongs to the class of models with separable interaction studied by Bogolubov Jr. z°) (see also ref. 21).
376
J.G. BRANKOV AND N.S. TONCHEV
3. Calculation of the free-energy density Let us introduce the following b o u n d e d operators Jr sgn (J2 (IEk - ,,,I) c_~._.c~.~,
.f, = V-'
k. T
(14) .!:l
J:~ = V ~ V sen v). (IE¢, - ,.I) - k,G
c_~
~c~.~.
12,1;•
and rewrite hamiltonian (13) in the form 2
+
+
J / ( i ) = v,2.. (E~ - / ~ ) c~.~c,,., - ".g,v X Y , J i + i-V(le/gd) k,~
(15}
i=1
]t can be easily shown that the above hamiltonian satisfies all conditions of a general theorem proved by Bogolubov jr.2°). According to this theorem the free-energy density for the model system with hamiltonian (15) converges in the t h e r m o d y n a m i c limit to the approximating free-energy density .t;~ ( I ; S~, -g2} = - k u T V - '
In Tr exp [ - ~ 0
(:1"
SI, $2) ]
(16)
calculated with the use of a trial hamiltonian* 2
,;eo( l ; s , . s , )
_
~_.
~(e~-v,)cZ~e~,,
m
_
~ v y~ (. i s, + # , s * ~ "]6~ +
•
i=1
k,~
+ ~ g (g:-i V I&l-" + g~' 12 • ,
(17)
i=l
The parameters oel and ~'~2 in eq. (16) are determined from the absolute minimum condition l o r r y ( I ; Sz, $2) ,fy (=l; S I ,
$ 2 ) min = f v S't, 5"2
(I ; S,, $2).
We turn now to the calculation o f f v (./; S j , S,). By inserting (14) in (17) and performing some simple transformations, 24,% ( I ; $1, $2) can be written as
c3/(P0 (~1; S 1 , S 2 ) = /~ (ck. l C _ k , _ k
1) ~ _ ~ , ( E k
)
_
Ek -}- 1~ ~C+k.
+K, _1
(18) * In general, the variational parameters in eq. (17) are complex numbers.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
377
where
2(E~) = and K=
~V
Z (IE~ - ~'1)S(E~),
(2 g~
A Z'(Ek) = St + - - S 2 Ek
)
~ IS~I2 + gg1A 2 - 2# • i=1
Obviously, the one-particle spectrum of hamiltonian (18) is fi: [(E~ - / 0
2 + I-r(E~)12] + + K.
(19)
Hence, for the trial free-energy densityfv (A ; S~, $2) we get f r (-J; S~, Sa) = - 2 k B T V -~ ~ in 2 cosh [(Ek -- #)2 + I~(E~)t2]~ + V - 1 K . k 2kRT (20) Setting in the above expressions S, = IS~I e i~,
$2 = LS2[ e i~
we notice that the right-hand side of eq. (20) depends on IS1 [. DS2I and S1 + - j-
Ek
32 2 = [S~[2 + 2cos(9~1 - q~z) ~
Ek
IS, I IS2[ + ~A2 ISzl z. E;
It can be easily shown that f v (A ; $1, $2) reaches extremum for cos (~i - ~e) = + 1. Therefore, without loss of generality $1 and $2 may be considered as real parameters. Further, in the limit V ~ ~ , we replace the summation over k in eq. (20) by integration over the one-electron spectrum ek and taking into account eq. (9) we write l i m v ~ f v (A ; $1 ~ $2) as w
f ~ ( 1; S , , Sz) = - 2 k . T f ~Oo(e) de In 2 cosh [(E - /z)22k.T + -~2(E)]~ + V- 1K.
-w
(21)
Here % ( 0 is the density of states in the original electron band {e~}. For the sake of brevity we omit the arguments of the functional dependence of E on e and A, defined by eq. (9)~
378
J.G. BRANKOV AND N.S. TONCHEV
Thus, in the thermodynamic limit the free-energy density corresponding to the model hamiltonian under consideration [see (13)] is given by fr,~ (3 ; ~q,, ~ ) =
min f ~ (A ; S~, $2).
(22)
A , S 1, $ 2
Absolute minimum condition (22) yields a set of self-consistent equations for the trial parameters 8 --f~
(,J; $1, $2) = O,
(23a)
.f~ (d; S,, S2) = 0,
(23b)
8A 8
8S1
C f ~ ( ; l ; S , , $2) 0S2
0.
(23c)
To the above equations we have to add the equation for the chemical potential 8 ---f~
(/I; $1, $2) = /i,
(24)
where /~ denotes the average number of electrons per unit volume. However, it appears more convenient to work in terms of the Fermi energy in the undistorted electron band,/~o, (instead of/i) which is related to ~ as follows /Jo
2 S ~o(e) de = /i.
(25)
-w
4. Analysis of the self-consistent equations in the weak BCS-coupling limit Here we study the case of a fixed deviation of the Fermi energy from the band center (see ref. I 1). In this case the lattice distortion (doubling of the period) and the BCS pairing are quite compatible (in contrast with the half-filled band case TM) due to the fact that in the ground state of the distorted phase either the upper zone {Eg _> ,JJ} is partially filled with "excessive" electrons (# > _1 if/% > 0) or the lower zone {Ek <-- - - J } is incompletely filled (# < -=~1 if/Zo < 0). Therefore such a ground state is unstable against infinitesimally weak BCS interaction. For definiteness we consider further/z0 > 0. Thus, taking the weak BCS-coupling limit (g~/w "~ 1), we assume T~ ~ Td, SO that for any fixed temperature T _< T~
LATTICE DISTORTION A N D E N E R G Y GAP OF S U P E R C O N D U C T O R
379
the distortion parameter A has a fixed nonzero value. Under these conditions the asymptotic form of eqs. (23)-(24) as $1 ~ 0 and Sz ~ 0 is iv
E-#
f Oo(e) de 1 tanh - E 2kBT
1
(26)
gd '
-- w
w
f qo(e) de
2(E)
tanh [(E - #)z + 2~2(E)1~ _-_ --,S' (27a)
2kBT
[(E - / 0 2 + 22(E)1 ~
gs
-iv
"w
f ¢o(e) de
E [(E
2(E) --
/~)2
ql-
tanh [(E - /02 + 22(E)1 ~
2kBT
2-'(E)]~
-w
S 2
gs (27b)
",v
-.
f¢
o(e) de tanh
E -/z
2kBT
1 - ~.
(28)
Iv
We remind that here E=
s g n e ( e 2 + A 2 ) ~,
2 ( E ) = z (IE - # 1 ) z ( E ) ,
2(E) = & + (A/E)&.
Notice that in accordance with the above assumptions we have neglected the effect of the BCS pairing both on the structural distortion parameter A and on the chemical potential/~. Now consider the ground state of the system (i.e., T = 0) using the constant density of states approximation for the undistorted electron band
~2o(e)=
(2w)-1,
- w _< e _< w.
(29)
In this case eqs. (26) and (28) reduce to w
1 j de w 2 (82 + d2)¢ [1 + sgn [(82 + / ] 2 ) ~t - fl]} = - - , ga 0
w
1 2w
f
0
de{1 - sgn[(e 2 + A 2 ) ~ - # ] }
=~-
1,
380
J.G. BRANKOV AND N.S. TONCHEV
or, p e r f o r m i n g the integration, we find in view of (3) and (25) the following coupled equations for A and ~
{w+(w 2+ I-) ~][(/t~-A)~+/t]-~
[w+(w ~+A0~)+]Ao ',
(/z2 --"12) ½ = /~o' Thus, ignoring terms of the order of 2
Ao/w
2
~ 1,
(30)
we obtain the well-known expressions for the zero-temperature insulating gap**) A = [Jo (J0 - 2/*o)] ½
(31)
and the Fermi energy in the distorted electron band 3,'~) /t = (/*o + A2) + = / l o
-/to.
