# Spin Transport in Magnetically Ordered Systems: Ferromagnets,
Antiferromagnets and Frustrated Systems

Danh-Tai Hoang 1 and Hung T. Diep 2111corresponding author 1 Biological Data
Science Institute, College of Science, The Australian National University,
Canberra, Australia<EMAIL_ADDRESS>
2 Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris
Université, CNRS, UMR 8089
2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise, France<EMAIL_ADDRESS>

###### Abstract

In this review, we outline the important results on the resistivity
encountered by an electron in magnetically ordered materials. The mechanism of
the collision between the electron and the lattice spins is shown. Experiments
on the spin resistivity in various magnetic materials as well as theoretical
background are recalled. We focus on our works since 15 years using
principally Monte Carlo simulations. In these works, we have studied the spin
resistivity in various kinds of magnetic systems ranging from ferromagnets and
antiferromagnets to frustrated spin systems. It is found that the spin
resistivity shows a broad peak at the transition temperature in systems with a
second-order phase transition, while it undergoes a discontinuous jump at the
transition temperature of a first-order transition. New results on the
hexagonal-close-packed (HCP) antiferromagnet are also shown in extended
details for the Ising case in both the frustrated and non-frustrated parameter
regions.

spin resistivity; ferromagnets; antiferromagnets; frustrated spin systems;
Monte Carlo simulation

##  I Introduction

The resistivity encountered by the displacement of an electron driven by an
applied electric field in a material is due to its collisions with the
material constituents such as atoms. In general these collisions are caused
not by direct contacts but by various potentials, magnetic and electric fields
from different sources. As a matter of fact, the motion of the electron is
slowed down except in the superconducting regime. One can mention many simple
examples such as the motion of an electron in a magnet, or in a lattice with
vibrating atoms (phonons) under an applied electric field.

The investigation of the resistivity is one of the most important tasks in
condensed matter physics. Apart from the desire to understand the mechanisms
lying behind the resistivity, the numerous applications using the transport
properties of electrons in electronic devices have motivated an increasing
number of studies, experimentally, theoretically and numerically. The study of
the resistivity has started after the discovery of the electron more than a
century ago by the simple free-electron Drude theory taking into account the
relaxation time $\tau$ between two successive collisions due to atoms. The
following relation has been established

$\sigma=\frac{ne^{2}\tau}{m}$ (1)

where $\sigma$ is the conductivity, $e$ the electron charge, $\tau$ the
electron relaxation time, $m$ the electron mass, and $n$ the number of
electrons crossing a unit srface per unit of time.

Over the years, there has been a large number of more realistic theories of
resistivity which take into account different interactions. Nevertheless, this
relation is still valid if the electron mass $m$ is replaced by its effective
mass $m^{*}$ which includes effects of the interactions of the electron with
its environment. The relaxation time $\tau$ should also be modified, it is no
more a constant but it depends on the collision mechanism. In a crystal it is
known that the effective mass of the electron can be ”heavier” or ”lighter”
than its mass at rest $m_{0}$ because it contains the effects of various
interactions. This strongly modifies the mobility of the electron in crystals.
As for $\tau$, it has a strong effect on the temperature dependence of the
resistivity. It has been established that $\tau$ depends on a power of the
electron energy. This power depends on the collision process such as
collisions with charged impurities, neutral impurities, magnetic impurities,
phonons, magnons, etc. Thus, in a crystal the total resistivity $\rho_{t}(T)$
is a sum of the contributions coming from various collision processes. At low
temperature ($T$), it is given by

$\rho_{t}(T)=\rho_{0}+A_{1}T^{2}+A_{2}T^{5}+A_{3}\ln{\frac{\mu}{T}}$ (2)

where $A_{1}$, $A_{2}$ and $A_{3}$ are constants. The first term is
$T$-independent, the second term, proportional to $T^{2}$, stems from the
scattering of the conducting electron at low $T$ by lattice spin-waves. Note
that the resistivity caused by a Fermi liquid is also proportional to $T^{2}$
with another coefficient. The $T^{5}$ term which is observed in metals comes
from the diffusion of conduction electrons by atomic vibrations. However, the
resistivity in metals show a linear-$T$ dependence at high $T$. The last term
expresses the contribution from the quantum Kondo effect, namely the
scattering of conduction electrons by magnetic impurities at extremely low
$T$.

In this review, we focus our attention on the resistivity $\rho$ due to the
spin of the conduction electron in magnetically ordered materials. For short
let us call it the ”spin resistivity” hereafter. This spin resistivity has
been widely studied both experimentally and theoretically for more than five
decades. The rapid development of the field is due mainly to many applications
in spintronics.

We are interested in magnetic materials which show a phase transition from a
magnetically ordered phase, such as ferromagnets and antiferromagnets, to the
paramagnetic state. Let us mention in the following some experiments which
have been performed in magnetic materials including metals, semiconductors and
superconductors. These experiments carried out on various materials show
different shapes of the spin resistivity around the phase-transition
temperature: SrRuO3 thin films Xia , Ru-doped La0.4Ca0.6MnO3 Lu2009 ,
antiferromagnetic $\epsilon$-(Mn1-xFex)3.25Ge Du2007 , semiconducting
Pr0.7Ca0.3MnO3 thin films Zhang , superconducting BaFe2As2 single crystals
Wang-Chen , LaFeAsO McGuire2008  and La1-xSrxMnO3 Santos2009 . We see in these
works that depending on the nature of the compound, $\rho$ can have a
pronounced peak Matsukura  or a change of its slope, or a curvature change at
the transition temperature $T_{C}$. Note that in the last case, one has a
maximum of the differential resistivity $d\rho/dT$ Stishov ; Shwerer .

Theoretically, the $T^{2}$ magnetic contribution in Eq. (2) has been obtained
by Kasuya Kasuya  taking into account the scattering of the electron spin by
the spin waves at low $T$. However, at higher $T$, specially in the region of
the phase transition of the magnetic lattice, there has been no such clear
mechanisms explaining different experimental behaviors of the spin
resistivity. de Gennes and Friedel DeGennes  have conjectured that the spin
resistivity has the origin in the spin-spin correlation so it should behave as
the magnetic susceptibility. As a consequence, it should diverge at $T_{C}$.
However, Fisher and Langer Fisher , and Kataoka Kataoka  have made the
observation that the range of spin-spin correlation should not be infinite at
$T_{C}$ due to collisions. This changes the shape of $\rho$ with respect to
the magnetic susceptibility near the phase transition. Let us mention that the
resistivity due to magnetic impurities has been calculated by Zarand et al.
Zarand  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, thus
in agreement with the spin-spin correlation idea.

