Sensors and Materials, Vol. 8, No. 6 (1996) 373-387

MYUTokyo
S &M0252

Physical Principles of Thermal Sensors
Sander van Herwaarden
Xensor Integration
P. 0. Box 3233, 2601 DE Delft, the Netherlands(Received March 29, 1995; accepted March 25, 1996)
Key words:

thermal transduction, transduction physics

We give an introduction to the physical principles on which silicon thermal sensors are
based. We briefly describe the transductions and conversions involved in these sensors,and different thermal sensors and the physics on which they are based.

1.

The Functional Principle of Thermal Sensors

In this paper, we describe thermal sensors based on silicon and related technologies,
with an electrical output signal. The input signal can be any of the six signal types definedby Middelhoek and Audet,< 1J i.e., mechanical, magnetic, chemical, radiant, the1mal andelectrical signals. In thermal sensors, the transduction of the input signal to the output
signal is carried out in two steps. First the input signal is transduced into a thermal signal,and then the thermal signal is transduced into the eleetrical output signal.

1.1 Thermal power sensors

An important distinction among thermal sensors is that between power sensors andconductance sensors. In power sensors the input signal which is measured by the sensor isthermal power, which is used to generate the electrical power of the sensor output signal.
Therefore, there is no output signal if the input signal is zero. These self-generating sensors
have no offset, and need no biasing. They operate in three steps.
1) A nonthermal signal C is transduced into a thennal heat signal P, by the sensor-specificaction Q.
P=QC

· 373

( 1)

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2) The heat P is converted into a thermal signal temperature difference 11T, by a thermal
resistance R.
!1T=RP(2)3) The temperature difference !-,.Tis transduced into an electrical voltage U, by a tempera­ture difference sensor S.
U=St,.T(3)

The total transfer ratio UIC of a thermal power sensor is given by
UI C= QRS

(4)

A psychrometer is an exception to this three-step operation. In this sensor, the nonthermal
signal is transduced directly into a temperature difference.

1.2 Thermal conductance sensors
In the case of thermal conductance sensors, the input signal C changes the thermal
conductance G between the sensitive area and the ambient. To measure the thermal
conductance, the sensor is biased with a heating power P. The transfer in these modulating
sensors is as follows.
1) The nonthermal signal C is transduced into a thermal conductance signal G, by asensor-specific transduction action Q, G0 is the offset of the sensor.
G=QC+Go(5)2) The conductance G is converted into a temperature difference 11T, by the heating power
P.

!iT=PIG(6)3) The temperature difference !-,.Tis transduced into an electtical voltage U, by a temperature
difference sensor S.
U=S!1T(7)

For thermal conductance sensors, the total transfer cannot be written in a multiplicative
form. Instead we find
U = PS!(QC+Go)-(8)This equation indicates the offset character of modulating sensors. In some thermal375

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conductance sensors, the electrical signal in step 3) has another form, such as a current or
resistance, depending upon the type of temperature difference sensor used.
In this paper we concentrate on the first transduction step, the sensor-specific transduction
action Q. We use various thermal sensors as examples. Considering the variety of thermal
sensors described in the literature, a comprehensive summary is not given. The second andthird transduction steps are described elsewhere. cz-4)2.

Heat Transfer Mechanisms

In most thermal power sensors, the heat is generated, and the transduction takes place inor on the sensor itself. In some, the transport of nonthermal power to the sensor involves
heat transfer mechanisms. In thermal conductance sensors, on the other hand, the change
in the conductance due to the nonthermal signal is in the ambient directly around thesensor, not in or on the sensor itself. In this case, heat transfer mechanisms are essential tothe operation of the sensor. There are four heat transfer mechanisms, conduction, convection,
radiation and phase transition. C5l
Conduction always occurs, and in thermal conductance sensors the first transduction
step is based on the thermal conduction Gconct between the active area of the sensor and theambient. The thermal conductance between two parallel surfaces with area A separated bya distance D is given by
Gcond= KAID,(9)

where K is the thermal conductivity of the medium present between the surfaces. The
sensor action may be based on the dependence of the physical signal- on K (pressure dependence: vacuum sensor/Pirani gauge; fluid dependence: thermal
conductivity and overflow sensor),- on D (plate-spacing dependence: accelerometer), or- on all three parameters K, D and A (thermal properties sensor).
Convection, the second mechanism for thermal conductance sensors, is heat transfer tomoving fluids, as in flow sensors. Usually convection is negligible in sensors other thanflow sensors. Although convection conducts heat away from the active area, the physicalprinciple is different from that of conduction, since the heat is not transferred by stationary
molecules from neighbor to neighbor, but by a continuous supply of molecules flowing
past. In the simplest formula for laminar flow over a flat plate, the heat transfer Gconv isgiven by
Gconv =

