- Email: [email protected]
Magnetic anisotropy and magnetostriction of atom pairs in metallic alloys
Journal of Magnetism and Magnetic Materials 123 (1993) 169-174 North-Holland
Magnetic anisotropy and magnetostriction of atom pairs in metallic alloys K. Kulakowski Institute of Physics and Nuclear Techniques, Academy of Mining and Metallurgy, 30059 Cracow, Poland
and J. G o n z a l e z Departamento de Fisica de Materiales, Facultad de Ciencias Quimicas, Universidad del Pais Vasco, 20009 San Sebastian, Spain Received 13 May 1991
We investigate the influence of anisotropic hopping between atoms on magnetic anisotropy and magnetoelastic coupling. The partition function of atom pair is found for p-states for a small number of electrons and in the presence of saturating magnetic field. The relation is discussed of the results to the experimental data on two-ion interaction.
1. Introduction
Up to now, we have no consistent theory of amorphous metallic alloys which can predict the values of magnetic anisotropy and magnetoelastic coupling constants. Even for crystals, such a theory is not completed as yet [1]. It is useful, then, to investigate different mechanisms of anisotropy even within simplified models, to be able to discuss experimental data. The two-ions mechanism of magnetoelastic coupling was discussed up to now within localized models [2-4]. Our aim here is to investigate the effect of the electronic intersite hopping processes, which are typical for metallic systems. It is known that the indirect coupling of localized states on neighboring atoms leads to magnetic anisotropy [6]. This coupling is also known Correspondence to: K. KuIakowski, Institute of Physics and Nuclear Techniques, Academy of Mining and Metallurgy, 30059 Cracow, Poland.
to give a contribution to magnetoelastic coupling [7]. Here we consider the direct hopping mechanism as the possible source of the above effects. For the case of a degenerated band, such hopping is expected to be anisotropic as well. In fact, the initial picture is the same as the one leading to the Coqblin-Schrieffer interaction. The origin of the dependence on an angle is the anisotropy of orbitals, which are mixed and decoupled by the spin-orbit interaction. For simplicity, here we discuss only the case of a diluted substitutional alloy with cubic symmetry, where a limited number of local atomic configurations is possible. We investigate the contribution to magnetic anisotropy and magnetostriction from pairs of neighboring atoms which are distributed within the lattice. In particular, the effect of the directional ordering of pairs is of our interest. It is often very difficult to decide if a system is actually in equilibrium. The relaxation times of some processes within amorphous samples can be
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
170
K. Kutakowski, J. Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
of the order of a weak or even longer. For completeness of the discussion, we give also some arguments on the applicability of fundamental equations on the model description of relaxation processes.
2. The kinetics
The values of magnetic anisotropy K and magnetostriction A are to be calculated as the average values
K ( t , T, Tann) = ~ P i ( t , Tann)Ki(T ),
(1)
i
A(t, T, L , , ) = E P i ( t, T~,n,)Ai(T)' i
(2)
where ki (A~) are the local contributions to K (A) if an atom pair is parallel to an ith axis, T is the temperature of measurement, Z~., is the annealing temperature, t is the annealing time and Pi is the probability of the orientation of a pair along an ith axis. For the P i ' S w e have the usual equations of motion
If we adopt the Arrhenius formulae for wij, the roots A1, A 2 a r e real and negative. Actually, the distribution of relaxation times r is expected to be continuous, following the continuity of the distribution of the direction of local easy axis. The width of the distribution of 1 / r can be evaluated as the absolute value of A1 - A 2. In the notation of eq. (4), this is equal to (U l - 4U2 )W2. If the symmetry of the problem allows us to write Wij = Wji = W ~ x ,
Wik = I4~ik = Wxk ,
(9)
w~.i = wkj= wkx.
for at least one pair of directions (i, j), the second derivative in eq. (3) can be formally omitted. In such symmetric case the solution is
p,(t) -p,(~) = [p,(t)-p,(~)]
exp[-(Zwa.,.+w,.k)t ], (10)
p2(t) -p2(~) = ½[P2(0) + p , ( 0 ) - 1] e x p [ - (w.,.k + 2w,.,.)t] + l [ p , ( o * ) - p , ( 0 ) ] e x p [ - ( w ~ k + 2wkx)t ],
dpi
dt
- E (wj, p i - wijp,), j4-i
where
wxk p l ( ~ ) - 2wkx + wxk '
wk.~ P2(~) - 2wkx + w~k
(13)
dZp 1 dpl ~dt + U) dt + U2pl + U 3 = 0 ,
(4)
dPl U4p2 = d~- + U6P' + U~,
(5)
P3 = 1 - p ) - P 2 ,
(6)
for example giving (7)
and U2, U3 are of the order of w2.. The solution for p~ is of course
p i ( t ) - p i ( ~ ) = A , exp(A,t) + B , exp(A2t ).
