Science in China Series G: Physics, Mechanics & Astronomy 2006 Vol.49 No.4 421—429 421

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DOI: 10.1007/s11433-006-0421-8

A study of two-dimensional magnetic polaron LIU Tao1, ZHANG Huaihong1, FENG Mang1,2 & WANG Kelin1,3 1. School of Science, Southwest University of Science and Technology, Mianyang 621010, China; 2. Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430070, China; 3. Department of Modern Physics, University of Science and Technology of China, Hefei 230026, China Correspondence should be addressed to Liu Tao (email: liut@swust.edu.cn) Received December 21, 2005; accepted January 11, 2006

Abstract By using the variational method and anneal simulation, we study in this paper the self-trapped magnetic polaron (STMP) in two-dimensional anti-ferromagnetic material and the bound magnetic polaron (BMP) in ferromagnetic material. Schwinger angular momentum theory is applied to changing the problem into a coupling problem of carriers and two types of Bosons. Our calculation shows that there are single-peak and multi-peak structures in the two-dimensional STMP. For the ferromagnetic material, the properties of the two-dimensional BMP are almost the same as that in one-dimensional case; but for the anti-ferromagnetic material, the two-dimensional STMP structure is much richer than the one-dimensional case.

Keywords: self-trapped magnetic polaron, bound magnetic polaron, impurity potential, anti-ferromagnet.

1 Introduction

Recently, the study of colossal magnetoresistance has become a hot topic[1―5], and the related discussion about magnetic polaron has also drawn much attention. Most of the studies on magnetic polaron are based on double-exchange model[6―9], from which we obtain the result that self-trapped magnetic polaron (STMP) only exists in anti-ferro- magnetic materials, while no polaron would be found in a ferromagnetic material with polarized carriers due to dispersion of the probability density distribution. Only in the case of defects or doped atoms existing in a ferromagnetic crystal is there a bound mag-netic polaron (BMP) in the bound potential, which has been tested experimentally. How-ever, no agreement for STMP has been made experimentally. Nevertheless, both BMP and STMP have been found in Cd1-xMnx and Pb1-xMnxT[10―12].

Both experimentally and theoretically, STMP has got more considerations than BMP, because STMP plays its role in carrier transportation[13―15]. Based on previous work on the one-dimensional magnetic polaron[16], we will consider the two-dimensional case in

422 Science in China Series G: Physics, Mechanics & Astronomy

this paper to discuss STMP and its properties in anti-ferromagnetic material and BMP in ferromagnetic material with an impurity potential.

2 Model

Considering the interaction of local magnetic momentum with spin-1/2 carriers in a ferromagnetic or anti-ferromagnetic crystal film, we can simply denote such a film by a two-dimensional lattice square, and the Hamiltonian involving only the nearest-neighbor interactions is

1 1 1 1, ,

ˆ ( )i j ij i j ij ij ij ijiji j

H t c c c c c c c cσ σ σ σ σ σ σσσ

+ + + ++ − + −↑= + + +∑

11 1 1 1( )ij ij ij i j ij i j ij ij ij ij

ij ijJ s J s s s s s s s sσ + − + −− ⋅ ⋅ + ⋅ + ⋅ + ⋅∑ ∑∓ , (1)

where ( )ij ijc cσ σ+ is the annihilation (creation) operator of the carrier with spin σ, ijσ

denotes the spin operator of the carrier at lattice point ij, and ijs is for the local spin at

lattice point ij. The terms for jumping, for the interaction between the spin of the carrier and the local spin, and for the interaction between local spins are denoted by t, J1 and J, respectively. The signs “−” and “+” correspond to ferromagnetic and anti-ferromagnetic situation, respectively.

For convenience, we will change the local spins to bosons by Schwinger’s theory, i.e., by means of the relation between ijs and two independent boson operators

( ), ( ),ij ij ij ija a b b+ + 1 ( )2ijx ij ij ij ijs a b b a+ += + , ( )

2ijy ij ij ij ijis a b b a+ += − − , 1 ( )

2ijz ij ij ij ijs a a b b+ += − .

