- Email: [email protected]
Strong critical fluctuations in domain walls in Sr hexaferrite: Effect of magnetic fields
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1873-1874
ELSEVIER
~l~
|0|NIB| 0|
,~
magneUsnl ond magnetic m~erlals
Strong critical fluctuations in domain walls in Sr hexaferrite: Effect of magnetic fields M. Malang, D. Garanin, J. K6tzler * Institut fiir Angewandte Physik, Universitiit Hamburg, Jungiusstr. 11, D-20355 Hamburg, Germany Abstract
The domain wall (DW) relaxation rate F w of the uniaxial Sr hexaferrite in the region 0.9Tc _< Tc = 740 K, displays a deep minimum at T = 0.99T c due to a continuous 2D phase transition from Bloch to linear domain walls. In fields parallel to the easy axis, F w is reduced by the increase of the DW spacing d, whereas a perpendicular field increases F w. This feature is associated with the increase of the DW order parameter m B with the field near the transition.
Recent experiments on the domain wall (DW) dynamics near the Curie temperature of the hexaferrites BaFe12019 [1] and SrFe12Ow [2] provided the first evidence for the transformation of the Bloch wall (BW) to the so-called linear wall (LW). This phase transition was predicted by Bulaevskii and Ginzburg [3] to occur in uniaxial ferromagnets at some temperature T * close to Tc. Above T* the DW magnetization profile is linear ( M ( x ) I[ ex) passing to elliptic (m B ~ My(O)/M s < 1) for T < T * (see Fig. 1). In the mean field (MF) approximation [3] T * is determined by the condition 0 ( T ~ F ) = 1, where O(T) =- 4 x z ( T ) / x ± .xz(T) cz (1 - T / T c ) -1 and X± = M s / H A = const are the longitudinal and transverse susceptibilities, and m a = ( 1 - 0) 1/2 represents the DW order parameter below Tr~v. It is difficult to observe this reconstruction of DWs because their width 6 is very small. However, the DW mobility /x turns out to be very sensitive to changes in the DW structure. The DW kinetic coefficient Lw = 2M~(oz)iz/d can be determined from the DW relaxation rate Fw = L w / ( X Z 1 + N ) measured by the dynamic susceptibility X(to) parallel to the easy axis, N Z is the demagnetization coefficient. LWs were first reported on the uniaxial ferromagnets GdCl 3 [4] and LiTbF4 [5]. The characteristic divergence of L w at Tc could he described by the following expression [4]:
Lw=3Lz(6/d )
(T> T*).
sents the width of the LW, which is given by the correlation length ~z cx X 1/2 and d is the domain period shrinking as a power of 1 / 6 [4]. This speeding up of L w [4,5] contrasts the well-known result by Landau and Lifshitz [6] predicting L w to decrease continuously towards Tc. In Refs. [7,8] the DW mobility g was calculated over the whole temperature range for the first time, taking into account both transverse and longitudinal relaxation mechanisms. The result
L w = 3Lz(6/d ) .F(ma; L,, L±),
(2)
(6 = 6 a = const for T < T *) shows that for elliptic walls L w rises sharply with m B when lowering temperature, because the effect of the longitudinal relaxation reduces when a circular BW shape is approached, i.e. for rn n ~ 1. For linear walls F = 1 and Eq. (2) coincides with Eq. (1), and at low T the result of Landau and Lifshitz is recovered. Near T * the theory predicts a deep minimum of L w, which has been observed in the experiments on Ba and Sr hexaferrites [1,2].
-
w
zy\
(1)
Here L z is the kinetic coefficient of M z in domains being non-singular at Tc for a uniaxial magnet, 6 = 6L reprex =-
* Corresponding author. Fax +49 40-4123-6368; email: [email protected].
