We have studied spin relaxation in the spin ice compound Dy2Ti2O7 through
measurements of the a.c. magnetic susceptibility. While the characteristic spin
relaxation time is thermally activated at high temperatures, it becomes almost
temperature independent below Tcross ~ 13 K, suggesting that quantum tunneling
dominates the relaxation process below that temperature. As the low-entropy
spin ice state develops below Tice ~ 4 K, the spin relaxation time increases
sharply with decreasing temperature, suggesting the emergence of a collective
degree of freedom for which thermal relaxation processes again become important
as the spins become highly correlated

A. P. Ramirez

Ari Mizel

B. G. Ueland

G. Ehlers

H. Karunadasa

H. B. G. Casimir

J. S. Slusky

J. Snyder

K. Matsuhira

P. Schiffer

R. J. Cava

S. T. Bramwell

English

arXiv

We have studied spin relaxation in the spin ice compound Dy2Ti2O7 through measurements of the ac magnetic susceptibility. While the characteristic spin-relaxation time (τ) is thermally activated at high temperatures, it becomes almost temperature independent below Tcross∼13  K. This behavior, combined with nonmonotonic magnetic field dependence of τ, indicates that quantum tunneling dominates the relaxational process below that temperature. As the low-entropy spin ice state develops below Tice∼4  K, τ increases sharply with decreasing temperature, suggesting the emergence of a collective degree of freedom for which thermal relaxation processes again become important as the spins become strongly correlated

Snyder, J.

Ueland, B. G.

Slusky, Joanna

Karunadasa, H.

Cava, R. J.

Mizel, Ari

Schiffer, P.

KU ScholarWorks (Univ. of Kansas)

