Effect of high density of states due to localization on the superconducting transition temperature

Physica 104B (1981) 383-396 © North-Holland Publishing Company

EFFECT OF HIGH DENSITY OF STATES DUE TO LOCALIZATION ON THE SUPERCONDUCTING TRANSITION TEMPERATURE* O. ENTIN-WOHLMAN Department of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel

and Racah Institute of Physics, Hebrew University, Jerusalem, Israel

Received 15 July 1980

The parameter N(0)I2, where N(0) is the density of states at the Fermi energy and 12 is the electron-phonon coupling constant, is calculated in a model which gives rise to high N(0) due to localization. The calculation is based on the localized description of superconductivity formulated by Appel and Kohn. It is found that although N(0) can become high, N(0)I 2 can nevertheless be very low.

1. Introduction The A-15 compounds are known for their high superconducting transition temperature. The high T c was related to the high electronic density of states [1 ]. The high electronic density of states was attributed to edges of onedimensional bands from the linear chain model [2]. Subsequent calculations, based on the LCAO [3] as well as APW [4] techniques, indicated the existence of a large number of sharp peaks in the density of states. Therefore, it may be useful to try to investigate in general how peaks in the electronic density of states affect T c. In this work we investigate a peak due to localized electrons, which become delocalized due to hybridization with another wider band. The hybridization, if small, broadens the peak somewhat, but does not wash it out.

2. The model The electronic properties which affect the transition temperature are usually expressed [5] by the combination N(0)/2, where N(0) is the electronic density of states at the Fermi energy and 12 denotes the electron-phonon interaction constant. The density of states at the Fermi energy can become very high when the Fermi energy falls on a peak in the density of states; such a situation may occur when one of the energy bands is very narrow. However, the inclusion of a narrow band also changes 12. Model calculations of 12 for narrow-band metals [6, 7] show that as the band becomes narrower, the corresponding I 2 decreases. The question therefore arises whether the increase in N(0) due to the inclusion o f a narrow band will compensate for the decrease it presumably causes in I 2. To study this question we have considere'd a system of two coupled energy bands, one of which is infinitesimally narrow, and calculated the energy spectrum and the e l e c t r o n - p h o n o n interaction in the tight-binding approximation (TBA). The calculations were applied for V3Si, and the two particular atomic wave-functions chosen were vanadium 3d a and 81 orbitals. The tight-binding parameters we used were [3] : * Work supported by a grant from the National Council for Research and Development, Israel and KfK (Kernforschungszentrum, Karlsruhe, Germany). 383

384

O. Entin-Wohlman/Density o f states and superconducting transition temperature

(i) One-center integrals (crystal field parameters). E.g.,

(1)

I((,x) = r, ] * 6 1 x ( r ) v(, - R) qx("), R

where V(r) is the atomic potential. (ii) Two-center integrals between nearest neighbour vanadium atoms. We have assumed that these vanish for the 51 orbitals. (Indeed, they are much smaller than the o-orbital two-center integrals [3]. The assumption that they are zero means that the uncoupled 81-band is infinitesimally narrow.) The o-orbital two-center integrals are of the form

(2)

:(o) = fdroAr) 11(,) oar - ½~a), where a is the lattice constant. (iii) Three-center integrals. These give the coupling between the o and 51 bands. The o and 51 functions are centered around nearest neighbour vanadium atoms, while the atomic potential is centered around the nearest vanadium atom in the perpendicular chain. Thus, the three-center integrals are of the form

J3(ox61x) = £ fdrox(r - di) V(r)~lx(r di

d;),

(3)

where d i = ~fa + ¼fa +- ¼ka.