Note that the spin resistivity depends on the spin orientation of the
environment: the electron encounters less resistance in a ferromagnet with
spins parallel to its spin than in a ferromagnet with spins antiparallel to
its spin. Imagine a film composed of three ferromagnetic layers where the
middle one is a soft ferromagnet. In the layer-coupling configuration
$\uparrow-\uparrow-\uparrow$, the movement of an up spin perpendicular to the
film encounters a resistance $R_{\uparrow}$. One applies now an external
magnetic field in the negative direction, small enough to reverse the spins in
the (middle) soft layer: one has the three-layer configuration
$\uparrow-\downarrow-\uparrow$. The up spin is found to encounter much
difficulty to cross the three layers: the resistance is $R_{\downarrow}$ which
is much larger than $R_{\uparrow}$. This is the phenomenon Giant-Magneto-
Resistance (GMR) discovered in Refs. Fert1 ; Fert2 ; Grunberg  which has many
applications in spintronics.

The absence of Monte Carlo (MC) simulation in the literature on the spin
resistivity has motivated our works since 2007: we have studied the spin
current in a number of systems including ferromagnets Akabli ; Akabli2 ;
Akabli3  and antiferromagnets Akabli4 ; Magnin ; Magnin3 ; Magnin2012  by MC
simulations. The behavior of $\rho$ as a function of $T$ has been shown to be
in general agreement with experiments and theories mentioned above. In
addition to ferromagnets and antiferromagnets, we have also studied the spin
resistivity in frustrated spin systems Magnin2 ; Hoang2011 . These systems
discovered in the early 80’s have been intensively studied. Many unusual
properties have been found. The reader is referred to Ref. DiepFSS  for
reviews on various frustrated spin systems. In this paper, we will summarize
the most important aspects and results of works on the spin resistivity in
ferromagnets, antiferromagnets and in some frustrated magnets.

In section II, we present our generic model and the MC method which we employ
to study the spin resistivity. Section III is devoted to the presentation of
our main MC results since 2007. Comparison with some experiments is made in
this section. In section IV, we show new results in the case of a hexagonal-
close-packed (HCP) crystal where we tune a frustration parameter allowing to
study both the non-frustrated case and the frustrated case in the phase space.
Conclusing remarks are given in section V.

##  II Model and Method

###  II.1 Model

We have investigated the spin resistivity in magnetically ordered materials by
using a newly-deviced efficient MC simulation method Magnin ; Magnin3 . The
success of the method was demonstrated when we studied the semiconducting MnTe
where the agreement with experiment is excellent Magnin2012 . This case will
be reviewed below. In ferromagnets, we found that the spin resistivity has a
high peak at the lattice order-disorder transition tremperature $T_{C}$. As
said earlier, this anomaly comes from the spin-spin correlation DeGennes ;
Fisher ; Kataoka  but remains finite at $T_{C}$. In antiferromagnets, one
observes only a broad maximum Haas . In addition to ferromagnets and
antiferromagnets, we have investigated the spin resistivity in the following
two frustrated systems: the face-centered cubic (FCC) lattice with Ising spins
Magnin2  and the $J_{1}-J_{2}$ simple cubic (SC) lattice Hoang2011 . We found
that that the first-order phase transition in these frustrated systems causes
a discontinuity of the spin resistivity at $T_{C}$.

Let us recall here the model and method which have been used in our early
works shown in Refs. Magnin ; Magnin2 : simulations have been carried out to
calculate the current of itinerant spins moving in the system under the action
of an electric field $\vec{\epsilon}$ applied in the $x$ direction. The
itinerant spin $\sigma_{i}$ carried by a conduction electron interacts with
its surrounding lattice spins inside the sphere of radius $D_{1}$ centered at
its position at the time $t$ on its trajectory across the crystal. The
Hamiltonian is supposed to be

${\cal H}_{l}=-\sum_{j}I_{ij}\vec{\sigma}_{i}\cdot\vec{S}_{j}$ (3)

where the sum is carried over all lattice spins in the sphere centered at the
itinerant Ising spin $\vec{\sigma}_{i}$. $I_{ij}$ denotes the distance-
dependent interaction between $\vec{\sigma}_{i}$ and $\vec{S}_{j}$. To be
general, we also consider the following interaction between an electron spin
with neighboring conduction spins within a sphere of radius $D_{2}$

${\cal H}_{i}=-\sum_{j}K_{ij}\vec{\sigma}_{i}\cdot\vec{\sigma}_{j}$ (4)

For simplicity, we suppose the distance-dependent interactions are

$\displaystyle I_{ij}$ $\displaystyle=$ $\displaystyle I_{0}\exp(-Br_{ij})$
(5) $\displaystyle K_{ij}$ $\displaystyle=$ $\displaystyle
K_{0}\exp(-Cr_{ij})$ (6)

where $I_{0}$, $B$, $K_{0}$ and $C$ are constants to be chosen so that the
energy of a conduction electron spin is much smaller than that of a lattice
spin. This choice is made from a physical consideration: in choosing so, we
avoid the influence of itinerant spins on the ordering of the lattice spins.
Note that this choice is justified in the almost-free electron model where
$I_{0}\simeq K_{0}\simeq 0$, and in semiconductors where they are larger but
still weak with respect to the exchange intergrals of the lattice spins
$J_{1}$ and $J_{2}$. A discussion in details on the choice of these parameters
has been given for example in Ref. Magnin .

Let us assume in the folklowing a concentration of one itinerant electron per
two lattice cells. This concentration is thus of the order of electron
concentration in normal metals which is $~{}10^{23}/cm^{3}$. With this
concentration, it is obvious that the averaged distance between two conduction
electrons is much larger than the cutoff distance $D_{2}$ which is of the
order of the lattice constant. However, due to the attractive nature of the
electron-electron interaction, Eq. (6), it is neccessary to introduce a
chemical potential term to insure that itinerant spins are uniformmly
dispersed in the crystal and they do not form clusters. This chemical
potential is written as

${\cal H}_{p}=D[n(\vec{r})-n_{0}]$ (7)

where $D$ is a positive constant, $n(\vec{r})$ denotes the concentration of
conduction spins in the sphere of cutoff radius $D_{2}$ centered at the
position $\vec{r}$ of the conduction spin under consideration, and $n_{0}$ the
averaged electron concentration.

$I_{0}$ is defined in Eq. (5) which represents the magnitude of the
interaction between a conduction electron and a localized lattice spin [Eq.
(3)]. $K_{0}$ is the magnitude of the interaction between two conduction
electrons [see Eq. (4) and (6)]. $D$ is the magnitude of the chemical
potential [Eq. (7)]. When we simulate a real material, if we have experimental
data on the resistivity and the exchange interactions, such as in the case
MnTe presented in section III.3, we can estimate the values of these
coefficients.

###  II.2 Simulation Method

Let us study a film of size $N_{x}\times N_{y}\times N_{z}$ where $N_{z}$ is
the film thickness which is much smaller than the sizes $N_{x}$ and $N_{y}$ in
the $x$ and $y$, respectively. Usually, we use $N_{z}=4-8$ and
$N_{x}=N_{y}=20-60$. Itinerant spins move under the action of an electric
field acting on the electron charge, applied in the direction $x$. The
electric field energy is

$\mathcal{H}_{E}=-e\vec{\epsilon}\cdot\vec{\ell}$ (8)

where $e$ is the electron charge, $\vec{\epsilon}$ the applied electrical
field and $\vec{\ell}$ the displacement vector of the electron.

