770 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-IS, NO. 10, OCTOBER 1968

Noise Due to Generation and Recombination of Carriers in p-TZ Junction Transition Regions

Abstract-The theory for the current noise associated with carrier generation and recombination in a p-n junction transition region is presented. The noise model is derived directly from the SRH defect center model. In a reverse biased junction the noise from this mechanism varies from two-thirds to full shot noise, depending upon frequency, and in a forward biased junction the noise ranges from three-fourths to full shot noise, depending upon injection level. These results approximately agree with published experimental results, but the agreement is not conclusive, especially in forward biased junctions. The theory shows that the noise associ- ated with transition region generation-recombination current is not particularly significant in bipolar transistors.

T I. INTRODUCTION

HE STEADY-STATE dc current associated with generation-recombination centers located within the transition region of a p-n junction was de-

scribed in 1957 by Sah et al. [l]. This current flow mechanism is especially important in silicon semicon- ductor devices and has become a basic part of common semiconductor device theory. Current flowing by dif- fusion across a p-n junction gives a noise level equal to full shot noise. I t has been thought that the noise associ- ated with current through transition region generation- recombination centers should be less than full shot noise because it represents a two-step process (e.g., hole cap- ture followed by electron capture).

The noise associated with the recombination current in forward biased junction diodes and transistors has been investigated several times with somewhat different results reported by the various investigators [2]-[5]. Only recently has the noise associated with current from generation centers in a reversed biased junction been observed as equal to two-thirds shot noise [6]. This has been identified as the important noise source limiting the sensitivity of large area diodes used for nuclear radiation detectors. The theoretical explanation pro- posed by these investigators for the current noise associ- ated with forward and reverse biased diodes has been based upon different approximations regarding the sta- tistics of the electron and hole generation or recombina- tion process. This has led to conflicting theories describ- ing the noise and its significance in forward biased

Manuscript received February 1, 1968. This work was supported in part by the National Science Foundation under Grant NSF

The author is with the Department of Electrical Engineering, GK-1419.

University of Washington, Seattle, Wash. 98105

TRANSITION REGION

/ - -G- I SRH CENTER 1 I

x 4 0 x = w

X -

Fig. 1. Diagram o f p-n junction diode containing one SRH center. The 4 capture and emission events are shown.

junctions [2 ] , [3] and its importance in junction tran- sistors.

This paper presents a general theory for the noise associated with current flowing through single level generation-recombination centers in both forward and reverse biased junctions. I t is based upon the Shockley- Read-Hall (SRH) model [7], making no further ap- proximations regarding the statistics of the generation- recombination process beyond those used t o derive the original S R H model and apply it to a p-n junction [l]. This more basic approach tends to clarify the work of the previous investigators and to add some insight con- cerning the different results observed on forward biased junctions. I t also demonstrates the relationship be- tween the noise associated with current through SRH centers and that induced by charge fluctuations on the centers. The latter is the common low-frequency noise source in junction field-effect transistors [8], [9].

11. STATISTICAL MODEL FOR TRANSITION REGION GENERATION-RECOMBINATION CURRENT

If the concentration of S R H centers is low, the fluc- tuations associated with individual centers can be con- sidered independent and a model using only one center can be used to work out the power spectrum of the cur- rent fluctuations. Fig. l is a diagram of a p-n junction having one S R H center in its transition region. The arrows refer to the four possible emission and capture events. Only current flowing through the SRH center is considered. Diffusion current flowing through the junc- tion is statistically independent and may be included later if desired.

LAURITZEN: NOISE IN $-E JUNCTION TRANSITION REGIONS

If the potential across the p-n junction is held con- stant, then the current flowing through i t as a function of time will be as sketched in Fig. 2 . An electron or hole capture event (cN or cp) results in a pulse of positive current flow, while an electron or hole emission event (eN or e p ) results in a pulse of negative current, each event corresponding to a transfer of charge between the SRH center and the majority carriers at the edge of the transition region. The carriers are assumed to travel at a constant limiting velocity v d and thus the instantaneous pulse current flowing in the external leads il(t) is

q v d il(t) = - = ilo (at a capture event) w

= - ilo (at an emission event) (1)

= 0 (otherwise).

