Nonergodicity in DRADA deuteron glass
Authors: V. Hugo Schmidt & Nicholas J. Pinto
This is an Accepted Manuscript of an article published in Ferroelectrics on [date of publication], 
available online: http://www.tandfonline.com/10.1080/00150199408244748.
V. Hugo Schmidt & Nicholas J. Pinto (1994) Nonergodicity in drada deuteron glass, 
Ferroelectrics, 151:1, 257-262, DOI: 10.1080/00150199408244748
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Ferroelectrics, 1994, Vol. 151, pp. 257-262 
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NONERGODICITY IN DRADA DEUTERON GLASS 
V. HUGO SCHMIDT and NICHOLAS J. PINTO* 
Physics Department, Montana State University, Bozeman, MT 59717, USA 
*Now at Physics Department, Wichita State University, Wichita, KS 67208 
(Received August 9, 1993; in final form October 31, 1993) 
Abstract A field-heating (FH), field-cooling (FC), zero-field heating (ZFH) 
sequence was used to study nonergodic polarization response to dc electric 
field in x4.28 Rb,-,(ND4),D2As04 (DRADA) deuteron glass. Polarization 
change in the low-temperature nonergodic region varied with temperature T 
nearly as Ty, where 5SyS6. This result is compared with predictions of the 
bound charge semiconductor model for proton glass. 
INTRODUCTION 
For people with a ferroelectrics background, a proton glass like 
Rbl-x(NH4)xH2As04 (RADA) is a mixed crystal with femlectric (FE) RbH2As04 
and antiferroelectric (AFE) NH4H2As04 parent materials. Over a range of about 
0.16<x<0.40 the FE and AFE interactions are frustrated and the crystal exhibits no 
phase transition.',* Outside this range, both and deuterated4 RADA 
crystals exhibit coexistence of proton glass and ordered FE or AFE phases before 
the x=O (FE) and x=l (ME) pure phases are reached. For people with a magnetic 
spin glass background, proton glass is a pseudospin glass with frustrated electric 
instead of magnetic interactions. The pseudospin's f l  spin value depends on 
whether the H atom's off-center position in the 0 - H - 0  bond contributes a + or - 
dipole moment. There are additional differences from spin glasses. First, proton 
glass interactions are short-range FE and AFE pseudospin-pseudospin interactionss-' 
instead of intermediate-range magnetic spin-spin interactions. Second, spin glasses 
usually lack significant spin-lattice interactions, but proton glass pseudospins 
because of their unlike (Ttb' and NH,') cation neighbors feel a significant bias 
which eliminates any permittivity cusp," whereas a permeability cusp 
corresponding to the spin glass "transition" is seen in spin glasses." 
large spread in permittivity (permeability) time constants.12 Below some 
temperature corresponding to the onset of nonergodicity, the upper time constant 
limit becomes infinite or at least longer than the experimenter is willing to wait. In 
the nonergodic state, the crystal cannot reach a new state of minimum free energy 
Both proton glasses and spin glasses develop with decreasing temperature a 
257 
258 V.H. SCHMIDT AND N.J. P N O  
required by changing external conditions (temperature, field, pressure, etc.). To try 
to observe the onset of nonergodicity, dielectric permittivity techniques can be 
pushed down to the millihertz or even upper microhertz range, but such 
experiments are quite time-consuining. One can attempt to observe long 
polarization decays directly with an electrometer, but leakage and drift cause 
serious problems. 
Levstik gj circumvented these difficulties by applying to 
Rbl-,(NH4),H2P04 (RADP) a field cooling (FC) technique, to be described later, in 
which the onset and decay of nonergodicity can be clearly observed. We report 
here on a modified FC technique applied to RADA, and on an analysis of the 
results. The motivation for both experiments was investigation of the nature of 
nonergodicity, including the question of whether there is a well-defied temperature 
at which the upper time constant limit suddenly becomes infinite, or whether that 
upper limit increases continuously and only becomes infinite at 0 K. 
EXPERIMENTAL METHOD 
The crystal was grown by slow evaporation from an aqueous solution of the parent 
crystals RbD2As04 and ND4D2As04 by Z. Trybula. A sample 0.58 nun thick 
perpendicular to the g axis was cut from the crystal, polished and then electroded 
with evaporated silver contacts. The crystal was in series with a (much larger 
capacitance) 15 nF low-leakage capacitor connected across the input of a Model 
610R Keithley electrometer, whose voltage was thus proportional to the sample 
polarization. An Oxford Model ESR-900 continuous-flow liquid-helium cryostat 
provided and controlled the low temperatures, but temperature was measured 
independently of this system, with a chromel-alumel Type K thermocouple. 
as described in the following Section together with the experimental results. 