The interaction between the lattice spins is given by the Hamiltonian

$\mathcal{H}_{L}=-J\sum_{i,j}\vec{S_{i}}\cdot\vec{S_{j}}$ (9)

where $J$ is the exchange interaction between nearest neighbors (NN)
$\vec{S}_{i}$ and $\vec{S}_{j}$. For ferromagnets $J>0$, and for
antiferromagnets $J<0$.

We use the standard Monte Carlo (MC) method to thermalize the lattice alone at
$T$. Next, we introduce the electrons into the lattice. We suppost that each
electron spin interacts with the surrounding lattice spins inside a sphere
centered at its position, of radius $D_{1}$. The electron spin also interacts
with other electron spins within a sphere of radius $D_{2}$. Electrons move
under the applied electric field.

In order to obtain the spin current we have to thermalize the state of the
spins of conduction electrons in the lattice. This is done by the following
steps:

i) we take a conduction spin and calculate its actual energy $E_{old}$ using
the different interactions mentioned above,

ii) we make a trial move $\vec{\ell}$ for the electron in a random direction
between 0 and $a$ where $a$ is the lattice constant. If the move $\vec{\ell}$
takes the itinerant electron outside the sample, then we put it inside on the
other sample end by virtue of the periodic boundary condition,

iii) we then calculate the new energy $E_{new}$. If $E_{new}<E_{old}$, then
the trial position is accepted. Otherwise, it is accepted with the probability
$\exp[-(E_{new}-E_{old})/(k_{B}T)]$.

iv) we take another conduction electron and repeat the three steps above. We
continue with other electrons until all electrons are considered: this
accomplishes one MC step/spin. A large number of MC steps/spin is neccessary
to arrive at a steady current state. We next average physical quantities of
interest at the temperature $T$ under consideration.

iv) we take another $T$ and repeat the above four steps. We should cover the
temperature region of interest.

This paper is a review of our publications over the past 15 years. In each
publication, we used various sample sizes to detect finite size effects. We
increased the sample size until the results do not depend anymore on the size.
The results are considered as valid thermodynamically. The finite size effect
and the finite size scaling are what the simulators do before reaching
conclusions. This problem is particularly important when one wants to study
the criticality or the order of a phase transion. In the review, we do not
repeat these details which depend on the studied system. Rather, we emphasize
on the results. The reader interested on the details of the simulation methods
is referred to each original publication.

We calculate the spin resistivity $\rho$ as :

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{1}{N_{s}}$ (10)

where $N_{s}$ is the number of mobile electrons passing thru a unit surface
perpendicular to the direction of the applied electric field $x$, per MC time
unit. An application with a real material using real units is presented in
subsect. III.3.

For a good thermal average, we have to perform very long MC runs and we
proceed as follows: for each configuration of the lattice spins we average the
spin resistivity over $N_{1}$ MC steps, then we thermalize again the lattice
with $N_{2}$ MC steps to get rid of the correlation between lattice spin
configurations, before averaging again the resistivity for $N_{1}$ MC steps.
We repeat this $N_{1}+N_{2}$ cycle for a large number of times $N_{3}$. The
total MC steps of resistivity averaging is about $4\times 10^{5}$ steps per
spin in our runs. We found by comparison that this ”multi-step” averaging
method strongly suppresses statistical fluctuations seen in our earlier work
Akabli2 .

It is obvious that the larger $N_{1}$ and $N_{2}$ yields the better
statistics. We know that depending on $T$, the relaxation time $\tau_{L}$ of
the lattice spins varies. We have to compare $\tau_{L}$ with the relaxation
time $\tau_{I}$ of conduction electron spins in order to choose a right value
of $N_{1}$ in order to calculate the average of the resistivity with one
lattice spin configuration at $T$. We know the two limiting cases. The first
case is when $\tau_{L}\simeq\tau_{I}$. In this case, we have to take
$N_{1}=1$, namely the lattice spin configuration should change with each move
of itinerant spins. The second case is when $\tau_{L}\gg\tau_{I}$. In this
case itinerant spins can be scattered many times along its trajectory across
the same lattice configuration and for many times across the lattice.

In order to choose a right value of $N_{1}$, we consider the following
temperature dependence of $\tau_{L}$ in non frustrated spin systems. The
relaxation time is expressed in this case as Hohenberg ; Peczak ; Prudnikov

$\tau_{L}=\frac{A}{|1-T/T_{C}|^{z\nu}}$ (11)

where $A$ is a constant, $\nu$ the correlation critical exponent, and $z$ the
dynamic exponent which depends on the spin model and space dimension. For 3D
Ising model, $\nu=0.638$ and $z=2.02$. From this expression, we see that as
$T$ tends to $T_{C}$, $\tau_{L}$ diverges. In the critical region around
$T_{C}$ the system encounters thus the so-called ”critical slowing down”: the
spin relaxation is extremely long due to the divergence of the spin-spin
correlation. When we take into account the temperature dependence of
$\tau_{L}$, the shape of the resistivity is strongly modified near $T_{C}$
where $\tau_{L}$ is very long, in contrast to the paramagnetic phase where the
relaxation time is very short due to rapid thermal fluctuations. We should
emphasize that at low $T$, $\rho$ does not depend on $\tau_{L}$ since in that
temperature range where the ordering of the lattice spins is almost perfect:
the spin landscape from one microscopic state to another does not change
significantly, so the motion of the itinerant electron spin does not
significantly vary (see Ref. Magnin3 ).

Finally, we note that we have also used the Boltzmann’s equation combined with
MC data to study the spin resistivity Akabli3 . However, the shape of the
resistivity peak at the transition temperature does not agree well with
experiments, unlike that obtained from direct MC simulations as shown below.
This proves the efficiency of MC simulations for the calculation of the spin
resistivity in magnetically ordered materials. The present review therefore
aims at promoting this method.

##  III Review

###  III.1 Spin Resistivity in Ferromagnets and Antiferromagnets

Experiments mentioned above amongst numerous other data show that the spin
resistivity in ferromagnets has a sharp peak at the transition temperature
$T_{C}$ of the lattice. We know by the theory of phase transitions and
critical phenomena that in the region around $T_{C}$, the so-called ”critical
slowing-down” phenomenon happens, resulting in extremely large $\tau_{L}$. The
peak in $\rho$ is due to this phenomenon via Eq. (11) where $\tau_{L}$
diverges at $T_{C}$. Our MC simulations using the method described above in
the case of a ferromagnet where the lattice spins are of the Ising type show a
sharp peak at $T_{C}$ (see Fig. 1) in agreement with experimental data. We
note that the spin resistivity for ferromagnets (as well as for
antiferromagnets) increases with decreasing $T$ at low $T$. This can be
explained by several causes: the freezing or crystallization of conduction
electrons takes place at low $T$ so that just a small number of conduction
spins with decreasing $T$ is mobile, or it may be the classical counter effect
of the quantum Kondo electron-impurity scattering if one considers the few
excited lattice spins at low $T$ are independent impurities, see the last term
of Eq. (2). Note that the shape of $\rho$ depends on the lattice type,
interaction strengths encountered by the conduction electrons, electron
concentration, relaxation time, spin model, applied magnetic-field amplitude
etc. In Ref. Magnin , we have shown that a decrease in the interaction between
conduction spins, $K_{0}$, reduces the increase of $\rho$ as $T\rightarrow 0$,
an applied magnetic field reduces the height of the resistivity peak and the
larger electron density reduces $\rho$. Finally, we emphasize that $\rho$
depends on the material intrinsically via the critical exponents $\nu$ and
$z$, see Eq. (11).

