Introduction
Langasite (LGS, La3Ga5SiO14) is a single
crystalline piezoelectric material suited for high-temperature applications
.
It belongs to the point group 32 and exhibits the same crystal structure as
quartz, but exceeds its operation temperature limit significantly. LGS does
not undergo any phase transformation up to its melting point at
1470 ∘C and may be piezoelectrically excited up to at least
1400 ∘C provided that stable electrodes are available
. The crystals exhibit mixed electronic and ionic
conductivity, which contributes to the loss at high temperatures. The oxygen
partial pressure-dependent conductivity impacts the performance of the
resonators in a minor way as long as the oxygen partial pressure is kept, e.g., above 10-20 bar at 600 ∘C
.
LGS can be used as a resonant sensor. When operated in the microbalance mode,
small mass changes of a layer with affinity to specific gas particles cause a
shift of the resonance frequency . Anticipated applications
of such sensors include fuel cells and gas reformers
. The stability of LGS in harsh environments such
as high temperatures and low oxygen partial pressures makes it suited for
wireless sensors based on surface acoustic wave (SAW) devices
.
Here, an accurate set of LGS material constants determined for a wide
temperature range is strongly required. The availability of such data sets is
scarce, since the published materials data are either limited to a fixed
temperature or cover only a narrow range around room
temperature ∘C . provide the components of elastic compliance tensor in the 25
to 600 ∘C temperature range, determined using the pulse-echo
ultrasonic technique .
In this work, the full set of elastic, electric and piezoelectric properties
of langasite is determined at elevated temperatures using bulk acoustic wave (BAW)
resonators of different orientations. Special attention is drawn to the
mechanical and electrical losses, which play an important role for
high-temperature applications. Subsequently, the full set of
electromechanical data is used to predict the wave velocity of SAW resonators
for two selected cuts of LGS. Their Euler angles are
(0, 138.5, 26.6∘) and
(0, 30.1, 26.6∘). Finally, the calculated values are
compared with the corresponding data extracted from the high-temperature
measurements of specially designed SAW test structures.
The LGS crystals for BAW resonators are provided by the Institute for Crystal
Growth, Berlin Adlershof, Germany. The wafers for SAW measurements are
fabricated by FOMOS-Materials, Russia, and Mitsubishi Materials Corporation,
Japan.
Model for BAW and SAW
High-temperature losses
A piezoelectric resonator operated at room temperature can be described by a
one-dimensional physical model of vibration, where electrical losses are very
low and thus negligible. At elevated temperatures, this description is not
accurate due to, e.g., the finite electrical conductivity. In this case the
elastic, dielectric and piezoelectric coefficients must be extended by
imaginary parts which express the mechanical, electrical and piezoelectric
losses.
Mechanical loss
The mechanical loss at elevated temperatures depends on the resonance
frequency f and can be described by the viscosity η. The loss of the
resonator at the angular frequency ω, with ω=2πf, is related
to the imaginary part of the elastic stiffness according to
:
c^=c+jωη.
The elastic compliance s must be extended by a viscous contributions in a
similar way. The relation between the elastic stiffness and the viscosity can
be expressed using the inverse resonant quality factor, Q-1 and the
mechanical loss tangent tanδM, by
tanδM=ℑ(c^)ℜ(c^)=ωηc≡Q-1.
Electrical loss
A perfectly insulating resonator with two parallel electrodes, excited by a
harmonic electric field E=E0ejωt, acts as a capacitor with a
dielectric material described by the permittivity ε. In
case of materials exhibiting dielectric loss, the dielectric tensor must be
regarded as a complex property consisting of real and imaginary parts
ε^=ε-jσω,
with the loss tangent tanδE given by
tanδE=ℑ(e^)ℜ(e^).
The origin of dielectric loss can be attributed to the electrical
conductivity σ of the material. Calculating the admittance Y for a
capacitor with the thickness a, the electrode area A and the dielectric
coefficient described by the complex property ε^ results in
Y=Aa(σ+jωε).
