Quantum relaxation of magnetisation in magnetic particles

Transcription

Journal of Low Temperature Physi~u', Vol. 104, Nos. 3/4, 1996
Quantum Relaxation of Magnetisation in
Magnetic Particles
N. V. Prokof'ev* and P. C. E. Stamp
Physics Departmentand CanadianInstitute of Advanced Research,
University of British Columbia, 6224 Agricultural Rd.,
Vancouver B.C., Canada V6T 1Z1
(Received November 7, 1995; revised May 3, 1996)
At temperatures below the magnetic anisotropy energy, monodomain
magnetic systems (small particles, nanomagnetic devices, etc.) must relax
quantum mechanically--thermal activation is' ineffective. The discrete nature
of the spectrum is important. This quantum relaxation must be mediated by
the coupling to both nuclear spins and phonons (and electrons if either particle or substrate is conducting).
We analyze the effect of each of these couplings, and then combine them
to get results for the physical relaxation of magnetic particles at low
temperature and bias. This done for both conducting and insulating systems.
The effect of electrons and phonons can be handled using "oscillator bath"
representations," but the effect of environmental spins must be described using
a "spin bath" representation of the environment, the theory of which was
developed in previous papers.
Conducting systems can be modelled by a "giant Kondo" Hamiltonian,
with nuclear spins added in as well At low temperatures, even microscopic
particles on a conducting substrate will have their magnetisation frozen over
miIlenia by a combination of electronic dissipation and the "degeneracy
blocking" caused by nuclear spins. Raising the temperature leads to a sudden
unblocking of the spin dynamics at a well defined temperature. We analyze
in turn the 3 different cases of (a) conducting substrate, conducting particle
(b) conducting substrate, insulating particle, and (c) conducting particle,
insulating substrate.
Insulating systems are quite different. The relaxation is strongly enhanced
by the coupling to nuclear spins. At short times the magnetization of an
ensemble of particles relaxes logarithmically in time, after an initial very fast
decay--this relaxation proceeds entirely via the nuclear spins. At longer times
*Also at: Russian Science Center "Kurchatov Institute," Moscow 123182, Russia.
143
0022-2291/96/0800-0143S09.50/0 @) 1996 Plenum Publishing Corporation
144
N . V . Prokof'ev and P. C. E.
Stamp
phonons take over, but the decay rate is still governed by the temperaturedependent nuclear bias field acting on the particles--decay may be exponential
or power-law depending on the temperature. Depending on the parameters of
the particles and the environment, the crossover from nuclear spin-mediated
to phonon-mediated relaxation can take place after a time ranging between
fi'actions of a second up to months.
The most surprising feature of the results is the pivotal role played by
the nuclear spins. The results apply to any experiments on magnetic particles
in which interparticle interactions are unimportant (we do not deal with the
effect of intelTarticle interactions in this paper). They are also relevant to
future magnetic device technology.
1. I N T R O D U C T I O N
1.1. Quantum Relaxation
One of the most thoroughly explored subjects in all of science is that
of thermal relaxation of magnetization, in magnetic systems of all shapes
and sizes. Investigations in this area (which go back many centuries) 1 have
revealed many subtleties, and even today there are many unsolved puzzles
(e.g., the physics of "magnetic avalanches").
Very recently a whole new set of questions in this area has arisen, with
the advent of well-characterised "nanoscopic" magnetic structures. 2 Such,
structures include "made-to-order" magnetic grains, magnetic wires and
superlattice arrays, as well as thin films and spin chains. There are very
obvious applications of such nanomagnets in, e.g., the recording industry,
as well as in information transmission and computing. Theoretical activity
has been particularly intense on 1-dimensional spin systems, 3 and on the
tunneling and coherence phenomenon which should exist in both grains
(involving "giant spin ''4' 5 dynamics) and in the dynamics of domain walls. 6
Both the theory and the experimental activity in the latter field have been
reviewed recently] -9
In this paper we focus on something rather different--the "quantum
relaxation" of magnetisation that must occur in any small magnetic particle, once thermally-activated relaxation has ceased. Although there does
not seem to have been a serious theoretical analysis of such relaxation
before, the conventional view has been that it must proceed via a simple
tunneling process (dissipative or otherwise). The picture we shall derive
here is more subtle, and contains some rather fascinating new physics. Tunneling is certainly involved; but we find that what really controls relaxation
is the nuclear spin system inside the magnetic particle, often with the help
of phonons or electrons. One can put the result in the following way: the
nuclear spins provide the essential "switch" that turns on (or turns off) the
Quantum Relaxation of Magnetisation in Magnetic Particles
145
quantum transitions. As far as we are aware, there has been no previous
recognition that nuclei had any role whatsoever to play in magnetic relaxation, despite the enormous number of experiments in this area. Nor do we
know of any theoretical analysis of the influence of electrons on magnetic
relaxation. There has certainly been no attempt at a theory dealing with the
combined effects of nuclear spins, phonons, and electrons, which is what we
do here. As we shall see, it is impossible to give a realistic analysis without
putting them all together.
The theory in this paper is partly based on previous work on quantum
coherence in magnetic grains, t~ 26.25 In that work a detailed analysis was
given of the way nuclear spins control the dynamics of "giant spins", i.e.,
spins with quantum number S ~> 1. We also analysed the effect of phonons
and electrons in "decohering" the quantum motion of S. The problem of
quantum relaxation is different, in that even if coherence is destroyed,
relaxation may still proceed incoherently. In this paper we will be looking
at this relaxation under two assumptions, viz., (i) that the applied bias is
small, and (ii) that the temperature is low. In fact we will assume that the
bias and temperature energy scales are less than the energy gap separating
the lowest doublet of states of the particle from the higher excited states.
This gap is of order the single ion anisotropy energy-in small particles it is
typically a few Kelvin (although it can be considerably higher).
Considering first insulating particles, we find that in the absence of
nuclear spins, phonons can only relax the magnetisation extremely slowly,
in a small biasing field, because they couple weakly to it. In the absence of
mobile electrons the nuclear spins can then liberate S, by greatly increasing
the phonon-mediated relaxation rate, as well as providing a purely nuclearspin mediated relaxation mechanism-this latter purely nuclear relaxation
mechanism is most important at short times. The internal bias field e acting
on S, generated by the combined hyperfine fields of all the nuclei, can
easily be greater than 10 K, even for very small particles (e.g., for TbFe3
grains), and depending on the size of the grain may be equivalent to an
external field from 100 G up to well over 1 T ! This bias changes with time
in a diffusive way, at a rate governed by inter-nuclear dipolar interactions
(in the absence of electrons). Every so often, as e shunts around, it brings
two states corresponding to different orientations of S into near
degeneracy. At this point the system can tunnel, in the absence of any
phonons, provided the time variation of the nuclear bias field a(t) gives it
time to do so. The phonon-mediated transition rate is increased simply
because it increases rapidly with bias, and the internal nuclear bias field
can be large even in small applied field.
On the other hand, we shall also see that at low T the nuclear spins
can also further inhibit the relaxation of S. If the nuclear spins are allowed
146
N . V . Prokof'ev and P. C. E. Stamp
to come to equilibrium, then at temperatures below the hyperfine coupling
energy, they exert a large negative bias field on S by adjusting their
polarization to be opposite to S, and thus lowering the total system energy
(and likewise raising the energy of the "final state" where S has reversed);
This is analogous to the conventional self-trapped polaron state. Under
these circumstances, the thermal phonons do not have enough energy to
mediate a transition, and so then even in the absence of electrons, S can
again be trapped for astronomical times. Thus, depending on T and on the
nuclear polarisation, the nuclei can enhance or suppress relaxation. This is
for systems containing no mobile electrons.
On the other hand if there are conduction electrons around, either in
the substrate or in the particle, this picture is radically altered. We shall
find that if the concentration of electrons is high enough, they will freeze
the giant spin dynamics quite regardless of what the nuclei may be doing.
This can be regarded as a purely "dissipative" effect, arising from the
Ohmic coupling between the grain Spin and the electronic bath-if the concentration of conduction electrons is high (particularly in the case where
both particle and substrate are conducting), then the effective coupling can
be very large. Amusingly, at temperatures or external biases above the
single ion anisotropy energy, there will be a very dramatic and sudden
"unfreezing" of the giant spin dynamics. The final picture is summarized in
the abstract to this paper.
We shall not try here to address experiments in any detail, for two
reasons. First, each different experimental system has its own idiosyncracies, and we shall see that our theory has many different parameters--it
is simply not possible to give detailed results for all possible cases, and we
feel that it would be preferable to examine particular systems for which
parameters are well known, as test cases. This we intend to do elsewhere.
Second, in most experiments on magnetic grains, the grains are rather close
together. As we shall briefly explain in this paper, this makes it probable
that even the low-T relaxation of an experimental ensemble of grains
proceeds via dipolar interactions between the grains.
Finally, although our purpose in this paper is not to deal with the
implications for magnetic technology involving quantum relaxation, we feel
we should at least make some remarks here about what they might be.
Fig. 1 shows the way in which magnetic computer memory elements have
decreased exponentially in size in the last 40 years--such plots have figured
large in recent discussions of the need for new kinds of computer design,
incorporating elements which operate quantum-mechanically. At present
computers, as well as magnetic tapes, use elements which behave classically, and are stable over long periods of time. This stability over decades
exists because they are big--the energy barrier between two states of the
147
10 25
10
20
IME
1015
10
10
10
5
QUANTUM
THRESHOLD~_
\
\
QUANTUM REGIME
I
1950
I
1960
I
1970
f
I
1980
1990
_ _ J
2000
DATE
Fig. 1. The Atomic Weight (in Daltons) of a typical computer memory element, as
a function of time. Note that the position of the "Quantum Threshold" depends on
temperature. As we shall see in this article, it also depends strongly on what the
element is made of.
element usually exceeds 100kBT even at room temperature. The hypothesized "quantum threshold", below which tunneling is important even at
T = 0 , is usually supposed to be for grains containing roughly 102-104
spins, depending on the material involved. However we shall find that the
threshold between stable behaviour and quantum relaxation can be moved
around a lot, depending on hew the magnetic system is coupled to the
nuclear spins, and whether or not it is coupled to electrons. As noted
above, we shall see that in cases where there is a high concentration of
mobile electrons in the system, one can freeze the dynamics of even
microscopic magnetic systems (containing only 10 spins), for times of
thousands of years or more. Conversely, we shall see how it is possible for
much larger insulating systems (containing 105 electronic spins or more) to
148
N.V. ProkoFev and P. C. E. Stamp
relax very quickly (in #s). These theoretical insights may have some
relevance to the design of future nanoscopic magnetic devices.
In what follows we shall analyse the role of electrons (Sec. 2), phonons
(Sec. 3), and nuclei (Sec. 4), acting in isolation on ~ Then in Sec. 5 we
shall put them together, to give the final rather complex picture. As noted
above, we save detailed discussion of experiments for other papers; but in
Sec. 6 we will indicate the general nature of our predictions.
1.2. The Model, and Energy Scales
We briefly describe the giant spin model here, and the various energy
scales relevant to the physics. The basic idea behind this model has been
explained in detail elsewhere, by a number of authors. 4"5'7'1.'13 The
exchange couplings J~s between electronic spins in the grain are enormous
compared to the anisotropy couplings K= (typically Ju~0.1-0.01 eV,
whereas K~ ~ 1 0 - ~ 1 K). The mesoscopic grains can be treated as giant
rotators, keeping only states IS, m), with S>~m>~-S. The spectrum of
this giant spin is shown in Fig. 2 in a small bias field ~ . Sometimes, as a
model example, we shall use the easy-axis/easy-plane system, having
Hamiltonian
1
H { ( S ) = ~ [ -KII S 2 + K• Sy2 ] - yeS-/7 o
(1,1)
We will usually assume that an external magnetic field is acting along the
z~-direction, so t h a t / t o =zHo; the external bias energy is then ~ u = 7eSHo,
where yes is the grain magnetic moment.
The low energy physics of (1.1) can be understood in terms of the
truncated Hamiltonian
Ho(f) = (2Ao e, cos 7rS - ~ufz) = As'~x - {Hf~
(1.2)
where ex, y, z are the Pauli matrices, and the tunneling splitting Ao is
A o ~"~o e-A~
(1.3)
Other bare parameters are
f~o ~ 2(KliK>) 1/2
(1.4)
Ao ~ 2S(KII/K• I/2
(1.5)
The "bounce frequency" of the instanton transition ~o is also roughly the
distance to the next pair of excited levels; the total barrier height in (1.1),
Quantum ReLaxation of Magnetisation in Magnetic Particles
149
between the 2 semiclassical minima at S = z' zS,
~ is SKI L. Thus if, say, S =
10 4, we might easily have ~o ~ 0.1-1 K, a barrier height ~0.014).1 eV, but
a splitting Ao ~ 10 6 K, or less.
This is certainly not the whole story. Magnetic grains having more
complicated symmetries are not so easily described using WKB/instanton
methods. The "internal" magnon modes of the giant spin are also neglected,
except for those uniform precession modes corresponding to IS, m). There
will also be phonons (spectrally very weak at these low energies) and
possibly electrons (which will interact strongly with all states of the grain).
There will also be spins on the surface of the grain which will couple
to their neighbours differently from those in the bulk. Some of these
couplings may be considerably less than Jo (e.g., those associated with
"dangling bonds"), and so these spins will couple more loosely to S.
However, we shall assume in line with our previous work that these "loose
spins" still have couplings >>~o to g. If not, they can in principle be
treated as part of the "spin bath" environment, along with nuclei and
paramagnetic impurity spins. In some systems there may also be surface
spins that are more strongly coupled to S than J,j (see, e.g., Ref. 32); these
clearly will move rigidly with S. Thus we see that unless there are some
extremely loose surface spins, having couplings ~ ~o or less to ~q, then their
effect will be benign-they simply move rigidly with S. However it is important to bear in mind when thinking about real systems that the measured
value of grain magnetization may not be simply related to S; there may be
some complicated surface reduction factor. Since this will depend on
the particular system involved, and is not well understood, 33 we will
(in common with other authors) not try to give a treatment of this
here, but simply note where our results may be altered by these surface
corrections.
Finally and most importantly, the nuclei in the grain (and also
possibly outside) will couple via hyperfine interactions to each level shown
in Fig. 2. The net result of this is shown in Fig. 3, depicting the nuclear
levels around a single grain level. Typically ~2o > co~ >>A o, where ~ok is an
individual hyperfine coupling; however N nuclei will spread out a single
grain level into a mass of 2 N levels, of Gaussian half-width ~ NI/2cOo, where
COo is the principal hyperfine splitting. For rare earths, where coo ~ 1 GHz
or greater, this half-width can easily exceed ~o for mesoscopic grains, and
the whole "giant spin" model begins to get rather complex.
In this paper we will ignore all physics at energies ~ o or greater,
simply including it into renormalization of parameters of a low-T effective
Hamiltonian. For such a treatment to be valid, we then require
k,~ T, ~ , Nl/2o)o ~ ~~o
(1.6)
150
/IX
O
C
Fig. 2. Schematic diagram of the energy scales in the giant spin, biased by a longitudinal field
H z. The bias e ~ 7eSHz, and f~o is the gap to the next set of excited states, In reality, for a
giant spin, the top of the barrier will be m u c h higher, at energy ~ (Sf~o) above the lowest
states, and there will be a total of 2S + 1 levels associated with the different giant spin states,
between the top and the bottom of the potential.
Before beginning, it is useful to bear in mind three simple points concerning energy scales:
(i) Note first that in the presence of any bias ~/~ which is much
greater than As in (1.2), the system is essentially "frozen" in one or other
of the states ]~ ) or 1~ ) corresponding to the eigenstates of "~ in (1.2).
Suppose, e.g., the system starts in the higher-energy state ] ~ ) at t =0.
Then the probability P(~
~1-1) of finding it 11) ) at time t, in the absence
of any other couplings, is given by
P(~
A2
~/~) = 1 - ~ 5 sin2 Et
(1.7)
151
,u,=0.3
ft-1
O
-t0
-5
0
5
C
Fig. 3. The distribution function W(e) for spin bath levels, around each giant
spin level. Typically the parameter ff governing the half-width of Gaussian peaks
around each polarization line satisfies # > 1 except for very small numbers of
nuclear spins (see Sec. 4 of text).
where E 3 = ~H
2 + As2 Thus the coupling to phonons, electrons, and nuclei is
necessary if we are to have any relaxation at all. Usually one thinks of this
as a kind of spontaneous or stimulated emission, perhaps in the presence
of dissipations in which the requisite energy is taken up by a bath of
oscillators.
In fact this conventional picture of tunneling of a biased 2-level
system, coupled dissipatively to a bath of oscillators, 12 works reasonably
well for a giant spin coupled to either electrons or phonons. This we shall
see in detail in the next two sections. However as soon as we introduce the
spin bath of nuclei, etc., it breaks down completely; this is basically why
our final results are surprising.
(ii) The second point arises from the finite-dimensional nature of the
giant spin Hilbert space. As emphasized by van H e m m e n and Suto 4' 13 this
means that we must use W K B calculations of tunneling rates with great
care when discussing the motion of S. This point is especially clear in
recent work of Politi et aL,14 analysing the Mn~2012 acetate systems; here
152
tunneling appears to proceed via a 4th-order term ~ ( ~ 4 + od4_) in Ho(S).
