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Mesoscopic quantum tunneling in small ferromagnetic particles
Journal of Magnetism and Magnetic Materials 116 (1992) 67-69 North-Holland
Mesoscopic quantum tunneling in small ferromagnetic particles * C. P a u l s e n b, L.C. S a m p a i o a,l, R. T u c o u l o u T a c h o u ~ r e s A. M a r c h a n d a, J.L. T h o l e n c e b a n d M. U e h a r a d
a,
B. B a r b a r a a, D. F r u c h a r t c,
Laboratoire de Magndtisme Louis Ndel, CNRS, 166X, 38042 Grenoble Cddex, France b Laboratoire de Recherches sur les tr~s Basses Tempdratures, CNRS, 166X, 38042 Grenoble Cddex, France " Laboratoire de Cristallographie, CNRS, 166X, 38042 Grenoble Cddex, France a National Research Institute for Metal, Sengen Sakura-Mura, Niiharigun, Ibaraki 305, Japan
The magnetic relaxation of a set of small ferromagnetic particles of Tb0.sCe0.sFe 2 of mean diameter 150 ,~ has been m e a s u r e d at low temperatures (down to 50 mK) and with a field up to 6 kOe. A crossover temperature has been observed at Tc = 1.2 K between thermal activation and q u a n t u m tunneling regimes. The field dependence of the crossover temperature has been determined, Tca~fH. This is the first example of q u a n t u m tunneling of domain walls on a mesoscopic scale detected in small ferromagnetic particles in agreement with the theory.
In the early 1980's, Leggett et al. [1] attracted attention for the possibility of quantum effect observations on a macroscopic scale. Later, Voss and Weeb [2] at IBM Yorktown Heights, presented a study of Macroscopic Quantum Tunneling (MQT) at low temperatures in Josephson junctions. Up to now, a great advance has been made in this field in Josephson junctions [3]. M Q T effects have apparently also been observed in other systems (depinning of charge density wave in NbSe3, TaSe 3 [4], superconducting wires [5], etc.). Magnetic systems constitute a new and interesting tool of research with possible practical applications [6]. The tunneling of the magnetization vector of a small ferromagnetic particle through its anisotropy energy barrier or the tunneling of a domain wall crossing a small particle through its pinning energ barrier are the
Correspondence to: Dr. B. Barbara, Laboratoire de Magn~tisme Louis N~el, CNRS, 166X, 38042 Grenoble C~dex, France. * Paper presented at the International Conference on Magnetism (ICM '91), Edinburgh, Scotland, 2 - 6 September 1991. I On leave from Centro Brasileiro de Pesquisas Fisicas CNPq, R u a Dr. Xavier Sigaud, 150 Rio de Janeiro, 22 290 Brazil.
new effects to be observed in magnetic systems. Recently, Awschalom et al. [7] used a microsusceptometer to study the magnetic susceptibility frequency dependence of an array of small magnetic particles. They find that the peak in X(~o) grows and becomes independent of decreasing temperature. Our sample is made of Tb0.sCe0.sFe 2 particles of mean diameter 150 ,~ randomly oriented and dispersed in a polymer matrix (details of sample preparation will be published elsewhere [8]). The large anisotropy of Tb 3+ ions allows an increase of the energy barrier and also of the temperature range in which the M Q T could be observed [9,10]. The experimental apparatus consists of a SQUID magnetometer with a superconducting coil and a dilution refrigerator. We have measured the magnetization time dependence near to the coercive field in a hysteresis loop where the maximum applied field is 6 T. More precisely, the hysteresis loop is interrupted in a field H of the order of H c and the magnetization values are recorded during about 104 s. We measured carefully the time relaxation of M(t) for about 50 pairs of (H, T) values with 1.7 kOe < H < 6.3 kOe and 0.050 K < T < 300 K. In this procedure, the mean relaxation time r is the time for which
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
C. Paulsen et al. / Mesoscopic quantum tunneling in small ferromagnetic particles
68
the magnetization vanishes: 1 / 7 = C/.r' where -;' = (1/2Ms)(dM/dt)M_ o and C = 30 for quasistatic experiments [11,12]• This relaxation time is related to a mean energy barrier through the Arrhenius law at high temperatures and through a nearly t e m p e r a t u r e independent exponential at low temperatures [9-12]• Putting l n ( % / 7 ) = E ( H ) / k T * where T * - T at high t e m p e r a t u r e and T * - T c = constant, at low temperature, the consistently linear plot of log((1/2M,)(dM/ dt)M= 0) vs. 1 / H shows that the mean activation energy is proportional to l / H - 1 / H o where H 0 -- 10 kOe is the coercive field in the absence of thermal fluctuations or quantum tunneling (fig. 1). This figure also shows that Tc is of the order of 1 K and % = 310-~3 s. The cross-over temperature is more clearly seen (fig. 2) where we have plotted In H c vs. In T for two different time scales• In the low t e m p e r a t u r e regime, fig. 1 shows a shift of the 1 / H lines from the point of coordinates ( I / H 0 = 0 . 1 , 1/~-~= l o - i i " s ) . This shift is consistent with a field dependence of the crossover t e m p e r a t u r e of the form Tc =A~FH- with
15
, l i l t ' ,
[
[
r
,
,
,
10 :~ 5 :~ 0 0,1
2.0~11
1.6 ~1
m
2
i I
l'I
~ [
[i
i ]
[ '
i
r I
I
I [ i [ l l
r i l l
• log ( 1/-t)=-3
~'I
~ ~Y "
<>I o g ( 1 / ' t ) = - 6
-.