P H Y S I C A L R E V I E W L E T T E R S week ending5 SEPTEMBER 2003VOLUME 91, NUMBER 10Quantum-Classical Reentrant Relaxation Crossover in Dy2Ti2O7 Spin IceJ. Snyder,1 B. G. Ueland,1 J. S. Slusky,2 H. Karunadasa,2 R. J. Cava,2 Ari Mizel,1 and P. Schiffer1,*1Department of Physics and Materials Research Institute, Pennsylvania State University,University Park, Pennsylvania 16802, USA2Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08540, USA(Received 21 February 2003; published 5 September 2003)107201-1We have studied spin relaxation in the spin ice compound Dy2Ti2O7 through measurements of the acmagnetic susceptibility. While the characteristic spin-relaxation time () is thermally activated at hightemperatures, it becomes almost temperature independent below Tcross  13 K. This behavior, com-bined with nonmonotonic magnetic field dependence of , indicates that quantum tunneling dominatesthe relaxational process below that temperature. As the low-entropy spin ice state develops belowTice  4 K,  increases sharply with decreasing temperature, suggesting the emergence of a collectivedegree of freedom for which thermal relaxation processes again become important as the spins becomestrongly correlated.DOI: 10.1103/PhysRevLett.91.107201 PACS numbers: 75.50.Lktibility were consistent with J  15=2 Dy3 ions. We entire temperature range [10], indicating the existence ofGeometrically frustrated magnetic materials, in whichthe geometry of the spin lattice frustrates the spin-spininteractions, have been shown to display a wide range ofnovel ground states [1,2]. There has been especially strongrecent interest in the so-called ‘‘spin ice’’ pyrochlorematerials (such as Dy2Ti2O7 and Ho2Ti2O7) in whichthe rare-earth spins are highly uniaxial due to strongcrystal fields [3]. Ferromagnetic and dipolar interactionsbetween these spins on this lattice of corner-sharing tet-rahedra are frustrated in a manner analogous to that ofprotons in ice leading to a variety of exotic behavior [4–16]. While the spin entropy freezes out only below Tice 4 K in Dy2Ti2O7 [9], magnetic susceptibility studies showa strongly frequency-dependent spin freezing at T 16 K [10,11]. In contrast to traditional spin glasses, thespin-freezing transition is associated with a very narrowrange of relaxation times and thus represents a ratherunusual example of glassiness in a dense magnetic system.Here we report a study of the spin-relaxation processesin the spin ice compound Dy2Ti2O7. We find that, whilethe characteristic spin-relaxation time is thermally acti-vated at high temperatures, it becomes almost tempera-ture independent below Tcross  13 K. We interpret thedata in terms of a crossover from thermal to quantummechanical relaxation mechanisms, direct evidence forwhich is provided by data taken in an applied magneticfield. While most previous studies of such crossoversinvolve isolated moments, this material represents aunique example of quantum relaxation in a dense spinsystem with developing correlations. These correlationsresult in a highly unusual reemergence of thermally acti-vated behavior for T & Tice as the collective degrees offreedom dominate the spin dynamics at low temperatures.Polycrystalline Dy2Ti2O7 samples were prepared usingstandard solid-state synthesis techniques [10,17]. X-raydiffraction demonstrated the samples to be single phase,and Curie-Weiss fits done to the high temperature suscep-0031-9007=03=91(10)=107201(4)$20.00 study the magnetization (M) and the resultant dc suscep-tibility (dc  dM=dH) measured with a QuantumDesign SQUID magnetometer, as well as the real andimaginary parts (0 and 00) of the ac susceptibility(ac) measured with either the Quantum Design physicalproperties measurement system or a simple inductancecoil in a dilution refrigerator at low temperatures.In Fig. 1, we show the temperature dependent suscep-tibility at different frequencies. We find that dc increasesmonotonically with decreasing temperature, but 0Tand 00T display a frequency-dependent spin freezing[10,11]. While magnetic site dilution studies [10] indi-cated that this freezing was associated with the develop-ment of spin-spin correlations, recent data suggest that itmay instead be a single-ion effect [15]. As demonstratedin the inset of Fig. 1(b), the frequency dependence of Tfcan be fit to an Arrhenius law, f  f0eEa=kBTf wheref0  109 Hz and EA  200–300 K, which is the energyscale set by the crystalline field [3,10,11]. The Arrheniuslaw behavior indicates that the spin-relaxation processesare thermally driven for T * Tf, and an extrapolationsuggests that one should observe Tf  8 K for frequenciesof order 10 mHz, the time scale of the nominally staticmagnetization measurements. The absence of