Thermal vacuum sensors based on integrated silicon thermopiles

THERMAL VACUUM SENSORS

BASED ON

INTEGRATED SILICON THERMOPILES

SANDER VAN HERWAARDEN

THERMAL VACUUM SENSORS

BASED ON

THERMAL VACUUM SENSORS

BASED ON

INTEGRATED SILICON THERMOPILES

Thermische vacuum sensoren op basis van geïntegreerde silicium thermozuilen

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag

van de Rector Magnificus prof.dr. J.M. Dirken, in het

openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Dekanen, op donderdag 15 oktober, te 16.00 uur.

door

Alexander Willem van Herwaarden elektrotechnische ingenieur

geboren te Rotterdam

Dit proefschrift is goedgekeurd door de promotor, Prof.dr.ir. S. Middelhoek.

Leden van de promotiecommissie: Wnd Rector Magnificus Prof.dr.ir. S. Middelhoek Prof.dr.ir. J. Davidse Dr. C.J.P.M. Harmans Prof.dr. M. Kleefstra Prof.dr. J. Middelhoek Prof.dr. N.F. de Rooij Prof.dr. J.F. Verweij

Delft University of Technology Delft University of Technology Delft University of Technology Dutch National Service of Metrology Delft University of Technology Twente University of Technology University of Neuchatel

Stellingen behorende bij het proefschrift

"Thermal vacuum sensors based on integrated silicon thermopiles"

van

STELLINGEN

1 Geintegreerde silicium thermozuilen zijn beter geschikt voor het meten van on-chip temperatuurverschillen dan transistorparen of weerstandsbruggen.

Dit proefschrift, Sec. II.3.

2 Elektrochemisch wegetsen van het substraat onder de thermozuil van op silicium thermozuilen gebaseerde sensoren verhoogt de gevoeligheid met een factor 30.

Dit proefschrift, Sec. II.3.2.

3 De prestaties van de in dit proefschrift gepresenteerde thermische vacuümsensoren, gefabriceerd met standaard siliciumtechnologie, zijn superieur aan die van de huidige Pirani gauges.

Dit proefschrift, Sec. IV.5.3.

4 Het nadeel van de lage thermische weerstand van siliciumthermozuil infrarooddetectoren wordt grotendeels goedgemaakt door de mogelijkheid van zogenaamde "cantilever beam" structuren en het niet in speciale gassen hoeven verpakken van de sensor.

C. Chibata, C. Kimura and K. Mikami, Far Infrared Sensor with Thermopile Structure, Proc. 1st Sensor Symposium, Japan (1981) 2 2 1 - 2 2 5

5 De thermische methode om de r.m.s.-waarde van een wisselspanning te meten is nauwkeuriger dan de elektronische methode, zelfs indien men implementaties vergelijkt die zijn vervaardigd met de technologie van de geintegreerde elektronische circuits.

6 Een smart sensor voor nauwkeurige thermische flowmeting kan worden gebaseerd op een door etsen vastgelegde thermische weerstand. Hierdoor vervalt de noodzaak van een nauwkeurige on-chip absolute elektrische spanningsreferentie.

7 Het thermo-elektrisch effect in silicium leent zich veel minder goed

voor energieconversie dan het foto-voltaisch effect.

8 In de toekomst zullen sensoren met on-chip microprocessor standaard

worden. Dit zal zeker bij thermische sensoren eerder in de vorm zijn

van een sensor met - in een klein hoekje - een microprocessor, dan

omgekeerd.

9 Geintegreerde actuatoren zijn een interessant onderwerp voor

onderzoek. Gezien hun beperkte mechanische mogelijkheden zijn hun

toekomstperspectieven echter minder florissant dan die van

geintegreerde sensoren.

10 Het verdient aanbeveling dat de technische universiteiten hun

nieuwe wetenschappelijke onderzoekprojecten (vóór) financieren uit

de opbrengsten van succesvolle lopende projecten, zoals dat ook in het

bedrijfsleven gebruikelijk is.

11 Ten onrechte wordt in sommige ziekenhuizen het dragen van

broekpakken door vrouwelijke verpleegkundigen verboden. In die

ziekenhuizen waar zij dat recht wel hebben verworven moeten de

draagsters er echter voor waken zich tegenover de patiënten op te

stellen als de vrouw met de broek aan.

12 De appeltaarten van Nederlandse banketbakkers zouden dezelfde

hoge kwaliteit verkrijgen als hun overige baksel, indien zij van

onderstaand recept gebruik zouden maken.

Men neme 1 kg goudreinetten. Schil de appels, ontdoe ze van hun klokhuis, snijdt ze in schijven en leg ze te drogen. Doe 400 gram tarwebloem, 250 gram boter, 200 gram basterdsuiker en 1 ei in een kom, en kneedt het met de mixer totdat zich deeg vormt.

V e t een 26 cm springvorm in, bepoeder hem met zelfrijzend bakmeel, strooi paneermeel op de bodem en bekleed hem met tweederde van het deeg.

Vul de springvorm, laag v o o r l a a g , met appelschijven, kaneel, rozijnen (50 g r a m in t o t a a l ) en geraspte citroenschil. Bedek de vulling kriskras met stroken van het resterende deeg en vouw de bovenrand van het deeg los van de springvorm. Z e t de oven aan op 175 °C (stand 3 a 4 voor gasovens), bestrijk de taart met eiwit, bestrooi haar met amandelschaafsel, en bak haar mooi bruin in 90 minuten.

Eerst zien, dan geloven (Seeing is believing)

Ter nagedachtenis aan mijn ouders

CONTENTS

0 PREFACE xi

0.1 OUTLINE OF THE THESIS x i i i

I VACUUM MEASUREMENT 1

1.0 OUTLINE 1 L I INTRODUCTION 2 1.2 METHODS AND GAUGES 6

1.2.1 Measurement methods 6 1.2.2 Forces exerted by the gas 8 1.2.3 Kinetic properties of the gas 10 1.2.4 Interaction of the gas with a beam of particles 12

1.2.5 Vacuum-gauge calibration 13 1.2.6 Overview thermal vacuum gauges 14

1.3 REVIEW 16

n THE THREE TRANSDUCTION STEPS 17

n.0 OUTLINE 17 n . 1 MECHANICAL TO THERMAL 18

II. 1.1 Kinetic theory of gases 18 II. 1.2 Thermal conductance by gases 21

11.2 THERMAL TO THERMAL 30 11.2.0 Introduction 30 11.2.1 Cantilever-beam structure 31 11.2.2 Round-membrane structure 38 11.2.3 Floating-membrane structure 40 11.2.4 Time constants 44 11.2.5 Infrared radiation and convection 49

11.2.6 Effects left out of the analysis 54

11.3 THERMAL TO ELECTRICAL 56

11.3.1 Thermoelectric effects 56 11.3.2 Integrated silicon thermopiles 62

ra

ra.o

ra.i

ra.i.i

m.2

DESCRIPTION OF THE SENSORS

Introduction

Silicon bipolar process Etching of silicon Cantilever-beam sensors Round-membrane sensors Floating-membrane sensors Housings and roofs

Overview of the sensor characteristics

EXPERIMENTAL S E T - U P

Vacuum system

Electrical measurement set-up Experimental procedure Measurement inaccuracies

DATA ANALYSIS

Integral data analysis Computer programs

Significance of the analysis

REVIEW 72 72 73 73 75 78 81 87 87 89 92 93 94 95 96 97 101 101 103 104 104 IV EXPERIMENTAL RESULTS 107 I V . 0 OUTLINE 1 0 7 I V . 1 ZERO-PRESSURE OFFSET 1 0 8 IV. 1.1 Magnitude 108 IV. 1.2 Repeatability 110 IV. 1.3 Offset reduction 113

