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Detectivity limits for diffused junction PbSnTe detectors
Infrared Physics. 1975, Vol. IS. pp. 317.-329 Pcrgamon
DETECTIVITY
F’rcss. Printed
m Great
Bntain.
LIMITS FOR DIFFUSED PbSnTe DETECTORS?
JUNCTION
M. R. JOHNSON,R. A. CHAPMANand J. S. WROBEL Texas
Instruments
Inc.,
Dallas,
Texas
75222,U.S.A.
(Receiwd 5 December 1974) Abstract-The resistance-area product (R,A) of diffused junction PbSnTe photo-volt& detectors under conditions of zero bias voltage is calculated for linearly graded and one-sided abrupt junctions. Equations are developed to demonstrate the overall dependence of ROA on grading constant when minority carrier diffusion current, depletion layer current, and tunneling current are taken into account for the linearly graded junction. Similarly, for the one-sided abrupt junction the dependence of R,,A on the carrier concentration of the lightly doped side is shown for the diffusion. depletion, and tunneling mechanisms. The calculations are c:lrried out for two devices of practical interest: Pb o ,,,Sn,.r,,8Te diffused with Sb with a 5 Itm cutoff at an operating temperature of 170”K, and Pb,.,,Sn,.,,Te diffused with Cd with an 11 pm cutoff at 77°K. The junctions formed by Sb diffusion obey the linearly graded model, whereas the Cd junctions formed in unannealed substrates are one-sided abrupt. Upon comparing measured R,A products with calculated values, we have established approximate values for the lifetimes within the depletion layers for each of these devices. The consequences of these results for the thermal noise-limited detectivity (D*) of these detectors are shown by plotting D* vs RoA. Within this framework, it can be argued that the inherently short lifetimes of PbSnTe play the dominant role in placing the upper limits on achievable D*.
I. INTRODUCTION
Calculations of detectivity (D*) limits for PbSnTe photovoltaic detectors from empirical models based on measured detector parameters are presented in this study. These results provide an estimate on the excess carrier life-time within these devices, about which very little is quantitatively known. A!! other device parameters are known with accuracy sufficient to make valid calculations of &,A product: and D*. Operation at zero bias voltage provides optimum signal-to-noise ratio; the incremental resistance at zero bias determines the thermal noise level and the ultimate D* that can be obtained when noise from other sources is negligible. Two diffused junction detectors are considered. The first half of this paper deals with linearly graded Sb-diffused junctions in Pb o 982Sno.o,,Te; these have a cutoff wavelength of 5 w at an operating temperature of 170°K. We calculate the &,A product for these graded junctions. taking into account minority carrier diffusion, depletion layer generatiorr-recombination, and tunneling. The R,A product is plotted vs the grading constant. Theory and experiment are compared for the value of grading constant appropriate to Sb diffusion (a = 1.9 x 10” cmA4); values are compared also for Al-diffused junctions (a = 2.3 x IO” cm-4). The second half of the paper deals with abrupt Cd-diffused junctions in PbO.,&O.z,Te with 11 w cutoff at 77°K. These junctions are formed by diffusing Cd into as-grown PbSnTe (N, 2 3 x 10’ 9 cme3) and are seen to obey the one-sided abrupt junction mode!. The RoA product is calculated with carrier concentration on the lightly doped (n) side as a parameter. Theory and experiment are compared for net donor concentrations of 1-2 x 10” cme3, which typify Cd-diffused junctions. The RoA products of linearly graded junctions are limited primarily by generationrecombination within the depletion layer. The same is true of the best abrupt junctions for temperatures less than 90°K. The depletion layer lifetime in these devices is short, t Work partly supported by the U.S. Army Night Vision Laboratory. $ Ro is the incremental resistance of the photovoltaic detector at zero volts bias. A is the effective sensitive area. 317
318
M. R. JOHNSON. R. A. CHAPMA& and J. S. WROBEL
about lo-* sec. At the moment very little in PbSnTe. These centers almost surely are dielectric constant would force their energy be insignificant in determining the lifetime. to raise the theoretical limits of detectivity
2. THERMAL
LIMITS
is known about the recombination centers not hydrogenic in nature, because the large levels near a band edge, where they would A dramatic increase in lifetime is required above those calculated here.
FOR
PHOTODIODES
The importance of the RoA product and the quantum efficiency for a detector whose noise has contributions from both the thermal noise of the detector zero bias resistance (R,) and the background photon flux can be seen from the following equation:
(1) Equation (I) follows directly from the defining equation for D* in terms of the detector signal-to-noise ratio with the provision that the Johnson-Nyquist noise of the zero bias resistance and the shot noise of the current generated by the background photon flux are added in quadrature. The detector of equation (1) is assumed to have an ideal spectral response (response cc i) and a peak response as well as an abrupt cutoff at the wavelength I.. Additional noise contributions, such as l/f noise, are not included in equation (1). Equation (1) is valid only at zero bias voltage for a photovoltaic detector; hence the subscript “0” on the detector resistance. The symbols in equation (1) have the following meanings: r] is the quantum efficiency, q is the electronic charge, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, R. is the diode incremental resistance at 0 V, A is the detector sensitive area, $B is the photon flux incident on the detector from its surroundings, and T is the Kelvin temperature of the detector. Equation (1) has been plotted in Fig. 1 as a function of RoA and q for a detector with i = 5 ,um at T= 170°K and & = 1.32 x 1016 cm-’ set-’ (300°K background 180” field of view). For small values of RoA product, the thermal noise of R, dominates the detector noise and D* a q(RoA) ‘/’ . As the RoA product is increased, the importance of the background noise term increases. It is clear that for low RoA products, both q and R,,A are crucial factors. On the other hand, for large R,,A products, 0: is determined by 9 and field of view. For the parameters used in Fig. 1, D* will not increase
QL
51 cm
2
Fig. 1. Thermal noise-limited D? vs R,A product for a photodiode with I, = 5 pm at T = 170°K. The field of view is 180’ with a background temperature of 300°K. Four values of the quantum efficiency, q. arc shown. The equation plotted here is valid only at zero bias voltage. An R,,A product greater than 100 is needed for background-limited operation at 170°K. The solid points represent measured Df and R,,A data for Sbdiffused detectors in Pb,, ss2Sn, “,sTe, with I, = 4.9 pm at T = 170’K.
319
Diffused junction PbSnTe detectors
with increasing RO for R,A > 100. The data points represent data for Sb-diffused and Al-diffused Pbo.sazSno.o,sTe, and are discussed below in comparison of theory and experiment. Equation (1) is plotted in Fig. 2 for the case of the 11 ,um detector at 77°K. We see that background-limited operation requires ROA 2 1 R-cm for a quantum efficiency of 50%. The data points represent data for Cd-diffused Pb,.,,Sn0.2 ,Te, and are discussed later in the text. We shall return to Figs. 1 and 2 after calculating theoretical values of ROA for the 5 and 11 pm devices. I
’
“‘I
o*=g g+&)4* A
.
7
I ?
[
0
1
“‘1
’
‘I’_
-t
1
A-llpm
T- 77'K
16'- $-6,52x 10’7cm-2sec-’
1 ”E
0.01
0001
IO
I.0
0.1 2 Qt
Ll
cm
Fig. 2. Thama1 noise-limited 0: vs R,A product for a photodiode with 1< = 11 q at T = 77°K. An R,A product greater than 1R-cm’ is needed for background-limited operation. The solid points represent measured 0: and R,,Adatafor Cd-diffused detectors in Pb,.,,Sn,.,,Te, with i,= llpmat T=77’K. 3. THEORETICAL
PERFORMANCE
Pb,.,s,Sn,.,,sTe
3.1 Model for d@sed
LIMITS
WITH I, = 5~
FOR
Sb-DIFFUSED
AT T= 170°K
Sb junctions
A simple model of the diffused Sb junction permits a calculation of reasonably firm limits on R,A and D*, subject to the choice of carrier life-time. This section presents an empirical model of Sb junctions in Pb0+s2 Sn,.,,sTe at 170°K and proceeds with the calculation of ROA for minority carrier diffusion currents and depletion layer generation-recombination currents. Tunneling current is shown to be unimportant for these graded junctions. To form the model we have used measurements of the junction capacitance-voltage characteristic and the dependence of the zero bias resistance on temperature. Figure 3 is the C-V plot at 170°K of an Sb-diffused junction. The important result is that C ,x V-“3, where I/ is the junction bias voltage, and this behavior implies a linearly graded junction. The grading constant, a, is found from the slope of the curve”’ to be 1.9 x 10ZZcm-4. A dielectric constant of 610 has been assumed, taking the measured value for graded PbTe junctions at 170°K. (‘I The built-in voltage of 0.26 V is found by extrapolating the C-V curve to zero capacitance. We then calculate a zero bias depletion width (‘) of 0*4pm for this junction. Figure 4 is a plot of measured zero bias resistance as a function of temperature. The important result here is that the resistance follows primarily the temperature dependence of n; ’ for temperatures near 170°K. This implies that the resistance is dominated by generation-recombination processes within the depletion region of the junction. In calculating RoA, therefore, we should expect to find that the depletion layer mechanism results in a lower ROA than the minority carrier diffusion mechanism. We calculate here the RoA as a function of the grading constant, a, for a fixed temperature of 170°K and a fixed 3., = 5 pm.
