Interplay of disorder and geometrical frustration in doped Gadolinium Gallium Garnet

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by  N. Woo

Interplay of disorder and geometrical frustration in doped Gadolinium Gallium Garnet
N. Woo,1 D.M. Silevitch,1 C. Ferri,2 S. Ghosh,2 and T.F. Rosenbaum1,3,∗
1The James Franck Institute and Department of Physics,
The University of Chicago, Chicago, IL 60637, USA
2Department of Physics, University of California, Merced, CA 95343, USA
3Division of Physics, Math and Astronomy, California Institute of Technology, Pasadena, CA 91125, USA
(Dated: January 29, 2015)
Thegeometrically-frustrated,triangularantiferromagnetGGGexhibitsarichmixofshort-range
orderandisolatedquantumstates. Weinvestigatetheeffectsofupto1%Neodymiumsubstitution
forGalliumontheacmagneticresponseattemperaturesbelow1Kinboththelinearandnonlinear
regimes. Substitutional disorder actually drives the system towards a more perfectly frustrated
5 state,apparentlycompensatingfortheeffectofimperfectGadolinium/Galliumstoichiometry,while
1 at the same time more closely demarcating the boundaries of isolated, coherent clusters composed
0 of hundreds of spins. Optical measurements of the local Nd environment substantiate the picture
2 of an increased frustration index with doping.

n
a PACSnumbers: 75.50.Lk,75.40.Gb,75.45.+j,75.30.Hx
J

8
Geometrically frustrated magnets suppress the ability probable spin configurations at absolute zero.8
2
of spins to freeze to temperatures well below their nat- Several experimental studies of GGG have probed the

] ural ordering scales and, in so doing, open to experi- subtle nature of the ground state, with differing re-
l
e mental scrutiny low temperature fluctuations, extended sults. Early bulk thermodynamic measurements9 found
- quantumstates,andthermodynamicallyisolateddegrees evidence of spin freezing and an associated spin glass
r
t of freedom. The topology of the interspin interactions transition at T ∼ 130 mK. By contrast, muon spin
s
. makes it impossible to simultaneously minimize all of relaxation10,11 and Mossbauer12 measurements found
at the pairwise interaction energies1 and, in principle, a that the spins remained unfrozen to temperatures as low
m fullyfrustratedsystemcannotformalong-rangeordered as25mK.Powderneutronmeasurementsobservedshort-
- state,evenatabsolutezero. Short-rangeeffectscandom- range correlations at elevated temperatures followed by
d inate the macroscopic behavior, with self-organization of a 140 mK transition to a state with longer range cor-
n spins into such quantities as quantum protectorates: co- relations and spin-liquid type behavior.13 Subsequently,
o
herent states that are topologically isolated from many a combination of linear and nonlinear ac susceptibility
c
[ sources of environmental decoherence.2,3 The introduc- measurements found a state consisting of a set of “quan-
tion of structural or compositional disorder to such a tum protectorates”, extended coherent states that are
1 frustrated system has the potential to tune the frus- topologically decoupled from the background, coexisting
v
tration effects and their macroscopic expressions. It is with at least short-range antiferromagnetic order onset-
9
2 not clear a priori, however, whether additional disor- ting at T ∼100 mK.14 More recent high-resolution heat
2 der enhances self-organization by further isolating the capacity measurements15 did not observe any evidence
7 spin clusters, or actually relieves frustration by break- of a spin glass transition in the 120–200 mK range. The
0 ing the symmetry of the interspin couplings and hence interesting spin dynamics all occur at temperatures be-
1. degrading the isolation between any extended quantum low 200 mK, over an order of magnitude lower than the
0 states and the environment.4,5 Here, we study the ef- Weiss temperature, confirming the picture of a highly
5 fects of progressively increasing the degree of disorder in frustrated spin configuration.
1 a geometrically frustrated antiferromagnet by introduc-
One effect contributing to this range of results is the
: ing controlled amounts of a magnetic dopant.
v difficulty in growing purely stoichiometric crystals of
Xi The parent compound, Gadolinium Gallium Garnet GGG, as excess magnetic Gd3+ ions tend to randomly
(Gd Ga O or GGG), has a cubic lattice in which the substitute for nonmagnetic Ga3+ ions.16,17 Gadolinium
r 3 5 12
a magnetic Gd3+ ions are located on two interpenetrating ions occupying gallium sites have a nearest-magnetic-
sub-lattices composed of corner sharing triangles with a neighbordistancethatis9%smallerthanthenormalGd-
Heisenberg spin symmetry and a single ion anisotropy Gd distance in GGG, resulting in both lattice strain and
of less than 0.04 K.6 Nearest neighbors are connected a local increase in the magnetic interaction strength.15
by corner sharing triangles on each sub-lattice, coupled Againstthisbackdrop,weexamineaseriesofGGGcrys-
by a 1.5 K antiferromagnetic exchange interaction and a talswithupto1%ofNd3+dopantssubstitutingforGd3+
0.7 K dipolar interaction, and the two sub-lattices inter- ions. The Nd ions introduce a controlled degree of disor-
actvianext-nearest-neighborinteractions. Theresulting derintothemagneticlatticethatbreaksthelocalsymme-
Weiss temperature is of order 2.3 K.7 The combination try of the magnetic interactions. By controlling the level
of the antiferromagnetic interaction and the topology of this quenched disorder, we can study the role it plays
of couplings provides a macroscopic number of equally- in the formation and evolution of the low-temperature
2

