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The boson–fermion model in the mean-field approximation
Physica C 303 Ž1998. 273–286
The boson–fermion model in the mean-field approximation E. Piegari, V. Cataudella ) , G. Iadonisi I.N.F.M. Unita´ di Napoli, Dip. di Scienze Fisiche, Mostra D’ Oltremare pad. 19, 80125 Naples, Italy Received 8 November 1997; revised 10 April 1998; accepted 2 May 1998
Abstract The boson–fermion model is studied within the mean-field approximation including Pauli principle effects in the boson energy renormalization and Coulomb short-range repulsion among fermions and bosons. It is shown that there is a finite critical temperature at which Cooper pairing occurs accompanied by the divergence of the renormalized boson propagator at q s 0 and v s 0. The chemical potential, the entropy and the specific heat are also discussed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Boson; Fermion; Mean-field approximation
1. Introduction The boson–fermion ŽBF. model of superconductivity introduced by Ranninger and Robaszkiewicz w1x Žsee also Micnas et al. w2x. has received a renewed interest in connection with the high Tc superconductors. The model describes a mixture of hybridized Žlocalized or itinerant. bosons and itinerant fermions and it has been studied in two and three dimensions by different approaches: mean-field approximation w3–8x and a self-consistent diagrammatic conserving approximation w9,10x. Within these approximations a number of interesting results have been obtained: the model, in fact, is able to provide high critical temperature w3,4x, linear temperature dependence for the resistivity w11x, is also compatible with the phenomenological approach of Varma et al. w12x Žsee Alascio and Proetto w13x. and support the existence of a pseudo-gap w9,10,8x. The physical justification of this class of models is usually based on the polaron–bipolaron approach to the superconductivity where it is assumed that itinerant polarons bind together into bipolarons which can be either localized or itinerant boson-like particle. In this paper we study some thermodynamic properties of the boson–fermion model within the mean-field approximation and discuss some controversial aspects of this approximation. In fact, recently, this type of solution has been criticized w14,15x and the possibility of a simultaneous occurring of Cooper pairing of fermions and boson condensation has been strongly questioned. In particular, it has been suggested that the inclusion of short-range Coulomb repulsion combined with the introduction of the boson energy renormalization
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Corresponding author.
0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 2 3 7 - 8
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prevents the formation of Cooper pairs at any temperature no matter how weak the repulsion is. This conclusion steams from a somehow too crude choice of the boson energy renormalization which, when the boson energy is very large compared to the Fermi energy, leads to unphysical results. On the contrary we will show that the inclusion of a self-consistent renormalization for the boson energy as well as the introduction of a short-range repulsion among fermions and bosons do not change qualitatively the results obtained in the mean-field approximation neglecting both effects w3,4x. The repulsion, in fact, is only responsible for a smooth monotonic decrease of the critical temperature. The paper is organized in the following way. In Section 2 the boson–fermion model is introduced and both the boson renormalization and the short-range Coulomb repulsion are included and discussed. In Section 3 the results for the critical temperature are reported and discussed as a function of the renormalized boson energy and the boson–fermion hybridization parameter g 2 . In Sections 4 and 5 the chemical potential, the order parameter, the entropy and the specific heat are studied within the mean-field approximation.
2. The boson–fermion model In the boson–fermion model the underlying mechanism for superconductivity is assumed to be through the reaction 2e ™ f ™ 2e, in which e denotes either an electron or a hole and the local boson field f denotes a resonant pair state. The Hamiltonian defining the BF model is w3,4x
ž
H s Ý 2n 0 q k
"2 k 2 2M
/
y 2 m b†k b k q Ý k
ž
"2 k 2 2m
/
y m c†k c k q
g
'V kÝ, p
b†k c p r2qk ,≠ c p r2yk ,x q h.c. ,
Ž 1.
where the first term is the kinetic energy of the bosons Ž2 n 0 and M are the bare excitation energy and the mass of f, respectively., the second term is the kinetic energy of the fermions of mass m and the third term controls the decay of bound pairs Žbosons. into free charges Žfermions. and vice versa. In Eq. Ž1. the fermion and boson chemical potentials are related by the ‘chemical’ equilibrium condition 2 m s m B s 2 m F , where m B and m F are the boson and fermion chemical potentials, respectively. Following Friedberg and Lee w3,4x ŽFL., the Hamiltonian ŽEq. Ž1.. can be diagonalized considering the k s 0 bosonic state only. In this way renormalized bosons and fermions are coupled only via the macroscopic occupation number < B < 2 of the k s 0 boson state. Then, by using the zeroth-order expression of the partition function w3,4x, for each temperature T and total density n, it is possible to calculate the chemical potential m and Bose-condensate < B < 2 solving the following coupled equations:
msn0y
g2
1
ÝE 4V k
ns2< B<2q
tanh
k
b Ek
ž / 2
2
,
Ž 2.
1
Ý V k
eŽ2 n 0qwŽ "
2
2
k .rŽ2 M .xy2 m .
1 q y1
Ý V k
1q
m y vk Ek
tanh
b Ek
ž / 2
,
Ž 3.
where v k s Ž " 2 k 2 .rŽ2 m., b s 1rK B T and Ek is the renormalized free fermion energy Ek s
)ž
"2 k 2 2m
2
ym
/
qg2 < B<2 .
Due to the interaction, 2 n 0 is not a physical quantity and, then, it is useful to introduce the renormalized energy 2 n in such a way that the properties of the BF model can be studied as a function of the
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phenomenological physical parameter n . By using elementary second order perturbation theory, the renormalized boson energy is given by the solution of w3,4x 2n s 2n 0 q
g2
1
Ý P nyv 2V k
Ž 4. k
and the boson finite lifetime is given by
G s Ž g 2rp . m3r2
(
n 2
.
In Eq. Ž4. P denotes the principal value and V is the system volume. By using Eq. Ž4., we can rewrite Eqs. Ž2. and Ž3. in such a way to eliminate the dependence on n 0 :
msny
g2
1
Ý 4V
Ek
k
ns2< B<2q
tanh
2
b Ek
ž / 2
qP
1
n yvk
,
1
Ý V k
e
Ž2 n qwŽ " 2 k 2 .rŽ2 M .xy2 m .
Ž 5. 1
q y1
Ý V k
1q
m y vk Ek
tanh
b Ek
ž / 2
.
Ž 6.
The first equation is similar to the gap equation in the BCS theory, the second equation is the condition for the conservation of the total particle number. It is worthwhile to emphasize the twofold importance of the boson energy renormalization: at the same time it eliminates the divergence in the sum of Eq. Ž2. and provides a finite lifetime to the pair state. Eqs. Ž5. and Ž6. have been studied in some detail in Refs. w3,4,7x assuming n , m, M and g to be density-independent parameters. Defined nn s Ž 3p 2 .
y1
Ž 2 mn .
3r2
the fermionic density at which the Fermi energy ´ F equals n , Friedberg and Lee w3,4x showed that the system exhibits a BCS-like properties when n - nn Žor ´ F - n . and a behaviour typical of the Bose–Einstein condensation when n ) nn Žor ´ F ) n .. In both cases,however, the Bose condensate amplitude determines the gap energy of the fermion system. As one can see from Fig. 1, in this phenomenological approach the boson–fermion system goes from a BCS-like region Žin low densities limit. to a BEC-like region Žin high densities limit. with increasing total particle number Žsee also Ref. w7x.. It has also been shown that the phenomenological introduction of a dependence on the total carrier density for the boson energy n and for the boson–fermion coupling constant g 2 is important in order to obtain a satisfactory agreement with the experimental data of critical temperature versus carrier density w16x. Recently the approach described has been criticized in two aspects w14,15x. First of all the boson energy renormalization of Eq. Ž4. based on a simple second order perturbation theory completely neglects the Pauli
Fig. 1. A sketch of the Fermi energy and renormalized boson energy as functions of the total density in the Friedberg and Lee phenomenological approach.
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276
Fig. 2. The boson propagator at the lowest order within the RPA.
principle effect. Furthermore, the model does not take into account any intra-cell Žshort-range. Coulomb repulsion which should be present in a phenomenological approach to the problem and which is not contained into the boson–fermion interaction of Eq. Ž1.. For what concerns the boson renormalization the correct expression including the Pauli principle can be obtained at the lowest order within the RPA ŽFig. 2. w14x. In this approximation the renormalized boson propagator takes the form: 2
Dy1 Ž q,i v n . s i v n y 2 n 0 y Ž "q . r2 M q 2 m y S b Ž q,i v n .
