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Domain formation in Ising magnets near Tc
843
Journal of Magnetism and Magnetic Materials 54-57 (1986) 843-844 DOMAIN
FORMATION
IN ISING MAGNETS
N E A R T~
G.A. GEHRING
Dept. of Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK
The shape dependent dipolar energy dominates the critical fluctuations in a finite slab and leads to an oscillatory magnetization which evolves into domains.
1. Introduction The ordered phase of an anisotropic ferromagnet with easy direction normal to the slab consists of domains which reduce the dipolar energy. It follows that the critical fluctuations must be influenced by the shape dependent dipolar energy and that the ordered phase which develops should show the periodicity of the domain structure. There are two ways in which the dipolar energy enters the paramagnetic free energy. There is the bulk term [1], a magnetization fluctuation of total wave vector k contributes an energy proportional to ( k : / k ) 2 this tends to inhibit fluctuations for which k~'~ 0 i.e. a M / ~ z :~ 0 so favouring states in which the magnetization is constant across the slab. There is also the demagnetizing energy which arises explicitly from the finite geometry which depends on the square of the magnetization at the surface. The bulk effect is responsible for domains in thick slabs L >> L~ and the surface effect for thin slabs L _< L , [2] where L~ ~- 40rof -1/z and r 0 is the range of the exchange forces and f ( = PoM~/NkT~) is the ratio of the dipolar energy at saturation to the ordering energy. Both of these dipolar energies cause the magnetization to develop domains. In a thin slab the bulk term dominates and so the magnetization is roughly constant across the slab. The demagnetizing dipolar energy per unit volume of a fluctuation of magnetization Mk(x ) = M k cos kx is # o M Z / 4 k L (terms of order exp - ~r/kL can be neglected). The total energy of a magnetization fluctuation of this type can be calculated and the value of k found which has the minimum energy: this corresponds to the 'soft mode' of the phase transition and becomes the ordered state [3,4]. In a thick slab the surface effect is large and so the most favourable magnetization fluctuations in the paramagnetic phase have zero surface magnetization e.g. M ( z x ) = M, cos kx c o s ( v z / L ) ( z = +_L/2) [2,5]. The bulk dipolar energy is proportional to ( v / L ) 2 [ ( ~ / L ) 2 + k2] -~ and the soft mode of the system is again characterised by a non-zero value of k. Similar considerations apply just below To. In a thin slab the stripe domains have thickness D which is falling rapidly with increasing temperature [6,4] at the
same time the width of the walls ( is diverging so the magnetization tends to a sinusoidal distribution of period v / D * . A similar limiting value of the transverse wave vector k ( = v / D * ) at T~ is found from the low and high temperature theories [4]. In a thick slab below Tc the system can reduce the demagnetization energy by developing branch domains [7] at the expense of having walls which are not parallel to z (which involve the bulk dipolar energy). The evolution of heavily branched systems L >> L k [8] is such that the magnetization tends to the same profile at T~ as the soft mode found for T > T~ [2]. In the analysis here we have neglected the phase modulation and wall bending treated in [3] because we assume there are sufficient impurities to pin the walls. 2. Above T~ As the temperature is lowered towards Tc a cross-over occurs to bulk dipolar dominated behaviour at t - - f (t = ( T - T~)/Tc; it is convenient to measure all temperatures in these reduced units which always refer to the hulk T~). We are concerned only with those materials for which f is sufficiently large so that the dipolar cross over occurs well before the correlation length becomes of the order of L, f > > (ro/L) 1/~ (1/I, = 1.6). In the bulk dipolar regime the correlations are anisotropic, £,,(= grfrot-I ) is growing much faster than ~± ( = rot - I / z ) [9]. The thickness of the slab becomes relevant at t+ where ~,r=L1, t + = r o f f / L . It is convenient to measure L in the unit which has already appeared in the defination of L k. We define L = lrof-1/2 so t + = f / l . Above t+ the fluctuation spectrum peaks about k = 0, below t+ the fluctuations start to peak at
k = +_v/D*. At
t+
the
of
the
spectrum
is
k+=
in these units x = l-Z/2. We consider the free energy associated with the magnetization fluctuations of transverse wave vector K.
