- Email: [email protected]
The dependence ofTcon carrier concentration in high-Tcsuperconductors
Superlattices and Microstructures, Vol. 21, No. 3, 1997
Tc The dependence of Tc on carrier concentration in high-T superconductors S. Dzhumanov† Institute of Nuclear Physics, 702132, Tashkent, Uzbekistan
(Received 9 July 1996) The dependence of Tc on carrier concentration, n, and its variation in high Tc and conventional superconductors is studied beyond the standard Fermi-liquid (i.e. BCS)-scenario within the novel two-stage Fermi–Bose-liquid (FBL) theory and two FBL scenarios in socalled fermion superconductors (FSC) and boson ones (BSC). It is shown that the initial increase of Tc (n) and its subsequent saturation at any optimal n = n opt is controlled by the concentration dependence of the condensation temperature, TB , of an attracting Bose-gas of bipolarons, and then the decrease of Tc (n) for higher n > n opt is determined by the concentration dependence of the depairing temperature of bound polaron pairs. c 1997 Academic Press Limited
Key words: Tc ’s concentration dependence, two Fermi–Bose-liquid scenarios, fermion and boson superconductors.
1. Introduction One of the key features common for all high Tc superconductors (HTSC) is the high dependence of Tc on the carrier concentration, n. As doping takes place, Tc first rises and goes through a maximum, sometimes with saturation at an optimal n = n opt , then rapidly decreases for n > n opt [1–5]. The nature of such behaviour of Tc (n) is still not well understood, although much theoretical effort has been devoted to its examination. The solution of this problem undoubtedly plays a crucial role in the identification of the novel and key mechanisms of high-Tc superconductivity. Here an attempt has been made to develop a more consistent and adequate theory of the concentration dependences, Tc (n), within the novel two-stage Fermi–Bose-liquid (FBL) model of superconductivity, which is based on a modified BCS-like pairing concept for fermions together with an underlying single particle condensation (SPC) and pair condensation (PC) of attracting composite bosons (Cooper pairs and bipolarons) [5, 6]. We show that the above anomalous dependences, Tc (n), follow naturally from this theory. .
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2. Fermion and boson superconductors In a parent Fermi system, the phase transition to the superconducting (SC) state is accompanied, as a rule, by the formation of composite bosons (which results from the BCS-like k-space pairing of carriers) with their subsequent transition to the superfluid (SF) state by attractive SPC and PC [6, 7]. So, the superconductivity in .
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† E-mail: [email protected]
0749–6036/97/030363 + 04 $25.00/0
sm960401
c 1997 Academic Press Limited
364
Superlattices and Microstructures, Vol. 21, No. 3, 1997
such a system results from the coexistence of two types of order parameter [6, 7]: one of them, 1 F , characterizes the bond strength of bound fermion pairs and the other, 1 B − , is the bond strength of all condensed bosons (i.e. the existence of a SF state). This leads to superconductivity via two FBL scenarios [5, 6]: (i) Fermigas↔BCS-like Fermi-liquid (FL) or unstable ideal Bose-gas (BG)↔SF Bose-liquid (SFBL), and (ii) Fermigas↔BCS-like FL or normal BG↔SFBL which is realized both in so-called fermion superconductors (FSC) (with 1 SC = 1 F < 1 B , Tc = TF = TB ) and in boson superconductors (BSC), (with 1 SC = 1 B < 1 F , Tc = TB < TF in favour of which are resistive and magnetic anomalies observed in HTSC [8]), respectively. Here we have TF ' 1.14(E bB + h¯ ω D ) exp[−1/D F V˜ F ] which is the depairing temperature of bound fermion pairs, E bB and V˜ F are their binding energy and effective interaction potential, respectively, h¯ ω D is the Debye energy, and D F is the density of states at the Fermi level. Now we show that in BSC, as the concentration n (= n F or n B , which is the polaron or bipolaron concentration) is increased, TF decreases and TB increases. As a result, TF and TB tend towards each other and at some n = n opt we obtain Tc = TB = TF , which corresponds to the crossing of TB (n) and TF (n), accompanied by the conversion of a BSC into a FSC (where Tc (n) = TF (n) rapidly decreases for n > n opt ). .