/
",~ ",
0.4
. ........
O.O
I
3
I
ll~J
I
L
2
IJ
I
1
I
I
I
k[J
I
0 I nT(K)
I
I
1
I
[
[
I
i
I
J
I
1
2
Fig. 2. Log-log plot of He(T) versus T (K) for two different measuring timescales, log ~" = 3 and 6. Inset: enlargement of the low temperature region, indicating a lowering of the tunneling rate at the lowest temperatures (possible effect of dissipation).
A = 0.38 _+ 0.05 (T in K, H in kOe). Such a form can be deduced from the work of Stamp on the quant,am tunneling of a single domain wall pinned by a point defect [10]. It can also be evaluated in a phenomenological model of a domain wall crossing a small particle and pinned at the particle boundaries [12]. From these results, it is easy to show that each tunneling event is characterized by a jump of each domain wall involving about 103 magnetic moments• Although not extremely large, this number is sufficient to allow the study of macroscopic quantum tunneling, using magnetic materials.
o~-5 0
8K
@K
K 4K 5K 6K _K
-10 15
....
0
I ....
0.1
I
0.2
I
i
,
,
0.3 0.4 1/Hc(kOd 1 )
i
J
i
J
0.5
,
,
J
J
0.6
Fig. 1. Semi-log plot of the measured reciprocal relaxation time d M / d t versus the reciprocal magnetic field 1 / H at different temperatures. The shift between the lines at T > 3 K and T < 2 K constitutes a qualitative proof for the existence of a finite cross-over temperature. At low temperatures T = 50 mK ( zx), 66 mK ( • ) , 100 mK ( • ), 200 mK (o) and 400 mK (t3). If the field dependence of the cross-over temperature To(H) is taken into account (see the text) the data points measured at T = 2 K, 1.5 K and low temperatures are shifted to the dashed lines.
Acknowledgements We are pleased to thank E.M. Chudnovsky, P.C.E. Stamp and D. Mukamel for fruitful discussions on M Q T in magnetic systems, Ph. Nozieres and R. R a m m a l for their interest on this subject. One of us (L.C.S.) is financially supported by CNPq-Brasil.
References [1] For a recent review on MQT, see: A.J. Legget et al., Rev. Mod. Phys. 59 (1987) 1.
C. Paulsen et al. / Mesoscopic quantum tunneling in small ferromagnetic particles [2] R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47 (1981) 265. [3] J. Clarke, A. Cleland, M. Devoret, D. Esteve and J. Martinitis, Science 239 (1988) 992. [4] R.E. Thorne, J.H. Miller, W.G. Lyons, J.W. Lyding and J.R. Truker, Phys. Rev. Lett. 35 (1985) 1006. [5] N. Giordano and E.R. Schuler, Phys. Rev. Lett. 63 (1989) 2417. [6] L. Gunther, Phys. World 28 (December 1990). [7] D.D. Awschalom, M.A. Mc. Cord and G. Grinstein, Phys. Rev. Lett. 65 (1990) 783.
69
[8] D. Fruchart et al., to be published. [9] E.M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60 (1988) 661. [10] P.C.E. Stamp, Phys. Rev. Lett. 66 (1991) 2802. [11] B. Barbara and M. Uehara, J. de Phys. 47 (1986) 235; Proc. Rare Earth Conf., Durham, Inst. Phys. Conf. series 37 (1978) 203, and refs. therein. [12] C. Paulsen, L. Sampaio, R. Tucoulou Tachou~res, B. Barbara, D. Fruchard, A. Marchand, J.L. Tholence and M. Uehara, Eur. Phys. Lett. (in press).
