Localization and dephasing driven by magnetic fluctuations in colossal magnetoresistance materials

Abstract

Localization and dephasing of conduction electrons in a low-carrier-density ferromagnet due to scattering on magnetic fluctuations is considered. We claim the existence of the “mobility edge”, which separates the states with fast diffusion and the states with slow diffusion; the latter is determined by the dephasing time. When the “mobility edge” crosses the Fermi energy a large and sharp change of conductivity is observed. The theory provides an explanation for the observed temperature dependence of conductivity in ferromagnetic semiconductors and manganite pyrochlores.

Full text

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arXiv:cond-mat/9808065v7 [cond-mat.str-el] 3 Mar 1999
EPJ manuscript No.
(will be inserted by the editor)
Localization and Dephasing Driven by Magnetic Fluctuations in
Low Carrier Density Colossal Magnetoresistance Materials
Eugene Kogan1,2,a, Mark Auslender3, and Moshe Kaveh1,2
1Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ilan University, Ramat-Gan 52900,
Israel
2Cavendish Laboratory, Madingley Road, Cambridge CB3 OHE, UK
3Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev P.O.B. 653, Beer-Sheva 84105,
Israel
Received: date / Revised version: date
Abstract. Localization and dephasing of conduction electrons in a low carrier density ferromagnet due to
scattering on magnetic fluctuations is considered. We claim the existence of the ”mobility edge”, which
separates the states with fast diffusion and the states with slow diffusion; the latter is determined by
the dephasing time. When the ”mobility edge” crosses the Fermi energy a large and sharp change of
conductivity is observed. The theory provides an explanation for the observed temperature dependence of
conductivity in ferromagnetic semiconductors and manganite pyrochlores.
PACS. 75.50.Pp Magnetic semiconductors – 75.70.Pa Giant magnetoresistance – 72.10.-d Theory of elec-
tronic transport; scattering mechanisms – 72.10.Di Scattering by phonons, magnons, and other nonlocalized
excitations
1 Introduction
Colossalmagnetoresistance (CMR) materials attract nowa-
days considerable interest, associated mostly with the prop-
erties of double-exchange manganite perovskites [1]. Class
of CMR materials, however, is much wider and includes,
in particular, magnetic semiconductors [2] and mangan-
ite pyrochlores [3]. All these materials are characterized
by strong interaction between the localized spins and itin-
erant charge carriers. In all these materials CMR is as-
sociated with the sharp increase of the resistivity when
the temperature T approaches the Curie temperature Tc.
However, taking into account the large variety of the ma-
terials involved and diverse manifestations of the effect, it
is difficult to expect that any single theory can provide a
universal explanation of the phenomena.
We concentrate on low carrier density materials (mag-
netic semiconductors and manganite pyrochlores), where
the carriers do not affect the spin-spin interaction, and
magnetic d- or f-ions interact mainly via ferromagnetic
direct exchange (super-exchange). These materials being
deficient in chalcogen (oxygen) or being properly doped,
have at low temperatures quasi - metallic conductivity.
When the temperature approaches Tc they undergo the
metal-insulator transition.
In the previous publications [4,5,6] we suggested, that
such behavior of the conductivity is due to Anderson local-
ae-mail: kogan@quantum.ph.biu.ac.il
ization of the carriers driven by spin fluctuations of mag-
netic ions. We considered the spin fluctuations as static;
hence the scattering of electrons by the fluctuations can be
treated as elastic, and hence it leads to the existence of the
mobility edge Ec. (This mechanism is close to the phonon
scattering induced electron localization [7,8].) When the
temperature increases, so does the scattering intensity,
which leads to the upward motion of the mobility edge.
The temperature at which the mobility edge crosses the
Fermi level is identified with the temperature of the metal-
insulator transition (MIT). This view point on temperature-
induced MIT has also been recognized and legitimated in
several recent publications [9,10,11].
In this paper we consider the influence of the dynamics
in the spin subsystem on the transition by developing an
idea suggested in Ref. [12]
2 Hamiltonian and Approximations
The Hamiltonian of the system has the form
H =
?
kσ
Ekc†
kσckσ−I
N
?
kqσσ′
Sqσσσ′c†
kσck+qσ′ + HM,(1)
where Ekis the bare electron spectrum, c†
electron creation and annihilation operators, I is the pa-
rameter of Hund exchange between the electrons and lo-
calized spins, Sq is the Fourier components of the spin
k,σ, ck,σare the

