Hysteresis and Relaxation Effects in the Spin-Ice Compound Dy 2 Ti 2 O 7 studied by Heat Transport
Simon Scharffe, Gerhard Kolland, Martin Hiertz, Martin Valldor, Thomas Lorenz
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Hysteresis and Relaxation E ff ects in the Spin-IceCompound Dy Ti O studied by Heat Transport Simon S charffe , Gerhard K olland ∗ , Martin H iertz , Martin V alldor † , and Thomas L orenz II. Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 K¨oln, GermanyE-mail: schar ff [email protected] (Received September 30, 2013)The low-temperature thermal conductivity κ of the spin-ice compound Dy Ti O shows pronouncedhysteresis as a function of magnetic field. Here, we investigate how these hysteresis e ff ects dependon temperature, the magnetic-field direction, the rate of magnetic-field change, and on the directionof the heat current. In addition, the time-dependent relaxation of the heat conductivity is investigated.These measurements yield information about possible equilibrium states and reveal that in the low-field and low-temperature region extremely slow relaxation processes occur. KEYWORDS: spin ice, Dy Ti O , magnetic frustration, relaxation, magnetic heat transport
1. Introduction
The study of magnetic monopole excitations in the spin-ice compound Dy Ti O is of high inter-est in current solid-state research [1–10]. Dy Ti O is a geometrically frustrated spin system wherethe magnetic Dy + ions form a pyrochlore lattice of corner-sharing tetrahedra. A strong crystal fieldresults in an Ising anisotropy with the local easy axes along one of the { } directions. As a conse-quence, the Dy + moments point either into or out of each tetrahedron and allow several degeneratespin configurations. When two spins point in and two out of a tetrahedron the magnetic dipole energyis minimized, leading to a sixfold degenerate ground state in zero magnetic field [11–15]. This orien-tation of the Dy + moments corresponds to the hydrogen displacement of water ice. Excited states arecreated by flipping a single spin resulting in two neighboring tetrahedra with configurations 1in-3outand 3in-1out, respectively. Due to the ground-state degeneracy such a dipole excitation fractionalizesinto two individual monopole excitations, which can freely propagate in zero field.Recently, we investigated these exotic magnetic excitations by thermal-conductivity measure-ments and found experimental evidence for monopole heat transport in Dy Ti O [16, 17]. In zeromagnetic field, this leads to a pronounced magnetic contribution κ mag in the low-temperature ther-mal conductivity κ . For finite magnetic fields, κ mag decreases and we observed a correlation betweenthe magnitude of κ mag and the degree of degeneracy of the di ff erent magnetic-field induced groundstates for ~ H || [001], ~ H || [1¯10], and ~ H || [111]. For all three field directions pronounced hysteresis ef-fects occur between the κ ( H ) curves measured with increasing or decreasing magnetic field. Belowabout 0 . κ ( H →
0) does not recover the initial κ = κ ( H =
0) obtained by zero-field cooling. Moreover, we found that the reduced κ ( H →
0) valuesslowly relax towards the corresponding κ . Slow relaxation phenomena in the low-temperature regionhave also been observed in other physical properties of Dy Ti O , e.g. , in the magnetization [21],the a.c.-susceptibility [10] or the specific heat [15, 16, 22]. ∗ Present address: Deutsches Zentrum f¨ur Luft- und Raumfahrt, Linder H¨ohe, 51147 K¨oln, Germany † Present address: Max-Planck-Institut f¨ur Chemische Physik fester Sto ff e, Noethnitzer Str. 40, 01187 Dresden, Germany n additional hysteresis of κ ( H ) is observed in the so-called kagome-ice state, which is realizedat low temperature for ~ H || [111] below about 1 T. For this field direction, the pyrochlore structure canbe best visualized as alternating triangular and kagome planes of Dy spins, which are stacked along[111]. While the spins of the triangular planes are fully aligned already by small fields ~ H || [111], thecompetition between the 2in / M ( H ) up to about 1 T, where a transition to the fullypolarized state occurs. Surprisingly, κ ( H ) is strongly hysteretic within the kagome-ice phase [17, 20],whereas there is no hysteresis in the corresponding magnetization plateau. This unusual hysteresis of κ ( H ) has been observed in Ref. [20] in measurements with the heat current ~ along and perpendicularto the magnetic field, i.e. with ~ perpendicular and within the kagome planes, respectively, and hasbeen confirmed by our data [17] measured with ~ || [1¯10]. However, according to Ref. [20] no hys-teresis of the zero-field values κ ( H =