(32)
Next we analyze the superconducting gap equations (27a) and (27b) at T = 0. The quasiparticle spectrum [see (19)] indicates that in the weak coupling limit the effective superconducting gap appears at the Fermi surface Ek~ = /t and its value is given by
(33)
S =- X(/t) = S, + A / t - I S 2 .
Thus, given the solution S I , 82 of the set of self-consistent equations (27a) and (27b), the gap _r we are interested in can easily be found. Replacing in these equations the integration variable e by E = (E2 + A 2)~ and ignoring terms of the order of Al2/w 2 .~ t in the limits of integration, we obtain w,
j
2(E)
" oA(E)
Sl
(34a)
dE
- w w
A j 5)a(E d E
2(E) E [(E - / t ) 2
+ 2_,(E)]+
32
gs
w
where
*2A(E) is the electron density of states in the distorted phase
(34b)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
381
In general the above equations do not allow the complete integration to be performed. However, in two limiting cases, namely (i) ~oo > / ~ and (ii) O~D ~ /~0, rather simple asymptotic forms of the solution (when g J w ~ 0) can be found. (i) First consider the case coD > #. In this case one can neglect the difference between the cut-off functions [see (12)] centered at the levels Ek = # and Ek+o = --#, assuming them to be approximately equal to 2(IEI) = ~1, t 0,
[El < ~oo IEI > e)D-
Then equations (34a) and (34b) may be written as (35a)
s~1L~o~ (s~, s~) + ~ (sl + A ~ - ' & ) -coo 1(2~ ( s l , s2) = Slw/g~, " 1 ~ T ( 2D) ( S 1 , 5 2 ) "]- A 7(3) (S1, LA~"I11co ~ 2 C~'2"CO]D
$2)
=
Szw/gs,
(35b)
where coD
1 (1~(S1,$2) = 1 f
co°
dE
7
(E~-A~)~
A
E-/z S2(E)]~-
x
" [ ( E - #)2 +
+ [(E + #)2 + ~ 2 ( _ E ) 1 ¢
}
, (36a)
~D
(E~ ---~)~
~ A
x
[(E-
1 #)2 + X2(E)]~
1
.}
[(E + /02 + X 2 ( - E ) ~ ' (36b)
COD
dE coo k ~ l ~
E (E 2 - A2) ~ A
x
1 [ ( E - t,) ~ + 2 : ( E ) ] ~ +
1 [(E + #)~ + s 2 ( - E ) ]
} ~ " (36c)
Now, looking for a nontrivial solution of eqs. (35a), (35b) which splits from the trivial one ($1 = $2 = 0), we have to find the asymptotic behaviour of integrals
382
J.G. BRANKOV AND N.S. TONCHEV
(36) as Sa ~ 0 and S,_ ~ O. F o r 1~ coCt) D (S~, $2) one immediately obtains: lim
s~,s~-~o
1~,oo (1) ( S 1 ,
$2)
In o)i) + (oo~, - A2) + "~ In 2o)o I* + (~2 _ A2)~ Ao
=
(37)
Here we have used relation (32). Next we notice that the upper limit o f the integrals I (2, 3) (S~ $2) can be extended to infinity within accuracy o f terms /z
lim
[1~2)(S~, $2)
t(2~
--o,o
S 1 , $2"-+ 0
(&,&)]
~
~t2~ < 1 , (O D
(38) 4 2 ZJ 2
lim
[i~3) (S~ , $2) - _% , (a~ 1 (S~ , $2)] ~ ---5- '~ 1.
S 1, $2~0
(O D
The asymptotic forms o f I~ 2' 3~ (S~, $2) are given by (see appendix):
1(2'~ ( S I , $ 2 ) _~ 1 In 4/~2 /~o
A 0X '
1
4# 2
i
(39)
/~o
1(~ ~ (SI, $2) - - In + -arctg /*/~o Ao S A# A
Thus the set o f equations for the superconducting gap [see (35)] reduces to the following asymptotically equivalent one
in 2c°D +._~ (1 + ..~ A. S---2-2.)In 4#o2 = in 2°~o Ao
/~o
/~ S~
AoX
Zo '
(40) A
A
1 +----
Sz'~
In
4# 2
A
$2
+----arctg-----
I~o
$2
In
20)o
where we have introduced the parameter (4l)
So = 2cod exp ( - w/gs),
which corresponds to the zero-temperature gap o f a " p u r e " superconductor (i.e., with ga = 0). C o m b i n i n g the above equations we find with the use o f (32) and (33) (oD > #),
m
(42)
where zl=l+
A2 In Ao [ In 2oD /~2 2-o So
A ~
arctg
#o7-* . A A
(43)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
383
(ii) Now we turn to the case 0)0 "~/Zo. In this case the set of equations (34) takes the form It+¢~ D
f
w
~"
/~ - -
hi(E)
dE
[(E
--
1 /~)2 _~_~'2(E)]{
= $ I - -W ,
(44a)
=
(44b)
gs
(9D
]J+O) D
'~ f It
-
hz(E) dE
[(E - #)2 + 2~2(E)]~
S 2 --~
g,
(9 D
-
where h,(E)
=
E
S(E)
(E 2 -- A2) k
~'
,
h~(E)-
A
S(E)
(E 2 - A2) ~
X
(45)
It is readily seen that the variation of h,,2(E) over the narrow energy interval involved here may be neglected, as it leads to corrections at most of the order /
2
/~(OD,/ZO ~ 1.
(46)
Further we shall consider condition (46) fulfilled. Replacing in eqs. (45) the functions h,,2(E) by their values at the Fermi surface
h d t O = ~,/~,o,
&(~) = A/t,o
and using the asymptotic form of the resulting integral (see appendix)
f
dE ~- 2 In 2o)o , [(E -- /z) 2 + X2(E)] + X
(47)
tt -- ~ D
we get in view of (40) (#//~o) X In (2c%/X) = S, In (20)D/SO), (A/po) X In (20)D/X) =
Sz
In (20)o/Xo).
From the above equations and eq. (33) it follows immediately that S = 2~OD ~ k, 20)D ,1 where z2 = 1 + (z~2/~2).
(0)D "~ /~o),
(48)
384
J.G. BRANKOV AND N.S. TONCHEV
As a final point here we mention that under the conditions (in our notation) w >>/{ >> /to > ~0D,
(49)
Mattis and Langer have given the following result 3) ~' = ~')D
sinh-~ O*/~o,,2g~ , ' ~) .
A
Conditions (49) do not correspond to any of the limiting cases studied here, unless -) I -~ 1"/-~0,/ [ - -
Y - - (')D ~
(9 D >~> ' l ,
which implies [see (46)]
~:I,"D/~ <
(50)
1.
If (50) is satisfied we believe that our result (48) (where z_~ reduces to 2 and /~o/(3o - /~o) t o / ~ o / l ) holds true.
5. Discussion
In the present paper we have studied a model a) for a superconductor ~)hich displays a structural phase transition. This model has the advantage of providing a simple, mathematically tractable expression for the modification of the BCS interaction which occurs in the distorted phase [see (10)]. It seems useful to compare the results obtained here to those found for extensively studied in the literature additive type models4-S'11-13). First we need a few brief remarks on what we call an additive model. The essential feature which distinguishes such a model from the one considered here is the assumption that the singled-out phonon mode Q does not affect the procedure of deriving the BCS pairing interaction6). With this assumption, the corresponding reduced hamiltonian may be written as ~add
=
(ek -- /0 ak. ,~ak.,, + o)bob e + g V -~ ~ ak++O, ,ak,,~ (bo + b2o)
~ k,
__
a
k, a
g~,V-'
~ 2([ek k. k'.~
__
/~ol)).(le,,
__
/~ol)ak+ a a _ k+, _,,a_k,_,,ak,.,,. (51)
Then, introducing an insulating gap parameter A [see (5)] and performing canonical transformation (7), we can replace hamiltonian (51) by [compare with (13)] Jf~oaa(z]) = Z (Ek -- #)Ck+,,Ck.,~ + ½V(z]Z/go) -
gsV-~
V k, k', tr
A ( e k , A ) A(ek,, I) c+, ~ c -+k ,
_~c_~,, _~ck,,~,
(52)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
385
where 2
(53)
In the above expression for )f'~da(A) we have omitted all the terms which are not of the BCS type (e.g., Ck.,~e-k.--,e-k,--~'ek'+O.,) + + since they do not contribute to the leading-order results13). Consider now the limiting cases studied in section 4. (i) When (% >> /z (which implies (oo >> #o) for most of the interaction range both 2 ([ek -/Zol) and 2 (leg + /Zo[) equal unity, so that A (e~, A) _~
I1,
led < ~%
0,
led > coD.
