Spin-Ice and Other Frustrated Magnets on the Pyrochlore Lattice

Home , Dysprosium titanate

arXiv:cond-mat/0210230v1 [cond-mat.stat-mech] 10 Oct 2002 1 Introduction 1. compou titanate dysprosium the of identification recent The Abstract e words: Key pnaao frglr( regular of analog spin osbelctoso each two on the on locations protons possible of rearrangements as entropy, the the from of arising origin the The explained rapidly. Pauling of experiment model followed temperature theory low one experimental and indeed of physics, was triumphs 1936 first in the [1] of Stout experi- and calorimetry Giauque The of hav- state. ment of ground virtue of entropic by violation an Thermodynamics, apparent ing of its Law of Third view genera- the in several physicists fascinated of has tions Ice excitement. erable te opud htaepsil pnielike: ice r spin long possibly are and that exchange, obs compounds super other been ranged have short between plateaus t competition considerabl magnetization in ice Predicted created regular entropy. over has advantage of unique Pauling, a of gives fields ice magnetic entropic regular of oeigo h ytm ildsustebscmodel basic the discuss will I the system. discuss and the phenomena of basic modeling the give of will review I quick talk this a In community. the in consid- activity stimulated erable have recovery entropy and expected degeneracy, the the the and reduces energy spins this between field, the coupling magnetic one Zeeman gives the the further of of it realization handle bond, each spin on a variable fold provides two struc- only wurtzite not and the DTO protons on Oxygen ture. close each two for of protons rule far two ice Fowler Bernal the to rpitsbitdt T3Proceedings LT23 to submitted Preprint pnIeadOhrFutae ant ntePrclr Lat Pyrochlore the on Magnets Frustrated Other and Spin-Ice E-mail:[email protected] h eetraiainta T ( DTO that realization recent The pnIe;Zr on nrp c ue DTO Rule; Ice ; Entropy Point Zero ; Ice Spin I h H c,hscue consid- caused has ice, ) − a O ninIsiueo cec,Bnaoe504,INDIA 560042, Bangalore Science, of Institute Indian − H Dy od subject bond, 2 i T rrmShastry Sriram B 2 O 7 Ho sa is ) 2 i T 2 O 7 n h w stannates two the and , oal omu iaaeadtesants briefly I stannates, most the behaviour, and ice titanate spin holmium for notably candidates other observation some evaded situation of the has summarizing that After scattering. state neutron order by ground ranged true long the predicted unre- in the some mention mainly also issues, will solved I magnetization recently. var- the seen against plateaus sensitively tested most and us, experiments, by ious proposed been has that xieet h blt omnplt pn using spins manipulate to ability The excitement. e eesses n a enue osuyterecovery the study to used been has and systems, hese re,tsigteudryn oe ossigo a of consisting model underlying the testing erved, nd ne ioa neatosbtensis discuss I spins. between interactions dipolar anged Dy a , 1 2 i T 2 O i.1 h yohoelattice pyrochlore The 1. Fig. 7 sa“pnIe,ie h pnanalog spin the i.e. “Spin-Ice”, a as Ho 2 Sn 2 O 7 , Dy 1Nvme 2018 November 11 2 Sn 2 O tice 7 . discuss the dynamical susceptibility that has been used to probe the nature of precursor spin liquid, i.e. the state that the spin ice melts into, it appears to be non- trivial in its correlations.

2. Basic Phenomenon and History

Early data [1] showed the surprising feature of entropy in the ground state, although crystallographic transitions prevented the cooling of Ih ice in the wurtzite structure to very low temperatures. Bernal and Fowler pointed out the importance of minimizing dipolar energy in the proximity of each Oxygen by formulating the famous “ice rule”, i.e the rule that two protons are close by and two are further from each Oxygen so that there is a six fold manifold of states that dominate the configuration space. Pauling made the connection with experiments by estimating the global entropy arising from these local ice rules- Fig. 2. The specific heat and integrated entropy from Ref([6]) the configurations on a single Oxygen influence the neighbours so there is a highly non trivial many body ranged magnetic dipole-dipole interaction in addition problem here. Pauling’s estimate of the ground state to the short ranged superexchange interaction J, which 1 3 entropy as R 2 log( 2 ) turns out to be an inspired ap- provides the single parameter of the theory. The values proximation in that that numerical computation of the of the local moment are large for DTO (J=15/2 and entropy yield estimates that are close. In the world of p = gJµB = 10 µB ) so the dipolar interaction energy is models, this led to the celebrated two dimensional ice 2.30K for nearest neighbours. The dipolar interac- ∼ model, where the combinatorial problem was solved tion is long ranged but not a source of divergences in exactly by Lieb[2] using Bethe’s Ansatz. Anderson[3] this problem, since all the interesting physics is away showed that the identity between the Spinel B site from zero momentum (i.e. ferromagnetic region). An sublattice and the wurtzite structure leads to an in- interesting point is that the ice rule configurations are teresting connection with the configurational entropy increasingly important at lower temperatures, in Fig- in magnetite. ure(3) we plot the fraction of tetrahedra in the ice rule Harris et al[4] showed in 1997 that the pyrochlore configuration for three different values of J taken from compound HTO ( Ho2T i2O7) could be considered as a crude monte carlo sampling. It is infact easy to see “spin ice” by noting the absence of LRO in neutron scattering and from µSR data. Moessner[5] empha- sized the fact that ferromagnetism plus a non collinear Ising type easy axis arrangement can lead to frustra- tion, which is commonly associated with antiferromag- netism. In a definitive experiment in 1999 Ramirez et al[6] showed that the entropy of DTO has a shortfall from the expected R log(2) by an amount that is very close to the Pauling value, thus establishing it as the first clear spin ice system. In Figure(2) we see that the entropy in zero field powder DTO saturates nicely, and a finite field, as small as .5T releases some of the entropy. We return to discuss the details of “entropy recovery” later. The currently popular basic model to describe the Fig. 3. The fraction of tetrahedra in ice ruled configurations spin ice system was formulated in Siddharthan et al[7]. for three values of superexchange J = 1.9, 1.24, .52 for curves from bottom up. The middle curve corresponds to DTO, and It consists of local Ising variables pointing along or the other to two possible parameters for HTO. against the local easy axis found in each case by joining the sites to the center of either of the two tetrahedra it that the six configurations allowed by the ice rule pos- belongs to. These local moments interact via the long sess a net magnetic moment that points along or an-