320
M.
R.
JOHNSON. R. A. CHAPMAN
22 2Q-
,
(
,
Sb diffvsed
,
,
and J. S.
,
,
WROLIEL
,
,
W,,Sn,a,,Te
Bias valiag9,
V
Fig. 3. Measured capacitance (C-‘!3) vs bias voltage for Sb-diffused Pb,.,s,Sn,.,,, Te at 170°K. The pn junction was diffused at 700°C to a depth of 17 pm into material with p7rqK 2 2 x 1018cm-3. The grading constant, a, inferred from the linearly graded model is (I = 1.9 x 10” cme4. The depletion width at zero bias is W = 0.4pm. The built-in voltage determined from the voltage axis intercept is @26V.
3.2 ROA for diffusion current The minority carriers in a linearly graded junction diffuse in the presence of a built-in field in the quasi-neutral regions on either side of the depletion region. To be completely rigorous, one must include the electric field in the calculation of the current. Mo1V3) has argued, however, that with certain assumptions, the ideal diode law holds for graded junctions as well, and the saturation current density is given as usual in terms of the (non-degenerate) doping levels on either side of the junction. As the grading constant is made smaller, the space charge dipole layer is less well confined. and for very small I
I
1
I
,
I
o-
0
Sb ditfused
D
X,-5pm
Zero bias resistance vs temperature 5 IO -
o 0
’
.
*
*
.
.
0
c
.*
.
. *
*
* .
.
.
I.
a?
5
6
7
9
9
I IO
I II
12
103/T
Fig. 4. Measured zero bias resistance vs temperature for four Sb-diffused diodes. The straight lines represent depletion layer-limited resistance behavior (l/n,) and diffusion-limited resistance behavior (I/n;). In the vicinity of T= 170°K (103/T= 5.9), most diodes exhibit a depletionlimited resistance behavior. As shown in the text. this can be accounted for by a depletion layer lifetime of IO-* set or greater at T= 170°K.
Diffused junction PbSnTe detectors
321
grading constants the p-n junction effect is lost. In general, the condition
must be satisfied to have well-defined space charge in the graded junction, where LD = J(eok7’/2q2ni) is th e intrinsic Debye length. If we substitute the appropriate values of an Sb-diffused junction in Pb,.sszSno.oisTe at T= 170°K we find that k = 0.8 x 104. Thus, the Sb junction satisfies the linearly graded junction approximation fairly well at 170°K. The minority carrier diffusion length on either side of the junction is expected to beL = J(kTpT/q) z 8pmforT = 170”K,p = 5 x 103cm2V-1,se-‘,andr = 10-8sec. The depletion width for the Sb junction at T = 170°K is W = 0.4pm; the majority carrier density at the edge of the depletion region is approximately aW/2 = 3.8 x 10” cmm3. In the quasi-neutral regions on either side of the depletion region, the net rate of change of ND - N, with position can be less than that given by a = 1.9 x 10” cmm4. For calculation of the minority carrier diffusion current, we need some approximation to the number of minority carriers within one diffusion length of the depletion region. This number lies somewhere between the value at the edge of the depletion region, nf t aW/2, and the value within the bulk, n’ + N,, ND. The linearly graded junction concept would be expected to break down for grading constants large enough that aW/2 2: N,, N, where NA and ND are the majority carrier concentrations in the neutral regions on either side of the junction. There are obviously several complications to calculation of the RoA product due to minority carrier diffusion for the graded junction as a function of the grading constant. We adopt the following simplified procedure: (1) for small grading constants we assume that RoA = kT/qlo still holds, and in calculating the saturation current I0 we use for the minority carrier concentrations the values at the edge of the depletion region, i.e., nf + aW/2. Since for the graded junction WK a- li3, we have RoA a a213, as shown below. (2) For large grading constants we assume the junction may be treated as abrupt. We again take R,,A = kT/ql,, and assume the minority carriers are given by nf + N,, ND. Small grading constant. The RoA product at zero bias is found by differentiating the current voltage characteristic R,,A = 2.
(3)
The saturation current density lo/A is given by
2 = q(!$)“‘[n,k)l”
+ p.(!!!!y”].
(4)
For nP and p. we take the values at the edge of the depletion region,
np,pn=
&.
(5)
This assumption allows the RoA product to be put in terms of the grading constant a. We also assume for PbSnTe that pL,= pp and 7, = zp since the effective masses are nearly equal. The depletion width W for the graded junction at’zero bias is given w
= 0
1 3
l&o Vbi 1’3 qa ’
____
(6)
where E is the dielectric constant, lo is the permittivity of free space, V,i is the built-in voltage (diffusion potential of the junction), and a is the grading constant. Combining
M.
322
R. JOHNSON, R. A. CHAPMAK and J. S. WROBEL
equations (3)-(6), we have
The principal result is that ROA tc a 2’3. There is a slight dependence of l’bi on a which is important for small a, but this has been ignored. V*i has a maximum value equal to E$q = constant. The principal temperature variation of ROA is the term n; ‘, which varies as exp(E,JkT) in which E,, is the extrapolated value of E, at T = 0. Large grading constant. Equation (7) predicts that ROA increases without bound as the junction becomes steeply graded. The RJ product cannot exceed that value given by an abrupt junction, however. The maximum ROA would be obtained for a one-sided abrupt junction. The doping level on the lightly doped side can be made as high as l-2 x 1018cm-3 for Pb,.982Sn,.,,,Te at T = 170°K without causing a Burstein shift of the cutoff wavelength. We shall use equation (4) with the p-side as the lightly doped side. The minority carriers are given by np = niLIN,.* The maximum R,-J product is given from equation (3) after these substitutions through Equation (4), as R,,A = ($)‘:‘($‘:’
2.
In evaluating equation (8) we take NA = 2 x lo’*, rn = IO-* set, pL,= 5 x 103, and ni = 15 x 1014. The result is ROA = 95R -cm2. 3.3 ROA for depletion
layer current
The common practice of differentiating
2k7’) - l] and setting equation applies only et CL,(~)and proceeds for electrons and holes tion rate is integrated recombination current as
the current-voltage equation I = IO [exp(qV/ V= 0 to find the zero bias resistance is not correct. In fact, this for qV> 2kT The correct procedure has been outlined by Sah, along the following lines. Assuming that the quasi-Fermi levels are constant throughout the depletion region, the net recombinaacross the depletion region to give an equation for generationin terms of applied bias. The current density varies with voltage --I
qni W
A = zi;,-i
2 sinh (qV/2kT) f (b). q( V,,i - l’)/kT
(9)
This equation is valid for a single energy level recombination center located anywhere within the forbidden gap. The new symbols are rno, TV,,= electron and hole lifetimes within the depletion layer. The function f(b) is a definite integral depending on the position of the recombination centers within the gap and bias voltage V. For small applied bias, however, f(b) may be taken as independent of V. We also neglect the bias dependence of the depletion width W for small bias (V+ kT/q). The zero bias resistance is then found by differentiating equation (9) and setting V = 0, = ‘/bl \&I 4
4
‘PO
Wf (b)
).
(10)
In applying this equation to PbSnTe we take t,, = rpO for simplicity. We further assume, because of nearly equal electron and hole masses, that the dominant recombination centers are located at E, = li = intrinsic Fermi level. With the additional condition V = 0, the function f(6) = 1. Equation (10) then becomes RoA=----
"b, ‘50 Pi
* The use of non-degenerate taneously.
W
(11)
statistics is questionable since degeneracy and the Burstein shift occur simul-
Diffused junction PbSnTe detectors
323
This result is larger by a factor qV,,lkT than the result normally obtained by differentiating I0 [exp(qV/2k7J - 11. This factor may be 9 1. V,i is the diffusion potential of the graded junction, which we take as E$q for simplicity. Upon substitution of the depletion width from equation (6), the RoA product becomes RoA
=
(!$)iz?!(d-)“‘.
