A critique on the application of infrared photodetector theory

Infrared Phys. Vol. 26, No. 3, pp. 141-153, Printed in Great Britain

0020-0891/86

1986

$3.00 + 0.00

Pergamon Journals Ltd

A CRITIQUE ON THE APPLICATION OF INFRARED PHOTODETECTOR THEORY 0. M. WILLIAMS Weapons Systems Research Laboratory, Defence Science and Technology Organisation, Department of Defence, GPO Box 2151, Adelaide, South Australia 5001 (Received

in original form 22 March

1985; in final form 3 December

1985)

Abstract-The physical and mathematical framework underlying photodetector theory is appraised, with particular interest in those aspects relevant to practical device application. The mathematical assumptions and conventions on which signal and r.m.s. noise calculations are based are examined, together with the relationships between the characteristic parameters, responsivity and detectivity, and device spectral response, frequency response and linearity. An equivalent ramp-function spectral response based on the definition of an effective cutoff wavelength is derived and is shown to be useful for cases where device data are incomplete. The standard expressions for responsivity, detectivity, signal and noise currents, signal-to-noise ratio and noise-equivalent temperature difference are examined with particular interest in their application to device use and IR system design.

1. INTRODUCTION

The theory underlying the application of IR photodetectors has been well developed by a number of investigatorP4’ to the point where there are now several review articles in which the general theory is described and in which one or more aspects are given special attention. For example, the fundamental physical and material properties are well-covered by Long(“) and by Elliott,(‘6) detector characterization is given special attention in the article by Limperis and MudarC”) in the ONR Infrared Handbook, and Lloyd”” includes a critique on the noise-equivalent temperature difference (NETD) and its position within the overall tradeoff analysis required for thermal imager design optimization. The present study is directed towards a critical appraisal of the physical and mathematical framework underlying the photodetector characteristic parameters, responsivity and detectivity, and their relationships to the parameters of interest to the designer and the’device user; namely, the signal current, r.m.s. noise current and signal-to-noise ratio (SNR). It is not claimed that the results presented here are necessarily new. Rather, the study is directed towards those aspects relevant to the practical application of photodetectors. The expressions for IR signal and background irradiance are examined briefly in Section 2 as a precursor to the discussions in Sections 3 and 4 on photodetector responsivity and detectivity respectively, with particular interest in the underlying mathematical assumptions and conventions. In Section 5, the signal and noise formulae expressed in terms of the responsivity and detectivity are examined and the standard results for SNR and NETD are derived. The study is concluded by a short discussion on the difficulty of separating parameters in the calculation of SNR and NETD. 2. SIGNAL

AND

BACKGROUND

IRRADIANCE

We follow the radiation analysis of Wolfe(‘9*20) in writing the Lambertian spectral irradiance Ej. (in W cme2 pm-‘), arriving at a photodetector characterized by a field-of-view (FOV) half-angle

8,

as

EL = z (A, L)[i&

sin’ e1+ A4,,l (sin* 8 - sin’ @,)I+ [ 1 - r (A.,L)]A4,,,,Asin’ 0,

(1)

where T(A, L) is the product of the spectral transmission coefficients through the optical system and through the atmosphere over range L between the source and photodetector. The half-angle et is subtended at the detector by the specified target. The three principal terms in equation (1) correspond to the irradiances produced by the target, the local background in the vicinity of the 141

142

0. M.

WILLIAMS

target and the combined atmosphere and optical component emissions, all expressed in terms of the appropriate spectral exitance Mi. The last term arises from the detailed balance requirement that is necessary to ensure that dynamic equilibrium is maintained in the atmosphere. We note that it is assumed in equation (1) that efficient cold shields are employed in order to reduce thermal radiation from outside the detector FOV to negligible proportions. In photodetector studies, it is usual to simplify equation (1) by assuming that the local background and atmospheric emissions can be combined into a common background characterized by a single composite temperature Tb. With such an assumption, equation (1) simplifies to the form Ei = Mb,j.(~, Tb) sin* 0 + r(A, L)[M,,,(A, T) - MbJA, T,,)] sin* 8,,

(2)

where T represents the effective blackbody temperature of the target. The two terms in equation (2) correspond respectively to the background spectral irradiance E,&., Tb) and the signal spectral irradiance E&, T). 3. RESPONSIVITY In this section, the fundamental forms of the photodetector responsivity relevant to small-signal calculations are examined. The mathematical forms of the responsivity are derived and are compared under blackbody irradiance conditions to an equivalent ramp-function responsivity truncated at an effective cutoff wavelength. The form of the ramp function responsivity is derived from 500 K blackbody data and is shown to be useful for cases where device data may be incomplete. The magnitude of the responsivity, together with detector frequency response and linearity, are discussed in relation to the responsivity parameter, the photogain’“) G. 3.1. DeJinition and forms of responsivity Photodetector current responsivity is defined in standard manner as the signal current developed per unit signal power incident on the detector surface. That is, for a detector of area A, R,(T)=-

dZ i, =AdE AE,(T)

