Brazilian Journal of Physics, vol. 33, no. 4, December, 2003 713

A Two Band Model for Superconductivity:
Probing Interband Pair Formation

R. E. Lagosa) and G. G. Cabrerab)

a)Departamento de F´ısica, IGCE, Universidade Estadual Paulista (UNESP)

CP. 178, 13500-970, Rio Claro, SP, Brazil
b)Instituto de F´ısica ‘Gleb Wataghin’, Universidade Estadual de Campinas (UNICAMP)

CP. 6165 Campinas, SP 13083-970 Brazil

Received on 23 May, 2003

We propose a two band model for superconductivity. It turns out that the simplest nontrivial case considers
solely interband scattering, and both bands can be modeled as symmetric (around the Fermi level) and flat, thus
each band is completely characterized by its half-band width Wn (n=1,2). A useful dimensionless parameter is
δ, proportional to W2−W1. The case δ = 0 retrieves the conventional BCS model. We probe the specific heat,
the ratio gap over critical temperature, the thermodynamic critical field and tunneling conductance as functions
of δ and temperature (from zero to Tc). We compare our results with experimental results for MgB2 and
good quantitative agreement is obtained, indicating the relevance of interband coupling. Work in progress also
considers the inclusion of band hybridization and general interband as well as intra-band scattering mechanisms.

1 Introduction

Magnesium Diboride (MgB2) appears to be a rather “un-
conventional” conventional superconductor [1, 2]. Two
band effects observed as deviations of conventional BCS in-
clude: anomalous specific heat [3] and two gaps features (in-
cluding double peaked tunneling conductance spectra) [4-
10]. The superconductive mechanism, nevertheless seems
to be conventional phonon BCS-like [11]. In this short com-
munication we present a two band model based on the clas-
sical work by Suhl et al. [12] and on an extension of the
latter applied to high Tc compounds [13]. We mention other
multiband models in the literature [14-21], and some cal-
culations and fittings within a multiband and strong cou-
pling context include Ref. [22-25]. In section II we intro-
duce a two band model [12, 13] and within the usual BCS
scheme we compute the mean field expressions for the free
energy, entropy, critical field, conductance, and the selfcon-
sistent equations for the gaps functions. In particular we
consider the simplest case: solely interband pairing cou-
pling via phonons. In section III we compare our simple
model with some experimental results for the case of MgB2

[26, 1, 27], indicating that the interband pairing mechanism
is somehow relevant. Finally in section IV we present some
concluding remarks and future work.

2 The two band model

Our model follows Ref. [12, 13], with the Hamiltonian

H =
∑
k,m

Ek,m

(
c†k,mck,m − c−k,mc

†
−k,m

)

− 1
N

∑
kq,m

Vn,mc
†
k,nc

†
−k,nc−q,mcq,m (1)

where the c†k’s are the usual creation operators,Ek,m are the
bands dispersion (m = 1, 2), Vn,m are the positive pairing
coefficients (V12 = V21 and D = V11V22 − V 2

12 �= 0). We
have defined k = (k, ↑), −k = (−k, ↓), N is the number of
sites and the last summation is with the usual energy cutoff
ωD. The order parameters ∆n are defined as the expectation
values

∆n =
1
N

∑
k,m

Vn,m

〈
c†k,mc

†
−k,m

〉

The effective Hamiltonian is given by (within the
Hartree Fock scheme for anomalous pairing, see Ref. [13])

Heff = NE0 +
∑
k,m

Ψ†
k,m (Ek,mσz − ∆mσx)Ψk,m

where



714 R. E. Lagos and G. G. Cabrera

E0 =
1
D

(
V22∆2

1 + V11∆2
2 − 2V12∆1∆2

)
, Ψk,m ≡

(
ck,m

c†−k,m

)

�

and σx, σz are the usual Pauli matrices. The free energy per
site F is given by

exp (−βNF ) = Tr exp
(−βHeff

)

F = E0 +
T

N

∑
k,m

ln fk,m(1 − fk,m)

where f(ω) = (exp(βω) + 1)−1, ωk,m =
√
E2

k,m + ∆2
m

and fk,m = f(ωk,m). The relative free energy δF =
F − F (∆1 = ∆2 = 0), the thermodynamic critical field