The zero-temperature superconducting gap in the weak-coupling limit is 12)
Sacld = 4#~ (S0"~ ~°/(J°-u°)
(~OO >> #).
(54)
Ao \Ao /
It should be mentioned that result (54) has been found first by Rusinov, Do Chan Kat and Kopayev 11) for a two-band semimetal by .using Green function technique. Comparing results (54) and (42) we see that they differ only in the factor z~-i which appears in the exponent of expression (42). This factor corresponds to the following renormalization of the coupling constant g~ [see (41)]
and originates from a factor of 4 2
1 +
- -
(55)
EREk,
increase in the BCS coupling coefficients in hamiltonian (10). Note that in the considered case the effective enhancement of the coupling parameter gs is less than the value of coefficient (55) taken at the Fermi surface, i.e., zl = 1 + #2 In
So
I n - - + In Z'o Ao
/z
since In (2o~olAo) > 1
while
1-1
arctg--
A arctg /~.o. < 1 /z A
< 1 + #2
386
J.G. BRANKOV AND N.S. TONCHEV
for all ~:1 and/~o- This is probably due to the fact that A 2 / E k E k , changes sign in the interaction region. Apart from the above mentioned difference, there is an obvious similarity in expressions (42) and (54), which reflects a common mechanism for enhancement of the superconducting gap, namely, the increase in the electron density of states at tile Fermi surface. This mechanism gives rise to the exponent ,uo/(Ao - ,Uo) the reciprocal value of which equals the ratio of the corresponding densities of electron states in the distorted and undistorted phases
c,~(/z)/0o(~o) = ~,//Zo = (Ao -/,o)//Zo > 1.
(56)
(ii) A somewhat different situation can be observed in the other limiting case: ('~D <{ t~0- The result found for the additive model (51) is ~3) /Zo (Zo'~ s . ~ , = 2,,~, J o 7 ,,o \ ~ /
4"° <~°-'°>/~°=
(o~o ~ /Zo).
(57)
This expression can be easily understood if we take into account the following factors. First, from eq. (53) we see that now the effective BCS coupling is reduced in the neighbourhood of the Fermi surface by a factor of A 2 (ek~, A), which leads to the renormalization 4
gs ---' ukFgs --
j2
4 (A0 -- /~0)2
(58)
gs.
Second, the contribution to the exponent from the rise in the electron density of states is the same as in the previous case [see (56)]. Third, the energy interval of the BCS pairing in the distorted phase is reduced by a factor of {[(/~o + o,o) z + A 2 ] ~ -
[(/Zo -
e0D) 2 + A2]~}/20)D
~
I~°
///0
/Z
'~0 -- ~0
(59)
A comparison between (57) and (48) reveals that the model studied in the present paper is free from both factors (58) and (59) which compete with the increase in the electron density of states. Moreover, an additional increase in the BCS coupling g5 occurs from (55) gs --* z2gs
(O)o ~ /~o)
which considerably enhances the superconducting gap Z' when ~o/2OJD <{ 1.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
387
Summarizing we note that in both of the considered limiting cases factor (55) in hamiltonian (10) leads to an additional enhancement of the zero-temperature superconducting gap which is of the same order of magnitude as the effect of the increase in the electron density of states at the Fermi level. Tracing back the origin of this factor we arrive at the modified Frohlich-type interaction described by the last term in eq. (8). Thus we conclude that scattering of electrons simultaneously by a phonon of wave vector q and by the static distortion of wave vector Q, which is described by terms of the type
+
Ck+q, oCk+Q, ff
(bq + b_+q)
plays an important role in forming the effective BCS interaction in the distorted phase.
Acknowledgements The authors are greatly indebted to Prof. Dr. L.V.Keldysh for his interest in this work. We gratefully acknowledge many informative discussions with Drs. Yu. V. Kopayev, N. M. Plakida and V. A. Zagrebnov.
Appendix Here we present a method of proving asymptotic forms (39), (47). This method is based on ideas of Tolmachev's analysis of the BCS compensation equation 22) and generalizes the scheme proposed in ref. 11. First we verify expressions (39). Note that each of the integrals i~2,a) (S~, $2) [see eqs. (36)] represents a sum of two terms, the first of which diverges logarithmically as $1 and $2 tend to zero, while the second remains bounded. Denoting the asymptotically diverging integrals by A (2'3) ($1, $2) we have co
1f
1~") ($1, Sz) ~- A (2) ($1, $2) - -~
dE (E 2 - A2)¢ (E + / z ) '
A
1;
I(~a) ( S , , $2) "" A ~3' ($1, $2) + --~
zl
(A.1) dE E(E z-A2) ½(E+#)
388
J.G. BRANKOV AND N.S. TONCHEV
With the aid of the following integrals E J(2)(S)
=
f
dx
( x 2 -~-/~12) ~
[(x-
1
[~)2 ..}_ $214x '
A
(A.2)
E
J~a)(2J) =
f
x (x 2
- - ,~2)@
[(x - ~)2 + S 2 ] ~ '
A
which depend on the superconducting gap parameter JL~ = $1 can write A (2,3) (S~, $2) as
+
A I u - I S 2 , one
co
A (2"3, (S~, Sz) =
f q~(E; Sa, $2) - -d~ J(n2'3)(X) dE,
(A.3)
A
where ,p (E; S , , & ) =
[(E - ~)-" + S : ] ~ [(E - /~)2 .~_ Xg(E)]~ "
Then, from (A.3) one finds A (2'3) (81, 32) = ~ (E, 81, $2) JOe2' 3)(~,)[f ~o
-
f J(E2' 3)(Z') - ~ - (p (E; S~, Sa) dE
A
= j ~ , a)(Z ) _
j(2, a)(Z ) - - ~ ~ (E; S~, $2) d E .
(A.4)
A
We expect that the leading asymptotic form of the integrals A (2,a) ($1, $2) is given by j(2, o~ a)(rT~ ~-~. To prove this we evaluate the last term in the right-hand side of eq. (A.4) : c¢:,
iS
J(E2"3)(Z") ~
d
(p (E; Sa, Sz) dE
--
9) ( E ; S 1 , S 2 A
dE.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR Computing dq~ (E; $1,
Sz)/dE and
q~(E; $1, $2) I -<
ASz
obtaining the majorization
1
/z
+
389
E2
AS:
m
S [1+~
Z" I
E[(E-#)2 +~,P21 ~
one can easily show that co
f d
~p(E; $I,
(A.5)
d E ~ (9 ( S 2 In ~').
A
Further, we note that J ~ ' 3)(X) are related to the elliptic integrals of the first and third kind as follows
J~)(X) = (pq)~ 1 F(% k), j(3#X~= 4t~ {F(%k)_H(% " " (pq)~(q _ p)Z 1 '~ ~j ([~2 _]_ ~2)~-
(q-P)2,k)} 4pq
(/IA2 -~ X2) ~
arctg
A
where q = [(# + A) 2 + S : ) ~,
p = [(/z - A) 2 + Z':] ~,
cp = 2 a r c t g ( q ) ~,
k = I(q + P)2 -
The leading asymptotic behaviour of the corresponding complete elliptic integrals as k' -- (1
-
k2) ~
~ d - /r# o
2~
O,
is given by z3) F (½=, k) ,-~ In 4k'- 1, H (½=, n, k) ,-~ (n + 1) -1 {In 4k '-1 q- n ~ arctg n~}.