If we wish to compare simulated spin resistivity to experimental measurements
performed on a given material, we have to use in the simulation the available
experimental physical parameters of that material. An example of quantitative
comparison for semiconductor MnTe is shown in subsection III.3.

Figure 1:  Resistivity $\rho$ as a function of $T$, obtained by simulations
using the $T$-dependent relaxation time, Eq. 11, for ferromagnet (black
circles) and antiferromagnet (white circles) of BCC structure, with arbitrary
unit in zero magnetic field. Other parameters: $\epsilon=1$, $I_{0}=2$,
$K_{0}=0.5$, $A=1$.

Note that the magnetic field applied on the system reduces the peak height as
shown in our work Ref. Magnin , in agreement with experiments Lu2009 .

Unlike ferromagnets, antiferromagnets have not been much studied. Haas Haas
has shown that in contrat to ferromagnets where the resistivity $\rho$ has a
sharp peak at the order-disorder transition of the lattice spins, in
antiferromagnets there is no such a peak. Using MC simulations, we found that
the peak does exist in an antiferromagnet but it is less sharp compared to
that of a ferromagnet, as seen in Fig. 1. We think that the alternate change
of sign of the spin-spin correlation with distance may have something to do
with the absence of a sharp peak. We have tested this idea on the effect of
the cut-off distance $D_{1}$ Magnin2 : in an antiferromagnet, when we increase
$D_{1}$, we will include successively up-spin shells and down-spin shells in
the sphere of radius $D_{1}$. Consequently, the difference between the numbers
of up and down spins in the sphere oscillates with varying $D_{1}$, giving
rise to the oscillatory behavior of $\rho$ observed at small $D_{1}$, unlike
in ferromagnets.

At this stage, we note that the presence of an itinerant spin will break the
invariance between a ferromagnet and its antiferromagnet counterpart in the
local Mattis transformation ($J_{ij}\rightarrow-
J_{ij},\vec{S}_{j}\rightarrow-\vec{S}_{j}$).

###  III.2 Frustrated $J_{1}-J_{2}$ Model on a Simple Cubic Lattice

Let us consider the simple cubic lattice with NN and NNN interactions as shown
in Fig. 2. The Hamiltonian is written as

${\cal
H}=-J_{1}\sum_{(i,j)}\vec{S}_{i}\cdot\vec{S}_{j}-J_{2}\sum_{(i,m)}\vec{S}_{i}\cdot\vec{S}_{m}$
(12)

where the first sum $\sum_{(i,j)}$ is made over the NN Ising spin pairs
$\vec{S}_{i}$ and $\vec{S}_{j}$ with interaction $J_{1}$, and the second sum
$\sum_{(i,m)}$ is performed over the NNN pairs with interaction $J_{2}$.

We focus our attention on the region of parameters which gives rise to a
frustration. For that purpose, we assume that $J_{1}$ is an antiferromagnetic
interaction, namely $J_{1}=-J<0$ ($J>0$), and $J_{2}$ is also
antiferromagnetic. We put $J_{2}=-\eta J$ where $\eta$ is a positive
parameter. The ground state (GS) of this system can be obtained by minimizing
the energy, or by comparing the energies of different spin configurations. We
can also numerically minimize the energy by using the steepest descent method
Ngo2007 . We obtain the GS antiferromagnetic configuration shown Fig. 3a for
$|J_{2}|<0.25|J_{1}|$, and the GS spin configuration shown in Fig. 3b for
$|J_{2}|>0.25|J_{1}|$. This latter configuration is 3-fold degenerate because
we can choose the parallel NN spins either on $x$, or $y$ or $z$ axis. In
addition, with the permutation of black and white spins, we have the total
degeneracy equal to 6.

Figure 2: Simple cubic lattice where the NN and NNN interactions, $J_{1}$ and
$J_{2}$, are indicated.

Figure 3: Ground state (GS) of the simple cubic lattice with Ising spins: (a)
GS when $|J_{2}|<0.25|J_{1}|$; (b) GS when $|J_{2}|>0.25|J_{1}|$. White
(black) circles denote up (down) spins. See text for comments.

We note in passing that in the case of the Heisenberg model in the frustrated
region ($|J_{2}|>0.25|J_{1}|$) the phase transition has been shown to be of
first order Pinettes . The system is very unstable due to its large
degeneracy. In the case of the Ising spin on the SC lattice treated here, we
found that the first-order character of the phase transition is even stronger
Hoang2011 .

We use $J_{1}=-J=-1$ (AF interaction) for the coupling between NN lattice
spins in the simulations. The energy is thus measured in the unit of $J$ and
the temperature is in the unit of $J/k_{B}$. All distances ($D_{1}$ and
$D_{2}$) are in the unit of the lattice constant $a$.

Simulations have been carried out by using the temperature-dependent
relaxation time of the lattice spins given by Eq. (11) where we have taken
$A=1$ and $\tau_{L}=1$ at $T=2T_{C}$ far from $T_{C}$. Such a choice leads to
$\tau_{L}=1$ at that temperature expected for fast thermal fluctuations in the
paramagnetic phase far above $T_{C}$.

Since we suppose that the interaction between conduction electron spins is
attractive, a chemical potential is required to avoid the collapse of the
system, namely to avoid that all conduction spins form a cluster [cf. Eq.(7)].
The chemical potential in thermodynamics makes the particle uniformly
distributed in the space. Its strength is expressed by $D$ which has to be
chosen in accordance with $K_{0}$. Figure 4 displays the phase diagram in the
space $(K_{0},D)$ which shows the collapse region. This allows us to avoid
this region and choose an appropriate value of $D$ for a given $K_{0}$.

Figure 4: $D$ versus $K_{0}$ in the case where $\eta=0.26$. The collapse
region is in green. We have used $D_{1}=D_{2}=1$, $I_{0}=0.5$, $\epsilon=1$.

To see the effect of the nature of phase transition on the spin resistivity,
in the following we focus on two typical cases: $\eta=0.2$ and $\eta=0.26$
which belong respectively to the regions of second- and first-order
transition.

A. For $\eta=0.2$:
The spin resistivity at temperatures below $T_{N}$ oscillates with varying
$D_{1}$. By analyzing the ratio of the number of up spins to the number of
down spins in the sphere of radius $D_{1}$, we found that it oscillates with
varying $D_{1}$: the maxima (minima) of $\rho$ correspond to the case of
largest (smallest) numbers of parallel (antiparallel) spins in the sphere
Hoang2011 ; Magnin2 . At very high temperatures where the lattice spins are
disordered, the numbers of up spins and down spins in the sphere of radius
$D_{1}$ should be equal. There is however a very small oscillation if the
temperature is close to $T_{N}$ and if $D_{1}$ is small.