M ( z , x) = M~(0)[cos~ 0 + sink o cos v z / L ] cos ~VVx/r0, F = # 0 / 2 Y~ [ ( t / f + x z + 1 / 4 x l ) cos 2 0 K
0304-8853/86/$03.50
width
(f/lroZ)l/z(= ~ 1 ) . It is convenient to write k = xgtf/ro
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
(1)
G.A. Gehring / Domainformation in Ising magnets
844
+ ½(t/f+ x 2 + (,~/l)2 + ( ~ / ~ / ) 2 ) sin2~ 0 + 4 / ~ ( t / f + K2) cos 0~ sin 0K] M~(O) I*o/2Y'~(22(O)M2(O),
(2)
t~
The trial function (1) was chosen for its simplicity; the two terms are not orthogonal and so the mixing term appears in the free energy however they do have the behaviours expected for thin and thick slabs, respectively. The terms in the free energy 1/4Kl, ½('~/~l) ~ come from the surface demagnetizing and bulk dipolar energy respectively (they are equal for K= ~+=1-1/2 for l = 4"~4), the term in (nT//) 2 is negligible. The form of the magnetisation soft mode, and the transition temperature for the slab are found from the following minimization: 0n~(0)/0~
= ~n~(0)/~0
$2~2(0)=0
= 0,
,at K = ~ * , 0 = 0 * , a n d
t=t*.
(3)
The value of t* is negative corresponding to a lowering of the ordering temperature because of the additional dipolar energy that must be supplied and D* =
~,~0/~*~.
For a thick slab l_> 500, ~ * = (~/1) 1/2= ~ + ; t* = -2"~f/l= -21Tt+ and cot 0~ = 8('~/l)/2. Taking f-~
1 appropriate for dipolar ferromagnets e.g. LiTbF 4 we find that for l = 10 3, t* = 10 -3 which is experimentally accessible. The scattering function may be approximated by the form for cos 0 = 0.
= [ k , T ( t / f + ~/L + I ( K : - x ' 2 ) 2 / ' ~ ] - '
(4) The theory for a thin slab is valid for l _< 30 (since L >> r0 for the theory to make sense the thin slab limit is only accessible for f < < 1 e.g. f = 4 × 10 3 for Barium ferrite). In this case t* = - 3 f 1 2/3/4, x * = ½l-~/3, tan0* --(2'rr2/2/3) -1. Since f is constrained to be small the value of t* is smaller than for the thick slab case.
3. Theory below Tc Just below Tc the magnetization has an ordered component given by the soft mode at ~*. At lower temperatures the higher harmonics 3 K* will appear when
the inverse correlation range k > 3~* etc. This is necessary to evolve to a system with sharp walls. A complementary approach is to start with the low temperature theory and look at the way in which the free energy evolves close to T~. There is a characteristic length .Y(T)=ow(T)/t%M2(T) where ow is the wall energy, .y(t) = 4 t ( / 3 f = 4rotl/2/3f when ~ >> r o. The free energy in the low temperature phase assuming a domain spacing D is given below for the thin (stripe) [6,4] and thick (highly branched) [7,8,2] slabs, respectively.
Fs=I~oM2[t/f+.Y/Ds+Ds/LA]+(9(M4), F B =/~0 M2 [ t / f + "y/D, + 8(.YD•)l/Z/3L]
(5)
+ 0(M4).
(6) The term .y/D is the domain wall energy D/LA is the surface dipolar energy (A = @ / 4 × 1.05) and the term in (.yD) 1/2 arises from the walls which are not parallel to 2: it takes the place of the bulk dipolar energy in this regime. The equilibrium values of D are found by minimization D s = (AL.Y) 1/2 and D s = (3L/4)z/3.y 1/3 and the estimate for t* from the condition that the coefficient of m 2 should vanish. Using the way in which 7 depends on ~ it is straightforward to show that at the transition D~' = ~ and D~ = 4~; the values of t* are t~ = --f(16/3Aa) 2/3 and t~ = - 1 6 / ¢ 3 l in close agreement to that found from the high temperature theory in section 2, The condition D - ( has previously [6,4,8] been used to estimate D*. There is an inherent inconsistency in using ~= rot-1/2 (as the characteristic domain wall width) which diverges at t = 0 rather than the real correlation length for the finite slab which diverges at t = - t*. The free energies given in eqs. (5) and (6) are valid for well defined non-interacting domain walls. In the temperature region below Tcslab for which the (~lab > D a generalisation of the continuum theory discussed at the beginning of this section is more appropriate. [1] [2] [3] [4] [5] [6] [7] [8] [9]
A. Aharony and M.E. Fisher, Phys. Rev. B8 (1973) 3323. W.A. Barker and G.A. Gehring, J. Phys. C (1985) in press, T. Garel and S. Doniach, Phys. Rev. B26 (1982) 325. W.A. Barker and G.A. Gehring, J. Phys. C16 (1983) 6415. W. Wasilewski, Phys. Lett. 84A (1981) 80. D. Stauffer, AIP Conf. Proc. 10 (1972) 827. J. Kaczer, Soy. Phys. JETP 19 (1964) 1204. M. Gabay and T. Garel, J, de Phys. Lett. 20 (1984) 989. J. Als-Nielsen and R.J. Birgeneau, Am. J. Phys. 45 (1977) 554.
