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PANG, Tet Chong, 1929- APPLICATION OF THE SHOCKLEY-READ RECOMBINATION STATISTICS TO THE STUDY OF THE P*NN+ DIODE.
The Ohio State University, Ph.D., 1963 Engineering, electrical
University Microfilms, Inc., Ann Arbor, Michigan APPLICATION OF THE SHOCKLEY-READ RECOMBINATION
STATISTICS TO THE STUDY OF THE P+NN+ DIODE
DISSERTATION
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Ftiilosophy in the Graduate School of The Ohio State lhiversity
By
Tet Chong Pang, B.S., M.Sc.
The Ohio State lhiversity 1963
Approved by
Adviser Department of Electrical Engineering PLEASE NOTE: Figures are not original copy. Pages tend to "curl". Filmed in the best possible way.
UNIVERSITY MICROFILMS, INC. ACKNOWLEDGMENTS
This research was started when the author was a student of
Professor E. Milton Boone to whom he is ever grateful for his overall guidance and understanding over the years.
He is deeply indebted to Professor Marlin 0. Thurston for his encouragement and many elucidating discussions. It is a pleasure to acknowledge the assistance in computer programming given by the staff of the Numerical Computation Laboratory, The Ohio State Lhiversity, in particular, Professor Theodore W. Hildebrandt. CONTENTS
Page
ACKNOWLEDGMENTS...... ii
LIST OF ILLUSTRATIONS...... v
Chapter
I INTRODUCTION ...... 1
II RECOMBINATION OF ELECTRONS AND HOLES ...... 3
III SHOCKLEY-READ RECOMBINATION STATISTICS ...... 6
IV DETERMINATION OF CARRIER DENSITIES ...... 12
V CONDUCTIVITY MODULATION AND NEGATIVE RESISTANCE ...... 17
Conductivity modulation ...... 17 Negative resistance ...... 17 Lifetime variation with carrier density .... 18
VI ANALYTICAL APPROACH TO THE SOLUTION OF TRANSPORT EQUATION ...... 24
VII NUMERICAL APPROACH TO THE SOLUTION OF TRANSPORT E Q U A T I O N ...... 28
Runge-Kutta-Gill m e t h o d ...... 28 Simpson's three-point formula ...... 30 Computer-graphical m e t h o d ...... 31
VIII DERIVATION OF THE BOUNDARY VALUES ...... 33
IX DETERMINATION OF THE APPLIED POTENTIAL ...... 38
X DISCUSSION ...... 45
XI CONCLUSION ...... 60 OONTTJJTS (continued)
Appendix Page
I AN APPROACH TO NUMERICAL SOLUTION OF DIOEE EQUATIONS WIIH SPACE C H A R G E ...... 62
II INTEGRATION OF THE APPROXIMATE EQUATION ...... 65
III COMPUTER PROGRAM AND EXAMPLE ...... 67
BIBLIOGRAPHY ...... 73
AUTOBIOGRAPHY...... 75
iv LIST OF ILLUSTRATIONS
Figure Page
1. Electronic Transition Scheme ...... 6
2. Energy Diagram Showing Various Energy Levels in the Band Gap ...... 7
3. Excess Electron Density Versus Excess Hole Density ...... 14
4. Derivative of Electron Density with Respect to Hole Density as a Function of Excess Hole Density ...... 16
5. Effective Lifetime for Holes as a Function of Excess Carrier Density ...... 22
6 . Effective Lifetime for Holes as a Function of Excess Carrier Density ...... 23
7. Five Possible Carrier Distribution in the N-region of a Diode under Forward B i a s ...... 34
8 . A Planar N+NP+ Diode Structure ...... 37
9. Potential Diagram of a Forward Biased P-N Junction Diode ...... 40
10. Potential Diagram of a P^NN4 D i o d e ...... 43
11. Modified Carrier Distribution and Its G r a d i e n t ...... 46
12. Calculated V-I Curve of a P+NN+ Diode under Forward Bias (Sp=3xl0~^**t no=7xl010) .... 49
13a. Calculated Applied Voltage Components and V-I Curve (Sp=3xl0-14, n=7xlOil) ...... 51
13b. Calculated Voltage Drop in the Base and Total Applied Voltage ...... 52
v LIST OF ILLUSTRATIONS (continued)
Figure Page
14. Calculated V-I Curve (SpSlO”^*4, nQ=7xlO^-^)...... 53
15. Calculated Voltage Components and V-I Curve (Sp=10-15, a = 1 0 ) ...... 54
16. Calculated Voltage Components and V-I Curve (Sp=10-15, a = 2 0 ) ...... 55
17. Figure 14 Redrawn on Linear Scale ...... 57
18. Figure 15 Redrawn on Linear Scale ...... 58
vi CHAPTER I INTRODUCTION
The fabrication of semiconductor devices consists largely of the introduction of electronically active imperfections into the crystal line semiconductor. Of these imperfections, donor and acceptor im purities are added deliberately to control the conductivity, while others such as recombination centers, trapping centers, and scat tering centers are unavoidably present also. Sometimes these centers are the result of residual impurities in the crystal or of defects introduced during the crystal growing process. They may also be cre ated as a result of the device fabrication process or by bombardment with high energy particles such as beta particles, alpha particles, or fast neutrons. In general, increase in the number of active centers (other than donors or acceptors) in a semiconductor device tends to increase the effective resistance of the device through deterioration or change of one or more of the physical parameters as lifetime, mobility and free carrier density. The manifestation can take many forms such as higher
voltage drop, lower anplification and higher leaking current. Centers
that behave as traps nay affect the rise time, the decay time and the switching time of some devices.
The presence of such active centers in a device is not all of
negative value; they are sometimes purposely incorporated in the 2 semiconductor naterial to i m p r o v e the performance of devices such as switching diodes. Hie negative resistance characteristic of certain diodes is found to be due to active canters. The fabrication of par ticle detectors and radiation dosimeters is based on the fact that damage in semi conductor materials due to particle borrbardment are electronically active and are almost permanent.
The object of this work is to investigate theoretically the effect of electronically active centers on the current-voltage characteristic
of a wide base P+NN4 silicon diode. The Shockley-Read monovalent flaw model of the electronic transition process at the active center is applied in the analysis. Instead of a constant lifetime, the S-R re
combination rate as a function of injection and capture cross sections
is used* This adds considerable complexity to the analytical solution.
Because of the nonlinear nature of the differential equation involved,
numerical solution seems to be the only method if the full significance
of the active centers is to be revealed. For this purpose the Runge-
Kutta iterative numerical integration is chosen for its simplicity and
its adaptability to computer calculation.
The current-voltage characteristics of an assumed planar structure
have been calculated far several combinations of the flaw density, the
net donor density, and the capture cross section for holes and for
electrons. Diodes with a low equilibrium carrier density in the N-
region have been found to exhibit negative resistance characteristics.
Other irregularities that have been observed experimentally can be ex
plained satisfactorily by the Shockley-Read recombination mechanism. CHAPTER II
RECOMBINATION OF ELECTRONS AND HOLES
In a semiconductor, electronic transitions take plaoe continuously.
Electrons nay be thermally excited from the valence band into the con duction band, or into localized levels in the band gap; they may also be excited from localized levels into the conduction band. To main tain balance, the inverse process of de-excitation takes plaoe at the same time. In thermal equilibrium, the rate of each process and its inverse are equal and balance in detail.
To avoid confusion, a few words of explanation are given here of some of the terms used in connection with flaws and recombination.
The term * recombination1' suggests the oombining of two charged par ticles, an electron and a hole, When recombination takes place, the charges of the two particles cancel each other. On the other hand, capture suggests immobilization of a free carrier; the recombination process may take place at the same time, but this need not be. If the center is neutral to begin with, after oapturing an electron it would be negatively charged. Now if this same center captured a hole,
this hole would recoirbine with the previously captured electron. A
capture without reoonbination releases potential energy equal to the
energy difference of the initial and the final state of the carrier, while a reoonbination involves the potential energy of the electron- hole pair. 3 1+
The terns "flaw," "center," "states," and "level" refer to the same thing in one plaoe or another, although each term has a somewhat
different connotation. Loosely, a flaw denotes a disruption of the
crystal lattice either by impurity atoms in interstitial or substi
tutional states, dislocations or other defects. A center denotes a
location at which electronic activities can take plaoe. The term
"state" originates from the quantum theory of matter, while the term
"level" has significance in connection with the energy level of a
state.