(12)
The principal result here is that ROA x Q“3. theory predicts that ROA varies slowly with grading constant. In evaluating equation (12), the term of greatest uncertainty is zo, the lifetime associated with the Schockley-Read-Hall generationrecombination centers. These centers almost surely are not hydrogenic in nature, because the large dielectric constant would force them near a band edge, where they would be insignificant in determining the lifetime. We can only suggest at this point that these centers lie well within the gap and are not associated with electrically active defects. 3.4 RoA for tunneling current There is, of course, no net tunneling current in a diode at zero bias; but there is an RoA product for tunneling for small voltage swings about zero. In a graded junction the tunneling probability is usually negligible. This is especially true for PbSnTe, because the high dielectric constant, in effect, reduces the electric field in the forbidden region. Also, the Fermi level at the edge of the depletion region must lie above the band edge both the n- and p-sides of the junction in order to have appreciable tunneling probability. This occurs for Pb o 9s$n0.0,8Te at T = 170°K for a carrier concentration of about 1 x 101”cm-3. Thus, aW/2 2 1 x lOi* is the requirement for overlap. Using equation (6) and letting V*i= E$q as before, the condition on a becomes a 2 1O23cm-4. Even for a = 1024, however, the tunneling length (W) is 01 cam, and the tunneling probability is negligible. For graded junctions in PbSnTe, then, the RoA products expected from tunneling are so high that they can be ignored. 3.5 Numerical results of RoA calculations for Sb-diffused 5 pm detectors The foregoing calculations have been summarized in Fig. 5. The R,A products calculated from equations (7), (8) and (12) have been plotted vs grading constant. We have
Pb,,,Sn,,,,Te
@m,
170°K
1
Sb
IO”
1
1
IO”
IO= 0,
cm
IO’*
-4
Fig. 5. Calculated zero bias RoA product at T= 170°K for linearly graded junction vs grading constant. Tunneling current is shown to be insignificant for graded junctions in PbSnTe. All details of the calculations and the critical assumptions are explained in the text. The calculations predict R,,A = 2 R-cm’ for the case of Sbdiffision (a = 1.9 x 1Oz2cm-‘), and @8 R-cm’ for Al-diffusion (D = 2.3 x IO**cme4). The solid points represent measured RoA products for Sbdiffused diodes and Aldiffused diodes.
M. R. JOHNSON,R. A. CHAPMANand J. S. WROLIEL
324
used the value of ni for E, = O-25eV calculated by Bate.“) Likewise, the dielectric constant at T = 170°K is taken from measurements by Bate”’ of capacitance versus temperature of graded PbTe junctions. For I(. and pr, we have taken the value measured at 170°K for p-type material. We have assumed a value for minority carrier lifetime of lo- a sec. This is the value calculated for radiative lifetime in degenerate PbTe by Washwell and Cuff.@’ Both the p- and n-sides of the Sb junction are nearly degenerate at T = 170°K. The extrinsic radiative lifetime is an upper limit to the actual lifetime since the likelihood of any trapping effects is small. The lifetime r,, within the depletion region is more difficult to estimate than the lifetime in the neutral regions. There is not sufficient experimental data to calculate ?0 a priori, but we have taken r. = lo- * sec. With this choice of parameters we find in Fig. 5 that the RoA product due to depletion layer generation-recombination dominates the effective RoA product for grading constants greater than 10” cme4. The effective RoA is the parallel sum of RoA (diffusion) and R,A (depletion). This theoretical result is in agreement with experiment, namely, that the detector resistance at T = 170°K is depletion layer-limited. Therefore, the choice of r. = lo- * set is reasonable. Notice that RoA (effective) rises slowly with grading constant. Therefore, dramatic improvement in RoA cannot be expected unless the junctions are made abrupt. The limit for RoA of 95Rcm2 for an abrupt junction with h.‘A= 2 x lo’* cmm3 is shown as the dashed line at the top of the curve. This is the maximum possible R,A for a junction in PbSnTe with ,$ = 5 pm at T = 170°K. Referring to Fig. 1, we see that the maximum theoretical D* for RoA = 95 R-cm’ and 9 = 0.5 is 1 x 10” crnH~‘/~ W- ’ at T = 170°K and i, = 5 pm. As noted earlier, the grading constant for Sb diffusion is a = 1.9 x lO22 cmd4. Figure 5 predicts RoA = 2R-cm2 for Sb diffusion. Several data points representing measured RoA products for Sb-diffused diodes are shown in Fig. 5. These RoA products represent the best diodes fabricated in our laboratory with Sb diffusion. The highest RoA product measured at 170°K was 4.5 R-cm’. Returning to Fig. 1, we see that the calculated RoA product of 2R-cm2 is sufficient to provide D* = 2.5 x 10” cm Hz’!~ W- ’ for a quantum efficiency of 50%. The data points of Fig. 1 represent measured DX and RoA data for the best Sb-diffused diodes, with & = 4.9 pm and T = 170°K. We conclude that the agreement with theory is good; the Sb-diffused diodes are well modeled by a linearly graded junction whose RoA product is controlled by depletion layer generation-recombination with a lifetime of lo-* set or longer at a temperature of 170°K. A lifetime of 2.3 x lo-* set is implied in the case of the highest RoA found experimentally (4.5 R-cm’). Also shown in Fig. 5 are data points representing measured R,A products for Aldiffused junctions. Al junctions are also linearly graded but with a somewhat smaller grading constant of 2.3 x lo2 I cm- 4. The measured RoA products for Al diffusion are lower than those for Sb, being in the range of 0.5l.0R-cm2 for the best diodes. Theory predicts a lower RoA for Al due to the smaller grading constant. The highest possible RoA product for a PbSnTe p-n junction detector with a 5 pm cutoff at 170°K is 95 R-cm’. The corresponding DX limit is 9.5 x 10” cm Hz”’ W- ’ if the quantum efficiency is 50%. From Fig. 1 it is apparent that some increase in this value of 0; is possible with cold shielding. 4. THEORETICAL Cd-DIFFUSED
. =
PERFORMANCE LIMITS Pb,, ,gSn0.2,Te WITH 1 I pm AT T = 77°K
FOR
Consider now the performanie limits of detectors with a cutoff wavelength of 11 pm at T = 77°K. Although these devices have been fabricated by several different methods, only Cd-diffusion to form pn junctions is considered here. Our best results have been obtained by this method, and these devices closely approach the ultimate performance limits established by the intrinsic materials properties of 01 eV PbSnTe. 4.1 Model for difised
Cd junctions
Cadmium is a compensating ‘neutral’ impurity in PbSnTe, with the very important property that n-type layers with net carrier concentrations of 1 x 10” cme3 can be
Diffused junction PbSnTe detectors IO
I
I
I
Cd diffused
I
I
325
I
I
Wo,sSn,,,,Te
9-
r IA
6-
~-~” “N- l.7.7; _
7-
+G¶
p$
.
6-
/
m -0-
/ S-
Nv
/
4
./
3-
/
2-
0.1
I/’
0
-01
-02
-03
-04
Bias voltage.
-05
-0.6 V
Fig. 6. Measured capacitance (C-‘I’) vs bias voltage for Cd-diffused Pbc.,,S~.r,Te at 77°K. The junction obeys the one-sided abrupt model, with a net donor concentration on the ditfused side of 8.9 x lOI cmm3. The junction was diffused into material with ~77~~= 3 x IOr cmW3. The built-in voltage determined from the voltage axis intercept is 009 V.
formed by diffusing Cd into degenerate p-type material. The long wavelength cutoff of photoresponse on the lightly doped n-side of a diffused Cd p-n junction in Pbo.,9Sn0.zITe does not suffer the Burstein-Moss shift. The thickness of the n-layer is chosen to provide optimum quantum efficiency. Although tunneling current dominates the reverse diode characteristic of these devices, the zero bias resistance can be quite high; values of the RoA product at 77°K as high as 5R-cm2 have been measured for monolithic detector arrays. Another important feature of diffused Cd junctions is that they are abrupt (one-sided). The capacitance-voltage characteristic of a diffused Cd junction is shown in Fig. 6. I
I
I
Cd diffused
I
I
I
I
1
A,- llpm
Zen, bias resistance temperature
VI
103/T
Fig. 7. Measured zero bias resistance vs temperature for three CddiRirsed diodes. of T= 77°K (Id/r = 13), diodes exhibit a depletion-limited resistance behavior. can be accounted for by a depletion layer lifetime of lo- a set or less at T = 77°K. to ditkion-limited resistance behavior occurs at T= 90°K for the better
In the vicinity These results The transition diodes.
M.
326
R.
