The distribution of gains in uniformly multiplying avalanche photodiodes: Experimental

IEEE TIRANSACTIONS ON ELECTRON DEVICES, VOL. ED-19, NO. 6, JUNE 1972 713

[51 R., D. Baertsch, “Noise and ionization rate measurements in silicon photodiodes,” IEEE Trans. Electron Devices (Corresp.), vol. ED-13, p. 987, Dec. 1966.

photodiodes,” J . Appl . Phys., vol. 38, pp. 4267-4274, Oct. 1967. , “Noise and multiplication measurements in InSb avalanche

ticle detector with internal multiplication,” IEEEE Trans. R. H. Haitz and F. M. Smits, ‘<Noise analysis for a silicon par-

Nucl. Sci . , vol. NS-13, pp. 198-207, June 1966. J. Conradi, “The distribution of gains in uniformly multiplying avalanche photodiodes: Experimental,” this issue, pp. 713-718. C. A. Lee, R. 4. Logan, R. L. Batdorf, J. J. Kleimack, and VI. Wiegmann, Ionization rates of holes and electrons in silicon,” Phys. Rev., vol. 134, pp. A761-A773, May 1964.

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P. P. Webb and R. J . McIn tye , “A silicon avalanche photodiode for 1.06 pm radiation, presented a t the Solid State Sensors Symp., Minneapolis, Minn., June 18-19, 1970.

Bull. Amer . Phys. SOC., Ser. 11, vol. 15, p. 813, June 1970. , “Single photon detection with avalanche photodiodes,”

G. A. Morton, H. M. Smith, Jr., and H, R. Krall, “The per- formance of high-gain first-dynode photomultipliers, IEEE Trans. Nucl. Sci., vol. NS-16, pp. 92-95, Feb. 1269. F. S. Goulding, J. T. Walton, and R. H. ,phi, Recent results on the optoelectronic feedback amplifier, IEEE Trans. Nucl. Sci., vol. NS-17, pp. 218-225, Feb. 1270. H. E. Kern and J. M. McKenzie, Noise studies of ceramic encapsulated junction field effect transistors (JFETs),” IEEE Trans. Nucl. Sci., vol. NS-17, pp. 425-432, June 1970.

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The Distribution of Gains in Uniformlv Multiplying Avalanche Photodiodes: Experimental

JAN CONRAD1

Ai~~traci-Experimental me murements of th le gain distribution and noise spectral density of silicon avalanche photodiodes are presented and compared with McIntyre’s theories [7], 18). Excellent agreement is obtained using ker f , the effective ratio of the hole and electron ionization coefficients, as the only adjustable parameter.

s I. INTRODUCTION I N C E T H E first reported measurements of en- hanced signal-to-noise rat’io in the photodetection of silicon avalanche diodes [ l ] , similar work has

been reported for a variety of materials [ 2 ] - [ 5 ] . Im- proved understanding of the limitations imposed on the attainable signal-to-noise ratio by the nature of the fluctuations in the avalanche process was achieved by Tager [6] and McIntyre [7] in their theories of the multiplication noise in avalanche diodes. A more com- plete theoretical understanding of these fluctuations was reached with McIntyre’s theory on the gain klistribu- tions in uniformly multiplying avalanche diodes [8].

Several attempts at comparing experimental results with Tager’s and R‘IcIntyre’s noise theories have been made with some success [3]-[j] , [SI-[12], but most have suffered from one or more complicating effects due to the presence of micropIasmas, nonuniform gain over the device area, simultaneous injection of both holes and electrons into the multiplying region and the possi- bility of variable internal quantum efficiency as the bias

IManuscript received September 2, 1971; revised November 12, 1972. This work was supported in part by the Defence Research Board of Canada, DRB Grant 5501-55, DIR Project E-114.

’The author is with RCA Ltd., Research Laboratories, Ste, Anne- de-Bellevue, P. Q., Canada,

on the device is changed. Fit of experimental results to theory has also suffered from lack of agreement with published values of the ionization rates of electrons and holes.

The measurements reported in this paper consist of noise spectral density and gain distribution measurements performed on avalanche diodes in which the above complications are absent. The results are compared to McIntyre’s noise and gain-distribution theories and excellent agreement is obtained using “k”, the effective ratio of the hole and electron ionization coefficients, as the only adjustable parameter.