0.664P/!·33Re0·5 K/L,

(10)

where L is a characteristic length, for instance the length of the sensor, Pr is (thetemperature dependent material constant) the Prandtl number, and Re is the Reynolds
number, which is dependent on the flow velocity V, Land the kinematic viscosity v (in m2/ s), that is Re = VL/ v. The sensor action can be based on the flow velocity dependence of376

Sensors and Materials, Vol. 8, No. 6 (1996)

the Reynolds number (as in flow sensors), or on the pressure dependence of the viscosity
and the conductivity, which are also constituents of the Reynolds number (as in some
Pirani gauges for vacuum measurement near atmospheric pressure).
Radiation is heat transfer by electromagnetic waves, either as infrared radiation (thermalradiation), or other forms, such as microwaves or magnetic fields which generate heat in adissipative layer (hysteresis in a magnetic layer, for instance). It is a self-generating effect,
and is therefore utilized in thermal power sensors. Given below are two fo1mulae for
infrared radiation. One is for the heat transfer between parallel plates, which, when bothplates are at around room temperature, is equal to(11)
where e is the emissivity or absorptivity of one plate. The other plate is assumed to be blackwith emissivity e = 1. All materials emit infrared radiation, but usually it does notcontribute significantly to the heat exchange of a thermal sensor with the ambient.
The other formula describes the heat transfer between an infrared sensor with black
detecting area Au (as viewed from the object) and temperature Tu and an object of area Aobj
(as seen from the sensor), temperature Tobi and emissivity e. The exchanged power is(12)Here, a is the Stefan-Boltzmann constant (56.7 x 10-9 W /K4m2), and dis the distance
between the sensor and the object. When a sensor with a sensitive area of 1 mm2 at 300 K
is exposed to an object at 301 K with an emissivity of 55% at a solid angle of 0.1 steradian(i.e., A0b/d2 = 0.1), the exchanged power is 100 nW.
Phase transition (for example, the heat generated by evaporation) is the last mechanismof heat transfer. Phase transitions are often induced by forced convection, which influences
the thermal conductance. This is also a self-generating effect. Similar to radiation, phasetransitions generate thermal power (be it negative or positive). A peculiarity of heattransfer by phase transition is that in some situations heat can be transferred from a cold toa hot object.
Table 1 gives an overview of the heat transfer mechanisms described above, together
with typical magnitudes of the heat transfer for different mechanisms. It shows the large
variations in magnitude among heat transfer mechanisms. In general, heat transferred byconduction and convection is much greater than that transferred by radiation for room­
temperature structures. However, since the power emitted from objects in the form of
infrared radiation is proportional to the fourth power of absolute temperature, the situation
is different at high temperatures. Free convection is seldom significant for microstructures,
since it is a size-dependent phenomenon, and is insignificant in comparison with conduction
in small structures. However, at very high temperatures (> 300° C) there is significant freeconvection even in microstructures. Phase transitions such as evaporation should be
avoided in self-generating sensors used to measure other heat signals, since evaporation
can give rise to significant parasitic heating or cooling even in thermal equilibrium.

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Table 1
Overview of heat transfer mechanisms and the magnitude of heat transfer for typical 500 µm
rnicrosensors.
Type

Characteristic of effect

W/Km2

RemarkRadiation

between black surfaces

6

objects at around room temperature
objects at around 1000°C

500

Conduction between surfaces 500 µm apart1
air at vacuum
air
50
water
1,200
silicon
300,000
Convection for sensors 500 µm long
free convection
forced convection in air
forced convection in waterPhasetransition

evaporation of water150
15,0001 Pa pressure
atmospheric pressureoften negligible in rnicrostructures
I mis flow
1 m/s flow

1000-10,000* with forced convection*Temperature difference not required for phase transition heat transfer to occur.