JOHNSON. R. A. CHAPMAN and J. S. WHOBEL
The capacitance versus bias behavior is a straight line when plotted as l/c2 vs c this indicates an abrupt junction. (‘) If one assumes a value of 500 for the dielectric constant, then one finds from the slope of the C-l/ curve a net carrier concentration of 8.9 x 1016 cm- 3 in this case. This carrier concentration is assumed to apply to the n-side of the junction, since the acceptor concentration before Cd diffusion is 3 x lOi cm- ‘. Carrier concentrations of I x 10” cmm3n-type have been measured on thin Hall samples diffused with Cd. The R,,A product of diffused Cd junctions at 77°K is dominated by depletion layer current, although minority carrier diffusion current dominates the R,A product at higher temperatures. This can be seen in Fig. 7, which is a plot of zero bias resistance vs temperature for three diodes. The transition from diffusion current to depletion layer current occurs at T= 90°K for the better diodes. In choosing a model for Cd-diffused Pb0.7&,.,,Te, we expect the abrupt junction equations to apply and we expect depletion layer current to dominate the resistance. 4.2 ROA for difision current The minority carriers arising on the lightly doped (n) side of the p-n contribute the dominant part of the diffusion current (equation (4)). The minority carriers are similar in magnitude on both sides of the junction; minority carrier lifetimes are thought to be similar on both sides of the saturation current for the one-sided abrupt junction is
junction will mobilities for likewise, the junction. The
(13)
When .the n-side is non-degenerate, pn = $/IV, where ND is the net donor concentration. Making these substitutions in equation (3), we arrive at R,A
=
(7)“’($‘;‘$
(14)
for minority carrier difision of the one-sided abrupt junction. The principal temperature variation of equation (14) is the term ni’ . Equation (14) fails when N, becomes so large that pnnn # nf ; for 01 eV PbSnTe at 77°K. this occurs for ND Z 2 x 1017
cm-3.u33
The RoA product can be increased by raising the carrier concentration on the lightly doped side. For Pb,,.7PSn,.2,Te, however, a practical limit of ND = 2 x 10” is set by the Burstein shift. At this point, then, the intrinsic properties of the material itself (r pi,, ni) establish the RoA limit. Cd-diffused PbSnTe, for which ND = 1 x 10” cm -S , therefore represents the optimum design choice for p-n junction detectors in 01 eV PbSnTe. 4.3 ROA for depletion layer current The discussion preceding equation (11) applies here also, and we shall make the same assumptions for the @l eV case. The RoA product is given by equation (11) in which we must put (15) W = &%, V,,IqN,)
for the one-sided abrupt junction. Equation (11) then becomes R,A
=
7O
I’,‘/’ NC2
tli(2EEOq)1’2 ’
(16)
Comparing equations (14) and (16), we can see that large values of ND are required for large RoA products in both cases.
Diffused junction PbSnTe detectors
327
In evaluating equation (16) we take Vb,= Edq, which is a constant, independent of doping level. The variation of Vr,,with ND is slight over the range of ND of interest. For large values of N, we again assume that the maximum value of the built-in voltage is E$q. 4.4 ROA for tunneling current Computing the contribution of tunneling current to the diode resistance can be a formidable task, even for the case of zero zpplied bias. Taking into account the bias dependence of the tunneling probability is essential for the case of small applied bias (c/6 kT/q). We shall follow here the method of Karlovsky.“’ The tunnel current-voltage relation for small applied bias (Ve k7’/q) is
I =
cz g 6% + E, - 4’)‘.
(17)
C is a constant given by (18) and m* is the average effective mass for electrons and holes me
_
2m:mt
(19)
rn:+ mh*'
The electron and hole masses of equation (19) are the density of states masses, which we take as O-0728m. for electrons and O-082m0 for holes from calculations by Bate.(s) These values lead to m* = 0.0771 m,, in equation (19) for O-1eV PbSnTe. The terms E, and E, in equation (17) are the positions of the Fermi level on either side of the junction. For the diffused junction at zero bias, the dominant contribution is from the p-side; to a very good approximation we may take E, + E, independent of N,. The term Z in equation (17) is the tunneling probability, for which the shape of the potential barrier must be chosen. We differentiate equation (17) with respect to V and find the very simple result at v= 0: (R 0 A)-’
=
czq 6%+ E,)‘. kT
The main variation of RoA with doping level N,, is through the tunneling probability. The potential barrier will be assumed to be parabolic, in which case the probability is (21)
The quantity F is the electric field throughout the depletion region, which approximate by a constant. The maximum value of F is ZV,JW(V = 0) for degenerate material. Following Karlovsky, we take for zero bias the average F m.x ad 112 Fmx, i.e. F = 3/2 Vbi/W(V = 0). W( V= 0) is just the depletion zero bias. In the spirit of earlier approximations we again take V,, = E$q. The tunneling probability at zero bias can now be written as n(eeo m*)‘j2 E, 3qh N;”
when w(V = 0) is substituted
from equation
1
(15). The RoA product
we must strongly between width at
(22) may be written
M. R.
328
JOHKSON, R. A. CHAPMAN and
J. S.
WROBEL
finally as R
(23)
0
from equation (20) after substitutions are made from equations (18) and (22). We have computed R,A for several values of N,. 4.5 Numerical results of RoA calculations for Cd-difised
11 pm detectors
The RoA products due to diffusion current, depletion layer current and tunneling current have been plotted vs ND in Fig. 8 from equations (14), (16) and (23), respectively.
_ Pt+,,Sn,,,Te
(1Ipm.
77-K)
Fig. 8. Calculated zero bias RoA product at T= 77°K for one-sided abrupt junction vs doping level on the lightly-doped side. For the parameters chosen to typify Cd-diffused Pb,.,,Sno.l,Te, the effective R,,A product is limited by depletion layer current for the maximum doping permitted without a Burstein shift of the photo-response (ND 5 2 x 10” cmd3). The solid points represent measured R,A products for Cd-diffused diodes corresponding to values of ND determined from C--V measurements.
The minority carrier lifetime has been taken as I x 10-s set, and is further assumed to be independent of doping level N,. An upper limit on 7p would be the extrinsic radiative lifetime. The radiative lifetime for 10” cme3 material is 7.8 x lo- * set, calculated by Hall’s method. (‘O) The value of 2 x lo4 for mobility lies within the range of mobilities measured for 1017 cm- 3 material. The dielectric constant has been taken as 500, in agreement with our earlier assumptions. In evaluating the depletion layer contribution we must again make a choice for the lifetime z. within the depletion region of the junction. The discussion following equation (12) applies to the 0.1 eV case also. The lifetime ~~ is best found empirically by measuring R. for a junction which is clearly depletion layer current-dominated. We shall again choose a value of 1 x lo- ’ set for to. The assumptions leading to equation (23) have been discussed above. The value of m* is taken as O-0771mo. The value of E,,, the position of the Fermi level on the p-side, is calculated for a p-type doping level of 3 x lOI cmm3 and is 0165 eV. With this choice of parameters we find in Fig. 8 that the depletion layer establishes the effective detector R,A product for values of ND of practical interest. This result agrees with experiment, namely, that the diode resistance is controlled by the depletion layer. For our choice of parameters, the diffusion contribution is important only for N, < 1016 cmw3. To increase the RoA product above the theoretical value of 5R-cm2 for ND = I x 1Ol7crne3 requires that the depletion layer lifetime, TV, must be greater
Diffused junction PbSnTe detectors
329
than 1 x lOma sec. Very little is known about r0 for PbSnTe devices, so the prognosis for higher theoretical RoA and Df remains unclear. Tunneling current leads to very small R,,A products for doping levels greater than 5 x 10” cmm3 (recall that the p-side is degenerate). This is academic, however, since the Burstein shift occurs for ND Z 2 x 10” cm-‘. The solid points in Fig. 8 represent measured R,,A products for Cd-diffused diodes, for which the corresponding N, was determined from C-I’mmeasurements. By comparing theory with experiment_ we conclude that T,, for these devices is 10m8 set or less. The best R,,A produced for ND = 1 x 10” from Fig. 8 is 5Rcm’. According to Fig. 2, this is adequate to permit background-limited operation, and with a quantum efficiency of 50”’ ,0 the Df is about 3.5 x 1O’Ocm Hz”~ W- ‘. A realistic field of view for these detectors is 60”; a diode with RoA = 5 R-cm2 would definitely show improved Df with cold shielding. The solid points in Fig. 2 represent measured Df with corresponding R,,A for good diffused Cd p-n junction detectors. The measured quantum efficiencies are 50% or less, and the long wavelength 50% cutoff is 11 pm. These values of DX are flat with frequency down to 1OOHz. We conclude that the agreement with theory is good; the Cd-diffused devices are well modeled by an abrupt junction whose RoA product at 77°K is controlled by depletion layer generation-recombination at zero bias, with a lifetime of 10-8sec or less. The best DX that can be obtained in principle is set by minority carrier diffusion current, since the depletion layer contribution could, in principle, be minimized. If we assume for the moment that 7. could be made very large, then the RoA (diffusion) from Fig. 8 is about 23R-cm2 for ND = 1 x 10”. If the minority carrier lifetime were set by the radiative process (TV = 7.8 x lo-*see for N, = 1 x 10’7cm-3), then the maximum theoretical R,A product would be 64Rcm’. This is a firm theoretical limit for 0.1 eV PbSnTe p-n junctions. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
SZE. S. M. Physics of Semiconductor DeEices, p. 92. Wiley. New York (1969). BATE.R. T. Texas Instruments Incorporated. unpublished data. MOLL. J. L. Physics o/Semiconductors, p. 123. McGraw-H;Il, New York (1964). SAH. C. T.. R. N. NOYCF..and W. SHOCKJXY.Proc. IRE 45, 1228 (1957). BATE.R. T. Texas Instruments Incorporated unpublished data. WASHWELL E. R. and K. F. Cm. Pmt. Seventh Inc. Conf on Physics of Semiconductors. p. 11 (1969). Szt, S. M. Physics ojSemiconductor Devices. p. 90. Wiley, New York (1969). JOHNSON,M. R. unpublished data. KARLOVSKY. J. Solid-sl. Electron. 10, 1109(1967). HALL. R. N. Proc. IEE lMB, 923 (1960).