Hc, entropy (per site) and specific heat are given, respec-
tively by

δF (T ) = E0 − T

N

∑
k,m

ln
(1 + coshβωk,m)
(1 + coshβEk,m)

= − 1
8π

H2
c

(2)

S = − 2
N

∑
k,m

((1 − fk,m) ln(1 − fk,m) + fk,m ln fk,m)

(3)

�
�

CV = T

(
∂S

∂T

)
V

=
2β2

N

∑
k,m

fk,m(1 − fk,m)
(
ω2

k,m +
1
2
β
∂∆2

m

∂β

)
(4)

The condensation energy is given by

δF (T = 0) = WC = E0 − 1
N

∑
k,m

(ωk,m − Ek,m) (5)

and the superconductor- normal tunneling differential conductance (conveniently scaled) is defined by

G(V ) = −
∑
m

∫
dερm,S(ε)

∂f(ε+ V )
∂ε

(6)

ρm,S(ε) = ρm

(
sign(ε)

√
ε2 − ∆2

m

)
Real

(√
(ε + iΓ)2

(ε+ iΓ)2 − ∆2
m

)
, Γ → 0+

Minimization of the free energy with respect to the gaps functions, yields a coupled nonlinear system of integral equations
for the gaps, to be solved selfconsistently, and given by

(V22 −DR1(∆1, T ))∆1 − V12∆2 = 0
(7)

−V12∆1 + (V11 −DR2(∆2, T ))∆2 = 0

where

Rm(∆m, T ) =
∫ +ωD

−ωD

dερm(ε)S
(√

ε2 + ∆2
m

)
, S(x) =

1
2x

tanh
( x

2T

)

�

and with ρm(ε) the density of states associated to the respec-
tive band. The transition temperature is the highest temper-
ature Tc = β−1

c , solution of

(V22 −DR1(0, Tc)) (V11 −DR2(0, Tc)) = V 2
12

3 Results

We compute the observables presented in the previous sec-
tion. In particular we consider only interband scattering



Brazilian Journal of Physics, vol. 33, no. 4, December, 2003 715

V11 = V22 = 0, V12 = λ, the simplest relevant case
[12, 13]. We consider two flat symmetric bands, with
ρm(ε) ≡ ρm(0) = ρm.

The gaps equations (7) now read

∆m = λρn∆nR(∆n, T ), n �= m = 1, 2

R(∆, T ) =
∫ ωD

0

dε√
ε2 + ∆2

tanh
(
β

2

√
ε2 + ∆2

)

At zero temperature the gaps equations are given by
(in conveniente units)

φ1 =
2∆1(T = 0)

3.53Tc
= exp

1
ξ

(1 − a)

(8)

φ2 =
2∆1(T = 0)

3.53Tc
= exp

1
ξ

(
1 − 1

a

)

where ξ2 = λ2ρ1ρ2 and a satisfies a selfconsistent equation.
An excellent approximate solution is given by

ln a =
ξθ

2 − θ
, θ = ln

√
1 + δ

1 − δ

with

−1 < δ =
ρ1 − ρ2

ρ1 + ρ2
< 1

Notice that all the above mentioned observables will
yield the standard BCS expressions [28] in the limit δ = 0.

The critical temperature is given by Tc =
1.13ωD exp(−ξ−1).

We label the bands such that δ > 0. If we consider
MgB2, from Ref. [1, 27] we have Tc 
 40 0K , ωD 

8000K yielding ξ 
 0.32. From Ref. [26] we approx-
imate W1 ≈ 5.6eV (ρ1 ≈ 0.179eV −1), W2 ≈ 14.eV
(ρ2 ≈ 0.071eV −1) yielding δ ≈ 0.432.

In Fig. 1 we plot the normalized gaps φm at zero tem-
perature, Eq.(8), and (minus) the condensation energy −Wc

Eq.(5), both as function of δ. The condensation energy is
normalized to the BCS reference state i.e. Wc = δF (δ, T =
0)/δF (δ = 0, T = 0) (see Eq.2). The chosen normalization
yields the standard BCS (weak coupling) value of unity for
the gaps and the condensation energy. As δ is varied away
from zero the condensation energy is less than the standard
BCS. One gap will depart from weak to ‘a medium coupling
regime’ (φ2 > 1), conversely the other gap will dive towards
‘a less than weak coupling regime’ (φ1 < 1), with the geo-
metrical average

√
φ1(δ)φ2(δ) ≡ 1 always in the standard

weak coupling regime. These features seem consistent as
we fit the parameter δ with experimental data [1]-[10].