3.

Overview of Thermal Sensors

Below we discuss the physical transduction p1inciples of various thermal sensors inmore detail. Table 2 gives an overview of the thermal sensors discussed below.
In Table 2 the first five sensors are self-generating thermal power sensors, and theothers are modulating thermal conductance sensors, which use an electrical resistance togenerate heat. With the exception of the EM-field sensor and the psychrometer, all of thesesensors have been realized using both silicon and thin-film technology.

3.1 Microcalorimeter

A microcalorimeter measures the heat generated during chemical reactions. Bataillard
describes various applications.<6l A chemical reaction occurs between two solutions in thereaction volume of the sensor (near the active area of the thermal sensor) after they aresupplied via two tubes. Alternatively, a catalyst or enzyme is immobilized on the active
area of the microcalorimeter, which initiates a chemical reaction when a single solution
comes into contact with it. (See Fig. 1). This setup ensures that the transfer bf reaction heatto . the sensor is optimized. It is even possible to measure the heat generated by
microorganisms such as bacteria immobilized near the active area of the microcalorimeter.
The heat produced by the bacteria (of the order of 1 pW per bacterium) is a measure of theconcentration of nutrients in the solution, such as glucose, or of the growth rate of bacteria

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Table 2
Overview of thermal sensors, their measurands and operation principles.
SensorMeasurand

Operation principle

Microcalorimeter
Psychrometer
Infrared sensor
RMS converter
EM-field sensor

concentration
humidity
infrared radiationelectrical power
EM fields

heat of chemical reaction - catalyst/enzyme areaheat of evaporation - wet wickblack-body radiation - black coating
Joule heating - heater resistancedissipation - resistive termination

Flow sensor
Vacuum sensor
Conductivity sensor
Mechanical sensor
Thermal properties sensor

fluid flow
vacuum pressure
fluid type
accelerationmaterial properties

convection
conduction - pressure dependence
conduction - type dependence
conduction - seismic massconduction and capacitanceconnectors for tubes
liquid channels [O-ring seal

PGA

Fig. 1. Liquid microcalorimeter in a flow-injection analysis (FIA) setup.

when sufficient glucose is present.
In practice, in an enzyme-based liquid microcalorimeter for determining the concentrationof glucose the transduction to the thermal domain takes place in the following two steps. o.s)First, the concentration C (in mol/m 3) of glucose in water is converted into a reaction
rate M (in mol/s) by bringing the mixture into contact with the enzyme.

Sensors and Materials, Vol. 8, No. 6 (1996)

379
(13)

M= C(dK!dt)

The chemical conversion efficiency dKI dt (in m3 / s) depends on the chemically active layer(catalyst or enzyme) and the particular experimental setup. For instance, one glucose
oxidase molecule can convert a maximum of 1000 glucose molecules per second, butconverts less if the solution is not replenished. When the solution is replenished, thechemical conversion efficiency depends solely on the chemically active layer (a material
property) and not on the experimental setup.
Next, the reaction rateMof the concentrate is transduced into heat of reaction P (in W)
by the change in enthalpy.
P=M(-Ml)(14)

The enthalpy change -Aff is the energy released during the chemical reaction (in J /mol)and is determined experimentally. For enzymatic oxidation of glucose, a value of 80 kJ /mol is found. According to eq. (13) a monolayer of glucose oxidase could produce a
maximum power of approximately 3 W /m2 /mol/m3 .

3.2 PsychrometerIn this sensor the relative humidity of air is determined by measuring the psychrometrictemperature reduction !iTp,ych of a wetted thermometer due to evaporation. This sensor istherefore based on a physical phase transition. The first transduction step can be describedbyf;.Tp sych = Q (pdry - Pwer)/ Parm·(15)