observedirreversibility in the magnetization in this temperaturerange implies that the characteristic relaxation time doesnot continue to follow the Arrhenius law to lower tem-perature [10,11].The spectrum of spin-relaxation times can be deter-mined from the frequency dependence of 00, which wehave measured as a function of temperature and field(Fig. 2). For a single characteristic relaxation time, ,00f has a well-defined form with a maximum at f 1= [18]. Experimentally, in a glassy system, one typi-cally finds broadening associated with a spread of relaxa-tion times, but 00f for Dy2Ti2O7 in zero magnetic fieldis only slightly broader than the single- form for our2003 The American Physical Society 107201-110 100 1k 10k10 100 1k 10k0.00.10.20.30.40.510 100 1k 10kT = 12 KFrequency  [Hz] 0 Oe 2.5 kOe 5 kOe 8 kOe 10 kOecbaT = 7.5 K χ ''  [emu/Oe molDy] FWHM single  T = 16 K 10 100 1k 10k0.040.060.080.100.120.140.160.18dH = 5 kOeDy1.6Y0.4Ti2O7 χ ''  [emu/Oe molDy]Frequency  [Hz] 9 K 10 K 11 K 12 K 13 KFIG. 2 (color online). (a)–(c) The frequency dependence ofthe imaginary part of the ac susceptibility (00). The peakmagnitude decreases monotonically with increasing field. Thebar in the 16 K plot shows the full width at half maximum ofthe theoretical response of a system with a single relaxationtime. (d) Similar data at H  5 kOe in a sample diluted with20% Y on the Dy site. The data show two distinct peaks,confirming the existence of two separate relaxation processes.The colored arrows indicate the temperature dependent (clas-sical relaxation) peak which moves down in frequency withdecreasing temperature.5 10 150.11100 5 101101001000τ   [ms]Temperature  [K] 0 Oe 4 kOe 10 kOe  τ  [ms]Temperature  [K]FIG. 3 (color online). The characteristic relaxation time, , asa function of temperature at various applied fields. Inset: Tat H  0 for a sample potted in epoxy down to T < 2 K [19].0.05 0.06 0.07 0.08468100123450 5 10 15 20 25 300.00.30.60.91.21.5012345  ln [ f (Hz)]1/T  [K-1]0            1            2           3aχ '  [emu/Oe molDy] χ dc f = 10 Hz f = 100 Hz f = 1 kHz f = 10 kHzb χ ''  [emu/Oe molDy]Temperature  [K]  M  [µ B/Dy]Field  [Tesla] 5 K 10 KFIG. 1. Temperature dependence of the magnetic susceptibil-ity for H  0. (a) The dc susceptibility and real part of the acsusceptibility (0). Inset: the dc magnetization versus appliedfield. (b) The imaginary part of the ac susceptibility (00).Inset: the measurement frequency versus inverse freezing tem-perature, fit to an Arrhenius law.P H Y S I C A L R E V I E W L E T T E R S week ending5 SEPTEMBER 2003VOLUME 91, NUMBER 10a temperature dependent characteristic spin-relaxationtime with a narrow distribution at any given temperature.The spectrum does broaden somewhat in a magnetic field(especially near H  0:5 T and T  12 K, as discussedbelow), but there is a clear local maximum to 00f foralmost our entire temperature and magnetic field range,which we take to indicate the characteristic relaxationtime, , where   1=fmax. While the form of the peakin 00f is somewhat broadened relative to the ideal caseand there is a broad background underneath the peak atthe lowest temperatures and highest fields, the resultantuncertainty in the determination of the peak frequencydoes not qualitatively change the temperature or magneticfield dependence of , and thus does not affect the dis-cussion below.As shown in Fig. 3, the temperature dependence of for T > Tice is quite striking. Above a crossover tempera-ture, Tcross  13 K,  is thermally activated, as expectedfrom the Arrhenius behavior of Tf. Below Tcross, however,we find that  is almost temperature independent (5 ms at H  0) down to T  Tice. The weak temperaturedependence raises a new question of how the spins arerelaxing for T < Tcross, since T is not consistent with a107201-2thermally activated process (which would require an un-physically weak energy barrier of EA < 2 K). Since theHamiltonian of this pure ordered system is relativelysimple, it is difficult to imagine how any combination ofthermal processes could result in the observed T. Weconclude instead that the relaxation process for Tice <T < Tcross is through quantum tunneling between the107201-2P H Y S I C A L R E V I E W L E T T E R S week ending5 SEPTEMBER 2003VOLUME 91, NUMBER 10two accessible Ising states, as has been suggested fordifferent Dy systems [20], other systems in which therelaxation rate changes from being thermally activatedto almost constant [21], and for Ho2Ti2O7 from neutronspin echo studies [15].As indicated by the three T curves in Fig. 3, theapplication of a magnetic field [22] significantly affectsthe spin-relaxation process. Importantly, the applicationof a field also confirms the coexistence of classical