Table Of Content

Temperature Dependence of the Spin Resistivity in Ferromagnetic Thin Films K. Akabli and H. T. Diep∗ Laboratoire de Physique Théorique et Modélisation, CNRS-Université de Cergy-Pontoise, UMR 8089 2, Avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France Themagneticphasetransitionisexperimentallyknowntogiverisetoananomaloustemperature- dependence of the electron resistivity in ferromagnetic crystals. Phenomenological theories based ontheinteraction betweenitinerant electron spinsandlattice spinshavebeensuggested toexplain theseobservations. Inthispaper,weshowbyextensiveMonteCarlo (MC) simulation thebehavior 8 oftheresistivityofthespincurrentcalculatedasafunctionoftemperature(T)fromlow-T ordered 0 phase to high-T paramagnetic phase in a ferromagnetic film. We analyze in particular effects of 0 film thickness, surface interactions and different kinds of impurities on the spin resistivity across 2 the critical region. The origin of the resistivity peak near the phase transition is shown to stem n from the existence of magnetic domains in the critical region. We also formulate in this paper a a theorybasedontheBoltzmann’sequationintherelaxation-timeapproximation. Thisequationcan J besolved using numerical data obtained by our simulations. Weshow that our theory is in a good 9 agreement with our MC results. Comparison with experiments is discussed. 2 PACSnumbers: 72.25.-b;75.47.-m ] i c s I. INTRODUCTION the singularity results mainly from ”short-range” inter- l- action at T >∼Tc where Tc is the transition temperature r ofthemagneticcrystal. FisherandLangerhaveshownin t The electric resistivity in magnetic metals has been m particular that the form of the resistivity cusp depends studied for years. The main effect of spin-independent on the interaction range. An interesting summary was t. resistivity is unanimously attributed to phonons. As published in 1975 by Alexander and coworkers15 which a far as spin-dependent resistivity is concerned, we had to m highlighted the controversial issue. To see more details wait until de Gennes and Friedel’s first explanation in onthemagneticresistivity,wequoteaninterestingrecent d- 19581 which was based on the interaction between spins publication from Kataoka.16 He calculated the spin-spin of conduction electrons and magnetic lattice ions. Ex- n correlation function using the mean-field approximation o perimentshaveshownthattheresistivityindeeddepends and he could analyze the effects of magnetic-field, den- c on the spin orientation.2,3,4,5,6 Therefore, the resistivity sity of conduction electron, the interaction range, etc. [ was expected to depend strongly on the spin ordering Although many theoretical investigationshave been car- of the system. Experiments on various magnetic ma- 1 riedout,todateveryfewMonteCarlo(MC)simulations v terials have found in particular an anomalous behavior have been performed regarding the temperature depen- 4 of the resistivity at the critical temperature where the dence of the dynamics of spins participating in the cur- 4 systemundergoesthe ferromagnetic-paramagneticphase rent. In a recent work,17 we have investigated by MC 4 transition.3,4,5,6 The problem of spin-dependent trans- simulations the effects of magnetic ordering on the spin 4 port has been also extensively studied in magnetic thin . currentin magnetic multilayers. Our results are in qual- 1 films and multilayers. The so-called giant magnetoresis- itative agreement with measurements.18 0 tance(GMR)wasdiscoveredexperimentallytwentyyears 8 ago.7,8 Since then, intensive investigations, both exper- Due to a large number of parameters which play cer- 0 imentally and theoretically, have been carried out.9,10 tainly important roles at various degrees in the behavior : v The so-called ”spintronics” was born with spectacular ofthespinresistivity,itisdifficulttotreatallparameters i rapid developments in relation with industrial applica- atthe same time. The firstquestionisofcoursewhether X tions. For recent overviews, the reader is referred to the explanation provided by de Gennes and Friedel can r a Refs. 11 and 12. Theoretically, in their pioneer work,de beusedinsomekindsofmaterialandthatbyFisherand Gennes and Friedel1 have suggested that the magnetic Langercanbeappliedinsomeotherkindsofmaterial. In resistivityis proportionaltothe spin-spincorrelation. In otherwords,wewouldliketoknowthevalidityofeachof other words, the spin resistivity should behave as the thesetwoarguments. Wewillreturntothispointinsub- magnetic susceptibility. This explained that the resis- section IIID. The second question concerns the effects tivity singularity is due to ”long-range” fluctuations of of magnetic or non-magnetic impurities on the resistiv- the magnetization observed in the critical region. Craig ity. Note that in 1970, Shwerer and Cuddy3 have shown et al13 in 1967 and Fisher and Langer14 in 1968 criti- and compared their experimental results with the differ- cized this explanation and suggested that the shape of ent existing theories to