Ž 7.
where
S b Ž q,i v n . s y
g2
Ý H G 0 Ž k , V n . G 0 Ž k q q,i V n q i v n . d k X
Vb
Ž 8.
X
X
n
and G 0 Ž k, V nX . is the free fermion propagator. v n and V nX are Matsubara frequencies for bosons and fermions, respectively. This expression for the self-energy is valid only for temperatures T G Tc since the free fermion propagator is used. After performing the sum over Matsubara frequencies the retarded boson self-energy becomes:
S b Ž q, v . s g 2
dk
H Ž 2p .
1 y nf Ž k . y nf Ž k q q . 3
" v y v k y v kqq
,
Ž 9.
where n f is the Fermi function. The renormalized boson energy at q s 0 can be obtained looking at the roots of the analytical continuation of Eq. Ž7. 2n s 2n 0 q
g2
ÝP 2V k
tanh b Ž v k y m . r2
n y vk
.
Ž 10 .
With the help of Eq. Ž10., we can rewrite Eq. Ž2. for T G Tc in terms of the renormalized energy n only. We get
msny
g2
Ý 4V
tanh b Ž v k y m . r2
vk y m
k
qP
tanh b Ž v k y m . r2
n y vk
.
Ž 11 .
The previous equation should be compared to equation 5 proposed in Refs. w3,4x: the inclusion of the Pauli principle effect has introduced the hyperbolic tangent in the numerator of the second sum. By inspection it is easy to see that a solution of Eq. Ž11. is given by m s n . However, as we will show numerically later, Eq. Ž11. can have more than a solution. The solution m s n corresponds to the case considered in the Alexandrov’s paper w14x where a different boson energy renormalization was used, i.e., 2n s 2n 0 q
g2
ÝP 2V k
tanh b Ž v k y m . r2
m y vk
Ž 12 .
E. Piegari et al.r Physica C 303 (1998) 273–286
277
and Eq. Ž11. was replaced by
msny
g2
Ý 4V
tanh b Ž v k y m . r2
vk y m
k
qP
tanh b Ž v k y m . r2
n y vk
.
Ž 13 .
The renormalization ŽEq. Ž12.. can be viewed as an approximate solution of the self-consistent renormalization ŽEq. Ž10... In fact Eq. Ž12., which replaces Eq. Ž11. when the renormalization ŽEq. Ž12.. is assumed, is obtained by Eq. Ž10. replacing n with m in the sum over k in Eq. Ž10.. In other words this approximation is equivalent to replace S b Ž k, v . with S b Ž k,0. in the analytical continuation of Eq. Ž7.. Eq. Ž12. has the advantage to provide an explicit expression for n but spoils the validity of condition Ž11.. In fact, Eq. Ž13. has only the solution m s n while Eq. Ž11. can have a different solution m - n depending on the values assigned to the parameters n and g. As we will show numerically in Section 3, the existence of a solution of Eq. Ž11. for a chemical potential m - n assures the possibility of a BCS-like solution. In fact, we expect, as in the mean-field approximation, that while m follows n for small n , i.e., n - e F Žboson condensation regime., in the opposite limit, i.e. n ) e F , m tends to e F ŽBCS-like regime.. The other criticism is related to neglecting of Coulomb repulsion among fermion and bosons in the model described by Eq. Ž1. w14x. Although a general description of the Coulomb repulsion effects has been reported in Ref. w17x, here, we wish analyse in particular how the introduction a short-range Coulomb repulsion effects the possibility of Cooper pairing and how the critical temperature is modified. In order to clarify this point, following Ref. w14x, we model the short-range repulsion as a repulsive delta function whose strength is Vc . With this choice Eq. Ž5. becomes 1s
ž
g2 2Ž n 0 y m .
y Vc
/
1
1
Ýv 2V k
kym
tanh
ž
b Ž vk y m. 2
/
Ž 14 .
where Vc is the strength of the on site repulsion. This equation can be easily obtained by the linearized BCS gap equation considering the effective fermion–fermion interaction due to the boson–fermion coupling and the delta-function repulsion Vff s
g2 V
D 0 Ž 0,0 . q
Vc V
Ž 15 .
where D 0 Ž q, v . is the free boson propagator ŽFig. 3.. It is useful to note that Eq. Ž14., as well as Eq. Ž2., coincides with the Thouless criterion for Tc , i.e., the requirement that the T matrix at q s 0 and v s 0 diverges at Tc . In fact, in the present case, T Ž 0,0 . s
Vff 1 y Vff S b Ž 0,0 . rg 2
Ž 16 .
and T Ž0,0. s 0 is equivalent to Eq. Ž14..
Fig. 3. The effective fermion–fermion interaction due to the boson hybridization and local Coulomb repulsion. D 0 Ž0,0. is the bare boson temperature Green’s function.
E. Piegari et al.r Physica C 303 (1998) 273–286
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Fig. 4. Boson self-energy taking into account the repeated scattering of intermediate fermions on each other via the Coulomb potential.
As in the previous case in which we have not considered the on-site repulsion, we wish to introduce the correct renormalized boson energy. It has been shown w15,18,19x that in this case the renormalized boson propagator becomes: Dy1 Ž q,i v n . s Dy1 0 Ž q,i v n . y
S b Ž q,i v n .
Ž 17 .
1 y Vc S b Ž q,i v n . rg 2
which corresponds to the renormalized boson self-energy of Fig. 4. As stressed in Refs. w18,19x this choice allows the divergence of the T matrix at q s 0 and v s 0 to be accompanied by the condensation of a boson mode Ž DŽ0,0. ¨ `. in agreement with a general property of the Hamiltonian ŽEq. Ž1... Again we can define the renormalized boson energy as a solution of 2n s 2n 0 q
g2
ÝP 2V k
tanh b Ž v k y m . r2
n y vk
1y
Vc
ÝP 2V k
tanh b Ž v k y m . r2
n yvk
Ž 18 .
and, then, replace n 0 in Eq. Ž14.. As in the case without the on-site repulsion we get the solution n s m , but, as we will also see in Section 3, in the present case there is another solution which allows the model to exhibit BCS-like properties. In Section 3 we shall show the behaviours of the critical temperature of the system Tc Ž n. and Tc Ž n . obtained taking or not into account the short-range Coulomb repulsion and the Pauli principle effects.
3. Critical temperature The critical temperature Tc of an interacting boson–fermion system based on the Hamiltonian ŽEq. Ž1.. is defined as the temperature at which the macroscopic occupation number of the k s 0 boson state < B < 2 vanishes. Consequently, it can be calculated putting < B < s 0 in Eqs. Ž2. and Ž3. and solving for m ŽTc . and Tc . In Fig. 5a we plot Tc versus n for two different values of the renormalized boson energy n and in Fig. 5b Tc versus n for two different values of the particle density n. The value of boson–fermion coupling constant is g s 1. The continuous lines show the behaviour of the critical temperature obtained by Eqs. Ž5. and Ž6. with n given by Eq. Ž4.. The dashed lines correspond to behaviour of Tc obtained taking into account the Fermi sphere presence only, therefore solving Eqs. Ž11. and Ž6.. Finally the dotted lines correspond to the behaviour of Tc obtained taking into account the short-range Coulomb repulsion too, therefore solving Eqs. Ž14. and Ž6. with n 0 given by Eq. Ž18.. As one can see from the plots, the Fermi sphere presence does not change the qualitative dependence of Tc on n and n . Instead the short-range Coulomb repulsion systematically reduces the critical temperature. Actually, when Vc ¨ `, m ŽTc . ¨ n and Tc reaches the minimum value.
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Fig. 5. Ža. Behaviour of Tc Ž n. for two different values of the boson energy n . The continuous line shows the behaviour of the critical temperature obtained by Eqs. Ž5. and Ž6. with n given by Eq. Ž4.. The dashed line corresponds to behaviour of Tc obtained solving Eqs. Ž11. and Ž6.. The dotted line corresponds to behaviour of Tc obtained solving Eqs. Ž14. and Ž6. with n 0 given by Eq. Ž18.. In this last case the chosen value for the parameter Vc is 1. Žb. Behaviour Tc Ž n . for two different values of the total particle density n. The curves have been obtained in the same way as described in Ža.. The values of n are in units of ´ 0 , the temperatures in Kelvin. The curves shown correspond to g 2 s1.