Our temperature and field control cycle consisted in four steps or processes, 
EXPERIMENTAL RESULTS 
Our technique consisted in first cooling the crystal from room temperature at zero 
field, so of course no polarization developed and nothing interesting happened in 
this f'i'it step. Then at the lowest temperature of 10 K, we began the field heating 
(FH) process by suddenly applied a dc field near 800 V/cm along the g axis and 
started raising the temperature at one of two rates, 1 Wmin or 4 Wmin. No 
noticeable rate dependence was seen in our results; with hindsight we should have 
used a much larger rate ratio because the time dependence of polarization is very 
weak compared to the temperature dependence in this type of experiment. Results 
for the 1 Wmin run appear in Fig. 1; results for the 4 Wmin run as well as both g- 
and paxis ac dielectric results will appear e1~ewhere.l~ The 4 K/min results show 
that the field application generated an immediate response from the electronic and 
direct ionic displacements. ("Direct" means not mediated by any hydrogen 
intrabond motion.) Following this direct response, no further increase occurred 
until the temperature rose to about 20 K, at which point there was an increase that 
NONERGODICITY IN DRADA DEUTERON GLASS 259 
-urn 1 
50 
00 
r 
r 
1 1 0  
1 E 0 
\ 
FIGURE 1 Temperature dependence of polarization, and of 
permittivity in ergodic range, along 8 axis of x9.28 DRADA. Open 
and fiied circles and open diamonds indicate field heating, field 
cooling, and zero field heating results respectively. Solid and dashed 
Lines show fits to polarization change proponional to T6 and T3 
respectively. 
became faster and faster until it approached the value required for minimum free 
energy, at which point the response flattened out more or less exponentially. The 1 
K/min results in Fig. 1 are similar, except that data-taking only began at 22 K, so 
the flat initial portion found in the 4 K/min run is missing. The temperature was 
raised further at the same 1 Wmin rate after the polarization curve flattened out, 
stopping near 180 K which is the limit required to prevent ionic (p ro ton i~ ) '~? '~  
conductivity from causing the electrometer voltage to drift upward. 
In the third step, the field cooling (FC) process, the rate of temperature increase 
was immediately changed to the same 1 Wmin rate of temperature decrease, 
keeping the applied field unchanged at 800 V/cm. In the ergodic regime the 
polarization retraced the FH curve, increasing according to a Curie-Weiss law and 
then deviating below the extrapolated Curie-Weiss curve as is typical for proton 
glasses. Near 60 K (the ergodic limit for this heatingkooling rate) the FC curve 
separates from the FH curve, the polarization can no longer respond appreciably, 
and the curve becomes flat as the lowest temperature of 10 K is approached. 
At this lowest temperature the last, zero-field heating (ZFH), step is begun by 
removing the applied field and heating at the same rate as in the FH step. The 
polarization immediately drops by the amount of the electronic and direct ionic 
response, then stays unchanged until near 20 K it begins dropping, slowly at fitst 
260 V.H. SCHMIDT AND N.J. PINTO 
and then more rapidly, but fiialIy approaches zero more or less exponentially. The 
ZFH and FH curves are seen to be nearly mirror images of each other. 
ANALYSIS 
We analyze our results based on ideas related to the bound charge semiconductor 
model for dielectric response of proton glass." As a simplest illustration of that 
model, consider RDA in the framework of the Slater-Takagi with 
dynamics provided by the effective diffusion of HAsO, and H3As0, "bound charge 
carriers" by means of successive hydro en intrabond transfers, a process first 
deduced from NMR relaxation results.2 121 These transfers can convert zero-energy 
polar Slater groups into nonpolar groups of energy ~0 and vice versa, or in other 
words, phonon energy is converted into Slater group codigurational energy and 
vice versa. In equilibrium, the diffusion (hydrogen transfers) should on the average 
not change the configurational energy, so the diffusion occurs along a path which is 
level on the average. In proton glass, one must also consider AFE and random bias 
interactions, but the basic premise of a level path average remains unchanged. 