Returning to the main discussion, in the band-to-band de-excita-
tion process, the potential energy of the electron-hole pair released
in the process trust be dissipated in one of three ways. It can be re leased as electromagnetic radiation, it can be given to a third carrier
as kinetic energy, it can also be released in the form of phonons
through interaction with the lattice atoms. The first process is known
as radiative recombination, and the second is described as three body
recombination or the Auger process. When localized states are present,
the same three processes can take place between the conduction band and
the localized states, and between the localized states and the valence
band. These localized states can be due to crystal defects or impuri ties.
The relative importance of the various recombination processes
depends to a large extent on the forbidden gap width and the temperature.
The presence of localized states in general favors recombination through
these states over the direct or indirect band-to-band process even 5 though the band gap may be smaller than 0.5 ev at room temperature.
Though radiative capture is always present in sane degree, it is not the dominant process at low injection or low excess carrier density.
It is quite significant at very high excess carrier density as in laser diodes. Of the other two, phonon-aided capture or recombination is much more probable[1 ], especially if the capture of the carrier is not by a strongly repulsive-charged center. In the phcncn-aided cap ture process, the energy is released by a cascade of one-phonon tran sitions until the carrier reaches the ground state. CHAPTER m
SHOCKLEY-READ RECOMBINATION STATISTICS
A brief discussion of the Shockley-Read model of the recombination through localized states is presented here to facilitate understanding of the problem ahead. This model assumes only one type of monovalent flaws or centers at one energy level in the band gap. A monovalent flaw is one whose empty and full states differ by only one unit of charge. For clarity we assume that when the flaw is empty it is in the neutral state. Thus four electronic transition processes can be associated with a single flaw. According to the S-R recent)ination scheme[2 ], a neutral or empty center can (a) capture an electron from the conduction band, or (b) emit a hole into the valence band. A neg atively charged center can (c) capture a hole, or (d) emit an electron.
Process (c) is a case of capture with recombination. Figure 1 shows the four electronic transition process connected with a single mono valent flaw.
CONDUCTION BAND ______Fig. 1— Electronic transition d scheme. The arrows indicate di rections of electron motion. The - 1 1 - 1 charge on the center before the transition is also shown.
VALANCE BAND An energy level can be assigned to the flaw corresponding to the electron ionization energy of the flaw. It is evident that only flaws with energy levels lying in the energy band gap would be effective in the transition process. Figure 2 depicts the relative positions of the various energy levels in the band gap. As our concern is in the non equilibrium recaibinaticn rate, it is therefore necessary to employ the quasi Fermi energy level for holes and electrons.
CONDUCTION BAND — Ec - conduction band edge „. _ q $n - electron quasi Fermi level
— ----- — ----- — q ♦r - Fermi level associated with flaw occupancy permi level
- — - q ♦p - hole quasi Fermi level
- Ep - flaw energy level
1 Ey - valence band edge VALfUCE BAND
Fig. 2— Energy diagram showing various energy levels in the band gap
To begin with, let c^CE) be defined as the average probability that an electron from an occupied conduction band state at energy E will be captured by an empty flaw in unit time. If the speed of the electron is v, and the capture cross section for an electron at energy
E is on , then c^CE) can be expressed as
c^CE) = average of v for states of energy E. (1) The rate of electron capture will be
r = Nr f rp Cn(E) fe (E) N(E) dE. (2)
As usual N(E) dE is the total number of quantum states in the energy range dE per unit volume, fe (E) is the fraction of states of energy E that are occupied, frp represents the probability that a flaw is empty, and Nr the flaw density.
Similarly the net rate of electron emission from the flaws into the oonduction band can be derived with the same reasoning and is given by
g = Np fr e ^ E ) f fp = 1 - fe , fraction of empty states at E. We define a mean capture coefficient j" c^ E ) fe (E) N(E) dE fe (E) N(E) dE Ec Note that fe (E) is thus far an unspecified function of energy. For carrier lifetime very much greater than the mean free time, fe (E) is assumed to have the farm fe (E) s ______i______(6) 1 ♦ exp(E - q ♦n )k/T 9 In other words even though the system is not in thermal equilibrium with injected carriers, Fermi-Dirac statistics are still acceptable in considering the electronic transition process, if the carrier life time is long and if the carriers are scattered nany times before de- excitation. For the same reason the probability that the flaw is occupied is f_ = ------1------(7) 1 + expCEp - q ♦pl/kT It is obvious at thermal equilibrium condition fr = fe , and ^ = 4^. As assumed, CnCE) (and enCE)) is a function of energy, hence injection. For nondegenerate cases fe(E) = expC^ - E)/kT, therefore most of the electrons still conoentrate close to the band edge, which means the capture probability is approximately the same for all elec trons in the conduction band. Now the application of the principle of detailed balance, which requires the rate of electron capture to be equal to the rate of elec tron emission at thermal equilibrium, leads to the relationship ^|1 = expCEp - E)/kT . (8 ) Thus the total rate of electron capture is obtained by integrating (r - g) over dE to give m Ucn = [ 1 - e x p C v - ^J^kT] fpp ^ fe Ucn s [ 1 * exP(*r “ ♦nJ/KT] fpp n = Nr Following the same procedure, an expression for the total rate of hole capture can be obtained as ucp s Nr By equating the total rate of hole capture to the total rate of electron capture for the steady state condition, we can solve for fr to give < ^ > ( 0 +7 1 1 ) + or in terms of capture cross sections f = ^"^SpPl . (15) r Sn (n'tn1) ♦ Sp(p+pi> Substitution of (15) into (10) yields for the net rate of recom bination U _ - PiV t (16) V ^ i ) + ^(p+px) For an illustration of the use of fr* let us take the case of appreciable injection in a nondegenerate silicon crystal. The ratio of the empty centers to the filled centers is given by 11 If the individual terms are known, the ratio is determined. For J^n >> SpPi and SpP >> S^ni, a situation usually met with if the energy level of the recombination centers is located close to the center of the band gap, (17) reduces to "r ~ "r = . (18) "r Snn And for large injection such that p=n, (18) reduces to "r'nr = . d9) Hr ^n Both (18) and (19) are identical to those of Rose[3] who obtained the expressions by neglecting the re-emission as small compared with cap ture rate of electrons and holes. The net rate of capture expression is used in the investigation of the current-voltage characteristic of a wide base F’W 1' silicon diode in the following discussion. CHAPTER IV DETERMINATION OF CARRIER DENSITIES In the steady state condition, the densities of both mobile elec trons and mobile holes in semiconductors are determined by the density and the energy level as well as the ratio of the capture cross section for holes to the capture cross section for electrons of the reconbina- tion canters. We shall consider the presence of only one type of re combination centers and only the nondegenerate case so that CIS) and (16) are applicable. As is customary, it is assumed that quasi-neu trality in space charge exists irrespective of the mode of excess carrier creation whether by injection througi contacts or a pi junc tion, or by electromagnetic radiation. It is further assumed that both the donor and the aoceptor impurities are fully ionized and that a reoonbination center carries a negative charge if filled. We thus have, by equating positive charges to negative charges including both the mobile and the immobile, Nd + p = Na + n + * <20> where N MSrin+SpPj.) Nd + P = N- + n + — ------* (21) ^(n-Hii) + Sp(p+px> This expression gives the relationship between mobile holes and elec trons explicitly if the constants in (2 1 ) are known. 12 13 Solving for n in terras of p, we get n = WQ + w^p + [W2 + W 3 P + W^p2]^ , (22) where WQ = -i.( Np-N^+^+ap^ +n-j_) , W-, = -I(a-1) , ^ 2 W 2 = ^(Nr+Nd-Na+api+ii! ) 2 - Nr(Nd-Na+aPl) , W 3 = ^.