Here Pct and Pwer are the partial vapor pressures of water at the air temperature and at the
wet-thermometer temperature, and Pa1m is the atmospheric pressure. The sensor-specifictransduction factor Q is of the order of 1700 K.<2) !iTp,ych in eq. (15) cannot simply beconverted to the relative humidity, because it depends on temperature and humidity, and ismuch smaller than the dew-point temperature reduction. An estimate is obtained bymultiplying Q by 30% x Pctry / Parm· At room temperature with Parm 100 k:Pa and Pctry 2
k:Pa, the sensitivity is typically of the order of 0.1 K / %RH.
Note, that in this sensor the nonthermal signal is transduced directly into a temperature
difference. Th�re is no conversion in the thermal domain, as there is in all other thermal
sensors. Accurate transfer only occurs if good thermal isolation is maintained, andtherefore design of an appropriate thermal resistance is necessary anyway. Some othermeasures, such as forced convection across the wetted thermometer of an appropriatemagnitude are also essential for accurate transduction. No silicon version of this sensor is
known to the author.
ry

=

""

3.3 Infrared Sensor
From the transduction point of view this is a fairly simple sensor. The transduction
from radiation to heat is carried out by a black absorber, which can have an efficiency of up

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to 99%. The first transduction step from incident radiation density P'\nc (in W /m2) tothermal power Pis given by
P= QP"mcQ = aAo'ffilter·

(16)
(17)

The absorptivity a is between O and 1, and denotes the fraction of infrared radiation power
which is absorbed by the black coating. Various black coatings are used for silicon infrared
sensors. A simple and efficient method is to use the silicon oxide and silicon nitride layers
produced in all semiconductor production processes. <9, 10> Lenggenhager<9> observed anabsorption of the order of 50% for radiation with wavelengths of 7 -14 µm. Using theabsorption of PVDF and metal electrodes is another way to implement this methodY 1 >Porous metal coatings are used to fabricate very black layers. < 12> Gold black will absorb
more than 99% of radiation over the entire infrared spectrum. A different method is usedat Xensor Integration. The sensor is spin-coated with a black polymer. Using techniquesvery similar to normal lithographic and RIE processes, a pattern in a 5-µm-thick coating is
obtained on the wafer, which absorbs about 90% of the infrared radiation. This layer ispatterned in a similar way to other thin films on silicon, and can withstand further
processing of the wafer. This is not the case for coatings such as gold black, which are very
vulnerable and cannot be handled once they have been applied.
A0 is the sensitive area of the sensor (usually the area that is coated with black), while'fm,e, is the transmission of the filter that is usually incorporated when the sensitive area is
encapsulated. This filter can be broadband, transmitting infrared radiation with wavelengths
between 2 and 14 µm, high pass, transmitting wavelengths between 7 and 14 µm, fordetection of objects at room temperature which emit radiation with a typical wavelength of
10 µm (intrusion alarm), or band-pass, transmitting, for instance, radiation at a wavelength
which is in the absorption band of a gas. Using appropriate filters, a gas sensor can beconstructed in which the radiation intensity is measured in a reference path and a path inwhich the gas mixture under investigation is present. The difference in intensity is due tothe presence of the radiation-absorbing gas. This method is used for detection of CO2 andCO.

3.4 RMS converter
In the RMS converter, the first transduction step, from electrical to thermal, is simply
performed by dissipation in an electrical resistor. Complications arise due to parasitic
thermoelectric Thomson and Peltier effects in DC signals, and skin effects and parasitic
capacitances and inductances at high frequencies, which cause differences in the heat
actually generated in the heater. These problems have been studied by calibrationengineers,< 13l and will not be discussed here.

3.5 EM-field sensor

The EM-field sensor has also been studied by calibration engineers. The EM-fieldsensor is in between the infrared sensor and the RMS converter. The infrared sensor
detects optical signals with very high frequencies of around 10 15 Hz. The RMS converter

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381

detecs electrical signals with frequencies of up to 109 Hz and the EM-field sensor measuresintermediate frequencies. The EM-field sensor transduces from radiant to electricalsignals, as does the infrared sensor.
No semiconductor EM-field sensors are known to the author, but current research is
focused on the termination of waveguides by metal patterns on glass plates, which convert
the EM-field energy into heat, that is subsequently detected using a bolometer. Here, thefirst transduction step, from radiant to thermal signals, is carried out using a specificallydesigned metal pattern which yields a lossless, reflectionless termination of the waveguide.
The actual design depends on the wave frequency and waveguide geometry_<I4l

