In the case of layered nanostructures, the delicate interplay between the local magnetic character and the structural organization calls for the use of suitable theoretical frameworks. To meet the challenge of a realistic description of bonding and structure at an equal level of reliability, these approaches have to be sustained by powerful computation tools. By relying on density functional theory (DFT)1, 2 the aim is to characterize layered organic-inorganic materials by providing : a) when needed, effective geometry optimization to go beyond incomplete experimental structural characterization b) an affordable description of the electronic structure and local magnetic properties and c) a viable playground for linking total energies and spin topologies to the values of the exchange coupling constants. A few considerations are in under to understand the assumptions underlying the use of DFT in the area of magnetic, molecule-based multi-layers materials. First and foremost, the performances of DFT are highly related to the quality of the exchange-correlation part of the energy functional. Second, the choice of the basis set has to be made by considering the desired level of manageability and computational convenience. For isolated magnetic molecules or magnetic crystals made of weakly interacting solids, the above criteria correspond to the choice of the hybrid B3LYP3, 4 exchange correlation functional in conjunction with localized orbitals as a basis set. This strategy is in principle well suited to find minima in the energy if the starting configurations are close to local structural minima. Such situation contrasts the DFT plane-waves method (DFT-PW)5 that is usually implemented to handle effectively complex structural searches, as it is the case in a large number of hybrid organic-inorganic layered systems. For these materials, the knowledge of the atomic structure is often incomplete, thereby making legitimate the methodological choice of relying on the DFT-PW approach. In what follows the DFT-PW strategy has been applied to the hydroxi-nitrate Cu2(OH)3(NO3) and to the hydroxi-acetate Cu2(OH)3(CH3COO).H2O. In the first case, we have taken advantage of a set of experimental coordinates available for the full unit cell to gain information on the spin topology and the electronic properties.6 In addition, a separate calculation carried out within the DFT-B3LYP approach has allowed us to link the calculated values of the exchange coupling constants to the macroscopic behaviour of the magnetic susceptibility.7
Electronic structure and magnetism in layered nanostructures: a density functional approach
Fig. 1 : Calculated spin densities distributions in two different planes of Cu2(OH)3(NO3). The vertical rows of Cu atoms along the b axis can be clearly distinguished, all featuring alternate spin polarizations (on the left, adjacent red and green colours) ) or having different spin polarization (on the right, where the rows exhibit either the same polarization (red or green) along the vertical b axis or alternate spin polarizations).
The case of Cu2(OH)3(CH3COO).H2O proved more challenging, due to the incompleteness of the set of coordinates available experimentally. Both Cu2(OH)3(NO3) and Cu2(OH)3(CH3 COO).H2O have a global S=0 magnetic character. In particular, the in-place magnetic nature of Cu2(OH)3(CH3 COO).H2O is found to manifest itself in a large variety of microscopic (local) forms, lying very close in energy. From the technical point of view, common to these two sets of calculations is the relaxation of the electronic structure to the ground state by minimization of the total energy with respect to the coefficients of the plane-waves expansion. Optimization of the structures (carried out for Cu2(OH)3(NO3) to ensure the correct reproduction of the experimental coordinates) is accomplished by means of first-principles molecular dynamics with a damping factor.8
We keep in mind that the measured magnetic susceptibility of Cu2(OH)3(NO3) is compatible with an in-plane antiferromagnetic character (Σ=0). Does this mean that single-layer models are unsuitable since they do not take into account variations in the spin distributions among different layers ? Our DFT approach proves that this is indeed the case. We display in Fig. 33 the projections of the spin density distribution on the two (a,b) planes of our simulation cell. This orientation allows to capture the detail of the magnetic interactions which are expected to occur among Cu atoms on the same (a,b) plane. As shown in Fig. 1 and in agreement with experiments, our calculated distribution of spin densities yields