0) seems to be present for both directions of ~ , in contrast to ourresults of Ref. [17] for ~ || [1¯10].In this report, we present a study of these unusual hysteresis e ff ects of κ ( H ) for the magneticfield directions along [001] and [111]. In order to derive in how far the di ff erent κ values representequilibrium values, we studied κ ( H ) for di ff erent magnetic-field sweep rates and the influence ofdi ff erent cooling procedures. Moreover, we present relaxation measurements κ ( t ) performed afterdi ff erent field-sweep cycles and we discuss the influence of the direction of the heat current on thehysteresis e ff ects for ~ H || [111].
2. Experimental
Large single crystals of Dy Ti O were grown by the floating-zone technique in a mirror furnace.Oriented crystals of approximate dimensions 3 × × were used to measure the thermal con-ductivity by the standard steady-state method. A heater produced a temperature gradient within thesample which was measured by two calibrated RuO thermometers. The heat current ~ was drivenalong the longest sample dimension, while the magnetic field was applied either parallel or perpen-dicular to ~ . Note that the standard steady-state method is a step-by-step technique where every datapoint needs several minutes to stabilize the temperature gradient and the average temperature. Thisthen typically results in e ff ective field-sweep rates of ≈ .
01 T / min. Most of the measurements wereperformed using this step-by-step technique, but, in addition, we also performed some measurementswhere the field was continuously varied with a larger rate of 0 . / min.Demagnetization e ff ects were taken into account for all measurements. Due to the geometry with ~ along the longest sample dimension this correction is large (up to ∼ . ~ , whereas much smaller corrections ( . .
08 T) are present for a field parallel to ~ .The demagnetization field is calculated on the basis of experimental magnetization data which weremeasured with a home-built Faraday magnetometer on thin samples to minimize the demagnetizatione ff ects within the magnetization measurements.
3. Results [001] and heat current along [1¯10]Figures 1(a)-(c) display field dependent measurements κ ( H ) for di ff erent magnetic-field-sweeprates normalized to κ obtained by zero-field cooling. As already discussed in Ref. [16], figure 1(a)reveals that after cycling the magnetic field up and down at 0 .
35 K the final κ ( H →
0) only recoversabout 90% of its initial zero-field κ , see curves (1) and (2). A subsequent field cycle results in κ ( H )curves (3) and (4) with equal endpoints, where curves (2) and (4) perfectly match each other. As is .0 0.1 0.2 0.3 0.4 0.50.50.60.70.80.91.0 0 1 2 3 454565860 (H-DM) (T)T= 0.35 K / ( m W / K m ) relaxation fit Time (h) ~ 0.01 T/min ~ 0.01 T/minT = 0.8 K(c) Fig. 1. (Color online) Field dependence κ ( H ) /κ for ~ H || [001] and ~ || [1¯10] for di ff erent field-sweep direc-tions (marked by arrows) and field-sweep rates. The order of the successive field sweeps is marked by thenumbers 1–4, where the initial sweep was started after cooling in zero field. The inset shows the relaxation ofthe reduced zero-field value towards the initial κ . The data of (a) are from Ref. [16]. shown in the inset of figure 1(a), the reduced zero-field κ ( H →
0) slowly relaxes back to the zero-field-cooled κ . The fit yields large relaxation times τ ≃ τ ≃
100 min. In contrast, no suchslow relaxation e ff ects are observed in the field range above 0 . κ ( H ) is non-hysteretic.This slow relaxation in the hysteretic region raises the question, in how far the measured κ ( H )curves also depend on the field-sweep rate. In Fig. 1(b), κ ( H ) is shown for di ff erent field-sweep ratesat T = . ∼ .