(3)

where the slope dZ/dE, which is measured at the value of the background irradiance, is constant for a linear detector. In general, however, dZ/dE and hence R,(T) may depend on the irradiance value. Voltage responsivity is obtained directly from current responsivity by multiplication by the dynamic resistance r = dV/dZ. It is of interest to develop some of the underlying photodetector physics by writing the signal current flowing in a device of depth d as is = eGgsAd,

(4)

where G is the photogain, (Is)the form of which is discussed below in Subsection 3.5, and where g, is the electron-hole photogeneration rate density due to the signal within the photodetector volume Ad. This latter parameter may be written directly in terms of the spectral quantum efficiency ~(1) and the photon spectral irradiance

thus allowing the current responsivity [equation (3)] to be written in the fundamental eG

4(T) = E,(T)

s m

(4 o tl @ )Eq,s,,

T) a.

form

(6)

The spectral or monochromatic form of the responsivity may be obtained directly from equation (6) by replacing Eq,s,l by the product of a d-function and the total photon flux Eq,s, and by substituting for Es from equation (5), in which case we obtain (7)

143

IR photodetectors-1 Step-function

equivalence

5

1

4

0

8

Wavelength

12

(pm 1

Fig. 1. Illustration of CMT spectral quantum efficiencies and spectral responsivities calculated from the results of Anderson.‘*‘) The dashed curves represent the step- and ramp-function equivalences.

The familiar wavelength dependence of the responsivity, as illustrated in Fig. 1, has therefore been derived. The non-linear behaviour near cutoff is due to the wavelength dependence of the quantum efficiency ~(2) which, for illustrative purposes, has been derived here from the formula r(n) = ~(1 -e-“‘)

(8)

for the particular case of an unbloomed cadmium mercury telluride (CMT) detector of depth d = 10 pm, using the results of Anderson v’) for the wavelength-dependent photoabsorption coefficient a. It is noted that it is assumed in equation (8) that internal reflections are neglected, as would occur for CMT mounted on a substrate of similar refractive index, for example CdTe. When internal reflections are important, the alternative formula(‘@ vr(nj _ (1 - P)U -e-? 1 - pe-“’

where p is the vacuum/detector reflection coefficient, should be used. In this case, the curves of Fig. 1 more closely approximate the case of a sudden wavelen~h cutoff, consistent with the increased absorption efikiency. Formulae (8) and (9) represent limiting cases. In practice, the form of q(l) may be derived by the inverse use of equation (7) from the spectral responsivity or detectivity curves supplied by the device manufacturer, and would be expected to lie between the limiting cases, depending on the exact nature of the device design. The standard relationship between the general and monochromatic forms, equations (6) and (7) respectively, may be derived immediately as

R(T)=

Q,R(J)&,O, s*

T) dl

(D%,i@, T) dJ f0

(10)

144

0. M. WILLIAMS

This result applies equally to current and voltage responsivity. Thus, the responsivity R(T) is just the average of the spectral responsivity over the signal spectral irradiance. Equation (10) therefore represents the general result for converting from the manufacturer’s spectral responsivity curves to the responsivity particular to the given photodetector application. 3.2. Cutoj‘ wavelength A number of characteristic wavelengths defined near to the photodetector cutoff appear in the literature. These include the true cutoff wavelength & where the quantum efficiency and hence responsivity fall to zero, an empirical cutoff wavelength defined as the point where the responsi~ty falls to half of its maximum value,1”***)and the loosely defined peak wavelength. The true cutoff wavelength equals hcfE, where EB is the gap energy for the photodetector. In examining the mathematical framework of photodetector theory, we have found it convenient to introduce an alternative definition of an efictive cutofl wavelength A: that is related directly to the general form [equation (6)] of responsivity appropriate to a blackbody signal irradiance. Thus, & is defined according to the mathematical equivalence mrl (J)&,a,,(& T) dl = tlo&(& T), (11) I0 where q0 is the asymptotic value of quantum efficiency and where the subscript B refers to blackbody emission, The function &&f, T) represents the fractional blackbody photon irradiance over the range [0, A:] incident on a photodetector characterized by a sudden cutoff at 2;. Thus

#BK, Tl =

i; s

Eb,&,

T) d;l = f,(ng T)a,T3sin25,

(12)