Mesoscopic quantum tunneling in small ferromagnetic particles * C. P a u l s e n b, L.C. S a m p a i o a,l, R. T u c o u l o u T a c h o u ~ r e s A. M a r c h a n d a, J.L. T h o l e n c e b a n d M. U e h a r a d
a,
B. B a r b a r a a, D. F r u c h a r t c,
Laboratoire de Magndtisme Louis Ndel, CNRS, 166X, 38042 Grenoble Cddex, France b Laboratoire de Recherches sur les tr~s Basses Tempdratures, CNRS, 166X, 38042 Grenoble Cddex, France " Laboratoire de Cristallographie, CNRS, 166X, 38042 Grenoble Cddex, France a National Research Institute for Metal, Sengen Sakura-Mura, Niiharigun, Ibaraki 305, Japan
The magnetic relaxation of a set of small ferromagnetic particles of Tb0.sCe0.sFe 2 of mean diameter 150 ,~ has been m e a s u r e d at low temperatures (down to 50 mK) and with a field up to 6 kOe. A crossover temperature has been observed at Tc = 1.2 K between thermal activation and q u a n t u m tunneling regimes. The field dependence of the crossover temperature has been determined, Tca~fH. This is the first example of q u a n t u m tunneling of domain walls on a mesoscopic scale detected in small ferromagnetic particles in agreement with the theory.
In the early 1980's, Leggett et al. [1] attracted attention for the possibility of quantum effect observations on a macroscopic scale. Later, Voss and Weeb [2] at IBM Yorktown Heights, presented a study of Macroscopic Quantum Tunneling (MQT) at low temperatures in Josephson junctions. Up to now, a great advance has been made in this field in Josephson junctions [3]. M Q T effects have apparently also been observed in other systems (depinning of charge density wave in NbSe3, TaSe 3 [4], superconducting wires [5], etc.). Magnetic systems constitute a new and interesting tool of research with possible practical applications [6]. The tunneling of the magnetization vector of a small ferromagnetic particle through its anisotropy energy barrier or the tunneling of a domain wall crossing a small particle through its pinning energ barrier are the
Correspondence to: Dr. B. Barbara, Laboratoire de Magn~tisme Louis N~el, CNRS, 166X, 38042 Grenoble C~dex, France. * Paper presented at the International Conference on Magnetism (ICM '91), Edinburgh, Scotland, 2 - 6 September 1991. I On leave from Centro Brasileiro de Pesquisas Fisicas CNPq, R u a Dr. Xavier Sigaud, 150 Rio de Janeiro, 22 290 Brazil.
new effects to be observed in magnetic systems. Recently, Awschalom et al. [7] used a microsusceptometer to study the magnetic susceptibility frequency dependence of an array of small magnetic particles. They find that the peak in X(~o) grows and becomes independent of decreasing temperature. Our sample is made of Tb0.sCe0.sFe 2 particles of mean diameter 150 ,~ randomly oriented and dispersed in a polymer matrix (details of sample preparation will be published elsewhere [8]). The large anisotropy of Tb 3+ ions allows an increase of the energy barrier and also of the temperature range in which the M Q T could be observed [9,10]. The experimental apparatus consists of a SQUID magnetometer with a superconducting coil and a dilution refrigerator. We have measured the magnetization time dependence near to the coercive field in a hysteresis loop where the maximum applied field is 6 T. More precisely, the hysteresis loop is interrupted in a field H of the order of H c and the magnetization values are recorded during about 104 s. We measured carefully the time relaxation of M(t) for about 50 pairs of (H, T) values with 1.7 kOe < H < 6.3 kOe and 0.050 K < T < 300 K. In this procedure, the mean relaxation time r is the time for which
0304-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
C. Paulsen et al. / Mesoscopic quantum tunneling in small ferromagnetic particles
68
the magnetization vanishes: 1 / 7 = C/.r' where -;' = (1/2Ms)(dM/dt)M_ o and C = 30 for quasistatic experiments [11,12]• This relaxation time is related to a mean energy barrier through the Arrhenius law at high temperatures and through a nearly t e m p e r a t u r e independent exponential at low temperatures [9-12]• Putting l n ( % / 7 ) = E ( H ) / k T * where T * - T at high t e m p e r a t u r e and T * - T c = constant, at low temperature, the consistently linear plot of log((1/2M,)(dM/ dt)M= 0) vs. 1 / H shows that the mean activation energy is proportional to l / H - 1 / H o where H 0 -- 10 kOe is the coercive field in the absence of thermal fluctuations or quantum tunneling (fig. 1). This figure also shows that Tc is of the order of 1 K and % = 310-~3 s. The cross-over temperature is more clearly seen (fig. 2) where we have plotted In H c vs. In T for two different time scales• In the low t e m p e r a t u r e regime, fig. 1 shows a shift of the 1 / H lines from the point of coordinates ( I / H 0 = 0 . 1 , 1/~-~= l o - i i " s ) . This shift is consistent with a field dependence of the crossover t e m p e r a t u r e of the form Tc =A~FH- with
15
, l i l t ' ,
[
[
r
,
,
,
10 :~ 5 :~ 0 0,1
2.0~11
1.6 ~1
m
2
i I
l'I
~ [
[i
i ]
[ '
i
r I
I
I [ i [ l l
r i l l
• log ( 1/-t)=-3
~'I
~ ~Y "
<>I o g ( 1 / ' t ) = - 6
-.
