Collective Spin Dynamics in the "Coherence Window" for Quantum Nanomagnets

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by  I. S. Tupitsyn

Collective Spin Dynamics in the ”Coherence Window” for Quantum Nanomagnets
I. S. Tupitsyn1,2
1 Pacific Institute for Theoretical Physics, University of British Columbia,
6224 Agricultural Rd., Vancouver, B.C. V6T 1Z1, Canada
2 Theoretical Division of the Institute of Superconductivity and Solid State Physics,
Russian Research Centre ”Kurchatov Institute”, Kurchatov Sq.1, Moscow 123182, Russia
5
0
Thespincoherencephenomenaandthepossibilityoftheirobservationinnanomagneticinsulators
0
attract moreandmoreattention inthelast severalyears. Recentlyit hasbeenshown thatinthese
2
systems in large transverse magnetic field there can be a fairly narrow ”coherence window” for

n phononandnuclearspin-mediateddecoherence. What kindofspindynamicscan thenbeexpected
a in this window in a crystal of magnetic nanomolecules coupled to phonons, to nuclear spin bath
J
and to each other via dipole-dipole interactions? Studyingmultispin correlations, we determine the

3 region of parameters where ”coherent clusters” of collective spin excitations can appear. Although
two particular systems, namely crystals of Fe8-triazacyclonane and Mn12-acetate molecules, are
] usedinthisworktoillustratetheresults,herewearenottryingtopredictanexistenceofcollective
l
l coherent dynamics in some particular system. Instead, we discuss the way how any crystalline
a
system of dipole-dipole coupled nanomolecules can be analyzed to decide whether this system is
h
- suitable for attempts to observe coherent dynamics. The presented analysis can be useful in the
s search for magnetic systems showing thespin coherence phenomena.
e
m
PACSnumbers:
.
t
a
m I. INTRODUCTION ment and the longitudinal bias acting on i-th molecule,
this two-state representation is valid only if ∆ is small
- i
d Inthelastdecadethequantumtunnelingphenomenon in comparison with the spin gap Eg to the next levels.
n in nanomagnetic insulators has been attracting exten- For example, in two well-known central spin |S~| = 10
o
c sive interest. Many experiments have been done to systems, Fe8-triazacyclonane (Fe8) and Mn12-acetate
[ study the tunneling relaxation in the ensembles of mag- (Mn12), this condition is met since Eg ∼ 5 K and
netic molecules with central molecular spins S~ .1,2,3,4 ∼11K respectively while the values of a zero-field tun-
3 i neling splitting are ∼10−7 K and ∼10−11 K.
These molecules couple to each other via dipole-dipole
v
0 interactions,5,6,7,8 to phonons4,9,10,11,12 and to nuclear Suppose that molecules do not interact with each
2 spins.5,13,14 The early study of these systems in a low other. Then the central spin of any molecule can os-
2 temperatureregimehasbeenconcentratedmainlyonthe cillate between states |↑i and |↓i (A3) and this process
8 incoherent tunneling in low transverse fields, when the is described by the probability P↓↑(t) (A4). If ∆ >> ξ,
0 magnitude of the ground state tunneling splitting 2∆ the amplitude of these oscillations is ≈ 1. If the central
4 (produced by the tunneling between two potential wellos spinofeachmoleculeisalsoisolatedfromitsnuclearsub-
0
separated by a barrier of magnetic anisotropy; ∆ is the system and from the phonon thermostat, the tunneling
/ o
t tunneling matrix element) is small in comparison with oscillations,beingcoherent,canlastforaninfinitelylong
a
m the parametersdescribinginteractionswithenvironment time. Interactions with the nuclear spin and the phonon
providing anomalously high decoherence. During last thermostats lead to decoherence and, after the so-called
-
d several years more attention has been paid to the spin decoherence time τφ, coherence will be suppressed and
n coherence phenomena.15,16,17,18,19 oscillations will disappear.
o Asithasbeenshownrecently,20 innanomagneticinsu- Thedecoherence”qualityfactor”,givinganestimation
c latorsinlargetransversefields,where∆ (H⊥)increases, for the number of coherentoscillations in the system be-
: o
v therecanbeafieldregion(”coherencewindow”)inwhich fore coherence will be suppressed, is Qφ ∼ 1/γφ, where
i γ = h¯/(∆ τ ) is the dimensionless decoherence rate.
X both phonon and nuclear spin-mediated decoherence are φ o φ
The contributions to the decoherence time τ from in-
drastically reduced (electronic decoherence in magnetic φ
r teractions with the nuclear spins and phonons (τnu and
a insulators is absent). The existence of such coherence φ
window is important both for fundamental physics (at- τph, respectively) are:20,21
φ
temptstofindmaterialsshowingcoherentspintunneling
phenomenon) and for quantum device engineering (at- 1 E2 1 S2Ω2∆3
= o ; = o o coth(∆ /k T), (1)
tempts to make a solid-state qubit). τφnu 2∆o¯h τφph Θ4D¯h o B
Atverylowtemperatureeachmoleculewithlargecen-
tral spin S~i can be modelled as a two-level system (Ap- where Eo is the half-width of the Gaussian distribution
pendix A), whose HamiltonianHi =−∆iτˆix−ξiτˆiz oper- of the hyperfine bias energies; ΘD is the Debye energy;
ates in a subspace of only two lowest states of S~ . With and Ω ∼ E is the energy of small oscillations in the
i o g
∆ and ξ being the ground state tunneling matrix ele- potential wells.
i i
2
The goal of the present work is to study the spin dy- (1) The ”central spin” Hamiltonians for Fe
8
namics in ensembles of dipole-dipole coupled magnetic and Mn molecules. Below ∼ 10 K for Fe and
12 8
molecules in the coherence window for the nuclear spin below∼40K forMn thesemoleculesaredescribedby
12
and phonon degrees of freedom at times t<τ , twosimilarS =10Hamiltoniansofmagneticanisotropy:
np
τnp =min{τφnu,τφph}. (2) HS(Fe) =−DSz2+ESx2+K4⊥(S+4 +S−4)−geµBH~S~, (5)
with24 D/k =0.23K, E/k =0.094K, and K /k =
Namely, in this work we would like to study the inter- B B 4 B
−3.28×10−5 K; and
nal dynamics of a temperature equilibrated system, but
not the dynamics induced by the artificialpreparationof
H(Mn) =−DS2−K||S4+K⊥(S4+S4)−g µ H~S~, (6)
a system at t = 0, say, in state | ↑↑ ... ↑i (the anal- S z 4 z 4 + − e B
ysis of the latter problem will be presented separately).
From now on, for the sake of brevity, the coherence win- with25 D/kB = 0.548 K, K4||/kB = 1.17 × 10−3 K,
dow will be called the NPC-window (nuclear spins and K4⊥/kB =2.2×10−5 K.
phononscoherencewindow). Toillustratetheresults,all Note that in the Fe8 system the tunneling splitting
particular calculations will be based on the parameters ∆o(H~⊥) and its period of oscillations with H~⊥ have
for two systems, namely, for crystals of Fe8 and Mn12 been measured26,27 while in the Mn12 system these pa-
molecules. rameters have never been measured. The latter makes
it rather problematic to verify the value of the tunnel-
ing splitting obtained directly fromthe Hamiltonian(6).