2 tiparallel to the three crystallographic axes, so at low 1200 B || [1 1 1] temperature one expects a renormalization of the ef- 1000 2 fective moment by a factor of 3 , and hence the uniform 4 800 B || [1 −1 1.414] susceptibility by 9 . Data from Lawes and Ramirez[10] on susceptibility is consistent with this expectation, as 600 seen in Figure(4). M (arb. units) 400

200

0 0 0.5 1 1.5 2 2.5 3 B (Tesla) 5

4 Dy2Ti2O7

3 H // [111] /Dy) B µ 2 0.48 K M (

0.99 K

1 1.65 K Fig. 4. The low temperature susceptibility of HTO follows a ∼ 4 Curie law but the Curie constant renormalizes to 9 below 0 the Ice rule scale of temperature. 0 5 10 15 20 25 30 H (kOe) The agreement between experiments and theory on specific heat is quite good with the basic model[7,8], Fig. 5. Figure on top is the theoretical prediction[11], the dotted curve should be compared with the one below from and one can fine-tune the basic model by adding su- experiment[12]. perexchange at further neighbour distance to improve the agreement further[9]. We note that the most strin- T=0.4K, [111] direction gent test is the behaviour of the magnetization versus magnetic field for different directions of the field. The J_s = 0.52 J_s=1.24 0.5 prediction of Siddharthan, Shastry and Ramirez[11] of J_s=1.92 magnetization plateaux for fields along < 111 > direc- 0.4 tion represent the best test, since the plateau at 1T ∼ corresponds to some tetrahedra breaking the ice rule M 0.3 configurations, and going into a 3-in 1-out configura- 0.2 tion. Recent experiments by Matsuhira et al[12] bear out this prediction very well, as seen in Figure(5). The 0.1 onset of the second plateau is sharp at low tempera- 0 0 0.5 1 1.5 2 tures, and can be used to experimentally define the ice B (Tesla) rule energy scale. It should be regarded as the finger- print of the ice rule physics. This scale arises from the Fig. 6. Three values of J and the resulting range of the plateau. competition beween the (known) ferromagnetic dipolar the values of J are 1.92, 1.24,.52 from left to right. energy and the (unknown) antiferromagnetic superexchange. In Figure(6) we show the range over which the depending upon the direction of the field relative to the ice rule scale changes by changing the exchange J, this symmetry axes. For example a double peak structure can be used ( with linear extrapolation) to deduce J if was predicted for fields along < 1, 1, √2 > direction − we know the location of the plateau. The values of the which looks rather close to the double peak seen very magnetic moment at the first and second plateaux are recently in experiments[12]. 1 1 easy to understand, the saturation values are 3 and 2 We next mention holmium titanate HTO, where of the maximum. Ho3+ has an identical magnetic moment as Dy+3, The problem of entropy recovery in the presence of and hence the same dipolar interaction, but possibly a uniform magnetic field was alluded to earlier. While different exchange J. Initial experiments [6] showed a the first experiment on this in Ref[6] was on poweder rising specific heat down at .50K, at which point the ∼ samples, the monte carlo data [11] already gave indica- system fell out of equilibrium. One can interpret this tions of considerable fine structure in the specific heat in one of two possible ways. It is imaginable [7,9,11]