andquantum spin-relaxation processes near Tcross, as can beseen in the 00f data taken at T  12 K and H  5 kOe[Fig. 2(b)]. At that field, the relative efficiency of the tworelaxation processes is such that the peak of 00f hasboth a maximum and a broad shoulder, indicating theexistence of two different relaxation processes. This co-existence is even more apparent in the data of Fig. 2(d)taken on a sample diluted with Y on the Dy sites [10] inwhich the two relaxation modes have similar efficienciesat H  5 kOe but better-separated frequencies (presum-ably due to the altered spin-spin and spin-lattice interac-tions, not chemical inhomogeneity since the shoulder isobserved in the pure compound [23]). These data clearlydemonstrate the evolution with decreasing temperaturefrom a temperature dependent (classical relaxation) peakto a temperature independent (quantum relaxation) peak.As shown in Fig. 4, the relaxation time has a non-monotonic dependence on magnetic field. For smallfields (H < 5 kOe),  decreases with increasing field,but for larger fields H increases and then saturates(the field required for saturation increases with decreas-ing temperature). We interpret the measured H interms of a simplified spin Hamiltonian: H  DS2z gB ~S  ~Happlied  gB ~S  ~Hdipole where D represents thelocal anisotropy, ~Happlied is the applied dc field, ~Hdipoleis the local dipolar field, and we neglect the relativelyweak exchange interactions [5]. Given the four different0 2 4 6 8 100510155 10 150.00.30.60.9τ   [ms]Field  [kOe] 6 K 9 K 10.5 K 12 K  χ'' max  [emu/Oe molDy]Temperature [K] 0 Oe 10 kOeFIG. 4 (color online). The field dependence of the character-istic spin relaxation time, . Inset: the magnitude of the peak in00f as a function of temperature at H  0 and 10 kOe.107201-3	111 directions of the Dy spins in Dy2Ti2O7 and that oursamples are polycrystalline, the applied field has com-ponents both transverse and along the axis of almostall spins in the sample. The low field decrease in Hcan be partially attributed to enhanced quantum tunnel-ing through coupling to the transverse components of~Happlied [24]. The applied field also can enhance tunnelingby locally canceling the longitudinal component of~Hdipole, and thus making the Sz states degenerate. Thislatter effect is evident in a numerical evaluation [25] ofthe quantum tunneling barrier (averaging over the direc-tions of both ~Hdipole and ~Happlied), which indicates thatH should exhibit a minimum when the cancellation isoptimized. Indeed, the field at which we observe a mini-mum ( 5 kOe) corresponds approximately to an energyscale of 1.7 K— only slightly smaller than the dipolarinteraction of 2:35 K [5,14]. The corresponding mini-mum in the data of Fig. 4 shifts to lower field at highertemperatures, as expected from the reduction of the meandipolar field with increasing thermal fluctuations.The increasing H at higher magnetic fields is at-tributable to the longitudinal component of ~Happlied sepa-rating the energies of the Sz states. The weaker fielddependence of H at the highest fields (at 12 K, forexample) presumably marks the return to thermal relaxa-tion processes when the Sz states are highly separatedby ~Happlied. A difference between the physics of tunnelingat low and high fields is suggested by the magnitude of themaxima in 00f, shown in the inset of Fig. 4, whichincrease monotonically with decreasing temperature inzero field (as expected from the reduced dissipation atlower temperatures), but decrease monotonically at10 kOe. The temperature dependences suggest strongcoupling of phonons to the quantum spin relaxation ashas been previously observed in Mn-12 acetate [21]. Therich behavior of H;T in Dy2Ti2O7 suggests that de-tailed modeling including the phonon spectrum will benecessary for a complete description of the quantumtunneling phenomena.While a crossover between thermal and quantum spinrelaxation has been observed in many systems, the case ofDy2Ti2O7 is particularly interesting because it is a densesystem in which spin-spin correlations are developing asT ! Tice. Below Tice, a maximum in 0T and irrever-sibility in the dc magnetization [11,14] suggest that again increases with decreasing temperature as the spinice correlations develop at low temperatures.We have alsoperformed direct measurements of T for T < Tice in apowder sample of Dy2Ti2O7 potted in epoxy for thermalcontact (Fig. 3, inset) [19]. Surprisingly, while T isalmost temperature independent for Tice < T < Tcross, itincreases extremely sharply below Tice. This low tempera-ture increase in T indicates the existence of an unusualdouble crossover in spin relaxation upon cooling: firstchanging from thermally activated to quantum tunnelingat Tcross, and