C o n d i t i o n s o f f e r r o m a g n e t i c o r d e r i n g i n a m o d e l w i t h A n d e r s o n - H u b b a r d c e n t e r s : e f f e c t i v e H a m i l t o n i a n a p p r o a c h
O K r a m a r , L D i d u k h , a n d Y u S k o r e n k y y
Ternopil National Technical University, Physics Department, 56 Ruska Str., Ternopil, Ukraine E-mail: kramarOtu. edu .te.ua
Abstract. A model of strongly correlated electron system in which magnetic impurity levels are hybridized with conduction band has been considered. The effective Hamiltonian has been constructed for the case of strong Coulomb correlation on basis of configurational representation of Hamiltonian with Hubbard X-operators describing the localized spin subsystem. Criteria for the ferromagnetic ordering stabilization have been found in partial case of partially filled band for arbitrary temperatures and Curie temperature has been calculated. For the partial case of weak effective exchange the formula for the Curie temperature reproducing the well-established results has been calculated analytically. The electron concentration region favorable for ferromagnetic ordering is determined by hybridization through effective exchange integral.
1. I n t r o d u c t i o n
2. C o n f i g u r a t i o n a l r e p r e s e n t a t i o n of H a m i l t o n i a n
Following papers [6, 10], we start from the model of Anderson-Hubbard material which describe localized (d—) subsystem hybridized with conduction (c—) band. Coulomb a n d exchange interactions within the localized subsystem are the most conveniently described in the configurational representation of H u b b a r d X-operators.
H = HQ + Hh + H^ + H2h + H^ + Hh^ (1) i i k<r ija Hn = ^ a V i k i y ^ x t + h.c), Ha = Y.^V^)ctxi2 + h.cl H2h = 2^(V(ijk,-k)x}°xfckrc-ki + h.c.), ijV. H2d = 2j2(V(ijK-^)X^X^c.^c^ + h.c.), ijV. Hhd = 2 ^ ( F ( u k , - k ) ( x f x f Vk tc1 4- XJ 2 t 0x / ° c _k tck 4) + /l.c.). ijV.
Here operator Xfl describes transition of site i from state |/) to state \k), (^{cy^) are creation (annihilation) operators for band electrons. Energy parameters of the model are the chemical potential //, the energy of intra-site Coulomb repulsion of electrons U, the direct inter-site exchange interaction J(ij), hybridization parameters V(ik) and V(ijk, — k ) . Let us introduce the dimensionless parameters which describe relative hybridization:
EF-Ed ftV " Ed + U-EF V(k,-kij) _ ., . . . . 2(EF-Ed)=V^K-kl^ V(ij,-kk) _ ... . . . = u2 d ( y '- k k) ' 2{Ed + U-EF) V(ij,-kk) _ = vhd(Vi -k k) -2Ed + U- EF
If one of the parameters vx (x = h, d, 2h, 2d, hd) satisfies the condition vx « 1 t h e n one can apply the p e r t u r b a t i o n theory to the hybridization interaction terms Hx (we note, t h a t t h e configuration representation of the Hamiltonian is most appropriate for this purpose).
In the case when Ed + U — EF » EF — Ed (or opposite case) one can neglect corresponding translation processes in the Hamiltonian. These conclusions are in accordance with estimation of the hybridization m a t r i x elements in the model of heavy fermions (see monograph [11]). T h e X—operator representation of the Anderson-like Hamiltonian is also suitable for mathematical treatment within Green function method.
3 . C a n o n i c a l t r a n s f o r m a t i o n a n d effective H a m i l t o n i a n
transformation which excludes t h e terms of t h e first order in hybridization parameters V(ik) and V(ijk, —k)
H = e^sh+sd+sih+sid) He~^sh+Sd+Sih+Sid) 5 (2)
where t h e unitary operator constituents are determined from equations
[S~h'Ho]+Hh = 0, [Si,H0]+Hi = 0, [S2-h, H0] + - [S-h, H-h] - H2-h = 0,
[S2d> H°] + 2 ft' Hd ~ H2d = °>
which exclude t h e negligible processes. In above equations a prime by t h e Poisson bracket means t h a t t h e terms \ [Sj,Hj\ having t h e same operator structure as H2, are included.
In this way t h e equation (2) u p to t h e forth order of magnitude has t h e following form (we take V{ik) to be first order of magnitude, V(ijk, —k) t h e second order, HQ of t h e zeroth order)
H H+[ShH] + \ [SK, [SK,H0]] + I [Sh, [Sh, [S~h,H]]] +
+
24 Sk> ^ . * * . 6 ft. ft. ft. *<>]]] + . . . (3)
Let us take into account t h a t spin-spin interaction between localized magnetic moments a n d indirect hopping in localized subsystem a t t r i b u t e only to terms of t h e fourth order of magnitude. Thus, we can neglect t h e processes of double creation or annihilation of electrons on t h e same site and t h e interaction of BCS-type in t h e itinerant subsystem. T h e resulting effective Hamiltonian has t h e form H — HQ + H1 + Hcd, (4) where Hn r<j2 ^3 Hcd = H0+j2'to(ij)xfx°°+"Em)*?*! ija ija ija = E ' * 0 2 ( u ) ( ^ x f - X f t x f )+ /l. c . , = E J i ( * k k ' 0 ( c + . c *tp t f + X?) + c^cu^xf + X?) -ikk' - < ck,tX ^ - c+^c^xf) + £ J2( i k k ' i ) ( c +tck,t( 4 + Xf) + ikk' + c^c^(XJ + Xf) + c + c v t * ? * + < ck,tx f ) .