11. DIODE STRUCTURE The diodes used in these experiments have the so-

called “reach-through” structure [13]-[15] (see Fig. l) and were fabricated in these laboratories. Under the application of reverse bias the depletion-region “reaches through” to the high resistivity T region a t a voltage such that the average gain is still low. Application of further bias extends the depletion region to the back p+ contact, and establishes a wide drift region, which is the photosensitive region. This ensures that all of the photogenerated carriers reaching the high-field region are of the same type, provided the test illumination is chosen such that the absorption length is short compared to the width of the drift region. Furthermore, by having a fairly wide n-type shoulder on the device, a nonmultiplying p-i-n diode is formed which fully de- pletes a t a relatively low bias, and which has substantially the same quantum efficiency as the avalanche

n OXjDE 'OW t------------ n---------------l

DISTPNCE

Fig. 1. Sketeches of reach-through avalanche-diode structure, impurity-concentration profile, and electric-field distribution.

region. Thus this structure has some distinct advar- tages over other structures from the point of view c i studying the avalanche processes in the diode: the quantum efficiency is substantially independent of bias and hence of avalanche gain, and also independent (cf wavelength, a t least over a substantial range of wave- lengths; the avalanche gain can be measured accuratel:/ and the noise in the injected and multiplied photocurrents measured in the same diode with confidencm: that in all cases the internal quantum efficiency is con., stant and that the primary or injected photocurrent entering the avalanche region is composed entirely of electrons.

The diodes used have a 200-pm wide depletion region; the pf contact diffusion is -1 pm deep.

111. ~ I E A S U R E M E N T OF AVERAGE GAIN The average gain of the diodes, taken as the ratio o i

the multiplied to injected photocurrents, was obtainetl by scanning a small light spot (-135-pm diameter:l across the device and locating the region of maximunl gain. As can be seen from Fig. 2 there is, a t biases neal. 90 V , a plateau coincident with the nonmultiplying par': of the diode where the gain is substantially constan:. over a distance more than five times the diameter of thc: light spot. At 90 V, the depletion region under this, plateau extends to within much less than a minorit4 carrier diffusion 1engt.h of the p+ contact, and i t is saff to assume that the internal quantum efficiency is thc same as that which prevails when the diode is operating in the avalanche mode and that under these conditions the plateau corresponds to unity gain. The light Spoi was then positioned on the region of maximum gair and the bias adjusted to give the desired gain.

i'

POSITION (mm)

Fig;. 2. Gain profile for the diode used in gain-distribution and noise measurements,

From Fig. 2 i t is evident that at the unity-gain plateau the photocurrent becomes somewhat position dependent at higher biases. This dependence can be accounted for by assuming that a small fraction of the light from the microscope is scattered into the avalanche region. By working back from the increase in photocurrent with voltage at the center of the unity gain- plateau one (can estimate that this effect is caused by creating approximately 0.5 percent of the photoelectrons in the high-gain region. This introduces negligible error when measuring gain, provided the injected current is measured a t low bias, but i t has a strong effect on the measurement of noise in the primary photocurrent. This effect is discussed further in Section V.

The diode was illuminated using a microscope with reflecting optics that was modified to allow viewing of the illuminated region through the microscope stage. With this arrangement the image of the aperture at the front of the system can be focused onto the diode using white light after which a narrow pass filter can be in- serted or the white light replaced by a GaAs LED, which was done for the gain-distribution measurements.

IV. GAIN DISTRIBUTION

The quantity of interest in these measurements is the probability PE,m that an avalanche event initiated by n primary electrons (holes) will result in an output pulse consisting of m electrons (holes). The apparatus used for these experiments is shown i n Fig. 3, and is basically the same as that used i n nuclear spectroscopy, with the silicon avalanche photodiode substituted for the semi-

CONRADI: GAIN DISTRIBUTION OF PHOTODIODES 715

2-i AVALANCHE DIODE

Fig. 3. Schematic of apparatus used for measuring gain distributions,

conductor diode detector. The charge-sensitive amplifier-:linear amplifier combination produces a shaped output pulse the height of which is linearly propor- tional to the number of electrons coming in to the charge-sensitive amplifier in a time determined by integration and differentiation time constants employed in the pulse shaping stages of the linear amplifier. The output from the linear amplifier goes to a multichannel pulse height analyzer via a biased amplifier. The overall linearity of the system is better than 1 percent. The entire system was calibrated using a germanium lithium drifted y-ray detector and standard y-ray sources ranging in energy from 60 keV to 1.33 MeV. With the discriminator set to reduce spurious pulses due to amplifier noise to an acceptable level, the mini- mum detectable pulse height was approximately 1400 electrons; thus single-electron-initiated avalanche events with gains of 1400 or more could be detected. When measuring the gain distribution in the case where the resulting pulses were to be considered as having origi- nated from single photons, the light intensity was re- duced to the lowest practical level to ensure that the mean time between the generation of electron-hole pairs was sufficiently longer than the 1 p s integration/differentiation times of the amplifier so that the probability of two primary pairs being generated within 1 p s was negligible.