Figure 5 displays the resistivity versus $T$ for $D_{1}=1.2$. The spin
resistivity shows a rounded maximum at the transition temperature. This is in
agreement with the curve experimentally observed in La0.4Ca0.6MnO3 by Lu et
al. Lu2009  (see Fig. 6, right panel).

Figure 5: Spin resistivity versus $T$ in the case $\eta=0.2$, $D_{1}=1.2$. We
have used the lattice size $N_{x}=N_{y}=20,N_{z}=6$. Other variables are
$I_{0}=K_{0}=0.5$, $D_{2}=1$, $D=1$, $\epsilon=1$. Figure 6: Experimental spin
resistivity versus $T$ are shown for several applied magnetic fields in the
compound La0.4Ca0.6MnO3. These data are in the figure 2 of Ref. Lu2009 .

For $D_{1}=0.8$ or $D_{1}=1$, the resistivity is smaller below the transition
temperature, as seen in Fig. 7. This shows the importance of the effect of the
interaction range on the spin resistivity in materials.

Figure 7: $\rho$ as a function of $T$ for $\eta=0.2$ with $D_{1}=0.8$ and $1$
((black circles and blue open circles, respectively). For simulations, we used
$N_{x}=N_{y}=20,N_{z}=6$, $I_{0}=K_{0}=0.5$, $D_{2}=1$, $D=1$, and
$\epsilon=1$.

B. For $\eta=0.26$:
At this value of $\eta$, the transition of the lattice spins is of first
order. The dependence of the resistivity on $D_{1}$ is very similar to that of
the second-phase transition, namely the resistivity at a given $T$ oscillates
as $D_{1}$ varies. The physical meaning of the oscillation has been given
above. More details can be found in Refs. Hoang2011 ; Magnin2 . We found that
the resistivity $\rho$ in the frustrated regime can go downward or upward at
the transition temperature depending on $D_{1}$ Hoang2011 , unlike in non-
frustrated ferromagnets and antiferromagnnets shown earlier. This is displayed
in Fig. 8 for two values of $D_{1}$ where one observes the discontinuity of
$\rho$ at the transition temperature. The discontinuity of $\rho$ has also
been found in other frustrated antiferromagnets such as the FCC
antiferromagnet Magnin2 .

Figure 8:  $\rho$ as a function of $T$ for $\eta=0.26$ where $D_{1}=0.8$
(black circles) and $D_{1}=1$ (blue open circles). We have used
$N_{x}=N_{y}=20,N_{z}=6$, $I_{0}=K_{0}=0.5$, $D_{2}=1$, $D=1$ and
$\epsilon=1$. See text for comments.

From the results shown above for the $J_{1}-J_{2}$ model, we come to the
conclusion that the behavior of the spin resistivity is a consequence of the
nature of the lattice transition. If the lattice transition is of second
order, then the resistivity of itinerant spins has a rounded peak, while if
the lattice transition is of first order, the resistivity is discontinuous at
the transition temperature.

###  III.3 The case of MnTe

Figure 9: Structure of MnTe of NiAs type: black and white circles present
respectively opposite spins. Interactions between NN, between next NN and
between third NN are indicated respectively by $J_{1}$, $J_{2}$, and $J_{3}$.
See values given in the text.

The pure semiconductor MnTe has two kinds of structures: the zinc-blend
structure or the hexagonal NiAs one shown in Fig. 9 Hennion . We focus on the
second structure where the Néel temperature is $T_{N}=310$ K Hennion2 , and
where many other experimental data are available. MnTe is a semiconductor with
a large gap (1.27 eV) and a carrier concentration $n=4.3\times 10^{17}$cm-3 at
room temperature Mobasser ; Allen . Without doping, MnTe is non degenerate.
The crystal is formed by ferromagnetic $xy$ hexagonal planes
antiferromagnetically stacked in the $c$ direction. The NN distance in the $c$
direction is $c/2\simeq 3.36$ Å, and the in-plane NN distance is $a=4.158$ Å.
From neutron scattering experiments, it was found that the main exchange
interactions between Mn spins in MnTe are the interaction between NN along the
$c$ axis with the value $J_{1}/k_{B}=-21.5\pm 0.3$ K, the ferromagnetic
exchange $J_{2}/k_{B}\approx 0.67\pm 0.05$ K between in-plane neighboring Mn
(they are next NN by distance), and the third NN antiferromagnetic interaction
$J_{3}/k_{B}\simeq-2.87\pm 0.04$ K (see Fig. 9). The spins are lying in the
$xy$ planes perpendicular to the $c$ direction with a small in-plane easy-axis
anisotropy $D_{a}$ Hennion2 . Let us emphasize that the values of the exchange
integrals given above were deduced from experimental data by fitting with a
free spin-wave theory Hennion2 . Other fittings with mean-field theories give
slightly different values: $J_{1}/k_{B}=-16.7$ K, $J_{2}/k_{B}=2.55$ K and
$J_{3}/k_{B}=-0.28$ K Mobasser . Note that the Mn spin is experimentally known
to be of the Heisenberg model with magnitude $S=5/2$ Hennion2 .

We write the following Hamiltonian for the lattice spins

$\displaystyle{\cal H}$ $\displaystyle=$ $\displaystyle-
J_{1}\sum_{(i,j)}\vec{S}_{i}\cdot\vec{S}_{j}-J_{2}\sum_{(i,m)}\vec{S}_{i}\cdot\vec{S}_{m}-J_{3}\sum_{(i,k)}\vec{S}_{i}\cdot\vec{S}_{k}$
(13) $\displaystyle-D_{a}\sum_{i}(S_{i}^{x})^{2}$

where the first sum is performed over the NN spin pairs, the second sum over
the NNN pairs and the third one over the third NN pairs. $D_{a}>0$ is an
anisotropy constant which favors the in-plane $x$ easy-axis spin
configuration.

The behavior of $\rho$ in MnTe as a function of $T$ has been experimentally
shown in several works Chandra1996 ; Li ; Russe ; Efrem ; He . We have studied
using MC simulations the spin resistivity in MnTe with the above Hamiltonian
Magnin2012 . Let us summarize this work here.

For MC simulations, we suppose the following Hamiltonian of the itinerant
spins:

${\cal H}_{i}=-\sum_{n}I(\vec{r}-\vec{R}_{n})\vec{\sigma}\cdot\vec{S}_{n}$
(14)

where the sum is performed by counting all the lattice spins $\vec{S}_{n}$
inside the sphere of radius $D_{1}=a$ centered at $\vec{r}$.
$I(\vec{r}-\vec{R}_{n})>0$ is the ferromagnetic distance-dependent interaction
between the itinerant electron spin $\vec{\sigma}$ at $\vec{r}$ and the Mn
spin $\vec{S}_{n}$ at $\vec{R}_{n}$ .

