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Amorphization of Heisenberg magnets
AMORPHIZATION OF HEISENBERG MAGNETS E. V. K U Z M I N , G. A. P E T R A K O V S K I I , S. S. A P L E S N I N Institute of Physics USSR, Academy of Sciences Siberian Branch, 660036 Krasnoyarsk, USSR
The a m o r p h o u s f e r r o m a g n e t is c o n s i d e r e d as a H e i s e n b e r g m o d e l w i t h s t o c h a s t i c a l l y d i s t r i b u t i o n e x c h a n g e p a r a m e t e r s J > 0, K < 0. M a g n e t i z a t i o n as f u n c t i o n of ?, = K / J a n d K - b o n d c o n c e n t r a t i o n is found. The r a n g e of a m o r p h o u s f e r r o m a g n e t existence, density of m a g n o n s t a t e s a n d Curie t e m p e r a t u r e are d e t e r m i n e d b y c o h e r e n t p o t e n t i a l a p p r o x i m a t i o n .
There are some experimental data [1, 2] on a nontrivial amorphization effect of nonmetal magnets: it is possible for a magnet after amorphization to change the type of magnetic order and to increase the magnetic phase transition temperature T¢. Similar effects arise in the case of amorphization of magnetic systems with structurally sensible exchange interactions. In this paper we develop the theory of such phenomena. The magnetic properties of the nonconducting magnets are described by the spin-Hamiltonian
(s =½): =
-
E
fm
AI, ~ =
Aml~ Aft=
O,
(1) where AI,,, are the exchange parameters, which are unequal to zero only for z nearest neighbours. In the general case the value as well as the sign of As,. fluctuate. For the ideal crystal the parameters As., Ao(f-m) m a y have the anisotropy distribution. For example, Ao(hl)=--J describes the exchange within the plane and Ao(hz) ~ K the interplane exchange. The ground state of the crystal is the ferromagnet (FM) for J > 0, K > 0 and the antiferromagnet (AFM) for other cases. The magnetic phase transition temperature T ° = T°(k), so that T ° ( h ) ~ 0, when 2~~ 0. Thus, the amorphization of the quasi low-dimensional crystals (P,I << 1) is characterized by some peculiarities. In this work the amorphization is performed only with respect to a statistical distribution of T and K exchange bonds. For the description of the amorphous magnet we will postulate that amorphization of the crystal leads to the isotropy of the material macroscopic properties and that in the average there is a short range order in the amorphous magnet as well as in the initial crystal. The ground state of the system with the spin-Hamiltonian (I) is a F M for Aj,, > 0
and an A F M for AS., < 0 (for the alternate lattices). When the value and sign of Aim are fluctuated the problem of determination of the ground state becomes complicated (the frustration problem). Let us consider the case when the exchange bonds fluctuate stochastically with the distribution function
K) +
p(A) = nKS(A --
(I - nK)8(A -- J),
(2) where n K and 1 - n K are the concentration of K and J-bonds, respectively. We can state that the ground state of the system is characterized by the FM long range order with the reduced magnetizacr (2~, z) [3], here n~r is tion 5(0) < omax for n K < n K the concentration for 5(0) = 0. We have determined 5(0)------~L-l~/of = 5(2,, nK, z) with 0I = _+ 1 by means of the MonteC a r l o m i n i m i z a t i o n of the e n e r g y W = - ¼Y.1=A~oiom with the distribution function (2). For the calculation we used the square (20 × 20) and simple cubic (10 × 10 × 10) lattices with the periodical boundary conditions. The results of these calculations are shown in fig. 1.
_~ 06
5 Z
3
~
b
06
04
04
_ 02
O~
, 06
08
_ rl~
OZ
0';
06
Og
~
I/K
Fig. 1. (a) The r e d u c e d m a g n e t i z a t i o n 0(0) at the various conc e n t r a t i o n s nx of the a n t i f e r r o b o n d K for the s q u a r e lattice with p a r a m e t e r s ~ = K / J : (1) = 0, (2) = - 0.25, (3) = - 0.5, (4) = - !, (5) = - 5; Co) T h e s a m e as (a) for the c u b i c lattice with p a r a m e t e r s ),: (1) = 0, (2) = - 1, (3) = - 10.
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1347-1348 ©North Holland
1347
1348
E. V. K u z m i n et a l . / Amorphization o f Heisenberg magnets
We used the coherent potential approximation (CPA) method for the calculation of the main characteristics of amorphous FM [4, 5]. This method is based on the approximation of the random material of some effective translational symmetry medium in which the exchange bonds are represented by the self-consistent coherent exchange parameter Ac( f rn, E ) which is a function of the excitation energy Green function, in this approximation: ((S;
Is.;
-
))E
k
6
E-
,
6zAc(E)e °
o ek
1-
=
7k, (3)
where parameter A c is the same for all z nearest neighbours and 8 = 8(T). Parameter Ac is determined from the equation ( T } = 0 where T is the scattering matrix connected with the fluctuations ~I,,, = A / m -- A ~ ( f - m), ( • • • ) is the configuration average. If there are no correlations of the exchange interaction fluctuations, then T X , t , , where t~ is the partial scattering matrix for any pair a of the nearest neighbours. In this case the equation ( T ) = 0 is equivalent to ( t ~ ) = 0. The t,-matrix is exactly calculated. After the configuration averaging of t, with function (2) we have found for parameter x = A J J:
q~(x, o~)
=
/
,.,r,;
.