(3a)

Using appropriate tight-binding parameters [3], the energy spectrum and the density of states were computed in the method described in ref. [3]. For the calculation o f I 2 we have used the localized description of narrowband superconductors, developed by Appel and Kohn [8, 9] and subsequently by Birnboim and Gutfreund [6, 7]. In this approach the equation for the vertex part of a Cooper pair is expressed in the Wannier representation. The approach was applied [6, 7] to V, Nb and Ta, assuming that the d electrons occupy the threefold degenerate F~5 band. We present an extension of the method in the next section, to include the contribution from the threecenter integrals. 3. The self-consistency equation In this section we first outline the formalism of Appel and Kohn (AK) [8, 9] for the special case of our model, and then derive an expression for the electron-phonon interaction constant. The self-consistency equation in the formalism of AK, in the vicinity of Tc, reads

p(no~) = - k BT ~ ~ P(n'~') <~7'~'lKIr/co>.

(4)

r/' tO'

Here P is the vertex part, K is the electron-electron interaction kernel, ~o is the imaginary frequency and r/ describes an electron-pair state. We shall use the contact approximation [8], which means that the two members of the pair are assumed to occupy the same state. Then, in our case,

O. Entin-Wohlman/Density of states and superconducting transition temperature

385

6

~?/ = X/~

~

~/(l)(r - rl) ~b/(l)(r - rl),

(5)

] = 1, 2,

l=1 where we have introduced the following notations: ~/(l)(r - rl) is an atomic orbital, r l is a vector to the lth V atom in the unit cell, l = 1 . . . . . 6 a n d ] = 1, 2 refers to o and 51 orbitals, respectively. E.g. [8], ~l(3)(r -- 7"3) = [(5/16)l[2u(R)(3y 2 - R2)/R2]R=r_r3 ,

r 3 = ½fa + t{a,

and a is the length of the unit cell. Using these notations, we have 1

~ G(n~'(l')l';nl/l(ll)ll)G(n~'(l')l';n2]2(12)l 11' n1111l1 n21212

2)

= ~ ~

X .

(6)

For simplicity, we have omitted the frequency indices. G denotes the electronic Green's functions, (1II) is the electron-electron interaction, and n is a vector to the center of the nth unit cell. We now introduce the transformation connecting the Bloch and the Wannier representations

~)kv(r) = E A nfl

(7)

eik'n b/(l)l(Pk) ~)/(1)(r - n - 7"l),

where v is the band index and bj(l)l(Pk ) are the elements of the unitary transformation,

bj(1)t(~ -k) = b?(l)l(~k),

(8a)

b ~(l)l(U l k ) b/ ( l)l(vk ) = 6 mq ,

(8b)

Y b?(l),(~k)b/,(q)l, (~k) = ~z, ~.~

(8c)

/t p

Using the form of the normal-state Green's function (since we are operating in the vicinity of To),

(9)

c(k'v'; k~) = ~,,~(k - k') - eAk ) '

we obtain 1

(rl/,lKlrl/) = Z E E - - e - i k ' ( n l - n 2 ) bf(l')l'(Plk)b}l(lOll(Vlk) 6N ll' fir2 VlkU2 1

× b / , c ) r ( V 2 - k ) b~(t2)t2(v2 - k )

w - Evl(k ) -co -

!

E

" (fl;f21IIf;f), ~2(k)

(10)

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O. En tin- Wohlman/Density of states and superconducting transition temperature

where f stands for (nf(l)l). This expression is simplified further by noting that the sum in (10) attains its maximum for n 1 = n2, so following AK we keep only this contribution. Next we note that the matrix elements bj(t)t(vk ) satisfy in our model the following relations (these are proved in appendix A) 6

E bj'(l')l'(Vlk)b?(l')l'(V2 k) = Fj, (v 1 k) 8Vl u2

(1 la)

l'=1

6

E

(11 b)

b]l(l')l'(Vlk) b~(l')l'(Vlk) = Fjl(Vlk)S]l]2.

l'=1

These relations express "partial orthogonality" of the eigenvectors which diagonalize the hamiltonian in the TBA, with 2

F/.(vk) = 1.

