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Precursor non-superconducting pairing and novel superconductivity in underdoped and optimally doped cuprates
Physica C 460–462 (2007) 1131–1132 www.elsevier.com/locate/physc
Precursor non-superconducting pairing and novel superconductivity in underdoped and optimally doped cuprates S. Dzhumanov Institute of Nuclear Physics, Uzbek Academy of Science, 702132 Tashkent, Uzbekistan Available online 28 March 2007
Abstract The precursor non-superconducting pairing of carriers and the novel superconductivity in high-Tc cuprates are studied within the large (bi)polaron model and the new BCS-like fermion and boson mean-field approximations by treating a two-component charge carrier picture. It is argued that pre-formed pairs (large bipolarons) reside in underdoped cuprates between the CuO2 planes and remain localized in the z-direction. With lowering temperature, these large bipolarons condense into a superfluid quasi-one-dimensional Bose-liquid state. In the intermediate electron–phonon coupling regime, the Cooper-pair formation and condensation in the CuO2 layers occur also at different temperatures. Large bipolarons and polaron Cooper pairs in optimally doped cuprates condense into a superfluid Bose-liquid state almost at the same temperature due to the doping-induced dimensional crossover. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Large polarons; Precursor non-superconducting pairing; Large real-space bipolarons; Polaron Cooper pairs; Novel superconductivity
1. Introduction Despite considerable progress both in experimental and theoretical studies of the precursor pairing of carriers in the normal state of underdoped and optimally doped cuprates and its relation to high-Tc superconductivity, there is little understanding of these phenomena [1]. Many experimental results [2–4] strongly suggest that the precursor pairing is unrelated to superconductivity as was first predicted theoretically [5,6] (i.e., it was postulated that the BCS-like pairing of carriers is only a necessary but not a sufficient for the occurrence of superconductivity) long before these observations and the electron–phonon interactions play an important role in high-Tc cuprates. In this paper we study the precursor non-superconducting (SC) pairing of carriers in real (r)- and k-space and the novel superconductivity in underdoped and optimally doped cuprates within the large (bi)polaron model and the new BCS-like fermion and
E-mail address: [email protected] 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.239
boson mean-field approximations by considering a twocomponent carrier picture (see also Ref. [7]). 2. Precursor pairing of carriers in real space The undoped cuprates are charge-transfer (CT) type insulators with long-range antiferromagnetic order. Upon p-type doping, the holes introduced into the oxygen valence band interact with acoustic and optical phonons and they are self-trapped with the formation of (bi)polaronic states in the CT gap of the parent cuprates. The doped carriers are assumed to form the large (bi)polarons [8,9]. The ground state energies of the (bi)polaronic carriers are calculated in the three-dimensional (3D) continuum model and adiabatic approximation by using variational method [9]. Our results show that the charge carriers in doped cuprates are large polarons and bipolarons with binding energies Ep 0.1 eV and EbB 6 Ep, respectively. These layered materials have anisotropic 3D crystal structure, where large polarons bound into r-space pairs (large bipolarons) between the CuO2 planes and remain localized in the z-direction (i.e., along the c-axis). The binding