DETECTIVITY
F’rcss. Printed
m Great
Bntain.
LIMITS FOR DIFFUSED PbSnTe DETECTORS?
JUNCTION
M. R. JOHNSON,R. A. CHAPMANand J. S. WROBEL Texas
Instruments
Inc.,
Dallas,
Texas
75222,U.S.A.
(Receiwd 5 December 1974) Abstract-The resistance-area product (R,A) of diffused junction PbSnTe photo-volt& detectors under conditions of zero bias voltage is calculated for linearly graded and one-sided abrupt junctions. Equations are developed to demonstrate the overall dependence of ROA on grading constant when minority carrier diffusion current, depletion layer current, and tunneling current are taken into account for the linearly graded junction. Similarly, for the one-sided abrupt junction the dependence of R,,A on the carrier concentration of the lightly doped side is shown for the diffusion. depletion, and tunneling mechanisms. The calculations are c:lrried out for two devices of practical interest: Pb o ,,,Sn,.r,,8Te diffused with Sb with a 5 Itm cutoff at an operating temperature of 170”K, and Pb,.,,Sn,.,,Te diffused with Cd with an 11 pm cutoff at 77°K. The junctions formed by Sb diffusion obey the linearly graded model, whereas the Cd junctions formed in unannealed substrates are one-sided abrupt. Upon comparing measured R,A products with calculated values, we have established approximate values for the lifetimes within the depletion layers for each of these devices. The consequences of these results for the thermal noise-limited detectivity (D*) of these detectors are shown by plotting D* vs RoA. Within this framework, it can be argued that the inherently short lifetimes of PbSnTe play the dominant role in placing the upper limits on achievable D*.
I. INTRODUCTION
Calculations of detectivity (D*) limits for PbSnTe photovoltaic detectors from empirical models based on measured detector parameters are presented in this study. These results provide an estimate on the excess carrier life-time within these devices, about which very little is quantitatively known. A!! other device parameters are known with accuracy sufficient to make valid calculations of &,A product: and D*. Operation at zero bias voltage provides optimum signal-to-noise ratio; the incremental resistance at zero bias determines the thermal noise level and the ultimate D* that can be obtained when noise from other sources is negligible. Two diffused junction detectors are considered. The first half of this paper deals with linearly graded Sb-diffused junctions in Pb o 982Sno.o,,Te; these have a cutoff wavelength of 5 w at an operating temperature of 170°K. We calculate the &,A product for these graded junctions. taking into account minority carrier diffusion, depletion layer generatiorr-recombination, and tunneling. The R,A product is plotted vs the grading constant. Theory and experiment are compared for the value of grading constant appropriate to Sb diffusion (a = 1.9 x 10” cmA4); values are compared also for Al-diffused junctions (a = 2.3 x IO” cm-4). The second half of the paper deals with abrupt Cd-diffused junctions in PbO.,&O.z,Te with 11 w cutoff at 77°K. These junctions are formed by diffusing Cd into as-grown PbSnTe (N, 2 3 x 10’ 9 cme3) and are seen to obey the one-sided abrupt junction mode!. The RoA product is calculated with carrier concentration on the lightly doped (n) side as a parameter. Theory and experiment are compared for net donor concentrations of 1-2 x 10” cme3, which typify Cd-diffused junctions. The RoA products of linearly graded junctions are limited primarily by generationrecombination within the depletion layer. The same is true of the best abrupt junctions for temperatures less than 90°K. The depletion layer lifetime in these devices is short, t Work partly supported by the U.S. Army Night Vision Laboratory. $ Ro is the incremental resistance of the photovoltaic detector at zero volts bias. A is the effective sensitive area. 317
318
M. R. JOHNSON. R. A. CHAPMA& and J. S. WROBEL
about lo-* sec. At the moment very little in PbSnTe. These centers almost surely are dielectric constant would force their energy be insignificant in determining the lifetime. to raise the theoretical limits of detectivity
2. THERMAL
LIMITS
is known about the recombination centers not hydrogenic in nature, because the large levels near a band edge, where they would A dramatic increase in lifetime is required above those calculated here.
FOR
PHOTODIODES
The importance of the RoA product and the quantum efficiency for a detector whose noise has contributions from both the thermal noise of the detector zero bias resistance (R,) and the background photon flux can be seen from the following equation:
(1) Equation (I) follows directly from the defining equation for D* in terms of the detector signal-to-noise ratio with the provision that the Johnson-Nyquist noise of the zero bias resistance and the shot noise of the current generated by the background photon flux are added in quadrature. The detector of equation (1) is assumed to have an ideal spectral response (response cc i) and a peak response as well as an abrupt cutoff at the wavelength I.. Additional noise contributions, such as l/f noise, are not included in equation (1). Equation (1) is valid only at zero bias voltage for a photovoltaic detector; hence the subscript “0” on the detector resistance. The symbols in equation (1) have the following meanings: r] is the quantum efficiency, q is the electronic charge, h is Planck’s constant, c is the speed of light, k is Boltzmann’s constant, R. is the diode incremental resistance at 0 V, A is the detector sensitive area, $B is the photon flux incident on the detector from its surroundings, and T is the Kelvin temperature of the detector. Equation (1) has been plotted in Fig. 1 as a function of RoA and q for a detector with i = 5 ,um at T= 170°K and & = 1.32 x 1016 cm-’ set-’ (300°K background 180” field of view). For small values of RoA product, the thermal noise of R, dominates the detector noise and D* a q(RoA) ‘/’ . As the RoA product is increased, the importance of the background noise term increases. It is clear that for low RoA products, both q and R,,A are crucial factors. On the other hand, for large R,,A products, 0: is determined by 9 and field of view. For the parameters used in Fig. 1, D* will not increase
QL
51 cm
2
Fig. 1. Thermal noise-limited D? vs R,A product for a photodiode with I, = 5 pm at T = 170°K. The field of view is 180’ with a background temperature of 300°K. Four values of the quantum efficiency, q. arc shown. The equation plotted here is valid only at zero bias voltage. An R,,A product greater than 100 is needed for background-limited operation at 170°K. The solid points represent measured Df and R,,A data for Sbdiffused detectors in Pb,, ss2Sn, “,sTe, with I, = 4.9 pm at T = 170’K.
319
Diffused junction PbSnTe detectors
with increasing RO for R,A > 100. The data points represent data for Sb-diffused and Al-diffused Pbo.sazSno.o,sTe, and are discussed below in comparison of theory and experiment. Equation (1) is plotted in Fig. 2 for the case of the 11 ,um detector at 77°K. We see that background-limited operation requires ROA 2 1 R-cm for a quantum efficiency of 50%. The data points represent data for Cd-diffused Pb,.,,Sn0.2 ,Te, and are discussed later in the text. We shall return to Figs. 1 and 2 after calculating theoretical values of ROA for the 5 and 11 pm devices. I
’
“‘I
o*=g g+&)4* A
.
7
I ?
[
0
1
“‘1
’
‘I’_
-t
1
A-llpm
T- 77'K
16'- $-6,52x 10’7cm-2sec-’
1 ”E
0.01
0001
IO
I.0
0.1 2 Qt
Ll
cm
Fig. 2. Thama1 noise-limited 0: vs R,A product for a photodiode with 1< = 11 q at T = 77°K. An R,A product greater than 1R-cm’ is needed for background-limited operation. The solid points represent measured 0: and R,,Adatafor Cd-diffused detectors in Pb,.,,Sn,.,,Te, with i,= llpmat T=77’K. 3. THEORETICAL
PERFORMANCE
Pb,.,s,Sn,.,,sTe
3.1 Model for d@sed
LIMITS
WITH I, = 5~
FOR
Sb-DIFFUSED
AT T= 170°K
Sb junctions
A simple model of the diffused Sb junction permits a calculation of reasonably firm limits on R,A and D*, subject to the choice of carrier life-time. This section presents an empirical model of Sb junctions in Pb0+s2 Sn,.,,sTe at 170°K and proceeds with the calculation of ROA for minority carrier diffusion currents and depletion layer generation-recombination currents. Tunneling current is shown to be unimportant for these graded junctions. To form the model we have used measurements of the junction capacitance-voltage characteristic and the dependence of the zero bias resistance on temperature. Figure 3 is the C-V plot at 170°K of an Sb-diffused junction. The important result is that C ,x V-“3, where I/ is the junction bias voltage, and this behavior implies a linearly graded junction. The grading constant, a, is found from the slope of the curve”’ to be 1.9 x 10ZZcm-4. A dielectric constant of 610 has been assumed, taking the measured value for graded PbTe junctions at 170°K. (‘I The built-in voltage of 0.26 V is found by extrapolating the C-V curve to zero capacitance. We then calculate a zero bias depletion width (‘) of 0*4pm for this junction. Figure 4 is a plot of measured zero bias resistance as a function of temperature. The important result here is that the resistance follows primarily the temperature dependence of n; ’ for temperatures near 170°K. This implies that the resistance is dominated by generation-recombination processes within the depletion region of the junction. In calculating RoA, therefore, we should expect to find that the depletion layer mechanism results in a lower ROA than the minority carrier diffusion mechanism. We calculate here the RoA as a function of the grading constant, a, for a fixed temperature of 170°K and a fixed 3., = 5 pm.