Here q is the electronic charge and re, is the transition region width. The length of a current pulse is equal to the electron or hole transit time, t , or t,, which is in- versely proportional to the distance between the S R H center and the transition region edge. The total amount of charge flowing in the external leads due to an electron or hole event qn or qp is

(W - x) qn -- = q---

W

qn + qp = q.

The smaller pulses associated with hole events in Fig. 2 are due to the location of the SRH center closer to the f i side rather than the n side of the transition region. The analysis will be limited to frequencies that are low compared to the inverse transit times. Therefore, the significant quantity is the amount of charge in the pulses and not the particular pulse shape.

771

TABLE I CAPTURE AND EMISSION PROBABILITIES

Hole capture Electron capture

CP

Hole emission eP Electron emission

CN PCP = P ncN= 1

e p = P l c , = M e N e p = nlcn = A

The SRH center fluctuates between two charge states as each emission or capture event occurs: Sp denotes the more positive and S.v the more negative of the two charge states. The probability of transferring from one state to the other depends upon the electron and hole capture and emission probabilities that are given in Table I.

The events themselves are denoted by capitalized sub- scripts. The probabilities per unit time are those given by the SRH model. They are expressed both in terms of the conventional notation used by Shockley and Read [7] and a shorter notation defined here (p, X, M , A) to facilitate frequent handling of these particular quan- tities.l

From the above model, the current power density spectrum can be determined using an extension of the technique employed by hlachlup [lo] that is outlined in the Appendix. The current spectral density function & ( w ) for this single SRH center is

- - < u < $ ~

where

1 1 T t = ___

P f X .f A c n ( n + %I) + cp(P + PI)

The notation is also defined in the list of symbols a t the end of

I - (4)

the paper.

772 IEEE TRANSACTIONS ON ELECTROS DEVICES, OCTOBER 1958

SIL = q 2 r t [ c p c n ( p n + n?) - 2 ~ ? c , ~ c , ~ ( p n - P Z ~ ) ~ ] (6)

S ~ H = 7c[(qn2 + q p 2 ) C p C n ( 9 n + %i2) + 2qn2cn2n%1 ( 7 )

+ 2qP2cP2PP11.

The time constant that appears is the SRH center fluctuation time constant r t . I t is identical to the charge fluctuation time constant that determines the frequency spectrum of generation noise in junction field-effect transistors [SI, [9], as well as the trap relaxation time constant discussed in very general terms by Sah [ll 1. The power density spectra are defined over both posi- tive and negative frequencies - t~ <w < + t~ and thus are a factor of two smaller than power density spectra defined over positive frequencies alone.

The dc term of the power density spectrum Sl0, when integrated over frequency, gives the square of the dc or average current flowing through the junction. The average current I , coming from a junction of cross-sec- tional area A , is conventionally [I] expressed in terms of the net recombination rate U.

(8)

U = N t r t c p c n ( p n - ni2)). (9)

The concentration of SRH centers is N t . If only one SRH center is involved, then I = p ~ , c ~ c , ( p n - n ~ ~ ) as is given in ( 5 ) .

For low frequencies (w < < l /r t) , the power spectrum has the constant value SIL, while for high frequencies (w> > 1 / ~ , ) , it has the constant value SIH. Note that SIH depends upon the position of the SRH center in the transition region (through ql, and q, ) while SIL may be independent of position if r t is constant.

Equations (3) through (7) can be integrated across the transition region to give the total current noise spectrum Si(w) from a concentration of centers N,.

Si(W) = J o 3 ~ ( w ) A t A d r (10)

I t is assumed that N t is low enough so that the SRH centers fluctuate independently. The specific limits for this to be valid will be discussed later. Si(w) will next be evaluated for specific reverse, zero, and forward bias conditions.