12)
P3 = 1 - - P l - - P 2 ,
with the condition 52i p / = 1. Here, wij is the probability of a local rotation of a pair from the ith direction to the jth one. For i, j, k = 1,2,3 eqs. (3) can be rewritten as
U 1 = wi2 -1- w13 + w31 -[- w21 -{- w23 -1- w32
(11)
(3)
(8)
Note that we do not need the equality P2 = P3. However, usually the local symmetry is absent. It is possible that the order of eq. (4) could be even higher than two. In Bernal holes [8], we have four local configurations and, then, the leading equation is of third order. We expect, however, that the kinetic behaviour of such a system is similar to the one described above, with some possible modifications at early times. Moreover, the analysis of the roots of an appropriate secular equation points out that the distribution of the relaxation times remains practically unchanged [9].
t5 Kutakowski, J. Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
3. I~cai free energy Our aim is to obtain the values of K/(T), A i ( T ) for an atom pair parallel to an ith axis. The average value of K(T) will determine the macroscopic easy direction. We consider the model system of magnetic atoms diluted in the non-magnetic band system. The concentration of magnetic component is c. For small values of c we have
K(c)
= cK 1 + c2K2 + c3K3 + • • • ,
A ( C ) = C A 1 q- c 2 A 2 - F c3A3 q-- • • • ,
(14) (15)
where K , , /~n a r e the contributions to K, A from clusters of n atoms. We truncate the series, neglecting the terms for n > 2. On the other hand, the configurational average of K 1 vanishes if we do not discuss topological ordering. The Hamiltonian is chosen to be
H=Hso+Hcf+Hah+Hel+Hu-tzN,
(16)
where //so is the spin-orbit coupling, Hcf is the energy of the deformation of local environment of an atom which is due to the tensile stress, Hah is the anisotropy hopping term which depends on the orientation of an atom pair with respect to the effective magnetic field, Hel is the usual isotropic elastic term, H U is the Coulomb interaction between electrons at the same site. The last term is the chemical potential. We limit ourselves to the case of a small number of electrons (no more than one per atom). If the Coulomb interaction is large enough, the contribution of the electronic configurations with higher number of electrons is small [10] and here it will be neglected. For such a case the spin-orbit term reduces to
Hso =Ao ~(~ t .Lt,
(17)
l
i.e. the sum over sites I. Here, t~ and L, are the Pauli matrix operator and the angular momentum operator, respectively, and A 0 is the one-electron spin-orbit constant which is positive. The initial basis of electron states is chosen to
171
be the p-band, which is the simplest degenerated one with possible preservation of the isotropic symmetry of the three-dimensional space. Also, we limit ourselves to the case of strong ferromagnetism, i.e. we consider only one spin state. Then the Coulomb term in the Hamiltonian may be omitted. The spin-orbit term is substituted by its Ising part 8ZLZ. The energy spectrum is now much more simple [12] than its more complete version [11]. We believe, however, that the essential features of the model solution are preserved, except the dependence of the energy spectrum on the Stoner gap, which exceeds the frames of this paper. The anisotropic hopping term Hah is introduced here and investigated as the source of the anisotropic ion-ion interaction. Its physical interpretation is the orbital-dependent direct hopping process, which is assumed here to preserve spin. We write H a h a s Hah =
E
Tu'(a, a',
o')at+a,,ar,,,~
(18)
ll'hh' o"
which is assumed to preserve spin. Here T is a constant, A, A' count orbitals, and 1, 1' are site indices. The sum over l, l' is limited to nearest neighbours. The dependence of T on A, a' has its origin in the anisotropy of electronic clouds of orbital wave functions. The same origin leads to the indirect exchange mechanism of anisotropy, which is known as the Coqblin-Schrieffer interaction [6]. The Hamiltonian H a h is postulated here as the simplest possible term of two-ion anisotropic interaction. We assume the angular dependence of the constants T to be the same as given in [13]. For cubic structure and if magnetization is parallel to OZ, it reduces to T(A, A', ~r) = ( t + b2e)aa,a, aa,e(,_,, ),
(19)
where • is a tetragonal strain, t and b are material constants and ? ( l - l') is the pair axis. For the same tetragonal mode Hcf is diagonal and it is proportional to bl•. The constants b 1 and b 2 c a n be seen as the one-ion and two-ions magnetoelastic coupling constants.
