Temperature Chaos, Memory Effect, and Domain Fluctuations in the Spiral Antiferromagnet Dy

The spiral antiferromagnetic phase of polycrystalline dysprosium between 140 K and the Néel temperature at 178 K and its domain wall (DW) dynamics were investigated using high-resolution ultrasonic spectroscopy. Two kinetic processes of quasi-static DW motion occur under non-isothermal and isothermal conditions. A “fast” process is proportional to the rate of the temperature change and results in a new category of anelastic phenomena: magnetic transient ultrasonic internal friction (IF). This IF, related to fast moving magnetic DWs, decays rapidly after interruptions of cooling/heating cycles. A second, “slow” kinetic process is seen as logarithmic IF relaxation under isothermal conditions. This second process is glass-like and results in memory and temperature chaos effects. Low-frequency thermal fluctuations of DWs, previously detected by X-ray photon correlation spectroscopy, are related to critical fluctuations with Brownian motion-like dynamics of DWs.


Materials and Methods
Internal friction, IF, is defined as the logarithmic decrement W W 2 δ = ∆ , with W the maximum stored elastic energy and ∆W the energy dissipated in a cycle of oscillations. The decrement and the Young's modulus, E, were measured by piezoelectric ultrasonic composite oscillator technique 22 over a temperature range 80-210 K at frequencies near 90 kHz, which is determined by the type of the quartz transducer used. The frequency range close to 10 5 Hz corresponds (at least in ferromagnets) to the highest sensitivity to the local displacements of magnetic DWs, producing micro-eddy current damping 23 . Temperature spectra of the internal friction and Young´s modulus were taken either in a continuous mode (uninterrupted scans) or in interrupted mode with isothermal holdings of 40 min. at selected temperatures with consequent resuming of the cooling/heating scan. The oscillatory strain amplitude was stabilized in all experiments at 10 0 5 ε = − , sufficiently low not to produce detectable non-linear effects. Dy samples of 13 × 2 × 1 mm 3 measured half the ultrasonic wavelength and were excited by a standing wave. Two series of samples of different origins were tested. The samples from the first series were spark-cut from 1 mm thick rolled plate, supplied by Sigma Aldrich (99.9% purity). The second series (as cast samples) was cut from an ingot produced by arc melting from 99.0% purity (99.9% purity with respect to rare earth elements) raw material purchased from Suzhou Chemical Co, China. Samples from the first series were studied in as received (rolled) state and after vacuum annealing at 920 K for 2 hours. Samples of all types and heat treatments showed essentially similar results. Annealing of rolled samples eliminated minor differences with as cast samples. If not specified otherwise, the data are presented for the as-cast samples. Figure 1 shows the IF (a) and Young's modulus (b) between 195 and 140 K for an as-cast sample. A Young's modulus minimum concomitant with an IF peak at T N = 178 K is typical for an AFM transition 24,25 . The IF spectra were registered under constant cooling/heating rate and differ notably below T N from the IF behaviour under isothermal conditions dT dt / 0 = 24,25 . The difference is that the IF below T N does not fall off rapidly for dT dt / 0 ≠ but remains at a similar level as the absorption peak at T N . Broad maxima of the IF below T N in Dy for longitudinal and especially for transverse waves were observed under non-isothermal conditions in 26 but were not commented upon. Strikingly, the non-isothermal IF decreases dramatically, (e.g. by a factor of 3 at 166 K) after interrupting a cooling run and subsequent isothermal relaxation ( Fig. 1(a)). The relaxed IF spectrum in Fig. 1(a) (shown by the black line) corresponds to the classical IF pattern in Dy under isothermal conditions 24 . The magnitude of relaxation during isothermal dwelling shows a maximum near the Villari point at 166 K 20,21 .

