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New type of antiferromagnetic polaron and bipolaron in high-Tc superconductors
Physica B 284}288 (2000) 437}438
New type of antiferromagnetic polaron and bipolaron in high-¹ superconductors Ewelina M. Hankiewicz*, Ryszard Buczko, Zbys"aw Wilamowski Institute of Physics, Polish Academy of Sciences, Al. Lotniko& w 32/46, PL 02-668 Warsaw Poland
Abstract The possibility of formation of a new type of polaron based on the quantum aniferromagnet (AF) model is reported. We take into account exchange interactions between localized d}d spins of the AF, as well as the p}d interaction of the AF with p-carriers. The energy minimum is found when maximum charge density occurs on every second spin. The formation of such `comb-likea polarons results from the damping of quantum #uctuations and the appearance of Van Vleck-like staggered magnetization. Such polarons tend to form pairs coupled by an AF `gluea. 2000 Elsevier Science B.V. All rights reserved. Keywords: Bipolaron; Quantum #uctuations
There are some convincing arguments that occurrence of bipolarons is responsible for the formation of a Bose liquid in high-¹ superconductors [1,2]. The mechanism of polaron and bipolaron formation, however, still remains an open question. Theoretical models of this e!ect invoke the phonons, plasmons as well as a magnetic `gluea [3,4]. In this paper we show that the spin exchange interactions alone can lead to the formation of magnetic polarons, called by us the `comb-likea polarons, which are characterized by a strong tendency to pair formation. For CuO -based superconductors binding energy of polarons is comparable to d}d coupling, J , i.e., of the \ order of 7.5 meV. We solve numerically the spin Hamiltonian which contains the term of d}d coupling between neighboring d-spins within the antiferromagnetic (AF) cluster, 2J } S ) S , and the term of p}d exchange be G H tween the p-carrier and the d-spins, J (i)S ) r. We \ G assume, that also in the case of a strong p}d hybridization, we can treat the carrier spin, r as a separate from d-spins. The p}d exchange constants J (i) are assumed \ to be proportional to the carrier density at the ith spin.
* Corresponding author. E-mail address: [email protected] (E.M. Hankiewicz).
The normalization condition of the carrier wave function guarantees that the sum of J (i) does not depend on \ the distribution, but is equal to a parameter N a which describes the strength of the p}d coupling. We examine various polaron shapes. We found that `comb-likea polarons on 1d and 2d clusters, where the electron density is distributed on spins from only one of Neel sublattices, are the most energetically favorable (see Fig. 1(a)). The total magnetic moment of such polaron is equal to and does not change with an increase of N a. The net magnetic moment of d-spins remains zero but, because of a nonvanishing correlation between p and d-spins, an energy gain due to p}d coupling occurs. The correlators 1p"S 2, i.e., the polarization G of d-spins seen by the carrier spin, is shown in Fig. 1(b). For a weak N a this staggered magnetization increases linearly with N a. Then the energy gain originating from the balance of p}d and d}d couplings, *E , increases with the square of N a, re#ecting Van Vleck character of the AF magnetization (see Fig. 2). The induction of the staggered magnetization is related to stabilization of quantum #uctuations of the AF cluster. In the inset in Fig. 2, *E is plotted as a function of inverse size of the AF chain. The weak dependence of *E on the AF cluster size shows that the spin exchange does not lead to any localization of the `comb-likea polarons. *E tends to a constant value for large size of cluster.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 0 0 5 - 0
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E.M. Hankiewicz et al. / Physica B 284}288 (2000) 437}438
Fig. 1. (a) Spatial distribution of carrier density in a `comb-likea bipolaron; (b) spin polarization of localized d-spins, S , as seen G by the carrier p-spins, p. For the monopolaron the polarization has a similar character.
The quadratic dependence of *E (N a) shows the possibility of formation of polaron pairs. In Fig. 2 the energy of the ground singlet, *E , and the "rst excited triplet of the bipolaron are shown. The binding energy, E , of such a pair is equal to *E !2 *E . For a weak p}d coupling, E is equal to of the singlet-triplet energy distance J . The p-spins are antiferromagnetically cor\ related at the singlet, and ferromagnetically at the triplet state. Thus, one can conclude that two carrier spins are e!ectively coupled by an indirect Heisenberg exchange, J r ) r, where the AF cluster plays a role of a `gluea. \ In CuO -based superconductors N a +10 J and \ the linear response approximation fails. The energies: *E , *E , E , and J , all saturate. As it is shown in \ the inset in Fig. 2 the saturation value of J is of the \ order of J for a small cluster and it tends to zero with \ a cluster increase. This allows us to conclude that a short AF coherence range (observed in HTSC) is the necessary condition for the existence of an AF `gluea. Simultaneously, the short AF coherence limits the range of bipolaron localization } an extension of the bipolaron
Fig. 2. Energy levels of the polaron (dashed) and of the bipolaron (solid curves) as a function of p}d coupling N a for M"16 spins. Dotted line: energy of ground state of antiferromagnet. The inset shows the dependence of the energy gain of the polaron, *E , for a small N a, and the e!ective indirect exchange, J , between carrier spins in the bipolaron for very \ large N a as a function of inverse cluster size.
beyond the AF coherence range would be accompanied with a decrease of the p}d energy gain.
Acknowledgement This work was supported by KBN grant 2 P03B 0007 16.
References [1] A.S. Alexandrov, J. Supercond. 12 (1999) 49. [2] S. Zhang, Science 275 (1997) 1089. [3] E. Salje et al., (Eds.), Polarons and Bipolarons in High-¹ Superconductors, Cambridge University Press, Cambridge, 1995. [4] N.D. Mathur et al., Nature 394 (1998) 39.