an antiferromagnetic character on each (a,b) plane, this character being different from one (a,b) plane to the other. We note that both parallel and antiparallel alignment between spin densities occur for Cu pairs along the b direction. This feature is confirmed by the observation of the four possible exchange mediated interactions among Cu centres, i.e. Cu(1)-Cu(1) via a NO3– group and a OH– group, Cu(2)-Cu(2) via two OH-groups, Cu(1)-Cu(2) via a NO3– group and a OH– group and Cu(1)-Cu(2) via two OH– groups (each row of Cu atoms being made of Cu(1) atoms or Cu(2) atoms, Cu(1) and Cu(2) being crystallographically different). All of them exhibit either parallel or antiparallel spin alignments, as it can be deduced by following the different rows of Cu sites forming the triangular lattice on the (a,b) planes, along and across the b direction. This indicates that there is no clear correlation between the nature of the bridging groups and the sign of the exchange interactions. In this context we note that, due to the triangular configuration of the exchange pathways within the layers, frustration effects may occur. However, these are not described by these specific set of calculations, since they are limited to the consideration of collinear magnetic effects. This means that the exchange-correlation functional does not depend on off-diagonal terms of the spin density matrix. The different topology of spin densities between the two Cu planes, which in our calculations was found to be unaffected by the structural optimization, confirms that models for magnetism relying on only one single, isolated layer, are insufficient for these compounds.
Fig. 2 : Intralayer exchange pathways and spin distributions corresponding to the calculated ground state. The copper atoms are represented by large spheres ; the white ones correspond to the chains with NO3 and OH bridging ligands while the black spheres have only OH bridging ligands. The oxygen, nitrogen and hydrogen atoms are represented by small gray spheres, from dark to bright, respectively. Exchange coupling pathways are identified with labels (from J1 to J6) and with dashed lines joining the corresponding pairs of Cu atoms.
The existence of a non unique distribution of spin densities along the c direction has important consequences on the magnetic dimensionality of Cu2(OH)3(NO3) Indeed, this implies that the distribution of local magnetic moments on the (a,b) layers has a periodicity along the c direction which differs from the one of the lattice. To obtain the link between atomic-scale and macroscopic behaviour one needs to calculate the exchange coupling constants by exploiting the electronic structure framework. We have achieved this goal by resorting to the most accurate (from the quantitative point of view) expression for the exchange-correlation functional, the B3LYP one. The calculated exchange coupling constants have been obtained (via DFT-B3LYP calculations) from the energies of seven spin arrangements of a supercell. They are related to the eigenvalues of the corresponding Ising Hamiltonian and used to obtain a system of six equations with six unknows, the Jα values. One obtains J1=+28.6 cm-1, J2=-22.9 cm-1, J3=+6.9 cm-1, J4=+10.7 cm-1, J5=-3.5 cm-1, J6=+7.9. cm-1. The largest values for the exchange coupling correspond to the two different Cu chains, J1 being ferromagnetic and J2 antiferromagnetic (Fig. 2). The interchain couplings are relatively weaker, featuring positive and negative values. The calculation of the energy corresponding to all possible spin distributions shows that the spin arrangement indicated in Fig. 2 is the most stable one. It should be noted that the pattern of the magnetic interaction for the exchange coupling J6 (Fig. 2, antiparallel alignment) does not correspond to the positive sign for J6. This is an unambiguous indication of competition among the different magnetic interactions resulting in a frustration effect typical of triangular-like networks. To extract the magnetic susceptibility from the calculated values of the exchange coupling constants one takes advantage of an appropriate statistical mechanics framework.9
Fig. 3 : Temperature variation of the magnetic susceptibility. The black circles are the experimental data, the white circles are the results using the J values obtained directly from DFT calculations. The white squares correspond to the results of the fit to the experimental data. Note that the optimization of the J values began from those obtained with the B3LYP functional. Inset : Corresponding temperature variation of the χT product (symbols are the same as above).
