Influence of Structure Fluctuations on Ferromagnetism in the Band Model

Short Notes K125 phys. stat. sol. (b) 74,K125 (1976) Subject classification 2 and 18.2 Sektion Physik d e r Technischen Universitat Dresden Influence of Structure Fluctuations on Ferromagnetism in the Band Model BY J . RICHTER, J. SCHREIBER, and K. HANDRICH The study of amorphous magnetism in itinerant electron models is necessary chiefly for transition metals (1).In many amorphous systems the width of the nearest neighbour maximum in the radial distribution function is much l a r g e r than the change of the averaged nearest neighbour separation in comparison to the c r y s - tal. Hence the fluctuations of the distances of atoms a r e very important and we have t o include off-diagonal randomness. We want to discuss the influence of structure fluctuations on ferromagnetism (FM) within the lattice model starting with an ide- alized density of states (DS) f o r crystalline Ni (cf. (2) and see Fig. l a ) . In analogy t o (3) we s t a r t with a Hartree-Fock approximation of Hubbard’s Ha- miltonian f o r a single tight-binding band and use the extended Lloyd model (4)f o r the treatment of the off-diagonal disorder problem. Thereby we apply the same notations a s in ( 3 ) . In the extended Lloyd model the hopping integrals t.. fluctuate 1J according t o a Lorentzian distribution with the width A and the mean value to (3). In difference to (3) we include also fluctuations of the U. t e r m in the spin dependent 1 site energy Eid = Ei + Uin-d. We choose E i u a s a linear function of surrounding r 1 In this case the exact structure averaged Green’s function<(;> is obtained by re- placing t . .+ t - i sign(A + Bn The DS is given by formula (5) 1J 0 -U A in Gcrystal. in (3) i f we substitute U-+U A - + A +Bn . In further discussions we choose t < 0 0’ -d 0 and distinguish between three cases: case I , positive coupling (A, B > 1); c a s e 11, only diagonal fluctuations (A + 0 , I MI,]BAI= const); case 111, negative coupling (A, B < - 1). K126 physica status solidi (b) 74 Fig. 1. Density of states of an idealized Ni system f o r various fluctuation parameters A and 1AA1 ( A in units of It I ). F o r all curves is i B A l = 0 (no U; fluctuations). 0 a ) A =O(correspondingtocrystal):b)caseI,A = 0.03, A = 13/12; c) case I, A = 0 . 0 6 , A =13/12; d) case 11, A+O, IAAI =0.03.13/12; e) case II,A+O, IAal= =0.06.13/12; f ) case II1,A = O . O $ , A = -13/12; g) case 111, A =0.06, A = -13/12 A qualitative insight in the effect of structure fluctu- ations on the DS yields the discussion of its first moments. We assume M = <c.> = e = 0. The second moment (con- 1 1 0 tributions ( e . 2>, < t2.>) is increased by fluctuations. The 1 '3 3 2 third moment (contributions<E.>, < E t >, < t t t > ) 1 i ij i j jk ki i s in general different from zero, since with certainty an E. - t . . correlation (second t e r m ) and a finite number of 1 1J threefold links (third term) in amorphous systems a r e realized; the DS then will be non-symmetrical. Furthermore one can argue that the lattice model for amor- phous solids yields an appropriate description a t least of the first moments of the DS, if the amorphous state has the same short-range o r d e r a s the crystal. In Fig. 1 the influence of structure,fluctuations on the DS e ( 0 )i s plotted. We note the strong influence of the E. - t.. coupling on the peak and on the tails. The 1 4 arrows indicate the position of the F e r m i level for the number of electrons n = 1.88. The diagonal matrix element c contains fluctuations in U., which we can divide id 1 n m n into two parts: U i z , -dU.- , where m is the magnetization. The t e r m U.