I V . 2 LOW- AND MID-PRESSURE PERFORMANCE 1 14

IV.2.1 Sensitivity and accommodation coefficients 115

IV.2.2 Cantilever-beam samples 118 IV.2.3 Round-membrane samples 121 IV.2.4 Floating-membrane samples 122

IV.2.5 Repeatability 125

IV.3 HIGH-PRESSURE PERFORMANCE 1 2 5

IV.3.1 Transition pressures 126 IV.3.2 Short thermopiles and extra pieces 127

IV.3.3 Constant-temperature measurement and roofs 128

IV.3.4 Repeatability 130

I V . 4 SPECIAL EXPERIMENTS 1 3 1

IV.4.1 Time constants 131 IV.4.2 Infrared radiation and convection 132

IV.5 PRESSURE PREDICTION 133

IV.5.1 Mid-pressure range 134 IV.5.2 High-pressure range 136 IV.5.3 Comparison with conventional Pirani gauges 138

IV.6 REVIEW 139

DISCUSSION AND CONCLUSIONS 142

THERMAL-TRANSPIRATION CORRECTION 145 PARAMETER-ESTIMATION ROUTINES 147 SAMENVATTING 1 51

ACKNOWLEDGMENT 153 ABOUT THE AUTHOR 154 LIST OF PUBLICATIONS AND PRESENTATIONS 155

REFERENCES 156 LIST OF SYMBOLS 160

o

PREFACE

This thesis is one of several theses that have resulted thus far from the project on integrated silicon sensors started in Delft in 1973 by Prof. S. Middelhoek. It all began with A.F.P. van Putten's design of an integrated silicon anemometer [0.1]. An integrated silicon sensor is the first element of an electronic data-processing system. Electronic information systems, which can be used to acquire information about our environment, can be divided into three elements, see Fig. 0.1 [0.2].

any signal Sensor electr. signal Signal Processor electr. signal Output transducer optical signal

Figure 0.1: An electronic information system.

They consist of an input transducer (sensor) to transduce non-electric signals into electric signals [0.3]. A signal processor (for instance a computer) modifies the electric signals into a convenient form. An output transducer, for instance a monitor, is then used to visualize the processed signals or the information contained in them. Or, alternatively, another output transducer may be used to store or transmit the information or to act upon it, as in robot systems. A few of the reasons for carrying out research on sensors, and in particular on integrated silicon sensors, are listed below.

The information-systems motive. As electronic computer systems have become very powerful, the bottleneck in information systems has shifted to the field of data acquisition. This is due to a lack of suitable sensors, as the sensor development has not kept pace with the rapid advancements in the field of electronic computers.

The economic motive. Because of the lack of suitable sensors matching the price-performance ratio of computers, a demand for sensors adapted both technically and economically for use with the computers of today is likely to arise.

The technical motive. As integrated silicon sensors are fabricated in the same highly advanced processes as electronic computer circuits, they may reap the benefits of this rapidly advancing technology, and are perfectly suited to be linked to these electronic circuits.

The academic motive. Doing research on sensors, which are devices operating in the fascinating interface between electronics and physics (or chemistry), is an attractive and interesting pastime, especially when it is carried out within the framework of integrated-circuit technology, with its many possibilities and novelties.

Review

These and other reasons draw many electrical-engineering students with a liking for physics, including me, to the silicon-sensors project group of Prof. S. Middelhoek. In 1982 I joined the sensor group as an undergraduate student, and took up the subject of integrated silicon thermopiles (sensors that transduce a temperature difference into an electrical voltage, exploiting the Seebeck effect). This project had been started three years earlier by Prof.S. Middelhoek and Ger Nieveld [0.4]. Integrated thermopiles are attractive transducers because their self-generating transduction principle allows a temperature-difference measurement without offset.

Towards the end of 1983 LinaSarro and I jointly started Ph.D. work on the applications of integrated silicon thermopiles. The characteristics of the thermopiles relevant to their use in sensors had been studied fairly well by then. One of the conclusions was that the high thermal conductivity of silicon prohibits the use of integrated thermopiles for off-chip temperature-difference measurement, because in that case thermal contact resistances introduce intolerably large errors. The emphasis of our work was therefore put upon studying the application of the thermopiles in thermal sensors. In thermal sensors the temperature difference is created on the chip itself, and thermal contact resistances have no influence.

LinaSarro first developed an etching process fully compatible with the bipolar integrated-circuit process run in Delft. The objective was to improve the thermal characteristics of the integrated thermopiles by etching away the superfluous silicon of the substrate under the thermopile, which forms a tremendous thermal short-circuit. She then started a project

on integrated infrared sensors, and developed the first fully integrated silicon thermal infrared sensor and sensor array using the Seebeck effect. The results of her work are set down in her Ph.D. thesis [0.5].

I first studied the applications of thermopiles in thermal sensors. In the course of this study Pieter Hochstenbach and I developed a thermal true RMS converter, which can very accurately measure the value of an AC signal [0.6]. This project was carried out in cooperation with the Science & Industry group of N.V., Philips' Gloeilampenfabrieken, and the Van Swinden Laboratory of the Dutch National Service of Metrology. After that I started a project on integrated thermal vacuum sensors, and developed various types of thermal vacuum sensors based on silicon thermopiles.

0.1 Outline of the thesis

The results of the thermal-vacuum-sensor project are reported in this thesis. Major contributions to the project have been made by LinaSarro, who successfully developed the etching process, and by Mr .E. Smit and Ir.P.K. Nauta, who are responsible for the successful design of the modified bipolar process.

The thermal-vacuum-sensor project had four main objectives.

Firstly, obtain a theoretical understanding of the operating principle of thermal vacuum sensors, and construct models of the structures feasible with integrated-circuits technology.

Secondly, show the feasibility of thermal vacuum sensors based on integrated silicon thermopiles, and design such sensors optimized with the results of the theoretical models.

Thirdly, experimentally verify the theoretical models of the sensors, and generate experimental results to improve the theoretical models.

Finally, get an impression of the performance of the sensors as a vacuum gauge, and compare the optimal structures with presently available thermal vacuum gauges.

Chap. I gives a short introduction about the significance of vacuum and on vacuum measurement. Some basic concepts used in the thesis are also explained.

Chap. II deals with the first objective, by giving a theoretical description of the transduction principle of the sensors. The effects determining the three transduction steps in thermal vacuum gauges are described.

The first step, the transduction from the mechanical to the thermal domain, depends upon the pressure dependence of the thermal conductance of a gas. The second step, the conversion in the thermal domain, is carried out by the influence of the thermally conductive gas on the temperature

distribution in the sensor structure. The third step, the transduction from the thermal to the electrical domain, is performed by the integrated silicon thermopiles.

Chap. Ill describes the exper imentaLiools used to show the feasibility of integrated thermal vacuum sensors (the second objective). The feasibility of making useful devices is shown in the description of the sensors used during the experiments. After exposing the experimental set-up some computer programs used for the data analysis are described.

Chap. IV describes the experimental results. First, the performance of the sensors as a function of pressure is described in three parts, examining the zero-pressure offset, the low- and mid-pressure performance and the high- pressure performance. By means of the data-analysis programs of Chap. Ill the theory of Chap. II is verified with the experimental results (the third objective). The results of some special experiments carried out to check some specific points then follow.