Consequently transitions may only proceed between states differing by
AS~ = + 4 (at least if we ignore dipolar interactions between the different
molecules in the actual experiments, 23 which will almost certainly play a
role there). To correctly handle such selection rules in the instanton
calculus is not a trivial problem.
Even more important, there is no continuum of final states in this tunneling problem, and tunneling may only proceed if there is near resonance
between initial and final states. In fact, as we have seen, the separation of
levels in the giant spin model is governed by ~o only, whether one is near
the bottom of the wells or near the top-this is true also for a biased giant
spin. This means that if one couples the giant spin to a set of oscillators
(e.g., phonons), one may not in general simply apply the Caldeira-Leggett
tunneling formalism 15 to calculate tunneling, since this formalism assumes
a continuum of final states. For this reason calculations such as those of
Garg and Kim, 16 of the tunneling rate of a strongly biased grain interacting
with phonons, can only be correct when the level spacing is comparable to,
or less than, the linewidth caused by phonon damping. Now the phonon
linewidths depend on the energy difference between initial and final states,
and are very typically very small- as we shall see in Sec. 3, for the biases
~-~ ~ ~o discussed in this paper, the phonon linewidths are typically inverse
hours, and the phonons are an extremely small perturbation on the discrete
level structure. Even in the the strongly-biased case (i.e., where the barrier
is small), one will need quite a large grain in order to make the phonon
linewidth as large as ~o. To estimate how big we note that the phonon
decay rate increases as roughly the cube of the relevant energy difference;
this energy difference, as well as the number of available final states in the
strongly-biased case, will be proportional, in the giant spin model, to S.
Thus in this case we expect the phonon linewidth at the top of the barrier
to go as S 4, and the calculations of Sec. 3 indicate that the linewidth will
be equal to ~o when S is
104 or greater; at this point the calculations
of 16 become valid (provided, that is, one ignores the effect of the nuclear
spins-in fact they will have a rather strong effect). If on the other hand one
tries to do phonon-mediated tunneling calculations in this strongly-biased
limit for smaller S, the discrete level nature of the spectrum is important.
We do not study this problem here; a successful theory might start from
the framework of, 17 and add phonons.
~
(iii) The third point has been a major theme of our work since 1992,
and concerns the correct way to deal with the "spin bath" of nuclei and
paramagnetic impurities, which is coupled to the grain. From Fig. 3 one
sees that if the number N of spins in the spin bath is large, then the level
153
spacing inside each broadened giant spin level complex will be very small.
Now even though it then makes sense to treat these levels in a continuous
spectrum approximation, one should not make the mistake of mapping the
problem onto an oscillator bath model. This is because the fundamental
postulate involved in the mapping to an oscillator bath is that the
couplings to the environment are very small compared to all other energy
scales, so that they can be handled by 2nd-order perturbation theory (see
for example Feynman and Vernon, 18 pp. 153-159). However the nuclear
spins are coupled to S by interactions which are in general much stronger
than the tunneling energy scale, because the latter is exponentially small;
and any paramagnetic impurity spins in the immediate environment will
couple even more strongly to the giant spin. Thus we believe that calculations of grain tunneling in the presence of nuclear spins which use an
oscillator bath representation of the spin environment (such as Ref. 19) are
invalid for any S, and are fundamentally misleading. We explain this point
more fully at the end of IVA. To deal with the effects of the spin environment we therefore resort to a "spin bath" description of this environment-the theory of this was developed in Refs. 10, 24-27. In this theory one
rewrites the coupled grain/spin environment sytem at low energies in terms
of a low-energy effective Hamiltonian, whose form can be derived by
standard instanton methods for a given microscopic Hamiltonian. This
low-energy effective Hamiltonian couples a two-level system (representing
the magnetic particle dynamics) to a set of N 2-level systems which represent the individual environmental spin modes. In the process of truncation
of the high-energy microscopic Hamiltonian to this low-energy form, the
spin environment degrees of fi:eedom acquire a rather complicated interaction with the 2-level system representing the grain. However the fact that
we are dealing only with 2-level systems means that the Hamiltonian is
tractable, and in fact can be dealt with exactly over most of the parameter
range. The method of solution is of course rather different from the
influence functional methods used for the oscillator bath models of the
environment. In the present paper we do not repeat any of the development
of this theory, but simply use it to deal with the magnetic relaxation
problem.
We now turn to the detailed treatment of the different relaxation
mechanisms. Readers wishing to see only a summary of the results should
proceed directly to Sec. 6.
2. G I A N T S P I N S A N D T H E E L E C T R O N
BATH
In this section we develop a somewhat crude model for the coupling
between the giant spin and the electron bath only, and use it to calculate
154
N.V. Prokof'ev and P. C. E. Stamp
relaxation rates for the grain magnetisation. Since, as far as we are aware,
no attempt has ever been made to deal with the coupled giant spin/electron
system before, it is clear that we must begin from first principles. The full
development of our model, including detailed discussion of the multiplescattering of the electrons by the individual spins inside the grain, the
effects due to surface spins, to the discreteness of the electronic spectrum
inside the grain if it is on an insulating substrate, the interaction between
electrons and the internal degrees of freedom (magnons) in the grain, etc.,
will not appear in this paper. We shall also set aside questions about how
conduction electron dynamics outside the grain, in a conducting substrate,
will be influenced by the grant spin dynamics. This is a very important
question for any experiment (which can use the substrate magnetoresistance as a probe of the grain dynamics), but is too complicated to deal
with here.
There are 3 physical situations which we will address here. The first
involves conducting electrons both inside and outside the grain; we shall
see that this problem can be analysed fairly clearly. Likewise the problem
of an insulating grain interacting with conducting substrate can be
modelled in a reasonably treatable way. The problem of a conducting grain
and an insulating substrate is more messy; however we can fairly easily
extract the main results. Our essential result is that unless the substrate is
insulating, electrons simply freeze S, unless S is very small. This result is
sufficiently clear-cut that we do not believe it will be changed by
refinements of the model. As stressed in the introduction, the calculations
here are a preliminary to the full calculation which include nuclear spins
(Sects. 4 and 5).
2.1. Conducting Grain and Conducting Substrate
We shall start by assuming that the giant spin, contained within a
volume Vo ~R3o, interacts with an electron fluid which permeates freely
through the boundary between grain and substrate; i.e., a Hamiltonian
Z ei4"~ct1~+g,~cA,/3
H = Oo(*~g) + E ~kC~, o-Clc, cr -}- 1 E J i s i " (~<xflE+
:c, a
i E Vo
k :1
(2.1)
describing short-range exchange interaction between the localized individual spins si, located at positions ri inside the grain, and conduction
electrons in momentum eigenstates f ) ; here ~i is the on-site spin operator,
the ~-~ are spin-l/2 operators, and the c).~ operator creates a conduction electron eigenstate. This model is crude--a more refined treatment
155
would couple grain electrons in eigenstates I/~) to substrate electrons in
f ) , via a transfer matrix ~ across the boundary. However it is intuitively
obvious that this is unnecessary once I~P~;]> Ae~, where Ae~ is the typical
level spacing between internal electron states of the grain. If the electronic
bandwidth is D, then Ae~ ~ D/S (we assume here that the number of unit
cells in the grain and S are of the same order of magnitude-clearly if the
grain happens to be a dilute magnetic system, or if one is dealing with a
magnetic macromolecule in which the unit cell contains a large number of
other non-magnetic groups, then Ae~ ~ xD/S, where x is the effective dilution). Now for most metals D ~ 1-5 eV; thus if S ~ 103, Ae~ ~ 10-100 K, so
that an insulating layer at least 10 ~, thick would be required to make such
a model necessary.
We assume in using (2.1) that the internal grain spins are locked
together to form a giant spin, and that the Ji already take account of the
virtual mixing between internal excited states of the magnetic ions and of
the giant spin (magnons, surface modes) and the electron bath. Thus, as
usual, we shall be working at low T and small /~o. However we shall
assume that the Ji can vary; near the surface of the grain boundary they
may be a little weaker. The present model does not include "loose spins",
i.e., surface spins which are more weakly coupled to the motion of S (the
reader should not confuse the Kondo coupling Ji here with the inter-spin
exchange J~ between spins inside the grain). These will be discussed in
Sec. 4 onwards. Typically the Kondo couplings Ji ~0.1-1 eV. Notice,
incidentally, that the physics of the interaction here between conduction
electrons and the giant spin is really quite different from that occuring
between a magnetic domain wall and conduction electrons. 34-36 The latter
problem, which is treated by looking at the coupling of the electrons to the
slow spacetime variation of magnetisation in the domain wall, leads to very
small dissipation. Here the opposite is true-the dissipative coupling will be
found to be very strong. The reason is of course that in the domain wall
problem the electron states change almost adiabatically to adjust for the
presence of the wall, whereas here they experience a very abrupt change in
environment upon meeting the grain. The same point, incidentally, applies
to phonons, which have a small effect on domain wall tunneling. 6' 7
We now rewrite (2.1), incorporating the giant spin hypothesis, to get
a volume averaged interaction
H~,, ~ ~ ]S. ~-~ Z Z F4c~+~.:,cL ~
F
~ d3F F
4=jG~o p( )e ~'7'=
(2.2)
(2.3)
156
N . V . P r o k o f ' e v and P. C. E.
Stamp
where the form factor F 4 integrates the number density p(f) of microscopic
spins over the grain volume, and f is the mean value of the Ji. H)~t is a
sort of "giant spin Kondo Hamiltonian", and as such contains a lot of
interesting physics, much of which we will ignore in this paper. Recall that
the Kondo coupling for microscopic spins is just (2.2) with p(f) in (2.3)
restricted to a single lattice cell; for this problem one conventionally defines
a dimensionless coupling g = IN(O) ~ 0.1 for most metals, where N(0) ~ p/D
is the Fermi surface density of states. The Kondo problem can be mapped
onto an Ohmic spin-boson problem of the kind discussed by Leggett et
al.; 12 even for this spin-l/2 system, there is a large dimensionless Ohmic
coupling to the electron bath; 0~x ~ O(1).
It is possible to start with (2.1) and build up an analogous description
by considering the individual Kondo scattering from each spin, in many
different orbital channels. Here we shall finesse this move by instead
integrating out the electrons directly, starting from (2.2). We first truncate
directly to a biased spin-boson Hamiltonian of form (for Ho(~) = Hoa(S) in
(1.1)):
N
b - - A o~x--~H~z+~1 E
Hell--
N
m~( 22+ c%xk)
2 2 + ~7
_ ~ CkXk
k=l
(2.4)
k=l
where the bias field is parallel to the easy axis (we shall ignore transverse
fields here), and {xk} describe a set of harmonic oscillators, in the usual
way; 12 notice that ~H is the external bias energy as before, and also that the
coupling constants ck are not to be confused with the conduction electron
creation operators defined previously. The Caldeira-Leggett spectral function for this problem is
Jb(co)= rc ~ [ckl2 g(e)--C0k)--=rC~b0~
2 x= ~ mkO~k
(2.5)
and the Ohmic coupling constant % is determined by a standard Fermi
surface average 2~ 21 over electrons, i.e.,
% = 2g2S2( [FI;_~' t2 )e.s.
(2.6)
The naive guess that % ~ S 2, apparently confirmed by (2.6), is however
wrong; carrying out the average, one gets
o~b=2g2S2j;d3Fd3tT' {sinkFlt'--Ftl~z
v ~ \ k~lV-V'l )
f
gS
~2
g2S4/3
(2.7)
\~)
This at first surprising result is a consequence of interference between
Kondo scattering at different sites--notice that k i I ~ Ro, where R o is the
157
particle radius, so that the problem is a multiple scattering one. A related
point is that the estimate of ~b in (2.7) looks perturbative--it is not clear
it should work for large %. To understand how this works, notice first that
for a grain of radius Ro, many orbital electron channels, having t ~<I. . . .
where /~ax ~ (kFRo) 2, are playing a role in the Kondo scattering. If we
again ignore a possible "magnetic dilution" of spins in the grain, then
(kFRo) 2 ~ S. If S ~ 10 3, the phase shifts in each channel are small (one has
~1 ~ gkeRo ~ gSa/3), making it possible to apply a perturbation theory,
even though the sum over all the channels can be rather large. The case of
large phase shifts, ie., for S > 10 3, is more complicated, and may result in
a giant spin dynamics different from that described by the standard Ohmic
model. We address this possibility in Appendix A.
For mesoscopic spins (S ~> 10 3) the dimensionless coupling :% in (2.7)
is very large; we may take over standard results from the spin-boson
problem to discuss its effect on the dynamics of S. For any 0%> 1,
coherence at zero bias is destroyed at any temperature; at T = 0 the grain
dynamics are rigidly locked, and the spin S cannot move. At finite T and
~,~ = 0, the system fluctuates incoherently between the states [ {}) and ]ll )
at a rate ~ (k~T/~o) 2~b- ~; on the other hand if ~.v > T, one has incoherent
relaxation at a rate ~({/~/f~o) 2~b-~. The general result for the relaxation
rate is ~2"2~
G -~ =2Ao ~oo
~
cosh \ 2 k B r /
r[2~b]
(2.8)
where F [ z ] is the Gamma function; for ~H= 0, this gives
~Ao~F2[o~b] ~2rckBT~ 2abv;'(T) = A o \ n o ] F [ 2 % ] \ ~ ]
(2.9)
On the other hand for very low T, such that ~H/kBT>> 1, it gives
5~eI(~H)= 2~'A~(-~/ ~ ~1o o ~H~
/ ] 2e:b-
(2.1o)
The crucial thing to notice here is that for microscopic spins (S ~ 100),
there will be easily observable relaxation of S. On the other hand if S is
much greater than 100, the giant spin will be frozen completely, even over
astronomical times, unless either bias or temperature is large (~H, k , T ~
f~o). Moreover, in this model we would expect that applying either a large
158
Stamp
bias or temperature would lead to a sudden "switching on" of the relaxation at some critical bias or temperature (however recall that our model is
not strictly valid for ~/~, k , T ~ ~2o, since higher levels of S will then enter
into the picture).
To see this, consider 2 examples. First. suppose that g 2 10 2 and
S ~ 3 0 , so eb ~ 1. Then (2.9) shows that for bias ~ u = 0 , we have r - l ( T )
A2kBT/~2o, and ifk.T:O, we have r I(T)~A2o~n/~ 2. Since Ao2/~o might
be something like 100 Hz when A o ~ 106 Hz, this implies low T or low ~H
relaxation times of fractions of a second.
On the other hand, consider a second example where g 2 10-2 but
S ~ 3 0 0 , so ~ b ~ 2 0 . Then if ~ H = 0 , we find that z l ( T ) ~ 10-12A2/~ o
(2~k~T/f~o)4~ and if T = 0, we find v - l ( ~ n ) ~ 10-46A~/~2o(~/f~o) 4~ Then
if ~,~= 0 we see that the relaxation very rapidly switches on at a temperature kBT~D.o/2~ (if we are interested in a relaxation time of order
seconds); for temperatures below this, relaxation times increase catastrophically (if T~ Tel2, we find r(T) ~ 106 years!). The same happens as a function of bias when T ~ in- Obviously if we go to even larger grains, with
S~> 1000, this switching process becomes almost a step function of temperature. Although our two-level model starts to break down once ~k8 T,
~H ~ ~2o this will not affect the validity of this switching result, or our
calculation of the switching temperature (we further note that the switching
temperature is also roughly the temperature at which the crossover to
classical overbarrier transitions takes place; it shows that the relaxation at
higher temperatures will be classical).
We have not included nuclear spins in these calculations--in Sec. 5 we
shall see how a combination of nuclear spins and electrons affects the
dynamics of S.
2.2. Insulating Grain and Conducting Substrate
The insulating grain can be analysed in a similar way, provided we
gloss over some of the complications arising from the variation of both the
electron density and the spin properties in the vicinity of the surface. Thus
we naively extend the previous model by writing an effective interaction
Hamiltonian
ZZ
s t
(2.11)
k q
d3V
(2.12)
159
where now the form factor takes account of the decay of the conduction
electron density 1%(,~)12 as one penetrates into the sample. By the same
manoeuvres as before this leads to an Ohmic coupling
o:s ~ g ~ S 2/3
(2.13)
where g~ ~ g, but is multiplied by some "scale factor" which describes the
depth to which the electrons penetrate into the grain; clearly it can vary
widely.
The reduction to only near surface spins makes ~, ~ ab for large S, but
from (2.13) we see that cq~>O(1) if S>~103. For smaller grains, i.e.,
S~< 100, one can have % < 1, and then results different from the "frozen
spin" behaviour of (2.8)-(2.10) are obtained. In the usual way 12' 2a we may
define a renormalised splitting As, and a damping rate F,., by
A s = A o ( A o / ~ o ) ~a/(1 C~s)
(2.14)
F s = 2~rcqke T
(2.15)
At T = 0 one can get ~2 coherent motion if % < 1/2; however any such
coherence is destroyed once Fs ~>As, which even for microscopic values of
S (i.e., S ~ 10) will occur at extremely low temperatures, usually much less
then 1 mK. For higher T w e find, when ~ U = 0 and ~,.< 1, that r -1 is still
given by (2.9), which in terms of A~ is
Tel(T) = 2 A s F2[c~s] (27~kBT'~ 2~
For finite ~
(2.16)
and T, the form (2.8) is still valid, as is (2.10) when
~H>~ T.