(A.6)
390
J.G. BRANKOV AND N.S. TONCHEV
Taking into account (A.6) and computing the remaining bounded integrals after setting in them S~ = $2 = 0, we obtain expressions (39). Relation (47) can be justified by the same method. Let us introduce E
G~(S) =
f
dx [(x - / ~ ) 2 + S_~]~
I~ - - ¢9 D
and write It+CO D
It+09
D
dE [(E - i 0 = + s 2 ( E ) ] + ,ll - - O) D
It - - O) D I t + ~0 D
= ~ (/~ + ~,,~; s , , s e ) G , + ~ , o ( S ) -
G~(S) - ~ E ~ (E; Sl, 82) dE #-
e) D
= G , + ~ , , ( S ) [1 + 0 (S~ in _;)].
Here we have expanded q (# + (oD; $1, $2) in power series of $1,2/0)D and used estimate (A.5) as well. Now direct integration gives G,,+~,o(S) = 2 I n c,)o + (~o~ + $2)~ -~ 21n 2coo
i.e., the same as (47).
Note added in proof Our result for the superconducting gap (42), (43) remarkably resembles the corresponding one obtained in ref. 11 for a two-band semimetal with interband pairing interaction. The relation between the first-principle model 3) studied here and the model of ref. 11 is discussed in our following paperZ4).
References 1) A.C.Lawson, Phys. Lett. 36A (1971) 8. A. D.C. Grassie and A.Benyon, Phys. Lett. 39A (1972) 199. W. Buckel and B. Stritzker, Phys. Lett. 43A (1973) 403. R.Inone and K.Tachikawa, Japan J. Appl. Phys. 12 (1973) 161. T.Takashima and H. Hayashi, Phys. Lett. 47A (1974) 209.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)
391
K. Levin, D.L. Mills and S.L. Cunningham, Phys. Rev. 10B (1974) 3821. D.C. Mattis and W.D.Langer, Phys. Rev. Letters 25 (1970) 376. Yu.V.Kopayev, Zh. Eksp. Teor. Fiz. 58 (1970) 1012. Yu. V. Kopayev and R. Kh. Timerov, Fiz. Tverd. Tela 14 (1972) 86. Yu.V.Kopayev and R.Kh.Timerov, Zh. Eksp. Teor. Fiz. 63 (1972) 290. A.G.Aronov and E.B.Sonin, Zh.Eksp. Teor. Fiz. 63 (1972) 1059. S.C.Lo and K.W.Wong, Nuovo Cim. 10B (1972) 361,383. L.K. Testardi, Phys. Rev. 5 (1972) 4342. P.B.Allen and M.L.Cohen, Phys. Rev. Letters 29 (1972) 1593. A.I. Rusinov, Do Chan Kat and Yu.V.Kopayev, Zh. Eksp. Teor. Fiz. 65 (1973) 1984. J.G.Brankov and N.S.Tonchev, Commun. JINR, P4-8150, Dubna, 1974. J. G. Brankov and N. S.Tonchev, Commun. JINR, P4-8907; P4-8940, Dubna, 1975. Yu.V.Kopayev, Fiz. Tverd. Tela 8 (1966) 2633. H.Fr6hlich, Proc. Roy. Soc. A215 (1952) 291. J. Bardeen and D.Pines, Phys. Rev. 99 (1955) 1140. N.N.Bogolubov, Izv. Akad. Nauk SSSR 11 (1974) 77. J. G, Brankov, N. S. Tonchev and V. A. Zagrebnov, Physica 79A (1975) 125. V.A.Zagrebnov, J.G.Brankov and N.S.Tonchev, preprint JINR, E4-8818, Dubna, 1975; Dokl. Akad. Nauk SSSR 225 (1975) 71. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 106 (1957) 162; 108 (1957) 1175. N. N. Bogolubov, Physica 26 (1960) S 1. N.N.Bogolubov Jr., A Method for Studying Model Hamiltonians (Pergamon Press, London, New York, 1972). P.A.l.Tindemans and H. W. Capel, Physica 72 (1974) 433; 75 (1974) 407. V.V.Tolmachev, The Theory of Fermi Gas (Moscow State Univ. Press, Moscow, 1973), p. 242. E.Jahnke, F.Emde and F.L6sch, Tafeln H6herer Funktionen (B.G.Teubner, Stuttgart, 1960). J.G. Brankov and N. S. Tonchev, Preprint IC/76/8, Trieste, 1976; to be published in Physica.
EFFECTS OF LATTICE DISTORTION ON THE ENERGY GAP O F A BCS S U P E R C O N D U C T O R J.G. BRANKOV and N.S. TONCHEV Joint Institute for Nuclear Research, Dubna, USSR
Received 6 January 1976
The model of Mattis and Langer for superconductors with a structural distortion associated with doubling of the lattice periodicity is studied. New results for the zero-temperature superconducting gap are foufid in the case of more than a half-filled band. The modification of both the electron density of states and the reduced BCS interaction is taken into account. A comparison with the results found for the additive type model is given.
I. Introduction
Recently, for some superconductors with high transition temperature Ts, a p r o n o u n c e d correlation between Ts and the temperature Td at which they undergo a structural distortion has been observed1). This fact inspires an increasing n u m ber of works on model systems* which display b o t h a structural and a superconducting phase transition a - la). In the case o f three-dimensional systems, the possibility for enhancement o f Ts due to a structural distortion has been pointed out in refs. 3 and 4. Further this problem has been extensively investigated s - l a ) and different approaches have been p r o p o s e d 9' 1o). One of the expected mechanisms for influence o f structural phase transitions on superconductivity is based on the change in the electron density o f states4). A particular model for the lattice distortion has been exploited and the effect o f the corresponding modification in the electron density o f states on Ts has been c o m p u t e d in the weak coupling limit 6' 1 l - 13). A n o t h e r a p p r o a c h has been p r o p o s e d in1°), which circumvents the p r o b l e m o f identifying the precise model for the structural transition and attempts to c o n * Here we confine ourselves to the three-dimensional case. For discussion of one-dimensional models see2) and references therein. 371
372
J.G. BRANKOV AND N.S. TONCHEV
struct instead a direct relation between the amount of phonon softening and the enhancement of the electron-phonon coupling parameter. The model we investigate in this paper has been suggested by Mattis and Langera). It allows one to study in a rather simple way the effects due to the modification in both the effective Bardeen-Cooper-Schrieffer (BCS) coupling and the electron density of states, associated with the lattice distortion. The model system under consideration describes a single-band metal with electron dispersion ek having the symmetry property
ek = - e k + a ,
Q = = (__1, __1, + 1 ) ,
(1)
(d
where a is the lattice spacing (2Q is a vector in the reciprocal lattice). However, the results obtained here apply equally to the case of a two-band semimetal with band extrema separated by a vector Q in momentum space (see e.g. ref. 11). It has been found 3,6) that the crystal structure of a half-filled single-band metal with an electron spectrum ek, satisfying relation (1), is unstable under infinitesimally small electron-phonon coupling. At a certain temperature To such a system undergoes a metal-insulator phase transition. The lattice has been found unstable also when the deviation of the chemical potential ~o from the half-filled band case (/~o = 0) is sufficiently small in comparison with the interaction parameter Ao11.13)
I~ol ~ ½&.
