Coexistence of paraelectric/proton-glass and ferroelectric (antiferroelectric) orders in Rb

This paper reviews the results of dielectric studies of the proton glass state in a mixed crystal Rubidium Ammonium Dihydrogen Arsenate (RADA) The coexistence of paraelectric/proton glass and ferroelectric or antiferroelectric orders, confirmed by other studies, has been discussed in detail. The phase diagram of RADA is asymmetric. The proton glass state exists for ammonium concentration in the range of 0.1 < x < 0.5 . The phase diagram of deuterated DRADA is presented. The proton glass state region in DRADA, 0.2 < x < 0.35 is narrower than that for non-deuterated RADA. The effects of hydrostatic pressure on the dielectric properties in the proton glass state are presented. The glass temperature Tg decreases with pressure and is expected to vanish at 5 kbar. Low temperature behaviour is still an open question, since there is no experimental evidence of Tg(p) dependence below 4K for proton glass systems.


Introduction
The studies of the proton glass state in a mixed crystal of Rubidium Ammonium Dihydrogen Phosphate { Rb 1?x (NH 4 ) x H 2 PO 4 (RADP) were launched by Courtens 1] in 1982.At low ammonium concentration x RADP has a distorted ferroelectric structure with a typical paraelectric-ferroelectric (P-FE) transition, while at high ammonium concentration (x close to 1) RADP exhibits a distorted paraelectric/aniferroelectric transition (P-AFE).Temperature of the phase transitions: T c of P-FE transition as well as T N of P-AFE transition signi cantly lowers with the increase of the concentration of the second component of a mixed crystal.
For NH 4 concentration 0:22 6 x 6 0:75 2], below the \freezing" temperature T g , c Z.Trybuła, J.Stankowski the competition between ferroelectric and antiferroelectric orderings leads to frustration of the protonic system and appearance of regions with a local short-range order but without any evidence of a long-range order (ferroelectric or antiferroelectric).The proton glass state was also detected in an isomorphous crystal of Rubidium Ammonium Dihydrogen Arsenate-Rb 1?x (NH 4 ) x H 2 AsO 4 (RADA) by Trybu la et al. 3] in the dielectric study.This paper will focus essentially on the properties of the RADA crystal.Earlier dielectric studies gave some insight into the properties of the proton glass state [4][5][6][7].The phase diagram for RADA obtained by those studies is unlike that for RADP, and reveals the glass state existence in the range of 0:1 6 x 6 0: 5 4,5].
Further studies showed [8][9][10][11][12][13][14][15][16] that the phase diagram is more complex, since paraelectric/proton-glass-ferroelectric or antiferroelectric phases coexist.This paper presents the up to date understanding of the proton-glass state.First, after a short introduction, dielectric results in the mixed RADA crystal will be reviewed.Then, the problem of phase coexistence in this crystal based on dielectric studies, discussion of the phase diagram and the e ect of pressure and electric eld on the proton glass state will be presented.Crystals of the KH 2 PO 4 (KDP) family were the most extensively studied ferroelectrics 17,18].Ferroelectric properties of these crystals originate from the ordering of the protons in hydrogen bonds linking tetrahedral PO4 or AsO4 units in the crystal structure.The unit cell is tetragonal.Spontaneous polarization is the result of proton ordering due to the shift along the c axis of the centre of Rb + or NH + 4  tetrahedrons with respect to phosphorus or arsenium atoms in H 2 PO 4 or H 2 AsO 4 groups, while hydrogen bonds are formed almost in (ab) plane ( gure 1).In an ordered ferroelectric phase, below T c , up-down con guration of hydrogen bonds ( gure 3) is favoured.The change from up to down con guration leads to a switch in the polarization of the crystal.The next-lying excited state corresponds to lateral congurations -two protons are in a lateral position.These con gurations are called Slater con gurations 19].Higher energies of the hydrogen bond network correspond to Takagi con gurations 20] related to the ordering of a distorted tetrahedron with one, three, four protons or lack of protons at all ( gure 2).Antiferroelectrically ordered ADP or ADA crystals, be-low N eel temperature T N , have lateral con gurations in the ground state.The rst excited state corresponds to up-down con guration.In mixed crystals of RADP or RADA there is a small energy di erence between the ground and excited states, so the multiplet ground state is possible, hence the proton glass state can occur ( gure 3).

