Table Of ContentWill spin-relaxation times in molecular magnets permit quantum information
processing?
Arzhang Ardavan, Olivier Rival, John J.L. Morton, and Stephen J. Blundell
Clarendon Laboratory, Department of Physics, University of Oxford, OX1 3PU, United Kingdom
Alexei M. Tyryshkin
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, United States
Grigore A. Timco and Richard E.P. Winpenny
Department of Chemistry, University of Manchester,
Oxford Road, Manchester, M13 9PL, United Kingdom
(Dated: February 1, 2008)
7
0 Using X-band pulsed electron spin resonance, we report the intrinsic spin-lattice (T1) and phase
0 coherence(T2)relaxationtimesinmolecularnanomagnetsforthefirsttime. InCr7M heterometallic
2 wheels,withM=NiandMn,phasecoherencerelaxationisdominatedbythecouplingoftheelectron
n spin to protons within the molecule. In deuterated samples T2 reaches 3 µs at low temperatures,
which is several orders of magnitude longer than the duration of spin manipulations, satisfying a
a
J prerequisitefor thedeployment of molecular nanomagnets in quantuminformation applications.
6
1
Certain computational tasks can be efficiently imple- axial Mn12-acetate [14], to the diamagnetic ring
5 mented using quantum logic, in which the information- Cr8F8Piv16 [15, 16]. A key recent chemical advance
v carrying elements are permitted to exist in quantum su- is the the development of procedures for magnetically
3 perpositions [1]. To achieve this in practice, a physi- ‘doping’ a diamagnetic cluster to synthesise paramag-
4
cal system that is suitable for embodying quantum bits neticmoleculesinasystematicandcontrollableway[17].
1
(qubits) must be identified. Some proposed scenarios Thus,substitutingaCr3+(s=3/2)byaMn2+(s=5/2)
9
0 employ electron spins in the solid state, for example or a Ni2+ (s = 1) generates the S = 1 Cr7Mn or the
6 phosphorous donors in silicon [2], quantum dots [3], het- S =1/2 Cr7Ni respectively.
0 erostructures [4] and endohedral fullerenes [5, 6], mo- Manyclustershavebeeninvestigatedusing thermody-
/
h tivated by the long electron-spin relaxation times ex- namicprobessuchasmagnetization[18]andheatcapac-
p hibited by these systems. An alternative electron-spin ity [19], and spectroscopic probes such as neutron scat-
- based proposal exploits the large number of quantum tering [20, 21, 22] and electron spin resonance [23, 24].
t
n states and the non-degenerate transitions available in These strategies have been very successful in determin-
a
high spin molecular magnets [7, 8]. Although these ad- ingtheenergyspectraandmagneticstructures,butapart
u
vantages have stimulated vigorous research in molecular from a limited number of observations of ‘demagnetisa-
q
: magnets [9, 10, 11], the key question of whether the in- tion tunnelling’ there are few [25] reports of definitive
v
trinsic spin relaxation times are long enough has hith- measurementsofrelaxationtimes. However,the feasibil-
i
X erto remained unaddressed. Here we show, using pulsed ityofmanyofthemostinterestingproposedapplications,
r electron spin resonance experiments on heterometallic in particular those involving classical or quantum infor-
a
wheels, that the relaxation times in molecular magnets mationprocessing,iscriticallydependentontheintrinsic
cansignificantlyexceedthedurationofcoherentmanipu- spin-lattice (T1) and phase coherence (T2) times.
lations,aprerequisiteforthedeploymentofthesesystems Measurements were performed using commercial
in quantum information applications. Bruker Elexsys 580 X-band pulsed ESR spectrometers,
Molecular magnets comprising clusters of exchanged employing 4He flow cryostats for temperature control.
coupled transition metal ions have been studied exten- The relaxationtimes were obtainedusing standardtech-
sively in recent years [12]. They can exhibit a substan- niques [26]: T2 from the decay of a 2-pulse Hahn echo
tial ground state spin with a large and negative zero- sequence,
field splitting (ZFS), leading to a spontaneous magnetic
π/2−τ −π−τ −echo, (1)
moment parallel to the easy axis. In the absence of a
magnetic field, the configurations in which the moment withτ varying;T1fromtherecoveryofthemagnetisation
is ‘up’ or ‘down’ relative to the easy axis are degener- (measured with a spin echo) after an inversion pulse,
ate, and this bistable nature has stimulated interest in
π−T −π/2−τ −π−τ −echo, (2)
the application of magnetic clusters as classical [13] or
quantum [7, 8, 9, 11] information elements. with T varying and τ fixed and short.