3.6 Flow sensor

Flow sensors are based on the transfer of heat to moving fluids. This effectively
increases the overall thermal conductance between the sensor and the ambient. For flow
sensors, the physics of the second and third steps in the transduction process are as simple
as those for all thermal sensors, but the similarity ends here. The physics of the firsttransduction step, from flow to thermal signals, and the encapsulation of the sensor, aremuch more complicated than those for most other sensors. Also, the encapsulation has agreat influence on the first transduction step, because it influences the type of flow. In
laminar flow, the fluid flows along straight lines, and in turbulent flow it flows in irregularpatterns and the local flow direction is unrelated to the average flow direction. In
micro structures laminar flow often occurs, although in thermal windmeters, micro turbulent
flow is also encountered. c2l The dependence of the heat transfer on flow velocity isdifferent for laminar and turbulent flow. For laminar flow, the first transduction step can beapproximated by(18)
where G is the total thermal conductance from the sensor to the ambient, G0 is the offset(the conduction part), and His the heat transfer normalized by the flow velocity V. The
convective part of the thermal conductance is proportional to the square root of the flow
velocity. The variable H includes such parameters as the sensor size, flow history, andfluid characteristics (viscosity, thermal conductivity, temperature).
The fluid flow upstream of the sensor can greatly influence the exact flow pattern overthe sensor and its heat transfer characteristics. That is why the encapsulation of the sensor
is important (see Fig. 2). The temperature profile of the flow sensor also influences the heattransfer to the flow.<2- • l
5 15

3.7 Vacuum sensor

This sensor is used to measure gas pressures below atmospheric pressure by measuring
the pressure-dependent thermal conductivity of gases. At very low pressures, when themean distance travelled by a molecule between collisions is much larger than the distance
between two surfaces, heat transfer between the surfaces occurs via individual molecules.
The rate of heat transfer is therefore proportional to the rate at which molecules hit thesurface, which is the absolute pressure. At higher pressures (atmospheric for surfaces 500

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Sensors and Materials, Vol. 8, No. 6 (1996)
flow
encapsulated bonding wires
sensor chip""'CJ==--�
carrier
,-I

( a)

0

01\

\ solder bumps( b)th.in carrier

CJ�O=------=U'--.Q
( c)

Qg;O
-

0�
-

( <l)
�doped region

---

CJ

( c)
Fig. 2.

Different methods of encapsulating silicon flow sensors.

µm apart), molecules transfer heat not from surface to surface, but by collisions with othermolecules. Doubling the pressure doubles the number of molecules transferring heat, but
the distance over which they transport the heat (the mean free path between collisions) ishalved, and the thermal conductivity of the gas is then independent of pressure. For low
pressures P (in Pa) the sensor-specific transduction action Q is given by
G = QP = (G"0 / P0) P.

(19)Here, G" 0 is the low-pressure thermal conductance, typically of the order of 1 W/m2K for383

Sensors and Materials, Vol. 8, No. 6 (1996)

gases such as nitrogen and helium, for a reference pressure P0 of 1 Pa. For the entirepressure range, the relation between thermal conductance and pressure becomes
G = (G" 0P1d D)l(G"0P + P0 KID),(20)

where Kl D is the thermal conductance between two plates separated by a distance D for agas with thermal conductivity K(see Fig. 3 and eq. (9)).<2. 15)

3.8 Conductivity sensor

This sensor is similar to the vacuum sensor, since it measures the thermal conductanceof a gas surrounding the sensor. However, in this case the conductance does not depend
upon the pressure, but, at atmospheric pressure, upon the gas type (or liquid type) instead,since all gases have different the1mal conductivities. For instance, for air the thermal
conductivity is K= 26 mW/Km at room temperature. Hydrogen has the highest conductiv­
ity at more than 180 mW /Km. For helium it is about 150 mW /Km while gases such asargon (18 mW /Km) and xenon (6 mW /Km) are even less conductive than air. The
conductivity sensor can be used in various ways. By measuring the thermal conductance it
can be used to determine the type of gas (see Fig. 3), or the composition of a binary gas
mixture. For the latter case, a complex formula describes the thermal conductivity of a gas

10-1--,---------------------------

HeliumNitrogen
Cl)
()
C:
ca

····· Argon

:::,
-0
C:
0
()

10-s-i----1----i----------+---+----1

10-1
Pressure (Pa)Fig. 3. Thermal conductance of different gases between two parallel plates saparated by a distanceof 0.5 mm versus pressure.

