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Dependence of the magnetization law on structural disorder in amorphous ferromagnets
Journal of Magnetism and Magnetic North-Holland, Amsterdam
Materials
127
79 (1989) 127-130
DEPENDENCE OF THE MAGNETIZATION IN AMORPHOUS FERROMAGNETS
LAW ON STRUCTURAL
DISORDER
E.M. CHUDNOVSKY Physics Depariment, Tufts University Medford, MA 02155, USA and Physics Department, New York, Bronx, NY 10468-1589, USA I Received
8 November
Lehman College, City University of
1988
Magnetic properties of some amorphous ferromagnets are well described within the random anisotropy real-space model. In that model the spatial distribution of local magnetic anisotropy axes is characterized by the correlation function C(x) which reflects a short-range structural order in a solid. We have found the explicit dependence of the magnetization law on C(x) in approaching saturation. It allows one to extract information about the structural disorder from the magnetization curve.
1. Introduction Considering the exchange interaction in amorphous magnets, one can divide them into two groups. The first group includes materials (spin glasses) where the sign of the exchange interaction between neighboring spins is random. For the second group (amorphous ferromagnets) the ferromagnetic exchange dominates. Unlike the crystalline ferromagnet, the amorphous ferromagnet has no global directions of the magnetic anisotropy. The direction of the anisotropy randomly fluctuates from one magnetic atom to another one [l]. If the anisotropy energy is larger than the exchange energy, then the orientation of any spin is determined by the direction of local anisotropy [2]. In the limit of weak random anisotropy, the exchange interaction favors long-range ferromagnetic order, so there must be ferromagnetic clusters of spins at small distances. At large distances, weak fluctuations in spin orientation due to the local random anisotropy destroy long-range order [3-51. In the absence of an external field magnetization smoothly and stochastically rotates over the solid, forming the magnetic phase which has been called correlated spin glass (CSG) phase [6]. A
’ Present
address.
0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
magnetic field transforms the CSG into a ferromagnetic state which still has some amount of disorder [7]. It has been called a ferromagnet with wandering axes (FWA) [8]. Both CSG and FWA correspond to extremal configurations of the energy functional E=jd3x{A(v*a)*-K(n*o)‘-H*M},
(1)
where M(x) is the local magnetization with fixed length M,,, u = M/M,, n is a unit vector of random anisotropy, H is the magnetic field, and A and K are exchange and anisotropy constants correspondingly. It is assumed that the random field n(x) is characterized by a correlation function C(x) which obeys c(0)
= 1,
c(x
X=-R,)
--j 0.
(2)
The parameter R, corresponds to the distance over which the local anisotropy axes are correlated, i.e. to the length of the short range structural order in the amorphous solid. Usually R, is of the order of several atomic distances. Recently it has been realized that this simple model explains numerous experimental results on amorphous ferromagnets [9]. In particular it predicts a square-root magnetization law &t4/~, cc l/a in approaching saturation for weak anisotropy systems [7], which has been observed in B.V.
128
E.&f. Chudnousky
/ Structural
Gd-Fe-Ga [lo], Gd-Er-Co Ill], HoFe, [12], Fe-Ni-Pb-B [13], and Mn-B [13,14] amorphous alloys. The square-root law applies when H < H, = 2A/MoR~. In the large field regime the model gives [7] &V/M, a l/H*. Approaching saturation is of specific interest for amorphous ferromagnets because this part of the magnetization curve is defined by the general characteristics of the amorphous structure, and does not contain metastable states [15,16]. That is why some information about structural disorder can be extracted from the magnetization curve. In this paper we show, basing upon the approach developed in ref. [8], that the magnetization law for the whole region of approaching saturation is o”dx
_-
e-~/R,
2
x C(x),
(3)
where R, = i/M,. Thus, the short range order correlation function C(x) can be, in principle, obtained from 6M(H) by the inverse Laplace transformation. 2. Magnetization
law in approaching saturation
It is convinient to use dimensionless variables x -+ x/R,, h = H/H,. Then the energy functional (1) becomes equivalent to E’=
d3x 1
disorder in amorphous ferromagnets
The CSG state corresponds to zero net magnetization in the absence of the applied magnetic field. At small distances the spins are ferromagnetically correlated, but at large distances the correlation between directions of spins exponentially decays with the distance between the spins [6-8,17-191 (u(x1)*u(x2))
=exp(
-
Here R, is the ferromagnetic
Proceeding to the saturation u in the form
r (4)
-&w~2}
where A = (KRz/A), and the last term is introduced to take into account the condition u* = 1. The dimensionless parameter A plays an important role in distinguishing between the cases of strong and weak anisotropy [6]. Our consideration applies to the latter case, when A is small. Extremal configurations u(x) satisfy
(7)
correlation
Q=jd
length:
3x c(x).
