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Thermodynamic properties of antiferromagnetic clusters
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31 www.elsevier.com/locate/jmmm
Thermodynamic properties of antiferromagnetic clusters F. Lo´pez-Urı´ asa,, G.M. Pastorb a
Advanced Materials Department, IPICYT, Camino a la Presa San Jose´ 2055, Col. Lomas 4a Seccio´n, 78216 San Luis Potosı´, SLP, Mexico b Laboratoire de Physique Quantique, Centre National de la Recherche Scientifique, Universite´ Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France Available online 15 April 2005
Abstract The finite temperature magnetic properties of antiferromagnetic (AF) clusters are investigated in the framework of the Hubbard model at half-band filling. The ground- and excited-state electronic energies En, the corresponding wave functions jCni, and the total spin Sn are calculated exactly by using a full many-body diagonalization method. A complete optimization and sampling of the topological structures of the cluster is performed. Representative results are presented for the specific heat C(T), magnetic susceptibility wðTÞ, and spin correlation functions gij ðTÞ between sites i and j of clusters having N ¼ 7 atoms. Low-temperature peaks in C(T) are observed and they are interpreted in terms of the spin-resolved excitation spectrum. wðTÞ shows a typical AF-like behavior of the form w 1=ðT þ T N Þ from which the cluster ‘‘Ne´el’’ temperature TN is derived. Finally, the effects of temperature-induced fluctuations of the cluster structure are discussed. r 2005 Elsevier B.V. All rights reserved. PACS: 36.40.Cg; 75.10.Jm Keywords: Atomic clusters; Antiferromagnetism; Finite-temperature; Exact diagonalization
1. Introduction The finite-temperature properties of magnetic clusters motivate a remarkable interest in current research on low-dimensional magnetism [1–3]. Antiferromagnetic (AF) clusters are a major
Corresponding author. Tel.: +52 444 8342000;
fax: +52 444 8342040. E-mail address: fl[email protected] (F. Lo´pez-Urı´ as).
theoretical challenge in this field, in particular due to the subtle competition between itinerant and localized behaviors and the resulting strong electron-correlation effects. Transition-metal (TM) and rare-earth (RE) clusters have attracted a considerable attention from both theoretical and experimental standpoints [4,5]. Experiments have revealed that the magnetic properties of TM and RE clusters are often very different from those of the corresponding solids. TM elements at the beginning of the series, i.e., for configuration
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.03.048
ARTICLE IN PRESS F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31
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3dn4sm with np5, one observes weak or vanishing magnetic moments and a clear tendency to AF correlations. From the point of view of theory, the study of AF metal clusters poses an interesting challenge due to the difficulty in describing the delicate balance between electron delocalization and local charge fluctuations. As in solids, a precise treatment of electron correlations and the implementation of sophisticated many-body techniques are necessary in order to determine the ground-state properties, the electronic excitations, and the resulting finite-temperature behavior. Moreover, in the case of small clusters, additional effects are expected to play a role, in a way that is specific to the extremely large surface-to-volume ratio and to the presence of relatively loose surface bonds. In particular, one should take into account that temperature-induced changes or fluctuations of the cluster geometry [6].