- is fully 12 12 analogous to usual G. fluctuations. The second t e r m causes a modification of formulae 1 for the Curie temperature T and the criterion f o r the existence of ferromagnetic C solutions, which now is written a s The index p denotes the paramagnetic s t a t e , p is the chemical potential, z the number of nearest neighbours, and eo the DS f o r the corresponding crystal. The quantity R i s plotted in Fig. 2 for a simple elliptic e 0 (cf. (3)). The sign of R Short Notes K127 Fig. 2 . Difference between L. and U. fluctuations 1 1 characterized by the t e r m R from formula (2) for a simple elliptic Q and the parameter A = 0 . The 0 dotted line represents case I , the solid line case 11, the dashed line case 111 i s essentially dependent on n and the magnitude on -0.1 IB4 0 0.05 0.70 0.75 In Fig. 3 and 4 results for the idealized DS of Ni (Fig. l a ) a r e shown (parameters n = 1.88 and U /It 0 0 I = 15 from reference (2)). In agreement with Fig. 2 the E. 1 fluctuations a r e more restrictive to ferromagnetism (FM). F o r case I1 and case 111 we find a mono- tonic weakening of F M by fluctuations. Both c a s e s a r e much more unfavouiable f o r F M than case I (cf. peaks in Fig. 1). F o r positive coupling (case I) there is a minimum in U i . e . ferromagnetic solutions a r e stabilized in this region ocritical' (Fig. 3). The reason is a shifting of the F e r m i level t o the top of the peak (see Fig. 1). The occurrence of the minimum i s dependent on the slope of the peak. F o r u02min(Uocritical ) FM does not exist in the corresponding crystal ( A = 0 ) but in the disordered case in a finite region of fluctuations around the minimum ferro- magnetic solutions occur (cf. (5)). In Fig. 4 the situation for a g r e a t e r U is represented. Here also for c a s e I T 0 C and m(T = 0 ) a r e monotonousiyweakened by fluctuations. Important in Fig. 4 is the relation between the lowering of T and m(T = 0 ) . Experiments show that in N i sys- C terns m(T =0) is more decreased than T in the amorphous phase in comparison to C the crystalline phase (1).This tendency is reproduced for positive coupling (case I ) . Only diagonal fluctuations (case 11) and negative coupling (case 111), which a r e only indicated by dotted lines in Fig. 4 , fail and yield the inverse tendency. The character- istic flattening of the m(T/T )/m(O) curve is not obtained within our model (cf. (5)). C This is not surprising, since the Stoner model is not appropriate for description of the temperature dependence of m . Summarizing we can say: The investigation of the influence of structure fluctu- ations on ferromagnetism (FM) starting with a density of states e0 with a sharp peak n e a r the upper band edge is more reasonable than calculations with a simple elliptic K128 physica status solidi (b) 74 I I I I 0 005 0.70 Fig. 3 Fig. 4 F i g . 3 . Criterion for the existence of ferromagnetic solutions f o r electron number n n = 1 . 8 8 . Case I, A = 13/12 respectively B- = 13/12. Case 11,A---* 0. n Case 111, A = -13/12 respectively B- = -13,?12, ---A = 0 , B # 0 (Ui fluc- 2 tuations), - A f 0 , B = 0 ( E ~ fluctuations) Fig. 4. Magnetization m for T = 0 K and Curie temperature T for the parameters n I U /It = 1 5 . 0 and n = 1 . 8 8 . Case I , A = 13/12 respectively B- = 13/12. C:seoII,A+O. Case 111, A = -13/12,---A = 0 , B 0 0 (U. huctuations), 1 - A # O , B = O ( E . fluctuations), 1 mforB=O,A+O eo in ( 3 ) . Therefore we get some new aspects a s the small stabilizing of F M f o r a certain type and magnitude of fluctuations. F u r t h e r m o r e the effect of the off-diagonal randomness and especially the E .-t.. coupling a r e m o r e important in comparison to 1 1J ( 3 ) . The difference in the decreasing of T and m(T = 0) in amorphous Ni systems C we can qualitatively reproduce within o u r model calculations. The authors a r e indebted to P r o f . K . Elk and D r . W . Loser for helpful discussions. References (1) J . G. WRIGHT, Amorphous Transition Metal F i l m s , 7th Internat. Coll. Magnetic Thin F i l m s , Regensburg, April 1975. - (2) H . HASEGAWA and J . KANAMORI, J . Phys. SOC.Japan 3 1 , 382 (1971). (3) J. RICHTER, K . HANDRICH, and J . SCHREIBER, phys. s t a t . sol. (b) 6 3 K61 (1975). (4) W. JOHN and J . SCHREIBER, phys. stat. sol. (b) 3, 1 9 3 (1974). (5) J . RICHTER, J . SCHREIBER, and K. HANDRICH, Preprint 05-27-76 and 05-28-76, TU Dresden, 1976. (Received February 3 , 1976)