Using silicon disk resonators to measure mechanical losses caused by an electric field



View

Online



Export

Citation

CrossMark

RESEARCH ARTICLE

JANUARY 04 2022

Using silicon disk resonators to measure mechanical losses

caused by an electric field



Y. Yu. Klochkov

;

L. G. Prokhorov

;

M. S. Matiushechkina

;

R. X. Adhikari

;

V. P. Mitrofanov



Rev

. Sci. Instrum.

93, 014501 (2022)

https://doi.org/10.1063/5.007631

Articles Y

ou May Be Interested In

Measurement of the temperature dependence of mechanical losses induced by an electric field in undoped

silicon disk resonators

Appl. Phys. Lett.

(April 2023)

Low magnetic Johnson noise electric field plates for precision measurement

Rev

. Sci. Instrum.

(November 2016)

A Pockels cell enabled heterodyne phase contrast imaging diagnostic for detection of ion cyclotron

emission

Rev

. Sci. Instrum.

(August 2022)

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

Using silicon disk resonators to measure

mechanical losses caused by an electric field

Cite as: Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

Submitted: 23 October 2021

•

Accepted: 13 December 2021

•

Published Online: 4 January 2022

Y. Yu. Klochkov,

L. G. Prokhorov,

M. S. Matiushechkina,

R. X. Adhikari,

and V. P. Mitrofanov

1, a)

AFFILIATIONS

Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia

Institute for Gravitational Wave Astronomy, University of Birmingham, Birmingham B15 2TT, United Kingdom

Laser Interferometry and Gravitational Wave Astronomy, Max Planck Institute for Gravitational Physics, 30167 Hannover,

Germany

LIGO Laboratory, California Institute of Technology, Pasadena, California 91125, USA

Author to whom correspondence should be addressed:

vpmitrofanov@physics.msu.ru

ABSTRACT

Several projects of the next generation gravitational-wave detectors use the high purity monocrystalline silicon test masses. The electric

field of the actuator that is applied to correct the position of the silicon test mass causes additional mechanical losses and associated noise.

Disk mechanical resonators are widely used to study mechanical losses in multilayer optical coatings that are deposited on the test masses

of gravitational-wave detectors. We use silicon disk resonators to study losses caused by an electric field. In particular, the dependence of

mechanical losses on the resistivity of silicon is investigated. The resonator is a thin commercial silicon wafer in which a low frequency nodal

diameter mode is excited. A DC voltage is applied between the wafer and a nearby electrode. We use two measurement configurations. In the

first configuration, the dependence of losses on the resistance in the voltage supply circuit is investigated. The dependence of losses on the

resistivity of silicon is investigated in the second configuration. We propose a model that relates the electric field induced mechanical loss in

disk resonators to the resistivity of the material. Measurements are carried out for low and high resistivity silicon wafers. The measurement

results are compared with calculations. Based on these studies, it is possible to estimate the loss and noise of the test masses of gravitational-

wave detectors associated with electrostatic actuators.

Published under an exclusive license by AIP Publishing.

https://doi.org/10.1063/5.0076311

I. INTRODUCTION

Several years have passed since the first direct detection of

gravitational waves.

Laser interferometric gravitational-wave (GW)

detectors LIGO and Virgo are working successfully.

2–4

Recently,

Japan’s KAGRA detector joined the network of GW observatories.

Researchers were faced with the challenge of improving the cur-

rent generation of GW detectors and developing more sensitive

next generation detectors. In one of the possible design options for

the implementation of the Einstein Telescope and Cosmic Explorer

projects of future GW detectors, test masses of detectors and their

suspensions will be made of high purity monocrystalline silicon and

cooled to low temperatures.

6,7

Silicon was chosen for its low internal mechanical losses, suit-

able thermal properties, and low optical absorption at wavelengths

of 1.5–2.5

To correct the position of the test masses of the

GW interferometric detectors, electrostatic actuators are used which

create an electric field between the test mass and electrodes of the

actuator.

Unlike the dielectric fused silica test masses of the current

GW detectors, the application of an electric field to the oscillating

semiconductor silicon test mass results in the additional mechanical

loss and thermal noise associated with this loss.

This loss is caused

by electric currents flowing in the oscillating conductive body due to

the change in the capacitance between it and the electric field elec-

trode. This mechanism of electrical or Ohmic losses was especially

evident in MEMS/NEMS systems, for example, in electromechan-

ical resonators fabricated from graphene, carbon nanotubes, and

nanowires due to their very small effective masses and strong charge

coupling.