We define ,ija ij ijN a a+= ,ijb ij ijN b b+= ,ij ija ijbN N N= + and 1 ,2ij ijs N=

1 ( ).2ijz ija ijbs N N= − So eq. (1) can be rewritten as

1 1 1 1 1

1 1 1 1 1

1

ˆ (

)

[(2 2 )2

( )( )]

4

ij ij ij ij ij i j ij i jij

ij ij ij ij ij i j ij i j

ij ij ij ijij ij ij ijij

ij ij ij ijij ij ij ij

H t c c c c c c c c

c c c c c c c c

J c c a b c c b a

c c c c a a b b

J

+ + + +↑ + ↑ ↑ − ↑ ↑ + ↑ ↑ − + ↑

+ + + +↓ + ↓ ↑ − ↓ ↓ + ↓ ↑ − + ↓

+ + + +↓ ↑ ↑ ↓

+ + + +↑ ↑ ↓ ↓

= + + +

+ + + +

− +

+ − −

∑

∑

∓ 1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

[2 2

2 2

2 2

2 2

(

ij i j ij i j ij i j ij i jij

ij ij ij ij ij ij ij ij

ij i j ij i j ij i j ij i j

ij ij ij ij ij ij ij ij

ij ij i j i j i j i j

a b b a a b b a

a b b a a b b a

b a a b b a a b

b a a b b a a b

a a a a a a

+ + + ++ + − −

+ + + ++ + − −

+ + + ++ + − −

+ + + ++ + − −

+ + ++ + − −

+

+ +

+ +

+ +

+ + +

∑

1 1 1 1 )ij ij ij ija a a a+ ++ + − −+

A study of two-dimensional magnetic polaron 423

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

( )

( )

( )].

ij ij i j i j i j i j ij ij ij ij

ij ij i j i j i j i j ij ij ij ij

ij ij i j i j i j i j ij ij ij ij

b b b b b b b b b b

b b a a a a a a a a

a a b b b b b b b b

+ + + + ++ + − − + + − −

+ + + + ++ + − − + + − −

+ + + + ++ + − − + + − −

+ + + +

− + + +

− + + +

(2)

In terms of the coherent states we successfully used in treating polarons in the Frohlich model and Holstein model[17―20], we suppose the solution of static states in our varia-tional method to be a coherent state

2 21 1( )

2 2( ) 0lm lm lm lm lm lm

lma b

ij ijij ijij

c c eα β α β

φ ϕ+ ++ − −

+ +↑ ↓

∑= +∑ . (3)

From normalization of the coherent state, we have 2 2( )ij ijij

φ ϕ= +∑ . So the en-

ergy expectation of the system can be calculated by eqs. (2) and (3):

12 2

1 1

1 1 1 1 1 1

12 2 2 2

1 1 1

( ) 2 (

)

[4 ( )( )]2

[4 (4

ij ij ij i j ij i jij ij

ij ij ij ij ij i j ij i j ij ij ij ij

ij ij ij ij ij ij ij ijij

ij ij i j i j i j iij

HE

t

J

J

φ ϕ φ φ φ φ

φ φ φ φ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

φ ϕ α β φ ϕ α β

α β α β α β

−

+ −

+ − + − + −

+ + − −

=

⎡ ⎤ ⎧⎪= + +⎢ ⎥ ⎨⎢ ⎥ ⎪⎣ ⎦ ⎩

+ + + + + +

⎫⎪− + − − ⎬⎪⎭

+

∑ ∑

∑

∑∓ 1 1 1 1 1

2 2 2 2 21 1 1 1

2 2 2 2 21 1 1 1

2 2 2 2 21 1 1 1

)

( )

( )

2 ( )].

j ij ij ij ij

ij i j i j ij ij

ij i j i j ij ij

ij i j i j ij ij

α β α β

α α α α α

β β β β β

α β β β β

+ + − −

+ − + −

+ − + −

+ − + −

+ +

+ + + +

+ + + +

− + + +

(4)

Since the local spin is related to j, and 2ijα and 2

ijβ are the mean numbers of the

bosons a and b at lattice point ij, we have 22ij ijjβ α= − , which yields 1

2 21 1

1 1 1 1 1 1

1 2 2 2 2

2 21 1

( ) 2 (

)

[2 2 ( )( )]

(2 )(2 )

ij ij ij i j ij i jij ij

ij ij ij ij ij i j ij i j ij ij ij ij

ij ij ij ij ij ij ijij

ij i j ij i jij

HE t

J j j

J j j

φ ϕ φ φ φ φ

φ φ φ φ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ

φ ϕ α α φ ϕ α

α α α α

−

+ −

+ − + − + −

+ +

⎡ ⎤ ⎧⎪= = + +⎢ ⎥ ⎨⎢ ⎥ ⎪⎣ ⎦ ⎩

+ + + + + +

⎫⎪− − + − − ⎬⎪⎭

⎡ − − +⎢⎣

∑ ∑

∑

∓ 2 21 1

2 2 2 21 1 1 1

(2 )(2 )

