# Acoustoelectric Effect for Rayleigh Wave in ZnO Produced by an Inhomogeneous In-Depth Electrical Conductivity Profile

## Abstract

**:**

## 1. Introduction

_{3}substrate adsorbs hydrogen, thus varying its electrical conductivity. The changed wave velocity and propagation loss represent the SAW sensor response to different hydrogen concentrations. In reference [3], a shear horizontal acoustic plate mode (SHAPM) sensor is studied, which measures the electrical conductivity of a liquid environment contacting one side of the quartz plate. The electric field associated with the acoustic wave extends several micrometers into the liquid and interacts with ions and dipoles in solution. Changes in the conductivity of the solution perturb the propagation velocity of the plate mode. In reference [4], the SHAPM devices are studied for sensing potassium ion concentrations in water. In reference [5], the SAW velocity and attenuation are measured as a nickel film is deposited onto a quartz SAW device: the AE response causes a rapid drop in velocity and a peak in attenuation over the 10 to 30 Å thickness range. The valley AE effect is described in reference [6]: the electric current densities in LiNbO

_{3}/dielectric/MoS

_{2}were calculated and their magnitudes were compared with the conventional diffusive current.

## 2. The AE Effect

_{ij}represents the stress vector (N/m

^{2}), c

_{ijkl}the elastic stiffness matrix (N/m

^{2}), e

_{ijk}the piezoelectric stress matrix (C/m

^{2}), ε

_{ij}the permittivity matrix (F/m), ${E}_{k}=-\raisebox{1ex}{$\partial \u0424$}\!\left/ \!\raisebox{-1ex}{$\partial {x}_{k}$}\right.$ the electric field vector (V/m), ${S}_{kl}=\frac{1}{2}\left(\frac{\partial {u}_{k}}{\partial {x}_{l}}+\frac{\partial {u}_{l}}{\partial {x}_{k}}\right)$ the strain component, ${u}_{k}$ the mechanical displacement component along the Cartesian axis ${x}_{k}$ (x

_{1}= x, x

_{2}= y, x

_{3}= z), and D

_{i}is the electrical displacement (C/m

^{2}). The superscripts E and S denote that the constants are evaluated at a constant electric field and strain, respectively.

_{i}(for i = 1, 2, 3) and β

_{4}are the amplitudes of the particle displacement components and of the electric potential. The boundary conditions for SAWs traveling along the surface of the half-space (z = 0 is the surface and the z axis points toward the bulk) are the following: 1. the normal components of the stress tensor ${T}_{3i}=0$must be zero at the free surface (z = 0); 2. the displacement components and electric potential must vanish at large depths (z →∞); 3. continuity of the potential and of the normal component of the electric displacement across the free surface (z = 0) of the piezoelectric half-space (${D}_{3}=$ ${D}_{3}^{air}$ and Ф = Ф

^{air}for z = 0). Details on the theoretical aspects of acoustic wave propagation in solid media are discussed in references [9,10].

_{s}= σ · h the sheet conductivity, h the thickness, σ the bulk conductivity (the running parameter) of the conductive over-layer covering the piezoelectric half-space; K

^{2}is the electroacoustic coupling coefficient of the piezoelectric half-space (about 0.96% for ZnO), ε

_{0}and ε

_{s}the air and half-space dielectric permittivity, k = 2π/λ is the wavenumber, $\Delta v=v-{v}_{0}$, ${v}_{0}$ is the velocity of the SAW traveling along the bare surface of the piezoelectric half-space, $v$ is the SAW velocity perturbed by the layer conductivity change, σ

_{c}= ${v}_{0}\left({\epsilon}_{0}+{\epsilon}_{s}\right)$is the critical conductivity corresponding to the attenuation peak.

#### Simulation Methodology and Results

_{0}= 253.5 MHz); a sweep parameter study was performed to calculate the real and imaginary parts of the mode eigenfrequency ${f}_{\sigma}$ at different Al layer electrical conductivity. Figure 2a shows the phase velocity $v=Real\left({f}_{\sigma}\right)\xb7\lambda $ and propagation loss $\alpha =40\pi \xb7lo{g}_{10}e\xb7\frac{Imag\left({f}_{\sigma}\right)}{Real\left({f}_{\sigma}\right)}=-54.6\xb7Imag\left({f}_{\sigma}\right)/Real\left({f}_{\sigma}\right)$ vs. Al conductivity curves.