The electron spin is supposed of the Ising type. We neglect therefore the
quantum effects which may be important at very low $T$ but our attention is
focused on the region of high-enough $T$ where quantum effects may be
neglected. We assume the following form of $I(\vec{r}-\vec{R}_{n})$ :

$I(\vec{r}-\vec{R}_{n})=I_{0}\exp[-\alpha(\vec{r}-\vec{R}_{n})]$ (15)

where the constants $I_{0}$ and $\alpha$ are chosen in such a way that the
interaction ${\cal H}_{i}$ yields an energy much smaller than the lattice
energy given by ${\cal H}$ (see the guide for the choice of different
constants given below Eq. (6) and in Ref. Magnin ). It is noted that the cut-
off distance $D_{1}$ is rather short so that only the first few neighbors are
inside the sphere, the results shown below do not therefore depend
significantly on the choice of the value of $\alpha$ in the exponential.
Finally, note that the concentration of conduction electrons in MnTe is
$n=4.3\times 10^{17}$cm-3 which is five orders lower than the concentration of
its surrounding lattice spins which is $\simeq 10^{22}$cm-3. This observation
justifies that the interaction between conduction electrons for MnTe can be
neglected. We have assumed this in the simulations shown in the following.

As mentioned above, the exchange interactions deduced from experimental data
have slightly different values, they depend on the theoretical Hamiltonian and
the approximations used to deduce it (often the mean-field approximation is
used, see a detailed example in Ref. samia ). Note that in semiconductors, the
carrier concentration varies with $T$ but since this concentration is very
low, we do not take into account its variation. Consequently, the number of
conduction electron spins used in the simulation is important only for the
statistical average. The current obtained is proportional to the number of
itinerant spins but there are no extra effects within our assumptions
mentioned above.

We have calculated $\rho$ of MnTe, using the exchange integrals slightly
modified with respect to the ones given above in order to obtain the best fit.
The obtained resistivity $\rho$ is shown in Fig. 10. Let us note that we have
taken $J_{3}$ slightly larger in magnitude than the value deduced from
experiments by mean-field approximation. Our value of $J_{3}$ was chosen in
order to obtain $T_{N}=310$ K which is in excellent agreement with
experiments. However the most striking feature is that the simulated $\rho$
shows a sharp maximum at $T_{N}$ and coincides with the experimental data over
the whole temperature range. Note that we have used $A=1$ and the well-known
Heisenberg critical exponents $\nu=0.707$, $z=1.97$ Peczak  for the lattice
spins. It is remarkable that with the same set of param:eters, we obtain an
excellent agreement with experiments in the temperature regions below $T<140$
K and above $T_{N}$. We note that we have tried earlier to use the Boltzmann’s
equation Akabli4  but the obtained result is not as good as the MC result
presented above.

From the simulated $\rho$, we can calculate the relaxation time of conduction
spins, we obtain $\tau_{I}\simeq 0.1$ ps. The mean free path can be also
estimated, it is equal to $\bar{l}\simeq 20$ Å, at the critical temperature.

Figure 10: Comparison between the simulated spin resistivity and the
experimental data of MnTe: Black circles are results from the Monte Carlo
simulation, white circles are experimental data taken from He et al.He . We
used for the simulation $J_{1}=-21.5$K, $J_{2}=2.55$ K, $J_{3}=-9$ K,
$I_{0}=2$ K, $D_{a}=0.12$ K, $D_{1}=a=4.148$ Å, $\epsilon=2\times 10^{5}$ V/m,
$L=30a$ (lattice size: $L^{3}$). See text for comments.

##  IV Phase transition and spin resistivity in the Ising HCP lattice

###  IV.1 Hamiltonian and Ground State

The lattice we consider is the HCP structure illustrated in Fig. 11. The $xy$
planes are triangular (hexagonal) and the stacking direction is $z$. We
suppose the following Hamiltonian

Figure 11: HCP lattice: the in-plane NN interaction is denoted by $J_{1}$ and
the inter-plane NN interaction is denoted by $J_{2}$.

${\cal H}=-\sum_{(i,j)}J_{i,j}\vec{S}_{i}\cdot\vec{S}_{j}$ (16)

where $J_{ij}$ is the AF interaction between nearest-neighbors (NN)
$\vec{S}_{i}$ and $\vec{S}_{j}$. We denote $J_{ij}=J_{1}$ if the NN are on the
$xy$ triangular plane, and $J_{ij}=J_{2}$ if the NN are on two adjacent planes
(see Fig. 11). The GS can be determined by minimization the local energy of
each spin and doing this for all spins, then repeating many times until the
total energy converges to a minimum. Normally, with a system without bond
disordering, this method needs just a small number of iterations. The GS can
be checked by looking at the final snapshot: it should be periodic. This
procedure of local energy minimization is called in the literature ”the
steepest-descent method”. The implementation of this method is very simple
Ngo2007  (i) we first create an initial random configuration (ii) we then
calculate the local field acting at a spin by its neighbors using (16) (iii)
we align the spin under consideration along the calculated local field, in
doing so its energy is minimum (iv) we take another spin and repeat the three
preceding steps until all spins are considered: this step completes one sweep
(v) we start again another sweep and we realize a large number of sweeps until
the total energy is minimm.

One can also analytically minimize the interaction energy as shown below to
find the GS. Let us assume that both interactions $J_{1}$ and $J_{2}$ are
antiferromagnetic. For simplicity we fix $J_{2}=-1$ and vary $J_{1}$.

The case of isotropic interaction, namely $J_{1}=J_{2}$ has been studied in
Ref. Diep1992 . We summarize the result here: for the HCP structure, each spin
is common for eight tetrahedra (four in the upper half-space and four in the
lower half-space along the $z$ axis) and a NN bond is shared by two
tetrahedra. The GS spin configuration of the system is formed by stacking
neighboring tetrahedra. In the GS, one has two pairs of antiparallel spins on
each tetrahedron. Their axes form an arbitrary angle $\alpha$. The GS
degeneracy is therefore infinite (see Fig. 2a of Ref. Diep1992 ). Note tthat
the periodic boundary conditions will reduce a number of the GS
configurations, but the degeneracy is still infinite. One particular family of
configurations of interest for both XY and Heisenberg cases is when
$\alpha=0$. The GS is then collinear with two spins up and the other two down.
The stacking sequence is simple because there are three equivalent
configurations due to the fact that there are three ways to choose the
parallel spin pair in the original tetrahedron.

The case where $J_{1}\neq J_{2}$ has been studied in Ref. Hoang2012  for the
Ising and XY cases. Let us recall some results concerning the Ising case which
allow us to understand the new results on the spin resistivity presented
below.

We use the steepest descent method described above with varying $J_{1}$
($J_{2}=-1$): we find two kinds of GS spin configuration: the first consists
of $xy$ ferromagnetic planes stacked antiferromagnetically along the $z$
direction, while the second one is the stacking of $xy$ AF planes such that
each tetrahedron has two up and two down spins. The transition between the two
configurations is determined as follows: one simply writes down the respective
energies of a tetrahedron and compares them

$\displaystyle E_{1}$ $\displaystyle=$ $\displaystyle 3(-J_{1}+J_{2})$ (17)
$\displaystyle E_{2}$ $\displaystyle=$ $\displaystyle J_{1}+J_{2}$ (18)

One sees that $E_{1}<E_{2}$ when $J_{1}>0.5J_{2}$, i.e. $|J_{1}|<0.5|J_{2}|$.
Thus the first configuration is more stable when $|J_{1}|<0.5|J_{2}|$.