Review of dielectric studies of mixed RADA
Since Courtens discovered the proton glass state in RADP 1-2], the temperature dependence of real " 0 and imaginary " 00 parts of electric permittivity has been extensively studied [21][22][23][24][25][26][27][28][29].We have contributed to this work with the observation of the proton glass state in a RADA mixed crystal 3].First measurements were carried out in a microwave X-band range for measuring electric eld frequency of 9.2 GHz.The methodology of " 0 and " 00 measurement in a microwave resonator is described in 33].As this method does not require to paste electrodes, it is especially useful for small crystals and polycrystalline samples.The studied crys-tal of RADA x = 0:31 was placed in the maximum of electric eld in cylindrical microwave resonator T M 010 .The c-axis was parallel to electric eld lines.The orientation of the sample in the microwave resonator is shown in gure 4. Thermal contact of the sample with the resonator was provided by special support.The microwave resonator was mounted to the heat exchanging facility in the helium ow cryostat.Assuming that the sample diameter is much smaller than the diameter of the resonator, " 0 and " 00 were determined from the following expressions: where r 0 is the radius of the cylindrical resonator; r is the radius of the sample; l and h -the height of the resonator and the sample, respectively; f 0 -and Q 0 -the frequency and quality factor of the empty resonator; f and Q are the frequency and quality factor of the loaded resonator.Figure 5 presents the results of the temperature dependence of " 0 and " 00 for RADA, x = 0:31.There is a clear cusplike maximum on " 0 (T) dependence around 65 K and a maximum on losses " 00 (T) dependence at T g = 48 K, typical of proton glass.
In proton glass the electric eld E i and polarization P i are strongly related to ordering.Local polarization P i is given by: where J ij is the interaction energy of pseudospins.There is no long-range order in proton glass, hence the average value of polarization vanishes, i.e. hPi = 0.
According to Edwards-Anderson, there is only a short range order within the clusters.The order parameter q E?A can be regarded as square of the average polarization: q E?A (T) = 1 N X i hp i i 2 ; ( where N is the number of dipoles in the cluster and p i is the polarization of a single dipole.In the proton glass state q E?A 6 = 0. Di erent models have been developed to describe the proton glass state.The model of clusters by Prelov sek and Blinc 51], extended by Matsushita and Matsubara 52], plays an important role in understanding phase diagrams in proton glass systems.The con guration energy is assumed to be given by a Hamiltonian of the form: where pseudospin i (i =1, 2, 3, 4) is +1 or -1 depending on each of the two possible positions along the hydrogen bond occupied by i-th proton.For a ferroelectric crystal the energy di erence between the lateral and up-down ground states equals " 0 and depends on J.For an antiferroelectric crystal, besides the energy di erence between the two con gurations, { " o , another parameter should be introduced.This extra parameter w accounts for the interaction between two neighbouring parallel hydrogen bonds, which is necessary to establish an antiferroelectric long-range order.For the mixed crystal RADP, considered as a system of clusters (j is the number of clusters), the Hamiltonian (2) can be written as 52]: (3) j = ( j 1 ?j 3 ) + ( j 2 ?j 4 ); j = ( j 1 + j 2 + j 3 + j 4 ); " j has a di erent value for each cluster in RADP.A simple Gaussian distribution function for random variable " j in the Hamiltonian (3) is assumed: where " is the average of energy " depending on the concentration x and is the energy parameter: = p 2h(" ?") 2 i: For the mixed proton-glass crystal RADP " can take the following values depending on the ammonium concentration x 52]: " > 0 for 0 < x < 0:5; " = 0 for x = 0:5; " < 0 for 0:5 < x < 1: The glass transition temperature T g is given by 52]: k B T g = 1 8 1 + 2y2 (1 + 2y) 2 ; where: y = exp ??" k B T : The average of "(x) is given by 52]: "(x) = "P + (x) + 0 P o (x) ?"P ?(x); where P + (x), P o (x) and P ?(x) are three x-dependent probability functions for nding a cluster in three groups: ferro, neutral and antiferro group, respectively.This function is plotted in gure 6a.The broken lines are "(x) dependences for di erent RADA crystals.The root mean square as a function of x ( gure 6b) is de ned as 52]: Kwun 7] in dielectric measurements at 700 Hz to 1 MHz and the existence of the proton glass state was limited to the concentration of 0:13 < x < 0:49 ( gure 9).Further studies of the proton glass state in RADA revealed dispersion of electric permittivity in the transition from the paraelectric to the proton glass phase.The dielectric properties of the RADA crystal (x = 0:35) were investigated for two perature to T f > 70 K.Deviation of " 0 (T) from the Curie-Weiss law determines   samples 6]: along a and c axis of the crystal.Figure 10 shows a real part of electric permittivity data for both crystal orientations.For the both directions a typical proton glass behaviour is observed.As the temperature lowers from the room temperature to about 40 K, " 0 initially increases.After a cusp-like maximum " 0 decreases to the value of approximately 10 at temperature T = 3 K.Electric permittivity " 0 can be well described by the Curie-Weiss law in the paraelectric phase from the room tem-temperature T f .At this temperature a short-range order is established in certain regions of the crystal { clusters start to be formed.On decreasing the temperature, the volume of each cluster increases to Vogel-Fulcher temperature T o (equation 5), according to the following equation rst applied to proton glass by Courtens 36]: where T o is Vogel-Fulcher freezing temperature, E c is a cut-o energy in the temperature unit, o is an attempt frequency.The temperatures T g of the freezing electric dipoles reorientation within the cluster have been determined for each measuring frequency from the maximum of " 00 (T).At T o , the reorientation of electric dipoles within the cluster becomes frozen.Dielectric dispersions of " 0 (T) and " 00 (T) have been demonstrated for T < 40 K ( gure 10).As the frequency of the measured electric eld increases, the maximum of " 00 (T) shifts to a higher temperature.
The proton glass state was also observed in K 1?x (NH 4 ) x H 2 AsO 4 x = 0:40, (KADA x = 0:40) 35].The substitution of Rb by K does not in uence the properties of the proton glass state signi cantly.The temperature dependence of " 0 a for KADA x = 0:40 is very similar to that of RADA.