Fig. 5a shows Tc as an increasing function of the total density n. The dependence on n has, instead, a different behaviour; we find that Tc Ž n . first increases even if smoothly and then decreases to a vanishing value. The reason of this non monotonic behaviour can be understood with the following argument. For fixed density, when the excitation boson energy n increases, the boson density decreases. Then, when n ) ´ F in the system there are almost only fermions and Tc drops because it is proportional to the gap energy of the fermion system which is controlled by the Bose condensate. This explains why Tc vanishes for n ) ´ F . On the other hand, when n - ´ F , Tc depends on the particle density n and on the difference n y m ŽTc .. Therefore, when n increases Žfor fixed n., the increase of n y m ŽTc . leads to a smooth increase of Tc Ž n .. We have checked that the maximum value for Tc is reached when 2 n , ´ F . We take this as an indication that the critical temperature of the superconducting state is essentially controlled by a BCS-like mechanism ŽBose condensate opens up a BCS-like gap in the fermion spectrum. for 2 n ) ´ F and it is essentially determined by the BEC of the bosons in the system in the opposite case, 2 n - ´ F .
280
E. Piegari et al.r Physica C 303 (1998) 273–286
Fig. 6. Tc r Tcma x Ž n. with n increasing function of the total density, n A n 3r 2 . The continuous line shows the behaviour of the critical temperature obtained by Eqs. Ž5. and Ž6. with n given by Eq. Ž4.. The dashed line corresponds to behaviour of Tc r Tcma x obtained taking into account the Fermi sphere presence only, therefore solving Eqs. Ž11. and Ž6.. The dotted line corresponds to behaviour of Tc obtained taking into account the short-range Coulomb repulsion too, therefore solving Eqs. Ž14. and Ž6. with n 0 given by Eq. Ž18.. In this case the chosen value for the parameter Vc is 1.
Now let us see how the behaviour of Tc Ž n. is modified if we introduce a density dependence of n . This phenomenological density dependence, introduced in Ref. w16x, has been chosen in such a way that the resonant energy n , being an increasing function of n, crosses ´ F . By using n A n 3r2 one can see from Fig. 6 that the ratio TcrTcmax clearly reproduces the typical behaviour found in many high Tc superconductors: the critical temperature first increases, then reaches a maximum and decreases to a vanishing value. The vanishing of Tc for high densities is due to the crossing of n with ´ F . In Fig. 6 Tc Ž n. is obtained by using the three approaches discussed in Section 3: respectively Eqs. Ž5. and Ž6. Žcontinuous line., Eqs. Ž11. and Ž6. Ždashed line. and Eqs. Ž14. and Ž6. with n given by Eq. Ž18. Ždotted line..
4. Chemical potential and order parameter In this section and in that following we will discuss some of the thermodynamic properties of the boson–fermion model. This analysis will be based on the pair of Eqs. Ž5. and Ž6. neglecting both the Pauli principle effects and the on-site repulsion. This choice can be justified on the basis of the results presented in Section 3 which do not show qualitative differences on the critical temperature due to the inclusion of Pauli principle effects and for small values of the on-site repulsion. For fixed n, n and g we have calculated the chemical potential and the order parameter as functions of the temperature T by Eqs. Ž5. and Ž6.. In Fig. 7a and b we plot m versus T and < B < versus T, respectively, for four different values of the excitation energy n . The curves shown correspond to n s 10 21 cmy3 and g 2 s 1. From the first plot it can be seen that, for fixed n and n , the behaviour of m ŽT . is almost constant for T - Tc while it decreases for T ) Tc . From Fig. 7a it is also evident that, when n - ´ F , the maximum of m ŽT ., m ŽTc ., is approximately n and when n ) ´ F , m ŽTc . s ´ F . In Fig. 7 the values n s 0.15´ 0 , n s 0.25´ 0 , n s 0.40 ´ 0 and
E. Piegari et al.r Physica C 303 (1998) 273–286
281
Fig. 7. Ža. Behaviour of m ŽT . for the following values of n : n s 0.15´ 0 , n s 0.25´ 0 , n s 0.4´ 0 and n s 0.6 ´ 0 . The corresponding values of the critical temperature are Tc s80 K, Tc s90 K, Tc s81.5 K and Tc s10 K. Žb. Behaviour of < B <ŽT . for the same values of n . The dashed line is attained for n s 0.15´ 0 : for this n value < B < is maximum at low temperatures and shows the fastest decrease at temperatures near Tc . The values of m , n and B are in units of ´ 0 , the temperatures in degrees Kelvin. The curves shown correspond to ns10 21 cmy3 and g 2 s1.
n s 0.60 ´ 0 have been used to which correspond the critical temperatures Tc s 80 K, Tc s 90 K, Tc s 81.5 K and Tc s 10 K, respectively. 1 The curves in Fig. 7a show that, with increasing n , the system goes gradually from the BEC-like regime in which m is continuous at the transition, to the BCS-like regime in which the chemical potential has a discontinuous change in slope at Tc . We must notice that for our choice of n the Bose region is not fully achieved because also for n s 0.15´ 0 the small fermions fraction produce a finite jump Ževen if very small. in the slope of the chemical potential at Tc . As concerns the behaviour of the order parameter as a function of T, from Fig. 7b we observe that the maximum value for < B < at T s 0 and the fastest decrease near Tc are attained for n s 0.15´ 0 Ždashed line.. Identifying the product g < B < as the energy gap D of the superconductors, we find that the ratio 2 DŽ0.rŽ K B Tc . as function of n , for fixed n, decreases and then increases w21x. Also using different values of n, we have checked that the minimum value for the ratio is reached when n , ´ F . With increasing n the values of the ratio
1
The energies are in units of ´ 0 s 70 meV, the effective boson and fermion masses in units of the band mass m ) s 5m e , the lengths in units of " r 2 m )´ 0 ; this choice is in agreement with the values used in Ref. w20x.
'
E. Piegari et al.r Physica C 303 (1998) 273–286
282
Table 1 Values of ´ F , n Ž n., m ŽTc ., Tc and 2 DŽ0.r k B Tc for 12 values of the total particle density n n Ž10 20 cmy3 .
´ F Ž´0.
n Ž´0.
m ŽTc . Ž ´ 0 .
Tc ŽK.
2 DŽ0.r k B Tc
4 5 6 7 8 9 10 11 14 15 18 20
0.287 0.333 0.376 0.416 0.451 0.485 0.517 0.563 0.661 0.692 0.781 0.838
0.096 0.134 0.176 0.222 0.266 0.313 0.362 0.437 0.628 0.696 0.915 1.07
0.095 0.131 0.171 0.215 0.256 0.301 0.347 0.417 0.591 0.651 0.781 0.838
47 58 66 73 78 82 85 88 79 69 5 0.3
2.45 2.20 2.03 1.91 1.81 1.73 1.67 1.57 1.40 1.36 3.24 3.53
The energies are in units of ´ 0 s 70 meV; the temperatures are in Kelvin.
Fig. 8. Ža. Behaviour of m ŽT . for the values of n in Table 1. For each value of the total density n is marked the corresponding value of the critical temperature. Žb. Behaviour of < B ŽT .< for same values of n in Table 1. The dashed lines correspond to first seven values of the total density n. The values of m ŽT . and < B ŽT .< are in units of ´ 0 , the temperatures in degrees Kelvin. The curves shown correspond to g 2 s1.
E. Piegari et al.r Physica C 303 (1998) 273–286
283
Table 2 Values of ´ F , n and Tc for two different values of the effective boson mass M, the boson energy n and the coupling constant g 2 M
n Ž10 20 cmy3 .
´ F Ž´0.
n Ž´0.
g
Tc ŽK.
2 DŽ0.r k B Tc
2m 2m 2m 5m 5m 5m
7 7 7 7 7 7
0.416 0.416 0.416 0.416 0.416 0.416
0.222 0.122 0.222 0.222 0.122 0.222
1 1 4.53 1 1 4.53
73 65 97 46 41 78
1.91 2.45 4.03 3.04 3.90 5.04
The energies are in units of ´ 0 s 70 meV; the temperatures are in Kelvin.