^{2}(about 1%) of the Rayleigh wave in c-ZnO half-space. If α/k is plotted vs. the relative velocity change $\raisebox{1ex}{$\Delta v$}\!\left/ \!\raisebox{-1ex}{${v}_{0}$}\right.$ (being ${v}_{0}$ the wave velocity along the bare ZnO surface) with the conductivity as the variable parameter, as shown in Figure 2b, the AE interaction assumes the form of a semi-ellipse centered at ($-\raisebox{1ex}{${K}^{2}$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, 0), the angular position along the semiellipse corresponding to the conductivity of the overlayer, in accordance with [16].

## 3. The Volume AE Effect

_{0}is the surface conductivity, δ

_{sd}is the skin depth, the distance into the ZnO material at which the conductivity has dropped by a factor of 1/e. The method adopted to account for the inhomogeneous conductivity distribution in the ZnO is to consider it as a stratified material with characteristics slowly varying over the layers. The wave propagating medium consists of a stacking sequence of 60 ZnO layers of equal thickness (δ = 40 nm), each layer having a different complex permittivity value: the imaginary part of the complex permittivity is frequency dependent according to $j\xb7\frac{{h}_{i}\sigma}{\omega {\epsilon}_{0}}$ where the electrical conductivity σ is the sweep parameter and h

_{i}is a dimensionless weighting factor that accounts for the in-depth exponentially degrading conductivity according to the expression ${h}_{i}={e}^{-\left(i-1\right)\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{${\delta}_{sd}$}\right.}$· $\left(\frac{1+{e}^{-\raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{${\delta}_{sd}$}\right.}}{2}\right)$, for i = 1 to 60, and δ the thickness of the layers, as shown in Figure 3. The stacking sequence of 60 ZnO layers covers a ZnO half-space, 7·λ thick, having real permittivity. Figure 3 shows, as an example, the column plot of ${e}^{-y/{\delta}_{sd}}$ vs. depth curve for δ

_{sd}= 100 nm: the black points show the mean value h

_{i}of the ordinates, which corresponds to two successive layers of width 40 nm.

#### Simulation Methodology and Results

^{−7}to 10

^{16}S/m) and for different δ

_{sd}values (from 100 to 500 nm). The δ

_{sd}values were chosen among those reported in the available literature: for incident UV light at 365 nm, the penetration depth in ZnO ranges from 80 to a few a hundred nanometers [17,18,19,20,21,22] since it is strongly related to the structural properties of the photo-conducting material.

_{sd}the running parameter. The black curves in Figure 5 represent the case of ZnO half-space covered by a thin Al layer, as described in the previous paragraph.

_{sd}(which varies in the range from 100 to 500 nm); the black curve represents the case of ZnO half-space covered by a thin Al layer. The curves can be resolved into two semi-ellipses, contrary to the single arc found in the previous case; the overlapping of the semi-ellipses increases with increasing the δ

_{sd}. The propagation loss $\alpha /k=-54.6\xb7Imag\left({f}_{\sigma}\right)/\left(k\xb7Real\left({f}_{\sigma}\right)\right)$ referred to as the surface AE effect (the black curve in Figure 5b) reaches an amplitude larger than that referred to as the volume AE effect since the imaginary part of the resonant frequency (0.58 MHz) is larger than that referred to the volume AE effect (from 0.48 to 0.55 MHz); with increasing the δ

_{sd}value, the maximum value of Imag(f) increases.

## 4. Discussions

_{sd}values; (2) the amplitude of the first velocity drop is similar to that of the surface AE effect; (3) the first drop is moved toward abscissa values lower than those referred to the surface AE effect; and (4) the magnitude of the total velocity drop of the volume AE effect exceeds that of the surface AE effect.

_{sd}and quite merges to the first peak at δ

_{sd}= 500 nm; (4) the amplitude of the large peaks increases with increasing the δ

_{sd}.

_{sd}= 200 nm, and the electric potential Ф shape, for some conductivity values, as an example.