In order to solve for the gaps, Eq.(7), we can use the
available low temperature and near the critical temperature
expansions [28]. These allow us to nicely interpolate, for
the full temperature regime 0 ≤ τ = T/Tc ≤ 1. Once this

is done we can readily compute the specific heat, Eq.(4), en-
tropy, Eq.(3), and the thermodynamic critical field, Eq.(2).

0.0 0.2 0.4 0.6 0.8
0.0

0.5

1.0

1.5

2.0

-w
C

φ
2

φ
1

δ

Figure 1. Gaps φm at zero temperature and (minus) the condensa-
tion energy −Wc versus δ. Convenient units φm = 2∆0

m/3.53Tc

and Wc = δF (δ, T = 0)/δF (δ = 0, T = 0). See text.

In Fig. 2 we plot the specific heat CV (normalized to
the normal state value at Tc) versus the temperature τ for
several values of δ. The standard BCS result is represented
by the curve δ = 0. The anomalous behavior of CV consists
in going under the BCS value in the region 0.5 < τ < 1,
and going over the BCS value in the region 0 < τ < 0.5.
This feature is in very good agreement with Ref. [3]. In
Fig. 3 we plot the entropy S (normalized to the normal state
value at Tc) versus the temperature τ for several values of δ.
The standard BCS result is represented by the curve δ = 0.
As δ departs from zero (bands are less ‘identical’) the sys-
tem increases its entropy. In Fig. 4 we plot the thermody-
namic critical field (normalized to the reference state δ = 0,
T = 0) versus the temperature τ for several values of δ. The
standard BCS result is again represented by the curve δ = 0.
As δ increases the critical field is reduced when compared
to the BCS value. This is in agreement with experimental
results [3].

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0

0.5

1.0

1.5

2.0

2.5

δ=0.7
δ=0.6

δ=0.4

δ=0.2
δ=0.0

CV

τ

Figure 2. Specific heat CV versus τ = T/Tc for several δ val-
ues; normalized to the normal state specific heat at Tc: Cn(Tc) =
4π2(ρ1 + ρ2)/6. See text.



716 R. E. Lagos and G. G. Cabrera

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0

0.2

0.4

0.6

0.8

1.0

0.7
0.6
0.4
0.2
0.0

δ=

S

τ

Figure 3. Entropy S versus τ = T/Tc for several δ values; nor-
malized to the normal state entropy at Tc: Sn(Tc) = 4π2(ρ1 +
ρ2)Tc/6. See text.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0

0.2

0.4

0.6

0.8

1.0

0.0
0.2
0.4
0.6
0.7

δ=

Hc

τ
Figure 4. Thermodynamic critical field Hc versus τ = T/Tc for
several δ values; normalized to the reference (BCS) state T = 0,
δ = 0, H2

c (τ ) = F (τ, δ)/F (0, 0). See text.

-20 -16 -12 -8 -4 0 4 8 12 16 20
0

1

2

3

τ=0.5

τ=0.1

τ=0.2

τ=0.0δ=0.5

G(V)

V(meV)
Figure 5. Tunneling conductance for several temperatures, at
δ = 0.5, versus applied voltage. A small dispersion is included
Γ = 0.1 meV [22]. See text.

In Fig. 5 we plot the conductance, Eq.(6) versus applied
voltage, for a fixed value of δ = 0.5, and for several tem-
peratures, and where a small dispersion is included, Γ = 0.1

meV [22].The double peaked form is in very good agree-
ment with observations (see for example Ref. [7]).

4 Concluding Remarks

We presented the simplest relevant two band model for
superconductivity, based on a standard BCS-like pairing
mechanism. We computed the gaps equations at zero tem-
perature. Also the specific heat, entropy, critical field and
conductance as function of temperature. We considered the
simplest interband scattering mechanism (one pairing pa-
rameter) and two planar symmetrical bands (one parame-
ter band model). Our results seems to be in very good
agreement with some experimental results on the compound
MgB2, indicating that interband pairing is somehow rele-
vant for this compound. These results are being investigated
further. Work in progress incorporates intraband pairing
mechanisms, an hybrid-like interband pairing mechanism
[13], absent in most of the theoretical models, and a more
involved band structure.

