DSpace at VNU: Magnetic polaron in quantum well

ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 310 (2007) e340–e342
www.elsevier.com/locate/jmmm

Magnetic polaron in quantum well
Phung Thi Thuy Honga, Nguyen Hoang Longa,b, Bach Thanh Conga,Ã
a

Faculty of Physics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam
Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, Japan

b

Available online 7 November 2006

Abstract
The zero temperature state of electron strongly interacting with localized spins in quantum well (QW) (magnetic polaron in QW) is
studied using double-exchange model and variational method. The numerical calculation shows that the ground state binding energy of
magnetic polaron in QW well is lower than that in one dimension (1D) case.
r 2006 Elsevier B.V. All rights reserved.
PACS: 71.23; 75.70
Keywords: Magnetic polaron; Quantum well; Double exchange

1. Introduction
Problem of phase separation in colossal magnetoresistance perovskites involves much attention of researchers
from both theoretical and experimental points of view [1].
In bulk magnetic perovskite materials, the Mott’s magnetic
polaron (MP) concept is the simplest physical explanation
for phase separation phenomenon. The exact solutions for
MP in one dimension (1D) and bulk three dimension (3D)

cases were studied by the works [2,3]. The work [4]
considers influence of Coulomb repulsion on stability of
MP. The aim of this contribution is to study the Mott’s
large MP in quantum well (QW).
2. Model and calculation
We consider a thin film having thickness l as a realization
of QW. We chose the OZ axis of the Cartesian coordinates
system OXYZ to be paralleled to the thin film XOY plane.
The confining potential at z ¼ 0, l is supposed to be infinity
and equals to zero for 0ozol. We use the Kondo lattice
model (KLM) with strong onsite Hund interaction JH
between localized t2g and narrow band eg electron spins.
ÃCorresponding author. Tel.: +84 4 8584069; fax: +84 4 8584069.

E-mail addresses: (N.H. Long),
(B.T. Cong).
0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jmmm.2006.10.318

For simplicity the localized t2g spins are considered as
classical vectors with module S ¼ 3/2. In strong coupling
limit, KLM reduces to the double-exchange model with the
Hamiltonian:
X
X
2
H ¼ À2t
cosðwij =2Þcþ
cosðwij Þ.
(1)

i cj þ JS
hi;ji

hi;ji

Here, t is the hopping integral supposed to be the same
for the electron at surfaces and inside the film. The cþ
i (cj) is
creation (annihilation) operator of electron having spin
parallel to the localized spin at the same site. wij is angle
between nearest neighbor localized t2g spin vectors.
Averaging (1) over ground state function and going to
the continuous limit for large MP case in cubic crystal
similarly to Ref. [2], we have:
Z

d~
rfZcð~
rÞ2 þ cð~
rÞDcð~
rÞg cosðwð~
rÞ=2Þ
Z
2Zt
cos2 ðwð~
þ
rÞ=2Þ d~
r,
a

E ¼ À 2t

ð2Þ

where Z is coordination number and a ¼ t=JS 2 . Here, the
lattice constant a is equal to 1, and D is Laplace operator.
Trial normalized wave function for MP, cð~
rÞ should be
satisfied the boundary condition, to be zero for z ¼ 0, l.
Furthermore, cð~
rÞ is chosen to have cylindrical symmetry


ARTICLE IN PRESS
P.T.T. Hong et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e340–e342

e341

with coordinate z, r:
rffiffiffiffiffi
npz
2l
2
cðz; ~
rÞ ¼
sin
eÀlr .
pl
l

(3)



wðz; ~
rÞ
aZc2 ðz; ~
rÞ
aZc2 ðz; ~
rÞ
¼
y 1À
cos
2
4
4


2
aZc ðz; ~
rÞ
À1 .
þy
4

yðxÞ ¼

1

if

x40;

0

if

xo0:

ð4Þ

Z

1

1

α=1
α=2
α=5

0.0

En ¼ À

0.00

-0.01
Energy

1.5

2.0

2.5

3.0

-0.02
n=1
α=1
α=2

-0.03

-0.04

6
8
10
Quantum well width (l )