From all of these results we conclude that for !ow T and ~g, the
relaxation dynamics of mesoscopic grains will be extremely slow in the
presence of a conducting substrate. Thus, although our models have certainly not included all possible nuances of the surface physics, one conclusion is very clear and unlikely to be affected by further refinements, i.e.,
that for k~T, ~H ~f~o, mesoscopic magnetic grains on conducting substrates will have their spin dynamics completely frozen. However once
either kg T or ~,~ is of order ~o or greater these dynamics will rapidly be
liberated-again, because this is also the crossover temperature to classical
relaxation, above this switching temperature the system relaxes classically.
Again, these statements will be modified if nuclear spin effects are important (Sec. 5).
160
2.3. Conducting Grain and Insulating Substrate
In the opposite extreme case when the substrate is insulating, the
character of the grain electron states is quite different--they form a set of
discrete states l/a, #} inside the grain, with mean spacing Ae, ~ D/S for
bandwidth D (again, we ignore magnetic dilution here). We describe these
states via creation and annihilation operators A~.~, A,,, ~; these states will
be rather complicated, even in the absence of spin-orbit coupling. It is very
useful to separate the exchange scattering interaction into bulk and surface
parts, where now the bulk part sums over the whole sample, assuming that
all the Ji are the same, whilst the surface part describes the difference
between -/4
b~'~ and the true interaction Hamiltonian H~,,t; thus if
~ int
Hint
= 2' 2
Ji( ~i 9~/~)
(2.17)
c~ci,/~
i~ vo
with the cll ~ creating electron on site i, we write
int
iE Vo
*
= ~S (S'~-~/~) Z c~ci.#==-~0v ( g ~#) ~ Ai,.~A~,/J
i e Vo
(2.18)
p
and the surface term
HiS = 1 ~ (Ji - ] ) ( ~ #~#) c~ ~c~. #
(2.19)
iE Vo
We notice that the bulk term is diagonal in both the site representation and the exact eigenstate representation, because of completeness of
states. This is important, because the condition ] >>~o means that the electron spins inside the system will rotate rigidly with S, and from the last
form in (2.18) we see that there is no way this adiabatic rotation can excite
the low energy orbital states 1/~,or) of the system. Thus the effect o f /~4- b"lk
int
is simply to renormalise the moment of inertia of the giant spin.
On the other hand the surface term does have off-diagonal matrix
elements (/1] HS, ]/~') and this may result in infrared electron-hole pair
production when S flips. The calculation of the corresponding coupling
parameter ~, proceeds in exactly the same way as in the previous subsection (the corrections to the plane-wave calculation are small in a large
grain with k r R o >> 1). Thus at high temperatures the internal conduction
electrons constitute an Ohmic bath with
% ~ g2 $2/3
(2.20)
161
(here gs is given by ( J i - ] ) N(0) averaged over the surface). Now we
notice that this result holds only at T>Aev. At lower temperatures the
effective coupling constant goes to zero exponentially with T:
~fr ~ %e -a~/ker
(2.21)
and the infrared renormalisation of the tunneling rate stops at
A, = A o
\ao/
,
(for Ae,,
f2o)
(2.22)
If Ace, >>f~o one has A, ~ Ao, and we do not have to worry about conduction
electrons in the grain at all because they will rotate adiabatically with o~.
The onset of coherent motion in zero bias can be found by comparing
the damping rate Fs = 2~oc~rkBT with A,. Expressing all the parameters in
terms of S we find the criterion of coherence in the form
kBT skBT
Ael, ~
1
ln(2~g2SZ/3D/SAs)
(2.23)
For experiments at mK temperatures and small bias ~.,v< Ae~, this means
that electronic dissipation effects can be ignored in such grains provided
S~< 105. On the other hand for k ~ T and/or ~ u > Ae~,, we return to the case
of strong Ohmic dissipation which looks very much the same as that for an
insulating grain on a metallic substrate.
3. G I A N T S P I N S A N D T H E P H O N O N
BATH
In the absence of electrons (and of nuclear spins) relaxation in a
biased system proceeds via phonons. This is a very slow process--typical
quantum relaxation times will be seen to be of order months. A number of
related previous studies have appeared. Garg & Kim ~6 and Chudnovsky22
have discussed grain tunneling in the strongly-biased (very small barrier)
limit, within a straightforward Caldeira-Leggett approach, assuming a continuum of final states available for the tunneling--we have already
explained potential problems with this approach in the introduction. Much
closer to the present theme is the work of Politi et al.~4 who have given a
thorough discussion of the effects of phonons on tunneling in the Mn~2012
acetate system, including the effects of discrete initial and final states, and
the nondiagonal couplings (to be defined below). They also take proper
account of the symmetries and selection rules in the tunneling transitions.
Whether their theory explains the relaxation measurements in Mn12012
acetate 23 is another matter--in our opinion the explanation may have more
162
Stamp
to do with dipolar interactions between the molecules. The treatment of
phonon effects we shall give is rather different from theirs, in that we are
dealing with a biaxial giant spin Hamiltonian; otherwise it is similar in that
our tactic is, as usual, to first find a low-energy effective Hamiltonian for
the problem. We first discuss the form this effective Hamiltonian should
take, depending on the symmetries of Ho(S) and the direction of the
applied bias field, and determine the relaxation of the giant spin dynamics
for the different possible couplings. The values of the relevant couplings are
then estimated -no attempt is made at exact calculations, since these
would depend so much on the detailed nature of the system in question.
Our main result is that these couplings are so weak that phonons by themselves are incapable of causing anything but very slow relaxation of S, even
for the biaxial giant spin Hamiltonian. This result is perhaps not too surprising; but we shall also need the quantitative expressions as "input" when
we come to deal with the realistic calculations of relaxation in the presence
of both phonons and nuclear spins.
3.1. Effective Hamiltonians, and Relaxation Rates
We start by assuming, for ease of discussion, a simple easy-axis, easyplane bare giant spin Hamiltonian in a bias (Eqs. (1.1), (1.2)). The
phonons are described by He, = ~4 v 4( b ~b r + 1/2), with frequencies v 4 = cq,
where c is the sound velocity, and b~ is the Bose creation operator (we take
account of polarization later, and ignore optical modes). Then the general
form of the truncated interaction is, up to linear couplings to the phonons,
given by
[
A~
(3.1)
q
1
x 4 (2m4v4)172 (b*q+b 4)
(3.2)
where Cq(fflo) and C~ are effective spin-phonon coupling constants-henceforth we call C a the "diagonal coupling", and C~- the "non-diagonal"
coupling. These couplings will be determined below. Notice that this coupling form is more general than the standard spin-boson coupling, 12 which
corresponds to the diagonal coupling only.
It is unusual to consider non-diagonal couplings (i.e., couplings in e_+),
since in general their effects are reduced, compared to those of diagonal
properties, by a factor (Ao/~o) 2 (or (~H/~o) 2 for large bias); cf. Ref. 21.
The reason for this is simply that any coupling nondiagonal in the site
index of the tunneling variable should be proportional to the overlap
163
integral ( t~ [ll ) ~ Ao/f~o. The diagonal coupling contains no such factor,
and must be rather viewed (in the framework of perturbation theory) as
providing transitions between the eigenstates of the two-level Hamiltonian
(1.2). Since the matrix element of f~ between the two eigenstates of (1.2) is
equal to Ao/E (or Ao/~ H for large bias) we arrive at the above estimate for
the relative role of diagonal and non-diagonal couplings in the standard
spin-bosun model assuming comparable C~(Ito) and C~-.
However we shall see that in magnetic systems the diagonal couplings
are zero unless we apply a "tipped field," which contains both a component
along the easy axis and a component away from it. The reason for this is
very simple--time reversal symmetry implies that the magnetoacoustic
interaction cannot distinguish between grain states $1 and $2 when S~ =
-$2 (a point which also arises in the interaction of phonons with domain
walls), 6 and diagonal couplings describe interactions which do distinguish
such states. Applying an external field allows such a coupling, at least in
principle, by breaking time reversal symmetry-in fact we will get a diagonal
term if the external field projects a component of S1 on S2- Consider our
easy-axis/easy-plane system in a weak magnetic field /Jo = (Hx, 0, H,).
Calculating the equilibrium directions of ~ and $2 in external field, and
then the difference [ ( S ~ ) 2 - ( S ~ ) 2 ] / S 2, we find the diagonal coupling in
terms of the non-diagonal coupling. In the case where the magnetoacoustic
coupling is ~ (S~) 2, then
3
2
- 7'e/4xH-
4(KII)3
c§
q
(3.3)
where C ) is independent of field; the form of C ) , in terms of the
microscopic Hamiltonian, is derived in the next subsection.
On the other hand if the magnetoacoustic coupling contains a term
~ S z S X then the difference [(S1)(S~ ) - ( S ~ ) ( S ~ ) ] / S 2 will be much larger
than that following from the (S-~)2 coupling, and we will get instead
c;_-
2(KII) 2 C 4i
(3.4)
Although we do not believe such a diagonal coupling will ever compare in
strength with the non-diagonal one, we will keep it non-zero in the present
section, to compare with the non-diagonal results (in subsequent sections
it will be ignored-readers wishing to see the modifications to the phononmediated relaxation in these sections can refer to the expressions given just
below).
Before evaluating the couplings in (3.1), we first give the results for
relaxation rates that derive from it. Just as for the electron bath, these rates
164
are very low, but for the opposite reason--instead of being frozen by a very
strong coupling, S is frozen, except over rather long time scales because the
effective coupling is so weak!
The relaxation rates are derived by first defining Caldeira-Leggett
spectral functions
2"s
= B~(~/OD)
12 6(co- v~)
(3.5)
"~
(3.6)
where ~ = z, L; this introduces the low-frequency couplings B~. These spectral functions describe the phonon effects on grain dynamics via 2nd order
perturbation theory. It has of course been known for decades that they are
very weak for co ~ OD, the Debye frequency (note that m ~>3 in 3 dimensions); this is the tgndamental reason for the coherent motion of polarons
and defects in insulators at low temperatures. 21 We will see below that
m = 3 in our case.
We start with the case where there is no transverse field Hx, so that
C ~ = 0 in (3.1). The relaxation dynamics for this case are described as
damped motion between S = +_2S with a relaxation rate (see, e.g., Ref. 21)
Z'--I(r
E 2
=
~
A ;v
2 ~o2 ~/= J•
coth(flE/2) ~ 4B• ~oo (~H/OD)2 G
~
+. cX,
(3.7)
(3.8)
where in the absence of nuclear fields the longitudinal bias ~H= 7~SH~ in
agreement with. 14 This rate is of course very small ( r - t 4 Ao, ~a). Notice
that r - ~ ( ~ / ) is quite different from that rate which would be calculated via
a naive application of the spin-boson results; 12 in fact it is smaller by a factor (~H/f~o) 2 (compare with the more standard "spin-boson" calculation
leading to Eq. (3.9) below), just because we deal with a non-diagonal
coupling.
On the other hand application of a tipped external field /-Io gives a
diagonal coupling, which can be incorporated into a spin-boson model; one
thus finds a relaxation rate
r - ' ( H o, ~ ) ~ 2 ~5 J~(E) coth(fiE/2) ~ 4Bz(Ho) \|
|
(3.9)
which, at a first glance, is independent of ~ as long as ~ / ~ T, but since
C~ ~ H z , we still have r l(Ho,~H)~2H if the bias energy is entirely due
Quantum Relaxation of Magnetisationin Magnetic Particles
to the applied field. Using (3.4) we may express Bz in terms of B•
write
2 (Ao,
r ~(Ho)~4B• \ 2(KII) 2 ] \|
2 r
0-~
165
and
(310
Thus we have reduced the problem of calculating the phonon-induced
relaxation to the determination of the coupling constant B• (or equivalently, of Ca) which we do in the next sub-section.
3.2. Magnetoacoustic Couplings
The magnet|
coupling is of course very well understood; 43 its
effect on the WKB tunneling action of a grain has been analysed by Garg
& Kim ~6 and by Chudnovsky. 22 There has been some confusion about the
relation between Chudnovsky's "angular momentum coupling" and the
usual magnet|
coupling, which we try to resolve here.
The essential point we wish to make in this sub-section is that the
couplings C~, in (3.1), and the corresponding B• in (3.6), are given to
within dimensionless constants ~ O(1) by
IC~l ~ 2S~o 141
(3,11)
B• ~ $2n2/| D
(3.12)
so that the relaxation rate in a longitudinal field//~ is finally
r-~(~/) ~ S2Ao A~ (TeSH~)3 coth \ ~ ]
(3.13)
on
whereas the relaxation rate in a field with non-zero transverse components
Hx has an additional term
z21(H~ ~ S2A~163 \2(KII) 2 |
~coth
\ 2-~jT]
(3.14)
By comparing (3.13) and (3.14) we find their ratio to be r-~(~u)/
r21(Ho) ~ [2S(Kll)2/y~H~o]9-. This ratio is large unless the external field
is almost parallel to ~ and the Zeeman energy overcomes the anisotropy
barrier. We do not consider this exotic possibility and thus ignore the
diagonal coupling in what follows.
To get some ideas of the relaxation rates implied by these formulae, let
us consider a situation where S = 103, A o ~ 1 MHz, T ~ 50 mK, | ~ 300 K,
166
Stamp
and ~H "~ y~SH~ ~ 5 0 mK. Then we get V(~H)~ 106 sec (nearly 2 weeks).
Thus, if we ignore nuclear spin effects, the relaxation times at 50 mK are
already very long; and Ao ~ 1 MHz is actually a large value for Ao. If the
bias energy is further reduced to the Ao scale we get astronomically large r.
To get (3.11) and (3.12) we start by noting that Chudnovsky's angular
momentum coupling, given in terms of the lattice displacement field ~7(7) by
HAM=J dV .(Vxff(~'))
(3.15)
G
integrated over the grain
in the magnetoacoustic
angular momentum to
coupling has the general
volume, is nothing but that set of transverse terms
coupling which is responsible for transporting
the transverse phonons. This magnetoacoustic
form
HMe = (~e/S) 2 f~ dFa,klS, SjUk,(i:)
(3.16)
where ukl is the strain tensor, and where the coefficients ~eaijk~
2
are energy
densities, which are linearly related to the magnetic anisotropy constants
(i.e., to K H and K=), since this is the only energy scale that can be involved.
From this we see that for the grain, HME ~ SK Iql' However we also have
~ = iE s, Ho(S)] - n o S
(3.17)
and so HAM ~ S f o 14[ also; the direct relation to the a~kl can be easily
determined for any particular Ho(S ) and lattice type from this commutator.
It follows now that C~ ~ 2 S f o 141 in (3.1) because ~ is non-zero only
when the giant spin undergoes transitions between the states I~ } and 111 }
and thus may generate only transverse coupling, in agreement with symmetry considerations presented above.
It is instructive to rederive this by estimating the correction 6A to the
bare tunneling action in a given strain field ukl, i.e., finding
Ao( { Ukl} ) = floe A({~k~})~ Aoe-d,A({uk,} )
(3.18)
This is determined trivially from the interaction HAM by integrating it over
time (assuming a negligible change in the instanton path); writing
~A({UKI})=~ C~ xr
q fo
(3.19)
167
we get
= 2Sno
e*4
•
2
= 2Sf~oF~(#l • ~+~).~
(3.20)
where F e is the form factor, and ~v is the polarization of the relevant
phonon; for long wavelength phonons ( q R o ~ 1) this gives us equation
(3.11); and B• in (3.12) immediately follows using (3.5).
It is sometimes useful to write all these couplings in terms of the relevant sound velocity, using 0 4 ~ p c s, where p is the host mass density;
usually this will mean the transverse sound velocity since it is usually lower
than the longitudinal one. Then without specifying exactly which of the
many combinations of the aukt enter into our expression for some particular
symmetry, but just calling this combination K a, we write
JJ_(~)
~
$2 K~ co'
pc 5
(3.21)
Finally, we remind the reader that none of the analysis in this section
has taken account of nuclear spins.
4. G I A N T S P I N S A N D T H E S P I N B A T H
We now turn to a very different kind of bath from the phonon and
electron bath, viz., the "spin bath" made up principally from nuclear spins
both inside and outside the grain. As noted in the introduction, and in our
previous work, lO.24-a7 we do not believe this spin bath can be mapped onto
an oscillator bath (a point which we shall explain in more detail below),
and so a completely different kind of theoretical framework is necessary to
describe its effects on mesoscopic or macroscopic quantum objects.
The main point of this section is to show how the spin bath, interacting in isolation with a giant spin ~, will basically block any relaxation
at all, unless either the grain happens to be in a total field (produced by
the sum of the external field and the "internal bias field" generated by the
nuclei) which is very small, less than an "energy bias window" of width
roughly equal to Ao; or the nuclear spin diffusion mechanism of "energy
bias diffusion" allows the system to find the resonance window. In what
follows we first briefly recall the 4 mechanisms by which the spin bath
controls the giant spin dynamics, and set up the formalism required to
analyse magnetic relaxation in the presence of nuclear spins. We then
derive the relaxation dynamics of S, first ignoring the internuclear spin
168
N.V. Prokorev and P. C. E. Stamp
diffusion (to see how S is frozen), then including it (to show how S can
then relax). We shall then find the rather amusing result that a small fraction of grains in an ensemble can actually relax rather quickly; the overall
relaxation of the ensemble is found to be roughly (but not exactly)
logarithmic in time, until very long times.