11–13

There is also electrostatic damping that depends on

the surface properties of the vibrating body and the electrodes.

Disk mechanical resonators in which nodal diameter modes are

excited are widely used to study mechanical losses in multilayer high

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-1

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

reflectivity thin film optical coatings deposited on the test masses

of current and next generation GW detectors.

15–20

We propose to

use disk resonators made of high-purity silicon to study mechani-

cal losses caused by an electric field. Note that it is also possible to

investigate these losses in silicon disks with optical coatings on their

surface.

In this article, we describe an experimental setup developed

for studying losses in silicon disk mechanical resonators caused by

an electric field. The resonator is a thin commercial silicon wafer

in which a low-frequency nodal diameter mode is excited. A DC

field is created when an electric voltage is applied between the wafer

and nearby electrodes. In order to reduce anchor losses in disk res-

onators, a nodal support technique was used.

16,21

An even greater

reduction in anchor losses was achieved using the so-called Gentle

Nodal Suspension (GNS) technique.

22,23

Unfortunately, we could not use the GNS technique for fixing

disk resonators since the electric field creates the instability of the

equilibrium position of the resonator.

For this reason, we clamp

the disk between two stems in the center of the disk. In particu-

lar, the dependence of mechanical losses on the resistivity of silicon

is investigated. We use two measurement configurations. The first

measurement configuration, called the two-electrode configuration,

allows us to study the dependence of the electric field induced fre-

quency shift and mechanical loss on the parameters of the electric

voltage supply circuit. In the second four-electrode measurement

configuration, we study the dependence of the frequency shift and

loss on the resistivity of silicon. We propose a model that relates the

loss measured in the four-electrode configuration to the resistivity of

silicon. The measurement results are compared using calculations.

II. EXPERIMENTAL SETUP AND MEASUREMENT

PROCEDURE

As resonators, we used commercial (Virginia Semiconductor,

Inc.) double side polished silicon wafers with a diameter of 50.8 mm

and a thickness of 0.27 mm. The wafer has two flats as shown in

Fig. 1(a). The flats with lengths of 16 and 8 mm cut into sides of

the wafer indicate the crystallographic plane (100) and the type of

silicon. A lot of vibration modes can be excited in the wafer.

chose a bending mode with two nodal diameters so that one of the

diameters passed through the middle of the longest length flat. This

mode has a natural frequency of about 800 Hz.

The silicon wafer was clamped between two stems mounted in

an aluminum frame as shown in Fig. 2. The wafer was grounded

through contact with the spherical surface of the upper stem. The

bottom stem was mounted in a spring-loaded manner and had a

hemispherical PTFE tip. The radius of curvature of the spherical

surface was about 20 mm. This value provided a small contact area

necessary to reduce frictional clamping losses and sufficient for the

convenience of installing a silicon wafer. This issue was discussed in

Refs. 21 and 22.

The ring-shaped electrode divided into four sectors was located

over the silicon wafer. The electrodes were etched on a fiberglass

copper clad plate and gold-plated, as shown in Fig. 1(b). A pattern

of electrodes creating an electric field corresponds to the distribu-

tion of displacements in the vibration mode. The orientation of the

plate with the electrodes was set so that the cuts in the ring electrode

coincided with the middle of the flats of the wafer through which

FIG. 1.

(a) Vibration mode of the wafer simulated through COMSOL. The color

indicates the value of the displacement amplitude in a direction orthogonal to the

wafer surface. The typical displacement amplitude used in this work is shown in

the color bar. (b) Schematics of the fiberglass copper clad plate with etched and

gold-plated electrodes.

the nodal diameter of the selected mode passes. Voltage could be

applied separately to every sector. A separation gap of about 0.3 mm

between the wafer and the electrode plate was adjusted using three

spring-loaded screws, as shown in Fig. 2.

To estimate the gap size, the capacitance meter Mastech

MS6013 was connected to the wires attached to the electrodes. The

capacitance was measured for the working separation gap and for

the 2.4 mm gap set using a calibrated plate. These measurements

allowed us to subtract the parasitic capacitance and calculate the

capacitance between the wafer and the electrode. The gap size was

calculated using the formula for the capacitance of a flat capacitor.

The measurement uncertainty of the gap size was about 0.02 mm.

FIG. 2.

Schematic of the experimental setup.