understand the impurity effect on the magnetic resistivity. However, the interpretation was not clear enough at the time to understand the real physical mechanism lying behind. The third question ∗Correspondingauthor,E-mail: [email protected] concerns the effects of the surface on the spin resistivity 2 inthinfilms. Thesequestionshavemotivatedthepresent work. InthispaperweuseextensiveMCsimulationtostudy Hm =− Ki,jsi·sj, (2) X the transport of itinerant electrons traveling in a ferro- hi,ji magnetic thin film. We use the Ising model and take where s is the itinerant Ising spin at position ~r , and into account various interactions between lattice spins i i denotes a sum over every spin pair (s ,s ). The and itinerant spins. We show that the magnetic resistiv- hi,ji i j itydependsonthelatticemagneticordering. Weanalyze iPnteractionKi,j dependsonthedistancebetweenthetwo this behaviorby using a new idea: instead of calculating spins, i.e. rij = |~ri−~rj|. A specific form of Ki,j will be the spin-spincorrelation,wecalculatethe distributionof chosen below. The interaction between itinerant spins clusters in the critical region. We show that the resis- and lattice spins is given by tivity depends on the number and the size of clusters of oppositespins. WeestablishalsoaBoltzmann’sequation H =− I s ·S , (3) which can be solved using numerical data for the cluster r i,j i j X distribution obtained by our MC simulation. hi,ji The paper is organized as follows. Section II is de- where the interaction I depends on the distance be- voted to the description of our model and the rules that i,j tween the itinerant spin s and the lattice spin S . For govern its dynamics. We take into account (i) interac- i i the sake of simplicity, we assume the same form for K tionsbetweenitinerantandlatticespins,(ii)interactions i,j and I , namely, between itinerant spins themselves and (iii) interactions i,j betweenlatticespins. Weincludeathermodynamicforce K = K exp(−r ) (4) i,j 0 ij duetothegradientofitinerantelectronconcentration,an I = I exp(−r ) (5) applied electric field and the effect of a magnetic field. i,j 0 ij For impurities, we take the Rudermann-Kittel-Kasuya- where K and I are constants. Yoshida (RKKY) interaction between them. In section 0 0 III, we describe our MC method and discuss the results we obtained. We develop in section IV a semi-numerical B. Dynamics theory based on the Boltzmann’s equation. Using the results obtained with Hoshen and Kopelman’s19 cluster- The procedure used in our simulation is described as counting algorithm, we show an excellent agreement be- follows. First we study the thermodynamic properties tweenourtheoryandourMC data. Concludingremarks of the film alone, i.e. without itinerant spins, using Eq. are given in Section V. (1). WeperformMCsimulationstodeterminequantities as the internal energy, the specific heat, layer magnetizations, the susceptibility, ... as functions of tempera- II. INTERACTIONS AND DYNAMICS ture T.20 From these physical quantities we determine the critical temperature T below which the system is in c A. Interactions the orderedphase. We showin Fig. 1 the lattice magnetization versus T for N =8, N =N =20. z x y We consider in this paper a ferromagnetic thin film. We use the Ising model and the face-centered cubic 1 (FCC) lattice with size 4N ×N ×N . Periodicbound- 0.9 x y z 0.8 ary conditions (PBC) are used in the xy planes. Spins 0.7 localized at FCC lattice sites are called ”lattice spins” 0.6 hereafter. Theyinteractwitheachotherthroughthe fol- M/M0 0.5 lowing Hamiltonian 0.4 0.3 0.2 0.1 Hl =−J Si·Sj, (1) 0 5 6 7 8 9 10 11 12 13 14 15 hXi,ji T where Si is the Ising spin at lattice site i, hi,ji indi- 8F.IGT.c1i:sL≃a9tt.5ic8eimnaugnniettoizfaJtio=n1v.ersustemperatureT forNz = catesthesumovereverynearest-neighbor(NNP)spinpair (Si,Sj), J(>0) being the NN interaction. Once the lattice has been equilibrated at T, we inject In order to study the spin transport in the above sys- N itinerant spins into the system. The itinerant spins 0 tem,weconsideraflowofitinerantspinsinteractingwith moveintothesystematoneend,travelinthexdirection, eachotherandwiththelatticespins. Theinteractionbe- escapethesystemattheotherendtoreenteragainatthe tween itinerant spins is defined as follows, firstend under the PBC.Note that the PBC areused to 3 ensurethattheaveragedensityofitinerantspinsremains The parameters we use in most calculations, except oth- constant with evolving time (stationary regime). The erwisestated(forexample,insubsectionIIIBforN )are z dynamics of itinerant spins is governed by the following s=S =1andN =N =20andN =8. Otherparam- x y z interactions: eters are D = D = 1 (in unit of the FCC cell length), 1 2 i) an electric field E is applied in the x direction. Its K = I = 2, L = 17, N = 8×202 (namely one itin- 0 0 0 0 π n energy is given by erant spin per FCC unit cell), v =1, k =( )( 0)1/3, 0 F a 2 HE =−eE·v, (6) r0 =0.05. AteachT theequilibrationtimeforthelattice spins lies around106 MC steps per spinandwe compute wherevisthevelocityoftheitinerantspin,eitscharge; statistical