0
0
~
Fig. 1. Magnetization profile of m(x)= M ( x ) / M s for Bloch (m~ = 1), elliptic (m B < 1) and linear (m B = 0) domain walls in unia×ial ferromagnets.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 3 9 1 - 8
M. Malang et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1873-1874
1874
The further interesting features of the DW relaxation observed in [1,2] are (i) a significantly lower transition temperature T * = 0.990Tc than T~F = 0.996Tc, and (ii) a very steep increase in L w with lowering temperature for T < T *. The latter could be described by a temperature variation of the order parameter m Bct (T* -T)/3B with the exponent /3a = 0.08 being much smaller than /3~ F = 1 / 2 [3]. These two features could be attributed [2] to strong two-dimensional (2D) fluctuations of m B [9] along the y-axis of the DW, whereas the fluctuations along the x-axis should be supressed by the effective in-plane anisotropy of magnetostatic origin e = M s / H A = X ± (see Fig. 1). In order to gain a deeper insight into the physics of this phase transition, we investigate here the DW relaxation in magnetic fields applied parallel and perpendicular to the easy axis (see Fig. 1) of the cylindrical single crystal of SrFe12019 (N z = 0.4) used in Ref. [2]. The experimental technique outlined in Refs. [1,4] was improved by introducing a heat pipe for measurements in finite magnetic fields. The X(oJ) data were evaluated as in Ref. [2]. For the parallel field H z the reduction of L~ is in agreement with previous results on GdC13 for T > T * [4]. This effect and also the behaviour below T * can be explained in terms of Eq. (2); using the magnetostatic result for the period d = d(1 - H z / ( N ~ M s ) ) [11], and the fact that due to the screening of H z by the DW motion, and F remain at their zero-field values. For the first time we report here the results on the kinetic coefficient L w obtained in a field H x = 610 Oe applied perpendicular to the easy axis. In this configuration, we couple directly to the DW order parameter m B (see Fig. 1) and thus strong effects on the mobility/x may be expected on the basis of Eq. (2). The striking result is depicted in Fig. 2, showing that L w is enhanced by H± which is in contrast to the effect of the parallel field H z. This enhancement is largest near T *. In the field H j_ the minimum of L w shifts significantly towards Tc, which suggests that an intrinsic mechanism is responsible for the 'rounded' shape of L~(T) about T*. ........
108 Lw (S.1)
e
........
e
........
SrFe12019
i
,
/ .
O H =0 [2]
• .,,=8oOe
. / ~
~ H±=610Oe
~
EW
"
lOS~ 4
A AA
•
~
"r'~l
ln4
"-10-4
...........
10-3
/ ~
T"
y_...r . . . . . . . . . .
I_T/T c
10-1
Fig. 2. Temperature variations of the domain wall kinetic coefficient L w in SrFel2019 in magnetic fields about the DW phase transition.
To discuss these novel features qualitatively, we note that the component Hy of the perpendicular field (Fig. 1) should have a maximum effect on m B just at the transition, where a root-form dependence ms(T*, Hy)otH~/~ is expected. Indeed, in the MF approach the factor F in Eq. (2) for m B << 1 near the DW phase transition has the form F = [1 + (31r/4)mB(T, H y ) ] ,
(3)
where m a is the solution of the MF equation of state in a perpendicular field Hy
m3 + ( O - 1 ) m B = ( ~ r / 2 ) ~ ,
~ - H y / H A <
(4)
and values ~ and d remain unchanged in small fields. Thus the field dependence of Lw(T, Hy) at T * could be explained by the increase in the order parameter m B with the critical index A, which should be significantly greater than 3, if 2D fluctuations in a DW are properly taken into account. Our first experimental data on Lw(T, H±), which are reported here, show however no distinct root-form dependence on magnetic field at T *. This could be attributed to the same mechanism that is responsible for the rounded shape of L w at H = 0. For the quantitative comparison with the experiment one should generalize Eqs. (3) and (4) to the case of strong magnetic fields ( H ± ~ HA). Another point of concern is the distribution of the domain wall orientations with respect to the applied field H ± . In small fields H ± both components: Hy and H z are equally represented, the component H x being non-critical and having less influence on m B because of the effective in-plane anisotropy e = X ± = 0.2. But in stronger fields H ± the whole DW plane can rotate to the direction of H ± to reduce its energy (if the DW pinning is weak enough), which should increase the influence of H ± on the DW order parameter m B and hence on the DW kinetic coefficient L w. References [1] J. K6tzler, M. Hartl and L. Jahn, J. Appl. Phys. 73 (1993) 6263. [2] J. Krtzler, D.A. Garanin, M. Hartl and L. Jahn, Phys. Rev. Lett. 71 (1993) 177. [3] L.N. Bulaevskii and V.L. Ginzburg, Sov. Phys. JETP 18 (1964) 530, [4] M. Grahl and J. Krtzler, Z. Phys. B 75 (1989) 527. [5] J. Kiftzler, M. Grahl, I. Segler and J. Ferrd, Phys. Rev. Lett. 64 (1990) 2446. [6] L. Landau and E. Lifshitz, Z. Phys. Sowjetunion 8 (1935) 153. [7] L.V. Panina, D.A. Garanin and I.G. Ruzavin, IEEE Trans. Magn. 26 (1990) 2826. [8] D.A. Garanin, Physica A 178 (1991) 467. [9] I.D. Lawrie and M.J. Lowe, J. Phys. A 14 (1981) 981. [10] D.A. Garanin, M. Malang and J. Krtzler, Phys. Rev. B, to be submitted. [11] C. Kooy, U. Enz, Philips Res. Rep. 15 (1960) 7.