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-2

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

The wafer vibration was excited resonantly with the AC volt-

age (up to 300 V) at a half of the mode resonant frequency using a

capacitive electrostatic drive, the plate of which was located under

the wafer of about 1 mm below its surface. The dependence of the

force between the capacitor plates on the square of the applied volt-

age leads to the appearance of the doubled frequency of the applied

AC voltage in the force spectrum. The wafer vibration was moni-

tored using the optical sensor. The laser beam hits the surface of the

sample at the point spaced 3–5 mm from the edge near the nodal

diameter of the investigated mode where the local tilt of the surface

is maximal during vibration. The local tilt of the wafer produced

by its vibration results in a deflection of the laser beam reflected

from the wafer surface. The reflected beam passes through a system

of mirrors and is detected by a split quadrant photodiode (QPD)

placed outside the vacuum chamber. Custom-developed software

based on LabVIEW was used to obtain the mode frequency

and

the decay time

from the photodiode signal measured during the

wafer free vibration decay. The frame with the clamped wafer was

placed in a vacuum chamber with a residual gas pressure of about

−

Torr. The residual gas damping gave a negligible contribution

to the measured mechanical loss.

A circuit for applying the DC voltage

to electrodes is shown

in Fig. 3. The resistor

is a variable external resistor (we can

neglect the contribution of the small internal resistance of the volt-

age source). We used this resistor in order to study a mechanism

of the loss caused by an electric field.

is a parasitic capacitance

that does not change with the wafer vibration. The contribution to

it mainly comes from the capacitance between the wires connect-

ing the electrodes to the voltage source and surrounding grounded

conductors.

is the electrical contact resistance between the sili-

con wafer and the metal stem. Through it, the wafer is connected

to a terminal of the DC voltage source and is grounded. Ground-

ing prevents the accumulation of static electric charges on the wafer,

which can be a source of additional non-controlled mechanical

losses.

The contact resistance

depends on the contact materi-

als, state of the contacting surfaces, the clamping force, and other

factors.

We used two measurement configurations. In the first configu-

ration called the two-electrode configuration, one pair of electrodes,

for example, “1” (see Fig. 3), was connected to the DC voltage source.

FIG. 3.

Circuit for applying the DC voltage

to electrodes.

Another pair of electrodes was grounded. In this case, when the sili-

con wafer vibrates, the capacitances between each of the two sectors

and the wafer change in phase. These changes add up and induce

a current in the external circuit. Operation in the two-electrode

configuration allows us to study the loss mechanism associated

with the relaxation process of the change in the electric charge on

the wafer. The vibration of the wafer leads to the change in the

capacitance between the wafer and the electrodes, which results

in the change in the electric charge and the electric current flow-

ing through the resistance in the voltage supply circuit (including

the external resistor

, the contact resistance

, and the wafer

resistance

). This leads to the loss of the vibration energy. The

relaxation time can be changed by changing the value of the external

resistor

In the second measurement configuration called the four-

electrode configuration, the DC voltage

was applied to all four

electrodes and the change in the total capacitance between the sil-

icon wafer and electrodes caused by the wafer vibration was close

to zero due to antiphase vibrations in the adjacent quarter sectors

of the wafer. Hence, with the full compensation, there is no cur-

rent in the external circuit and therefore no associated loss. The loss

in the four-electrode configuration is caused by the redistribution

of the induced electric charge in the wafer and the corresponding

electric current flowing through it. This loss depends on the wafer

resistivity.

The mechanical loss

−

is calculated from the decay time

the vibration mode with the resonant frequency

according to the

relation

−

(

)

−

. The electric field induced mechanical loss

−

was determined as a difference between

−

measured when the

voltage

was applied to the electrodes and

−

measured when all

the electrodes were grounded,

−

−

−

(1)

III. CALCULATION OF THE ELECTRIC FIELD INDUCED

LOSS AND FREQUENCY SHIFT IN THE MECHANICAL

OSCILLATOR

To analyze the mechanism of mechanical losses caused by the

relaxation process in the DC voltage supply circuit, consider the

electromechanical system with lumped parameters shown in Fig. 4.

A mass

of the mechanical oscillator with a natural frequency

√

(

)

, where

is a stiffness of the spring, performs a peri-

odical motion

(

)

. The change in the gap

(

)

+

(

)

leads to

the change in the capacitance between the mass and the electrode

(

)

+

(

)

. Using the condition

(

)

≪

in the first

order approximation in

(

, we obtain the equations for the total

charge

(

)

in the total capacitance

(

)

+

and the equation for

the mass displacement

(

)

, whereas on the right side, there is an

expression for the electrostatic force acting between the plates of the

capacitor,

−

+

+

−

[

−

(

−

)

]

(2)

where

+

and

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-3

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

FIG. 4.