then reverting to thermally activated as the107201-3P H Y S I C A L R E V I E W L E T T E R S week ending5 SEPTEMBER 2003VOLUME 91, NUMBER 10developing spin correlations require that groups of spinsmust evolve coherently. The authors are unaware of simi-lar behavior in other systems exhibiting spin relaxationthrough quantum tunneling, suggesting that this phe-nomenon is associated with the development of correla-tions and the associated emergence of a collective degreeof freedom [26].Aside from the novelty of a double-crossover in therelaxation mechanism, the observation of quantum re-laxation in Dy2Ti2O7 indicates the importance of suchprocesses in frustrated rare-earth magnets, which displaya range of novel low temperature states [27]. The resultsimply that even such large spin systems, where the spinsare typically treated classically, must be considered withthe effects of quantum dynamics. As the emergence ofthermal relaxation of the correlated spins at low tempera-tures indicates, the combination of quantum and ther-mally activated processes in a strongly correlated spinsystem can form the basis for unanticipated physicalphenomena associated with the collective properties ofsuch spin systems.We acknowledge helpful discussions with D. Huse andR. Moessner, and support from the Army Research OfficeGrant No. DAAD19-01-1-0021. R. J. C. was supported inpart by NSF Grant No. DMR-9725979. A. M. acknowl-edges support from the Packard Foundation.10720*Email address: schiffer@phys.psu.edu[1] A. P. Ramirez, in Handbook of Magnetic Materials,edited by K. J. H. Buschow (Elsevier, Amsterdam,2001), Vol. 13.[2] P. Schiffer and A. P. Ramirez, Comments Condens.Matter Phys. 18, 21 (1996).[3] The first excited state crystal field in Dy2Ti2O7 has anenergy scale 200–300 K. S. Rosenkranz et al., J. Appl.Phys. 87, 5914 (2000).[4] R. Siddharthan et al., Phys. Rev. Lett. 83, 1854 (1999);R. Siddharthan, B. S. Shastry, and A. P. Ramirez, Phys.Rev. B 63, 184412 (2001).[5] B. C. den Hertog and M. J. P. Gingras, Phys. Rev. Lett. 84,3430 (2000); R. G. Melko, B. C. den Hertog, and M. J. P.Gingras, Phys. Rev. Lett. 87, 067203 (2001).1-4[6] S.T. Bramwell and M. J. P Gingras, Science 294, 1495(2001).[7] M. J. Harris et al., Phys. Rev. Lett. 79, 2554 (1997).[8] M. J. Harris et al., Phys. Rev. Lett. 81, 4496 (1998);S.T. Bramwell and M. J. Harris, J. Phys. Condens.Matter 10, L215 (1998).[9] A. P. Ramirez et al., Nature (London) 399, 333 (1999).[10] J. Snyder et al., Nature (London) 413, 48 (2001); Phys.Rev. B 66, 064432 (2002).[11] K. Matsuhira, Y. Hinatsu, and T. Sakakibara, J. Phys.Condens. Matter 13, L737 (2001).[12] K. Matsuhira et al., J. Phys. Condens. Matter 12, L649(2000).[13] A. L. Cornelius and J. S. Gardner, Phys. Rev. B 64,060406 (2001).[14] H. Fukazawa et al., Phys. Rev. B 65, 054410 (2002).[15] G. Ehlers et al., J. Phys. Condens. Matter 15, L9(2003).[16] S.T. Bramwell et al., Phys. Rev. Lett. 87, 047205 (2001).[17] We have also obtained qualitatively equivalent data on asingle crystal sample.[18] H. B. G. Casimir and F. K. du Pré, Physica (Amsterdam)5, 507 (1938).[19] The spin-relaxation times for samples potted in epoxy(nonmagnetic Stycast 1266) are offset relative to those inloose powder samples, presumably resulting from thepressure induced by the differential thermal contractionof the epoxy and the Dy2Ti2O7. J. Snyder (unpublished).[20] N. Vernier and G. Bellessa, J. Magn. Magn. Mater. 177,962 (1998), and references therein.[21] B. Barbara et al., J. Magn. Magn. Mater. 200, 167 (1999),and references therein.[22] The applied dc field is parallel to the excitation field usedfor the ac magnetic susceptibility measurements.[23] The full effects of dilution on the relaxation time arediscussed elsewhere. J. Snyder, Ph.D. thesis, PennsylvaniaState University, 2003; (to be published).[24] For example, see J. Brooke, T. F. Rosenbaum, andG. Aeppli, Nature (London) 413, 610 (2001); J. Brookeet al., Science 284, 779 (1999), and references therein.[25] Ari Mizel (unpublished).[26] O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys.Rev. Lett. 88, 067203 (2002); S.-H. Lee et al., Nature(London) 418, 856 (2002).[27] See, for example, Y. K. Tsui et al., Phys. Rev. Lett. 82,3532 (1999).107201-4

Quantum-Classical Reentrant Relaxation Crossover in Dy2Ti2O7 Spin Ice

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Quantum-Classical Reentrant Relaxation Crossover in
                    
                      
                        
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Quantum-Classical Reentrant Relaxation Crossover in Dy2Ti2O7 Spin-Ice

Quantum-Classical Reentrant Relaxation Crossover in Dy2Ti2O7 Spin-Ice

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