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2Kogan et al.: Localization and Dephasing Driven by Magnetic Fluctuations in CMR Materials
density, σ is the vector of the Pauli spin matrices and
HMis the direct exchange interaction described by super-
exchange integral J(Q).
Let us state the relations between the parameters of
the problem. We consider the case of wide conduction
band W ≫ 2IS, where W ∼ 1/ma2is the width of the
conduction band (a is the lattice constant and and m is
the electron mass), S is the spin of magnetic ion. This in-
equality is certainly applicable to such magnetic semicon-
ductors as EuO and EuS [13]. For manganite pyrochlores
we do not have strong inequality [14], but we believe that
the approximation still works in this case, at least semi-
quantitatively.1Due to low carrier density considered (not
in excess of 10−19cm−3), the Fermi energy EF is at least
an order of magnitude less then 2IS (which is larger than
0.5 eV in the materials considered [13,14]). We consider
ferromagnetic phase and temperatures such, that the spin
splitting of the conduction band is larger than EF(estima-
tions show, that this will be true up to the temperatures
very close to Tc). All our assumptions can be thus reduced
to inequalities
W ≫ 2IS
2ISz≫ EF≫ T,(2)
where Sz, is the average spin of magnetic ion.
In the wide conduction band case the electron-spin ex-
change can be treated as a perturbation, leading to elec-
tron scattering. The conduction electrons being fully spin-
polarized and spin-flip processes thus being forbidden, the
scattering (in the Born approximation) is connected only
with the longitudinal spin correlator < δSz
is argued [15], that for the wavevector q small enough
(qa < const(Sz)2) the correlator is dominated by contri-
bution of weakly interacting spin waves with the disper-
sion law
ωQ= 2Sz[J(0) − J(Q)]
and quasi-classical occupation numbers
qδSz
−q>. It
(3)
nQ=
T
ωQ
(4)
As a result the static correlator is [15]
< δSz
qδSz
−q>=
T2
8Sz2C2
1
qa,
(5)
where C is the spin stiffness (for nearest-neighbor ex-
change in a cubic lattice C ≃ Tc/2S(S + 1)).
For the transport relaxation time we obtain
1
τ=2π
NI2?
q
< δSz
qδSz
−q>k · q
k2δ(Ek− Ek+q)
=
ma2I2T2
16πSz2C2∼I2S(S + 1)
W
T2
T2
c
S2
Sz2
(6)
1We emphasize here the difference between the manganite
pyrochlores, which are n-type low carrier density intermediate-
band materials, and the manganite perovskites, which are p-
type high carrier density narrow-band materials; the latter thus
being definitely outside the scope of the theory presented in the
paper.
(a)(b)
0t
0tt0
0t12
12 t-1t-2
21
Fig. 1. Diagrams for the Diffuson (a) and the Cooperon (b).
Solid line is dressed electron propagator, dashed line connect-
ing times t and t′corresponds to Φzz(t − t′).
We see that for temperatures high enough, τEF < 1.
Hence we need some kind of strong scattering theory. As
such we shall use the self-consistent localization theory
by Vollhard and W¨ olfle (VW) [16], extended in Ref. [17]
to systems without time-reversal invariance. But first we
should calculate the crucial parameter in our approach -
the dephasing time τϕ.
3 Dephasing Time
The inverse dephasing time can be defined as the mass
of the Cooperon [18,19]. (An alternative, but essentially
equivalent view on dephasing see in Ref. [20].) For the
Cooperon C(R,t) we obtain equation
?∂/∂t − D∇2+ [f(0) − f(t)]?C(R,t) = 0,
f(t) =2π
NI2?
and Φzz(q,t) is the temporal longitudinal spin correlator
(Φzz(q,t = 0) ≡< δSz
Eq. (7) can be easily understood if we compare dia-
grams for the Diffuson and the Cooperon on Fig. 1. The
Diffuson does not have any mass because of Ward identity.
In the case of the Cooperon, the Ward identity is broken:
interaction line which dresses single particle propagator
is given by static correlator, and interaction line which
connects two different propagators in a ladder is given by
dynamic correlator. The difference [f(0)−f(t)] shows how
strongly the Ward identity is broken and, as we’ll see be-
low, determines the mass of the Cooperon. Solving Eq. (7)
we get [21]
(7)
where
q
Φzz(q,t)δ(Ek− Ek+q)(8)
qδSz
−q>).
C(t) = Cel(t)exp
??t
0
[f(t′) − f(0)]dt′
?
,(9)
where Cel is the Cooperon calculated ignoring the in-
elasticity of scattering.
Using the spin-wave picture described above, we obtain
Φzz(q,t) =
1
N
?
Q
nQnQ+qexp[i(ωQ+q− ωQ)t]. (10)
Performing integration in Eq. (9) we get
C(t) = Cel(t)exp?−t3/τ3
ϕ
?,(11)

