Order-parameter coupling in the improper ferroelectric lawsonite

This article has been downloaded from IOPscience. Please scroll down to see the full text article.

2012 J. Phys.: Condens. Matter 24 255901

(http://iopscience.iop.org/0953-8984/24/25/255901)

Download details:IP Address: 131.111.164.128The article was downloaded on 08/02/2013 at 18:16

Please note that terms and conditions apply.

View the table of contents for this issue, or go to the journal homepage for more

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 24 (2012) 255901 (6pp) doi:10.1088/0953-8984/24/25/255901

Order-parameter coupling in theimproper ferroelectric lawsonite

E K H Salje1,2, K Gofryk3, D J Safarik3 and J C Lashley3

1 Department of Earth Sciences, Cambridge University, Cambridge CB2 3EQ, UK2 Center for Nonlinear Science, Los Alamos National Laboratory, Los Alamos, NM 87545, USA3 Physical Metallurgy Group MST-6, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

E-mail: ekhard@esc.cam.ac.uk

Received 19 March 2012, in final form 28 April 2012Published 25 May 2012Online at stacks.iop.org/JPhysCM/24/255901

AbstractLow-temperature specific heat and thermal expansion measurements are used to study thehydrogen-based ferroelectric lawsonite over the temperature range 1.8 K ≤ T ≤ 300 K. Thesecond-order phase transition near 125 K is detected in the experiments, and thelow-temperature phase is determined to be improper ferroelectric and co-elastic. In theferroelectric phase T ≤ 125 K, the spontaneous polarization Ps is proportional to (1) thevolume strain es, and (2) the excess entropy �Se. These proportionalities confirm the impropercharacter of the ferroelectric phase transition. We develop a structural model that allows theoff-centering of hydrogen positions to generate the spontaneous polarization. In thelow-temperature limit we detect a Schottky anomaly (two-level system) with an energy gap of� ∼ 0.5 meV.

1. Introduction

Improper ferroelectricity has important ramifications forapplications because it eliminates Schottky barrier(s) forcharge injection and paraelectric dead layers. Lawsonite,CaAl2Si2O7(OH)2H2O, has been reported to show aferroelectric hysteresis below 125 K with an estimatedspontaneous polarization of 3 µC cm−2 at 40 K. In light ofsparse data, Salje et al predicted that the Pmcm–P21cn phasetransition leads to the formation of an improper ferroelectricstate [1]. There are no symmetry constraints preventingthe spontaneous polarization from being the driving orderparameter, and on the basis of this alone, one could conjecturethat the transition is proper or pseudo-proper ferroelectric [1].Other improper ferroelectrics, such as perovskites, have beenused to generate ferroelectricity [2]. This result demonstratesthat improper ferroelectric behavior is not always controlledby the highest possible symmetry of the order parameter [2].

Ferroelectric polarization in lawsonite is related tohydrogen hopping. Dynamical measurements have shownthat hopping persists to low temperatures with an activationenergy of 0.03 eV [3]. It has been shown that globalorder/disorder behavior can be excluded as the mechanism

because the excess entropy of the ferroelectric phase transitionis much smaller than the order/disorder limit, with an excessentropy of Se = 3.75 J K−1 mol−1 [4–7]. Lawsonite isa natural mineral [8–17]. Since the pioneering structuralwork by Libowitzky and Armbruster [11], and subsequentspectroscopic work [9], the original work on lawsonite waspublished in the mineralogical literature where ferroelectricand antiferroelectric minerals are rare [18, 19].

We report thermodynamic data for lawsonite in theferroelectric phase. Our results allow comparisons betweenthe temperature evolution of the spontaneous polarizationPs, and (1) the excess entropy �Se, and (2) the excessvolume strain es. These observations show that (i) thespontaneous polarization is not the driving order parameter ofthe ferroelectric phase transition, (ii) the polarization dependson the square of the driving order parameter Q, and (iii) belowT � 2 K a Schottky anomaly (two-level system) is present inthe specific heat data with an energy gap of � = 0.5 meV.

The paper is organized by presenting a descriptionof experimental techniques followed by a discussion andresults for polarization, specific heat, and thermal expansionmeasurements. A description of the ferroelectric transition isgiven using a Landau potential.