## 5. Conclusions

_{o}, with the ZnO electrical conductivity as a variable parameter, gives the representation of the double-relaxation response as opposed to the single-relaxation response which characterizes the well-known AE effect due to surface conductivity changes onto piezoelectric media. The calculated volume AE response showed that initially, it is only the more superficial layers of the stacked region that become conductive and therefore contribute to the AE effect in a manner quite similar to the surface AE effect: the configuration can be schematized as that of a surface layer (with variable conductivity) coated on a piezoelectric half-space. Subsequently, with the further increase of the conductivity (with the further increase in the absorbed UV power), the more superficial layers of the stacked region remain conductive while the underlying layers begin to become conductors and contribute to the AE effect: the configuration can be schematized as that of a conductive layer coated on a layer (with variable conductivity)/piezoelectric half-space.

_{sd}, suggests that, other than the velocity and attenuation changes arising from perturbed electrical conditions, also a velocity dispersion effect takes place due to the increased thickness of the ZnO conductive-overlayer while increasing the conductivity.

_{sd}: by using N from 10 to 60, δ from 100 to 300 nm, and δ

_{sd}from 240 to 1000 nm the double-relaxation effect is always clearly visible; the choice of δ = 40 nm, N = 60 represents a good compromise if a δ

_{sd}value ranging from 100 to 500 nm is selected.

_{3}SAW sensor showed a frequency shift of 170 kHz under 40 mW/cm

^{2}UV power illumination. In reference [24], a ZnO layer as thin as 71 nm is used as a UV sensing element: the authors say that this choice is suitable to avoid the perturbation of the wave propagation in the LiNbO

_{3}substrate from the ZnO mass loading. In reference [25], the ZnO(250 nm)/128°yx-LiNbO

_{3}showed a sensitivity of 6000 ppm/(μW/cm

^{2}) in a wide UV power range (from 0.010 to 40 mW/cm

^{2}): despite the wide UV power range, the double-relaxation phenomenon is not observed, probably due to the small ZnO layer thickness. In reference [26], however, the UV-induced frequency shifts of the Sezawa wave in ZnO (3.23 µm)/Si-based SAW oscillator were measured in a wide range of UV light intensities: the frequency shift vs. UV power curve exhibits two sensitivities which are 8.12 ppm/(μW/cm

^{2}) in the low power region (up to about 50 µW/cm

^{2}) and 1.62 ppm/(μW/cm

^{2}) in the high power region (from about 50 to 551 µW/cm

^{2}); unfortunately, the authors did not study the effect of the UV power on the wave propagation loss. The existence of two slopes in the frequency shift vs. the UV power source curve is attributed by the authors to the saturation of photogenerated carriers. It would have been useful to have some more measurements of both frequency shift and propagation loss over a wider range of UV powers (>551 µW/cm