3 that the system undergoes a transition to an long susceptibility show the same features, and these seem range ordered (LRO) state which is anyway predicted to be consistent above 100K with a dynamical sus- χ0 ∼ 1 by theory (see below), and that the slow dynamics ceptibility α , where χ0 and α departing (1−iωτ) ∼ T hides this transition. One has to then explain the dif- from the Debye value of unity, and further being rather ference between DTO and HTO as possibly arising temperature dependent, with α .544 at 170K and ∼ from differences in the ice rule scale, or equivalently dα/dT .06/0K. Thus the non Debye relaxation with ∼ J, so that HTO would have a much lower ice rule temperature dependent α, and the origin of the relax- temperature ( hence larger J ) whereas DTO with a ation rate τ τ0 exp(ǫc/kT ) demand a fundamental | | ∼ higher ice rule temperature would be stuck in a subset understanding that is missing at the moment. of configurations that would prevent it from under- going a transition to a state with lower free energy. The other view point[14] ascribes the difference in 3. behaviour to the significant hyperfine coupling constants in HTO, as known from early work of Blote et al[13]. Subtracting the nuclear component makes the data for HTO look similar to DTO, and the recent first principles calculations of the hyperfine constants Acknowledgements It is a pleasure to acknowledge of these two compounds [16] lends weight to this view my collaborators: Art Ramirez and Gavin Lawes at point, as does the absence of LRO as indicated by Los Alamos National Lab, and Rahul Siddharthan at neutron scattering [14,15]and other experiments at IISc/ENS Paris. low temperatures 500mK. ∼ Staying with the issue of LRO, we mention that even in the case of DTO one expects LRO at low enough temeratures. Siddharthan et al [9,11] were the first to show, by explicit enumeration of ground states for finite References clusters, that the model for DTO should have LRO of [1] W E Giauque and J W Stout, J Chem Phys 1 (1933), 515. a certain type. This has been corraborated by Melko 18 et al[17], who use a loop algorithm to equilibriate the [2] E H Lieb , Phys Rev Letts, ,(1967) 1046. system rather than single spin flips, and find the same [3] P W Anderson, Phys Rev 102,( 1956), 1008. structure, with a transition at .20K. In contrast, [4] M J Harris et al, Phys Rev Letts, 79(1997), 2554. ∼ neutron scattering[18] sees no signs of LRO down to [5] R Moessner, Phys Rev B 57 (1998), 5587. low temperatures 500mK! Therefore it seems to me ∼ [6] A P Ramirez, A Hayashi, R J cava, R Siddharthan, B S that DTO and possibly HTO may yet have a surprise Shastry, Nature 399 ( 1999), 333. in store at low temperatures. [7] R Siddharthan, B S Shastry, A Ramirez, A Hayashi, R J Turning to other possible spin ice compounds, the Cava , S Rosenkranz, Phys Rev Letts 83,( 1999) 1854. stannates M2Sn2O7 with M = Dy,Ho have been re- [8] B C den Hertog, M Gingras, Phys Rev Letts84( 2000), 3430. cently proposed [19]. These have small differences in [9] R Siddharthan, Ph D Thesis, IISc Bangalore, July (2000). lattice constants, and hence slightly different dipolar as well as superexchange interactions, and should be [10] G Lawes, A Ramirez, to be published (2002). useful in fixing several details of the models. [11] R Siddharthan, B S Shastry and A Ramirez, Phys Rev Finally on the topic of spin ice, I would like to men- B63 ( 2001) 184412, and cond-mat 0009265. tion two very recent and nice experiments on the AC [12] K Matsuhira, Z Hiroi, S Takagi, T Sakakibara, J Phys susceptibility of DTO. Schiffer et al [20] and Matsuhira Cond Mat 14 ( 2002) L559, H Fukazawa, R G Melko, R B 65 et al [21] have found that the curie type divergence of Higashinaka, Y Maeno, M Gingras, Phys Rev (2002) 054410. the low field susceptibility ( as in Figure(4)) is infact 43 cut off at a low temperature 150K that depends [13] H Blote, R Wielinga, W Huiskamp, Physica (1969), 549. ∼ 87 upon the frequency of the probe field, resulting in two [14] S Bramwell et al, Phys Rev Letts ( 2001), 47205. maxima, one at a higher temperature 180K and the [15] H Kadowaki, Y Ishii, K Matsuhira, Y Hinatsu, Phys Rev ∼ other at a lower temperature 1.50K. Each of these B65 (2002), 144421. ∼ maxima is freqency dependent, and the from the Ar- [16] Y Jana, D Ghosh, Phys Rev B61( 2000), 9657; J Mag mag rhenius dependence of these we can extract a character- Materials 248( 2002),7. istic energy scale ǫc corresponding to these peaks. The [17] R Melko, B C den Hertog, M Gingras, Phys Rev Letts87( higher one yields an energy scale 2200K, very nearly 2001), 67203. ∼ the crystal field splitting seen in [7], and the lower one [18] T Fennell et al, to be published (2001), cond-mat 0107414. give 100K, presumably a multiple of the ice rule en- ∼ [19] K Matsuhira, Y Hinatsu, K Tenya, T Sakakibara, J Phys ergy scale. The real as well as imaginary parts of the Cond Mat 12 ( 2000) L649.

4 [20] J Snyder, J Slusky, R Cava, P Schiﬀer, Nature 413( 2001) 48. [21] K Matsuhira, Y Hinatsu, T Sakakibara, J Phys Cond Mat 13 ( 2001) L737.