(2 )(2 ) (2 )(2 )

ij i j ij i j

ij ij ij ij ij ij ij ij

j j

j j j j

α α α α

α α α α α α α α

− −

+ + − −

− −

+ − − + − −

∑

2 2 2 2 2 2 21 1 1 1( ) ( / 2) 6 4i j i j ij ij ij ijj j jα α α α α α+ − + −

⎤+ + + + ⋅ − − + ⎥⎦. (5)

We will below make a variational solution from eq. (5). For convenience of discussion,

424 Science in China Series G: Physics, Mechanics & Astronomy

we set 1t

Jρ ≡ , 1

EJ

ε ≡ , and 1JgJ

≡ . Our following calculation consists of two cases:

(1) Anti-ferromagnetic case without impurity potential, i.e., −J in eq. (5); (2) the ferro-magnetic case involving impurity potential at lattice point (i0, j0). In the latter case, be-sides choosing +J, we have to add a term

0 0 0 01[ ]i j i jc c σ σΔ +− in Hamiltonian, where Δ1 is

the strength of the impurity potential. By repeating the deduction from eqs. (2) to (5), we obtain a new form for the energy expectation with an additional term

0 0 0 0

2 2 1[ ]i j i jΔ φ ϕ− +

with respect to eq. (5). We set 11J

ΔΔ ≡ in our discussion below.

3 Results and discussion

Following the treatment for one-dimensional polaron, we employ again the variational method and the anneal simulation for our solution.

3.1 Anti-ferromagnetic case

Figs. 1―3 are three-dimensional and two-dimensional probability density distribu-tions for spin-up carrier (ψ2), spin-down carrier (φ2) and the local magnetic momentum, respectively. In the two-dimensional distribution of the local magnetic momentum Sz, we denote the spin-down by black square ■ and the spin-up by white square □; in the two-dimensional probability density distribution of the carrier spin, we denote maximum probability by black square ■ and probability zero by white square □. In the case of ρ = 0.1, when g changes from 10−7 to 0.005, there is STMP with distribution from 7×7 lat-tices (see Fig. 1) to 3×3 lattices. When g is small, there is only the single-peak structure in the probability density distribution of carriers with either spin-up or spin-down, and the distribution region for the same magnetic momentum (i.e., Sz ≈ −2) of the local spin is also shrinking with g.

When g increases to 0.0007, the distribution of the probability density for the spin-down carrier becomes multi-peaked, whereas it is still of single-peak for the spin-up carrier. When g becomes 0.0011, a multi-peak structure appears in the distribution of the probability density for the spin-up carrier, and this multi-peak structure becomes very obvious for the spin-down carrier. With further increase of g, for example, to 0.1, the re-gion for the multi-peak structure is enlarging from 4 lattices to 11×11 lattices (for spin-up) and to 9×9 lattices (for spin-down) in Fig. 2.

When g = 0.0003 and ρ changes from 8×10−7 to 6, the distributions of the probability density for both spin-down and spin-up carriers become multi-peaked from the sin-gle-peak structure. Meanwhile, their local regions are first shrinking, and then expanding, while the distribution region of the local spin with the same magnetic momentum keeps enlarging.

The multi-peak structure only appears in two-dimensional STMP. In the one-dimen- sional case, the magnetic momentums of the local spin and the carrier spin are always the

A study of two-dimensional magnetic polaron 425

Fig. 1. The probability density distributions of carriers with spin-up (ψ2) and spin-down (φ2), and the distribution of local magnetic momentum Sz, in the case of ρ = 0.1 and g = 0.00001.

Fig. 2. The probability density distributions of carriers with spin-up (ψ2) and spin-down (φ2), and the distribution of local magnetic momentum Sz, in the case of ρ = 0.1 and g = 0.10. same. So the interaction between the up and down components of the local magnetic momentums is restricted in one dimension, which results in the only distribu-tion:…↓↑↓↑↓↑↑↑↑↓↑↓↑↓…, i.e., the probability density distribution of the carrier with

426 Science in China Series G: Physics, Mechanics & Astronomy

different local magnetic momentums on both sides and the same local magnetic momen-tum in the middle. This distribution region is different for different parameters. In the two-dimensional case, the variation is much richer than that in the one-dimensional case, as shown in Figs. 1―3. Besides, we can see from Fig. 4 that the equipotential contour changes from circular distribution to the distribution with hexagons. Shortly speaking, the anti-ferromagnetic material in two dimensions has more degrees of freedom for variation. So we have structures with single-peak or multi-peaks for different parameters.

Fig. 3. The probability density distributions of carriers with spin-up (ψ2) and spin-down (φ2), and the distribution of local magnetic momentum Sz, in the case of g = 0.0003 and ρ = 0.0000008.