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energies of large polarons and bipolarons are manifested as the two temperature independent pseudogaps in the excitation spectrum of the underdoped cuprates (e.g., see Ref. [10]). 3. Precursor pairing of carriers in k-space The in-gap polaronic band in doped cuprates merges with the valence band when the system is doped into the metallic regime and the mobile polaronic carriers are confined in the quasi-two-dimensional (2D) CuO2 layers and they have well-defined momentum k at Wp > Ep, where Wp is the bandwidth of polarons. In this case the unusual form of BCS-like theory [6] can describe the precursor pairing of large polarons in the CuO2 layers above Tc naturally. This BCS formalism, extended towards the intermediate electron–phonon coupling strength, leads to the following equation for the pairing pseudogap (PG) DPG and the PG temperature TPG qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z eA e2 þ D2PG 1 de qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh ; ð1Þ ¼ kF 2k B T 0 e2 þ D 2 PG
where kF ¼ V~ F N ðeF Þ; V~ F is the effective pairing interaction potential between two fermions, N(eF) is an elliptic density of states (for the anisotropic hole dispersion), eF is the polaron Fermi energy, e is the hole energy measured from eF, eA is the cutoff energy chosen as eA = Ep for eF < Ep + hxD and eA = Ep + hxD for eF > Ep + hxD, xD is the Debye frequency. At kF 6 1 the doped cuprates are in the strong- and intermediate-coupling regimes. The pairing PG DPG increases with decreasing doping and temperature. 4. Novel superconductivity Usually, the BCS-like pairing of carriers and Bose– Einstein condensation of an ideal Bose-gas of charged bosons are considered as the basic mechanisms of superconductivity in high-Tc cuprates. However, these two traditional approaches might be inadequate for the cuprates. Instead, as argued in Refs. [5,6], one should focus on the alternative approach in which one considers the novel superconductivity as the result of two distinct phenomena, binding of polarons into composite bosons (bipolarons and zero- and non-zero momentum Cooper pairs [5,11]) and condensation of these bosons into a superfluid (SF) Boseliquid state. The localized c-axis large bipolarons with density nB and mass mB in underdoped cuprates will interact and condense into a SF quasi-one-dimensional (1D) Bose-liquid state at their SC transition temperature TC1.
While the polaron Cooper pairs with density nc and mass mc condense into a SF Bose-liquid state in the CuO2 layers at a second SC transition temperature TC2. Such 1D and 2D SF Bose-liquids described by the BCS-like theories of boson pairing would exhibit the new SC properties with half-integer h/4e magnetic flux quantization. According to these theories the unconventional SC order parameters DC1, DC2 and transition temperatures TC1, TC2 of bipolarons and Cooper pairs are determined from the equations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Z eB coth ðe þ j~ lB jÞ D2C1 2k B T 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ de ð2Þ pffiffi 2 V~ B DB 0 lB jÞ D2C1 e ðe þ j~ and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Z eC coth ðe þ j~ lC jÞ D2C2 2k B T 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ de; ð3Þ 2 V~ c Dc 0 ðe þ j~ lC jÞ D2C2 pffiffiffiffiffiffi pffiffiffi where DB ¼ mB =2 2ph; Dc ¼ mc =2ph2 ; V~ B and V~ c are the effective interbipolaron and intercooperon potentials ~B and l ~C are chemical [5], eB and ec are the cutoff energies, l potentials of bipolarons and cooperons, respectively. Eqs. (2) and (3) are solved together with the appropriate equations for particle numbers nB and np = ni + 2nc, where ni is the density of unpaired polarons in the CuO2 layers. Large bipolarons and polaron Cooper pairs in optimally doped cuprates are assumed to have almost the same masses and condensation temperatures and unusual SC states described by 3D SF Bose-liquid theory [5,6] due to the doping-induced dimensional crossover. Acknowledgments This work is supported by the Science and Technology Center of Uzbekistan (Grant F2.1.11) and by the STCU Grant 3505. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. K. Gorny et al., Phys. Rev. Lett. 82 (1999) 177. V.M. Krasnov et al., Phys. Rev. Lett. 84 (2000) 5860. J.L. Tallon et al., Phys. Rev. B 68 (2003) 180501. S. Dzhumanov et al., Physica C 235–240 (1994) 2339. S. Dzhumanov, P.K. Khabibullaev, Pramana J. Phys. 45 (1995) 385. J. Ranninger, J.M. Robin, Physica C 253 (1995) 279. D. Emin, M.S. Hillery, Phys. Rev. B 39 (1989) 6575. S. Dzhumanov et al., Phys. Rev. B 54 (1996) 13121. D. Mihailovic et al., Phys. Rev. B 60 (1999) R6995. S. Fujita, J. Supercond. 5 (1992) 83.