Next an impression is given of the accuracy of the pressure prediction by the sensors, and a comparison with conventional Pirani gauges is made to get an impression of the performance of the sensors as a vacuum gauge (the fourth objective). With this step the objectives of the thermal-vacuum-sensor project are accomplished and the project is concluded.

In Chap. V the main results of the project are discussed and summarized. The thesis ends with two sections on the thermal-transpiration correction and the parameter-estimation routines, and the references, list of symbols and other customary appendices.

It was (on the whole) a joy to write this thesis and perform the research described in it. I hope that those who receive this thesis may derive as much pleasure from it as I did.

Sander van Herwaarden Delft, 24 June 1987.

I

VACUUM MEASUREMENT

1.0 OUTLINE

This thesis describes the development of thermal vacuum sensors. Chap. I is a brief introduction to the concepts of sensor, vacuum and thermal vacuum measurement.

In Sec. 1.1 the chapter start out with a short introduction on transducers, and on the six domains in which physical signals may be classified. Then a brief description of the thermal vacuum sensor is given, and the significance of vacuum is sketched. Finally, for those unfamiliar with electrical circuits, the concepts of resistance and sheet resistance are elucidated.

In Sec. 1.2 some basic principles of vacuum measurement methods will be discussed to get a good insight into the detailed principle of operation of the devices.

Then insight into the various ways to measure vacuum will be given by reviewing some important vacuum gauges, including several vacuum gauges used as reference in the experiments. Three groups of gauges can be distinguished [1.1]. We will first discuss the gauges measuring the force exerted by the gas on a surface of known area, and thus directly measure pressure. Next the gauges measuring a pressure-related kinetic property of the gas such as viscosity or thermal conductivity are reviewed, and finally we will consider the group of gauges which measure the interaction between the gas and a beam of particles.

After that the methods for vacuum-gauge calibration are briefly treated, including the one used in our experiments.

The section ends with an overview of thermal vacuum gauges. This will provide an impression of the work which has already been done on these types of gauges, and serve as a reference for comparing the work presented in this thesis.

Sec. 1.3 concludes the chapter with a review of the main results and conclusions.

1.1 INTRODUCTION Signal domains

Whenever we want to increase our knowledge about our environment, we quickly feel the need for devices capable of providing quantitative data on physical or chemical quantities. As we probe deeper into the secrets of nature and the amount of data increases, the need to modify and interrelate the available data arises. This is the only way in which the underlying information may be extracted from the mass of data to increase our insight into the world around us.

The highly developed technology for making electronic data processors has enabled us to manipulate data in nearly any way we want. To do this it is, however, necessary that the data be available in electrical form.

In general, the information we require can be encoded in signals of various types. For practical purposes it is convenient to distinguish six types of signal domains: the mechanical, chemical, magnetic, radiant, thermal and electrical signal domain, as shown in Fig. 1.1.

To enable an efficient processing of data there is a need for devices capable of transducing signals of one of the five non-electrical domains into an electrical signal. Devices capable of transducing signals from one signal domain to another are called sensors.

There is a myriad of possible transductions between the signal domains. A transduction can be carried out from any of the six domains directly to any of the other five domains, but it can also be done via one or more intermediate domains, as in so-called tandem transducers.

Thermal vacuum sensor

This thesis describes the development of a thermal vacuum sensor, a device which transduces the mechanical signal pressure into an electrical voltage with a known relation between the two signals.

In this sensor the transduction from pressure to electrical voltage is carried out in three steps, as is indicated in Fig. 1.1. First, the mechanical signal pressure is transduced into the thermal signal thermal conductance of the gas. Then the thermal signal thermal conductance is converted into another thermal signal, temperature difference. Finally, the thermal signal temperature difference is transduced into the electrical voltage.

Figure 1.1: The six signal domains, with the transduction of pressure by a capacitance manometer (dashed line) and by a thermal vacuum gauge (solid lines).

The first step is mainly a physical phenomenon, which we exploit for measuring. vacuum. Its description is basic physics. The second step involves the creation of a suitable thermal configuration, designed for optimum modification in the thermal domain. The third step is the transduction of the temperature difference by the thermopile. This step has been studied in detail before incorporating the other two steps. The three transduction steps are treated extensively in Chap. II.

Vacuum

"Although the sensor measures pressure, it is called a vacuum sensor, because it only measures pressures below atmospheric pressure, and the name for this pressure range is vacuum.

Formerly the unit in which vacuum was expressed was the Torr, defined as the pressure exerted by a 1 mm-high mercury column. This unit is derived from the height of the mercury column pushed up by the atmospheric pressure, which is approximately 760 mm. In honor o( the Italian Evangelista Torricelli (1608-1647), who first measured the atmospheric pressure (with a mercury column), the unit of pressure was named Torr. Presently the Si-unit Pascal, named after the Frenchman Blaise Pascal (1623-1662) and abbreviated as Pa, is becoming standard: l P a = l N / m2 and 1 Torr = 133.32 Pa. Also in wide use is the millibar

(mbar), which is 100 Pa. Atmospheric pressure is thus around 760 Torr, 1013 mbar and 101,325 Pa.

Significance of vacuum

Depending upon their physical characteristics, various types of vacuum can be distinguished [1.2].

Just below atmospheric pressure (100 kPa-1 kPa) the pressure difference is of importance, as in vacuum cleaners. Human respiration also depends upon a small underpressure.

At lower pressures (1 kPa - 1 mPa) the reduced gas density is of importance, as in thermos bottles in which cold or hot fluids are thermally isolated by the vacuum between the inner and the outer bottle. The thermos bottle in fact relies on the same physical principle as the thermal vacuum sensors, the decrease in the thermal conductance of a gas when the pressure decreases.

At still lower pressures (1 mPa - 1 /xPa) the long mean free path between collisions becomes the objective. Consider a television set in which the electron beam can only reach the screen and illuminate it because there are no gas molecules to scatter the beam.

The lowest pressures presently attainable (1 /xPa - 1 pPa), some twenty orders of magnitude below atmospheric pressure, have mainly been used up to now for scientific purposes.

For all these pressure ranges one often needs vacuum sensors (or vacuum gauges as vacuum technologists usually call them) to measure the degree of vacuum actually obtained. The methods and gauges to measure vacuum will be the subject of this chapter.

Resistances

In this thesis signals of various physical natures will be treated. It is often convenient to describe non-electrical systems in terms of electrical systems, because of the advanced description and analysis tools available for electrical circuits. In particular, the concept of resistance is useful in describing the relation between various quantities within any signal domain, whether electrical or not. The resistance of an object gives the relation between the through quantity flowing through the object in the presence of an across quantity existing between both ends of the object. In this thesis three types of resistances will be distinguished in particular. Firstly, the electrical resistance, or just resistance R, which describes the relation between the electrical voltage U (in Volts) across the resistance and the electrical current / (in Amperes) through the resistance, see Fig. 1.2a:

U = IR " (1.1) Eq. (1.1) is the famous law of Ohm, and the unit of electrical resistance is named Ohm after the German Georg Simon Ohm (1789-1854), who first formulated this law.

Secondly, there is the thermal resistance Rth (in K/W) describing the ratio of the temperature difference AT (in Kelvin) to the heat flow P (in Watt) of any physical object, see Fig. 1.2b.

AT = PR _th (1.2)

■' I ■' p 0

hT A P

/?th /?fl

Figure 1.2: Resistances: a) electrical b) thermal c) flow.