(a+1) (Njn+N^-Na+ap^+ri^) - Nr , and Wu = i.Ca+1) 2 , where a is defined as the ratio of the capture cross section for a hole to the capture cross section for an electron. The relation given by (22) is rather involved and not suitable for analytical solution of the transport equations of a semiconductor de vice. Vte can remove the square root by series expansion of the square root terra. In doing so, we are restricting the relationship to a par ticular situation only. For example, if Nj, and Nd - Na differ by a factor of 40 times, (2 2 ) can be reduced to a simpler expression with out sacrificing too much in accuracy: Nr(Nd-Ka»ap1 4p) ^ (23) n = Nd-Na+p- Nr+Nd-Na+ap1 +n1 +(a+1)p * The last term on the right is nothing but the density of the filled centers. From (22) and (23), it is evident that very significant error can result if the simple relation n = N + p is used. Figure 3 shows the excess electron density plotted against excess hole density for a few representative cases of flaw density and capture cross section ratio. / Slope of !*>)/* (P - - (P 0.058 20 6.9U1013 2.9xl0u 7x1013 3x10^ 3xl0‘U 10"15 TxlO11 100 P lj. 3.—Icm listd bM M Electron Density as A Function of Nomadized Excess Hole Density. 10-5 lo-fc ifl-3 ICT2 lorl i io 102 The effect of the capture cross section ratio is evident on viewing Figure 3, i.e. the excess electron density is larger for a large ratio compared to that of a smaller ratio for the sane amount of excess hole density when the excess carrier densities are less than the flaw den sity. Putting it in another way, we can say that the excess holes are more effective in releasing electrons from the filled flaws. There is not much difference in the effectiveness when the exaess hole density approaches the flaw density, for then the flaws are mostly empty. To the knowledge of the writer, in all analysis the gradient of hole density is taken to be equal to that of the electron density. This is a good approximation if the equilibrium density of one type is very much greater than the flaw density and also at high injection, Uhder this condition, the flaws cure either approximately all filled or all empty. For moderate excess minority carrier density compared with the equilibrium majority carrier density, the occupancy of the flaws will not change much with increase or decrease of excess minority car rier density. That means any increase in minority carriers would be compensated by an equal increase in majority carriers. For the analysis to follow, the net impurity density is of the same order of magnitude as the flaw density so that the free majority carrier density is quite small. Differentiation of (22) with respect to p will show that it is far from unity, if numerical values are used. Some representative curves are plotted in Figure to show the variation of dn/dp with ex cess hole density for two cases of background impurity. The approxima tion of letting dn/dp = 1 may not alter the gross features of a diode V-I characteristic, but it would certainly obscure the finer features like the irregularity of a V-I curve. 100 20 PS«. km— Ttao Soto of Chufo of U m Pi •• Kloetron Dondtp with Rospoct to tho Ft h Rolo Donaity u A Function CHAPTER V CONDUCTIVITY MODULATION AND NEGATIVE RESISTANCE Conductivity modulation Conductivity is generally defined as the coefficient a in J = a E, where J is the current density and E the electric field. In semicon ductor devices, the drift current is given by J = q Cv^n + UpP) E. The equivalent conductivity is then o = q (i^n + ppp). Any change in one of the factors in the bracket at one point in the semiconductor would vary the value of the conductivity at that point. In the bulk material of a pn junction diode, the carrier concentrations n and p increase with injection, thus increasing the conductivity. This is said to be conductivity modulated through increase in carrier density. Evidence of conductivity modulation is exhibited in the low voltage drop of a wide base PvN or PirN diode, or the negative resistance PIN diode. Negative resistance In a P+NN+ diode containing a large flaw density of the right type, the increase in effective lifetime is as important to conduc tivity modulation as the increase in carrier densities alone. As a matter of fact, the increase in lifetime and the increase in carrier densities reinforce and compound the conductivity modulation effect. As a result increase in current flow through the diode causes decrease 17 18 in' voltage drop across the diode. We have then a negative resistance diode. This effect has been observed and reported[4]. For the observation of negative resistance characteristics, it is necessary that 1he equilibrium carrier density be snail, and that the lifetime at snail injection be snail also. This is because of the requirement of non-negligible voltage drop in the base region (the N- region) so as not to be masked by the junction voltage drop. In other words, the presence of modulation would not show up if the increase in junction drop is greater than the decrease in voltage drop in the base region. The low level lifetime of holes in the N-region is given in gen eral by tpjto = (Nr v Sp)”l, assuming all the flaws are filled[5]. Thus in addition to large flaw density it is necessary to have a large cap ture cross section for holes also. It is found that Nr should be of the same order of magnitude as the net donor density in order to have small equilibrium carrier densities. This is the result obtained with (22). Lifetime variation with carrier density In connection with conductivity modulation, it would be profitable to examine the variation of lifetime with excess carrier density. For this purpose, we can nake use of the S-R recombination rate U. The effective lifetime of a carrier can be generally defined as the ratio T ^ = ______Excess carrier density # Net difference of reoombination and generation rate Thus we have for hole carriers xeff = P ~ Po = (p - PpXnj+apx+n+ap) (24) U (npwi-j_Pi) SpNrv If n is much larger than p, a sli^it increase in p would not change n too much; therefore we can set np - = np - n? * n (p - pQ) . Thus for an n-type crystal at low level of injection, the effective lifetime of the hole carriers is approximately given by (n^+api+n+ap) (25) n If for the purpose of comparison we let n = n^ + ap^ + ap , (26) then teff i q becomes 2/(NrvSp). At high level of injection, n * p >> (n^ + api); (24) reduces to Teff hi = (27) Nr v Sp Hence in order to have large increase in lifetime with injection, the capture cross section ratio must be large. This is one reason neg ative resistance phenomenon in germanium diodes was first observed at low temperature[6 ], For certain impurities in germanium and silicon, the capture cross section ratio can be as large as three thousand at liquid nitrogen temperature [1 ]. It has been noted above that the equilibrium carrier density of the majority species should be snail. The maximum allowable magnitude 20 depends on the ratio of the base width to the low level diffusion length. For large base width the majority carrier density can be much larger than the intrinsic carrier density and still give appreciable voltage drop in the base region. From (26) we see that as n^ (and also p^) is a function of the flaw energy level, the location of the flaw energy level for large lifetime variation should be close to the intrinsic Fermi level. It can be on the same half of the energy gap as the majority carrier Fermi energy level, or on the other half so long as (26) is approximately satisfied. Negative resistance can occur in a high resistivity p-type base diode according to this analysis. The reason it has not been observed so far perhaps lies in the fact that most impurities have capture cross section ratios greater than unity. For a p-type base diode, the equation corresponding to (26) is p = nx + apx + n. The general requirement for large lifetime variation at a low level of injection can be suimarized as follows: (a) large capture cross section ratio; (b) small equilibrium carrier density; and (c) flaw energy level close to the intrinsic energy level. Figures 5 and 6 show the variation of carrier lifetime as a func tion of the excess hole carrier density calculated far various sets of parameters. All the curves have the same shape, and the greatest rate of change of the lifetime occurs in the range 0 . 1 to 1 0 of the normal ized excess carrier densities. At the lower end of the excess carrier density, the lifetime is not included in the graph. In all cases it is much lower than the apparent asymptotic value shown in the graphs. 