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Millimetre-wave investigation of the soft mode in ferroelectric PbHPO4.

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Ann. Physik 1 (1992) 399-408
der Physik
0 Johann Ambrosius Barth 1992
Millimetre-wave investigation of the soft mode in
ferroelectric PbHP04
M. Briskot and H. Happ
11. Physikalisches Institut, UniversitBt zu Koln, Ziilpicher Stral3e 77, W-5000 Koln 41, Germany
Received 18 May 1992, revised version I3 July 1992, accepted I3 July 1992
Abstract, The complex dielectric function C(w) along the a axis of PbHPO, single crystals is
determined from reflection measurements at frequencies between 70 and 225 GHz in the paraelectric and
the ferroelectric phase near the transition temperature T,. = 310 K. The data are complemented by a
remeasurement of the properties of the lowest far-infrared mode. The mm-wave data can be described
by a simple Debye formula. The relaxation time exhibits critical slowing down, and the corresponding
relaxation frequency varies between about 2 and 60 GHz near T,. Proton tunneling appears to play no
essential role in this dielectric mechanism related to proton ordering.
Ferroelectrics; Permittivity; mm-waves.
1 Introduction
In the group of ferroelectrics with hydrogen bonds in which the protons play an essential
role in the phase transition, P b H P 0 4 (LHP) is of particular interest. In this system the
hydrogen bonds are arranged in a simple way linking PO, groups into linear chains
along the c axis of the monoclinic structure. There are no hydrogen bond cross-links
between the chains, see Fig. 1. The spontaneous polarisation P , develops nearly along
the projection of the hydrogen bonds in the ac plane and can be primarily attributed to
the ordering of the protons which are assumed to move in a double well potential along
their bonds [ 11. This picture is confirmed by results of neutron diffraction measurements.
Fig. 1 Unit cell of PbHPO,.
0 0. 0 H.
0 Pb,
Ann. Physik 1 (1992)
As reported by Nelmes and coworkers [2, 31 the degree of proton ordering below T, =
310 K closely follows the curve of the spontaneous polarisation P,(T). No soft lattice
mode is found in the far-infrared (FIR) spectrum of LH P which could account for the
Curie-Weiss type temperature dependence of the static dielectric constant &,(T)near T,
[4, 51. These findings and the results of dielectric measurements at a few frequencies in
the sub mm-wave range [4] and the microwave range [5] indicated that the dielectric
mechanism involved in the phase transition is located in the microwave region and is
primarily due to proton motion. Hence LHP with its simple arrangement of hydrogen
bonds is especially suited for the application of the pseudo-spin model, originally
introduced by de Gennes [6] to describe collective motions of protons in double well
potentials which takes into account the possibility of tunneling.
In a previous paper [7] we reported on measurements of the temperature dependence
of the complex dielectric function C(w) of LHP in the range between 0.3 and 10 GHz in
the paraelectric and ferroelectric phase using a method of time domain spectroscopy.
These measurements refer to the crystallographic a axis which is near to the direction of
spontaneous polarisation P,.The results clearly show that the main dielectric dispersion
near T, occurs in this frequency range and is of relaxation type. An effort to describe the
data by a simple Debye formula or that of an overdamped oscillator met with difficulties
to obtain a satisfactory fit to both, E' and E" in the dispersion wings above 7 GHz. As
in the method used larger systematic errors in the determination of absolute values at
these frequencies cannot be excluded a definite conclusion as to the type of dielectric
mechanism could not be given.
On the other hand, the sub mm-wave data [4] and data measured below 1 GHz [8] could
be well described by a Debye formula. Hence it seemed worthwhile to provide additional
experimental information of the dielectric behavior of LHP below the FIR. In the present
work we have determined E(w)and its temperature dependence in the region of the phase
transition at selected frequencies between 70 and 225 GHz and in the FIR from 17 to 120
cm- ' including the lowest FIR mode. The results confirm the picture of a simple
relaxator and are discussed in view of current theories based on the pseudo-spin model.
