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Fluctuation-induced pseudogap in high-Tc cuprates
Physica C 408–410 (2004) 422–423 www.elsevier.com/locate/physc
Fluctuation-induced pseudogap in high-Tc cuprates Kirill Mitsen, Olga Ivanenko
*
Lebedev Physical Institute, Department of Solid State Physics, Leninskii pr. 53, Moscow 119991, Russia
Abstract The pseudogap in HTSC is treated in the framework of the model assuming that the interaction of band electrons with negative-U centers (NUC) is responsible for anomalous properties of HTSC compounds. It is shown that the pseudogap is the superconducting gap arising in under- and optimum-doped HTSC at T > Tc due to the large fluctuations of NUC occupation in small clusters. The temperature of pseudogap opening T is calculated for YBa2 Cu3 O6þd as a function of doping. The results are in a good agreement with experiment. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Pseudogap; Negative-U center; Percolation
Earlier [1] we have proposed the mechanism of NUC formation in HTSC. In accordance with [1] these NUC are formed at certain conditions on the pairs of neighboring Cu ions. The most of anomalous HTSC properties can be regarded as a consequence of the interaction of electrons from oxygen px;y -band with NUC. This interaction results in strong renormalization of the effective electron–electron interaction if the scattering processes with the intermediate virtual bound states are taken into account [2–4]. NUC in HTSC play a role of pair acceptors resulting in the generation of the hole pairs localized in a vicinity of NUC. Conductivity and superconductivity in such a system arise if the percolation over the hole pair localization areas or (that is the same) over the NUC chains takes place. Earlier [1] we have assumed, that the pseudogap observed in HTSC, is nothing else than the superconducting gap but developing at T > Tc due to the large fluctuations of NUC occupation in small NUC clusters because of electron pairs transitions between NUC and px;y -band. The matter is that in BCS superconductor with electron– phonon interaction the superconducting gap vanishes due to the thermal excitations over the Fermi surface that decrease the number of unoccupied states available for
*
Corresponding author. E-mail address: [email protected] (O. Ivanenko).
the electron pair scattering. In the model [1] the mechanism of superconducting gap suppression is the occupation of NUC by electrons. Therefore fluctuation-induced reduction of pair electronic level occupation will amplify the superconducting interaction and can result in fluctuation-induced turning on superconductivity at T > T > Tc0 (here Tc0 is equilibrium value of Tc ). Such large fluctuations can happened in the underdoped and optimum-doped samples when the significant number of NUC belong to finite nonpercolative clusters. The average size of finite clusters decreases with doping and relative fluctuations of NUC occupation increase in these clusters (i.e. T goes up). On the other hand, in overdoped samples when practically all Cu-ions belong to infinite percolation cluster, large fluctuations become impossible. Here we are going to deduce the dependence of pseudogap opening temperature T on doping d for YBa2 Cu3 O6þd on the ground of the proposed model [1]. According to [1] NUC is formed in a given YBa2 Cu3 O6þd unit cell when three consecutive oxygen sites in CuO3 chain (Fig. 1) are occupied. The concentration of such cells at random distribution of oxygen ions in chains is equal to d3 . The percolation comes when the percolation threshold over NUC on the square lattice is exceeded. Supposing that NUC distribution over square lattice sites is random the percolation threshold pc ¼ d3c ¼ 0:593 [5], i.e. dc ¼ 0:84 according to YBa2 Cu3 O6þd phase diagram [6].
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.017
K. Mitsen, O. Ivanenko / Physica C 408–410 (2004) 422–423
423
Fig. 1. The negative-U center in YBa2 Cu3 O6þd is formed when 3 consecutive oxygen sites in a CuO3 -chain are occupied.
At d < dc the combined NUC form the chain clusters of various sizes. The NUC occupation numbers and, consequently, hole concentration in px;y -band per cell for every finite cluster, g, is equal [1]: g ¼ 2T =ðT þ T0 Þ; where T0 is temperature independent constant. Hall measurements [7] give T0 400 K. Let us consider a cluster, containing S of NUC. The number of electrons on NUC in given cluster at temperature T ispequal N ¼ 2TS=ðT þ T0 Þ. This number can ffiffiffiffi decrease on N ¼ ð2TS=ðT þ T0 ÞÞ1=2 due to fluctuations. A necessary condition for the superconductivity to arise at temperature T because of fluctuations in given cluster is pffiffiffiffiffiffiffiffiffiffiffiffiffi N ðT Þ NðT Þ ¼ Nc ; ð1Þ where Nc ¼ 2Tc S=ðTc þ T0 Þ––the number of electrons on NUC at the superconducting transition temperature Tc . Solving Eq. (1) we find T as a function of S. Fig. 2 shows T ðSÞ dependences for T0 ¼ 300 and 800 K. It is seen that T value weakly depends on T0 . The range of S values is limited at the left by the value S ¼ 2 that cor i.e. the coherence responds to the cluster size 16 A, length in CuO2 planes. In order to find T ðdÞ-dependence it is necessary to know the average size of finite clusters S as a function of d. For this purpose we use the results of work [8], where
Fig. 3. The temperature of pseudogap opening T as a function of oxygen concentration d for YBa2 Cu3 O6þd (solid line). Solid squares correspond to the experimental data from [9].
