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Dielectric dispersion of ferroelectric ceramics and single crystals at microwave frequencies.

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Ann. Physik 3 (1994) 578-588
der Physik
0 Johann Ambrosius Barth 1994
Dielectric dispersion of ferroelectric ceramics
and single crystals at microwave frequencies
G. Arlt, U. Bottger, and S. Witte
Institut fiir Werkstoffe der Elektrotechnik, Aachen University of Technology, D-52056 Aachen
Received 4 February 1994, 1st revision 9 August 1994, 2nd revision 25 August 1994,
accepted 25 August 1994
Abstract. The dielectric constant of ferroelectric ceramics shows strong Debye-like relaxation at GHz
frequencies. This relaxation is caused by sound generation inside the crystallites. The ferroelastic domain walls which vibrate in the electric ac field are very effective shear wave transducers. The shear
wave emission has a maximum when the wavelength of the sound wave is comparable to the width of
the domains. Above this frequency the domain walls can no longer vibrate due to inertia and do not
contribute to the dielectric constant. The observed dependence of the microwave dispersion on temperature, doping and grain size of the ferroelectric ceramic is in good agreement with the proposed
mechanism of the emission of acoustic shear waves.
Keywords: Ferroelectricity; Domain walls; Acoustic shear waves.
1 Introduction
Strong dielectric dispersion in ferroelectrics at microwave frequencies has been observed
and discussed since about 40 years ago. Von Hippel [l] in 1952 has found this dispersion
in ceramics of Barium Titanate (BT). Gerson and Petersen [2] have measured similar
relaxation effects in Lead Zirconate Titanate (PZT). In ceramics of Lithium Niobate
(LN) Yao, McKinstry and Cross [3] have found a comparatively small ralaxation step
of the dielectric constant. They explained this effect with' sound emission by the
piezoelectric grains. Many more detailed dielectric measurements on ceramics of BT
and PZT were reported by several investigators Poplavko et al. [4], Kersten et al. [5, 61,
n r i k et al. [7, 8, 91, B6ttger et al. [ 10, 1 I] and Li et al. [12]). Common results in most
of these investigations were the following observations: 1) The relaxation curves in the
Cole-Cole diagram were almost semicircular, i.e. the relaxation is Debye-like. 2) The
relaxation step in ceramics of BT and in PZT is very large up to A E 1000. It is increasing with temperature. 3) The relaxation frequency is almost independent of temperature
up to the Curie temperature. Above the Curie temperature the relaxation step decreases
strongly and the relaxation frequency is dependent on temperature. 4) On different
samples of the same material the relaxation frequencies and steps are different.
Turik et al. [9] and Magfione et d. [I31 have investigated the dielectric dispersion
in single crystals of BT. Whereas Turik's results do not give a clear answer whether
the relaxation effect also exists in single crystals, Maglione et al. have found large relaxation steps near 10' Hz in plain and in Fe doped BT single crystals above and below
G. Ark et al., Dielectric dispersion of ferroelectric ceramics at microwave frequencies
the Curie temperature. They attribute the effect to the hopping of off center Ti ions.
Nothing is known about the domain configuration of their single crystals.
Ferroelectric ceramics of the perovskite type have a characteristic domain configuration caused by the minimization of energy [141. The configuration is well known in the
case of tetragonally distorted ceramics: the domains form regular stacks with 90 domain walls (DWs) between the laminar domains as shown in Fig. 1. Single crystals contain in most cases regions with similar more coarse 90" stacks. The width of the domains is small compared with their lateral size. In rhombohedrally distorted perovskites wedge shaped domains with approximately 70 O and 110 " domain wails form a
moreaor less irregular configuration. The width of the domains in ceramics is of the
order of 1 pm.
We have shown in a former paper [I51 that 90" DWs in an electric ac field emit
elastic shear waves. All calculations were carried out using the coordinate system
shown in Fig. 1. Correspondingly the rotated material constants e.g. dielectric,
piezoelectric and elastic constants have to be used. The matrix of the elastic constants
in the rotated system indicates that shear and longitudinal waves propagating in x3
direction are coupled. For simplicity this mode coupling is neglected in the treatment.
