Physica E 9 (2001) 374–379www.elsevier.nl/locate/physe

Localization and dephasing driven bymagnetic uctuationsin colossal magnetoresistancematerials

Eugene Kogana;b; ∗, Mark Auslenderc, Moshe Kavehd;e

aJack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, IsraelbCavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK

cDepartment of Electrical and Computer Engineering, Ben-Gurion University of the Negev P.O.B. 653, Beer-Sheva 84105, IsraeldJack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel

eCavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK

Abstract

Localization and dephasing of conduction electrons in a low-carrier-density ferromagnet due to scattering on magnetic uctuations is considered. We claim the existence of the “mobility edge”, which separates the states with fast di�usion andthe states with slow di�usion; the latter is determined by the dephasing time. When the “mobility edge” crosses the Fermienergy a large and sharp change of conductivity is observed. The theory provides an explanation for the observed temperaturedependence of conductivity in ferromagnetic semiconductors and manganite pyrochlores. ? 2001 Elsevier Science B.V. Allrights reserved.

PACS: 75.50.Pp; 75.30.Vn; 72.10.−d.Keywords: Manganites; Magnetic semiconductors; Colossal magnetoresistance

1. Introduction

Colossal magnetoresistance (CMR) materials at-tract nowadays considerable interest, associatedmostly with the properties of double-exchange man-ganite perovskites [1]. Class of CMR materials,however, is much wider and includes, in particular,magnetic semiconductors [2,3] and manganite py-rochlores [4,5]. All these materials are characterizedby the strong interaction between the localized spins

∗ Corresponding author. Fax: +972-3-535-3298.E-mail address: kogan@quantum.ph.bin.ac.il (E. Kogan).

and itinerant charge carriers. In all these materials,CMR is associated with the sharp increase of theresistivity when the temperature T approaches theCurie temperature Tc. However, taking into accountthe large variety of the materials involved and diversemanifestations of the e�ect, it is di�cult to expect thatany single theory can provide a universal explanationof the phenomena.We concentrate on low-carrier-density materi-

als (magnetic semiconductors and manganitepy-rochlores), where the carriers do not a�ect thespin–spin interaction, and magnetic d- or f-ionsinteract mainly via ferromagnetic direct exchange

1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved.PII: S 1386 -9477(00)00231 -9

E. Kogan et al. / Physica E 9 (2001) 374–379 375

(super-exchange). These materials being de�cient inchalcogen (oxygen) or being properly doped, have atlow-temperatures quasi-metallic conductivity. Whenthe temperature approaches Tc they undergo themetal–insulator transition (MIT).In the previous publications [6–8] we suggested,

that such behavior of the conductivity is due toAnderson localization of the carriers driven by spin uctuations of magnetic ions. We considered the spin uctuations as static; hence, the scattering of electronsby the uctuations can be treated as elastic, and henceit leads to the existence of the mobility edge Ec. (Thismechanism is close to the phonon scattering-inducedelectron localization [9,10].) When the temperatureincreases, so does the scattering intensity, which leadsto the upward motion of the mobility edge. The tem-perature at which the mobility edge crosses the Fermilevel is identi�ed with the temperature of the MIT.This view point on temperature-induced MIT hasalso been proposed independently in several recentpublications [11–13].In this article, we reformulate the results of

Ref. [14], where such behavior of the conductivitywas ascribed to Anderson localization of the carriersdriven by spin uctuations of magnetic ions, the in- uence of the dynamics of the spin subsystem beingtaken into account.

2. Hamiltonian and approximations

The Hamiltonian of the system has the form

H =∑k�Ekc

†k�ck�

− IN∑kq��′

S̃q���′c†k�ck+q�′ + HM; (1)

where Ek is the bare-electron spectrum, c†k;�; ck;� are

the electron creation and annihilation operators, I isthe parameter of Hund exchange between the elec-trons and localized spins, Sq is the Fourier componentsof the spin density, � is the vector of the Pauli spinmatrices and HM is the direct exchange interaction de-scribed by super-exchange integral J (Q).Let us state the relations between the parameters of

the problem. We consider the case of wide conduc-tion band W/2IS, where W ∼ 1=ma2 is the width

of the conduction band (a is the lattice constant andm is the electron mass), S is the spin of magneticion. This inequality is certainly applicable to suchmagnetic semiconductors as EuO and EuS [15]. Formanganite pyrochlores we do not have strong in-equality [16], but we believe that the approximationstill works in this case, at least semi-quantitatively.Due to low-carrier-density considered (not in excessof 10−19 cm−3), the Fermi energy EF is at least anorder of magnitude less then 2IS (which is larger than0.5 eV in the materials considered [15,16]). We con-sider ferromagnetic phase and temperatures such,that the spin splitting of the conduction band islarger than EF (estimations show, that this willbe true up to the temperatures very close to Tc).All our assumptions can be thus reduced to in-equalities

W/2IS;

2ISz/EF/T; (2)

where Sz, is the average spin of magnetic ion.In the wide conduction band case the electron–spin

exchange can be treated as a perturbation, leading toelectron scattering. The conduction electrons beingfully spin-polarized and spin- ip processes thus beingforbidden, the scattering (in the Born approximation)is connected only with the longitudinal spin correlator〈�Szq�Sz−q〉. It is argued [17], that for the wave vec-tor q small enough (qa¡ const(Sz)2) the correlator isdominated by contribution of weakly interacting spinwaves with the dispersion law

!Q = 2Sz[J (0)− J (Q)] (3)

and quasi-classical occupation numbers

nQ =T!Q: (4)

As a result, the static correlator is [17]

〈�Szq�Sz−q〉=T 2

8Sz2C2

1qa; (5)

where C is the spin sti�ness (for nearest-neighbor ex-change in a cubic lattice C ' Tc=2S(S + 1)).