Model of the electromechanical system used for the calculation of the

electric field induced loss and frequency shift in the mechanical oscillator.

An approximate solution to these differential equations is given

in the Appendix. It has been found that the electric field introduces

the additional negative stiffness

and loss

−

into the oscillator.

The additional negative stiffness results in the relative change in the

oscillator’s frequency

(

)

(

)

−

(

)

[

+

(

)

+

(

−

)]

(3)

−

(

)

+

(

)

(4)

Equation (4) and the first term in square brackets of Eq. (3)

describe the loss and the relative change in the resonant frequency,

which are typical for the relaxation process with the relaxation time

rel

The silicon disk mechanical resonator is a system with dis-

tributed parameters. We can introduce the effective mass

and the

effective capacitance

for the working mode of the wafer vibration

in order to use these equations for the analysis of the experimental

data. The circuit shown in Fig. 4 does not take into account the resis-

tance of the silicon wafer

and the electrical contact resistance

between the wafer and the metal stem that is used to clamp the wafer.

The contact resistance is difficult to control. The contact resistance

and the wafer resistance are not shunted by the parasitic capaci-

tance

for the electric current generated due to the wafer vibration.

We separated the loss associated with these resistances from the loss

associated with the external resistor

When operating in the four-electrode configuration, the losses

of the resonator associated with the resistance in the DC voltage sup-

plying circuit and the contact resistance are significantly reduced.

This allows us to investigate the electric field induced loss caused by

the silicon resistivity. The silicon wafer vibration leads to the change

in the local capacitances between the disk and the electrode. When

FIG. 5.

(a) Distribution of the wafer displacement amplitude in a direction orthog-

onal to its surface. (b) Model of a distributed RC line used for the calculation of

the redistribution of electric charges on the wafer. (c) Equivalent electrical circuit

with varying capacitances between the wafer and the electrodes. The distributed

capacitances and resistances of the wafer are replaced with equivalent lumped

elements

, and

. Arrows near capacitors

and

show

antiphase changes in their capacitances.

the voltage

is applied to the electrode, the vibration causes a

redistribution of electric charges in an annular strip of the silicon

disk in accordance with the distribution of the vibration amplitude

of the wafer [see Fig. 5(a)].

Taking into account the symmetry of the wafer vibrations, the

disk can be partitioned into four identical sectors so that each sector

is located between the maximum values of the vibration amplitude

at points A and C. At the nodal point B, the deflection of the sample

changes the sign. In each sector, an annular strip of a silicon disk and

an electrode located above it form an electrical resistive–capacitive

line, as shown in Fig. 5(b). In the annular strip of each sector, a redis-

tribution of charges occurs, caused by an antiphase change in the

capacitance of two distributed capacitors

∗

and

∗

with per-unit-

length capacitance

and a distributed resistor

∗

with

per-unit-length resistance

, where

is the vacuum permit-

tivity,

is the electrical resistivity of silicon,

is the width of the

strip,

is a separation gap between the electrode and the silicon

disk, and

is the disk thickness.

is a total length of the strip on the disk along which the

current flows, then the total resistance of the sector strip is

+

and the capacitance of the strip on a half of

the sector is

. The equivalent electrical circuit

with varying capacitances between the wafer and the electrodes is

shown in Fig. 5(c). A value of

is defined as the length of a circle

drawn in the middle of the annular electrode.

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-4

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

A distributed RC line can be characterized by a number of

time-constants. We choose the dominant one as an approxima-

tion. It has the maximum value and is expressed through the

lumped values of the total capacitance C and resistance R:

≈

(

)

≈

0.405RC. Taking into account that the capacitances

of half-sectors are connected in series between themselves and

the strip resistance, we obtain the following for the considered

sector:

≈

(5)

We assume that the main mechanism of the electric field

induced mechanical loss of the silicon disk is the relaxation pro-

cess associated with the gap capacitance and the disk resistivity. It

is described by the equations similar to Eqs. (3) and (4) with the

replacement of

, and

to the corresponding parameters of

the wafer and electrodes. The parasitic capacitance

for this sys-

tem is assumed to be negligible. In this case,

rel

≪

1, and we

obtain

−

−

(

)

rel

(6)

Substituting in Eq. (6) the measured value of the relative fre-

quency shift and

rel

, we obtain an estimate of the electric field

induced mechanical loss of the disk resonator, caused by the silicon

resistivity.