2 DŽ0.rŽ K B Tc . obtained are 2.34, 1.90, 1.60, 2.66, respectively. We have also checked w21x that for n s 0.70 ´ 0 the value of the ratio is 3.53, as in the BCS prediction, and a further increase of n does not change this value. So far, we have examined the BF model assuming the parameter g, n and n as independent parameter. Now, fixing g 2 s 1, we assume that the excitation boson energy n increases with the density. As we have seen in Section 3, this phenomenological choice allows us to reproduce the typical behaviour of Tc with the charge carrier density w16x. The dependence on the total density n of n Ž n A n 3r2 . is taken in such a way that n f ´ F for n s 1.5 = 10 21 cmy3 In Table 1 we give the values of ´ F , n Ž n., m ŽTc ., Tc and 2 DŽ0.rK B Tc for some values of the particle density n. The corresponding values of Fermi temperature go from ; 200 K to ; 700 K. In Fig. 8a and b we plot m versus T and < B < versus T respectively for those values of n given in Table 1. As it can be observed the behaviours of m ŽT . and < B ŽT .< are qualitatively similar to the ones shown in Fig. 7. As shown in Fig. 8b, < B ŽT s 0.< is not a decreasing function of n Žas happens instead when n increases for fixed n. but < B ŽT s 0.< first increases Ždashed lines. and then decreases Žfull lines., furthermore the maximum value of B Ž0. does not correspond to the maximum value of Tc ŽTable 1.. The values of the ratio 2 DŽ0.rK B Tc , shown in Table 1, say that the increase of the Bose condensate at T s 0 is less than the corresponding increase of Tc because when DŽ0. s g < B <Ž0. increases, the ratio 2 DŽ0.rK B Tc decreases. We have checked w21x that for n ) ´ F the maximum value of the ratio is the BCS value 3.53, while for n - ´ F higher values of the ratio can be achieved either increasing the difference between the values of the boson and fermion effective masses or using lower values of n for fixed n. We note that boson masses much higher than fermion masses Ž M 4 2 m. produce lower critical temperature, while excitation energy values much lower than Fermi energy produce both an increase of the Bose condensate and a decrease of Tc . In this discussion, so far, we have assumed g 2 s 1. Introducing a dependence on the particle density n in g, higher values of the ratio 2 DŽ0.rK B Tc can be achieved for n - ´ F . Then assuming that g Ž n. is a decreasing function of the density we find that an increase of g, for fixed n, produces both a decrease of the Bose condensate and an increase of Tc . We have discussed our choice for g Ž n. previously w16x. The above results are summarized in Table 2.
5. Entropy and specific heat The entropy of the boson–fermion system can be readily calculated by the zeroth-order expression of the pressure p as a function of T, m and B w3,4x. One obtains SsV
Ep
ž / ET
1 y
m ,T
s2 K B Ý log Ž 1 q ey b E k . q k
EB
Ý T k Ž e bE
B
y 1.
.
2
E
k y K B Ý log Ž 1 y e b E . Ý T k Ž 1 q ey b E . k B
k
Ž 19 .
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Using the curves of the m ŽT . and < B ŽT .< shown in Fig. 7 we obtain the entropy S as a function of T for four different values of n Žsee Fig. 9.: we used n s 10 21 cmy3 and g 2 s 1. From the plots it is evident that, when n ) ´ F , SŽT . decreases of almost two orders of magnitude and shows a linear dependence on T above Tc . For n s 0.15´ 0 , the dashed line corresponds to the entropy of an ideal Bose gas with Tc s 80 K. As one can see from Fig. 9, with increasing n , the increase of the fermion density leads to smaller values of SŽT .. Above Tc , the more n is greater than ´ F , the more the behaviour of SŽT . is similar to the one of an ideal Fermi gas; vice versa the more n is lower than ´ F , the more the behaviour of SŽT . is similar to the one of an ideal Bose gas. The vice versa, however, holds only for T near Tc because, with increasing T, the number of bosons, distributed according to the principles of statistical mechanics, strongly decreases. If we use the curves of the m ŽT . and < B ŽT .< shown in Fig. 8, we obtain the behaviour of SŽT . for those values of n listed in Table 1 ŽFig. 10.. The calculated curves confirm that with increasing n Žand n Ž n.. the number of fermions in the system increases; in fact the behaviour of SŽT . above Tc tends to be linear in T at higher densities. From the expression of entropy as function of T, m and B, the specific heat of the boson–fermion system at constant volume is given by cv s T
dS dT
sT
ES
ž / ET
qT m, B
ES
Em
ES
EB
ž / ž / ž / ž / Em
T,B
ET
qT
r
EB
m ,T
ET
.
Ž 20 .
r
We calculate the derivatives of m and B with respect to the temperature at fixed n taking into account the condition Žd nrdT s 0. and bearing in mind that below Tc is EmrET f 0 and above Tc , EBrET becomes meaningless. Using the curves of m ŽT . and < B <ŽT . shown in Fig. 7, we obtain the specific heat c v as a function
Fig. 9. Behaviour of SŽT . for the following values of n : n s 0.15´ 0 , n s 0.25´ 0 , n s 0.40 ´ 0 , n s 0.60 ´ 0 . The corresponding values of the critical temperature are Tc s 80 K, Tc s 90 K, Tc s 81.5 K and Tc s 10 K. For n s 0.15´ 0 , the dashed line is the behaviour of SŽT . for an ideal Bose gas with Tc s 80 K. The curves shown correspond to n s 10 21 cmy3 and g 2 s 1. The temperatures are in degrees Kelvin.
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Fig. 10. Behaviour of SŽT . for the first 10 values of n and n in Table 1. The curves are obtained fixing g 2 s1. The temperatures are in degrees Kelvin.
Fig. 11. Behaviour of c v ŽT . for the following values of n : n s 0.15´ 0 , n s 0.25´ 0 , n s 0.40 ´ 0 , n s 0.60 ´ 0 . The corresponding values of the critical temperature are Tc s 80 K, Tc s 90 K, Tc s 81.5 K and Tc s 10 K. For n s 0.15´ 0 , the dashed line is the behaviour of c v ŽT . for an ideal Bose gas with Tc s 80 K. The curves shown correspond to n s 10 21 cmy3 and g 2 s 1. The temperatures are in degrees Kelvin.
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of T for four different values of n Žsee Fig. 11.. The result shows that the behaviour of c v is similar to the one expected by the BCS theory. The only exception is in the case n < ´ F Ž n s 0.15´ 0 . where the effect of the hybridizations of fermions and bosons is more evident. As one can see from the plot, for this value of n , the temperature region T ) Tc exhibits a positive curvature which can be considered as the signature of a system made essentially of bosons. However, even in this case, the presence of a very small number of fermions is able to produce a jump of c v at Tc .
6. Conclusions In this work we have discussed the mean field approximation of the boson–fermion model with special emphasis on the boson energy renormalization and on the possibility of including the Coulomb short-range repulsion among fermions and bosons. We have obtained a pair of coupled equations which allows for a finite critical temperature at which bosons condensate and Cooper pairs form. These equations are the extension of the mean field equations studied by Friedberg and Lee to the case where the Pauli Principle is fully taken into account in the boson energy renormalization and a delta function repulsion among fermions and bosons is considered. We have shown that for small values of the delta function repulsion Ž Vc , g . the critical temperature is only slightly lowered while it vanishes when Vc 4 g. In any case there is a large region in the parameter plane Ž g y Vc . in which the critical temperature is finite. Finally we have calculated the chemical potential, the entropy and the specific heat emphasizing the cross-over from BCS-like to Bose–Einstein condensation properties.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x
J. Ranninger, S. Robaszkievwicz, Physics B 135 Ž1985. 468. R. Micnas, J. Ranninger, S. Robaszkiewicz, Rev. Mod. Phys. 62 Ž1990. 113, and references therein. R. Fridberge, T.D. Lee, Phys. Lett. A 138 Ž1989. 423. R. Fridberge, T.D. Lee, Phys. Rev. B 40 Ž1989. 6745. Y. Bar Yam, Phys. Rev. B 43 Ž1991. 359. Y. Bar Yam, Phys. Rev. B Ž1991. 2601, and references therein. J. Ranninger, J.M. Robin, Physica C 253 Ž1995. 279. J. Ranninger, J. Robin, preprint to be published on Phys. Rev. B Ž1997.. J. Ranninger, J.M. Robin, M. Eschrig, Phys. Rev. Lett. 74 Ž1995. 4027. J. Ranninger, J. Robin, Phys. Rev. B 53 Ž1996. 962. G.M. Eliashberg, Pisma Zh. Eksp. Teor. 46 Ž1987. 94. C.M. Varma, P.B. Littlewood, S. Schmitt-Rink, E. Abrahams, A.E. Ruckenstein, Phys. Rev. Lett. 63 Ž1989. 1996. B.R. Alascio, C.R. Proetto, Sol. St. Commun. 75 Ž1990. 217. A.S. Alexandrov, J. Phys.: Condensed Matter 8 Ž1996. 6923. A.S. Alexandrov, Physica C 274 Ž1997. 237. G. Iadonisi, V. Cataudella, D. Ninno, M.L. Chiofalo, Phys. Lett. A 196 Ž1995. 359. T. Kostyrko, J. Ranninger, Phys. Rev. B 54 Ž1996. 13105. R. Friedberg, H.C. Ren, O. Tchernyshyov, private communication. T. Kostyrko, Acta Phys. Polonica A 91 Ž1997. 399. G. Iadonisi, M. Chiofalo, V. Cataudella, D. Ninno, Phys. Rev. B 48 Ž1993. 12966. E. Piegari, Degree Thesis, University of Naples, Naples, 1996.