^{2}) to verify if the hypothesis of an approaching plateau due to a double-relaxation phenomenon is realistic.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Caliendo, C.; Verona, E.; D’Amico, A. Surface Acoustic Wave (SAW) Gas Sensors. In Gas Sensors; Sberveglieri, G., Ed.; Springer: Dordrecht, The Netherlands, 1992. [Google Scholar] [CrossRef]
- Fisher, B.H.; Malocha, D.C. Study of the acoustoelectric effect for SAW sensors. IEEE Trans. Ultrason. Ferroelectr. Freq. Control.
**2010**, 57, 698–706. [Google Scholar] [CrossRef] [PubMed] - Niemczyk, T.M.; Martin, S.J.; Frye, G.C.; Ricco, A.J. Acoustoelectric interaction of plate modes with solutions. J. Appl. Phys.
**1988**, 64, 5002–5008. [Google Scholar] [CrossRef] - Caliendo, C.; Verona, E.; D’Amico, A.; Mascini, M.; Moscone, D. Acoustic love-wave sensor for K
^{+}concentration in H_{2}O solutions. Sens. Actuators B Chem.**1992**, 7, 602–605, ISSN 0925-4005. [Google Scholar] [CrossRef] - Martin, S.J.; Ricco, A.J. Effective utilization of acoustic wave sensor responses: Simultaneous measurement of velocity and attenuation. In IEEE Symposium (IUS) Ultrasonics, Proceedings of the 1989 Ultrasonics Symposium, Montreal, QC, Canada, 3–6 October 1989; IEEE: New York, NY, USA, 1989; pp. 621–625. [Google Scholar]
- Kalameitsev, A.V.; Kovalev, V.; Savenko, I.G. Valley Acoustoelectric Effect. Phys. Rev. Lett.
**2019**, 122, 256801. [Google Scholar] [CrossRef] [Green Version] - Takahashi, Y.; Kanamori, M.; Kondoh, A.; Minoura, H.; Ohya, Y. Photoconductivity of Ultrathin Zinc Oxide Films. Jpn. J. Appl. Phys.
**1994**, 33, 6611. [Google Scholar] [CrossRef] - Sharma, P.; Sreenivas, K. Analysis of ultraviolet photoconductivity in ZnO films prepared by unbalanced magnetron sputtering. J. Appl. Phys.
**2003**, 93, 3963. [Google Scholar] [CrossRef] - Mason, W.P.; Thurston, R.N. Physical Acoustics: Principles and Methods; Academic Press: London, UK, 1970. [Google Scholar]
- Farnell, G.W.; Adler, E.L. Elastic Wave Propagation in Thin Layers. In Physical Acoustics; Mason, W.P., Thurston, R.N., Eds.; Academic Press: New York, NY, USA, 1972. [Google Scholar]
- Ricco, J.; Martin, S.; Zipperian, T.E. Surface Acoustic Wave Gas Sensor Based on film conductivity changes. Sens. Actuators
**1985**, 8, 319–333. [Google Scholar] [CrossRef] - Ingebrlgtsen, K.A. Linear and nonlinear attenuation of acoustic surface waves in a piezoelectric coated with a semiconducting film. J. Appl. Phys.
**1970**, 44, 454–459. [Google Scholar] [CrossRef] - Kino, G.S.; Reeder, T.M. A normal mode theory for the Rayleigh wave amplifier. IEEE Trans. Electron. Devices
**1971**, 18, 909–920. [Google Scholar] [CrossRef] - Adler, R. Simple theory of acoustic amplification. IEEE Trans Sonics Ultrason.
**1971**, 18, 115–118. [Google Scholar] [CrossRef] - Datta, S. Surface Acoustic Wave Devices; Prentice-Hall: Englewood Cliffs, NJ, USA, 1985. [Google Scholar]
- Ballantine, D., Jr.; White, R.; Martin, S.; Ricco, A.; Zellers, E.; Frye, G.; Wohltjen, H. Acoustic Wave Sensors: Theory, Design and Physico-Chemical Applications; Levy, M., Stern, R., Eds.; Academic Press: New York, NY, USA, 1997. [Google Scholar]
- Paulauskas, I.; Jellison, G.; Boatner, L.; Brown, G. Photoelectrochemical Stability and Alteration Products of n-Type Single-Crystal ZnO Photoanodes. Int. J. Electrochem.
**2011**, 2011, 563427. [Google Scholar] [CrossRef] [Green Version] - Alsaad, A.M.; Ahmad, A.A.; Qattan, I.A.; Al-Bataineh, Q.M.; Albataineh, Z. Structural, Optoelectrical, Linear, and Nonlinear Optical Characterizations of Dip-Synthesized Undoped ZnO and Group III Elements (B, Al, Ga, and In)-Doped ZnO Thin Films. Crystals
**2020**, 10, 252. [Google Scholar] [CrossRef] [Green Version] - Venkatachalam, M. Structural and Optical Properties of Sol –Gel Spin Coated Zno Thin Films. Int. J. Recent Sci. Res.
**2014**, 5, 1773–1776. [Google Scholar] - Ghusoon, M.A.; Chakrabarti, P. ZnO-based interdigitated MSM and MISIM ultraviolet photodetectors. J. Phys. D Appl. Phys.
**2010**, 43, 415103. [Google Scholar] - Aslan, E.; Zarbali, M. Tuning of photosensitivity and optical parameters of ZnO based photodetectors by co-Sn and Ti doping. Opt. Mater.
**2022**, 125, 112030. [Google Scholar] [CrossRef] - Yi, F.; Liao, Q.; Yan, X.; Bai, Z.; Wang, Z.; Chen, X.; Zhang, Q.; Huang, Y.; Zhang, Y. Simple fabrication of a ZnO nanorod array UV detector with a high performance. Phys. E Low-Dimens. Syst. Nanostructures
**2014**, 61, 180–184, ISSN 1386-9477. [Google Scholar] [CrossRef] - Sharma, P.; Sreenivas, K. Highly Sensitive Ultraviolet Detector Based on ZnO/LiNbO
_{3}Hybrid Surface Acoustic Wave Filter. Appl. Phys. Lett.**2003**, 83, 3617. [Google Scholar] [CrossRef] - Kumar, S.; Sharma, P.; Sreenivas, K. Low-intensity ultraviolet light detector using a surface acoustic wave oscillator based on ZnO/LiNbO
_{3}bilayer structure. Semicond. Sci. Technol.**2005**, 20, L27. [Google Scholar] [CrossRef] - Karapetyan, G.Y.; Kaydashev, V.E.; Kutepov, M.E.; Minasyan, T.A.; Kalinin, V.A.; Kislitsyn, V.O.; Kaidashev, E.M. Passive wireless UV SAW sensor. Appl. Phys. A
**2020**, 126, 794. [Google Scholar] [CrossRef] - Wei, C.-L.; Chen, Y.-C.; Cheng, C.-C.; Kao, K.-S.; Cheng, D.-L.; Cheng, P.-S. Highly sensitive ultraviolet detector using a ZnO/Si layered SAW oscillator. Thin Solid Film.
**2010**, 518, 3059–3062. [Google Scholar] [CrossRef] - Caliendo, C. Acoustoelectric effect for Rayleigh and Sezawa waves in ZnO/fused silica produced by an inhomogeneous in-dept electrical conductivity profile. in progress.