(2)
Here the parameter J o corresponding to the value of the insulating gap for a half-filled band is related to the band width 2w, the electron-phonon coupling strength g(Q) and the phonon energy e)(Q) as follows 3'6) .d o = w sinh -~ - w -, g~
gd = -2g -2 (Q) c'~(Q)
(3)
The phase transition in this case is of the type of metal-degenerate semiconductor3). In general, the essential condition for the lattice instability to exist is the presence of fiat regions of the Fermi surface which coincide (or "almost" coincide3'6)) under translation by some fixed wave vector14). This three-dimensional instability is analogous to the well-known Peierls instability of one-dimensional metals*. * Notice that the one-dimensional instability appears for Q = 2kv, kv being the Fermi wave vector, whereas in the case of three-dimensional systems the corresponding instability wavevector is always Q = ~a- 1 (+ l, + 1, + 1) and it coincides with 2kv only if the electron band is half-filled.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
373
The idea proposed by Mattis and Langer in 3) consists in deriving the effective electron-electron pairing interaction from the Frohlich hamiltonian in the spirit of the usual BCS theory but with the appearance of a macroscopic lattice distortion taken into account. It should be emphasized that when performing the conventional procedure 15) for eliminating (in second order perturbation theory) the linear electron-phonon coupling from the hamiltonian, one does not presuppose a macroscopic filling of any phonon mode to take place. However, a lattice distortion changes both the electron and phonon dispersions and, in particular, causes a shift in the Fermi level. These effects, if taken into account, may lead to an explicit dependence of the BCS-type interaction on the order parameter of the distorted phase A. The aim of this paper is to provide a more detailed analysis of the possible effects of a structural phase transition on superconductivity, based on the explicit solutions for the zero-temperature superconducting gap found here for a relevant model problem.
2. The model hamiltonian
It seems useful to treat first the simplest case of Einstein phonons and momentum-independent electron-phonon coupling parameter3). Thus the hamiltonian we start with is ~/ = Z (~k - ~) ak,~,ak,¢, + + o ) ~ (b+b~ + ½) k, ~
+ gv-
tl
(bq + b+q) 2 E ak+,,~ak,~ + q
(4)
k,o"
Here aj,,+ ~, ak,~ are creation and annihilation operators, respectively, for electrons in states of wave vector k and spin ~ (o = _ 1); b +, bq are the corresponding operators for phonon modes of wave vector q. In eq. (4) V denotes the volume of the system and # is the chemical potential by means of which we adjust the number of electrons. The one-electron energies ek are measured relative to the band center and satisfy eq. (1). Due to the appearance of a macroscopic occupation number of the phonon _x _~ + mode of wave vector Q (below Td), the operators V b+Q and V b+Q can be replaced by a c-number (following Bogoliubov's ideat6))
V-~b*+o --* - ( 2 g ) -1 A,
(5)
where the trial parameter A has to be determined from the absolute minimum principle for the free energy density (see ref. 17). It should be noted that under
374
J.G. BRANKOV AND N.S. TONCHEV
some general conditions the substitution (5) has been rigorously justified in ref. 18. Further we consider instead of hamiltonian (4) the thermodynamically equivalent one (6)
H ( A ) = Ho(A) + H , ,
where Ho(A) = ~ [(eL - ~) a k+, a (l k , cr - ~la+o , oak, o] + ½V(J2/'ga) k,a
H~ = '" Z
(b+bq + ½) + g V -
Z ak+~, + ,ak,~ (bq + b+q).
~ ~
qvaO
qq-O
k, a
Following ref. 3, we diagonalize the term Ho(A) using the canonical transformation ak,a
=
UkCk,a
"1- L l k ( ' k + O , a ~
(7) ak+Q, a ~
- - U k C k , a -q- l l k C k + Q , a ~
with ck = s g n e k ~
u~ = - 1 I 1 + 2)~ ,]2 (e~ +[~:~1 _:1 1 ~ ,
1
(~ +[ek[.J~)~ 1 ~.
The resulting hamiltonian is +
1
k,a
q~Q
~ (bo + b_+O[l(k + q, k) cLo, oc,,o
+ gg-~ y q~-Q
+p(k
2
k, a
+ q,k)
C k++ q , a C k + Q , a ] ~
where ! (k, k') = u~uk, + vkv~,, p (k, k') = ukv~, - vku,,.
(8)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
375
In eq. (8) the dispersion Ek for electrons in the distorted lattice is given as ER = sgn eh (e~ + A2) ~.
(9)
Next, applying the procedure of Bardeen and Pines ~5) for eliminating the linear electron-phonon coupling from the hamiltonian, Mattis and Langer 3) have shown that the effective hamiltonian of the reduced BCS type is
yc(~)
y~ (E~
=
-
~) c k+, ° e ~ . .
+ -1-v(A:lg~)
k, ex
+ 2g2oJV_ 1 ~ 1 + A2/EkEk, + + k,k' (Ek -- Ek') 2 -- o) 2 Ck. l C - k , - l C - k , , - l e k , , 1 .
(10)
It should be mentioned that in eq. (10) effects of renormalization of the phonon frequencies have been neglected: Now, in contrast with ref. 3, we make explicitly the BCS approximation ag) g2(o (E~ -
-g~2 (lEg -- /~l) 2 ([Ek, -- ,ul),
Ek,) 2 - - (,~2
(11)
where lEg - # [ < OgD IEk--#l >o'o.
I1,
2(IEk--,ul)=
0,
(12)
The only difference between the model studied here and the one originally proposed by Mattis and Langer a) is that we introduce an independent interaction parameter gs (of the order of magnitude g2/o)) and a cut-off energy (Oo (of the order of (o) which are in principle phenomenological parameters. The model of ref. 3 corresponds to the particular case g~ = g2/oJ and (oD = o). Thus, with the aid of approximation (11) hamiltonian (10) reduces to the following exactly solvable one ~(A)
y~ ( E k -- ~,)
=
ck,,ck,~ + ½V(Zl2/g.) +
R, ~r
j2 )
-
-
gs v - 1 ~,~,,',,, 2(IEk -- #11 2 (IE,,, -- #1) (1 + - EkEk, +
+
X Ck, a t - k ,
- o C - - k ' , --aCk,,~r.
(13)
The system defined by eq. (13) belongs to the class of models with separable interaction studied by Bogolubov Jr. z°) (see also ref. 21).
376
J.G. BRANKOV AND N.S. TONCHEV
3. Calculation of the free-energy density Let us introduce the following b o u n d e d operators Jr sgn (J2 (IEk - ,,,I) c_~._.c~.~,
.f, = V-'
k. T
(14) .!:l
J:~ = V ~ V sen v). (IE¢, - ,.I) - k,G
c_~
~c~.~.
12,1;•
and rewrite hamiltonian (13) in the form 2
+
+
J / ( i ) = v,2.. (E~ - / ~ ) c~.~c,,., - ".g,v X Y , J i + i-V(le/gd) k,~
(15}
i=1
]t can be easily shown that the above hamiltonian satisfies all conditions of a general theorem proved by Bogolubov jr.2°). According to this theorem the free-energy density for the model system with hamiltonian (15) converges in the t h e r m o d y n a m i c limit to the approximating free-energy density .t;~ ( I ; S~, -g2} = - k u T V - '
In Tr exp [ - ~ 0
(:1"
SI, $2) ]
(16)
calculated with the use of a trial hamiltonian* 2
,;eo( l ; s , . s , )
_
~_.
~(e~-v,)cZ~e~,,
m
_
~ v y~ (. i s, + # , s * ~ "]6~ +
•
i=1
k,~
+ ~ g (g:-i V I&l-" + g~' 12 • ,
(17)
i=l
The parameters oel and ~'~2 in eq. (16) are determined from the absolute minimum condition l o r r y ( I ; Sz, $2) ,fy (=l; S I ,
$ 2 ) min = f v S't, 5"2
(I ; S,, $2).
We turn now to the calculation o f f v (./; S j , S,). By inserting (14) in (17) and performing some simple transformations, 24,% ( I ; $1, $2) can be written as
c3/(P0 (~1; S 1 , S 2 ) = /~ (ck. l C _ k , _ k
1) ~ _ ~ , ( E k
)
_
Ek -}- 1~ ~C+k.