The equation describes the admittance of an electric circuit with capacitor and resistance connected in parallel.
Piezoelectric loss
In analogy to the mechanical and electrical loss, the imaginary part of the
piezoelectric coefficient can be introduced. The nature of this
“piezoelectric loss” is however not clear. It may be explained by, e.g.,
jumping of lattice defects or the movement of domain walls in polycrystalline
materials . However, for most materials
this term is negligible and assumed to be zero . The
validity of this assumption for langasite is proven in previous studies,
where calculations with complex piezoelectric coefficient are used to
evaluate the data . Therefore, in this article the
imaginary part of the piezoelectric coefficient is omitted.
Piezoelectric equations
In case of piezoelectric materials, the relation between mechanical and
electrical properties is described by the piezoelectric equation
c^eTeε^⋅SE=TD,
where the complex stiffness c^ and the piezoelectric
constant e are 4th and 3rd rank
tensors, respectively. The complex dielectric coefficient
ε^, strain S and stress
T are 2nd rank tensors. The electric field
E and electric displacement D are vectors. In
order to differentiate between tensors and tensor components, the former is
written in bold.
Due to the tensor symmetry and the crystal symmetry of LGS many tensor
components vanish, so that the number of independent coefficients in
Eq. () is reduced to 10
:
c^11c^12c^13c^1400e1100c^12c^11c^13-c^1400-e1100c^13c^13c^33000000c^14-c^140c^4400e14000000c^44c^140-e1400000c^14c^660-e110e11-e110e1400ε^11000000-e14-e110ε^11000000000ε^33,
with c^66=c^11-c^12/2. Here, the
c^ij, e^ij and ε^ij represent the
components of stiffness tensor c, piezoelectric tensor
e and dielectric coefficient tensor
ε, respectively.
Equation () applies, when the strain
S and electric field E are independent
variables. This is the case where a resonator is operated in the thickness
shear mode of vibration. In case of length-extensional mode of vibration,
where the elastic stress T and the electric field
E are independent variables, an alternative notation applies
s^dTdε^⋅TE=SD.
Here, the complex elastic compliance s^ is used instead
of the stiffness c^, and the piezoelectric constant
d replaces the coefficient e. The relation
between piezoelectric tensors e and d and the
relation between elastic compliance and stiffness are shown, e.g., in
. The latter is used to
calculate the component s12(T) of the elastic stiffness tensor from the
coefficient c66(T) as follows:
s12(T)=s11(T)-12c66(T)-2s14(T)2s44(T).
The determination of all components of the elastic stiffness and compliance
tensors requires different crystal cuts. The shear components of those
tensors determine two different modes of vibration. The stiffness
c^66 describes the thickness shear (TS) vibration of a partially
electroded Y-cut resonator, and the elastic compliance s^44
describes the face shear (FS) vibration of a rectangular Y-cut plate. The
remaining four components of the elastic compliance tensor may be determined
using differently oriented rods excited in length-extensional (LE) mode of
vibration. Here, the effective compliance seff is a
superposition of several components as function of the angle φ
between the rod and the Y axis
:
seff(φ)=s11cos4φ+s33sin4φ-2s14cos3φsinφ+s44+2s13cos2φsin2φ.
Several angles φ have to be chosen in such a way that for every
φ at least one term in Eq. () dominates. The
effective piezoelectric coefficient deff is a superposition of
two different coefficients as a function of the φ:
deff(φ)=-d11cos2φ+d14cosφsinφ.
From Eq. () it may be seen that four differently
oriented rods with angles φ of -30, 0,
30 and 45∘ are required to determine the majority
of components of the elastic compliance s^. The
coefficient s12 remains to be determined and is obtained from the
thickness shear mode of vibration as shown in Eq. ().