The boson–fermion model in the mean-field approximation E. Piegari, V. Cataudella ) , G. Iadonisi I.N.F.M. Unita´ di Napoli, Dip. di Scienze Fisiche, Mostra D’ Oltremare pad. 19, 80125 Naples, Italy Received 8 November 1997; revised 10 April 1998; accepted 2 May 1998
Abstract The boson–fermion model is studied within the mean-field approximation including Pauli principle effects in the boson energy renormalization and Coulomb short-range repulsion among fermions and bosons. It is shown that there is a finite critical temperature at which Cooper pairing occurs accompanied by the divergence of the renormalized boson propagator at q s 0 and v s 0. The chemical potential, the entropy and the specific heat are also discussed. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Boson; Fermion; Mean-field approximation
1. Introduction The boson–fermion ŽBF. model of superconductivity introduced by Ranninger and Robaszkiewicz w1x Žsee also Micnas et al. w2x. has received a renewed interest in connection with the high Tc superconductors. The model describes a mixture of hybridized Žlocalized or itinerant. bosons and itinerant fermions and it has been studied in two and three dimensions by different approaches: mean-field approximation w3–8x and a self-consistent diagrammatic conserving approximation w9,10x. Within these approximations a number of interesting results have been obtained: the model, in fact, is able to provide high critical temperature w3,4x, linear temperature dependence for the resistivity w11x, is also compatible with the phenomenological approach of Varma et al. w12x Žsee Alascio and Proetto w13x. and support the existence of a pseudo-gap w9,10,8x. The physical justification of this class of models is usually based on the polaron–bipolaron approach to the superconductivity where it is assumed that itinerant polarons bind together into bipolarons which can be either localized or itinerant boson-like particle. In this paper we study some thermodynamic properties of the boson–fermion model within the mean-field approximation and discuss some controversial aspects of this approximation. In fact, recently, this type of solution has been criticized w14,15x and the possibility of a simultaneous occurring of Cooper pairing of fermions and boson condensation has been strongly questioned. In particular, it has been suggested that the inclusion of short-range Coulomb repulsion combined with the introduction of the boson energy renormalization
)
Corresponding author.
0921-4534r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 2 3 7 - 8
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prevents the formation of Cooper pairs at any temperature no matter how weak the repulsion is. This conclusion steams from a somehow too crude choice of the boson energy renormalization which, when the boson energy is very large compared to the Fermi energy, leads to unphysical results. On the contrary we will show that the inclusion of a self-consistent renormalization for the boson energy as well as the introduction of a short-range repulsion among fermions and bosons do not change qualitatively the results obtained in the mean-field approximation neglecting both effects w3,4x. The repulsion, in fact, is only responsible for a smooth monotonic decrease of the critical temperature. The paper is organized in the following way. In Section 2 the boson–fermion model is introduced and both the boson renormalization and the short-range Coulomb repulsion are included and discussed. In Section 3 the results for the critical temperature are reported and discussed as a function of the renormalized boson energy and the boson–fermion hybridization parameter g 2 . In Sections 4 and 5 the chemical potential, the order parameter, the entropy and the specific heat are studied within the mean-field approximation.
2. The boson–fermion model In the boson–fermion model the underlying mechanism for superconductivity is assumed to be through the reaction 2e ™ f ™ 2e, in which e denotes either an electron or a hole and the local boson field f denotes a resonant pair state. The Hamiltonian defining the BF model is w3,4x
ž
H s Ý 2n 0 q k
"2 k 2 2M
/
y 2 m b†k b k q Ý k
ž
"2 k 2 2m
/
y m c†k c k q
g
'V kÝ, p
b†k c p r2qk ,≠ c p r2yk ,x q h.c. ,
Ž 1.
where the first term is the kinetic energy of the bosons Ž2 n 0 and M are the bare excitation energy and the mass of f, respectively., the second term is the kinetic energy of the fermions of mass m and the third term controls the decay of bound pairs Žbosons. into free charges Žfermions. and vice versa. In Eq. Ž1. the fermion and boson chemical potentials are related by the ‘chemical’ equilibrium condition 2 m s m B s 2 m F , where m B and m F are the boson and fermion chemical potentials, respectively. Following Friedberg and Lee w3,4x ŽFL., the Hamiltonian ŽEq. Ž1.. can be diagonalized considering the k s 0 bosonic state only. In this way renormalized bosons and fermions are coupled only via the macroscopic occupation number < B < 2 of the k s 0 boson state. Then, by using the zeroth-order expression of the partition function w3,4x, for each temperature T and total density n, it is possible to calculate the chemical potential m and Bose-condensate < B < 2 solving the following coupled equations:
msn0y
g2
1
ÝE 4V k
ns2< B<2q
tanh
k
b Ek
ž / 2
2
,
Ž 2.
1
Ý V k
eŽ2 n 0qwŽ "
2
2
k .rŽ2 M .xy2 m .
1 q y1
Ý V k
1q
m y vk Ek
tanh
b Ek
ž / 2
,
Ž 3.
where v k s Ž " 2 k 2 .rŽ2 m., b s 1rK B T and Ek is the renormalized free fermion energy Ek s
)ž
"2 k 2 2m
2
ym
/
qg2 < B<2 .
Due to the interaction, 2 n 0 is not a physical quantity and, then, it is useful to introduce the renormalized energy 2 n in such a way that the properties of the BF model can be studied as a function of the
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phenomenological physical parameter n . By using elementary second order perturbation theory, the renormalized boson energy is given by the solution of w3,4x 2n s 2n 0 q
g2
1
Ý P nyv 2V k
Ž 4. k
and the boson finite lifetime is given by
G s Ž g 2rp . m3r2
(
n 2
.
In Eq. Ž4. P denotes the principal value and V is the system volume. By using Eq. Ž4., we can rewrite Eqs. Ž2. and Ž3. in such a way to eliminate the dependence on n 0 :
msny
g2
1
Ý 4V
Ek
k
ns2< B<2q
tanh
2
b Ek
ž / 2
qP
1
n yvk
,
1
Ý V k
e
Ž2 n qwŽ " 2 k 2 .rŽ2 M .xy2 m .
Ž 5. 1
q y1
Ý V k
1q
m y vk Ek
tanh
b Ek
ž / 2
.
Ž 6.
The first equation is similar to the gap equation in the BCS theory, the second equation is the condition for the conservation of the total particle number. It is worthwhile to emphasize the twofold importance of the boson energy renormalization: at the same time it eliminates the divergence in the sum of Eq. Ž2. and provides a finite lifetime to the pair state. Eqs. Ž5. and Ž6. have been studied in some detail in Refs. w3,4,7x assuming n , m, M and g to be density-independent parameters. Defined nn s Ž 3p 2 .
y1
Ž 2 mn .