0

where

crs= 1.5202 x 10” photons cm-’ s-’ is the photon equivalent of the Stefan-Boltzmann constant (see, for example, Wolfe(‘9*20)),and wheref,(&T) is the fraction of photons in the blackbody radiation field within the range 10, A:]. Equation (12) may be manipulated readily by use of the photon formulation(‘9) of blackbody radiation theory. It may be shown in particular that & is a function of AT alone. The derived functional form is shown below in Fig. 3. It is of interest to note for ~culational purposes that for IT 6 5000 pm K, f&IT) may be written to a good approximation in the analytical form f,(LT) = (81/a4)e-x(1 + x +$X2),

(13)

where x = hc/(lkT). This approximate form is valid provided that eX>>1 and incorporates the coincidence that the constant u, is only 0.04% smaller than the expression 4n5k3/(81h3c2), in direct analogy to the fundamental expression 2n5k4/(15h3c2) for the Stefan-Boltzmann constant 6. 3.3. Evaluation of the eflective cutof wavelength The equivalence defined above by equation (11) is illustrated by the dashed lines in Fig. 1. Clearly, the definition corresponds to a step-function equivalence of the spectra1 quantum efficiency q(n), and to a r~~function equivalence of the spectral ~ponsi~ty R(A). It is ~ph~i~ that equation (11) is a mathematical equivalence and is not an approximation. Given r&X), either directly or through R(1) via equation (7), then equations (11) and (12) may be applied together with the photon form(20) of the Planck blackbody function in order to derive lzf as a function of blackbody temperature. The results obtained by using the idealized curves of Fig. 1 for CMT are shown in Fig. 2. It is clear that in this case, & is not a strong function of T, nor is it much smaller than the true cutoff wavelength A, for typical values in the range A, < 15 pm. The difference between & and & is even smaller for the case when internal reflections are important, as discussed above in Subsection 3.1. The particular significance of the use of the effective cutoff wavelength to the designer or device user is that calculations may be simplified with minimal loss of precision. That is, if the value of ni is known or may be either calculated or estimated, then responsivity calculations as in equation (IO) over blackbody spectra or over portions of blackbody spectra may be simplism by use of the

145

II? photodetector.+-I

t 1

f

I

f

rrlttf m

I

trltill loo

X,fpmi

Fig. 2. Calculated variation af &/A, for CMT with true cutoff wavelength A, and blackbody temperature.

equivalent ramp-function responsivity in place of the actual specified detector spectral response. For cases where accurate values of nf are required, then the equivalence [equation (1 l)] should be evaluated numerically, using equation (12) and the known form off$&T) together with the spectral variation n(J) of quantum efficiency, as obtained from the available spectra1 responsivity or detectivity curve. However, when lesser precision is required, then the value of ,?gmay be estimated without detailed calcufation. In ci~umstances where the cutoff wavelength is known, then curves such as those shown in Fig. 2 may be useful for estimating 2:. Alternatively, when the spectral ~~nsi~ty or detectivity curves are available, then a method based on simp1e inspection is suggested. It is noted from the example of Fig. l(a) that the step-function equivalent curves cross the quanta efficiency curves at approx. 0.6 of the peak quantum efficiency qo. Similarly, as seen from Fig. l(b), the curves cross at about 0.7R@3, where R(&) is the ramp-function equivalent responsivity extrapolated to L = &. It is suggested without proof that these crossover points can be used to define a reasonable estimate of &, thus avoiding the need for detailed calculation. Although the value obtained for Jg is necessarily approximate, it is maintained that since it has been derived essentiahy by inspection of the mathematical equivalence [equation (1 I)], more precise estimates of photodetector signal and noise currents may be obtained than through the alternative use of either the true or empirically derived cutoff wavelengths or through use of the often-quoted responsivity R(&,) at the peak wavelength. For many cases required by the designer or device user, the precision attained is expected to be satisfactory. A note of caution must be added. In evaluating signal levels and SNR, as discussed below in Subsection 5.2, the spectral responsivity (or detectivity) must be averaged over the spectral product of the differential signal exitance and the transmission coefficient. Significant errors can be introduced in the case where the detector cutoff wavelength lies close to the short-wavelength end of a transmission band if the effective cutoff wavelength concept is used with undue care. In such cases, the use of the simplified ramp-function equivalent spectral form is perhaps better avoided.

It can be the case that the photodetector data available for a particular application are incomplete. For example, only the cutoff wavelength and the blackly responsivity at 500 K, as specified by the device manufacturer or measured by the user, may be known. In other cases, a designer may be selecting on an initial basis amongst several photodetector types and may have no need for complete data. In such circumstances, a spectral-to-blackbody conversion formula is useful for transforming from the specified 500 K blackbody responsivity or detectivity data to the ramp-function equivalent values R(L) or D*(n), prior to spectral averaging aalculations such as are required in the evaluation of equation (10) above. The spectral-to-blackbody conversion formula may be derived from the above analysis. We return to equation (10) and specify a blackbody source, recognizing further from the general irradiance equation (2) that in a laboratory or manufacturer’s test system the steady background

146

0.