172
K. Kutakowski, ,L Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
T h e local free energy is to be calculated as usual [10]: 1
Ft = ---In
/3
gt,
(20)
where Z~ is the local partition function and ~ is the Fermi energy. T h e latter is to be d e t e r m i n e d from the condition 1
0
n =----In /3 at*
Zef f
(21)
and it is c o m m o n for the system as a whole. T h e r e f o r e , Zet.r should include the information on the probabilities Pi:
(22)
Ze ff =Z~P ~Z,,.P~Z:p-~.
H e r e Z x is the local partition function for a pair parallel to OX. If two electrons are on a pair, the hopping is frozen by C o u l o m b interaction. If the latter is large enough, the contribution from these electronic configurations are effectively reduced to its one ion part. As we are mainly interested in the two-ions effect, we shall not include the configurations with two electrons.
In the limit of small t, K(c) tends to zero. As ~ T >~11' we see that the local easy plane is perpendicular to an a t o m pair. This agrees with what we know about pseudodipolar interaction. In the limit of high t e m p e r a t u r e s , K tends to zero. Actually it is zero at finite t e m p e r a t u r e s . This difference is the artificial consequence of our assumption of strong ferromagnetism. O u r results are therefore valid only for low t e m p e r a tures and saturating magnetic fields. At zero temperature, K is just the difference of ground state energies
K ( c ) = c 2 [ m a x ( A , t) - R ( + ) ] .
(27)
T h e limit of small A is unphysical for this model, as A contains the information on magnetism: A = A 0 - h,
(28)
where h is the orbital Z e e m a n term. H e r e we do not consider the case of A < 0. T h e magnetoelastic coefficient a is calculated
as aF = -
-
-
(29)
4. R e s u l t s
T h e local anisotropy free energy K z is calculated as the difference of local free energies of two configurations, w h e r e an a t o m pair is parallel and p e r p e n d i c u l a r to the effective magnetic field. T h e latter is assumed to saturate spins, but not the orbital m a g n e t i c moments. T h e two-ions anisotropy energy K(c) (see eq. (14)) is
c2 K(c) = ---In /3
1 +XII exp(/3/,) 1 + - v T exp(/3p.)
,
(23)
where "~'rl = 4cosh /3A + 2cosh fit,
(24)
~ v v = 2cosh / 3 R ( + ) + 2cosh / 3 R ( - ) + 2 ,
(25)
R( + ) =
in saturating magnetic field parallel to the tensile strain. H e r e we have a = na(ins),
(30)
where a(ins) is calculated for insulators. From now on, we will calculate a(ins) and we will denote it as A. Below, the contributions to a are given from two atomic configurations, i.e. for an atom pair parallel and p e r p e n d i c u l a r to the tensile strain (and to magnetization). If a pair is p e r p e n d i c u l a r to the strain (easy direction), we have (see eq. (15))
~'2 = bl)tm (1) + b 2 a r ( 2 ) ,
(31)
and if it is parallel to the strain,
[ l ( t 2 + 2A 2) + (114 + A 2 t 2 ) l / 2 ] I/2 (26)
a2 = b,all(1 ) + b2Ail(2 ) .
(32)
K. Kutakowski, J. Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
There,
Aan(r) = A l l ( r ) - - A T ( r ) ,
c o s h / J R ( + ) + cosh/JR( - ) - 2 A t ( l ) = cosh /JR(+) + c o s h / J R ( - )
+ 1' (33)
At(2) -
t (1+S 2 Z r R ~ - ~ -sinh / J R ( + ) 1-S ) + R---~_) sinh / J R ( - ) ,
(34)
(35)
sinh/Jt All(2) = 2cosh/JA + c o s h / J t '
(36)
and t 2 + 2A 2
(t 4 + 4A2t2) 1/2"
(37)
At the limit of small temperature, the cases A > t and A < t should be considered separately, at least for All. If A < t, AT(l) cancels partially with A,(1), and it changes its sign when the sample is rotated. As this is not the case of our samples, we prefer to admit that A > t. Then, Av(2) vanishes at zero temperature and it is positive for finite temperatures. On the contrary, Av (2) is negative in low temperatures. This means that the two-ion contribution to A cancels partially in disordered samples. In T = 0, however, its sign is opposite to the sign of b 2. To extract the contribution from the ordering 1 of pairs, let us rewrite PI as ~ - x , p z + p 3 as 2 g + x, where x, if positive, means the tendency of pairs to be parallel to magnetization. The contributions from one-ion and two-ions mechanisms are
a(1) = bl[ais(1) -'l-xaan(1)],
(38)
a(2) = bz[Ais(2) +xaan(2)],
(39)
where the isotropic and anisotropic parts A~s, Aan are
AiS(r) = [All(r ) + 2 A T ( r ) ] ,
(41)
and r = 1,2. If A > t, Aan(l) vanishes at T = 0. On the contrary, if A < t, Ais(1) vanishes at the same limit.
