Bolometer noise: nonequilibrium theory

John C. Mather

New theoretical results for noise in cryogenic bolometers are derived. Johnson noise is reduced by as muchas 60% by electrothermal feedback from the bias supply. Phonon noise in the thermal link is reduced by asmuch as 30% relative to the usual equilibrium formula. Photon noise in the Rayleigh-Jeans limit is comput-ed with attention to the attenuation of the photon correlations in the light beam. Basic results on bolometerresponsivity, time constant, and thermal properties are presented in a new and convenient form. Excess1/f and contact shot noise are also discussed.

1. Introduction

The theory of bolometer operation has been treatedby several authors. 1 -7 The typical bolometer is a tem-perature-sensitive resistor, heated by incoming radia-tion and cooled by thermal conduction through itselectrical lead wires. Figure 1 shows the electrical biascircuit, the thermal structure, a sample current-voltageplot, and the equivalent electrical circuit.

In this paper we first present the results of the smallsignal analysis of the bolometer as a transducer. Theseresults are equivalent to the results of Jones4 but areexpressed in terms of directly observable parameters:the bolometer resistance R = E/I at the operating point,the low frequency dynamic impedance Z = dE/dI, andthe apparent time constant Te,

Subsequent sections present new analyses for noisein the bolometer in the presence of electrical bias power.We find that the major limiting noise sources have allbeen overestimated, so that the ultimate sensitivity ofbolometers should be closer to the intrinsic photon noiselimit. Johnson noise from the bolometer is reduced byelectrothermal feedback by as much as 60%, since thebias current does work on the Johnson noise source.We also present a new analysis for the phonon noise inthe thermal conductance from the bolometer to the heatsink. The result is a function of the temperature de-pendence of the thermal conductivity of the link ma-terial, but the detailed geometry of the link need not beknown.

We also discuss photon noise in the Rayleigh-Jeanslimit. We find that several previous authors have notaccounted for the reduction in the Boson coherencefactor suffered when a beam is attenuated or only par-tially absorbed by the detector. We conclude with acomputation of the effect of amplifier and load resistornoise and brief discussion of excess bolometer noise.

The author is with NASA Goddard Space Flight Center, Laboratoryfor Extraterrestrial Physics, Greenbelt, Maryland 20771.

Received 4 September 1981.

We will use the following notation: E = potentialacross the bolometer; I = current through the bolome-ter; R = resistance = ElI; ZL (w) = dynamic load im-pedance; w = angular frequency; time factor = exp(iwt);P = electrical power into bolometer = EI; Q = radiationpower absorbed by bolometer element; W = total power= P + Q; T = ambient temperature; T = bolometertemperature; AT = temperature rise = T - T,; Z =dynamic resistance = dE/dI at zero frequency; Z(co) =complex dynamic impedance at angular frequency ;G = dynamic thermal conductance = dP/dT at zerofrequency; S = zero frequency voltage responsivity =dE/dQ; S(c0) = responsivity = dE/dQ at angular fre-quency co; SD = spectral density operator; C = heatcapacity at operating temperature; -re = effective timeconstant; = C/G = physical time constant; kB =Boltzmann constant; RN,L = parameters of the equiv-alent electrical circuit; TL = temperature of load re-sistor; ax = absorptivity of detector; = emissivity ofthermal radiation source; f = transmissivity of opticalsystem; A Q = throughput of optical system; Ts = tem-perature of background radiation source.

II. Small Signal Analysis

We give the small signal parameters in terms of themeasured time constant T e, the bolometer resistance R,and the impedance Z. These formulas will be used inthe noise analysis that follows. Many bolometers havea maximum in voltage at a particular bias current atwhich point Z = 0. This will be used as a frequent ex-ample where the electrothermal feedback from the biasbattery significantly changes the external properties ofthe bolometer.

The small signal voltage responsivity of the simplebolometer is given by

SW = 1 (Z/R)-1 1

2I (Z/ZL) + 1 1 + jwTe(1)

We note the limiting case for Z = 0, co = 0 is simply S(0)= -1/2I.

