colossal magnetoresistance and phase separation in magnetic semiconductors || back matter

Appendix A

Double exchange Hamiltonian for quantum spins

We reproduce here calculations leading from s-d model Hamiltonian (2.1) and Eq. (2.23) for the single-site wave function to the double exchange effective Hamiltonian for quantum spins (2.24). It will be assumed that, if a ^-electron is located at a magnetic atom, its total spin is S - 1 /2 (as was explained in Subsect. 2.3.2, the condition A < 0 is not a necessary condition for this). In order to construct such a double exchange Hamiltonian in case of an arbitrary spin configuration, equal effective spin magnitudes are attributed to all atoms, the difference in real spin magnitudes of atoms with or without a s-electron being accounted for by the structure of the Hamiltonian. The spin S of free atoms not occupied by electrons has been chosen as such an effective spin. The effective spin thus introduced coincides with the true spin of free atoms, and has no actual physical meaning for atoms housing an electron. However, all physical quantities can be expressed in terms of the effective spin by applying familiar rules. Here is the simplest example: when the true spin of an atom with an electron at it is equal to S - 1/2, the projection of its true spin T is connected with the projection of its effective spin Sz by means of the equality T = Sz -1/2.

In order to introduce the effective spin for A < 0, we alter the description of an atom carrying an electron: instead of the projection of its total moment M (2.23) we shall use the value of the projection of the

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426 Colossal Magnetoresistance and Phase Separation

intrinsic atomic spin possible for a specified M . Namely, we put M = S"g o - 1/2 in Eq. (2.23) where the new index of state S"g 0 can assume 25+7 different values. Their number will obviously exceed the number S"g o - 1/2 in Eq. (2.23) where the new index of state S"g 0 can assume 2S+1 different values. Their number will obviously exceed the number of allowed M values by unity. Actually, the extra degree of freedom is a fictitious one, because when S"g 0 = - S, (i. e., M = - S -1/2), the wave function (2.23) vanishes.

Eq. (2.23) can be rewritten with account taken of the changes in the indices of states by introducing the operators S~ which decrease the spin projection:

VsM=y'g(S*g0) = Lz(S^)d(S*,Sl0)\0>,

L.(SL)= l g* go V2S + 1 ^ < i - ^ = r s ^

(A.l)

When writing down Eq. (A.l), we made use of Eq. (1.40). As is seen from Eq. (A.l), the extra wave function with S*0= - S really disappears.

Now the wave function of the system consisting of a s-electron and d-spins is expanded in the eigenfunctions of the Hamiltonian HA. Since the electron interacts only with atom it occupies, these functions in the single-electron case can be written down in the form

X(S*0) = y,s(S>0)YlS(S*,S*0) = Lg(S*0)\0>\SZ >, (A.2) f*g

is0*>n#s/,s«>) The set of the functions %[Sg0) (A.2) is obviously complete in the

space of functions dependent on the co-ordinate g of the electron and on the spins on all magnetic atoms. Accordingly, the wave function of the system can be presented as an expansion in these functions:

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Appendix A Double exchange Hamiltonian for quantum spins All

V=J,cs(S*)xe(SZ). (A.3)

The summation over all the electron co-ordinates and atomic spin projections is assumed in Eq. (A.3). Because of the delta-like nature of the basis functions Xg(^o )> m e expansion coefficients c ( S0

Z ) have the meaning of the envelope of the wave function.

By substituting the expansion (A.3) into the Hamiltonian (2.1) we are able to obtain the wave equation in a matrix form:

c,(Sf) A(S + l)

E (A.4)

^ct(Sz )[Xg(S? ),(H, +HM )Xt(S

z )]. t.s;

In writing Eqs. (A.4) we made use of the expression (2.23) for the eigenvalue of HA corresponding to the total moment S -1/2.