(9)
where 1u1 1 e 1, u,, = 1 - :a:. The tion law in this region is given by
magnetiza-
z =t(u:)_
(10)
Let us now write the equation for u which follows from eq. (6) to the first order in A +Z 1, (v* -~=)a,
= -An,,n,
n’n,,~
h
Solution
. n,,
(11) and
n I are compo-
+n,.
(12)
of eq. (12) is
Correspondingly, v*a+An(n~u)+h+yu=O. Multiplying we get v2a+An(n*u)
eq. (5) by u and expressing
(5) y via u
=u[uv2u+A(n.u)=+h*u]. This equation gives u(x) for fixed h and configuration of the random field n(x).
(u:) = (-&)2/jd3x’ x exp( -p
+h (6) fixed
(8)
region let us write
h u = U,,h + CT1)
In this equation p2 = h, nents of the vector
;(~*o)2-+o)2-h*o
‘x1irx2’)_
d3x”
1x - x’ 1-p ] x - x” 1) 1x - x’ 1 1x - x” 1
E.M. Chudnovsky
The correlation be written as
function
for anisotropy
axes can
(n,,cx’,n,,cx”,~,c~‘)~~(~“)) = &c(
Ix’-I”]),
(15)
which is the definition of C(x). The coefficient & is obtained at x’ =x”. Substituting eq. (15) into eq. (14) we get (16)
I(p)
= --$//d’x
d3ye-;;+%(
over the angle variables
I(p) = ~P~wdx
M,=
gives
dy e-p(x-y)~‘~~‘dr X
Integrating obtain I(P)
e-P”~Udu/Udz ”
by parts
= f i”dx
zC( z).
(19)
over u, and then over U, we
e-pxx2C(x),
A2 15PO
for C(x)
= edX,
(20)
from which follows eq. (3). The dependence of fM/M,, on the magnetic field is universal in two limits: H-=X H, and H > H,. The first limit corresponds to p < 1. In this limit I(p) = s2/41rpRi and eqs. (lo), (16) give the square-root magnetization law in approaching saturation: (21)
(22)
(23)
+Pj3
A2 -ep2’4U($, 30P
p)
for C(x)
=e
-x=/2
,
(24
zC( z).
Noticing that the expression under the integral is symmetric with respect to the bisectrix of the coordinate system (x, y), and turning to a new coordinate system (u, u) rotated by a/4 with respect to (x, y), eq. (18) can be written as = ipdU
SM -= MO
08)
1(P)
This magnetiation law does not have any dependence on A and R,, and coincides with the magnetization law for an amorphous ferromagnet where the anisotropy is large in comparison with exchange [2]. The crossover regime H - H, corresponds to p - 1. In that range of the magnetic field, the magnetization curve is defined by the explicit form of the correlation function C(x). Explicit formulae for approaching saturation are listed below for two typical forms of C(x): 6M
Ix-y]). (17)
Integration
129
/ Structural disorder in amorphous jerromagnets
where
U($, p) is a parabolic
cylindrical
function.
3. Discussion We have shown that for H - H, the magnetization law depends on the explicit form of C(x). Thus C(x) can, in principle, be obtained from SM( H) by the inversed Laplace transformation of eq. (3) if measurement of M(H) includes H - H,. This would allow one to extract information about the structural disorder in amorphous solids from the magnetization curve alone. The crossover from the intermediate field regime described by eq. (21) to the large field regime described by eq. (22) has not yet been found experimentally. There may be two reasons for this. First, in amorphous materials with large exchange and R, of atomic scale, the crossover field H, is of the order of the exchange field in crystalline ferromagnets, so it can hardly be reached experimentally. Second, in a large field 6M becomes so small that it is difficult to establish the magnetization law. To study the crossover, one should use a material with not too large an exchange and/or extended correlations in structural order (large R,). Note that the parameter A must still be smaller than 1 for such a material, otherwise it will not display the intermediate field regime at all. Good candidates might be the gado-
130
E.M. Chudnovsky
/ Structural disorder in amorphous ferromagnets
linium amorphous alloys studied by Sellmyer et al. [lO,ll].