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Kogan et al.: Localization and Dephasing Driven by Magnetic Fluctuations in CMR Materials3
where
1
τ3
ϕ
=
π
3N2I2?
Qq
nQnQ+qδ(Ek− Ek+q)(ωQ+q− ωQ)2.
(12)
Calculating the integrals in Eq. (12) we obtain
1
τϕ
=
?I2T2(W − 1)ma5k3
F
18π
?1/3
≃
?
I2T2E3/2
W5/2
F
?1/3
(13)
where W is the Watson integral. It is worth noting that
dephasing time is defined by the second time derivative
of Φzz(q,t) at t = 0 which can be calculated via second
moment of corresponding spectral density; the result turns
out to be essentially the same as Eq. (12). So the spin-wave
picture, being physical one, is not crucial for obtaining
1/τϕ.
It should be noticed that the form of the Eq. (11) for
the Cooperon is quite general, provided the scatterers are
in a ballistic motion, irrespective of whether they are point
particles [21], phonons [8], or spin waves, like in our case.
The result for the dephasing time can be understood
using simple qualitative arguments. If all the collisions
lead to the same electron energy change δE, the dephasing
time could be obtained using relation [18]
τϕδE
?τϕ
τout
∼ 2π,(14)
where τϕ/τoutis just the number of scattering acts during
the time τϕ(τoutis the extinction time). So in this case
1
τ3
ϕ
∼(δE)2
τout
. (15)
If we rewrite the formula for the extinction time
1
τout
=2π
NI2?
q
< δSz
qδSz
−q> δ(Ek− Ek+q)(16)
in the form
1
τout
=2π
N2I2?
Qq
nQnQ+qδ(Ek− Ek+q) (17)
and notice that (ωQ+q−ωQ) is just the energy change of
the electron when scattering on a spin wave, we immedi-
ately see that Eq. (12) is just Eq. (15) with the integration
with respect to different collision induced energy changes
built in.
4 Conductivity Calculation
The time-reversal invariance in the system we are con-
sidering is broken for two reasons. First, because we are
considering ferromagnetic system, it is naturally to expect
that the magnetic field is present in the system. Even
more important is that the dephasing itself breaks the
time-reversal invariance. We have shown in the previous
Section, that due to dephasing the diffusion pole of the
particle-particle propagator disappears, although particle-
hole propagator still has a diffusion pole, which is guaran-
teed by particle number conservation. Inserting Eq. (11)
into the self-consistent equations proposed in Ref. [17], for
the (particle-hole) diffusion coefficient D and the particle-
particle diffusion coefficient˜D we obtain system
D0
D
= 1 +
1
πν
?∞
0
?
k
e−˜ D(k+2e
cA)
2t−t3/τ3
ϕdt(18)
D0
˜D
= 1 +
1
πν
?∞
0
?
k
e−Dk2tdt,(19)
where ν is the density of states at the Fermi surface, D0
is the diffusion coefficient calculated in Born approxima-
tion and the momentum cut-off |k| < 1/ℓ, where ℓ is the
transport mean free path, is implied. The conductivity is
connected to the diffusion coefficient in a usual way
σ = ne2(3D/2E), (20)
where E is the Fermi energy, and n is the concentration.
For simplicity, we will make an analysis of self-consistent
equations only in the absence of magnetic field (A = 0).
In our case (τϕ ≫ τ), like in the case of purely elastic
scattering, the conductivity drastically differs in the re-
gions E > Ecand E < Ec, where the mobility edge Ecis
obtained from the equation [16]
Ecτ =
?
3/4π. (21)
More exactly, we have essentially three regions:
1. metallic region (E > Ec) with fast diffusion
D ∼ D0, (22)
where dephasing is irrelevant;
2. ”dielectric region” (E < Ec) with slow diffusion
D ∼ D0(kℓ)2(τ/τϕ), (23)
determined by the dephasing time;
3. critical region around Ec, (|E/Ec− 1| ≪ (τ/τϕ)1/3)
D ∼ D0(τ/τϕ)1/3. (24)
When the ”mobility edge” crosses the Fermi level (it is
achieved by tuning the temperature) the resistivity changes
sharply, which looks like a metal-insulator transition.
If we want to take into account the magnetic field in
Eq. (18), it must be noticed, that the vector potential A
does not commute with the momentum k. So the equation
takes the form
?∞
D0
D
= 1 +
1
2πν
0
?
E⊥
e−E⊥t−t3/τ3
ϕdt(25)
where
E⊥=˜D
?
k2
H+
4
l2
H
?
N +1
2
??
(26)
(lHis the magnetic length).