Results
After resuming the interrupted cooling run the IF reaches rapidly the previous level of continuous cooling. The IF spectrum, registered under consequent uninterrupted heating, shows a local IF dip at the temperature of the isothermal relaxation during cooling. This is the memory effect, exemplified by the data at 166 K in Fig. 1(a), see the inset in Fig. 1(a) for details. The memory effect is determined as a difference between the IF values in an uninterrupted scan and in the local minimum at 166 K, the inset in Fig. 1(a). The memory effect corresponds to the part of the overall IF relaxation, which is represented by its isothermal component slow δ (inset in Fig. 1(a)). Heating the sample above T N erases this memory effect ( Fig. 1(a)): it does not reappear in a consecutive uninterrupted cycle. Figure 1(c) depicts similar results for as received (rolled) sample. Several minor differences with the data for as-cast samples, Fig. 1(a,b), can be noted. First, the IF above T N is higher in rolled sample and the peak at T N cannot be clearly discerned, presumably due to the high density of defects (dislocations). Second, a minor difference in the YM values of the two samples can likely be accounted for by different textures. For the as received (rolled) sample, the lowest temperature of the thermal cycle was 95 K and cooling/heating scans were interrupted at several temperatures: 170, 160, 150, 140 and 120 K, showing the same overall IF trend and the same relaxation on cooling and heating. Thus, the relaxation is a generic effect in helical AFM polycrystalline dysprosium, which persists in samples of different origin and thermomechanical history. The relaxation process observed on cooling/ heating depends on the cooling/heating rate just before the interruption, but is not sensitive to the previous history of the sample (previous interruptions or variations of the cooling rate) as shows Fig. 1(c). This feature is a hallmark of 'temperature chaos' 27,28 and, together with the memory effect, describes a glassy state [27][28][29] . Figure 2 shows the IF relaxation during interruptions of cooling at 166 K (a,b) and 140 K (c). Time dependences of the temperature T and cooling rate  T dT dt / = during interruption of cooling at 166 K are shown in Fig. 2(a), corresponding variations of the IF are compared with the absolute value of cooling rate dT dt / in Fig. 2 Figure 2(c) depicts the IF kinetics for interruption of cooling at 140 K. Two distinct dynamic processes are found in IF kinetics: firstly a slow relaxation when the temperature is constant (  T 0 = ) and, secondly, a component T  δ~ detected during temperature stabilization. The latter is dominant in the IF kinetics after interruption of cooling at 166 K but is almost absent at 140 K. The IF and temperature data from Fig. 2(b,c) are re-plotted in Fig. 3(a,b) as function of T  and confirm a strong decrease of the IF proportional to  T during interruption of cooling at 166 K (a) and its absence at 140 K (b). The proportionality between  T and the IF extends over a large interval of negative values of T  and is kept for positive T  during heating from the small negative overshoot ( Fig. 3(a)). The proportionality to  T creates an IF minimum in Fig. 2(b) near 5700 s. Thus, the overall IF is a function of temperature, T, time t (seen as "slow" isothermal relaxation component), and cooling/heating rate T  (responsible for the "fast" relaxation). In a simplest form, assuming independence of slow, δ slow , and fast, δ fast , IF components, the overall IF T t T ( , , ) δ  is: is the temperature spectrum of fully relaxed IF. The black line in Fig. 1(a) is an approximation to such a spectrum obtained after 2400 s of relaxation.
www.nature.com/scientificreports www.nature.com/scientificreports/ The isothermal IF relaxation component, δ slow during complete stoppage at 166 K is estimated in Fig. 3(a) as a distance between the line  T ( ) δ at =  T 0 and the lowest IF value achieved during relaxation. The crossing point of Fig. 3(a). δ slow is small by comparison with fast δ at 166 K. δ slow defined from the data of Fig. 3(a) is shown in Figs 2(b) and 1(a). According to Fig. 1(a), the magnitude of the memory effect coincides with the degree of isothermal relaxation. Hence, only this sluggish, isothermal relaxation contributes to the memory effect, but not the  T -dependent kinetics. Figure 2(b) indicates that, during resuming of cooling, the jerky recovery of the steady state IF value occurs once the IF gradually increases to the level observed before the isothermal relaxation, i.e. increases by the same value of slow δ . The T  dependence is extremely small at 140 K, Fig. 3(b), so that the only variation in δ exists at the isothermal point  = T 0. The inset in Fig. 2(c) shows δ slow and fast δ versus temperature. slow δ gradually declines on cooling, whereas δ fast has a clear maximum at 166 K, the temperature of Villari point in Dy 20,21 . www.nature.com/scientificreports www.nature.com/scientificreports/ Figure 2(b,c) shows fittings of the IF sluggish isothermal relaxation at 166 K and 140 K with a logarithmic law. Excluding the initial transitory period of temperature stabilization, the data agree well with logarithmic trends.

Discussion
The distinguishing feature of the IF spectra of Dy are their low values. The IF increases nearly an order of magnitude upon ordering in the helical AFM phase with δ ≈ − 10 4 , Fig. 1(a). This value is still small compared with the IF of Dy in the ferromagnetic state 26,30 . The increase of IF in the helical phase compared with the paramagnetic state indicates that anelastic phenomena below T N originate from an ordered magnetic structure. No resonance phenomena are expected because our measurement frequency 10 5 Hz is much smaller than relaxation frequencies of individual spins or spin waves. The frequency near 10 5 Hz corresponds to the maximum sensitivity for collective spin rearrangements represented by DW-related magnetomechanical (microeddy current) IF in ferromagnets 23 . This interpretation of FM domain wall related IF is not directly applicable to pure AFM structures. In helical AFM Dy, however, DWs perpendicular to the c-axis possess a net magnetic moment and their motion can be the origin of DW-related magnetic IF. Therefore, we consider DWs as relevant structure contributing to the IF at ~10 5 Hz.
