Multi-parameter sensing is examined for thickness shear mode (TSM) resonators that are in mechanical contact with thin films and half-spaces on both sides. An expression for the frequency-dependent electrical admittance of such a system is derived which delivers insight into the set of material and geometry parameters accessible by measurement. Further analysis addresses to the problem of accuracy of extracted parameters at a given uncertainty of experiment. Crucial quantities are the sensitivities of measurement quantities with respect to the searched parameters determined as the first derivatives by using tentative material and geometry parameters. These sensitivities form a Jacobian matrix which is used for the exemplary study of a system consisting of a TSM resonator of AT-cut quartz coated by a copper layer and a glycerol half-space on top. Resonant and anti-resonant frequencies and bandwidths up to the 16th overtone are evaluated in order to extract the full set of six material–geometry parameters of this system as accurately as possible. One further outcome is that the number of employed measurement values can be extremely reduced when making use of the knowledge of the Jacobian matrix calculated before.

For many years thickness shear mode (TSM) resonators such as the quartz crystal microbalance (QCM) measuring systems are in use for direct recording of mechanically varying situations on a small geometric scale. The essential part of such devices is a piezoelectric plate with electrodes on both sides, enabling the excitation of thickness shear mode vibrations by applying an alternating current (a.c.) voltage. The configuration implies the occurrence of resonant behavior, i.e., a small frequency range with strongly increased oscillation amplitude at a given applied voltage amplitude. The peak frequency (resonant frequency) is shifted under the influence of changed boundary conditions at the plate surface, such as the case at the deposition of a thin layer or at an impact of an adjacent fluidic (gaseous and liquid) medium on one or both plate surfaces. Just these plausible exemplary situations have been crucial for the first relevant publications on this matter by Sauerbrey (1959) and King Jr. (1964).

The considered configuration of a TSM resonator also includes the appearance of higher harmonics. Thus, the examination of different harmonic resonant frequencies can enlarge the sensing issues of the device (Johannsmann, 2001; Q-Sense E4 Operator Manual, 2010). To an increasing degree, biological configurations are also explored (Li et al., 2005; Eisele et al., 2012; Schönwälder et al., 2014) which are lossy as a rule. For a long time, the widths of resonance peaks have also been used for the mechanical evaluation of a lossy material system under study (Rodahl and Kasemo, 1996; Johannsmann et al., 2009; Oberfrank et al., 2016). A common parameter to describe that bandwidth is the so-called full width at half-maximum (FWHM), given as the difference of frequencies belonging to one-half of the squared maximum of admittance amplitude. Due to the existence of so-called anti-resonant frequencies for piezoelectric plates, i.e., the frequency of minimum admittance amplitude, the bandwidths are influenced not only by the mechanical properties but also by the piezoelectric strength of a TSM plate. We define the FWHM of anti-resonance as the difference of frequencies belonging to one-half of the squared maximum of impedance amplitude.

Obviously, a considerable number of measuring data can be taken from TSM resonator experiments which are worth being examined as a function of mechanical properties of the surroundings of a TSM plate. The approach to monitor more than one changing material parameter has been made by several authors (see, for example, Martin et al., 1991; Lucklum et al., 1999; Johannsmann, 2008), but the aim of this work is to treat the problem in a more general manner.

We extend the procedure of sensing properties to many parameters and search for suitable combinations of experiments in order to achieve accurate results as much as possible. The ability for the design of an optimal experimental strategy is the final goal of our study. The theoretical treatment of that problem is based on non-approximated relations between the experimental quantities, i.e., admittance and impedance vs. frequency, and the material parameters (Weihnacht et al., 2007; Bruenig et al., 2008). With this, the popular way of performing that process in terms of equivalent circuits, as introduced by Butterworth (1915) and Van Dyke (1926) for the first time, will be avoided. Having the mentioned relations, all derivatives of significant frequencies (resonant, anti-resonant frequencies, and FWHM frequencies) with respect to the searched material parameters can be calculated. This will be done on the basis of a set of material parameters assumed to be reasonable. The mentioned derivatives have the character of sensitivities and form a Jacobian matrix.

