# Spin Resistivity in Thin Films

In this chapter, we show the recent results on the spin resistivity in magnetically ordered materials obtained by Monte Carlo simulations. We discuss its behavior as a function of temperature in various types of crystal: ferromagnetic, antiferromagnetic and frustrated spin systems. In the model used for simulations, we take into account the interaction between itinerant spins and that between lattice spins and itinerant spins. We also include a chemical potential term, as well as an electric field. We study in particular the behavior of the spin resistivity at and near the magnetic phase transition where the effect of the magnetic ordering is strongest. In ferromagnetic crystals, the spin resistivity shows a sharp peak very similar to the magnetic susceptibility. This can be understood if one relates the spin resistivity to the spin-spin correlation as suggested in a number of theories. The dependence of the shape of the peak on physical parameters such as carrier concentration, magnetic field strength, relaxation time, etc., is discussed. In antiferromagnets, the peak is not so pronounced and in some cases it is absent. Its direct relationship to the spin-spin correlation is not obvious. As for frustrated spin systems with strong first-order transition, the spin resistivity shows a discontinuity at the phase transition. To show the efficiency of the simulation method, we compare our results with

*Physics of Magnetic Thin Films: Theory and Simulation *Hung T. Diep

Copyright © 2021 Jenny Stanford Publishing Pte. Ltd.

ISBN 978-981-4877-42-8 (Hardcover), 978-1-003-12110-7 (eBook) www.jennystanford.com recent experimental data performed on semiconducting MnTe of NiAs structure. We observe a very good agreement with experiments on the spin resistivity in the whole range of temperature.

The results of this chapter are part of the results which have been published in Refs. [3-6,151, 218-220].

## Introduction

The study of the behavior of the resistivity is one of the fundamental tasks in materials science. This is because the transport properties occupy the first place in electronic devices and applications. The resistivity has been studied since the discovery of the electron a century ago by the simple Drude theory using the classical free particle model with collisions due to atoms in the crystal. The following relation is established between the conductivity *a* and the electronic parameters *e* (charge) and *m* (mass):

where r is the electron relaxation time, namely the average time between two successive collisions. In more sophisticated treatments of the resistivity where various interactions are taken into account, this relation is still valid provided two modifications (i) the electron mass is replaced by its effective mass which includes various effects due to interactions with its environment (ii) the relaxation time r is not a constant but dependent on collision mechanisms. The first modification is very important, the electron can have a "heavy” or “light” effective mass which modifies its mobility in crystals. The second modification has a strong impact on the temperature dependence of the resistivity: r depends on some power of the electron energy, this power depends on the diffusion mechanisms such as collisions with charged impurities, neutral impurities, magnetic impurities, phonons, magnons, etc. As a consequence, the relaxation time averaged over energy, < r >, depends differently on *T* according to the nature of the collision source. The properties of the total resistivity stem thus from different kinds of diffusion processes. Each contribution has in general a different temperature dependence. Let us summarize the most important contributions to the total resistivity *p _{t}[T)* at low temperature (Г) in the following expression:

where *A _{b} A*

*and A*

_{2}_{3}are constants. The first term is

*T*-independent, the second term proportional

*to T*

*represents the scattering of itinerant spins at low*

^{2}*T*by lattice spin waves. Note that the resistivity caused by a Fermi liquid is also proportional to

*T*The Г

^{2}.^{5}term corresponds to low-Г resistivity in metals. This is due to the scattering of itinerant electrons by phonons. Note that at high

*T,*metals show a linear-Г dependence. The In term is the resistivity due to the quantum Rondo effect caused by a magnetic impurity at very low

*T.*

We are interested here in the spin resistivity *p* of magnetic materials. This subject has been investigated intensively both experimentally and theoretically for more than five decades. The rapid development of the field is due mainly to many applications in particular in spintronics.