II. HAMILTONIAN AND INTERACTIONS However,wewouldliketostudytheregionoflargetrans-
verse fields where ∆ (H~⊥) is already large (although
o
∆ << E ) and is less sensitive to some variations of
At very low temperatures a set of molecules with cen- o g
tral molecular spins |S~ | = S coupled to each other via the anisotropy constants28 (moreover, at some stage we
i starttomakeestimationsratherthanexactcalculations).
the dipole-dipole interaction can be described by the ef-
Thus, in what follows we use the Hamiltonians (5) and
fective Hamiltonian:
(6) for Fe and Mn molecules.
8 12
1 (2) Dipolar interactions. For the sake of defi-
H = (−∆ τˆx−ξenτˆz)+ Vˆ (~r ), (3)
i i i i 2 dd ij niteness we apply a transverse magnetic field along the
Xi Xij x-axis, so that only the Siz and the Six projections of
where τˆz and τˆx are the Pauli matrixes; ∆ is the tun- the total molecular spin S~i are nonzero. Therefore, the
neling matrix element; and ξen is the bias aciting on i-th interaction term Vˆdd can be rewritten as:
i
molecule from externaland nuclear fields. The last term
Vˆ (~r )= Vαβ(~r )τˆατˆβ, (7)
in (3) describes the dipolar coupling between pairs of dd ij dd ij i j
molecules, separated by distance |~rij|=|~ri−~rj|: {α,βX}={x,z}
where all Vαβ(~r ) can be obtained from Eq.(4). The
E (~τˆ ~r )(~τˆ ~r ) dd ij
Vˆ (~r )= D ~τˆ ~τˆ −3 i ij j ij , (4) i-th bias energy ξen in (3), as it is written, contains con-
dd ij |~r |3 i j |~r |2 i
ij ij ! tributions only from the longitudinal external and nu-
clear fields. The dipolar contribution to the total bias
whereE =(µ /4π)g2µ2S2;µ /4π =10−7N/A2(inthe ξ acting on i-th molecule can be written in the form
D 0 e B 0 i
SI system of units); g is the electronic g-factor; and µ ξd =−g µ SzHz(dip) and the longitudinaldipolar filed
e B i e B i i
is the Bohr magneton. Note that Hamiltonian (3) does Hz(dip) at i-th site is:
i
not include the interactions with phonons and nuclear
spins. Instead, the known results20,21 for the phonon Hˆz(dip)= FD 3(τˆjzzij +τˆjxxij)zij −τˆz , (8)
and nuclear spin decoherence rates, Eq.(1), will be used. i |~r |3 |~r |2 j
j6=i ij (cid:18) ij (cid:19)
Coherence window for the nuclear spin and phonon X
channels of decoherence opens up at high transverse
where F = (µ /4π)g µ S and z , x are the corre-
D 0 e B ij ij
fields,wherethe valueofthetunnelingsplitting becomes
sponding components of vector~r .
ij
large in comparison with the parameters describing in-
The distributions of the dipolar bias energies created
teractions of the central spin with the environment. At
by molecular spins in polarized and depolarized samples
these conditions all ∆ in a sample are approximately
i are different in the low transverse field limit and similar
the same22 and for brevity can be replaced (where it is in the high transverse fields limit (where S~ is oriented
i
reasonable)by one parameter∆ , whose transversefield
o nearlyalongthetransversefielddirection). Atlowtrans-
dependence ∆o(H~⊥) can be calculated using the corre- verse fields the half-width WD of the dipolar bias dis-
sponding molecular Hamiltonian for the central spin S~. tribution in a completely depolarized sample is several
For both the Fe and the Mn molecules these Hamil- times larger than in a polarized sample. At high trans-
8 12
tonians are (approximately) known. verse fields the half-width W in both samples is nearly
D
3
the same (comparing the longitudinal field distributions TounderstandhowW behavesatlargetransversefields,
D
for polarized and depolarized samples at Hx = 4.8 T in it is sufficient to calculate this parameter in a model de-
Fig.5ofAppendix Bonecanseethattheyarenearlythe polarized sample where all molecules are in states | ⇑i,
same). This parametercanbe calculatednumericallyfor but Sz/|Sz| = 0.32 The transverse field dependence
i i i
any sample. of important parameters for crystals of Fe and Mn
8 12
P
(3) Hyperfine interactions. The interactions molecules is presented in Figs.1 and 2 (the description
of the central molecular spin S~ with the nuclear spin of our calculation procedure is given in Appendix B).33
i
Deviations from the results of Figs.1 and 2 for nonzero
bath lead to the ”spread” of each molecular spin state
characterized by the half-width E of the Gaussian dis- populationsofstates|⇓iareinsignificantuptothelimit
o
of equipopulation - this is clear, for example, from Fig.5
tribution of the hyperfine bias energies ξ . For N nu-
N n
clear spins I~ in each molecule, one finds20,21,29 E2 = (Appendix B).
k o
Nn (I +1)I (ω||)2/3, where {ω||} are the (longitudi- Depending on the crystal structure and the sample
k=1 k k k k geometry, the dipolar fields distribution can be shifted
nal) couplings between the central spin and each k-th
Pnuclear spin.13,14 Knowledge of all nuclear moments and (such a shift can be rather large, see, for example, Fig.5
in Appendix B). This shift changes with the transverse
positionsofallnucleiinthemolecule30 allowsonetocal-
field. The larger the shift, the slower both the W (H⊥)
culate all these coupling constants and E .8,20,21,29,31 D
o and the E (H⊥) decrease with H⊥. This effect can be
(4) The transverse magnetic field behavior of o
seen in the high-field part of Fig.1. Two upper curves
important parameters. The ground state | ⇑i
for both W andE representthe results ofcalculations
(symmetric) and the excited state | ⇓i (antisymmet- D o
in our Fe cluster ”as it is” (with no longitudinal field
ric) (A2) of Hamiltonian (A1) are separated by the en- 8
compensation). To obtain the two lower curves for both
ergy gap 2ε = 2(∆2 + ξ2)1/2. At low temperatures
i i i W andE ,thecorrespondingexternallongitudinalfield
(k T < ∆ ) in the limit ∆ >> ξ in a temperature D o
B o i i wasappliedforeachvalueoftheexternaltransversefield
equilibrated sample most of molecules are in states |⇑i.
to shift a position of the longitudinal fields distribution
Then, calculatingmatrixelements h⇑|τˆz|⇑i=ξ /ε and
i i i backtozero(thelongitudinalfieldcompensatedsample).
h⇑|τˆx|⇑i=∆ /ε ,for∆ >>ξ onefindsh⇑|Sz|⇑i→0
i i i i i i The shift in our Mn sample is small.
and h⇑ |Sx| ⇑i → S. Thus, as ∆ increases with the 12
transverseifield, both W and E shoould decrease. 1x1011 (Mn12 crystal)
D o
1x1011 (Fe8 crystal) 1x1000 WD(K) d e psaomlaprilzeed D o (K)
depolarized
1x1000 sample D o (K) 1x10--11
K)1x10--11 WD(K)p soalmarpizleed nergy (K)1x10--22 Eo(K) p osalamrizpeled
gy (1x10--22 Eo(K) E1x10--33
er
n
E
1x10--33 1x10--44
Field
1x10--44 comspaemnpslaeted 1x10--55
0 1 2 3 4 5 6 7 8 9
1x10--55 Hx (T)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 FIG. 2: Mn12 crystal (cluster of 503 Mn12 molecules; no
Hx (T) ”fasterrelaxingspecies”22,23). Thecurvesare: solidlinewith
circles - WD(Hx) for ”depolarized” sample; stars - WD(Hx)
FIG. 1: Fe8 crystal (cluster of 503 Fe8 molecules, for de- forpolarizedsample;diamonds-Eo(Hx);solidline-∆o(Hx).
