- Email: [email protected]
Importance of hybrid pairs in superconductors
0038-1098/86 $3.00 + .00 Pergamon Press Ltd.
Solid State Communications, Vol. 57, No. 10, pp. 825-828, 1986. Printed in Great Britain.
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS O.L.T. de Menezes Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, 22290.- Rio de Janeiro, Brasil
(Received 18 November 1985 by B. Miihlschlegel) The role of hybrid pairs in a system of two hybridized bands in the presence of electron-phonon interaction is studied. In the superconducting regime we can obtain a general gap equation for the Cooper pairs as function of the hybridization and of the ratio between the bandwidths. The importance of the interband pairs compared to the intraband pairs is discussed.
IT HAS RECENTLY been discussed that unusual behaviolars exhibited by some heavy fermion systems are mainly due to the effect of the one electron hybridization between states with quite different degrees of localization [1 ]. On the other hand, the importance of the electronphonon (EP) interaction has been argued in heavyfermions, mixed valence, and Kondo systems. Several theoretical works have dealt with the possible mechanisms responsible for the superconductivity behaviour in these systems, as well as the character of the electron pairs: singlet or triplet, intraband or interband pairing [2]. We intend in this letter to consider the effect of the EP interaction in the case of two hybridized bands with different dispersion relations and different EP coupling parameters. For the sake of simplicity, we will not consider Coulomb repulsions. This means that we will treat a non-magnetic case and that clearly a more general discussion has to take these repulsions into account. In particular we intend to obtain information on the relevance of hybrid pairs suggested previously [3]. Let us write the purely electronic part of the Hamiltonian:
e" electron scatterings (e', e " = a, b), and bq is the phonon operator, for a phonon with wavevector q and energy h~%. The pure phononic part is:
~ph = ~ hwqb~bq.
In order to decouple electrons and phonons in equation (2), one can perform a canonical transformation [4], which is approximated up to second order in the EP interaction expansion, or the displacement operator [5 ]. Another way to treat this problem is to orthogonaiize the electronic part of the Hamiltonian defining orthogonal a and/3 states, and see how the jfel--ph is modified [6]. Using the canonical transformation, ako = U k a k - Vk#ka,
bka = Z ~ k o + Uk/dka, Vk /d k
b t = E 6~atkoako + Z ekbkabka k,a
~a
=
where a~o(b~o) creates an electron with energy e~(b), wavevector k and spin a in the a(b) band. Vk is the one electron k-dependent hybridization. The EP part of the Hamiltonian can be considered in a generalized form, which in short notation reads:
E
(5)
[(e~t - - E ~ ) 2 --~ VIii 1 / 2 '
2
(1)
k, o"
e', e" k, q, (3"
[(ff~ - - E ~ ) 2 q- V 2 ] 1/2'
and
+ Z Vk(atkabko + h.c.),
ae'-." = Z
(4)
where
•"Ok = ~el
(3)
q
t ,t ,, b_a)ek+qoeko.
(2)
Here g~e,eq,, is the EP coupling parameter involving e' and 825
(+)
+ V
,
(6)
one obtains a new electronic Hamiltonian:
j£el = Z E~tatkaO~ka+ Z E~tko~ko ' ka
(7)
k,a
with ak(/dk) denoting the lower (upper) states in energy. Applying the same transformation [equation (5)], to the EP part of the Hamiltonian [equation (2)] one gets:
826
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS
~e~--ph =
E
k ' ~ (b Z k q c* "Ye'e" q
+ -Dt - q )~C k' +t q
" c~eko~
(8)
Vol. 57, No. 10
The energy E i = ( t ~ i l H l t ~ i ) (i=o~a, /3/~, and a/3) can be minimized, and one can get the possible solutions through a gap equation:
with e' and e" now representing ~ and/3 electrons. The new EP coupling are defined as: kq
:
+ gkaqu¢+ qVk + g £ q v ¢ +qUk, Tk~q
×
:
__g~q/./~+qVk +
=
k q * gb'b Vl~+qb/k
k,q * k,q * - - g b a b/k+q/Ak + g a b Vk+q ~k'
--g~qZ~+qUk + g bk,q . + qVk b L/k
'~'~aq :
k'qv* 7.) __ ~ k , q , , * //. +gab k+q k gab U k + q K"
(9)
After some algebra it is possible to obtain the renormalized Hamiltonian. We will consider here only terms which contribute to the formation of two-particle pair states. The effective electron-electron attractive interactions are:
1Dk, qat ~o + t/3~ P - k - q , - f f P - k
k,q
r~.