10953-8984/12/255901+06$33.00 c� 2012 IOP Publishing Ltd Printed in the UK & the USA

J. Phys.: Condens. Matter 24 (2012) 255901 E K H Salje et al

2. The experiment

The polarization data used in the analysis in the subsequentsections of this paper are taken from [1]. All experiments wereperformed on natural samples of polycrystalline lawsonite.The specific heat was measured by a thermal-relaxationtechnique on a physical properties measurement systemplatform (PPMS) from Quantum Design. The holder wasattached to the platform with a thin layer of Apiezon Ngrease and the samples were short-circuited by applying a thinlayer of gold. Measurements were performed over the interval1.8 K ≤ T ≤ 300 K. The accuracy of the PPMS specificheat data is determined here by comparing data measuredfor copper (for conducting samples) and synthetic sapphiresamples (for insulating samples) with standard literaturevalues. The system exhibits an overall accuracy of betterthan 1% for temperatures between 100 and 300 K, while theaccuracy diminishes to 3% at lower temperatures.

We have modified the temperature scale (and in somecases the platform itself) using fixed points and a calibrationof the Cernox (from Lakeshore Cryotronics) thermometersagainst a RhFe thermometer calibrated in the UnitedKingdom at the National Physical Laboratory against the3He and 4He vapor-pressure scales. A full review of thistechnique and its application to the PPMS platform is givenin [20, 21]. The relaxation curves fitted within the presentversion of the Quantum Design software (version 3.8) arebased on an analysis given in [20]. The first examplesof the thermal-relaxation technique can be found in [22].Thermal-relaxation calorimetry is designed to make absolutemeasurements of the heat capacity on small samples. Onelimitation is that thermal broadening can occur near Tc. Thisis a standard problem owing to the �T of the heat pulse, inthis case 1% of the absolute temperature. These issues arediscussed at length in [20].

The coefficient of linear thermal expansion (CTE)was measured using a three-terminal capacitive dilatometer.The resolution of this apparatus is determined as 5 pm.This apparatus operates with several kinds of cryogenicarrangements—in vacuum, in helium exchange gas, and whileimmersed in liquid—and can reach temperatures of 30 mKand magnetic fields to 45 T. We record the capacitance withan AH2700A variable frequency capacitance bridge fromAndeen-Hagerling. Measurement frequencies are dictated bythe noise floor and are typically 1–3 kHz. The relativeaccuracy and precision analysis for this arrangement arereviewed by Schmiedeshoff et al [23].

3. Results and discussion

The temperature evolution of the spontaneous polarization Psand the remanent polarization Pr is linear between 40 and115 K [1]. At higher temperatures the transition behavior isthermally broadened. If this smearing of the transition wasrelated to small random conjugated fields, their strength wouldbe very weak, on the order of 0.01 B [1]. Such small fieldsare common and may be associated with structural impurities

Figure 1. Temperature evolution of the normalized length change.

or they could result from coupling to a second structuralinstability as we will discuss below.

Polarization measurements below 40 K showed in-complete hysteresis curves because the applied field wasinsufficient to move all ferroelectric domain boundaries. Arough estimate of the absolute values of Ps is 3 µC cm−2

at 40 K [1]. This value is larger than the estimated valueof Ps from structural data [9, 24]. In this case only thedisplacive component of the bias (static) hydrogen along thecrystallographic a-direction was obtained. Large additionalfield-induced shifts must exist which are superimposed withthermal fluctuations in the a-direction. Not surprisingly, thecrystallographic displacement parameters, U11, along thecrystallographic a-axis are largest in the ferroelectric phasein lawsonite (U11 = 17 (×100) at 20 K [24]). This resultneeds to be viewed in conjunction with domain formationin the ferroelectric phase. Crystals used in previous x-rayand neutron scattering experiments were not poled andtherefore showed a high non-uniformity in the macroscopicpolarization.

The linear temperature dependence of Ps alreadysuggested the improper character of the ferroelectric state [1].Our new thermodynamic measurements also confirm thesecond-order character of the transition. We examine thetemperature evolution of the volume strain evol in figures 1and 2. Symmetry dictates that the volume strain scales asthe square of the order parameter Q, and we find thisproportionality. We also scale the spontaneous polarization asa function of the excess volume strain over the temperaturerange 40 K ≤ T ≤ 125 K as shown in figure 3. We find thatthe two quantities are proportional to each other, apart from ina small interval near the transition point.

Structurally, the transition is not the result of theformation of spontaneous polarization, and is unrelated tothe condensation of a ferroelectric polar soft mode. Rather,the driving force appears to stem from stearic instabilities ofhydroxyl groups and water molecules. The structure containsone water molecule and two hydroxyl groups, yet only therotation of the water molecule seems to contribute to theferroelectric structural distortion [9–11]. This rotation seems

2

J. Phys.: Condens. Matter 24 (2012) 255901 E K H Salje et al

Figure 2. Temperature dependence of the excess volume strain inferroelectric lawsonite.