the statistics of finite clusters SðpÞ was found by Monte Carlo method for site percolation problem for hard-core particles on the triangular 2D-lattice. In this task the percolation threshold value pc ¼ 0:85 equals to that in our case. Here p is a probability that the given site is occupied and pc is the value of p appropriate to the percolation threshold. Taking into account the scaling behavior of SðpÞ in a vicinity of pc and the coincidence of percolation thresholds in both cases the statistics of finite clusters should be expected to be identical, too. For 0:1 < p < 0:82 the curve SðpÞ (Fig. 5b from [8]) can be approximated by the dependence SðpÞ 0:9 ð0:95 pÞ. Considering the oxygen distribution over the sites in chains to be random, we have the occupation probability of these positions p ¼ d. The substitution of SðdÞ in the Eq. (1) gives T ðdÞ dependence for YBa2 Cu3 O6þd , shown in Fig. 3 by solid line. The solid squares on Fig. 3 are the experimental data from [9] where the temperature of a pseudogap opening was determined as the temperature of deviation of resistance temperature dependence from linear behavior. Taking into account the conditional character of such determination the agreement should be considered as good.
References
Fig. 2. T as a function of S for two values T0 ¼ 300 and 800 K.
[1] K.V. Mitsen, O.M. Ivanenko, JETP 91 (2000) 579. [2] E. Simanek, Solid State Commun. 32 (1979) 731. [3] C.S. Ting, D.N. Talwar, K.L. Ngai, Phys. Rev. Lett. 45 (1980) 1213. [4] G.M. Eliashberg, JETP Lett. 46 (1987) S81. [5] R.M. Ziff, Phys. Rev. Lett. 69 (1992) 2670. [6] R. Beyers et al., Nature 340 (1989) 619. [7] N.G. Ong et al., Physica C 185–189 (1991) 34. [8] C.-K. Hu, K.-S. Mak, Phys. Rev. B 39 (1989) 2948. [9] B. Wuyts, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 53 (1996) 9418.
Fluctuation-induced pseudogap in high-Tc cuprates Kirill Mitsen, Olga Ivanenko
*
Lebedev Physical Institute, Department of Solid State Physics, Leninskii pr. 53, Moscow 119991, Russia
Abstract The pseudogap in HTSC is treated in the framework of the model assuming that the interaction of band electrons with negative-U centers (NUC) is responsible for anomalous properties of HTSC compounds. It is shown that the pseudogap is the superconducting gap arising in under- and optimum-doped HTSC at T > Tc due to the large fluctuations of NUC occupation in small clusters. The temperature of pseudogap opening T is calculated for YBa2 Cu3 O6þd as a function of doping. The results are in a good agreement with experiment. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Pseudogap; Negative-U center; Percolation
Earlier [1] we have proposed the mechanism of NUC formation in HTSC. In accordance with [1] these NUC are formed at certain conditions on the pairs of neighboring Cu ions. The most of anomalous HTSC properties can be regarded as a consequence of the interaction of electrons from oxygen px;y -band with NUC. This interaction results in strong renormalization of the effective electron–electron interaction if the scattering processes with the intermediate virtual bound states are taken into account [2–4]. NUC in HTSC play a role of pair acceptors resulting in the generation of the hole pairs localized in a vicinity of NUC. Conductivity and superconductivity in such a system arise if the percolation over the hole pair localization areas or (that is the same) over the NUC chains takes place. Earlier [1] we have assumed, that the pseudogap observed in HTSC, is nothing else than the superconducting gap but developing at T > Tc due to the large fluctuations of NUC occupation in small NUC clusters because of electron pairs transitions between NUC and px;y -band. The matter is that in BCS superconductor with electron– phonon interaction the superconducting gap vanishes due to the thermal excitations over the Fermi surface that decrease the number of unoccupied states available for
*
Corresponding author. E-mail address: [email protected] (O. Ivanenko).