averages over 106 MC steps per spin. Taking ii)achemicalpotentialtermwhichdependsonthecon- J =1,wefindthatT ≃9.58forthecriticaltemperature c centrationofitinerantspins(”concentrationgradient”ef- of the lattice spins (see Fig. 1). fect). Its form is given by We define the resistivity ρ as H =Dn(r), (7) c 1 ρ= , (10) where n(r) is the concentration of itinerant spins in a n sphere of radius D centered at r. D is a constant taken 2 where n is the number of itinerant spins crossing a unit equal to K for simplicity; 0 area perpendicular to the x direction per unit of time. iii)interactionsbetweenagivenitinerantspinandlat- tice spins inside a sphere of radius D (Eq. 3); 1 iv) interactions between a given itinerant spin and A. Effect of thickness and effect of magnetic field. otheritinerantspinsinsideasphereofradiusD (Eq.2). 2 Let us consider the case without an applied magnetic We show in Figs. 2 and 3 the simulation results field. The simulationis carriedoutas follows: ata given for different thicknesses. In all cases, the resistivity ρ T we calculate the energy of an itinerant spin by taking is very small at low T, undergoes a huge peak in the into account all the interactions described above. Then ferromagnetic-paramagnetic transition region, decreases we tentatively move the spin under consideration to a slowly at high T. new position with a step of length v in an arbitrary di- 0 We point out that the peak position of the resistivity rection. Note that this move is immediately rejected if follows the variation of critical temperature with chang- the new position is inside a sphere of radius r centered atalatticespinoranitinerantspin. Thisexclu0dedspace ingthickness(seeFig.2)andρatT >∼Tc becomeslarger whenthethicknessdecreases. Thisisduetothefactthat emulates the Pauli exclusion principle in the one hand, surface effects tend to slow down itinerant spins. We re- and the interaction with lattice phonons on the other turn to this point in the next subsection. hand. If the new position does not lie in a forbidden regionof space, then the move is accepted with a proba- The temperature of resistivity’s peak at a given thick- bility given by the standard Metropolis algorithm.20 ness is always slightly higher than the corresponding Tc. Tostudythecasewithimpurities,wereplacerandomly Letus discussthe temperaturedependence ofρ shown a number of lattice spins S by impurity spins σ. The in Fig. 3: impurities interact with each other via the RKKY inter- i) First, ρ is very low in the ordered phase. We can action as follows explain this by the following argument: below the transition temperature, there exists a single large cluster of HI =− L(ri,rj)σi·σj (8) lattice spins with some isolated ”defects” (i. e. clusters hXi,ji of antiparallel spins), so that any itinerant spin having the parallel orientation goes through the lattice without where hindrance. The resistance is thus very small but it in- L(r ,r )=L cos(2k |r −r |)/|r −r |3 (9) creases as the number and the size of ”defect” clusters i j 0 F i j i j increase with increasing temperature. L0beingaconstantandkF theFermiwavenumberofthe ii) Second, ρ exhibits a cusp at the transition temper- lattice. The impurity spins also interact with NN lattice ature. We present here three interpretations of the exis- spins. However,to reduce the number of parameters,we tence of this cusp. Note that these different pictures are take this interaction equal to J =1 as that between NN not contradictory with each other. They are just three lattice spins (see Eq. 1) with however σ 6= S. We will different manners to express the same physical mecha- consider two cases σ =2 and σ =0 in this paper. nism. The first picture consists in saying that the cusp is due to the critical fluctuations in the phase transition region. We know from the theory of critical phenom- III. MONTE CARLO RESULTS ena that there is a critical region around the transition temperature T . In this region, the mean-field theory c We let N itinerant spins travel through the system shouldtakeinto accountcriticalfluctuations. The width 0 several thousands times until a steady state is reached. of this region is given by the Ginzburg criterion. The 4 limit of this ”Ginzburg” region could tally with the re- high fields. This is whatwe observedin simulations. We sistivity’speakandGinzburgtemperature.15 Thesecond show results of ρ for several fields in Fig. 4. pictureisduetoFisher-Langer14andKataoka16whosug- gestedthattheformofpeakisduemainlytoshort-range 1 (a) spin-spin correlation. These short-range fluctuations are 0.9 (b) (c) known to exist in the critical region around the critical 0.8 point. ThethirdpicturecomesfromourMCsimulation17 0.7 0.6 whichshowedthattheresistivity’speakisduetothefor- M/M0 0.5 mationofantiparallel-spinclustersofsizesofafewlattice 0.4 cells which are known to exist when one enters the crit- 0.3 ical region. Note in addition that the cluster size is now 0.2 0.1 comparablewiththeradiusD ofthe interactionsphere, 1 0 whichinturnreducesthe heightofpotentialenergybar- 5 6 7 8 9 10 11 12 13 14 15 T riers. We have checked this interpretation by first creat- inganartificialstructureofalternateclustersofopposite FIG. 2: Lattice magnetization versus temperature T with spins and then injecting itinerant spins into the system. different thicknesses (Nz) of the film: crosses, void circles Weobservedthatitinerantspinsdoadvanceindeedmore and black triangles indicate data for Nz=5, 8 and bulk, slowly than in the completely disordered phase (high-T rTecs(pNezct=ive5l)y.