(12)

In what follows we shall consider only the phonon contribution to the electron-electron interaction ([II) in eqs. (6) and (10). In that case, the explicit calculation of ( II I) in our model (see appendix B) yields that l 1 = l 2 and that each l implies a certain I 1. The function ([I1) thus calculated, does not depend on l. Therefore, using these facts and eqs. (1 1) we obtain 1

(~fiKD~i)= ~

~

~ 6o2

-1 E2(k )

Ff(vk)I~l(vk) ~ (fl;fll'lf;f).

(13)

nl

Our next task is to calculate the quantity (1II). With the transformation (7) this is = ~ Afl(Vlkl)Afl(V2k2)h~(v3k3)A~(v4k4) (Vlkl; vi ki

v2k21IIv3k3; v4k4>,

(14)

with

A f(vk) = V//~ eik" n b/(l)l(Vk)"

(14a)

We now have [on the lhs of (14)] in the Bloch representation. This quantity is = E O h ( k l - k 3 ) g x ( V l k l ; v 3 k 3 ) g h ( v 2 k 2 ;

v 4 k 4 ) 6 ( k l - k3 - k2 + k4),

(15)

where Dx(q) is the phonon Green's function for wavevector q and polarization X (we remind the reader that frequency indices are omitted for brevity) and g is the matrix element of the electron-phonon interaction,

gx(Vlkl; v3k3) =

E

A~a(Vlkl)Af4(v3k3)
-- Rn414 )) 10qx>,

q = k 1 - k 3.

hf4 (16)

O. Entin-Wohlman/Density of states and superconducting transition temperature

387

Here (lqxl, (0qxl denotes a state with 1,0 (q~k) phonon, respectively, and Rnl = n + r l. Inserting eqs. (15) and (16) into eq. (14) and making use of eqs. (8), we obtain

D~(q)~ ~

=~ qh

~(n 1 - n 3 + n 4 - n ) / 5 ( n 1 - n 5 + n 6 - n )

n 3 ...n 6

X e iq. (n6--n4) (lqh I(~]l(ll)(r - R n 3 l ] ) I-1]~/(l)(r - Rn4i))10ax) (17)

X (Oqxl(~b/l(lO(r - Rnsll)lH]t~/(l)(r -- Rn6l))llqx).

The matrix element of H is calculated in the modified tight-binding approximation (MTBA) [6, 10]. In the MTBA, the atomic orbitals centered around the instantaneous positions of the ions. Namely, the hamfltonian and the atomic orbitals ~k/(l)(r - Rnl ) are expanded to first order in the ion displacements. The detailed calculation of the matrix element is given in appendix B. Using the result eq. (B.10) in eq. (17), we obtain


(18)

where a = x, y, z (we have used the cubic symmetry of the lattice [9] ) n' = n 1 - n, and

1}l/(qn ll/) = (1 - e -iq (n'+rll-rt)) Ot

I

°

3 ~(n' + rll - r/) ~ Ji(2)(n' + rll - rI)8/il

+ ~,, 8(n' + rll -- 1"/)(1 -- e-iq'(m+rll -ri)) O(m Ki(1)(m + mi + rll - 7"i)a 7"11- 7i)bjjl + ~

I( 1 _ e_iq.(m+rll_,i))

mi

_ e-iq.(m+rtl-ri))

0 + (e_iq.(n,+rll_rl) O(m + Tit -- ri) ~

3 ] 3 ( m - r t ' + r I -7"i) a' Jl!3/(m +~'I1 - r i ' m - n '

+ r l - 7"i)"

(19)

The terms appearing in eq. (19) are the contribution of the two-center integrals, one-center integrals and threecenter integrals, respectively. The first two terms are proportional to 8ii 1 while the third, from eq. (3), implies that/1 = 1,] = 2 o r ] l = 2,] = 1. Therefore, the product which appears in eq. (18) splits into two terms, i.e.,