2. Theory The development of the theory of cluster magnetism is hindered by two main difficulties: an accurate treatment of electron correlations, which are fundamental for a profound understanding of magnetism, and the optimization of the cluster geometry, on which the magnetic properties of itinerant electrons are known to depend strongly. Since these two problems are formidable, and their combination even more so, it is natural that most studies available hitherto have attempted to deal at best with one of them at a time. In this paper we consider a model, which is simple enough to allow an exact solution of the many-body problem and of the geometry sampling in topological space, and which at the same time contains enough complexity to be able to shed light on the physics of real systems (e.g., 3d-TM). The most popular and probably the most simple physical model for describing correlated itinerant electrons on a lattice is given by the well-known Hubbard Hamiltonian [7], X þ X H ¼ t c^ is c^ js þ U (1) n^ i# n^ i" . ðiajÞ s
i
The first term is the kinetic-energy operator that describes electronic hoppings between nearest neighbors sites i and j (t40). The second term takes into account the intra-atomic Coulomb repulsions that are the dominant contribution from the electron–electron interaction (UX0) [7]. The model is characterized by the number of the electrons n and by the dimensionless parameter U/t that measures the importance of correlations. The Hubbard model for small clusters is solved numerically by expanding its eigenfunctions jCli in a complete set of basis states jFm i which have definite occupation numbers nm is at all orbitals is, m i.e., n^ is jFm i ¼ nm is jFm i with nis ¼ 0 or 1. The expansion coefficients alm of jCl i ¼ Sm alm jFm i are determined by standard numerical diagonalization procedures [6]. In this way, both ground- and excited-state properties are obtained exactly within the framework of the Hubbard Hamiltonian [Eq. (1)]. The finite temperature properties are derived from the canonical partition function Q over electronic and structural degrees of freedom. Q is given by XX Q¼ eb l ðgÞ ; (2) g
l
where l ðgÞ is the lth eigenenergy corresponding to a cluster geometry g. b ¼ 1=T refers to the temperature of the cluster source that defines the macroscopic thermal bath with which the small clusters are in equilibrium before expansion. Thermal averages correspond then to the ensemble of clusters in the beam. Moreover, keeping n and N fixed (canonical ensemble) corresponds to the experimental situation in charge and size selected beams [4]. Taking into account only NN hoppings with fixed bond lengths results in a discretization of the configuration space. The sampling of cluster geometries can be performed within the graph space. Notice that in spite of these simplifications the number of site configurations increases extremely rapid with N [6]. The properties of interest, such as the specific heat C(T), its spin-resolved contributions CS(T), the magnetic susceptibility wðTÞ, and the spin correlation functions gij ðTÞ ¼ hS i S j i, are obtained straightforwardly from the derivatives of Q.
ARTICLE IN PRESS F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31
3. Results Fig. 1 shows the temperature-dependent magnetic properties of a Hubbard cluster having N ¼ 7 atoms, one electron per site (n ¼ N), and Coulomb repulsion strength U=t ¼ 64. Here we focus on the effect on electronic excitations by considering the same cluster structure for all T, which corresponds to the most stable topology at T ¼ 0 [see inset of Fig. 1(b)]. The specific heat C (dashed-line) exhibits a low-temperature narrow peak followed a broader peak at higher T. The first one corresponds to doublet–doublet excitations (i.e., within the minimum-spin S ¼ 1=2 manifold) while the second one involves spin excitations with a major contribution from S ¼ 12 to 32 transitions. It is interesting to observe that this double-peak structure does not appear in clusters with and even number of atoms (half-band filling). For even N ¼
Fig. 1. Temperature dependence of (a) specific heat C, (b) NN spin-correlation functions gij ¼ hSi S j i, and (c) inverse of the magnetic susceptibility 1=w for a cluster having N ¼ 7 atoms, n ¼ N ¼ 7 electrons, and Coulomb repulsion strength U=t ¼ 64. The inset in (b) illustrates the structure which yields the optimum ground-state energy.