~
//
/
,
CPA
1 • = - ~ E eXk'O~-")
x =
/ ( &/" 2
+
(1
× [1
-
-
,,,,)n,,(1
(2/z)(l/.~L)
Fig. 2. The existence range of amorphous FM the various concentration n x of the K-bond parameter X. The coherent parameter Im x(0) between opt and ~2 and lm x ( 0 ) = 0 in the under line 9)2.
spin-wave states corresponding to t ° is approximated by the half ellipsis. The density of spin-wave states f o r a m o r p h o u s FM gam(o~, 2~) = S P ( - ~ r - 1 I m ( ( S I + S-\\CPA.,//,o+0J~ has a complex behaviour and it strongly varies depending on value 2~. For the region of small ~0 and when ?~ < 0, p,l<
-
-
E t ° / (~o - xe°),
(4)
where ~ - - 1 - n K +~knx; c = nK + ~ ( 1 - nx), o~ = E / f z J . The dispersion law C~m for spin-waves in the effective medium is determined by the equation o~ - t ° Re x(o~) = 0.
(line drawing) at and the various ~ 0 in the range range which lies
(6)
and the main result is the fact that Tam(0) 4: 0. Thus, we have shown that the results of the amorphization of the quasi low-dimensional magnets (l?q< Zc0 and for the case )~ < 0 the transition A F M ~ amorphous F M takes place.
(5) References
The existence region of amorphous F M (fig. 2) is determined by the condition RE x(0) > 0. F r o m eqs. (4) and (5) for small ~o (and when RE x(0) > 0) it is obvious that C/~m CC k 2, and that the damping of spin-waves is oc k 5. The total solution for x(to) has been found by the numerical calculation with z = 6 and nK = ½. The density of
[I] F. J. Litterst, J. de Phys. 36 (1975) LI97. [2] C. A. Sablina, G. A. Petrakovskii, E. N. Agartanova and V. P. Piskorskii, JETF Lett. 24 (1976) 357. [3] E. V. Kuzmin, G. A. Petrakovskii, JETF 75 (1978) 265. [4] B. Velicky, S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1978) 747. [5] R. Kloss, Intern. J. Magnetism 5 (1972) 251.
The a m o r p h o u s f e r r o m a g n e t is c o n s i d e r e d as a H e i s e n b e r g m o d e l w i t h s t o c h a s t i c a l l y d i s t r i b u t i o n e x c h a n g e p a r a m e t e r s J > 0, K < 0. M a g n e t i z a t i o n as f u n c t i o n of ?, = K / J a n d K - b o n d c o n c e n t r a t i o n is found. The r a n g e of a m o r p h o u s f e r r o m a g n e t existence, density of m a g n o n s t a t e s a n d Curie t e m p e r a t u r e are d e t e r m i n e d b y c o h e r e n t p o t e n t i a l a p p r o x i m a t i o n .
There are some experimental data [1, 2] on a nontrivial amorphization effect of nonmetal magnets: it is possible for a magnet after amorphization to change the type of magnetic order and to increase the magnetic phase transition temperature T¢. Similar effects arise in the case of amorphization of magnetic systems with structurally sensible exchange interactions. In this paper we develop the theory of such phenomena. The magnetic properties of the nonconducting magnets are described by the spin-Hamiltonian
(s =½): =
-
E
fm
AI, ~ =
Aml~ Aft=
O,
(1) where AI,,, are the exchange parameters, which are unequal to zero only for z nearest neighbours. In the general case the value as well as the sign of As,. fluctuate. For the ideal crystal the parameters As., Ao(f-m) m a y have the anisotropy distribution. For example, Ao(hl)=--J describes the exchange within the plane and Ao(hz) ~ K the interplane exchange. The ground state of the crystal is the ferromagnet (FM) for J > 0, K > 0 and the antiferromagnet (AFM) for other cases. The magnetic phase transition temperature T ° = T°(k), so that T ° ( h ) ~ 0, when 2~~ 0. Thus, the amorphization of the quasi low-dimensional crystals (P,I << 1) is characterized by some peculiarities. In this work the amorphization is performed only with respect to a statistical distribution of T and K exchange bonds. For the description of the amorphous magnet we will postulate that amorphization of the crystal leads to the isotropy of the material macroscopic properties and that in the average there is a short range order in the amorphous magnet as well as in the initial crystal. The ground state of the system with the spin-Hamiltonian (I) is a F M for Aj,, > 0
and an A F M for AS., < 0 (for the alternate lattices). When the value and sign of Aim are fluctuated the problem of determination of the ground state becomes complicated (the frustration problem). Let us consider the case when the exchange bonds fluctuate stochastically with the distribution function
K) +
p(A) = nKS(A --
(I - nK)8(A -- J),
(2) where n K and 1 - n K are the concentration of K and J-bonds, respectively. We can state that the ground state of the system is characterized by the FM long range order with the reduced magnetizacr (2~, z) [3], here n~r is tion 5(0) < omax for n K < n K the concentration for 5(0) = 0. We have determined 5(0)------~L-l~/of = 5(2,, nK, z) with 0I = _+ 1 by means of the MonteC a r l o m i n i m i z a t i o n of the e n e r g y W = - ¼Y.1=A~oiom with the distribution function (2). For the calculation we used the square (20 × 20) and simple cubic (10 × 10 × 10) lattices with the periodical boundary conditions. The results of these calculations are shown in fig. 1.