If configurational energy is plotted vertically against number of diffusion steps 
horizontally, one obtains a random walk plot of energy against step number, similar 
to the well-known "drunkard's walk" plot of position z. step number. The 
maximum barrier encountered in diffusing N steps (in a given sense along the 
path, without retracing and ignoring side trips) will be proportional to Nln. The 
maximum barrier that can be crossed (in, say, one minute during which T changes 
appreciably) will be proportional to temperature T. This means that in this time the 
number of steps diffused will be roportional to T2. This diffusion then occurs 
over a "volume" of size about N , proportional to T6. In the ZFH process one 
could think of this diffusion as destroying the polarization throughout the diffusion 
region, so the amount of polarization drop from its initial value should be 
proportional to T6. We see in Fig. 1 that the fit to this T6 law is surprisingly good. 
Similar arguments can be applied to the polarization rise in the FH process, and the 
fit is similarly good. 
&13 for RADP and 
obtained equally good fit.14 Their method had no FH process. Instead, they 
employed what they called a zero-field cooling process and observed the equivalent 
of the high-temperature end of our FH curve. Their curve extended down to a 
temperature only slightly below the separation point from their FC curve, so it was 
not possible to try to fit this curve to the T6 law. 
It is tempting to end the story here and to be proud of our good fit. 
However, a more careful analysis must take more details of the bound charge 
semiconductor model into account. For instance, the number of diffusion steps of 
an HASO, or H3As04 bound charge carrier is not proportional to the radius 
diffused in real space, because the diffusion path itself is a thermally activated 
random walk in real space. We then must consider whether the maximum barrier 
height encountered depends on real space radius or on diffusion path "radius." At 
the low temperatures for which nonergodicity occurs, the carrier is confined to a 
small region and its path will include sections that retrace and thus do not change 
the configurational energy, SO we consider that the pertinent radius is the real space 
6 
P 
We applied this T6 "law" to the ZFH data of Levstik 
NONERGODlCITY IN DRADA DEUTERON GLASS 261 
radius. Also, the FH and ZFH processes involve time as well as temperature, and 
an accurate analysis must include time also as a parameter. We take these 
considerations into account in the following simple model, which leads to a T3 iaw 
for the polarization decay or rise. 
The polarization change in the FH and ZFH processes is the carrier 
concentration n multiplied by the mean carrier displacement a> and carrier 
effective charge q. In the nonergodic regime n is temperature-independent because 
the carriers are trapped and cannot find annihilation partners. We assume the mean 
carrier energy relative to the trap site is €d(r/rl)l'. The 112 exponent gives the 
energy barrier height spatial dependence discussed above and provides the 
temperature/the limitation on carrier diffusion. Ed is the rms energy change per 
path step, and rl is the length of one path step. The electrical energy of a carrier is 
-qEx=-qErcose, where E is the electric field strength and 8 is the polar angle of the 
carrier relative to the field direction. Integration over r and solid angle yields 
AP=nq<x>=(9!/5 !)qE(kT)3r12/3~2= 1 008qE(kT)3r12/~,4 
From the T3-law fit in Fig. 1, the polarization change equals the initial polarization 
of 10 nC/cm2 for the ZFH curve at the "er odic temperature" Te=44 K. For rl we 
use 3.24 A, for q we use the c-axis valuel'of 0.08 proton charge, and we assume a 
fractional carrier concentration of 0.001, a typical low-temperature value from 
Monte Carlo simulations. Then Eq. (1) will balance if the rms energy Ed is chosen 
as 26 K in temperature units. This is lower than expected by about a factor of 3. 
Another problem with this simple model is that, as seen in Fig. 1, the data could fit 
a ? law, but one cannot force the exponent much lower. It is puzzling that the T6 
law which is based on naive ideas gives a better fit than the T3 law based on what 
seems a more physically sound model. 
DISCUSSION 
The "nonergodic" type experiments such as the FH and ZFH runs reported here and 
elsewhere provide a different type of test for any theory of proton glass dynamics 
than is provided by fits to ac dielectric permittivity data, Nh4R or Brillouin 
scattering results, etc. Variations of these nonergodic experiments should be made, 
for instance using a much broader range of heatinglcooling rates, and interrupting 
FH or ZFH runs by removing or applying a field, respectively, as well as 
employing different field strengths. A successful theory must be able to explain 
both the nonergodic and permittivity type experimental data, and it must be based 
on well-defmed local interactions rather than having merely a phenomenological 
basis. Providing such a theory-remains a difficult and exciting challenge. 
ACKNOWLEDGEMENTS 
This work was supported by NSF Grant Dh4R-9017429. Dr. Stuart Hutton 
automated the apparatus used in this experiment. Prof. John Drumheller kindly 
allowed use of his EPR liquid helium cryostat and temperature control apparatus. 
262 V.H. SCHMIDT AND N.J. PINTO 
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