Undeuterated glass RADA
Detailed dielectric studies of mixed RADA crystals for di erent x concentrations have shown that for very small, as well as for very large x concentrations the crystals are exclusively in a ferroelectric /or antiferroelectric phase.The rst evidence of the coexistence of paraelectric, glass and ordered ferroelectric phases was given by Trybu la , Schmidt and Drumheller 8] for RADA x = 0:12; 0.15 and 0.20.This result was con rmed by Eom et al. 9] in dielectric and laser optical studies.The latter shows that the orthorhombic symmetry { optically biaxial characteristic of a ferroelectric phase is preserved up to the lowest temperature at which the glass state is present.The crystal in the glass state is tetragonal (optically uniaxial), like in a paraelectric phase.It seems that the glass state exists independently of the ferroelectric one.
Temperature dependences of the real part of electric permittivity " 0 a in RADA for di erent ammonium concentrations x, revealing the coexistence of glass and ferroelectric phases, are shown in gure 11 and 12 8].The ammonium concentration was determined by measuring the rubidium content with ame atomic-absorption spectroscopy.In contrary to RADP, in RADA the concentration of rubidium (1?x) in solution equals the concentration in the crystal.The following concentrations were studied: x = 0, x = 0:12 0:01, x = 0:15 0:01 and x = 0:20 0:01.For x = 0:12 and x = 0:15 there are two transitions: at T c to a ferroelectric phase  and at T g to a glass phase.The transition to the ferroelectric phase is not frequency dependent, while that to the glass state displays dispersion both in " 0 and " 00 ( gure 12).The measurements along the a axis of the crystal, perpendicular to ferroelectric axis c, excluded the possibility of dispersion, typical of KDP ferroelectrics related to the freezing of the dynamics of the domains walls.Such dispersion does not exist along the a axis.The RADA results in gure 11 clearly show two phase transitions.In the case of the coexistence of ferroelectric and glass phases, Tg for the given frequency of the measured electric eld is lower than Tg at the same frequency for RADA exhibiting only the glass state.This observation suggests that the volume of the cluster with a short-range order in crystals with phase coexistence is smaller than the volume of the clusters in crystals in the glass state only.Moreover, because of the shorter correlation length, the dynamics of the clusters is less hindered.that such coexistence can be observed even for x = 0:01.The part of the RADA phase diagram presenting the coexistence of ferroelectric/glass phases is shown in gure 13.
The coexistence of paraelectric/proton glass and ferroelectric orders below the glass transition temperature T g was con rmed by spontaneous polarization measurements in deuterated DRADA an undeuterated RADA crystals by Pinto et al. 12].The temperature dependence of P s for a ferroelectric crystal RDA and mixed RADA x = 0:08 was determined from a saturated hysteresis loop in the standard Sawer-Tower circuit.Figure 14 presents the results for a nondeuterated crystal.A RDA crystal exhibits a distinct jump of spontaneous polarization at T c = 110 K, typical of the rst order phase transition.The spontaneous polarization reaches the value of 3:6 0:5 C/cm ?2 at the temperature far below T c .For a mixed crystal RADA x = 0:8, below T = 94 K, there is a gradual increase of P s .The maximum value of P s attained in a mixed c rystal is lower than that of a RDA crystal.Spontaneous polarization of a mixed crystal can be thus described by the following expression 12]: P sd (T) = P so " 0 1 (T) " 0 1 (T) + " 0 2 (T) ; (6) where P sd is the spontaneous polarization of the mixed crystal, P so is the maximum spontaneous polarization of the pure crystal RDA well below T c ; " 0 1 and " 0 2 electric permittivity values marked in gure 15 which gives temperature dependences of " 0 for proton-glass RADA x = 0:40 and RADA x = 0:08.Spontaneous polarization of the mixed crystal is equal to the spontaneous polarization in the pure RDA crystal well below T c multiplied by the fraction of the mixed crystal that becomes ferroelectric below T c .Similar results were obtained by Pinto et al. 12] for the deuterated D-RADA x = 0:08 crystal.The maximum value of the spontaneous polarization in mixed RADA x = 0:8, and its deuterated counterpart is lower than that of the pure crystal.This indicates that at lower temperatures there are still paraelectric clusters closely interlocked with ferroelectric clusters.The complex dielectric permittivity " 0 a (T) and " 00 a (T) of DRDA for x = 0:28 in the deuteron glass range, typical of the glass state, are presented in gures 16 and 17.The dielectric permittivity " 0 a (T) of deuterated DRADA for x = 0:39 studied by Trybu la et al. 11] is shown in gure 18.A typical behaviour attributed to the transition from the paraelectric to antiferroelectric state is marked at T N = 127 K.This phase transition takes place at the same temperature for di erent frequencies of the measured electric eld.Precise measurements for the temperature below 100 K show the occurrence of dispersion of the permittivity " 0 a (T) in the temperature range from 20 K to 90 K ( gure 19).Analysis of the shape of the temperature   dependence of " 00 a (T) ( gure 20) proves the existence of phases of a short-range order.In the antiferroelectric phase, the values of " 00 do not change with temperature, as a result of the opposite dipolar moments of the two sublattices of the crystal due to the ordering of deuterons in the hydrogen bond O-D O and the formation of the lateral Slater con guration.The temperature and frequency dependence of " 00 , two orders of magnitude smaller than those characteristic of the concentration range in which deuteron glass can exist, provides information that the regions of the glass state are formed where a long-range antiferroelectric order disappears.A complex temperature Phase diagram of Rb 1?x (NH 4 ) x H 2 AsO 4 (DRADA).
dependence of " 0 a (T; ) indicates two kinds of electric dipolar relaxation.One of these mechanisms can be described by the Vogel-Fulcher temperature (equation 5) with the parameters: T 0 = 25:7 K, E c = 105:6 K and o = 1:16 10 8 Hz, and is typical of clusters with a short-range order characteristic of proton (deuteron) glass.The second relaxation mechanism is the thermally activated Arrhenius dipolar reorientation with activation energy E c = 1105 K and frequency o = 1:44 10 12 Hz.The Arrhenius-type relaxation is related to free dipoles which are released from the melting long-range ordering but have not managed to form a cluster yet.Similar behaviour of relaxation, described by the Arrhenius equation, was reported by Hutton et al. 42] for proton glass which is a mixture of antiferroelectric betaine phosphate (BP): (CH) 3 NCH 2 COO H 3 PO 4 , and ferroelectric betaine phosphite (BPI): (CH) 3 NCH 2 COO H 3 PO 3 and in which the hydrogen bonds between PO 3 or PO 4 groups form quasi-one dimensional chains.The Arrheniustype relaxation is typical of strong glass 43] characterized by a low density of con gurational states in their potential energy.
Figure 21 presents a new phase diagram of DRADA displaying the coexistence of paraelectric/proton glass and ferroelectric (antiferroelectric) phases.