In a separate paper [16] the dielectric dispersion will be treated which is caused by
the emission of sound waves from piezoelectric domains.
2 Sound generation by domain walls
The single crystalline domain stack shown in Fig. 1 in the paraelectric state is a perfect
cube, in the ferroelectric state with spontaneous polarization Poit has serrated edges
with a = So due to the spontaneous deformation So. In equilibrium the polarization
charge is screened by free carriers such that no electric fields are present. The stack can
be deformed e.g. by an electric field E , which increases type I domain and decreases
type I1 domain. If type11 domains were perfectly driven out of the stack its shape
would be a parallelogram as indicated by dotted lines. Then a shift of mass in x1direction would occur. Fig. 2 shows in the upper part more detailed the mass shift u1 and
Fig. 1 Stack of four domains with
ferroelastic domain walls.
Ann. Physik 3 (1994)
adjacent grains
adjacent grains
Fig. 2 Upper part: two domains in equilibrium (full lines) and with displaced domain
walls (dashed lines). The mass shift above
and below the wall becomes visible. Lower
part: the domain wall is schematically replaced by a transducer and the environment
of the two domains is indicated.
u2 above and below the displaced DW. Although the displacement A1 alternates in
sign in a stack from wall to wall, the mass shift u1 and u2 above and below each wall
remains as in the example in the upper part of Fig. 2. These shifts add up to a gross
shear of the stack. The reversible displacement of the DWs from their equilibrium positions is assumed to be governed by a force constant per area k and by a force per area
due to an electric field El or an elastic stress field T5which was calculated before [17]
A lk = force per area = f i P o E , + 2S0T5 .
The shifted DWs cause polarization charges at the serrated edges which induce free
charges on the distant electrodes (not shown). These charges are the source of a dielectric displacement AD, in the stack which is
with the domain width d. The two static displacement Eqs. (1) and (2) also hold in alterl
nating fields, though, with a dynamic force constant k. A relation between A l and ul
and uq can be derived for the geometric conditions shown in Fig. 2. The transducer
equations for the DWs which we derived previously [is] follow from (1) and (2)
u, - uz = j w C,,,NE, j w C,,,T5
G.Arlt et al., Dielectric dispersion of ferroelectric ceramics at microwave frequencies
58 1
Here em is the intrinsic dielectric constant of the ferroelectric material in the xi direction. Cmand N are the following abbreviations
Cm= 4 S i / k
and N = P O / ( f i S o ).
The Eqs. (3) and (4) contain neither terms of inertia nor terms of attenuation. They
describe the reciprocal transducer properties of the DW as a lumped element. The
displacement amplitude k A 1 of the DW motion is much smaller than the wavelength
of the emitted sound wave. The DW mass therefore can be neglected. In the lower part
of Fig. 2 the DW is schematically replaced by a transducer. The acoustic properties of
the matter above and below the domain wall is characterized by the complex acoustic
impedances Z1 and Z2 which are defined as
Z, =
- T5/ti1
and Z, =
+ T5/u2 .
If the impedances are known from theory, ti,, ti2 and Ts can be eliminated from the
transducer Eqs. (3) and (4). In this case a direct relation between dl and El is found
from which the complex dielectric constant E = &'-j&'' can be derived
dl/El = j o e o ( E ' - j c f f )
In ceramics the regions above and below the DW contain grains which all have
similar elastic constants for shear waves. The domain wall as transducer emits shear
waves into these regions where they are scattered and attenuated. If the DW has a
diameter larger than the wavelength of the emitted shear wave the impedance is the
acoustic impedance 2, for plane shear waves
with the density e of the ferroelectric and its average elastic shear constant cTs. The
dielectric constant derived from the transducer Eq. (7) then is
which is a Debye relaxation with a relaxation step
and a relaxation time
The relaxation step is identical with the domain wall contribution to the dielectric
constant determined formerly [18]. But a correction factor 113 has to be introduced
because in the unpolarized ceramic only about 1/3 of all grains are oriented such that
the electric field excites shear waves.