The sum of coefficients s44+2s14 is separated using the
face shear mode of vibration described by s44. The stiffness tensor
c^ is calculated using the relation between stiffness and
elastic compliance. Additionally, two rectangular X- and Z-cut plates are
required to determine the two independent complex components of the
dielectric tensor, ε^11 and ε^33.
The imaginary part of dielectric tensor provides the information about
conductivity σ1 and σ3 (commonly denoted as
σX and σZ, respectively) of langasite.
All crystal cuts used in this work are visualized in Fig. .
The crystal cuts of langasite
used to determine the full set of electromechanical data
. Four differently rotated rods with electrodes
perpendicular to X axis vibrate in length extensional mode (LE). The
rectangular Y-cut plate vibrates in face shear (FS) and thickness shear (TS)
modes. The rectangular X- and Z-cut plates are used for conductivity
measurements.
Models for different crystal orientations
Length-extensional mode
In case of a rod, where the width b, length l and thickness a are
chosen to be parallel to the x3, x2 and x1 axes, respectively,
and the applied electric field is parallel to the x1 axis, the
piezoelectric relation shown in Eq. ()
is reduced significantly. Due to the boundary conditions chosen here, only
the stress and strain in x2 direction are relevant. All other components
of the tensors T and S vanish. Similarly, only
the electric displacement D1 and electric field E1 are taken into
account resulting in the following equation:
S2=s^22T2+d12E1D1=d12T2+ε^11E1.
The strength of the coupling between the mechanical and electrical properties of a piezoelectric medium is
expressed as coupling factor k^, defined by
k^122=d122s^22ε^11.
With the coupling factor, Eq. () can be transformed in
T2=1s^22∂u2∂x2-d12s^22E1,D1=d12s^22∂u2∂x2+ε^111-k^122E1.
Solving Newton's equation of motion with T2 described by
Eq. () and under the assumption of harmonic time
dependence ejωt of the electric field E1, the electric
impedance of the resonator is calculated
Y=jωlbaε^1-k^2+k^22lbρs^22tanlbρs^222.
The parameter of the tangent function reflects the phase shift and a wave velocity in the bulk of a resonator:
ν^=lbρs^22.
The capacity C0 of the resonator equals
C0=ε^11lba.
From Eqs. (), () and () the admittance Y of a resonator is found to be
Y=jωC01-k^2+k^22ν^tanν^2.
Thickness shear mode
In case of a partially electroded Y-cut resonator with the electric field
applied parallel to the x2 axis, only the strain S6 and stress
T6 do not vanish. The piezoelectric relation shown in
Eq. () can be expressed as
T6=c^66S6+e26E2,D2=e26S6+ε^22E2.
Under the assumption of a harmonic time dependence of an electric field,
Eq. (), Newton's equation of motion may be transformed
analogously to the case of length extensional mode of vibration. The
resulting equation is found to be
c^66+e262ε^22∂2u1∂x22+ω2ρu1=0.
Here, a new property, the piezoelectrically stiffened shear modulus
c¯ is introduced
c¯=c^66+e262ε^22=c66+e262ε22+σ2/ε22ω2+jωη+σ21+σ22/e262ω2.
This shear modulus depends on the piezoelectric and dielectric coefficients,
as well as on the electromechanical losses, i.e., electric conductivity and
viscosity. The consequences of these relations are discussed elsewhere
.
Using these equations, the electric impedance of the thickness shear resonator is obtained
Z=1jωC01-k^t22ν^tanν^2,
with coupling factor k^t2=e262/c^66ε^22 and the reduced wave velocity ν^=ωaρ/c^.
Here, a denotes the thickness of the resonator.
Face shear mode
The face shear vibration of a Y-cut resonator is determined by the elastic
compliance s44. This two-dimensional motion cannot be described by a
physical model which has a straightforward analytical solution. Therefore,
only an approximate solution for the resonance frequency fr of such a
resonator as presented in
is used:
fr=κΘπl1ρs44.