2 Experimental details
The complex dielectric funtion E(w ) was determined from the amplitude reflection
coefficient i. = r exp [i(n + p ) ] of large single crystals measured at normal incidence to
the ac plane with E parallel to the crystallographic a axis. This axis includes an angle of
about 22" with the direction of P,. Measurements perpendicular to the a axis revealed
no essential temperature dependence in the range from 273 to 328 K investigated and are
LHP single crystals with dimensions of about 8 x 2 x 6 mm3 (a x b x c ) were grown
by the gel method [9] using tetramethoxysilane as gel former [I 01. Still larger crystals with
dimensions of about 15 x 4 x 10 mm3 could be obtained by the continuation of the
growth process in newly prepared gels [ I 11. The temperature was kept slightly above the
transition point, i. e. in the paraelectric phase. Crystals grown in this phase appear to have
less defects than those grown in the ferroelectric phase as was concluded from an NMR
study [ 121. Nevertheless, an essential improvement of crystal quality can be achieved by
a suitable heat treatment at about 473 K [ 131. E. g., the peak value of E , at T, increases
from about t 000 of a virgin crystal to 5 000 or more. This heat treatment was applied to
all samples before measurement.
M. Briskot, H. M. Happ, Millimetre-wave investigation
Fig. 2 Layout of the mm-wave interferometer. S: source, FP: Fabry-Perot filter, A: analysator, BS:
beamsplitter, S/R: sampleheference, MM: Michelson-mirror, M 1 - M5: spherical mirrors.
Two methods of measurement were used in the mm-wave range. A Gunn-oscillator,
various Impatt-oscillators and frequency multipliers served as radiation generators. At
70 and 89 GHz standard waveguide technique was applied using an E-band hybrid tee
to measure the reflection coefficient of a sample. The sample was composed of three
polished crystals carefully mounted to achieve a common plane surface. The sample
terminated an oversized X-band waveguide which was connected to one arm of the hybrid
tee by an appropriately tapered waveguide section. The temperature of the sample holder
was controlled by a peltier element and measured by a PT 100 resistor with an error
G0.2 K . 2(0) was calculated from the equation
This equation refers to normal incidence of plane waves. In a standard waveguide
operated in the H I omode 2. has to be substituted by the expression (2 - p ) / ( 1 - p ) ,
p = (A/Ac)2, where A is the free space wavelength and Ac the cutoff wavelength [14]. For
and the correction could be neglected within
the oversized waveguide usedp < 8 *
the errors of measurement, Ar < 3%, Ayl G2". Eq. (1) was also used in the methods
described below.
The mm-wave measurements above 90 GHz were performed with a dispersive
Michelson interferometer using Gaussian beam mirror optics and wire grids as polarising
beam splitters. This instrument, schematically shown in Fig. 2, was built in our
laboratory [15] according to the principles published by Parker et al. [I61 with some
modifications of the optics including phase modulation of the optical path difference
produced by the moving Michelson mirror. The sample was again composed of several
Ann. Physik 1 (1992)
large crystals as described above and given a size appropriate to the Gaussian beam waist.
It was mounted in a heatable holder and could be precisely interchanged with the fixed
Michelson mirror.
The behaviour of 2 in the transition to the FIR was determined from reflection
measurements in the range from 17 to 120 cm using a dispersive FIR Fourier spectrometer. This instrument was built [I71 in form of a tilt-compensated Michelson interferometer according to the principles published by Genzel and Kuhl [18]. The beams are
focussed on the Michelson mirrors, one of which can be precisely interchanged with a
sample. In this way both, the modulus and phase of i- enter the interferogram and can
be determined after Fourier transformation [ 191. Thus the complications of Kramers
Kronig analysis in determining the phase of i- from a limited spectrum are avoided.