Finally, there is the flow resistance Rih expressed in the SI unit s/m3 or,

more conveniently, in seconds/liter. This resistance gives the ratio of the pressure difference AP (in Pascal) to the gas throughput Q (in Pam3/s or

in Pal/s) for a vacuum tube or any other type of flow resistance, see Fig. 1.2c.

AP = QRn (1.3) It is perfectly possible to extend the electrical analog further. In Chap. II this will be done for the thermal domain to elucidate the thermal characteristics of the sensors. Vacuum systems may also be described in terms of electrical circuit parameters, enabling a vacuum system to be analyzed by Spice, a well-known and advanced computer program for simulating electrical circuits [1.3].

One other type of resistance often used in this thesis is the so-called sheet resistance. Consider a layer of material of uniform thickness D and with a thermal conductivity K ( the same reasoning applies for the electrical sheet

resistance). If one calculates the thermal sheet resistance RBt of a square slab of this layer with length and width X (see Fig. 1.3), then the result is:

A (length) _ 1 _ 8 t *(width) **> KD

(1.4)

Eq. (1.4) shows the important characteristic of the sheet resistance: the actual size of the square X is of no importance for the value of RBt. If the length and width of the square are both increased equally, RBt remains the same. The thermal sheet resistance is expressed in K/WG, the electrical sheet resistance RBe in fi/D (o means 1 square long). The sheet resistance enables the resistance of a geometry to be calculated by multiplying the length-to-width ratio of the geometry by the sheet resistance. This makes calculations independent of the specific dimensions, and makes the results more generally applicable. Later on we will encounter the length-to-width ratio of a thermopile, denoted by A, and calculate the thermal resistance of a thermopile using Rst.

AT

Figure 1.3: Sheet resistance.

1.2 METHODS AND GAUGES 1.2.1 Measurement methods

In this section the various vacuum gauges - devices capable of measuring absolute pressures below atmospheric pressure: vacuum - will be briefly reviewed to give an impression of how vacuum can be measured. For a more extensive treatment of vacuum gauges and vacuum measurement in

general the reader is referred to the book of Berman [1.4], which deals extensively with these subjects.

Pressure is the force per unit area.exerted by a gas on its surroundings by the momentum transfer of the individual gas molecules colliding with the surroundings. The measurement of the pressure itself is practically restricted to pressures above 1 mPa, because below 1 mPa the forces become too small to be measurable. Gauges for lower pressure ranges therefore rely on the measurement of a non-mechanical property of the gas, related to the mechanical property pressure through basic physics. Berman [1.5] distinguishes three main principles on which the transduction process of vacuum gauges can be based: measurement of the forces exerted by the gas; of the kinetic properties of the gas; and of the interaction between the gas and a particle beam.

The first principle is that the pressure measurement is carried out by converting the force exerted by the gas on a surface of known area into a displacement, which is then measured optically or electrically. The operating range of vacuum gauges working in this way is generally restricted to the higher pressures, between 100 kPa (= atmospheric pressure) and 1 Pa, except for measurements with McLeod gauges and specially constructed primary standards, which are not used for the continuous monitoring of pressure.

The second principle of vacuum measurement allows routine measurement of pressures between 100 kPa and 1 /xPa (considerations of cost and accuracy determine the choice of gauge in the range above 1 Pa). In this principle the pressure is measured by measuring a pressure-dependent kinetic property of the gas. This can be the momentum exchange between the gas and a moving object immersed in the gas as in the so-called viscosity gauges, or the energy transfer by the gas between two surfaces at different temperatures, as in the thermal gauges.

A very special gauge is the Brownian-motion gauge, which measures the fluctuations in the momentum transfer of the gas molecules to a particle suspended in the gas by observing the motion of the particle. This gauge is not yet commercially available.

For even lower pressures, between 1 Pa and 1 pPa (1 picoPascal = 10"12Pa)

or even 1 f Pa (1 femtoPascal = 10~15 Pa), gauges based on the third

principle are required. This principle is based on measuring the interaction between the gas and a beam of particles.

Various types of these gauges exist. Some are based upon the attenuation or scattering of a light beam or a beam of charged particles (such as electrons or ions) by the gas molecules. In other types the gas molecules are excited or ionized by an electron beam, and either the number of photons emitted by the molecules falling back to their ground state, or the number of gas molecules being ionized is a measure of the pressure.

1.2.2 Forces exerted by the gas

Mercury-filled tube

In 1643 Torricelli invented the first vacuum gauge when he performed the experiment with the mercury-filled tube (see Fig. 1.4). In this tube the mechanical quantity pressure is converted into the mechanical quantity height (of the mercury column) with the specific mass of mercury and the earth's gravity as conversion parameters. A quantitative vacuum gauge was obtained by the Frenchman René Descartes (1596-1650) when he attached a paper scale to a mercury-filled tube in 1647 [1.6].

zero pressure

atmospheric pressure

mercury

Figure 1.4: Measurement of the atmospheric pressure with a mercury-filled tube.

McLeod gauge

In 1874 the Englishman H. McLeod invented an important improvement

by making an apparatus in which the (very low) pressure of■■ a known volume of gas was converted to a much higher pressure by compressing the gas into a very small volume. The invention of McLeod uses Boyle's law, which states. that pressure times volume is a constant at constant temperature. It is actually a (pressure) amplifier in the mechanical domain which amplifies a very low pressure up to a level where it can be

measured with an ordinary mercury column (although one could also measure the amplified pressure with another vacuum gauge [1.7]). In this way the lowest pressure that could be measured was lowered from approximately 1 mm of mercury column (100 Pa) to several decades below that pressure. Presently McLeod gauges are capable of measuring absolute pressures of 1 mPa and lower.

Bourdon gauge

A different pressure gauge is the Bourdon gauge, in which a spiralled hollow tube changes shape when pressure differences exist between the inside and the outside of the tube. A pointer connected to the tube indicates its current shape. When the tube changes shape as the pressure difference between the inside and the outside changes, the pointer turns to another position and indicates the new pressure difference. The range of the Bourdon gauge is in general from 100 Pa to atmospheric pressure.

Diaphragm (capacitance) manometers

A pressure gauge closely related to the Bourdon gauge is the diaphragm manometer, in which the unknown pressure is measured by admitting it to one side of a thin diaphragm, while on the other side a known reference pressure exists, such as zero or atmospheric pressure. The pressure difference will cause the diaphragm to deflect, which can be measured by attaching a. pointer to the diaphragm, as is done in barometers. Alternatively, one can obtain an electrical output signal by using the diaphragm as one of the plates of an electric capacitor. Deflection of the diaphragm will cause a change in capacitance, which can be measured electrically. After the conversion step within the mechanical domain (pressure into deflection) the transduction is towards the electrical-signal domain instead of the radiant-signal domain.

Diaphragm manometers exist in many implementations. The most expensive ones can have a resolution lower than 1 mPa at 100 Pa full scale, with an inaccuracy of less than 0.1%. There also exist many microelectronic implementations of the diaphragm manometer. These can be divided into three types.

The first is the capacitive type in which the deflection of the membrane results in a change in an electrical capacitance. The second is the piezoresistive type, in which the deflection of the membrane results in a change in resistance of the diffused resistors integrated in the membrane. The third is the oscillating type, in which the deflection of the membrane results in a shift of the mechanical resonance frequency of the membrane. These devices are often made of monocrystalline or polycrystalline silicon using etching techniques, and have silicon membranes [1.8-1.12]. The operating range of these devices can be from approximately lOOmPaand lower to atmospheric pressure, but high-sensitivity devices need to be protected against overpressure damaging the membrane.