M.

WILLIAMS

A,‘TIpm

K)

Fig. 3. Cutoff wavelength variation of the photon fraction j$:T) and the spectral-to-blackbody conversion constant for typical photodetector effective cutoff wavelengths.

term is removed by chopping the signal irradiance and subsequent phase-sensitive detection of the current or voltage signal.“‘) Furthermore, the close proximity of source and detector allow the transmission coefficient to be set to unity. Under these special conditions, we may apply the second term of equation (2) to equation (lo), and by dividing by the spectral form [equation (711, obtain the spectral-to-blackbody conversion formula (14) corresponding to source blackbody temperature T and background blackbody temperature Tb. Here, the step-function equivalence [equation (1 l)] has been applied such that the numerator may be derived immediately from equation (12) using either Fig. 3 or alternatively the approximation (13) for j&IT). The denominator is of course immediately calculable from the standard blackbody equation E,(T)

= aT4sin28.

(15)

It is seen from Fig. 3 that the spectral-to-blackbody conversion constant evaluated at & is to first order a function of l:T alone, the second-order dependence on cutoff wavelength occurring only at low values of it:T when the influence of the background terms in equation (14) becomes significant. Given the manufacturer’s value or a laboratory-measured value for R(T), usually at 500 K, then R(L;) may be derived directly from the curves of Fig. 3 and hence the ramp-function equivalent values R(1) appropriate to all wavelengths may be derived as M(n:)/ng. It is noted in passing, that since the responsivity and detectivity D * are characterized by identical spectral forms, as is proven below in Section 4, the conversion formula (14) and the corresponding curves of Fig. 3 apply equally to R and D*. It is of interest, for completeness, to explain the shape of the conversion curves. At low AT, few photons are capable of producing electron-hole photopairs and hence the blackbody responsivity is low. As the spectral forms of the blackbody radiation and the detector become better matched, the blackbody responsivity rises. The peak conversion efficiency that is apparent is consistent with the nature of the blackbody spectrum where the mean photon energy at high AT exceeds the gap energy EB, thus leading towards diminished values of energy transfer efficiency and hence of the conversion constant.

IR p~ot~ete~tors-I

147

3.5. Photogain In completing the discussion on responsivity, it is of interest to examine the form of the photogain c5) G introduced in equation (4) above, a parameter that embodies such design features of interest as the magnitude of the responsivity, detector linearity and frequency response. In the case of photodiodes at midband frequencies where minority carriers are swept rapidly across the junction and no gain processes occur, G = 1. However, for photoconductors in which the excess carrier motion is governed by the ambipolar transport equations,(24) G can assume values of the order of 100. The most revealing form for photoconductors may be derived from the review by Elliott,c’6) following the earlier analysis by Williams~) on minority-carper sweepout effects. It may be shown simply for an n-type phot~onductor in the absence of diffusion, and under the assumption that the excess-carrier density falls to zero at the positive electrode, that the photogain for a photoconductor of length 1 in an electric field L assumes the form

(16) where K and c(h are the electron and hole mobilities respectively (b = H/P,,), and where 1 -f

qff=z

(

*

1 -exp(-t,/r)

[

II

(17)

represents the mean time that an excess carrier resides within the phot~onductor volume. Thus, in low electric fields z,e equals the r~ombination lifetime r and in high electric fields reffapproaches it,, or the time that that excess-carrier population appears to be swept under ambipolar conditions from the average position at the centre of the photoconductor to the collecting electrode. The reader is referred to Adler et al. cz3)for a discussion of ambipolar motion in semiconductors, and to ElliotP) or Williams”) for discussions on sweepout effects in photoconductors. In an n-type photoconductor such as CMT, the ambipolar mobility is not too different from the hole mobility and hence, under conditions of strong minority-carrier sweepout, the form (16) for G tends towards the value gb + 1). Since b can be of the order of 50-200 for photoconductors such as CMT, large values of G are attainable. This result is consistent with the physical nature of the photoconductive process where, in order to preserve charge quasi-neutrality, electrons arriving at the collecting electrode are replaced by electrons at the source electrode during the entire period ree that the excess charge exists in the phot~onductor. A single photoelectron thus has the appearance of making many traversals over the phot~onductor length. It is clear from the above discussion that there is a close relationship between the photogain G and the frequency response of a photoconductor. Indeed, the ambipolar transport equation for the excess carrier may be solved in such a way that the frequency effects are included in the definition of G (see, for example, van der Zie104)).The frequency effects in photodiodes, which occur because of the finite RC product across the junction, may similarly be embodied in the definition of G. It is not, however, within the scope of the present work to comment further on photodetector frequency response. The results [equations (16) and (17)] quoted above have been included in order to illustrate the dependence of the photogain G on the fundamental photoconductor parameters and are not intended for calculation. Indeed, enhanced values of photogain by factors of up to 6 have been reported in recent work(‘2-‘4)and att~buted to the development of potential barriers near to the collecting electrode caused by excess doping levels in the vicinity of the metallic contacts. In such cases, the boundary conditions are different from those underlying the derivation of the simple form (16) for G. The form of G given by equations (16) and (17) also embodies the linearity of the photoconductor. The recombination lifetime r is dependent on the number densities of charge carriers within the semiconductor volume (see, for example, Kinch and Borrello(‘)) and hence, through the excess-photocarrier density, on the magnitude of the irradiance; (“) G is therefore dependent also on the irradiance. While many detectors such as photodiodes are characterized by an inherently linear current response, and furthermore, under uniform background irradiance a small-signal