01 T / min the κ ( H →
0) valuerecovers about 95% of the initial κ . Increasing the field-sweep rate of the subsequent field cycleto 0 . / min causes additional features in κ ( H ). The κ ( H ) curve (3) measured with increasing fieldshows a minimum around 0 . κ ( H ) curve (1) around 0 .
55 T.Decreasing the field again results in the κ ( H ) curve (4), which is almost constant and close to curve (2)until the field falls below 0.25 T, where an even larger hysteresis opens and after this faster field cycle κ ( H →
0) only recovers about 80% of the initial κ .The data of Figs. 1(a) and (b) clearly show that even small field-sweep rates of ∼ .
01 T / min maybe too large to reach equilibrium states in the low-temperature range of spin ice. As may be naturallyexpected, this slow equilibration vanishes towards higher temperatures, as is shown in Fig. 1(c). Onthe other hand, the hysteresis e ff ects also disappear in the low-temperature range when the magneticfield is increased. This is not only suggested by the data of Figs. 1(a) and (b), but also follows fromour additional data measured to higher fields, which are non-hysteretic and do not show such slowrelaxation e ff ects [16, 17]. The fact that the relaxation / hysteresis e ff ects rather rapidly vanish towardshigher fields appears also natural, because the magnetization of spin ice is essentially saturated in thisfield range. One might suspect that the slow equilibration in the low-temperature / low-field region is asingle-ion property of the large Ising spins of the Dy + ions, which only slowly equilibrate because ofthe rather large splitting to the higher-lying crystal-field states. This can be ruled out, however, fromour measurements on the related half-doped material (Dy Y ) Ti O , which does not show suchhysteresis e ff ects [17]. Thus, we conclude that the slow equilibration is a particular spin-ice feature. [111] and heat current along [1¯10]Figure 2(a) displays κ ( H ) /κ for ~ H || [111] with a heat current ~ measured along [1¯10] at varioustemperatures. The kagome-ice phase is clearly seen in a plateau-like feature of κ ( H ) below ∼ .0 0.5 1.0 1.5 2.0 2.50.40.71.01.31.61.9 -0.8 -0.4 0.0 0.4 0.845505560 (H-DM) (T)T 0.4 K ( m W / K m ) H || [111](b)j || [110] 234 5 6 relaxation fit1010 (c)(d) (H-DM) 0.58 T time (h) M ( B / D y ) H (T)Magnetization
Fig. 2. (Color online) Field dependence κ ( H ) for ~ H || [111] and ~ || [1¯10] for di ff erent field-sweep directions(marked by arrows). In (a), the κ ( H ) /κ curves for di ff erent temperatures are shifted with respect to each otherand the inset displays a corresponding magnetization curve (data from [17]). Low-field hysteresis loops of κ ( H ) are displayed in (b), where the order of the successive field sweeps is marked by the numbers 1–6. Anexpanded view of κ ( H ) in the kagome-ice region is displayed in (c), which also shows the κ ( t ) curves (7) and(10) and the additional field sweeps (8) and (9); see text. The relaxation curves κ ( t ) with corresponding fits asa function of time are displayed in (d). As already mentioned, below 0 . κ ( H ) data show a clear hysteresis in the kagome-ice phase,whereas no such hysteresis is present in the plateau region of the corresponding magnetization curveas is shown in the inset of Fig. 2(a). Moreover, compared to the initial κ obtained by zero-fieldcooling, the low-temperature field cycles result in reduced zero-field values κ ( H → κ (not shown). This zero-field relaxation is present for allthree field directions ~ H || [001], ~ H || [110], and ~ H || [111], where in all cases the heat current was drivenalong [1¯10], see