Here, κ is a solution of transcendental equation tanκ+κ=0, and Θ is a correction factor defined as
Θ=1-12κκ2-2κ2+23/2s11+s332s44.
Parameters of SAW propagation
The parameters of SAW propagation, like velocity v, propagation loss
α and coupling coefficient k2, are calculated from the tensor
data, obtained from the impedance measurements of BAW resonators as described
in Sect. .
For this purpose the equations of motion
ρ∂2ui∂t2=∂Tij∂xj,∂Di∂xi=0,
with i,j=1,2,3, have to be solved together with the piezoelectric Eq. ()
for the semi-infinite substrate crystal of
the given cut. For this, the Green function method described in
is used. The poles and zeros of the
Green function component G44 on the complex slowness space s
correspond to surface acoustic waves, propagating on a free and on a
metallized surface, respectively. A complex pole
G44p=Ks-s0
where the value s0=s0′+js0′′ is associated with a SAW, which propagates on the free
surface with the velocity v0=1/s0′. The attenuation is
α=2πf⋅s′′s′⋅20⋅lge[dB/µs].
Here, f and e are the frequency of the SAW and the Euler number, respectively.
Parameters of SAW propagation extracted from the SAW measurements
Special test SAW devices are designed for the experimental determination of
the surface acoustic wave parameters v, α and k2. These are
four delay lines of two different lengths. Two of them have free propagation
paths, the other two propagation paths are metallized as shown in
Fig. .
Delay lines of different length.
Thereby, identical input and output interdigital transducers (IDTs) with eight electrodes per electrical period are used (see Fig. ).
IDT with eight electrodes per period.
They can be regarded as multi-electrode transducers with double spatial
sampling. As shown in , the transfer function S21 of
such IDT has four pass bands (four harmonics) with central frequencies
corresponding to 1/4f0, 3/4f0, 5/4f0, 7/4f0,
where f0=v/(2p). The pitch p corresponds to the period of the electrode
in the IDT. In our case the transfer function S21 shows the
1st, 3rd, 5th and 7th
harmonics at about 150, 450, 750 and 1050 MHz (Fig. ).
Transfer function S21 of a delay line with four
harmonics (after signal processing).
During signal processing the difference between time delay of long and short
delay lines at each harmonic is obtained very precisely .
This allows the determination of the group velocity based on the known length
difference between both delay lines. The influence of the IDTs is, therefore,
automatically eliminated. The results for the group velocity of SAWs on the
free surface and on the metallized surfaces are extrapolated to zero
frequency. This way, the phase velocity for SAW on the free LGS surface
v0 and on the short-circuited LGS surface vm (i.e., metallized
surface without mass loading by the metal film) is determined. Further, the
electromechanical coupling coefficient k2 is obtained from the phase
velocities v0 and vm using the Ingebrigtsen relation
k2=-2⋅vm-v0v0.
The propagation losses α at the harmonic frequencies are calculated to be
α=1Δt⋅20lg|S21long||S21short|[dB/µs],
where S21long and S21short are the transfer
function of the long and short delay line at the harmonic frequency,
respectively, and Δt is the difference of delay times of the two
delay lines for this harmonic.
Results and discussion
Materials parameters
Electric properties at elevated temperatures
The conductivity of the X- and Z-cut langasite plates is summarized in Fig. . The anisotropy of the conductivity is clearly visible.
Conductivity of LGS as function of
temperature.
The full set of elastic stiffness
coefficients of langasite as function of temperature calculated from BAW
measurements (solid line) in comparison with the data obtained by (dotted line).
The piezoelectric
coefficient of langasite as function of temperature.
The inverse resonant quality factor Q-1 and the electric conductivity σX of langasite rod operated at 220 kHz in length extensional mode of vibration.