:
#4#
1
r,k,q a3 :
+
(10)
-k,-q k,q %~e~ 733 /hO~q,
and (11)
We will adopt here a very simple approach [7]. We will analyze the presence of bound states for electrons pairs (Cooper-like pairs). Since a and/3 states are orthogonal the solutions are easier to be obtained. Three possible "basis states" can be considered it one applies creation operators of two electrons for k > kF (wavevector at the Fermi level): ~
:
-
gN/~bbUk+q)2
~ x / ~ O ~,
2 2 2 . gaagbb(Uk+q -- Z~+ q ) ( U k - ~k)]
(14)
Let us transform the a and/3 states back to a and b states. One gets, as expected and recently [ 7 ] assumed, pairs functions in terms of linear contributions of intraband pairs (aa and bb pairs) and interband (ab pairs). This yields
e : ~ and/3,
p,,k,q -k,-q k,q c~3 ---- 7c~3 73 a /hcoq.
(x/~v~-
-
2 [(g __gbb)2Uk+qVk+qUkVk rk'# = h%
with
for
(gv'~/~k+,~ + g4~-gVk+~) ~
F ~ q = hoo--~(~r~-aaVk+q
+ p tk, qr~" t ~ a ~ P- k - q , - 0"~- k, - o-O~k+ q, ffO~k, o
r ~ '~ = ~ ' - % ~ 2 / h %
1 hCOq
x
o P k + q oPk ff
_{_ p ' k , q~,~ r~ f4~ ~, l xO~fl ~ - k - q, OP- k, - oPk÷ q, o'~tk, o] ,
(13)
where ?~ is the bound state energy of the pair. Equation (13) gives, as expected, three solutions: a a pairs, /33 pairs, and the hybrid a3 pairs. For simplicity, and since no detailed calculations are available for the EP coupling parameters, one adopts a reasonable assumption: g2 b = g 2 ~ g~agbb with g's constants and equation (11) can be simplified.
E [L~oeoe r ~ k , q t^Xt - k - q , - o O ~ - k , - t T O g kt+ q , o O ~ k , o k,q,o
~Cr = - -
= o,
k,q ~k
__ ~ k , q ~ , ,, __ k , q , , , o, ,Sba Vk+qt~k 5 a b t~k+qVk, k,q 7a{3
- 2~.+u~_x
k,q , k,q , gaa ~ + q ~ k + g b b Hk+qHk
~ac~ =
E
k
aa ~ ,* * A k [ U ~ 2 ~ k + Z~ 2 ~ kbb + ~ : ~ k U k c k
ab
1,
, , ab Ct~t~ = E Bk[~l~ 2 @~a at_ /./~2 e b b - - N//~-UkVkl//k ] , k and
~
=
k
+ ( u f - v~2) ~ ' 1 .
(15)
where
E A k[ t~kt t Oe_ tk,) , k
¢~b = Ib*k~b-*k,), k
¢Ib
and ~b~3 = ~ Ck atktP-k+°t -- a~k+3*k*). k
(12)
1 V ~ l a'k* b*_k~ - a*_k~ b*k~>,
are the singlet intraband and interband pairs.
(16)
Vol. 57, No. 10
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS
I(llk 12
827
I(~k I= I0
I.(
•"° ", ..
0.8 0.6
l /
;
0
.
/
/1
// 04
4
I/" "\
O.2 O
0
I/2
I
it
i/z
k/kM
i
k/kN
Fig. I. Probability amplitudes I~kl 2, in the @aa states, are plotted against k/kM. For I ~ l ~ . . . , for I¢~bl e , and for 14~b]~ , withA = 0 . 0 1 and V = 0.1.