References [l] R. Harris, M. Plischke and M.J. Zuckermann, Phys. Rev. Lett. 31 (1973) 160. [2] E. Callen, Y.I. Liu and J.R. Cullen, Phys. Rev. B16 (1977) 263. [3] Y. Imry and S. Ma, Phys. Rev. Lett. 35 (1975) 1399. [4] R.A. Pelcovits, E. Pytte and J. Rudnick, Phys. Rev. Lett. 40 (1978) 476. R.A. Pelcovits, Phys. Rev. B19 (1979) 465. [5] R. Alben, J.J. Becker and M.C. Chi, J. Appl. Phys. 49 (1978) 1653. [6] E.M. Chudnovsky and R.A. Serota, Phys. Rev. B26 (1982) 2697. [7] E.M. Chudnovsky and R.A. Serota, J. Phys. Cl6 (1983) 4181.
[8] E.M. Chudnovsky, W.M. &slow and R.A. Serota, Phys. Rev. B33 (1986) 251. [9] E.M. Chudnovsky, Magnetic properties of amorphous ferromagnets (invited talk at the 4th Joint MMM-Intermag Conference, Vancouver, Canada, 12-15 July 1988), J. Appl. Phys. 64 (1988) 5770. [lo] D.J. Sellmyer and S. Nafis, J. Appl. Phys. 57 (1985) 3584. [ll] D.J. Sellmyer, S. Nafis and M.J. O’Shea, J. Appl. Phys. 68 (1988) 3743. [12] J.J. Rhyne, IEEE Trans. Magn. MAG-21 (1985) 1990. [13] M.J. Park, SM. Bhagat, M.A. Manheimer and K. Moorjani, Phys. Rev. B33 (1986) 2070. [14] W.A. Bryden, J.S. Morgan, T.J. Kistemnacher and K. Moorjani, J. Appl. Phys. 61 (1987) 3661. [15] R.A. Serota and P.A. Lee, Phys. Rev. B34 (1986) 1806; J. Appl. Phys. 61 (1987) 3965. [16] W.M. Saslow, Phys. Rev. B35 (1987) 3454. [17] A. Aharony and E. Pytte, Phys. Rev. B27 (1983) 5872. [18] E.M. Chudnovsky, Phys. Rev. B33 (1986) 2021. [19] VS. Dotsenko and M.V. Feigelman, J. Appl. Phys. Cl6 (1983) L803.
Materials
127
79 (1989) 127-130
DEPENDENCE OF THE MAGNETIZATION IN AMORPHOUS FERROMAGNETS
LAW ON STRUCTURAL
DISORDER
E.M. CHUDNOVSKY Physics Depariment, Tufts University Medford, MA 02155, USA and Physics Department, New York, Bronx, NY 10468-1589, USA I Received
8 November
Lehman College, City University of
1988
Magnetic properties of some amorphous ferromagnets are well described within the random anisotropy real-space model. In that model the spatial distribution of local magnetic anisotropy axes is characterized by the correlation function C(x) which reflects a short-range structural order in a solid. We have found the explicit dependence of the magnetization law on C(x) in approaching saturation. It allows one to extract information about the structural disorder from the magnetization curve.
1. Introduction Considering the exchange interaction in amorphous magnets, one can divide them into two groups. The first group includes materials (spin glasses) where the sign of the exchange interaction between neighboring spins is random. For the second group (amorphous ferromagnets) the ferromagnetic exchange dominates. Unlike the crystalline ferromagnet, the amorphous ferromagnet has no global directions of the magnetic anisotropy. The direction of the anisotropy randomly fluctuates from one magnetic atom to another one [l]. If the anisotropy energy is larger than the exchange energy, then the orientation of any spin is determined by the direction of local anisotropy [2]. In the limit of weak random anisotropy, the exchange interaction favors long-range ferromagnetic order, so there must be ferromagnetic clusters of spins at small distances. At large distances, weak fluctuations in spin orientation due to the local random anisotropy destroy long-range order [3-51. In the absence of an external field magnetization smoothly and stochastically rotates over the solid, forming the magnetic phase which has been called correlated spin glass (CSG) phase [6]. A
’ Present
address.
0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
magnetic field transforms the CSG into a ferromagnetic state which still has some amount of disorder [7]. It has been called a ferromagnet with wandering axes (FWA) [8]. Both CSG and FWA correspond to extremal configurations of the energy functional E=jd3x{A(v*a)*-K(n*o)‘-H*M},
(1)
where M(x) is the local magnetization with fixed length M,,, u = M/M,, n is a unit vector of random anisotropy, H is the magnetic field, and A and K are exchange and anisotropy constants correspondingly. It is assumed that the random field n(x) is characterized by a correlation function C(x) which obeys c(0)
= 1,
c(x
X=-R,)
--j 0.