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3. Carrier concentration dependence of Tc in fermion and boson superconductors It is clear that the binding energy of k-space bipolarons, 21 F ' 4(E bB + h¯ ω D ) exp[−1/D F V˜ F ] cannot exceed the binding energy of real (r )-space bipolarons, E bB + h¯ ω D (see also Ref. [7]). Then the above BCS-like expression for TF is valid only at D F V˜ F < 0.7 and the dependence TF (n) is determined by the concentration dependence of D F . In the carrier localization model corresponding to polaron formation, D F increases rapidly as n → 0 (see [9]) and a more suitable approximation is D F ∼ 1/E F , where E F = (h¯ 2 /2m ∗ )(3π 2 n)2/3 is the Fermi energy, and m ∗ is the effective mass of carriers. According to the two-stage FBL scenarios of superconductivity, the decrease of Tc at higher carrier concentration is controlled by the behaviour of TB (n) and TF (n) crossing. This takes place near n ' n opt (Fig. 1 and 2). Then TF can be expressed as TF (n) ' 1.14(E bB + h¯ ω D ) exp[−an 2/3 ] (which is tenable for 1/D F V˜ F = an 2/3 > 1.43), where a = (h¯ 2 /2m ∗ )(3π 2 )2/3 /V˜ F . The system (i.e. HTSC) approaches a half filled (i.e. Bose–Einstein condensate (BEC)) at the limit an 2/3 < 1.43. This corresponds to the formation of r -space polarons (bipolarons) with nearly zero bandwidth and we have a dilute polaron (bipolaron)-gas, i.e. polaronic (bipolaronic) insulators. While at an 2/3 > 1.43 the broadening of a polaron band occurs, where BCS-like k-space pairing of polarons takes place with the formation of the k-space bipolarons. So an 2/3 ' 1.43 corresponds to a localized driven metal-insulator transition. Hence, TF rapidly drops as an 2/3 > 1.43 and n starts to increase further. The concentration dependence of TB for an attracting 3D- and 2D-BG has the form TB (n B ) ' TBEC (n B )[1 + √ 1.42γ B TBEC (n B )/ξBA ] (where the Boltzmann constant k B = 1 and γ B 1) and TB (n B ) = −T0 (n B )/ ln[1− 2/3 exp(−2γ B /(γ B + 2))], where TBEC (n B ) = 3.31h¯ 2 n B /m B is the BEC temperature of an ideal 3D-BG. m B is the mass of free bosons, T0 (n B ) is the BEC-like temperature for an ideal 2D-BG, γ B is the interboson coupling constant, and ξBA is the cutoff energy for the attractive part of an interboson interaction potential VB (k, k 0 ). According to Luban’s k-space effective mass approximation [10] the interaction between particles of a 3D-BG causes changes in their k-space masses. Then the redefined, or renormalizable, mass of attracting bosons and their BEC temperature is m ∗B = m B [1 − n B VB (0)/ξBA ]−1 (in our context with n B VB (0)/ξBA 1) 2/3 and TBEC (n B ) = 3.31h¯ 2 n B /m ∗B ). We have also constructed such a k-space effective mass approximation for an attracting 2D-BG. So, the redefined TBEC (n B ) and T0 (n B ) for a non-ideal 3D- and 2D-BG, respectively, will contain the multiplier [1 − n B VB (0)/ξBA ]. It should be noted that the real condensation temperature TB of a non-ideal 3D-BG was not determined by Luban but its expression for redefined TBEC (n B ) is part of the TB 2/3 expression which has been derived in Refs [6, 11] for an attracting 3D-BG. Thus, TB first rises nearly as ∼ n B (for a 3D-BG) and ∼ n B (for 2D-BG), then saturates at some n B ∼ n opt and starts to decrease as n B > n opt .
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Superlattices and Microstructures, Vol. 21, No. 3, 1997
365
T TF(n) > 2Tc
~n2/3
Bipolaronic insulator
Polaronic metal
Bipolaronic metal with the BCS-like pseudo-gap ∆F
TB(n) = Tc
TB(n) > Tc 2∆F ≅8 Tc
FSC TF(n) = Tc
2∆B(=∆SC) γ ≅ B Tc
2∆F(≅∆SC) = 3.52 Tc
BSC
n = nopt
n
Fig. 1. 3D-Insulator-metal-superconductor. .