Molecules in this class have been synthesised with Two factors conspire to complicate the measurement
widely varying properties, from the S = 10 highly of intrinsic lifetimes in anisotropic magnetic clusters.
2
Firstly, in a crystal the magnetic cores are typically sep-
arated by rather small distances of the order of 1 nm
to 10 nm, and are therefore coupled by dipolar interac-
tions. Fortwofreeelectronspins,the dipolarinteraction
is of the order of 100/r3 MHz·nm3 [26], so this dipolar
coupling is the dominant relaxation mechanism limiting
T2 in crystals of clusters. Secondly, clusters tend to ex-
hibit strongly axial behaviour with significant zero-field
splittings; this lifts the degeneracy of the ∆m = ±1
S
transitions in high spin (S > 1/2) molecules, and leads
to a strong dependence of the transition energies on the
orientation with respect to the external magnetic field.
Thus if we are to study a particular transition in an
ensemble of identical molecules, they must be orienta-
tionally ordered. The standard approach to solving the
FIG. 1: (Color online) X-band echo-detected ESR as a func-
first problem, dissolving the crystal (thereby increasing
tion of magnetic field for (A) Cr7Ni and (B) Cr7Mn mea-
the averageseparationof the clusters), leads to a second
suredat4.5K(blue),andthesimulatedpowderspectrumfor
problem, an ensemble of randomly oriented, highly axial
a species with S = 1, g = 2, D = 21 GHz, E = 1.9 GHz
molecules. (green). Inthisexperiment,theintensityofaHahn-echosig-
We chose the two compounds studied here with these nal with short (τ =300 ns) delays is measured as a function
factors in mind. Cr7Ni has a ground state spin of of the applied magnetic field. Selective pulses, of 64 ns for
a π/2-pulse and 128 ns for a π-pulse, ensure that only spins
S =1/2,soitexhibitsasingleESRtransitionand,there-
within awindow ofabout 0.3 mTare excited. Usingabroad
fore,nozero-fieldsplitting;theanisotropyoftheg-factor, integration window suppresses 1H ESEEM effects. The echo
which is small, is the only contribution to a dependence
intensityisproportionaltotheESRexcitationspectrum. The
of the transition energy on external magnetic field ori- finestructurein thedata close to 0.33 T is an artifact of the
entation. A dilute dissolved sample of this material is cavity.
therefore amenable to ESR measurements without caus-
ing problems associated with the orientational disorder.
with the Hamiltonian
Cr7Mnisaverycloselyrelatedcompoundwithaground
state spin of S = 1; the similarity in its structure leads H=gµB·S+DS2+E(S2−S2) (3)
us to expect that it should share relaxation mechanisms z x y
with Cr7Ni, but its higher spin allows us to examine the withparametersg =1.9,D =21GHzandE =1.9GHz,
consequences of the zero-field splitting. For example, a which reproduces the main features of the data (green
modulation ofthe zero-fieldsplitting, which mightoccur line in Figure 1(B)); deviations are probably due to the
as a result of coupling with thermally excited mechani- fact that in the experiment, the pulses deviate from per-
cal deformations of the molecule, provides further spin- fectπandπ/2rotationsatdifferentpartsofthespectrum
lattice and phase-coherence relaxation mechanisms [27]. because the transition probabilities depend strongly on
Sampleswerepreparedasreportedelsewhere[17], and orientation. Both spectra show some low-amplitude fine
dissolved in toluene. The solution was diluted progres- structure close to 0.33 T; these lines arise from impuri-
sively until the results reported below were no longer ties in the cavity and are present in the absence of the
dependent on concentration, indicating that the dipo- sample.
lar coupling between clusters had become negligible; Figure 2(A) shows the intensity of Hahn echos mea-
this occurred for concentrations below approximately sured in Cr7Ni as a function of the delay time τ (de-
0.2 mg/ml, corresponding to a mean separation of clus- fined in equation 1), for long, selective pulses (blue)
ters in excess of about 25 nm. and short, broadband pulses (cyan). The echo gener-
Figure1showsecho-detectedESRasafunctionofthe ated by broadband pulses exhibits very strong modula-
applied magnetic field measured at 4.5 K for (A) Cr7Ni tions (electron spin echo envelope modulation, or ES-
and (B) Cr7Mn. The spectrum for Cr7Ni shows a single EEM) with some second harmonic content, associated
narrow line, as expected for an S = 1/2 species. The with coupling of the electron spin to nearby nuclear mo-
line shape is approximately gaussian, with a width of ments [26]. The nuclear frequency extracted from the
about 0.01 T, suggesting that the broadening is due to modulation, 16.6±0.1 MHz, indicates that the relevant
inhomogeneity of the cluster environment. In contrast, nuclei are protons. This ESEEM can be suppressed by
the spectrum for Cr7Mn is broad and contains structure usinglonger,selectivepulses,asshownbythebluelinein
thatischaracteristicofanS =1specieswithazero-field Figure2(A);thisechodecayiswell-describedbyamono-
splitting exceeding the microwave energy. Also shown is exponential fit (green dashed line), yielding a coherence
a simulated powder spectrum [28] for an S = 1 species time of T2 =379±1 ns at 4.5 K.