320
M.
R.
JOHNSON. R. A. CHAPMAN
22 2Q-
,
(
,
Sb diffvsed
,
,
and J. S.
,
,
WROLIEL
,
,
W,,Sn,a,,Te
Bias valiag9,
V
Fig. 3. Measured capacitance (C-‘!3) vs bias voltage for Sb-diffused Pb,.,s,Sn,.,,, Te at 170°K. The pn junction was diffused at 700°C to a depth of 17 pm into material with p7rqK 2 2 x 1018cm-3. The grading constant, a, inferred from the linearly graded model is (I = 1.9 x 10” cme4. The depletion width at zero bias is W = 0.4pm. The built-in voltage determined from the voltage axis intercept is @26V.
3.2 ROA for diffusion current The minority carriers in a linearly graded junction diffuse in the presence of a built-in field in the quasi-neutral regions on either side of the depletion region. To be completely rigorous, one must include the electric field in the calculation of the current. Mo1V3) has argued, however, that with certain assumptions, the ideal diode law holds for graded junctions as well, and the saturation current density is given as usual in terms of the (non-degenerate) doping levels on either side of the junction. As the grading constant is made smaller, the space charge dipole layer is less well confined. and for very small I
I
1
I
,
I
o-
0
Sb ditfused
D
X,-5pm
Zero bias resistance vs temperature 5 IO -
o 0
’
.
*
*
.
.
0
c
.*
.
. *
*
* .
.
.
I.
a?
5
6
7
9
9
I IO
I II
12
103/T
Fig. 4. Measured zero bias resistance vs temperature for four Sb-diffused diodes. The straight lines represent depletion layer-limited resistance behavior (l/n,) and diffusion-limited resistance behavior (I/n;). In the vicinity of T= 170°K (103/T= 5.9), most diodes exhibit a depletionlimited resistance behavior. As shown in the text. this can be accounted for by a depletion layer lifetime of IO-* set or greater at T= 170°K.
Diffused junction PbSnTe detectors
321
grading constants the p-n junction effect is lost. In general, the condition
must be satisfied to have well-defined space charge in the graded junction, where LD = J(eok7’/2q2ni) is th e intrinsic Debye length. If we substitute the appropriate values of an Sb-diffused junction in Pb,.sszSno.oisTe at T= 170°K we find that k = 0.8 x 104. Thus, the Sb junction satisfies the linearly graded junction approximation fairly well at 170°K. The minority carrier diffusion length on either side of the junction is expected to beL = J(kTpT/q) z 8pmforT = 170”K,p = 5 x 103cm2V-1,se-‘,andr = 10-8sec. The depletion width for the Sb junction at T = 170°K is W = 0.4pm; the majority carrier density at the edge of the depletion region is approximately aW/2 = 3.8 x 10” cmm3. In the quasi-neutral regions on either side of the depletion region, the net rate of change of ND - N, with position can be less than that given by a = 1.9 x 10” cmm4. For calculation of the minority carrier diffusion current, we need some approximation to the number of minority carriers within one diffusion length of the depletion region. This number lies somewhere between the value at the edge of the depletion region, nf t aW/2, and the value within the bulk, n’ + N,, ND. The linearly graded junction concept would be expected to break down for grading constants large enough that aW/2 2: N,, N, where NA and ND are the majority carrier concentrations in the neutral regions on either side of the junction. There are obviously several complications to calculation of the RoA product due to minority carrier diffusion for the graded junction as a function of the grading constant. We adopt the following simplified procedure: (1) for small grading constants we assume that RoA = kT/qlo still holds, and in calculating the saturation current I0 we use for the minority carrier concentrations the values at the edge of the depletion region, i.e., nf + aW/2. Since for the graded junction WK a- li3, we have RoA a a213, as shown below. (2) For large grading constants we assume the junction may be treated as abrupt. We again take R,,A = kT/ql,, and assume the minority carriers are given by nf + N,, ND. Small grading constant. The RoA product at zero bias is found by differentiating the current voltage characteristic R,,A = 2.
(3)
The saturation current density lo/A is given by
2 = q(!$)“‘[n,k)l”
+ p.(!!!!y”].
(4)
For nP and p. we take the values at the edge of the depletion region,
np,pn=
&.
(5)
This assumption allows the RoA product to be put in terms of the grading constant a. We also assume for PbSnTe that pL,= pp and 7, = zp since the effective masses are nearly equal. The depletion width W for the graded junction at’zero bias is given w
= 0
1 3
l&o Vbi 1’3 qa ’
____
(6)
where E is the dielectric constant, lo is the permittivity of free space, V,i is the built-in voltage (diffusion potential of the junction), and a is the grading constant. Combining
M.
322
R. JOHNSON, R. A. CHAPMAK and J. S. WROBEL
equations (3)-(6), we have
The principal result is that ROA tc a 2’3. There is a slight dependence of l’bi on a which is important for small a, but this has been ignored. V*i has a maximum value equal to E$q = constant. The principal temperature variation of ROA is the term n; ‘, which varies as exp(E,JkT) in which E,, is the extrapolated value of E, at T = 0. Large grading constant. Equation (7) predicts that ROA increases without bound as the junction becomes steeply graded. The RJ product cannot exceed that value given by an abrupt junction, however. The maximum ROA would be obtained for a one-sided abrupt junction. The doping level on the lightly doped side can be made as high as l-2 x 1018cm-3 for Pb,.982Sn,.,,,Te at T = 170°K without causing a Burstein shift of the cutoff wavelength. We shall use equation (4) with the p-side as the lightly doped side. The minority carriers are given by np = niLIN,.* The maximum R,-J product is given from equation (3) after these substitutions through Equation (4), as R,,A = ($)‘:‘($‘:’
2.
In evaluating equation (8) we take NA = 2 x lo’*, rn = IO-* set, pL,= 5 x 103, and ni = 15 x 1014. The result is ROA = 95R -cm2. 3.3 ROA for depletion
layer current
The common practice of differentiating
2k7’) - l] and setting equation applies only et CL,(~)and proceeds for electrons and holes tion rate is integrated recombination current as
the current-voltage equation I = IO [exp(qV/ V= 0 to find the zero bias resistance is not correct. In fact, this for qV> 2kT The correct procedure has been outlined by Sah, along the following lines. Assuming that the quasi-Fermi levels are constant throughout the depletion region, the net recombinaacross the depletion region to give an equation for generationin terms of applied bias. The current density varies with voltage --I
qni W
A = zi;,-i
2 sinh (qV/2kT) f (b). q( V,,i - l’)/kT
(9)
This equation is valid for a single energy level recombination center located anywhere within the forbidden gap. The new symbols are rno, TV,,= electron and hole lifetimes within the depletion layer. The function f(b) is a definite integral depending on the position of the recombination centers within the gap and bias voltage V. For small applied bias, however, f(b) may be taken as independent of V. We also neglect the bias dependence of the depletion width W for small bias (V+ kT/q). The zero bias resistance is then found by differentiating equation (9) and setting V = 0, = ‘/bl \&I 4
4
‘PO
Wf (b)
).
(10)
In applying this equation to PbSnTe we take t,, = rpO for simplicity. We further assume, because of nearly equal electron and hole masses, that the dominant recombination centers are located at E, = li = intrinsic Fermi level. With the additional condition V = 0, the function f(6) = 1. Equation (10) then becomes RoA=----
"b, ‘50 Pi
* The use of non-degenerate taneously.
W
(11)
statistics is questionable since degeneracy and the Burstein shift occur simul-
Diffused junction PbSnTe detectors
323
This result is larger by a factor qV,,lkT than the result normally obtained by differentiating I0 [exp(qV/2k7J - 11. This factor may be 9 1. V,i is the diffusion potential of the graded junction, which we take as E$q for simplicity. Upon substitution of the depletion width from equation (6), the RoA product becomes RoA
=
(!$)iz?!(d-)“‘.
