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Surface Effects in a MultilayerWe see so far that when there is a magnetic phase transition, the spin resistivity undergoes an anomaly. In magnetic thin films, when there is a surface phase transition at a temperature T_{s} different from that of the bulk one (T_{c}), we expect two peaks of p one at T_{s} and the other at T_{c}. We show here an example of a thin film composed of three subfilms: The middle film of four atomic layers between two surface films of five layers. The lattice sites are occupied by Ising spins interacting with each other via nearest neighbor ferromagnetic interaction. Let us suppose the interaction between spins in the outside films be Js and that in the middle film be J. The interfilm interaction is J. The lattice structure is face centered cubic. In order to enhance surface effects, we suppose in addition Js << J ■ We show in Fig. 17.17 the layer magnetization as well as the layer susceptibility. We observe that the outside films Figure 17.17 Top: Magnetization versus T in the case where the system is made of three films: The first and the third have five layers with a weaker interaction ]_{s}, while the middle has four layers with interaction / = 1. We take J_{s} = 0.2]. Black triangles: magnetization of the surface films, stars: magnetization of the middle film, void circles: total magnetization. Bottom: Susceptibility versus T. Black triangles: susceptibility of the surface films, stars: susceptibility of the middle films, void circles: total susceptibility. See text for comments. Figure 17.18 Resistivity p in arbitrary unit versus T of the system described in the previous figure’s caption. undergo a phase transition at a temperature far below the transition temperature of the middle film. As stated above, a phase transition induces an anomaly in the spin resistivity: The two phase transitions observed in Fig. 17.17 give rise to two peaks of p shown in Fig. 17.18. The surface peak of p has been also seen in a frustrated film [219]. The Case of MnTeThe pure MnTe has either the zinc blende structure [148] or the hexagonal NiAs one shown in Fig. 17.19. We confine ourselves in the latter case where the Neel temperature is T_{N} = 310 К [339]. Hexagonal MnTe is a crossroad semiconductor with a big gap (1.27 eV) and a roomtemperature carrier concentration of n = 4.3 x 10^{17}cm^{3} [8, 236]. Without doping, MnTe is nondegenerate. The behavior of p in MnTe as a function of T has been experimentally shown [13, 59, 101, 142, 209]. The hexagonal is composed of ferromagnetic xy hexagonal planes antiferromagnetically stacked in the c direction. The NN distance in the c direction is c/2 ~ 3.36 A shorter than the inplane NN distance which iso = 4.158 A. Neutron Figure 17.19 Structure of MnTe of NiAs type is shown. Antiparallel spins are shown by black and white circles. NN interaction is marked by J_{u} next NN interaction by J_{2}, and third NN one by J_{3}. scattering experiments show that the main exchange interactions between Mn spins in MnTe are (i) interaction between NN along the c axis with the value Ji/k_{B} = 21.5±0.3 K, (ii) ferromagnetic exchange]г/к_{в} ~0.67 ± 0.05 К between inplane neighboring Mn (they are next NN by distance), (iii) third NN antiferromagnetic interaction J_{3}/k_{B} ~ —2.87 ± 0.04 K. The spins are lying in the xy planes perpendicular to the c direction with a small inplane easy axis anisotropy D [339]. We note that the values of the exchange integrals given above have been deduced from experimental data by fitting with a formula obtained from a free spin wave theory [339]. Other fittings with meanfield theories give slightly different values: Ji/k_{B} = —16.7 K, y_{2}//fe = 2.55 К and J_{3}/k_{B} = 0.28 К [236]. The Hamiltonian is given by
where S, is the Heisenberg spin at the lattice site /, *^{s made }over the NN spin pairs S, and S_{y} with interaction /1, while £](;,/») ^{and} Ep, are made over the NNN and third NN neighbor pairs with interactions J_{2} and J_{3}, respectively. D_{a} > 0 is an anisotropy constant which favors the inplane x easyaxis spin configuration. The Mn spin is experimentally known to be of the Heisenberg model with magnitude S = 5/2 [339]. The interaction between an itinerant spin and surrounding Mn spins in semiconducting MnTe is written as