A SEM photograph of a very ingeneous device is shown in Fig. 1.5 [1.12]. It consists of a diaphragm with two hills on top, which are connected by an oscillating structure (the floating membranes in the middle). If due to a pressure difference the diaphragm is deformed, the hills on the diaphragm tend to move away from each other, stretching the oscillating structure. The stress induced by this changes the resonance frequency of the oscillator, which in this way is a measure of the pressure.

Figure 1.5: Diaphragm manometer with frequency output based on a resonator structure (photograph courtesy of J. Greenwood f 1.12J).

1.2.3 Kinetic properties of the gas

Under 1 Pa the forces exerted by the gas become so low that effects other than the momentum transfer by the gas have to be used for routine pressure measurement. In the low, medium and high vacuum range, from

100 kPa to 1 /iPa, gauges measuring the kinetic properties of the gases are routinely used.

Two main types exist, those measuring the viscosity and those measuring the thermal conductance of the gas. A third type is the Brownian-motion gauge, which measures fluctuations in the momentum transfer by the gas to ultrafine particles (1 nm) suspended in the gas [1.13]. It is named after

the British botanist R. Brown, who first studied the movement of large particles suspended in a fluid. This gauge is still in the experimental stage, and will not be discussed further.

Viscosity gauges

There are various ways in which the viscosity of a gas, which is closely related to the thermal conductivity of the gas, can be put to use in vacuum measurement. Very delicate gauges are those in which a disk or sphere is set to oscillate in the presence of the gas. The damping action of the gas is related to its pressure, both in the free-molecular state and the viscous state.

This damping action may be measured in several ways. The first are the constant-power and constant-energy measurement modes, in which the shift or rate of decrease of oscillation frequency is a measure of the viscosity of the gas, and thus of the pressure. Another approach is the const ant-frequency measurement mode, in which the oscillation frequency is kept constant by supplying extra power to overcome the damping action of the gas. The amount of extra power required to maintain the oscillation frequency is then the pressure-dependent output signal. The devices can also be constructed in various ways. They can be designed to experience the viscosity of the gas itself, by suspending a disk in mid-air, or to experience a viscous drag by letting it rotate or oscillate just above an immobile surface.

Many microelectronic vacuum sensors have been made using the viscosity effect: some based on oscillating membranes, actuated piezoelectrically or thermally [1.14-1.15], and others based on oscillating bridges, beams or forks [1.16-1.17]. The operating range of these devices is usually between

lOOkPaand lOOmPa.

Spinning rotor gauge

A viscosity gauge which has received a lot of attention during the last decades is the spinning rotor (SR) gauge, which is becoming an important standard in the high and ultra-high vacuum range [1.18]. It consists of a steel ball, magnetically levitated, which is rotated in the presence of the gas. The gauge is operated in the constant-energy mode, in which the steel ball is magnetically accelerated to a rotation frequency of 400 Hz. Then the acceleration is stopped and the deceleration caused by the pressure-dependent viscous damping is detected. Inaccuracies of as low as

1% are attained within parts of its pressure range of 1 Pa to 1 nPa. Because of its high accuracy, the absence of any pumping action, and its theoretically very predictable operation, the gauge is expected in the future to become the primary pressure standard at very low pressures [1.19].

Thermal gauges

Thermal gauges belong to the most widely used vacuum gauges. This applies in particular to the Pirani gauge, developed in 1906 by the Italian Marcello Pirani. In applications where accuracy is not the main objective, such as the monitoring of the low and medium vacuum between the backup pump and the high-vacuum pumps, Pirani gauges are very popular because of their ruggedness, reliability and low cost.

The conventional Pirani gauge consists of a hot wire, whose pressure-dependent heat loss is measured. Just as with viscosity gauges two operating modes are used. One is the constant-power measurement mode, in which the temperature of the hot wire is a function of pressure, when heating with a constant power. This temperature is measured by measuring the temperature-dependent resistance of the hot wire, as in a Pirani gauge, or by measuring the hot-wire temperature with a thermocouple, as in the thermocouple gauge. The other method uses the constant-temperature mode (comparable to the constant-frequency mode of viscosity gauges), in which the temperature of the hot wire is kept constant by means of feedback, and the required heating power is the pressure-dependent output signal. Both the Pirani and the thermocouple gauge may also be operated in this mode.

Miniaturized implementations of thermal vacuum gauges often employ a different structure, namely a hot surface instead of a hot wire. Again the operating mode may be either the constant-power or constant-temperature mode, and the temperature measurement may be performed by either a temperature dependent resistance or a thermocouple or thermopile [1.20-1.24]. The devices presented in this thesis all belong to the hot-surface thermopile type of thermal vacuum gauges, some of which are suitable for constant-power measurement, while yet others lend themselves better to constant-temperature measurement.

In contrast to viscosity gauges, which often range from 100 mPa to 100 kPa, thermal vacuum gauges only range from 100 mPa to 1-10 kPa, because the thermal conductivity of gases becomes pressure independent at high pressures. In specially designed gauges the upper pressure limit may be increased to 100 kPa, either by convective heat transfer or by reducing the characteristic dimensions of the gauge.

1.2.4 Interaction of the gas with a beam of particles

Various gauges based on the interaction of moving particles with the gas molecules exist. Many of these are only put to use on special occasions. In the fluorescence gauge an electron beam excites the gas molecules, which emit ultraviolet photons when falling back to their ground state. The number of photons counted by a Geiger-Muller photon counter is then a measure of the pressure. In another gauge the photons of a laser beam are

scattered by the gas molecules, and either the intensity of the scattered photons or the attenuation of the laser beam provides information about the gas density. Similar gauges use an electron beam to obtain an electrical output signal without the transduction step from optical to electrical. Alternatively, a radioactive source of a, p or 7-particles may be used in very special applications such as satellites.

As exotic as these gauges are, the cold-cathode and, in particular, the hot-cathode ionization gauge are very important. Together with the Pirani gauge the hot-cathode gauge is probably the most widely used gauge, with the Pirani measuring the medium vacuum and the hot-cathode gauge the ultra-high vacuum, between 1 Pa and 1 nPa and lower.

The hot-cathode ionization gauge functions as follows. A hot filament emits a well-controlled electron current. The electrons ionize gas molecules, which are collected by an electrode at a negative electrical potential. The ratio of the resulting ion current to the emitted electron current is a measure of the pressure; this ratio is typically of the order of 0.02/Pa for nitrogen. The best-known implementation of this gauge is the Bayard-Alpert gauge, but there are many modifications, especially to improve the low-pressure resolution.

The cold-cathode ionization gauge is a less complicated, but also less accurate, implementation of the ionization gauge. It also relies on the ionization of the gas molecules by electrons. Because the cold cathode emits fewer electrons, a magnetic field is created around the gauge to prolong the time of flight of the electrons before they reach the anode, which increases the chance of an electron ionizing a gas molecule. The range of the cold-cathode gauge is of the order of 10 mPa to 100 pPa.

1.2.5 Vacuum-gauge calibration

There are four main methods to calibrate vacuum gauges like the ones presented in the previous section. A short description of these methods will be given below. For an extensive treatment of the subject the reader is once more referred to Berman [1.25].