tails see Appendix B). The curves are: circles - half-width
WD ofthedipolarbiasdistributionvstransversefieldHx (in
Tesla) for model depolarized sample (see the text); stars - (5)NPC-window. Whenstudyingthespindynam-
WD(Hx)forpolarizedsample;triangles-sameascircles,but ics in the NPC-window, one needs to know the regionof
for the field-compensated sample; diamonds - half-width Eo thefieldwherethiswindowissituated. Thiswindowcan
ofthehyperfinebiasdistribution;squares-sameasdiamonds be rather narrow and for our examples of the Fe and
8
but for the field-compensated sample; solid line - tunneling the Mn systems this can be seen in Fig.3. Since we
12
matrix element ∆o(Hx). All molecules are in a state | ⇑i. are not going to discuss here a coherence optimization
(WD, Eo, ∆o are in Kelvins.) strategy,20 in this Figure we present the transverse field
behaviorof the dimensionless decoherence rates γnu and
When ∆o << WD, in a depolarized sample the value φ
ofW is severaltimes largerthan ina polarizedsample. γph, Eq.(1), for external field along the x axis only and
D φ
4
for molecules containing only natural isotopes.34 (Note pendix C). Such a pair is described by the Hamilto-
that both rates are almost insensitive to changes in the nian (C3) (ξ is the bias energy; in general, it is the
i
populationsofstates|⇑iand|⇓i.) Thesmalloscillation time-dependent parameter) and can be found in four
energy Ω is ∼ E and, like E (H⊥), slowly decreases states: | ⇑ ⇑ i; | ⇑ ⇓ i; | ⇓ ⇑ i; | ⇓ ⇓ i. In the limit
o g g 1 2 1 2 1 2 1 2
with H⊥. In zero field Ω =2SC (DE)1/2 (C ≈1.56) ∆ >> |Vαβ| two central ”flip-flop” states | ⇑ ⇓ i and
o ⊥ ⊥ o dd 1 2
for Fe and Ω ∼ 2SD for Mn .4 The Debye energy |⇓ ⇑ i, linked by the effective tunneling matrix element
8 o 12 1 2
Θ for Fe and Mn is known experimentally.35
D 8 12
∆ ∆
∆ ∼|Vzz(R)| 1 2 (9)
1x1011 Natural isotopes ff dd ε ε
1 2
T=0.05 K
1x1000 (only the largest term in B(R~) Eq.(C6) is shown - its
formissimilartothatknownfromthetheoryofdielectric
1x10--11 glasses36,37), can be considered as an effective two-level
g nu system with the asymmetry
f
g 1x10--22 (Mn1 2 ) ξ ≈|ε −ε |∼|ξ2−ξ2|/2∆ <<∆ (10)
f (Fe8 ) ff 1 2 1 2 o o
1x10--33 (q D =38K) (recall that all ∆i are supposed to be the same). Note
(q D =33K) that all Vdαdβ(R) are independent of the external field.
1x10--44 Two other states are separated from the two flip-flop
g ph statesbytheenergygaps>∆o andintheregionoffields
1x10--55 f where∆ >|Vαβ|,theireffectontheflip-floptransitions
o dd
is small. If ∆ > ξ , two flip-flop states are in reso-
ff ff
1x10--66 nance and the amplitude of oscillations (with frequency
0 1 2 3 4 5 6 7 8 9 E ∼ (ξ2 +∆2 )1/2, (C8)) between them is ≈ 1. At
H x (T) thfefsameftfime, tfrfansitions between other states are not
FIG. 3: The transverse magnetic field behavior of the di-
inresonance(theyareaccompaniedbytheenergychange
mensionless decoherence rates γnu (shown by squares and
circles) and γph (diamonds and φtriangles) at azimuthal an- >∆o) and their amplitude is <∼(Vdαdβ)2ξi2/∆4i.
gle ϕ = 0 (i.eφ., along the x-axis) for Fe8 (squares and di- In the field region where ∆o <∼ |Vdαdβ| it is more con-
amonds) and Mn (circles and triangles) systems at T = venient to solve the problem for the dynamics of a pair
12
0.05 K. The results are presented for the natural isotopes of interacting spins in the basis set (A3).38 However, in
(Fe56, H1, Br79, N14, C12andO16speciesforFe molecule this limit it can be rather difficult (or even impossible)
8
and Mn55, H1, C12 and O16 species for Mn12 molecule). to observe coherent dynamics in an ensemble of spins.
First of all, in this limit WD also can be >∼ ∆o (like in
Fe andMn ). Moreover,thevarietyofdifferentcollec-
8 12
tiveprocessesleadsto additionalphaserandomnessand,
III. MULTIMOLECULAR PROCESSES IN THE
consequently, to suppression of coherence.
LIMIT ∆o >>{WD,Eo,¯h/τnp} Inwhatfollowswesupposeto workonlyinthe partof
the NPC-window where ∆ >|Vαβ| and the probability
If in the NPC-window the half-width W of the dipo- o dd
D to observe coherent spin dynamics is larger.
larbiasdistributionislargerthan∆ ,anyspindynamics
o At low temperatures (k T < ∆ ) a number of the
B o
is, in general, incoherent. Outside of this window, inde-
moleculesintheexcitedstate|⇓iwithenergy+εcanbe
pendently ofthe ratio∆o/WD,the spindynamics is also estimatedasN (T)∼N e−ε/kBT/(e−ε/kBT+eε/kBT)∼
ex o
incoherent. In this Section we study the multimolecu- N e−2∆o/kBT (N is the total number of molecules) and
o o
larcorrelationsinduced by the dipole-dipole interactions
is small compared to the number of the ground state
between molecules in states | ⇑i and | ⇓i in the NPC-
molecules. These excited molecules are uniformly dis-
window. We assume that in the region of fields of our
tributedoverthesample,andeachofthemissurrounded
interest ∆ >> {W ,E ,¯h/τ } (like, for example, in
o D o np by the ∼ N /N ground state molecules. The ex-
the region 3.5 < Hx < 4.2 T for Fe and in the region o ex
8 cited molecule can be in a ”flip-flop resonance” with
6.7<Hx <7.4T for Mn12, see Fig.3). the ground state molecule only if ξff <∼ ∆ff. The
time needed for the flip-flop transition to happen is
∼ ¯h/∆ ∼ O(R3) and the fastest transitions are ex-
ff
A. One-pair processes pected to be between the nearest-neighbor molecules.
For the effective two-level systems composed of the two
Atverylowtemperatures,whenonlytwoloweststates nearest-neighbormoleculesweintroducethecorrespond-
of each molecule are occupied, in the limit ∆ >> ingeffectivetunnelingmatrixelement∆nnandtheasym-
o ff
{W ,E ,¯h/τ } we work in the representation (A2) metry ξnn.