The determination of uncertainties of extractable parameters from TSM resonator measurements are found in the literature only in exceptional cases (Lucklum et al., 1998). A thorough treatment of that problem for the complex situation of multi-parameter sensing is the content of the final part of this study. It will be carried out in a quite similar manner by using Jacobian matrices as was calculated for surface acoustic waves (SAW) in literature (Kovacs et al., 1988; Weihnacht et al., 2017).

The general aim of the present study is consistent with accentuation formulated in current literature (see, for example, Rupitsch, 2019): “simulation-based material characterization” is “of great interest to science and industry”.

In view of the complexity of the addressed task, we start with the electromechanical basics of TSM resonators. The configuration of the material system under study is shown in Fig. 1. It consists of a piezoelectric single crystal plate, such as quartz, with electrodes, stacks of layers, and half-spaces on the bottom and on top. According to the TSM concept it is one-dimensional in the direction normal to the interfaces. The application of an a.c. voltage at the electrodes creates frequency-dependent oscillations in the system.

One-dimensional material configuration consisting of a piezoelectric plate, such as quartz, with electrodes, stacks of layers, and half-spaces on both sides.

The sensing behavior of TSM resonators is based on the correct mathematical description of the dynamic behavior of the material system of Fig. 1. The one-dimensionality results in a treatment which comprises the propagation of bulk acoustic waves (BAWs) in the direction normal to the surfaces for each part of the material system and the fulfillment of boundary conditions at all interfaces and surfaces. The electromechanical material properties of quartz plate, layers, and of fluidic half-space result in specific BAW parameters in each case. BAW parameters are the phase velocity which can have an imaginary part in lossy materials, besides the particle displacements and the electric potential of wave.

For simplicity we assume isotropic symmetry for the layers and for the
half-spaces. We use one more simplification: due to the coverage of quartz
plate by electrodes on both sides, the electric potential is constant
outside the plate. So we focus the discussion on the more
complicated BAW behavior in the quartz plate for now. According to the point group
symmetry 32, quartz has six independent elastic stiffness constants
(

Two preferential orientations of quartz crystals for TSM
applications: AT and SC cuts represent directions with small first- and
second-order temperature coefficients of frequency constant (TCF).
Besides this, the SC cut is stress-insensitive. The initial (

Equation of motion:

Equation (9) will be simplified for certain crystal orientations, e.g., in our
case of the 32 point group symmetry, for so-called singly rotated
orientations (

A lot of orientations of the quartz plate exist that have especially suitable properties for TSM applications. Exemplarily, we consider here the so-called AT and SC cuts (Fig. 2). Both are temperature-stable at room temperature, and the SC cut additionally features stress insensitivity. The AT cut is singly rotated and can be described by the simpler Eq. (9a) in contrast to the doubly rotated SC cut that follows the general case of Eq. (9). It can be assumed that the main features elaborated in the next sections qualitatively may also apply for doubly rotated TSM resonator plates.

Now we confine the further studies on singly rotated orientations because of
expectable straightforward relationships. The solutions of the eigenvalue
problem of Eq. (9a) are given by the following:

The TSM resonant phenomenon as the primary object of this publication is the result of interference of forward and backward BAWs reflected at the boundaries of the plate which can be constructive with maximum vibration amplitude at certain frequencies. The piezoelectricity of the plate material enables one to understand the complete dynamic behavior by the measurement of electric quantities vs. frequency using the surface electrodes. In order to get mathematical expressions for that, the results of BAW propagation of Sect. 2.1 and 2.2 will be combined in the following with the boundary conditions.

We continue with the assumption of a singly rotated piezoelectric crystal plate
of 32 symmetry in the middle of the material structure of Fig. 1. All other
parts (layers and half-spaces) should be isotropic. The wave propagation is
described by Eq. (9a) in cases of isotropy with

At present we reduce the material structure under study to the
piezoelectric plate of thickness

Before admittance calculation as the eventual aim, the full solution of
shear BAWs will be elaborated. Rewriting Eqs. (3) and (4) for our case of the
piezoelectric shear BAW in singly rotated 32 crystals delivers the following:

In order to get the relationship between voltage

The current

The admittance

With a filled half-space on one side (

In the next step layers on the side of the material configuration of Fig. 1
will be incorporated. Continuity for particle displacements and stresses
applies on all layer interfaces. In our special case of pure shear wave
propagation normal for surfaces of isotropic layers, we have only one
component

At the end of the same transformations of the system of equations as
calculated above but now with more complicated expressions one realizes the comfortable
circumstance that Eq. (28) still applies but with effective acoustic
impedances

Here it should be noted that many of obtained relations concerning the admittance of a TSM resonator coated with layers and that is in contact with a liquid half-space have already been derived in literature, but mostly in the frame of equivalent circuitry and Mason transmission line (Mason, 1948) treatment (see e.g., Lucklum, 2002; Johannsmann, 2014).

