Athermal domain wall creep near a ferroelectric quantum critical point ARTICLE Received 21 Nov 2015 | Accepted 11 Jan 2016 | Published 16 Feb 2016 Athermal domain wall creep near a ferroelectric quant[.]
ARTICLE Received 21 Nov 2015 | Accepted 11 Jan 2016 | Published 16 Feb 2016 DOI: 10.1038/ncomms10675 OPEN Athermal domain-wall creep near a ferroelectric quantum critical point Fumitaka Kagawa1, Nao Minami2, Sachio Horiuchi3 & Yoshinori Tokura1,2 Ferroelectric domain walls are typically stationary because of the presence of a pinning potential Nevertheless, thermally activated, irreversible creep motion can occur under a moderate electric field, thereby underlying rewritable and non-volatile memory applications Conversely, as the temperature decreases, the occurrence of creep motion becomes less likely and eventually impossible under realistic electric-field magnitudes Here we show that such frozen ferroelectric domain walls recover their mobility under the influence of quantum fluctuations Nonlinear permittivity and polarization-retention measurements of an organic charge-transfer complex reveal that ferroelectric domain-wall creep occurs via an athermal process when the system is tuned close to a pressure-driven ferroelectric quantum critical point Despite the heavy masses of material building blocks such as molecules, the estimated effective mass of the domain wall is comparable to the proton mass, indicating the realization of a ferroelectric domain wall with a quantum-particle nature near the quantum critical point RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, 1-1-1 Higashi, Tsukuba 305-8565, Japan Correspondence and requests for materials should be addressed to F.K (email: fumitaka.kagawa@riken.jp) NATURE COMMUNICATIONS | 7:10675 | DOI: 10.1038/ncomms10675 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10675 T he nonequilibrium creep dynamics of elastic objects under small external forces is often described in reference to a phenomenological, multidimensional potential landscape with multiple minima Typical examples include isolated/latticeformed flux lines in superconductors under electric currents1, charge-/spin-density waves under electric currents2,3 and ferroelectric/ferromagnetic domain walls under electric/ magnetic fields4–16 From a phenomenological perspective, when a moderate external force is applied, the system begins to travel through this multidimensional potential landscape by successively overcoming the potential barriers separating distinct metastable states, resulting in creep motion in real space This creep motion is induced by thermal activation, or thermal fluctuations An experimentally elusive question is how the creep kinetics would be affected when quantum fluctuations develop while thermal fluctuations are suppressed to the greatest possible extent In this work, to address this question, we targeted a recently discovered organic ferroelectric charge-transfer complex, TTFQBr2I2 (ref 17), where TTF and QBr2I2 denote tetrathiafulvalene and symmetrically diiodo-substituted quinone, respectively A combination of nonlinear permittivity and polarization-retention measurements reveals that the domain-wall creep motion manifests athermal, quantum-mechanical behaviour as a result of quantum fluctuations developing towards a ferroelectric quantum critical point (QCP) The estimated effective mass of the ferroelectric domain wall is found to be much lighter than the mass of the material building blocks, which are molecules in this case, and this situation is reminiscent of the electrically mobile bond solitons in conducting polymers, such as polyacetylene a Results Phase diagram TTF-QBr2I2 consists of alternating stacking of TTF and QBr2I2 with a uniform spacing, and it is paraelectric at ambient pressure down to the lowest temperature (Fig 1a,b) Nevertheless, there exists a second-order ferroelectric transition accompanied by a QCP at a moderate pressure17, pcE0.25–0.26 GPa (Fig 1c), and above that pressure, the dimerization of TTF and QBr2I2 occurs and results in a ferroelectric polarization oriented approximately along the stacking direction (Fig 1b) TTF-QBr2I2 thus provides an ideal platform to explore the kinetics of ferroelectric domain walls under systematically controlled quantum fluctuations In fact, when the pressure approaches pc from above, the ferroelectric transition temperature Tc continuously decreases to zero as TcB(p pc)0.5 (Fig 1c), consistently with previously reported results17; moreover, the spontaneous polarization Ps at the lowest temperature decreases as PsB(p pc)0.5 (Fig 1d) The squareroot decreases in Tc and Ps are consistent with theoretical predictions that consider quantum