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PM-FM transition in a DE model
Eugene Kogan , Mark Auslender ^ ' *
^Jack and Pearl Resnick Institute of Advanced Technology, Department of Physics, Bar-Ran University, Ramat-Gan 52900,
Israel
^Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84^05
Israel
Abstract
We study paramagnetic - ferromagnetic transition due to exchange interaction between classical localized mag-
netic moments and conduction electrons. By solving the Dynamical Mean Field Approximation equations we find
explicit formula for the transition temperature Tc for arbitrary electron dispersion law, concentration and relation
between exchange coupling and the electron band width. We present the results of calculations of the for the
semi-circular electron density of states.
Key words: , strong correlations, double-exchange model, magnetism, ferromagnetic order
PACS: 75.10.Hk, 75. 30. Mb, 75.30.Vn
The double-exchange (DE) model [1,2,3] is one of the
basic ones in the theory of magnetism. Magnetic or-
dering appears in this model due to exchange coupling
between the core spins and the conduction electrons.
The Hamiltonian of the model is
H
-na'-n' a
na(3
(1)
where c and are the electrons annihilation and cre-
ation operators, S„ is the operator of a core spin,
is the electron hopping, J is the exchange coupling be-
tween a core spin and n electrons, a is the vector of the
Pauli matrices, and a, 13 axe spin indices.
We calculate the temperature of a paramagnetic-
ferromagnetic transition Tc in a double-exchange
model for arbitrary electron dispersion law, concentra-
tion and relation between the exchange coupling and
the electron band width by formulating and solving
the DMFA equations. We treat the core spins as clas-
autho
Tel: +972 8 6461583 fax: +972 8
Corresponding
6472949
Email addresses: [email protected] (Eugene Ko
gan), [email protected] (Mark Auslender).
sical vectors. The DE Hamiltonian in a single electron
representation can be presented as
(2)
Let us introduce Green's function and local Green's
function
G{E) = {E-H)-\ G'ioc(£) = (g'„„(S)). (3)
In the last equation, the averaging is with respect to
random configurations of the core spins. In the frame-
work of the DMFA approach to the problem (see [4]
and references therein) the local Green's function is
expressed through the the local self-energy E by the
equation
Gioc(S) =30 [E~T.{E)
(4)
where go{E) — Ylw ~ ^k)^^ the bare (in the ab-
sence of the exchange interaction) local Green's func-
tion. The self-energy satisfies equation
Gioc(-E')
G^^l{E) + ±{E) + Jni-a
(5)
Preprint submitted to Elsevier Science
2 February 2008
where {X{m)) = J X(m)P(m), and P{m) is a proba-
bility of a given spin orientation (one-site probability).
The quantities G and E are 2x2 matrices in spin space.
The DMFA assumption for the probability P{m) is
based on the Equation
AD{E,m) = -ilmlndet [l + (^Jma + £j Gioc] ,(6)
where the argument of both doc and S is i? + iO. So
the change in thermodynamic potential is [6,5,7]
AQ{m) = / f{E)AD{E, m)dE,
(7)
where f(E) is the Fermi function. The DMFA approx-
imation for the one-site probability P{m) is:
P(m) oc exp [-/3An(m)] .
(8)
Eqs. (5) and (8) are the system of non-linear (inte-
gral) equations. In the ferromagnetic (FM) phase near
the Curie temperature, Eqs. (5) and (8) can be lin-
earized [5] with respect to M. Thus we reduce the
DMFA equations to a traditional MF equation [7]
P(m) oc exp (-3/3TcM • m)
(9)
The parameter Tc is formally introduced as a coeffi-
cient in the expansion of An(m) with respect to M.
Non-trivial solution of the MF equation M = (m) can
exist only for T < Tc, hence Tc is the ferromagnetic
transition temperature.
For the Tc, after straightforward, though lengthy al-
gebra, we obtain
2J^ 7
37r J
Im
(Sg'-g)(l+Sg)
FTP
2J^g
3
dE, (10)
where E and g are determined by the properties of the
system in the PM phase (P(m) = const). Eq. (10) is
the main result of our paper. Let us apply Eq. (10) to
the case of semi-circular bare density of states No (e) =
{2/W)^E^/W^ - 1, for which Eq. (10) takes the form
Tc = Im
6
(11)
For arbitrary exchange, the integral in Eq. (11) can
be calculated only numerically. Note that that our main
result (equation for the Tc) indicates it's own limits of
validity. In part of the J/W — n plane, Eq. (11) gives
Tc < 0. Negative value of Tc means, that at any tem-
perature, including T = 0, the paramagnetic phase
is stable with respect to appearance of small spontar-
neous magnetic moment, and strongly suggests that
the ground state in this part of the phase plane is non-
ferromagnetic (NFM) [8].
0.05
0.04
0.03
0.02
0.01
Fig. 1. Tc as a function of electron concentration n: J/W — .25
(dotted line), J/W = 1 (dash-dotted line), J/W = 2 (dashed
line), and J/W = 20 (solid line)
/
FM
/
/
NFM
0.4
0.6
n
0.8
[1]
[2]
[3]
[4]
[5]
[6]
[7]
Fig. 2. The curve Tc = on the J/W - n plane
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