5. Discussion
sinh /J_A- cosh/Jt All(1) = 2 2cosh /JA + c o s h / J t '
S=
173
(40)
The results on magnetic anisotropy and magnetostriction, when combined with the kinetic equations, should allow us to discuss the correlations between both quantities as dependent on temperature and time. As yet, however, the dependence on magnetization is not included. Therefore we limit our discussion to a brief description of the case of low temperatures. The time dependence of anisotropy and magentostriction were shown experimentally to be correlated [14]. At early times, both quantities show variations which cannot be described simply as Aexp(-yt). Therefore we believe that our generalization of kinetic equations is justified. The possibility of temperature-induced interionic magnetoelastic coupling can be relevant for an explanation of magnetostriction in FeCoSib, FeNiSiB amorphous alloys [15-17]. There, the one-ion mechanism was found to dominate up to a certain temperature. In amorphous state, strong fluctuations of interatomic distance are possible. The effect can be particularly important in as-quenched state. We believe that our model can be reinterpreted as to describe the pairs of atoms where the distance is shorter than for other pairs. With such interpretation, the limitation of our description to the case of a small concentration of magnetic atoms is not necessary. Instead, we might expect the dependence of the two-ions magnetostriction on annealing processes, which should reduce the fluctuation of the ion-ion distance. Such a reduction of two-ions interaction was found [18] for CoMnB alloys. We should add also that the twoions magnetostriction seems to be very sensitive to inhomogeneity of a system [16,17]. The contribution from pair ordering can be substracted from experimental data as the difference between the results for longitudinal- and
174
K. Kutakowski, Z Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
transversal-field a n n e a l i n g . This difference should be c o m p a r e d with the expression for Aan(2). W e should stress, however, that w i t h o u t the d e p e n d e n c e o n the spin c o n t r i b u t i o n to m a g n e t i z a t i o n , any c o m p a r i s o n with the Callens theory is not reliable. C o n c l u d i n g , the anisotropic h o p p i n g b e t w e e n orbital states of n e i g h b o r i n g atoms is f o u n d to lead to m a g n e t i c a n i s o t r o p y of a t o m pairs a n d m a g n e t o e l a s t i c coupling. T h e g e n e r a l i z a t i o n of the kinetic e q u a t i o n s is also done. T h e results, o b t a i n e d with some severe assumptions, reflect however some features of the e x p e r i m e n t a l data.
Acknowledgements O n e of the a u t h o r s (K.K.) is grateful to Vicerr e c t o r a d o del C a m p u s del G u i p u z c o a for the kind invitation a n d hospitality.
References [1] G.H.O. Daalderop, P.J. Kelly and M.F.H. Schuurmans, Phys. Rev. B 44 (1991) 12054.
[2] H.B. Callen and E. Callen, Phys. Rev. 132 (1963) 991. [3] H. Szymczak, Phys. Stat. Sol. (b) 88 (1978) K97. [4] E. du Tremolet de Lacheisserie, J. Magn. Magn. Mater. 67 (1987) 102. [5] F. Gautier, in Magnetism of Metals and Alloys, ed. M. Cyrot (North-Holland, Amsterdam, 1982). [6] B.R. Cooper, R. Sieman, D. Yang, P. Thayamballi and A. Banerjea, in Handbook on the Physics and Chemistry of the Actinides, ed. A.J. Freeman and G.H. Lander (North-Holland, Amsterdam, 1985) vol. 2. [7] Duong hal Trieu, J. Magn. Magn. Mater. 75 (1988) 7. [8] J.D. Bernal, Proc. Roy. Soc. A 280 (1964) 299. [9] V. Smirnov, Cours de Mathematiques Superieures, tome I (Editions Mir, Moscou, 1969) p. 471. [10] G. Beni, P. Pincus and D. Hone, Phys. Rev. B 8 (1973) 3389. [11] K. Kutakowski and J. Wenda, J. Magn. Magn. Mater. 94 (1991) 247. [12] K. Kutakowski and E. du Tremolet de Lacheisserie, J. Magn. Magn. Mater. 81 (1989) 349. [13] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [14] M. Vazquez, J. Gonzalez and A. Hernando, J. Magn. Magn. Mater. 53 (1986) 323. [15] J. Gonzalez and E. du Tremolet de Lacheisserie, J. Magn. Magn. Mater. 78 (1989) 237. [16] E. du Tremolet de Lacheisserie and J. Gonzalez, J. Physique 50 (1989) 949. [17] J. Gonzalez and E. du Tremolet de Lacheisserie, Phys. Stat. Sol. (a) 115 (1989) K233. [18] J. Gonzalez and E. du Tremolett de Lacheisserie, J. Phys. Cond. Matter. 2 (1990) 6235.