###  IV.2 Phase Transition in the case of Ising Spins on the HCP Lattice

In the following, we present the results of simulations using the Hamiltonian
Eq. (16). We use the sample size $N_{x}\times N_{y}\times N_{z}$ with
$N_{x}=N_{y}=18$ and $N_{z}=8$, namely 16 atomic planes along the $z$ axis,
and the periodic boundary conditions in all directions. We use the first
$10^{6}$ MC steps per spin to reach equilibrium and we average physical
quantities with the next $10^{6}$ MC steps per spin. The energy is expressed
in the unit of $|J_{2}|=1$.

Let us define $\eta=J_{1}/J_{2}$. We have seen that the GS changes at
$\eta_{c}=0.5$, so we show below the properties of the system on both sides of
this value. Figure 12 displays the averaged energy per spin, the order
parameter (staggered magnetization), the specific heat and the susceptibility
for $\eta=0.3$. As seen, the transition is of second order since there is no
discontinuity of the energy and the order poarameter at the transition
temperature.

Figure 12: The case of Ising spin on the antiferromagnetic HCP lattice: (a)
energy per spin $E$, (b) order parameter $M$, (c) specific heat $C_{V}$ and
(d) susceptibility $\chi$, versus temperature $T$ for $\eta=J_{1}/J_{2}=0.30$.
See text for comments.

For $\eta=J_{1}/J_{2}>0.5$, Fig. 13 for $\eta=0.85$ and $1$ shows that the
discontinuity of $E$ and $M$ at the transition is very large, a signature of a
strong first-order transition in both cases.

Figure 13: The case of Ising spin on the antiferromagnetic HCP lattice: (a)
energy per spin $E$; (b) order parameter $M$ versus temperature $T$ for
$\eta=J_{1}/J_{2}=0.85$ (blue open circles) and $1$ (red triangles). See text
for comments.

In order to confirm the order of the phase transition, we measure the energy
histogram taken during the averaging MC time. Figure 14 shows the energy
histogram taken at the transition temperature for $\eta=0.3$ (black), $0.85$
(blue) and $1$ (red). We observe here that the first case is a Gaussian
distribution indicating a second-order transition, in contrast to the last two
cases which show double-peak histograms confirming a first-order transition.

Figure 14: Energy histogram $P(E)$ for $\eta=0.3$ (black circles), $0.85$
(blue open circles), and $1$(red triangles). See text for comments.

Figure 15 displays the phase diagram in the space $(T_{C},\eta)$ where zone
(1) and zone (2) denote the ordering of the first, and second kinds,
respectively; (P) indicates the paramagnetic phase. Note that the transition
line between (1) and (P) is a second-order line, while that between (2) and
(P) is a first-order line.

Figure 15: Transition temperature $T_{C}$ versus $\eta$. (1) denotes the
second-order region, (2) the first-order region and (P) the paramagnetic
phase.

Note that in the $XY$ case, the change of the GS takes place at
$\eta_{c}=1/3$. We have studied this case in details in to Ref. Hoang2012 .

Finally, let us emphasize that all 3D frustrated systems we know so far
undergo a first-order transition DiepFSS  including the much-studied
antiferromagnetic stacked triangular lattice Itakura2003 ; Bekhechi2006 ;
Ngo2008a ; Ngo2008b ; Delamotte2005 , the FCC antiferromagnets Diep-Kawamura ,
the simple cubic fully frustrated lattices Blankschtein1984 ; Ngo2010 ;
Ngo2011a ; Ngo2011b , helimagnets Diep1989 , and antiferromagnetic HCP lattice
studied here (see more details in Refs. Diep1992 ; Hoang2012 ).

###  IV.3 Spin resistivity in the HCP lattice with Ising spins

The results in this subsection are new, they are not published so far. Using
the method which has been described in subsection II, we carry out MC
simulations to study the spin resistivity in the Ising case. We show in Fig.
16 the resistivity at two temperatures, below and above the transition
temperature, as a function of $D_{1}$ for the GS belonging to phase (1). We
show in Fig. 17 the case of a GS belonging to phase (2) (see Fig. 15). Similar
to the case of $J_{1}-J_{2}$ model on the simple cubic lattice considered in
Ref. Hoang2011 ; Magnin2 , we find here an oscillation of $\rho$ at low
temperature. Note that $\rho$ is always smaller at low temperature than at
high temperature, whatever the value of $D_{1}$ is. The physical origin of the
oscillation has been discussed above.

Figure 16: $\rho$ versus $D_{1}$ for $\eta=0.3$ at $T=1.6<T_{C}$ (black
circles) and at $T=2.8>T_{C}$ (open circles) where $T_{C}\simeq 2.4$. Other
parameters are $N_{x}=N_{y}=18$, $N_{z}=8$, $D_{2}=1$, $I_{0}=2$, $K_{0}=0.5$,
$C_{1}=C_{2}=1$, $A=1$, $D=0.5$, $\epsilon=1$. Figure 17: $\rho$ versus
$D_{1}$ for $\eta=1$ at $T=1.5<T_{C}$ (black circles) and at $T=1.9>T_{C}$
(open circles) where $T_{C}\simeq 1.7$. Other parameters are $N_{x}=N_{y}=18$,
$N_{z}=8$, $D_{2}=1$, $I_{0}=2$, $K_{0}=0.5$, $C_{1}=C_{2}=1$, $A=1$, $D=0.5$,
$\epsilon=1$.

The spin resistivity $\rho$ for $\eta=0.3$ and $1$ is shown in Fig. 18 as a
function of $T$, here the distances $D_{1}$ and $D_{2}$ are in unit of the
distance between the NN lattice spins, and $I_{0}$, $K_{0}$ and $D$ which have
the energy dimension are in the unit of $|J_{2}|=1$. As in the frustrated
$J_{1}-J_{2}$ model shown above, one finds here that $\rho$ has a broad peak
in the second-order region, in contrast to the first-order region where it
undergoes a discontinuous jump at the phase transition. Some remarks are in
order:

Figure 18: Spin resistivity $\rho$ of the Ising HCP model versus temperature
$T$ for (a) $\eta=0.3$; (b) $\eta=1$. $N_{x}=N_{y}=18$, $N_{z}=8$ (namely $16$
planes in the $z$ direction), $D_{1}=D_{2}=1$, $\epsilon=1$, $I_{0}=2$,
$K_{0}=0.5$, $C_{1}=C_{2}=1$, $A=1$, $D=0.5$. All distances are in unit of the
NN distance, energy constants are in unit of $|J_{2}|=1$. See text for
comments.