12

ð5Þ

Both l and Z are chosen for minimizing of the energy
function E. Using complex integral representation for y(x),
we obtain minimum energy of MP and value of parameters
corresponding to this minimum after complicated calculation:


3
taZ3
p2 n2
1À 2 ,
648pl
Zl

(6)



2Z
p 2 n2
1À 2 .
9
Zl

(7)

Z ¼ 2l ¼

4

1.0



aZl
sin2 ðnpzÞeÀ2y

Ây 1À
2pl


aZl
sin2 ðnpzÞeÀ2y À 1 .
þy
2pl



dx

2

0.5

Fig. 2. Angle between t2g spins vectors (w) in ground state (n ¼ 1) as a
function of distance in film plan r (in unit of lattice constant a), z ¼ l/2.

dy sin2 ðnpxÞeÀ2y Z À 4l
0
0

p 2 n2
aZl
sin2 ðnpxÞeÀ2y y
þ 4ly À 2
2pl
l



aZl
 1sin2 ðnpxÞeÀ2y
2pl


aZl
sin2 ðnpxÞeÀ2y À 1
þy
2pl

Z 1 (
Z
2Ztpl 1
aZl 2 4
þ
dz
dy
sin ðnpzÞeÀ4y
al 0
2pl
0

E ¼ À 4t

n=1

Distance in film plan (ρ)

ρ)

Putting Eqs. (3), (4) in Eq. (2), we have
Z

3.0

2.9

In Eq. (4), y(x) is a step function,


Angle (χ)

Here n is integer number, r2 ¼ x2 þ y2 and l is the first
variational parameter. We note that the length quantities r,
l are measured in the unit of lattice constant a. The
angle wðz; ~
rÞ between localized spins vectors is also trial
and containing a second variational parameter Z (see also
Ref. [2]):

3.1

14

Fig. 1. Ground state (n ¼ 1) MP energy (in unit t) as a function of QW
width l (in unit of lattice constant a).

Fig. 1 shows the dependence of the energy of MP

in cubic crystal (Z ¼ 6), En (measured in unit of t) on
the width of QW well l for the first (n ¼ 1) quantized level.
The minimum of En(l) curves expresses the maximum
binding energy of MP, DE max
n . Further calculation for
n ¼ 2, 3 reveals that DE max
is largest for ground state
n
(n ¼ 1). When a ¼ 1, we have DE max
% 0:02t at l % 4a and
1
DE max
%
0:01t
at
l
%
7a.
The
value
of
DE max
is lower than
2
1
that in 1D case (0.09t, see Ref. [2]). For very thin film,
the MP state is unstable and the binding energy tends to
be zero.
Fig. 2 illustrates the dependence of the angle between t2g
spins vectors w in ground state as a function of distance in

film plan r for z ¼ l=2 and several values of a. Fig. 2 shows
that there is the canting configuration of t2g spins near
center of MP and far from it, the antiferromagnetic order
occurs.


ARTICLE IN PRESS
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P.T.T. Hong et al. / Journal of Magnetism and Magnetic Materials 310 (2007) e340–e342

Acknowedgements
The authors thank the QG 05-06 and fundamental
research programs for support.
References
[1] E. Dagotto, Nanoscale Phase Separation and Colossal Magnetoresistance, Springer, New York, 2002.

[2] S. Pathak, S. Satpathy, Phys. Rev. B 63 (2001) 214413.
[3] M. Yu. Kagan, A.V. Klapsov, I.V. Brodsky, K.I. Kugel, A.O.
Sboychakov, A.L. Rakhmanov, cond-mat/0301626 vol. 1, 1 Jan 2003.
[4] B.T. Cong, P.T.T. Hong, B.H. Giang, in: Proceedings. of the ninth
Asian-Pacific Physics Conference (APPC), Hanoi, October 26–31,
2004, p. 501.