Static methods

The first two methods are called static methods, because the calibration pressure is obtained in a static way, as the result of a fixed amount of test gas and the vacuum chamber volume. In these methods the reading of the gauge to be calibrated is compared with the reference gauge by subjecting them to the pressure exerted by a fixed amount of the test gas.

The first method is the direct method, in which the reference gauge and the gauge to be calibrated are connected to the same vacuum chamber and experience the same pressure. This method is used for the pressure range of 10 mPa to 10 Pa, and inaccuracies of ^ 1 % or lower are attainable.

The second method uses so-called static expanders, in which the reference gauge measures the pressure exerted by a fixed amount of test gas in a small vacuum chamber. Then the test gas is expanded into a big vacuum chamber and the resulting pressure is measured by the gauge to be calibrated. If the volume ratio of the small and big vacuum chambers is 1 : n - l , then the pressure experienced by the gauge to be calibrated is n times as low as that measured by the the reference gauge. This enables calibrations to be made at low pressures while still operating the reference gauge at a high pressure, where it is more accurate. The pressure range of this method is 100 /zPa to 1 kPa. A large number of effects influence the accuracy of this method, which is very intricate and time consuming.

Dynamic methods

In the two other methods, the dynamic methods, the calibration pressure is the result of the dynamic balance of a gas throughput Q of the test gas and the pumping speed S of the vacuum pump.

In the direct method the reference gauge and the gauge to be calibrated are both attached to the same vacuum chamber in which the calibration pressure is created, and the gauge to be calibrated is directly compared with the reference gauge. The operating range is between 100/xPa and 100 Pa, with inaccuracies of the order of 10% at a 100/zPa calibration pressure.

In the indirect method, the dynamic expansion method, the reference gauge and the gauge to be calibrated are placed in two separate vacuum chambers which are connected by a flow resistance with a known value Rn. In this way a pressure difference QRn is created between the reference gauge and the gauge to be calibrated, enabling calibration to be made at a low pressure while comparing it with a reference gauge operated in its accurate high-pressure range. The dynamic expansion method offers many possible operation procedures, and allows calibration at pressures even lower than 1 /iPa. The inaccuracies of this method strongly depend upon the actual procedure and calibration pressure.

The method used for the experiments in this thesis is the direct static method. This method is the simplest, and is perfectly suited to the calibration of thermal vacuum gauges in the 0.1 Pa to 100 kPa range.

1.2.6 Overview thermal vacuum gauges

To give an impression of the development of miniaturized thermal vacuum gauges an overview of these devices is given in Table 1.1. Unfortunately, the presentation of the experimental results is not very uniform, which makes it impossible to draw quantitative conclusions. The figures listed should therefore be considered as indicative only. Some data about

conventional Pirani gauges (hot wire) are also listed. The operating ranges listed in Table 1.1 are those listed in the papers describing the experimental results.

Table 1.1: Thermal vacuum gauges.

type technology range sensitivity ref. hotwire conventional 0.1 Pa-100 Pa <100mV/Pa [1.26] hotwire conventional l P a - l k P a <2mV/Pa [1.26] hotwire conventional 0.5/iPa-lPa 3%/Pa [1.27] hot surface magnetite O.lmPa-lOOPa 5 mW/Pa [1.20] hot surface TaN thin film 1 mPa-100 kPa 5 mW/Pa [1.21] hot surface GaSb thin film 1 mPa-100 kPa <0.3°/o/decade * [1.22]

hot surface BiSb thermopile 0.5 mPa-0.5 kPa [1.23] hot surface silicon resistor ImPa-lOOkPa <0.5%/decade * [1.24]

* decade = factor of ten pressure change

Poulter et al. [1.26] have investigated the reproducibility of off-the-shelf commercial Pirani gauges. The zero-pressure instability was found to be of the order of 20% and 7% over a six months period, for low- and high-pressure-range gauges, respectively. After resetting the zero indication the reproducibility of both gauges was found to be better than 6% over the range listed in Table 1.1.

Oguri [1.27] studied an ultra-low-noise Pirani gauge under optimized conditions, thermostatic bath, 12 hours warm-up time, etc. He found a resolution of 0.7 /iPa, with a zero-pressure-output drift of approximately

1 /xPa/minute, and a sensitivity of approximately 30 mV/Pa for a I V supply voltage of the bridge.

Varicak and Saftic [1.20] studied magnetite-thermistor sensing elements on large foils to improve the resolution of their gauge. They achieved a mid-range sensitivity of approximately 7%/Pa, ~600 mV change between 6 and 60 mPa at a 150 V supply voltage.

Shioyama et al. [1.21] made Pirani gauges based on TaN thin films. At low pressures, below 100 Pa, they observed a heat loss of approximately 45 /xW/KPa for a 2 x 2 mm TaN sensor on a glass substrate, when operating the device at constant temperature. At high pressures a sublinear relation between the pressure and the sensor output was obtained.

Patel and Mahajan [1.22] used GaSb thin films and found a logarithmic relation between the pressure and the sensor output at low pressures, with a slope of approximately 30 mV per tenfold change in pressure, at a 9 V bridge supply. The instability of the thin films decreased the output curve of the sensor by a factor of 2 over a thirty-day period, indicating ageing effects in the thin-film resistor.

BiSb heater to improve the resolution of the thermal gauge, and obtained a resolution of approximately 0.5 mPa.

Bretschi [1.24] used the silicon planar technology to overcome the ageing problems of the thin film technology.

Summarizing, it may be stated that a number of interesting sensors have already been developed with miniaturization techniques.

1.3 REVIEW

The main conclusions and results of Chap. I are briefly listed below. Introduction (1.1)

Six signal domains are introduced in which physical signals may be classified: mechanical, chemical, magnetic, radiant, thermal and electrical. Thermal vacuum sensors, transducing the mechanical signal vacuum into an electrical output voltage via thermal signals (temperature differences) are briefly described, and the importance of vacuum - absolute pressures lower than atmospheric pressure - is emphasized.

The concept of resistance and sheet resistance for electrical, thermal and vacuum configurations is introduced.

Vacuum measurement (1.2)

Three methods of measuring vacuum are introduced: measure directly the forces exerted by the gas on a diaphragm; measure the viscosity or thermal conductance of the gas; and measure the interaction between the gas molecules and a beam of particles, an electron beam for instance.

Vacuum gauges relying on these methods are described. Theses include the capacitance manometer for the lOOmPa to 100 kPa range which measures the force on a known area and is independent of gas type. The Pirani gauge measures the thermal conductance and is widely used for low-cost measurement between lOmPa and lOkPa. The hot-cathode ionization gauge measures the ionization of gas molecules by an electron beam and is widely used for the range 1 nPa and 1 Pa.

Four main principles of calibrating untested gauges are briefly described. They are based on the comparison of the unknown gauge with a reference gauge with known characteristics. This is done either at equal pressures (the direct method), or with a calculable pressure difference (the indirect method) to enable calibration at reduced pressures below the operating range of the reference. In addition, the calibration pressure may be the result of a fixed amount of test gas in a fixed volume (the static method), or of the dynamic balance of a test-gas leak and the pumping speed of the vacuum pumps (the dynamic method).

Finally, an impression of miniaturized versions of thermal vacuum gauges is given in Sec. 1.2.6. This shows that a number of experimental devices have already been realized using different technologies.

