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Spatial distribution of Gaussian fluctuations of the molecular field and magnetization in the pyramid-like Ising nanoscopic system interacting with the substrate

Abstract We study thermodynamic properties of an Ising model of a ferromagnetic nanoscopic pyramid deposited onto a ferromagnetic bulk substrate. The influence of the interaction between the pyramid and the substrate is calculated in terms of the equilibrium reduced-state (density) operator used for description of thermodynamic properties of nanoscopic systems. The spatial distribution of the fluctuations of molecular field and magnetization in the nanoscopic pyramid is obtained in the Gaussian fluctuations approximation. Experimental consequences for the magnetic force measurements are briefly discussed.


INTRODUCTION
During the past two decades we have witnessed significant advances in the ability to synthesize nanoscale structures as well as development of novel experimental method allowing exploration of their physical properties. [1] This is exciting for two reasons. Nanoscopic magnetic systems [2] have always been very attractive from the theoretical point of view. On the other hand, studies of nanoscopic systems are characterized by a close coupling between theory and experiment because of rapidly increasing number of experimental works on real nanoscopic materials. [1] These materials often correspond remarkably close to certain idealized spin models and are great challenges not only for physicists, but also for chemists and engineers.
There is a great interest in producing two-dimensional arrays of magnetic nanodots which may serve e.g. as magnetic recording media. Due to small dimensions of these particles quantum-mechanical effects cannot be ignored. Effects observed in quantum dots are relevant also in quantum computing and spin electronics. It was shown recently that nanoscopic structures in the form of piramids or similar shapes can be fabricated on a bulk substrate. [3,4,5] In experiments aiming at the investigation of nanoscopic systems we deal frequently with such systems deposited on a bulk substrate. However, most often in the theoretical description of such experiments the interaction between the nanosystem and the substrate is neglected. The main purpose of this work is to find a way to include this and calculate its effect on the thermodynamic properties of the Ising model of a ferromagnetic nanoscopic system. To achieve this we apply the equilibrium reduced-state (density) operator (ERSO) [6] and carry out the calculations within the Gaussian fluctuations approximation (GFA).
ERSO is the most general equilibrium state operator. It has been derived from the generalized Schrödinger variational principle and not on the quantum statistical mechanics, i.e. without statistical hypotheses. When the term describing the interaction between the system and its environment is neglected ERSO takes the form of the statistical operator of the Gibbs canonical distribution. ERSO applied to an exactly solvable microscopic model leads to exact results.
GFA is a modified version of the high density expansion method as has been proposed in refs. [7,8]. GFA is an improvement over the molecular field approximation (MFA) due to the self-consistent inclusion of Gaussian fluctuations of this field. The essential new element of GFA is the summing up of the partial sums of Feynman diagrams of the same structure of recurrent formulae at each stage of the calculations. GFA is based on a classification of the Feynman diagrams in terms of 1/z, where z is the effective number of spins interacting with any given spin. Owing to this procedure the theory becomes internally consistent and does not lead to unphysical results such as, for example, a complex Curie temperature. [9] 2. THE MODEL In this paper we consider magnetic properties of a nanoscopic pyramid (nanopyramid).
They are well described by the simple model of localized and ordered spins with the following spin-1/2 Hamiltonian where I is the coupling parameter and stands for summation over pairs of different simple cubic (sc) lattice points. Our considerations are restricted to nearest-neighbor interaction only.
In expression (1) f denotes the two-dimensional position vectors of a spin belonging to a given monoatomic layer l = 1, 2, 3, 4 of the pyramid. There are 4 spins in the l = 1 layer, 16 spins in the l = 2 layer, 36 spins in the l = 3 layer and 64 spins in the l = 4 layer. The total number of spins in the pyramid is 120 (see Fig. 1).
We assume that the nanoscopic pyramid is deposited on a bulk ferromagnetic substrate sufficiently well described by the spin-1/2 Ising model for a simple cubic lattice and the Hamiltonian Here g denotes the two-dimensional position vectors of spin belonging to a given monoatomic layer. The summations always run over different sites. The substrate is di-vided into monoatomic layers parallel to the planes (100) of a sc-lattice. The position of each layer is given by the number r = 1, 2, ...
Let us assume that the interaction of the nanoscopic pyramid with the bulk substrate is described by the Heisenberg Hamiltonian of the form In order to take into account the interaction of the nanoscopic pyramid with the bulk substrate we shall apply ERSO [6] suitable for description of a physical situation similar to the one we are concerned with. In the derivation of this particular form of ERSO we use the fact that although the Universe as a whole is in the pure state, its arbitrary multiparticle parts are inevitably in mixed states. This is purely a quantum effect following from the holistic properties of the quantum theory, formally related to the fact that the Universe In our case ERSO takes the following form [6]: where F is the free energy and is an effective term describing the interaction between the pyramid and the bulk substrate, Tr s [...] is the partial trace over the substrate states, K is the correlation operator, β = (k B T ) −1 , and is the free energy of the substrate. Substituting expresssion (3) into Eq. (5) with the assumption that and K ≃ 0, we arrive at where is the average bulk substrate spin moment in the layer r = 1.