Y Y q/(qn'6Oqj(-qn'tlO --

i,jlIljl 2 +

+ j,2 i )lz212,

(2o)

nt

where Ii/represents the contribution of the first two terms in (19) to the sum over n', and 12 is the contribution of the third term there. In the explicit calculation of Ill/. 12 and 1/212(see appendix C) it is found that these quantities, to leading order, do not depend on q. From eqs. (13), (18) and (20) we thus obtain that the kernel
O. Entin-Wohlman/Densityof states and superconductingtransition temperature

388

qx

Dx(qco)

2NMwx (q)

= L(co),

(21)

and an electronic factor. To calculate the electronic factor we note that the terms in eq. (18) do not depend on the frequency [7, 8], or on k. (Here we use the BCS approximation and assume that the electron-phonon interaction vanishes for co/> co0, where coo is some typical phonon energy.) Leaving these terms aside for the moment we have (see eq. (4))

]¢BT Z

1 1 ~ k~v icon[2 _~E2(k)F.i'(vk)Fil(Pk)=kB

C-On

1 (g)iconl2 + E2

(.On

1

"~

1

T ~ -6~ d'EN(E) Ff'(E)F.il

~N(O)FI.,(EF)6.1(EF) In

coo /¢B T

.

(22)

Here [conl = lrkBT(2n + 1), N(0) = N(EF) is the density of states at the Fermi energy and E F denotes the Fermi energy. Collecting the results (13), (18), (20), (21) and (22) we find that the self-consistency equation (4) becomes l-'1 = ~N(0)LIn coO

kBT

['2 = ~N(O)L In

[PIFI(EF) + P2F2(EF)] (FI(EF) Ill112 + F2(EF)II2[2),

coO . . [['IFI(EF) + P2F2(EF)] (F2(EF) II1212 + FI(EF)II212).

kBT

(23)

The solution of these equations yields coO In ~ =

kBT

[L~N(O)(F2(EF)]111[2 + F2(EF) II1212 +

2FI(EF ) F2(EF)i1212)] -1.

(24)

Thus, the terms in the square brackets in (24), apart from the phonon term L, play the role o f N ( 0 ) I 2 in our model [6]. This is our basic result and we now turn to discussing its meaning. From eqs. (19) and (20) we see that 111112represents the contribution of the o-band alone to the electronphonon interaction constant. It is multiplied by F12(EF) which gives (see eq. (1 lb)) the weight of the o contribution. Similarly for the second term in the square brackets of (24), which describes the contribution of the 81 band. (We remind the reader that F 1 + F 2 = 1.) The third term gives the effect of the coupling between the two bands, since 1/212, from eqs. (19) and (20) is related to the coupling J3, eq. (3). Now, the peak in the density of states N(E) becomes higher as the coupling decreases and we approach the limit in which the bands are not coupled. But then the weight of the 81 contribution, F2(E ) (at the energy where the peak occurs) approaches unity and the weight of o, FI(E ) tends to zero. Even if the Fermi energy falls on that peak, we see that the contribution of the coupling term may tend to zero (note that as J3 decreases, so does 11212, see appendix C). We have computed the expression

~N(O)(F2(EF)Ill112 + 2FI(EF)F2(EF)II212) (N(O)I2)eff, =-

(25)

neglecting the contribution of the o and 61 one-center integrals. It is believed that their contribution to (N(0)/2)eff is small [7]. The computation was carried out for various values of the three-center integral J3(o81), i.e., the

O. Entin-Wohlman/Density of states and s u p e r c o n d u c t i n g transition temperature

389

coupling parameter, and for various values of n i - the occupation number of electrons per V atom. This last quantity determines the Fermi energy. The results are depicted in fig. 1. Fig. 2 portrays the density of states at the Fermi level as a function of the coupling. It is seen that in the case n i = 1.8, N(0) can be extremely high; nevertheless, (N(0)I2)eff is very low. There are recent experiments in which the effect of disorder on the transition temperature of A-15 compounds was investigated. The data when analyzed for [11 ] V3Si , using McMillan's formula [5], reveal that N(0)I decreases slower than N(0), as obtained from specific heat and critical field data. Our results show that N(0)I= may even increase as N(0) decreases.

