Interdigital sensors and transducers

doi:10.1109/JPROC.2004.826603

Interdigital Sensors and Transducers

ALEXANDER V. MAMISHEV, MEMBER, IEEE

,

KISHORE SUNDARA-RAJAN

, STUDENT MEMBER, IEEE

, FUMIN YANG

, STUDENT MEMBER, IEEE

,

YANQING DU

, MEMBER, IEEE

,

AND

MARKUS ZAHN, FELLOW, IEEE

Invited Paper

This review paper focuses on interdigital electrodes—a geo-

metric structure encountered in a wide variety of sensor and

transducer designs. Physical and chemical principles behind the

operation of these devices vary so much across different fields of

science and technology that the common features present in all

devices are often overlooked. This paper attempts to bring under

one umbrella capacitive, inductive, dielectric, piezoacoustic, chem-

ical, biological, and microelectromechanical interdigital sensors

and transducers. The paper also provides historical perspective,

discusses fabrication techniques, modeling of sensor parameters,

application examples, and directions of future research.

Keywords—Dielectric measurements, interdigital sensors, non-

destructive testing (NDT), sensor design, sensor modeling, spec-

troscopy, surface acoustic waves (SAWs), transducers.

I. INTRODUCTION

A. Motivation for This Paper

Interdigital electrodes are among the most commonly used

periodic electrode structures. Recent advances in such fields

as nondestructive testing (NDT), microelectromechanical

systems (MEMS), telecommunications, chemical sensing,

piezoacoustics, and biotechnology involve interdigital elec-

trodes in very different ways. At the same time, a number of

Manuscript received February 16, 2003; revised December 30, 2003. This

work was supported in part by the National Science Foundation under CA-

REER Award 0093716, in part by the Center for Process Analytical Chem-

istry, in part by the Electric Power Research Institute, in part by the Univer-

sity of Washington Royalty Research Fund, in part by the Link Foundation,

in part by the Air Force Office of Sponsored Research, and in part by the

American Public Power Association.

A. V. Mamishev, K. Sundara-Rajan, and F. Yang are with the Sensors,

Energy, and Automation Laboratory (SEAL), Department of Elec-

trical Engineering, University of Washington, Seattle, WA 98195 USA

(e-mail: mamishev@ee.washington.edu; kishore@ee.washington.edu;

fuminy@ee.washington.edu).

Y. Du is with Underwriters Laboratories, Santa Clara, CA 95050 USA

(e-mail: Yanqing.Du@us.ul.com).

M. Zahn is with the Department of Electrical Engineering and Computer

Science, Massachusetts Institute of Technology, Cambridge, MA 02139

USA (e-mail: zahn@mit.edu).

Digital Object Identifier 10.1109/JPROC.2004.826603

common features are shared among these applications. The

purpose of this paper is to outline common features and to

highlight the differences of sensor geometry, manufacturing

techniques, choice of materials, analytical and numerical

modeling, design optimization, system integration, and

data analysis. It is difficult and perhaps even excessive to

maintain equally deep and comprehensive treatment of all

these subjects. Instead, the fringing electric field sensors are

given the deepest emphasis in this manuscript. Significant

aspects of other types of sensors are discussed, while repe-

tition is avoided. References are provided to major review

papers and books in each section devoted to a particular

field of interdigital electrode applications, such as dielectric

imaging, acoustic sensors, and MEMS.

It is not possible to develop a universal sensor and a uni-

versal parameter estimation algorithm that would provide

the maximum information about material properties in all

applications. Each application requires a judicious choice

of sensor design and associated parameter estimation algo-

rithms. As a technical system develops, the requirements for

each element become clearer and affect the requirements for

each element of the trinity shown in Fig. 1. For example, it

may become clear during the development stage that the one-

sided access to the material under test (MUT) is not necessary

due to the specifics of the manufacturing process. In this case,

the electrode layout design should not be limited to interdig-

ital structures only. The dual-sided access has advantages of

larger, easily measurable capacitances and a more uniform

field distribution. The examples of appropriate matching of

sensors and parameter estimation algorithms with different

applications are encountered throughout this paper.