_~ 06
5 Z
3
~
b
06
04
04
_ 02
O~
, 06
08
_ rl~
OZ
0';
06
Og
~
I/K
Fig. 1. (a) The r e d u c e d m a g n e t i z a t i o n 0(0) at the various conc e n t r a t i o n s nx of the a n t i f e r r o b o n d K for the s q u a r e lattice with p a r a m e t e r s ~ = K / J : (1) = 0, (2) = - 0.25, (3) = - 0.5, (4) = - !, (5) = - 5; Co) T h e s a m e as (a) for the c u b i c lattice with p a r a m e t e r s ),: (1) = 0, (2) = - 1, (3) = - 10.
Journal of Magnetism and Magnetic Materials 15-18 (1980) 1347-1348 ©North Holland
1347
1348
E. V. K u z m i n et a l . / Amorphization o f Heisenberg magnets
We used the coherent potential approximation (CPA) method for the calculation of the main characteristics of amorphous FM [4, 5]. This method is based on the approximation of the random material of some effective translational symmetry medium in which the exchange bonds are represented by the self-consistent coherent exchange parameter Ac( f rn, E ) which is a function of the excitation energy Green function, in this approximation: ((S;
Is.;
-
))E
k
6
E-
,
6zAc(E)e °
o ek
1-
=
7k, (3)
where parameter A c is the same for all z nearest neighbours and 8 = 8(T). Parameter Ac is determined from the equation ( T } = 0 where T is the scattering matrix connected with the fluctuations ~I,,, = A / m -- A ~ ( f - m), ( • • • ) is the configuration average. If there are no correlations of the exchange interaction fluctuations, then T X , t , , where t~ is the partial scattering matrix for any pair a of the nearest neighbours. In this case the equation ( T ) = 0 is equivalent to ( t ~ ) = 0. The t,-matrix is exactly calculated. After the configuration averaging of t, with function (2) we have found for parameter x = A J J:
q~(x, o~)
=
/
,.,r,;
.
~
//
/
,
CPA
1 • = - ~ E eXk'O~-")
x =
/ ( &/" 2
+
(1
× [1
-
-
,,,,)n,,(1
(2/z)(l/.~L)
Fig. 2. The existence range of amorphous FM the various concentration n x of the K-bond parameter X. The coherent parameter Im x(0) between opt and ~2 and lm x ( 0 ) = 0 in the under line 9)2.
spin-wave states corresponding to t ° is approximated by the half ellipsis. The density of spin-wave states f o r a m o r p h o u s FM gam(o~, 2~) = S P ( - ~ r - 1 I m ( ( S I + S-\\CPA.,//,o+0J~ has a complex behaviour and it strongly varies depending on value 2~. For the region of small ~0 and when ?~ < 0, p,l<
-
-
E t ° / (~o - xe°),
(4)
where ~ - - 1 - n K +~knx; c = nK + ~ ( 1 - nx), o~ = E / f z J . The dispersion law C~m for spin-waves in the effective medium is determined by the equation o~ - t ° Re x(o~) = 0.
(line drawing) at and the various ~ 0 in the range range which lies
(6)
and the main result is the fact that Tam(0) 4: 0. Thus, we have shown that the results of the amorphization of the quasi low-dimensional magnets (l?q<
(5) References
The existence region of amorphous F M (fig. 2) is determined by the condition RE x(0) > 0. F r o m eqs. (4) and (5) for small ~o (and when RE x(0) > 0) it is obvious that C/~m CC k 2, and that the damping of spin-waves is oc k 5. The total solution for x(to) has been found by the numerical calculation with z = 6 and nK = ½. The density of
[I] F. J. Litterst, J. de Phys. 36 (1975) LI97. [2] C. A. Sablina, G. A. Petrakovskii, E. N. Agartanova and V. P. Piskorskii, JETF Lett. 24 (1976) 357. [3] E. V. Kuzmin, G. A. Petrakovskii, JETF 75 (1978) 265. [4] B. Velicky, S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175 (1978) 747. [5] R. Kloss, Intern. J. Magnetism 5 (1972) 251.