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,à 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 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 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) 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 8584069; fax: +84 8584069 E-mail addresses: longnh@presto.phys.sci.osaka-u.ac.jp (N.H Long), congbt@vnu.edu.vn (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 H ẳ 2t coswij =2ịcỵ coswij ị (1) i cj ỵ 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 cỵ 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~ rfZc~ rị2 ỵ c~ rịDc~ rịg cosw~ rị=2ị Z 2Zt cos2 w~ ỵ rị=2ị d~ r, a E ẳ 2t 2ị where Z is coordination number and a ¼ t=JS 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 ẳ 0, l Furthermore, c~ rị 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 cz; ~ rị ẳ sin elr pl l (3) wðz; ~ rÞ aZc2 ðz; ~ rÞ aZc2 z; ~ rị ẳ y cos 4 aZc ðz; ~ rÞ À1 ỵy yxị ẳ if x40; if xo0: 4ị Z 1 =1 =2 =5 0.0 En ẳ 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 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 , 648pl Zl (6) 2Z p n2 1À Zl (7) Z ¼ 2l ¼ 1.0 aZl sin2 ðnpzÞeÀ2y Ây 1À 2pl aZl sin2 npzịe2y ỵy 2pl dx 0.5 Fig 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 ẳ l/2 dy sin2 npxịe2y Z 4l 0 p n2 aZl sin2 npxịe2y y ỵ 4ly À 2pl l aZl  1À sin2 ðnpxÞeÀ2y 2pl aZl sin2 ðnpxÞeÀ2y À þy 2pl Z ( Z 2Ztpl aZl ỵ dz dy sin npzịe4y al 2pl 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 ẳ x2 ỵ y2 and l is the rst 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 Ground state (n ¼ 1) MP energy (in unit t) as a function of QW width l (in unit of lattice constant a) Fig 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, 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 DE max % 0:01t at l % 7a The value of DE max 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 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 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 e342 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, 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 ... α=2 -0.03 -0.04 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... thin film, the MP state is unstable and the binding energy tends to be zero Fig illustrates the dependence of the angle between t2g spins vectors w in ground state as a function of distance in. .. spins vectors is also trial and containing a second variational parameter Z (see also Ref [2]): 3.1 14 Fig Ground state (n ¼ 1) MP energy (in unit t) as a function of QW width l (in unit of lattice