21 Note that the effective lifetime in Figure 5 is defined differ ently from (2*4), and is based on the simplified equation (42). If we define an effective lifetime according to the expression & . , (28) dx2 Teff then (42) gives *eff = 2(P ~ P° )Cfr _ J i J M ______(29) U [bf(p) + 1] 24 I II E i-E p 0.058 eV 0.058 e7 » r 2.Jxl0*4 6.94*10*3 20 I 3x10*4 7x10*3 10-*5 SP 3xl0“*4 10*3 7X10*1 a 20 100 1 6 ^ (p - Po) 2Dpbf(p) 4* (bf(p) 4- 1) l 2 4»£ s t:e 4 0 (p - p )/N Pig. 5.— Effective Lifetime reyy as A Function of Normalized Injected Excess Hole Carrier Density. ■l-Xr-0.050 tT, I-JxlO1*, no-1015 Ir-2.9x10*4, sp-10_15 10 a-10 1 10 (P “ Pb)/* Fig. 6.— Effective Lifetime ^9 f f m m A Function of Normalized Injected Excesa Hole Carrier Denaitj. CHAPTER VI ANALYTICAL APPROACH TO THE SOLUTION OF TRANSPORT EQUATION The chief feature of this discussion lies in using the S-R recom bination rate in the continuity equations. In this respect, it is similar to the analysis of a pn junction diode by Sah, Noyce and Shockley[7], It differs from Sah et al. in the inclusion of the re- conbination centers in the space-charge equation. A truly rigorous solution is too complicated to be treated analytically, hence the following assumptions are made in the discussion; (a) A planar P+NN++ diode with abrupt junctions (b) Femd-Dirac statistics valid in the base region of the diode and nondegenerate conditions (c) Mobilities and oapture cross sections independent of elec tric field and injection level under nondegenerate condition (d) Quasi-neutrality outside transition regions. The basic equations for steady state d-c condition are the current transport equation (30) (31) the continuity equations div Jp = -qU , (32) div Jn = q U , (33) the Poisson's equation div E = & (p + Nd - Na - n - nr ) (34) and the Shockley-Read steady-state recombination rate NrvSp(pn - nj) (35) n + + ap + ap^ where a is the capture cross section ratio. The task ahead is to solve for and express the total diode current as a function of the applied voltage. To begin with it is necessary to express all equations in terns of one species of carrier. This can be accomplished by invoking the quasi-neutrality which leads to (22). The electron density gradient can easily be obtained in terms of the hole density and its gradient by differentiating (22). In functional form, (36) In like manner (37) where f’ is the derivative of f(p) with respect to p. Now combining and rearranging (30) and (31), we get w s tJ + ^ cf; - b I?3/ qDP (P+hn) ’ (38) differentiation of which gives 26 Subtracting (30) from (31) and then differentiating the resultant expression, obtain *!" . 2 k . 2 qU dx dx = qDu {(bn-p)- 1 & L + 9L & *L - & ] kT dx kT dx dx + b + ^2.) . (40) dx2 dx2 Substituting (39) into (40) and making use of (36) and (37), we obtain after some manipulation the following expression: _U _ _2_bf(E) £p + bT ___ [dp-!2 q rdEjb[n-f(p)p] (ul) Dp Dbf(p)+1]dx* [bf(p)+l] dx kT dx [bf(p)+l] In order to solve the above equation analytically, we have to discard the field gradient tern on the ground of quasi-neutrality condition. For ordinary diodes with moderate equilibrium majority carrier density, the square term is usually snail compared with the remaining term on the right of (41). These two approximations will be discussed further. (In Appendix I, an approach to the numerioal solution of the transport equations without approxination is presented.) Equation (41) thus simplifies to d2p . U [bf(p)+l] £ ? " Dp 2 bf(p) ’ = NrvSp [bf(p)+l] (pi - rj) ^ 2 Dp bf(p) (n+n^+ap+api) This expression can be integrated if the approximate relation (23) is used for n in (42). This is done in Appendix II. The resultant equation after the first integration is fairly complex and cannot be 27 further integrated without making drastic approximations. It is therefore deemed worthwhile to do a numerical solution of (41) with out the field gradient tern. This is discussed in the next chapter. CHAPTER V II NUMERICAL APPROACH TO THE SOLUTION OF THE TRANSPORT EQUATION Runge-Kutta-Gill method The method employed for the numerical solution of (41) without the field gradient term is the Runge-Kutta fourth order method[8 ], In order to apply this method on a digital computer, the calculation pro cedure developed by Gill was used[9]. This method was chosen for its simplicity and its self-starting feature; i.e. only the functional value at a single point is required to obtain the functional value a step ahead. The essential feature of the solution of - f (x,y) with the dx initial condition yCx^ = y0 is the calculation of the k's of Runge's formulas: = hf(5^ , yQ ) , k 2 = hfCxo + y h , yQ + | kx) , k 3 = hfCxo + y h, y0 + y k2) , (43) ki* = hf(xo + h, yQ + k3) , k = y(xo ♦ h) - y0 = i 28 29 of k = c1 k1 + c2 k 2 + C3 k 3 + c4 k4 , through terms of order h4. Hence the error in the solution is of order 1 1 5 . The Gill procedure is a refinement of the method with round-off error condensation at each step to yield greater accuracy. The expres sions are: kx = hf(xo, yo), yi = y0 + 7 Ckx - 2 q0 ), k2 = hfCxb + \ h, yi>, y2 = yi + Cl - /j>Ck2 - qi>, k3 = hf(Xq + |h , y2), y3 = y2 + Cl +/j)Ck3 - q2), k4 = hf(xo + h, y3), y4 = y3 + £ (Jo* - 2 q 3), qi = qo + 3 C? Ckx - 200)] - 7 k1# q2 = qi + 3 CCl -/j)(k2 - qi>] - (1 k2, (44) q3 = q2 + 3 [(1 +/})(k 3 - q2)] - (1 + 7}) k 3 , q 4 = q 3 + 3 [ 7 Oq* - 2q3)] - 7 k4 . As the q's are the correction terms, initially Qq is zero; q4 becomes q0 and y4 becomes yQ for the next step. We see that to evaluate y at Xq + h involves four operations each consisting of finding an increment in y (which is the k), a tentative value for y, and a correction term q. The step size is chosen to give the accuracy desired. This de pends on the behavior of the function f(x,y). If this function varies slowly with x, then the step size can be large, and the converse is true. One criterion is that (k2 - k 3 )/(k^ - k2 ) should not exceed a few per cent of h. For the integration of a second order differential equation, it is necessary to apply the Gill procedure twice at each point. The first 30 round is to obtain dy/dx from d^y/dx^; and with this as the intial value the second round gives y. These values of dy/dx and y are sub- stituted into d^y/dx^ to be integrated again. S imps on's three-point formula After having found the carrier distribution, its gradient and the corresponding current, vie can put these values into (38) to find the electric field at each point in the base region. To find the voltage drop over a region it is necessary to integrate (38). This integra tion is quite straight forward, and the three-point Simpson's nile[103 is adequate for this purpose. The three-point formula is given by (45) Y 2 - yQ where the primes indicate derivatives with respect to x, and s in the remainder term is some value between Xq and X2 » In terms of electric field and voltage without the remainder term we have V2 j " V2 (j-1 ) = | CE2 (j-l) + 4 E2 (j-^) + • (46) SunntLng over the interval 2Mi, we obtain the total voltage drop, V2 M - V0 , in terms of the electric field at each point in the interval M M where 2 M is the total number of intervals. 31 COmputer-graphi ca 1 method Solving (41) without the dE/dx term and to obtain the V-I char acteristic by computer alone requires Hatching the boundary conditions at the terminals of the diode. The matching procedure can be done by the computer: it involves comparing the integrated values with the final boundary values, correcting the initial values for any discrep ancy, then integrating again with the new initial values. This is done until the final boundary conditions are satisfied. Obviously this procedure requires considerable computer time. The process to be described can give the same accuracy with far less computer time. We see that (41) without the dE/dx term involves dp/dx, n and p. Thus we can start the integration with one of the two sets of initial conditions: (a) dp/dx = 0 , p = z > pQ , at x = 0 . (b) d 2 p/dx2 = 0 , p=z>pt), atx=Q. where z is a numerical value and p0 is the thermal equilibrium hole carrier density in the N-region of the P+NN+ diode, and x is somewhere in this region. The selection of these conditions is discussed in the following chapter. The integration with the Gill procedure is carried through x = d, the base width of the diode. At each point xj the values of Ja (xj) and Jb(xj) are calculated using (54) and (55) developed in the follow ing chapter. The correct value of J for the particular value of z is determined by plotting graphically the few values of Ja (xi) and Jfc(xj) which