(12)
The principal result here is that ROA x Q“3. theory predicts that ROA varies slowly with grading constant. In evaluating equation (12), the term of greatest uncertainty is zo, the lifetime associated with the Schockley-Read-Hall generationrecombination centers. These centers almost surely are not hydrogenic in nature, because the large dielectric constant would force them near a band edge, where they would be insignificant in determining the lifetime. We can only suggest at this point that these centers lie well within the gap and are not associated with electrically active defects. 3.4 RoA for tunneling current There is, of course, no net tunneling current in a diode at zero bias; but there is an RoA product for tunneling for small voltage swings about zero. In a graded junction the tunneling probability is usually negligible. This is especially true for PbSnTe, because the high dielectric constant, in effect, reduces the electric field in the forbidden region. Also, the Fermi level at the edge of the depletion region must lie above the band edge both the n- and p-sides of the junction in order to have appreciable tunneling probability. This occurs for Pb o 9s$n0.0,8Te at T = 170°K for a carrier concentration of about 1 x 101”cm-3. Thus, aW/2 2 1 x lOi* is the requirement for overlap. Using equation (6) and letting V*i= E$q as before, the condition on a becomes a 2 1O23cm-4. Even for a = 1024, however, the tunneling length (W) is 01 cam, and the tunneling probability is negligible. For graded junctions in PbSnTe, then, the RoA products expected from tunneling are so high that they can be ignored. 3.5 Numerical results of RoA calculations for Sb-diffused 5 pm detectors The foregoing calculations have been summarized in Fig. 5. The R,A products calculated from equations (7), (8) and (12) have been plotted vs grading constant. We have
Pb,,,Sn,,,,Te
@m,
170°K
1
Sb
IO”
1
1
IO”
IO= 0,
cm
IO’*
-4
Fig. 5. Calculated zero bias RoA product at T= 170°K for linearly graded junction vs grading constant. Tunneling current is shown to be insignificant for graded junctions in PbSnTe. All details of the calculations and the critical assumptions are explained in the text. The calculations predict R,,A = 2 R-cm’ for the case of Sbdiffision (a = 1.9 x 1Oz2cm-‘), and @8 R-cm’ for Al-diffusion (D = 2.3 x IO**cme4). The solid points represent measured RoA products for Sbdiffused diodes and Aldiffused diodes.
M. R. JOHNSON,R. A. CHAPMANand J. S. WROLIEL
324
used the value of ni for E, = O-25eV calculated by Bate.“) Likewise, the dielectric constant at T = 170°K is taken from measurements by Bate”’ of capacitance versus temperature of graded PbTe junctions. For I(. and pr, we have taken the value measured at 170°K for p-type material. We have assumed a value for minority carrier lifetime of lo- a sec. This is the value calculated for radiative lifetime in degenerate PbTe by Washwell and Cuff.@’ Both the p- and n-sides of the Sb junction are nearly degenerate at T = 170°K. The extrinsic radiative lifetime is an upper limit to the actual lifetime since the likelihood of any trapping effects is small. The lifetime r,, within the depletion region is more difficult to estimate than the lifetime in the neutral regions. There is not sufficient experimental data to calculate ?0 a priori, but we have taken r. = lo- * sec. With this choice of parameters we find in Fig. 5 that the RoA product due to depletion layer generation-recombination dominates the effective RoA product for grading constants greater than 10” cme4. The effective RoA is the parallel sum of RoA (diffusion) and R,A (depletion). This theoretical result is in agreement with experiment, namely, that the detector resistance at T = 170°K is depletion layer-limited. Therefore, the choice of r. = lo- * set is reasonable. Notice that RoA (effective) rises slowly with grading constant. Therefore, dramatic improvement in RoA cannot be expected unless the junctions are made abrupt. The limit for RoA of 95Rcm2 for an abrupt junction with h.‘A= 2 x lo’* cmm3 is shown as the dashed line at the top of the curve. This is the maximum possible R,A for a junction in PbSnTe with ,$ = 5 pm at T = 170°K. Referring to Fig. 1, we see that the maximum theoretical D* for RoA = 95 R-cm’ and 9 = 0.5 is 1 x 10” crnH~‘/~ W- ’ at T = 170°K and i, = 5 pm. As noted earlier, the grading constant for Sb diffusion is a = 1.9 x lO22 cmd4. Figure 5 predicts RoA = 2R-cm2 for Sb diffusion. Several data points representing measured RoA products for Sb-diffused diodes are shown in Fig. 5. These RoA products represent the best diodes fabricated in our laboratory with Sb diffusion. The highest RoA product measured at 170°K was 4.5 R-cm’. Returning to Fig. 1, we see that the calculated RoA product of 2R-cm2 is sufficient to provide D* = 2.5 x 10” cm Hz’!~ W- ’ for a quantum efficiency of 50%. The data points of Fig. 1 represent measured DX and RoA data for the best Sb-diffused diodes, with & = 4.9 pm and T = 170°K. We conclude that the agreement with theory is good; the Sb-diffused diodes are well modeled by a linearly graded junction whose RoA product is controlled by depletion layer generation-recombination with a lifetime of lo-* set or longer at a temperature of 170°K. A lifetime of 2.3 x lo-* set is implied in the case of the highest RoA found experimentally (4.5 R-cm’). Also shown in Fig. 5 are data points representing measured R,A products for Aldiffused junctions. Al junctions are also linearly graded but with a somewhat smaller grading constant of 2.3 x lo2 I cm- 4. The measured RoA products for Al diffusion are lower than those for Sb, being in the range of 0.5l.0R-cm2 for the best diodes. Theory predicts a lower RoA for Al due to the smaller grading constant. The highest possible RoA product for a PbSnTe p-n junction detector with a 5 pm cutoff at 170°K is 95 R-cm’. The corresponding DX limit is 9.5 x 10” cm Hz”’ W- ’ if the quantum efficiency is 50%. From Fig. 1 it is apparent that some increase in this value of 0; is possible with cold shielding. 4. THEORETICAL Cd-DIFFUSED
. =
PERFORMANCE LIMITS Pb,, ,gSn0.2,Te WITH 1 I pm AT T = 77°K
FOR
Consider now the performanie limits of detectors with a cutoff wavelength of 11 pm at T = 77°K. Although these devices have been fabricated by several different methods, only Cd-diffusion to form pn junctions is considered here. Our best results have been obtained by this method, and these devices closely approach the ultimate performance limits established by the intrinsic materials properties of 01 eV PbSnTe. 4.1 Model for difised
Cd junctions
Cadmium is a compensating ‘neutral’ impurity in PbSnTe, with the very important property that n-type layers with net carrier concentrations of 1 x 10” cme3 can be
Diffused junction PbSnTe detectors IO
I
I
I
Cd diffused
I
I
325
I
I
Wo,sSn,,,,Te
9-
r IA
6-
~-~” “N- l.7.7; _
7-
+G¶
p$
.
6-
/
m -0-
/ S-
Nv
/
4
./
3-
/
2-
0.1
I/’
0
-01
-02
-03
-04
Bias voltage.
-05
-0.6 V
Fig. 6. Measured capacitance (C-‘I’) vs bias voltage for Cd-diffused Pbc.,,S~.r,Te at 77°K. The junction obeys the one-sided abrupt model, with a net donor concentration on the ditfused side of 8.9 x lOI cmm3. The junction was diffused into material with ~77~~= 3 x IOr cmW3. The built-in voltage determined from the voltage axis intercept is 009 V.
formed by diffusing Cd into degenerate p-type material. The long wavelength cutoff of photoresponse on the lightly doped n-side of a diffused Cd p-n junction in Pbo.,9Sn0.zITe does not suffer the Burstein-Moss shift. The thickness of the n-layer is chosen to provide optimum quantum efficiency. Although tunneling current dominates the reverse diode characteristic of these devices, the zero bias resistance can be quite high; values of the RoA product at 77°K as high as 5R-cm2 have been measured for monolithic detector arrays. Another important feature of diffused Cd junctions is that they are abrupt (one-sided). The capacitance-voltage characteristic of a diffused Cd junction is shown in Fig. 6. I
I
I
Cd diffused
I
I
I
I
1
A,- llpm
Zen, bias resistance temperature
VI
103/T
Fig. 7. Measured zero bias resistance vs temperature for three CddiRirsed diodes. of T= 77°K (Id/r = 13), diodes exhibit a depletion-limited resistance behavior. can be accounted for by a depletion layer lifetime of lo- a set or less at T = 77°K. to ditkion-limited resistance behavior occurs at T= 90°K for the better
In the vicinity These results The transition diodes.
M.
326
R.