The theory in this section does not take account of interactions
between the nuclear sub-system and paramagnetic impurities, which can
lead to T~ processes. In fact the physics governing the T~ relaxation in
small magnetic particles, and indeed any insulating magnet at very low T,
is not well understood at all. Theoretically one expects literally astronomical values of Tx, and although the measured values can often be months or
years, they are still many of orders of magnitude too short for the theory. 42
Given the mystery surrounding the low-T longitudinal relaxation, we will
ignore it until we we come to discuss physical results-at this point we will
deal with it phenomenologically. At very long times this means that our
theory will need to be supplemented by experimental measurements of Tt,
since it will be an adjustable parameter in the theory.
4.1. Effective Hamiltonian, and the 4 Coherence-Blocking Mechanisms
In an earlier series of papers 1~ 24-26 we have constructed a theoretical
description of a giant spin interacting with a more or less arbitrary spin
bath, and solved this model for the quantum dynamics of S in the absence
of any external bias. Elsewhere we have also used this model to deal with
coherence experiments in SQUID's, and discussed the role of the spin bath
in the quantum theory of measurement. 2v
Here we generalise this work to deal with a biased giant spin. As in the
previous papers, we shall assume that the giant spin can be truncated
to two levels, and that k~T, ~ o ;
all effects of higher levels will be
absorbed into the parameters of an effective Hamiltonian. This Hamiltonian
was derived in Refs. 10, 24-26, and is written for the biased case as
Heff= 2Ao f ~+ COSl OP+ ~ l (%ffk + l~kVk)'~-k] + H.c.}
N
N
_ye .tL+ 2z ky, (Oklk
,,4. ~kq-k ~= l
= 1
• " "ak.
(4.1)
There is also an inter-nuclear interaction of dipolar form; thus in
general we have to add to (4.1) a term
Ha,p({g~}) = E
k~k"
,Vkk,
' ' ~akak
"*" ^e
(4.2)
169
We shall not need to write down the matrix elements V2~, in detail;
their strength is ~ T 21, the transverse nuclear relaxation rate. Typically we
shall assume t h a t / t o is parallel to the easy axis of our easy-axis/easy-plane
system, so that the total effect of/-]o will be written as
~/~S/Io ~ ~ :
(4.3)
where ~H = '/eSHo. The spin bath is described in (4.1) by spin-l/2 variables
{gk}; the vectors Nk, b'~,, ~, and rfik are unit vectors.
Equation (4.1) may seem rather intimidating, but is not too difficult to
understand. Let us first recall the meaning of the various terms. The
"static" couplings between S and the {ff~} are described by the last two
terms, and the dynamic effects are given by the terms in the curly brackets.
The longitudinal static coupling co~ tells us the change in energy of ~k
when S flips between •
thus the vector co~l-k represents the difference
between the effective field acting on o~k before S flips and that field after
flips. If the {ak} describe nuclei, this field will be almost entirely hyperfine
in origin, unless the { ~ } are produced by truncating the levels of a higherspin nucleus (i.e., a nuclear moment ~ with [fk] > 1/2); in this latter case
the total field acting on the {gk} will also include, e.g., quadrupolar contributions. The transverse vector co~•
represents the sum of the fields
acting on ~k, before and after S flips.
The dynamic terms in (4.1) originate as follows. The phase ~ = ~S +
Zk ~bk is the sum of the "Kramers phase" ~S (see Introduction--notice that
in (4.1) we have not yet included cos ~S into the definition of Ao) and a
renormalisation ~ ~bk of this phase, caused by the {d~}. The term containing
c~k describes the amplitude for gk to flip, under the influence of ~, when
flips; both ~b~ and ~k partly derive from the unitary "transfer matrix" irk
describing the effect of a tunneling S on #k:
],~k
.,~q~,,
i~ -=T~
~ Zki~}
/ = e -il d~~,,(~) IZk}
i?~ = e _+i(~,~k.~+~)
(4.4)
(4.5)
Here Hi,,t(r ) describes the microscopic interaction between S and ~k; JZxi'~}
is the environment state before S flips, and Iz{~'} after S flips. The signs
+ refer to the path traced by S whilst flipping. The term in ~k and further
contributions to ~bk and 0c~ are necessary because environmental spins also
have an effect on the tunneling action for S (in exact analogy with the nondiagonal phonon terms discussed in Sec. 3).
The derivation of (4.1) has been described in detail elsewhere.l~ Here
we shall simply quote the values of the parameters in (4.1) that arise if we
start with the microscopic hyperfine Hamiltonian
170
]
N
HHyp = H~o ( ~) + ~ k~=, coke. [k + Hd+
=
(-KLIS~+K•
+ 2 COkS'fk +Hdip
(4.6)
k=l
i.e., our usual easy-axis/easy-plane model coupled to nuclei via hyperfine
couplings cog. One then finds, to order ~coh/~o
= ~S
(r
= O)
~h = c ~ k / ~ = rccoh/2f~o
(if cob < F2o)
(4.7)
CO~ ~---COk
•
COb
= (2,
ffk = --2
(4.8)
The assumption in (4.7), that the hyperfine coupling cob ~g2o, is
almost always true. In any case ~h, ~h, and Ch can be calculated as general
functions of cob/fro, for any initial microscopic Hamiltonian (in many cases
analytically). ~~ It is useful to bear in mind the physical meaning of the
parameter c~h in this regime; ~h is the amplitude for ffh to coflip with S.
Note also that more general interactions than (4.6) will produce further
renormalisations of Ao.
Working with (4.1) (which is of course nothing but a low-energy effective Hamiltonian, in the usual spirit which also led to (2.4) and (3.1))
allows us to bring out the four essential physical mechanisms operating in
the dynamics of S. These are
(i) Even when/]o = 0, a longitudinal internal field bias e = ~fi~ Nk = l (Dh(Tk
II z
acts on S, whose effects were first discussed in Ref. 25 (and more recently
in).lo,26, 27,41 Typically for nuclei we expect the hyperfine couplings to be
tightly clustered around a principal value coo, with coo ~>A o. There will
however be a spread dcoh, due not only to inter-nuclear couplings (principally
via dipolar or Nakamura-Suhl interactions, with dcoh ~ 10 4-106 Hz) but
also other couplings such as transfer hyperfine couplings. One may then
define a "density of states" W(e) for internal bias e (Fig. 3); if we first define
a "hyperfine spread" parameter tt by
# = N 1/2 dcok/coo
(4.9)
then if/~
=
171
N 1/2 (~(Dk/(Do ~ l, W(e) consists of Gaussian peaks
1
G,(e-(DoAN/2)=r~drl/2 exp{-(e-(DoAN/2)/C2 }
(4.10)
F , = it(Do
(4.11)
of width
around e = (DoAN/2, where A N = N T- N l is the total nuclear polarization,
inside a Gaussian envelope of width N~/2(Do; this envelope extends out to
a maximum bias _+N(Do, and so
W( e ) = 2
~v y , CO~u + AN)/2 G,u(,~ _ O,)o AN/2)
(4.12)
AN
where C~ is the binomial coefficient (Here we assume that N = e v e n
integer; otherwise A N = 0 is impossible! The modifications required for
N = odd are simple, and irrelevant to the ensuing discussion).
In most cases, however,/~ > 1, and the different "polarisation groups"
(i.e., the different peaks) completely overlap, and we end up with a
Gaussian envelope for W(e), i.e.,
W(e)-*fexp{-2e2/((D2oN)};
(#>1)
(4.13)
(D o
where f = x/~/TrN (cf. Ref. 25).
Now for tunneling to occur at all, the total bias (e + ~H) must be ~ A o
or less; if (e + ~ ) ~>Ao, the grain is simply trapped. The fraction of grains
in an ensemble having bias ~<Ao is roughly A ~Ao/N&o k if/~ ~ 1; in the
more usual case where tt > 1, we have a fraction A ~ Ao/NI/2(D o. The result
for / t ~ 1 arises because only a fraction ~Ao/NI/2c~(Dk of that portion
f = ~ / r c N of grains having AN = 0 can flip when the different polarisation
groups do not overlap.
Thus only a small fraction A of spins are not "degeneracy blocked"
from tunneling (in the absence of nuclear dynamics, the effects of which are
discussed below and in a Sec. 5). This degeneracy blocking mechanism
operates just as easily whether there is an external bias or n o t - - i n both
cases only grains in the small window of bias can make transitions.
(ii) Even in the absence of an external thermostat, the nuclear
system can change the internal bias e. This occurs via dipolar interactions
between the nuclei, at a rate ~ T [ 1 (Ref. 10). This pairwise flipping of
172
nuclei conserves AN but allows the nuclear bath to "wander in bias space"
over the full range of bias energy associated with a particular polarization
group, i.e., over an energy range ~#mo N1/2 (~09k.If the spread &ok is due
entirely to dipolar interactions, so &% ~ T 2 1 then the fluctuating bias
covers the whole energy range in a time T2. This fluctuating bias e(t) can
destroy coherence, 1~ but it can help magnetic relaxation, by helping the
system find the small bias window. We shall call this second mechanism the
"energy bias diffusion" mechanism.
=
(iii) The third mechanism, called "orthogonality blocking, ''~~
arises because the transverse fields cok~ acting on the environmental spins
(which are typically <~r,~ll~cause a mismatch between the initial and final
nuclear wave-functions, in a way reminiscent of Anderson's catastrophe. 28
Defining sin20k . . .~.k • I ~ k,ll (assuming Ok ~ 1), this blocking effect can be
parametrised by ~c, where
N
e ~= [ I cos Ok
(4.14)
k=l
so that ~c~ 1/2 Zk 02. Orthogonality blocking inhibits both coherence and
relaxation of ~; it spreads out the high-frequency (~Ao) response of the
grain to lower frequencies.
(iv) A final interaction mechanism arises simply because the giant
spin, whilst making a transition, can flip the nuclear spins.~~ 25, 26 This process is parametrised by the ~k in (4.5); for the whole spin bath one finds
that on average a number 2 of spins will be flipped each time o~ flips, where
= 1/2 Zk ~ for 0~k ~ 1. This causes phase decoherence; we have previously
called this "topological decoherence" because it adds a random winding
number to the effective action for S. It has a decoherence effect on tunneling;
we shall see that its effect on quantum relaxation is actually to allow for
transitions mediated by the nuclear system only, and this process increases
the short-time relaxation rate of an ensemble of grains.
The formal discussion of all these mechanisms has appeared in our
previous work-the task of this section is to elaborate their consequences for
magnetic relaxation. In the next two sub-sections we will deal first with the
dynamics of S in the absence of spin diffusion between the nuclei, to show
how the nuclei keep all but a tiny fraction of grains frozen; then we show
how spin diffusion changes this picture, and allows roughly logarithmic
long-time relaxation, and fast short-time relaxation.
Finally, let us return to the question (noted in the introduction) of the
relation between the spin bath description of the nuclear (or other)
environmental spins, and a possible oscillator bath description. To justify
our belief that a spin bath can almost never be mapped to an oscillator
173
bath in this way, we examine the conditions under which this mapping can
be made. As discussed in Refs. 15, 18, this mapping certainly can be made
if the couplings to the environmental modes are small enough to be handled by 2nd-order perturbation theory. In many physical systems this is
assured since the couplings are ~ O(N- 1/2), because the relevant modes are
delocalised. However this is not true for the spins in the spin bath; not only
is each spin localised, so that the coupling strengths to the quantum system
are independent of N, they are also rather strong. In fact for perturbation
theory to be strictly applicable, we would require
(a)
the weak coupling condition col! ~ co~, and
(b)
that ~b~, 0~, and ~ be extremely small (more precisely, these
parameters have to be ~ N-1/2).
Under these conditions 27 one can write down a Caldeira-Leggett spectral function, describing a coupling to a set of oscillators having masses mk
and frequencies f ~ , and having the form
J(co)='2 ~-m-~s
7c v ~ ( ( O k )
o~
2-"----v--OtCO--COk
k CO~
•
(4.15)
Now in the usual Feynman Vernon/Caldeir~Leggett formalism, one
requires the weak coupling condition Ck ~ k ; here we have the correspondence Ck=CO~, and f ~ k__- - ~ k• , which gives the condition (a) above.
Clearly it is highly unlikely that this condition will be satisfied (unless there
are very strong quadrupolar contributions to the effective field acting on
the nuclei, one will instead have co~ >>co~-!); and as N becomes larger it
clearly becomes more and more difficult to satisfy condition (b) (notice
that condition (b) requires, inter alia, that the number 2 of nuclei flipped
during a transition of S be much less than unity).
Thus we see that the perturbative treatment of the couplings to the
bath almost inevitably will fail in the case being studied in this paper. One
might object at this point that we have only shown that any mapping to
an oscillator bath must be highly non-trivial; one could still try and argue
that there may exist a mapping to some set of oscillators (only very distantly
related to the nuclear spin degrees of freedom by some highly non-linear
canonical transformation) which couple only weakly (indeed, ~ 0(N-1/2))
to ~q). This is certainly possible in principle (it actually works for the case
of magnons coupling to a magnetic domain wall, < 7 where one can make
a mapping from the original magnon states to triplets of magnons).
However we have not the slightest idea what such a transformation might
be, nor what the relevant spectral function J(co) would look like. In fact as
we shall see, the effect of the spin bath on the dynamics of S is so unlike
174
that of the oscillator baths we are aware of that it seems to us very unlikely
that any reasonable mapping could reproduce them. We do not belabour
this point any more here, as it has been made previously in a number of
different forms. We note also that some similar remarks have been made in
a quite different context by Shimshoni and Gefen. 4~
4.2. Nuclear Spin-mediated Tunneling Rate: Formal Expression
We assume that neither the grain nor the spin bath is connected to an
external thermostat except at time t ~<0 (in reality both are coupled to
phonons and possibly electrons--see next section). We also neglect nuclear
spin diffusion for now, and estimate its role later. Then relaxation of ~q, in
a bias, can only occur if the bias field due to the {#k} allows states I ~ ;
{ a k } ), with the {ak } in some polarisation state AN i", to overlap in energy
(within ~ A o ) w i t h some states l i t ; ' l o~k" ) ) , where the polarisation state
AN ~" ~ AN s" in general. Thus we expect some nuclear spins to flip, if ~ is
to relax, although if r is large, and the applied bias is small, one can even
have energy overlap between initial and final states having the same AN ~ 0
(this does not, of course, mean that no spins are flipped during the transition, but only that none have to be flipped).
The formal treatment of this problem is a generalisation of our treatment of the unbiased case. We write, for the time correlation function P(t)
giving the probabality of finding the state [1} ) at time t if it was 10 ) at
t = O, the form derived in Ref. 10:
VI/'(g) e
P(t; T; ~r) = 1 + f de
Z(~)
fie
N/2
~ = -N/2
[P~(t,e +~-
MC~
l]
(4.16)
where Z(fl) is the partition function, so that P(0) = 1, and fi = 1/kT is the
inverse temperature.
This formula is crucial, and so we now spend a little time explaining
it (the derivation was given previously). 1~ The average ~ de is over initial
internal bias, with both thermal and density of states weightings, for an
ensemble of grains. PM(t, a) is a function defined in Ref. 10; it describes a
single grain in an ensemble having bias e and for which, every time S flips,
the polarisation state of the {ak} changes by 2M:
PM(t) =
dy e y . . . .
~dcp
F ~ , ( v ) e 2ivy* ~P~l(t,a,q),y)
(4.17)
175
In this last equation, P~) describes a simple biased 2-level system (cf.
Eq. (1.7))
2
A~(q~, y) sin2(EM(q~ ' y) t)
2 =
EM
e2
(4.18)
(4.19)
+AM2
Av(c,o, y ) = 2 Ao Icos cpJv(2 ~ { 2 - 2') Y)I
(4.20)
and )~= 1/2 Y~k %,2 2' = 1/2 52~ 0~(n;) 2 (so that 2 >~2').
Eq. (4.17) for P ~ ( t ) can be understood as combining a "phase average"
2 f ~ F~,(v)e2iV(*-~l;
F ~ , ( v ) = e -42v2
(4.21)
v
over a phase q~ and a winding number v), with an "orthogonality average"
f ;dy e-Y
(4.22)
These averages are performed over the biased 2-level correlation function
P(~~ e, q~, y), in which the tunneling amplitude AM(q~, y) depends on
M, cp, and y via (4.20). Notice that AM ~Ao(2-2')M/2/M!, which falls off
very steeply once M exceeds (2--2')1/2; the tunneling amplitude for processes involving higher M is negligible. The meaning of these averages has
been explained elsewhere. 1~
We see that Eq. (4.16) for P(t) involves an independent sum over processes in which the polarization state of the nuclei changes by 2M each
time S flips. For those who find this form peculiar, a more lengthy physical
discussion is given in Appendix B.
We may simplify Eq. (4.16) by changing variables from ~ + ~ MCOo/2-~ e, and write it as
P(t; T; ~/r
1 + ~ f de
W(e - ~H + MCOo/2)e-~(~-~"+ M~oo/al
M
• [PM(t, e ) - 1]
z(fl)
(4.23)
Then because the function [PM(t, e ) - 1 ] ~ ( A o / e ) 2 for large bias, and in
most cases W(e) is a smooth function on a Ao scale (for i~coo > Ao, where
/~ was defined in Eq. (4.9)), we have
176
N . V . Prokof'ev and P. C. E,
P(t; T; ~H)= 1 + ~ W(Ma)~
Stamp
e ~(~Ve~Oo/2-r
Z(~)
xfde[PM(t,e)--l],
(#COo >>Ao)
(4.24)
(we assumed here that flA o ~ 1). In the opposite limiting case # = 0, when
the Gaussian in (B1) is a 6-function, we have
1
P( t) - 2Nz(fl) ~, c~N- + MI/2e ~M~176
t, O)
(4.25)
M
where C,~ is again the binomial coefficient.