Ann. Physik 3 (1994)
When the diameter of the emitting domain wall is not larger than the wavelength of
the emitted shear wave Z,,, in (1 1) has to be replaced by a complex impedance Z, = Z,
which is strongly frequency dependent. Whereas the above solution of ~ ( o
) the
acoustic transmission line impedance 2, is purely of the Debye type, the solution for
small emitters (diameter IA ) shows a decrease oft; in a much broader frequency range.
The magnitude of the step is, however, the same in both cases.
The model used above for the derivation of the Debye equation has the following
limitation: When the DW is displaced the environment of the wall is elastically stressed.
In the model the reversible elastic energy of this effect is stored exactly at the site of
the DW, in nature it is stored as elastic energy up to a depth d/2 above and below the
DW. The dielectric response therefore can moderately deviate from the Debye response.
3 The force constant k
In a free laminar single crystal as that shown in Fig. 1 the DWs can be displaced reversibly by very weak fields. Larger fields cause irreversible wall motion. The nature of
the restoring force in a perfect crystal is not quite clear yet. The walls are probably kept
in equilibrium position by point defects inside the crystal and at the surface of the
crystal. Details are not known. The defects may thus lead to a force constant ksc. At
frequencies below the resonance frequency of the sample the DW displacement contributes not only to the piezoelectric constants but it also contributes to the dielectric
and to the elastic constants. In single crystals these contributions have not yet been
studied neither theoretically nor experimentally. At frequencies above the acoustic
resonance of the sample the force constant is much higher mainly due to elastic deformations inside the crystal which counteract the deformations caused by the displacement of the DWs. It is the same effect which prevails in ceramics at high frequencies.
Let us, for example, consider the laminar crystal shown in Fig. 1 as a grain of a
ceramic. Two different degrees of clamping can be considered.
At frequencies below the acoustic resonance of the grain the grain can deform only
as much as the adjacent grains allow. The force constant for this effect was calculated
elsewhere [17] by assuming the grain being embedded in an elastic matrix having the
same elastic constants as the grain. The result was
kgrain = ZCssSi/d
with z = 2. Actually the environment of the grain is not only elastic, it is also piezoelectric and it also has DWs which also try to deform their environment. This can lead to
a stronger clamping with z between 2 and 4. Perfect clamping of the grain is reached
for z = 4. The electric part of the force constant [I71 need not be considered since it
is coupled to the electrodes. Adjacent grains according to their orientation can either
clamp each other perfectly or they can be excited to a common vibration mode. The
distribution of the resonance frequencies therefore will be much wider than the distribution of the grain size.
At frequencies above the acoustic resonance of the grain the domains can still be
deformed, the gross shape of the grain, however, has to be preserved. Fig. 3 shows the
positions of the DWs in equilibrium and displaced. The deformation caused by the
DWs is counteracted by the elastic shear deformation of the domains. The force constant for this case can be derived from the elastic deformation shown in Fig. 3 and is
G. Arlt et al., Dielectric dispersion of ferroelectric ceramics at microwave frequencies
Fig. 3 Deformation of the domains under preservation of the
gross shape of the grain. The
equilibrium condition is drawn in
full lines, the deformed domains
are drawn in dashed lines.
Well above the domain resonance the inertia of the domains does no longer allow DW
and mass motion.
Another physical mechanism [18] can enhance the force constants: acceptor defects
in the ferroelectric can reduce the DW mobility considerably. It is the same kind of
defects which cause electric or elastic internal bias fields Ei and Ti (fields which shift
the hysteresis curve on the field axis). Theterm kdefwith the wall thickness a is given
and has to be added to the force constant (13). The relaxation frequency f, (11) is
shifted to a higher frequency by this effect.