Pressure dependence of a proton glass phase
The rst high-pressure studies in RADP mixed crystals were carried out by Samara et al. [45][46].The e ect of pressure was expected to in uence the glassy state via the pressure dependence on the hydrogen-bond length.These studies have led to a much better understanding of the nature of the competing inter-molecular and intra-molecular interactions which are responsible for the establishment of a long-range order.Pressure modi es interactions responsible for a short-range correlation; therefore, the results of pressure dependence provide new insights into the formation and properties of the glassy state.In the family of KDP crystals the proton moves in a double-well potential along the hydrogen bond.In the high-temperature tetragonal paraelectric phase the protons are disordered in the potential wells leading to an e ectively symmetric hydrogen bond.In the glass state the proton freezes in one or another potential minimum; this results in an elongated asymmetric hydrogen bond.Pressure reduces the H-bond length and favours a more symmetric bond with a lower energy barrier, which in turn leads to a lower glass transition temperature.For a su ciently high pressure, the Hbond will become e ectively symmetric, so that there will be no order and the glassy state will vanish.The results of Samara for the RADP with x = 0:48, 72 % deuterated crystal 45] and RADP with x = 0: 50 46] show the lowering of the glass temperatures T g , as presented in gure 22.The results for c and a crystallographic axes were qualitatively similar.Comparison of a pressure-induced suppression of the glassy state in a RADP mixed crystal with the suppression of the ferroelectric state in RDP and the antiferroelectric state in ADP shows an important di erence between the glassy transition and ferroelectric or antiferroelectric transitions.For pure ferroelectric or antiferroelectric crystals the magnitude of slope dT c;N =dp increases with pressure at high pressure values.The data strongly suggest that the transition vanishes at an in nite slope, i.e. dT c;N =dp !?1 as T c;N !0. The results of glassy RADP x = 0:5 are different.Glass transition temperature T g decreases linearly with pressure up to temperature 5 K with no hint of any impending increase in the magnitude of dT g =dp at a lower temperature.There are no data below 4 K, and only linear T g (p) extrapolation to higher pressures is given.Samara supposes that the proton glass phase will disappear at 5 kbar.Up to this time there is no experimental evidence of a linear or nonlinear T g (p) response below 4 K. Samara believes that this linear dependence is most likely the evidence of the nonequilibrium nature of the glass transition in RADP glass crystals.There is a large hydrogen-isotope e ect not only on T g but also on its pressure derivative.The magnitude of dT g =dp decreases from -3.6 to -2.0 K/kbar on deuteration.The higher T g and smaller dT g =dp for deuterated glass are due to the fact that a deuteron is located deeper in the potential well than a proton, and there is a much lower probability for tunneling between two potential wells along the O-D O bond.
The e ects of hydrostatic pressure on the dielectric properties and phase transitions were investigated in a RADA crystal by Samara and Schmidt 16] for the compositions in the coexistence region of proton-glass and ferroelectric or antifer- roelectric orders.Figure 23 shows that this crystal is inhomogeneous, containing a su ciently large region (about 4 % of the sample's volume) of pure RDA, as indicated by T c1 = 110 K.The second transition at T c2 = 90 K is related to the presence of RADA for x = 0:1.Glass transition temperature T g is marked at 30 K. The pressure derivatives of T c for the paraelectric-ferroelectric transitions in RDA (-4.6 K/kbar) are about twice as large as those of T N for the paraelectricantiferroelectric ones in ADA (-1.97 K/kbar).The pressure derivative of T g for the paraelectric-proton glass transition (-2.2 K/kbar) is of about the same magnitude as for ADA suggesting that the compressibility of ADA clusters in RADA determines the glass transition.The T g is weakly dependent on the composition over most of the region of a proton glass phase.Samara's results indicate that pressure derivative dT g =dp is also essentially independent of the composition.The decrease of T c , T N , and T g with pressure results from an increase in the tunnelling frequency and a decrease in the dipolar interaction which is long-range in the case of ferroelectric and antiferroelectric phases and short-range (probably antiferroelectric) in proton glass.