148

0. M. WILLIAMS

value of G may be assigned, nevertheless, it should be remembered that photoconductors can be inherently non-linear. Strictly, a specific photogain Gb should be assigned for the background photocurrent in order to highlight the potential difference from the value G corresponding to the small-signal photocurrent. 4.

DETECTIVITY

In this section, the standard expressions for the detectivity D* as specified by the device manufacturers from measurements of the r.m.s. noise fluctuations are examined, with particular interest in the blackbody background convention under which D * is defined. The 1: 1 correspondence between D* and the responsivity is highlighted. 4.1. Noise current Expressions for photodetector noise have been derived previously by a number of investigators (see, for example, Hill and Vliet,@ Pruett and Petritz,‘4) Long,uS) van der Zie1(24)and Elliott et a1.(25)), and are not rederived here. Instead, a short outline is given in a less rigorous manner in order to highlight the origins of the noise sources in the familiar terms of electronic noise theory. The noise contributions in a photoconductor comprise generation-recombination noise applying to both photon-generated and thermally-generated carriers, Johnson-Nyquist noise and l/f or flicker noise. The latter contribution is dependent on the surface properties of the photoconductor and can be confined by suitable design to be significant only at low frequencies. In the present analysis, we consider only the midband frequency region where the squared noise fluctuation, neglecting l/f noise, may be written as 2” = 2Gi(2eiAf /G,,) + 4kT,Af /r 0

(18)

for device temperature T,, resistance r, and noise bandwidth A$ Here, i is the total current including both photon and thermal contributions, and Gb is the background-dominated photogain, consistent with the above discussion on detector linearity. Thus the thermal- and photon-generated primary current is i/Gb. The shot noise nature of the first term of equation (18) is immediately apparent, the additional factor of 2 accounting for the equal contributions from electron-hole generation and recombination. Williams”) has shown that this factor should be reduced to 4/3 under conditions of strong minority-carrier sweepout. The standard form of the Johnson noise term in equation (18) is also immediately apparent. By using the forms (3) and (6) as applied to the photocurrent iq, and form (4) as applied to the thermally generated current iih, we may rewrite equation (18) in the useful form, generalized for both photoconductors and photodiodes, as 2 = ue2GtAAf

* v (1 )E&, [S 0

T) dl + &

1

+ &hd

(19)

This equation is identical to that given previously by Long, (Is) although it has been expressed in a slightly different form. The factor u equals 2 for photodiodes (where also Gb = l), and is normally assumed as 4 for photoconductors except under the strong sweepout conditions discussed above in which case u = 8/3. In the photodiode case, the thermal generation rate is a function of both device temperature and voltage,‘“) and the resistance r, is the dynamic resistance at zero voltage bias. It is of interest to comment on several features of equation (19). Since Gb is large for photoconductors, the Johnson noise term is normally small compared to the remaining terms. In contrast, for photodiodes, Johnson noise can be significant. In order to reduce this term, it is important that the photodiode be characterized (26)by a large value of the product rd. Thermal generation, which is strongly temperature dependent, may be minimized by keeping the device depth as small as possible within the obvious limitation that the quantum efficiency defined by equations (8) or (9) above should not be significantly reduced. The device temperature T,, as specified by the manufacturer, is defined in order to keep both the Johnson noise and thermal noise terms as small as practicable.