ni= 1.2

O0

00

n i = 1.5

n i = 1.8

Q 0

0

OQ

O

O

00000

0.01

0.01

0.01

0

0 000

0

0

el A

o z O o

o o

I

I

I

.

i

0.03

0.01

I

OOl

I

,

Q03

Js (Ry)

o

°

oo

0

o

I

I

I

0.01

J~ (Ry)

0.03 JI (Ry)

Fig. 1. (N(0)12)ef f as a function of the coupling parameter J3 for various values of the occupation number n i (electrons per vanadium atom). Parameters used in the computation are given in table C.I (see appendix C).

n i = 1.8. n i -i.2

n i =1.5

o ¢3 QO O0

v

is

IOO

5 0 0

U

Q

Q

•

Q

m

~1

..e O

1; "o

d z

.

I

I

001

I

0.03 J I (Ry)

:.

I

I

0.01

I

0.03 JilRy)

;

I

I

"

°

I

0.03 ~

0.01 JIIRy)

Fig. 2. N(0) as a function of the coupling pa.rameter J 3 for various values of the occupation number n i.

.

390

O. En tin- Wohlman/Density of states and superconducting transition temperature

4. Discussion

Sharp peaks in the density of states may arise from several causes, localization being only one. Another possibility for a sharp peak is destructive interference due to multiple scattering. In this case, the various transfer integrals are big, but a cancellation effect reduces the group velocity to close to zero. Since the transfer integrals are large, their modulation by the lattice is large, giving rise to a large 12. The Labbe model [2] is a special case of such a situation, and indeed yields a high Tc. We are currently investigating models of this type.

Acknowledgements The author wishes to thank Prof. M. Weger who proposed the model for many useful discussions. The help of Dr. I. Goldberg in the computation process is greatly appreciated.

Appendix A

Properties of the transformation b](l)l(vk ) The transformation (7) relates the Wannier and the Bloch representation. Using it, the matrix elements of the Hamiltonian in the Bloch representation are given by

(~ku(r)lHl~k,d(r)) = 5(k - k') E ~ E e-ik'n b;*(l)l(l)k) b]l(ll)ll (U,k) (t~](l)(r _ ,i.l)~.11~]1(11) (r q-n - 7"11~. n ]]1 111 (A.1) Explicit calculation of the matrix element (~](l)(r - T/)IHI ff-, (l,)(r + n - 1"l,)), in the framework of our model (eqs. (1)-(3)) gives a 12 X 12 matrix which consists of three 2t ~, 4 matrices, one for each direction. E.g., the matrix for the x-direction reads ]

•

2n e-ik'n(t~](l)(r --Tl)lIt]~]l(ll)(r-t- n

~la(kx)

--T/l))= kff]3(kx)

,,

I

l~3(kx) 1

/~/51 d

.

,

l,l 1= 1, 2,

.

(A.2)

where

• Io(kx) =

37/3(kx) =

~K(o)

J(l + e-ikxa)]

[_J(1 + eikxa)

K(o)

I°

J3(1- e-V'xa)3

J3(1 - e ikxa)

=

M~I

0

.

K(51 )

,

(A.3)

,

(A.4)

(A.5)

O. Entin- Wohlman/Density of states and superconducting transition temperature

391

It is convenient to apply the unitary transformation

-~2[~

;]'

0=[:

l iI '

(A.6)

which transforms (A.2) into

~a(kx)

lCl3(kx)] ,

(A.7)

with

=

Ma(kx)

~K(°) + J( l + c°s kxa)

J sin kxa

LJ sin kxa

K(o) - J(1 + cos kxa )

Fl - cos kxa

-sin kx a

L - sin kxa

- 1 + cos

1

,

(A.8a)

]