/
",~ ",
0.4
. ........
O.O
I
3
I
ll~J
I
L
2
IJ
I
1
I
I
I
k[J
I
0 I nT(K)
I
I
1
I
[
[
I
i
I
J
I
1
2
Fig. 2. Log-log plot of He(T) versus T (K) for two different measuring timescales, log ~" = 3 and 6. Inset: enlargement of the low temperature region, indicating a lowering of the tunneling rate at the lowest temperatures (possible effect of dissipation).
A = 0.38 _+ 0.05 (T in K, H in kOe). Such a form can be deduced from the work of Stamp on the quant,am tunneling of a single domain wall pinned by a point defect [10]. It can also be evaluated in a phenomenological model of a domain wall crossing a small particle and pinned at the particle boundaries [12]. From these results, it is easy to show that each tunneling event is characterized by a jump of each domain wall involving about 103 magnetic moments• Although not extremely large, this number is sufficient to allow the study of macroscopic quantum tunneling, using magnetic materials.
o~-5 0
8K
@K
K 4K 5K 6K _K
-10 15
....
0
I ....
0.1
I
0.2
I
i
,
,
0.3 0.4 1/Hc(kOd 1 )
i
J
i
J
0.5
,
,
J
J
0.6
Fig. 1. Semi-log plot of the measured reciprocal relaxation time d M / d t versus the reciprocal magnetic field 1 / H at different temperatures. The shift between the lines at T > 3 K and T < 2 K constitutes a qualitative proof for the existence of a finite cross-over temperature. At low temperatures T = 50 mK ( zx), 66 mK ( • ) , 100 mK ( • ), 200 mK (o) and 400 mK (t3). If the field dependence of the cross-over temperature To(H) is taken into account (see the text) the data points measured at T = 2 K, 1.5 K and low temperatures are shifted to the dashed lines.
Acknowledgements We are pleased to thank E.M. Chudnovsky, P.C.E. Stamp and D. Mukamel for fruitful discussions on M Q T in magnetic systems, Ph. Nozieres and R. R a m m a l for their interest on this subject. One of us (L.C.S.) is financially supported by CNPq-Brasil.
References [1] For a recent review on MQT, see: A.J. Legget et al., Rev. Mod. Phys. 59 (1987) 1.
C. Paulsen et al. / Mesoscopic quantum tunneling in small ferromagnetic particles [2] R.F. Voss and R.A. Webb, Phys. Rev. Lett. 47 (1981) 265. [3] J. Clarke, A. Cleland, M. Devoret, D. Esteve and J. Martinitis, Science 239 (1988) 992. [4] R.E. Thorne, J.H. Miller, W.G. Lyons, J.W. Lyding and J.R. Truker, Phys. Rev. Lett. 35 (1985) 1006. [5] N. Giordano and E.R. Schuler, Phys. Rev. Lett. 63 (1989) 2417. [6] L. Gunther, Phys. World 28 (December 1990). [7] D.D. Awschalom, M.A. Mc. Cord and G. Grinstein, Phys. Rev. Lett. 65 (1990) 783.
69
[8] D. Fruchart et al., to be published. [9] E.M. Chudnovsky and L. Gunther, Phys. Rev. Lett. 60 (1988) 661. [10] P.C.E. Stamp, Phys. Rev. Lett. 66 (1991) 2802. [11] B. Barbara and M. Uehara, J. de Phys. 47 (1986) 235; Proc. Rare Earth Conf., Durham, Inst. Phys. Conf. series 37 (1978) 203, and refs. therein. [12] C. Paulsen, L. Sampaio, R. Tucoulou Tachou~res, B. Barbara, D. Fruchard, A. Marchand, J.L. Tholence and M. Uehara, Eur. Phys. Lett. (in press).