4 Experimental results and discussion
The calculation of the dielectric constant is based on the emission of acoustic waves
from one isolated DW. In a stack many DWs are arranged in series. They all emit and
receive acoustic waves and therefore strong interaction and strong resonances are expected in the stacks. At the upper and lower face (orientation according to Fig. 1)
waves are launched into the neighboring grains. Using network theory we have
calculated the resonance inside the cascade and the emission from both ends of the
cascade. In spite of these strong resonances in the grain the relaxation curve of the real
ceramic sample shows the same relaxation as with only one DW. Sharp resonances,
however, disappear due to the distribution of the domain width d. In the calculation
a shear elastic constant c, was used. It was mentioned in the introduction that due to
the existence of the elastic constant cS3the waves are quasi-shear waves. For the interaction it seems to be important that the sign of c53alternates from domain to domain.
"* T
Ann. Physik 3 (1994)
Fig. 4 Measured a) and calculated b)
dielectric dispersion of Barium Titanate
If (5) and (8) and the force constant k = kdomare introduced into the relaxation
Eqs. (10) and (11) one gets the following results for the relaxation step and for the
relaxation frequency
Measurements were performed on ceramics of Barium Titanate (BT) and Lead Zirconate Titanate (PZT). The measuring method, the equipment and some results were
presented previously [10, 111.
In BT the measured relaxation and calculated relaxation curves are presented in
Fig. 4. The following values [I91 were used for the calculation: Po= 0.26C/m2,
So = 0.01, c55= c g = 34.02GN/m2 and e = 6.02.103kg/m3. The domain width is
assumed to be d = 1 pm and the intrinsic dielectric constant E , = 1100. There seems
to be a good qualitative agreement between the two curves.
Fig. 5 shows the increase of the relaxation step with temperature for tetragonal PZT.
The relaxation frequency is independent of temperature. The increase of A & is confirmed by the above formula for A&:in BT the spontaneous deformation So is dependent on the spontaneous polarization by So = QP;where Q is an electrostrictive constant which is not strongly dependent on temperature. Introduced in (15) A & then
becomes proportional 1/ P i . Between room temperature and the Curie temperature Po
decreases by a factor of 1/2. The factor 4 for the increase of A &in this temperature
range is consistent with measurements. In Fig. 6 this increase of he with temperature
is shown for coarse grained BT. The Curie temperature of the sample is Tc= 415 K.
In single crystals [13] and in ceramics [7, 211 strongly temperature dependent dielectric
relaxations are observed also near the Curie temperature in the paraelectric phase. We
believe that fast and slow polarization fluctuations exist in this phase. The boundaries
of these fluctuations will also emit acoustic waves if they are forced to vibrate by electric or elastic fields.
For fine grained tetragonal and rhombohedral PZT ceramics the measured relaxation curves are given in Fig.7. The elastic constants cS5are not known for these
ceramics. In rhombohedral ceramics the domain width was about 0.5 pm. The relaxation frequency near 1 GHz then indicates that the elastic constant of the rhombohedral
ceramic is similar to that of BT. This is different in the tetragonal ceramic. Here the
G. Ark et al., Dielectric dispersion of ferroelectric ceramics at microwave frequencies
Fig. 5 Temperature dependence of the
relaxation curves of tetragonal Lead Zirconate Titanate
Fig. 6 Dielectric dispersion in coarse grained Barium Titanate ceramics [tl]. The
relaxation frequency is independent of temperature, the relaxation step increases with
temperature. Above the Curie temperature
T, = 415 K the dispersion is still present, it
decreases with further increase of the temperature.
' I . " ' :
0, I
domain width is of the order of 0.1 pm. Either the elastic constant in this material
(which is near the morphotropic phase boundary) is much smaller than in BT or an additional still unknown effect lowers the relaxation frequency.
Well aged BT which was doped with different amounts of Ni has relaxation curves
as shown in (Fig. 8). Ni-doping introduces internal bias fields Ei which according to
(14) increase the relaxation frequencies and lower the relaxation steps. Both of these
Ann. Physik 3 (1994)
... ...