Figure 3. Proportionality between the volume strain and thespontaneous polarization. This proportionality demonstrates that thespontaneous polarization is not the driving order parameter of theferroelectric phase transition. The dashed line represents a linear fitof the data.

hindered by the close distance between the calcium-bearingdense structural layers at the crystallographic plane. Onlya distance increase between these layers, relative to thelow-temperature extrapolation of the paraelectric phase,allows the water molecule to rotate. A possible scenariois that all hydrogen related groups show rotations as partof the structural relaxation during the phase transition.These rotations and those of the other polyhedra widen thedistance between water molecules along the crystallographica-direction and allow the proton position to tilt away from themirror plane.

The specific heat data were measured over intervals areshown in figures 4–6. The improper ferroelectric transitionat 125 K and the structural transition at 275 K are evident.We describe the fitting procedure from the lowest temperatureto the paraelectric phase 150 K. Below T � 5 K there is apronounced upturn in C/T versus T as shown in figure 7. Thespecific heat data are characteristic of a two-level system inthis temperature range, Cp ∼ T

−2. Below T ≤ 20 K the data

Figure 4. Temperature dependence of the specific heat inlawsonite. The two phase transitions are clearly visible. Thelow-temperature transition near T ∼ 125 K is the ferroelectrictransition investigated in this paper.

Figure 5. Excess specific heat of the ferroelectric phase transition.The extrapolated specific heat has been subtracted from the datashown in the previous figure.

can be expressed by

C(T) =�

n

BnTn + CSchottky(T) (1)

where the coefficients in the harmonic lattice are Bn and n

increases with temperature, and the Schottky expression is

CSchottky (T) = kR�2 exp(−�/T)

T2(1 + exp(−�/T))2 (2)

where k is a concentration of Schottky centers per formulaunit (CaAl2Si2O7(OH)2H2O), R is the universal gas constant,and � is the energy gap separating the two levels. From the fit(shown as the dashed line in figure 7, we obtain k = 0.06 and� = 0.5 meV).

At higher temperatures T ≥ 25 K, the phonon heatcapacity Cph dominates. The phonon contribution to thespecific heat in the vicinity of the 125 K transition is estimated

3

J. Phys.: Condens. Matter 24 (2012) 255901 E K H Salje et al

Figure 6. Dependence of the spontaneous polarization on theexcess entropy. In proper ferroelectric systems this correlation ispurely quadratic, while a large linear regime extends over the largesttemperature interval in lawsonite. The line represents a linear fit ofthe data.

Figure 7. Temperature evolution of the low-temperature specificheat of lawsonite showing a Schottky anomaly near 2 K.

using the expression

Cph(T) = pCDebye(T) + (1 − p)CEinstein(T) (3)

where p is a constant, CDebye(T) is the Debye heat capacity,and CEinstein(T) is the Einstein heat capacity. The resultingexcess specific heat (Ce = Cp(T) − Cph(T)) in the vicinity ofthe improper ferroelectric transition is shown in figure 7.

The excess volumetric strain evol (excess over the para-electric phase) was obtained from dilatometry measurementsdescribed in the experiment section. The CTE α(T) isobtained from our measurements using the equation

α(T) = 1L

dL

dT= 1

L

dL

dT

����cell+sample

− 1L

dL

dT

����cell+Cu

+ αCu(T)

(4)

where L is the sample length, and the thermal expansionof copper αCu is taken from the literature. The secondterm represents measurements with a copper single-crystal

standard installed in the dilatometer, and we call this term thecell effect. The volumetric strain relative to the paraelectricphase is obtained by integration:

�L

L=

�α(T) dT. (5)

The excess volumetric strain is obtained by subtracting thecontribution of the paraelectric phase. The excess volumestrain as a function of temperature is shown in figure 3. Thetemperature dependence of the volume strain is approximatelylinear below 125 K.

3.1. Landau theory of the improper ferroelectric transition

Our results lead to an unusual scenario found in lawsonite thatcan be characterized as follows.