376 E. Kogan et al. / Physica E 9 (2001) 374–379

Fig. 1. Diagrams for the Di�uson (a) and the Cooperon (b). Solidline is dressed electron propagator, dashed line connecting timest and t′ corresponds to �zz(t − t′).

For the transport relaxation time we obtain

1�=2�NI 2∑q〈�Szq�Sz−q〉

k · qk2�(Ek − Ek+q)

=ma2I 2T 2

16�Sz2C2∼ I 2S(S + 1)

WT 2

T 2c

S2

Sz2 : (6)

We see that for temperatures high enough, �EF¡ 1.Hence, we need some kind of strong scattering theory.As such, we shall use the self-consistent localizationtheory by Vollhard and W�ol e (VW) [18], extendedin Refs. [19,20] to systems without time-reversalinvariance. But �rst we should calculate the crucialparameter in our approach – the dephasing time �’.

3. Dephasing time

The inverse dephasing time can be de�ned asthe mass of the Cooperon [21,22]. (An alternative,but essentially equivalent view on dephasing see inRef. [23].) For the Cooperon C(R; t) we obtain equa-tion

{9=9t − D32 + [f(0)− f(t)]}C(R; t) = 0; (7)

where

f(t) =2�NI 2∑q�zz(q; t)�(Ek − Ek+q) (8)

and �zz(q; t) is the temporal longitudinal spin corre-lator (�zz(q; t = 0) ≡ 〈�Szq�Sz−q〉).Eq. (7) can be easily understood if we compare

diagrams for the Di�uson and the Cooperon onFig. 1. The Di�uson does not have any mass be-cause of Ward identity. In the case of the Cooperon,the Ward identity is broken: interaction line whichdresses single-particle propagator is given by staticcorrelator, and interaction line which connects twodi�erent propagators in a ladder is given by dynamic

correlator. The di�erence [f(0)− f(t)] shows howstrongly the Ward identity is broken and, as we willsee below, determines the mass of the Cooperon.Solving Eq. (7) we get [24]

C(t) = Cel(t) exp{∫ t

0[f(t′)− f(0)] dt′

}; (9)

where Cel is the Cooperon calculated ignoring thein-elasticity of scattering.Using the spin-wave picture described above, we

obtain

�zz(q; t) =1N∑QnQnQ+q

×exp[i(!Q+q − !Q)t]: (10)

Performing integration in Eq. (9) we get

C(t) = Cel(t) exp[− t3=�3’]; (11)

where

1�3’=

�3N 2

I 2∑QqnQnQ+q

×�(Ek − Ek+q)(!Q+q − !Q)2: (12)

Calculating the integrals in Eq. (12) we obtain

1�’=(I 2T 2(W − 1)ma5k3F

18�

)1=3

'(I 2T 2E3=2FW 5=2

)1=3; (13)

whereW is the Watson integral. It is worth noting thatdephasing time is de�ned by the second-time deriva-tive of �zz(q; t) at t = 0 which can be calculated viasecond moment of corresponding spectral density; theresult turns out to be essentially the same as Eq. (12).So the spin-wave picture, being physical one, is notcrucial for obtaining 1=�’.It should be noticed that the form of the Eq. (11) for

the Cooperon is quite general, provided the scatterersare in a ballistic motion, irrespective of whether theyare point particles [24], phonons [10], or spin waves,like in our case.The result for the dephasing time can be under-

stood using simple qualitative arguments. If all thecollisions lead to the same electron energy change �E,

E. Kogan et al. / Physica E 9 (2001) 374–379 377

the dephasing time could be obtained using relation[21]

�’�E√�’�out

∼ 2�; (14)

where �’=�out is just the number of scattering acts dur-ing the time �’ (�out is the extinction time). So in thiscase

1�3’

∼ (�E)2

�out: (15)

If we rewrite the formula for the extinction time

1�out

=2�NI 2∑q〈�Szq�Sz−q〉�(Ek − Ek+q) (16)

in the form

1�out

=2�N 2I 2∑QqnQnQ+q�(Ek − Ek+q) (17)

and notice that (!Q+q − !Q) is just the energy changeof the electron when scattering on a spin wave, weimmediately see that Eq. (12) is just Eq. (15) with theintegration with respect to di�erent collision-inducedenergy changes built in.