≃9T.4c(7b.ulk) ≃ 9.79, Tc(Nz = 8) ≃ 9.58 and paramagnetic phase). We have next calculated directly the cluster-size distribution as a function of T using the Hoshen-Kopelman’salgorithm.19Theresultconfirmsthe effect of clusters on the spin conductivity. The reader is referredtoourpreviouswork17 forresultsofamultilayer B. Effect of surface case. We willshow inthe nextsectiona cluster distribution for the film studied in this paper. The picture suggested above on the physical mecha- iii)Third,ρislargeintheparamagneticphaseandde- nism causing the variation of the resistivity helps to un- creaseswith anincreasing temperature. Above T in the derstandthe surfaceeffect shownhere. Since the surface c paramagnetic phase, the spins become more disordered spins suffer more fluctuations due to the lack of neigh- as T increases: small clusters will be broken into single bors,weexpectthatsurfacelatticespinswillscattermore disordered spins, so that there is no more energy barrier strongly itinerant spins than the interior lattice spins. between successive positions of itinerant spins on their The resistivity therefore should be larger near the sur- trajectory. The resistance, though high, is decreasing face. This is indeed what we observed. The effect how- with increasing T and saturated as T →∞. ever is very small in the case where only a single surface layerisperturbed. Toenhancethesurfaceeffect,wehave iv) Let us touch upon the effects of varying D and 1 perturbedanumberoflayersnearthesurface: weconsid- D at a low temperatures. ρ is very small at small D 2 1 eredasandwichofthreefilms: themiddlefilmof4layers (D < 0.8): this can be explained by the fact that for 1 is placedbetween two surface films of 5 layerseach. The such small D , itinerant spins do not ”see” lattice spins 1 in-plane interaction between spins of the surface films is in their interaction sphere so they move almost in an taken to be J and that of the middle film is J. When empty space. TheeffectofD2 is onthe otherhandqual- s J =J one has one homogeneous14-layerfilm. We have itatively very different from that of D1: ρ is very small s simulated the two cases where J =J and J =0.2J for at small D2 but it increases to very high value at large s s sorting out the surface effect. In the absence of itiner- D . We conclude that both D and D dominate ρ at 2 1 2 their smallvalues. However,atlargevalues,onlyD has 2 astrongeffectonρ. Thiseffectcomesnaturallyfromthe 140 criterionon the itinerant spins concentrationusedin the (a) (b) movingprocedure. Also,wehavestudiedtheeffectofthe 120 (c) electric field E both above and below T . The low-field 100 c spin current verifies the Ohm regime. These effects have 80 been also observed in magnetic multilayer.17 The reader ρ 60 is referred to that work for a detailed presentation of 40 these points. 20 Let us show now the effect of magnetic field on ρ. As 0 it is well known, when a magnetic field is applied on a 0 2 4 6 8 10 12 14 16 18 20 T ferromagnet, the phase transition is suppressed because themagnetizationwillnevertendtozero. Criticalfluctu- FIG. 3: Resistivity ρ in arbitrary unit versus temperature ations arereduced, the number ofclusters ofantiparallel T for different film thicknesses. Crosses (a), void circles (b) spins diminishes. As a consequence, we expect that the and black triangles (c) indicate data for bulk, Nz=8 and 5, peak of the resistivity will be reduced and disappears at respectively. 5 1 (a) 0.9 (b) (c) 0.8 0.7 0.6 M/M0 0.5 0.4 0.3 0.2 0.1 0 3 4 5 6 7 8 9 10 11 12 M/M0 12 FIG.4: Resistivity ρin arbitrary unit versustemperature T, ((ab)) 10 (c) for differentmagnetic fields. Voidcircles, stars, blackrectan- gles, void triangles and black triangles indicate, respectively, 8 data for (a) B = 0,(b) B = 0.1J,(c) B = 0.3J,(d) B = 1J 6 and (e) B=2J. χ 4 2 ant spins, the lattice spins undergo a single phase tran- 0 sition at T ≃9.75 for J =J, and two transitions when -2 c s 3 4 5 6 7 8 9 10 11 12 J = 0.2J: the first transition occurs at T ≃ 4.20 for T s 1 ”surface”films andthe secondatT2 ≃9.60for ”middle” FIG. 5: Upper figure: Magnetization versus T in the case film. This is seen in Fig. 5 where the magnetization of wherethesystemismadeofthreefilms: thefirstandthethird the surface films drops at T1 and the magnetization of have 5 layers with a weaker interaction Js, while the middle the middle film remains up to T2. The susceptibility has has4layerswithinteractionJ =1. WetakeJs=0.2J. Black two peaks in the case J = 0.2J. The resistivity of this triangles: magnetization of the surface films, stars: magne- s case is shown in Fig. 6: at T < T the whole system is tization of the middle film, void circles: total magnetization. 