**Figure 1.**Schematic of the SAW delay line together with the detail of the unit cell (not in scale) including the ZnO domain (7λ thick), the Al domain (0.01 µm thick), and the air domain (2λ thick).

**Figure 2.**(

**a**) The phase velocity and propagation loss vs. Al conductivity curves; (

**b**) the parametric representation of the AE response, the α/k vs. $\raisebox{1ex}{$\Delta v$}\!\left/ \!\raisebox{-1ex}{${v}_{0}$}\right.$ curve with the conductivity as parameter, being k = 2π/λ the wavevector.

**Figure 3.**The column plot of the ${e}^{-y/{\delta}_{sd}}$ vs. depth curve, assuming δ

_{sd}= 100 nm: the black points show h

_{i}, the mean value of the ordinates corresponding to two successive layers of width 40 nm.

**Figure 4.**(

**a**) Cross-section of the SAW delay line illuminated by UV radiation; (

**b**) FEM unit cell of the ZnO half-space (partially discretized) and air (the picture is not in scale).

**Figure 5.**(

**a**) Relative phase velocity change $\raisebox{1ex}{$\Delta v$}\!\left/ \!\raisebox{-1ex}{${v}_{0}$}\right.$ and (

**b**) propagation loss α vs. ZnO conductivity curves for different values of the UV penetration depth ${\delta}_{sd}$ (from 100 to 500 nm).

**Figure 6.**The α/k vs. $\raisebox{1ex}{$\Delta v$}\!\left/ \!\raisebox{-1ex}{${v}_{0}$}\right.$ curves, with σ as the variable parameter for different values of δ

_{sd}(which varies in the range from 100 to 500 nm); the black curve represents the case of ZnO half-space covered by a thin Al layer.

**Figure 7.**(

**a**) Velocity and propagation loss vs. the ZnO conductivity curves; the electric potential profile for σ equal to (

**b**) 0.0001, (

**c**) 1, and (

**d**) 10

^{20}S/m; all the plots refer to δ

_{sd}= 200 nm.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Caliendo, C.
Acoustoelectric Effect for Rayleigh Wave in ZnO Produced by an Inhomogeneous In-Depth Electrical Conductivity Profile. *Sensors* **2023**, *23*, 1422.
https://doi.org/10.3390/s23031422

**AMA Style**

Caliendo C.
Acoustoelectric Effect for Rayleigh Wave in ZnO Produced by an Inhomogeneous In-Depth Electrical Conductivity Profile. *Sensors*. 2023; 23(3):1422.
https://doi.org/10.3390/s23031422

**Chicago/Turabian Style**

Caliendo, Cinzia.
2023. "Acoustoelectric Effect for Rayleigh Wave in ZnO Produced by an Inhomogeneous In-Depth Electrical Conductivity Profile" *Sensors* 23, no. 3: 1422.
https://doi.org/10.3390/s23031422