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Spin tunneling in a swept magnetic field
View the table of contents for this issue, or go to the journal homepage for more
1997 Europhys. Lett. 39 1
(http://iopscience.iop.org/0295-5075/39/1/001)
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EUROPHYSICS LETTERS July
Europhys Lett pp
Spin tunneling in a swept magnetic eld
L Gunther
Department of Physics and Astronomy Tufts University Medford MA USA
Laboratoire de Magnetisme Louis Neel CNRS BP Grenoble France
received September accepted in nal form May
PACS w Quantum mechanics
PACS Ck Nonmetals
PACS q General theory of resonances and relaxations
Abstract We present a theory whose goal is to account for the recent exciting hysteresis
experiments on the spin cluster crystal Mn
acetate that indicate the occurrence of spin tun
neling We have found the tunneling probability P for the tunneling of spin from one spin state
into another at a spin level crossing in response to a swept applied magnetic eld H It is shown
that P is a function of only one parameter the dimensionless sweep rate given by rh
where r is the rate at which the two energy levels approach each other as the eld is swept being
proportional to the sweep rate dHdt
is the minimum energy gap at the anticrossing
h
being the tunneling rate thereat Numerical integration leads to P exp
We nd
that contrary to widespread belief the experiments cannot be understood in terms of tunneling
of individual spin clusters in the presence of a static applied transverse eld Rather the required
transverse eld must have dynamic nuclear spins and other dynamic spin clusters as its source
Recently there have appeared a number of experimental articles that report the appearance
of steps in the hysteresis of the spin cluster Mn
acetate henceforth referred to as Mn
which has been shown to be in a spin state at the low temperatures K of the
experiments On the practical side such systems are very exciting since they are the prototype
for a highly dense computer hard drive In addition such a spin system provides us with a
rst step in trying to study the phenomenon of quantum tunneling of magnetization QTM
since such a system can be well characterized and consists of an ensemble of identical weakly
coupled units The molecular crystal Mn
has received quite a bit of attention in recent years
and has been very well characterized The composite systems that have been studied are
complicated by having a distribution of parameters such as size and energy barrier upon which
the tunneling rate is exponentially dependent As a consequence it has been very dicult to
characterize the samples so as to be able to interpret the experimental results with condence
Now the steps in the hysteresis curve appear at specic applied longitudinal magnetic elds
H
n
which can be shown to correspond to the energy level crossings produced by a combination
of the splittings due to energy anisotropy and Zeeman splitting In addition the relaxation
time of the magnetization in a static eld exhibits dips at the same elds H
n
however as the
temperature is varied the overall trend of is Arrhenius behavior with an energy barrier that
corresponds to the anisotropy energy As a result of a previous series of experiments on Mn
it was pointed out that thermally assisted resonant tunneling is an essential feature of
the behavior of such systems In agreement with this idea is the fact that which steps appear
in the hysteresis loop is also temperature dependent This is understood to be a result of the
various degrees of excitation of the spin clusters among the spin states since the probability
to tunnel increases rapidly with degree of excitation However taken as a group steps appear
at all level crossings which is indicative of the presence of an internal transverse magnetic
c
Les Editions de Physique
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EUROPHYSICS LETTERS
eld H
T
A central issue is the nature of this transverse eld As to its origin we have the
dipole eld of neighboring spin clusters as well as the nuclear spins A major question is the
extent to which this eld acts eectively as a static eld Another issue is the role of other
interactions in the relaxation process both at xed applied eld as well as in the swept eld of
a hysteresis experiment As has been discussed by Politi et al the interaction of the spin
cluster with phonons may play a signicant role in the tunneling rate This paper does not
deal with relaxation rates at xed eld it relates only to the hysteresis experiments