Journal of Magnetism and Magnetic Materials 54-57 (1986) 843-844 DOMAIN
FORMATION
IN ISING MAGNETS
N E A R T~
G.A. GEHRING
Dept. of Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK
The shape dependent dipolar energy dominates the critical fluctuations in a finite slab and leads to an oscillatory magnetization which evolves into domains.
1. Introduction The ordered phase of an anisotropic ferromagnet with easy direction normal to the slab consists of domains which reduce the dipolar energy. It follows that the critical fluctuations must be influenced by the shape dependent dipolar energy and that the ordered phase which develops should show the periodicity of the domain structure. There are two ways in which the dipolar energy enters the paramagnetic free energy. There is the bulk term [1], a magnetization fluctuation of total wave vector k contributes an energy proportional to ( k : / k ) 2 this tends to inhibit fluctuations for which k~'~ 0 i.e. a M / ~ z :~ 0 so favouring states in which the magnetization is constant across the slab. There is also the demagnetizing energy which arises explicitly from the finite geometry which depends on the square of the magnetization at the surface. The bulk effect is responsible for domains in thick slabs L >> L~ and the surface effect for thin slabs L _< L , [2] where L~ ~- 40rof -1/z and r 0 is the range of the exchange forces and f ( = PoM~/NkT~) is the ratio of the dipolar energy at saturation to the ordering energy. Both of these dipolar energies cause the magnetization to develop domains. In a thin slab the bulk term dominates and so the magnetization is roughly constant across the slab. The demagnetizing dipolar energy per unit volume of a fluctuation of magnetization Mk(x ) = M k cos kx is # o M Z / 4 k L (terms of order exp - ~r/kL can be neglected). The total energy of a magnetization fluctuation of this type can be calculated and the value of k found which has the minimum energy: this corresponds to the 'soft mode' of the phase transition and becomes the ordered state [3,4]. In a thick slab the surface effect is large and so the most favourable magnetization fluctuations in the paramagnetic phase have zero surface magnetization e.g. M ( z x ) = M, cos kx c o s ( v z / L ) ( z = +_L/2) [2,5]. The bulk dipolar energy is proportional to ( v / L ) 2 [ ( ~ / L ) 2 + k2] -~ and the soft mode of the system is again characterised by a non-zero value of k. Similar considerations apply just below To. In a thin slab the stripe domains have thickness D which is falling rapidly with increasing temperature [6,4] at the
same time the width of the walls ( is diverging so the magnetization tends to a sinusoidal distribution of period v / D * . A similar limiting value of the transverse wave vector k ( = v / D * ) at T~ is found from the low and high temperature theories [4]. In a thick slab below Tc the system can reduce the demagnetization energy by developing branch domains [7] at the expense of having walls which are not parallel to z (which involve the bulk dipolar energy). The evolution of heavily branched systems L >> L k [8] is such that the magnetization tends to the same profile at T~ as the soft mode found for T > T~ [2]. In the analysis here we have neglected the phase modulation and wall bending treated in [3] because we assume there are sufficient impurities to pin the walls. 2. Above T~ As the temperature is lowered towards Tc a cross-over occurs to bulk dipolar dominated behaviour at t - - f (t = ( T - T~)/Tc; it is convenient to measure all temperatures in these reduced units which always refer to the hulk T~). We are concerned only with those materials for which f is sufficiently large so that the dipolar cross over occurs well before the correlation length becomes of the order of L, f > > (ro/L) 1/~ (1/I, = 1.6). In the bulk dipolar regime the correlations are anisotropic, £,,(= grfrot-I ) is growing much faster than ~± ( = rot - I / z ) [9]. The thickness of the slab becomes relevant at t+ where ~,r=L1, t + = r o f f / L . It is convenient to measure L in the unit which has already appeared in the defination of L k. We define L = lrof-1/2 so t + = f / l . Above t+ the fluctuation spectrum peaks about k = 0, below t+ the fluctuations start to peak at
k = +_v/D*. At
t+
the
of
the
spectrum
is
k+=
in these units x = l-Z/2. We consider the free energy associated with the magnetization fluctuations of transverse wave vector K.