ELSEVIER
~l~
|0|NIB| 0|
,~
magneUsnl ond magnetic m~erlals
Strong critical fluctuations in domain walls in Sr hexaferrite: Effect of magnetic fields M. Malang, D. Garanin, J. K6tzler * Institut fiir Angewandte Physik, Universitiit Hamburg, Jungiusstr. 11, D-20355 Hamburg, Germany Abstract
The domain wall (DW) relaxation rate F w of the uniaxial Sr hexaferrite in the region 0.9Tc _< Tc = 740 K, displays a deep minimum at T = 0.99T c due to a continuous 2D phase transition from Bloch to linear domain walls. In fields parallel to the easy axis, F w is reduced by the increase of the DW spacing d, whereas a perpendicular field increases F w. This feature is associated with the increase of the DW order parameter m B with the field near the transition.
Recent experiments on the domain wall (DW) dynamics near the Curie temperature of the hexaferrites BaFe12019 [1] and SrFe12Ow [2] provided the first evidence for the transformation of the Bloch wall (BW) to the so-called linear wall (LW). This phase transition was predicted by Bulaevskii and Ginzburg [3] to occur in uniaxial ferromagnets at some temperature T * close to Tc. Above T* the DW magnetization profile is linear ( M ( x ) I[ ex) passing to elliptic (m B ~ My(O)/M s < 1) for T < T * (see Fig. 1). In the mean field (MF) approximation [3] T * is determined by the condition 0 ( T ~ F ) = 1, where O(T) =- 4 x z ( T ) / x ± .xz(T) cz (1 - T / T c ) -1 and X± = M s / H A = const are the longitudinal and transverse susceptibilities, and m a = ( 1 - 0) 1/2 represents the DW order parameter below Tr~v. It is difficult to observe this reconstruction of DWs because their width 6 is very small. However, the DW mobility /x turns out to be very sensitive to changes in the DW structure. The DW kinetic coefficient Lw = 2M~(oz)iz/d can be determined from the DW relaxation rate Fw = L w / ( X Z 1 + N ) measured by the dynamic susceptibility X(to) parallel to the easy axis, N Z is the demagnetization coefficient. LWs were first reported on the uniaxial ferromagnets GdCl 3 [4] and LiTbF4 [5]. The characteristic divergence of L w at Tc could he described by the following expression [4]:
Lw=3Lz(6/d )
(T> T*).
sents the width of the LW, which is given by the correlation length ~z cx X 1/2 and d is the domain period shrinking as a power of 1 / 6 [4]. This speeding up of L w [4,5] contrasts the well-known result by Landau and Lifshitz [6] predicting L w to decrease continuously towards Tc. In Refs. [7,8] the DW mobility g was calculated over the whole temperature range for the first time, taking into account both transverse and longitudinal relaxation mechanisms. The result
L w = 3Lz(6/d ) .F(ma; L,, L±),
(2)
(6 = 6 a = const for T < T *) shows that for elliptic walls L w rises sharply with m B when lowering temperature, because the effect of the longitudinal relaxation reduces when a circular BW shape is approached, i.e. for rn n ~ 1. For linear walls F = 1 and Eq. (2) coincides with Eq. (1), and at low T the result of Landau and Lifshitz is recovered. Near T * the theory predicts a deep minimum of L w, which has been observed in the experiments on Ba and Sr hexaferrites [1,2].
-
w
zy\
(1)
Here L z is the kinetic coefficient of M z in domains being non-singular at Tc for a uniaxial magnet, 6 = 6L reprex =-
* Corresponding author. Fax +49 40-4123-6368; email: [email protected].
0
0
~
Fig. 1. Magnetization profile of m(x)= M ( x ) / M s for Bloch (m~ = 1), elliptic (m B < 1) and linear (m B = 0) domain walls in unia×ial ferromagnets.