+K, _1
(18) * In general, the variational parameters in eq. (17) are complex numbers.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
377
where
2(E~) = and K=
~V
Z (IE~ - ~'1)S(E~),
(2 g~
A Z'(Ek) = St + - - S 2 Ek
)
~ IS~I2 + gg1A 2 - 2# • i=1
Obviously, the one-particle spectrum of hamiltonian (18) is fi: [(E~ - / 0
2 + I-r(E~)12] + + K.
(19)
Hence, for the trial free-energy densityfv (A ; S~, $2) we get f r (-J; S~, Sa) = - 2 k B T V -~ ~ in 2 cosh [(Ek -- #)2 + I~(E~)t2]~ + V - 1 K . k 2kRT (20) Setting in the above expressions S, = IS~I e i~,
$2 = LS2[ e i~
we notice that the right-hand side of eq. (20) depends on IS1 [. DS2I and S1 + - j-
Ek
32 2 = [S~[2 + 2cos(9~1 - q~z) ~
Ek
IS, I IS2[ + ~A2 ISzl z. E;
It can be easily shown that f v (A ; $1, $2) reaches extremum for cos (~i - ~e) = + 1. Therefore, without loss of generality $1 and $2 may be considered as real parameters. Further, in the limit V ~ ~ , we replace the summation over k in eq. (20) by integration over the one-electron spectrum ek and taking into account eq. (9) we write l i m v ~ f v (A ; $1 ~ $2) as w
f ~ ( 1; S , , Sz) = - 2 k . T f ~Oo(e) de In 2 cosh [(E - /z)22k.T + -~2(E)]~ + V- 1K.
-w
(21)
Here % ( 0 is the density of states in the original electron band {e~}. For the sake of brevity we omit the arguments of the functional dependence of E on e and A, defined by eq. (9)~
378
J.G. BRANKOV AND N.S. TONCHEV
Thus, in the thermodynamic limit the free-energy density corresponding to the model hamiltonian under consideration [see (13)] is given by fr,~ (3 ; ~q,, ~ ) =
min f ~ (A ; S~, $2).
(22)
A , S 1, $ 2
Absolute minimum condition (22) yields a set of self-consistent equations for the trial parameters 8 --f~
(,J; $1, $2) = O,
(23a)
.f~ (d; S,, S2) = 0,
(23b)
8A 8
8S1
C f ~ ( ; l ; S , , $2) 0S2
0.
(23c)
To the above equations we have to add the equation for the chemical potential 8 ---f~
(/I; $1, $2) = /i,
(24)
where /~ denotes the average number of electrons per unit volume. However, it appears more convenient to work in terms of the Fermi energy in the undistorted electron band,/~o, (instead of/i) which is related to ~ as follows /Jo
2 S ~o(e) de = /i.
(25)
-w
4. Analysis of the self-consistent equations in the weak BCS-coupling limit Here we study the case of a fixed deviation of the Fermi energy from the band center (see ref. I 1). In this case the lattice distortion (doubling of the period) and the BCS pairing are quite compatible (in contrast with the half-filled band case TM) due to the fact that in the ground state of the distorted phase either the upper zone {Eg _> ,JJ} is partially filled with "excessive" electrons (# > _1 if/% > 0) or the lower zone {Ek <-- - - J } is incompletely filled (# < -=~1 if/Zo < 0). Therefore such a ground state is unstable against infinitesimally weak BCS interaction. For definiteness we consider further/z0 > 0. Thus, taking the weak BCS-coupling limit (g~/w "~ 1), we assume T~ ~ Td, SO that for any fixed temperature T _< T~
LATTICE DISTORTION A N D E N E R G Y GAP OF S U P E R C O N D U C T O R
379
the distortion parameter A has a fixed nonzero value. Under these conditions the asymptotic form of eqs. (23)-(24) as $1 ~ 0 and Sz ~ 0 is iv
E-#
f Oo(e) de 1 tanh - E 2kBT
1
(26)
gd '
-- w
w
f qo(e) de
2(E)
tanh [(E - #)z + 2~2(E)1~ _-_ --,S' (27a)
2kBT
[(E - / 0 2 + 22(E)1 ~
gs
-iv
"w
f ¢o(e) de
E [(E
2(E) --
/~)2
ql-
tanh [(E - /02 + 22(E)1 ~
2kBT
2-'(E)]~
-w
S 2
gs (27b)
",v
-.
f¢
o(e) de tanh
E -/z
2kBT
1 - ~.
(28)
Iv
We remind that here E=
s g n e ( e 2 + A 2 ) ~,
2 ( E ) = z (IE - # 1 ) z ( E ) ,
2(E) = & + (A/E)&.
Notice that in accordance with the above assumptions we have neglected the effect of the BCS pairing both on the structural distortion parameter A and on the chemical potential/~. Now consider the ground state of the system (i.e., T = 0) using the constant density of states approximation for the undistorted electron band
~2o(e)=
(2w)-1,
- w _< e _< w.
(29)
In this case eqs. (26) and (28) reduce to w
1 j de w 2 (82 + d2)¢ [1 + sgn [(82 + / ] 2 ) ~t - fl]} = - - , ga 0
w
1 2w
f
0
de{1 - sgn[(e 2 + A 2 ) ~ - # ] }
=~-
1,
380
J.G. BRANKOV AND N.S. TONCHEV
or, p e r f o r m i n g the integration, we find in view of (3) and (25) the following coupled equations for A and ~
{w+(w 2+ I-) ~][(/t~-A)~+/t]-~
[w+(w ~+A0~)+]Ao ',
(/z2 --"12) ½ = /~o' Thus, ignoring terms of the order of 2
Ao/w
2
~ 1,
(30)
we obtain the well-known expressions for the zero-temperature insulating gap**) A = [Jo (J0 - 2/*o)] ½
(31)
and the Fermi energy in the distorted electron band 3,'~) /t = (/*o + A2) + = / l o
-/to.
(32)
Next we analyze the superconducting gap equations (27a) and (27b) at T = 0. The quasiparticle spectrum [see (19)] indicates that in the weak coupling limit the effective superconducting gap appears at the Fermi surface Ek~ = /t and its value is given by
(33)
S =- X(/t) = S, + A / t - I S 2 .
Thus, given the solution S I , 82 of the set of self-consistent equations (27a) and (27b), the gap _r we are interested in can easily be found. Replacing in these equations the integration variable e by E = (E2 + A 2)~ and ignoring terms of the order of Al2/w 2 .~ t in the limits of integration, we obtain w,
j
2(E)
" oA(E)
Sl
(34a)
dE
- w w
A j 5)a(E d E
2(E) E [(E - / t ) 2
+ 2_,(E)]+
32
gs
w
where
*2A(E) is the electron density of states in the distorted phase
(34b)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
381
In general the above equations do not allow the complete integration to be performed. However, in two limiting cases, namely (i) ~oo > / ~ and (ii) O~D ~ /~0, rather simple asymptotic forms of the solution (when g J w ~ 0) can be found. (i) First consider the case coD > #. In this case one can neglect the difference between the cut-off functions [see (12)] centered at the levels Ek = # and Ek+o = --#, assuming them to be approximately equal to 2(IEI) = ~1, t 0,
[El < ~oo IEI > e)D-
Then equations (34a) and (34b) may be written as (35a)
s~1L~o~ (s~, s~) + ~ (sl + A ~ - ' & ) -coo 1(2~ ( s l , s2) = Slw/g~, " 1 ~ T ( 2D) ( S 1 , 5 2 ) "]- A 7(3) (S1, LA~"I11co ~ 2 C~'2"CO]D
$2)
=
Szw/gs,
(35b)
where coD
1 (1~(S1,$2) = 1 f
co°
dE
7
(E~-A~)~
A
E-/z S2(E)]~-
x
" [ ( E - #)2 +
+ [(E + #)2 + ~ 2 ( _ E ) 1 ¢
}
, (36a)
~D
(E~ ---~)~
~ A
x
[(E-
1 #)2 + X2(E)]~
1
.}
[(E + /02 + X 2 ( - E ) ~ ' (36b)
COD
dE coo k ~ l ~
E (E 2 - A2) ~ A
x
1 [ ( E - t,) ~ + 2 : ( E ) ] ~ +
1 [(E + #)~ + s 2 ( - E ) ]
} ~ " (36c)
Now, looking for a nontrivial solution of eqs. (35a), (35b) which splits from the trivial one ($1 = $2 = 0), we have to find the asymptotic behaviour of integrals
382
J.G. BRANKOV AND N.S. TONCHEV
(36) as Sa ~ 0 and S,_ ~ O. F o r 1~ coCt) D (S~, $2) one immediately obtains: lim
s~,s~-~o
1~,oo (1) ( S 1 ,
$2)
In o)i) + (oo~, - A2) + "~ In 2o)o I* + (~2 _ A2)~ Ao
=
(37)
Here we have used relation (32). Next we notice that the upper limit o f the integrals I (2, 3) (S~ $2) can be extended to infinity within accuracy o f terms /z
lim
[1~2)(S~, $2)
t(2~
--o,o
S 1 , $2"-+ 0
(&,&)]
~
~t2~ < 1 , (O D
(38) 4 2 ZJ 2
lim
[i~3) (S~ , $2) - _% , (a~ 1 (S~ , $2)] ~ ---5- '~ 1.