D o np ff
{|⇑i,|⇓i}ofthe Hamiltonian(A1)(Appendix A). Con- In a simple cubic lattice each excited molecule can
sideronepairofinteractingmoleculesinthesample(Ap- make a flip-flop transition with any of its six nearest-
5
neighborgroundstatemoleculeswiththesameprobabil- ∆ >>{W ,E ,¯h/τ } and ∆ >|Vαβ| both these res-
o D o np o dd
ities. In a generic lattice these probabilities can be dif- onant pairs experience mainly the flip-flop transitions.
ferent since ∆nn depends on the lattice structure. The If R >> a, the strength |Vαβ(R)| of the interac-
ff dd
averageoverthreecrystallographicaxesvalueofthe∆nn tions between molecules belonging to different pairs is
ff
is ∼ WD(H⊥ = 0) for polarized sample, see Figs.1 - 2 <∆nffn. Thene, using the same arguments as for one pair
(W (0) for polarized sample is ∼ E /V(1), where V(1) ofmolecules,inthecaseoftworesonantpairswecanalso
D D o o
consider only corresponding collective ”flip-flop” transi-
is the volume per one molecule).
If ∆nn > ξnn for the overwhelming majority of the tions between the eigenstates of eachresonantpair. The
ff ff effective tunneling matrix elements connecting these col-
nearest-neighbormolecules(thisissueisdiscussedinSec-
lective flip-flop states (separatedfrom allother states by
tion IVA), it is unlikely that at low temperatures any
the energy gaps >∆nn) is
resonant pair of the nearest-neighbor molecules (say, i- ff
th and j-th molecules) will remain in resonance for a ∆nn∆′nn
long time. Instead, since the total probability for the ∆(2)(R~)∼|Vzz(R~)| ff ff , (12)
excited molecule (either i-th, or j-th, as a resultof oscil- ff dd EnnE′nn
ff ff
lations (C7)) to create a resonance with one of the other
five nearest-neighbor molecules is larger than the prob- whereEfnfn ∼((ξfnfn)2+(∆nffn)2)1/2. Similarly to the case
ability to remain in resonance with the same molecule of one pair of molecules, these collective flip-flop states
all the time t ∼ τ , the fastest flip-flop transitions can can also be considered as an effective two-level system
np
”propagate”throughthecrystalinvolvingmoreandmore with the asymmetry
new molecules. Of course, not only the nearest-neighbor ξ(2) ∼|Enn−E′nn|. (13)
molecules can be involved, but also the ”lengthy” pairs ff ff ff
(with ∆ff(R)<∆nffn). However,flip-flop transitions be- Note that we deliberately consider two pairs of nearest-
tween the nearest-neighbor molecules are faster. neighbor molecules since transitions between such
Inwhatfollows,forbrevity,these”mobile”(or”poten- molecules are faster, and in the limit of our interest the
tiallymobile”)flip-floptransitionsbetweenthestates|⇑i probability to find them in resonance is larger.
and | ⇓i in the nearest-neighbor molecules will be called The effective matrix element ∆(2) describes the flip-
”flipons” (a kind of magnon). The number of flipons is ff
flop transitions between the eigenstates of two resonant
determined by the number of excited molecules N (T).
ex pairs(i.e., oftwoeffective TLS). If∆(2) >ξ(2), tworeso-
In a generic lattice ∆nn can be different along different ff ff
ff nantpairsareinresonancewitheach other. Theflip-flop
crystallographicaxes. However,iffliponmovesalongone
transitions between the eigenstates | ⇑i and | ⇓i of each
axis and for this axis {(∆nn) } are large in comparison
ff i moleculeinside ofone resonantpair aredescribedby the
with {(ξnn) }, such a movement is, in some sense, ”co-
ff i matrix element ∆nn. Then, since ∆nn > ∆(2), the fre-
herent” since flipon leaves site i only because of equal ff ff ff
quency of oscillations between states | ⇑i and | ⇓i of
probabilities for the excited spin to create a resonance
with both of its nearest neighbors along this axis. eachmolecule insucha resonantgroupoffour molecules
Itisworthmentioningthat,ifthereisawholedistribu- is ∼∆nn, but the group correlationtime is ∼¯h/∆(2).
ff ff
tionof∆ (say,ifthereare”fasterrelaxingspecies”22,23) Inagenericlattice,ifthenearest-neighbormoleculesin
o
in a sample, the fraction of resonant flip-flop molecules twopairsarelocatedalongdifferentaxes,theasymmetry
decreases. This is because in such a sample for some ξ(2) can be ∼ ∆nn >> ∆(2) and such two pairs can be
ff ff ff
fraction of pairs the asymmetry ξff can be ∼ ∆o (and outofresonance. However,ifmoleculesinbothpairsare
∆o increases with H⊥). These impurities can essentially located along the same axis (with a lattice constant a)
limit (or even completely block) the motion of flipons. and if ∆nn >>ξnn, the asymmetry is
ff ff
ξ(2) ∼|Vzz(a)|f(ξ /∆ ), f(ξ /∆ )∼O(ξ2/∆2), (14)
ff dd i o i o i o
B. Multi-pair processes
where the average value of f(ξ /∆ ) can be estimated
i o
Onaverage,twonearest-neighborexcitedmoleculesare roughly as ∼ WD2/∆2o. Then, for the average asym-
separated by the distance metry one gets ξ(2) ∼ |Vzz(a)|(W /∆ )2 << |Vzz(a)|
ff dd D o dd
Rex(T)∼(Vo(1)No/Nex)1/3. (11) and for such paiers the condition ∆(f2f) >∼ ξf(2f) can be, in
principle, fulfilled (actually, the difference of two mean
At kBT << ∆o, Rex(T) is large compared to a ≡ energies A and A′ (C6) also contributes to ξ(2) and
ff
(Vo(1))1/3 (in a cubic lattice a ≡ a; a is the lattice this gives a similar effect). The term neglected in (14)
constant). Consider two pairs of the resonant neaerest- is ∼ (∆nn/W (H⊥))2 times smaller than the retained
ff D
neighbor molecules and let thee distance between these one - in the field region of our interest in most systems
pairs be R. This group of four molecules contains two ∆nn(a)/W (H⊥)>>1.
′ ff D
excited molecules (with energies ε2, ε2) and two ground Note that for resonant pairs composed of the flip-flop
state molecules (with energies ε , ε′). In the limit moleculeswith∆ (R)<<∆nn,theglass-likescenario37
1 1 e ff ff
6
canberealized. Inthiscasetwopairscanbeinresonance distance between flipons resulting in the suppression of
only if ∆(2) ∼∆ (for most of such pairs ξ(2) ∼∆ ). correlations in clusters (if they appear).