Ref. [17].In order to further investigate the relaxation processes within the kagome-ice phase, completelow-field hysteresis loops of κ ( H ) for positive and negative magnetic field were performed, whichare shown in Fig. 2(b). Concerning the reduced zero-field values, the observed systematics of κ ( H )is analogous to that already discussed above for ~ H || [001]. In addition, we find that the hysteresis inthe kagome-ice phase in the initial field cycle is less pronounced than in the subsequent cycles. Inparticular, the minima in the κ ( H ) curves (3) and (5) are more pronounced than the minimum in theinitial curve (1). Therefore, we also performed relaxation studies in this field region, which are shownin Figs. 2(c) and (d). First, we measured the time dependence κ ( t ) at constant field and temperaturestarting from the κ ( H ) curve (5). Similar to the zero-field case, we observe a slow relaxation with atime constant τ ≈
100 min and the κ ( t ) curve seems to relax towards the initial κ ( H ) curve (1). Thus,the relaxation measurement was stopped after 4.5 hours and the field was cycled up to 0.8 T andback to ∼ .
58 T. The corresponding κ ( H ) curves (8) and (9) do, however, not follow the initial fielddependent κ ( H ) curves (1) and (2). Instead, the κ ( H ) curve (8) approaches curve (5) and the field-decreasing κ ( H ) curve (9) essentially follows curve (6). At ∼ .
58 T, we then again measured κ ( t )and observed a weak, almost linear increase of κ ( t ) yielding a very large relaxation time τ ≈
520 min.These data suggest that the relaxation within the kagome-ice phase is not directly related to the zero-field relaxation and also reveal that, depending on the field and temperature region, a true thermal .0 0.5 1.0 1.5 2.00.40.60.81.0
21 (c) 0.015 T/min
H || j || [111] (H-DM) (T)T = 0.6 K 2(a) (b) relaxation fit ( m W / K m ) Time (h) (H-DM) = 0.25 T (H) after 3 T fc (0.6 K,H) fc Fig. 3. (Color online) Field dependence κ ( H ) /κ for ~ H || [111] and ~ || [111] for di ff erent field-sweep direc-tions (marked by arrows). The curves for di ff erent temperatures in (a) are shifted with respect to each other.Successive field sweeps (1) – (3) at 0.6 K together with a relaxation curve (4) are shown in (b). The insetdisplays the κ ( t ) curve (4) as a function of time, while in (c) the κ ( H ) /κ curve (2) is compared to di ff erent κ ( H ) /κ values obtained by field cooling and to a field-decreasing run measured after field cooling in 3 T. equilibrium can be hardly reached under typical experimental conditions. [111] and heat current along [111]Figure 3(a) displays the field-dependence κ ( H ) /κ for the same magnetic-field direction [111] asin the previous subsection, but now with the heat current ~ || [111]. Again, there is a hysteresis of κ ( H )within the kagome-ice phase which has no analogue in the magnetization curves, but there are alsosystematic di ff erences between κ ( H ) /κ for the same field but di ff erent heat-current directions. Firstof all, the hysteresis width with respect to the di ff erent absolute values of κ ( H ) is significantly largerfor ~ || [111] than for ~ || [1¯10]. Secondly, the upper critical field, where the hysteresis closes, stronglydecreases with increasing temperature for ~ || [111], while it is essentially temperature-independent for ~ || [1¯10]. Finally, for ~ || [111] the intial zero-field κ is recovered in the field-decreasing runs despitethe fact that there is a finite remnant magnetization, when M ( H ) is measured with a comparable sweeprate, see inset of Fig. 2(a).Relaxation and zero-field-cooled measurements of κ ( H ) reveal that in the hysteresis region of κ ( H ) /κ the upper branches, which are obtained with decreasing field, are closer to thermal equi-librium than the lower ones. For example, κ ( t ) relaxes from the lower κ ( H ) curve (1) to the uppercurve (2) at T = . τ ≈