Magnetic anisotropy and magnetostriction of atom pairs in metallic alloys K. Kulakowski Institute of Physics and Nuclear Techniques, Academy of Mining and Metallurgy, 30059 Cracow, Poland
and J. G o n z a l e z Departamento de Fisica de Materiales, Facultad de Ciencias Quimicas, Universidad del Pais Vasco, 20009 San Sebastian, Spain Received 13 May 1991
We investigate the influence of anisotropic hopping between atoms on magnetic anisotropy and magnetoelastic coupling. The partition function of atom pair is found for p-states for a small number of electrons and in the presence of saturating magnetic field. The relation is discussed of the results to the experimental data on two-ion interaction.
1. Introduction
Up to now, we have no consistent theory of amorphous metallic alloys which can predict the values of magnetic anisotropy and magnetoelastic coupling constants. Even for crystals, such a theory is not completed as yet [1]. It is useful, then, to investigate different mechanisms of anisotropy even within simplified models, to be able to discuss experimental data. The two-ions mechanism of magnetoelastic coupling was discussed up to now within localized models [2-4]. Our aim here is to investigate the effect of the electronic intersite hopping processes, which are typical for metallic systems. It is known that the indirect coupling of localized states on neighboring atoms leads to magnetic anisotropy [6]. This coupling is also known Correspondence to: K. KuIakowski, Institute of Physics and Nuclear Techniques, Academy of Mining and Metallurgy, 30059 Cracow, Poland.
to give a contribution to magnetoelastic coupling [7]. Here we consider the direct hopping mechanism as the possible source of the above effects. For the case of a degenerated band, such hopping is expected to be anisotropic as well. In fact, the initial picture is the same as the one leading to the Coqblin-Schrieffer interaction. The origin of the dependence on an angle is the anisotropy of orbitals, which are mixed and decoupled by the spin-orbit interaction. For simplicity, here we discuss only the case of a diluted substitutional alloy with cubic symmetry, where a limited number of local atomic configurations is possible. We investigate the contribution to magnetic anisotropy and magnetostriction from pairs of neighboring atoms which are distributed within the lattice. In particular, the effect of the directional ordering of pairs is of our interest. It is often very difficult to decide if a system is actually in equilibrium. The relaxation times of some processes within amorphous samples can be
0304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
170
K. Kutakowski, J. Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
of the order of a weak or even longer. For completeness of the discussion, we give also some arguments on the applicability of fundamental equations on the model description of relaxation processes.
2. The kinetics
The values of magnetic anisotropy K and magnetostriction A are to be calculated as the average values
K ( t , T, Tann) = ~ P i ( t , Tann)Ki(T ),
(1)
i
A(t, T, L , , ) = E P i ( t, T~,n,)Ai(T)' i
(2)
where ki (A~) are the local contributions to K (A) if an atom pair is parallel to an ith axis, T is the temperature of measurement, Z~., is the annealing temperature, t is the annealing time and Pi is the probability of the orientation of a pair along an ith axis. For the P i ' S w e have the usual equations of motion
If we adopt the Arrhenius formulae for wij, the roots A1, A 2 a r e real and negative. Actually, the distribution of relaxation times r is expected to be continuous, following the continuity of the distribution of the direction of local easy axis. The width of the distribution of 1 / r can be evaluated as the absolute value of A1 - A 2. In the notation of eq. (4), this is equal to (U l - 4U2 )W2. If the symmetry of the problem allows us to write Wij = Wji = W ~ x ,
Wik = I4~ik = Wxk ,
(9)
w~.i = wkj= wkx.
for at least one pair of directions (i, j), the second derivative in eq. (3) can be formally omitted. In such symmetric case the solution is
p,(t) -p,(~) = [p,(t)-p,(~)]
exp[-(Zwa.,.+w,.k)t ], (10)
p2(t) -p2(~) = ½[P2(0) + p , ( 0 ) - 1] e x p [ - (w.,.k + 2w,.,.)t] + l [ p , ( o * ) - p , ( 0 ) ] e x p [ - ( w ~ k + 2wkx)t ],
dpi
dt
- E (wj, p i - wijp,), j4-i
where
wxk p l ( ~ ) - 2wkx + wxk '
wk.~ P2(~) - 2wkx + w~k
(13)
dZp 1 dpl ~dt + U) dt + U2pl + U 3 = 0 ,
(4)
dPl U4p2 = d~- + U6P' + U~,
(5)
P3 = 1 - p ) - P 2 ,
(6)
for example giving (7)
and U2, U3 are of the order of w2.. The solution for p~ is of course
p i ( t ) - p i ( ~ ) = A , exp(A,t) + B , exp(A2t ).