149

IR photodetectors--I 4.2. Definition of D*

The definition of the specific detectivity D* is related closely but not exactly to the bracketed term in the noise current equation (19). By convention, D * is evaluated from separate measurements of the blackbody signal response and of the r.m.s. noise current obtained when the photodetector is exposed in the absence of signal to a blackbody background at a well-defined ambient temperature, usually in the range 290-300 K. In order to enforce consistency with such conditions, the first term within the square brackets of equation (19) must be replaced by the equivalent blackbody term; that is, within the present formalism, by the step-function equivalence defined by equations (11) and (12) above. The noise-equivalent power (NEP) is defined by dividing the r.m.s. noise current given by equation (19) by the current responsivity, and D* is thence defined in a standard manner as D*

=

(AAf )I’*

NEP or R,(T)

D*(T, 0) =

Geu ‘/* w#QK, Td + &

(20) + gad]“*

where, for convenience, we have made the usual tacit assumption that the photodetector responds linearly to all sources of irradiance. That is, Gb in equation (19) is set equal to the small signal photogain G. Equation (20) represents a general form of D* consistent with device specifications. The importance of adopting a standard convention for the photon irradiance term on the denominator is immediately apparent. Only by application of this convention is the denominator independent of both the input signal and the detector spectral response. Indeed, the spectral response is embodied solely in the numerator, as specified by the current responsivity R,(T). The 1: 1 correspondence between responsivity and detectivity has therefore been highlighted. Both obey the same transformation between the general and spectral forms and, accordingly, we may write mD*@, T) &,(A, T) dl D*(T)=

s’

m s0

9

(21)

4 ,(A T) dl

where D*(l) is the spectral form of D* obtained by replacing R,(T) in equation (20) by &(A) as given above according to equation (7). We note further that both the responsivity and the detectivity must necessarily satisfy the same spectral-to-blackbody conversion formula, given either by equation (14) or Fig. 3 above. The spectral detectivity D*(A) under background-limited (BLIP) conditions in which the Johnson and thermal noise terms are negligible is obtained simply from equation (20), using the spectral form (7) for Ri(A), as

Mn) Dimly 0) = hc~“~[~,&,(I;, TJ”*’

(22)

We note in passing that D& clearly varies as sin-‘@, where 8 is the half-angle defining the FOV of the photodetector. For a non-BLIP detector where D* is specified by equation (20), the FOV dependence is also a function of sine but is of course complicated by the presence of the Johnson and thermal noise terms. The familiar spectral dependence of D& is illustrated in Fig. 4, as obtained from the responsivity curves of Fig. 1 and from equation (12) for &(A:, T,,). The examples shown are specific to the photoconductor case with u set equal to 4. The identical spectral dependences of responsivity and detectivity are clearly evident. The dashed curves represent the ideal photodiode and photoconductor limits calculated from equation (22) as D& (I.:, z/2) with the quantum efficiency set equal to unity. Thus, the ideal limit represents the maximum detectivity that can be attained appropriate to any given cutoff wavelength.

150

0.

t

M.

WILLIAMS

/ I1111111

I

I

1

Wavelength

. .

Fig. 4. Vanatlon of D&

III

10

(pm

)

with wavelength for the calculated CMT photoconductor in Fig. 1.

5. SIGNAL

AND

NOISE

In the above sections, we have examined the characteristic parameters, the underlying physical based and the methods for estimating the spectral values are available. In this section, we comment IR signal and r.m.s. noise calculations.

responsivities shown

CALCULATIONS

derivation of the fundamental photodetector principles on which the detector properties are characteristics when only the 500 K blackbody on the use of the characteristic parameters for

5.1. Noise calculation The calculation of the r.m.s. noise current is simple for the usual case defined by convention; that is, where the photon noise contribution is dominated by background blackbody radiation filling the FOV of the optical system. In this case, the r.m.s. noise current is obtained simply as the product of the current responsivity and the NEP, or in terms of D*, as (23)

Clearly, all dependences on the signal cancel, including both the spectral and temperature characteristics. This result is of course consistent with the standard convention that effectively forces the noise fluctuation to be constant, consistent with the specified blackbody background radiation. Equation (23) is applicable to most standard photodetector applications. It is, however, useful to write the more general form for conditions where the convention is not obeyed. This may occur, for example, when a hot target fills the FOV of the optical system, or alternatively, when an intense spot from an IR laser target designator lies within the FOV. In this case, the r.m.s. noise current should be evaluated from equation (19) which, by suitable mathematical manipulation using definitions (20) and (22), may be written in terms of the photodetector characteristic parameters as

(24)

where DQE = D *ID ;L,p

(25)