II

THE THREE TRANSDUCTION STEPS

11.0 OUTLINE

In this chapter the operation of the thermal vacuum sensors presented in this thesis will be considered from a theoretical point of view. In practice, the theoretical and experimental results undergo a simultaneous development; sometimes the theory is developed first and is borne out by experimental results, while in other instances the experimental results are explained by a theoretical model derived afterwards. In this thesis, however, theory and experiment are kept strictly separate. The theoretical results are presented first, with two objectives in mind. The first is to give a well-founded insight into the operating principle of thermal vacuum sensors by thoroughly explaining the underlying physical effects. The second objective is to create a well-organized and quickly accessible set of quantitative theoretical predictions against which the experimental results may be checked.

A theoretical treatment of the three steps in the transduction process will be given. The first step, the transduction from the mechanical to the thermal domain (from gas pressure to gas thermal conductance), is treated in Sec. II.l. The second step, the conversion from the thermal to the thermal domain (from thermal conductance to temperature difference), is dealt with in Sec. II.2. The last step, the transduction from the thermal to the electrical domain (from temperature difference to thermopile output voltage), is treated in Sec. II.3.

characteristics necessary to explain the thermal conductivity of gases are described. Then the thermal conductance by gases is derived for various physically different situations, and a synthesis of the phenomena is given (Sec. II. 1.2). This is done for the thermal conductance between two parallel plates, because that is the structure of the sensors. Sec. II. 1.2 concludes with a practical formula describing the gas thermal conductance in our sensor structures for the entire pressure range.

Sec. II.2 describes the temperature distribution in the sensor structures in the presence of a thermally conducting gas. This is done with one-dimensional models of a cantilever-beam structure, a round-membrane structure, and a floating-membrane structure.

Sec. II.2.4 is devoted to the calculation of the time constants of the sensor structures, to see how quickly they respond to pressure variations.

In Sec. II..2.5 two heat transfer mechanisms other than conduction are reviewed, namely infrared radiation and convection, in order to estimate their possible influence on the performance of the vacuum sensors.

Sec. II.2.6 gives an overview of some of the effects left out of the one-dimensional analysis.

Sec. II.3 discusses the properties of the thermopile, used for the third step of the transduction, the temperature-difference measurement.

First the thermoelectric effects: the Seebeck effect, the Peltier effect, and the Thomson effect, will be treated. Then the properties of intregrated silicon thermopiles will be examined.

In Sec. II.4 the chapter ends with a review in which the main theoretical results are recapitulated.

H.1 MECHANICAL TO THERMAL

n.1.1 Kinetic theory of gases

The vacuum sensors described in this thesis are all based on the pressure dependence of the thermal conductivity of gases. The classic kinetic theory of gases provides a good description of the physical effects leading to this dependence [2.1]. Three centuries have already passed since the Englishman Robert Boyle (1627-1691) made his contribution to the theory of gases which led to the general gas law:

PV = NmR0T (2.1)

which states that the pressure P (in Pa = N/m2) of any perfect gas is the

same at thermal equilibrium if the molecular densities are equal (atoms will also be called molecules for convenience) [2.2]. That is, it is the same if the number of molecules Nm divided by the volume V (in m3) is equal.

Avogadro's number NA = 6.022 x 102 3 molecules/mole. JR0 is the universal

gas constant and amounts to R0 = 8.314 J/Kmole. Boltzmann's constant k = R0/NA= 1.3805 x l O '2 3 J/K follows from R0 and NA. T is the absolute temperature in Kelvin. For Eq. (2.1) to be true the gas has to be perfect, which implies that the attractive forces between the gas molecules, and the net volume occupied by the molecules in comparison to the total volume V, are negligible. At low pressures and room temperature this is true for most gases of simple composition. According to the kinetic theory Eq. (2.1) can be written as:

P = nkT = jnmv* (2.2) In Eq. (2.2) we move from the macroscopically observable variable P to the

variables of the kinetic theory, since the right side of this equation directly links the pressure to the rate of momentum transfer as exerted by a gas of density n (in molecules/m3), molecular mass m (in kg) and

root-mean-square velocity vr (in m/s). From Eq. (2.2) it follows that the

average kinetic energy £a v per gas molecule is:

that is, the average kinetic energy of a gas molecule of any gas is the same at the same temperature, and it is proportional to the absolute temperature. From Eq. (2.3) we may conclude that for two surfaces with a temperature difference of A7* the heat transfer per molecule from one surface to the other will be of the order of 3kAT/2.

In general, the velocity of the gas molecules ranges between zero and infinity, and the distribution of gas molecules as a function of velocity is given by the Maxwell-Boltzmann distribution:

n dv / v -i ^2kT> vexp[^2kTJ ^A)

giving the fraction fv of all molecules with velocity between v and v+dv. With Eq. (2.4) we can calculate the root-mean-square velocity vr and the

average velocity vav by evaluating the following integrals:

[ " v ^ d v / f /vdv - v r- ( - ^ ) * (2.5)

8*r l * (2.6)

\,,*l\,* -

Based on the distribution function it is possible to evaluate the molecular incidence rate <j>, which is the number of molecules striking a square meter of (imaginary) surface per second [2.3]:

* = | « vB V (2.7)

Because heat is eventually transferred by molecules, the incidence rate, describing the number of heat-transporting molecules that hit a surface each second, is a key parameter in the description of the thermal conductivity of gases. Another quantity of interest is the mean free path between collisions of the molecules A, which gives the mean distance over which an average molecule can travel and transport heat before colliding with other molecules. A is approximated by the equation:

1 kT (2.8)

2±7Trt£2 2±7T£2/>

in which £ is the diameter of the molecules. Eq. (2.8) clearly shows that A is inversely proportional to the pressure. Sutherland found experimentally that A is also temperature dependent at constant n, and he used the following relation to describe A:

A

T-TïT^rT

in which A^ is the mean free path at very high temperature, c is Sutherland's constant, and AT is the mean free path at temperature T.

Table II. 1 lists the approximative values for c as given by Roth [2.4] and Dushman [2.5], and for A^ as given by Roth [2.4] and A3 0 0 K as given by

Dushman [2.5] at 1 Pa, for the relevant gases.

These gases are argon, helium and nitrogen (which were used as test gases), oxygen (which together with nitrogen, i.e. air, enters the vacuum chamber through leaks), and water vapor (which is the main residual gas at high vacuum, 1 mPa pressure).

The spread in the data presented in Table II. 1 is probably an indication of the difficulty of experimentally determining A^ and c. The data of A ^ have been recalculated on the assumption that the molecular density for these data was'based on 1 Torr and 0 °C [2.4].

Table 11.1: Mean free paths and Sutherland's constant. gas type argon helium nitrogen oxygen water vapor Sutherland's constant c [2.4] 169 79 112 132 600 c [2.5] 133 98 102 110 659

(K) mean free path at 1 Pa (mm) >oo [2-4] 10.2 23.4 8.9 10.1 13.9 ^300 K [2.5]-7.1 19.8 6.7 7.3 4.6

II.1.2 Thermal conductance by gases

The thermal conductivity of a gas is rather difficult to calculate accurately [2.6]. A good approximation, however, can easily be obtained with the kinetic theory of gases. Because of the structure of the sensors presented in this thesis, we will do that below for the heat transfer between two parallel plates with a gap of size M (in m) between them, plate 1 being at temperature Tx and plate 2 at temperature T2. We can distinguish three cases, which will be dealt with separately. First we will consider the two extreme cases in which the effects which occur can easily be explained.

The first extreme is the free-molecular thermal conductance, which occurs at low pressures when the mean free path between collisions A is more than a hundred times the gap size M. In this pressure range the thermal conductance of gases is proportional to pressure. The factor one hundred is chosen because the remaining error due to a non-infinite A is in that case less than 1%.