are approximately of the same magnitude and whose coordinates x^ 32 and xj are related by x^ + Xj ■ d. If there is no point where this is satisfied, the initial conditions have to be changed. That is, if initial condition (a) does not give a value which satisfies the final boundary conditions, initial condition (b) has to be used. With the current J detemined, J and z form a pair and are used in (38) to get the voltage drop in the base region. A skeletal program for the nu merical integration of d^p/dx^ written in the OSU SCATRAN language for the IBM 7090 digital computer is given in Appendix III, together with an exanple of the graphical boundary matching procedure. CHAPTER V I I I DERIVATION OF THE BOUNDARY VALUES In selecting the boundary values far the numerical solution of the second order differential equation, it is profitable to consider the general farm of the minority carrier distribution in the base region. Figure 7 shows five likely distributions. The first and the second derivatives with respect to distance of each distribution are sketched for conparison. (a) and (b) apply to an ordinary pn junction diode with ohmic contact x = d, where p and n decay to the equilibrium values. Whereas (c), (d) and (e) apply to a diode with nejority car rier contact, (e) is a possible carrier distribution for large leak age current. It is interesting to note that (b), (d) and (e) would not be pos sible if the (dp/dx) ^ term and the dE/dx term were dropped from (41); for then the second derivative could never be negative. The drop in the carrier density to the right of is a oonsequenoe of the presence of leakage current at the HL junction (HL for high-low), and the con dition of zero field intensity at x = d. The distribution adopted far our analysis is (c). This is the proper distribution if the electric field is not zero at the boundaries of the base region, and if the leakage current is not large. 33 (a) (b) ( Po 0 0 0 d £ fiS 0 Fig. 7.— Rough Sketches of Fire Possible Hols Carrier Distributions Together with Their First and Second Derivatives in the N-reglon of A Diode under Forward Bias Condition. 35 Configuration (c) has a basic property which enables us to simplify our computation, that is at x = Xq » dp/dx = 0 and p is a minimum. This is our initial condition in integrating (41) in both directions, to the left and to the right, from Xq . The boundary conditions at the edges of the transition regions of the pn junction and the HL junction respectively are now to be deter mined. The following notations are used: n-pp^ = Free carriers in N+-region. n 2 ,P2 = Free carriers in N-region. n 3 »P3 = Free carriers in P+-region. ni(0),pi(0) = Free carriers in N+-region just outside the transition region of the NN4 junction. n 2 (d),p2 (d) = Free carriers in N-region just outside the transition of the NP4 junction. n 2 (0 ),p2 (0 ) = Free carriers in N-region just outside the transition region of the NN4 junction. n(d),p(d) = Free carriers in P+-regian just outside the transition region of the NP4 junction. nio.Pio = Equilibrium carrier densitiesin N4 -region. n2o»P2o 3 Equilibrium carrier densitiesin N-region. n3o*P3o = Equilibrium carrier densitiesin P4 -region. It is to be assumed that recombination in the transition region is not significant in the range of the current density concerned. Instead, a leakage current at both junctions is to be included in the calculation. The approximation of constant quasi Fermi levels in the transition region 36 is also assumed, thus enabling us to relate the carrier densities on one edge of the transition region to those on the other edge. This is shown in the next chapter. Other assumptions are that Lp and 1^ in the heavily doped N+-regicn and P+-region respectively are snail but still large compared with the width s of the region; and that the flaws being mostly empty in the P+-regian and mostly filled in the N+-region, the hole carrier density gradient and the electron carrier density gradient are the same. These permit us to use the approxina- tions t pj.(0>.- pip (1)8) dx s and — * - ~ ftjo (49) dx s From (30), (31) and (38), we get the general expressions j p = ' {Jp ‘ ^ Cpf(p)+n] ^ > <50) and Jn = —i (Jhn + bqDh [pf(p)+n] (51) p + bn dx Applying (48) and (49) to (50) and (51) respectively we obtain, in the heavily doped regions, JPX = * { J ? 1 " (Pl+nl) CPl<0>“Plo3> . <52) 311(1 Jn3 = \ lT— • < ^ 3 - ^ Cp3 +n3) 0 1 3 (0 )^1 3 -,]} , (53) P3 + m 3 s As there is no recombination current in the transition regions, we can equate the hole current densities on each side of the NN+ 37 junction and the electron current densities on each side of the NP+ junction to obtain the following expressions: __ {(p1 +n1 )(p2 +bn2) (pl~plo) nlP 2 wl 2 Pl s - Cp^+bni) Cp2 f(p2 )+Ti2 ^ (54) J = Jb = ^ — 2 -- {(p3+Ti3)(p2+bn2> ^3*^30^ n 3 P 2 Wi 2 P 3 s + (p3 +bn3) Cp2 f(P2 )+n 2 ;i (55) dx The former applies to the NN+ junction, while the latter to the NP+ junction. The appendages (0) and (d) have been left out in the above equations; all p's and n's refer to the values at the edges of the transition regions. Expressions (54) and (55) are the boundary conditions we must satisfy simultaneously. Note that the terms pi, n^, p3, and n 3 can be put in terms of p 2 and n 2 with the help of the relations n 2 p 2 * niPi “ NdPi* at NN 4 junction, and n 2 P 2 * n 3 P 3 * n 3 Na* at NP+ junction. Ohmic contact Lightly doped Chmic contact N-region ni Pi n 2 P2 n 3 P 3 d d+s Heavily doped Heavily doped N+-region P+-regicn Fig. 8 — Planar N+NP+ diode structure with ohmic contacts at both ends CHAPTER IX DETERMINATTCN OF THE APPLIED POTENTIAL There are expressions for the determination of the applied voltage across a transition region in terms of the carrier densities at the edges of the transition region. In most cases, they are derived for high conductivity material and for long diffusion length compared with the transition region width. These expressions must therefore be applied with reservation for high base resistivity P+NN+ diode and for short diffusion length. In this chapter an expression relating the applied voltage across the transition region to the carrier densi ties outside the transition region is derived for a diode with a mod erately doped P-region and lightly doped N-region. One inescapable assumption we have to make is that under the for ward bias condition the diode is in a quasi thermal equilibrium state. This, together with the imposition of the level of mobile carrier den sities under 2 x 10^/cm3, enables us to make use of the Boltzmann distribution function. Thus we have the well-known relations p = n^ exp (*p - *)/VT , (56) and n = n^ exp (♦ - . (57) These can be put into another form using the Fermi potential + and the thermal equilibrium carrier densities pQ and iTq to give p = p0 exp (♦p - *)/VT , (58) 38 39 and n = nQ exp (♦ - ♦n )/Vr • (59) At thermal equilibrium ^ = 4 ; and, since Hq = rip and pc = pp in a p-type semiconductor while Hq = and po = pn in an n-type semi conductor, the diffusion potential at a pn junction is given by VD = VT In Ee = Vx In UlI . (60) °p Figure 9 is a potential diagram showing the relative positions of the various potential levels in a diode under the forward bias condi tion with the transition width enlarged. The applied potential is ob tained by taking the difference 4>(a) - $(b), which is exactly the same as (Vp - Vg) in magnitude. At the edges of the transition region n(a) = rip exp (♦(a) - ^ ( a ) ) / ^ , (61) n(b) = nn exp U(b) - ^ ( b ) ) / ^ , (62) p(a) = pp exp (♦p(a) - $(a))/V«r , (63) and p(b) = p^ exp (♦p(b) - $(b))/VT . (64) Dividing (58) by (59) and (61) by (60), we obtain = ilex p CVj - (♦„<«> - ♦„ = exp [Vj - VD - (^(a) - ^(b))]/Vj , (65) 3nd p(a) S " (#p(b) - $p(a))3/V^> , (66) where Vj is the applied potential across the junction. Multiplying (61) by (63), and (62) by (64), we get another set of equations. n(a) p(a) = n£ exp (tp(a) - ^ ( a ) ) / ^ , (67) and n(b) p(b) = n? exp (♦p(b) - ^ ( b ) ) / ^ . (6 8 ) Positive K-Region Transition P-Region Potential Region ♦v Chmic Ohmic Contact Contact ♦nOO-^ntb) 4 = ^ Original Fermi Level Drawn Relative to the Now Electrostatic Potential Electrostatic Potential without Applied Bias Fig. 9•— Potential Diagrsn of A p-n Junction Diode under Forward Bias, Shewing the Relative Position of Various levels. 