3r2
the fermionic density at which the Fermi energy ´ F equals n , Friedberg and Lee w3,4x showed that the system exhibits a BCS-like properties when n - nn Žor ´ F - n . and a behaviour typical of the Bose–Einstein condensation when n ) nn Žor ´ F ) n .. In both cases,however, the Bose condensate amplitude determines the gap energy of the fermion system. As one can see from Fig. 1, in this phenomenological approach the boson–fermion system goes from a BCS-like region Žin low densities limit. to a BEC-like region Žin high densities limit. with increasing total particle number Žsee also Ref. w7x.. It has also been shown that the phenomenological introduction of a dependence on the total carrier density for the boson energy n and for the boson–fermion coupling constant g 2 is important in order to obtain a satisfactory agreement with the experimental data of critical temperature versus carrier density w16x. Recently the approach described has been criticized in two aspects w14,15x. First of all the boson energy renormalization of Eq. Ž4. based on a simple second order perturbation theory completely neglects the Pauli
Fig. 1. A sketch of the Fermi energy and renormalized boson energy as functions of the total density in the Friedberg and Lee phenomenological approach.
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Fig. 2. The boson propagator at the lowest order within the RPA.
principle effect. Furthermore, the model does not take into account any intra-cell Žshort-range. Coulomb repulsion which should be present in a phenomenological approach to the problem and which is not contained into the boson–fermion interaction of Eq. Ž1.. For what concerns the boson renormalization the correct expression including the Pauli principle can be obtained at the lowest order within the RPA ŽFig. 2. w14x. In this approximation the renormalized boson propagator takes the form: 2
Dy1 Ž q,i v n . s i v n y 2 n 0 y Ž "q . r2 M q 2 m y S b Ž q,i v n .
Ž 7.
where
S b Ž q,i v n . s y
g2
Ý H G 0 Ž k , V n . G 0 Ž k q q,i V n q i v n . d k X
Vb
Ž 8.
X
X
n
and G 0 Ž k, V nX . is the free fermion propagator. v n and V nX are Matsubara frequencies for bosons and fermions, respectively. This expression for the self-energy is valid only for temperatures T G Tc since the free fermion propagator is used. After performing the sum over Matsubara frequencies the retarded boson self-energy becomes:
S b Ž q, v . s g 2
dk
H Ž 2p .
1 y nf Ž k . y nf Ž k q q . 3
" v y v k y v kqq
,
Ž 9.
where n f is the Fermi function. The renormalized boson energy at q s 0 can be obtained looking at the roots of the analytical continuation of Eq. Ž7. 2n s 2n 0 q
g2
ÝP 2V k
tanh b Ž v k y m . r2
n y vk
.
Ž 10 .
With the help of Eq. Ž10., we can rewrite Eq. Ž2. for T G Tc in terms of the renormalized energy n only. We get
msny
g2
Ý 4V
tanh b Ž v k y m . r2
vk y m
k
qP
tanh b Ž v k y m . r2
n y vk
.
Ž 11 .
The previous equation should be compared to equation 5 proposed in Refs. w3,4x: the inclusion of the Pauli principle effect has introduced the hyperbolic tangent in the numerator of the second sum. By inspection it is easy to see that a solution of Eq. Ž11. is given by m s n . However, as we will show numerically later, Eq. Ž11. can have more than a solution. The solution m s n corresponds to the case considered in the Alexandrov’s paper w14x where a different boson energy renormalization was used, i.e., 2n s 2n 0 q
g2
ÝP 2V k
tanh b Ž v k y m . r2
m y vk
Ž 12 .
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and Eq. Ž11. was replaced by
msny
g2
Ý 4V
tanh b Ž v k y m . r2
vk y m
k
qP
tanh b Ž v k y m . r2
n y vk
.
Ž 13 .
The renormalization ŽEq. Ž12.. can be viewed as an approximate solution of the self-consistent renormalization ŽEq. Ž10... In fact Eq. Ž12., which replaces Eq. Ž11. when the renormalization ŽEq. Ž12.. is assumed, is obtained by Eq. Ž10. replacing n with m in the sum over k in Eq. Ž10.. In other words this approximation is equivalent to replace S b Ž k, v . with S b Ž k,0. in the analytical continuation of Eq. Ž7.. Eq. Ž12. has the advantage to provide an explicit expression for n but spoils the validity of condition Ž11.. In fact, Eq. Ž13. has only the solution m s n while Eq. Ž11. can have a different solution m - n depending on the values assigned to the parameters n and g. As we will show numerically in Section 3, the existence of a solution of Eq. Ž11. for a chemical potential m - n assures the possibility of a BCS-like solution. In fact, we expect, as in the mean-field approximation, that while m follows n for small n , i.e., n - e F Žboson condensation regime., in the opposite limit, i.e. n ) e F , m tends to e F ŽBCS-like regime.. The other criticism is related to neglecting of Coulomb repulsion among fermion and bosons in the model described by Eq. Ž1. w14x. Although a general description of the Coulomb repulsion effects has been reported in Ref. w17x, here, we wish analyse in particular how the introduction a short-range Coulomb repulsion effects the possibility of Cooper pairing and how the critical temperature is modified. In order to clarify this point, following Ref. w14x, we model the short-range repulsion as a repulsive delta function whose strength is Vc . With this choice Eq. Ž5. becomes 1s
ž
g2 2Ž n 0 y m .
y Vc
/
1
1
Ýv 2V k
kym
tanh
ž
b Ž vk y m. 2
/
Ž 14 .
where Vc is the strength of the on site repulsion. This equation can be easily obtained by the linearized BCS gap equation considering the effective fermion–fermion interaction due to the boson–fermion coupling and the delta-function repulsion Vff s
g2 V
D 0 Ž 0,0 . q
Vc V
Ž 15 .
where D 0 Ž q, v . is the free boson propagator ŽFig. 3.. It is useful to note that Eq. Ž14., as well as Eq. Ž2., coincides with the Thouless criterion for Tc , i.e., the requirement that the T matrix at q s 0 and v s 0 diverges at Tc . In fact, in the present case, T Ž 0,0 . s
Vff 1 y Vff S b Ž 0,0 . rg 2
Ž 16 .
and T Ž0,0. s 0 is equivalent to Eq. Ž14..
Fig. 3. The effective fermion–fermion interaction due to the boson hybridization and local Coulomb repulsion. D 0 Ž0,0. is the bare boson temperature Green’s function.
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Fig. 4. Boson self-energy taking into account the repeated scattering of intermediate fermions on each other via the Coulomb potential.
As in the previous case in which we have not considered the on-site repulsion, we wish to introduce the correct renormalized boson energy. It has been shown w15,18,19x that in this case the renormalized boson propagator becomes: Dy1 Ž q,i v n . s Dy1 0 Ž q,i v n . y
S b Ž q,i v n .
Ž 17 .
1 y Vc S b Ž q,i v n . rg 2
which corresponds to the renormalized boson self-energy of Fig. 4. As stressed in Refs. w18,19x this choice allows the divergence of the T matrix at q s 0 and v s 0 to be accompanied by the condensation of a boson mode Ž DŽ0,0. ¨ `. in agreement with a general property of the Hamiltonian ŽEq. Ž1... Again we can define the renormalized boson energy as a solution of 2n s 2n 0 q
g2
ÝP 2V k
tanh b Ž v k y m . r2
n y vk
1y
Vc
ÝP 2V k
tanh b Ž v k y m . r2
n yvk
Ž 18 .
and, then, replace n 0 in Eq. Ž14.. As in the case without the on-site repulsion we get the solution n s m , but, as we will also see in Section 3, in the present case there is another solution which allows the model to exhibit BCS-like properties. In Section 3 we shall show the behaviours of the critical temperature of the system Tc Ž n. and Tc Ž n . obtained taking or not into account the short-range Coulomb repulsion and the Pauli principle effects.
3. Critical temperature The critical temperature Tc of an interacting boson–fermion system based on the Hamiltonian ŽEq. Ž1.. is defined as the temperature at which the macroscopic occupation number of the k s 0 boson state < B < 2 vanishes. Consequently, it can be calculated putting < B < s 0 in Eqs. Ž2. and Ž3. and solving for m ŽTc . and Tc . In Fig. 5a we plot Tc versus n for two different values of the renormalized boson energy n and in Fig. 5b Tc versus n for two different values of the particle density n. The value of boson–fermion coupling constant is g s 1. The continuous lines show the behaviour of the critical temperature obtained by Eqs. Ž5. and Ž6. with n given by Eq. Ž4.. The dashed lines correspond to behaviour of Tc obtained taking into account the Fermi sphere presence only, therefore solving Eqs. Ž11. and Ž6.. Finally the dotted lines correspond to the behaviour of Tc obtained taking into account the short-range Coulomb repulsion too, therefore solving Eqs. Ž14. and Ž6. with n 0 given by Eq. Ž18.. As one can see from the plots, the Fermi sphere presence does not change the qualitative dependence of Tc on n and n . Instead the short-range Coulomb repulsion systematically reduces the critical temperature. Actually, when Vc ¨ `, m ŽTc . ¨ n and Tc reaches the minimum value.