3 Results
Fig. 3 shows the temperature dependence of
and E” evaluated from the average values
of i- at frequencies between 70 and 225 GHz. It corresponds to the variation of 2 with
temperature expected at fixed frequencies in the wing of the dielectric dispersion the
characteristic frequency of which shifts to lower values in approaching T,, cf. Figs. 4
and 5.
v [GHz]
40 20-
Temperature [K]
Fig. 3 Temperature dependence of c’
and E“ in the mm-wave range, T, =
310 K. The scale refers to the curves
at 225 GHz. The upper curve5 are
shifted by a multiple of five uniis.
Fig. 4 presents the frequency dependence of E’ in the mm-wave range and in the range
of the lowest FIR mode at selected temperatures in the paraelectric and the ferroelectric
phase, respectively. Fig. 5 refers to the corresponding data of E ” . Also inserted are the
data of the sub mm-wave measurements [4]. All results in the temperature range from 273
M. Briskot, H. M. Ham, Millimetre-wave investigation
--- -
- - . - I
- . * I
' '
' . ' . I
. . I
to 328 K can be well described by a superposition of a Debye relaxation term and that
of a damped FIR oscillator according to the equation
represents the contribution to E' of all modes at higher frequencies. The parameters
were determined by least square fits of the calculated dispersion curves to our data using
a stochastic search method [20]. These curves are shown in Figs. 4 and 5 as broken lines.
Trials to describe the mm-wave data by a heavily overdamped oscillator gave no significant improvement. The parameters of the oscillator term in (2) for the temperatures
denoted in Figs. 4 and 5 are compiled in Tab. I . They exhibit only normal temperature
Ann. Physik I (1992)
Fig. 5 Frequency dependence of E" at selected temperatures in the mm-wave range (data points) and
in the range of the lowest FIR-mode (full curves). The broken curves are calculated from Eq. (2). The
points L ' are from [4].
dependence of this mode. The temperature dependence of the reciprocal relaxation
parameters, A e i ' and T - ' , is depicted in Fig. 6. In the paraelectric phase A E follows
a Curie-Weiss law
with a Curie constant C = 1 400 f 100. The relaxation time exhibits critical slowing down
and can be described by
M. Briskot, H. M. Happ, Millimetre-wave investigation
Table 1 Parameters of the FIR oscillator term in Eq. (2).
27 3
Estimated errors:
So < 1 % ;
< 5%.
Fig. 6 Temperature dependence of the
T-' of the
reciprocal parameters E ~ and
relaxation term in Eq. (2).
Temperature [K]
with 7 0 = (1.67 ? 0.06) *
s. A relation of this type for the relaxation time, with T
in the numerator, is obtained in the mean field approximation of the dynamic
susceptibility of the Ising model (see e.g. Blinc and Zeki [21]). Near T, this can be
simplified to the form (4).In the present case ro would correspond to the relaxation time
of single proton flips between the potential minima of the hydrogen bonds. In the
ferroelectric phase AcR deviates from a Curie-Weiss law close to T,. This might be due
to the influence of domain formation. The relaxation time again shows critical slowing
down. In applying (4)to the ferroelectric phase we have to substitute T, with the somewhat smaller value T = 308 K. We obtain 7 0 = (0.79 0.03)
s, i.e. about half
of the value in the paraelectric phase.
Ann. Physik 1 (1992)
Table 2 Comparison of relaxator parameters evaluated from different dielectric measurements;
paraelectric phase.
1400 f 100
1 600
2 200
1.67 k 0.06
I .9
this work
4 Discussion
The dielectric behaviour in the frequency range between 0.3 and 10 GHz expected
according to Eqs. (2) to (4) is in qualitative agreement with the experimental results of
our previous work [7] on different crystals within the uncertainties of the method used.