41 It has been shown in the literature that the drop in the quasi Fermi level in the transition region is quite small[7,11], Thus (65) and (6 6 ) can be approximated to give 2 $ = exp (Vj - VD)/VT . £ g > . (69) This is the most important formula in diode theory. It takes other farms depending on the resistivity of the material employed in the making of the device and the level of injection. For fairly high con ductivity material n(b) * and p(a) * Pp. (69) then reduces to n(a) = np exp (V j/V -p) , (70) and p(b) = pn exp (V j/V p ) . (71) For low conductivity material, the correct expression is (69). This involves more calculation, because we have to solve the diode equations on both sides of the transition region. In the literature[12,13] equations (67) and (6 8 ) are expressed as n(a) p(a) = n? exp (V j/V T ) , (72) and n(b) p(b) = n£ exp (V j/V T ) , (73) rexpectively. This is acceptable only if the excess majority carrier densities are snail compared with the equilibrium carrier densities, so that ♦n(b) * 4(b), and *p(a) = *(a). But then (72) and (73) are iden tical to (70) and (71) respectively. Using (72) or (73) gives a false sense of accuracy at low injection, and a sizable error at very high injection, especially in the low conductivity side of the diode. Mother useful expression can be obtained by applying the approx imation of constant quasi Fermi level to (67) and (6 8 ) to give 42 n(a) p(a) = n(b) p(b) . (74) It is customary to neglect the voltage drop outside the transition, and this is a good approximation at low injection. For low conductivity material this is no longer permissible. Referring to the potential dia gram, Figure 9, the voltage drop in the N-region is given by Vtj = i|»(b) - g/(d); and in the P-region it is given by Vp = <|»(c) - This can be obtained analytically by integration of the electric field in each re gion. In order for this to be possible, the carrier distributions must first be determined. For the P+NN+ diode the voltage drops in the P+- and the N*-regions are negligible for most applications compared with the voltage drop in the N-region and across the F*N junction. Again it requires the inte gration of the electric field given by (38). Figure 10 shows the po tential diagram of the P^NN* diode with and without applied forward bias. The center figure is drawn without showing the change in the electrostatic potential in the N-region. To get a better understanding of how the various potential levels outside the transition regions oome to assume the depicted shapes, let us consider the diode equations in one dimension J = qup {E (ptbn) ♦ [b f(p) - 1 ] & ) , (75) Jn = qUn [Bn + £ 1 * -qutf ^ , (76) and Jp = qup Gp " = -qUpP • (77) As current flowing in one direction cannot change sign midway in the ■*-» tSIOl Af n ■ii ■ m r m n b ia s — nuw i m n «n un au a aew a Fig. 10.— Potential Diagram of a Diode diode, (76) and (77) require d ^ /dx and d^/dx be monotcnically increas ing or decreasing with distance. And as both n and p and their gra dients in the N-region of a pn junction diode decrease in magnitude toward the ohmic contact on the right, a natural consequence of dif fusion and recombination, (75) demands the electric field to increase to the right so as to keep J constant. E being the negative gradient of the electrostatic potential, we therefore have the electrostatic potential sloping increasingly to the right. For a P+NN+ diode, some modification of the level is required. Both n and p and their gradients decrease toward the oenter from both junctions. This is due to injection from both junctions. Hence the electric field should be largest at the point where the carrier den sities are mininum. We have so far determined oily the general shapes of the various levels. CHAPTER X DISCUSSION In this analysis, several approximations were made. Of these the most drastic one was the assumption of quasi-neutrality. This assump tion was nade use of in finding a relationship between the electron carrier density and the hole carrier density in the presence of reoonv- bination centers; and again in simplifying the second order differ ential equation (41). This assumption would have been perfectly ac ceptable had it not been for the low equilibrium carrier density used in some of the calculations. The electric field gradient term in (41) estimated with data obtained from the integration shows that this neglected term is by no means snail ccnpared with the other two terms in (41). The inclusion of this term would shift the carrier distribution minimum towards the P+N junction, at the same time reducing the carrier density to the left of the minimum (Figure 11) thus increasing the voltage drop. However, this shift would increase the carrier density to the right, thus decreasing the voltage drop. It is not easy to determine the net change in voltage drop. It seems that for the same voltage drop across the diode, the current density should be reduced on account of the smaller hole density gradient at the edge of the P*N junction. The change due to the inclusion of possible spaoe charge is roughly sketched in Figure 11. 45 N-R«gion 0 P N Junction UN Junction Pig. 11.—Hole Carrier Distribution and Ita Qradlont. The Daah Curves are Probable Configuration for the Spaoe Charge Caae. 47 In connection with the assumption of quasi-neutrality, Lampert has pointed out the inaccuracy which would occur at low injection level in a PIN diode, and stated that the V-I characteristics should approach those of a space-charge limited diode [14], This is not entirely satis factory in some of the cases calculated in view of the fact that there is no lack of majority carriers to neutralize the injected excess hole carriers for the formation of appreciable spaoe charge. The limiting factor in the N-region of the P+NN diode is in the low lifetime of the minority hole carriers. The spaoe-charge limited current flow is pre sent only in symmetric diodes such as NIN or FNP diodes where one junction is always reverse biased. Instead of the injection of majori ty carriers as at the NN+ junction of the P+NN+ diode, the IN or the NP junction actually extracts carriers from the middle (or base) region. For Chse 1 in Table I, the calculated spaoe-charge limited current at the threshold voltage (SCL^) is closer to the threshold diode cur rent than the rest. This is because this particular case has a large capture cross section for holes and a snail equilibrium majority car rier density. The large capture cross section contributes to a shorter diffusion length, and thus a greater base width to diffusion length ratio. Figure 12 shows the deviation of the calculated V-I curve from the SCL curve for this case before the breakover point. Even in this case, the current is far from the V2 dependency as expected by Lampert. One reason has already been mentioned above; another is in the exclu sion of diffusion current in the SCL case. Therefore for a P+NN+ diode with a snail low level lifetime and an almost intrinsic N-region, the correct V-I characteristics before the breakover point should be TABLE I Calculated Values for a Planar P+NH+ Diode at 300° K. d s 0.08 cm., Up = 500 cm2/volt-sec., b = 3 «r N n o *pl0 tpth sp 15 a V’nan • Vth xth SCLth x 10-15 x 1013 x 1013 x 1013 x 10-6 x 10-6 1. 30 7.14 7.0 0.007 100 3.7 300 0.047 0.043 0.28 9.9 x 10’2 2. 30 6. 94 7.0 0.07 100 3.77 54 0.048 0.237 0.33 3.2 x 10“3 3. 10 7.14 7.0 0.007 100 0.95 10.9 0.14 1.17 0.022 1.3 x 10'* 4. 1 29.0 30.0 1.0 20 1.85 5.2 0.345 2.46 0.45 2.98 x 10"5 5. 1 29.0 30.0 1.0 10 4.75 7.6 0.345 1.68 0.7 6.35 x 10*5 Formulas used to calculate some of the values above: SCLth * 9 ~ ~ 7 " h2 8 d 3 Density, Amp/cm2 T CM rH 'o i H rH rH o rH O Fig. 12.— Calculated V-I Curve of A P*NN* Diode under Forward Bias. Forward under Diode P*NN* A of Curve V-I Calculated 12.— Fig. 1 12 1C>3 102 10 1 Slope of 1 of Slope / ple otg, Volta Voltage, Applied Np-T.LUIO1^ tV S^*lf*0*C58 0 1 + 0 1 X 7 - O 1 I 0 0 1 * * Sp-3xlO-^ N-7XICA3 SCL Current SCL *♦9 50 similar to the modified SCL characteristics of Shockley and Prim[15], The modified SCL characteristics are derived with the inclusion of the diffusion current, and approach the SCL at high voltage. Figures 13, m , 15, and 16 show the current deviates progressly from any V0 (c= a constant) as the capture cross section decreases and the equilibrium majority carrier increases. The junction voltage drop also becomes increasingly dominant as the voltage drop in the base region decreases. As mentioned earlier, the calculated V-I characteristic curves shew the occurrence of negative resistance. This agrees well with Lampert's reasoning [I1*], i.e., the increase in the lifetime of the minority carriers due to an increase in the carrier density leads to large conductivity modulation and negative resistance in a double injection diode. Between the threshold voltage (or the breakover voltage) and the voltage minimum there appears an irregularity exhibiting a decrease in the current with voltage. Fhysically this could be explained as fol lows. With the collapse of the high field due to an increase in the carrier densities, the density gradients decrease as a consequence. Since the diffusion current dominates at the edge of the transition regions, this decrease in density gradients reduces the current flow. Thus we have the phenomenon of a positive resistance region in between two negative resistance regions. The fact that no positive resistance irregularity has been ob served could be attributed to two possible factors. First, the usual setup for displaying the volt-anpere curve of a diode consists of a Slop* of 2 /Slop* of 1 10~A 1 io 10‘ Applied Voltage, Volta Pig. 13e.