We point out that the maximum of susceptibility at T = 12 K is particularly well reproduced (Fig. 3). Due to the difference in the height of the susceptibility maximum, the χT product does not follow the experimental trend, indicating that the ferromagnetic interaction J1 is likely to be overestimated within our DFT-B3LYP approach. A new refinement based on the optimization of the agreement with the experimental data yields as final J values : J1 = +14.0 cm-1, J2 = -34.9 cm-1, J3 = +6.9 cm-1, J4 = +10.4 cm-1, J5 = -4.9 cm-1 and J6 = +2.0 cm-1. Note that the employed value g=2.235 was also optimized using as starting point g=2.227 obtained from the Curie’s constant of the experimental data. The main difference comparing with the values obtained with the B3LYP functional is the relative strength of the two strongest interactions, being now the antiferromagnetic J2 interaction the predominant one.
An X-ray powder diffraction study of Cu2(OH)3(CH3COO).H2O has clearly stressed that the water molecules are located in between the inorganic layers, although they can be easily and reversibly removed by moderate heating. In spite of the availability of this kind of structural information, the set of atomic coordinates is incomplete, this shortcoming being due to the lack of synthesized single crystal of suitable quality.10 By using our first-principles approach, we were able to obtain optimized structures for S=0 and S=4 total values of the spin, and for two system sizes, N=72 and N=288 (one and four unit cells, respectively). The structure of Cu2(OH)3(CH3COO).H2O, presented in Fig. 4, is characterized by the alternation between OH- and CH3COO- groups within the first coordination shell of Cu atoms and by the presence of water molecules ensuring cohesion via hydrogen bonds (H-Bonds). The structures obtained for the two classes of spin topologies (partial spin Σ = 0 on each single layer, and finite non-zero partial spin on each layer, Σ ≠ 0, referred to as AF-in and F-in hereafter, respectively) are very close and feature very limited changes within each class.
Also, the interatomic distances in the various molecular groups (OH–, CH3COO– and H2O) replicate quite regularly within the cell, the standard deviations being quite small. This holds true also when considering the different spin topologies found for N=72 (S = 0 AF-in, S = 0 F-in and S = 4), as well as the two cases studied for N=288. Cu atoms are found in a highly distorted octahedral environment. Two patterns of coordination shells can be distinguished. In the first pattern, two O atoms belonging to the CH3COO– groups are located rather close to the Cu atom, having distances in the range 2.02 – 2.18 Å, the remaining four nearest neighbours being O atoms of OH– groups. Interestingly, these could also be found at rather large distances ( 3 Å). In the second pattern, the O atom that belongs to the CH3COO– group is the farthest apart among the six nearest neighbours of the Cu atom, the five remaining lying in OH– groups at distances ranging from 1.93 Å and 2.59 Å. The full set of the relevant interatomic distances can be found elsewhere.11
The energy differences separating distinct spin topologies within each one of the two AF-in and F-in classes are quite small, namely 0.15 eV for AF-in and 0.20 eV for F-in at most. We found that the total energy in the ground state AF-in structure is lower by 0.50 eV than its F-in counterpart. In turn, this same total energy for the ferromagnetic case is higher by 0.9 eV with respect to the F-in one, leading to E(S = 0, AF-in) < E(S = 0, F-in) < E(S = 4). From the energy difference between different realizations of the same macroscopic magnetic behaviour one can extract an estimate of the exchange coupling constant. This can be done in the framework of an Ising spin Hamiltonian approximation, in which the magnetic centres located on the Cu atoms interact through a law of the kind H = |J| SASB, Sk being a two value (1/2, -1/2) spin variable located on the k = A and k = B magnetic centres. For instance, the energetic cost ΔE = 0.019 eV separating the two lowest spin configurations can be readily associated with the number of spin flip exchanges necessary to convert one spin configuration into the other. Typical values for |J| are of the order of 100/200 cm-1.