Imagine that the system of spins starts out in the presence of a large negative applied
longitudinal eld Then a given spin can start out in any one of states of denite S
z
ranging from m to Let us suppose that initially the populations of the spin states
are thermally equilibrated and that in the course of sweeping the eld through zero and on to
positive values the spins can only make transitions to other S
z
states via quantum tunneling
If we further assume that the spins tunnel independently of one another in a static transverse
eld our theory leads to a specic shape for a step corresponding to a given level crossing its
width in particular
Because there are many level crossings at a given eld H
n
the observed step will be a
weighted average of the steps produced at each level crossing In particular the size of a
step M in magnetization will be given by the saturated magnetization M
s
multiplied by
a sum over level crossings m to m
of the following product of factors the probability
Bm that a spin will be found in the state S
z
m when H reaches the value H
n
the
probability P mm
that the spin will tunnel at the crossing and the size of a jump in
magnetization g
B
m
m To the extent that the occupation numbers of the levels are not
thermally equilibrated in between elds H
n
the probability function Bm is not given by a
Boltzmann factor since the populations of the levels at the level crossings will change as a
result of tunneling that has taken place at previous level crossings This issue requires further
investigation
In this paper we focus on hysteresis in a swept magnetic eld as a result of spin tunneling
alone in the presence of a static transverse eld The problem becomes a straightforward
quantumtheoretical problem Its analog is Zener tunneling of a Bloch electron from one
band into another by a swept electric eld The swept magnetic eld corresponds to the
swept wave vector that is induced by the applied electric eld A major dierence lies in the
terminology itself While Zener tunneling refers to the passing of the system ie the electron
from one band into another spin tunneling refers to the remaining of the spin system in the
same energy level Another dierence is that the spin case is vastly simplied in its being
reducible to a twolevel process rather than an interband process involving many energy levels
Our basic result eqs and holds generally for tunneling at a single level crossing that
is converted into an anticrossing by odiagonal matrix elements
Thus we consider a simple twostate system with Hamiltonian
H
A h C
C B h
where generally the parameters A B and C are constant in time In the absence of the
odiagonal element C we have two levels that cross as the parameters h and h
are swept
from to increasing linearly in time The odiagonal elements introduce a gap
C at the crossing so that the level crossing becomes an anticrossing The energy level
structure changes because of the gaps To be specic we will use the spin system of interest
as our example in which case we have
h g
B
S
z
Ht rt and h
t g
B
S
z
Ht r
t
The basic problem is as follows Suppose that in the distant past the system is in the lower
-
l gunther spin tunneling in a swept magnetic field
Fig The level probabilities jb
j
as a function of the reduced time variable The tunneling
probability P is given by the value of jb
j
for innite time
level corresponding to the spin state S
z
m with H equal to Then we are interested
in the probability P that the system is found in the same energy level at innite time after
having passed the anticrossing P is then our tunneling probability In the tunneling process
the spin state changes to S
z
m
Failure to tunnel brings the system into a dierent energy
level but keeps the system in the original spin state S
z
m In this paper we will show that
P is a function of a single parameter the eective dimensionless sweep rate given by
rh
where r jr
rj and
is the minimum energy gap at the anticrossing
We obtain an excellent t to the numerically derived results to within a couple percent for
all with the function
P exp
h
i
We see this function plotted in g As expected the smaller the sweep rate the more
adiabatic the behavior and correspondingly the greater the tunneling rate in our context In
the socalled sudden approximation we switch levels corresponding to no change in the state of
S
z
and an absence of tunneling The probability for Zener tunneling to take place corresponds
to our function P The general qualitative behavior is totally expected what is new is