Fclr this same reason the measurements were performed on a diode cooled to near 77 K to reduce the leakage current in the device to negligible proportions; from the background count rate it was estimated that the "dark" current in the avalanche region was of the order of A (before gain). I t was not clear whether this 'was thermally generated leakage current, or photogenerated due to minute light leaks in the optical system.

I n the case where the input pulse consists of n primary electrons with n > 1, a pulsed GaAs LED was used as the light source with a pulse width of 250 ns. The average num'ber of photoelectrons in each pulse @ was determined from

I . . - m a - q R j (1)

where l i n j is the dc photocurrent measured on the unity-gain plateau of the diode and f is the pulse repeti- tion frequency.

NOW, since the photoelectrons in each pulse are dis- tributed in a Poissonian fashion and the theory of gain distributions assumes each pulse is initiated by exactly tz photoelectrons, a modification to the theory must be made. From [8, eq. (16a)l we get

NUMBER OF ELECTRONS IN THE OUTPUT PULSE (m)

Fig. 4. Calculated gain distributions with (solid lines) and without (dashed lines) corrections for the Poisson distribution of photoelectrons.

(&+I) P7k.m =

where n is the number of photoelectrons in a pulse; m = n + r is the number of electrons in the output pulse; M is the average gain; and k is the ratio of ionization coefficients of holes to electrons.

If @ is the average number of phiotoelectrons in a pulse then p(n) , the probability of obtaining n photoelectrons in a pulse is given by

and P,(m), the probability of obtaining an output pulse of height rn when the average number of photoelectrons is @, is given by

M I

Pn(m) = P(n)Pn,m. (4) n= 1

Unfortunately no simple and reasonably accurate approximations to this expression have been found and it has been necessary to evaluate it numerically. The effect of the correction due to the Poisson distribution of the photoelectrons on the calculated gain distribution is shown in Fig. 4. I t is these corrected gain dis-

71 6 WEE TRAXSACTIONS OK ELECTROY DEVICES, JU'NE 1972

tributions against which the measured gain distribl.l.. tions are compared.

The results of measurements performed with sing:e photoelectrons are shown in Fig. 5. The solid lines a.re theoretical curves evaluated using (4) and an effectiT,,e k of 0.028. This K gave the best fit of experiment l o theory; a 5 percent variation in K causes a n0ticeab.e change in the fit. I t should perhaps be emphasized th:r-t both the ordinate and abscissa of the experiment,il points and theoretical curves are absolute. In view I ; f

the wide range of values of M , m, and PI,,,, over which the experiment was performed the, agreement with theory is excellent.

Results of measurements performed with a pulsed GaAs LED in which the average number of photoelectrons per pulse ranged from 5 to 50 are shown i n Figs. 6-8, and compared with theory, using k =0.0%(3. Again, the agreement with theory is quite good.

17. NOISE SPECTRAL DENSITY The noise spectral density of silicon avalanche diod1:s

was measured at 7 7 K with the same diode as used i n the gain-distribution measurements. The apparatus i!j

the same as that used in the gain-distribution measur(:- ments except that the biased amplifier and pulse height analyzer are replaced by a true rms voltmeter. The bandwidth in which the noise power was measured (-350 KHz) is determined by the time constants i n the linear amplifier.

For a comparison of experiment with McIntyre's noise theory [ 7 ] one needs to obtain a measure of tf e noise current in the primary and multiplied photocurrents.

Now, the noise spectral density in the multiplied current in the case where only electrons enter tf!e avalanche region is

where 2 pI inj is the shot noise spectral density of t h e injected or primary photocurrent. As mentioned in Se:- tion 111, because of scattering in the microscope, ai)- proximately 0.5 percent of the photoelectrons are ge I-

erated in the high-gain region when the light is incidellt on the center of the unity-gain plateau. The measurement of noise in the primary photocurrent was done (it 90 V where the average gain of the diode is appro3i- mately 4.5; when 0.5 percent of the photoelectrons are generated in a gain 4.5 region with k =0.028, then from (5) we find that the measured noise spectral densi-::v will increase by approximately 18 percent. A correctiom for this effect has been made tQ the results shown i n Fig. 9, where 1 / M 2 times the measured ratio of noise power in the multiplied current to that in the i '1-

jected current is plotted as a function of gain. T'l le solid lines are plots of the excess noise factor or ,U[I - ( I - k ) ( ~ f - - 1 ) ~ / . ! ~ P ] versus ;IF for d i f fe re~~t

* o - l l L ' 1 ' " 1 ' 1 ' ' 0

103 10 4 10 5 106 1

NUMBER OF ELECTRONS IN OUTPUT PULSE (m)

1

Fig. 5 . Experimental and theoretical gain distributions B = 1.