ff ff ff ff
Knowing the sample average value of the asymmetry Suppose that at t = 0 there is a correlated cluster
ξ(2), from the requirement ∆(2)(R~) > ξ(2) one can esti- (TM < T < ∆o/kB) composed of nearly equidistant
ff ff ff flipons (with distance ≈R ). If ∆nn >>ξnn and if the
matethe average”resonant”distancebetweentwopairs: ex ff ff
flipons move along the same axis, correlations between
e
R(2)(H⊥)∼[V(1)V /ξ(2)(H⊥)]1/3, (15) them will not necessarily be destroyed immediately af-
res o dd ff
ter the first ”jump”. Of course, if at t > 0 the flipons
where39 Vdd ∼ ED/Vo(1). IfeR <eRr(2e)s, two pairs could start to move along different axes, in a generic lattice
be, in principle, in resonance with each other. However, any correlations can be destroyed almost immediately
evenifξf(2ef) →0,notanytwopairsareinresonancesince (i.e., at t >∼ ¯h/∆nffn) since in this case ξf(2f) can become
if R>Rph, where ∼ ∆nn >> ∆(2). Note, however, that the flipons have
ff ff
e larger probability to move along the axis with shortest
R (T,H⊥)∼[V(1)V τph(T,H⊥)/¯h]1/3, (16)
ph o dd φ latticeconstant. Then,ifweconsideronlyaquasi-1dmo-
the two-pairs correlation time is longer than incoherent tionofflipons(i.e.,alongthesameaxis),wecanestimate
phonon-assisted transitiones in each molecule. Only the the longest ”motional” dephasing time τdm. Comparison
pairssatisfyingtheconditionR<R =min{R(2),R }, ofthistimewiththeclustercorrelationtimetc (tc <τnp)
m res ph
shows whether the correlated cluster with the average
canbeinresonance. Thus,ifR (T,H⊥)<R ,mostof
ex m distances R between the nearest-neighbor flipons can
the closest pairs of resonant molecules are able to come ex
appear.
intoresonancewitheachother. Thishappensattemper-
For the sake of simplicity, we approximate the flipon
atures T >T ,
M
centers of mass motion by the discrete ”random walks”
T =max{T(2),T }, (17) model (Appendix D). At t > 0 the distances R(t) be-
M res ph
tween the nearest-neighbor flipons in the whole cluster
with Tr(e2s) and Tph given by (kBT <∆o): become distributed around Rex with nonzero half-width
δR(t). Thus,insteadofasingle”line”∆(2)(R )onealso
k T(2)(H⊥)∼2∆ /ln[V /ξ(2)], (18) ff ex
B res o dd ff gets a whole distribution of values ∆(2)(R) with nonzero
ff
e e mean-squaredeviationδ∆(2)(t)=h(∆(2))2−h∆(2)i2i1/2.
k T (H⊥)∼2∆ /ln[Θ4 V /S2Ω2∆3]. (19) ff ff
B ph o D dd o o Knowing δ∆(2)(t), the motional dephasing time can be
If at these conditions τ >t , where obtained from the condition
np c e
tc ∼¯h/|Vdzdz(Rex)|∼(h¯/Vdd)(Re3x/Vo(1)), (20) Ndm
τm =t Nm; t δ∆(2)(N)∼¯h; t =h¯/∆nn (21)
at t>∼tc the whole hierarchyoef (more or less) correlated d f d N=0 f f ff
flip-flop clusters of the increasing ”size” n (the number X
of involvedresonant flip-flop pairs) can, in principle, ap- for N =t/t (or from the condition τdmdtδ∆(2)(t)∼¯h
pear. The time t estimates the cluster correlationtime. f 0
c at large values of N). Obviously, the correlations in the
Note, however, that if ∆nn > ξnn for most of the R
ff ff wholeclusterwillbedestroyedtogetherwiththedestruc-
nearest-neighbor molecules, instead of interactions be-
tion of resonances between the nearest-neighbor flipons.
tween fixed pairs of resonant molecules, in the limit of
(2)
Since in each pair both flipons can move, for h∆ i we
our interest we have a set of flipons moving through the ff
have
sampleandinteractingwitheach-other. AtT <T they
M
participateinthecollectiveprocessesveryrarely(interac-
N
tionscanbeneglected). Whentemperatureincreases,the ∼ |Vzz(R +(r −r )a)|P (r )P (r ) (22)
dd ex 1 2 N 1 N 2
number of flipons also increases and collective processes
r1,rX2=−N
become more frequent. At T >T correlationsbetween
M e e e e e
flipons, in principle, maystill leadto the creationof cor-
with the condition (R +(r −r )a)/R ≥η(T). Here
related clusters. However, due to various decorrelation e e ex 1 2 ex
η(T) is the dimensionless (in units of R (T)) minimally
ex
(dephasing) processes, these clusters can be destroyed
possible distance between the centers of mass of two
e e e
rapidly (or they will not be able to appear at all). Such
flipons. Each distribution P (r ) gives the probability
N i
dephasing processes will be considered in Section IIIC.
to find the i-th flipon at the distance r a (r < N) from
i i
its t=0 position after total N steps (Appendix D).41
e
The solutionof Eq.(21) depends onη and ρ (Eq.(D4))
C. Decorrelation ee e
and can be found numerically. For p = q = s = 1/3
(Eq.(D1)) and ρ=2/3 we get
(1) Flipon motion. If flipons are delocalized, the
effectivetunneling matrixelement∆(2) changeswiththe τm =Nmt ≡3λ(η)[R /a]2t ∼3λ(η)[a/R ]t . (23)
ff d d f ex f ex c
e e
7
If η = 2a/R , for 0.05 ≤ η ≤ 2/3 we get λ ≡ λ ≈ contains contributions from all individual spins in the
ex 2
1.6η2 + 0.35η + 0.045. If η = a/R , for 0.05 ≤ η < sample. When any j-th flipon makes a transition, the
ex
1/2 we geet λ ≡ λ1 ≈ 0.34η2 +0.2η +0.03. Note that change of the bias, acting on i-th flipon, is δifj(Rij) ∼
the configurations of the nearlyeequidistant flipons with Vdzdz(Rij)(ξfnfn)i(ξfnfn)j/(Efnfn)i(Efnfn)j. (Here (ξfnfn)i,j can
Rex =adonotexist,incontrasttothosewithRex =2a. bebothpositiveandnegativeandthetermVzxτzτx does
dd i j
However, if flipons move along the same axis, but in the not change its sign when j-th flipon makes transition.)
neareste-neighborrows,thecentersofmassofsomeflipones Then, if Nt flipons make a transition (max(Nt) = Nex),
can be separated by the distance a. To take this effect the total change of the bias acting on i-th flipons is
intoaccount,onecanuseλ=(λ2+λ1)/2forestimations. δf = Nt δf(R ).
The answer for τm can be found in the equivalent di- i j=1 ij ij
d e Depending on the degree of ”polarization” Mt =
mensionless form Dfτdm/Re2x = λ(η,ρ), where Df is the (Ngs P− Nes)/(Ngs + Nes) of the group of Nt flipons
flipon effective diffusion coefficient (D4) and at ρ=2/3, (N and N are the numbers of flipons in their ground
gs es
λ(η,ρ) ≡ λ(η) from Eq.(23). Tehe ρ-dependence of τm and excited states), the total bias change |δf| can vary
d i
is roughly ∼ ρ−1/3. Then, τdm/tc can become larger ei- roughly from δif ∼ |Vdzdz(Rex)| [(ξfnfn)i/(∆nffn)i][ξfnfn/∆nffn]
teher (i) at T → ∆ /k , when flipons are in their dense
o B to∼δfln(N )(∆nnisthesampleaveragevalueof(∆nn)
phase and essentially localized (δR(t) → 0); or (ii) if i t ff ff j
e e e
s = 1−p−q → 1, ρ → 0 and flipons are almost immo- and ξfnfn is the sample average absolute value of (ξfnfn)j).