Fig. 2. Probability amplitudes 14)kl2, in the ~a~ state, are plotted against k/kM. For I ~ 1 = = 14)~bl2 . . . . . . , and for l¢~bl 2 , with A = 0.01 and V = 0.1.
The role of hybrid pairs, ~ b , was recently [8] considered as less important that the other pairings, when localized states are in the presence of conduction states, in agreement with [9]. We want to discuss here, using a simple argument, that if hybridization between two band states is important the presence of hybrid pairs should not be negligible. If the states in equation (15) are assumed to be normalized the relative amplitudes, ICk[2, for pair states in the aa pair wave function ~ba~ are:
however, are stabilized, only in a short range of the parameters. In a limiting case, A = 1 and h ( . O D e b y e = 0.02 eV, a/~ states are to be considered only tor very low hybridization (V < 10 -2 eV). It is to be noted that, due to the hybridization effect, even if one considers EP coupling only in one of the band states (eg. gaa d=O, gbb =gab = 0), the interband and the intraband pairs can be stabilized. However, the contribution of each pair is very sensitive to the parameters chosen, as well as to the k-dependent hybridization, supposed in the above discussion to be constant. We conclude that the effect of hybrid pairs can be important and should not be neglected in an adequate physical description of the problem involving superconductivity in two band systems.
[qS~al2 = [ukl n, I@~bl2 _-- IVkl4, I~bl 2 = 2[VklZlUk[2.
(17)
In f f ~ , the probability amplitudes are obtained changing a -~ b in equation (17), and in ffa~ one has: t ~ l 2 = I¢~bl 2 = 21Ukl=lval 2, I¢~bl a --- lUkl4 + Irk] 4 -- 21uklzlVkl 2.
REFERENCES (18)
We do not intend here to calculate the energy gap. It depends on the strength of the hybridization, on the g's coupling parameters, and on the bandwidth ratio A, if the two band ( a , b ) states are supposed, for simplicity, to be homothetic. Let us discuss some aspects of the probability amplitudes as function of k/kM (kM - the k maximum). Figure 1 shows ICkl2 against k/kM for the c~c~ states. For k/kM < ½, the aa pairs are the dominant ones, whereas for k/kM > ½ the bb pairs are the most relevant contribution. ~llae hybrid, ab pairs, are important for k/kM ~ ½, I@~bl2 attaining a maximum value equal to the sum of the amplitudes ot the aa and bb pairs. Again, similar considerations can be made for/~3 states, by changing a ~ b. Figure 2 shows the probability amplitudes in the ~/3 states. These,
1.
2. 3.
4.
5.
A.W. Overhauser & J. Appel, Phys. Rev. B31,193 (1985); J.L. Smith & P.S. Riseborough, J. Magn. Magn. Mat. 47 & 48, 545 (1985); J.L. Smith, Z. Fisk & S.S. Hecker, Physica 130B, 151 (1985); D.D. Koeling, B.D. Koeling & G.W. Crabtree, Phys. Rev. B31,4699 (1985). See for instance in the Proceedings of the International Conference on Valence Fluctuations, J. Magn. Magn. Mat. 47 & 48 (1985). O.L.T. de Menezes & A. Troper, Valence Instabilities, (Edited by P. Wachter & H. Boppart), p. 53, North Holland, (1982); A.J. Fedro & S.K. Sinha, ibM, p. 371. A. Avignon, F. Brouers & K.H. Bennemann, J. Physique C5, 377 (1979); F. Brouers & O.L.T. de Menezes, Phys. Status Solidi (b) 104, 541 (1981). P. Entel, H.J. Leder & N. Grewe, Z. Phys. B30, 277 (1978).
828 6. 7.
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS N. Grewe, P. Entel & H.J. Leder, Z. Phys. B30, 393 (1978). O.L.T. de Menezes, Solid State Comrnun. 56, 799 (1985).
8. 9.
Vol. 57, No. 10
P. Entel, J. Zielinski & M. Matschke, preprint (1985). E.C. Valadares, A. Troper & O.L.T. de Menezes, J. Magn. Magn. Mat. 47 & 48,400 (1985).