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T
TF(n) >> Tc
~n
Bipolaronic insulator
Polaronic metal
Bipolaronic metal with the BCS-like pseudo-gap ∆F
TB(n) = Tc
TB(n) > Tc
BSC
FSC
TF(n) = Tc
n = nopt
n
Fig. 2. 2D-Insulator-metal-superconductor. .
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(Fig. 1 and 2) in accordance with the observations [1–4]. The crossing of TB (n) and TB (n) close to n ' n opt also takes place. If such crossing of TB (n) and TF (n) takes place up to, or after, the saturation of TB (n), then the dependence of Tc (n) exhibits a sharp maximum, or plateau. These features of Tc (n) were also observed in HTSC [1–4]. So not only low-Tc metal superconductors, with very low concentration of pairing carriers and E bB = 0 [5, 6], but also HTSC, with very high concentration of pairing carriers, belong to FSC systems. .
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Superlattices and Microstructures, Vol. 21, No. 3, 1997
References .
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
}Y. I. Uemura, Physica 185–189C, 733 (1991). } F. Mott, Physica 205C, 191 (1993). N. } Ushio and H. Kamimura, Physica 185–189C, 1699 (1991). H. } N. R. Rao and A. K. Ganguli, Physica 235–240C, 9 (1994). C. } X. Zhao and W. I. Zhu, Physica 235–240C, 75 (1994). Z. } Dzhumanov, Physica 235–240C, 2269 (1994). S. } Dzhumanov, Preprint ICTP IC/95/221. Trieste (Italy, 1995). S. } L. Tholence, L. Puech, A. Sulpice et al. Physica 235–240C, 1545 (1994). I. } H. Kim, K. Levin, and A. Auerbach, Phys. Rev. B 39, 11633 (1989). Ju. } Luban, Phys. Rev. 128, 965 (1962). M. } Dzhumanov, P. J. Baimatov, and P. K. Khabibullaev, Uzb. Zh. Phys. 6, 24 (1992). S.
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Tc The dependence of Tc on carrier concentration in high-T superconductors S. Dzhumanov† Institute of Nuclear Physics, 702132, Tashkent, Uzbekistan
(Received 9 July 1996) The dependence of Tc on carrier concentration, n, and its variation in high Tc and conventional superconductors is studied beyond the standard Fermi-liquid (i.e. BCS)-scenario within the novel two-stage Fermi–Bose-liquid (FBL) theory and two FBL scenarios in socalled fermion superconductors (FSC) and boson ones (BSC). It is shown that the initial increase of Tc (n) and its subsequent saturation at any optimal n = n opt is controlled by the concentration dependence of the condensation temperature, TB , of an attracting Bose-gas of bipolarons, and then the decrease of Tc (n) for higher n > n opt is determined by the concentration dependence of the depairing temperature of bound polaron pairs. c 1997 Academic Press Limited
Key words: Tc ’s concentration dependence, two Fermi–Bose-liquid scenarios, fermion and boson superconductors.
1. Introduction One of the key features common for all high Tc superconductors (HTSC) is the high dependence of Tc on the carrier concentration, n. As doping takes place, Tc first rises and goes through a maximum, sometimes with saturation at an optimal n = n opt , then rapidly decreases for n > n opt [1–5]. The nature of such behaviour of Tc (n) is still not well understood, although much theoretical effort has been devoted to its examination. The solution of this problem undoubtedly plays a crucial role in the identification of the novel and key mechanisms of high-Tc superconductivity. Here an attempt has been made to develop a more consistent and adequate theory of the concentration dependences, Tc (n), within the novel two-stage Fermi–Bose-liquid (FBL) model of superconductivity, which is based on a modified BCS-like pairing concept for fermions together with an underlying single particle condensation (SPC) and pair condensation (PC) of attracting composite bosons (Cooper pairs and bipolarons) [5, 6]. We show that the above anomalous dependences, Tc (n), follow naturally from this theory. .
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.
.
.