The second extreme is the viscous thermal conductivity at high pressures, where A is more than a hundred times as small as M. In this pressure range the thermal conductance of gases is independent of pressure. The factor one hundred makes the remaining error because of the non-zero value of A less than 1%.

Secondly we will investigate the thermal conductance in the transition region, where both effects play a role. The formula for the thermal conductivity in this region will appear to be valid to a good approximation in the entire pressure range, and the formulas found earlier will prove to be the limit of this formula for the extreme situations.

From the treatment below it will be made clear that the thermal conductance of a gas between two plates is somewhat dependent upon the specific properties of the materials and configuration. Therefore, we will talk in terms of thermal conductance rather than of thermal conductivity, as the latter is usually associated with a material property, and the former with the characteristics of a specific configuration.

Free-molecular thermal conductance

If the mean free path A is much longer than the distance M between the plates, the heat will be transferred by individual molecules. We can distinguish two streams of (non-interacting) molecules, one stream consisting of molecules traveling from plate 1 to plate 2, and one stream the other way, see Fig. 2.1.

Figure 2.1: Free-molecular-gas thermal conductance between parallel plates.

We can calculate the heat transferred per square meter H (in W/m2) to be

equal to the molecular incidence rate <j> (and thus pressure) times the heat capacity of a molecule times the temperature difference. Because we are considering the molecules crossing a plane instead of the gas as a whole, the average heat capacity of a monatomic molecule is 2kT instead of HkT, since the faster molecules not only carry more heat but also cross the gap faster. This yields as the thermal conductance G (in W/m2K) at low

pressures:

G=

T

r^\

"

-

In practice, molecules do not reach full thermal equilibrium with a surface when they collide with it, but they exchange on the average only a part of thejr heat excess (or deficit) with that surface. The extent to which they do so is called the accommodation coefficient, and in Fig. 2.1 it is indicated by assigning effective temperatures T{ and r2' to the streams

reflected from plates 1 and 2 [2.7]. For the situation of Fig. 2.1 we can write:

(T^-T^^a^T^T^) (2.11) Eq. (2.11) states that the heat transfer from stream 2 to plate 1 is lower

than that expected for a temperature difference {T1 - 7"2'), it is only a

fraction av In Eq. (2.11)"T2' is the temperature which can be associated

with the average energy of the molecules in stream 2. A similar equation applies for the heat exchange at plate 2, yielding as the effective accommodation coefficent a:

^ _ 7 Y - 7 y _

a

T1-T2 ax + a2 - a ^

This means that, although the temperature difference of the plates is T2 - Tlt the heat transfer will only have the size as if it were JT2' - T{.

Usually, a depends upon the temperature, the type of gas, the type of surface, the gas coverage of the surface, and the pressure.

When the accommodation coefficient and the increase in the specific heat due to rotational and vibrational energy modes of polyatomic molecules is included, the thermal conductance of gases in the free molecular case is given by:

cssa

^=h

y

^ -

-

^P^

P

(2.i3)

in which the free-molecular thermal conductivity (7moj (the gas property)

and the low-pressure thermal conductance G0 (the actually measured -value) are expressed in W/m2KPa, and 7 = cp/cv, the ratio of the specific

heat at constant pressure to that at constant volume (7 is approximately 5/3 for monatomic gases, 7/5 for diatomic gases and approaches unity for multiatomic gases).

Viscous thermal conductivity

In this section we use the term conductivity instead of conductance, which we use everywhere else. This may seem inconsistent. However, unlike the other situations, the heat transfer by a gas in the viscous state is determined solely by the gas itself and is not influenced by external factors. It is therefore a true gas property, which is why we, for once, talk in terms of conductivity.

The heat transfer is determined by the properties of the gas as a viscous medium if the mean free path A is much smaller than the gap M. In the

case of Fig. 2.2 the temperature difference T2 - Tx imposes a temperature gradient VT = (T2 - T^)/M. The thermal conductivity of the gas may now be calculated by determining the heat flux H (in W/m2) crossing any

reference plane parallel to the plates.

Figure 2.2: Viscous-gas thermal conductance between parallel plates.

Again, the number of molecules crossing a square meter per second is given by Eq. (2.7), and for the heat capacity of an average molecule crossing the plane we will once more take 2kT. The situation differs from the free-molecular case now in that the average temperature of a molecule crossing the reference plane is not T{ or T2\ It is only XVT higher or lower than the temperature at the reference plane, because a molecule crossing the reference plane originates, on the average, one mean free path away from the reference plane [2.8]. The rate of heat transfer is therefore given by:

H~jnvay2k[\VT-(-WT)) (2.14) which yields for the thermal conductivity (7vUc of a viscous gas:

The dimension of C7visc is W/mK. Taking into account the increase of the

specific heat due to the rotational and vibrational energy modes of polyatomic molecules, and all the subtleties of the collision processes, the final approximation of Cvisc becomes:

in which r\ is the viscosity of the gas, 77 ~ j«wvavA. If we use the classic

formula for the specific heat cv ~ k/mi^-l) (see also Eq. (2.3), (7-1) = 2/3

for monatomic gases), and if for simplicity we leave out Sutherland's effect, we find:

G

. „(97-5) (Ml)iJL

2 i 7 )

°V 1 S C _ 4 ( 7 - 1 ) L7TWJ 7r£2 K '

Unlike Eq. (2.13), the pressure P is absent in Eq. (2.17)

Thermal conductance in the transition region

There exists an intermediate region, for those pressures where the mean free path A has approximately the same order of magnitude as the gap M. In that case the so-called "temperature jump" may be used to explain the pressure dependence of the thermal conductance.

Consider the situation at plate 1 (see Fig. 2.3). Molecules colliding with the surface have an average energy corresponding to the temperature T\ one mean free path away from the surface. Molecules reemitted from the surface have an energy corresponding to an average temperature T{, which for an accommodation coefficient of 1 is Tv At the surface, therefore, the average temperature of the molecules is not lower than (T\ + r1) / 2 , which must be somewhat higher than Tt if any heat transfer is to take place.

This temperature discontinuity (temperature jump) at the surface can be taken into account by increasing the effective gap between the plates by "jump distances" J1 and J2, such that the temperature gradient in the middle, where the thermal conductance is determined by the viscous character of the gas, equals the external temperature difference divided by the effective gap (see Fig. 2.3). Because the heat flux H is equal to V r Gv i s c, where VT is the gradient in the middle of the gap where Cvisc

applies, this leads to the following equation [2.9]:

*~!r

f

~,

(2.i8)

Figure 2.3: Thermal conductance of a gas in the transition region. Calculations show that for a = 1 the jump distance / is approximately 2X, and that / increases as a decreases. In general, the jump distance may be approximated as:

/ = 2 - a 97-5

a 2(f+l) (2.19) J is inversely proportional to the pressure via A. For very low pressures J»M, and when J^ = J2 is taken, Eq. (2.18) simplifies to Eq. (2.13) for a1 = a2, using Eq.(2.12). For high pressures where J«M, Eq.(2.18)

yields GviBC/M as the thermal conductance.

The thermal conductance over the whole pressure range

Equation (2.18) may be written slightly differently to arrive at the following expression, which gives the thermal conductance G of a gas between two parallel plates for all pressures and integrates the formulas for the three cases above into:

H ppt

7-,-r,

° P + p

In this equation G0 is the actually measured low-pressure thermal conductance (in W/m2KPa).