Monte Carlo study of magnetic structures in rare-earth amorphous alloys

Using the Monte Carlo method, we studied magnetic properties of the models of Re-Tb and ReGd amorphous alloys as well as pure amorphous Tb and Gd. The magnetic phase diagrams for the models of amorphous Tb and Gd were constructed. We determined the values of 0 D J (for Tb) and 1 2 J J (for Gd) at which the transition into spin-glass-like state takes place. The local magnetic structure was studied by the spin correlation functions and the angular spin correlation functions. The difference in magnetic structures in amorphous alloys with the random anisotropy and with fluctuations of exchange interaction was revealed.


Introduction
Spin glasses are of great interest as a new class of magnetic materials having the unique physical properties [1,2].
Binary amorphous alloys of heavy rare-earth metals (REM) with nonmagnetic transition metals are very little studied, although in this case the transition to the spinglass state takes place. In amorphous alloys of the rhenium-terbium (Re-Tb) and rhenium-gadolinium (Re-Gd) systems the typical for spin glasses maximum on the temperature dependence of dynamic magnetic susceptibility (T) and irreversibility of magnetization M(T) were experimentally revealed [3].
However, the nature of the spin-glass state on the microscopic level is studied insufficiently. This encourages us to construct and analyze computer models of atomic structure and magnetic properties of these materials.
As an object of our study, we chose the Re-Tb system with the strong random anisotropy and the Re-Gd system with fluctuations of exchange interactions. The choice of these alloys is caused by the prospects of their application as memory elements.

Simulation Technique
Using the molecular dynamics method, we constructed the models of atomic structure of the Re-Tb and Re-Gd amorphous alloys and of pure amorphous Tb and Gd in the wide compositional region. Each model consisted of 100 000 atoms. Interatomic interaction was described by an empirical polynomial potential [4]. The radial distribution functions (RDFs) calculated for the models are in good agreement with the results of the X-ray diffraction experiment [5].
Using the Monte Carlo method in the frame of the Heisenberg model, we carried out simulation of magnetic properties of Re-Tb amorphous alloys and of pure amorphous Tb. The standard Metropolis algorithm of the Monte Carlo method was used [6]. Exchange interaction between the neighbouring magnetic ions as well as the random magnetic anisotropy play the most important role in forming of the magnetically ordered structures in amorphous alloys containing REM. Therefore, we used the following model Hamiltonian [7]: where ij J is exchange integral between the spins i and j; D is anisotropy constant; i S is classical Heisenberg spin; i n is unit vector determining the direction of the local anisotropy axes.
Dependence of the exchange integral on the interatomic distance r was chosen as a linear function: where 0 19, 26 K J  is average value of the exchange integral which is equal to the corresponding value for crystalline Tb; r1 is position of the first peak on the partial pair RDF gTb-Tb(r); rmin is position of the first minimum of the gTb-Tb(r) function.
Directions of local anisotropy axes were chosen randomly at each site. The value of anisotropy constant in this model varied in a wide region ( We chose the random anisotropy constant for the Re-Tb alloys linearly dependent on the concentration of Tb atoms: where x is concentration of Tb atoms, %. For pure terbium (x = 100 at. %) this formula yields 0 6,6 D J  . For amorphous Gd and Re-Gd amorphous alloys the Hamiltonian was chosen as follows [8]: The value of the exchange integral in the second coordination sphere J2 varied in a wide range ( 1 2 2 16 J J   ).

Magnetic phase diagram for amorphous Tb
For the model of pure amorphous Tb we calculated the temperature dependencies of spontaneous magnetization ( Fig. 1)   As a result, we obtained the magnetic phase diagram for amorphous Tb in the 0 D J -T coordinates (Fig. 2). It allows one to determine the phase state of the system depending on the emperature and the value of anisotropy constant.
In all the models the transition to the spin-glass-like state was observed (Fig. 3), except of Re95Tb5 and Re90Tb10 where the paramagnetic phase exists in the entire temperature region up to 1 K. Magnetization at low temperatures for all compositions did not exceed 0,1Ms. With increasing concentration of Tb atoms the transition temperature linearly increases. This linear dependence qualitatively agrees with the experimental results [3].
The spin-glass transition takes place only if the concentration of Tb atoms x 13  at. %, i.e. above the percolation threshold in this system. x, at. % Tb Fig. 3. Compositional dependence of the spin-glass transition temperature for the models of Re-Tb amorphous alloys amorphous Tb For the models of pure amorphous Tb, the spin correlation functions were calculated (Fig. 4): where K0, А, r0 are constants;  is correlation length. We calculated the dependence of the correlation length  from the approximation of the   K r function by the eq. (6) on 0 D J for amorphous Tb at Т = 1 К (Fig. 5). In Fig. 5 dTb is the diameter of the terbium atom (dTb = 0.354 nm). The temperature dependence of the correlation length for the model of amorphous Tb at 0 6,6 D J  is shown in Fig. 6. Near the phase transition temperature Tf the correlation length sharply decreases, and in the paramagnetic phase it is near one diameter of Tb atom.
For these models we also calculated the angular spin correlation functions   P  which are the distribution functions of the angles between directions of spins arranged within the first coordination sphere (Fig. 7).

Magnetic phase diagram for amorphous Gd
For the model of pure amorphous Gd, we calculated the temperature dependencies of spontaneous magnetization M at variuos values of the 1 2 J J ratio ( 1 2 10 J J  , 11, 12, 13 и 14). In the Fig. 8 we present the dependence maximum spontaneous magnetization (at T = 1 К) on the 1 2 J J ratio for the model of amorphous Gd.