111. REVERSE BIAS CONDITION In a reverse biased junction, the transition region is

depleted and the free electrons and holes are swept out. The SRH centers fluctuate from one charge state to another alternately generating a hole and an electron. The power density spectrum of the current produced is given by (3) through ( 7 ) with = n = 0.

SlH = (4,’ + qp2)Ttepe, (11) SIL = q2rt3epe,(ep2 + en2) (1 2 )

(13)

Integrating the high-frequency term (1 1) over the transition region using ( 2 ) and (10) gives

2 2 SiH = q 2 r t e p e n N t A w - = U Aw-

3 4 ‘ ‘ 3 (14)

for the high-frequency term of the current noise spec- trum SiH. N , has been taken as constant. It is con- venient to express the current noise spectrum s ( ( ~ ) in terms of equivalent shot noise current I,, where p leg= S;(w). Taking the ratio of Ieq and the average current I = q U A w , one obtains

This gives the ratio of the noise observed to full shot noise for the same dc current. Following the same steps for the low-frequency term ( 1 2 ) gives

1 1(16) = - when ep = e,

2 = 1 when ep >> e, or ep << e,.

Equation (15) is identical with that given by Scott and Strutt [6]. The factor of two-thirds shot noise was verified by them on measurements made on large area silicon diodes designed for nuclear radiation detectors. Their measurements were made a t frequencies frorn 20 kHz to 50 kHz. These frequencies are well above the values of 1/27r7, observed on field-effect transistors [9] (approximately lo2 to lo4 Hz). The low-frequency noise as given by (16) would be expected to always give: full shot noise since i t would be extremely unlikely for any SRH center to have its emission coefficients e, and e p approximately equal.

I t is interesting to note that the shot noise reduction a t high frequencies (15) is not due to any special values for the emission constants, but due instead to the: fact that the size of the current pulses produced by a SRH generation center in the depletion region is a fraction of the electronic charge q. They appear to be generated independently (from a statistical point of view) for w > > 1 /Tt . On the other hand, at low frequencies any shot noise reduction is a function of the emission time constants alone (16) and has no relationship to the position of the centers in the transition region. At low frequencies the alternate emission of a hole and an electron appear as a single event and any possible shot noise reduction would then come from a given SRH center’s tendency to “smooth” the emission of carriers. Since the center must go through a cycle of two charge states each time one unit charge crosses the entire junc-

LAURITZEN: NOISE IN p-n JUNCTION TRANSITION REGIONS 773

tion, the probability of two charge units following in close succession (without a wait time) is small. This could only occur if en A ep.

The SRH centers that generate the dc reverse bias current must have energy levels close to the center of the band gap [ l ] ! [ 8 ] , [ 9 ] . Thus T t has a relatively large value (Tt = 0,001 s observed for the gold center [SI).

IV. ZERO BIAS CONDITION

Under zero bias conditions, pn = ni2 and the low-fre- quency power spectrum (6) becomes

S1L = q22rtcpcnn~2. (1 7)

The hole and electron concentrations p and n will vary with distance across the junction, so ~t will be a function of x . If we consider only SRH centers located in an in- cremental region Ax, then the low-frequency current noise spectrum SiL)A, becomes

S i L ) ~ 2 = SILhTtAAx = 2kTG. (18)

This is equal to the thermal noise spectrum 2 KTG from the diode conductance G, which can be evaluated from (8) and (9) letting p and n vary as pn = ni2exp(q V / k T )

Thus (6) gives thermal noise for equilibrium conditions, as should be expected.

V. FORWARD BIAS CONDITIONS In a forward biased junction pn> >niz1 the SRH

center relaxation time T t becomes very small and the low-frequency noise spectrum is the one of primary in- terest. For forward bias conditions (6) becomes

S I & = q7r1CpCnpIZ[1 - 27t2CpC,tpn]. (20)

Again as in the zero bias example, T t is a function of x and i t will be simpler to confine our attention only to SRH centers located within an incremental region Ax. The noise current spectrum in this incremental region S i L ) ~ Z = S ~ L N t A A x and the dc current I ) a X = q UAAx. The ratio of equivalent shot noise current I,, to dc cur- rent becomes

Again the only quantity that varies with x in this ex- pression is T ~ , which can be plotted as a function of x when certain general assumptions are made about the junction.