2. Fermion and boson superconductors In a parent Fermi system, the phase transition to the superconducting (SC) state is accompanied, as a rule, by the formation of composite bosons (which results from the BCS-like k-space pairing of carriers) with their subsequent transition to the superfluid (SF) state by attractive SPC and PC [6, 7]. So, the superconductivity in .
.
.
.
† E-mail: [email protected]
0749–6036/97/030363 + 04 $25.00/0
sm960401
c 1997 Academic Press Limited
364
Superlattices and Microstructures, Vol. 21, No. 3, 1997
such a system results from the coexistence of two types of order parameter [6, 7]: one of them, 1 F , characterizes the bond strength of bound fermion pairs and the other, 1 B − , is the bond strength of all condensed bosons (i.e. the existence of a SF state). This leads to superconductivity via two FBL scenarios [5, 6]: (i) Fermigas↔BCS-like Fermi-liquid (FL) or unstable ideal Bose-gas (BG)↔SF Bose-liquid (SFBL), and (ii) Fermigas↔BCS-like FL or normal BG↔SFBL which is realized both in so-called fermion superconductors (FSC) (with 1 SC = 1 F < 1 B , Tc = TF = TB ) and in boson superconductors (BSC), (with 1 SC = 1 B < 1 F , Tc = TB < TF in favour of which are resistive and magnetic anomalies observed in HTSC [8]), respectively. Here we have TF ' 1.14(E bB + h¯ ω D ) exp[−1/D F V˜ F ] which is the depairing temperature of bound fermion pairs, E bB and V˜ F are their binding energy and effective interaction potential, respectively, h¯ ω D is the Debye energy, and D F is the density of states at the Fermi level. Now we show that in BSC, as the concentration n (= n F or n B , which is the polaron or bipolaron concentration) is increased, TF decreases and TB increases. As a result, TF and TB tend towards each other and at some n = n opt we obtain Tc = TB = TF , which corresponds to the crossing of TB (n) and TF (n), accompanied by the conversion of a BSC into a FSC (where Tc (n) = TF (n) rapidly decreases for n > n opt ). .
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3. Carrier concentration dependence of Tc in fermion and boson superconductors It is clear that the binding energy of k-space bipolarons, 21 F ' 4(E bB + h¯ ω D ) exp[−1/D F V˜ F ] cannot exceed the binding energy of real (r )-space bipolarons, E bB + h¯ ω D (see also Ref. [7]). Then the above BCS-like expression for TF is valid only at D F V˜ F < 0.7 and the dependence TF (n) is determined by the concentration dependence of D F . In the carrier localization model corresponding to polaron formation, D F increases rapidly as n → 0 (see [9]) and a more suitable approximation is D F ∼ 1/E F , where E F = (h¯ 2 /2m ∗ )(3π 2 n)2/3 is the Fermi energy, and m ∗ is the effective mass of carriers. According to the two-stage FBL scenarios of superconductivity, the decrease of Tc at higher carrier concentration is controlled by the behaviour of TB (n) and TF (n) crossing. This takes place near n ' n opt (Fig. 1 and 2). Then TF can be expressed as TF (n) ' 1.14(E bB + h¯ ω D ) exp[−an 2/3 ] (which is tenable for 1/D F V˜ F = an 2/3 > 1.43), where a = (h¯ 2 /2m ∗ )(3π 2 )2/3 /V˜ F . The system (i.e. HTSC) approaches a half filled (i.e. Bose–Einstein condensate (BEC)) at the limit an 2/3 < 1.43. This corresponds to the formation of r -space polarons (bipolarons) with nearly zero bandwidth and we have a dilute polaron (bipolaron)-gas, i.e. polaronic (bipolaronic) insulators. While at an 2/3 > 1.43 the broadening of a polaron band occurs, where BCS-like k-space pairing of polarons takes place with the formation of the k-space bipolarons. So an 2/3 ' 1.43 corresponds to a localized driven metal-insulator transition. Hence, TF rapidly drops as an 2/3 > 