External dc electric field dependence of electric permittivity in RADA
The temperature dependence of the eld-cooled (FC), zero eld-cooled (ZFC) and eld-heated (FH) static permittivity " 0 was studied in deuterated DRADP crystals by Levstik et al. 47] and in Deuterated Rubidium Ammonium Dihydrogen Arsenate DRADA x = 0:28 by Pinto et al . 44].Unlike magnetic spin glass, where only random-bond type interactions exist 48], proton and deuteron glass are characterized by the presence of random bonds and a random eld [49][50].The random bias electric eld is due to the random sites of the NH 4 (or ND 4 ) groups; this leads to a random freeze-out of the acid proton (or deuteron) in the hydrogen bonds O-H O as temperature is lowered below T g .Because of this eld the Edwards-Anderson order parameter q EA (see equation 1) is nonzero in the whole temperature range.In magnetic spin glass systems the q EA is zero above glass transition temperature and nonzero below T g .
The static dielectric response of deuteron glass to the external dc electric eld depends on the history of the sample.It is important how the low temperature glass phase (nonergotic) is reached.Above the glass phase the eld-cooled " 0 FC and zero eld-cooled " 0 ZFC electric permittivities are the same, whereas below T g , in general " 0 FC > " 0 ZFC .The DC electric eld increases the value of a static permittivity.The eld-cooled electric permittivity " 0 FC retains the same value as temperature is lowered below T g and is constant at the temperature decrease due to the gradual freeze-out of the acid proton or deuteron in the hydrogen O-H O bond.On switching the external dc eld o , remanent polarization is observed which vanishes as temperature is raised above T g .