J¢43(kx) = J3

•

(A.8b)

lCxaA

We now investigate the properties of the eigenvectors of (A.7). The four-dimensional eigenvector can be written as [ ~ ] , where a and/3 are two.dimensional vectors. Then we find, from (A.7), that 1

/ 3 - - -

x - K(~ 1)

~3 a,

(A.9)

where X is the eigenvalue. Noting that 2143 • ~3 = 2J2(1 - cos kxa),

(A.10)

(proportional to the unit matrix) we obtain - + [ 2 J 2 ( 1 - c ° s kxa) ] ot+Otv, +[3+13v, -ava v, 1 + (Xv -~-l~(--~d_~-(61) ) .

(A.11)

This quantity should be equal to gin,,. It therefore follows that ot+otv, is equal to 6w, , times a quantity which depends on kv and the eigenvalue. It can be easily seen that the same conclusion holds for the eigenvectors of (A.2), since d(eq. (A.6)) is orthogonal. This proves relation (1 la). Next we calculate otv(3 + u. Using (A.9), we find. +

J3

otv[3v = ~

((1 -- cos kxa)(ot 2 - off) - 2CZla2 sin kxa),

- K(,1)

where CZl,cz2 are the two components of the vector cz. From eqs. (A.7) and (A.9) we obtain

(A.12)

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O. En tin- Wohlman/Density of states and superconducting transition temperature

K(o)+J(1 +coskxa )+

~2 = - a l J s i n k x a

X v - K ( 6 1 ) ...... Xv .

(A.13)

Inserting this into (A.12) and using the secular equation of (A.7),

1 J sin kxa

×

IK

[K

J sin kxa

2J~O-cosk~a)

(o) - J ( l + cos kxa ) +

(o)+J(1 +coskxa)+

2J2(1 - cos kxa )

-X,

Xu - K(61)

]

] - Xv

=0,

(A.14)

we find that a+13u = O. This proves relation (1 lb).

Appendix B

The calculation o f the matrix elements o f H in the MTBA We consider here the matrix element

(lqxJ(~Jl(ll)(r -- Rn311) [HI ~j(l)(r -- Rn4l))lOqx) ,

(B. 1)

which appears in eq. (17). Here Rnl denotes the instantaneous position of the ion, and H is the electron hamiltonian

(B.2)

H = T e + ~ V(r - Rnl),

nl where T e denotes the electron kinetic energy (contributions from the kinetic energy of the ions, which do arise in the MTBA, are negligible [12] ). We introduce the expansion to first order in the ion displacements, (B.3)

Rnl "-*n + r l + Unl = Rnl + Unl , into eq. (B.1) and separate the result into three parts. The first is

6(Rn31,- Rn4l) ~ ~', c~ mi

× <~Oh(q)(r-

gn~l)lV(r

(

°)

U sn311 - + Umi ORn311 OY~i

- gmi)le//(lO(r

-

Rnaq)h

(B.4)

where a = x, y , z and the prime denotes that the term Rmi = R n , is excluded from the sum. Eq. (B.4) represents the contribution of the one-center integrals (so-called crystal field'parameters) to the electron-phonon matrix element. In our model these imply 11 = 1" We thus obtain from (B.4) that this contribution is t

8JhS(Rn311 _ Rn4t) ~ a

~ (U~3q _ U~i) 3 K)I)(Rn3ll - Rmi ). mi 0(R~311 -- Rmi)

(B.5)

393

O. Entin-lCohlman/Density of states and superconducting transition temperature The second contribution to eq. (B.1) arises from the two-center integrals and is

a U~

"3ll 8R~ l + Un41 -~n41)(~g/l
E

a

a(Rn3l, - R~aI) J:2)(Rn31'

c~

(B.6)

Rn4t)"

Here we have neglected derivatives of overlap integrals, arising from

(~]]l(ll)(r- Rn311)lT e + V(r - Rn41)l~/
(B.7)