New type of antiferromagnetic polaron and bipolaron in high-¹ superconductors Ewelina M. Hankiewicz*, Ryszard Buczko, Zbys"aw Wilamowski Institute of Physics, Polish Academy of Sciences, Al. Lotniko& w 32/46, PL 02-668 Warsaw Poland
Abstract The possibility of formation of a new type of polaron based on the quantum aniferromagnet (AF) model is reported. We take into account exchange interactions between localized d}d spins of the AF, as well as the p}d interaction of the AF with p-carriers. The energy minimum is found when maximum charge density occurs on every second spin. The formation of such `comb-likea polarons results from the damping of quantum #uctuations and the appearance of Van Vleck-like staggered magnetization. Such polarons tend to form pairs coupled by an AF `gluea. 2000 Elsevier Science B.V. All rights reserved. Keywords: Bipolaron; Quantum #uctuations
There are some convincing arguments that occurrence of bipolarons is responsible for the formation of a Bose liquid in high-¹ superconductors [1,2]. The mechanism of polaron and bipolaron formation, however, still remains an open question. Theoretical models of this e!ect invoke the phonons, plasmons as well as a magnetic `gluea [3,4]. In this paper we show that the spin exchange interactions alone can lead to the formation of magnetic polarons, called by us the `comb-likea polarons, which are characterized by a strong tendency to pair formation. For CuO -based superconductors binding energy of polarons is comparable to d}d coupling, J , i.e., of the \ order of 7.5 meV. We solve numerically the spin Hamiltonian which contains the term of d}d coupling between neighboring d-spins within the antiferromagnetic (AF) cluster, 2J } S ) S , and the term of p}d exchange be G H tween the p-carrier and the d-spins, J (i)S ) r. We \ G assume, that also in the case of a strong p}d hybridization, we can treat the carrier spin, r as a separate from d-spins. The p}d exchange constants J (i) are assumed \ to be proportional to the carrier density at the ith spin.
* Corresponding author. E-mail address: [email protected] (E.M. Hankiewicz).
The normalization condition of the carrier wave function guarantees that the sum of J (i) does not depend on \ the distribution, but is equal to a parameter N a which describes the strength of the p}d coupling. We examine various polaron shapes. We found that `comb-likea polarons on 1d and 2d clusters, where the electron density is distributed on spins from only one of Neel sublattices, are the most energetically favorable (see Fig. 1(a)). The total magnetic moment of such polaron is equal to and does not change with an increase of N a. The net magnetic moment of d-spins remains zero but, because of a nonvanishing correlation between p and d-spins, an energy gain due to p}d coupling occurs. The correlators 1p"S 2, i.e., the polarization G of d-spins seen by the carrier spin, is shown in Fig. 1(b). For a weak N a this staggered magnetization increases linearly with N a. Then the energy gain originating from the balance of p}d and d}d couplings, *E , increases with the square of N a, re#ecting Van Vleck character of the AF magnetization (see Fig. 2). The induction of the staggered magnetization is related to stabilization of quantum #uctuations of the AF cluster. In the inset in Fig. 2, *E is plotted as a function of inverse size of the AF chain. The weak dependence of *E on the AF cluster size shows that the spin exchange does not lead to any localization of the `comb-likea polarons. *E tends to a constant value for large size of cluster.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 0 0 5 - 0
438
E.M. Hankiewicz et al. / Physica B 284}288 (2000) 437}438
Fig. 1. (a) Spatial distribution of carrier density in a `comb-likea bipolaron; (b) spin polarization of localized d-spins, S , as seen G by the carrier p-spins, p. For the monopolaron the polarization has a similar character.
The quadratic dependence of *E (N a) shows the possibility of formation of polaron pairs. In Fig. 2 the energy of the ground singlet, *E , and the "rst excited triplet of the bipolaron are shown. The binding energy, E , of such a pair is equal to *E !2 *E . For a weak p}d coupling, E is equal to of the singlet-triplet energy distance J . The p-spins are antiferromagnetically cor\ related at the singlet, and ferromagnetically at the triplet state. Thus, one can conclude that two carrier spins are e!ectively coupled by an indirect Heisenberg exchange, J r ) r, where the AF cluster plays a role of a `gluea. \ In CuO -based superconductors N a +10 J and \ the linear response approximation fails. The energies: *E , *E , E , and J , all saturate. As it is shown in \ the inset in Fig. 2 the saturation value of J is of the \ order of J for a small cluster and it tends to zero with \ a cluster increase. This allows us to conclude that a short AF coherence range (observed in HTSC) is the necessary condition for the existence of an AF `gluea. Simultaneously, the short AF coherence limits the range of bipolaron localization } an extension of the bipolaron
Fig. 2. Energy levels of the polaron (dashed) and of the bipolaron (solid curves) as a function of p}d coupling N a for M"16 spins. Dotted line: energy of ground state of antiferromagnet. The inset shows the dependence of the energy gain of the polaron, *E , for a small N a, and the e!ective indirect exchange, J , between carrier spins in the bipolaron for very \ large N a as a function of inverse cluster size.
beyond the AF coherence range would be accompanied with a decrease of the p}d energy gain.
Acknowledgement This work was supported by KBN grant 2 P03B 0007 16.
References [1] A.S. Alexandrov, J. Supercond. 12 (1999) 49. [2] S. Zhang, Science 275 (1997) 1089. [3] E. Salje et al., (Eds.), Polarons and Bipolarons in High-¹ Superconductors, Cambridge University Press, Cambridge, 1995. [4] N.D. Mathur et al., Nature 394 (1998) 39.