15 March 1982 / Vol. 21, No. 6 / APPLIED OPTICS 1125

LOADZL

EI(a) IAS - R.E/I

[T ;tBolornterO (RADIAT ION)

(b) T IGT Tc

P- E-I |R(T)|

C-HEAT CAPACITYG-DYNAMIC THERMAL CONDUCTANCE

BIAS CIRCUIT

THERMAL CIRCUIT

Both the bolometer responsivity and impedance arerepresented by the equivalent circuit given in Fig. 1(d).The effective voltage source is

eg = Q/2I,

while the circuit parameters are

RN = RZ/(R-Z),

L = T R+Z2 R-Z

(5)

(6)

(7)

(c) E BIAS CURVE

(d) N R DYNAMIC EQUIVALENT CIRCUIT

e'-Q/21

(e) Im(Z) COMPLEX EFFECTIVE IMPEDANCE

REAL Z)

Fig. 1. Bolometer definitions: (a) bias circuit; (b) thermal design;(c) sample current-voltage plot; (d) equivalent small signal circuit;

(e) complex impedance as a function of frequency.

The apparent time constant -re describes the respoiof the bolometer to modulated radiation. It also is Itime constant for the decay of the transient part of response to a change in the bias battery voltage. physical time constant X is related to the apparent ticonstant Te by

C 2R Z+ZLT = 're - -G Z+RR+ZL

For Z = 0, ZL -, we find that = 2, so that bolometer appears to respond twice as fast as it woiwithout electrothermal feedback.

The bolometer has an effective electrical impedaiZ(cw) which is independent of the bias impedance.is given by

Z+R1 + jlT-

2Z

Z+R2R

This function has the values Z(0) = Z, Z(co) = R.intermediate frequencies it traces a semicircle in Icomplex impedance plane, centered on (Z + R)/2,shown by the equivalent form

1- j WTZ+Z( Z+RZ Z-R 2R2 2 Z+R

1+jWoT 2R

where the second term has a constant modulus as afunction of c. This is shown in Fig. 1(e).

We note that the effective inductance L is very large. Ifr = 0.01 sec, R = 2 X 106 Q, Z = 0, we find L = 104 H.

To determine the effective dynamic thermal con-ductance, the temperature coefficient of bolometer re-sistance must be measured separately by direct mea-surement. Then the thermal conductance is givenby

G = I2dRZ+R .dTZ-R (8)

The heat capacity can then be determined from Eq.(2).

Ill. NoiseThe noise in a bolometer is produced by a variety of

uncorrelated sources analyzed in the following sections.We summarize the results here. The electrical noiseequivalent power (NEP) is given by

NEP 2 = NEp30 h2nson + NEp2 ermal + NEphoton + NEP2oad

rise + NEPampifier + NEPexcess,

the where

'he NEP302,f = kT I + 2w),rhe NEPJohnson = 4kBT |Z-+R|(+ T)me I

NEP2h erma = 4kBGT2f7 T X

(2) X T [(t) I dt

;he J.[()

uld NEP hoton = 4 A+kh3 x l (1+eX-f)(aEf),

ice whereIt x = hv/kBTs,

p2 4kBTL 2Z1(3) NEPload - ZL Z

R

2

[1 + W2T2 (Z+ R)21 ,

2p 2 2Z1 (+ R21 eANEPamp= i I1+ W2 r2 1+ I2Wl

R

(9)

(10)

(11)

(12a)

(12b)

(13)

(14)

IV. Johnson Noise AnalysisMost authors have computed bolometer Johnson

noise from the resistance R, assuming the bolometer tobe a simple resistor at temperature T. Other authorshave assumed that the Johnson noise arises from thereal part of Z(cv), which clearly cannot be true since Z

1126 APPLIED OPTICS / Vol. 21, No. 6 / 15 March 1982

(a)()ej

(b)

JOHNSON NOISE SOURCE

BOLOMETER WITHJOHNSON NOISE

(C) RN' BOLOMETER EQUIVALENT(C) RN R NOISE CIRCUIT

e=- e2I

Fig. 2. (a) Johnson noise of a resistor or bolometer with thermalfeedback neglected; (b) equivalent circuit including feedback; (c)

Thevenin equivalent of (b).

can be zero or negative. We give here the argument fora new formula including electrothermal feedback fromthe bias source.

The bolometer is now considered to be a resistor inseries with a fluctuating noise voltage generator ej, asshown in Fig. 2(a). If the value of G (the thermal con-ductance) were very large, the full equivalent circuit fornoise would be the same regardless of dc bias current.Electrical work done on the Johnson noise source wouldbe dissipated in the bolometer but would not affect thebolometer temperature or resistance. This noise hasa mean value of zero and so produces no average heatflow to the bolometer.