Precursor non-superconducting pairing and novel superconductivity in underdoped and optimally doped cuprates S. Dzhumanov Institute of Nuclear Physics, Uzbek Academy of Science, 702132 Tashkent, Uzbekistan Available online 28 March 2007
Abstract The precursor non-superconducting pairing of carriers and the novel superconductivity in high-Tc cuprates are studied within the large (bi)polaron model and the new BCS-like fermion and boson mean-field approximations by treating a two-component charge carrier picture. It is argued that pre-formed pairs (large bipolarons) reside in underdoped cuprates between the CuO2 planes and remain localized in the z-direction. With lowering temperature, these large bipolarons condense into a superfluid quasi-one-dimensional Bose-liquid state. In the intermediate electron–phonon coupling regime, the Cooper-pair formation and condensation in the CuO2 layers occur also at different temperatures. Large bipolarons and polaron Cooper pairs in optimally doped cuprates condense into a superfluid Bose-liquid state almost at the same temperature due to the doping-induced dimensional crossover. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Large polarons; Precursor non-superconducting pairing; Large real-space bipolarons; Polaron Cooper pairs; Novel superconductivity
1. Introduction Despite considerable progress both in experimental and theoretical studies of the precursor pairing of carriers in the normal state of underdoped and optimally doped cuprates and its relation to high-Tc superconductivity, there is little understanding of these phenomena [1]. Many experimental results [2–4] strongly suggest that the precursor pairing is unrelated to superconductivity as was first predicted theoretically [5,6] (i.e., it was postulated that the BCS-like pairing of carriers is only a necessary but not a sufficient for the occurrence of superconductivity) long before these observations and the electron–phonon interactions play an important role in high-Tc cuprates. In this paper we study the precursor non-superconducting (SC) pairing of carriers in real (r)- and k-space and the novel superconductivity in underdoped and optimally doped cuprates within the large (bi)polaron model and the new BCS-like fermion and
E-mail address: [email protected] 0921-4534/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.239
boson mean-field approximations by considering a twocomponent carrier picture (see also Ref. [7]). 2. Precursor pairing of carriers in real space The undoped cuprates are charge-transfer (CT) type insulators with long-range antiferromagnetic order. Upon p-type doping, the holes introduced into the oxygen valence band interact with acoustic and optical phonons and they are self-trapped with the formation of (bi)polaronic states in the CT gap of the parent cuprates. The doped carriers are assumed to form the large (bi)polarons [8,9]. The ground state energies of the (bi)polaronic carriers are calculated in the three-dimensional (3D) continuum model and adiabatic approximation by using variational method [9]. Our results show that the charge carriers in doped cuprates are large polarons and bipolarons with binding energies Ep 0.1 eV and EbB 6 Ep, respectively. These layered materials have anisotropic 3D crystal structure, where large polarons bound into r-space pairs (large bipolarons) between the CuO2 planes and remain localized in the z-direction (i.e., along the c-axis). The binding
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energies of large polarons and bipolarons are manifested as the two temperature independent pseudogaps in the excitation spectrum of the underdoped cuprates (e.g., see Ref. [10]). 3. Precursor pairing of carriers in k-space The in-gap polaronic band in doped cuprates merges with the valence band when the system is doped into the metallic regime and the mobile polaronic carriers are confined in the quasi-two-dimensional (2D) CuO2 layers and they have well-defined momentum k at Wp > Ep, where Wp is the bandwidth of polarons. In this case the unusual form of BCS-like theory [6] can describe the precursor pairing of large polarons in the CuO2 layers above Tc naturally. This BCS formalism, extended towards the intermediate electron–phonon coupling strength, leads to the following equation for the pairing pseudogap (PG) DPG and the PG temperature TPG qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z eA e2 þ D2PG 1 de qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tanh ; ð1Þ ¼ kF 2k B T 0 e2 þ D 2 PG
