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Quantum tunneling in magnetic systems of various sizes
(invited)
B. Barbara, L C Sampaio, J. E. Wegrowe, B A Ratnam, A. Marchand, C.
Paulsen, M. Novak, J. L. L Tholence, M. Uehara, D. Fruchart
To cite this version:
B. Barbara, L C Sampaio, J. E. Wegrowe, B A Ratnam, A. Marchand, et al.. Quantum tunneling in
magnetic systems of various sizes (invited). Journal of Applied Physics, American Institute of Physics,
1993, 73 (10), pp.6703 - 6708. �10.1063/1.352508�. �hal-01659995�
Quantum tunneling in magnetic systems of various sizes (invited)
B. Barbara, L. C. Sampaio, J. E. Wegrowe, B. A. Ratnam, and A. MarchandC. Paulsen, M. A. Novak, and J.
L. TholenceM. UeharaD. Fruchart
Citation: Journal of Applied Physics 73, 6703 (1993); doi: 10.1063/1.352508
View online: http://dx.doi.org/10.1063/1.352508
View Table of Contents: http://aip.scitation.org/toc/jap/73/10
Published by the American Institute of Physics
Quantum tunneling in magnetic systems of various sizes (invited)
B. Barbara, L C. Sampaio, J. E. Wegrowe, B. A. Ratnam, and A. Marchand
Laborutoire Louis N&l, BP166, 38042, Grenable, France
C. Paulsen, M. A. Novak, and J. L. Tholence
Centre de Recherches sur Les Tr& Busses Tempgratures, BP1 66 33042, Grenoble, France
M. IJehara
National Research Institute for Metals, 1-2-I. Sengen, Tsukuba, Ibaraki 305, Japan
D. Fruchart
Laboratoire de Cri~tallographie, BP166, 38042, Grenobie, France
hfagnetic relaxation experiments constit.ute a unique method of determining the nature of
fluctuations in dissipative magnetic systems. At high temperatures these fluctuations are thermal
and strongly temperature dependent. At low temperatures, where quantum fluctuations
dominate, magnetic relaxation becomes independent of temperature. Such behavior has been
observed in many systems. In this review we emphasize the study of low temperature relaxation
in ferromagnetic nanoparticles, layers, and multilayers (including “domain wall junctions”),
and large single crystals. The results of magnetic relaxation experiments are shown to agree with
theoretical predictions of quantum tunneling of the magnetization. When dissipation becomes
important, in large and complex systems, a time dependent WKB exponent needs to be
introduced.
1. INTRODUCTION
More than sixty years aft.er the creation of quantum
mechanics, the analysis of its foundations still continues to
be a very active field of research.’ Recent advances in me-
soscopic physics, made possible by new technologies of
miniaturization and by the development of highly sensitive
SQTJID magnetometry, have contributed to a renewed in-
terest in this field of research. Some years ago, mesoscopic
physics was limited to transport measurements. Now me-
soscopic physics deals more and more with magnetism.
After the discovery of persistent currents,2’3 which can be
viewed as coherent diamagnetism, we are now on the road
to the discovery of similar quantum-coherent phenomena
in ferromagnetism, ferrimagnetism, antiferromagnetism.
Few attempts have been made to investigate quantum
tunneling of the magnetization (QTM) in the spirit of me-
soscopic physics (see D. Awschalom et af.” and this sym-
posium). This approach is certainly the most interesting,
though rather difficult. The effect of environmental spins
OJI
the large (tunneling) moment has been considered re-
cently by Leggctt’ and Stamp” in the framework of the
difference associated with the parity of these moments.697 It
seems that the large number of environmental spins might
destroy the quantum coherence.
A different approach to QTM is the measurement of
the temperature dependence of the magnetic relaxation
right after an abrupt change in the applied magnetic field.
This technique known as “magnetic after-effect,” is more
than 50 years oldsPy
and is now commonly used to deter-
mine particle size distributions, energy barrier distsibu-
tions, activation volumes, and more generally, to analyze
the process of magnetization reversal (see e.g., Ref. 10).