IV. RESULTS OF MEASUREMENTS AND THEIR

DISCUSSION

Vibration modes of the silicon wafers were identified from the

measured eigenfrequencies by comparing them with the calculated

ones using the finite element modeling with COMSOL Multiphysics

modeling software. We present the measurement results for two sil-

icon wafers with different resistivities. The low resistivity (LR) wafer

No. 1 was fabricated from boron doped silicon with a resistivity of

(

1.0

±

0.5

)

cm. It had the resonant frequency of

≈

873 Hz for

the working mode with two nodal diameters. The high resistivity

(HR) wafer No. 2 was fabricated from undoped silicon with a resis-

tivity of

(

1.6

±

0.1

)

×

cm. It had the working mode frequency

≈

832 Hz. In the absence of an electric field, the decay time

of the freely decaying vibration amplitude of the silicon resonators

was found to be of about 11 s. This corresponds to the mechani-

cal loss

−

3.5

×

−

. One of the main mechanisms of losses is

the thermoelastic loss associated with heat flow through the disk as

it vibrates. The calculated thermoelastic loss for such a resonator is

−

≈

×

−

It can be assumed that the remaining losses were

mainly the clamping loss and the surface loss. When an electric volt-

age was applied to the electrodes, the mode frequency and the decay

time were measured in both measurement configurations. The elec-

tric field induced mechanical loss

−

was calculated according to

Eq. (1).

The electric field induced relative frequency shift and loss

measured for both wafers in the two-electrode measurement con-

figuration as a function of a square of the applied voltage are

shown in Figs. 6(a) and 6(b). The separation gap of

(

0.32

±

0.02

)

mm was set for the HR wafer and

(

0.27

±

0.02

)

for the LR wafer. The experimental data are well approximated

by a square function of the voltage in accordance with Eqs. (3)

and (4). Notice that the loss measured for the HR wafer is much

FIG. 6.

Electric field induced relative frequency shift

(

)

(a) and loss

−

(b) of HR (red triangles) and LR (blue circles) silicon wafers measured as a function

of a square of the applied voltage

in the two-electrode measurement configura-

tion. Fitted results are shown by solid lines. The separation gap is

=

0.32 mm

for the HR wafer and

=

0.27 mm for the LR wafer. The resistor

=

17 k

was included in the voltage supply circuit.

higher than that measured for the LR wafer. We will discuss this

in more detail below. The electric field induced relative frequency

shift and loss of both wafers measured in the two-electrode mea-

surement configuration as a function of a value of the external

resistor

included in the voltage supply circuit (

150 V)

are shown in Figs. 7(a) and 7(b). The results do not depend

on the polarity of the applied voltage within the measurement

errors.

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-5

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

FIG. 7.

Electric field induced relative frequency shift (a) and loss (b) of HR (red

triangles) and LR (blue circles) silicon wafers measured in the two-electrode mea-

surement configuration as a function of a value of the external resistor

in the

voltage supply circuit. Fitted results are shown by solid curves. The separation gap

=

0.32 mm for the HR wafer and

=

0.27 mm for the LR wafer. The applied

voltage is

=

150 V.

The measured data for the relative frequency shift were fitted

with the model Eq. (3) and for the loss with Eq. (4). A constant term

that takes into account the loss associated with the contact resistance

and the wafer resistance

has been added to Eq. (4). Thus, the

fitting formulas look the same although the constant terms in them

had different meanings,

(

)

+

(

)

+

(7)

−

+

(

)

+

(8)

We used the Origin 8.5 software to calculate fitting parameters.

The fitted results are shown by the solid lines in Figs. 6 and 7.

The calculated fitting parameters are presented in Table I. The

standard errors of fitting parameters are small although the relation-

ships between parameters

−

2 and

are found to have

large errors. Apparently, this is a consequence of the inaccuracy of

the model while taking into account the influence of the contact

resistance

and the resistance of the wafer

. Nevertheless, we

can consider that the model satisfactorily describes the experimental

data.