12)
P3 = 1 - - P l - - P 2 ,
with the condition 52i p / = 1. Here, wij is the probability of a local rotation of a pair from the ith direction to the jth one. For i, j, k = 1,2,3 eqs. (3) can be rewritten as
U 1 = wi2 -1- w13 + w31 -[- w21 -{- w23 -1- w32
(11)
(3)
(8)
Note that we do not need the equality P2 = P3. However, usually the local symmetry is absent. It is possible that the order of eq. (4) could be even higher than two. In Bernal holes [8], we have four local configurations and, then, the leading equation is of third order. We expect, however, that the kinetic behaviour of such a system is similar to the one described above, with some possible modifications at early times. Moreover, the analysis of the roots of an appropriate secular equation points out that the distribution of the relaxation times remains practically unchanged [9].
t5 Kutakowski, J. Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
3. I~cai free energy Our aim is to obtain the values of K/(T), A i ( T ) for an atom pair parallel to an ith axis. The average value of K(T) will determine the macroscopic easy direction. We consider the model system of magnetic atoms diluted in the non-magnetic band system. The concentration of magnetic component is c. For small values of c we have
K(c)
= cK 1 + c2K2 + c3K3 + • • • ,
A ( C ) = C A 1 q- c 2 A 2 - F c3A3 q-- • • • ,
(14) (15)
where K , , /~n a r e the contributions to K, A from clusters of n atoms. We truncate the series, neglecting the terms for n > 2. On the other hand, the configurational average of K 1 vanishes if we do not discuss topological ordering. The Hamiltonian is chosen to be
H=Hso+Hcf+Hah+Hel+Hu-tzN,
(16)
where //so is the spin-orbit coupling, Hcf is the energy of the deformation of local environment of an atom which is due to the tensile stress, Hah is the anisotropy hopping term which depends on the orientation of an atom pair with respect to the effective magnetic field, Hel is the usual isotropic elastic term, H U is the Coulomb interaction between electrons at the same site. The last term is the chemical potential. We limit ourselves to the case of a small number of electrons (no more than one per atom). If the Coulomb interaction is large enough, the contribution of the electronic configurations with higher number of electrons is small [10] and here it will be neglected. For such a case the spin-orbit term reduces to
Hso =Ao ~(~ t .Lt,
(17)
l
i.e. the sum over sites I. Here, t~ and L, are the Pauli matrix operator and the angular momentum operator, respectively, and A 0 is the one-electron spin-orbit constant which is positive. The initial basis of electron states is chosen to
171
be the p-band, which is the simplest degenerated one with possible preservation of the isotropic symmetry of the three-dimensional space. Also, we limit ourselves to the case of strong ferromagnetism, i.e. we consider only one spin state. Then the Coulomb term in the Hamiltonian may be omitted. The spin-orbit term is substituted by its Ising part 8ZLZ. The energy spectrum is now much more simple [12] than its more complete version [11]. We believe, however, that the essential features of the model solution are preserved, except the dependence of the energy spectrum on the Stoner gap, which exceeds the frames of this paper. The anisotropic hopping term Hah is introduced here and investigated as the source of the anisotropic ion-ion interaction. Its physical interpretation is the orbital-dependent direct hopping process, which is assumed here to preserve spin. We write H a h a s Hah =
E
Tu'(a, a',
o')at+a,,ar,,,~
(18)
ll'hh' o"
which is assumed to preserve spin. Here T is a constant, A, A' count orbitals, and 1, 1' are site indices. The sum over l, l' is limited to nearest neighbours. The dependence of T on A, a' has its origin in the anisotropy of electronic clouds of orbital wave functions. The same origin leads to the indirect exchange mechanism of anisotropy, which is known as the Coqblin-Schrieffer interaction [6]. The Hamiltonian H a h is postulated here as the simplest possible term of two-ion anisotropic interaction. We assume the angular dependence of the constants T to be the same as given in [13]. For cubic structure and if magnetization is parallel to OZ, it reduces to T(A, A', ~r) = ( t + b2e)aa,a, aa,e(,_,, ),
(19)
where • is a tetragonal strain, t and b are material constants and ? ( l - l') is the pair axis. For the same tetragonal mode Hcf is diagonal and it is proportional to bl•. The constants b 1 and b 2 c a n be seen as the one-ion and two-ions magnetoelastic coupling constants.
172
K. Kutakowski, ,L Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
T h e local free energy is to be calculated as usual [10]: 1
Ft = ---In
/3
gt,
(20)
where Z~ is the local partition function and ~ is the Fermi energy. T h e latter is to be d e t e r m i n e d from the condition 1
0
n =----In /3 at*
Zef f
(21)
and it is c o m m o n for the system as a whole. T h e r e f o r e , Zet.r should include the information on the probabilities Pi:
(22)
Ze ff =Z~P ~Z,,.P~Z:p-~.