Experiments have been performed in many magnetic materials including metals, semiconductors and superconductors. One interesting aspect of magnetic materials is the existence of a magnetic phase transition from a magnetically ordered phase to the paramagnetic (disordered) state. Very recent experiments such as those performed on the following compounds show different forms of anomaly of the magnetic resistivity at the magnetic phase transition temperature: ferromagnetic SrRu0_{3} thin films [368], Ru- doped induced ferromagnetic Lao.4Cao._{6}Mn0_{3} [215], antiferromagnetic e-(Mni__{x}Fe_{x})_{3}.25Ge [98], semiconducting Pr_{0}.7Cao._{3}Mn0_{3} thin films [377], superconducting BaFe_{2}As_{2} single crystals [352], and Lai__{x}Sr_{x}Mn0_{3} [308]. Depending on the material, *p* can show a sharp peak at the magnetic transition temperature *T _{c}* [227] or just only a change of its slope, or an inflexion point. The latter case gives rise to a peak of the differential resistivity

*dp/dT*[269, 324].

As for theories, the *T** ^{2}* magnetic contribution in Eq. (17.2) has been obtained from the magnon scattering by Kasuya [178]. However, at high

*T*in particular in the region of the phase transition, much less has been known, de Gennes and Friedel [69] proposed the idea that the magnetic resistivity results from the spin-spin correlation so it should behave as the magnetic susceptibility, thus it should diverge at

*T*Fisher and Langer [117], and Kataoka [179] have suggested that the range of spin-spin correlation changes the shape of

_{c}.*p*near the phase transition. The resistivity due to magnetic impurities has been calculated by Zarand et al. [375] as a function of the Anderson's localization length. This parameter expresses in fact a kind the correlation sphere induced around each impurity. Their result shows that the resistivity peak depends on this parameter, in agreement with the spin-spin correlation idea.

The absence of Monte Carlo (MC) simulation in the literature on the spin transport has motivated our recent works: We have studied the spin current in ferromagnetic [3-5] and antiferromagnetic [6, 218-220] materials by MC simulations. The behavior of *p* as a function of *T* has been shown to be in agreement with main experimental features and theoretical investigations mentioned above.

In this chapter, we give a review of these works and outline the most important aspects and results. We consider in some details the case of MnTe where our simulation is in excellent agreement with experiments.

In Section 17.2, we show our basic model. We describe our MC method in Section 17.3. Results on ferromagnets and antiferromagnets are shown and discussed in Section 17.4. The spin resistivity of frustrated spin systems is presented in Section 17.5. Surface effects on the spin resistivity in a multilayer are shown in Section 17.6. The case of MnTe is considered in Section 17.7. Concluding remarks are given in Section 17.8.

## Model

The model used in our MC simulation is very general. The itinerant spins move in a crystal whose lattice sites are occupied by localized spins. The itinerant spins and the localized spins may be of Ising, XY or Heisenberg models. Their interaction is usually limited to nearest neighbors (NN) but this assumption is not necessary. It can be ferromagnetic or antiferromagnetic.

Our purpose here is to study the effect of the magnetic transition on *p.* This transition occurs at a high temperature where it is known that the quantum nature of itinerant electron spins does not make significant additional effects with respect to the classical spin model. Therefore, to simplify the task, we consider here the classical spin model.

### Interactions

We consider a crystal of a given lattice structure where each lattice site is occupied by a spin. The interaction between the lattice spins is given by the following Hamiltonian:

where S, is the spin localized at lattice site / of Ising, XY or Heisenberg model, *J*,,y the exchange integral between the spin pair S, and S*j* which is not limited to the interaction between nearest- neighbors (NN). Hereafter, except otherwise stated, we take * J_{it}j* =

*for NN spin pairs, for simplicity.*

**J***]*> 0 (< 0) denotes ferromagnetic (antiferromagnetic) interaction. The system size is

*L*where

_{x}xL_{y}xLz*Lj[i*= x,

*y, z)*is the number of lattice cells in the

*i*direction. Periodic boundaiy conditions (PBC) are used in all directions.