To the best of our knowledge, Bean and Livingstone” and
Weil” were the first to propose QTM as a possible alter-
native explanation of an upturn of the particle size distri-
bution at low energies (leading to a bimodal distribution).
The present understanding of QTM, however, suggests
that it would have been difficult to demonstrate quantum
effects in nickel due to its weak anisotropy. A few years
later, magnetic after-effect experiments have been per-
formed on several rare-earth-based (highly anisotropic)
systems.*3 These experiments showed two important fea-
tures (i) fast magnetic relaxation at low temperature, at-
tributed to very narrow domain walls (thickness of a few
interatomic distances) and (ii) energy barriers propor-
tional to the reciprocal applied field,
E
al/H, attributed to
domain wall motion via a-dimensional nucleations on the
wall surface. This latter mechanism was motivated by anal-
ogy with ferroelectric domain wall motion (Ref. 13 and
references therein). In the same year, 1973, Egami’” for-
malized this interpretation in two models of 2-d nucleation:
thermal or quantum nucleation of narrow domain walls
with intrinsic pinning. Since 1975, several papers devoted
to the static and dynamic properties of SmCo&ui,, single
crystal have appeared. In particular a crossover tempera-
ture from thermal activation to quantum tunneling of
about 10 K (Ref. 15) have been interpreted by a phenom-
enological model in which domain walls are pinned by
point defects. l6 Similar results showing a crossover temper-
ature of 5 K on Dy,A& single crystal (dominated by in-
trinsic pinning) was interpreted” in terms of Egami’s the-
ory (see Fig. 1). These early results on Dy,Al, are no less
suggestive of QTM than several of the more recent ones.
However, as recently mentioned by Stamp,‘s the modern
story of QTM really starts in 1986 with the detailed anal-
ysis of the SmCo,,&tt,, (Ref. 15) experiments. The effects
of dissipation and possible sample heating in these bulk
systems have been considered later.&
With regard to QTM theory, we should mention that
theory is now more advanced than the experiments. Ex-
cepting the two papers of Egami and a paper of Chud-
novsky (1979),” theory really started in 1986-88 with the
papers of Chudnovsky and Gunther’e and Enz and Schill-
ing.*’ Decisive breakthroughs were made by Stamp with
6703
J. AppLPhys. 73 (IO), 15 May 1993
0021-8979/93/106703-06$06.00
0 1993 American
institute of Physics
6703
01 ’
I
1
%
FIG. 1. Thermal variation of the coercive field of Dy,Al, at the time scale
of 10’ s. The point (0) has been taken at 4.2 K and 10 6 s, The low
temperature plateau with a crossover temperature at 5 K is time depen-
dent (from Ref. 17).
the evaluation of the QTM probability for a domain wail
with dissipation.2” In general, theoretical predictions con-
verge on the idea of extremely weak dissipation eff~ts in
magnetic systems qf nanometric dimensions (see, e.g. Ref.
23).
hdlly, one
must say that the general ideas of mac-
roscopic quantum tunneling (MQT), developed by Leggett
in the early SO’S,‘~ led the way for most, if not all of the
theoretical studies, and certainly stimulated experimental
searches for MQT.
In this paper we show that nonthermal relaxation ef-
fects, similar to those obtained in Dy,Al, and SmCo3,5Cu,,S
single crystals’“‘17 exist in a large number of diverse sys-
tems. In Sec. II we give a comprehensive study of two
systems of nanoparticles (TbCeFe,, 150 A and FeC, 20 A)
in which variations of the energy barrier with field, mag-
netization level, and temperature are determined down to
50 mK. Section IIT discusses some of the amorphous layer
and multilayer results, and in particular, presents first re-
sults on domain wall junctions. In this progression from
the nanoscopic to the mac.roscopic scale, we finally de-
scribe in Sec. IV, the dynamical magnetic and thermal
behavior of some large ferromagnetic single crystals. ‘These
systems are extremely interesting to study the effects of
dissipation in compIex systems. They might show real
MQT events at very short time scale.