Solid State Communications, Vol. 57, No. 10, pp. 825-828, 1986. Printed in Great Britain.
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS O.L.T. de Menezes Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150, 22290.- Rio de Janeiro, Brasil
(Received 18 November 1985 by B. Miihlschlegel) The role of hybrid pairs in a system of two hybridized bands in the presence of electron-phonon interaction is studied. In the superconducting regime we can obtain a general gap equation for the Cooper pairs as function of the hybridization and of the ratio between the bandwidths. The importance of the interband pairs compared to the intraband pairs is discussed.
IT HAS RECENTLY been discussed that unusual behaviolars exhibited by some heavy fermion systems are mainly due to the effect of the one electron hybridization between states with quite different degrees of localization [1 ]. On the other hand, the importance of the electronphonon (EP) interaction has been argued in heavyfermions, mixed valence, and Kondo systems. Several theoretical works have dealt with the possible mechanisms responsible for the superconductivity behaviour in these systems, as well as the character of the electron pairs: singlet or triplet, intraband or interband pairing [2]. We intend in this letter to consider the effect of the EP interaction in the case of two hybridized bands with different dispersion relations and different EP coupling parameters. For the sake of simplicity, we will not consider Coulomb repulsions. This means that we will treat a non-magnetic case and that clearly a more general discussion has to take these repulsions into account. In particular we intend to obtain information on the relevance of hybrid pairs suggested previously [3]. Let us write the purely electronic part of the Hamiltonian:
e" electron scatterings (e', e " = a, b), and bq is the phonon operator, for a phonon with wavevector q and energy h~%. The pure phononic part is:
~ph = ~ hwqb~bq.
In order to decouple electrons and phonons in equation (2), one can perform a canonical transformation [4], which is approximated up to second order in the EP interaction expansion, or the displacement operator [5 ]. Another way to treat this problem is to orthogonaiize the electronic part of the Hamiltonian defining orthogonal a and/3 states, and see how the jfel--ph is modified [6]. Using the canonical transformation, ako = U k a k - Vk#ka,
bka = Z ~ k o + Uk/dka, Vk /d k
b t = E 6~atkoako + Z ekbkabka k,a
~a
=
where a~o(b~o) creates an electron with energy e~(b), wavevector k and spin a in the a(b) band. Vk is the one electron k-dependent hybridization. The EP part of the Hamiltonian can be considered in a generalized form, which in short notation reads:
E
(5)
[(e~t - - E ~ ) 2 --~ VIii 1 / 2 '
2
(1)
k, o"
e', e" k, q, (3"
[(ff~ - - E ~ ) 2 q- V 2 ] 1/2'
and
+ Z Vk(atkabko + h.c.),
ae'-." = Z
(4)
where
•"Ok = ~el
(3)
q
t ,t ,, b_a)ek+qoeko.
(2)
Here g~e,eq,, is the EP coupling parameter involving e' and 825
(+)
+ V
,
(6)
one obtains a new electronic Hamiltonian:
j£el = Z E~tatkaO~ka+ Z E~tko~ko ' ka
(7)
k,a
with ak(/dk) denoting the lower (upper) states in energy. Applying the same transformation [equation (5)], to the EP part of the Hamiltonian [equation (2)] one gets:
826
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS
~e~--ph =
E
k ' ~ (b Z k q c* "Ye'e" q
+ -Dt - q )~C k' +t q
" c~eko~
(8)
Vol. 57, No. 10
The energy E i = ( t ~ i l H l t ~ i ) (i=o~a, /3/~, and a/3) can be minimized, and one can get the possible solutions through a gap equation:
with e' and e" now representing ~ and/3 electrons. The new EP coupling are defined as: kq
:
+ gkaqu¢+ qVk + g £ q v ¢ +qUk, Tk~q
×
:
__g~q/./~+qVk +
=
k q * gb'b Vl~+qb/k
k,q * k,q * - - g b a b/k+q/Ak + g a b Vk+q ~k'
--g~qZ~+qUk + g bk,q . + qVk b L/k
'~'~aq :
k'qv* 7.) __ ~ k , q , , * //. +gab k+q k gab U k + q K"
(9)
After some algebra it is possible to obtain the renormalized Hamiltonian. We will consider here only terms which contribute to the formation of two-particle pair states. The effective electron-electron attractive interactions are:
1Dk, qat ~o + t/3~ P - k - q , - f f P - k
k,q
r~.