Figure 1 .
Figure 1.Ordered hydrogen-bonds system that gives spontaneous polarization.

Figure 2 .
Figure 2. Con gurational energies of 16 types of phosphate PO 4 or arsenate AsO 4 tetrahedra, represented in the pseudospin formalism and in terms of Slater energy " o and Takagi energy " 1 52].

Figure 4 .
Figure 4. Diagram of the TM 010 resonator: a) con guration of the electromagnetic eld in the resonator, b) resonator loaded with a sample 33].

Figure 7 .
Figure 7.The temperature and concentration x dependence of the electric permittivity " 0 c and " 00 c in RADA x=0.31 at 9.2 GHz of the measured electric eld 4].

Figure 8 .
Figure 8.The phase diagram of RADA.PE, FE, AFE, and PG denote paraelectric, ferroelectric, antiferroelectric and proton glass phases, respectively.The open circles and crosses denote maximum of " 0 c and " 00 c , respectively.The full circles denote the middle of the phase transition region from the " 0 c (T ) dependence 4].

Figure 9 .
Figure 9.The phase diagram of RADA done by Kim and Kwun 7] from electric permittivity " 0 a and " 00 a .
Howell, Pinto and Schmidt 10] have shown that the coexistence of ferroelectric / glass phases exists even for lower concentrations, i.e. x=0.05.Recent measurements up to 0.4K by Trybu la et al. 37] reveal

3. 2 .
Deuterated glass DRADA Deuteration shifts the transition temperature T c upward from 110 K for RDA to 170 K for DRDA or, in antiferroelectrics, the N eel temperature T N from 216 for ADA to 304 K for DADA and increases the value of the spontaneous polarization P s compared with the undeuterated crystal 40].The studies of deuteron glass have revealed that, like in a undeuterated crystal, the state of glass may coexist with the ferroelectric or antiferroelectric order 10-14].Deuteration leads to the narrowing of glass existence range to 0:2 6 x 6 0:35 14-15,41].

Figure 18 .
Figure 18.Temperature dependence of the real part of the electric permittivity e>a for DRADA x = 0:39, at the frequency of the measuring electric eld = 10 kHz 11].

Figure 20 .
Figure 20.The t of the experimentally obtained " 00 a (T; ) data for DRADA x = 0:39 using two Gaussian-shape lines (at = 10 kHz).The contributions of two di erent relaxation mechanisms are marked 11].