fluctuation effects18,19, indicating that the quantum fluctuations evolve towards the pressure-driven ferroelectric QCP and are, therefore, tunable Nonlinear permittivity arising from domain-wall creep In pursuing the nontrivial domain-wall dynamics arising from quantum fluctuations, we first focus on the domain-wall creep under an a.c electric field, Eac, using permittivity measurements as a sensitive probe (for the schematics of the domain-wall structure, see Supplementary Fig 1) Figure 2a displays the profile measured at permittivity-temperature (e0 T) 0.34 GPa, which is relatively far from the ferroelectric QCP c QBr2I2 TTF 100 I Br S S Br I O b Paraelectric 0.34 GPa 0.31 400 ′ S Temperature (K) S 0.26 GPa 0.28 600 O 50 200 0 20 40 60 Temperature (K) 80 Ferroelectric d Ps Ps 2.5 Ps 2.0 –2 Ferroelectric P (μC cm ) QBr2I2 Spontaneous polarization (μC cm–2) TTF 1.5 Ec 0.34 GPa 0.31 GPa 0.28 GPa 0.26 GPa –1 1.0 –2 0.5 0.0 –40 –20 20 –1 Electric field (kV cm ) 0.1 40 0.2 0.3 Pressure (GPa) 0.4 Figure | Quantum fluctuations developing near the ferroelectric quantum critical point in TTF-QBr2I2 (a) The chemical forms of the TTF and QBr2I2 molecules TTF and QBr2I2 denote tetrathiafulvalene and symmetrically diiodo-substituted quinone, respectively (b) Schematic diagrams of the structural changes occurring upon the ferroelectric transition In the ferroelectric phase, the TTF and QBr2I2 molecules are dimerized, and the small arrows indicate the shifts in the molecules that occur upon dimerization Two dimerization patterns are possible, and they are degenerate in energy The red arrows indicate the direction of the spontaneous electric polarizations, Ps (c) The pressure–temperature phase diagram The inset shows the temperature dependence of the permittivity (measured at 100 kHz) at the pressures investigated in this study The ferroelectric transition temperatures can be determined from the permittivity peaks (d) The pressure dependence of the spontaneous polarization at the lowest temperature The inset shows the pressure variations in the polarization hysteresis loop measured at the lowest temperature: T/Tc ¼ 0.1 for 0.28; 0.31 and 0.34 GPa; and T/Tc ¼ 0.3 for 0.26 GPa NATURE COMMUNICATIONS | 7:10675 | DOI: 10.1038/ncomms10675 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10675 a which is near pc and hence yields a low Tc (E13 K), nonlinear permittivity is observed in a broad T/Tc range, and its magnitude depends only weakly on temperature (Fig 2b), implying that the domain-wall creep at 0.26 GPa is an athermal process The possible crossover from thermal to athermal domain-wall creep is further supported by the temperature dependence of the coercive electric field, Ec, in the polarization hysteresis loop (Fig 2c); although at 0.34 GPa, Ec increases towards lower temperatures, it exhibits less significant variation and eventually becomes nearly temperature-invariant as the pressure approaches pc (E0.25–0.26 GPa) Because the polarization reversal process is generally accompanied by domain-wall creep, the pressure variation of the Ec-T profile again suggests that although the domain-wall creep at 0.34 GPa is facilitated by thermal fluctuations, this is not the case at pressures near the ferroelectric QCP 1,500 P = 0.34 GPa (Tc = 64 K) 1,000 ′ Eac = kV cm−1 500 Eac = 0.13 kV cm−1 b 1,500 P = 0.26 GPa (Tc = 13 K) 1,000 ′ Eac = kV cm−1 Eac = 0.13 kV cm−1 500 0.0 c 0.5 T/Tc 1.0 Coercive field (kV cm–1) 20 0.34 GPa (Tc = 64K) 0.31 GPa (Tc = 52K) 0.28 GPa (Tc = 30K) 0.26 GPa (Tc = 13K) 15 10 0.0 0.2 0.4 0.6 0.8 1.0 T/ Tc Figure | Probing the ferroelectric domain-wall creep via dielectric properties (a,b) The permittivity-temperature profiles measured under a.c electric fields of different magnitudes: at 0.34 GPa (TcE64 K) (a) and at 0.26 GPa (TcE13 K) (b), the latter of which is close to the ferroelectric QCP (c) The temperature dependence of the coercive electric field at different pressures In a–c the temperatures are represented in the form of reduced temperatures, T/Tc (E0.25–0.26 GPa) and hence, yields a high Tc of B64 K (regarding possible ambiguity of Tc, see Supplementary Note 1) We note that near Tc, the e0 measured at EacE1 kV cm–1 exhibits larger values than that measured at EacE0.13 kV cm–1; that is, the permittivity exhibits highly nonlinear characteristics Given that domain-wall creep is innately a nonlinear response to an external electric field4,6,7,9, the origin of this nonlinear permittivity is attributable to the domain-wall creep8 occurring under the application of a field of EacE1 kV cm–1 (the same conclusion can also be drawn from the frequency dependence of the permittivity; see Supplementary Fig 2) For T/Tco0.6, the nonlinear