Full set of piezoelectric and elastic properties
As a result of the procedure described in Sect. ,
the complete elastic compliance tensor and the piezoelectric tensor as a
function of temperature, s(T) and d(T), are
available. They are used to calculate the stiffness tensor
c(T) and the piezoelectric tensor e(T) using
the respective relations. The elastic stiffness and the piezoelectric
coefficient as function of temperature are summarized in
Figs. and , respectively.
A comparison of elastic data from this work with data obtained by . show a very good agreement except for
the stiffness coefficient c33 which differs by about 5 % from the
referenced work. In case of the purely acoustic method as used by it is possible to measure the phase velocity along the x3
axis directly. However, in case of resonant measurement, the excitation of
resonant vibration in this direction is not possible. As shown in
Eq. (), all components of piezoelectric tensor
d or e related to the x3 direction equal
zero. Therefore, the stiffness coefficient c33 as well as the elastic
compliance s33 can be determined using the effective stiffness or
compliance resulting from a vibration of a rotated crystal. In the related
Eq. (), the term sin4φ ,
dominates only for 45∘ rotated rod and nearly vanishes for all
other rotation angles. For this reason the coefficients s33 and c33
are most error-prone which explains the small discrepancy mentioned above.
Temperature-dependent loss
As already mentioned in Sect. , the
electrical conductivity and mechanical properties such as viscosity
contribute to the losses in high-temperature piezoelectric materials. The
viscosity of the langasite rod with the angle φ=0 (see
Fig. ) expressed in form of the inverse resonant
quality factor Q-1 (see Eq. ) is calculated
from the physical fit. These data and the electric conductivity
σX are shown in Fig. . Here, a
maximum of loss around 500 ∘C is clearly visible.
SAW properties
The characterization of SAW devices is carried out up to about
730 ∘C where the Zr / Pt-metallization failed through de-wetting
and decomposition into droplets as visualized in
Fig. .
Dewetted metal layer after heating
up to 800 ∘C.
Phase velocity of
Rayleigh waves on free surface v0 (solid line) and on metallized surface vm (dotted line), cut (0, 138.5, 26.6∘).
Comparison of our measurement results
with literature data. In case of , the values are calculated using tensor data.
cut
v0, m s-1
TCF1, ppm K-1
TCF2, ppb K-2
(0, 138.5, 26.6∘)
2741.9
1.5
-41
(0, 30.1, 26.6∘)
2464.6
28
-39
results of other authors
(0, 138.5, 26.6∘)
2741.8a
-7a
-51a
2734b
1b
–
2743c
–
–
(0, 30.1, 26.6∘)
2462.1a
31a
-69a
2392.1c
–
–
a , b ,
c .
Coupling coefficient k2
for Rayleigh waves on cut (0, 138.5, 26.6∘), calculated with tensor
data from BAW measurements (dashed lines) and extracted from SAW measurements (points).
The propagation loss in
[dB/µs] of Rayleigh waves on free surface on both cuts
(0, 138.5, 26.6∘) and
(0, 30.1, 26.6∘), measured at different frequencies.
The phase velocities v0 and vm on free and metallized surfaces for
the cut (0, 138.5, 26.6∘) are shown in
Fig. . The obtained phase velocities at room
temperature and coefficients of frequency for a second order polynomial fit
are given together with literature data in
Table .
Figure shows the electromechanical coupling
coefficient calculated from Eq. () for the propagation geometry
with Euler angles (0, 138.5, 26.6∘) and from the
tensor data of BAW resonators. The difference between BAW and SAW data for
k2 are caused by the uncertainty of the SAW method. As shown in
Eq. (), k2 results from subtraction of two large numbers
which were obtained by an extrapolation procedure. Furthermore, room
temperature data are in satisfactory agreement with the corresponding
electromechanical coupling coefficient, k2=0.33, calculated from the
tensor data given in , and with data
obtained by other authors, k2=0.5 and
k2=0.34 .
The propagation loss in
[dB/µs] of Rayleigh waves on metallized surface on cut
(0, 138.5, 26.6∘) for different frequencies.