FIG. 1. Imaginary part of the ac magnetic susceptibility χ(cid:48)(cid:48)
as a function of frequency f at a series of temperatures T
FIG. 2. Real part of the susceptibility χ(cid:48) as a function of
for GGG:Nd with x = 0, 0.1%, and 1%. The susceptibility
x temperatureT ataseriesoffrequenciesf forGGG:Nd with
wasmeasuredusinga40mOeprobefieldtoensurelinearre- x
x = 0, 0.1%, and 1%. The functional form and tempera-
sponse. Alow-frequencymodeat50Hzandahigh-frequency
ture scale for all three doping levels is similar; however, the
modethatpeaksabovethe10kHzmeasurementwindowchar-
static incoherent background is suppressed for x = 1.0% as
acterize the global response of all three materials. For x=0
frustration is enhanced by doping.
and 0.1%, a flat low-frequency response develops at 85 to 95
mK, corresponding to fluctuations associated with ordering.
Atx=1.0%,thisfeatureisnotobserved,indicatingthatany
Finally, we exploited the coupling of the Nd ions to
ordering is suppressed below 30 mK.
theGdionstoprobeopticallythelocalmagneticenviron-
ment. Lifetimedatawascalculatedfromthehomogenous
linewidthoftheR →Z transitionofthe4F →4I
1 5 3/2 9/2
magnetic phases.
ground to first excited state emission multiplets of the
We performed ac susceptibility measurements on a se- Nd3+ ion. The samples were mounted in a liquid helium
ries of (5 × 5 × 10) mm3 single crystals of GGG:Ndx, flow cryostat and excited with a Ti:Sapphire laser tuned
with x = 0, 0.1, and 1.0% (Princeton Scientific). X- to 808 nm. The excitation wavelength was chosen to be
ray lattice constant measurements showed that all three as close to the absorption band edge as possible for the
crystalsexhibitedasimilardegreeofoff-stoichiometryof 4F → 4I emission lines. The emission is collected
3/2 9/2
approximately 3%. We employed a gradiometric suscep- in reflection and dispersed by an Acton 300i spectrome-
tometer attached to the cold finger of a helium dilution ter onto a thermoelectrically cooled CCD with spectral
refrigerator,usingsapphirerodspressedagainstthefaces resolution of 0.18 nm.
of the crystals to provide a thermal link to the cryostat. The complex ac susceptibility at audio frequencies de-
Thecomplexacsusceptibilitywasmeasuredinthelinear- lineatesthenatureandtimescalesofthelow-energymag-
responselimitusing40mOeofprobefieldandmappedas netic modes of the material. Peaks in the imaginary
functionsoffrequency(1Hzto10kHz)andtemperature componentofthesusceptibility,χ(cid:48)(cid:48),yieldinformationon
(30 to 300 mK) for all three dopings. characteristicrelaxationtimes. Thepresenceofaflatre-
Thenonlinearresponseofthesystemprovidesthemost sponseinthef →0limitcorrespondsto1/f noiseinthe
acute insight into the behavior of the local, isolated spin magnetization, a signature of a spin state with scale in-
clusters. To that end, we measured the susceptibility as variance and hence a complex hierarchy of energies.19,20
a function of drive amplitude (mOe to Oe) at fixed fre- We plot in Fig. 1 imaginary susceptibility spectra for
quency. In addition to the direct effect on the GGG GGG:Nd (x=0,0.1,1%)for30≤T ≤300mK.Forall
x
crystals, large ac magnetic fields, particularly at kilo- threesamples,weobservetwopeaksinthesusceptibility,
hertz frequencies, also induce eddy currents in metallic one at 50 Hz and one that appears to lie above the 10
mounts, resulting in significant Ohmic heating. To al- kHz upper limit of the measurement. This corresponds
leviate this issue, we mounted the susceptometer on a totwocharacteristicrelaxationtimescalesinthesystem,
carbon-fiber/sapphireframework18 designedtominimize one at 20 msec and one shorter than 100 µsec.
eddy currents while maintaining a high thermal conduc- The characteristic long-time relaxation at 50 Hz does
tivity for efficient thermal linkage to the cryostat. The not depend strongly on either T or x. By contrast, the
ac probe field was applied using a duty cycle of 1–2% to behaviorofthesysteminthelow-frequencylimitchanges
further reduce the heat load. markedly with x. For pure GGG and GGG with 0.1%
3
Nd, we observe a clear f → 0 plateau in the imaginary
susceptibility at T ∼ 85 to 95 mK. No such plateau ap-
pears in the 1.0% doped material. The appearance of
such a plateau is often correlated with the entry into
a spin glass state.19,21 If that were the case here, the
plateau should remain robust for all T below the glass
transition temperature. Instead, the plateau behavior is
only stable over a finite temperature band, below which
the susceptibility approaches zero with finite slope. We
associatetheplateauwithantiferromagneticordering,at
least on the 100 A˚scale.13 The introduction of Nd sup-
presses the ordering to T < 30 mK, effectively restoring
a higher degree of geometric frustration.
The relative frustration index as a function of doping
also can be discerned by looking at the real component,
χ(cid:48), of the linear susceptibility, as shown in Fig. 2 for all