M ( z , x) = M~(0)[cos~ 0 + sink o cos v z / L ] cos ~VVx/r0, F = # 0 / 2 Y~ [ ( t / f + x z + 1 / 4 x l ) cos 2 0 K
0304-8853/86/$03.50
width
(f/lroZ)l/z(= ~ 1 ) . It is convenient to write k = xgtf/ro
© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
(1)
G.A. Gehring / Domainformation in Ising magnets
844
+ ½(t/f+ x 2 + (,~/l)2 + ( ~ / ~ / ) 2 ) sin2~ 0 + 4 / ~ ( t / f + K2) cos 0~ sin 0K] M~(O) I*o/2Y'~(22(O)M2(O),
(2)
t~
The trial function (1) was chosen for its simplicity; the two terms are not orthogonal and so the mixing term appears in the free energy however they do have the behaviours expected for thin and thick slabs, respectively. The terms in the free energy 1/4Kl, ½('~/~l) ~ come from the surface demagnetizing and bulk dipolar energy respectively (they are equal for K= ~+=1-1/2 for l = 4"~4), the term in (nT//) 2 is negligible. The form of the magnetisation soft mode, and the transition temperature for the slab are found from the following minimization: 0n~(0)/0~
= ~n~(0)/~0
$2~2(0)=0
= 0,
,at K = ~ * , 0 = 0 * , a n d
t=t*.
(3)
The value of t* is negative corresponding to a lowering of the ordering temperature because of the additional dipolar energy that must be supplied and D* =
~,~0/~*~.
For a thick slab l_> 500, ~ * = (~/1) 1/2= ~ + ; t* = -2"~f/l= -21Tt+ and cot 0~ = 8('~/l)/2. Taking f-~
1 appropriate for dipolar ferromagnets e.g. LiTbF 4 we find that for l = 10 3, t* = 10 -3 which is experimentally accessible. The scattering function may be approximated by the form for cos 0 = 0.
= [ k , T ( t / f + ~/L + I ( K : - x ' 2 ) 2 / ' ~ ] - '
(4) The theory for a thin slab is valid for l _< 30 (since L >> r0 for the theory to make sense the thin slab limit is only accessible for f < < 1 e.g. f = 4 × 10 3 for Barium ferrite). In this case t* = - 3 f 1 2/3/4, x * = ½l-~/3, tan0* --(2'rr2/2/3) -1. Since f is constrained to be small the value of t* is smaller than for the thick slab case.
3. Theory below Tc Just below Tc the magnetization has an ordered component given by the soft mode at ~*. At lower temperatures the higher harmonics 3 K* will appear when
the inverse correlation range k > 3~* etc. This is necessary to evolve to a system with sharp walls. A complementary approach is to start with the low temperature theory and look at the way in which the free energy evolves close to T~. There is a characteristic length .Y(T)=ow(T)/t%M2(T) where ow is the wall energy, .y(t) = 4 t ( / 3 f = 4rotl/2/3f when ~ >> r o. The free energy in the low temperature phase assuming a domain spacing D is given below for the thin (stripe) [6,4] and thick (highly branched) [7,8,2] slabs, respectively.
Fs=I~oM2[t/f+.Y/Ds+Ds/LA]+(9(M4), F B =/~0 M2 [ t / f + "y/D, + 8(.YD•)l/Z/3L]
(5)
+ 0(M4).
(6) The term .y/D is the domain wall energy D/LA is the surface dipolar energy (A = @ / 4 × 1.05) and the term in (.yD) 1/2 arises from the walls which are not parallel to 2: it takes the place of the bulk dipolar energy in this regime. The equilibrium values of D are found by minimization D s = (AL.Y) 1/2 and D s = (3L/4)z/3.y 1/3 and the estimate for t* from the condition that the coefficient of m 2 should vanish. Using the way in which 7 depends on ~ it is straightforward to show that at the transition D~' = ~ and D~ = 4~; the values of t* are t~ = --f(16/3Aa) 2/3 and t~ = - 1 6 / ¢ 3 l in close agreement to that found from the high temperature theory in section 2, The condition D - ( has previously [6,4,8] been used to estimate D*. There is an inherent inconsistency in using ~= rot-1/2 (as the characteristic domain wall width) which diverges at t = 0 rather than the real correlation length for the finite slab which diverges at t = - t*. The free energies given in eqs. (5) and (6) are valid for well defined non-interacting domain walls. In the temperature region below Tcslab for which the (~lab > D a generalisation of the continuum theory discussed at the beginning of this section is more appropriate. [1] [2] [3] [4] [5] [6] [7] [8] [9]
A. Aharony and M.E. Fisher, Phys. Rev. B8 (1973) 3323. W.A. Barker and G.A. Gehring, J. Phys. C (1985) in press, T. Garel and S. Doniach, Phys. Rev. B26 (1982) 325. W.A. Barker and G.A. Gehring, J. Phys. C16 (1983) 6415. W. Wasilewski, Phys. Lett. 84A (1981) 80. D. Stauffer, AIP Conf. Proc. 10 (1972) 827. J. Kaczer, Soy. Phys. JETP 19 (1964) 1204. M. Gabay and T. Garel, J, de Phys. Lett. 20 (1984) 989. J. Als-Nielsen and R.J. Birgeneau, Am. J. Phys. 45 (1977) 554.