To summarize, we may understand (4.16) in a fairly simple way as
including all those effects of the spin bath on the dynamics of S which arise
in the absence of any internal dynamics of the spin bath itself. These effects
come from the averages over phase (Eq. (4.21), over the orthogonality
mismatch between initial and final spin bath states (Eq. (4.22)), and over
the internal bias e, acting on S, caused by the spin bath-one finally then
sums over all possible changes of the bath when S flips. It is perhaps worth
noting, for those who may be used to the theory of "oscillator bath"
environments,18 that the averages appearing here for the spin bath environment are very different in form, for the simple reason that in the case of the
spin bath, most of the dynamics of the spins comes fi, om their coupling to the
macroscopic system itself (in our case, to S). By contrast in the oscillator
bath models, the coupling to the macroscopic system is weak ( ~ 0(N-I/2),
for each oscillator), and the dynamics of the individual oscillators is only
weakly perturbed by the system. This is why, in the theory of the oscillator
bath, one can first calculate the weakly perturbed oscillator motion as a
function of the system coordinates, and then integrate out the oscillators by
functional averaging. No such manoeuvre is possible for the spin bath (or
indeed any other environment where the couplings are not weak).
We now turn to what has been left out of Eq. (4.17) for PM(t), i.e.,
everything coming from the independent dynamics of the spin bath (independent, that is, from S). Formally this can be put in by including these
dynamics in P~) in (4.18). As we argue below this corresponds to allowing
a time-dependent bias e(t) in (4.16) and (4.17).
We start by neglecting the effect of coupling of the nuclei to phonons,
which is actually an incredibly small effect in most cases. In reality the
nuclear spin dynamics comes almost entirely from nuclear spin diffusion,
parametrized by the transverse relaxation time T2, and caused by the
dipolar interaction in (4.3). As we shall see presently, if the spin bath is
177
mainly composed of nuclei, this spin diffusion is crucial, since without it ~q
cannot relax at all. Spin diffusion processes, in which AN remains
unchanged but pairs of nuclei flip, cause e(t) to fluctuate in time because
of the variation gCOkin the coupling of each nuclei to ~q (thus, for a process
]1"~) ~ IST) involving two nuclei, the total change in internal bias ~ c~CO~;
if N nuclei flip, the change in e is ~3COkN1/2). Now we shall assert that this
fluctuation is fast, in the sense that for large values of M the time it takes
e(t) to change by an energy A ~ is usually much less than the time A ~ ~
required for ~qto flip if it is in the coherence window (of energy width A~);
this assertion will be justified below. Consequently it is not necessary (or
even useful) to go through the elaborate detour of recalculating P~) in
Eq. (4.18), including a coupling to some effective "oscillator bath", intended
to model the effects of spin diffusion. We simply write instead that for this
fast diffusion problem, where e(t) varies over an energy range #coo, one has:
c(
N + M --
M.)/2 e
-- fl( M - -
M H ) c'Jv/2
2Nz(fl)
M
x I (P~(t,e(r)+3~,))~(~)-~]
(4.26)
where we introduced the notation ~ f / - M~rcoo/2 + d~u with M u = integer
and [d~/[ ~<coo/2 to define the shift in the polarisation change enforced by
applying the external bias. The time average in (4.26) is over a fluctuating
bias:
_
=
j ~Fx,(v)
v=
x (P(~
e 2i~(*-~,
--oo
e(r) + ~ - , , q), Y))~(:)
(4.27)
and in the case of fast diffusion this average is given by the usual
incoherent expression
( Pl~ t, e(v) + 6~H, cp, y) ) ~(~)- 89= 89e -'/~M(ar
2
rM~ (c~/j) = 2A~tG~(g4u)
(4.28)
(4.29)
and Av(rp, y) is given by (4.20). In the usual case where r > 1, or if the
external bias is close to a multiple of COo, the expression for the effective
relaxation rate simplifies to
2A2
zM1 _ Fi ' zc1/2
(4.30)
This concludes our formal discussion of the problem of quantum
relaxation of ~q when it is coupled to a spin bath.
178
4.3. Nuclear Spin-mediated Relaxation
We now use the formal results just derived to find P(t, ~H) for a few
interesting cases, We start by ignoring spin diffusion, in order to
demonstrate the way in which relaxation is blocked in its absence. We then
include it to get the physical form of the relaxation for sufficiently short
times (how short depends on how fast is the phonon--mediated relaxation-see the next section).
(a) N o Spin Diffusion: Let us first consider a really pathological case,
in which there is no degeneracy blocking at all, i.e., r = 0. Then the function W(e) simply becomes a set of sharp lines, and all states in a grain
ensemble may resonantly tunnel, if (5~H = 0 (i.e., if the external bias field is
a multiple of the splitting between the different hyperfine-split line groups).
If6~/f ~ 0, then no resonance is possible. Initially there will be fast relaxation,
involving processes where M is not large. At longer times the higher-M
processes take over--recall that for large M, the transition amplitude A~t
is very small, since AM~Ao(2-2')M/2/M!, which collapses when M >
( 2 - 2 ' ) 2/2. To get some idea of the resulting relaxation, consider what
happens if ~u = 0, and let us ignore topological decoherence for simplicity
(which makes no difference for the long-t asymptotics of the result). Then
we have
P~(t) =
=1
dy e YP~)(t, y)
dye Y[ 1 + cos(2A~u(y) t]
(4.31)
where AM(y) = AoJM(2 ~ ) ,
assuming 2' = 0 (note if 2' = 0, then the
topological phase average (4.21) collapses to c~(~o-(I)) and we get (4.31)
anyway). The resulting curves are shown for 2 = 5 in Fig. 4, for various
PM(t). It is easy to then find the behaviour of P(t) by substituting (4.31)
into (4.25). The steepest-descent integral over M gives an accurate answer
for long times. However we more quickly derive this long time behaviour
from dimensional arguments. The sum over M contains some x//2N polarization groups inside the broad envelope function C(NN+ M)/2; thus the crossover from P(t)m 1 to the equilibrium P(t)~ 1/2 occurs around a time tc
such that A , / ~ t c ~ 1, i.e.,
ln(Ao t c) ~ x / ~ In (~eN2)
(4.32)
Quantum
Relaxation
of
Magnetisation
in
Magnetic
179
Particles
Pll
M=IO
0.8
X=5
M=6
0.6
M=0
0.4
I
i
i
i
,
I
i
,
50
,
i
I
i
,
f
100
T
I
,
150
,
,
i
Aot
Fig. 4. The time dependence of different contributions PM(t) to the time
correlation function P(t) for an ensemble of grains interacting only with nuclear
spins, ignoring the effects of nuclear spin diffusion. We assume that the
parameter 2 = 5, i.e., roughly 5 nuclei out of N = 1000 are flipped each time
flips.
For shorter times the relaxation is roughly logarithmic, viz.
1 - P(t)
f~_
ln(~ot)
X/Nln[ln([~ot)/ev/~) ]
;
(t~tc)
(4.33)
(t >>6)
(4.34)
whereas for longer times one has
P(t)-- l/2~exp
-3
ln2/3(A~
N1/~
.;
J
with logarithmic accuracy.
However even the tiniest degeneracy blocking will upset these
results--in fact if 5o)k-,//N> AM, the result (4.31) fails completely. Since
the proverbial "thunderstorm on Jupiter" is enough to give a &% exceeding
180
AM/v/N for the large values of M governing long time relaxation, we see
that for relaxation (just as for coherence), 1~ it is the limit of strong
degeneracy blocking that is experimentally meaningful. Let us therefore
again consider the zero external bias case, but now assuming that/~ > Ao/
COo. We go back to (4.24), again assume 2 ' = 0 for simplicity, and, noting
that the oscillatory sin 2 EMt term in p~/ (Eq. (4.18)) gives an integrated
contribution
:,
A 2
9)
g -[- L~M
l ae
2
M(Y)
"~rt~/82
, -2
sm-t
+
A2(y)]
gAM(y)[2AM(Y)'NzJo(z )
2
(4.35)
aO
we find
W(MCOo/2) e -,8M~ ~ dye y ~AM(y)
1- P(t)= •
M
~-fl]
O0
2
"2AM(y) t
]o
az Jo(
)
(4.36)
This result describes an ensemble of grains in which a fraction A of grains
relaxes, leaving a fraction 1 - A completely unrelaxed in the infinite-time
limit; A is given by
~7~o W(Mcoo/2) e-flM~176 I -~~
A = 1-P(t~ oo)=T~
Z(fl)
Jo dye -y [JM(2 x/@)[
M
(4.37)
Since terms for M > ~ make almost no contribution to A due to the
collapse of AM, it can be approximated for large 2 by
A = ,~1/4 -A- oj , fc--N~~176
(4.38)
O)o
where f = ~2/rcN is the number of states in the AN = 0 polarisation group
(in Ref. 10 we showed that individual terms in the sum over M decreased
as 2 1/4; but the sum over all M gives a factor 21/2).
Thus we see that apart from the small number of grains in near
resonance, if we ignore the effects of nuclear spin diffusion (see below for
these), most grains will be frozen for eternity by the nuclear bias field, even
in the absence of an external bias. Adding an external bias makes essentially no difference to this (since this bias is physically indistinguishable
from the internal bias). In fact the external bias will have no effect at all
on (4.36) until ~
the energy scale of variation of W(e). For H ~ 1, this
means that we will see oscillations in the decay rate as a function of ~r,
181
with period COo.For ~ > 1, the decay rate and decay function A will change
very little until ~H ~ COo, / N ; for ~H >>C~ ~
the relaxed fraction A --, 0
since not even the nuclei can bring the system into resonance. We derive
also a useful formula for the bias and temperature effects by noting
that only small M ~ x / ~ contribute to the sum in Eq. (4.24) and thus
W(Mcoo/2 - ~iJ) exp{ -fl(Mcoo/2 - ~r)} ~ IV( - ~.~/) exp{ fl~-H)} giving
P(t; T; ~ ) = P(t; oo; 0)
W(~H) e ~H
w(o) z(~)
(4.39)
(b) IncludingSpin Diffusion: Physical Relaxation: None of the results
(4.31)-(4.39) is physical, because we have ignored the time variation of e
caused by nuclear spin diffusion. Let us now demonstrate that nuclear spin
diffusion is fast, so that equations (4.27)-(4.29) are justified. Notice first
that since dipolar pairwise flips occur at a rate T f i for a given pair, so that
we have at least N T ~ 1 nuclear flips per second in the sample. Then in a
time AM~ we have N(T2 AM) 1 such flips, and in this time the bias will
change by at least de ~ 6COo ~/N/(T 2 AM). The condition for fast diffusion is
6~ >>AM (so the system has no time to tunnel), i.e., fast diffusion requires I~
N
A 3 <~
(50%) 2
(4.40)
If the spread &Oo arises from the dipolar interactions between the same
kind of nuclei only, then fiCOo~ T2-~ (but usually 5coo > TZ 1, because of
other nuclei and fields) then this criterion becomes AM ~ T [ 1N1/3. Now in
fact T~-1 will be typical 103-106 Hz (depending on the isotopic concentration of nuclear spins, etc.). For any mesoscopic spin ~o < 106 Hz (at least
for most bare Hamiltonians), so that even if (4.40) is violated for M = 0
(i.e., the zero polarisation change process), it is obeyed for large values of
M. Since the bulk of the relaxation involves values of M ~ 2N ~/2, we see
that (4.40) will always be the relevant condition. In this case Eqs. (4.26)(4.30) follow immediately.
Let us now therefore evaluate P(t) for the coupled grain/spin bath
system, taking full account of the dipolar interactions between the spins,
and assuming (4.42) to be satisfied; using (4.26)-(4.28), with 2', this means
we must evaluate, for/x > Ao/coo
1 -- L ~, (~(N+M Mtt)/2 e--fl~~
P(t) ---~ - 2 M ~ U
2NZ(fl)
Xfodye-'expf-2
~~tJ2M(2,,//~)}
v
,a
(4.41)
182
The integral is very easily evaluated; calling it I, we have
i~j~ ~ dye-Yexp{_2
~j
ooo
0
z7~2t
1/2Ct iM2)/2e2\a~__yM l~
__7~
dye-YO(yM(t)--y)
where the step-function
O(yM(t ) -y),
(4.42)
with
M2
~
Y~a(t) = )~e2 (n 1/2F~/2A2ot)1/M
(4.43)
arises again because values of M range up to roughly V / ~ , and are thus
typically large; thus
1 1 c(N+M
P( t) -- ~ ~ ~ ~M~ N
MH)/2 e
flOJo(M
-~IH)/2
2Nz(fl)
{lexpl
M2(nl/2F/j27~2t)l/Ml}
(4.44)
Noting again that ifct > 1, the dependence of P(t) on ~,v in (4.44) will
be rather small until ~H ~OOox/-N, we begin by analysing this result for
~ r = 0. This zero bias case has of course been previously analyzed in our
coherence paper, l~ but there we were only interested in the very short time
behaviour of P(t), i.e., that part involving frequencies ~ Ao, connected with
possible coherent oscillations (note that coherent behaviour can only arise
in the M = 0 sector of (4.16), even if one ignores spin diffusion).
However in analysing the quantum relaxation properties of S, we need
to sum over all x / ~ important terms in (4.44); the effect of the spin diffusion will be to unblock the longtime relaxation. In evaluating (4.41) or
(4.44), a steepest-descent integration over M is possible, but just as with
the case of no spin diffusion, dimensional analysis is sufficient; the crossover time from unrelaxed to almost relaxed behaviour occurs at a time t,.
given by
ln(2
A-~ to) ~ x / ~ l n
n 1/2F~
(~eN2)
(4.45)
(compare (4.32)), and the short-time relaxation is roughly logarithmic, as
in (4.33), viz.,
X/2
1--P(t,~H=O)~
ln(2A2t/n'/2C~ )
nJv
-..ln[(le/ ~/2~_)ln(2AOtnl/2F~)_
o/
J
.
(t~t.)
(4.46)
183
The long-time behaviour is, analogously to (4.34), given by
P(t, ~ / = 0) - 1/2 ~ exp f
3 1n2/.3(2~o2t/~.l/2Fu) };
(2N)1/3
(t >~tc)(4.47)
with In In corrections in the exponent. Thus, amusingly, the results with
spin diffusion included look just like those without spin diffusion, provided
/~ = 0 in the latter. The physical reason for this is simple--rapid spin diffusion basically eliminates the effects of degeneracy blocking, by allowing the
bias e(t) to cover the whole energy range of each polarisation group. This
is then the fundamental reason why spin diffusion "unlocks" S and allows
it to relax.
In Fig. 5 we show a plot of P(t) against ln(2Aot), which clearly brings
out the 3 relaxation regimes. For very short times ~ A o 1, there is a sudden
P1,
1
X=IO
0.9
0.8
0.7
0.6
/
0.5
~
0
,
,
~
I
50
,
~
1
,
100
,
~
~
150
Ln(2Aot)
Fig. 5. The time correlation function P(t) for an ensemble of insulating grains,
now plotted against In t; all contributions PM(t) are included. The relaxation is
mediated by nuclear spin bath-mediated transitions only. The figure describes
either (a) P(t) neglecting nuclear spin dit'fusJom and having zero degeneracy
blocking (/~ =0), or (b), including both spin diffusion and finite degeneracy
blocking; both situations give the same curve (see text). In this plot we assume
that 2 = l 0.
184
relaxation involving only processes with M ~ 21/2 or less. For times t such
that Ao ~ ~ t ~ t~, we have roughly logarithmic relaxation. Then for t ~> t C,
we have the behaviour in (4.34) or (4.47).
At this point it is important to realise that for all but microscopic
spins, t~ will be astronomically long. F r o m (4.45) we have
tc~ 2A~ \2e 2]
(4.48)
Consider now, as in Fig. 5, the case of a mesoscopic spin where, e.g.,
Ao~ 1 MHz, F ~ = 100 MHz, N = 1000, and 2 = 10. We then find that
tc ~ 105o s, i.e., more than 10 ~~ times the age of the universe! Thus for
mesoscopic grains we always get roughly logarithmic relaxation at long
times, from nuclear spins, in the quantum regime (although we emphasize
that it is not exactly logarithmic-the slope of the graph changes slowly with
time). Readers familiar with the experimental measurements of magnetic
relaxation in grains may not be surprised at this result, since it is almost
a central dogma of magnetism that logarithmic relaxation must arise from
a distribution of energy barriers. 39 This is roughly what has happened here,
in that the distribution of barriers is provided by the distribution of nuclear
polarizations; however we emphasize that the results derived here have not
been derived for a set of static barriers, since we have taken into account
the essential role of the "energy diffusion" mechanism.