References

[1] C. Buzea and T. Yamashita; Supercond. Sci. Tech 14, R115
(2001).

[2] M. Angst and R. Puzniak; arXiv:cond-mat/0305048.

[3] F. Bouquet, R. A. Fisher, N. E. Phillips, D. G. Hinks, and J.
D. Jorgensen; Phys. Rev. Lett. 87, 047001 (2001).

[4] S. Tsuda, T. Yokoya, T. Kiss, Y. Takano, K. Togano, H. Kito,
H. Ihara, and S. Shin; Phys. Rev. Lett. 87, 177006 (2001).

[5] S. Tsuda, T. Yokoya, Y. Takano, H. Kito, A. Matsushita, F.
Yin, H. Harima, and S. Shin; arXiv: cond-mat/0303636.

[6] G. Karapetrov, M. Iavarone, W. K. Kwok, G. W. Cabtree, and
D. G. Hinks; Phys. Rev. Lett. 87, 4374 (2001).

[7] M. Iavarone, G. Karapetrov, A. E. Koshelev, W. K. Kwok, G.
W. Cabtree, and D. G. Hinks; arXiv:cond-mat/0203329.

[8] A. Sharoni, I. Felner, and O. Millo; Phys. Rev. B63, 220508
(2001).

[9] H. Schmidt, J. F. Zasadzinski, K. E. Gray, and D. G. Hinks;
Rev. B 63, 220504 (2001).

[10] Y. Bugoslavsky, Y. Miyoshi, G. K. Perkins, A. V. Berenov,
Z. Lockman, J. L. MacManus-Driscoll, L. F. Cohen, A. D.
Caplin, H. Y. Zhai, M. P. Paranthaman, H. M. Christen, and
M. Blamire; Supercon. Sci. Tech. 15, 526 (2002).

[11] X. K. Chen, M. J. Konstantinović, J. C. Irwin, D. D. Lawrie,
and J. P. Franck; Phys. Rev. Lett. 87, 157002 (2001).

[12] H. Suhl, B. T. Matthias, and L. R. Walker; Phys. Rev. Lett. 3,
552 (1959).

[13] R. E. Lagos, Solid State Commun. 79, 323 (1991).

[14] T. Örd and T. N. Kristoffel, Physica C370, 17 (2002).

[15] K. Yamaji; J. Phys. Soc. Jap. 59, 677 (1990), ibid. 70 1476
(2001).

[16] M. Imada; J. Phys. Soc. Jap. 70, 1218 (2001).

[17] N. Nakai, M. Ichioka, and K. Machida; J. Phys. Soc. Jpn. 71,
23 (2002).



Brazilian Journal of Physics, vol. 33, no. 4, December, 2003 717

[18] J. E. Hirsch and F. Marsiglio, Phys. Rev. B 64, 144523
(2001).

[19] V. P. Ramunni, G. M. Japiassú, and A. Troper; Physica C364-
365 , 190 (2001)

[20] I. N. Askerzade and A. Gencer; J. Phys. Soc. Jpn. 71, 1637
(2002)

[21] T. Mishonov and E. Penev; arXiv:cond-mat/0206118.

[22] H. J. Choi, D. Roundy, H. Sun, M. L. Cohen, and S. G. Louie;
arXiv:cond-mat/0111183, Nature 418, 758 (2202).

[23] C. Joas, I. Eremin, D. Manske, and K. H. Bennemann; Phys.
Rev. 65, 132518 (2002).

[24] A. A. Golubov, J. Kortus, O. V. Dolgov, O. Jepsen, Y. Kong,
O. K. Andersen, B. J. Gibson, K. Ahn, and R. K. Kremer; J.
Phys. Condens. Matter 14, 1353 (2002).

[25] A. Brinkman, A. A. Golubov, H. Rogalla, O. V. Dolgov, and
J. Kortus; Phys. Rev. B65 180517 (2002).

[26] J. M. An and W. E. Pickett; Phys. Rev. Lett. 86, 4366 (2001).

[27] J. Kortus, I. I. Mazin, K. D. Belashchenko, V. P. Antropov,
and L. L. Boyer; Phys. Rev. Lett. 86, 4656 (2001).

[28] M. Crisan; ‘Theory of Superconductivity’ World Scientific,
1989.