B. Terminology

The explosion in the quantity of scientific information has

brought its share of confusion to the subject of interdigital

sensors. As a result, lack of coordination of research efforts

is not uncommon in this field. For example, theoretical

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Fig. 1. Every sensing application requires an optimum

combination of inherently dependent elements of the measurement

system comprising sensor design and parameter estimation

algorithms.

expressions obtained for capacitance of interdigital piezo-

electric sensors are rarely mentioned in papers published in

the fields of dielectrometry or MEMS; researchers in one

country are often unaware of efforts in other countries; and

sensor designers repeat mistakes of previous generations.

Ambiguity of terminology does not help this situation: not

everyone associates periodic microstrip electrode structures

with interdigital patterns. Moreover, the word “interdigital”

itself does not have direct analogs in other languages. Thus,

this most frequent term is often replaced by such equivalents

as “periodic,” “microstrip,” “comb,” and “grating,” as well

as such variations as “interdigitated” and “combed.” The

overlap of the terms is not complete, for example, not every

microstrip circuit is interdigital, and not every interdigital

circuit is strictly periodic.

The term

interdigital, selected for use throughout this

paper, refers to a digitlike or fingerlike periodic pattern of

parallel in-plane electrodes used to build up the capacitance

associated with the electric fields that penetrate into the

material sample or sensitive coating.

Another term frequently misunderstood in context of

interdigital sensors is wavelength. One should distinguish

between the radiation wavelength of electromagnetic waves

and the wavelength of the spatial periodicity, or spatial

wavelength, of the geometrical structure. The former is the

wavelength

with the frequency equal to the frequency of

the voltage source applied to the sensor electrodes

(1)

where

is the speed of light and is the frequency of the

voltage source. This variable is typically discussed in con-

text of radio frequency (RF) and microwave circuits. For ex-

ample, a 600-MHz electromagnetic wave has the wavelength

of 50 cm. The spatial wavelength of the periodic interdigital

structure is the distance between the centerlines of the adja-

cent fingers belonging to the same electrode.

Using the terms capacitance and conductance for the de-

scription of terminal characteristics of an interdigital sensor

may be misleading, especially when one encounters effec-

tive negative values of capacitance or conductance. Strictly

speaking, these are mutual capacitance (or transcapacitance)

and mutual conductance (or transconductance), as defined

in multielectrode circuits. For the overwhelming majority of

cases, this distinction is not important enough to justify the

use of longer, more cumbersome terms.

Sensors, transducers, and detectors, as explained below,

are related; hence, these terms are often used interchange-

ably. The choice of the term usually depends on the function

of the device that one wants to emphasize.

A sensor is a device whose output can be quantified and

changes with one or more physical phenomena. This output

information can be used for process monitoring and control.

A transducer is a device that converts one form of energy into

another form. The measurement of physical variables asso-

ciated with the resulting form of energy allows estimation of

the physical variables associated with the input energy. A de-

tector is a device indicating presence, absence, or change of

the signal qualitatively, either as a binary signal or as a low

resolution signal with several states.

C. Historical Perspective

Historically, the first and still the most common reason

for making an interdigital electrode structure is to increase

the effective length, and, therefore, the capacitance between

the electrodes. Perhaps the earliest example of interdigital

electrode design is found in the patent of N. Tesla, issued

in 1891 [1]. In this example, each “finger” is a rectangular

plate, immersed in an insulating liquid. The total capaci-

tance of the “electrical condenser” proposed by Tesla in-

creases approximately linearly with the number of plates.

This principle is sometimes used in modern capacitors as

well. Theoretical expressions for calculation of capacitance

between coplanar strips appeared in the 1920s [2]. Antenna

designers use such periodic strip patterns to control the radia-

tion patterns. The technology, later named microdielectrom-

etry, started more than 20 years ago as a modification of the

charge-flow transistor (CFT), originally developed for mon-

itoring the sheet resistance of thin-film materials [3]. The

CFT was an MOS-compatible device based on contempo-

rary principles of transistor electronics [4], [5]. In this device,

the time delay between the application of the gate voltage

and a complete inversion of the channel region is dependent

on the sheet resistance of the MUT. Subsequent experiments

demonstrated that bulk conduction effects can be separated

from surface conduction. The separation of these two phe-

nomena gave additional valuable insights into the conduction

properties of the materials under test. Extensive use of inter-

digital electrodes for sensing applications started in the 1960s

[6] along with other forms of coplanar electrode structures

[7]. Later, independent dielectrometry studies with single [3],

and multiple [8], [9] penetration depths using interdigital

electrodes have continued in several countries. Commercial-

ization of multiple penetration depth dielectrometry in the

United States had been initiated in the 1980s and 1990s [10],

[11].