permittivity becomes vanishingly small, thus highlighting the vital role of thermal fluctuations in the domain-wall creep at 0.34 GPa By contrast, when the pressure is decreased to 0.26 GPa, Polarization decay Athermal domain-wall creep invokes successive quantum tunnelling between adjacent minima in the multidimensional potential landscape (Fig 4, inset) In general, quantum tunnelling within an isolated double-well-potential system has been empirically verified by observing the crossover behaviour of the relaxation-type dynamics, in response to the development of quantum fluctuations, from the Arrhenius-type (classical) regime to the temperature-invariant (quantum) regime20–22 However, because creep dynamics are inherently a form of kinetics that is distinct from relaxation behaviour23,24, it is not obvious how to define the relaxation time associated with creep dynamics In this study, we focus on a relaxation phenomenon originating from the creep motions of ferroelectric domain walls, that is, the decay of the net macroscopic polarization25,26 In realistic ferroelectrics, the remnant polarization after polarization switching may decay with time as a result of the nucleation of domains with the opposite polarization and their subsequent growth via domain-wall creep; in fact, such polarization decay towards a multidomain state has been observed in TTF-QCl4 (or TTF-CA), which is composed of stacking of TTF and QCl4 similarly to the case of TTF-QBr2I2 (ref 27) As will be discussed below (Fig 3d and Supplementary Fig 3), we find that the polarization decay in TTF-QBr2I2 can, in fact, be described within the framework of relaxation behaviour, allowing us to extract the relaxation time associated with the creep dynamics and, thus, discuss the classical-to-quantum crossover We measured two successive polarization hysteresis loops with a delay time of td, following the electric-field protocol illustrated in Fig 3a (for details, see the Methods section) and found that the remnant net polarization, Pr, appreciably decayed with time (Fig 3b) The observed td dependences of Pr at various values of T/Tc under 0.34 GPa are summarized in Fig 3c As expected from the thermal domain-wall creep, dramatic prolongation of the polarization retention was observed towards low temperatures, and at T/Tc ¼ 0.1, no polarization decay was discernible on the considered timescale (1 10–4–3 101 s) In extracting the relaxation time t from the decay profile, we exploited the standard relaxation equation: n o 1ị Pr td ị ẳ P1 ỵ P0 P1 ịexp td =tịb ; where P0 and PN denote the values of Pr in the limits of td-0 (indicated in Fig 3b) and td-N, respectively, and b is the phenomenological stretching parameter that represents the distribution of t We assumed that the value of b does not depend on temperature to avoid fitting ambiguity because of the limited td range of the data set (for the pressure dependence of b, see Supplementary Fig 4); in fact, when {Pr(td) PN}/(P0 PN) is plotted as a function of the normalized time, td/t, for each T/Tc, the NATURE COMMUNICATIONS | 7:10675 | DOI: 10.1038/ncomms10675 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10675 c 2.0 Time Measurement Preset 0.3 1.0 0.5 0.5 0.0 Pr 10 s –1.0 0.2 T/Tc= 0.9 0.0 10–7 f 1.0 T/Tc = 0.3 T/Tc = 0.5 0.5 0.0 T/Tc = 0.9 T/Tc = 0.5 P = 0.34 GPa –10 10–5 10 20 –0.5 10–6 10–3 100 td –1 Electric field (kV cm ) 103 10–4 10–1 10–2 Delay time td (s) 10–5 1.5 P = 0.26 GPa (Tc = 13 K) {Pr(t d) – P∞}/(P0–P∞) {Pr(t d) – P∞}/(P0–P∞) 10 ms 0.0 –2.0 –20 T/Tc = 0.3 T/Tc = 0.5 P = 0.34 GPa (Tc = 64 K) td = 0.1 ms 1.0 10–4 10–1 10–2 Delay time td (s) 1.5 P0 ・・・・・・ Electric polarization (μC cm–2) d 2.0 P = 0.26 GPa (Tc = 13 K) T/Tc= 0.9 10–7 b 0.4 T/Tc = 0.1 1.5 Pr (μC cm–2) Electric field Delay time td e P = 0.34 GPa (Tc = 64 K) Pr (μC cm–2) a 106 1.0 T/Tc = 0.3 T/Tc = 0.5 T/Tc = 0.9 0.5 0.0 –0.5 10–6 10–3 100 td τ(T/Tc) 103 106 τ(T/Tc) Figure | Polarization-decay properties under the influence of thermal and quantum fluctuations (a) The measurement protocol for probing the polarization decay (b) Typical polarization hysteresis loops with various delay times, measured at 0.34 GPa and T/Tc ¼ 0.5 (c) The delay-time dependences of the remnant polarization at select temperatures and 0.34 GPa (TcE64 K) (d) The normalized relaxation behaviour of the polarization decay at 0.34 GPa (e) The delay-time dependence of the remnant polarization at 0.26 GPa (TcE13 K), which is near the ferroelectric QCP (f) The normalized relaxation behaviour of the polarization decay at 0.26 GPa The lines in c–f represent fits to the standard relaxation equation; see equation Classical-to-quantum crossover In Fig 4, three key observations can be highlighted: