Piezoelectric Stiffening

Abstract

The electrical impedance between the front and the back electrode of the resonator affects the frequency shift by a mechanism called piezoelectric stiffening. If the front electrode is not grounded well (that is, if electric fringe fields permeate the sample), the sample’s electric and dielectric properties enter this impedance. They can be probed by switching between a grounded front electrode and a grounded back electrode. For the parallel plate, the formalism can be cast into a form, where the effects of a nonzero electric displacement are equivalent to a stress exerted onto the two resonator surface. Its influence on the frequency shift can be treated within the frame of the small load approximation (SLA).

Glossary

Variable

Definition (Comments)

26

As an index: component of piezoelectric coupling tensor applicable to AT-cut quartz (Table 5.1)

A

(Effective) area of the resonator plate

c

Stiffness tensor (Table 5.1)

c _p, c _V

Specific heat capacity at constant pressure and constant volume

C _L

Load capacitance (inserted with the aim of pulling the resonance)

C ₀

Parallel capacitance

C ₁

Motional capacitance

$\bar{C}_{ 1}$

Motional capacitance taking piezoelectric stiffening into account

d

One of the tensors quantifying piezoelectric coupling (Table 5.1, piezoelectric strain coefficient)

d ₂₆

Relevant component of the d-tensor (piezoelectric strain coefficient, d ₂₆ = 3.1 × 10^-12 m/V for AT-cut quartz)

d _q

Thickness of the resonator (d _q  = m _q /ρ _q  = Z _q /(2ρ _q f ₀) )

$\hat{D}, \, D$

Electric Displacement (if bold: a vector; if not bold: z-component of ${\hat{\mathbf{D}}}$ or D)

D

As an index: at constant electric Displacement

e

One of the tensors quantifying piezoelectric coupling (Table 5.1 (piezoelectric stress coefficient))

e ₂₆

Relevant component of the e-tensor (piezoelectric stress coefficient, e ₂₆ = 9.65 × 10^-2 C/m² for AT-cut quartz)

$\hat{E}, \, E$

Electric field (If bold: a vector. if not bold: z-component of ${\hat{\mathbf{E}}}$ or E)

E

As an index: at constant Electric field

$\hat{f}$

Force density (Do not confuse with frequency)

f

Frequency

f _r

Resonance frequency

f _s

Series resonance frequency

f ₀

Resonance frequency at the fundamental (f ₀ = Z _q /(2m _q ) = Z _q /(2ρ _q d _q ))

g

One of the tensors quantifying piezoelectric coupling (Table 5.1)

G

Shear modulus

G _q

Shear modulus of AT-cut quartz (G _q  ≈ 29 × 10⁹ Pa G _q  is the piezoelectrically stiffened modulus)

h

One of the tensors quantifying piezoelectric coupling (Table 5.1)

h _q

Half the thickness of a the resonator plate

k

Wavenumber (k = ω/c)

k _t

Piezoelectric coupling coefficient (k _t  = (e ₂₆/(ε _q ε₀ G _q ))^1/2)

$\hat{I}$

Electric current

m _q

Mass per unit area of the resonator (m _q  = ρ _q d _q  = Z _q /(2f ₀))

n

Overtone order

OC

As an index: the resonance condition of the unloaded plate under Open-Circuit conditions. With no current into the electrodes (more precisely, with vanishing electric displacement everywhere), piezoelectric stiffening is fully accounted for by using the piezoelectrically stiffened shear modulus.

p

Pressure

P̂

Electric polarization

PE

As an index: PiezoElectric stiffening

PESC

As an index: PiezoElectric stiffening with Short-Circuited electrodes

q

As an index: quartz resonator

$\hat{Q}_{S}$

Electrical surface charge density (Unit: C/m²)

ref

As an index: reference

R

Resistance

R _ex

External resistance

R _out

Output resistance of the driving electronics

s

Compliance tensor (Table 5.1)

S

Entropy (Do not confuse with infinitesimal strain tensor)

S

As an index: Surface or at constant Strain (At constant strain: in Table 5.1)

S _ij

Infinitesimal strain tensor (Table 5.1, Eq. 5.4.2)

T

Temperature

T

As an index: at constant stress (Table 5.1)

T _ij

Stress tensor (Table 5.1)

u

Tangential displacement

u _i

Displacement (component of vector)

$\hat{u}_{S}$

Tangential displacement at the resonator surface

V

Volume

x _i

Spatial coordinate (component of vector)

z

Spatial coordinate perpendicular to the surface of a layer or to the resonator plate

$\tilde{Z}_{ex} , \, Z_{ex}$

External electrical impedance

$\tilde{Z}_{PE}$

Stress-velocity ratio at the two resonator surfaces resulting from of PiezoElectric stiffening

$\tilde{Z}_{PESC}$

Same as $\tilde{Z}_{PE}$ with Short-Circuited electrodes

$\tilde{Z}_{q} , \, Z_{q}$

Acoustic wave impedance of AT-cut quartz (Z _q  = 8.8 × 10⁶ kg m⁻² s⁻¹)

Δ

As a prefix: A shift induced by the presence of the sample

ε

A small distance (Eq. 5.3.10)

$\tilde{\upvarepsilon },\;\upvarepsilon$

Dielectric permittivity (A tensor, Table 5.1)

$\tilde{\upvarepsilon }_{q} ,\;\upvarepsilon_{q}$

Dielectric constant of AT-cut quartz (ε _q is the clamped dielectric constant. ε _q  = 4.54 for AT-cut quartz)

ε₀

Dielectric permittivity of vacuum (ε₀ = 8.854 × 10⁻¹² C/(Vm))

φ

Electric potential

ϕ

Factor converting between mechanical and electric quantities in the Mason circuit (ϕ = Ae ₂₆/d _q )

κ _R

Resonator’s motional stiffness

ρ _q

Density of crystalline quartz (ρ _q  = 2.65 g/cm³)

$\tilde{\upsigma },\upsigma$

Tangential stress

$\tilde{\upsigma }_{PE} ,\upsigma_{PE}$

Piezoelectrically-induced apparent tangential stress

ω

Angular frequency