The existence of surface acoustic waves (SAWs) in an

isotropic material was first demonstrated by Lord Rayleigh

in 1885 [12]. SAW technology, as applied to modern elec-

tronic systems, arose originally from radar technology. In

1965, White and Voltmer [13] demonstrated the basic SAW

delay line structure by depositing two thin-metal interdigital

transducers (IDTs) on a polished piezoelectric plate. Until

the 1980s, this technology has been mainly used for military

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applications. Starting from the 1980s, SAW devices began to

be used for consumer electronics and in telecommunications.

SAW devices have been made to meet the needs of acoustic

filter technology [6], [14] and later spanned fields of commu-

nication, signal processing, chemical sensing, nondestructive

evaluation, and biomedical applications. Extensive reviews

of acoustic sensor technology have been written [15], [16]

and somewhat reduce the need for the acoustic sensor tech-

nology coverage in this paper.

D. Common Features

One-side access. Several inherent advantages of the

planar interdigital geometry attract device designers. One

of the most important ones, especially for the NDT sensors

and piezoacoustic transducers, is that only a single-side

access to the test material is required. One can penetrate

the sample with electric, magnetic, or acoustic fields from

one side of the sample, leaving the other side open to the

environment which can allow absorption of gas, moisture,

or chemicals, which can change electrical properties of the

MUT. A sensitive layer of chemical or biological nature

deposited over the electrodes can also interact with a gas or

liquid environment, allowing monitoring of concentration

of chemicals in such materials as air, transformer oil, or

the human body. In some situations, the other side of the

material sample may be too far away or inaccessible due to

design limitations for an electrode so that one-sided access

is essential.

Control of signal strength. By changing the area of the

sensor, the number of fingers, and the spacing between them,

one can control the strength of the output signal. A tradeoff

between the signal-to-noise ratio and the minimum sensing

area is selected based on the application requirements. In

microchip sensors, the size of the sensitive area is usually

of little consequence, whereas in imaging devices it plays a

major role.

Imaging capability. A relatively new field of research

is interdigital frequency-wavelength dielectrometry, a

close relative of electrical impedance tomography. This

technology employs interdigital electrode pairs of variable

spatial periodicity. The depth of penetration of quasi-static

electric field lines into the material is frequency independent

and proportional to the spatial wavelength, defined as the

distance between the centerlines of neighboring electrodes

of the same polarity. The differing penetration depths

of multiple wavelengths make possible spatial profiling

of dielectric and conduction properties and geometry of

semi-insulating materials. That is, properties of individual

layers across the thickness of a stratified medium can be

determined without direct access to each layer. Movement of

the sensor along the material surface with subsequent signal

processing makes possible, in principle, three-dimensional

(3-D) imaging of dielectric properties of insulating materials.

The dielectric properties can be related to many other phys-

ical properties, such as porosity, density, structural integrity,

and chemical content of materials under test. At very high

radian frequency

, the skin depth further

limits the penetration of electric and magnetic fields into a

medium with conductivity

and magnetic permeability .

Multiple physical effects in the same structure.In

addition to sensing applications, fringing electric fields are

used increasingly to generate mechanical forces, especially

in MEMS. Scaling of motors and actuators to the distances

on the order of tens of micrometers make electric fields a

feasible choice for actuation of moving parts, whereas mag-

netic fields dominate the traditional macro scale designs. A

strong electric field enhancement at the ends of interdigital

electrodes generates a lateral force, which is used for

comb displacement. Electrical force levitation is achievable

using fringing fields in the direction perpendicular to the

interdigital pattern plane.