first, whereas the t 1/T profile exhibits Arrhenius-type behaviour at 0.34 and 0.31 GPa, it is temperatureinvariant within the experimental accuracy at 0.26 GPa; second, at an intermediate pressure of 0.28 GPa, the crossover behaviour from the Arrhenius regime to the temperature-invariant regime is evident; and third, the saturated t value in the temperature-invariant regime, tquantum, is smaller at 0.26 GPa (Epc) than at 0.28 GPa The thermal-to-athermal crossover of t towards the QCP is consistent with the empirical evidence used to validate quantum tunnelling20–22, thus leading us to conclude that near the ferroelectric QCP, the domain-wall creep is facilitated exclusively by successive quantum tunnelling rather than by thermal activation Discussion Finally, to quantitatively evaluate the validity of this quantum-mechanical domain-wall creep, we estimate the effective mass of the ferroelectric domain wall, meff, using 0.34 GPa 105 0.31 GPa 0.28 GPa Relaxation time (s) polarization-decay profiles yield a universal relaxation curve (Fig 3d) The fitting using equation is also successful for the data at 0.31 and 0.28 GPa (Supplementary Fig 3), thereby validating the application of the relaxation equation to the polarization-decay properties The results at 0.26 GPa (Fig 3e) were also analysed (Fig 3f), but in this case, the relaxation process is nearly independent of temperature, corroborating the athermal nature of the domain-wall creep at 0.26 GPa In this way, the t 1/T profiles were derived at various pressure values, that is, at varying proximity to the QCP (at 0.25–0.26 GPa), as shown in Fig 102 0.26 GPa 10–1 10–4 10–7 0.1 0.2 1/T (K–1) 0.3 Figure | Thermal–athermal crossover of the domain-wall dynamics The lines represent fits to Matthiessens rule: 1/t ẳ 1/tclassical ỵ 1/tquantum, where tquantum is constant at a given pressure and tclassical follows the Arrhenius law The appreciable deviation of the simplest Matthiessen’s rule from the experimental data at 0.28 GPa is partly attributable to the pressure inhomogeneity that arises from the pressure medium and the sample imperfection The error bars represent the numerical ambiguity of t obtained from the fitting results in Fig The inset is a schematic diagram comparing the classical thermally activated creep (solid arrows) with the quantum athermal creep (broken arrows) of a domain wall (modelled as a particle) The multi-valley schematic represents a multidimensional potential landscape the simplest Wentzel–Kramers–Brillouin approximation21,22: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tquantum ¼ t0 exp 2w 2meff D=‘ ; ð2Þ NATURE COMMUNICATIONS | 7:10675 | DOI: 10.1038/ncomms10675 | www.nature.com/naturecommunications ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms10675 where w is the domain-wall tunnelling distance (where we adopt a value ofE7.29 Å (6.5 K, 0.2 GPa) as the unit-cell length along the stacking direction), 2pt0 is the inverse of the attempt frequency, and D represents the activation barrier relevant to the Arrhenius behaviour The values of the three quantities tquantum, t0, and D are available from the data at 0.28 GPa (for the details of the analysis, see Supplementary Note 2); from these values, the order of magnitude of meff at 0.28 GPa is estimated to be E6–8 102 me, where me denotes the electron mass Given that the proton, which is anticipated to be capable of quantum tunnelling in solids28,29, has a mass of E1.8 103 me, the estimated meff is reasonably small Moreover, meff is much smaller than the masses of the TTF and QBr2I2 molecules (E3.7 105 and E9.4 105 me, respectively), although the domain-wall creep entails the displacement of TTF and QBr2I2 molecules (Fig 1b; see also Supplementary Fig 1) A similar situation can also be observed in polyacetylene, in which the mass of the building blocks (carbon atoms) is E2.2 104 me, whereas the meff of the bond soliton (or, equivalently, a misfit in the C ¼ C paring) is much lighter, onlyE10 me (ref 30) The small meff of the bond soliton in this material has been rationalized in terms of a large soliton width of E14 unit cells31, and it can therefore be expected that in TTF-QBr2I2, the domain-wall width should further broaden as quantum fluctuations develop; in fact, although the amount of quenched randomness is understood to be preserved upon a change in pressure, D decreases towards the ferroelectric QCP (Fig and Supplementary Fig 5), suggesting an increase in the domain-wall width We expect that such a broadened domain-wall width underlies the unexpectedly small effective mass and, thus, leads to the quantum-particle nature of the domain wall The direct detection of the domain-wall width and domain structure32 