1.43 and n starts to increase further. The concentration dependence of TB for an attracting 3D- and 2D-BG has the form TB (n B ) ' TBEC (n B )[1 + √ 1.42γ B TBEC (n B )/ξBA ] (where the Boltzmann constant k B = 1 and γ B 1) and TB (n B ) = −T0 (n B )/ ln[1− 2/3 exp(−2γ B /(γ B + 2))], where TBEC (n B ) = 3.31h¯ 2 n B /m B is the BEC temperature of an ideal 3D-BG. m B is the mass of free bosons, T0 (n B ) is the BEC-like temperature for an ideal 2D-BG, γ B is the interboson coupling constant, and ξBA is the cutoff energy for the attractive part of an interboson interaction potential VB (k, k 0 ). According to Luban’s k-space effective mass approximation [10] the interaction between particles of a 3D-BG causes changes in their k-space masses. Then the redefined, or renormalizable, mass of attracting bosons and their BEC temperature is m ∗B = m B [1 − n B VB (0)/ξBA ]−1 (in our context with n B VB (0)/ξBA 1) 2/3 and TBEC (n B ) = 3.31h¯ 2 n B /m ∗B ). We have also constructed such a k-space effective mass approximation for an attracting 2D-BG. So, the redefined TBEC (n B ) and T0 (n B ) for a non-ideal 3D- and 2D-BG, respectively, will contain the multiplier [1 − n B VB (0)/ξBA ]. It should be noted that the real condensation temperature TB of a non-ideal 3D-BG was not determined by Luban but its expression for redefined TBEC (n B ) is part of the TB 2/3 expression which has been derived in Refs [6, 11] for an attracting 3D-BG. Thus, TB first rises nearly as ∼ n B (for a 3D-BG) and ∼ n B (for 2D-BG), then saturates at some n B ∼ n opt and starts to decrease as n B > n opt .
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Superlattices and Microstructures, Vol. 21, No. 3, 1997
365
T TF(n) > 2Tc
~n2/3
Bipolaronic insulator
Polaronic metal
Bipolaronic metal with the BCS-like pseudo-gap ∆F
TB(n) = Tc
TB(n) > Tc 2∆F ≅8 Tc
FSC TF(n) = Tc
2∆B(=∆SC) γ ≅ B Tc
2∆F(≅∆SC) = 3.52 Tc
BSC
n = nopt
n
Fig. 1. 3D-Insulator-metal-superconductor. .
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T
TF(n) >> Tc
~n
Bipolaronic insulator
Polaronic metal
Bipolaronic metal with the BCS-like pseudo-gap ∆F
TB(n) = Tc
TB(n) > Tc
BSC
FSC
TF(n) = Tc
n = nopt
n
Fig. 2. 2D-Insulator-metal-superconductor. .
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(Fig. 1 and 2) in accordance with the observations [1–4]. The crossing of TB (n) and TB (n) close to n ' n opt also takes place. If such crossing of TB (n) and TF (n) takes place up to, or after, the saturation of TB (n), then the dependence of Tc (n) exhibits a sharp maximum, or plateau. These features of Tc (n) were also observed in HTSC [1–4]. So not only low-Tc metal superconductors, with very low concentration of pairing carriers and E bB = 0 [5, 6], but also HTSC, with very high concentration of pairing carriers, belong to FSC systems. .
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Superlattices and Microstructures, Vol. 21, No. 3, 1997
References .
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
}Y. I. Uemura, Physica 185–189C, 733 (1991). } F. Mott, Physica 205C, 191 (1993). N. } Ushio and H. Kamimura, Physica 185–189C, 1699 (1991). H. } N. R. Rao and A. K. Ganguli, Physica 235–240C, 9 (1994). C. } X. Zhao and W. I. Zhu, Physica 235–240C, 75 (1994). Z. } Dzhumanov, Physica 235–240C, 2269 (1994). S. } Dzhumanov, Preprint ICTP IC/95/221. Trieste (Italy, 1995). S. } L. Tholence, L. Puech, A. Sulpice et al. Physica 235–240C, 1545 (1994). I. } H. Kim, K. Levin, and A. Auerbach, Phys. Rev. B 39, 11633 (1989). Ju. } Luban, Phys. Rev. 128, 965 (1962). M. } Dzhumanov, P. J. Baimatov, and P. K. Khabibullaev, Uzb. Zh. Phys. 6, 24 (1992). S.
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