the electron pair scattering. In the model [1] the mechanism of superconducting gap suppression is the occupation of NUC by electrons. Therefore fluctuation-induced reduction of pair electronic level occupation will amplify the superconducting interaction and can result in fluctuation-induced turning on superconductivity at T > T > Tc0 (here Tc0 is equilibrium value of Tc ). Such large fluctuations can happened in the underdoped and optimum-doped samples when the significant number of NUC belong to finite nonpercolative clusters. The average size of finite clusters decreases with doping and relative fluctuations of NUC occupation increase in these clusters (i.e. T goes up). On the other hand, in overdoped samples when practically all Cu-ions belong to infinite percolation cluster, large fluctuations become impossible. Here we are going to deduce the dependence of pseudogap opening temperature T on doping d for YBa2 Cu3 O6þd on the ground of the proposed model [1]. According to [1] NUC is formed in a given YBa2 Cu3 O6þd unit cell when three consecutive oxygen sites in CuO3 chain (Fig. 1) are occupied. The concentration of such cells at random distribution of oxygen ions in chains is equal to d3 . The percolation comes when the percolation threshold over NUC on the square lattice is exceeded. Supposing that NUC distribution over square lattice sites is random the percolation threshold pc ¼ d3c ¼ 0:593 [5], i.e. dc ¼ 0:84 according to YBa2 Cu3 O6þd phase diagram [6].
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.03.017
K. Mitsen, O. Ivanenko / Physica C 408–410 (2004) 422–423
423
Fig. 1. The negative-U center in YBa2 Cu3 O6þd is formed when 3 consecutive oxygen sites in a CuO3 -chain are occupied.
At d < dc the combined NUC form the chain clusters of various sizes. The NUC occupation numbers and, consequently, hole concentration in px;y -band per cell for every finite cluster, g, is equal [1]: g ¼ 2T =ðT þ T0 Þ; where T0 is temperature independent constant. Hall measurements [7] give T0 400 K. Let us consider a cluster, containing S of NUC. The number of electrons on NUC in given cluster at temperature T ispequal N ¼ 2TS=ðT þ T0 Þ. This number can ffiffiffiffi decrease on N ¼ ð2TS=ðT þ T0 ÞÞ1=2 due to fluctuations. A necessary condition for the superconductivity to arise at temperature T because of fluctuations in given cluster is pffiffiffiffiffiffiffiffiffiffiffiffiffi N ðT Þ NðT Þ ¼ Nc ; ð1Þ where Nc ¼ 2Tc S=ðTc þ T0 Þ––the number of electrons on NUC at the superconducting transition temperature Tc . Solving Eq. (1) we find T as a function of S. Fig. 2 shows T ðSÞ dependences for T0 ¼ 300 and 800 K. It is seen that T value weakly depends on T0 . The range of S values is limited at the left by the value S ¼ 2 that cor i.e. the coherence responds to the cluster size 16 A, length in CuO2 planes. In order to find T ðdÞ-dependence it is necessary to know the average size of finite clusters S as a function of d. For this purpose we use the results of work [8], where
Fig. 3. The temperature of pseudogap opening T as a function of oxygen concentration d for YBa2 Cu3 O6þd (solid line). Solid squares correspond to the experimental data from [9].
the statistics of finite clusters SðpÞ was found by Monte Carlo method for site percolation problem for hard-core particles on the triangular 2D-lattice. In this task the percolation threshold value pc ¼ 0:85 equals to that in our case. Here p is a probability that the given site is occupied and pc is the value of p appropriate to the percolation threshold. Taking into account the scaling behavior of SðpÞ in a vicinity of pc and the coincidence of percolation thresholds in both cases the statistics of finite clusters should be expected to be identical, too. For 0:1 < p < 0:82 the curve SðpÞ (Fig. 5b from [8]) can be approximated by the dependence SðpÞ 0:9 ð0:95 pÞ. Considering the oxygen distribution over the sites in chains to be random, we have the occupation probability of these positions p ¼ d. The substitution of SðdÞ in the Eq. (1) gives T ðdÞ dependence for YBa2 Cu3 O6þd , shown in Fig. 3 by solid line. The solid squares on Fig. 3 are the experimental data from [9] where the temperature of a pseudogap opening was determined as the temperature of deviation of resistance temperature dependence from linear behavior. Taking into account the conditional character of such determination the agreement should be considered as good.
References
Fig. 2. T as a function of S for two values T0 ¼ 300 and 800 K.
[1] K.V. Mitsen, O.M. Ivanenko, JETP 91 (2000) 579. [2] E. Simanek, Solid State Commun. 32 (1979) 731. [3] C.S. Ting, D.N. Talwar, K.L. Ngai, Phys. Rev. Lett. 45 (1980) 1213. [4] G.M. Eliashberg, JETP Lett. 46 (1987) S81. [5] R.M. Ziff, Phys. Rev. Lett. 69 (1992) 2670. [6] R. Beyers et al., Nature 340 (1989) 619. [7] N.G. Ong et al., Physica C 185–189 (1991) 34. [8] C.-K. Hu, K.-S. Mak, Phys. Rev. B 39 (1989) 2948. [9] B. Wuyts, V.V. Moshchalkov, Y. Bruynseraede, Phys. Rev. B 53 (1996) 9418.