So far the deduction of the Eqs. (A.4) was straightforward. Now a non-trivial expedient will be used. The transformation of the wave equation in the matrix form (A.4) into a wave equation in an operator form is accomplished by introducing the operators Sg0 acting in the space of functions of indices of states Sz

g0 in the same way as the corresponding operators Sg act in the space of true variables Sz

g . We limit ourselves to calculation of the expression

*Lct(SZ)[Xt(S?),HlXt(Sn] = Xi+Xz (A.5)

In the nearest neighbour approximation, like in Eq. (1.1), using Eq. (A.2) we write for Xj, which is generated by the first term in Eq. (A. 1):

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^ + 1 AX

— I ^ + i ^ f ^ + i ^ j ^ f X , 2 ) , (A.6) 2 5 ~r l AX

since

< S* 1S0Z >= I I Z 3 ( S t >Sn mSt,Sf

z0 ) = US(Sn. S'tQ ). (A.7)

f s, f

When calculating X2, generated by the second term in Eq. (A.l), we employ a very useful relationship

SS(Sz,St) = S+S(Sz,SZ), (A.8)

which just means that the index of the state S0Z is considered as a

variable now. Using also Eq. (A.7), one obtains

_ . x .<5 1z l5 g

+ 5 g + A IS 0z >c g + A rS^

A 2 2S + 1 ~ AX ,l(S + s^)(s + sz

g+Afi)

(2S + i)p + s^ AX Js + szs+A,0

Here the relationship (A.8) is used, as well as the fact that operators S0

do not act upon the wave function 15/ >. The square brackets separate out the terms upon which the corresponding operators act. After repeated use of Eq. (A.8) with respect to 8( 5 / ,Sg ) , replacing Sg by 5 / etc., one obtains:

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Appendix A Double exchange Hamiltonian for quantum spins 429

A 2 _ 1 Zi Jgl Jg+A,l^L I :

(IS + Dp + S^ A 5; V S + 5 « + A . O

= - - ^ X T = L = 5 g ^ - + A , 1 ^ I f L i . (A.10)

The divergence of the last term in Eq. (A. 10) at S*+A] = -S is spurious

as, according to Eq. (1.40), the radicals in the nominator and denominator compensate. In reality, this term goes to zero, as should be the case. Obviously, the wave equation (A.4) with account taken for Eqs. (A.6) and (A. 10) corresponds to the double exchange Hamiltonian for quantum spins (2.24).

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Index

Anderson localisation, 6, 22, 29 35,69, 156,203,216,406

antiferromagnetic CE type ordering, 307, 333, 360, 372, 420

antiferromagnetic layered ordering, 16,49, 140, 150, 154, 163, 170, 171,331,372

antiferromagnetic staggered ordering, 49, 50, 57, 138, 146, 147, 150, 153, 162, 163, 170, 171, 176,252,254,255,259

antiferromagnetic semiconductor, 2 ,3 ,4 ,8 ,9 , 10, 12,89,90, 139, 141, 142, 154, 170, 172, 173, 176, 185, 186, 188, 189,244, 245, 249, 256, 258, 259, 260, 277, 295, 345

antimetamagnetism 53, 57

bound ferron, 8, 11, 132, 174,175, 176, 177, 178, 180, 181, 183, 184, 185, 186, 187, 204,244,244,259,309, 319, 320, 321,322,323,327,331, 366,416,427

canted antiferromagnetic ordering, 9, 10,40,49,56, 143, 159, 172,173, 175, 177, 186, 252, 253, 254, 255, 256, 263, 320, 321,322,323,324,325, 326,327,328,329,331,363, 364,366,377,403,418,426

colossal negative isotropic magnetoresistance (CMR), 2, 3, 4, 11, 12, 13, 15, 16, 18, 19,89, 94, 100, 101, 121, 122, 124, 127, 133, 198,202,203,205, 217,220,221,268,287,289, 290,294,347, 349,351,366, 371,374,375,377,379, 386, 387, 403,404,406,415,425, 426

crystallographic anisotropy, 42, 87, 307,

charge ordering, 334, 335, 336, 337,338,339,340,344,351, 362, 362, 364, 365, 366, 367, 368,371,373,374,375,384, 385,389,396,397,399,412, 413,417,418,419,420,421, 426

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double exchange, 5, 6, 7, 8, 9, 16, 17, 18,63,65,66,68,70,71, 73,74,75,76,77,81,84,88, 104, 106, 107, 106, 109, 111, 117, 119, 122, 125, 130, 139, 140, 141, 143, 144, 152, 163, 168, 171, 173, 181, 192, 193, 215,26,217,223,224,228, 249,251,252,253,262,271, 272,276,20,316,358,404, 405,406,407,410,411,416, 417,419,420,423