We define the interaction between itinerant spins and localized lattice spins as follows:

where **I _{it}j** denotes the interaction which depends on the distance between electron / and spin

*Sj*at lattice site

*j.*For simplicity, we suppose the following interaction expression:

where r,_{;}- = |r, - r_{;}|, *I** _{0}* and

*a*are constants. We use a cutoff distance

*D*for the above interaction. In the same way, the interaction between itinerant electrons is defined by

with *Kjj* being the interaction between electrons *i* and *j,* limited in a sphere of radius Z)_{2}. The choice of the constants *K** _{0}* and

*ft*will be discussed below.

Note that the choice of an exponential law does not affect the general feature of our results presented in this chapter because the short cut-off distance used here limits the interaction to a small number of neighbors, typically to next nearest neighbors (NNN), so the choice of another law such as a power law, or even discrete interaction values, for such a small cut-off will not make a qualitative difference in the results.

Itinerant electrons move under an electric field applied along the *x* axis. The PBC ensure that the electrons that leave the system at one end are to be reinserted at the other end. These boundary conditions are used in order to conserve the average density of itinerant electrons. One has

where *e* is the electronic charge, e an applied electric field and *l* a displacement vector of an electron.

Since the interaction between itinerant electron spins is attractive, we need to add a kind of "chemical potential” in order to avoid a possible collapse of electrons into some points in the crystal and to ensure a homogeneous spatial distribution of electrons during the simulation. The chemical potential term is given by

where n(r) is the concentration of itinerant spins in the sphere of *D** _{2 }*radius, centered at r,

*n*

*the average concentration, and*

_{0}*D*a constant parameter.

### Choice of Parameters and Units

As said earlier, our model is very general. Several kinds of materials such as metals, semiconductors, insulating magnetic materials, etc., can be studied with this model, provided an appropriate choice of the parameters. For example, non-magnetic metals correspond to *li j = Kjj* = 0 (free conduction electrons). Magnetic semiconductors correspond to the choice of parameters *K** _{0}* and

*1*

*so as the energy of an itinerant electron due to the interaction*

_{0}*H*should be much lower than that due to

_{r}*H*namely itinerant electrons are more or less bound to localized atoms. Note that

_{m},*H*depends on the concentration of itinerant spins: For example, the dilute case yields a small

_{m}*H*We make simulations for typical values of parameters which correspond to semiconductors. The choice of the parameters has been made after numerous test runs. We describe the principal requirements which guide the choice:

_{m}-- (i) We choose the interaction between lattice spins as unity, i.e., 1/1 = 1-
- (ii) We choose the interaction between an itinerant and its surrounding lattice spins so as its energy E, in the low
*T*region is the same order of magnitude with that between lattice spins. To simplify, we take*a =*1. This case corresponds to a semiconductor, as said earlier. - (iii) The interaction between itinerant spins is chosen so that this contribution to the itinerant spin energy is smaller than E, in order to highlight the effect of the lattice ordering on the spin current. To simplify, we take
*p = 1.* - (iv) The choice of
*D*is made in such a way to avoid the formation of clusters of itinerant spins (agglomeration) due to their attractive interaction [Eq. (17.7)]. - (v) The electric field is chosen not so strong in order to avoid its dominant effect that would mask the effects of thermal fluctuations and of the magnetic ordering.
- (vi) The density of the itinerant spins is chosen in a way that the contribution of the interactions between themselves is much weaker than E, , as said above in the case of semiconductors.

Within the above requirements, a variation of each parameter does not change qualitatively the results shown below. Only the variation of *Di* in some antiferromagnets does change the results (see Ref. [219]).

The energy is measured in the unit of *|J* |. The temperature is expressed in the unit of *J /кв-* The distance (Dx and Z)_{2}) is in the unit of the lattice constant *a.* Real units will be used in Section 17.7 for comparison with experiments.