H e r e Z x is the local partition function for a pair parallel to OX. If two electrons are on a pair, the hopping is frozen by C o u l o m b interaction. If the latter is large enough, the contribution from these electronic configurations are effectively reduced to its one ion part. As we are mainly interested in the two-ions effect, we shall not include the configurations with two electrons.
In the limit of small t, K(c) tends to zero. As ~ T >~11' we see that the local easy plane is perpendicular to an a t o m pair. This agrees with what we know about pseudodipolar interaction. In the limit of high t e m p e r a t u r e s , K tends to zero. Actually it is zero at finite t e m p e r a t u r e s . This difference is the artificial consequence of our assumption of strong ferromagnetism. O u r results are therefore valid only for low t e m p e r a tures and saturating magnetic fields. At zero temperature, K is just the difference of ground state energies
K ( c ) = c 2 [ m a x ( A , t) - R ( + ) ] .
(27)
T h e limit of small A is unphysical for this model, as A contains the information on magnetism: A = A 0 - h,
(28)
where h is the orbital Z e e m a n term. H e r e we do not consider the case of A < 0. T h e magnetoelastic coefficient a is calculated
as aF = -
-
-
(29)
4. R e s u l t s
T h e local anisotropy free energy K z is calculated as the difference of local free energies of two configurations, w h e r e an a t o m pair is parallel and p e r p e n d i c u l a r to the effective magnetic field. T h e latter is assumed to saturate spins, but not the orbital m a g n e t i c moments. T h e two-ions anisotropy energy K(c) (see eq. (14)) is
c2 K(c) = ---In /3
1 +XII exp(/3/,) 1 + - v T exp(/3p.)
,
(23)
where "~'rl = 4cosh /3A + 2cosh fit,
(24)
~ v v = 2cosh / 3 R ( + ) + 2cosh / 3 R ( - ) + 2 ,
(25)
R( + ) =
in saturating magnetic field parallel to the tensile strain. H e r e we have a = na(ins),
(30)
where a(ins) is calculated for insulators. From now on, we will calculate a(ins) and we will denote it as A. Below, the contributions to a are given from two atomic configurations, i.e. for an atom pair parallel and p e r p e n d i c u l a r to the tensile strain (and to magnetization). If a pair is p e r p e n d i c u l a r to the strain (easy direction), we have (see eq. (15))
~'2 = bl)tm (1) + b 2 a r ( 2 ) ,
(31)
and if it is parallel to the strain,
[ l ( t 2 + 2A 2) + (114 + A 2 t 2 ) l / 2 ] I/2 (26)
a2 = b,all(1 ) + b2Ail(2 ) .
(32)
K. Kutakowski, J. Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
There,
Aan(r) = A l l ( r ) - - A T ( r ) ,
c o s h / J R ( + ) + cosh/JR( - ) - 2 A t ( l ) = cosh /JR(+) + c o s h / J R ( - )
+ 1' (33)
At(2) -
t (1+S 2 Z r R ~ - ~ -sinh / J R ( + ) 1-S ) + R---~_) sinh / J R ( - ) ,
(34)
(35)
sinh/Jt All(2) = 2cosh/JA + c o s h / J t '
(36)
and t 2 + 2A 2
(t 4 + 4A2t2) 1/2"
(37)
At the limit of small temperature, the cases A > t and A < t should be considered separately, at least for All. If A < t, AT(l) cancels partially with A,(1), and it changes its sign when the sample is rotated. As this is not the case of our samples, we prefer to admit that A > t. Then, Av(2) vanishes at zero temperature and it is positive for finite temperatures. On the contrary, Av (2) is negative in low temperatures. This means that the two-ion contribution to A cancels partially in disordered samples. In T = 0, however, its sign is opposite to the sign of b 2. To extract the contribution from the ordering 1 of pairs, let us rewrite PI as ~ - x , p z + p 3 as 2 g + x, where x, if positive, means the tendency of pairs to be parallel to magnetization. The contributions from one-ion and two-ions mechanisms are
a(1) = bl[ais(1) -'l-xaan(1)],
(38)
a(2) = bz[Ais(2) +xaan(2)],
(39)
where the isotropic and anisotropic parts A~s, Aan are
AiS(r) = [All(r ) + 2 A T ( r ) ] ,
(41)
and r = 1,2. If A > t, Aan(l) vanishes at T = 0. On the contrary, if A < t, Ais(1) vanishes at the same limit.