1 ordered, ρ is therefore small. When T ≤ T ≤ T the Lowerfigure: SusceptibilityversusT ofthesamesystemasin 1 2 theupperfigure. Blacktriangles: susceptibilityofthesurface surface spins are disordered while the middle film is still films, stars: susceptibility of the middle films, void circles: ordered: itinerant spins encounter strong scattering in total susceptibility. Seetext for comments. the two surface films, they ”escape”,after multiple colli- sions, to the middle film. This explains the peak of the surface resistivity at T . Note that already far below T , 300 1 1 (a) (b) anumberofsurfaceitinerantspinsbegintoescapetothe (c) 250 middle film, making the resistivity of the middle film to decrease with increasing T below T up to T , as seen in 200 1 2 Fig. 6. Note that there is a small shoulder of the total ρ 150 resistivity at T . In addition, in the range of tempera- 1 tures between T and T the spins travel almost in the 100 1 2 middle film with a large density resulting in a very low 50 resistivityofthemiddle film. ForT >T ,itinerantspins 2 0 2 4 6 8 10 12 14 16 18 20 flow in every part of the system. T FIG.6: ResistivityρinarbitraryunitversusT ofthesystem described in the previous figure’s caption. Black triangles: C. Effect of impurity resistivity of the surface films, void circles: resistivity of the middlefilm,blacksquares: totalresistivity. Seetextforcom- ments. In this subsection, we take back N = N = 20 and x y N =8. z concentration. We understandthatlarge-spinimpurities must reinforce the magnetic order. 1. Magnetic impurities In Figs. 8, 9 and 10, we compare a system without impurity to systems with respectively 1 and 2 and Totreatthecasewithimpurities,wereplacerandomly 5 percents of impurities. The temperature of resistiv- a number of lattice spins S by impurity spins σ =2. We ity’s peak is a little higher than the critical temperature suppose an RKKY interaction between impurity spins and we see that the peak height increases with increas- (see Eq. 8). Figure 7 shows the lattice magnetization ing impurity concentration (see Fig. 10). This is easily for several impurity concentrations. We see that crit- explainedbythefactthatwhenlarge-spinimpuritiesare ical temperature T increases with magnetic impurity’s introduced into the system, additional magnetic clusters c 6 1 0.9 ((ba)) 140 ((ba)) (c) 0.8 (d) 120 0.7 100 0.6 M/M0 0.5 ρ 80 0.4 60 0.3 40 0.2 20 0.1 0 0 5 6 7 8 9 10 11 12 13 14 15 0 2 4 6 8 10 12 14 16 18 20 T T FIG.7: LatticemagnetizationversustemperatureT withdif- FIG. 10: Resistivity ρ in arbitrary unit versus temperature ferentconcentrationsofmagneticimpuritiesCI. Voidcircles, T. Two cases are shown: without (a) and with 5% of mag- crosses, starsandblack diamondsindicate,respectively, data netic impurities (b) (void circles and crosses, respectively). for(a)CI =0%,(b)CI =1%,(c)CI =2%and(d)CI =5%. Our result using the Boltzmann’s equation is shown by the continuous curves (see sect. IV): thin and thick lines are for (a) and (b),respectively. Tc ≃10.21 for 5%. 140 ((ba)) 2. Non-magnetic impurities 120 100 For the case with non-magnetic impurities, we replace 80 ρ randomly a number of lattice spins S by zero-spin im- 60 purities σ = 0. Figures 11, 12 and 13 show, respec- 40 tively, the lattice magnetizations and the resistivities for 20 non-magnetic impurity concentrations 1% and 5%. We 0 0 2 4 6 8 10 12 14 16 18 20 observe that non-magnetic impurities reduce the criti- T cal temperature and the temperature of the resistivity’s FIG. 8: Resistivity ρ in arbitrary unit versus temperature peak. This can be explained by the fact that the now T. Two cases are shown: without (a) and with 1% (b) of ”dilute” lattice spins has a lower critical temperature so magnetic impurities (void circles and crosses, respectively). thatthe scattering ofitinerantspins by lattice-spin clus- Our result using the Boltzmann’s equation is shown by the ters should take place at lower temperatures. continuous curves (see sect. IV): thin and thick lines are for (a) and (b),respectively. Notethat Tc ≃9.68 for CI =1%. 1 (a) 0.9 (b) (d) 0.8 0.7 around these impurities are created in both ferromag- 0.6 netic and paramagnetic phases. They enhance therefore M/M0 0.5 0.4 ρ. 0.3 0.2 0.1 0 5 6 7 8 9 10 11 12 13 14 15 T 140 ((ba)) FIG.11: Latticemagnetization versustemperatureT fordif- 120 ferent concentrations of non-magnetic impurities: (a) CI = 100 0% (void circles), (b) CI = 1% (crosses) and (c) CI = 5% 80 (black diamonds). ρ 60 40 20 0 D. Discussion 0 2 4 6 8 10 12 14 16 18 20 T FIG. 9: Resistivity ρ in arbitrary unit versus temperature DeGennesandFriedel1haveshownthattheresistivity T. Two cases are shown: without and with 2% of magnetic ρisrelatedtothespincorrelation<Si·Sj >. Theyhave impurities (void circles and crosses, respectively). Ourresult suggested therefore that ρ behaves as the magnetic sus- using the Boltzmann’s equation is shown by the continuous ceptibilityχ. However,unlikethesusceptibilitywhichdi- curves(seesect. IV):thinandthicklinesarefor(a) and(b), vergesatthe transition,the resistivity observedin many respectively. Tc ≃9.63 for 2% . experimentsgoesthroughafinitemaximum,i. e. acusp, 7 periments). We think that all systems are not the same 140 ((ba)) because of the difference in interactions, so one should 120 not discard a priori one of these two scenarios. 