where kF ¼ V~ F N ðeF Þ; V~ F is the effective pairing interaction potential between two fermions, N(eF) is an elliptic density of states (for the anisotropic hole dispersion), eF is the polaron Fermi energy, e is the hole energy measured from eF, eA is the cutoff energy chosen as eA = Ep for eF < Ep + hxD and eA = Ep + hxD for eF > Ep + hxD, xD is the Debye frequency. At kF 6 1 the doped cuprates are in the strong- and intermediate-coupling regimes. The pairing PG DPG increases with decreasing doping and temperature. 4. Novel superconductivity Usually, the BCS-like pairing of carriers and Bose– Einstein condensation of an ideal Bose-gas of charged bosons are considered as the basic mechanisms of superconductivity in high-Tc cuprates. However, these two traditional approaches might be inadequate for the cuprates. Instead, as argued in Refs. [5,6], one should focus on the alternative approach in which one considers the novel superconductivity as the result of two distinct phenomena, binding of polarons into composite bosons (bipolarons and zero- and non-zero momentum Cooper pairs [5,11]) and condensation of these bosons into a superfluid (SF) Boseliquid state. The localized c-axis large bipolarons with density nB and mass mB in underdoped cuprates will interact and condense into a SF quasi-one-dimensional (1D) Bose-liquid state at their SC transition temperature TC1.
While the polaron Cooper pairs with density nc and mass mc condense into a SF Bose-liquid state in the CuO2 layers at a second SC transition temperature TC2. Such 1D and 2D SF Bose-liquids described by the BCS-like theories of boson pairing would exhibit the new SC properties with half-integer h/4e magnetic flux quantization. According to these theories the unconventional SC order parameters DC1, DC2 and transition temperatures TC1, TC2 of bipolarons and Cooper pairs are determined from the equations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Z eB coth ðe þ j~ lB jÞ D2C1 2k B T 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ de ð2Þ pffiffi 2 V~ B DB 0 lB jÞ D2C1 e ðe þ j~ and
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Z eC coth ðe þ j~ lC jÞ D2C2 2k B T 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ de; ð3Þ 2 V~ c Dc 0 ðe þ j~ lC jÞ D2C2 pffiffiffiffiffiffi pffiffiffi where DB ¼ mB =2 2ph; Dc ¼ mc =2ph2 ; V~ B and V~ c are the effective interbipolaron and intercooperon potentials ~B and l ~C are chemical [5], eB and ec are the cutoff energies, l potentials of bipolarons and cooperons, respectively. Eqs. (2) and (3) are solved together with the appropriate equations for particle numbers nB and np = ni + 2nc, where ni is the density of unpaired polarons in the CuO2 layers. Large bipolarons and polaron Cooper pairs in optimally doped cuprates are assumed to have almost the same masses and condensation temperatures and unusual SC states described by 3D SF Bose-liquid theory [5,6] due to the doping-induced dimensional crossover. Acknowledgments This work is supported by the Science and Technology Center of Uzbekistan (Grant F2.1.11) and by the STCU Grant 3505. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. K. Gorny et al., Phys. Rev. Lett. 82 (1999) 177. V.M. Krasnov et al., Phys. Rev. Lett. 84 (2000) 5860. J.L. Tallon et al., Phys. Rev. B 68 (2003) 180501. S. Dzhumanov et al., Physica C 235–240 (1994) 2339. S. Dzhumanov, P.K. Khabibullaev, Pramana J. Phys. 45 (1995) 385. J. Ranninger, J.M. Robin, Physica C 253 (1995) 279. D. Emin, M.S. Hillery, Phys. Rev. B 39 (1989) 6575. S. Dzhumanov et al., Phys. Rev. B 54 (1996) 13121. D. Mihailovic et al., Phys. Rev. B 60 (1999) R6995. S. Fujita, J. Supercond. 5 (1992) 83.