Moreover for microscopic spins ( S ~ 10), one finds that tc can be
short. For example, consider a situation where S = 2 0 , ~o = 10 ~~GHz,
Ao~ 100 MHz, F~ ~ 100 MHz, N = 10, and 2 = 0 . 1 (this should roughly
describe a particle containing 10 Tb atoms). Then one finds tc ~ 1 s, and
any direct relaxation experiment (typically conducted over time periods
1 ms < t < 104 s) would be mostly in the long-time relaxation regime, where
the relaxation is no longer logarithmic. Thus one should probably avoid a
naive interpretation of our results in terms of a distribution of static energy
barriers.
Let us now recall that the results (4.46)-(4.48) are by no means comp l e t e - a l t h o u g h they are correct to within In In corrections for a coupled
grain/spin bath system, a realistic calculation must incorporate all three of
the relaxation mechanisms we have discussed, in Secs. 2, 3, and this one.
This is done in the next section. However one important result has in fact
emerged from our analysis. As we shall see in the next section, in the case
of insulating grains on an insulating substrate, the phonon-mediated
relaxation will turn out to be rather slow, even in the presence of nuclear
spins. What this means is that at short times the only mechanism of relaxation for these systems will be via the nuclear spins, in the way discussed
185
here, and so equations (4.46)-(4.48) should actually describe the short-time
relaxation of an ensemble of independently relaxing grains.
We do not consider the case of large bias (~,>O9ov/N in (4.26)(4.29)) here, but deal with it in the context of the general case. We now
proceed directly to this general case.
5. COMBINING THE MECHANISMS---PHYSICAL RESULTS
We now come to the crux of the paper, which involves putting together
the various mechanisms discussed in Secs. 2-4 to give a physically
realistic picture of quantum relaxation in magnetic particles. As mentioned
in the introduction, we do not attempt a comprehensive discussion of all
possible cases. Such a discussion would involve consideration of a large
class of "giant spin" bare Hamiltonians, with widely varying behaviour,
and with each being considered throughout the whole range of field strength,
field orientation, and temperature. This would only be the beginning--we
would then have to go on to examine the large variety of couplings to
nuclear spins (with hyperfine couplings ranging over 3 orders of magnitude,
plus quadrupolar couplings), the variety of different magnetoacoustic
couplings, etc., etc.
We adopt the tactic here of showing how the mechanisms combine,
and giving some details for one model. In other papers, in preparation, we
analyse some specific experimental systems in much greater detail. We
emphasize once again here that, in our view, magnetic relaxation in many
experiments must involve the dipolar interactions between grains in an
essential way.
We begin by showing how phonons and nuclear spins combine to give
a rather surprising form for the relaxation at low T. This analysis is
appropriate to the case where both grain and substrate are insulating. We
then go on to include electrons--in this case phonons become irrelevant,
and only the combination of electrons and nuclei are important. In both
cases we remark, in a phenomenological way, on the effect that T 1 processes in the nuclear sub-system will have on our results.
Before discussing the details, let us first state the method we shall use.
We have already seen how the dynamics of S is given in the presence of a
spin bath (Eqs. (4.26) or (4.41)). When we couple in a bath of oscillators
which allow transitions between states [S~I;)(l{(Tk}> and 1S2;~2{(~k}> of
the combined grain/spin bath system, in which there is no restriction on the
difference 2M between the polarizations of the spin bath states ]ZI> or
IZ2>, or on their biases e~ and ea (apart from those imposed by energy
conservation), the form of P(t) must change-it is no longer given by (4.16).
In this paper we will not attempt to give a general expression covering all
186
N . V . ProkoPev and P. C. E. Stamp
mechanisms simultaneously; this is more than we need. Instead we will use
2 expressions, which apply in the 2 limiting cases of interest. These are
(a) For short times, when the spin-mediated relaxation dominates;
then we ignore the oscillator bath, and use Eq. (4.16).
(b) For longer times the oscillator bath-mediated relaxation will take
over. In this regime, each time S flips, some of the bias energy is taken up
by an oscillator mode, and the rest by some number r of flipped spins in
the spin bath. There is no restriction on either r or the polarization change
2M, for this process to work. The complete calculation of P(t) here is very
complicated; however we will claim that as soon as we reach times where
the typical oscillator-mediated transition rate, at a typical bias, is faster
than the spin bath-mediated transition rate (for those grains in an ensemble
which have not yet relaxed), then a good approximation for P(t) turns out
to be
P(t; T, ~H) = fJ de W(e-~H) e-p(~-e.){ p(eq)(T, e)
Z( fl )
+ [ 1 - p ( e q ) ( T , e ) ] e ,/:(~,r)}
(5.1)
where r l(e, T) is the oscillator-mediated relaxation rate, and p(eq)(T, e)=
e-~/r/(2 cosh(e/T)) is the equilibrium population of the state $1 in a given
bias (in our previous discussion of the nuclear spin effects we assumed that
this bias was actually much less than T). We note that this expression is
incoherent because it applies to the majority of grains in a large bias, which
cannot relax via the spin bath as discussed in the previous Section. The
relaxation rate in this incoherent case can be calculated as a second order
perturbation theory expression in the tunneling amplitude A o. Thus the
only effect of the spin bath which is left is the distribution over the bias.
The formal justification of (5.1) is easiest to follow for the case of pure
orthogonality blocking. In a typical bias e (.OoN1/2>~A o the second order
perturbation theory expression for the tunneling rate is given by
~
- '(e) : 2~ Z pe~
ij
I U~.I 2 ~ p Z~ i V2~h) [z ~(e + E ~ - E, + oo( M i - M f )/2 )
oc7
(5.2)
where the sums are over the initial and final states of the spin bath (i, f )
and oscillator bath (0~,y) with the equilibrium density matrices peq. Here
V (ph) describes the interaction with the phonons (we do not even need its
explicit form to prove the point), and the orthogonality rotation operator
U is given by: 10,26
N
U = F[ e-iak~X
k=l
(5.3)
187
The crucial point here is that in a large bias one may drop the energy
~Oo(Mi-Mj.)/2 transferred to the spin bath from the d-function argument
in (5.2), because only transitions for which M i - M F ~ N 1/2 contribute to
the answer. After that we have the sum over the complete set of states f,
so that U~fU)i = 1 because the of the unitarity of U, and the final answer
is that for the oscillator bath alone, provided we ignore the processes
discussed in the last section, involving nuclear spin flips; this is why (5.1)
is a long-time approximation.
The generalization of this argument to include topological decoherence
changes the result in only a minor way. Indeed, with nonzero ~ , we have: to
N
U= 1~[ e i~k~X(ei|
e--i|
(5.4)
k=l
and the result for the relaxation rate in the oscillator bath is renormalized
by the factor
D = (UU*)
= 2 + 2 c o s ( 2 ~ ) F~.(1)
(5.5)
Unless 2 = 0 and the topological phase qb is a multiple of n, this factor is
of order ~ 1, and clearly has no essential effect on the results. Thus, apart
from the renormalization factor D, the relaxation of a single grain in a
nuclear bias field e is that already given in sections 2 and 3, and all that
remains for a grain ensemble is to average over the bias field.
5.1. Insulators: Spin Bath Plus Phonons
The case of insulating grains and an insulating substrate is the one
where there is most obvious competition and interplay, between the
nuclear spin and phonon relaxation mechanisms. They are both slow. We
have already seen (Fig. 5, and Eq. (4.46)) what the short-time nuclearmediated relaxation will look like. Now consider the implications of (5.1)
for the long-time phonon-mediated relaxation. We worked out v-l(e) for
phonon relaxation in the absence of nuclear spins; for a longitudinal bias
this was given by (3.13), and will be the same in the presence of the spin
bath (apart from the renormalisation factor (5.5) above). Thus we write
v ~(e, T) in the form
r-~(e, T ) = r o ~ Eoo
coth(e/2keT)
(5.6)
where 7"o I is a "typical" normalising relaxation rate, defined as
7"o1=7.-l(e=Eo, kBT=Eo)~-'X2Ao(-A~176
3
\o~/\oo/
(5.7)
188
Pll
Phonon
0.9
0.8
Nuclear
~
'
'
,
T, c r o s s o v e r
0.7
0.6
0.5
10
20
30
40
Ln(2Aot)
Fig. 6. The same plot of P(t) against In t as in Fig. 5, for an ensemble of insulating grains, but now including the effect of phonon- mediated relaxation, as
well as nuclear-mediated relaxation. The curve marked "phonon" shows the
relaxation of the ensemble due to phonons acting in the internal bias field of the
nuclei, for small applied bias. The curve marked "Nuclear" shows the relaxation
mediated by the nuclear bath only (as in Fig. 5), but including also a
phenomenological crossover to TI-mediated relaxation. In reality the system
will initially relax along the Nuclear curve, and then crossover to the phonon
curve at some crossover time-here this crossover time occurs when ln(A ot) ~ 30.
We assume that N = 1000 and that S = 1000, and that 2 = 10. Notice that if T 1
is short enough the T1 crossover can also occur before the crossover to phononmediated relaxation.
and
Eo is t h e w i d t h o f the G a u s s i a n p e a k in W(e), i.e.,
E o =~ooN1/2/2
(5.8)
N o t i c e t h a t for a fixed bias e = Eo, t h e a c t u a l r e l a x a t i o n r a t e r-l(Eo, T)
still d e p e n d s o n t e m p e r a t u r e as T - ' ( e , T ) = r o ' coth(Eo/2kBT). A t l o w T
this scales d o w n to t h e c o n s t a n t v a l u e ~ o ~, w h e r e a s at h i g h T it i n c r e a s e s
linearly, i.e., z - ' (Eo, T) = 2~ ~ ' k~ T/Eo.
It is useful to get s o m e p r e l i m i n a r y i d e a o f t y p i c a l t i m e scales for To.
L e t us c o n s i d e r 3 e x a m p l e s , viz., (i) a T b O x i d e grain, w i t h S ~ 1000,
189
containing N ~ 1000 Tb nuclei; (ii) a particle of Er As (such particles are
apparently insulating), 29 containing N ~ 1000 Er nuclei, with S ~ 1000; and
(iii) Ni O grains with S ~ 1000, but with N ~ 10 Ni nuclear spins only
(coming from the 1% of Ni 59 nuclei). Let us also assume that | ~ 100 K
and Ao ~ 1 MHz for each example. However Eo varies dramatically--for
the Tb O grain, coo ~ 5 GHz, and Eo ~ 4 K ; for the Er As grain,
coo ~ 1 GHz, and E o ~ 0.8 K; whilst for the Ni O grain, coo ~ 28 MHz, and
E o ~ 2.3 x 10-3 K. We thus get a wide range of time scales; for the Tb O
grains one has r o ~ l - 6 x l 0 2s, for the Er As grains, one has r o ~ 5 s ,
whilst for the Ni O grains, one has zo ~ 4.5 years!
We now consider the general behaviour of P(t, T; ~r). We start by
noticing that the weighting function for the initial states, of bias e, contributing to (5.1), will be
w(e, T; ~ ) :
W(e- ~/~)
1
~ x/2~r
: e xE ~p
e-P(~
~)
z(~)
{-~ [Eo
k e T J }"
(5.9)
Ignoring ~ for the moment, we see that in zero external field this distribution peaks at negative bias e ~ -E2o/k~T. This is natural; as we lower
T, more and more of the nuclei align with S, thereby lowering the energy
of the combined system in the initial state. This process continues until ~ N
nuclei are aligned with S, i.e., when e ~ - N c o o / 2 , at a temperature
k e T ~ c o o / 2 . We have then reached the bottom edge of the distribution
W(e), way out in the lower Gaussian tail. If ~ va O, the distribution peaks
at e ~ (~Lr-- E2o/k~ T).
Thus we write
P( t, I; ~H) : f de w(e, T; ~H){ P(eq)( T, e) + [1 -- P(*q)( T, e)] e -'/~(~' T)}
_ [ dx e - ln~x 4~+ ~/T? e-X/T + eX/Te-(t/~o) x3 ooth(xnT)
-- J ~
2 cosh(x/T)
(5.10)
where we have normalised all energies by E o :
x=s
o
4~:4~/Eo
T = k~ T/Eo
(5.11)
190
We now look at some interesting limits of (5.11).
(a) High Temperature, zero External Field: We consider the regime
k,T~>Eo when T~> 1. Then (5.10) is easily evaluated; one finds
P(t, T ; ~ H ) - - I / 2
1
1
{ ~2 2Tt ;
2(l +4Tt/ro)l/2ex p _ - ~ H ro-~Tt j
1
(5.12)
1
2 1 +4Tt/ro) ~/2;
(k'T~>E~
(5.13)
(We recall that this result is correct only when k~T~f~o--otherwise
higher levels of the grain will come into play.)
This surprising power-law behaviour is actually easily understood.
The initial fast relaxation (at t ~ ~o) comes from grains with large bias; the
slower relaxation (r ~>to) comes from the grains with smaller bias. The
typical relaxation occurs at a rate Tro 1, i.e., it is faster for higher temperatures, in accordance with the remarks just after Eq. (5.8). We may
think of the power law as a "grain-ensemble" sum of a lot of different
exponential decays, or as a funny kind of "stretched exponential". Note
however that the decay is not logarithmic in time.
(b) Low Temperature, zero External Bias: We now assume that
T ~ 1; this is a little more complicated. From (5.10) we now have
P ( t , T ; ~ H ) ~ I + ~) dx e ~/2(x-g~+l/T)2
v/~
x (e x3coth(x/2T)t / T o
1 + eiH/T
X ( e -x3
-
eXIT
2 cosh(x/T)
--
1)
dx e - ( X (H)2/2
J x / ~ 2 cosh(x/T)
1/2T2 (
(5.14)
coth(x/2r)t/ro _ 1)
Let us consider first the behaviour of (5.14) when (H = 0. Using the fact
that T ~ 1 one may further simplify this expression to
P(t, T; ~ H = 0 ) ~ 1 + ~ e -
~/2~
~
- - e
cosh(x)
-~3
(5.15)
This clearly defines a temperature-dependent relaxation rate
(A~
3
(5.16)
191
which goes to zero at low T. We note however that the validity of (5.15)
requires that the calculation is not affected by the edges of W(e), which
means that at a temperature kBT~COo/2, Eq. (5.16) crosses over to a constant value, i.e.,
( A o ~ ( 6% ~3
/
Z'ejs~(T--, 0) -+ N3/2To 1 ~ S 2 A o \ | 1 7 4
(5.17)
but the magnitude of the relaxing component at this temperature is already
extremely small; in fact 1 -P(t--+ c~)~N-I/2e N. It is again useful to consider what is the the maximum relaxation time at low T. For the examples
previously mentioned, we have (i) for the Tb O grain, r~yf(T--,O)~ 10
minutes; (ii) for the Er As grain, reo~(r--+0 ) ~ 2 days; and (iii) for the Ni
O grain, Teff("( --'+ 0 ) ~ 4 x 10 7 years.
Defining the exponentially small total amplitude of the relaxing component as
A ~27-~
(5.18)
we find the long time asymptotics of P(t) as
P(t, T; ~/t= 0) --* 1 - A + A (2Veff(T)~ '/2
\
rot
/
(5.19)
The reason for the T 3 decrease in the relaxation rate at low T is simply
that as we lower the temperature, the typical bias energy decreases linearly
with T, because only a small fraction of grains in the Gaussian tail which
have very small bias e ~ k , T contribute to the relaxation. The majority of
the grains are actually trapped by the hyperfine interaction in a negative
bias energy e ~ EZ/k, T>>k,~ T. Thus at this low temperature even phonons
will not help to liberate the giant spin.
(c) Finite External Bias: For ~/~ ~ 1/T the previous answer (5.19)
hardly changes except that the amplitude of the relaxing component is
given now by
A~~
e-1/2(c-H-1/r)2,
((/_/~ l/T)
(5.20)
We observe that already for rather small bias (/~ ~ 5P~ 1 we have an
exponential dependence on {~/. A negative applied bias will simply further
suppress an already exponentially small fraction of relaxing grains and we
192
N.V. Prokof'ev and P. C. E. S t a m p
shall not go into more details here. The case of positive ~H is nmch more
intriguing and surprising. After an exponential increase in the amplitude up
to A ~ 1 for (H ~ l/T, the answer changes drastically. For ( H > 1/T one
finds the magnetization function to be
P(t, T; ~H)~ j _V_dx
/ ~ e_V2(X_(H+ l/~)2
ixl3 r/~o
,
(~h-> l/T) (5.21)
which has different behaviour depending on whether we look at short
times, where
P(t, T; ~H) ~,e ~/~er
L
( t/rely< ( 4 . - l/T) 2)
r),
T e f f ( ~ t t , T ) = ( ~ H - - l / T ) 3 To 1
(5.22)
(5.23)
or long times, where
P(t, T; 4H) ~ ~
r(1/3) e-(g-- ~/~)2/2
3x/~
(~) 1/3
(t/Zeff>((H-1/T)2)
(5.24)
Note that Eqs. (5.21)-(5.24) are derived in the approximation that ~H-I/T>> 1. As before, the bias is restricted to be much less than flo. Finally,
at t > To, this behaviour changes yet again to P(t)~ exp{--(~H--l/T)2/2}
(ro/4tT) 1/2. We note the anomalous temperature and bias dependence of
the effective relaxation rate (5.23) near the crossover.