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n the ground state is a singlet and the specific heat presents only one clearly detached low-temperature peak. An analysis in terms of the spinresolved contributions to C shows that in this case the relevant low-lying excited states involve changes in the total spin, mainly from S ¼ 0 to 1 (singlet-triple excitations). The different NN spin-correlation functions gij show interesting dependences as a function of temperature which can be related to the local environment of the atoms i and j [see Fig. 1(b) and the labeling of the atoms in the illustration of the structure]. One observes that as T increases, the thermal fluctuations progressively destroy the ground-state spin correlations that are strongly AF (gij o0 at low T). This is for example the case for i ¼ 7 and j ¼ 2, or i ¼ 6 and j ¼ 4. More remarkable effects are found among the atoms i ¼ 1, 3, 5, and 7, due to magnetic frustrations. At low temperatures, the AF correlations are strong between sites 7 and 1, while the correlations between 7 and 5 (or equivalently between 1 and 5) are weak. However, as T increases another type of AF-like order is favored, in which the AF coupling between sites 7 and 5 as well as between 1 and 5 are strengthened at the expense of a quite rapid reduction of the correlation at the bond between 7 and 1. Let us finally recall that in the strongly correlated limit the charge fluctuations are almost completely suppressed. Therefore, for U=t ¼ 64 one finds gij ¼ S 2i ¼ S i ðS i þ 1Þ ’ 34which corresponds to well-localized spin moments at all atoms. The magnetic susceptibility wðTÞ allows us to quantify the stability of the AF within the cluster. For large T, one observes that the temperature dependence of w has the form 1=w ffi ðT þ T N Þ, where TN can be interpreted as the cluster ‘‘Ne´el’’ temperature [see the dashed line in Fig. 1(c)]. Comparison with Fig. 1(b) also shows that the short-range spin-correlation functions are significantly reduced for T4T N . Systematic calculations of TN as derived from wðTÞ, as a function U/t provide a consistent physical picture. One observes that TN first increases for small U/t as AF order sets in reaching a maximum for U=t ffi 10. Then, in the strongly correlated regime (U=tb1) one obtains that T N aJ eff ¼ 4:88t2 =U as corresponds to the Heisenberg limit [8].
ARTICLE IN PRESS e30
F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31
Fig. 2. Illustration of the most stable geometries of clusters with N ¼ 7 atoms, n ¼ 7 electrons, and U=t ¼ 64. The numbers indicate the isomerization energies in units of the hopping integral t.
In Fig. 1(a), the solid line shows results for the specific heat C(T) including all contributions from excitations of both electronic and structural degrees of freedom [gX0 and lX0 in Eq. (2)]. Comparison is made with the case where only electronic excitations are taken into account [g ¼ 0 and l ¼ 0 in Eq. (2)]. (dashed-line). One observes that structural fluctuations introduce an additional low-temperature peak in C(T), which is the result of structural changes or isomerizations. The variety of low-lying isomers shown in Fig. 2 demonstrates the strong competition in the stability of cluster structures. Indeed, the energy differences between the first eight structures are of the order of only 103t. The first excitation implies a structural change since Dstruct ¼ 0:0033t, while the first electronic excitation in the most stable T ¼ 0 structure is much larger, namely, Delect ¼ 0:0132t. A more detailed account of the present investigations including a discussion of the effects of structural fluctua-
tions on other cluster properties will be published elsewhere [8].
Acknowledgments Financial supports from CONACyT (Mexico) through grants J36909-E and 39643-F and from the EU GROWTH Contract No. G5RD-CT2001-00478 are gratefully acknowledged.
References [1] See, for instance, G. M. Pastor, in: C. Guet, P. Hobza, F. Spiegelman, F. David (Eds.), Atomic Clusters and Nanoparticles, Lectures Notes, Les Houches Summer School of Theoretical Physics, EDP Sciences, Les Ulis/ Springer, Berlin, 2001, p. 335ff. [2] G.M. Pastor, J. Dorantes-Da´vila, Phys. Rev. B 52 (1995) 13799. [3] F. Lo´pez-Urı´ as, G.M. Pastor, J. Appl. Phys. 87 (2000) 4909.
ARTICLE IN PRESS F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31 [4] I.M.L. Billas, A. Chaˆtelain, W.A. de Heer, Science 265 (1994) 1682; S.E. Apsel, J.W. Emert, J. Deng, L.A. Bloomfield, Phys. Rev. Lett. 76 (1996) 1441. [5] S. Pokrant, J.A. Becker, J. Magn. Magn. Mater. 226–230 (2001) 1921.
e31
[6] G.M. Pastor, R. Hirsch, B. Mu¨hlschlegel, Phys. Rev. Lett. 72 (1994) 3879. [7] J. Hubbard, Proc. R. Soc. London A 276 (1963) 238; J. Hubbard, Proc. R. Soc. London A 281 (1964) 401. [8] F. Lo´pez-Urı´ as, G. M. Pastor, to be published.
Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31 www.elsevier.com/locate/jmmm
Thermodynamic properties of antiferromagnetic clusters F. Lo´pez-Urı´ asa,, G.M. Pastorb a
Advanced Materials Department, IPICYT, Camino a la Presa San Jose´ 2055, Col. Lomas 4a Seccio´n, 78216 San Luis Potosı´, SLP, Mexico b Laboratoire de Physique Quantique, Centre National de la Recherche Scientifique, Universite´ Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France Available online 15 April 2005
Abstract The finite temperature magnetic properties of antiferromagnetic (AF) clusters are investigated in the framework of the Hubbard model at half-band filling. The ground- and excited-state electronic energies En, the corresponding wave functions jCni, and the total spin Sn are calculated exactly by using a full many-body diagonalization method. A complete optimization and sampling of the topological structures of the cluster is performed. Representative results are presented for the specific heat C(T), magnetic susceptibility wðTÞ, and spin correlation functions gij ðTÞ between sites i and j of clusters having N ¼ 7 atoms. Low-temperature peaks in C(T) are observed and they are interpreted in terms of the spin-resolved excitation spectrum. wðTÞ shows a typical AF-like behavior of the form w 1=ðT þ T N Þ from which the cluster ‘‘Ne´el’’ temperature TN is derived. Finally, the effects of temperature-induced fluctuations of the cluster structure are discussed. r 2005 Elsevier B.V. All rights reserved. PACS: 36.40.Cg; 75.10.Jm Keywords: Atomic clusters; Antiferromagnetism; Finite-temperature; Exact diagonalization
1. Introduction The finite-temperature properties of magnetic clusters motivate a remarkable interest in current research on low-dimensional magnetism [1–3]. Antiferromagnetic (AF) clusters are a major
Corresponding author. Tel.: +52 444 8342000;
fax: +52 444 8342040. E-mail address: fl[email protected] (F. Lo´pez-Urı´ as).
theoretical challenge in this field, in particular due to the subtle competition between itinerant and localized behaviors and the resulting strong electron-correlation effects. Transition-metal (TM) and rare-earth (RE) clusters have attracted a considerable attention from both theoretical and experimental standpoints [4,5]. Experiments have revealed that the magnetic properties of TM and RE clusters are often very different from those of the corresponding solids. TM elements at the beginning of the series, i.e., for configuration
0304-8853/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2005.03.048
ARTICLE IN PRESS F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31
e28
3dn4sm with np5, one observes weak or vanishing magnetic moments and a clear tendency to AF correlations. From the point of view of theory, the study of AF metal clusters poses an interesting challenge due to the difficulty in describing the delicate balance between electron delocalization and local charge fluctuations. As in solids, a precise treatment of electron correlations and the implementation of sophisticated many-body techniques are necessary in order to determine the ground-state properties, the electronic excitations, and the resulting finite-temperature behavior. Moreover, in the case of small clusters, additional effects are expected to play a role, in a way that is specific to the extremely large surface-to-volume ratio and to the presence of relatively loose surface bonds. In particular, one should take into account that temperature-induced changes or fluctuations of the cluster geometry [6].