In t h e above formulae to(ij), £2(2.7), £02 ( u ) are t h e integrals of indirect hopping through t h e sites with localized electrons (cation subsystem in transition metal compounds, q u a n t u m dots, etc), J i ( i k k ' i ) and J2(ikk'i) are hybridization exchange integrals.
to2(ij) = ^(to(iJ) + h(ij)) Ji(*kk'») J2(*kk'*) + V(ik)V(k'i) 2 \ e k — Ed ek' — ^ d / V(*k)V(k'») ( 1 1 + ek-Ed-U ek,-Ed-U
T h e magnitudes of these parameters can essentially renormalize the bare band hopping integral and enhance localization effects. In fig. 1 t h e mechanisms of t h e b a n d a n d hybridization hoppings are shown. Due to the substantial overlapping of the wave functions of conduction electrons one should expect t h a t not only indirect hopping renormalizes the b a n d hopping b u t also hybridization exchange has greater magnitude (of order of tA/U3) t h a n direct exchange interactions. In the case of strong correlation U » u>d (here Wd is d—band halfwidth) a n d
'ofe') t2(y)
v(i) v(Lj) v(tk) v(iic)
- o - ( ) o I | ( J h f
-4 ) t(v)
F i g u r e 1. Hybridization and band hopping pro-cesses.
n < 1 the effective Hamiltonian of localized electron subsystem has the form:
Heff = (Ed - M) E ( 4 + 4 ) + E Uij)xfx^ -J-f-Y! ( x r * r + * r * r ) • (5) i ija ijff
In distinction from s t a n d a r d t — J Hamiltonian the hopping amplitude is substantially renormalized as it represents a n indirect hopping here. In other respects the Hamiltonian (5) is very simple and allows for analytical calculations. T h e simplest analytical approach of choice is the decoupling of equation of motions, similar to the first step of work [3].
4. C o n d i t i o n of f e r r o m a g n e t i c o r d e r i n g
T h e energy spectrum obtained within projection procedure [12] in the Green function m e t h o d Ek = —fj, — iok + znaJeff allows us to calculate the mean numbers of spin-up and spin-down electrons n^ 1 - n | , 2wd I -Wd dt ££+*•. e x p ( m ) + l H Wd I dt 1 — n^ ~^TJ e x p ( ^ ) + l ' (6)
At n —> 0 the above equation reproduces the corresponding molecular field equation. T h e obtained equation has the ferromagnetic solution determined by the condition
zJeff > 2(1 -n)wd (2 - n )2 coth (1 - n)wd ( 2 - n ) 6 — coth Wd 2 6 (1 - n)wd ( 2 - n ) 6 (8) 4(l-n),
For zero t e m p e r a t u r e this yields the inequality z Jeff > (2-n)2wd which is in agreement with t h e condition of ferromagnetic ordering stabilization in polar model with strong interaction ([13]). Equalizing left and right sides of the inequality (8) we obtain the equation for Curie t e m p e r a t u r e . Let us take @c <^wd- T h e n
6 c wd
( 1 - n )
( 2 - n ) l n ^ ' (9)
where the ground state system magnetization is
m0 (2 - n )2
-4(1 -n)wd
zJ. eff (10)
If the b a n d is less t h a n half-filled, one has 6 c = zJeff/2 from eq. (9), in agreement with the above considerations. It is interesting to note t h a t in eq. (9) Curie t e m p e r a t u r e value is proportional to the conduction band width, though ferromagnetic ordering is stabilized by exchange mechanism. One can see from eq. (10) t h a t the saturation can be reached only for half-filled band. From figs. 2,3 one can see t h a t the ferromagnetic ordering is quite stable for electron
1.00 0.50 0.40 0.30 • Oc/Wd 0.20 • 0.10 • 0.00 1 i' 11• i i IIIIIIIIIII i 0.00 1 i i i lit ii 11 h i i 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.50 0.60 0.70 0.80 0.90 1.00 0 / wd
F i g u r e 2. T e m p e r a t u r e dependence of FiS^re 3 . Concentration dependence of magnetization at n = 0.9. zJeff/wd = 1 C u r i e t e m p e r a t u r e . zJeff/wd = 1 for solid for solid curve, zJeff/wd = 0.5 for dashed c u r v e' zJeff/™d = 0.75 for dashed curve, curve, zJeff/w = 0.4 for dotted curve. zJeff/™ = 0-5 for dotted curve.
R e f e r e n c e s
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