The IF under continuous cooling/heating contains two kinetic processes. The first is a sluggish isothermal relaxation with logarithmic kinetics, and the second is a fast  T -dependent component. The "sluggish" isothermal relaxation with logarithmic kinetics contributes to the memory effect. It involves ageing of a glassy system, which moves it towards a global energy minimum and reflects the internal restructuring of the magnetic microstructure (DW configurations) at T > 140 K. In contrast, our fast relaxation IF component scales linearly with  T over a www.nature.com/scientificreports www.nature.com/scientificreports/ wide temperature range and is typical in the behaviour of transient IF, trans δ . δ trans accompanies macro-and microplastic deformation of crystals [31][32][33] and first order structural transitions [34][35][36] . Non-magnetic trans δ is approximately inversely proportional to the oscillation frequency ω 31-36 : (2) trans n with < n 1. The nearly inverse frequency dependence makes non-magnetic δ trans detectable only at low frequencies (<~10 2 Hz) and negligible at ultrasonic frequencies 31,36 .
We now discuss possible structural origins of δ trans in polycrystalline Dy. Firstly, the helical AFM state shows negative thermal expansion along the hexagonal axis 37,38 and positive thermal expansion in the basal plane 37 . Temperature changes are then expected to provoke significant variations of exchange energies and change the pitch of the helical structure 39 . Variations of pitch result in the rotation of the magnetic moments at DWs during temperature changes. The rotation of magnetic moments affects the energy of their dipolar interactions and provokes a continuous rearrangement of the DW structure over a wide temperature range. Secondly, the strong anisotropy of thermal expansion may result in intense thermal stresses in polycrystalline Dy, also affecting DW configurations over the range centred around the Villari point at ca. 166 K. Experiments with single crystals should allow one of the two scenarios to be chosen. Since the motion of DWs in AFM Dy is accompanied by hysteresis 21,40 , the combination of translational and hysteretic oscillatory motion of DWs with magnetic moments may induce eddy currents with δ ω trans via the Faraday law. This proportionality cancels out the conventional inverse frequency dependence of δ trans , Eq. (2), and thus makes the magnetic transient term detectable at ultrasonic frequencies as a fast IF component, proportional to  T . The magnetic transitory IF term is expected to be nearly frequency-independent up to a frequency of microeddy current relaxation and fall off rapidly above this frequency limit. For ferromagnets, the frequency of microeddy current relaxation ranges from ca. 10 5 Hz 23 to several MHz 41 . Very low IF levels in the AFM Dy do not permit verification of the IF dT dt / dependence at very low frequencies due to poor experimental resolution 30,42,43 . Figure 2(c) and the inset show that the "fast" IF component nearly disappears below approximately 140 K with blocking of the translational motion of DWs. We suggest that the origin of this blocking is the emergence of ferromagnetically ordered nuclei. The following observations support this hypothesis.
• The IF level increases abruptly on cooling below ca. 140 K, Fig. 1(c).  Fig. 2(c). t stop and t min in (a,b) are the same as in Fig. 2(a) for the temperature of 166 K, and Fig. 2(c) for the temperature of 140 K.
• The YM started to soften, Fig. 1(c), as precursor to the ferromagnetic phase 44 .
• The IF is promoted by weak magnetic fields below 140 K (not shown). This effect is consistent with the emerging net magnetization provoking macroeddy current damping.
Ferromagnetic nuclei interact strongly with antiferromagnetic DWs and block efficiently their translational motion while this leaves the possibility of local isothermal rearrangements (represented by slow δ ). The motion of DWs and the two kinetic processes are confirmed by two-stage recovery of the steady state IF after resumed cooling. Just after the restart of cooling, the IF increases gradually until the IF level reaches the same value as before the slow relaxation (inset in Fig. 2(b)). During this stage, DWs recover unrelaxed "de-pinned" states without notable large-scale movement and no T  dependency before the IF level reaches the "de-pinned" level. Once the "de-pinned" state is reached, the IF increases abruptly towards the steady state value due to the fast recuperation of the transient IF term, i.e. movements of DWs. Their fast initiation results in jerky behaviour under constant  T and stems from the undercooling required to "de-pin" DWs before they start to move. In order to additionally confirm the existence and hierarchy of the two steps in the recovery of the steady-state DW dynamics, experiments with variable time of isothermal dwelling were performed. Figure 4 shows the temperature and dT dt / versus time (a), the absolute value of dT dt / and the IF versus time (b), and IF versus dT dt / (c) www.nature.com/scientificreports www.nature.com/scientificreports/ during several interruptions of cooling at 166 K. Isothermal segments from 1 to 15 min were pre-set. The shortest isothermal segment (1 min) results in a brief decrease of the cooling rate: the system does not reach the pre-set isothermal dwelling. Under these circumstances, the IF changes reversibly and is essentially proportional to T  , Fig. 4(b). The increase of the duration of the isothermal segment provokes a time-dependent decrease of the IF level (data in Fig. 4(b)) and the emergence of the IF hysteresis versus dT dt / , Fig. 4(c). For longer dwelling times, the separation of the IF into the slow and fast components becomes even more evident, Fig. 4(b). During renewed cooling the "fast" IF increases rapidly (in a jerky manner with a small overshoot, Fig. 4(b)) to the steady state level after undercooling by ca. 0.5 K. This tendency confirms that the occurrence of the sluggish relaxation progressively impedes transient IF, associated with the motion of DWs.