S 1, $2~0
(O D
The asymptotic forms o f I~ 2' 3~ (S~, $2) are given by (see appendix):
1(2'~ ( S I , $ 2 ) _~ 1 In 4/~2 /~o
A 0X '
1
4# 2
i
(39)
/~o
1(~ ~ (SI, $2) - - In + -arctg /*/~o Ao S A# A
Thus the set o f equations for the superconducting gap [see (35)] reduces to the following asymptotically equivalent one
in 2c°D +._~ (1 + ..~ A. S---2-2.)In 4#o2 = in 2°~o Ao
/~o
/~ S~
AoX
Zo '
(40) A
A
1 +----
Sz'~
In
4# 2
A
$2
+----arctg-----
I~o
$2
In
20)o
where we have introduced the parameter (4l)
So = 2cod exp ( - w/gs),
which corresponds to the zero-temperature gap o f a " p u r e " superconductor (i.e., with ga = 0). C o m b i n i n g the above equations we find with the use o f (32) and (33) (oD > #),
m
(42)
where zl=l+
A2 In Ao [ In 2oD /~2 2-o So
A ~
arctg
#o7-* . A A
(43)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
383
(ii) Now we turn to the case 0)0 "~/Zo. In this case the set of equations (34) takes the form It+¢~ D
f
w
~"
/~ - -
hi(E)
dE
[(E
--
1 /~)2 _~_~'2(E)]{
= $ I - -W ,
(44a)
=
(44b)
gs
(9D
]J+O) D
'~ f It
-
hz(E) dE
[(E - #)2 + 2~2(E)]~
S 2 --~
g,
(9 D
-
where h,(E)
=
E
S(E)
(E 2 -- A2) k
~'
,
h~(E)-
A
S(E)
(E 2 - A2) ~
X
(45)
It is readily seen that the variation of h,,2(E) over the narrow energy interval involved here may be neglected, as it leads to corrections at most of the order /
2
/~(OD,/ZO ~ 1.
(46)
Further we shall consider condition (46) fulfilled. Replacing in eqs. (45) the functions h,,2(E) by their values at the Fermi surface
h d t O = ~,/~,o,
&(~) = A/t,o
and using the asymptotic form of the resulting integral (see appendix)
f
dE ~- 2 In 2o)o , [(E -- /z) 2 + X2(E)] + X
(47)
tt -- ~ D
we get in view of (40) (#//~o) X In (2c%/X) = S, In (20)D/SO), (A/po) X In (20)D/X) =
Sz
In (20)o/Xo).
From the above equations and eq. (33) it follows immediately that S = 2~OD ~ k, 20)D ,1 where z2 = 1 + (z~2/~2).
(0)D "~ /~o),
(48)
384
J.G. BRANKOV AND N.S. TONCHEV
As a final point here we mention that under the conditions (in our notation) w >>/{ >> /to > ~0D,
(49)
Mattis and Langer have given the following result 3) ~' = ~')D
sinh-~ O*/~o,,2g~ , ' ~) .
A
Conditions (49) do not correspond to any of the limiting cases studied here, unless -) I -~ 1"/-~0,/ [ - -
Y - - (')D ~
(9 D >~> ' l ,
which implies [see (46)]
~:I,"D/~ <
(50)
1.
If (50) is satisfied we believe that our result (48) (where z_~ reduces to 2 and /~o/(3o - /~o) t o / ~ o / l ) holds true.
5. Discussion
In the present paper we have studied a model a) for a superconductor ~)hich displays a structural phase transition. This model has the advantage of providing a simple, mathematically tractable expression for the modification of the BCS interaction which occurs in the distorted phase [see (10)]. It seems useful to compare the results obtained here to those found for extensively studied in the literature additive type models4-S'11-13). First we need a few brief remarks on what we call an additive model. The essential feature which distinguishes such a model from the one considered here is the assumption that the singled-out phonon mode Q does not affect the procedure of deriving the BCS pairing interaction6). With this assumption, the corresponding reduced hamiltonian may be written as ~add
=
(ek -- /0 ak. ,~ak.,, + o)bob e + g V -~ ~ ak++O, ,ak,,~ (bo + b2o)
~ k,
__
a
k, a
g~,V-'
~ 2([ek k. k'.~
__
/~ol)).(le,,
__
/~ol)ak+ a a _ k+, _,,a_k,_,,ak,.,,. (51)
Then, introducing an insulating gap parameter A [see (5)] and performing canonical transformation (7), we can replace hamiltonian (51) by [compare with (13)] Jf~oaa(z]) = Z (Ek -- #)Ck+,,Ck.,~ + ½V(z]Z/go) -
gsV-~
V k, k', tr
A ( e k , A ) A(ek,, I) c+, ~ c -+k ,
_~c_~,, _~ck,,~,
(52)
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
385
where 2
(53)
In the above expression for )f'~da(A) we have omitted all the terms which are not of the BCS type (e.g., Ck.,~e-k.--,e-k,--~'ek'+O.,) + + since they do not contribute to the leading-order results13). Consider now the limiting cases studied in section 4. (i) When (% >> /z (which implies (oo >> #o) for most of the interaction range both 2 ([ek -/Zol) and 2 (leg + /Zo[) equal unity, so that A (e~, A) _~
I1,
led < ~%
0,
led > coD.
The zero-temperature superconducting gap in the weak-coupling limit is 12)
Sacld = 4#~ (S0"~ ~°/(J°-u°)
(~OO >> #).
(54)
Ao \Ao /
It should be mentioned that result (54) has been found first by Rusinov, Do Chan Kat and Kopayev 11) for a two-band semimetal by .using Green function technique. Comparing results (54) and (42) we see that they differ only in the factor z~-i which appears in the exponent of expression (42). This factor corresponds to the following renormalization of the coupling constant g~ [see (41)]
and originates from a factor of 4 2
1 +
- -
(55)
EREk,
increase in the BCS coupling coefficients in hamiltonian (10). Note that in the considered case the effective enhancement of the coupling parameter gs is less than the value of coefficient (55) taken at the Fermi surface, i.e., zl = 1 + #2 In
So
I n - - + In Z'o Ao
/z
since In (2o~olAo) > 1
while
1-1
arctg--
A arctg /~.o. < 1 /z A
< 1 + #2
386
J.G. BRANKOV AND N.S. TONCHEV
for all ~:1 and/~o- This is probably due to the fact that A 2 / E k E k , changes sign in the interaction region. Apart from the above mentioned difference, there is an obvious similarity in expressions (42) and (54), which reflects a common mechanism for enhancement of the superconducting gap, namely, the increase in the electron density of states at tile Fermi surface. This mechanism gives rise to the exponent ,uo/(Ao - ,Uo) the reciprocal value of which equals the ratio of the corresponding densities of electron states in the distorted and undistorted phases
c,~(/z)/0o(~o) = ~,//Zo = (Ao -/,o)//Zo > 1.