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 0 4 - 8 8 5 3 ( 9 4 ) 0 1 3 9 1 - 8
M. Malang et al. /Journal of Magnetism and Magnetic Materials 140-144 (1995) 1873-1874
1874
The further interesting features of the DW relaxation observed in [1,2] are (i) a significantly lower transition temperature T * = 0.990Tc than T~F = 0.996Tc, and (ii) a very steep increase in L w with lowering temperature for T < T *. The latter could be described by a temperature variation of the order parameter m Bct (T* -T)/3B with the exponent /3a = 0.08 being much smaller than /3~ F = 1 / 2 [3]. These two features could be attributed [2] to strong two-dimensional (2D) fluctuations of m B [9] along the y-axis of the DW, whereas the fluctuations along the x-axis should be supressed by the effective in-plane anisotropy of magnetostatic origin e = M s / H A = X ± (see Fig. 1). In order to gain a deeper insight into the physics of this phase transition, we investigate here the DW relaxation in magnetic fields applied parallel and perpendicular to the easy axis (see Fig. 1) of the cylindrical single crystal of SrFe12019 (N z = 0.4) used in Ref. [2]. The experimental technique outlined in Refs. [1,4] was improved by introducing a heat pipe for measurements in finite magnetic fields. The X(oJ) data were evaluated as in Ref. [2]. For the parallel field H z the reduction of L~ is in agreement with previous results on GdC13 for T > T * [4]. This effect and also the behaviour below T * can be explained in terms of Eq. (2); using the magnetostatic result for the period d = d(1 - H z / ( N ~ M s ) ) [11], and the fact that due to the screening of H z by the DW motion, and F remain at their zero-field values. For the first time we report here the results on the kinetic coefficient L w obtained in a field H x = 610 Oe applied perpendicular to the easy axis. In this configuration, we couple directly to the DW order parameter m B (see Fig. 1) and thus strong effects on the mobility/x may be expected on the basis of Eq. (2). The striking result is depicted in Fig. 2, showing that L w is enhanced by H± which is in contrast to the effect of the parallel field H z. This enhancement is largest near T *. In the field H j_ the minimum of L w shifts significantly towards Tc, which suggests that an intrinsic mechanism is responsible for the 'rounded' shape of L~(T) about T*. ........
108 Lw (S.1)
e
........
e
........
SrFe12019
i
,
/ .
O H =0 [2]
• .,,=8oOe
. / ~
~ H±=610Oe
~
EW
"
lOS~ 4
A AA
•
~
"r'~l
ln4
"-10-4
...........
10-3
/ ~
T"
y_...r . . . . . . . . . .
I_T/T c
10-1
Fig. 2. Temperature variations of the domain wall kinetic coefficient L w in SrFel2019 in magnetic fields about the DW phase transition.
To discuss these novel features qualitatively, we note that the component Hy of the perpendicular field (Fig. 1) should have a maximum effect on m B just at the transition, where a root-form dependence ms(T*, Hy)otH~/~ is expected. Indeed, in the MF approach the factor F in Eq. (2) for m B << 1 near the DW phase transition has the form F = [1 + (31r/4)mB(T, H y ) ] ,
(3)
where m a is the solution of the MF equation of state in a perpendicular field Hy
m3 + ( O - 1 ) m B = ( ~ r / 2 ) ~ ,
~ - H y / H A <
(4)
and values ~ and d remain unchanged in small fields. Thus the field dependence of Lw(T, Hy) at T * could be explained by the increase in the order parameter m B with the critical index A, which should be significantly greater than 3, if 2D fluctuations in a DW are properly taken into account. Our first experimental data on Lw(T, H±), which are reported here, show however no distinct root-form dependence on magnetic field at T *. This could be attributed to the same mechanism that is responsible for the rounded shape of L w at H = 0. For the quantitative comparison with the experiment one should generalize Eqs. (3) and (4) to the case of strong magnetic fields ( H ± ~ HA). Another point of concern is the distribution of the domain wall orientations with respect to the applied field H ± . In small fields H ± both components: Hy and H z are equally represented, the component H x being non-critical and having less influence on m B because of the effective in-plane anisotropy e = X ± = 0.2. But in stronger fields H ± the whole DW plane can rotate to the direction of H ± to reduce its energy (if the DW pinning is weak enough), which should increase the influence of H ± on the DW order parameter m B and hence on the DW kinetic coefficient L w. References [1] J. K6tzler, M. Hartl and L. Jahn, J. Appl. Phys. 73 (1993) 6263. [2] J. Krtzler, D.A. Garanin, M. Hartl and L. Jahn, Phys. Rev. Lett. 71 (1993) 177. [3] L.N. Bulaevskii and V.L. Ginzburg, Sov. Phys. JETP 18 (1964) 530, [4] M. Grahl and J. Krtzler, Z. Phys. B 75 (1989) 527. [5] J. Kiftzler, M. Grahl, I. Segler and J. Ferrd, Phys. Rev. Lett. 64 (1990) 2446. [6] L. Landau and E. Lifshitz, Z. Phys. Sowjetunion 8 (1935) 153. [7] L.V. Panina, D.A. Garanin and I.G. Ruzavin, IEEE Trans. Magn. 26 (1990) 2826. [8] D.A. Garanin, Physica A 178 (1991) 467. [9] I.D. Lawrie and M.J. Lowe, J. Phys. A 14 (1981) 981. [10] D.A. Garanin, M. Malang and J. Krtzler, Phys. Rev. B, to be submitted. [11] C. Kooy, U. Enz, Philips Res. Rep. 15 (1960) 7.