2. Theory The development of the theory of cluster magnetism is hindered by two main difficulties: an accurate treatment of electron correlations, which are fundamental for a profound understanding of magnetism, and the optimization of the cluster geometry, on which the magnetic properties of itinerant electrons are known to depend strongly. Since these two problems are formidable, and their combination even more so, it is natural that most studies available hitherto have attempted to deal at best with one of them at a time. In this paper we consider a model, which is simple enough to allow an exact solution of the many-body problem and of the geometry sampling in topological space, and which at the same time contains enough complexity to be able to shed light on the physics of real systems (e.g., 3d-TM). The most popular and probably the most simple physical model for describing correlated itinerant electrons on a lattice is given by the well-known Hubbard Hamiltonian [7], X þ X H ¼ t c^ is c^ js þ U (1) n^ i# n^ i" . ðiajÞ s
i
The first term is the kinetic-energy operator that describes electronic hoppings between nearest neighbors sites i and j (t40). The second term takes into account the intra-atomic Coulomb repulsions that are the dominant contribution from the electron–electron interaction (UX0) [7]. The model is characterized by the number of the electrons n and by the dimensionless parameter U/t that measures the importance of correlations. The Hubbard model for small clusters is solved numerically by expanding its eigenfunctions jCli in a complete set of basis states jFm i which have definite occupation numbers nm is at all orbitals is, m i.e., n^ is jFm i ¼ nm is jFm i with nis ¼ 0 or 1. The expansion coefficients alm of jCl i ¼ Sm alm jFm i are determined by standard numerical diagonalization procedures [6]. In this way, both ground- and excited-state properties are obtained exactly within the framework of the Hubbard Hamiltonian [Eq. (1)]. The finite temperature properties are derived from the canonical partition function Q over electronic and structural degrees of freedom. Q is given by XX Q¼ eb l ðgÞ ; (2) g
l
where l ðgÞ is the lth eigenenergy corresponding to a cluster geometry g. b ¼ 1=T refers to the temperature of the cluster source that defines the macroscopic thermal bath with which the small clusters are in equilibrium before expansion. Thermal averages correspond then to the ensemble of clusters in the beam. Moreover, keeping n and N fixed (canonical ensemble) corresponds to the experimental situation in charge and size selected beams [4]. Taking into account only NN hoppings with fixed bond lengths results in a discretization of the configuration space. The sampling of cluster geometries can be performed within the graph space. Notice that in spite of these simplifications the number of site configurations increases extremely rapid with N [6]. The properties of interest, such as the specific heat C(T), its spin-resolved contributions CS(T), the magnetic susceptibility wðTÞ, and the spin correlation functions gij ðTÞ ¼ hS i S j i, are obtained straightforwardly from the derivatives of Q.
ARTICLE IN PRESS F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31
3. Results Fig. 1 shows the temperature-dependent magnetic properties of a Hubbard cluster having N ¼ 7 atoms, one electron per site (n ¼ N), and Coulomb repulsion strength U=t ¼ 64. Here we focus on the effect on electronic excitations by considering the same cluster structure for all T, which corresponds to the most stable topology at T ¼ 0 [see inset of Fig. 1(b)]. The specific heat C (dashed-line) exhibits a low-temperature narrow peak followed a broader peak at higher T. The first one corresponds to doublet–doublet excitations (i.e., within the minimum-spin S ¼ 1=2 manifold) while the second one involves spin excitations with a major contribution from S ¼ 12 to 32 transitions. It is interesting to observe that this double-peak structure does not appear in clusters with and even number of atoms (half-band filling). For even N ¼
Fig. 1. Temperature dependence of (a) specific heat C, (b) NN spin-correlation functions gij ¼ hSi S j i, and (c) inverse of the magnetic susceptibility 1=w for a cluster having N ¼ 7 atoms, n ¼ N ¼ 7 electrons, and Coulomb repulsion strength U=t ¼ 64. The inset in (b) illustrates the structure which yields the optimum ground-state energy.