Equation (32) for the electrical admittance of acoustic pure shear waves propagating through a material system, as shown in Fig. 1, results in a specific periodicity caused by the trigonometric functions being contained. This is the well-known resonant and anti-resonant behavior, being characteristic of piezoelectric structures. Due to the interference of up and down waves, one has special vibration situations at certain frequencies. Two examples for the admittance amplitude depicted over a wide frequency range are shown in Fig. 3a.

Calculated admittance amplitude as a function of frequency according
to Eq. (32) for two cases of TSM material configurations: (1) AT-cut quartz
plate with a liquid half-space of glycerol and (2) the same system as in (1),
but with an additional copper layer below glycerol. The frequency range is
extended over 16 resonances and anti-resonances;

The remaining data applied for shear BAWs in AT-cut quartz are the piezoelectric
stress constant 0.0958

In contrast to Fig. 3a, many more distinctive curves are obtained when
subtracting the linearly frequency-dependent floor as seen in Fig. 3b. Such a
procedure is simply realized and usually done in experiments, especially
when keeping in mind that the floor originated from the capacitance

Figure 3b exhibits 16 resonance maxima of admittance amplitude accompanied by
minima of anti-resonant frequencies positioned tightly above. The red curves
(without any layer on the quartz plate) demonstrate rising resonances
only at odd frequency harmonics, whereas when adding a 5

Many attempts were made to find expressions for identifying material
parameters directly from resonant frequency shifts. For example
the well-known formulas of Sauerbrey (1959) and of Kanazawa and Gordon (1985) exist,
but in each case with a limited range of validity. For example, according to
Kanazawa the relative shift

Admittance amplitude vs. frequency after removing the floor signal at fundamental resonance and anti-resonance. Considered material systems: quartz plate with thin Cu layer and water half-space (red curve) and quartz plate with thick Cu layer and glycerol mixture half-space (blue curve).

Two other examples of resonant behavior are depicted in Fig. 4. The curves are restricted to the fundamental resonance (1st harmonic). It is seen that there are not only reductions of resonant and anti-resonant frequencies by thickening the copper layer, but the widths of peak and anti-peaks are also increased when replacing natural water with a glycerol mixture in the upper half-space of the material system. The evaluation of the corresponding bandwidths by the parameter FWHM was introduced above.

Corresponding to the width of the resonance peak, the FWHM of
anti-resonance is found using the curve of impedance instead of admittance
amplitude. In sum we can extract

Equation (32) enables us to come to significant conclusions of parameter
extraction from TSM measurements. In connection with Eq. (33a, b, and c) the
dependence on acoustic impedances

One decisive question in the procedure of multi-parameter sensing is the following: how
many and which parameters can be extracted from the
measurements, and under which conditions can they be extracted? It is obvious from the admittance curves of Fig. 3b that a
considerable number of experimental data can be supplied to find out the
related material and geometry parameters that produce such specific
behavior. The essential point is the influence of the material parameters
according to Eq. (39) on the list of measured frequencies, i.e.,
resonant and anti-resonant frequencies

The used data of studied exemplary material systems.

The number of involved quantities produces some complexity in the problem.
Forming the first derivatives of experimental data with respect to the
searched parameters is a preferential method to elucidate the situation.
Again, Eq. (32) is hereby appropriate for performing such numerical analysis
and facilitates the approach. The aforesaid first derivatives represent the
sensitivities of experimental data against extractable parameters. We start
with values for the material–geometry parameters which are assumed to be
reasonable, keeping in mind that for the possibility of parameter extraction at the
end of the procedure, the use of sensitivities for that has to be considered
in connection with their certainty or uncertainty. This means that it is
advantageous to form a Jacobian matrix

Examples for sensitivities as functions of

Numbering

The structures depicted in Fig. 5 exhibit some features which can be employed
for further refinement of multi-parameter search. The dependences of
sensitivities

Example of the environment of SSQ

Measurements at higher overtones seem more beneficial for the extraction of
these parameters. Viscosities (parameter 3 for Cu and 6 for glycerol) are
extractable from measured FWHM data, preferentially at high overtones.
The sensitivities of resonant and anti-resonant frequencies vs.