IR photodetectors--I

151

is the detective quantum efficiency, as defined by Limpet-is and Mudarc’) in the ONR Infrared Handbook. It is noted that DQE may be evaluated by comparing D*(&) with D&,(&), the latter value obtained by multiplication of the appropriate value of the idealized photodetector curve shown in Fig. 4 by the square root of the quantum efficiency. The form of the correction factor in equation (24) is quite apparent for the BLIP case where DQE is set equal to unity. In this case, the correction has the effect of replacing the assumed blackbody noise fluctuation by the actual photon noise fluctuation. The correction factor is then proportional to the square root of the photon flux capable of generating electron-hole pairs. The remaining correction terms in equation (24) account for the finite Johnson and thermal noise contributions in the more general non-BLIP case. 5.2. Signal and SNR calculation Signal current may be written immediately by combination general form i,(T) = A s0

OD Ri(~)&(J,

of equations (3) and (10) in the

T) d4

(26)

where the signal spectral irradiance is defined according to the second term of equation (2). Evaluation of the signal current therefore reduces to the problem of determining the signal spectral irradiance and integrating it over the known responsivity spectrum. In the present analysis, we direct the assessment towards the SNR obtained by dividing is, given by equation (26), by i,,.,. given by equation (23). Following some manipulation using equations (2) and (20), we obtain the standard result (27) for small temperature differences AT between the specified target and background. The overall angular dependence for a BLIP detector is therefore sin28,/sin6. Thus, SNR may be improved by reducing the angular FOV, but only to within the limit set by the target filling the entire aperture. This limitation may be seen more clearly by rewriting SNR, following appropriate manipulation of the geometry factors in equation (27), in the alternative form SNR = n(j$),,2

AT Irn r(J, L) aM$’ 0

T, / D*(1,0) dlZ,

(28)

Tb

where A0 is the effective collection area (including obscuration) of the optical system and Q is the solid angle subtended by the target. In practice, A, will be fixed by optical and mechanical design constraints and 8 can therefore only be reduced by increasing the system focal length. This also is usually fixed by design constraints, so that there is little practical scope for the improvement of SNR by FOV reduction. As a final comment, it is noted that the comparison between detectivities of different detectors, as illustrated in Fig. 4, can be misleading. At first sight, it would appear that low cutoff wavelength detectors are inherently better than high cutoff wavelength detectors. This is, of course, not the case. The problem occurs when it is assumed incorrectly that D *, rather than SNR, is a subsystem figure-of-merit. On examining either equations (27) or (28), it is seen that in order to evaluate SNR, D*(n) must be integrated over the differential spectral exitance function and transmission coefficient across the atmospheric transmission band of interest. The magnitude of the integral depends critically on the extent of spectral matching between D*(2) and M&, T). Thus, an 8-14 pm photodetector is well-matched to 300 K blackbody radiation, whereas a 3-5 pm photodetector is well-matched to 500 K radiation. It is therefore deceptive to regard D* as an all-embracing photodetector figure-of-merit. 5.2.1. Practical SNR calculation. In closing the present discussion, we point out that equations (27) and (28) are related closely to two parameters that are used to varying degrees by workers in the field. The term M* is used(i6*22) as a figure-of-merit defined equal to the integral in equations

152

0. M.

WILLUMS

(27) and (28). That is,

M*(L, Tt,,0) =

s

aM,i(AT) m ~(1,L)

D *(A, 0) dlZ.

aT

0

(29)

Tb

Thus M* is an effective SNR parameter that is useful in overcoming the above criticism of D*. It does, however, have the disadvantages that there is no inherent spectral form and also there is a lack of generality due to the integration over the spectral atmospheric transmission coefficient that is necessarily dependent on range and atmospheric conditions. A more widely used and closely related parameter is the NETD which is defined as the temperature difference at which SNR = 1; that is, NETD = AT/SNR

(30)

which may be expressed in alternative forms by appropriate substitution from equations (27) or (28). The reader is referred to the excellent critique by Lloyd(“) on the position of NETD as a subsystem figure-of-merit in thermal imaging system design. The disadvantage of both M* and NETD, and for that matter SNR, for quick design calculations is that they each involve the same integration of the spectral transmission coefficient r(J, L) over the differential spectral exitance and D*(A)product. It is possible to mathematically define an average transmission coefficient, as discussed by Lloyd, (18)but this does not lead to any obvious simplification of the required calculational procedure. The exact spectral form of D*(A) may be replaced by the ramp function equivalent form, as discussed for the responsivity in Subsection 3.4 above, but this must still be integrated over the product of the spectral transmission coefficient and the spectral exitance. In those cases where a reasonable level of precision is required, the designer would still resort to the detailed evaluation of the integral over the spectral band of interest. The major reason for the difficulty lies in the fact that the emission, transmission and photodetection of IR radiation are all inherently photon transfer rather than energy transfer processes. In this respect, it may be shown simply that appropriate manipulation of the integral in equations (27) and (28) leads quickly to photon parameters and to a simplification whereby the source radiation, the transmission and the detection factors may be separated. There is, however, little scope for simplification within the conventional irradiance formulation of photodetector theory that has been examined in the present study. An alternative photon formulation is presented in the following paper in which it is shown that a complete separation of the parameters defining SNR may be successfully achieved. These results highlight the fact that the emission, transmission and detection of IR radiation by detectors of the photon type is more naturally described in terms of photon parameters. The conventional irradiance approach must therefore be seen as somewhat restrictive in that simplified forms of the important figure-of-merit parameters SNR and NETD are not readily available. 6.