H. FERROMAGNETIC NANOPARTICLES
A. Experimental procedure
General aspects of relaxation measurements in mag-
netic systems with thermal and quantum fluctuations are
considered in Ref. 25. When the experimental system is
sufficiently disordered, energy barriers are uniformly dis-
tributed (as in, e-g., amorphous alloys). We then measure
the magnetic viscosity S=dM/d In t. In general, for non-
uniform distributions, it is important to perform relaxation
experiments at constant magnetization M so that llrp does
not interfere with the evaluation of the energy bar-
rier.‘5~2s~2” We usually use &f=O and plot dM/dt vs time.
This defines the “mean” relaxation time T,
6704
J. Appl. Phys., Vol. 73, No. 10, 15 May 1993
f/-r= (l/‘zrw,) (dM’dr)M~o.
(1)
This relaxation time T coincides with the “median”
relaxation time of the distribution, “most probable” relax-
ation time for a symmetrical distribution.
B. Quantum tunneling of domain walls in Tb,,5Ce0.5Fe2
particles
Tb,.,C!~,,Fe, particles were obtained by a reduction in
H, gas. They have a relatively large size distribution
around a mean value of 150 A and a cubic symmetry.
Crystal field effects acting on the total angular momentum
give a large anisotropy of the free energy (5 x lo7 erg/
cm”). The Fe-Fe exchange ene’rgy is also important ( 10”
erg/cm3). At zero field and in thermodynamical equilib-
rium, most of the particles will be single domain (R,- y/
&l.? - lo3 A). Near H,, however, metastable configura-
tions will introduce, on average, one domain wall (of
thickness S- 50 A) per particle (details will be published
elsewhere) .26
In our experiment the magnetization of the
sample was first saturated in a field of 8 T. The field was
then decreased to zero, reversed, and stabilized at a given
value of H close to the coercive field He. The magnetic
relaxation was then measured as a function of the applied
field aild temperature. The low temperature measurements
were made using a high field/low temperature SQUID
magnetometer and a moveable miniature dilution refriger-
ator. The recording of the decay of the magnetization was
continuously measured for 1 to 2 h and the median relax-
ation time defined above was determined. Experimentally T
is directly related to the slope (dM/dt) - ’ of the iW( t)
curve at M=O.
1. Field dependence of fhe median energy barrier
More than 70 different values of T(H,T) were mea-
sured. These allow us to determine the field and tempera-
ture dependencies of the corresponding energy barrier, the
median energy barrier (MEB). We have found that the
field and temperature variations of T(H,T) are well de-
scribed by
T(H,T) =T~ exp [A(l/H- l/Ho)/kT*],
(2)
where Ho is the maximum coercive field, A is a constant,
and T* is the effective temperature defined in Ref. 15
( T*= T at high temperature and T*= T, at low temper-
ature). This expression is illustrated in Fig. 2. The straight
lines represent the values of I/T calculated from the sum of
the thermal activation (TA) and quantum t.unneling (QT)
rates [the T*(T) curve, which is extracted from this plot,
is very close to the one given by the “quantum harmonic
oscillator’* formula].‘5,2’ The best fitting parameters are
1/H0=0.15&0.02 and log 1/~~=8.5*0.5 (coordinates of
the focal point) as well as the crossover temperature
T,=tiOO mK were obtained by a rather sensitive scaling
plot of log(dM/dt) vs (l/H- 1/Ho)/k7’ (this plot was
very similar to the one given here in Fig. 5 for the magnetic
viscosity). The measured data points for T < 400 mK are
clustered on a line very close to the calculated QTM limit
(labeled MQT in Fig. 2). Note that these plots and their
Barbara et al.
6704
-‘ix1
0.2. 0.3
0.4 0.5 0.6
0.7
l/H(koe-l)
FIG. 2. Log(If-T) vs l,Ii in Tb,j&eX,51?e2. The temperature range is
0.05=< T! K) <c IO and the low temperature data are at T-400, 200, 100,
66, and 50 IT&.
extrapolations are very similar to those which establish our
present belief of MQT in Josephson junctions.