— Calculated Applied Voltage Components and V-I Curve. Density, Aiip/c* 10- 10 id Vlae or a Cret est Poto o te rvos i e. re u fig Previous the of ortion P Density Current Lai r fo Voltage Diode - Fig. 13b.-~Caloulated Voltage Drop in the Base Base the in Drop Voltage 13b.-~Caloulated Fig. Slop* of 2 of Slop* Applied Voltage, Volts Voltage, Applied total 1 and and T otal otal T Applied Applied 10 52 Current Density, Amp/co^ 10 10 10 10 10 10 rl * r3 r5 -2 " ‘ ' 10-1 r / / / / / lp o 2 of Slope / t / Slope of 1 1 of Slope / / / i. i. acltdVI Curve. V-I Calculated lit.— Fig. / / / Applied Voltage, Volts Voltage, Applied 1 total \ 10 n 0 - - 0 n Ei-Ej^C.058 Sp = 10*14 10*14 = Sp Txiol3 H » « 7.1Axl013 Nr 100 0 -1 a 0 1 0 1 x 7 «V 10' 53 CM Current Density. Air.p/cr. 10 ' 10"' it 1• acltdApidVlseCmoet adVI Curve. V-I and Components Voltsye Applied Calculated Fift* 15•— lp of Slope lp of Slope Applied Voltage, Volts Volts Voltage, Applied Current Density, Amp/cm^ K) z Z «I A W vn or. ro D. » a 0* < o< wo I o I I < MI a cn tn 56 variable rectified sinusoidal voltage source and two resistors in series with the diode under test; one resistor is for monitoring the current through the diode and the other acts as a current limiter. Thus an increase in the applied voltage must be accompanied by an increase in the current. In other wards the usual setup cannot shew the positive resistance region in between two negative resistance regions. Second, transient effects such as trapping could have rrasked this irregularity in the retrace. Experimentally the V-I curve would take the shape of the dash line in Figure 17. This curve then looks strikingly similar to a V-I curve reported by Holonyak (Figure 3)[*+]. It is interesting to note that Figure 18 looks very much like Figure 6 of Holonyak*s. As in all cases, the current goes up almost vertically, the same as reported. Lamport*s conclusions that (a) the ratio of the threshold voltage to the minimum voltage is approxinately equal to the ratio of the cap ture cross sections, and (b) the hole transit time in the base region is equal to twice the low level lifetime, are not unequivocally sub stantiated by the calculated curves and data, nor by Holonyak*s experi ments. Table I shows some of the calculated values. From these values, it seems that Lanpert's analysis is applicable to wide-base diodes and to diodes with a near intrinsic base region, as in Cases 1 and 2 of Table I. This is expected considering that his analysis neglects com pletely the diffusion components of the diode current, and that no junction voltage is included. Figure 12 and 13 also show the sensitivity of the V-I characteris tics to changes in the equilibrium majority carrier density, especially Density, Amp/co' 0.1 0.2 0.3 o . How V-I Curve Would Appear on a Qunre-Treoor. a on Appear Curve Would V-I How 0 a 0 Pig. 17.— v-i Curve In Figure Figure In Curve v-i 17.— Pig. 2 4 Applied Voltage, Volts Voltage, Applied U 1 6 Redrawn on Linear Seale. The Deah line Shove line Deah The Seale. on Linear Redrawn 0 1 8 12 57 14 Current Density, Amp/cm' 20 12 16 0 4 8 2 6 8 6 6 2 0 Fig. 18.— V-I Curve in Figure 15 Redrawn on Linear Seale. Linear on Redrawn 15 Figure in Curve V-I 18.— Fig. ple otg, Volts Voltage, Applied Vtotal 58 59 when it is not much above the intrinsic level. Hence diodes containing recombination centers of this nature and having very low carrier den sity would exhibit very high photosensitivity [4,16 3. It must be mentioned that Stafeev has qualitatively shown how the change in the hole diffusion length can lead to negative resistance[163. His result is obtained by solving the linearized second order differ ential diode equation, then plotting V-I curves for various hole dif fusion lengths. Negative resistance is shown by moving progressively from one curve to the other as the diode current is increased. CHAPTER XI CONCLUSION To recapitulate, the modified nonlinear second order differential equation obtained by applying the Shockley-Read recombination mechanism to a P+NN+ diode containing monovalent recombination centers has been solved numerically by the Runge-Kutta method with a digital oomputer. The result shews that for a low equilibrium majority carrier density in the N-region, the hole density gradient is very much smaller than the electron density gradient at a low level of injection, and that the effective hole carrier lifetime increases by a factor approx imately equal to the capture cross section ratio. Physically, with the injection of excess hole carriers many electrons are released from the recombination centers per hole carrier; and as less filled centers are present to capture and recombine with the excess hole carrier, the lifetime of the hole carrier increases. This increase in the excess hole and electron densities in the low conductivity region of the P+NN+ diode both through injection by the junction and through the release of trapped electrons, results in conductivity modulation leading to the negative resistance effect. The calculated V-Icurves exhibit negative resistance character istics for several combinations of the flaw density, the capture cross sections far holes and for electrons, and the net donor density. The curves do not show the dependency before or after the negative 60 61 resistance region as expected by Lanpert. Their correctness in the general characteristics is vindicated by the experimental V-I curves of Holonyak. There is more than one factor in determining the threshold voltage, the capture cross section for holes and the equilibrium majority carrier density being the dominant factors. In general a large capture cross section for holes and a small equilibrium density give a large threshold voltage. Whereas the capture cross section for electrons (or the cap ture cross section ratio) and the density of the recombination centers are dominant in determining the minimum voltage, i.e., a large capture cross section for electrons (or snail capture cross section ratio) and a large density of recombination centers would give large minimum volt age. All this is a natural consequence of the relationships Tp»(nrSpV)-^ and Tn *[(Nr-nr )Snv]~^, where nr is large before breakover and snail after breakover. Che might ask whether a P+wN+ diode (n far lightly doped p-type) would give negative resistance characteristic if it contains monovalent flaws with capture cross section ratio much greater than unity, or much less than unity. The analysis points to an affirmative answer to both. It should be mentioned that for a p-n junction diode containing recom bination oenters having very large capture cross section ratio, it is not necessary to have double injection for the occurrence of ne^tive resistance. On the other hand, if the capture cross section ratio is not very large, double injection is a prerequisite in order to have a large conductivity modulation. APPENDIX I AN APPROACH TO NUMERICAL SOLUTION OF DIODE EQUATIONS WITH SPACE CHARGE If quasi neutrality cannot be assumed to exist in a diode under forward bias condition, the electron carrier density cannot be put in terms of the hole carrier density. It therefore requires the simul taneous integration of two coupled second order differential equations. The following expressions can be obtained from (30) to (34): (1-1) (1-2) where p + bn Nr(n+api) Dp (n+n^+ap+api) kTe n+nj+ap+ap^ and _ _U _ n^ dE Nr (n+api) Dn (n+ri]_+ap+ap]_) kTe n+n^+ap+ap^ N here is the net donor density, N 62 63 These two equations contain J, n and p onlyj they can be integrated from the minimum carrier density point as done in the integration of the approximate equation. One assumption to be irade here is that at a point where (dE/dx) = 0, dp/dx and dn/dx are also zero at that point. This is certainly