When the value of G is now reduced to its normalvalue, two new effects appear: the effective bolometerimpedance becomes Z(cv) instead of R, and the biascurrent dissipates a power in the Johnson noise source,which may actually be detected by the bolometer. Forthe usual bolometer with a negative temperature coef-ficient, this second effect opposes the original noisevoltage, since an increase in ej increases the powerdissipated in the bolometer, increases its temperature,decreases its resistance, and decreases the outputvoltage. The equivalent circuit is now given by Fig.2(b). We will use the shorthand that

ej( ) = [SD(ej) .01/2, (15)

where the units are V/Hzl/2.By Thevenin's theorem, Fig. 2(b) may be converted

to Fig. 2(c), with the new noise source ej(c) given by

ej(w) = ej(co) j * (16)

jor+2RR + Z

We see that for c = 0, Z = 0, ej = ej/2. Even greaterreductions are possible for Z < 0, which can occur forbolometers with very large negative temperature coef-ficients. When the bolometer is inserted in the biascircuit, the load resistor modifies the noise further,giving an actual noise output voltage of

+1

ZL jreJ(w) = eJ(w) R+ -

R +ZL 1e + .

j*r,

(17)

The noise voltage ej has of course the spectral den-sity

SD(ej) = 4kBTR. (18)

It is interesting to compute the NEP implied by thisnoise source alone. It is given by

NEP = e6(0)/S(w)

IZ+RI= V4kBTP IZR 1 +jwrl, (19)

which is independent of the load resistance. This im-portant result shows that changing the electrothermalfeedback effect by varying ZL has no effect on the NEP,as one would expect. The NEP also depends on the realphysical time constant CIG rather than the effectivetime constant Te.

V. Phonon Noise

Spontaneous random energy flows into and out of thebolometer occur at all times. On the microscopic scalethis is produced by the random propagation of energycarriers, be they electrons, phonons, photons, orquasi-particles. On a microscopic scale it is readilyshown from thermodynamics that this power flow (callit Pn) has the spectral density

SD(Pj) = 4kBT 2 G, (20)

where G is the thermal conductance between bolometerand heat sink. This result applies strictly to an iso-thermal system. We again define the shorthandnotation that

P (W) = [SD(P.)I1. 1/2. (21)

In this section we will generalize the result to a bo-lometer with a temperature gradient in the thermal link.The result will be independent of the geometricalproperties of the link, providing that the mean free pathof the energy carriers is short compared with the length(defined as X) of the link. This is the opposite limitfrom that considered by Boyle and Rogers. 5 Essentiallythe result is a weighted mean temperature to use in Eq.(20), with the weighting function depending on thethermal conductivity of the link as a function of tem-perature.

The approach is to imagine the link divided into smallelements in series. Each element is nearly isothermaland produces a random power flow given by applyingEq. (20) to the small element. Most of the power flowis immediately conducted back again through the ele-ment, but a small amount reaches the bolometer. Wecompute this amount and then add the contributionsfrom the various elements.

We assume that the thermal link is narrow, so thattemperature t, thermal conductivity k, and cross-sec-tional area A are all functions of a single spatial coor-

15 March 1982 / Vol. 21, No. 6 / APPLIED OPTICS 1127

T,

ti tj

A X X q IIi q

sX5 =O Xb =X

IJ

Fig. 3. Elements of thermal noise analysis. Terms are defined intext.

dinate x, defined as the distance from the heat sink.The bolometer temperature is T, and the heat sinktemperature is Tc We further assume for simplicitythat the link is of homogeneous material so that k is afunction of t only. Under steady state conditions, thepower flow through the link is W(T,TC). The equationfor the temperature gradient in the link is

W(T,TC) = k(t)A(x) d (22)dx

which is immediately integrable toT X

W(T,TC)= SA k(t)dt [A(x)]-'dx, (23)

and then differentiable to give

G = dW/dT = k(T) . 3' [A(x)]-1 dx. (24)

This definition of the thermal conductance G is thesame one used in the small signal analysis. Equation(24) shows that the variation of G with T is the same asthe variation of k, justifying the arguments made byLow6 in determining the scaling laws for bolometerproperties as a function of temperature.