i) At very low temperature, the resistivity increases with decreasing
temperature. This behavior can be understood by the freezing of the itinerant
spins due to low $T$: The energy of itinerant spins is low, they occupy the
low-energy positions in the periodic lattice, it is difficult to move them out
by the insufficient thermal energy. They are somewhat frozen in almost
periodic positions; namely a pseudo crystallization occurs. Note that the
increase of resistivity with decreasing $T$ at very low $T$ was observed in
many experiments on various materials and is not limited to ferromagnets
Du2007 ; Wang-Chen ; Santos2009 ; Li . This increase of $\rho$ with decreasing
$T$ in the quantum case has been explained by J. Kondo using a third-order
perturbation theory Kondo : the scattering of $s$-electrons by $d$-electrons
of localized magnetic impurities gives rise to a resistivity minimum at a
finite $T$. We have also found here this minimum of $\rho$ at low $T$ with the
classical spin model. The similarity with the quantum Kondo effect can be
explained by the fact that an excited localized lattice down-spin (in a very
small number at low $T$) can be viewed as an impurity which captures nearby
conduction up-spins.

ii) Outside this low-$T$ region, when $T$ increases, the thermal energy
progressively unfreezes the itinerant spins. As a consequence, $\rho$
decreases and passes through a minimum (see discussion above). However, at
higher $T$, the scattering with the lattice spins is stronger, $\rho$
increases up to the transition temperature.

iii) At the transition temperature, $\rho$ shows a peak. The physical
mechanism leading to the peak can be explained: in a previous work Akabli3 ,
it was found from our simulations that the peak is due to scattering of the
itinerant spins by antiparallel-spin clusters which are numerous in the
transition region. When one gets close to the transition point, the number of
clusters of down spins are the most numerous, giving rise to the peak in
$\rho$. Note that the ”defects” clusters (i. e. clusters of antiparallel
spins) have an energy barrier to resist the passage of itinerant spins. This
is also the origin of the extremely long relaxation time in the critical
region.

iv) Well above the transition temperature, in the paramagnetic phase, as
temperature increases, clusters of down and up spins will be broken more and
more into independent disordered spins, namely spins with zero energy,
itinerant spins move easily on their trajectory, making a decrease of $\rho$
with increasing $T$.

Note that we have also varied the radius $D_{1}$ to see its effect on $\rho$
at the transition in the present frustrated HCP model. We found the same
effect seen in other antiferromagnets we studied previously Magnin2 ;
Hoang2011 : at a given temperature, an oscillation of $\rho$ with varying
$D_{1}$. oscillates slightly with distance. The origin of this oscillation has
been analyzed avove in the $J_{1}-J_{2}$ model.

Finally, let us look at some experimental data obtained for ferromagnets and
antiferromagnets. Figure 19 shows experiments by Du et al. performed on
$\varepsilon$-(Mn1-xFe${}_{x})_{3.25}$Ge antiferromagnets Du2007 , experiments
by McGuire et al. performed on antiferromagnetic superconductors LaFeAsO
McGuire2008 , by Chandra et al. on thin Cd1-xMnxTe films Chandra1996 .
Experiments by Santos et al. on antiferromagnetic La1-xSrxMnO3 Santos2009  are
shown in Fig. 20. We see here that our results on the shape of the spin
resistivity are in agreement with these experiments. In the lack of physical
data on these experimental materials, we cannot make a quantitative comparison
as we did in the MnTe case presented above.

Figure 19: Experiments on the resistivity as a function of $T$ iperformed by
(a) by Du et al on $\varepsilon$-(Mn1-xFe${}_{x})_{3.25}$Ge antiferromagnets
Du2007 , (b) by McGuire et al. on antiferromagnets LaFeAsO McGuire2008 , and
(c) by Chandra et al. on thin films of Cd1-xMnxTe Chandra1996 . Figure 20:
Resistivity versus temperature on antiferromagnetic La1-xSrxMnO3. The figures
presented are taken from Fig. 7 of Santos2009 .

##  V Conclusion

In this paper, we have reviewed some important works published on the spin
resistivity in magnetically ordered systems. We have focused on our works
published over the past 15 years using mainly Monte Carlo simulations. These
works were motivated by the absence of Monte Carlo works, even at the present
time except ours, in spite of the fact that this method of simulation has
proven to be very efficient when comparing its results with experimental data.
In the case of MnTe where there are sufficient experimental data, we have made
a quantitative comparison between experimental and simulated spin
resistivities. The agreement between experiments and simulations is excellent.
This review therefore aims at promoting this method to study more realistic
cases.

As demonstrations, we have used this method to study the spin resistivity in
generic ferromagnets and antiferromagnets. The cases of frustrated systems
have also been presented: the $J_{1}-J_{2}$ model and the antiferromagnetic
HCP lattice.

Let us summarize the results on the two frustrated systems.

The $J_{1}-J_{2}$ model is a simple cubic lattice with Ising spins interacting
with each other via NN and NNN antiferromagnetic interactions, $J_{1}$ and
$J_{2}$ respectively. The GS of this model is determined by the ratio
$\eta=J_{2}/J_{1}$. We have shown that the GS changes at the critical value
$\eta_{c}=0.25$. For the non-frustrated region in the phase space, namely
$\eta<0.25$, the GS is simply composed periodically of two interpenetrated
tetrahedra formed by the NNN sites. In the frustrated region, namely
$\eta>0.25$, the GS can be described as composed of one line of spin up, one
line of spin down, alternately, in one crystal direction. The degeneracy is
three because there is a freedom to choose one direction among three. The
total degeneracy is 6 if we count the statesw of reverse spins. The transition
in the frustrated region is theoretically of first order since the present
6-fold GS is equivalent to the $q$-state Potts model with $q=6$. We know that
in three dimensions, the transition of the Potts model is of first order from
$q=3$. We have found this directly from the simulation. In the non-frustrated
region, namely $\eta<0.25$, the transition is found to be of second order. We
have performed MC simulations to obtain $\rho$ of the conduction spins. We
found that $\rho$ displays a broad maximum at the second-order phase
transition while it undergoes a discontinuous change at the first-order
transition.

The Ising model on the antiferromagnbetic HCP lattice has been also studied in
this review. We assumeed an in-plane interaction $J_{1}$ and an inter-plane
interaction $J_{2}$, both antiferromagnetic. We found that the GS changes at
the critical value $\eta_{c}=0.5$. Below (above) which the spins in the $xy$
planes are ferromagnetic (antiferromagnetic). The nature of the transition
changes in these two regions: it is of second order below $\eta_{c}$ and of
first order above $\eta_{c}$. The spin resistivity has been simulated in both
regions of $\eta$. In the second-order region, it shows a broad maximum while
in the first-order region, the resistivity $\rho$ makes a discontinuous jump
at the transition. This feature is what we also found in other frustrated spin
systems.

These findings reviewed in this paper show a close relationship between the
nature of the phase transition and the shape of the spin resistivity in real
materials. We hope that this review convinces the magnetic community on the
use of MC simulations for transport phenomena.

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  * (58) See ”Kondo Effect - 40 Years after the Discovery” - Special Issue of the Journal of the Physical Society of Japan 2015, 74, Issue 1, with reviews from world top scientists: Jun Kondo, Philippe Nozières, A. C. Hewson, Yukihiro Shimizu and Osamu Sakai, Ian Affleck amongst others.