3
FIG. 2: (Color online) (A) Decay of the Hahn echo inten-
sity in Cr7Ni as a function of delay time for long, selective
pulses (64 ns π/2-pulse, 128 ns π-pulse, blue) and for short
broadbandpulses(16nsπ/2-pulse,32nsπ-pulse,cyan). (B)
Decay of theHahn echo intensity for long, selective pulses in
per-deuteratedCr7Ni. Dashed green lines indicate fitsto the
data. Thefit in(B) assumes thattheESEEMeffect isdomi-
natedbyasingle harmonicatthe2DZeeman frequency[29]. FIG.3: (Coloronline)(A)T1 asafunctionoftemperaturein
(Notethe different horizontal axis scales.) Cr7Ni(blueopencircles)andCr7Mn(redopensquares). (B)
T2 asafunctionoftemperatureforCr7Ni(blueopencircles),
Cr7Mn (red open squares) and per-deuterated Cr7Ni (blue
The strong coupling to protons may provide an effi- filled circles).
cient phase decoherence path, reducing T2 [30]. A direct
test of whether this is indeed an important relaxation
mechanism is to measure the same Hahn echo decay in there are signs that it begins to saturate at tempera-
theper-deuteratedanaloguecompound,asshowninFig- tures below about 2 K. There are two interesting obser-
ure 2(B). 2D has a gyromagnetic ratio about six times vations: first, there is very little difference between the
smaller than 1H; the ESEEM frequency is correspond- hydrogenatedCr7NiandCr7Mn,despitethestrongmag-
ingly about six times smaller at 2.556±0.005 MHz, and neticanisotropyofthe latter;second,thefactorofabout
T2 is about six times longer, at 2210±20 ns. This con- sixbetweenthedecoherencetimesfortheper-deuterated
firms that the coupling to protons dominates the spin and hydrogenatedCr7Ni is retained over the whole tem-
decoherence in the hydrogenated sample. (Note that we perature range over which a Hahn echo is measurable.
wereunable to decreasethe bandwidth ofthe pulses suf- Both observations support the hypothesis that dipolar
ficiently to suppress the lower frequency ESEEM in the coupling with 1H or 2D nuclei (which belong to organic
deuterated compound.) ligands of the magnetic cluster and are well-distributed
Figure3showsthetemperaturedependencesofT1 and about the core [17]) dominates the phase decoherence in
T2 for each compound. T1 is comparable between the these materials. Phase decoherence arisingfrom fluctua-
Cr7Ni and Cr7Mn, and increases rapidly as the temper- tions of the zero field splitting in Cr7Mn is negligible in
ature falls. This suggests that thermal processes (such comparison over the temperature range studied here.
as couplings to phonons) are responsible for the longitu- In conclusion, we have measured intrinsic spin-lattice
dinal relaxation. At low temperatures, there is a varia- (T1) relaxation times and, for the first time (to the best
tion of a factor of about two in T1 measured at differ- of our knowledge), the phase coherence (T2) relaxation
ent points in the spectrum of Cr7Mn. Exciting different times in molecular magnets. We find that in the het-
parts of the powder spectrum corresponds to selecting erometallic clusters Cr7Ni and Cr7Mn, T1 is long and
sub-populations of molecules by orientation. These ori- somewhat dependent on the magnetic anisotropy of the
entationalsub-populationsrelaxatdifferentrates,show- cluster, but that T2 is dominated by the coupling to the
ingthatthe magneticanisotropyplaysatleastsomerˆole nuclearmomentsofprotonsinthevicinityofthe cluster.
inthelongitudinalrelaxation,probablythroughspectral There is no evidence of coupling between the magnetic
diffusion effects [31]. There is no such variation of T2. clusterandthefluorinenuclei,whichhadpreviouslybeen
In each compound, T2 also increases as the tempera- identified as a potential decoherence path [9]. Futher-
ture is decreased, though less dramatically than T1, and more, we find that the intrinsic phase coherence time T2
4
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