Fig. 3 is a plot of the electron energy band diagram for a forward biased F-PL junction. The SRH center energy level and intrinsic energy level are E , and Ei, respectively. The electron and hole quasi-fermi levels EF, and Epn are assumed to be constant across the transition region. The hole and electron concentrations can be obtained from the usual relationships, p = n i

c

DISTANCE x - Fig. 3. Electron energy band diagram versus distance for $-n junc-

tion under forward bias condition.

- I co PI

..........................................

DISTANCE x - (a )

- .................................................. I %PI

- " - w w .................................................... - I

F c.n, I

(c,c,np)'" W 9

DISTANCE x - (b)

Fig. 4. SRH center relaxation time constant versus distance in forward bias p-n junction transition region for (a) low bias level and (b) high bias level conditions. For these examples c,nl>>c&l.

exp[q/kT(Ei-EFp)] and n=niexp[q/kT(EFn-Ei)]. I f it is assumed that the intrinsic potential Ei varies approximately linearly in the region of interest, then T~ can be plotted as a function of x as shown in Fig. 4. For arbitrary capture coefficients cn and c, and trap level Et, two different plots of T~ are possible for different bias levels. These are shown as low bias and high bias levels in Fig. 4. (In the diagram it is assumed that cnnl> >c,pl.) Under low bias levels T t reaches a maxi- mum value of l / c n n l , while under high bias levels T~

reaches a maximum value of 1 / (4c , c ,pn ) t . The maxi- mum value of T t determines the amount of possible shot noise reduction indicated by (21). For low bias

and

7 74 IEEE TRANSACTIONS ON ELECTRON DEVICES, OCTOBER j.968

For high bias

and

The quantities p* and n* are commonly called the equal- ity carrier concentrations: p * =cnnl/cp and n* =cppI/cn. If pln*=p*nl=ni2, then the condition expressed in (22a) will not occur in forward bias operation. However, in general one must consider that a t low forward bias, full shot noise is produced by the current flowing through the SRH centers and at higher bias levels there will be some reduction from full shot noise. Under these high bias conditions the SRH centers that contribute to the current are located physically in the transition region close to where T~ has maximum value. The ap- proximation of a linearly varying intrinsic potential is quite accurate across this small region. If E is the con- stant electric field here, then the intrinsic potential Ed becomes Ei(x) =~(?41/2 -x) and the ratio leq/l for the whole junction can be evaluated in closed form using (8), (lo), and (20) for the high bias condition (23a).

SlLdX 3

Both SIL and U decay exponentially at a distance away from their maximum value, and thus the limits of in- tegration can be extended to infinity.

Equations (22b) and (24) indicate that for arbitrary SRH centers or forward bias levels, the low-frequency ratio of I e q / I must always be between 1 and 9 . Since minority carrier diffusion current across a p-n junction always produces full shot noise a t low frequencies, i t should be difficult in practice to observe the differences in the noise produced by the two mechanisms of dif- fusion current and recombination current.

Equations (22b) and (24) are valid for low frequencies f where f< l /2~7 ,=n i /~ (cPc , )bexp q V / k T A lO'Hz. 7t is taken at its maximum value assuming ni= lO'O~rn-~, (cpc,) 4 = 3.10-8 cm3/s, V = 0.6 volts. This is the primary frequency range of interest, Expressions valid a t higher frequencies can be derived from ( 7 ) by following the identical approach, They are somewhat more complex due to trapping effects at the edge of the transition re- gion and will not be considered here.