is currently not possible, and such an effort would be challenging because the experiments would need to be performed under pressure; nevertheless, our observations establish that even a ferroelectric domain wall composed of heavy building blocks such as molecules can acquire a quantum-particle nature when a system is located near a ferroelectric QCP Recently, the development of permittivity towards a ferroelectric QCP has been drawing renewed attention and being extensively studied over a wide region of the paraelectric phase33,34 In the present study, by contrast, we establish that domain-wall motions, which normally become frozen and thus negligible at low temperatures, can affect the macroscopic physical properties in a ferroelectric phase even at the lowest temperature when quantum fluctuations are developing Methods Sample preparation The purchased TTF was purified via repeated recrystallization and vacuum sublimation The QBr2I2 was synthesized and purified according to methods described in the literature17 Slow evaporation of a mixed acetonitrile solution of equimolar amounts of TTF and QBr2I2 under a stream of argon gas afforded 1:1 neutral TTF-QBr2I2 complex crystals as the minor product Elongated plates of TTF-QBr2I2 were selectively grown at °C in a second evaporation after seeding with a few crystals The single crystals obtained have a typical dimension of 0.3 0.3 1.5 mm3 Permittivity and polarization measurements under pressure A clamp-type dual-structure CuBe-NiCrAl cylinder cell was used for the pressure measurements Daphne 7373 oil was used as the pressure-transmitting medium Pressure inhomogeneity is likely present as inferred from the pressure dependence of b (Supplementary Fig 4) The permittivity was measured along the b axis (the stacking direction) at kHz, unless otherwise specified, by using an LCR meter (Agilent Technologies, E4980A) To remove the background contributions from the parasitic capacitance (E0.1 pF), which is approximately one-tenth of the sample capacitance at the lowest temperature at 0.34 GPa (Fig 2a), the open compensation was performed before the permittivity measurements on the sample The polarization hysteresis loops (Fig 1d, inset, and Fig 3b) were measured at 10 kHz by using a commercial ferroelectric tester (Radiant Technologies, Precision Premier II) The measurements were performed for three single crystals, and we confirmed the reproducibility Polarization-retention measurements The polarization-retention properties were measured according to the electric-field protocol illustrated in Fig 3a; during the preset process, a triangular electric pulse larger than the coercive electric field was applied to prepare a nearly monodomain ferroelectric state, and then, after a delay time td, a measurement pulse of the same magnitude was applied to probe to what extent the sample remained polarized as a function of td References Feigel’man, M V., 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Kobayashi for providing us with the lattice parameter under pressure at low temperature This work was partially supported by CREST, JST and JSPS KAKENHI (Grant nos 25220709 and 15H05459) Author contributions N.M and F.K performed all experiments N.M and F.K analysed the data S.H grew the single crystals used for the study Y.T and F.K planned and supervised the project F.K wrote the manuscript All authors discussed the results and commented on the manuscript Additional information Supplementary Information accompanies this paper at http://www.nature.com/ naturecommunications Competing financial interests: The authors declare no competing financial interests Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ How to cite this article: Kagawa, F et al Athermal domain-wall creep near a ferroelectric quantum critical point Nat Commun 7:10675 doi: 10.1038/ncomms10675 (2016) This work is licensed under a Creative Commons Attribution 4.0 International License The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ NATURE COMMUNICATIONS | 7:10675 | DOI: 10.1038/ncomms10675 | www.nature.com/naturecommunications ... motion manifests athermal, quantum- mechanical behaviour as a result of quantum fluctuations developing towards a ferroelectric quantum critical point (QCP) The estimated effective mass of the ferroelectric. .. of thermal fluctuations in the domain- wall creep at 0.34 GPa By contrast, when the pressure is decreased to 0.26 GPa, Polarization decay Athermal domain- wall creep invokes successive quantum tunnelling... activated creep (solid arrows) with the quantum athermal creep (broken arrows) of a domain wall (modelled as a particle) The multi-valley schematic represents a multidimensional potential landscape