The extraction of material–geometry parameters from TSM measurements requires a fitting procedure between theoretically and experimentally determined frequencies. A precedent examination of achievable results is important for avoiding unsuccessful calculation efforts originating from inappropriate selection of experimental results caused by marginal sensitivities. The outcome of such a study enables one to develop the strategy of fitting or, in other words, to optimize the multi-parameter sensing approach.

For that purpose the environment of the minimum of the sum of squared relative
differences (SSQ) between theoretical and experimental values

An illustration of the situation is given in Fig. 6. The considered sample
is an AT-cut quartz TSM resonator, with a thickness 185

However, in the very most cases

Flow chart of the approach to multi-parameter sensing of layers and half-spaces that are in contact with a TSM resonator.

A key question for the experimental strategy of identifying material–geometry parameters is how to achieve high accuracy with a low number of experiments. Answers can be found on the basis of supposed appropriate initial values for the searched parameters and by analyzing the environment of the SSQ.

A self-evident value for a given uncertainty

Case I corresponds to the full set of resonant and anti-resonant frequencies and FWHMs of 1st, 3rd, and 5th–16th harmonics.

Case II is, compared to I, restricted to the resonances.

Case III is the combination of 1st, 3rd, and 5th overtones.

Case IV is the full set of 5th and 6th overtones.

Case V is yielded by a combination of anti-resonant frequency of the 1st harmonic, resonant frequency of the 16th overtone, resonant FWHMs of the 1st and of the 16th overtone, and anti-resonant FWHMs of the 3rd, 5th, 6th, and 8th overtones.

There is only small worsening when restricting the full set of measurements to resonant frequencies and resonant FWHMs (case II compared to case I).

Cases III and IV suggest that a strong reduction of the measurement number can produce inadmissibly high uncertainties.

Case V, unlike III and IV, exhibits acceptable small uncertainties in the face of reduction of experimental efforts compared to I by a factor of 7.

Uncertainties of material–geometry parameters of the material system described in Sect. 3.3. The columns are related to five different combinations of measurements.

Based on an extra derived analytic expression (Eq. 32) for the frequency dependence of electrical admittance, the suitability of a TSM resonator for multi-parameter sensing of layers and half-spaces that are in contact with the resonator plate on both sides has been analyzed. Resonant and anti-resonant frequencies as well as related bandwidths (FWHMs) were considered to be dependent on material–geometry parameters to calculate sensitivities of these experimental values against the searched parameters. This Jacobian matrix was used to evaluate the environment of the minimum of fitting procedure between experimental and theoretical values as a function of all material–geometry parameter variations, with the aim of obtaining their uncertainties after extraction from the experimental results. The separation of uncertainties of searched parameters requires a back-and-forth orthogonal transformation in the parameter space for the diagonalization of the squared Jacobian matrix.

Figure 8 summarizes the procedure for realizing multi-parameter sensing of layers and half-spaces that are in contact with a TSM resonator as it was carried out in this study, using an analytic expression for the electrical admittance.

It was shown for the special case of a TSM resonator of AT-cut quartz coated
by a 5

The demonstrated procedure is suitable for developing an experimental strategy for multi-parameter sensing involving both the minimization of parameter uncertainties as well as of experimental effort.

No data sets were used in this article.

The publication is prepared solely by MW.

The author declares that there is no conflict of interest.

The author wishes to thank Gerald Gerlach, Technische Universität Dresden, for the encouragement in elaborating this study, for supporting progress, and for help with finishing the work by helpful and stimulating discussions. Thanks also to Andrey Sotnikov and Hagen Schmidt, Leibniz Institute for Solid State and Materials Research Dresden (IFW), for instructive views on the evaluation of experimental data in microacoustics and for a critical reading of the paper (Hagen Schmidt).

This paper was edited by Gerald Gerlach and reviewed by two anonymous referees.