This point was discussed in ref. (10) where we have argued that, at least partly, such contributions are accounted for by suitable parametrization of the two-center integrals. In our model we keep only the intra-band two-center integrals, therefore, j/(2) = 6/1/j/(2)" The third contribution to eq. (B.1) comes from three-center integrals. This contribution is

(

E E l : hall aRn I + Un41 c~ a a mi 31 aRn4l =E E

a mi

O~

(Un3ll

--

Ot

+

) (~fl(ll)(r_ Rn311)lV(r _ Rmi)l~i(1)(r_ Rn41))

a

(t~/l(ll)(r -- Rnatl)l V(r - Rmi)l~/(l)(r - Rn41))

m

Umi) OR~ l q- ( n4 l -- Umi ) 31

a .~

Ol

mi

3h

m

Umz. ) a

'~ q-(Un4 ' - Umi ) i)(R~4/ -__Rmi)]d 3~j311)"(Rn3l I - Rmi , Rn 4' - Rmi),

3/1 - Rrni)

(B.8) where the prime denotes that the terms Rmi

= Rn311 or Rnfl are excluded from the sum. Using

(lqxlU~llOqx)= eiq'Rnt eC'(q) g2NMh--x(q) ,

(B.9)

we finally obtain

(lqhl(~/l('l) (r -- Rn3'l)]l-~](l) (r - Rn4l))lOqx )= E

×

iq" Rnah _ eiq "Rn41 )

e~(q)N~'2N:(.o~.(q)

B

a(R~3ll - R~4l) J/2)(Rn3ll

+ Er mi E~(gn311 -- Rn4l)(eiq'Rn3ll -- elq " "Rmt)"

-

RnM ) 5/11

a(Rn3l 1 _ Rmi) at Kj(1)(Rn31t -- Rmi) ~J]l

O. Entin-Wohlman/Density of states and superconducting transition temperature

394

+

[(eiq'Rn3ll_ eiq'Rmi) a(Rn3/1

-

Rmi )

4" (elq'Rn41 -- e iq "Rmi) O(R~41 _ R~ni) (B.IO)

Inspection of eq. (B.10) reveals that the matrix element can be written as e iq "Rn311 times a function which depends on n 3 - n4, l, ll, j and Jl- Moreover, from the assumptions of our model, eqs. (1)-(3), we see that the value of l uniquely determines l 1 . That is, consider for example the case l = 1, i.e., r l = ½[~a+ ~[a (vanadium atom on the x-direction chain). In the case of K (1) we have l 1 = l. For j ( 2 ) w e have l 1 = 2, i.e., rll = ~[~a - ~[a, and for j(3) we again have l I = 2 (since in that case we also consider two nearest-neighbour V atoms on the same chain). The same situation occurs for the other two chains, in the y- and z-direction. Thus, the electron-phonon matrix element in our model gives the same result for any of the three directions, namely it is independent of l.

Appendix C

Calculation of the derivatives Here we outline the evaluation of the derivatives which appear in eq. (19), and from them the quantities Ill 2 of eq. (20). The derivatives of the two-center integrals are found in the way described in ref. [7]. We have the quantity (see eq. (19))

~(n' + "Q1 + 6(n'

-

-

TI) c~

J/(2)(n' + rll - rl) = [6(n')(6116112 + 612~111)

[a)~l12~ll q- ~(n'

+ :a)6111512] ~

j/(2)(pX),

px = ½[a.

(C.1)

From ref. [9] we have

B

j~2)(px) = J(o) = ~ e-QX(2 + Qx + ~(Qx)2),

x = ½a,

(C.2)

where B and Q are parameters which are found by fitting to an APW calculation of the band structure [13]. We introduce the notation J ' = - ~4 e-QX(6 + 4Qx + ~(Qx) 2 + ½(Qx)3),

x =

½a.

Then, when the derivatives of the one-center integrals are not taken into account, we obtain (see eq. (20)) 111112 = (j,)2 ~ ](1 - e ' iq. (n'+rll- *t))6(n') + 6(n' - ~))12811~ll 2 ~ 4(J') 2.

