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Spin tunneling in a swept magnetic field

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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

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

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

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

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