bile even at T <∆o/kB. The latter can be, in principle, Thenethe shorteest dephasing time τds is ∼ tf. However,
realized in a sample with impurities. the deipole-dipole interaction changes its sign with the
Ifτm(T)<<t (T),thecreationofacorrelatedcluster direction of R~ and, on average, in the case M → 1
d c ij t
at the average distance R (T) is virtually impossible. the ”surface” spins will mainly determine the maximum
ex
Solving either the equation Nm(R )t = t (R ), or the value of the bias change |δf|. For spins (flipons) in the
d c f c c i
equation bulk this essentially reduces |δf| and increases τs.
i d
D t (R )/R2 =λ(η ,ρ), (24) If Mt → 0, simultaneous transitions of many flipons
f c c c c nearly cancel the effect of each other resulting in |δif| <∼
one finds the average distance Rec and the temperature δif <<|Vdzdz(Rex)|<<∆nffn. In this casethe spectraldif-
k T ∼2∆ (H⊥)/ln[(R /a)3−1] (25) fusion mechanism cannot destroy the resonance neither
B c o c
ienside of the individual flipons, nor between them. In-
at which τdm ∼ tc and cluster can appear. For ρ = 2/3 deed, the contribution from the ξnn to the asymmetry
andη =2a/R wegetR ∼3aandkeT ∼0.6∆ (H⊥). ff
c c c B c o ξ(2) in the limit ∆nn > ξnn is ∼ (ξnn)2/2∆nn and its
For ρ = 2/3 and ηc = a/Rc we get Rc ∼ 2a and ff ff ff ff ff
T ∼∆ (H⊥). These estimations shows that, if the sce- change due to δf is ∼δf (ξnn)i/(∆nn)i.
c o e e i i ff ff
nario with p ≈ q ≈ s ≈ 1/3 is realized, τm(T) < t (T) Onaverage,inatemperatureequilibratedsample,itis
e d ec
almost everywhere except the flipons dense phase at plausible to assume M →0 and in the limit ∆nn >ξnn
t ff ff
T → ∆ /k , where τm(T) ∼ t (T) and where t (T) de- the spectraldiffusion effectis muchweakerthanthe mo-
o B d c c
creasesitself(aswellasτnp(T),seeEq.1). If,incontrast, tional dephasing effect (broadening of the ∆(2) distribu-
s → 1 and ρ → 0, τm >> t and correlated clusters can ff
d c tion due to the flipons motion). This remains valid also
appear even at T <T <T .
M c if the flipons are allowed to move. In this case the spec-
(2) ”Spectral diffusion”. The above described
traldiffusiondephasingtimeτs isroughly∼(∆nn/ξnn)3
pictureisvalidonlyif∆nn >ξnn formostofthenearest- d ff ff
ff ff times longer than the motional dephasing time τm (as it
neighbor molecules. In the opposite limit, at T > T d
M wasalreadynoted,iffliponsmovealongdiffereentaxeis,in
thecorrelatedclusterswillbecomposedofalmostimmo-
a generic lattice correlationscan be destroyedalreadyat
bile ”lengthy” flip-flop pairs with ∆ (R > a) < ∆nn,
ff ff t∼t ).
satisfying the condition ∆ ∼ ∆(2). This scenario is f
ff ff
very similar to that in dielectric glasses37 ande the clus-
ter dephasing time at t < τnp will be determined by the IV. DISCUSSION
process similar to the ”spectraldiffusion” in glasses.36,40
The change of states of fixed effective TLS results in the
In this Section we discuss a temperature equilibrated
bias fluctuations and, consequently,in the dephasing. In sample at kBT <∼∆o, where only the fraction Nex(T) of
this limit (∆nffn < ξfnfn) the cluster ”spectral diffusion” molecules are in the excited states | ⇓i. Thus, we are
dephasing time τs is ∼ t since the asymmetry ξ(2) for not going to discuss the problem of the magnetization
d c ff
most groups of two resonant pairs is ∼∆ff. relaxationinthelimit∆o >>{WD,Eo}. If,forexample,
In the limit ∆nn >ξnn the spectral diffusion-like pro- the sample is prepared at very high temperatures and
ff ff
cess can contribute as well. Instead of going deeply then rapidly cooled down to low temperatures, it will
into the details, here we only estimate the correspond- start to relax to its temperature equilibrated state - this
ing effects. Consider, for simplicity, the case of immo- relaxation process will not be discussed here either.
bile flipons. The bias (ξfnfn)i, acting on any i-th flipon, If,insomesystem,inalltheNPC-window∆o <∼|Vdαdβ|,
8
it can be very difficult (most probably, impossible) to respectively). Forthefield-compensatedFe sample,the
8
observe any coherent spin dynamics in an ensemble of corresponding values are about two times smaller.
interacting spins (Section IIIA). Here we consider only To summarize the results, we note that under the as-
thepartoftheNPC-window,where∆ >|Vαβ|. Inboth sumption of the absence of impurities with larger (or
o dd
the Fe8 and the Mn12 systems, which are used in this smaller) values of ∆i, in both systems in the NPC-
workto illustrate howthe problemcanbe analyzed,this window the average asymmetry ξnn (as well as δε) is
ff
field region is situated to the right of the minimum of
small in comparison with ∆nn for most of the nearest-
γφnu +γφph (see Fig.3). The collective processes, having neighbormolecules. Inthiscafsfeflieponscanmoveinboth
the largestamplitude inthis regionoffields, arethe flip-
systems. e
flop processes.
Note, however, that in the Mn crystal there are
12
faster relaxing (minor) species22,23 (about 5−10 % of
all molecules have lower potential barrier and larger val-
A. The ratio ∆nn/ξnn ues of ∆ ) - these species can change somehow both the
ff ff i
ξnn and the δε. In the NPC-window for major species
ff
The fastest pair flip-flop proceesseseare the processes thespindynamicsoftheseminorspeciesisincoherental-
between the nearest-neighbor molecules. The average reeady. SinceeventheHamiltonianfortheseminorspecies
strength of the nearest-neighbor dipole-dipole interac- is still unknown (or unpublished), we cannot describe
tions and the average value of the flipon effective tun- their effect quantitatively. Obviously, they can decrease
neling matrix element ∆nn are ∼ V (V ∼ 0.12 K the number of flipons and limit their motion.
ff dd dd
for Fe and ∼ 0.07 K for Mn ). To find the aver-
8 12
age asymmetry ξnn, weefirst calculaete thee distributions
ff
P (ε,H⊥) of the ε = ε − ε (ε = (∆2 + ξ2)1/2; B. Collective spin dynamics
12 1 2 i i i
all ∆ depend oen both the external and the dipolar
i
transverse fields and are obtained by the exact diago- From Eqs.(18)-(19) in the Fe crystal at Hx = 3.8 T
8
nalization of (5) and (6), Appendix B), where ε are (∆ ≈ 0.35 K) one finds T ∼ 0.1 K for the field com-
1,2 o M
for the nearest-neighbor molecules only. Then, we get pensated sample (T ∼ 0.11 K with no field compen-
M
ξfnfn = (1/2) −+εεoodε |ε|P12(ε,H⊥) (εo is the maximum scarytisotna)l.atTHhexc=or7re.0spTon(d∆ing≈te0m.3p6erKat)uirseTfor ∼the0.1M3nK12.
value of ε). o M
R AtT <T theaveragedistancebetweenfliponsislarge,
e Distribution of [e 1- e 2] (unnormalized) and so thMe collective multi-pair flip-flopprocesses are es-
nearest-neighbor molecules
5000 (a) Fe8 crystal 5000 (b) Mn12 crystal sentially ”frozen” and correlated clusters of flipons can
not appear.