We have to remark that this value is larger than the one extracted from the temperature behaviour of the magnetic susceptibility (of the order of 30-60 cm-1 as reported in Ref. 12), by confirming the overestimate of the exchange coupling constants calculated within the BLYP exchange-correlation functional.2
By referring to a specific spin configuration selected among those characterized by S=0,Σ ≠ 0 Figure 5 shows the spin density topology on one of the Cu2(OH)3(CH3COO).H2O layers. For the selected value of the isosurface (10-2 e/Å3), a predominant number of spin densities belonging to the same band (say, the majority one, α), are clearly visible on the Cu sites. In addition, α spin densities occur on some of the O atoms of the OH– groups, nearest neighbours of the Cu atoms. The question arises on the values of the Cu-O interatomic distances and on the number of Cu-O neighbours needed to trigger the presence of a significant spin-density on the O atoms. By focusing on the absolute values, the largest spin densities for the O atoms (in between 0.172 and 0.224) are found to correspond to a pair of Cu neighbours carrying spin densities of the same sign and separated by the distances in the range 1.91-2.21 Å. The effect of the third Cu neighbour on the spin densities of the O atoms is very much reduced, due to a larger interatomic Cu-O distance. Opposite signs on the Cu neighbours result in values lower than 0.08. The same analysis can be carried out for the O atoms in CH3COO– groups. Among the eight available O atoms, only the four closest to the Cu neighbours have non vanishing the spin densities. In analogy with the previous case (O atoms in OH– groups) largest spin densities are associated to pairs of neighbouring Cu atoms carrying spin densities of the same sign. However, larger interatomic distances concur to lower the absolute values, proving that from the standpoint of the magnetic interaction, the Cu-O—OH-Cu bridge is more effective than the Cu-O— CH3COO-Cu one. Overall, the two examples of application of the DFT-PW strategy show that both structural and electronic properties are accessible to modern first-principles methods, the only limitation being the performances of the exchange-correlation functionals. The combination of DFT-based functionals and of the hybrid ones can be considered a valuable strategy to confer a quantitative character to atomic-scale techniques aimed at the evaluation of the magnetic exchange coupling constants.
1. W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133-A1138.
2. E. Ruiz, S. Alvarez, A. Rodriguez-Fortea, P. Alemany, Y. Pouillon and C. Massobrio, in From Magnetism : Molecules to Materials II, eds. J. S.Miller and M. Drillon, Wiley-VCH Verlag GmbH, Weinheim, Germany, 2001, vol. II, pp. 227-279.
3. A. D. Becke, J. Chem. Phys., 1993, 98, 5648-5652.
4. P. J. Stephens, F. J. Devlin, C. F. Chabalowski and M. J. Frisch, J. Phys. Chem., 1994, 98, 11623-11627.
5. in, http://www.cpmd.org, eds. Within the DFT/plane waves framework, we have adopted the CPMD code, Copyright IBM Corp. 1990-2009, Copyright MPI fur Festkorperforschung Stuttgart 1997-2001, http://www.cpmd.org, 1990-2009.
6. C. Massobrio, P. Rabu, M. Drillon and C. Rovira, The Journal of Physical Chemistry B, 1999, 103, 9387.
7. E. Ruiz, M. Llunell, J. Cano, P. Rabu, M. Drillon and C. Massobrio, J. Phys. Chem. B, 2006, 110, 115-118.
8. R. Car and M. Parrinello, Phys. Rev. Lett., 1985, 55, 2471-2474.
9. J. Cano and Y. Journeaux, in Magnetism : Molecules to Materials V, eds. J. S. Miller and M. Drillon, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2005, pp. 189-239.
10. N. Masciocchi, E. Corradi, A. Sironi, G. Moretti, G. Minelli and P. Porta, Journal of Solid State Chemistry, 1997, 131, 252-262.
11. F. Yang, M. Boero and C. Massobrio, J. Phys. Chem. B, 2010, to be published.
12. M. I. Gîrtu, C. M. Wynn, W. Fujita, K. Agawa and A. J. Epstein, Phys. Rev. B, 2000, 61, 4117-4130.