:
#4#
1
r,k,q a3 :
+
(10)
-k,-q k,q %~e~ 733 /hO~q,
and (11)
We will adopt here a very simple approach [7]. We will analyze the presence of bound states for electrons pairs (Cooper-like pairs). Since a and/3 states are orthogonal the solutions are easier to be obtained. Three possible "basis states" can be considered it one applies creation operators of two electrons for k > kF (wavevector at the Fermi level): ~
:
-
gN/~bbUk+q)2
~ x / ~ O ~,
2 2 2 . gaagbb(Uk+q -- Z~+ q ) ( U k - ~k)]
(14)
Let us transform the a and/3 states back to a and b states. One gets, as expected and recently [ 7 ] assumed, pairs functions in terms of linear contributions of intraband pairs (aa and bb pairs) and interband (ab pairs). This yields
e : ~ and/3,
p,,k,q -k,-q k,q c~3 ---- 7c~3 73 a /hcoq.
(x/~v~-
-
2 [(g __gbb)2Uk+qVk+qUkVk rk'# = h%
with
for
(gv'~/~k+,~ + g4~-gVk+~) ~
F ~ q = hoo--~(~r~-aaVk+q
+ p tk, qr~" t ~ a ~ P- k - q , - 0"~- k, - o-O~k+ q, ffO~k, o
r ~ '~ = ~ ' - % ~ 2 / h %
1 hCOq
x
o P k + q oPk ff
_{_ p ' k , q~,~ r~ f4~ ~, l xO~fl ~ - k - q, OP- k, - oPk÷ q, o'~tk, o] ,
(13)
where ?~ is the bound state energy of the pair. Equation (13) gives, as expected, three solutions: a a pairs, /33 pairs, and the hybrid a3 pairs. For simplicity, and since no detailed calculations are available for the EP coupling parameters, one adopts a reasonable assumption: g2 b = g 2 ~ g~agbb with g's constants and equation (11) can be simplified.
E [L~oeoe r ~ k , q t^Xt - k - q , - o O ~ - k , - t T O g kt+ q , o O ~ k , o k,q,o
~Cr = - -
= o,
k,q ~k
__ ~ k , q ~ , ,, __ k , q , , , o, ,Sba Vk+qt~k 5 a b t~k+qVk, k,q 7a{3
- 2~.+u~_x
k,q , k,q , gaa ~ + q ~ k + g b b Hk+qHk
~ac~ =
E
k
aa ~ ,* * A k [ U ~ 2 ~ k + Z~ 2 ~ kbb + ~ : ~ k U k c k
ab
1,
, , ab Ct~t~ = E Bk[~l~ 2 @~a at_ /./~2 e b b - - N//~-UkVkl//k ] , k and
~
=
k
+ ( u f - v~2) ~ ' 1 .
(15)
where
E A k[ t~kt t Oe_ tk,) , k
¢~b = Ib*k~b-*k,), k
¢Ib
and ~b~3 = ~ Ck atktP-k+°t -- a~k+3*k*). k
(12)
1 V ~ l a'k* b*_k~ - a*_k~ b*k~>,
are the singlet intraband and interband pairs.
(16)
Vol. 57, No. 10
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS
I(llk 12
827
I(~k I= I0
I.(
•"° ", ..
0.8 0.6
l /
;
0
.
/
/1
// 04
4
I/" "\
O.2 O
0
I/2
I
it
i/z
k/kM
i
k/kN
Fig. I. Probability amplitudes I~kl 2, in the @aa states, are plotted against k/kM. For I ~ l ~ . . . , for I¢~bl e , and for 14~b]~ , withA = 0 . 0 1 and V = 0.1.