5. Discussion
sinh /J_A- cosh/Jt All(1) = 2 2cosh /JA + c o s h / J t '
S=
173
(40)
The results on magnetic anisotropy and magnetostriction, when combined with the kinetic equations, should allow us to discuss the correlations between both quantities as dependent on temperature and time. As yet, however, the dependence on magnetization is not included. Therefore we limit our discussion to a brief description of the case of low temperatures. The time dependence of anisotropy and magentostriction were shown experimentally to be correlated [14]. At early times, both quantities show variations which cannot be described simply as Aexp(-yt). Therefore we believe that our generalization of kinetic equations is justified. The possibility of temperature-induced interionic magnetoelastic coupling can be relevant for an explanation of magnetostriction in FeCoSib, FeNiSiB amorphous alloys [15-17]. There, the one-ion mechanism was found to dominate up to a certain temperature. In amorphous state, strong fluctuations of interatomic distance are possible. The effect can be particularly important in as-quenched state. We believe that our model can be reinterpreted as to describe the pairs of atoms where the distance is shorter than for other pairs. With such interpretation, the limitation of our description to the case of a small concentration of magnetic atoms is not necessary. Instead, we might expect the dependence of the two-ions magnetostriction on annealing processes, which should reduce the fluctuation of the ion-ion distance. Such a reduction of two-ions interaction was found [18] for CoMnB alloys. We should add also that the twoions magnetostriction seems to be very sensitive to inhomogeneity of a system [16,17]. The contribution from pair ordering can be substracted from experimental data as the difference between the results for longitudinal- and
174
K. Kutakowski, Z Gonzalez / Magnetic anisotropy and magnetorestriction of atom pairs
transversal-field a n n e a l i n g . This difference should be c o m p a r e d with the expression for Aan(2). W e should stress, however, that w i t h o u t the d e p e n d e n c e o n the spin c o n t r i b u t i o n to m a g n e t i z a t i o n , any c o m p a r i s o n with the Callens theory is not reliable. C o n c l u d i n g , the anisotropic h o p p i n g b e t w e e n orbital states of n e i g h b o r i n g atoms is f o u n d to lead to m a g n e t i c a n i s o t r o p y of a t o m pairs a n d m a g n e t o e l a s t i c coupling. T h e g e n e r a l i z a t i o n of the kinetic e q u a t i o n s is also done. T h e results, o b t a i n e d with some severe assumptions, reflect however some features of the e x p e r i m e n t a l data.
Acknowledgements O n e of the a u t h o r s (K.K.) is grateful to Vicerr e c t o r a d o del C a m p u s del G u i p u z c o a for the kind invitation a n d hospitality.
References [1] G.H.O. Daalderop, P.J. Kelly and M.F.H. Schuurmans, Phys. Rev. B 44 (1991) 12054.
[2] H.B. Callen and E. Callen, Phys. Rev. 132 (1963) 991. [3] H. Szymczak, Phys. Stat. Sol. (b) 88 (1978) K97. [4] E. du Tremolet de Lacheisserie, J. Magn. Magn. Mater. 67 (1987) 102. [5] F. Gautier, in Magnetism of Metals and Alloys, ed. M. Cyrot (North-Holland, Amsterdam, 1982). [6] B.R. Cooper, R. Sieman, D. Yang, P. Thayamballi and A. Banerjea, in Handbook on the Physics and Chemistry of the Actinides, ed. A.J. Freeman and G.H. Lander (North-Holland, Amsterdam, 1985) vol. 2. [7] Duong hal Trieu, J. Magn. Magn. Mater. 75 (1988) 7. [8] J.D. Bernal, Proc. Roy. Soc. A 280 (1964) 299. [9] V. Smirnov, Cours de Mathematiques Superieures, tome I (Editions Mir, Moscou, 1969) p. 471. [10] G. Beni, P. Pincus and D. Hone, Phys. Rev. B 8 (1973) 3389. [11] K. Kutakowski and J. Wenda, J. Magn. Magn. Mater. 94 (1991) 247. [12] K. Kutakowski and E. du Tremolet de Lacheisserie, J. Magn. Magn. Mater. 81 (1989) 349. [13] J.C. Slater and G.F. Koster, Phys. Rev. 94 (1954) 1498. [14] M. Vazquez, J. Gonzalez and A. Hernando, J. Magn. Magn. Mater. 53 (1986) 323. [15] J. Gonzalez and E. du Tremolet de Lacheisserie, J. Magn. Magn. Mater. 78 (1989) 237. [16] E. du Tremolet de Lacheisserie and J. Gonzalez, J. Physique 50 (1989) 949. [17] J. Gonzalez and E. du Tremolet de Lacheisserie, Phys. Stat. Sol. (a) 115 (1989) K233. [18] J. Gonzalez and E. du Tremolett de Lacheisserie, J. Phys. Cond. Matter. 2 (1990) 6235.