that we have an expression for the tunneling probability that is valid for all sweep rates
We rst proceed to present an outline of our method and then discuss further the con
sequences of our results We make use of the procedure presented in Bohms Quantum
Theory as follows We rst obtain the energies of the quasistationary states states
for given applied eld H treated as a constant They are well known in the form
E
t E
tt
where
E
AB h h
and t
AB h h
Here
C is the minimum gap between the two levels which occurs at the anticrossing
To obtain the transition probability in a swept eld we start with the timedependent
Schroedinger equation
H ji ih
t
ji
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EUROPHYSICS LETTERS
We let j ti be the set of eigenstates with label of the quasistationary Hamiltonian H
at time t We then expand ji in the form
ji
X
c
exp
i
Z
t
E
t
dt
h
j ti
The E
n
and coecients c t are the corresponding eigenvalues and coecients respectively
of the Hamiltonian In our case we have only two states
Let us next set
a
t
t
t
t
and make the substitution with a
a
c
t b
t exp
Z
t
a
t
dt
Then substitution of eqs and into eq and making use of orthogonality
lead to
b
t
X
b
exp
i
Z
t
E
E
dt
h
a
t
where E
E
ha
i a
can be shown generally to be pure imaginary In our case
a
vanishes and E
E
E
E
t
We now introduce a dimensionless shifted time variable as follows The time at which
the levels cross is at t
A Br The time scale t
r is the time interval over
which tunneling takes place between levels The dimensionless time parameter is given by
t t
t
with the levels crossing at
The amplitudes b
and b
for the two levels can be shown to obey the equations
db
d
b
exp
i
Z
d
p
a
and
db
d
b
exp
i
Z
d
p
b
Now assume that at t the eld is far to the left of the anticrossing and that the system is
in the level so that the amplitudes for the two levels are b
and b
respectively
These equations have been integrated numerically to yield the probability P that tunneling
takes place after the eld has been swept past the anticrossing being equal to jb
j
at t
See g for plots of the level probabilities jb
j
vs time The functional form of eq was
a guess in an attempt to t our results of the numerical integration Remarkably the t was
good to within a couple percent for all This may well be the exact solution to eqs
Let us now apply our results to the case of Mn
which has a spin S The reduced rate
will be seen to vary greatly depending upon the particular level crossing so that the tunneling
can range from being essentially certain to being essentially impossible We assume with Politi
et al the presence of a simple quadratic uniaxial anisotropy energy in a parallel eld H We
omit the quartic S
anistropy since the relaxation and hysteresis experiments indicate
tunneling with changes in S
z
We therefore focus on the eect of a transverse eld H
T
Thus the Hamiltonian reads
H DS
z
g
B
HS
z
g
B
H
T
S
x
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l gunther spin tunneling in a swept magnetic field
We will consider a crossing between the S
z
m level and the S
z
mn level The crossing
then occurs at an applied eld H
n
nDg
B
n
T independent of m We then have
r g
B
m
dH
dt
r
g
B
m n
dH
dt
We take for the minimal gap at the anticrossing
DS
H
T
H
A
mn
f
mn
where H
A
DSg
B
T is the anisotropy eld which we will take as T and
f
mn
S
mn
S mS m n
S mS m n
m n
Even for H
T
as large as G the critical dimensionless parameter H
T
H
A
is
so small as to lead to a great sensitivity of the tunneling probability on the particular level
crossing We have
m nhg
B
dH
dt
DSf
mn
H
A
H
T
mn
m n
f
mn
H
A
H
T
mn
where we have used g and a eld sweep rate of Gs typical in recent experiments
According to eq the transverse eld H
TC
necessary for is a sharply increasing
function of m n As a result P will often be essentially zero for H
T
H
TC
and
essentially unity when H
T
H
TC
which results in an eective discontinuity with respect to
mn For example for n is equal to for m and to
for m Since
the fraction of spins in a sample that occupy such high levels m is much less than unity
a jump at H due to tunneling in our model and in a static transverse eld of G would
not be observable The situation does not change even for a static transverse eld of G
nor for the steps at n
Discussion For small the tunneling probability in a swept eld P
while the tunneling rate at an anticrossing for xed eld is proportional to the rst power of
being given by
h We can understand this result in the case of a large energy gap
as follows Tunneling takes place only over the time scale t
r The probability for