(C.3)

H'

We have neglected the terms which depend on q since (see eqs. (18) and (21)), they lead to ~qxeiq'nDx(qw)h/ 2NMcoh(q) =~L(nw), which is neglected compared with L(w) [6], (i.e., L (0w)).

O. Entin-Wohlman/Density o f states and superconducting transition temperature

395

Next we consider the three-center integrals. We use for them the approximate form [13] e -Qldll e -QId2' 3 3d2x - d 2 d2y - d2z jj~3) = j3(a61 ) = 7SN/~"G - [dl[ [d2[ 8 d2 d2

(C.4)

where G is a parameter found by fitting to an APW calculation, and d 1 = ~ia + ½fa + ¼fca,

(c.5)

d 2 = - ~ [ a + ½/a + ~f~a,

for the three-center integral relating two-nearest-neighbours V atoms on the x-direction chain with a potential centered on a V atom on the z-direction chain. Similarly, when the potential is centered on a V atom on the ydirection chain, we have d I ~ d3, d 2 ~ d4, with d 3 = ~lia + ~/a + ½/ca,

(C.6)

d 4 = - ~ i a + ~fa + ½/ca.

From eq. (C.4) we can thus calculate the derivatives according to the components of the vectors d. We insert them into the last term in eq. (19), and then calculate the last term in eq. (20), again keeping only the terms which do not depend on q. The final result is

(~)

2

11212=

16X/~ 32

9

+4 (Qa)2 Qa

(C.7) ,

where J3 is [3] (see eq. (C.4))

(c,8)

J3 = - 7 5 x / ~ G e-Qax/'6/2/a2.

The parameters used in the computation are given in table C.I. It should be noted that the variation o f J 3 in our model means variation of G (the parameter describing the strength of the potential on next-nearest neighbour [13] ). Changing J3 from 0.001 to 0.03, changes - G from 0.1016 (a.u.) to 3.047 (a.u.). These values are in accordance with those fitted to APW calculations [13]. Table C.I The parameters used in the computation of the band structure, the density of states and (N(0)I2)eff (from refs. [3, 1 3] ) K(a)

K(81)

0.1883(Ry)

0.2944(Ry)

J(a)

-0.080(Ry)

a

Q

4.722 A

0.467(a.u.)

B

-10.344(a.u.)

References [1] F. J. Morin and J. P. Maita, Phys. Rev. 129 (1963) 1115. [2] J. Labbe and J. Friedel, J. Physique 27 (1966) 153, 303. [3] M. Weger and I. B. Goldberg, Solid State Phys. 28, H. Eherenreich, D. Turnbull and F. Seitz, ed. (Academic Press, New York, 1973) p. 1. [4] A. T. Van Kessel, H. W. Myron and F. M. Mueller, Phys. Rev. Lett. 41 (1978) 181; B. M. Klein, L. L. Boyer, D. A. Papaconstantopoulos and L. F. Mattlieis, Phys. Rev. B18 (1978) 6411. [5] W. L. McMilliam, Phys. Rev. 167 (1968) 441. [6] A. Birnboim and H. Gutfreund, Phys. Rev. B9 (1974) 139.

396

O. Entin- Wohlman/Density of states and superconducting transition temperature

[7] A. Birnboim and H. Gutfreund, Phys. Rev. B12 (1975) 2682. [8] J. Appel and W. Kohn, Phys. Rev. B4 (1971) 2162. [9] J. Appel and W. Kohn, Phys. Rev. B5 (1972) 1823. [10] O. Entin-Wohlman, Solid State Comm. 34 (1980) 879. [ 11 ] S. Alterovitz, to be published. [12] J. Ashkenazi, M. Dacorogna and M. Peter, Solid State Comm. 29 (1979) 181. [13] J. Ashkenazi and M. Weger, J. Phys. Chem. Solids 33 (1972) 631.