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4Kogan et al.: Localization and Dephasing Driven by Magnetic Fluctuations in CMR Materials
5 Discussion
Let us return to Eq. (12). The electron energy change in
a single scattering δE ∼ Tc
?EF/W ≪ T, though all the
in the dephasing. This quasi-elasticity of scattering gives
the opportunity to calculate the dephasing time the way
we did. (The quasi-elasticity condition holds even better
for Eq. (17); in this case only the spin waves with small
wave vectors contribute.)
When analyzing explicitly the CMR effect, we should
first and most take into account the influence of the mag-
netic field on the spin disorder. The static spin correlator
(in ferromagnetic phase) becomes [15]
spin waves (with the energies up to ∼ SzTc/S) participate
< δSz
qδSz
−q>=
T2
4πSz2C2
1
qatan−1qξ
2,(27)
where ξ ∼ a
the long wave spin fluctuations are suppressed, which de-
creases scattering and hence reduce the mobility edge.
This mechanism is appropriate for describing CMR ef-
fect in magnetic semiconductors [5], and can be applied
to manganite pyrochlores (these results will be presented
elsewhere).
Second, magnetic field shifts the mobility edge by cut-
ting off the Cooperon (see Eq. (25)). It is appropriate
here to explain, why dephasing, which also cuts off the
Cooperon, influences the localization in a totally differ-
ent way. Consider a case of no magnetic field and a sim-
plified version of the self-consistent localization theory,
when we ignore the difference between D and˜D, and
also consider the dephasing mechanism which leads to
C(t) = Cel(t)exp[−t/τϕ] time dependence.2Then instead
of equations (18) and (19) we have a single one
?
SzC/gµBH is the correlation length. Thus
D0
D
= 1 +
1
πν
?
k
1
Dk2+ 1/τϕ.(28)
We see, that due to presence of kd−1in the numerator in
this equation, the pole of the Cooperon is of no special
importance at d = 3. The dephasing leads to the delocal-
ization not because it leads to the disappearance of the
diffusion pole, but because there appears in the denomi-
nator the term, which does not depend on D.
Consider finally the paramagnetic (PM) phase. In the
absence of self-consistent localization theory which takes
into account the spin-flip processes, a very rough idea
about the localization in the PM phase we can get from
the Ioffe-Regel criterium for the position of the mobility
edge τEF ≈ 1. Using the well-known expression for spin-
disorder scattering rate at temperatures above, but not
too close to, Tc we arrive to two opportunities. For the
relatively high Fermi energy EF > E0 ∼ I4S4/W3the
increase of the temperature above Tc leads to a reverse
insulator-metal transition. In the opposite case the sys-
tem remains in the dielectric phase.
2This happens when the scatterers are in a diffusive motion
[21].
We are grateful to D. Khmel’nitskii for the useful discussions.
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