tunneling is then given by the product
ht
The width of a hysteresis step is given by H t
dHdt Since t
is inversely
proportional to the sweep rate the width is independent of the sweep rate being given by
H f H
A
m nH
T
H
A
mn
It would be interesting to study the extent to
which the width depends upon the sweep rate since such dependence would re!ect the eect
of dynamical interactions For the step at n m and a transverse eld as large as
G we obtain a width of about
G which is a minuscule fraction of the observed
width However for the step at n m we obtain a value of G T which is
not much smaller than the observed width
Our results reveal an extreme sensitivity of the tunneling probability with respect to the
level crossing and hence indirectly to the temperature We also note that the dependence of the
tunneling probability upon the sweep rate is not weak in contrast to the logarithmic dependence
that one often encounters in the case of hysteresis of mesoscopic magnetic systems
It is clear that our simple theory cannot account for the observed jumps in the hysteresis
experiments both as far as step height and width are concerned We believe that the dipole
eld acting on a given spin cannot be treated as a static eld spin tunneling is a cooperative
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EUROPHYSICS LETTERS
phenomenon wherein an avalanche process may be taking place among the ensemble of spins
some of which switch initially via thermal activation Secondly the hyperne of the nuclei
acting on the spins must be playing a central role
Additional remark After submitting this article for publication we became aware of an
article by Dobrovitski and Zvezdin DZ on the same problem The authors corroborate our
result for the probability eq with an analytical calculation We have objection to their use
of the inverse of the time derivative of the probabilities jb
j
as a measure of this time width
since the probabilities execute oscillations in their course towards their stationary values We
therefore believe that DZs time width thus underestimates the actual width t
and therefore
underestimates the actual width H
"""
The author is grateful toB Barbara for very worthwhile discussions and to the Laboratoire
de Magn#etisme Louis N#eel Grenoble for their warm hospitality
REFERENCES
Friedman J Sarachik M P Tejada J and Ziolo R Phys Rev Lett
Thomas L Lionti F Ballou R Gatteschi D Sessoli R and Barbara B Nature
Hernandez J Zhang X X Luis F Bartolem
e J Tejada J and Ziolo R
Europhys Lett
Gunther L Physics World Stamp P Nature Chudnovsky
E C Science
Gunther L andBarbara B Editors Proceedings of the rst workshop on Quantum Tunneling
of Magnetization QTM Kluwer Dordrecht
Sessoli R Gatteschi D Caneschi A and Novak M Nature London
Gatteschi D Caneschi A Pardi L and Sessoli R Science
Villain J HartmannBoutronF Sessoli R and RettoriA Europhys Lett
Novak M and Sessoli R in Quantum Tunneling of Magnetization ref p
Barbara B et al J Magn Magn Mater
Politi P RettoriA HartmannBoutronF andVillain J Phys Rev Lett
In actuality the exponent obtained in the t was which we have replaced by
Bohm D Quantum Theory J Wiley New York Chapt There is a misleading remark
therein connected with the transition from eq to eq with no eect on the nal results
The actual problem of spin tunneling involves a more complex Hamiltonian with a S
S matrix where S is the total spin We have proved that the relations eqs and hold
generally close to a crossing that is split by odiagonal terms when the Zeeman energies due to
the applied longitudinal eld here reected by h and h
are small perturbations to the diagonal
terms of the Hamiltonian
In fact our formula reduces exactly to that derived for Zener tunneling if we make the transcription
r
denergy
dt
Ek
k
dk
dt
h
k
m
eE
h
where e and m are the electron charge and mass respectively k is the wave vector to be set
equal to that at the band edge and E is the applied electric eld Our result holds for all sweep
rates while the result for Zener tunneling makes use of a WKB approximation that holds only
for small electric elds corresponding to low sweep rates See Smith R A Wave Mechanics
of Crystalline Solids Chapman and Hall London
Garanin D A J Phys A L
Barbara B and Gunther L J Magn Magn Mater
In a preprint received after this paper was rst submitted for publication Prokofev N V
and Stamp P C E claim that the relaxation time in xed eld is governed by the uctuating
hyperne elds and the spinphonon interaction See also their more comprehensive article on the
role of nuclear spins J Low Temp Phys