three sample concentrations. χ(cid:48) consists of a frequency-
dependentpeakedcomponentridingonafrequencyinde- FIG. 3. Real part of the susceptibility χ(cid:48) of GGG:Ndx in
pendent background. The peak reflects the freezing out the nonlinear response regime, measured at f = 5 kHz as a
of magnetic modes at the pertinent time scale of the in- function of the applied probe field H at a series of tempera-
tures. At elevated T, there is clear threshold behavior in the
terrogation frequency, but conspicuously does not go to
susceptibilityandthenonlinearresponsecanbefittoaBril-
zero (note the suppressed zero of Fig. 2). We attribute
louin function characteristic of coherent clusters with a large
this background to a Heisenberg spin bath made avail-
effective spin (see text for details). Below T = 175 mK, the
able by imperfections in the crystal and deviations from
clusterdynamicsfreezeoutandthesmallincreaseinχarises
stoichiometry. Rather than increasing the background from inductive heating of the susceptometer (and hence the
susceptibility, the Nd ions at the highest level of substi- sample) at large H.
tution effectively reduce the degree of disorder and push
the system towards a more perfectly frustrated state.
We can ask whether this apparently counter-intuitive are not bound up in clusters. Even in the presence of
result also applies to the quantum protectorates previ- doping-induced disorder, clusters acting as single large
ouslyreportedforundopedGGG.14 Anexperimentalsig- effective spins continue to form. For T ≤ 160 mK, the
nature of these coherent spin clusters in GGG occurs spinclustersfreezeoutandthenonlinearcontributionto
in the nonlinear susceptibility. As a function of drive the susceptibility disappears. The slow rise in χ(H) at
field amplitude, the magnetization traces out a Brillouin the highest drive fields is most likely due to Ohmic heat-
function, with approach to saturation above 1 Oe. This ing that is at worst 25 mK for a 2.75 Oe/5 kHz applied
behavior is characteristic of clusters that act as large ef- field at lowest T by reference to the linear-response data
fective spins excited between quantized states.14 Similar of Fig. 2.
field-induced formation of extended states has been ob-
The fit to the Brillouin form for T ≥ 175 mK helps
served as well in the dilute Ising magnet LiHo Y F ,
x 1−x 4 specify the nature of the cluster dynamics. We show in
where quantum fluctuations rather than geometric frus-
Fig. 4 the aggregate moment of the spins bound up in
tration preserve the isolated degrees of freedom.22–24
coherent clusters at f = 15 Hz, as well as the threshold
Here we explore the effects of the Nd doping on these
activation field for the onset of the nonlinear response as
cluster states in GGG.
afunctionofT. Thetypicalclustersizeis150–200spins,
We plot in Fig. 3 the nonlinear susceptibility at f =5 correspondingto6to8unitcellsoftheGGGlattice. The
kHz for the three dopant concentrations over almost a moment of the clusters only depends weakly on T, until
decade in T. The response at f = 15 Hz is qualita- the entire mode freezes out (Fig. 3). There is, however,
tively similar. There are two distinct regimes in tem- a definite compositional dependence (Fig. 4a). The 0
perature. For T ≥ 175 mK, the susceptibility is essen- and 0.1% Nd-doped samples have comparable moments,
tially constant at low applied field, followed by a con- whereasthe1%crystalhasanetclustermomentreduced
tinuous drop once the applied field is above a threshold by20%. TheNddoesnotacttoimprovetheclustercor-
value. The shape of the susceptibility above the thresh- relations (by, e.g., acting as nucleation sites for coherent
old can be described by the derivative of the Brillouin clusters),butinsteadservestoreducethedisorderinthe
function, χ(H) = NcNsµBgJddHBJ(y) + const., where GGGlatticeandmorecloselydemarcatestheboundaries
y = NsµBgJH, B = 2J+1coth(2J+1y)− 1 coth( 1 y), of the isolated spin clusters.
kBT J 2J 2J 2J 2J
andforGGGJ=7/2andg=2. N isthenumberofmi- While the net moment of the clusters does not vary
s
croscopic spins bound in an individual cluster, N is the appreciablywithT,thethresholdfieldrequiredtoexcite
c
total number of clusters per unit volume, and the con- thecoherentresponseisthermallyactivatedandobeysan
stant term accounts for the susceptibility of spins that Arrhenius form (Fig. 4b) with an energy barrier height
4
that spin-spin interactions are the dominant effect. As