. .. .. .. ..... . . . .
. .............
Fig. 7 Dielectric dispersion in
tetragonal (T) and rhombohedra1 (R)
Lead Zirconate Titanate ceramics [lo]
. "":
PZ,--x T,.
e ' .................................
0.05 mat?'. Ni
0.2 mot% Ni
300 .'
log (f I
Fig. 8 Dielectric dispersion in Ni-doped
Barium Titanate [ll]. Ni-doping shifts
the relaxation frequency to higher
values and decreases the relaxation step.
G. Arlt et al., Dielectric dispersion of ferroelectric ceramics at microwave frequencies
effects appear in the measurements Fig. 8. The equivalent effect was observed in Mndoped PZT by Kersten and Schmidt [6] and in Fe-doped PZT by Bijttger [21]. In addition the measurements were extended down to low frequencies. The slight decrease of
the dielectric constant from the levels at 106Hz to the levels before the relaxation
begins at lo8- lo9 Hz can be due to grain relaxation. It is noteworthy that grain relaxation constitutes only a very small or no contribution to the total relaxation.
Figs. 9 and 10 show measurements on coarse and fine grained PZT and BT. In the
case of PZT the grain size is different by a factor 20. The domain size in the fine
= 4.5 times smaller than in coarse
grained ceramic [14] then is expected to be
grained PZT. Accordingly the relaxation frequencies are expected to have the same
relationship which is confirmed by the measurements. Here both curves still are Debye
like indicating that the domain width which is about L / 2 at the relaxation frequency
is smaller than the domain diameter. In Fig. 10 the very fine grained BT has a very
broad relaxation curve. The grains having a diameter of about 0.2 pm have only one
or two domains. At 1 GHz the wavelength of the shear wave is about 2 pm. The emitting DWs or grains thus have a diameter much smaller than the wavelength of the
sound emitted which explains the broad relaxation curve.
Likewise many dielectric dispersion effects in relaxor ferroelectrics [20] can be attributed to sound emission from the vibrating domain walls and phase boundaries of
small ferroelectric clusters.
Fig. 9 Dielectric dispersion in coarse
and fine grained Lead Zirconate
Titanate P Zo,sz
Fig. 10 Dielectric dispersion in coarse
and very fine grained Barium Tiranate.
Ann. Physik 3 (1994)
In single crystals the domains are more irregular than in ceramics. In certain regions
of the crystals nests of parallel ferroelectric domains can be found. The DWs between
these domains can similarly emit shear waves which are scattered at other domains and
reflected at the surface before being absorbed.
5 Conclusions
Sound emission by ferroelastic domain walls is the main origin of the strong dielectric
dispersion in ferroelectric ceramics at high frequencies. The model of the domain wall
as shear wave emitter Fan explain all observed peculiarities of the relaxation step and
of the relaxation frequencies: the Debye like relaxation, the dependence on frequency,
on temperature and on doping.
In single crystals the wall displacement enhances the gross piezoelectric effect at frequencies below the resonance of the sample. Above this frequency ferroelastic domain
walls also cause dielectric dispersion by the emission of elastic waves.
In ceramics which are exposed to electric ac fields near the relaxation frequencies the
density of acoustic shear wave energy is expected to be high. A considerable fraction
of the incident electromagnetic energy is expected to be converted into acoustic waves.
It seems promising to detect these ultrasonic waves directly by light scattering.
Furthermore it appears technologically feasible to deposit a stable laminar domain
system as in Fig. 1 in an appropriate embedding as a very effective shear wave transducer.
We thank Dr. A.N. Pertsev, St. Petersburg and Dr. J. Fousek, Prague for stimulating discussions
and the Deutsche Forschungsgemeinschaft for financial support.
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[21] U. BBttger, Thesis RWTH Aachen, 1994
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crystals, microwave, dielectric, frequencies, ferroelectric, single, ceramic, dispersion
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