52 min, as is shown in Fig. 3(b). Moreover, the κ ( H ) values obtained either by field-cooling in various fixed fields or by field-cooling in 3 T anddecreasing the field to zero well agree with the initial field decreasing κ ( H ) curve (2), see Fig. 3(c).Qualitatively, this hysteresis and relaxation of κ ( H ) can be understood by assuming that field-induceddisorder within the kagome-ice phase causes an additional suppression of κ , because starting fromthe fully polarized high-field state will cause less disorder in the kagome-ice phase than entering thisphase from the entropic zero-field spin-ice ground state; see also the discussions in Refs. [17, 20].Finally, our finding that the presence or absence of a zero-field hysteresis of κ ( H →
0) for ~ H || [111]depends on the direction of ~ indicates that the strength and the direction of a heat current throughspin ice may be an additional parameter to influence its thermal equilibration. . Conclusion The thermal conductivity κ of the spin-ice compound Dy Ti O shows strong hysteresis andslow relaxation processes towards equilibrium states in the low-temperature and low-field regime.In general, the thermal conductivity in the hysteretic regions slowly relaxes towards larger valuessuggesting that there is an additional suppression of the heat transport by field-induced disorder inthe non-equilibrium states. The degree of hysteresis does not only depend on temperature and themagnetic-field direction, but also on the rate of the magnetic-field change and, for ~ H || [111] even onthe direction of the heat current. The observation that the rate of thermal equilibration in spin icecan be influenced by a finite heat current along certain directions may possibly yield important in-formation about the dynamics of monopole excitations and their interaction with phonons. However,further investigations of this e ff ect for other directions of the magnetic field and di ff erent directionsof the heat current are necessary. Acknowledgments
This work has been financially supported by the Deutsche Forschungsgemeinschaft via SFB 608 andthe project LO 818 / References [1] I. A. Ryzhkin, J. Exp. Theor. Phys. , 481 (2005)[2] C. Castelnovo, R. Moessner, and S. L. Sondhi, Nature , 42 (2008)[3] D. J. P. Morris et al. , Science , 411 (2009)[4] S. R. Giblin et al. , Nat. Phys. , 252 (2011)[5] C. Castelnovo, R. Moessner, and S. L. Sondhi, Phys. Rev. B , 144435 (2011)[6] H. Kadowaki et al. , J. Phys. Soc. Jpn. , 103706 (2009)[7] L. D. C. Jaubert and P. C. W. Holdsworth, J. Phys.: Condens. Matter , 164222 (2011)[8] S. T. Bramwell et al. , Nature , 956 (2009)[9] S. Blundell, Phys. Rev. Lett. , 147601 (2012)[10] L. Yaraskavitch et al. , Phys. Rev. B , 020410 (2012)[11] A. P. Ramirez et al. , Nature , 333 (1999)[12] S. T. Bramwell and M. J. Gingras, Science , 1495 (2001)[13] T. Sakakibara et al. , Phys. Rev. Lett. , 207205 (2003)[14] J. F. Nagle, J. Math. Phys. , 1484 (1966)[15] D. Pomaranski et al. , Nature , 353 (2013)[16] G. Kolland et al. , Phys. Rev. B , 060402(R) (2012)[17] G. Kolland et al. , Phys. Rev. B , 054406 (2013)[18] Y. Tabata et al. , Phys. Rev. Lett. , 257205 (2006)[19] K. Matsuhira et al. , J. Phys.: Condens. Matter , L559 (2002)[20] X. F. Sun et al. , Phys. Rev. B , 144404 (2013)[21] K. Matsuhira et al. , J. Phys. Soc. Jpn. , 123711 (2011)[22] B. Klemke et al. , J. Low Temp. Phys.163