100 Inthispaper,wesuggestanotherpicturetoexplainthe 80 cusp: when T is approached, large clusters of up (resp. ρ c 60 down) spins are formed in the critical region above T . c 40 As a result, the resistance is much larger than in the 20 ordered phase: itinerant electrons have to steer around largeclustersofoppositespinsinordertogothroughthe 0 0 2 4 6 8 10 12 14 16 18 20 entire lattice. Thermalfluctuations arenot largeenough T toallowtheitinerantspintoovercometheenergybarrier FIG.12: ResistivityρinarbitraryunitversustemperatureT createdbythe oppositeorientationoftheclustersinthis for two cases: without (a) and with 1% of non-magnetic im- temperature region. Of course, far above T , most clus- purities(b)(voidcirclesandcrosses,respectively). Ourresult c ters have a small size, the resistivity is still quite large using the Boltzmann’s equation is shown by the continuous curves(seesect. IV):thinandthicklinesarefor(a) and(b), with respect to the low-T phase. However, ρ decreases respectively. as T is increased because thermal fluctuations are more andmorestrongerto helpthe itinerantspintoovercome energy barriers. 140 ((ba)) What we havefound here is a peak of ρ, not a peak of 120 dρ/dT. So, our resistivity behaves as the susceptibility 100 although the peak observed here is not sharp and no di- vergenceisobserved. Webelievehoweverthat,similarto 80 ρ commonly known disordered systems, the susceptibility 60 peak is broadened more or less because of the disorder. 40 The disorder in the system studied here is due the lack 20 of periodicity in the positions of moving itinerant spins. 0 0 2 4 6 8 10 12 14 16 18 20 T FIG.13: ResistivityρinarbitraryunitversustemperatureT IV. SEMI-NUMERICAL THEORY for two cases: without (a) and with 5% of non-magnetic impurities(b)(voidcirclesandcrosses,respectively). Ourresult In this paragraph, we show a theory based on the using the Boltzmann’s equation is shown by the continuous Boltzmann’s equation in the relaxation-time approxima- curves(seesect. IV):thinandthicklinesarefor(a) and(b), tion. To solve completely this equation, we shall need respectively. some numerical data from MC simulations for the clus- tersizesaswillbeseenbelow. Usingthesedata,weshow withoutdivergence. Toexplainthis,FisherandLanger14 that our MC result of resistivity is in a good agreement and then Kataoka16 have shown that the cusp is due to with this theory. Let us formulate now the Boltzmann’s equation for short-rangecorrelation. Thisexplanationisinagreement our system. When we think about the magnetic resis- withmanyexperimentaldatabutnotall(seeRef. 15for tivity, we think of the interaction between lattice spins review on early experiments). Let us recall that <E >∝ <S ·S > where the and itinerant spins. We recognize immediately the im- i,j i j portant role of the spin-spin correlation function in the sum is taken over NN (or shoPrt-range) spin pairs while Tχ ∝< ( S)2 >= < S ·S > where the sum determinationofthemeanfree-path. Ifweinjectthrough is performPedi ovier all spPini,jpairs.iThijs is the reason why the system a flow of spins ”polarized” in one direction, namely ”up”, we can consider clusters of ”down” spins short-range correlation yields internal energy and long- in the lattice as ”defect clusters”, or as ”magnetic im- range correlation yields susceptibility. Roughly speaking, if <S ·S > is short-ranged, then purities”, which play the role of scattering centers. We i j thereforereducethe problemtothe determinationofthe ρ behaves as < E > so that the temperature derivative number and the size of defect clusters. For our purpose, of the resistivity, namely dρ/dT, should behave as the we use the Boltzmann’s equation with uniform electric specific heat with varying T. Recent experiments have field but without gradient of temperature and gradient found this behavior(see for example Ref. 5). Now, if < S ·S > is long-ranged, then ρ behaves as of chemical potential. We write the equation for f, the i j Fermi-Dirac distribution function of itinerant electrons, the magnetic susceptibility as suggested by de Gennes as andFriedel.1 Inthiscase,ρundergoesadivergenceatT c as χ. One should have therefore dρ/dT > 0 at T < Tc ¯hk.eE ∂f0 ∂f anddρ/dT <0atT >Tc. Insomeexperiments,thishas ( m )( ∂ε )=(∂t)coll, (11) been found in for example in magnetic semiconductors (Ga,Mn)As6 (see also Ref. 15 for review on older ex- wherekisthewavevector,eandmtheelectroniccharge 8 and mass, ǫ the electron energy. We use the following Notethatalthoughtherelaxationtimeτ isaveragedover relaxation-time approximation all states in the Fermi