(2)
The parameter R, corresponds to the distance over which the local anisotropy axes are correlated, i.e. to the length of the short range structural order in the amorphous solid. Usually R, is of the order of several atomic distances. Recently it has been realized that this simple model explains numerous experimental results on amorphous ferromagnets [9]. In particular it predicts a square-root magnetization law &t4/~, cc l/a in approaching saturation for weak anisotropy systems [7], which has been observed in B.V.
128
E.&f. Chudnousky
/ Structural
Gd-Fe-Ga [lo], Gd-Er-Co Ill], HoFe, [12], Fe-Ni-Pb-B [13], and Mn-B [13,14] amorphous alloys. The square-root law applies when H < H, = 2A/MoR~. In the large field regime the model gives [7] &V/M, a l/H*. Approaching saturation is of specific interest for amorphous ferromagnets because this part of the magnetization curve is defined by the general characteristics of the amorphous structure, and does not contain metastable states [15,16]. That is why some information about structural disorder can be extracted from the magnetization curve. In this paper we show, basing upon the approach developed in ref. [8], that the magnetization law for the whole region of approaching saturation is o”dx
_-
e-~/R,
2
x C(x),
(3)
where R, = i/M,. Thus, the short range order correlation function C(x) can be, in principle, obtained from 6M(H) by the inverse Laplace transformation. 2. Magnetization
law in approaching saturation
It is convinient to use dimensionless variables x -+ x/R,, h = H/H,. Then the energy functional (1) becomes equivalent to E’=
d3x 1
disorder in amorphous ferromagnets
The CSG state corresponds to zero net magnetization in the absence of the applied magnetic field. At small distances the spins are ferromagnetically correlated, but at large distances the correlation between directions of spins exponentially decays with the distance between the spins [6-8,17-191 (u(x1)*u(x2))
=exp(
-
Here R, is the ferromagnetic
Proceeding to the saturation u in the form
r (4)
-&w~2}
where A = (KRz/A), and the last term is introduced to take into account the condition u* = 1. The dimensionless parameter A plays an important role in distinguishing between the cases of strong and weak anisotropy [6]. Our consideration applies to the latter case, when A is small. Extremal configurations u(x) satisfy
(7)
correlation
Q=jd
length:
3x c(x).
(9)
where 1u1 1 e 1, u,, = 1 - :a:. The tion law in this region is given by
magnetiza-
z =t(u:)_
(10)
Let us now write the equation for u which follows from eq. (6) to the first order in A +Z 1, (v* -~=)a,
= -An,,n,
n’n,,~
h
Solution
. n,,
(11) and
n I are compo-
+n,.
(12)
of eq. (12) is
Correspondingly, v*a+An(n~u)+h+yu=O. Multiplying we get v2a+An(n*u)
eq. (5) by u and expressing
(5) y via u
=u[uv2u+A(n.u)=+h*u]. This equation gives u(x) for fixed h and configuration of the random field n(x).
(u:) = (-&)2/jd3x’ x exp( -p
+h (6) fixed
(8)
region let us write
h u = U,,h + CT1)
In this equation p2 = h, nents of the vector
;(~*o)2-+o)2-h*o
‘x1irx2’)_
d3x”
1x - x’ 1-p ] x - x” 1) 1x - x’ 1 1x - x” 1
E.M. Chudnovsky
The correlation be written as
function
for anisotropy
axes can
(n,,cx’,n,,cx”,~,c~‘)~~(~“)) = &c(
Ix’-I”]),
(15)
which is the definition of C(x). The coefficient & is obtained at x’ =x”. Substituting eq. (15) into eq. (14) we get (16)
I(p)
= --$//d’x
d3ye-;;+%(
over the angle variables
I(p) = ~P~wdx
M,=
gives
dy e-p(x-y)~‘~~‘dr X
Integrating obtain I(P)
e-P”~Udu/Udz ”
by parts
= f i”dx
zC( z).
(19)
over u, and then over U, we
e-pxx2C(x),
A2 15PO
for C(x)
= edX,
(20)
from which follows eq. (3). The dependence of fM/M,, on the magnetic field is universal in two limits: H-=X H, and H > H,. The first limit corresponds to p < 1. In this limit I(p) = s2/41rpRi and eqs. (lo), (16) give the square-root magnetization law in approaching saturation: (21)
(22)
(23)
+Pj3
A2 -ep2’4U($, 30P
p)
for C(x)
=e
-x=/2
,
(24
zC( z).