2. Temperature dependence of the effective
temperature P(7)
The effective temperature T* is determined from the
slopes ofthe lines representing each isotherm in Fig. 2, that
is l/I’*=& log( l/r)/d( I/R). The variations of l/T*,
versus the reciprocal temperature l/T, are represented in
Fig. 3. At high temperatures a TA regime is observed
where T* = T. At temperatures below 600 mK, T* = T, is
practically independent of temperature. This is exactly
what is expected in the general theory of MQT ( Leggett1”4
as well as in theories of QTM in ferromagnetic sys-
tems.“‘r.24328
This is also consistent with our earlier QTM
experiments on bulk ferromagnets (Ref. 15 and references
therem).
3. Temperature dependence of the tunneling volume
This volume, analogous to the well known TA volume
in thermal activation processes, can be evaluated at any
soo-
/
I ,
I
0
FE. 3. Ii?‘* vs 1fT. The crossover temperature is T,=O.6 K. Inset.:
high temprrxture ‘IA regime. Note that the TA regime extmpolates to
I/y-*-O.
a
b
FIG. 4. Schematic representation of the supposed motion of a domain
wall crossing a particle. A 2-d nucleation takes place on the domain wall
(bj and the nucleated domain wall deformation tnoves from the center of
the particIe to its boundaries (soliton) (b) and (c).
temperature by equating the eftergy barrier E=kT* ln(t/
to) to the Zeeman energy at the maximum coercive ficld.‘s
This volume, V= E/&H= 30k, T*/MsIi is proportional
to the temperature above 1 K (at T= 10 K, Vz.5~ 10”
A”) and is constant below 1 K, Y=2.10” A3). The exist-
ence of such small volumes clearly shows that the dynam-
ics of magnetization reversal is not uniform at the scale of
each particle. The magnetization reversal of each particle
proceeds by a succession of domain wall quantum jumps,
each jump being initiated by a 2-d quantum nucleation on
its surface, followed by classical soliton motion (scheme
Fig. 4).29,“0
4. Barkhausen noise
The
sum
of all such local magnetization jumps in vol-
umes V, leads, above the crossover temperature, to the well
known nonequilibrium Barkhausen noise (BN). Below the
crossover temperature, nonequilibrium BN dominates be-
cause local reversals of the magnetization are induced by
QTM rather by TA. Note that “equilibrium BQN” is cer-
tainly also present due to the many QT events with corre-
lation lengths smaller than the 2-d critical nucleation
length: if the tunneling from say, the A well to the B well
concerns a number of moments smaller than the critical
size required for 2-d nucleation, the elastic domain wall
tension should drive t.he inverse tunneling from B
to A.
Domain wall tension is equivalent here to negative dissipa-
tion and should increase the effects of coherence.
5. Dissipa t/on
The QTM/TA crossover time t,, defined for bulk sys-
tems in Sec. IV is dramatically increased in nanoparticles.
This is because the quasiparticles emitted at each QTM
event should immediately be absorbed in the surrounding
bath due to their extremely large surface/volume ratio.
The effect. of dissipation of different origins here is only
related to the coupling of the order parameter to the de-
grees of freedom of the bath. Our experiment.s on small
particles do not appear to show adverse effects of dissipa-
tion. Only a weak t.emperature dependence T*( T) in
Tb,&eesFe, which might well be due to t.he fact that the
experiment was performed close enough to the coercive
field, which may well have increased the effects of dissipa-
tion.31*32 The classical soliton-like motion which follows
each QTM event is certainly weakly dissipative. The “soli-
ton” will come to rest immediately (Fig. 4) ) unlike in bulk
samples where the classical motion, which takes the form
of a catastrophic avalanche comes to rest after a discret.e
magnetization (and heat) pulse.
6705
J. Appl. Phys., Vol. 73, No. IO, 15 May 1993
Barbara et
al.
6705