NUMBER OF ELECTRONS IN OUTPUT PULSE (m)

Fig. 6. Experimental and theoretical gain distributions f i = 5 .

CONRADI: GAIN DISTRIBUTION OF PHOTODIODES

5

NUMBER O F ELECTRONS IN OUTPUT PULSE lml

Fig. 7. Experimental and theoretical gain distributions =20.

- THEORY

10 1

I

J O 5

NUMBER O F ELECTRONS IN OUTPUT P U L S E (mi

Fig. 8. Experimental and theoretical gain distributions %=SO.

1.0 I I I 4 I l l 1

10 100 1000

G A I N IKi

Fig. 9. Excess noise factor versus gain.

values of k . As is apparent, the experimental poin1.s follow the theoretical curves for k =0.028 quite well, and the k value so found is in good agreement with th2.t found from the gain-distribution measurements.

VI. CONCLUSIONS The data reported in this paper provide exceller t,

experimental evidence for the validity of McIntyre s theories of multiplication noise and gain distributiors in avalanche photodiodes [ 7 ] , [SI. The good agreemert between theory and experiment with a single value (d”

the effective ratio of the hole and electron coefficienl:; over the wide range of avalanche gains used gives weig2: 1: to the validity of the constant k approximation. The actual value of keff is in good agreement with the ionization rate data of Lee et al. [16] after their data ‘ s transformed to 77 K using the approach of Crowell and Sze [I 7 1 to the analysis of Baraff [18], provided keff c3

chosen equal t o t ha t a t M = 00 and is calculated at(:-

cording to [S] for the particular electric-field distribl1,- tion in the avalanche region of the diode used.

ACKNOWLEDGMENT The author wishes to thank P. P. Webb for providirg

the diodes used in these experiments, and R. J . M I : - Intyre for many useful discussions.

REFERENCES

H. Melchior and LV, ’r. Lynch, “Signal and noise response of

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R. D. Baertsch, “Noise and multiplication measurements in InSb avalanche photodiodes,” J . Appl. Phys., vol. 38, pp.

W. T. Lindley, R. J. Phelan, Jr., C. M. Wolfe, a$ A. G. Foyt, “GaAs Schottky barrier avalanche photodiodes, Appl. Phys. Lett., vol. 14, pp. 197-199, Mar. 15, 1969. A. S. Tager, “Current fluctuations in a semiconductor (dielec- tric) under the conditions of impact ionization and avalanche breakdown,’] Sov. Phys.-Solid State, vol. 6, pp. 1919-1925, Feb. 1965.

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BUZZ. Amer. Phys. Soc., Ser. 11, vol. 15, p. 813, June 1970. , “Single photon detection with avalanche photodiodes,”

C. A. Lee, R. 4,. Logan, R. L. Batdorf, J. J . Kleimack, and W. Weigmann, Ionization rates of holes and electrons in silicon,” Phys. Rev., vol. 134, pp. A761-A773, May 4, 1964.

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[l] K. M. Johnson, “High-speed pktodiode signal enhancement , L E [I71 C. R. Ciowell and S. XI. Sze, “Temperature -dependence of avalanche breakdown voltage, IEEE Trans. Electron Devicts, avalanche multiplication in semiconductors,” Appl. Phys. Lett., vol. ED-12, pp. 55-63, Feb. 1965. vol. 9, 242-244, Sept. 15, 1966.

[2] G. Lucovsky and R. B. Emmons, “Avalanche multiplication ,n [18] G. A. Baraff, “Distribution functions and ionization rates for InAs photodiodes,’’ Proc. IEEE (Corresp.), vol. 53, p. 180, Fen. hot electrons in semiconductors,” Phys. Rev., vol. 128, pp. 2507- 1965. 2517, Dec. 15, 1962.

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The distribution of gains in uniformly multiplying avalanche photodiodes: Experimental