ferromagnetic semiconductor, 2, 3,4,5,12, 13,14, 18, 19,20, 22, 33, 34, 35, 37, 88, 89, 90, 92,93,94,96,97,98, 100, 101, 104, 105, 120, 129, 130, 131, 135, 139, 141, 15, 170, 171, 175, 176, 188, 189, 194, 195, 196,198,203,210,220,226, 227, 233, 234, 237, 238, 239, 24,244,287,289,299,317, 344, 345, 345, 346, 348, 349, 351,356,357,404,405,406, 407

ferron, 8, 9, 12, 152, 153, 155,156, 157, 158, 160, 161, 162, 163, 164, 165, 166, 167, 168, 170, 171, 172, 173, 174, 175,26,243,255,256,257, 296

ferron-polaron, 133, 158, 159, 411 ferron-string, 166, 168

giant blue optical absorption edge shift, 88, 139, 140, 141, 172

giant magnetoresistance, 2 giant red optical absorption edge

shift, 88, 90, 92, 93, 94, 95,

102,104,105,116,120, 121, 126, 127

granulated systems, 1, 2

indirect exchange, 5, 14, 15, 78, 80, 82, 84, 89, 120, 124, 125, 126, 130, 157, 175, 181, 185, 191, 192, 193, 195, 196, 197, 199,204,207,210,213,220, 234,238, 244,245,281, 293, 295,309,319,331,332,348, 399,414,416

isotope effect, 227, 228, 231, 232,233,391,392,393,394, 395, 396, 397, 398, 399, 400, 411,412,413,414,415

Hubbard model, 66, 67, 68 Heisenberg Hamiltonian, 41, 43,

49,50,51,52,55,57,58,59, 62,64,79,80,85,88, 135, 136, 138, 139, 155, 160, 190,214, 237,244,45,262,309,315

Jahn-Teller effect, 17, 18,43, 298, 300, 303, 348, 349, 393, 394, 400

Jahn-Teller phonons, 17, 18, 19, 393,394,407,411,412,415, 416,417,419,425.

magnetic anisotropy, 42, 43, 45, 55, 56, 87, 309, 326, 402

magnetic string, 143, 146, 147, 148, 149, 150, 151, 152

magnetoimpurity theory of CMR, 12, 13, 16, 120,203,317,348, 351,403,404,406,416

magnetic-electronic phase separation, 8, 10, 138, 168,

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Index 457

256,257,259,20,266,271, 272,275,276,277,290,291, 292,320,321,322,323,324, 328,331,332,337,344,354, 355,381,399,417,419

magnetic-impurity phase separation, 293, 294, 295, 296, 328

magnetic phase separation close to the interphase boundary, 333, 334, 335, 374, 375, 396, 397, 419

metamagnetism 4, 55, 56, 58, 59, 137, 138, 158,281,351,368, 369,372,374,384,421,423, 424

Mott transition at T =0, 2, 9, 30, 31,32,33,34,35,244,295, 321,351,394,407,412,422

Mott temperature-induced transition, 15,205,218,219, 221,223,225,226,228,317, 405

multilayered films, 1, 2.

non-Heisenberg exchange, 41, 55,82,84, 136, 138, 141, 191, 196, 197,244,253,281

oxygen-off-stoichiometry insulator-to-metal transition, 227, 228

phase-separated layered structure, 10,11, 152,254,257,

258, 259, 271, 272, 273, 274, 292

phase-separated droplet structure, 10, 11,257,258,259,260,261, 265,266,268,271,272,274,

275, 276, 277, 284, 287, 289, 290,291,323,387

photoinduced insulator-metal transition, 387, 388, 389, 390 photomagnetoresistance, 103

positive magnetoresistance, 100, 102, 128, 129

relative magnetoresistance , 1, 100, 127, 128,202,349

resistivity peak, 2, 3, 4, 12, 13, 16, 18, 19,99, 100, 120, 124, 127, 131, 196, 198,200,203, 204,205,210,211,226,228, 287,309,317,319,332,345, 346, 348, 349, 353, 383, 403, 404,405,406,407,410,411, 416

s-d model, 5, 6, 61, 62, 66, 71, 106

simple exchange, 6, 8, 63, 79, 82, 88, 104, 108, 109, 110, 122, 128, 130, 138, 140, 142, 152, 155, 165, 175, 183, 191,205, 215,216,223,224,226,233, 243,249,250,251,260,261, 267,401,403,405,409

small polaron 19, 26, 27, 28, 29, 30,31,316,321,339,341,346, 392,408,409,410,412,420

stripes, 331,334,335,336,337

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