We now compute the noise power q into the bolom-eter produced by random fluctuations in the intervalfrom xi to xj, with 0 < xi < xj < X, as shown in Fig. 3.Subscript s refers to the heat sink, i and j to the twoends of the element under consideration, and b to the.bolometer chip. Each part of the link is described byits own heat flow function W having the dimension ofpower. For example, Wis(ti,ts) = -Wsi(ti,t,). Wedefine Gi = WiS/ati,Gi = aWsi/Ot,, etc. as thermalconductances and Ris = Gj-' etc. as thermal resistances.We hold t and tb fixed and introduce a random noisepower source Pij across the link Gij. Using Ati,\Atj asthe fluctuating parts of t,tj we obtain the nodal equa-tions

q = GJbAtj, (25)

Pi = (Gjb + Gi)At - GijAti, (26)

-Pij = (Gi, + Gij)Ati - Gji/tj.

The solution to these equations is

q = Pij- [(Gib + Gji)(Rjb) + GijRis 1.

As x -xi - 0, Gij - , and therefore

q - pijRij1(Ris + Rib)-

From Eq. (24),

Ris+Rb=-'~' d.+ f" d'k(t ) [J A(x) xi A )

1 rx dx k(tb)

W~i) Jo A (x) (t ) Rs

Similarly,

R- (xj - xi)k(ti)A(xi)

(30)

(31)

Now the spectral density of Pij is SD(pij) = 4kBTVGijfrom the classical equilibrium fluctuation formula. Thespectral density of q is given by

SD(q) = e (R i; ) 4kBt 2Gij.elements Ri i

(32)

This sum may be transformed with the aid of Eqs.(22)-(24) to

fT tk(t)12 dt

SD(q) = 4kBGT2 Tk(T)

T k(t) dtKT k(T)

(33)

For typical materials at low temperature, k(t) t.For normal metals, f = 1, while for crystalline dielectricsor superconductors, 3 = 3. For conductivities of thisform, integrals in Eq. (33) may be done explicitly. Theresult may be expanded to lowest orders in A = 1 -(Tc/T), giving

SD(q) = 4BGT2 [1- + 1 A + (+ 2)(3/ 2 A2...], (34)

For a typical case, fl = 3, A = 0.2, this formula gives thevalue in brackets as 0.683, while the true value is 0.652.In this case, the effective mean value of T is close to TCrather than T. We remind the reader that G is definedas the derivative of W with respect to T.

VI. Photon NoisePhoton noise on bolometers arises from the random

fluctuations in the rate of absorption of radiationquanta. We discuss this briefly because many authorsignore the degradation of photon coherence by beamattenuation and by partial absorption.1-3 5-7 We citevan Vliet8 and Hauser9 for a better answer; van Vliet'sEq. (68) implies that the spectral density of fluctuationsin absorbed power is

SD(Q) = 2fdvQh[l + n(v)], (35)

where Q is the radiant power absorbed, Q is the samequantity per unit optical bandwidth, and is the ef-fective photon mode occupation number. The valueof i7 is given by

?(P) = aEf/(eX - 1), (36)

(27) where a is the detector absorptivity for the particulardetective process producing a measurable result, is theemissivity of the source, f is the transmissivity of the

(28) optical system connecting them, a, E, and f are evaluatedat v, and x = hlhBTs, where T, is the temperature ofthe source. Equations (12) are also derived by substi-

(29) tuting into Eqs. (35) and (36) the following expressionfor Q:

1128 APPLIED OPTICS / Vol. 21, No. 6 / 15 March 1982

b

__

NOISE EQUIVALENT CIRCUITOF AMPLIFIER

IDEALAMPLIFIER

Fig. 4. Equivalent noise circuit of bolometer and amplifier. Termsare defined in text.

hv 3 1Q = 2(aff)AQ -____ * (37)

c2 ex-1For far IR detectors operating in a high radiation

background from a room temperature optical system,the difference between aef and 1 may be very impor-tant. Note that the NEP in Eq. (12) is the electricalNEP; the optical detector NEP is Eq. (12) divided bya.

For completeness, similar terms should be includedfor all elements of the telescope and detector system,including one term for the self-emission of the detector.For cryogenic detectors this term is extremely small.

VIl. Load Resistor and Amplifier Noise

Amplifier and load resistor noise are typically rep-resented by the equivalent circuit in Fig. 4 with noisevoltage and noise current generators as shown. We findthat the total noise voltage from these sources at theamplifier input is given by

1 jTZ+R 24kBTL ZZL cv2 2Ze2 W)~~ i21 (ZZW)_____ +e (W).