SUMMARY

In this paper, the physical and mathematical framework underlying the application of IR photodetector theory has been appraised. The underlying assumptions and conventions have been identified in order to assist the user in understanding the properties and limitations of photodetectors, in terms of the typical responsivity and detectivity data available as device specifications. Equivalent ramp-function forms of the spectral responsivity and detectivity have been defined in terms of an effective cutoff wavelength in order to simplify spectral integration calculations of signal current and SNR, particularly for cases where device data are incomplete. The absence of a simplified and readily calculable form of SNR within the standard irradiance formulation of the theory has been highlighted. No attempt has been made to detail the underlying materials science, nor the fundamental physical principles of operation. For these, reference is made in particular to the review article by Elliott.“‘) Rather, the paper has been confined to a critical appraisal of those factors that influence both the calculation of signal and noise currents and the optimization of photodetector application. It must be realized that such an optimization is usually only part of an overall design tradeoff for

IR photodetectors--I

153

which numerous other design parameters must be considered, as is discussed in the wide-reaching work by Lloyd”‘) on thermal imaging systems. Acknowledgements-The author wishes to thank Dr G. G. O’Connor for the useful conversations during the course of this work, and also the referee who was instrumental in improving the author’s perception of the field. REFERENCES 1. E. S. Rittner, In Proc. Photoconductivity Conference, Atlantic City, Ga (Edited by R. G. Brekenbridge), p. 215. Wiley, New York (1956). 2. J. E. Hill and K. M. Vliet, Physicu 24, 709 (1958). 3. R. C. Jones, Proc. Inst. Radio Engrs 47, 1495 (1959). 4. G. R. Fruett and R. L. Petritz, Proc. Inst. Radio Engrs 47, 1524 (1959). 5. D. Long, Infrared Phys. 7, 121 (1967). 6. D. Long, Infrared Phys. 7, 169 (1967). 7. R. L. Williams, Infrared Phys. 8, 337 (1968). 8. R. D. Hudson, Infrured System Engineering. Wiley, New York (1969). 9. M. A. Kinch and S. R. Borrello, Infrared Phys. 15, 111 (1975). 10. M. A. Kinch, S. R. Borrello and A. Simmons, Infrared Phys. 17, 127 (1977). 11. F. Szmulowicz and R. Baron, Infrared Phys. 28, 385 (1980). 12. Y. J. Shacham-Diamand and I. Kidron, Infrared Phys. 22, 9 (1982). 13. C. T. Elliott and T. Ashley, Infrared Phys. 21, 105 (1981). 14. A. Kolodny and I. Kidron, Infrared Phys. 22, 367 (1982). 15. D. Long, In Optical and Infrured Detectors (Edited by R. J. Keyes), pp. 101-147. Springer-Verlag, Berlin (1977). 16. C. T. Elliott, In Handbook on Semiconductors, Vol. 4 (Edited by T. S. Moss), pp. 727-798. North-Holland, New York (1978). 17. T. Limperis and J. Mudar, In The Infrured Handbook, Chap. 11 (Edited by W. L. Wolfe and G. J. Zissis). OIhce of Naval Research, Washington, D.C. (1978). 18. J. M. Lloyd, Thermal Imaging Systems. Plenum press, New York (1975). 19. W. L. Wolfe, In Applied Optics and Optical Engineering, Vol. 8, Chap. 5 (Edited by by R. R. Shannon and J. C. Wyant), pp. 117-170. Academic Press, New York (1980). 20. W. L. Wolfe, In Ref. (17), Chap. 1. 21. W. W. Anderson, Infrared Phys. 20, 363 (1980). 22. J. A. Chiari and F. D. Morten, Electronic Components and Applications, Vol. 4. Mullard Tech. P&l. M82-0099, pp. 242-252 (1982). 23. R. B. Adler, A. C. Smith and R. L. Longini, Introduction to Semiconductor Physics. Wiley, New York (1964). 24. A. van der Ziel, Fluctuation Phenomena in Semiconductors. Buttenvorths, London (1959). 25. C. T. Elliott, D. Day and D. J. Wilson, Infrared Phys. 22, 31 (1982). 26. P. Knowles, GEC JI Res. 2, 141 (1984).