In particular, the relaxation frequency v R = ( 2 n r ) - ' in the paraelectric phase,
calculated from equation (4) shifts to 2 GHz close to T, in good agreement with the
corresponding value evaluated from the measurements in this region. The present results
are consistent with those obtained from measurements in the sub mm-wave range [4] and
the HF range [8], which were also interpreted in terms of Eqs. (2) to (4). In Tab. 2 the
relevant quantities of the relaxator are listed for comparison. Taking the now available
experimental dielectric results together it can be concluded that the soft mode of proton
dynamics in LHP is of relaxation type and can be described by a monodispersive Debye
formula. Near T, the relaxation frequencies are located in the range between about 2
and 60 GHz.
Current theories of proton dynamics in hydrogen bonded ferroelectrics are based on
the pseudo-spin model [21] in which the Ising type hamiltonian is supplemented by a
tunneling term - Q C Sf. Q is the tunneling integral and S: the x-component of the
pseudo-spin refering to the proton in the double well potential at site i. Tunneling is
introduced as a possibility to explain large isotopic shifts of T, on deuteration as Q
depends exponentially on the mass of the particles. In crystals like LHP in which
molecular groups are linked into linear chains by simple hydrogen bonds the theoretical
problem can be reduced to that of the dynamics of single chains with strong proton
coupling and weaker interchain coupling which can be taken into account by a mean field
approximation. The parameters of this model are, besides Q, the effective intrachain and
interchain coupling constants J , , and J , respectively [22]. According to this theory an
oscillatory soft mode is expected corresponding to collective in phase proton motion of
the chains, see e.g. [22, 231. However, if Q/Jll e 1 holds the dielectric susceptibility
takes on the form of a simple Debye relaxator [24, 251. Chaudhuri et al. [23] have
published a rather elaborate treatment of the LHP problem using Green's functions.
They also considered proton-phonon coupling. With a single set of model parameters
they obtained a very good fit to the experimental data of T,, the Curie constant C, the
static dielectric constant E , ( T )and the spontaneous polarisation P,( T). The parameters
representing the tunneling integral and the intrachain coupling turned out to be D =
2.17 c m - ' (65 GHz) and Jli = 172.4 cm-I. A corresponding fit to the data of the
deuterated crystal gave Q = 0.27 cm-' (8 GHz) and Jll= 251.3 c m - ' . Other theoretical
treatments of the static properties of LHP also arrive at D/Jli e 0.1 [26, 271. These
theoretical results indicate that the dielectric response of the LHP system should be of
relaxation type as is observed experimentally. Of course, a heavily overdamped oscillator
mode cannot be excluded [28].
M. Briskot, H. M. Happ, Millimetre-wave investigation
In view of the fact that tunneling appears to play no essential role in the dielectric
behaviour of LHP, recent results of a high-pressure neutron diffraction study in
deuterated LHP by Nelmes and coworkers [29] are of particular interest. It was found
that the Curie temperature of PbDPO, (452 K) takes on the same value as that of LHP
(310 K) within the uncertainty if the distance of the potential minima in the hydrogen
bonds is decreased under pressure to that observed in LHP. This indicates that geometric
effects of deuteration [30] have to be taken into account and the importance of tunneling,
introduced to explain the isotopic shift of T, has to be reconsidered.
5 Summary
The dielectric properties of LHP near to the direction of spontaneous polarisation P,
are investigated in the mm-wave range and the range of the lowest FIR mode. In the
temperature region of the ferroelectric phase transition this mode exhibits only normal
temperature dependence in agreement with earlier findings. The soft proton mode which
causes the Curie-Weiss anomaly of E, at T, is located in the MW and mm-wave range and
is of relaxation type. It can be described by a Debye formula. The relaxation time exhibits
critical slowing down, and the corresponding relaxation frequency varies between about
2 to 60 GHz near T,. The results are in general agreement with those of our previous
measurements in the MW range and in good quantitative agreement with published
results obtained in the HF and sub mm-wave range. A very small value of the tunneling
integral 52 found in theoretical treatments of the LHP problem based on the pseudo-spin
model supports the conclusion that tunneling plays no essential role in the dynamics of
the proton system.
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