JOHNSON. R. A. CHAPMAN and J. S. WHOBEL
The capacitance versus bias behavior is a straight line when plotted as l/c2 vs c this indicates an abrupt junction. (‘) If one assumes a value of 500 for the dielectric constant, then one finds from the slope of the C-l/ curve a net carrier concentration of 8.9 x 1016 cm- 3 in this case. This carrier concentration is assumed to apply to the n-side of the junction, since the acceptor concentration before Cd diffusion is 3 x lOi cm- ‘. Carrier concentrations of I x 10” cmm3n-type have been measured on thin Hall samples diffused with Cd. The R,,A product of diffused Cd junctions at 77°K is dominated by depletion layer current, although minority carrier diffusion current dominates the R,A product at higher temperatures. This can be seen in Fig. 7, which is a plot of zero bias resistance vs temperature for three diodes. The transition from diffusion current to depletion layer current occurs at T= 90°K for the better diodes. In choosing a model for Cd-diffused Pb0.7&,.,,Te, we expect the abrupt junction equations to apply and we expect depletion layer current to dominate the resistance. 4.2 ROA for difision current The minority carriers arising on the lightly doped (n) side of the p-n contribute the dominant part of the diffusion current (equation (4)). The minority carriers are similar in magnitude on both sides of the junction; minority carrier lifetimes are thought to be similar on both sides of the saturation current for the one-sided abrupt junction is
junction will mobilities for likewise, the junction. The
(13)
When .the n-side is non-degenerate, pn = $/IV, where ND is the net donor concentration. Making these substitutions in equation (3), we arrive at R,A
=
(7)“’($‘;‘$
(14)
for minority carrier difision of the one-sided abrupt junction. The principal temperature variation of equation (14) is the term ni’ . Equation (14) fails when N, becomes so large that pnnn # nf ; for 01 eV PbSnTe at 77°K. this occurs for ND Z 2 x 1017
cm-3.u33
The RoA product can be increased by raising the carrier concentration on the lightly doped side. For Pb,,.7PSn,.2,Te, however, a practical limit of ND = 2 x 10” is set by the Burstein shift. At this point, then, the intrinsic properties of the material itself (r pi,, ni) establish the RoA limit. Cd-diffused PbSnTe, for which ND = 1 x 10” cm -S , therefore represents the optimum design choice for p-n junction detectors in 01 eV PbSnTe. 4.3 ROA for depletion layer current The discussion preceding equation (11) applies here also, and we shall make the same assumptions for the @l eV case. The RoA product is given by equation (11) in which we must put (15) W = &%, V,,IqN,)
for the one-sided abrupt junction. Equation (11) then becomes R,A
=
7O
I’,‘/’ NC2
tli(2EEOq)1’2 ’
(16)
Comparing equations (14) and (16), we can see that large values of ND are required for large RoA products in both cases.
Diffused junction PbSnTe detectors
327
In evaluating equation (16) we take Vb,= Edq, which is a constant, independent of doping level. The variation of Vr,,with ND is slight over the range of ND of interest. For large values of N, we again assume that the maximum value of the built-in voltage is E$q. 4.4 ROA for tunneling current Computing the contribution of tunneling current to the diode resistance can be a formidable task, even for the case of zero zpplied bias. Taking into account the bias dependence of the tunneling probability is essential for the case of small applied bias (c/6 kT/q). We shall follow here the method of Karlovsky.“’ The tunnel current-voltage relation for small applied bias (Ve k7’/q) is
I =
cz g 6% + E, - 4’)‘.
(17)
C is a constant given by (18) and m* is the average effective mass for electrons and holes me
_
2m:mt
(19)
rn:+ mh*'
The electron and hole masses of equation (19) are the density of states masses, which we take as O-0728m. for electrons and O-082m0 for holes from calculations by Bate.(s) These values lead to m* = 0.0771 m,, in equation (19) for O-1eV PbSnTe. The terms E, and E, in equation (17) are the positions of the Fermi level on either side of the junction. For the diffused junction at zero bias, the dominant contribution is from the p-side; to a very good approximation we may take E, + E, independent of N,. The term Z in equation (17) is the tunneling probability, for which the shape of the potential barrier must be chosen. We differentiate equation (17) with respect to V and find the very simple result at v= 0: (R 0 A)-’
=
czq 6%+ E,)‘. kT
The main variation of RoA with doping level N,, is through the tunneling probability. The potential barrier will be assumed to be parabolic, in which case the probability is (21)
The quantity F is the electric field throughout the depletion region, which approximate by a constant. The maximum value of F is ZV,JW(V = 0) for degenerate material. Following Karlovsky, we take for zero bias the average F m.x ad 112 Fmx, i.e. F = 3/2 Vbi/W(V = 0). W( V= 0) is just the depletion zero bias. In the spirit of earlier approximations we again take V,, = E$q. The tunneling probability at zero bias can now be written as n(eeo m*)‘j2 E, 3qh N;”
when w(V = 0) is substituted
from equation
1
(15). The RoA product
we must strongly between width at
(22) may be written
M. R.
328
JOHKSON, R. A. CHAPMAN and
J. S.
WROBEL
finally as R
(23)
0
from equation (20) after substitutions are made from equations (18) and (22). We have computed R,A for several values of N,. 4.5 Numerical results of RoA calculations for Cd-difised
11 pm detectors
The RoA products due to diffusion current, depletion layer current and tunneling current have been plotted vs ND in Fig. 8 from equations (14), (16) and (23), respectively.
_ Pt+,,Sn,,,Te
(1Ipm.
77-K)
Fig. 8. Calculated zero bias RoA product at T= 77°K for one-sided abrupt junction vs doping level on the lightly-doped side. For the parameters chosen to typify Cd-diffused Pb,.,,Sno.l,Te, the effective R,,A product is limited by depletion layer current for the maximum doping permitted without a Burstein shift of the photo-response (ND 5 2 x 10” cmd3). The solid points represent measured R,A products for Cd-diffused diodes corresponding to values of ND determined from C--V measurements.
The minority carrier lifetime has been taken as I x 10-s set, and is further assumed to be independent of doping level N,. An upper limit on 7p would be the extrinsic radiative lifetime. The radiative lifetime for 10” cme3 material is 7.8 x lo- * set, calculated by Hall’s method. (‘O) The value of 2 x lo4 for mobility lies within the range of mobilities measured for 1017 cm- 3 material. The dielectric constant has been taken as 500, in agreement with our earlier assumptions. In evaluating the depletion layer contribution we must again make a choice for the lifetime z. within the depletion region of the junction. The discussion following equation (12) applies to the 0.1 eV case also. The lifetime ~~ is best found empirically by measuring R. for a junction which is clearly depletion layer current-dominated. We shall again choose a value of 1 x lo- ’ set for to. The assumptions leading to equation (23) have been discussed above. The value of m* is taken as O-0771mo. The value of E,,, the position of the Fermi level on the p-side, is calculated for a p-type doping level of 3 x lOI cmm3 and is 0165 eV. With this choice of parameters we find in Fig. 8 that the depletion layer establishes the effective detector R,A product for values of ND of practical interest. This result agrees with experiment, namely, that the diode resistance is controlled by the depletion layer. For our choice of parameters, the diffusion contribution is important only for N, < 1016 cmw3. To increase the RoA product above the theoretical value of 5R-cm2 for ND = I x 1Ol7crne3 requires that the depletion layer lifetime, TV, must be greater
Diffused junction PbSnTe detectors
329
than 1 x lOma sec. Very little is known about r0 for PbSnTe devices, so the prognosis for higher theoretical RoA and Df remains unclear. Tunneling current leads to very small R,,A products for doping levels greater than 5 x 10” cmm3 (recall that the p-side is degenerate). This is academic, however, since the Burstein shift occurs for ND Z 2 x 10” cm-‘. The solid points in Fig. 8 represent measured R,,A products for Cd-diffused diodes, for which the corresponding N, was determined from C-I’mmeasurements. By comparing theory with experiment_ we conclude that T,, for these devices is 10m8 set or less. The best R,,A produced for ND = 1 x 10” from Fig. 8 is 5Rcm’. According to Fig. 2, this is adequate to permit background-limited operation, and with a quantum efficiency of 50”’ ,0 the Df is about 3.5 x 1O’Ocm Hz”~ W- ‘. A realistic field of view for these detectors is 60”; a diode with RoA = 5 R-cm2 would definitely show improved Df with cold shielding. The solid points in Fig. 2 represent measured Df with corresponding R,,A for good diffused Cd p-n junction detectors. The measured quantum efficiencies are 50% or less, and the long wavelength 50% cutoff is 11 pm. These values of DX are flat with frequency down to 1OOHz. We conclude that the agreement with theory is good; the Cd-diffused devices are well modeled by an abrupt junction whose RoA product at 77°K is controlled by depletion layer generation-recombination at zero bias, with a lifetime of 10-8sec or less. The best DX that can be obtained in principle is set by minority carrier diffusion current, since the depletion layer contribution could, in principle, be minimized. If we assume for the moment that 7. could be made very large, then the RoA (diffusion) from Fig. 8 is about 23R-cm2 for ND = 1 x 10”. If the minority carrier lifetime were set by the radiative process (TV = 7.8 x lo-*see for N, = 1 x 10’7cm-3), then the maximum theoretical R,A product would be 64Rcm’. This is a firm theoretical limit for 0.1 eV PbSnTe p-n junctions. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
SZE. S. M. Physics of Semiconductor DeEices, p. 92. Wiley. New York (1969). BATE.R. T. Texas Instruments Incorporated. unpublished data. MOLL. J. L. Physics o/Semiconductors, p. 123. McGraw-H;Il, New York (1964). SAH. C. T.. R. N. NOYCF..and W. SHOCKJXY.Proc. IRE 45, 1228 (1957). BATE.R. T. Texas Instruments Incorporated unpublished data. WASHWELL E. R. and K. F. Cm. Pmt. Seventh Inc. Conf on Physics of Semiconductors. p. 11 (1969). Szt, S. M. Physics ojSemiconductor Devices. p. 90. Wiley, New York (1969). JOHNSON,M. R. unpublished data. KARLOVSKY. J. Solid-sl. Electron. 10, 1109(1967). HALL. R. N. Proc. IEE lMB, 923 (1960).