where 7(rR„) > 0 is a ferromagnetic exchange interaction between itinerant spin a at r and Mn spin S„ at lattice site R„. The sum on lattice spins S„ is limited at cutoff distance D = a. We use here the Ising model for the electron spin. In doing so, we neglect the quantum effects which are, of course, important at very low temperature but not in the transition region at room temperature where we focus our attention. We suppose the following distance dependence of /(r  R„):
where I_{0} and a are constants. We choose a = 1 for convenience. The choice of 1_{0} should be made so that the interaction 7У, yields an energy much smaller than the lattice energy due to V. (see the discussion on the choice of variables given above). Note that the cutoff distance is rather short so that the obtained results shown below still keep a general character which does not depend on the choice of exponential form. Since in MnTe the carrier concentration is n = 4.3 x 10^{17}cm^{3}, very low with respect to the concentration of its surrounding lattice spins ~ 10^{22}cm^{3}, we do not take into account the interaction between itinerant spins. As said before, the values of the exchange interactions deduced from experimental data depend on the model Hamiltonian, in particular the spin model, as well as the approximations. Furthermore, in semiconductors, the carrier concentration is a function of T. In our model, there is, however, no interaction between itinerant spins. Therefore, the number of itinerant spins used in the simulation is important only for statistical average: The larger the number of itinerant spins, the better the statistical average. The current obtained is proportional to the number of itinerant spins but there are no extra physical effects. Using the exchange integrals slightly modified with respect to the ones given above, we have calculated p of the hexagonal MnTe. The result of p is shown in Fig. 17.20. Note that with Уз slightly larger in magnitude than the value deduced from experiments, we find T_{N} = 310 К in excellent agreement with Figure 17.20 Spin resistivity p versus temperature T. Black circles are from Monte Carlo simulation, white circles are experimental data taken from He etal. The parameters used in the simulation are/ 1 = —21.5K,/_{2} = 2.55 K, У з = 9 К, /о = 2 К, D_{a} = 0.12 К, = a = 4.148 A, e = 2 x 10^{s} V/m, L = 30a (lattice size L^{3}). experiments. Furthermore, we observe that p shows a pronounced peak and coincides with the experimental data. The values we used to obtain that agreement are A = 1 and Heisenberg critical exponents v = 0.707, 2=1.97 [267]. In the temperature regions below T < 140 К and above T_{N} the MC result is also in excellent agreement with experiment, unlike in our previous work [6] using Boltzmann's equation. Using the value of p, we obtain the relaxation time of itinerant spin equal to r/ ~ 0.1 ps, and the mean free path equal to 7 ~ 20 A, at the critical temperature. ConclusionWe have shown in this chapter how MC simulations can be used to produce properties of spin transport in magnetic materials. The method is very general, it can be easily applied to a wide range of materials from ferromagnets to antiferromagnets of different lattices and spin models. The results of the spin resistivity p as a function of temperature under different situations can be obtained and compared to experiments. We were concentrated in the magnetic phase transition region where theories failed to predict correct behaviors of p. This is due to the fact that the magnetic resistivity is intimately related to the spinspin correlation which is very different from one material to another. This correlation, as we know in the domain of phase transition and critical phenomena, governs the nature of the transition: phase transitions of second order of different universality classes and phase transitions of first order. Needless to say, the nature of the phase transition affects the behavior of p as seen above: different shapes of p and discontinuity at T_{c}, etc. We have, for a good demonstration of the efficiency of our method, studied the case of MnTe where experimental data are recently available for the whole temperature range. Our result is in excellent agreement with experiments: It reproduces the correct Neel temperature as well as the shape of the peak at the phase transition. Part III 
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