Hx=3.8 T Hx=7.0 T
4000 4000 At TM < T <∼ ∆o/kB the collective multi-pairs pro-
cessesareunfrozenand,whenthetemperatureincreases,
3000 3000 the whole hierarchy of correlated clusters of increas-
ing size n (the number of involved flipons, max{n} =
2000 2000
N (T)) can, in principle, appear. The motion of flipons
ex
leads to the suppression of correlations. The spectral
1000 1000
diffusion effect in the limit ∆ >> {W ,E } is rather
o D o
weak (Section IIIC) and the spectral diffusion dephas-
0 0
ing time τs is longer than the motional dephasing time
-2x10--22 0x1000 2x10--22-5x10--33 0x1000 5x10--33 τm. Ifatcdertaintemperaturethe cluster dephasingtime
Bias (K) d
τms(T)=min{τm(T),τs(T)} is shorter than the cluster
d d d
nFeIiGgh.b4o:rTmheoledcisutlreisb:u(taio)nastoHf txh=e ε3.=8εT1i−nεt2hefocrrythstealneoafrFeset8- cteorrsrealattaiovnertaigmeeditsct(aTn)c,ecRreat(iTon) iosfvtihrteucaollryreilmatpeodsscilbulse-.
molecules (no field compensation); (b) at Hx = 7.0 T in the Instead, only short-living (ewxith τms < t < τ ) corre-
crystal of Mn molecules (no faster relaxing species). All d c nd
12 lations between molecules at distances R < R can be
cluster parameters are the same as in Figs.1 - 3. Here ε is in ex
present.
Kelvins. The states | ⇑i and | ⇓i are equipopulated. Both
distributions are averaged over threecrystallographic axes. If flipons can move, the correlated clusters at aver-
age distance Rex can appear only at temperatures T >∼
T > T (Eqs.(17, 25)), when τm becomes longer than
Forexample,twodistributions(atHx =3.8T forFe c M d
8 t and R (T) becomes shorter then R (Eq.(24)). In
andatHx =7.0T forMn )arepresentedinFig.4. For c ex c
12 Fe (for Hx = 3.8 T) and in Mn (for Hx = 7.0 T)
these distributions we get ξnn ≈ 1.5×10−3 K for Fe 8 12
ff 8 crystals this may happen already at T ∼ 0.25 K (as-
(withnofieldcompensation)andξnn ≈2.3×10−4K for suming p=q =s=1/3 and ρ=2/3, see Section IIIC).
ff
e
Mn (the corresponding mean-square deviations δε = Note,however,thatintheFe crystalforexample,where
12 8
hε2 −hεi2i1/2 are ≈ 4.7×10−3 Keand ≈ 6.4×10−4 K all lattice constants are different, only flipons oriented
9
(propagating) along the same crystallographic axis can the number of oscillations (C7) can be limited only by
create correlated cluster. At the same time, flipons have Q ∆nn/∆ .
φ ff o
larger probability to propagate along the axis with the
shortest inter-molecular distance (i.e., with larger ∆nn).
ff
In this case a quasi-1d motion of flipons is more proba- V. SUMMARY
ble than a 3d one. Nevertheless, if at t > 0 flipons will
start to change their orientation, this process will speed
In the present work the internal dynamics of
up decorrelation.
a temperature-equilibrated crystalline sample of the
If the cluster dephasing time τms is ∼ t << τ
d c np dipole-dipoleinteractingmoleculeswiththecentralspins
(Eq.2), during the time interval t = τnp the correlated S~ has been studied in the coherence window for nuclear
clusterscanbe createdanddestroyed(roughly)∼τ /t i
np c spin and phonon degrees of freedom.
times. During the cluster life-time τms all molecules,
d At large external transverse magnetic fields the tun-
belonging to cluster, can make ∼ ∆nnt /¯h coherent os-
ff c neling matrix element ∆ (Section I and Appendix A)
cillations (C7). At T ∼ 0.25 K the correlated clusters o
increases,whereasboththehalf-widthofthedipolarbias
can ”reappear” ∼ 50 times in Fe at Hx = 3.8 T (for
8 distributionW andthe half-widthofthe hyperfine bias
a cluster of flipons oriented along the same axis) and D
distribution E decrease (Section II and Appendix B).
∼ 10 times in Mn at Hx = 7.0 T. At these fields o
12 At a certain value of the transverse field, the coher-
one may expect ∼ 30 oscillations (C7) in both Fe and
8 ence window for phonon and nuclear spin-mediated de-
Mn . All these estimations do not take into account
12 coherence (the NPC-window) opens up (Sections I and
the coherence optimization strategy20 and we again as-
II). Outside of the NPC-window the spin dynamics is
sume p = q = s = 1/3 and ρ = 2/3. Note also that the
incoherent. If in the whole NPC-window the average
phonondecoherencerateγph increaseswithtemperature
φ strength of the dipole-dipole interactions between the
(see Eq.1), but at k T <∆ this increase is slow.
B o nearest-neighbor molecules V or W are larger than
dd D
In the case of Mn12, in the NPC-window for major ∆o,thespindynamicsisalsoincoherent. Intheopposite
species the spin dynamics of molecules belonging to mi- limit the coherent spin dynameics is possible.
nor species is incoherent already. Due to the difference
In the limit ∆ > V and if the effective matrix
o dd
in ∆ , the molecules of major and minor species cannot
i element ∆nn, describing transitions between two flip-
createcorrelatedflip-floppairsbetweeneachother. This ff
flop states of the neareest-neighbor molecules, Eq.(9),
results (i) in a decrease in the number of flipons; (ii) in
is large compared to the asymmetry ξnn of these two
partial localization of flipons (depending on the fraction ff
states,Eq.(10),thespincorrelationsbetweenthenearest-
ofimpurities);and(iii)inrandomizationofprocessesdue
neighbor molecules lead to the creation of resonant flip-
to interactions between molecules belonging to different
floppairs(SectionIIIAandAppendixC). Suchresonant
species. The last effect gives rise to the incoherent pair
pair experiences oscillations (C7) between states | ⇑⇓i
processes, leading to suppression of coherence.
and |⇓⇑i of two molecules with frequency ∼∆nn.
The larger the concentration of impurities with inco- ff
If∆nn >ξnnforthemostpairsofthenearest-neighbor
herent internal dynamics, the smaller the probability for ff ff
molecules, the resonant flip-flop transitions can ”prop-
correlated clusters to appear. However, at T > T the
c
agate” in the crystal, involving more and more new
sample can become covered by correlated clusters of the
molecules (one molecule in a pair remains in its ground
sizes smaller than the average distance between impuri-
state but another nearest-neighbormolecule creates new
ties R . For example, in Mn at Hx = 7.0 T and
im 12
T = 0.25 K one gets Rex ∼ 3a. If Rim >∼ 10a, at these resonance with the excited molecule). This ”mobile”
magnon-like process (a spin-excitation) between the
values of field and temperature the correlatedclusters of
states of two involved nearest-neighbor molecules in our
the radius ∼3R can, in principle, appear.
ex e e work is called ”flipon” (Section IIIA). The number of
If ∆nn < ξnn for most of the nearest-neighbor
ff ff flipons is limited by the number of excited molecules
molecules (i.e., no flipons), the correlated clusters com-
N (T).