From this analysis we conclude that unless the external bias is larger
than E o (or E2o/kBT at low temperature) the ensemble averaged results for
the magnetization relaxation are not described by the naive theory of a two
level system coupled to a bath of oscillators; interactions with the spin bath
completely change the answer. It is worth noting three effects which have
to be kept in mind when considering the evolution of the experimental data
in external bias:
(i) Since the dependence on ~H starts when ~H>E o or ~H>E~/
k~T, one cannot derive the value of the tunneling amplitude from this
dependence (as one can in the case of an isolated two-level system).
(ii) At low temperature the giant spin is "trapped" in its initial state
by the large and negative bias produced by the spin bath.
(iii) Only large and positive bias can liberate the giant spin and
allow complete magnetization relaxation. The temperature and bias dependence of the relaxation rate is anomalous near the crossover region
~H ~ 1/T. Of course, when ~H >>1/T we recover back the pure case of
oscillator bath relaxation.
193
At this point we would like to add 2 important cautionary remarks.
We first comment on our starting assumption that the initial state of
the spin bath is equilibrated with the giant spin direction S = S~. Experimentally this could be arranged by applying a very strong negative bias ~
during a time period much longer than the longitudinal NMR relaxation
time T~, and switching it off at t = 0. In some systems however this procedure may not work because of an astronomically long T~ (in fact, low
temperature longitudinal NMR is still something of an unsolved mystery;
spin-lattice relaxation times at mK temperatures ought to be many years in
insulating crystals, unless some gapless magnetic excitations are involved,
e.g., on the sample surface or crystal defects). It may be useful then to
introduce two different temperatures--one for the oscillator bath Tp~, and
the other for the spin bath Ts (which may even be negative, i.e., Ts < 0!).
Our basic equation (5.1) is still valid in this more general case, but now the
distribution over the initial bias is defined by the spin bath temperature Ts,
whereas the giant spin evolution toward equilibrium is governed by the
crystal temperature Tph, i.e.,
P(t, ~ ) = ~, da
D
HAl
~i~)~
Lt Ss)
e (~-e~)/rs{P(eq)(Tph, e)
-k [ 1 - p(eq)( Tph , g)] e -t/z(e" Tph)}
(5.25)
One may proceed with the analysis of this expression as before. At this
point we feel that considering more cases in this paper will not add much
to the physical picture. Depending on the particular experimental system
and sample preparation, the necessary formulae can be easily derived from
(5.25).
Our second cautionary remark concerns the role of longitudinal
nuclear relaxation in the nuclear spin bath. As noted above (and in Ref.
42), a major unsolved problem in this area is the physics governing the
low-T longitudinal relaxation, which theory predicts to be astronomically
long. Because this problem is unsolved, we will not try here to include T 1
processes in any microscopic way. What we shall do instead is simply
assume some time TI(T), to be determined by experiment, and ask how the
existence of longitudinal relaxation will affect the results we have derived.
The importance of T1 processes is simply that they allow the energy bias
diffusion to occur not just within a given polarisation group-they change
the polarization state of the nuclear bath. Thus the bias, instead of just
diffusing over an energy range F~ in a time ~ T2 (without changing the
polarization), can also diffuse over the whole range N~o o of bias, but now
in a very much longer time T 1 (which may be years at very low T). This
194
bias varies so slowly that we are in the limit of slow diffusion-the system
will have plenty of time to tunnel if it finds itself in a "coherence window"
of width A~, even if M is quite large.
The qualitative effect of this is very easy to see. Up to times of order
T1, the longitudinal relaxation will play a negligible role in the problem.
However for times of this order the results in Eqs. (4.45)-(4.47) will break
down, and we expect purely nuclear-mediated relaxation to continue at the
same rate as that in these equations, if we substitute t = T 1 into them i.e.,
at t ~ T 1 we expect the ensemble-averaged rate to become very roughly
exponential, with a rate T~-~. We will not attempt here to give a more
accurate calculation than this of the crossover to T~-driven relaxation.
What this means in practise for the relaxation of the magnetic
insulators is depicted in Fig. 6. At short times the relaxation is nuclearmediated, until times of order minutes or hours (varying widely with the
material and the size of the particles), at which point we switch to phononmediated relaxation, according to the theory just discussed. Usually the " T 1
crossover" to T~-driven relaxation will be later than this, and we will not
see it, but it is conceivable that it might be shorter, in which case we will cross
over, not to phonon-mediated relaxation, but instead to a "longitudinal" T 1driven relaxation. Since it seems likely that these processes are largely going
to be determined by paramagnetic impurities in the substrate, it will be
quite hard to deal with them, either on the theoretical or experimental
level, beyond the phenomenological level described here. Experimentally T1
can be months or even years in pure magnetic insulators at very low T.
Summarising what we have found for insulating systems, we see that
at short times, the relaxation proceeds entirely via the nuclear spin bath,
and is logarithmic in time (Eq. (4.46)). At longer times phonons take over,
and we get power law decay in time; at low T this goes as (Teff(T)/t) ~/2,
with re~! ~ T 3 (Eqs. (5.16) and (5.19)), but only a small fraction of grains
relax unless the bias ~H ~ Eo/k, T or larger. This bias is necessary to counteract the nuclear bias field. At higher temperatures we still get power-law
relaxation (Eq. (5.13)). The crossover between nuclear spin-mediated and
phonon-mediated relaxation can be understood by matching Eq. (4.46)
with the relevant phonon expression. Again, we remind the reader that
these results take no account of inter-grain interactions.
5.2. Spin Bath Plus Electrons
We now turn to the case where either the grain or the substrate is conducting. We use the giant Kondo model of Sec. 2 to describe interaction of
with electrons, with a dimensionless coupling c~. If both grain and substrate are conducting, then 0~= % ~ g2S4/3 (Eq. (2.7)); if only the substrate
195
conducts, then ~ = % ~ g2S2/3 (Eq. (2.13)); and if only the grain conducts,
~x~ ~Xse -~,`/'~'~r (Eq. (2.21)).
The formal analysis is almost identical to that just used for phonons-we start again from (5.1), now using the general result (2.8) for the electronic relaxation rate. The high temperature limit is most transparent
because %(T) in Eq. (2.9) is independent of the bias. Thus we find a pure
exponential relaxation
P ( t ) = l/2(l + e
,/~xr));
(o~keT, k B T > Eo)
(5.26)
at high temperature (note again that we still assume that k BT is
significantly less than ~o) with T~ given by (2.9).
For very small grains or conducting grains on an insulating substrate
at ke T ~ Ae~, we have a peculiar relaxation regime when the internal bias
is much less than temperature, but the electronic damping rate F~ =
2z~0&eT is already small, that is, for 0~~ 1 there is a temperature range
where F ~ E o ~ kBT. Now the electronic relaxation rate is inversely proportional to the bias energy
A 2
re~(T, e) = 8zro:keTTff,
(2zk s T r e ~ ke T)
(5.27)
Substituting this expression to (5.1) we find the time correlation function
as
1
dx e
1/2(x
(H)2e--t/(X2Ze(T.Eo)~
(5.28)
which yields
P(t) = 1/2(1 +e-'ST/~e(r'E~
1/2( 1 + e-t/~XT' r
({H= 0)
(5.29)
(~H >>Eo)
(5.30)
The temperature dependence of the effective relaxation rate is ~ k B T0~(T),
and again in the small external bias we have an unusual decay law- it is
neither simple exponential nor power-law.
Let us deal now with the low-temperature behaviour, k B T / E o ~ 1.
Using (2.10) we then get
eIt, v; r
1 + j dx e_i/2(X_(H + 1/~) 2
ex/r
2 cosh(x/T)
f ( x ) = cosh(x) I r [ ~ + ix/~]l 2
(e f(x)tire:If_ 1 ),
(5.31)
196
(compare (5.14)). If the bias ~H < 1/T, this defines a temperature dependent
relaxation rate
2
(Ao'~(2~rkBT~ 2~-1
ref](T) = 20~-]F[
A~
\no)\ no )
(5.32)
The answers crucially depend on the parameter 0~.We consider three limiting
cases:
(i) For 0~~ 1 we approximate the function f(x) by f(x) ~ x coth(x)
(~2 + (x/zr)2)- 1 and obtain
7rt
P(t, H; 3H) ~ 1 --A - - ;
(t < ~2~ez)
(5.33)
O(T eff
P( t, H; ~H) ~ I - RA ( rCt ] 1/2",
( O~272eff< t "< Y2eff)
(5.34)
\ T eff/
P(t,H;~H)~I-A+A
( - -t ~1/4;
e -2~ ~,/77~z; (reff> t)
(5.35)
\ Z eff /
(ii) For 0~= 1/2 we have f ( x ) = z c, and, as in the high temperature
limit, the relaxation is given by the simple exponential law
P(t) = 1 - A(1 - e-~t/wr);
(5.36)
(iii) Finally we consider the case of large e, which has a power-law
asymptotic at long times
P(t,T;~H)~I-A+2AF
~
\Trt/
In all these expressions A(T, gH) is given by (5.20).
The interpretation of these results also depends on the value of cc If
both grain and substrate are conducting, then the relaxation time in (5.32)
will be greater than the age of the universe unless S ~< 100 (depending on
the value of ga) This shows the astonishing power of the electron bath to
"freeze" the dynamics of S, unless it is quite microscopic in size. We shall
not analyze here the case of very small S, since once S ~ O(10), the detailed
structure of W(e) will no longer be Gaussian--there will be lots of fine
structure, depending on the particular system involved. Such studies are
best done on a case-by-case basis.
If only the substrate is conducting, then a = % will be small until S
10 3. For larger S then the giant spin is again frozen. For S ~ 10 3 the results
197
(5.33)-(5.36) can be applied if (H ~ 1/T. Again, as (H approaches l/T,
there will be an amusing crossover, with a relaxation rate given for finite
~/t > 1/T by
r~/(T, ~/t) = r~l( T, e = ~H-E2o/T)
(5.3s)
Finally, if only the grain is conducting, ~ can be exponentially small,
if the spacing Ae~ between the internal electron levels is large enough--this
typically requires that Ae~ ~>f~o, so that S~< 105 (cf. Eq. (2.23)). In this
case, unlike the cases where the substrate is conducting, the short-time
relaxation will be again dominated by nuclear spin-mediated transitions,
and the weak electron-mediated transitions only enter at later times in
possible competition with the phonon-mediated transitions; the typical
electron-mediated rate becomes
2
r ~ ( T ) = 4 ~ s ~o2 e A~,/k,r
(5.39)
Note that for low enough temperature, Teffin (5.39) becomes so long
that electrons in the grain become irrelevant, and we return to the calculations for the nuclear spin/phononmediated problem, i.e., the grain behaves
as an insulator.
Thus to summarize, we see that depending on whether the grain and/
or substrate are conducting, we get behaviour ranging from grains frozen
for all eternity (if e >> 1), to grains behaving essentially as insulators, with
short-time relaxation controlled by the nuclei (when c( ~ 0). In the former
case we see an extraordinary alliance between the electrons (with their
strong dissipative suppression of tunneling) and nuclear spins (which trap
~q in a negative bias e ~ -E2o/kBT) to block any motion of ~q even at low
T, where traditionally one expects tunneling. At low T the nuclear spins
have the effect of applying a strong negative internal bias field to ~q, as we
have already seen; but for large 0~the real reason for the freezing has more
to do with the straightforward dissipative localisation of ~q, already familiar
in the context of the Ohmic spin-boson problem.12
Notice, incidentally, that at large c~ even T1 processes in the nuclear
bath will not change this conclusion, again because the blocking is caused
by the dissipative freezing of the dynamics of ~q.
6. S U M M A R Y & C O N C L U S I O N S : PRACTICAL IMPLICATIONS
Let us first summarize our main results. At temperatures T such that
kBT~ ~o, we may employ a 2-level model with bias to describe the grain.
198
Coupling to electrons and/or phonons in this model can be treated by the
usual oscillator bath methods. However the crucial coupling turns out to
be to the nuclear spin bath, to which oscillator bath models do not apply.
The main effect of this coupling is to spread each of the 2 levels for S into
a Gaussian "band" of half-width E o ~ N~/2coo, and total width Ncoo, where
coo is the hyperfine coupling and N the number of nuclei in the grain. If N
and coo are small, Eo may only be a few m K (as in Ni or Fe grains). If N
and coo are large (as in mesoscopic rare earth grains), E o may be hundreds
of Kelvin or more (so that Eo > f~o). At low temperatures the nuclei begin
to line up with the grain vector ~, and then S finds itself in a negative internal nuclear bias field, with a mean bias e of roughly e ~ -E2o/k~T. Then
at short times the only way that the grain can relax is by taking energy
from the nuclear system. The inter-nuclear dipolar interactions allow this
to happen, by causing the bias e to become time-dependent--the result is
a slow relaxation, which for an ensemble of grains gives a fraction of
relaxed grains going roughly logarithmically in time (Sec. 4):
1 -P(t) ~
ln(27%2tl~l12F.)
frNln[(lle VI~)ln(2 A2t/~'i2C.)]
X/__2
(6.1)
This logarithmic relaxation, shown in Fig. 5, has nothing to do with a distribution of sizes of the grains (we assume all grains have the same size).
At longer times the grains can relax very slowly via either electron- or
phonon-mediated transitions, again in the time-varying nuclear field
(Fig. 6) If the grain is insulating (no electrons), we get power-law relaxation; a fraction A of grains can relax, where
--
2Eo
exp
- 1/2
Eo
keTj j
(6.2)
in a bias field ~ / , and these grains relax as A('ceff/t) 112 where
(Ao~(kBT~ 3
r~f~(T) ~ SaA~ \ |
OD )
(6.3)
if ~H ~ E2/ke T, and P(t) ~ 1/2(1 + e -'/~</) with
T~f)(~H,T)~_Q~ E~ ~3
(no~Eo~ 3
knT) S2A~\OD/~ODJ
(6.4)
once ~H > E2/k~ T. Thus very few grains can relax at until the external bias
~/~ compensates the internal bias e and "untraps" 57.
199
If there are electrons around, the results depend on whether the substrate is conducting or not. If it is, then all but microscopic spins (S ~ 100)
will be trapped in states of frozen magnetization for astronomical times,
once the temperature goes below f~o- This remarkable result is a combination of "degeneracy blocking" caused by the nuclear bias field, and the very
strong electronic dissipation. However if only the grain is conducting, and
the substrate is insulating, then for S~105, the electronic relaxation
becomes negligible, and we go back to the fully insulating case; the full
results for all values of electronic coupling appear in Sec. 4.2.
It is clear that these results have implications both for future magnetic
device design, and for the very many experiments that have been done on
magnetic grain relaxation over the years. 38 As far as devices are concerned,
perhaps the most interesting result is the freezing of S when both grain and
substrate are conducting, unless S ~ 100. This may have far-reaching
implications for future computers and information storage, since it implies
that if we are prepared to go to low temperatures, one should be able to
use even magnetic molecules 37 as permanent memory storage elements,
perhaps of the "M.R.A.M." type (cf. Ref. 30). Even more tempting is the
possibility that we may be able to manipulate S in such molecules
indirectly, by controlling the nuclear spin polarisation, since it is the
nuclear bias field that controls the dynamics of S.
As far as experiments on grain relaxation are concerned, we believe
that it is quite urgent that experimental tests of our results be done at low
T. Unfortunately most of the many relaxation measurements that have
already been done are at higher temperatures, where relaxation is dominated
by thermal activation. What we have found here is that the low-T
behaviour does not look like the conventional tunneling picture at all-there is no crossover to a temperature--independent "tunneling relaxation"
at temperatures k B T ~ f ~ o. The low-T behaviour of magnetic particles is
not governed by the external bias at all, unless it is very large; instead it
is governed by the random distribution of internal nuclear bias fields. These
results are very different from previous theories of the tunneling of
magnetic particles (see, e.g., Ref. 14).Thus we believe that it would be useful
to look again at those experiments that have been done at low T on ensembles of relaxing grains, 7-9' 23.29 particularly in an attempt to understand the
"plateau" that often appears at low T in the relaxation rate (the "magnetic
viscosity plateau").
However an important cautionary note is necessary, in virtually all of
these experiments, inter-grain interactions (mediated by the grain dipolar
fields) are very important. We believe this to be the reason for the common
occurence of "avalanche" magnetization reversal in grain ensembles (see,
e.g., Ref. 23); these avalanches are prima facie evidence for the importance
200
N . V . ProkoPev and P. C. E. Stamp
of inter-grain interactions (such avalanches also occur in multi-domain
magnets, because of dipolar interactions between the domains and
walls).7.31 Thus to interpret most experiments (including the remarkable
recent results on Mn ~20 12 molecules), 23 we must incorporate these interactions. This will be the subject of a future paper.
However some experiments may not suffer this problem. The recent
results of Coppinger et al., 29 on ErAs grains, with S ~ 10 3, may be a case
in point. It is not yet clear what is the effective coupling to the electron
bath for this system, but we believe that the theory in the present paper
should be applicable to this case.
This summarizes our results for the problem of the quantum relaxation of a single mesoscopic or nanoscopic magnetic particle, or for an
ensemble of n o n - i n t e r a c t i n g such particles. We have dealt with the regime
of small bias and low temperature, but excluded interactions between particles, and we have left the expressions in a general form applicable to an
"easy axis/easy plane" particle. The main purpose of the somewhat lengthy
analysis was to show the full variety of relaxation mechanisms for independent particles, and how they work together depending on coupling to
electrons, phonons, and nuclei, as general functions of temperature and
external field. There are 2 obvious extensions of this work which will be
necessary to deal with real experimental systems. First, we would like to
apply these results directly to experiments in which the magnetic particles
are relaxing independently. Second and perhaps more important, we need
to analyse the effect of interparticle interactions, which will modify the
results in the present paper. This will be the subject of future articles.