Fig. 2. Probability amplitudes 14)kl2, in the ~a~ state, are plotted against k/kM. For I ~ 1 = = 14)~bl2 . . . . . . , and for l¢~bl 2 , with A = 0.01 and V = 0.1.
The role of hybrid pairs, ~ b , was recently [8] considered as less important that the other pairings, when localized states are in the presence of conduction states, in agreement with [9]. We want to discuss here, using a simple argument, that if hybridization between two band states is important the presence of hybrid pairs should not be negligible. If the states in equation (15) are assumed to be normalized the relative amplitudes, ICk[2, for pair states in the aa pair wave function ~ba~ are:
however, are stabilized, only in a short range of the parameters. In a limiting case, A = 1 and h ( . O D e b y e = 0.02 eV, a/~ states are to be considered only tor very low hybridization (V < 10 -2 eV). It is to be noted that, due to the hybridization effect, even if one considers EP coupling only in one of the band states (eg. gaa d=O, gbb =gab = 0), the interband and the intraband pairs can be stabilized. However, the contribution of each pair is very sensitive to the parameters chosen, as well as to the k-dependent hybridization, supposed in the above discussion to be constant. We conclude that the effect of hybrid pairs can be important and should not be neglected in an adequate physical description of the problem involving superconductivity in two band systems.
[qS~al2 = [ukl n, I@~bl2 _-- IVkl4, I~bl 2 = 2[VklZlUk[2.
(17)
In f f ~ , the probability amplitudes are obtained changing a -~ b in equation (17), and in ffa~ one has: t ~ l 2 = I¢~bl 2 = 21Ukl=lval 2, I¢~bl a --- lUkl4 + Irk] 4 -- 21uklzlVkl 2.
REFERENCES (18)
We do not intend here to calculate the energy gap. It depends on the strength of the hybridization, on the g's coupling parameters, and on the bandwidth ratio A, if the two band ( a , b ) states are supposed, for simplicity, to be homothetic. Let us discuss some aspects of the probability amplitudes as function of k/kM (kM - the k maximum). Figure 1 shows ICkl2 against k/kM for the c~c~ states. For k/kM < ½, the aa pairs are the dominant ones, whereas for k/kM > ½ the bb pairs are the most relevant contribution. ~llae hybrid, ab pairs, are important for k/kM ~ ½, I@~bl2 attaining a maximum value equal to the sum of the amplitudes ot the aa and bb pairs. Again, similar considerations can be made for/~3 states, by changing a ~ b. Figure 2 shows the probability amplitudes in the ~/3 states. These,
1.
2. 3.
4.
5.
A.W. Overhauser & J. Appel, Phys. Rev. B31,193 (1985); J.L. Smith & P.S. Riseborough, J. Magn. Magn. Mat. 47 & 48, 545 (1985); J.L. Smith, Z. Fisk & S.S. Hecker, Physica 130B, 151 (1985); D.D. Koeling, B.D. Koeling & G.W. Crabtree, Phys. Rev. B31,4699 (1985). See for instance in the Proceedings of the International Conference on Valence Fluctuations, J. Magn. Magn. Mat. 47 & 48 (1985). O.L.T. de Menezes & A. Troper, Valence Instabilities, (Edited by P. Wachter & H. Boppart), p. 53, North Holland, (1982); A.J. Fedro & S.K. Sinha, ibM, p. 371. A. Avignon, F. Brouers & K.H. Bennemann, J. Physique C5, 377 (1979); F. Brouers & O.L.T. de Menezes, Phys. Status Solidi (b) 104, 541 (1981). P. Entel, H.J. Leder & N. Grewe, Z. Phys. B30, 277 (1978).
828 6. 7.
IMPORTANCE OF HYBRID PAIRS IN SUPERCONDUCTORS N. Grewe, P. Entel & H.J. Leder, Z. Phys. B30, 393 (1978). O.L.T. de Menezes, Solid State Comrnun. 56, 799 (1985).
8. 9.
Vol. 57, No. 10
P. Entel, J. Zielinski & M. Matschke, preprint (1985). E.C. Valadares, A. Troper & O.L.T. de Menezes, J. Magn. Magn. Mat. 47 & 48,400 (1985).