x (%)
5 0.1
1
)
s
p
(
τ
4.5
4
4 10 100
T (K)

FIG. 4. Cluster dynamics of GGG:Ndx at f = 15 Hz. (a) FIG. 5. Decay time of the R1 → Z5 emission line of Nd3+
Aggregate moment of all spins involved in coherent clusters in GGG:Ndx as a function of temperature for x = 0.1% and
as a function of temperature T for x = 0, 0.1, and 1.0%. 1.0%. Thefasterdecoherencetimewithincreaseddopantcon-
Dashed lines show the mean values for the three doping con- centration supports the notion of increased frustration.Lines
centrations. Increased doping reduces the cluster size. (b) are guides to the eye.
Activationfieldvs. inversetemperature. TheArrheniusform
establishes the germane energy scales for cluster spin flips as
a function of x. the Nd3+ ions fill in the Gd vacancies in the GGG lat-
tice, the local environment becomes less disordered, the
frustrationindexincreases,andthedecoherencetimede-
creases.
of 0.48 ± 0.05 K. The energy gJµ H to flip a cluster of
B Using a Nd doping series to vary the effective degree
200 spins with J = 7/2, g = 2 and a threshold field for
of disorder in GGG, we demonstrate that increasing the
Student Version of MATLAB
the nonlinear response ∼ 1 Oe is approximately 0.2 K,
Nd concentration actually reduces the effects of disorder
consonantwiththemeasuredbarrierheightandthescale
andenhancesthesignaturesoffrustration. Wepositthat
set by the dipolar coupling. Although the barrier height
the Nd compensates for the disorder arising from the off
isequalwithinerrorbarsforallthreeNdconcentrations,
stoichiometric growth process and, in so doing, permits
the threshold field for the 1% doped sample is reduced
the intrinsic geometric frustration of the lattice to sup-
by25%incomparisontothemorelightlydopedsamples
press the formation of an ordered state. The nonlinear
as the smaller clusters more easily change orientation in
susceptibility reveals quantitative aspects of the coher-
response to the driving field.
ent spin clusters and their tenability with disorder and
We show in Fig. 5 the decay time of the optical R → frustration. Inprinciple,theNdionscouldbemarshaled
1
Z emission line of Nd3+ in GGG:Nd as a function of to interrogate locally the isolated cluster dynamics via
5 x
temperature for x = 0.1 and 1%. Typically, as a system optical studies at lower T.
is cooled, the homogeneous linewidth (inversely propor-
tional to the decay time) of a dipole transition decreases
withthesuppressionofthephonondensityofstates. The ACKNOWLEDGMENTS
contrary trend observed here indicates the availability of
a different decoherence mechanism, and while this de- We thank Y.Feng for assistance with the single crys-
creasing trend in decoherence is not seen in other Nd3+ tal x-ray diffraction performed at sector 4 of the APS
garnets like YAG and YAP,25,26 it has been reported in at Argonne National Laboratory. Work at the Univer-
Mn4+ doped GGG, where it was attributed to spin-spin sity of Chicago was supportedbythe Department of En-
couplingbetweentheGdandthedopantions.27 Itisim- ergy Office Basic Energy Sciences, Grant No. DE-FG02-
portanttonotethatthattheexcitedstatelifetimeislong 99ER45789 and the work at the University of California,
in rare earth doped crystals,28 and thus, judging by the Merced was supported by the National Science Founda-
extremely fast decay time seen in the data, we can infer tion, Grant No. DMR-1056860.
5
∗ email: [email protected] Balakrishnan,O.A.Petrenko,A.Gomez,S.W.Kycia,M.
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