ZL LA(W)) Z LJ + jwre A

(38)

The term from the amplifier current noise and loadresistor noise Z is frequency dependent because of thefrequency dependence of Z(cv). Equations (13) and (14)are derived by dividing Eq. (38) by I S(W) 12.

Vil. Bolometer Excess Noise

Most bolometers have excess electrical noise, espe-cially at low frequencies. We will discuss two types,flicker ("1/f") noise and contact shot noise. Both aremodified by thermal feedback in the same way as theJohnson noise discussed above.

Van der Ziel10 describes McWhorter's theory offlicker noise. In this theory flicker noise is due totrapping centers on the surface of a semiconductingfilament, actually located within the first 2-4 nm of theoxide layer. The resistance of the element is modulatedby the slow trapping and release of charge carriers,giving a voltage noise which is proportional to the biasvoltage, with a 1/f1/2 dependence on frequency. Thismodel of a fluctuating resistance permits estimation ofthe NEP scaling law as the thermal conductance G isvaried. The result is that the NEP is proportional to+\- as usual with the provision that the modulationfrequency scales as G. A reduced G value allows asmaller bias voltage which reduces the flicker noise toexactly compensate for the increase caused by operatingat a lower frequency.

Another common noise source appears to be contactshot noise. Some germanium bolometers built in ourlaboratory have an electrical nonlinearity which is notdue to the bias power heating the bolometer. It is ob-servable in three ways: (1) in Jones's ac bridge method 4

for determining responsivity, spurious high readings areobtained at high ac modulation frequencies, especiallyat low dc bias. (2) A plot of resistance R vs bias powerP does not show a smooth nearly straight line approachto a constant as P is reduced. (3) The responsivitydeduced from Eq. (1) has a spurious high value at lowbias current and does not vary with bias current in thesame way as the optically determined responsivity.

This noise source may be modeled as a barrier havinga dynamic impedance Zbar with the full shot noisevoltage en = Zbar(2qI)1/2 V/Hz1 2 . If Zbar is more thana few percent of the bolometer impedance, the shotnoise may exceed the Johnson noise. For one bolometertested, approximate values were Zbar = 1 MQ, R = 20MQ, I = 30 nA, T = 2 K. With these values, the shotnoise is 98 nV/Hz1/2 , while the Johnson noise is 47nV/Hz1/ 2 . This shot noise becomes even more impor-tant at lower bias current, since Zbar seems to scale asI-1 up to -50 MQ for this particular bolometer.

We cannot distinguish shot noise from Johnson noiseby their spectral dependence, since both are white noiseand both are modified by thermal feedback in the sameway.

IX. Summary and Conclusions

We have given new estimates for noise in bolometers.Previous comparisons of measured bolometer perfor-mance to the ideal limit may need to be revised, andthere may be further room for improvement beforefundamental limits are achieved. Johnson, photon, andphonon noise sources are all reduced relative to com-monly used estimates due to the nonequilibrium stateof the bolometer and to the inclusion of the photon co-herence factors.

I greatly appreciate the comments and suggestionsmade by Paul L. Richards, who knew that bolometernoise theory was incomplete, and by Rainer Weiss,Michael Hauser, and Harvey Moseley. Hauser alsoindependently derived Eqs. (12).

References1. F. R. Arams, Ed., Infrared-to-Millimeter Wavelength Detectors

(Artech House, Dedham, Mass., 1973).2. E. H. Putley, Phys. Status Solidi 6, 571 (1964).3. F. J. Low and A. R. Hoffman, Appl. Opt. 2, 649 (1963).4. R. C. Jones, J. Opt. Soc. Am. 43, 1 (1953).

5. W. S. Boyle and K. F. Rogers, Jr., J. Opt. Soc. Am. 49, 66(1959).

6. F. J. Low, J. Opt. Soc. Am. 51,1300 (1961).7. S. Zwerdling, R. A. Smith, and J. P. Theriault, Infrared Phys. 8,

271 (1968).8. K. M. van Vliet, Appl. Opt. 6, 1145 (1967).9. M. G. Hauser, "Calculation of Expected Infrared Signals and

Background-Induced Noise Limitations," available from theauthor, NASA/GSFC, Greenbelt, Md. 20771.

10. A. Van der Ziel, Noise (Prentice-Hall, Englewood Cliffs, N.J.,1970), p. 109.

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I z (W) z i iL L A