Noticing that the expression under the integral is symmetric with respect to the bisectrix of the coordinate system (x, y), and turning to a new coordinate system (u, u) rotated by a/4 with respect to (x, y), eq. (18) can be written as = ipdU
SM -= MO
08)
1(P)
This magnetiation law does not have any dependence on A and R,, and coincides with the magnetization law for an amorphous ferromagnet where the anisotropy is large in comparison with exchange [2]. The crossover regime H - H, corresponds to p - 1. In that range of the magnetic field, the magnetization curve is defined by the explicit form of the correlation function C(x). Explicit formulae for approaching saturation are listed below for two typical forms of C(x): 6M
Ix-y]). (17)
Integration
129
/ Structural disorder in amorphous jerromagnets
where
U($, p) is a parabolic
cylindrical
function.
3. Discussion We have shown that for H - H, the magnetization law depends on the explicit form of C(x). Thus C(x) can, in principle, be obtained from SM( H) by the inversed Laplace transformation of eq. (3) if measurement of M(H) includes H - H,. This would allow one to extract information about the structural disorder in amorphous solids from the magnetization curve alone. The crossover from the intermediate field regime described by eq. (21) to the large field regime described by eq. (22) has not yet been found experimentally. There may be two reasons for this. First, in amorphous materials with large exchange and R, of atomic scale, the crossover field H, is of the order of the exchange field in crystalline ferromagnets, so it can hardly be reached experimentally. Second, in a large field 6M becomes so small that it is difficult to establish the magnetization law. To study the crossover, one should use a material with not too large an exchange and/or extended correlations in structural order (large R,). Note that the parameter A must still be smaller than 1 for such a material, otherwise it will not display the intermediate field regime at all. Good candidates might be the gado-
130
E.M. Chudnovsky
/ Structural disorder in amorphous ferromagnets
linium amorphous alloys studied by Sellmyer et al. [lO,ll].
References [l] R. Harris, M. Plischke and M.J. Zuckermann, Phys. Rev. Lett. 31 (1973) 160. [2] E. Callen, Y.I. Liu and J.R. Cullen, Phys. Rev. B16 (1977) 263. [3] Y. Imry and S. Ma, Phys. Rev. Lett. 35 (1975) 1399. [4] R.A. Pelcovits, E. Pytte and J. Rudnick, Phys. Rev. Lett. 40 (1978) 476. R.A. Pelcovits, Phys. Rev. B19 (1979) 465. [5] R. Alben, J.J. Becker and M.C. Chi, J. Appl. Phys. 49 (1978) 1653. [6] E.M. Chudnovsky and R.A. Serota, Phys. Rev. B26 (1982) 2697. [7] E.M. Chudnovsky and R.A. Serota, J. Phys. Cl6 (1983) 4181.
[8] E.M. Chudnovsky, W.M. &slow and R.A. Serota, Phys. Rev. B33 (1986) 251. [9] E.M. Chudnovsky, Magnetic properties of amorphous ferromagnets (invited talk at the 4th Joint MMM-Intermag Conference, Vancouver, Canada, 12-15 July 1988), J. Appl. Phys. 64 (1988) 5770. [lo] D.J. Sellmyer and S. Nafis, J. Appl. Phys. 57 (1985) 3584. [ll] D.J. Sellmyer, S. Nafis and M.J. O’Shea, J. Appl. Phys. 68 (1988) 3743. [12] J.J. Rhyne, IEEE Trans. Magn. MAG-21 (1985) 1990. [13] M.J. Park, SM. Bhagat, M.A. Manheimer and K. Moorjani, Phys. Rev. B33 (1986) 2070. [14] W.A. Bryden, J.S. Morgan, T.J. Kistemnacher and K. Moorjani, J. Appl. Phys. 61 (1987) 3661. [15] R.A. Serota and P.A. Lee, Phys. Rev. B34 (1986) 1806; J. Appl. Phys. 61 (1987) 3965. [16] W.M. Saslow, Phys. Rev. B35 (1987) 3454. [17] A. Aharony and E. Pytte, Phys. Rev. B27 (1983) 5872. [18] E.M. Chudnovsky, Phys. Rev. B33 (1986) 2021. [19] VS. Dotsenko and M.V. Feigelman, J. Appl. Phys. Cl6 (1983) L803.