Simplified modeling. When the aspect ratio of the

electrode finger length to the spatial wavelength is large,

the numerical simulation and theoretical modeling is greatly

simplified because it can be done in the approximate

two-dimensional (2-D) limit rather than in 3-D. For this

reason, interdigital electrodes are popular when extensive

modeling is required for the interpretation of experimental

results. For example, elaborate models of electrical or

acoustical interactions of interdigital sensors and trans-

ducers with stratified media are often developed using 2-D

approximations. There is a trend, however, toward full

3-D simulations, taking advantage of continuing dramatic

increases in computer speed.

Spectrum. Fig. 2 shows the frequency spectrum for

acoustic and electromagnetic phenomena and instrumenta-

tion ranges. The lowest frequencies, starting at 1

Hz are

used for laboratory measurement of conduction properties,

where low-level conduction phenomena are difficult to

otherwise measure. The frequencies below 1 Hz are not

suitable for real-time industrial monitoring and are used

mostly in the laboratory studies of physical phenomena.

The characteristics of wave propagation drive the selection

of frequency ranges above 1 Hz. Especially significant are

such properties as spatial resolution, penetration depth,

attenuation, and external interference.

II. P

HYSICAL PRINCIPLES

A. Electric Field Sensing

A fringing field dielectrometry sensor has the same prin-

ciple of operation as the more conventional parallel-plate

or coaxial cylinder dielectric sensor cell. The voltage is ap-

plied to the electrodes, and the impedance across the elec-

trodes is measured. However, unlike the parallel-plate cell,

the fringing field sensor does not require two-sided access

to the MUT. Fig. 3 shows a gradual transition from the par-

allel-plate capacitor to a fringing field capacitor. In all three

cases, electric field lines pass through the MUT; therefore,

the capacitance and conductance between the two electrodes

depends on the material dielectric properties as well as on the

electrode and material geometry.

Interdigital dielectrometry is a subset of interdigital elec-

trode sensor applications that relies on direct measurement

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Fig. 2. Frequency spectrum for acoustic and electromagnetic interdigital sensors (partly based on

[16], [192], and [193]).

Fig. 3. A fringing field dielectrometry sensor can be visualized

as: (a) a parallel-plate capacitor whose (b) electrodes open up to

provide (c) a one-sided access to the MUT.

of dielectric properties of insulating and semi-insulating ma-

terials from one side [17]–[19]. The basic idea is to apply

a spatially periodic electrical potential to the surface of the

MUT. The combination of signals produced by variation of

the spatial period of the interdigital comb electrodes com-

bined with the variation of electrical excitation frequency

potentially provides extensive information about the spatial

profiles and dielectric spectroscopy of the MUT. Since the

changes in the dielectric properties are usually induced by

changes in various physical, chemical, or structural proper-

ties of materials, the dielectrometry measurements provide

effective means for indirect nondestructive evaluation of vital

parameters in a variety of industrial and scientific applica-

tions [10], [20].

Usually, the capacitance between two coplanar strips, as

shown in Fig. 3(c), is comparable to the stray capacitance

of the leads (conductors that connect the electrodes with the

electrical excitation source). Therefore, in order to build up

an easily measurable electrode structure, the coplanar strip

pattern may be repeated many times.

The art and science of designing sensors with multiple in-

terdigital electrode pairs and processing the output signals

to gain knowledge about the MUT is the subject of multi-

wavelength interdigital frequency wavenumber (

- ) dielec-

trometry which has been under development for about three

decades. The penetration depth of the fringing quasi-static

electric fields above the interdigital electrodes is proportional

to the spacing between the centerlines of the sensing and the

driven fingers and is independent of frequency. Overviews of

important concepts related to this technology are available in

[11] and [20]–[25].