(1) All structural instabilities are related to the rotationaldegrees of freedom of all functional groups and polyhedra inthe structure. On cooling from room temperature, lawsoniteundergoes first a co-elastic Cmcm–Pmcn phase transition,which leads to a spontaneous strain e1 related to the structuralorder parameter Q1 with the expected correlation e1 = Q

21 [5,

25–27].(2) The ferroelectric Pmcn–P21cn transition is also

co-elastic and its active irreducible representation transformsas a polar vector. This means that Ps could be the orderparameter. This scenario can be ruled out by our data,where we find that Ps ≈ �S ≈ Q

211. The inverse electrical

susceptibility �−1, on the other hand, follows the expectedlinear dependence �−1 ≈ |T − Tc|. The Gibbs free energyof the transition reflects all structural instabilities, namely thepolar instability (Ps, �

−1), the dilatation of the lattice (�vol, C),and the structural order parameter related to molecularrotations (QII, χ

−1):

G(QII, Ps, evol, T) = 12 A�s[coth(�s/T) − coth(�s/Tc)]Q2

II

+ 14 BQ

4II + 1

2ε−1P

2s + 1

4 B�P

4s + 1

2 Ce2vol

+ 14 B

��e

4vol + �G(QII, Ps, evol, T). (6)

The last term describes the coupling between the observablesQIIPs, and evol. The irreducible representations of QII andPs are the same, while evol transforms as the identityrepresentation and couples in lowest order as Psevol. Thecoupling part of the Gibbs free energy is

G(QII, Ps, evol) = λ1QIIPs + γ �1Q

2IIPs

2 + λ2Q2IIevol

+ λ3evolP2 + λ�

3e2volP

2s . (7)

(3) The temperature dependence of evol is small, so weput C = constant and B

�� = 0. The role of evol is to enable thegeometrical coupling between QII and Ps. Relaxing

∂G

∂evol= 0, (8)

leads to evol ∼ (Q2II+λ3λ2P

2s )

C+λ�3P2

s. Replacing the volume strain in

G(QII, Ps, evol, T) introduces further coupling terms betweenQII and Ps. The observed proportionality between Ps and

4

J. Phys.: Condens. Matter 24 (2012) 255901 E K H Salje et al

evol can stem from identical transition temperatures in �−1

and C, even when the coupling is weak. The experimentalobservation evol ∼ Q

2II means that λ3λ2P

2s and λ�

3P2s are

both < C.(4) We can simplify the Gibbs free energy if we treat Ps,

evol as relaxation parameters without direct contributions tothe excess entropy (B� = B

�� = 0, λ1 = 0, λ3 = 0). For

∂G

∂evol= 0, (9)

and

∂G

∂Ps= 0, (10)

we find evol ∼ Q2II, Ps ∼ Q

2II, and for strong volume coupling,

evol ∼ Ps. The additional temperature dependence of �−1

requires the explicit consideration of finite values of B�. The

relevant Gibbs free energy can be approximated as

G(QII, Ps, evol, T) = 12

A�s

�coth

��s

T

�− coth

��s

Tc

��Q

2II

+ 14 BQ

4II + 1

2ε−1P

2s + 1

4 B�P

4s

+ λ/1Q

2IIP

2s + λ

/2Q

4IIP

2s . (11)

(5) The physical picture is, in either scenario, thatthe transition is driven by structural rotations which openthe distance between the dense x = 1/2 layers which,in turn, enable the water molecules to swing around thecrystallographic b-axis. The excess entropy of the transitionis given by the molecular movement while the entropy ofthe ferroelectric polarization remains small. The ferroelectricpolarization is mainly determined by the geometrical openingof the inter-plane distances, and we find that Ps = evol.

4. Conclusion

We note that the coupling schemes for the improperferroelectric transition in lawsonite are unusual becauseenergy terms, which are symmetry allowed in lowest order,seem to be small, while higher-order coupling dominates.Our observations confirm that (i) Ps is not the driving orderparameter of the ferroelectric phase transition, and (ii) thepolarization Ps depends on the square of the driving orderparameter Q. The strong geometric coupling between themolecular rotations opening the gap for the rotation of H2Oleads to the following conclusions.

(1) Domain structures of lawsonite are expected tocontain a large number of 180◦ ferroelectric domains butno 90◦ domains. The 180◦ domains may be strongly pinnedbecause the structural order parameter will provide templatesfor the pinning via the structural domains. The strain scalesas the square of the structural order parameter, so kinks(domain walls) in the order parameter become very narrowbreathers in strain. The kink in the order parameter followsthe tanh(x/w) profile where x is the vector perpendicular tothe wall and w is the domain wall width. This template can pinthe ferroelectric 180◦ walls, which are usually equally narrow.A similar scenario was described for chiral domain walls with

bi-quadratic coupling by Conti et al [28]. The strong pinningof the spontaneous polarization seen in [1] may reflect thismechanism.