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Fig. 5. Ža. Behaviour of Tc Ž n. for two different values of the boson energy n . The continuous line shows the behaviour of the critical temperature obtained by Eqs. Ž5. and Ž6. with n given by Eq. Ž4.. The dashed line corresponds to behaviour of Tc obtained solving Eqs. Ž11. and Ž6.. The dotted line corresponds to behaviour of Tc obtained solving Eqs. Ž14. and Ž6. with n 0 given by Eq. Ž18.. In this last case the chosen value for the parameter Vc is 1. Žb. Behaviour Tc Ž n . for two different values of the total particle density n. The curves have been obtained in the same way as described in Ža.. The values of n are in units of ´ 0 , the temperatures in Kelvin. The curves shown correspond to g 2 s1.
Fig. 5a shows Tc as an increasing function of the total density n. The dependence on n has, instead, a different behaviour; we find that Tc Ž n . first increases even if smoothly and then decreases to a vanishing value. The reason of this non monotonic behaviour can be understood with the following argument. For fixed density, when the excitation boson energy n increases, the boson density decreases. Then, when n ) ´ F in the system there are almost only fermions and Tc drops because it is proportional to the gap energy of the fermion system which is controlled by the Bose condensate. This explains why Tc vanishes for n ) ´ F . On the other hand, when n - ´ F , Tc depends on the particle density n and on the difference n y m ŽTc .. Therefore, when n increases Žfor fixed n., the increase of n y m ŽTc . leads to a smooth increase of Tc Ž n .. We have checked that the maximum value for Tc is reached when 2 n , ´ F . We take this as an indication that the critical temperature of the superconducting state is essentially controlled by a BCS-like mechanism ŽBose condensate opens up a BCS-like gap in the fermion spectrum. for 2 n ) ´ F and it is essentially determined by the BEC of the bosons in the system in the opposite case, 2 n - ´ F .
280
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Fig. 6. Tc r Tcma x Ž n. with n increasing function of the total density, n A n 3r 2 . The continuous line shows the behaviour of the critical temperature obtained by Eqs. Ž5. and Ž6. with n given by Eq. Ž4.. The dashed line corresponds to behaviour of Tc r Tcma x obtained taking into account the Fermi sphere presence only, therefore solving Eqs. Ž11. and Ž6.. The dotted line corresponds to behaviour of Tc obtained taking into account the short-range Coulomb repulsion too, therefore solving Eqs. Ž14. and Ž6. with n 0 given by Eq. Ž18.. In this case the chosen value for the parameter Vc is 1.
Now let us see how the behaviour of Tc Ž n. is modified if we introduce a density dependence of n . This phenomenological density dependence, introduced in Ref. w16x, has been chosen in such a way that the resonant energy n , being an increasing function of n, crosses ´ F . By using n A n 3r2 one can see from Fig. 6 that the ratio TcrTcmax clearly reproduces the typical behaviour found in many high Tc superconductors: the critical temperature first increases, then reaches a maximum and decreases to a vanishing value. The vanishing of Tc for high densities is due to the crossing of n with ´ F . In Fig. 6 Tc Ž n. is obtained by using the three approaches discussed in Section 3: respectively Eqs. Ž5. and Ž6. Žcontinuous line., Eqs. Ž11. and Ž6. Ždashed line. and Eqs. Ž14. and Ž6. with n given by Eq. Ž18. Ždotted line..
4. Chemical potential and order parameter In this section and in that following we will discuss some of the thermodynamic properties of the boson–fermion model. This analysis will be based on the pair of Eqs. Ž5. and Ž6. neglecting both the Pauli principle effects and the on-site repulsion. This choice can be justified on the basis of the results presented in Section 3 which do not show qualitative differences on the critical temperature due to the inclusion of Pauli principle effects and for small values of the on-site repulsion. For fixed n, n and g we have calculated the chemical potential and the order parameter as functions of the temperature T by Eqs. Ž5. and Ž6.. In Fig. 7a and b we plot m versus T and < B < versus T, respectively, for four different values of the excitation energy n . The curves shown correspond to n s 10 21 cmy3 and g 2 s 1. From the first plot it can be seen that, for fixed n and n , the behaviour of m ŽT . is almost constant for T - Tc while it decreases for T ) Tc . From Fig. 7a it is also evident that, when n - ´ F , the maximum of m ŽT ., m ŽTc ., is approximately n and when n ) ´ F , m ŽTc . s ´ F . In Fig. 7 the values n s 0.15´ 0 , n s 0.25´ 0 , n s 0.40 ´ 0 and
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Fig. 7. Ža. Behaviour of m ŽT . for the following values of n : n s 0.15´ 0 , n s 0.25´ 0 , n s 0.4´ 0 and n s 0.6 ´ 0 . The corresponding values of the critical temperature are Tc s80 K, Tc s90 K, Tc s81.5 K and Tc s10 K. Žb. Behaviour of < B <ŽT . for the same values of n . The dashed line is attained for n s 0.15´ 0 : for this n value < B < is maximum at low temperatures and shows the fastest decrease at temperatures near Tc . The values of m , n and B are in units of ´ 0 , the temperatures in degrees Kelvin. The curves shown correspond to ns10 21 cmy3 and g 2 s1.
n s 0.60 ´ 0 have been used to which correspond the critical temperatures Tc s 80 K, Tc s 90 K, Tc s 81.5 K and Tc s 10 K, respectively. 1 The curves in Fig. 7a show that, with increasing n , the system goes gradually from the BEC-like regime in which m is continuous at the transition, to the BCS-like regime in which the chemical potential has a discontinuous change in slope at Tc . We must notice that for our choice of n the Bose region is not fully achieved because also for n s 0.15´ 0 the small fermions fraction produce a finite jump Ževen if very small. in the slope of the chemical potential at Tc . As concerns the behaviour of the order parameter as a function of T, from Fig. 7b we observe that the maximum value for < B < at T s 0 and the fastest decrease near Tc are attained for n s 0.15´ 0 Ždashed line.. Identifying the product g < B < as the energy gap D of the superconductors, we find that the ratio 2 DŽ0.rŽ K B Tc . as function of n , for fixed n, decreases and then increases w21x. Also using different values of n, we have checked that the minimum value for the ratio is reached when n , ´ F . With increasing n the values of the ratio
1
The energies are in units of ´ 0 s 70 meV, the effective boson and fermion masses in units of the band mass m ) s 5m e , the lengths in units of " r 2 m )´ 0 ; this choice is in agreement with the values used in Ref. w20x.
'
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Table 1 Values of ´ F , n Ž n., m ŽTc ., Tc and 2 DŽ0.r k B Tc for 12 values of the total particle density n n Ž10 20 cmy3 .
´ F Ž´0.
n Ž´0.
m ŽTc . Ž ´ 0 .
Tc ŽK.
2 DŽ0.r k B Tc
4 5 6 7 8 9 10 11 14 15 18 20
0.287 0.333 0.376 0.416 0.451 0.485 0.517 0.563 0.661 0.692 0.781 0.838
0.096 0.134 0.176 0.222 0.266 0.313 0.362 0.437 0.628 0.696 0.915 1.07
0.095 0.131 0.171 0.215 0.256 0.301 0.347 0.417 0.591 0.651 0.781 0.838
47 58 66 73 78 82 85 88 79 69 5 0.3
2.45 2.20 2.03 1.91 1.81 1.73 1.67 1.57 1.40 1.36 3.24 3.53
The energies are in units of ´ 0 s 70 meV; the temperatures are in Kelvin.
Fig. 8. Ža. Behaviour of m ŽT . for the values of n in Table 1. For each value of the total density n is marked the corresponding value of the critical temperature. Žb. Behaviour of < B ŽT .< for same values of n in Table 1. The dashed lines correspond to first seven values of the total density n. The values of m ŽT . and < B ŽT .< are in units of ´ 0 , the temperatures in degrees Kelvin. The curves shown correspond to g 2 s1.