One of the most attractive features of multiwavelength

dielectrometry is the ability to measure complex spatially

inhomogeneous distributions of properties from one side. As

the complexity of spatial distribution grows and the number

of unknown variables in each experiment increases, the

parameter estimation algorithms become more complicated,

computationally intensive, and less accurate and reliable. Ul-

timately, every major type of spatial distribution of material

properties requires a different parameter estimation algo-

rithm. The types of spatial distributions include, but are not

limited to, homogeneous materials, multiple-layer materials,

local discontinuities (such as cracks and electrical trees),

global discontinuities of microstructure (such as grains or

fibers forming the material), and, finally, smoothly varying

properties. On the electrical properties side, materials under

test may be purely insulating or weakly conductive. Various

phenomena may affect sensor response, including frequency

dispersion, electrode polarization due to an electrochemical

double layer, quality of interfacial contact, and many others.

A conceptual schematic of an

-k dielectrometry sensor

measurement is presented in Fig. 4. For a homogeneous lossy

dielectric medium of semi-infinite extent, periodic variation

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Fig. 4. A generic interdigital dielectrometry sensor [25], [50].

of quasi-static electric potential along the surface in the di-

rection produces an exponentially decaying pattern of elec-

tric fields penetrating into the medium in the

direction. The

variation of shade in the MUT indicates the possible variation

of material properties and, thus, variations in the complex di-

electric permittivity

with the distance from the surface.

Concepts of the forward and the inverse problems are

widely used in the literature related to this technology. Here,

the forward problem is defined as the task of determining the

electric field distribution and the interelectrode admittance

matrix when the geometry, material properties, and external

excitations are given. Correspondingly, the inverse problem

requires determining either material properties or associated

geometry, or both, when the imposed excitations and experi-

mental values of the sensor admittance matrix are available.

Each application that involves theoretical modeling usually

requires solving the forward problem.

The forward problem can be solved using several ap-

proaches. One of them is to use a continuum model [26].

From the electro-quasi-static field point of view, in a ho-

mogeneous lossy dielectric, the electric scalar potential of

the field excited by the driven electrode is a solution to

Laplace’s equation. At any constant

position, the electric

field distribution far away from the sensor edges is periodic

in the

direction and assumed uniform in the direction.

For a homogeneous dielectric of semi-infinite extent, the

scalar potential can be written as an infinite series of sinu-

soidal Fourier modes of fundamental spatial wavelength

that decays away in the direction

(2)

where

is the wavenumber of each

mode. For

excitations at radian frequency , such

that

, the complex surface capac-

itance density

relates at a planar surface

Fig. 5. Half-wavelength cross section with a superimposed

equivalent



-circuit model.

constant to the potential at that surface for the th

Fourier mode of the homogeneous dielectric of semi-infinite

extent as

(3)

where

(4)

is the complex permittivity with

as material dielectric per-

mittivity and

as ohmic conductivity. Then, knowledge of

at the electrode surface lets us calculate the terminal cur-

rent from the potential distribution at that surface. It is also

possible to solve the forward problem with commercial fi-

nite-element software [27], with finite-difference techniques,

or by using analytical approximations [28].

Fig. 5 shows the equivalent circuit of the sensor superim-

posed onto the schematic view of a sensor half-wavelength.

Note that each wavelength has an opposite conducting guard

plane at the bottom of the substrate. For each wavelength,

a follower op-amp drives the guard plane at the substrate

bottom at the voltage

, thus eliminating any cur-

rent between the sensing and guard electrodes through the

substrate. Therefore, the effect of

and on circuit re-

sponse is eliminated, which simplifies response analysis and

improves device sensitivity.

The concept of imposed frequency wavenumber (

- )

goes beyond dielectrometry applications. Earlier studies led

by J. R. Melcher employed interdigital electrodes to study

electrohydrodynamic surface waves and instabilities [29],

effects on static electrification in insulating liquids [30],

[31], and electromechanics of electrochemical double layers

in liquid electrolytes [32].

The penetration depth of the fringing electric fields above

the interdigital electrodes is proportional to the spacing

between the centerlines of the sensing and the driven fin-

gers. Fig. 6 illustrates the idea of multiple penetration depths

for a three-wavelength sensor. The variation of the material

properties across the thickness of the MUT in the

direction

can be approximately found by simultaneously solving three

complex equations describing this three-wavelength experi-

mental arrangement as a piecewise three-layer system.

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Interdigital sensors and transducers

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