The four-electrode measurement configuration was used to

study the dependence of the electric field induced loss on the sil-

icon wafer resistivity. The electric field induced relative frequency

shift and loss of both wafers measured as a function of a square of

the applied voltage in the four-electrode measurement configuration

are shown in Fig. 8. The experimental data are well approximated by

a quadratic dependence on the applied voltage. Only in the case of

losses measured for the LR wafer, a large scatter of the experimen-

tal data is observed. This is due to small values of the electric field

induced losses, which are calculated as the difference between two

large values.

The electric field induced frequency shift and loss of the LR and

HR silicon wafers measured in the four-electrode measurement con-

figuration as a function of a value of the external resistor

in the

voltage supply circuit are shown in Fig. 9. In contrast to the two-

electrode configuration, the frequency shifts are almost independent

of the value of the external resistance

for both samples and the

loss of the HR wafer. Although for the LR wafer a small peak of losses

is still observed, it is about 50 times smaller than the peak observed

in the two-electrode measurement configuration and is associated

with the incomplete compensation of the local capacitance changes

caused by the wafer vibration. For the HR wafer, the peak is not visi-

ble due to high background losses, which are obviously caused by its

higher resistivity.

Substituting in Eq. (6) the measured relative frequency shift

for the HR wafer

(

)

(

1.8

±

0.1

)

×

−

and the calcu-

lated time-constant

rel

, we obtain an estimate of the electric

field induced mechanical loss of the HR silicon wafer, associated

with the silicon resistivity. The calculated estimate of

−

was

found to be 3

×

−

for the applied voltage

150 V. The value

TABLE I.

Fitting parameters calculated from experimental data.

Sample

−

, 10

−

(

−

)

, 10

−

, 10

−

, 10

−

(

−

)

, 10

−

, 10

−

LR wafer No. 1

(

1.07

±

0.02

)

(

8.6

±

0.3

)

(

0.990

±

0.002

)

(

2.067

±

0.007

)

(

9.24

±

0.03

)

(

0.17

±

0.02

)

HR wafer No. 2

(

0.86

±

0.02

)

(

9.6

±

0.5

)

(

0.910

±

0.002

)

(

1.5

±

0.2

)

(

12.5

±

2.0

)

(

9.6

±

0.4

)

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-6

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10

Review of

Scientific Instruments

ARTICLE

scitation.org/journal/rsi

FIG. 8.

Electric field induced relative frequency shift

(

)

(a) and loss

−

(b) of the HR silicon wafer (red triangles, left axis) and LR silicon wafer (blue cir-

cles, right axis) measured as a function of a square of the applied voltage

in the

four-electrode measurement configuration. Fitted results are shown by solid lines.

The separation gap is

=

0.32 mm for the HR wafer and

=

0.27 mm for the

LR wafer. The resistor

=

17 k

was included in the voltage supply circuit.

−

(

2.3

±

0.1

)

×

−

was measured for the HR wafer in the

four-electrode measurement configuration. This is a satisfactory

agreement between the measured and the estimated loss, given the

approximate calculation.

For the LR wafer, the calculated estimate of

−

associated with

the wafer resistivity was found to be 3

×

−

for the applied voltage

150 V. The loss value

−

(

±

)

×

−

was measured for

the LR wafer in the four-electrode measurement configuration with

the resistor

17 k

that was small in comparison with the loss

peak resistor. In this case, the calculated loss is much less than the

uncertainty of the measurement. It is also possible that the losses

are associated not with the resistivity of the silicon wafer, but, for

example, the properties of its surface

contribute to the measured

electric field induced losses.

FIG. 9.

Electric field induced frequency shift and loss of HR (red triangles) and

LR (blue circles) silicon wafers measured in the four-electrode measurement con-

figuration as a function of a value of the external resistor

in the voltage

supply circuit. Fitted results are shown by solid curves. The separation gap is

=

0.32 mm for the HR wafer and

=

0.27 mm for the LR wafer. The applied

voltage is

=

150 V.

In order to compare the measured electric field induced losses

with the calculated ones using Eq. (6), we made several measure-

ments for each silicon wafer using a new clamping of the wafer each

time. Since the size of the gap between the wafer and the plate with

electrodes was slightly different each time, the values of the losses

were reduced to a single gap of 0.32 mm using the electric field

induced frequency shift. For this purpose, the relationship between

the gap and the frequency shift obtained from the measurements was

Rev. Sci. Instrum.

, 014501 (2022); doi: 10.1063/5.0076311

, 014501-7

Published under an exclusive license by AIP Publishing

05 October 2023 23:47:10