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Sensors and Materials, Vol. 8, No. 6 (1996)

mixture with volume fractions a of the first component and (1 - a) of the second.< 17l(21)
In this formula, the auxiliary variable </J12 depends upon the viscosity and molecular weightof the gas molecules, as(22)
where M1 is the molecular weight of the first gas component, and µ1 is its dynamic
viscosity. In the equation for </)21 , the indices are reversed.
Finally, the conductivity sensor can be utilized as a level sensor, since the thermal
conductivity of a liquid is much higher than that of a gas. It can be used to detect overflow,or to measure level in a continuous manner, although the range is limited by the sensor
dimensions.

3.9 Mechanical Sensors

In mechanical thermal sensors such as the accelerometer, the mechanical signal accel­
eration a is converted into a force F by a seismic mass m. Then the force is converted intoa gap size change D by mechanical springs with a spring constant K using the damping
characteristics of the surrounding gas to obtain an appropriate frequency behavior. Finally,
the gap size is transduced into a temperature difference by the thermal conductance of thegas in the gap. (See Fig. 4). Therefore, for the accelerometer, the steady state transfer isgiven by
F=ma
D=KF
G= "KA/D+G0

•

(23)
(24)
(25)

Here, A is the area of the gas conducting heat from the sensor to the seismic mass, and G0
is the offset thermal conductance from the sensor to the ambient via other paths, parallel tothe conductance in the gap. The physical principle of the thermal accelerometer does notdiffer greatly from that of the capacitive accelerometer. The gap is measured thermally,rather than capacitively. This means that the first two conversion steps are very similar forthese two devices. Therefore I refer the reader to literature on capacitive accelerometers
and refs. (18) and (19).
3.10 Thermal properties sensor
Thermal sensor structures can also be used to measure the thermal properties of thinfilms and of the sensor materials themselves. In general, absolute measurements of thethermal resistance and the thermal time constants of a particular structure are made. For
modem micromachined structures such as the floating membrane or cantilever beam,measurements are relatively simple. Figure 5 shows a structure in which a silicon island is385

Sensors and Materials, Vol. 8, No. 6 (1996)
suspended mass�-�heat sink

substrateU\thermo ile

<==

q 1/2

m
0

�

c===:>

q 2=f(a)

x= L

7\J

q 1/2

X

Fig. 4. Schematic drawing and cross section of a thermal accelerometer.

suspended in a closed silicon nitride membrane. This is a variation of the floating­membrane structure.<20l The measurements are performed in vacuum, so that only conduc­tion in the solid material adds to the measured conductances (radiation effects are usually
not significant). In this way, the thermal conductance r<: and capacitance Cp of various
materials have been detennined, such as those for silicon (150 W /Km and 700 J /kgK),
silicon dioxide (1-1.5 W /Km and 730 J /kgK), low-stress silicon nitride SiNu (3-3.5· W/Km and 700 J /kgK), polysilicon (18-30 W /Km and 770 J /kgK) and aluminum with 1 %silicon (180-220 W /Km).<2, -2 )
20

2

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Sensors and Materials, Vol. 8, No. 6 (1996)

heating resistor

Si membrane

Fig. 5. Thermal properties sensor consisting of a silicon island suspended in a silicon nitride closed
membrane.

4.

Discussion

Many thermal sensors have been developed. Some of them are already used commercialy(such as infrared sensors for intrusion alarms), others are used for specific purposes(accelerometer), and some are not fabricated in silicon technology (psychrometer). Thereis a great difference in the applicability of various thermal sensors. Infrared sensors arevery simple devices because they can be hermetically sealed during encapsulation. The
same applies to RMS converters, the semiconductor version of which is becoming more
and more popular. Research on semiconductor flow sensors is extensive, but the difficulties
encountered in encapsulation and its influence on the sensor mean that the commercial
success of these sensors is limited.

Sensors and Materials, Vol. 8, No. 6 (1996)

387

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