sphere, i. e. states at T = 0, its temperature dependence comes from the parameters ∂f f1 ( k) =−( k), f1 =f −f0, (12) ξ and nc. These will be numerically determined in the ∂t coll τk k k k following. If we know τ we can calculate the resistivity by the Drude expression where τ is the relaxation time. Supposing elastic colli- k sions, i. e. k = k′, and using the detailed balance we m 1 have ρ = , (21) m ne2τ ∂f Ω ( ∂tk)coll = (2π)3 Z wk′,k(fk1′ −fk1)dk′, (13) LetususenowtheHoshen-Kopelman’salgorithm19 to determinethemeanvalueofξandthenumberoftheclus- where Ω is the systemvolume, wk′,k the transitionprob- ter’s mean size, for different temperatures. The Hoshen- ability between k and k′. We find with Eq. (12) and Kopelman’s algorithm allows to regroup into clusters Eq. (13) the following well-known expression spins with equivalent value for T < T or equivalent en- c ergy for T > T . Using this algorithm during our MC 1 Ω c (τ ) = (2π)3 Z wk′,k(1−cosθ) simulation at a given T, we obtain a ”histogram” repre- k senting the number of clusters as a function of the clus- ×sinθk′2dk′dθdφ, (14) ter size. For temperature T below T , we call a cluster c a group of parallel spins surrounded by opposite spins, where θ and φ are the angles formed by k′ with k, i. e. and for T above T a cluster is a groupof spins with the spherical coordinates with z axis parallel to k. c sameenergy. AtagivenT,weestimatetheaveragesizeξ We use now for Eq. (14) the ”Fermi golden rule” and using the histogramas follows: calling N the number of i we obtain spins in the cluster and P the probability of the cluster i 1 Ωm deduced from the histogram, we have ( ) = |<k′|V|k >|2(1−cosθ)sinθ τk ¯h32πk Z N P ×δ(k′−k)k′2dk′dθ (15) ξ = i i i, (22) P P i i We give for the potential V the following expression P which reminds the form of the interactions (4)-(5) In doing this we obtain ξ for the whole temperature range. We note that we can fit the cluster size ξ with −r V =V exp( ), (16) the following formula 0 ξ(T) ξ =A|T −T|ν/3, (23) where ξ(T) is the size of the defect cluster and V a con- c 0 stant. WeresolveEq.(15)withEq.(16)andwehavethe where ν is a fitting parameter and A a constant. These following expression parameters are different for T < T and T > T . Figure c c 1 32V2Ωmkπ sinθ(1−cosθ) 14 and Figure 15 show the averagesize and the average ( )=( 0 ) dθ, (17) τk ¯h3 Z (K2+ξ−2)3 number of cluster versus temperature. To simplify our approach we consider that the cluster’s geometry is a where K =|~k−~k′| is given by sphere with radius ξ. Note that due to the fact that our fitting was made separately for T < T and T > T , no c c K =|~k−~k′|=k[2(1−cosθ)]1/2, (18) efforthasbeenmadeforthe matchingatT =T exactly, c but this does not affect the behavior discussed below. Integrating Eq. (17) we obtain We distinguish hereafter temperatures below and 1 32(V Ω)2mπ above Tc in establishing our theory. We write ( ) = 0 n ξ2 τ (2k¯h)3 c k 1 4k2ξ2 1 ×[1− − ] (19) ρ =ρ (1+C n ξ2[−1+ 1+4k2ξ2 (1+4k2ξ2)2  m 0 inf c 1+(2kFξ)2 +ln(1+(2k ξ)2)]), F wEqh.er(e19n)cfiosrththeeniunmtebrevralo[f0c,lkuFst]earsndofwsiezeobξt.aWinefiinnatellgyrate T <Tc ξ =(31A6iπnf)1/3(Tc−T)νinf/3,   Binf −(T −TG)2 (1) = (4V0Ωm)2n ξ2  nc =(2απ )×exp[ 2α2 ]. τ π(h¯)3/2 c 1 ×[ +ln(1+(2k ξ)2)−1] (20) 1+(2k ξ)2 F (24) F 9 TABLEI: Various numerical values obtained byMC simulations which are used to plot Eqs. (24) and (25). Impurity T T ν ν α c G inf sup 0% S=1 9.58 7.4443 +/- 0.066 -0.9254 +/- 0.015 -0.2267 +/- 0.006 1.51875 +/- 0.07 1% σ=2 9.68 7.5006 +/- 0.054 -0.9253 +/- 0.017 -0.1449 +/- 0.006 1.62908 +/- 0.06 2% σ=2 9.63 7.7103 +/- 0.049 -0.9856 +/- 0.016 -0.1135 +/- 0.004 1.64786 +/- 0.05 5% σ=2 10.2 7.9658 +/- 0.094 -1.1069 +/- 0.016 -0.0747 +/- 0.002 2.13618 +/- 0.10 1% σ=0 9.47 7.2866 +/- 0.062 -0.9028 +/- 0.013 -0.2106 +/- 0.006 1.51766 +/- 0.06 5% σ=0 9.10 7.0105 +/- 0.054 -0.9381 +/- 0.015 -0.1607 +/- 0.006 1.47261 +/- 0.05 the critical region from below. In Eq. (24), α is the half-width of the peak of n shown in Fig. 15. ρ is the 1 c 0 ρm =ρ∞(1+Csupncξ2[−1+ 1+(2k ξ)2 resistivity at T =0 and ρ∞ is that at T =∞. F WesummarizeinTable1thedifferentresultsobtained +ln(1+(2k ξ)2)]), T >Tc ξ =(3Asup)1/3(TF−Tc)νsup/3, fOotrhtehrepacraasmesetsetursdAieidnfb,yBMinfC,Csiimnfu,laAtsiounp,sBshsuopwanndabCosvuep.  16π are fitting parameters which are not of physical impor-  n =B exp[−D(T −T )]+n .  c sup c 0 tance andtherefore notgivenherefor the sakeofclarity. Using the numerical values of Table 1 and the average (25) cluster size and cluster number shown in Fig. 14 and Fig. 15, we plot Eqs. (24) and (25) by continuous lines In Eq. (24) we call T the temperature on which the in Figs. 8-13to comparewith MC simulations shownin G cluster of small size are gathering to form bigger clus- these figures. We emphasize that our theory provides a ter. This temperature marks the limit when one enters good ”fit” for simulation results. 