Journal of Magnetism and Magnetic Materials 310 2007 e340–e342Magnetic polaron in quantum well Phung Thi Thuy Honga, Nguyen Hoang Longa,b, Bach Thanh Conga, a Faculty of Physics, Hanoi U
Trang 1Journal of Magnetism and Magnetic Materials 310 (2007) e340–e342
Magnetic polaron in quantum well Phung Thi Thuy Honga, Nguyen Hoang Longa,b, Bach Thanh Conga,
a Faculty of Physics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam
b Department of Physics, Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, Japan
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
r2006 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
magnetoresis-tance 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
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:
H ¼ 2tX
hi;ji
cosðwij=2Þcþi cjþJS2X
hi;ji
cosðwijÞ (1)
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:
E ¼ 2t
Z d~rfZcð~rÞ2þcð~rÞDcð~rÞg cosðwð~rÞ=2Þ
þ2Zt a
Z cos2ðwð~rÞ=2Þ d~r, ð2Þ
where Z is coordination number and a ¼ t=JS2 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
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Corresponding author Tel.: +84 4 8584069; fax: +84 4 8584069.
E-mail addresses: longnh@presto.phys.sci.osaka-u.ac.jp (N.H Long) ,
Trang 2with coordinate z, r:
cðz; ~rÞ ¼
ffiffiffiffiffi
2l
pl
r
sin npz
l
elr2 (3) Here n is integer number, r2 ¼x2þy2 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]):
coswðz; ~rÞ
2 ¼
aZc2ðz; ~rÞ
4 y 1
aZc2ðz; ~rÞ 4
þy aZc
2
ðz; ~rÞ
4 1
In Eq (4), y(x) is a step function,
yðxÞ ¼ 1 if x40;
0 if xo0:
Putting Eqs (3), (4) in Eq (2), we have
E ¼ 4t
Z 1
0
dx
Z 1 0
dy sin2ðnpxÞe2y Z 4l
þ4ly p
2n2
l2
aZl 2pl sin
2ðnpxÞe2yy
1 aZl
2pl sin
2
ðnpxÞe2y
þy aZl
2pl sin
2ðnpxÞe2y1
þ2Ztpl
al
Z 1
0
dz
Z 1 0
dy aZl 2pl
2
sin4ðnpzÞe4y (
y 1 aZl
2pl sin
2ðnpzÞe2y
þy aZl 2pl sin
2ðnpzÞe2y1
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 calcula-tion:
En¼ taZ3
648pl 1
p2n2
Zl2
3
Z ¼ 2l ¼2Z
9 1
p2n2
Zl2
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, DEmax
n Further calculation for
n ¼ 2, 3 reveals that DEmax
n is largest for ground state (n ¼ 1) When a ¼ 1, we have DEmax1 0:02t at l 4a and
DEmax2 0:01t at l 7a The value of DEmax1 is lower than 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 2illustrates 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 2shows that there is the canting configuration of t2g spins near center of MP and far from it, the antiferromagnetic order occurs
-0.04
-0.03
-0.02
-0.01
0.00
Quantum well width (l)
α = 1
α = 2
n = 1
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).
2.9 3.0 3.1
Distance in film plan (ρ)
α = 1
α = 2
α = 5
n = 1
Fig 2 Angle between t 2g 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.
Trang 3The authors thank the QG 05-06 and fundamental
research programs for support
References
[1] E Dagotto, Nanoscale Phase Separation and Colossal
Magnetoresis-tance, 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.