(56)
(ii) A somewhat different situation can be observed in the other limiting case: ('~D <{ t~0- The result found for the additive model (51) is ~3) /Zo (Zo'~ s . ~ , = 2,,~, J o 7 ,,o \ ~ /
4"° <~°-'°>/~°=
(o~o ~ /Zo).
(57)
This expression can be easily understood if we take into account the following factors. First, from eq. (53) we see that now the effective BCS coupling is reduced in the neighbourhood of the Fermi surface by a factor of A 2 (ek~, A), which leads to the renormalization 4
gs ---' ukFgs --
j2
4 (A0 -- /~0)2
(58)
gs.
Second, the contribution to the exponent from the rise in the electron density of states is the same as in the previous case [see (56)]. Third, the energy interval of the BCS pairing in the distorted phase is reduced by a factor of {[(/~o + o,o) z + A 2 ] ~ -
[(/Zo -
e0D) 2 + A2]~}/20)D
~
I~°
///0
/Z
'~0 -- ~0
(59)
A comparison between (57) and (48) reveals that the model studied in the present paper is free from both factors (58) and (59) which compete with the increase in the electron density of states. Moreover, an additional increase in the BCS coupling g5 occurs from (55) gs --* z2gs
(O)o ~ /~o)
which considerably enhances the superconducting gap Z' when ~o/2OJD <{ 1.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR
387
Summarizing we note that in both of the considered limiting cases factor (55) in hamiltonian (10) leads to an additional enhancement of the zero-temperature superconducting gap which is of the same order of magnitude as the effect of the increase in the electron density of states at the Fermi level. Tracing back the origin of this factor we arrive at the modified Frohlich-type interaction described by the last term in eq. (8). Thus we conclude that scattering of electrons simultaneously by a phonon of wave vector q and by the static distortion of wave vector Q, which is described by terms of the type
+
Ck+q, oCk+Q, ff
(bq + b_+q)
plays an important role in forming the effective BCS interaction in the distorted phase.
Acknowledgements The authors are greatly indebted to Prof. Dr. L.V.Keldysh for his interest in this work. We gratefully acknowledge many informative discussions with Drs. Yu. V. Kopayev, N. M. Plakida and V. A. Zagrebnov.
Appendix Here we present a method of proving asymptotic forms (39), (47). This method is based on ideas of Tolmachev's analysis of the BCS compensation equation 22) and generalizes the scheme proposed in ref. 11. First we verify expressions (39). Note that each of the integrals i~2,a) (S~, $2) [see eqs. (36)] represents a sum of two terms, the first of which diverges logarithmically as $1 and $2 tend to zero, while the second remains bounded. Denoting the asymptotically diverging integrals by A (2'3) ($1, $2) we have co
1f
1~") ($1, Sz) ~- A (2) ($1, $2) - -~
dE (E 2 - A2)¢ (E + / z ) '
A
1;
I(~a) ( S , , $2) "" A ~3' ($1, $2) + --~
zl
(A.1) dE E(E z-A2) ½(E+#)
388
J.G. BRANKOV AND N.S. TONCHEV
With the aid of the following integrals E J(2)(S)
=
f
dx
( x 2 -~-/~12) ~
[(x-
1
[~)2 ..}_ $214x '
A
(A.2)
E
J~a)(2J) =
f
x (x 2
- - ,~2)@
[(x - ~)2 + S 2 ] ~ '
A
which depend on the superconducting gap parameter JL~ = $1 can write A (2,3) (S~, $2) as
+
A I u - I S 2 , one
co
A (2"3, (S~, Sz) =
f q~(E; Sa, $2) - -d~ J(n2'3)(X) dE,
(A.3)
A
where ,p (E; S , , & ) =
[(E - ~)-" + S : ] ~ [(E - /~)2 .~_ Xg(E)]~ "
Then, from (A.3) one finds A (2'3) (81, 32) = ~ (E, 81, $2) JOe2' 3)(~,)[f ~o
-
f J(E2' 3)(Z') - ~ - (p (E; S~, Sa) dE
A
= j ~ , a)(Z ) _
j(2, a)(Z ) - - ~ ~ (E; S~, $2) d E .
(A.4)
A
We expect that the leading asymptotic form of the integrals A (2,a) ($1, $2) is given by j(2, o~ a)(rT~ ~-~. To prove this we evaluate the last term in the right-hand side of eq. (A.4) : c¢:,
iS
J(E2"3)(Z") ~
d
(p (E; Sa, Sz) dE
--
9) ( E ; S 1 , S 2 A
dE.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR Computing dq~ (E; $1,
Sz)/dE and
q~(E; $1, $2) I -<
ASz
obtaining the majorization
1
/z
+
389
E2
AS:
m
S [1+~
Z" I
E[(E-#)2 +~,P21 ~
one can easily show that co
f d
~p(E; $I,
(A.5)
d E ~ (9 ( S 2 In ~').
A
Further, we note that J ~ ' 3)(X) are related to the elliptic integrals of the first and third kind as follows
J~)(X) = (pq)~ 1 F(% k), j(3#X~= 4t~ {F(%k)_H(% " " (pq)~(q _ p)Z 1 '~ ~j ([~2 _]_ ~2)~-
(q-P)2,k)} 4pq
(/IA2 -~ X2) ~
arctg
A
where q = [(# + A) 2 + S : ) ~,
p = [(/z - A) 2 + Z':] ~,
cp = 2 a r c t g ( q ) ~,
k = I(q + P)2 -
The leading asymptotic behaviour of the corresponding complete elliptic integrals as k' -- (1
-
k2) ~
~ d - /r# o
2~
O,
is given by z3) F (½=, k) ,-~ In 4k'- 1, H (½=, n, k) ,-~ (n + 1) -1 {In 4k '-1 q- n ~ arctg n~}.
(A.6)
390
J.G. BRANKOV AND N.S. TONCHEV
Taking into account (A.6) and computing the remaining bounded integrals after setting in them S~ = $2 = 0, we obtain expressions (39). Relation (47) can be justified by the same method. Let us introduce E
G~(S) =
f
dx [(x - / ~ ) 2 + S_~]~
I~ - - ¢9 D
and write It+CO D
It+09
D
dE [(E - i 0 = + s 2 ( E ) ] + ,ll - - O) D
It - - O) D I t + ~0 D
= ~ (/~ + ~,,~; s , , s e ) G , + ~ , o ( S ) -
G~(S) - ~ E ~ (E; Sl, 82) dE #-
e) D
= G , + ~ , , ( S ) [1 + 0 (S~ in _;)].
Here we have expanded q (# + (oD; $1, $2) in power series of $1,2/0)D and used estimate (A.5) as well. Now direct integration gives G,,+~,o(S) = 2 I n c,)o + (~o~ + $2)~ -~ 21n 2coo
i.e., the same as (47).
Note added in proof Our result for the superconducting gap (42), (43) remarkably resembles the corresponding one obtained in ref. 11 for a two-band semimetal with interband pairing interaction. The relation between the first-principle model 3) studied here and the model of ref. 11 is discussed in our following paperZ4).
References 1) A.C.Lawson, Phys. Lett. 36A (1971) 8. A. D.C. Grassie and A.Benyon, Phys. Lett. 39A (1972) 199. W. Buckel and B. Stritzker, Phys. Lett. 43A (1973) 403. R.Inone and K.Tachikawa, Japan J. Appl. Phys. 12 (1973) 161. T.Takashima and H. Hayashi, Phys. Lett. 47A (1974) 209.
LATTICE DISTORTION AND ENERGY GAP OF SUPERCONDUCTOR 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24)
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