(2) Domain walls will have vanishing spontaneouspolarization which, in turn, can lead to a collapse of thestructural spacers and hence a reduction of the local strain.In this case novel structural elements, which are compatiblewith the smaller inter-planar distances, may appear. Thisscenario was recently observed in CaTiO3 [29, 30]. BelowT ≤ 5 K, a Schottky-type anomaly is observed with an energygap � ∼ 0.5 meV. The origin of this anomaly is unclear;one possible origin could be attributed to a superposition oftwo hydrogen states [31], or another could be the presenceof paramagnetic centers. The thermodynamic behavior isintriguing because there are few materials that do not followthe transformation mechanism which is symmetry allowed inlowest-order Landau theory. In lawsonite the phase transitionis continuous, so uncertainties due to stepwise transformationscan be excluded. There appears to be a partial compensationbetween dipolar moments and molecular rotations. Althoughthis mechanism is most likely, it is difficult to prove bymeans of macroscopic measurements. A way to address themechanism is to repeat the structure analysis using poledsingle crystals. The functional groups are related to theposition of hydrogen or deuterium, so a determination of theirexact positions could elucidate the origin of the spontaneouspolarization in lawsonite.

Acknowledgments

Part of this work was performed at the University ofCambridge, and the thermodynamic measurements wereperformed and the paper was written in Los Alamos. Thework at Los Alamos was performed under the auspices ofthe United States Department of Energy. JCL is grateful toAngus Lawson (the mineral name lawsonite originates fromhis family) for discussions on the Schottky anomaly.

References

[1] Salje E K H et al 2011 J. Phys.: Condens. Matter 23 222202[2] Fennie C J and Rabe K M 2005 Phys. Rev. B 72 100103[3] Salje E K H and Carpenter M A 2011 J. Phys.: Condens.

Matter 23 112208[4] Meyer H W et al 2000 Eur. J. Mineral. 12 1139[5] Hayward S A et al 2002 Eur. J. Mineral. 14 1145[6] Martin-Olalla J M et al 2001 Eur. J. Mineral. 13 5[7] Carpenter M A et al 2003 Am. Mineral. 88 534[8] Kornev I et al Phys. Rev. Lett. 93 196104[9] Libowitzky E and Armbruster T 1995 Am. Mineral. 80 1277

[10] Beran A and Libowitzky E 1999 NATO Advanced Study

Institute on Microscopic Properties and Processes in

Minerals vol 543 (Dordrecht: Springer) p 493[11] Libowitzky E and Rossman G R 1996 Am. Mineral. 81 1080[12] Schranz W et al 2005 Z. Kristallogr. 220 704[13] Libowitzky E 2009 Eur. J. Mineral. 21 915[14] Daniel I et al 2000 Eur. J. Mineral. 12 721[15] Sondergeld P et al 2000 Phase Transit. 71 189[16] Sondergeld P et al 2000 Phys. Rev. B 62 6143[17] Sondergeld P et al 2001 Phys. Rev. B 64 024105[18] Salje E, Schmidt C and Bismayer U 1993 Phys. Chem. Miner.

19 502

5

J. Phys.: Condens. Matter 24 (2012) 255901 E K H Salje et al

[19] Zhang M et al 1995 Phys. Chem. Miner. 22 41[20] Lashley J C et al 2003 Cryogenics 43 369

Hwang J S, Lin K J and Tien C 1997 Rev. Sci. Instrum. 68 94[21] Shi Q, Snow C L, Boerio-Goates J and Woodfield B F 2010

J. Chem. Thermodyn. 42 1107[22] Bachmann R, Kirsch H C and Geballe T H 1970 Rev. Sci.

Instrum. 41 547Hatta I 1979 Rev. Sci. Instrum. 50 292

[23] Schmiedeshoff G M et al 2006 Rev. Sci. Instrum. 77 123907[24] Kolesov B A, Lager G A and Schultz A J 2007 Eur. J.

Mineral. 20 63[25] Salje E K H et al 1993 Acta Crystallogr. B 49 387

[26] Bismayer U and Salje E 1981 Acta Crystallogr. A 37 145see also Salje E and Parlinski K 1991 Supercond. Sci. Technol.

4 93Bratkovsky A M et al 1994 J. Phys.: Condens. Matter

6 3679[27] Carpenter M A, Salje E K H and Graeme-Barber A 1998 Eur.

J. Mineral. 10 621[28] Conti S et al 2011 J. Phys.: Condens. Matter 23 142203[29] Van Aert S et al 2012 Adv. Mater. 24 523[30] Goncalves-Ferreira L et al 2008 Phys. Rev. Lett. 101 097602[31] Fukai M, Inaba A, Matsuo T, Suga H and Ichikawa M 1993

Solid State Commun. 87 939

6