The propagation losses α on free surfaces for both investigated cuts
of langasite are shown in Fig. . Similar to
the langasite rods operated in length-extensional mode of vibration, SAW
devices exhibit a local propagation loss maximum at around
520 ∘C for all measured frequencies.
The propagation loss on the metallized surface shown in
Fig. increases monotonically. However, they
exhibit sharp changes of the slope approximately at the same temperature as
for the free surface.
Stability of the electrodes for BAW and SAW resonators
LGS-based SAW and BAW devices exhibit an operation temperature limit caused
by the stability of the platinum electrodes. It is found, that their lifetime
drops drastically as their thickness decreases. This effect is described by,
e.g., . Due to this effect
the SAW devices used in this work failed at about 730 ∘C,
whereas the BAW resonators wearing thicker electrodes could be operated up to
900 ∘C. If more stable electrodes, obtained by, e.g., screen
printing, are used, even higher operation temperatures for BAW devices are
demonstrated .
Above the mentioned temperatures, de-wetting of the metal layer is observed,
as shown in Fig. for SAW structures. Several
attempts have been made to investigate and possibly reduce this effect. Da
Cunha et al. conducted a series of experiments with different layer
combinations for the metallization to extend the temperature limits for the
SAW devices . Most recent attempts to
minimize the degradation of electrodes and prolong the lifetime of SAW
elements include use of metal alloys as well as high-temperature ceramic
electrodes. Richter et al. report about successful operation of SAW
elements at 800 ∘C for several hours using Ti / Pt coated
langasite . Further attempts include application of
thin protective layers, like Al2O3. An example of successful
application of multi-layer stack metallization consisting of
Al / AlxOy / Pt for the high-temperature sensors is given in
.
Correlation of SAW-measured phase velocities with calculated BAW data
In order to compare the bulk and surface acoustic wave properties, the full
set of temperature-dependent materials data is determined using BAW
resonators and applied to calculate the phase velocity v0 of SAW devices
with Euler angles of (0, 138.5, 26.6∘) and
(0, 138.5, 26.6∘). As seen in
Fig. , the results of the calculation and the
measured data are in good agreement especially for the
(0, 138.5, 26.6∘) cut, which confirms the validity of
the fitted parameters and the calculated materials data.
Comparison of Rayleigh wave velocity on free
surface v0 obtained from SAW and BAW measurements (solid and dashed lines, respectively).
Conclusions
A method of materials data extraction from measured data of BAW resonators is
developed for piezoelectric crystals. All components of the stiffness,
elastic compliance and piezoelectric tensors for langasite are determined in
the temperature range from 20 to 900 ∘C. Furthermore, the
conductivity of the crystal in X and Y directions and the quality factor Q
for resonant deformations in the X direction are obtained as function of
temperature. For these experiments, BAW resonators with frequencies in the
range 200–5000 kHz are used. The obtained elastic stiffness tensor is in
good agreement with data published by other authors, except for the stiffness
coefficient c33, which exhibits a deviation of about 5 % from the
literature data.
Based on the obtained tensor data, the phase velocity and the coupling
coefficient for SAW propagation of the crystal cuts
(0, 138.5, 26.6∘) and
(0, 30.1, 26.6∘) is calculated. The same parameters
are extracted from measurements of SAW delay lines up to 730 ∘C,
which are fabricated using the crystal cuts mentioned above and operated at
frequencies from 150 up to 1050 MHz.
Values for the velocity of SAW propagation and coupling coefficients obtained
from SAW and from BAW measurements show a good agreement. SAW devices exhibit
a local maximum of propagation loss at around 520 ∘C. BAW rods
exhibit a maximum of losses at the same temperature.
Thin film Pt electrodes for BAW and SAW devices limit the maximum temperature
of measurement to 900 and 730 ∘C, respectively. The operating
temperature may be significantly increased by application of protective
layers and thick electrodes. The latter is, however, feasible in the case of BAW
resonators only.