NOTE ADDED: Since this paper was submitted for publication, a
number of papers have appeared which have some bearing on the subject
of the quantum relaxation of grains.
(i) A preprint of Friedman et al. 45 has found a remarkable structure
in the hysteresis curve of a sample of Mn acetate molecules, which they
argue is due to phonon-mediated resonant tunneling of the giant spin
between different energy levels. At the same time Burin et al., 44 on the basis
of the avalanche results in Paulsen and Park, 23 as well as the results of
Friedman, argue for a quite different picture, in which the dipolar interactions between the Mn molecules are essential in the tunneling and the
thermal relaxation; the final picture involves both the phonon processes
discussed by Politi et al.14 and intermolecular dipolar interactions, working
together. The crucial point is that the experiments show that the selection
rules which must operate if only phonon processes are involved, are clearly
being broken, and the only reasonable way for this to happen is via dipolar
interactions.
201
Another set of experiments by Thomas et al. 46 on Mn molecules in a
regular array confirms this picture of interaction driven relaxation. It seems
to us that these two beautiful experimental results, taken together, provide
very strong evidence for the role of dipolar interactions in the relaxation.
(ii) A paper by Levine and Howard 4v has analysed the changes
occurring in our discussion of the spin bath effects on grain tunneling, once
one takes account of the strong fluctuations in an antiferromagnetic particle about the mean-field Neel state. While we are in agreement with the
computer simulations of the low-frequency spectrum of the Neel antiferromagnet done by these authors, we do not agree with their discussion of the
coupling to the nuclei. They tried estimating the effect of the nuclear spins
on coherent grain tunneling perturbatively, in an expansion in the hyperfine coupling COo, assuming that the hyperfine coupling HhZ= --COo52i IjSi, i.e., that the hyperfine interaction is the same on each of the atomic sites
of the grain. They apply this to the ferritin system, 5~ for which there are
N ~ 100 nuclear spins with coo ~ 50 MHz each. Levine and Howard make
a perturbative expansion, to second order in the hyperfine coupling, taking
only the transverse couplings into account They thereby estimate an effect
of order coo/J~
2
40 kHz, for a single nuclear spin, where J is the very high
energy exchange coupling between the localised electronic moments. This is
to be compared with an experimental Ao ~ 1 MHz (see Ref. 50). This
estimate was not multiplied by N.
However their estimate ignores the far stronger longitudinal coupling
Hlong = --COo ~ i S~I;, which has a large matrix element V01 = ( {Is} [(0l
Hlo,,g ]l)[{Is} ) between the 2 lowest states. Moreover there is no need to
estimate V01; it can be calculated directly. In the semiclassical limit, ignoring
nuclear spin flips so that the nuclear field is static, we have already seen
that the longitudinal coupling is nothing but the internal bias field e of the
present paper, i.e., that Vol ~ Eo = x / N coo, which for Ferritin is ~ 500 MHz
(see discussion following Eq. (8) of Ref. 25, or the detailed analytic results
in Refs. 10, 26). In the "quantum delocalisation regime" of interest to ReE 47,
one can simply calculate Vol numerically, with the numerically produced
doublet states at hand. We have done this, in the same way as Levine and
Howard, for values of K/J ranging from 3 (semi-classical behaviour) to 0.1
(Quantum limit), and find that V01 decreases by a factor of only 1.5
between the 2 limits, and is given very accurately by the semiclassical result
when K / J = 3. One then finds for the ferritin system that instead of the
nuclear spin effects being ~ 25 times smaller than A ~ 1 MHz, as claimed
in, 47 they are ~500 times larger! Of course in general the detailed theory
is considerably more complex than this, since one must take account of the
202
other effects discussed in the main body of the present paper (i.e., topological decoherence, orthogonality blocking, and nuclear spin diffusion, as well
as the degeneracy blocking which arises from the longitudinal hyperfine
field); all of this is discussed in Refs. 10, 25, 26 in the context of coherence
(as opposed to relaxation). The cases of ferritin and Mn acetate are a little
bit special in that topological decoherence plays little role (for Mn acetate,
even orthogonality blocking is probably negligible).
We should note also that since the publication of Ref. 47, Levine and
Howard have issued an erratum 48 which corrects the nuclear spin calculations in Ref. 47.
(iii) A very recent preprint by Abarenkova and Angles d'Auriac has
appeared, 49 which studies the problem of a giant spin coupled to a few
nuclear spins, using exact diagonalisation and perturbative techniques. This
study essentially focusses on how accurate is the truncation of the giant
spin model coupled to a set of nuclei, to the effective Hamiltonian we use
in our papers. As far as we can tell their results for integer giant spin tunneling correspond to our analysis in the limit of pure degeneracy blocking.
(iv) we have also received a preprint from Politi et al. 5~ which is a
longer version of their earlier work, and contains a very detailed analysis
of the quantum relaxation of single Mn acetate molecules, following on
from their earlier work.14 In addition to a discussion of the phonon relaxation for this system, with which we are in agreement, they also make a
number of remarks which are apparently in disagreement with the work
reported in the present paper. In particular they give an analysis of
Chudnovsky's interaction form and claim that it must be zero-this explicitly
contradicts the discussion in our paper (Sec. 3.2). They also give an
analysis of nuclear spin effects which makes no sense to us-they claim that
since the spreading in the particle levels, caused by the nuclei, will leave
only a fraction A of molecules in resonance, then this will cause the relaxation rate to decrease by a factor A. This is incorrect-what it actually would
do is to cause only a fraction A of molecules to relax at the original rate.
The real picture, analysed in Secs. IV and V of the present paper, shows
that one can get roughly the reduction in the rate for an ensemble of
molecules that Politi et al. are looking for, provided one includes (a) fast
diffusion in bias space caused by nuclear spin diffusion, and (b) ignores the
influence of the nuclear spins on the phonon-mediated relaxation (which of
course one cannot do in general). We also do not understand the claim by
these authors that their estimate is in accord with the "oscillator bath"
model calculations of nuclear spin effects by Garg. 19 This is not just
because we disagree with Garg's method and results (see main text, particularly the end of Sec. 4.1 ), but also because the result given by Garg is
203
in fact quite different, both in numerical magnitude and form, from the one
given by Politi et al.
In any case, we now would argue that the experimental evidence that
interparticle dipole interactions play a crucial role in the relaxation of the
Mn acetate system, makes it untenable to use a theory without them (see
(i) above).
APPENDIX
IN
A. K O N D O - T Y P E R E N O R M A L I Z A T I O N S
T H E MULTI-ORBITAL SCATTERING
In truncating the initial Hamiltonian (2.2) to the 2-level system for the
giant spin one generates both the diagonal, i.e., proportional to "~#~ term,
and nondiagonal terms ~ ~_+#~. The latter are usually not present in the
spin-boson Hamiltonian because they are subdominant to the diagonal
ones. In general we may write
H e f f = A o f x + r z a. .-. .
~ Jlckl,~Ck,l,~+~+
~ ,
.~,-~
~
O"
T
kk'l
* "~Ck'l',
J l l•' Ckl,
o~
(A1)
kk'll"
Here I and 1' are the orbital channels for electron scattering off the grain
(lmax ~ k F R o >> 1), and we write the Hamiltonian to be diagonal in l for
z_~#z coupling. We observe that initially (before scaling) the transverse
coupling J~ is exponentially small as compared with jz. In fact j ~ / j z ~ Ao/
~o ~ ( ~ [ ~ ) because transverse matrix elements flip the giant spin as well
and are therefore proportional to the overlap between the [~ ) and I~ )
states.
Formally (A1) is a strongly anisotropic Kondo model with multichannel scattering in a magnetic field A o applied along the weak-scattering
direction. Since the J• are small we may ignore the renormalization of the
J~ and study the renormalization group equation for the J:- first. Introducing dimensionless couplings g~, = J~,N(O) and the same for gf, we write
the well-known scaling equations for the Kondo-problem as
~
lz,
-
1
% = ~75~2Z a2
~
(A2)
(A3)
l
where ~ = ln(f~o/T ) is the scaling variable, and 6~ is the scattering phase
shift due to gj. The crucial point is that for a large grain, when many scattering channels are equally coupled to the giant spin, the Anderson
orthogonality 2a term % will always dominate in the scaling of g• Thus,
204
N.V. ProkoPev and P. C. E. Stamp
quite generally, we never face the strong coupling regime when the g= are
of order gZ and large.
Naively one would ignore these transverse couplings completely, but
we have to remember that the giant spin dynamics is governed by the
transverse magnetic field Ao which is also subject to renormalization due to
Anderson orthogonality. The relevant renormalization group variable is
fugacity Ao/T, and
aA~ = A~(1 - as)
a~
(A4)
(the solution of this equation is explicitly used in our equations (2.8)(2.10)). Thus the question of whether we may neglect conduction electron
spin flips or not depends on the parameter
}
if 0 is less than unity we are back to (2.8)-(2.10), and electron spin
flips are irrelevant. If we find 0 > 1 then those correct expression for the
giant spin relaxation time will be (see, e.g., [21 ] )
"c l~2~zlJ~rN(O)l~T((27ET'g)max'~
2(~s O)'
\
/
Ol-I-(~"=O
-T-
(A6)
Note that for 0 > 1 we definitely have ~s ~ $2/3 >~ 1, SO the S dynamics are
definitely incoherent, and the above refinement is hardly different from
(2.8)-(2.10) (it is equivalent to going from 20~.- l to 2o%+ 1 - 20).
APPENDIX B.
PHYSICAL PROCESSES INVOLVED IN
T H E S P I N B A T H MEDIATED RELAXATION
In this Appendix we look a little more closely at the physical processes
involved in the formula (4.16) for the nuclear spin-mediated relaxation.
Why should the polarization state change by 2M each time in Eq. (4.16)?
After all, when S flips, the number of nuclei which flip is random (with an
average number 2)! The answer to this question comes from energy conservation considerations.
Consider a single grain that at t = 0 has its nuclear environment in the
state IX1, el, AN) corresponding to the polarisation group AN and the bias
energy el (see Fig. 7). Now S flips--what are the possible final states for
the combined system, which have energies in resonance with e 1? There are
three possibilities:
205
(1) N o n e o f the e n v i r o n m e n t a l spins flip; t h e n the final state is
iZ1, ej-, - - A N ) ) w i t h e j = - - e l , since vza~ c h a n g e s sign.
(2) S o m e s m a l l n u m b e r o f e n v i r o n m e n t a l spins, say r ~ N ,
flip
t o g e t h e r w i t h S, so t h a t the n e t c h a n g e in the n u c l e a r spin p o l a r i s a t i o n is
2 M (i.e., (r + M)/2 u p a n d (r-M)/2 d o w n spins are flipped). As a first
a p p r o x i m a t i o n the final state e n e r g y will be ef~,--e 1 +Moo; a s s u m i n g
t h a t the s p r e a d in n u c l e a r frequencies is small, we m a y neglect for the
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-M+2
13>
Fig. 7. Some possible transitions when S flips, for a grain in zero external field.
The initial state of the combined giant spin/nuclear spin system has energy el;
in the diagram this state l1 } has either initial nuclear polarisation AN = M or
AN= M + 2. Transitions to states 12) or [3) are accomplished by changing the
nuclear polarization by 2M (so the final polarisation is either - M or - M + 2).
The transition to state 14) is made by flipping a very large number ~ N of
nuclear spins, whilst still changing the polarization by 2M only. This allows us
to "fine tune" resonance with state I1), as described in the text. Only a few
polarization groups are shown in the Figure; the insert shows how these fit into
the distribution W(e)(shown for both initial and final states).
206
moment the correction 6er ~ 6cokxfr < (#coo, coo). We denote these states as
h2), and L3> in Fig. 7.
(3) An enormous number of nuclear spins flip together with S, with
r as large as N; in this case we can not neglect the correction 6e~_ u ~flcoo
any more, because it is comparable or even larger (for # > 1) than the typical difference min{M}(2e- Mcoo)~coo. Then we can use c~er to "fine tune"
a resonance (which is impossible, in general, for the case (2)). Thus, we
find in this case ef= -~1 + Mcoo + (~e~. The correction strongly depends, of
course, on the particular set of nuclei flipped; a "fine tuned" situation is
shown as state 14) in Fig. 7.
It seems at first, that the latter possibility is the best we can do for
tunneling in resonance. Recall, however, that the probability that any given
environmental spin flips during the transition is very small (0~ < 1 ). Now, if
we are going to tunnel between l1 ) and 14> then the amplitude of such a
transition will be ~ Ao(~k) u, and for large N will be even less then 2 - N 6C0~!
Thus such a narrow resonance is simply impossible to realise.
In fact, only a small number of environmental spins may be flipped
with a reasonable probability. Since the probability to flip none is just e-~
2 r
we find the probability to flip exactly r spins to be p~ ~ e 2 (0ok)
C~
e XU/r!, which peaks at r ~ 2. The parameter 2 may be large in some
systems, but still )~< N. One immediately recognizes that the case (1) plays
a role at small 2, while the case (2) will dominate when 2 >> 1 (formally, we
could include (1) in (2) as a particular transition with r = 0).
Now we make use of the inequality coo >>A o to notice that among all
possible transitions with different M, the energy mismatch between the
initial and final states e ~ - ef= 2e~-Mcoo is either much larger then Ao for
all M, or is close to the resonance e l - e l ~ Ao for only one specific value
M~, with all the other transitions being 1>coo away in energy. Once we
have a resonance the system will make transitions from ft to g by changing
the nuclear spin polarization by exactly _+2M~ each time to maintain this
resonance. As for all the other transitions with M r M~, they give only
very small corrections, of order (Ao/coo) 2 or less (cf. Eqs. (1.7) and (4.18))
to the main contribution to the dynamics of S, coming from the M = M~
term. Of course Eqs. (4.16) and (4.17) simply sum over all processes, and
the resonance value of M, which totally dominates in the sum, is included
automatically.
It is worth noting that nothing depend explicitly on the initial
polarization state. This point is further illustrated in Fig. 7. For given ea the
initial polarization state can be M, and if e~ is close to the center of the
Gaussian G~(e- Mcoo/2), then the resonance value is M~ = M. If however
el is, say, the down-tail state of another polarization group M + 2 , then
\\
207
\X
X
X
\
\
\
-Ma-2
j/
f
I1>
E
~j/~
/
\
\
Mz _~__
MI+M 2
-
-
2
--(')o
.
.
.
.
\ \\
.
.
.
.
.
/
a . . . .
\
L . . . . .
I
-M
\
2
\
s
Fig. 8. The same set of transitions as described in Fig. 4, but now for a grain in
an external bias ~H, acting on ST.The change in polarization is M 1+ M 2. The
insert shows the initial and final distributions W(e ++~H), displaced from each
other by 2~H.
transition to the up-tail state of - M + 2
has exactly the same energy
m i s m a t c h and polarization change 2M. That is w h y the only relevant
statistical average is over the bias energy. O f course, if/~ ~ 1, and different
polarisation groups do not overlap, the bias energy is related to the specific
g r o u p and Eq. (4.16) m a y be written as ( ~ H = 0 )
P(t)
- - 2 N z ( f l1)
~Mr(N+~/2fdeG~(e_Mcoo/2)
e_~PM(t,e+~i~_M~Oo/2)
~u
(B1)
It is easy to understand n o w that external bias, which is indistinguishable from the internal one as far as the giant spin is concerned, will simply
shift the resonance condition for given e to some other value M~ + ~,, as is
clearly seen from the diagram in Fig. 8, which is similar to Fig. 7, but now
with ~H r O.
208
ACKNOWLEDGMENTS
T h i s w o r k was s u p p o r t e d b y N S E R C i n C a n a d a , b y the I n t e r n a t i o n a l
Science F o u n d a t i o n ( M A A 3 0 0 ) , a n d b y the R u s s i a n F o u n d a t i o n for Basic
R e s e a r c h (95-02-06191a). we w o u l d also like to t h a n k I. Affleck, M. B l o o m ,
B. G. T u r r e l l , a n d W. H a r d y for d i s c u s s i o n s o f t h e t h e o r y , a n d B. B a r b a r a ,
A. Berkowitz, F. C o p p i n g e r , D. M a u d e , C. P a u l s e n , a n d J. P o r t a l for disc u s s i o n s c o n c e r n i n g e x p e r i m e n t s . M o r e g e n e r a l d i s c u s s i o n s c o n c e r n i n g the
spin b a t h , w i t h S. C o l e m a n , A. J. Leggett, R. P e n r o s e , a n d W. U n r u h , were
also very useful, as was c o r r e s p o n d e n c e w i t h E. S h i m s h o n i .
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(3.52) of this reference. Incidentally, these equations in the text are valid even when the
field goes to zero. Thus, the phonon relaxation rate goes to zero with the field-this is of
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in the text. Notice also that the discussion of the zero-field limit of the equations in Sec. 3
is entirely academic unless the magnetic grain has been completely isotopically purified.
22.
23.
24.
25.
26.
27.
28.
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44.
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49.
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209
This is because the hyperfine coupling of a single nuclear spin to the giant spin will almost
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simply an input to the complete theory in Sec. 5.
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Quantum relaxation of magnetisation in magnetic particles

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