Fig. 4. Equipotential contours for probability density distribution of carriers, where the parameters from left to right are 1) g = 0.0001, ρ = 0.1; 2) g = 0.0003, ρ = 0.08; 3) g = 0.0003, ρ = 0.0000008; 4) g = 0.10, ρ = 0.1.

3.2 Ferromagnetic case with impurity potential

The experience obtained in the treatment of one-dimensional case has told us that no magnetic polaron would appear in a ferromagnetic material without impurity potential. So we only discuss below BMP in a ferromagnetic material with impurity potential.

(1) Δ and ρ are fixed, but g is changed. In terms of our calculation, when polaron

A study of two-dimensional magnetic polaron 427

appears, as long as we fix the values of Δ and ρ, there is no significant change in the dis-tribution of polarons, no matter how we change g. We also find that the distribution of the carrier spins in polarons, particularly for the spin-up case, remains unchanged, and the change in the distribution of the local spin occurs only at the edge.

(2) Δ and g are fixed, and ρ is changed. With the increase of ρ, the localization of the polaron degrades with increased dispersion of the distribution. When ρ increases, for example, to 0.2, although there is impurity potential (Δ ≠ 0), the half-width of the distri-bution of the carrier probability density increases linearly with lattice, which results in no localization.

Fig. 5. The variation of the half-width of the probability density of the carrier with N, where g = 0, ρ = 0.2 and Δ = 0.781.

(3) g and ρ are fixed, but Δ is changed. When Δ is smaller than ρ, the polaron dis-tributes over the whole lattice, which implies no polaron. With the increase of Δ, the lo-calizations increase quickly. Our calculation shows that from Δ = 1 to 2, the localization remains almost unchanged. We use Fig. 5 to explain how to get the right results in the calculation which presents the details of the change for the half-width of the probability density distribution of the carrier via site number N in the case of ρ = 0.2, g = 0, and im-purity potential Δ = 0.781. It shows clearly that when N is larger than 20 the result should be taken as the right one.

Fig. 6 presents the variation of the polaron with Δ in the case that ρ and g are fixed,

Fig. 6. The variation of the half-width of the probability density of the carrier with Δ, where g = 0 and ρ = 0.2.

428 Science in China Series G: Physics, Mechanics & Astronomy

where the dashed and solid curves denote 9×9 and 15×15 lattices, respectively. We can find that the half-width remains unchanged when Δ≥1, which implies that further in-crease of Δ does not help for the localization of polaron and that the polarons in two dif-ferent lattices are approaching the same. So after Δ reaches this parameter region, the distribution of the polaron is irrespective of the lattice size. This means that when the polaron is localized within a very small region, we may replace the real lattice by a lim-ited lattice in our calculation. After Δ is smaller than 0.8, the dispersion of the polaron increases quickly, and no polaron exists any more.

Above results show that the variation of g has no effect on formation of the polaron. When ρ and g are fixed, the bigger the Δ, the better the localzation of the polaron, and the polaron vanishes when Δ is small enough. Moreover, when Δ and g are fixed, the smaller the ρ, the better the localization of the polaron. This means that ρ helps for dispersion of the polaron and Δ is helpful for carrier localization. Furthermore, since the BMP forms around the impurity potential, its structure is, of course, only of a single peak.

Before finishing our discussion, we have to mention two recent work[21,22] which from the continuous limit, discussed the dynamics of the ferromagnetic material and obtained the one- and two-dimensional soliton solutions. Although it is different from our discrete treatment, the work reminds us that whether there is a polaron in the ferromagnetic mate-rial without impurity needs further investigation.

4 Conclusion

The previous investigations for magnetic polaron are focused on the one-dimensional case, based on purely numerical calculation. We have also studied a one-dimensional case by the variational method[16], and the present work is an extension of ref. [16]. Al-though it would be very difficult and complicated, the same method can be directly ap-plied to a three-dimensional case. We have shown in this paper that the formation of BMP due to the impurity potential is almost the same in both one-dimensional and two- dimensional situations, but compared with the one-dimensional case, the two-dimen- sional STMP has shown much richer variations, such as multi-peak structure. This is a new result we had never found in previous studies, for example, the study of a big pola-ron by Frohlich model and a small polaron from Holstein model. To our knowledge, this multi-peak structure in polarons only appeared in ref. [23], in which another one-dimen- sional physical system was discussed. Since STMP is more important from the viewpoint of transportation, we believe that the multi-peak structure in two-dimensional STMP would be helpful in future studies.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 10474118), and the foundation from Sichuan Provincial Department of Educa-tion (Grant No. 041139).

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