Finally, the concept of low-frequency DW thermal fluctuations in Dy 19 is revisited. DW speckle variations were reported over a narrow temperature range 179.4-177.5 K 19 , reflecting low-frequency thermal DW fluctuations. Critical spin fluctuations were discarded as a possible origin of these fluctuations, due to a strong difference in the characteristic frequencies of critical fluctuations and time/frequency window of experimental observations 19 . The relaxation time of the spin system diverges approaching the critical point. The critical slowdown is represented by the critical ultrasonic attenuation peak at T N 25 . The temperature range of the critical attenuation (ca. 175-180 K, Fig. 1(a) and ref. 25 ). coincides with the range (179.4-177.5 K) where long-term variations of the speckle patterns were reported in 19 . Fluctuations may induce Brownian motion of DWs, which can be detected on a long time scale. Thus, low-frequency fluctuations of DWs in Dy close and just below the T N are related to critical fluctuations. Moreover, the absence of DW speckle variations outside the range of critical fluctuations does not mean that the DWs are frozen some 10 K below T N

19
. Our observations of magnetic transient IF well below T N (to approx. 150 K) and of the IF relaxation at lower temperatures indicate DW motion down to ca. 140 K.
Our observations of the slow DW dynamics in AFM Dy are consistent with an overall picture in various ferroics and multiferroics. Slow kinetics movements have been observed in antiferroelectric materials 45 and wall meandering and slow relaxations at high temperatures are dominant in LiNbO 3 where kinks in walls occur at very high temperatures 46 . When walls intersect in LaAlO 3 , they form tweed structures which also remain (meta-)stable unless heated above the ferroelastic transition point. This tweed is locally dipolar and the polarity persists again if the sample has not been heated 47 . Tweed structures are just one example of domain glass states 10,11 , which are quasi stable with relaxations towards the uniform equilibrium state being so slow that no macroscopic relaxation has yet been observed experimentally. Finally, extremely slow relaxations are predicted to occur in all jammed ferroelastic materials, such as SrTiO 3 where inter-boundary relaxations and jamming prevent fast kinetic processes towards equilibrium and where sluggish relaxations dominate at sufficiently low temperatures 48 .
As for the second, "fast" IF component or transient ultrasonic IF of magnetic origin, we predict its existence in ferromagnets and multiferroics under different experimental conditions, implying rearrangement of the magnetic DW structure. In particular, magnetic transient ultrasonic IF should exist, during temperature variations, in such multiferroics as ferromagnetic martensites. The microplastic straining of anisotropic martensitic variants through the displacement of twin boundaries 32 provokes the "mechanical" transitory IF which is negligible at ultrasonic frequencies 31,36 , Eq. (2). Rearrangement of magnetic DWs associated with twin boundaries motion under thermal stresses together with their oscillatory motion are expected to be mediated by magnetoelastic coupling and can lead to the transitory IF of magnetic origin. Another possible scenario of observations of magnetic transitory ultrasonic IF is the formation of tweed structure in ferromagnets, like premartensitic transition in Ni 2 MnGa 49,50 . Experimental studies of these predicted phenomena are ongoing.

Summary and Conclusions
• We observe in antiferromagnetic Dy a new category of anelastic phenomena -ultrasonic transitory internal friction, related to the fast rearrangement of domain wall structure -and predict its existence in other ferroics and multiferroics under different conditions. In polycrystalline Dy the maximum intensity of the transitory term is centered around Villari point at ca. 166 K. • The fast rearrangement of the domain wall structure is followed by the isothermal relaxation with logarithmic kinetics, which demonstrates features typical in glassy systems: temperature chaos and memory effects. • Low-frequency thermal fluctuations of DWs, previously detected by X-ray photon correlation spectroscopy close to the Néel temperature, are related to critical fluctuations with Brownian motion-like dynamics of DWs.