posed of lengthy flip-flop pairs with ∆ (R ∼ R ) << ex
∆nn can appear already at T < T <ffT . Theexasym- At T < TM = max{Tr(e2s),Tph}, Eqs.(18,19), the dis-
ff M c tances between flipons are long and the correlations be-
metry ξ(2) for two resonant pairs in such a cluster is
ff tween them are unimportant. At T > TM the correla-
∼ ∆ ∼ ∆(2) and the cluster life-time is ∼ t . This tions between flipons become crucial and at certain con-
ff ff c
means that all resonant pairs will be able to make only ditions can lead to the creation of correlated clusters of
one corelated flip-flop transition before correlations will flipons (Section IIIB). Each cluster represents a corre-
be suppressed (if t << τ , such a cluster can reap- lated group of molecules experiencing coherent oscilla-
c np
pear ∼τ /t times). If ∆nn >ξnn, but for some reason tions (C7) between their lowest states |⇑i and |⇓i.
np c ff ff
ρ→0(s→1),thecorrelatedclustersofimmobileflipons The flipons motion and the spectral diffusion process
canappear. Becauseoftheweakeningofthespectraldif- result in the suppression of correlations (Section IIIC).
fusion effect in the limit ∆ >> {W ,E }, the dephas- If ∆nn >> ξnn for most pairs of the nearest-neighbor
o D o ff ff
ingtimefortheseclusterscanbelimitedonlybyτ and molecules (Section IVA) and flipons can move, the cor-
np
10
related cluster can appear only at T >∼Tc, Eq.(25). The where τˆx,τˆz are the Pauli matrixes multiplied by 2; ∆o
average inter-flipon distance in this case is given by the is the tunneling matrix element; and ξ is the asymmetry
solution of Eq.(24) and the cluster dephasing time (i.e., between two states (i.e., the longitudinal bias). One can
its life-time) τms is ∼ t (Eq.(20)). Molecules that be- easily solve this problem for eigenfunctions:
d c
longtotheclustercanmake∼∆nnt /¯hoscillations(C7)
ff c
before the correlations will be suppressed. During the |⇑i = u|↑i+v|↓i; |⇓i=−v|↑i+u|↓i;
total phonon/nuclear spin coherence time τnp, Eq.(2), (u,v) = ((ε±ξ)/2ε)1/2; ε= ξ2+∆2, (A2)
the coherent clusters can ”reappear” <∼τnp/tc times. At o
T < Tc only random short-living (t < tc) correlations where the corresponding energies in pstates | ⇑i,| ⇓i are
within small groups of flipons, separated by distances given by E =∓ε and
⇑,⇓
R<R (T), Eq.(11), can appear.
ex
ThesmallertheeffectiveflipondiffusioncoefficientD , 1 0
f |↑i= ; |↓i= . (A3)
Eq.(D4), the longer the cluster life-time τms. If, for 0 1
d (cid:18) (cid:19) (cid:18) (cid:19)
some reason, the flipons are localized (D → 0), the
f
correlated clusters of immobile flipons can appear even Ifattimet=0systemwasinstate|↑i,theprobabilities
at T < T . Their life-time can be limited only by τ , to find system at time t in states |↑i or |↓i are
c np
and the number of oscillations (C7) can be limited only
by Q ∆nn/∆ . If ∆nn < ξnn, the correlated clusters ∆2 ∆2
φ ff o ff ff P =1− o sin2(εt/¯h); P = o sin2(εt/¯h). (A4)
of ”lengthy” resonant flip-flop pairs (molecules in these ↑↑ ε2 ↓↑ ε2
pairsareseparatedbythe distanceR∼R )canappear
ex This describes the oscillations with frequency ε between
also at T <T . Their life-time is limited by t . On aver-
c c states |↑iand |↓i. Inthe limit ∆ <<ξ the oscillations
age, all pairs in these clusters will be able to make only o
are suppressed since their amplitude is ∆2/ξ2 <<1.
one flip-flop transition before correlations will be sup- o
pressed. If t < τ , these clusters can reappear several
c np
times.
APPENDIX B: METHOD OF CALCULATIONS
It is worth mentioning also that various systems allow
thecoherenceoptimizationstrategy(orientationofexter-
(1) W (H⊥). To obtain the transverse field be-
nal transverse field in a plane, chemical replacement of D
havior of the dipolar bias distribution half-width W in
isotopes, etc.20) to be applied to get longer spin/phonon D
the crystals of Fe and Mn molecules, two clusters
coherencetime-intervalτ ortoshiftthecoherencewin- 8 12
np of different crystal symmetry are used. (a) The Fe
dow down to lower values of the transverse field. 8
crystal. The cluster for the Fe system contains 503
Thisconcludesourstudyofthe collectivespindynam- 8
unit cells arranged in a triclinic lattice array with lat-
icsofatemperatureequilibratedsampleinthecoherence
tice parameters:30 a = 10.522(7) ˚A; b = 14.05(1) ˚A; c =
window for phonon and nuclear spin-mediated decoher-
15.(1) ˚A with angles α = 89.90(6)o;β = 109.65(5)o;γ =
ence. The presentedanalysiscanbe appliedto any crys-
109.27(6)o. Each unit cell of volume V ≈ 1969 ˚A3
talline nanomagnetic insulator composed of the central o
contains eight spin-5/2 Fe+3 ions, correctly positioned
spin S~ molecules, and can be useful in the search for and oriented.30,43 (b) The Mn crystal. The cluster
12
magneticsystemsshowingthe spincoherenceandcollec- for the Mn system contains 503 unit cells, arranged
12
tive phenomena. The analysis for the induced dynamics in a tetragonal lattice array with lattice parameters:30
(when system at t = 0 is prepared in some initial state) a = b = 17.1627(6) ˚A; c = 12.2880(4) ˚A with angles
will be presented separately. α = β = γ ≈ 90o. Each unit cell of volume V ≈ 3619.5
o
ItisapleasuretothankA.Morellofornumeroushelp- ˚A3 containstwelvespins: fourspin-3/2Mn4+ ionsinthe
ful and motivating discussions. The author is grate- innershellandeightspin-2Mn3+ ionsintheoutershell,
ful to R. Sessoli and W. Wersdorfer for providing him correctly positioned and oriented.30
with complete files on the structure of the Mn -acetate
12 Thedistributionsofthedipolarbiasfieldsandenergies
molecule. The author is also indebted to S. Burmistrov, in the cluster are calculated taking into account all in-
L. Dubovskii and I. Polishchuk for useful discussions. ternalspins~s(p) of eachmolecule (S~ = ~s(p), p=8 in
This work is supported by Russian grant RFBR 04-02- i i p i
Fe and p= 12 in Mn ). The internal molecular spins
17363a. 8 12 P
~s(p) cannot flip independently - each molecule changes
i
its total spin orientation as a rigid object. Initially, all
molecules inthe sample are orientedalong the easy axis,
APPENDIX A: TWO-LEVEL SYSTEM eitheratrandomwithprojectionsSz =±S (fordepolar-
i i
ized sample with initial magnetization M = 0), or with
The effective Hamiltonin of a biased two-level system projectionsSiz =+Si(forpolarizedsamplewithM =1).
(TLS) has the form: To obtain the average longitudinal bias field acting
on i-th molecule, we calculate the longitudinal fields
H =−∆ τˆx−ξτˆz, (A1) hz(p), createdby all internalspins of all molecules in the
TLS o i