Assuming cnnl > >cppl condition (23a) can be ex- pressed in terms of the forward voltage V as follows

[12] on silicon transistor emitter base junctions. There should be a range of bias voltages above 0.3 volts where the effect of the shot noise reduction should be observed with very careful techniques. The bias voltage must be kept below the value where diffusion current begins to dominate. Instead of using a conventional diode struc- ture, it should be easier to detect this on a transistor emitter base junction where the diffusion current noise and transition region recombination noise can be sepa- rated by open circuiting the transistor base.

Chenette and Van der Ziel [j] have performed such measurements within an accuracy of 10 percent and have detected no shot noise reduction. Their measure- ments were performed on silicon alloy and XlADT tran- sistors both having alloy emitter junctions. Schneider and Strutt [2] have made measurements on diodes and have observed a noise level of + shot noise. I t is difficult to compare Schneider and Strutt's results, which are plotted in terms of equivalent noise resistance. I t would be necessary to know the diode conductances to relate their data to the equations presented in this article. In their theoretical explanations both investigators as- sumed simple physical models for the recombination process. Van der Ziel's model [3] gives full shot noise a t low frequencies while Schneider and Strutt's gives one-half shot noise [2].

I t would be significant to repeat the accurate mea- surements of Chenette and Van der Ziel on modern planar transistors to determine if the factor of three- fourths is observed. The fact that the predominant re- gion of recombination occurs at the semiconductor sur- face should not affect these calculations as long as the model of a single-level S R H recombination center is valid. Extensive measurements by Reddi [I31 indicate that the single-level S R H model is valid for surface recombination on planar transistors.

VI. LIMITS TO THE VALIDITY OF THE APPROACH In (10) the assumption was made that generation and

recombination a t various SRH centers in a junction could be considered statistically independent. This would mean that charge fluctuations on one SRH center would not affect other centers. This requirement is met when the trap concentration N, is low enough so that it does not induce fluctuations on the hole and electron quasi-fermi levels EFp and Epn. For a forward biased junction this implies that N t < < (pn) *, In a. reverse biased junction, fluctuations in E F p and E p n are limited by the doping concentration Nw at the more lightly doped edge of the junction depletion region. The limita- tion on N t would then be N , < < N w . This identical limitation would apply to the validity of the generation noise model for a junction field-effect transitor [Si, 191

KT cn having a high concentration of defect centers. _ I

> 2 ( E t - i- log- 0*3 (23) Sah [11 1 , in his general treatment of SRH center relaxation times, indicates a much lower limit Nt<<n;

The numerical value of 0.3 volts comes from the recom- for the validity of (4) for T t in a reverse biased depletion bination center coefficients measured by Bartholomew region. Sah's model does not take into account the fact

Q CP

LAURITZEK: NOISE IN p-n JUNCTION TRANSITION REGIONS 775

that in a strongly reversed biased junction the carriers generated by the SRH centers are swept to the deple- tion region edge. The depletion region edge will move laterally in response to SRH center fluctuations, but the effect on rt will be insignificant unless N t is of the same order of magnitude as the doping.

VII. CONCLUSIONS

By deriving the noise model directly from the original S R H defect center model i t was not necessary to make any assumptions regarding the capture and emission process as was done by previous workers in this field. The noise produced by this mechanism is usually so close in magnitude to shot noise that i t is very difficult to observe it as being different from the noise produced by diffusion current across the junction. In this sense the noise from the generation-recombination centers appears to be not particularly significant in modern p-n junction devices (with the striking exception of the field-effect transistors). This fact has been indirectly born out experimentally during the last 10 years be- cause most of the noise formulations originally derived for germanium devices were transferred over to silicon transistors with only minor modifications to account for the variation of gain with bias current [SI.

This noise model may have significant applications to noise from SRH centers at the surface, or to new struc- tures where this mechanism of current flow is especially important.