4. Conductivity calculation

The time-reversal invariance in the system we areconsidering is broken for two reasons. First, becausewe are considering ferromagnetic system, it is natu-rally to expect that the magnetic �eld is present in thesystem. Even more important is that the dephasingitself breaks the time-reversal invariance. We haveshown in the previous section, that due to dephasingthe di�usion pole of the particle–particle propagatordisappears, although particle–hole propagator stillhas a di�usion pole, which is guaranteed by parti-cle number conservation. Inserting Eq. (11) into theself-consistent equations proposed in Refs. [19,20],for the (particle–hole) di�usion coe�cient D andthe particle–particle di�usion coe�cient D̃ we obtainsystem

D0D= 1 +

1��

∫ ∞

0

∑ke−D̃(k+(2e=c)A)

2t−t3=�3’ dt; (18)

D0D̃= 1 +

1��

∫ ∞

0

∑ke−Dk

2t dt; (19)

where � is the density of states at the Fermi surface,D0is the di�usion coe�cient calculated in Born approxi-mation and the momentum cut-o� |k|¡ 1=‘, where ‘is the transport mean-free path, is implied. The con-ductivity is connected to the di�usion coe�cient in ausual way

� = e2�D: (20)

For simplicity, we will make an analysis ofself-consistent equations only in the absence of mag-netic �eld (A= 0). In our case (�’/�), like in thecase of purely elastic scattering, the conductivitydrastically di�ers in the regions E¿Ec and E¡Ec,where the mobility edge Ec is obtained from theequation [18]

Ec�=√3=4�: (21)

More exactly, we have essentially three regions:(1) metallic region (E¿Ec) with fast di�usion

D ∼ D0; (22)

where dephasing is irrelevant;(2) “dielectric region” (E¡Ec) with slow di�usion

D ∼ D0(k‘)2(�=�’); (23)

determined by the dephasing time;(3) critical region around Ec, (|E=Ec − 1|.

(�=�’)1=3)

D ∼ D0(�=�’)1=3: (24)

When the “mobility edge” crosses the Fermi levelEF (it is achieved by tuning the temperature) the re-sistivity changes sharply, which looks like a metal–insulator transition.If we want to take into account the magnetic �eld in

Eq. (18), it must be noticed, that the vector potentialA does not commute with the momentum k. So theequation takes the form

D0D= 1 +

12��

∫ ∞

0

∑E⊥e−E⊥t−t3=�3’ dt; (25)

where

E⊥ = D̃[k2H +

4l2H

(N +

12

)](26)

(lH is the magnetic length).

378 E. Kogan et al. / Physica E 9 (2001) 374–379

5. Discussion

Let us return to Eq. (12). The electron energychange in a single scattering �E ∼ Tc

√EF=W.T ,

though all the spin waves (with the energies upto ∼ SzTc=S) participate in the dephasing. Thisquasi-elasticity of scattering gives the opportunity tocalculate the dephasing time the way we did. (Thequasi-elasticity condition holds even better for Eq.(17); in this case only the spin waves with small wavevectors contribute.)When analyzing explicitly the CMR e�ect, we

should �rst and most take into account the in uenceof the magnetic �eld on the spin disorder. The staticspin correlator (in ferromagnetic phase) becomes [17]

〈�Szq�Sz−q〉=T 2

4�Sz2C21qatan−1

q�2; (27)

where � ∼ a√SzC=g�BH is the correlation length.

Thus, the long-wave spin uctuations are suppressed,which decreases scattering and hence reduce themobility edge. This mechanism is appropriate for de-scribing CMR e�ect in magnetic semiconductors [7],and can be applied to manganite pyrochlores (theseresults will be presented elsewhere).Second, magnetic �eld shifts the mobility edge by

cutting o� the Cooperon (see Eq. (25)). It is appropri-ate here to explain, why dephasing, which also cuts o�the Cooperon, in uences the localization in a totallydi�erent way. Consider a case of no magnetic �eldand a simpli�ed version of the self-consistent localiza-tion theory, when we ignore the di�erence between Dand D̃, and also consider the dephasing mechanismwhich leads to C(t) = Cel(t) exp[− t=�’] time depen-dence. Then, instead of Eqs. (18) and (19) we havea single one

D0D= 1 +

1��∑k

1Dk2 + 1=�’

: (28)

We see, that due to presence of kd−1 in the numeratorin this equation, the pole of the Cooperon is of no spe-cial importance at d= 3. The dephasing leads to thedelocalization not because it leads to the disappear-ance of the di�usion pole, but because there appearsin the denominator the term, which does not dependon D.Consider �nally the paramagnetic (PM) phase.

In the absence of self-consistent localization theory

which takes into account the spin- ip processes, avery rough idea about the localization in the PMphase we can get from the Io�e–Regel criterium forthe position of the mobility edge �EF ≈ 1. Using thewell-known expression for spin-disorder scatteringrate at temperatures above, but not too close to, Tcwe arrive to two opportunities. For the relatively highFermi energy EF¿E0 ∼ I 4S4=W 3 the increase of thetemperature above Tc leads to a reverse insulator–metal transition. In the opposite case, the systemremains in the dielectric phase.

Acknowledgements

This research was supported by The Israel Sci-ence Foundation founded by The Israel Academy ofSciences and Humanities.

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