GAUSSIAN FLUCTUATION OF THE MOLECULAR FIELD
As a starting point to GFA we choose the following decomposition of the Hamiltonian where the perturbative part H 1 is defined by the tranformation where is the fluctuation operator of the z-component of the spin and and F is the free energy.
According to the rules of the thermodynamic perturbation expansion we can write where and The right-hand side of expression (13) can be expanded into a series with respect to the perturbing term H 1 . From such an infinite series we now choose a certain partial sum which can be represented graphically in the following way[10] and symbols MM denote the renormalized interaction line, L(W f l ) = ln(2 cosh(W f l /2)), where we assumedh = 1 and f ′ l ′ denotes summation over nearest neighbours of the fl spin.
As a result of calculating the infinite sum (18) and (19) we obtain where Similarly, using eq. (9) for r = 1, 2, . . . , the molecular field V r of the bulk substrate in the GFA satisfies the equation where and δW f l , δV r are the mean Gaussian fluctuations of the molecular fields W f l and V r , respectively.
After introducing the following reduced magnitudes, equations (24) -(28) can be written in a compact form, where Equations (30)-(35) were solved numerically. In order to carry out the calculations we have to specify the magnetic lattices pyramid and substrate as a simple cubic one. Results are presented graphically.      and δV r for pyramid and the substrate respectively.
Finally, Fig. 10 shows Gaussian fluctuation of the molecular field as a function of temperature.
In Table 1 the Curie temperature t c of the ferromagnetic nanoscopic pyramid is given in relative units for different values of parameters a and b obtained in MFA and GFA.
We can see that for a nanopyramid the ratio of the Curie temperature in GFA to that obtained in MFA is 0.783. For an infinite simple cubic lattice this ratio is equal to 0.856 and for a square lattice Ising monolayer it has a value of 0.799. [8] The ratio of the Curie temperature of a monolayer obtained in GFA to the exact result is 1.409. The same quantity for the sc bulk Ising ferromagnet is approximately 1.138.

CONCLUSIONS
The results we obtained lead us to the following conclusions: • The interation of a ferromagnetic nanoscopic pyramid with its bulk ferromagnetic substrate may have essential influence on the properties of the ferromagnetic pyramid, • The distribution of Gaussian fluctuations is highly nonuniform, • lim t→0 δW fl = 0, as expected, • The maximum of molecular field fluctuations δW fl is reached at the Curie temperature t c , as expected, • The Curie temperature t c of the pyramid strongly depends on the Curie temperature of the bulk substrate, • We believe that ERSO (4) may be successfully applied in studies of the influence of the bulk substrate on the thermodynamic properties of nonmagnetic systems and other more complex nanoscopic systems. [4,5] The development of the magnetic exchange force microscopy (MExFM) [13]  it is also possible that the spin ordering of the tip is present but is very weak. Performing this experiment at lower temperatures, as authors of Ref. [14] suggest, may yield better results.
Atomically sharp magnetic tips are crucial in achieving lateral high spatial resolution.
Therefore the knowledge of spatial distribution of fluctuations and its temperature dependence might improve the calibration of the STM device and help in estimating experimental error of the MExFM method. For small distances between the tip and the surface it may be necessary to include interaction of more than one atom of the tip or of the surface. [15] In this case our model with nonuniform magnetization of the tip may provide useful insight. The temperature dependence of mean value of magnetization of the individual tip layers shown in Fig. 6 shows that the magnetization in the second layer can be significantly larger that the magnetization of the apex. This implies an increased range of tip-surface separation where the interaction with atoms in the next layer should be taken into account.
The ability to control magnetic properties of nanoclusters [16] is one of the central problems of nanotechnology. Arrays of nanoclusters are also intensively studied. [17,18] The theory presented here may be applied to various geometrical shapes used in experiments, [2] e.g. chains of particles, striped and cylindrical nanowires, nanodots, nanojunctions, surface steps. We intend to carry out calculations in a more realistic model of the substrate surface and the interface between the pyramid and the substrate.
Finally let us note that ERSO (4) couples the nanoscopic pyramid with the substrate.
This means that the system does not have a finite number of spins. Therefore the use of e.g. Monte Carlo simulations would have little justification.