APPENDIX DERIVATION OF THE POWER DENSITY SPECTRUM

The Greek letter symbols p, X, M , and A for the cap- ture probabilities per unit time will be used (Table I), as they are simpler to handle. The capture events tN

and c p will be considered as taking place during and for a time t, or t, after the SRH center changes state, as shown in Fig. 2. This implies that a hole or electron crosses the transition region to the SRH center after a previous one is captured. The exact timing of this charge transfer is not significant as we are going to con- sider only transit times that are very short compared to the wait time between events.

The probability for an electron to be in the process of moving across and being captured C N will be expressed as P [cl\r]. The probabilities for the four possible events shown in Figs. 1 and 2 are found from the coefficients in Table I.

P [ C N I = X t , P [ S P ] P [ C P ] = P t P P [ S N ] (26)

P [ ~ N ] = At, ,P[w] P [ e p ] = Mt,P[sp].

The probabilities for the SRH center being in states s p or S N are P [SP] and P [snr]. These can be determined by setting the net electron generation-recombination rate equal to the net hole rate : P [ s p ] (X + M ) = P [ s N ] ( p +A). Thus

P[sN] = 1 -- P [ s P ] = T~(F + A). ( 2 7 )

I n order to obtain the power density spectrum i t is first necessary to find the autocorrelation function R1(r) for the current &(t) where

RI(7) = Cil(t>il(t + .>P[il(t>, i l ( t + 711

= i l o q P [ C N ; CP, 71 + P [ c p ; C N , TI + . * 1 . (28)

Here P [ c M ; cp, T ] denotes the joint probability of an electron capture a t time t and a hole capture at time t f T . There are 16 such terms corresponding to the 16 possible joint events. They can each be evaluated by splitting into conditional probabilities as follows.

P [ c N ; cp, T I = P [ C P , T / C N ] P N [ C ] = P C C P , T I S N I P [ C N l . (29)

The conditional probability P [ C P , T / S N ] can be rew-itteu in terms of a conditional probability on the charge states p [ S N , 7 / 5 N I .

P [ C P , d S N 1 = P [ C P / S N , .IP[SN, d J N 1

= Pt,P [ S N , T/SNI.

The variation of this probability as r changes an incre- ment from T to T+AT can be expressed as

P [ s N , T + AT/SN] = P [ s N , T/sN](~ - PAT - AAT) (31) + P [ s P , T/sN](X + M ) AT.

Let AT approach zero and (31) can be solved as a differ- ential equation [ lo] with the boundary condition P [sN, r = O / s N ] = 1 to obtain

P [ S N , T / S N ] = 1 - P [ S P , +NI (3 2)

and

The 16 terms of (28) can each be evaluated using (26), (27 ) , (29); ( 3 0 ) , and (32). Two examples are

P [ c N ; C P , T I = Pt ,P[SN, 7 - / S N ] X t , P [ S P l

The (1 - 171 /La) term is valid only for 1 T I < 1 and must be included to account for the possibility of r being SO

small as to include the same event twice. All of the terms have been derived for r greater than zero. However

therefore P [ c P , rnega t ive ; 6x1 =P [ c p ; cJV, r p o s i t i v e l

776 IEEE TRANSACTIONS ON ELECTRON DEVICES, OCTOBER 1968

All the 16 terms in (28) are summed and since we are I/ bias voltage across junction only interested in frequencies much less than the transit ze, transition region width time, we take the limits tn-+O, t,-+O, while holding x distance through transition region i d , =pp and i& = p n . This gives the autocorrelation 6(w) dirac delta function function &(TI. The power density spectrum Sl(w) X, p, A , M electron and hole capture and emission given in (3) is found by taking the fourier transform. probabilities per unit time (defined in

Table I) ~ l ( w ) = J-1 ~ ~ ( 7 ) exp ( - j w ) d T . (35) I- offset time in autocorrelation function

rt SRH center relaxation time (4). w angular frequency.

LIST OF SYMBOLS A cross-sectional area of p-n junction ACKNOWLEDGMENT cnr c,, en, e p electron and hole capture and emission The author would like to thank E. B. Schibli for his

coefficients helpful comments on the manuscript. C N , c p , eN, e p electron and hole capture and emission

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