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Table 2 Values of ´ F , n and Tc for two different values of the effective boson mass M, the boson energy n and the coupling constant g 2 M
n Ž10 20 cmy3 .
´ F Ž´0.
n Ž´0.
g
Tc ŽK.
2 DŽ0.r k B Tc
2m 2m 2m 5m 5m 5m
7 7 7 7 7 7
0.416 0.416 0.416 0.416 0.416 0.416
0.222 0.122 0.222 0.222 0.122 0.222
1 1 4.53 1 1 4.53
73 65 97 46 41 78
1.91 2.45 4.03 3.04 3.90 5.04
The energies are in units of ´ 0 s 70 meV; the temperatures are in Kelvin.
2 DŽ0.rŽ K B Tc . obtained are 2.34, 1.90, 1.60, 2.66, respectively. We have also checked w21x that for n s 0.70 ´ 0 the value of the ratio is 3.53, as in the BCS prediction, and a further increase of n does not change this value. So far, we have examined the BF model assuming the parameter g, n and n as independent parameter. Now, fixing g 2 s 1, we assume that the excitation boson energy n increases with the density. As we have seen in Section 3, this phenomenological choice allows us to reproduce the typical behaviour of Tc with the charge carrier density w16x. The dependence on the total density n of n Ž n A n 3r2 . is taken in such a way that n f ´ F for n s 1.5 = 10 21 cmy3 In Table 1 we give the values of ´ F , n Ž n., m ŽTc ., Tc and 2 DŽ0.rK B Tc for some values of the particle density n. The corresponding values of Fermi temperature go from ; 200 K to ; 700 K. In Fig. 8a and b we plot m versus T and < B < versus T respectively for those values of n given in Table 1. As it can be observed the behaviours of m ŽT . and < B ŽT .< are qualitatively similar to the ones shown in Fig. 7. As shown in Fig. 8b, < B ŽT s 0.< is not a decreasing function of n Žas happens instead when n increases for fixed n. but < B ŽT s 0.< first increases Ždashed lines. and then decreases Žfull lines., furthermore the maximum value of B Ž0. does not correspond to the maximum value of Tc ŽTable 1.. The values of the ratio 2 DŽ0.rK B Tc , shown in Table 1, say that the increase of the Bose condensate at T s 0 is less than the corresponding increase of Tc because when DŽ0. s g < B <Ž0. increases, the ratio 2 DŽ0.rK B Tc decreases. We have checked w21x that for n ) ´ F the maximum value of the ratio is the BCS value 3.53, while for n - ´ F higher values of the ratio can be achieved either increasing the difference between the values of the boson and fermion effective masses or using lower values of n for fixed n. We note that boson masses much higher than fermion masses Ž M 4 2 m. produce lower critical temperature, while excitation energy values much lower than Fermi energy produce both an increase of the Bose condensate and a decrease of Tc . In this discussion, so far, we have assumed g 2 s 1. Introducing a dependence on the particle density n in g, higher values of the ratio 2 DŽ0.rK B Tc can be achieved for n - ´ F . Then assuming that g Ž n. is a decreasing function of the density we find that an increase of g, for fixed n, produces both a decrease of the Bose condensate and an increase of Tc . We have discussed our choice for g Ž n. previously w16x. The above results are summarized in Table 2.
5. Entropy and specific heat The entropy of the boson–fermion system can be readily calculated by the zeroth-order expression of the pressure p as a function of T, m and B w3,4x. One obtains SsV
Ep
ž / ET
1 y
m ,T
s2 K B Ý log Ž 1 q ey b E k . q k
EB
Ý T k Ž e bE
B
y 1.
.
2
E
k y K B Ý log Ž 1 y e b E . Ý T k Ž 1 q ey b E . k B
k
Ž 19 .
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Using the curves of the m ŽT . and < B ŽT .< shown in Fig. 7 we obtain the entropy S as a function of T for four different values of n Žsee Fig. 9.: we used n s 10 21 cmy3 and g 2 s 1. From the plots it is evident that, when n ) ´ F , SŽT . decreases of almost two orders of magnitude and shows a linear dependence on T above Tc . For n s 0.15´ 0 , the dashed line corresponds to the entropy of an ideal Bose gas with Tc s 80 K. As one can see from Fig. 9, with increasing n , the increase of the fermion density leads to smaller values of SŽT .. Above Tc , the more n is greater than ´ F , the more the behaviour of SŽT . is similar to the one of an ideal Fermi gas; vice versa the more n is lower than ´ F , the more the behaviour of SŽT . is similar to the one of an ideal Bose gas. The vice versa, however, holds only for T near Tc because, with increasing T, the number of bosons, distributed according to the principles of statistical mechanics, strongly decreases. If we use the curves of the m ŽT . and < B ŽT .< shown in Fig. 8, we obtain the behaviour of SŽT . for those values of n listed in Table 1 ŽFig. 10.. The calculated curves confirm that with increasing n Žand n Ž n.. the number of fermions in the system increases; in fact the behaviour of SŽT . above Tc tends to be linear in T at higher densities. From the expression of entropy as function of T, m and B, the specific heat of the boson–fermion system at constant volume is given by cv s T
dS dT
sT
ES
ž / ET
qT m, B
ES
Em
ES
EB
ž / ž / ž / ž / Em
T,B
ET
qT
r
EB
m ,T
ET
.
Ž 20 .
r
We calculate the derivatives of m and B with respect to the temperature at fixed n taking into account the condition Žd nrdT s 0. and bearing in mind that below Tc is EmrET f 0 and above Tc , EBrET becomes meaningless. Using the curves of m ŽT . and < B <ŽT . shown in Fig. 7, we obtain the specific heat c v as a function
Fig. 9. Behaviour of SŽT . for the following values of n : n s 0.15´ 0 , n s 0.25´ 0 , n s 0.40 ´ 0 , n s 0.60 ´ 0 . The corresponding values of the critical temperature are Tc s 80 K, Tc s 90 K, Tc s 81.5 K and Tc s 10 K. For n s 0.15´ 0 , the dashed line is the behaviour of SŽT . for an ideal Bose gas with Tc s 80 K. The curves shown correspond to n s 10 21 cmy3 and g 2 s 1. The temperatures are in degrees Kelvin.
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Fig. 10. Behaviour of SŽT . for the first 10 values of n and n in Table 1. The curves are obtained fixing g 2 s1. The temperatures are in degrees Kelvin.
Fig. 11. Behaviour of c v ŽT . for the following values of n : n s 0.15´ 0 , n s 0.25´ 0 , n s 0.40 ´ 0 , n s 0.60 ´ 0 . The corresponding values of the critical temperature are Tc s 80 K, Tc s 90 K, Tc s 81.5 K and Tc s 10 K. For n s 0.15´ 0 , the dashed line is the behaviour of c v ŽT . for an ideal Bose gas with Tc s 80 K. The curves shown correspond to n s 10 21 cmy3 and g 2 s 1. The temperatures are in degrees Kelvin.
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of T for four different values of n Žsee Fig. 11.. The result shows that the behaviour of c v is similar to the one expected by the BCS theory. The only exception is in the case n < ´ F Ž n s 0.15´ 0 . where the effect of the hybridizations of fermions and bosons is more evident. As one can see from the plot, for this value of n , the temperature region T ) Tc exhibits a positive curvature which can be considered as the signature of a system made essentially of bosons. However, even in this case, the presence of a very small number of fermions is able to produce a jump of c v at Tc .
6. Conclusions In this work we have discussed the mean field approximation of the boson–fermion model with special emphasis on the boson energy renormalization and on the possibility of including the Coulomb short-range repulsion among fermions and bosons. We have obtained a pair of coupled equations which allows for a finite critical temperature at which bosons condensate and Cooper pairs form. These equations are the extension of the mean field equations studied by Friedberg and Lee to the case where the Pauli Principle is fully taken into account in the boson energy renormalization and a delta function repulsion among fermions and bosons is considered. We have shown that for small values of the delta function repulsion Ž Vc , g . the critical temperature is only slightly lowered while it vanishes when Vc 4 g. In any case there is a large region in the parameter plane Ž g y Vc . in which the critical temperature is finite. Finally we have calculated the chemical potential, the entropy and the specific heat emphasizing the cross-over from BCS-like to Bose–Einstein condensation properties.
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