3 600 2.5 500 2 400 ξ 1.5 n_c 300 1 200 0.5 100 0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 T T FIG.14: Meansizeξofmagneticclustersversustemperature FIG. 15: Number of average cluster size nc for both above T for both aboveand below Tc. and below Tc. Based on those results, we can extract the resistivity Weseethattheformofρ ofthealloysNi-Fe(1%)and ρ corresponding only to the addition of impurities. ρ I I I Ni-Fe(0.5%) experimentally observed3 can be compared is defined as tothecurvesof1%and2%ofmagneticimpuritiesshown in Fig. 16. For Ni-Cr(1%) and Ni-Cr(2%), experimental ρ =ρ −ρ , (26) I m standard curvesareinagreementwithourresultsofnon-magnetic where ρ is the resistivity without impurities (see impurities shown in Fig. 17. standard the first line of Table 1). We compare now the resistiv- Finally, to close this section, let us show theoretically ityρ withexperimentsrealizedbyShwererandCuddy.3 fromtheequationsobtainedabove,theeffectsoftheden- I It is important to note that the change of behavior can sity of itinerant spins on the resistivity. Figure 18 shows be explained if we use the correct value for ν, T , T , that, as the density n is increased,the peak of ρ dimin- c G 0 etc. Figure 16 showsρ with magnetic impurities corre- ishes. It is noted that this behavior is very similar with I sponding to the Ni-Fe system , while Fig. 17 shows ρ thatobtainedbyKataoka.16InourMCsimulationshown I in the case of non-magnetic impurities corresponding to above,wehavechosenn =1/4. Suchaweakdensityhas 0 the Ni-Cr case. allowedus to avoidthe flip of lattice spins upon interac- 10 15 V. CONCLUSION 1% magnetic impurities 2% magnetic impurities 10 ρ 5 0 -5 0 2 4 6 8 10 12 14 16 18 20 WehaveshowninthispaperresultsofMCsimulations T onthetransportofitinerantspinsinteractingwithlocal- FIG. 16: Resistivity ρI in arbitrary unit versus temperature izedlatticespinsinaferromagneticFCCthinfilm. Vari- T. Void circles and black triangles indicate data for 1% and ousinteractionshavebeentakenintoaccount. We found 2% magnetic impurities, respectively. thatthespincurrentisstronglydependentonthelattice spinordering: atlowT itinerantspins whosedirectionis 15 1% non magnetic impurities paralleltothelatticespinsyieldastrongcurrent,namely 5% non magnetic impurities a small resistivity. At the ferromagnetic transition, the 10 resistivity undergoes a huge peak. At higher temperatures, the lattice spins are disordered, the resistivity is ρ 5 still large but it decreases with increasing T. From the discussion given in subsection IIID, we conclude that 0 the resistivity ρ of the model studied here behaves as the magnetic susceptibility with a peak at T . dρ/dT, c -5 0 2 4 6 8 10 12 14 16 18 20 differential resistivity is thus negative for T > Tc. The T peakoftheresistivityobtainedhereisinagreementwith experimentsonmagnetic semiconductors(Ga,Mn)As for FIG. 17: Resistivity ρI in arbitrary unit versus temperature example.6 Ofcourse,tocomparethe peak’sshapeexper- T. Void circles and black triangles indicate data for 1% and 5% non magnetic impurities, respectively. imentally obtained for each material, we need to refine our model parameters for each of them. This was not the purpose of the presentpaper. Instead, we were look- tionwithitinerantspins. Inthecaseofstrongdensity,we ing forgenericeffects to showphysicalmechanisms lying expect that a number of lattice spins, when surrounded behind the temperature dependence of the spin resistiv- by a large number of itinerant spins, should flip to ac- ity. In this spirit, we note that early theories have re- commodate themselves with their moving neighbors. So lated the origin of the peak to the spin-spin correlation, thelatticeorderingshouldbeaffected. Asaconsequence, while our interpretation here is based on the existence critical fluctuations of lattice spins are more or less sup- of defect clusters formed in the critical region. This in- pressed, so is the peak’s height, just like in the case of terpretationhas been verifiedby calculating the number an applied magnetic field. Kataoka16 has found this in and the size of clusters as a function of T by the use his calculation by taking into account the spin flipping: of Hoshen-Kopelman’s algorithm. We have formulated the resistivity’s peak disappears then at the transition. a theory based on the Boltzmann’s equation. We solved It would be interesting to perform more MC simulations thisequationusingnumericaldataobtainedforthenum- with varying n . This is left for a future investigation. berandthesizeofaverageclusterateachT. Theresults 0 on the resistivity are in a good agreement with MC results. 200 150 ρ 100 50 Finally, let us conclude by saying that the clear physi- calpictureweprovideinthispaperfortheunderstanding 0 0 2 4 6 8 10 12 14 16 18 20 ofthebehavioroftheresistivityinasingleferromagnetic T film will help to understand properties of resistivity in FIG.18: ResistivityρinarbitraryunitversustemperatureT morecomplicatedsystemssuchasantiferromagnets,non- for different densities of itinerants spins. Curves from top to Ising spin systems, frustrated spin systems, disordered bottom are for n0 =0.2, 0.33, 0.5, 0.7, 1 and 1.5. media, ... where much has to be done.