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n the ground state is a singlet and the specific heat presents only one clearly detached low-temperature peak. An analysis in terms of the spinresolved contributions to C shows that in this case the relevant low-lying excited states involve changes in the total spin, mainly from S ¼ 0 to 1 (singlet-triple excitations). The different NN spin-correlation functions gij show interesting dependences as a function of temperature which can be related to the local environment of the atoms i and j [see Fig. 1(b) and the labeling of the atoms in the illustration of the structure]. One observes that as T increases, the thermal fluctuations progressively destroy the ground-state spin correlations that are strongly AF (gij o0 at low T). This is for example the case for i ¼ 7 and j ¼ 2, or i ¼ 6 and j ¼ 4. More remarkable effects are found among the atoms i ¼ 1, 3, 5, and 7, due to magnetic frustrations. At low temperatures, the AF correlations are strong between sites 7 and 1, while the correlations between 7 and 5 (or equivalently between 1 and 5) are weak. However, as T increases another type of AF-like order is favored, in which the AF coupling between sites 7 and 5 as well as between 1 and 5 are strengthened at the expense of a quite rapid reduction of the correlation at the bond between 7 and 1. Let us finally recall that in the strongly correlated limit the charge fluctuations are almost completely suppressed. Therefore, for U=t ¼ 64 one finds gij ¼ S 2i ¼ S i ðS i þ 1Þ ’ 34which corresponds to well-localized spin moments at all atoms. The magnetic susceptibility wðTÞ allows us to quantify the stability of the AF within the cluster. For large T, one observes that the temperature dependence of w has the form 1=w ffi ðT þ T N Þ, where TN can be interpreted as the cluster ‘‘Ne´el’’ temperature [see the dashed line in Fig. 1(c)]. Comparison with Fig. 1(b) also shows that the short-range spin-correlation functions are significantly reduced for T4T N . Systematic calculations of TN as derived from wðTÞ, as a function U/t provide a consistent physical picture. One observes that TN first increases for small U/t as AF order sets in reaching a maximum for U=t ffi 10. Then, in the strongly correlated regime (U=tb1) one obtains that T N aJ eff ¼ 4:88t2 =U as corresponds to the Heisenberg limit [8].
ARTICLE IN PRESS e30
F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31
Fig. 2. Illustration of the most stable geometries of clusters with N ¼ 7 atoms, n ¼ 7 electrons, and U=t ¼ 64. The numbers indicate the isomerization energies in units of the hopping integral t.
In Fig. 1(a), the solid line shows results for the specific heat C(T) including all contributions from excitations of both electronic and structural degrees of freedom [gX0 and lX0 in Eq. (2)]. Comparison is made with the case where only electronic excitations are taken into account [g ¼ 0 and l ¼ 0 in Eq. (2)]. (dashed-line). One observes that structural fluctuations introduce an additional low-temperature peak in C(T), which is the result of structural changes or isomerizations. The variety of low-lying isomers shown in Fig. 2 demonstrates the strong competition in the stability of cluster structures. Indeed, the energy differences between the first eight structures are of the order of only 103t. The first excitation implies a structural change since Dstruct ¼ 0:0033t, while the first electronic excitation in the most stable T ¼ 0 structure is much larger, namely, Delect ¼ 0:0132t. A more detailed account of the present investigations including a discussion of the effects of structural fluctua-
tions on other cluster properties will be published elsewhere [8].
Acknowledgments Financial supports from CONACyT (Mexico) through grants J36909-E and 39643-F and from the EU GROWTH Contract No. G5RD-CT2001-00478 are gratefully acknowledged.
References [1] See, for instance, G. M. Pastor, in: C. Guet, P. Hobza, F. Spiegelman, F. David (Eds.), Atomic Clusters and Nanoparticles, Lectures Notes, Les Houches Summer School of Theoretical Physics, EDP Sciences, Les Ulis/ Springer, Berlin, 2001, p. 335ff. [2] G.M. Pastor, J. Dorantes-Da´vila, Phys. Rev. B 52 (1995) 13799. [3] F. Lo´pez-Urı´ as, G.M. Pastor, J. Appl. Phys. 87 (2000) 4909.
ARTICLE IN PRESS F. Lo´pez-Urı´as, G.M. Pastor / Journal of Magnetism and Magnetic Materials 294 (2005) e27–e31 [4] I.M.L. Billas, A. Chaˆtelain, W.A. de Heer, Science 265 (1994) 1682; S.E. Apsel, J.W. Emert, J. Deng, L.A. Bloomfield, Phys. Rev. Lett. 76 (1996) 1441. [5] S. Pokrant, J.A. Becker, J. Magn. Magn. Mater. 226–230 (2001) 1921.
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[6] G.M. Pastor, R. Hirsch, B. Mu¨hlschlegel, Phys. Rev. Lett. 72 (1994) 3879. [7] J. Hubbard, Proc. R. Soc. London A 276 (1963) 238; J. Hubbard, Proc. R. Soc. London A 281 (1964) 401. [8] F. Lo´pez-Urı´ as, G. M. Pastor, to be published.