not true, but the carrier density gradients are so small at that point that it should be a very good approxination. Any deviation should not affect the integrated result noticeably. With this assump tion, one obtains the following as the starting values: (1-3) ntn^+ap+api J = qDp £ £ (bn + p) , (1-4) kT S =* ~— — *t (1-5) to? Dp d?n _ U (1-6) d ? * One still has to choose a value far J (or E) for a particular in itial p. This means the solution cannot be determined by a one-pass process as done in Appendix III. Che way to choose an approximate value far J to start the integration is to use a value for J in between the space charge limited current and the solution of the approximate equation, for the low level case. While for the high level case, the solution of the approximate equation is as good a choice as any. Having chosen the initial values, one has to determine the final boundary conditions. From the electrostatic potential level shown in Figure 10, one would find that there are points at which d^/dx is zero. 64 If these points are taken as the two boundaries of the N-region, one would obtain for the final boundary condition from (38) Unfortunately this condition would not be permissible at very low injection level, on account of the fact that the transition region width of both junctions changes with injection. In other words, the point where diji/dx = 0 shifts towards the junction with increasing in jection at low level of injection. The amount of shift depends on the equilibrium carrier density in the N-region. At higher injection level, the shift would be insignificant compared with the step size of integration. For a wide base diode, the shift can be ignored even at extremely lew injection level. Offhand, it is hard to put a limit on the injection level or the minimum width for the shift to be negligible. The integration can be done as in Appendix III. At each point (1-9) is calculated. The calculated boundary for the NN* junction is at the point where J = J, and that of the NP+ junction is at the point X2 where J = -J. If x^ + X2 = d, the base width, the chosen pair of initial values, J and p, is correct. If (x^ + X2 > » d , J should be reduced, and the reverse is true, lb determine the exact value might require quite a few passes, but it can be programmed to do so if the amount of conputer time is of no concern. It should be pointed out this approach is also applicable to the transition region of a diode. APPENDIX II INTEGRATION OF THE APPROXIMATE EQUATION To integrate the approximate equation (*+2), it is necessary to put it in terns of p alone. An approximate expression for f(p) can be obtained by differentiating (22) with respect to p, and applying the binomial expansion as dene in obtaining (23). Thus we have Si!! = - — + - Nr+N»api+ni+ (a+1 )p (a+1) - 2Nr (Il-l) dP 2 2 [Nr+N+ap^+ti2 + (a+1 )p2 - 4Nr (N+ap^+p)]7 Using (23) and (II-l) in (**2), multiplying both sides of (42) by dp/dx, and integrating, we get 2 p 2+b1f *-b0 [g£] = Cl [p2 - p2(0)] + C2 Cp - p(0)]+ C3 In [■ p2(0)+B]p(0)+Bo P2 + 2I^P + D& + A p2(0)+2E^p(0)+qg+A [2p+B1-(Bj-4B0 >Y 2p( 0) +Bj + (B^-4Bq )y^ 2p+B1+(B2-4BQ4 2p(0)+B1-(B1-4Bo)^ p+E^- A p(0)+Cb+A ] + [ g <0>]2 , (II-2) P+Dq+ f k p ( o 7 + E b - A where the constants contain terms of flaw density, impurity density, mobility ratio, hole mobility, capture cross section ratio, capture cross section for holes, and flaw energy level. Further approximation 65 66 has to be made before (II-2) can be integrated. Even then the inte grated expression with the integration constants would still be too conplicated to be used in the electric field equation (38) from which the voltage drop in the base region can be obtained by further inte gration. This is one more reason, in addition to the limited range of applicability of the approximate equation (23), that the numerical so lution of the second order differential equation (41) is preferred. APPENDIX III COMPUTER PROGRAM AND EXAMPLE The chief features of the computer program are as follows: (a) Forma las to be used in the calculation are fed into and stored in the memory system of the computer. (b) Numerical coefficients are also fed into and stared in the memory. (c) The number of sets of initial values, the integration step size H, and the interval of integration HM are specified. (d) For the integration process, initial values are obtained from the memory, or calculated with the given data as the situation demands. (e) The integration is performed for each set of given data over the specified interval, and is repeated as many times as there are data. (f) Intermediate or final solutions are printed out as required. (g) The Runge-Kutta-Gill integration process consists of four rounds of calculation to extrapolate one set of values H unit of dis tance ahead from the starting point. This set of extrapolated values is used as the initial values for the next integration to get another set a step ahead of the last. This is carried on until the specified interval os covered. The skeletal source language statements presented below are writ ten in the OSU SCATRAN language[17] far the IRM 7090 digital computer. 67 68 Comments are inserted within square brackets to facilitate understand ing of the program. This program is written for the determination of the current density flowing througi a P+M + diode corresponding to the minority carrier density specified at the minimum point. Some of the symbolic representations are: Y=p/N, DRY=(d/N) dp/dx, DY2=(d2/N) *(d^p/dx2), ZA=f(p), EN=n/N, CJ=J the current density, *multipli cation operation. Skeletal Source Language Statements FORMULA (UI)=FDV.(Y,DRY)=(-GE*GJ+(GE(1)+GE(2)*ZA)*DRY)/ (Y+DD(1)*EN)- FORMULA (U2)=FDVA. (Y,DRY) = (GE*GJ+ (GECl )+GE(2) *ZA*DRY)/ (Y+DD(1)*EN)- FORMULA (U3)=FRT. CJ)=SQRT. (W(2)+Y(J)*W(3)+Y(J)*WOO))- FORMULA (W)=FXP. FORMULA FORMULA (U7)=FUC. (J)=DI(1)*Y(J)*EN-1/DD(2)>*(1/ZA+DD(1))/ (EN+DI(2)*Y(J)+DI(3))- FORMULA (U8)=FDY2. (UC)=UC-ZD- [In the above statements GE's, DD#s, DI*s, and W's are numerical coeffi cients. In the following statements, Afs, B's, C's, G's, FN, HN, M, NS and GA are also numerical coefficients. The integration process is to start with the calculation of sane initial values.] DO THROUGH (SOLVED),K=1,1,K,l£.NS- Y=YE(K)- J=0- OSUM=0- HM=0- Q(J)=0- QD(J)=0- DRY(J>=0- X=FRT.(J)- EN=FEN.(J)- XP=FXP.(J)- ZA=FZA.(J)- UC=FUC.(J)- ZD=FZD. (J)- DY2 (J)=FDY2. (UC)- H=HN- [The left side is the designation of the memory location where the numerical value corresponding to the right side is stored. The inte gration interval is specified by FN and also the maximum value of Y. ] DO THROUGH (SUM),M=1,1,(HM.I£.FN.AND.Y.LE.3.0.X.**)- GILL DO THROUGH (EXP0L),J=1,1,J.LE.«♦- HD(J)=H*DY2 (J-l)- DRYA(J)=A(J)*(HD(J)-B(J)*QD(J-1))- DRY(J)=DRY(J-1)+DKYA(J)- QD(J)=QD(J-1)+3*DRYA(J)-C(J)*HD(J)- HK(J)=H*DRY(J-1)- YA(J)=A(J)*HK(J)-B(J)*Q(J-1)*A(J)- Y(J)=Y(J-1)+YA(J)- Q(J)=Q(J-1)+3*YA(J)-C(J)*HK(J)- 70 X=FRT. (J>- EN=FEN.(J)- XP=FXP.(J)- ZAp FZA. (J>- u c =f u c .(J)- ZD=FZD.(J>- EXPOL DY2(J)=FDY2.(UC)- [The Gill integration process is from GILL to EXPOL. ] DY2(0)=DY2(4)- DRY (0 ) =DKY (4 )- Y(0)=Y(4)- QD(0)=QD(4)- Q(0)=Q(4)- [The above are resetting the initial values for the next step, the boundary values DB=Ja and RB=Jb are to be calculated next.] SIMP EP=Y*EN- EY=EN/Y- DB= ( (ZA+EY) * CEP/DD( 1 )+G( 3 ) ) *DRY/DD( 0 ) + (EP+G(3 ) ) * (EP-GCl) ) * U+DD(1)*EY)*G(2))/ RB=(DRY*(1+EX*ZA) *(DD(1)*EP+G(3))/DD(3)+(EP+G(3))*CEP-G(1))* (l+EX/EDCl))*GA)/(G(3)-Y.P.2)- WRITE OUTPUT,l,(HMtY,DB,RB,ENfDRY)- SUM CONTINUE- SOLVED CONTINUE- 71 Computation time can be shortened by putting bypass statements before DB and RB, so that only values of DB within the interval HM=0 to HM=l/2, and the values of RB within the interval HM=l/2 to HM=1, are to be calculated. This is because the matching point is likely to occur within these intervals respectively. With the calculated values of DB and RB, the exact value for the current J corresponding to the initial value of p can be obtained by the graphical method which will be demonstrated later. Assuming that we have determined the value for J, we can now find the voltage drop in the base region by the Simpson's three point formula. The above program can be modified to include this suimring process. Tor this purpose the following statements are totake the place of the state ments after SIMP. Modification Statements DV=0- TRANSFER TO (FA) PROVIDED (HM.G.P(K))- [This statement is to bypass the following after the required interval has been covered.] W=FDVA. (Y,DRY)- FA IV1=FDV.(Y,DRY)- TRANSFER TO (FD) PROVIDED (M.E.2*(M/2))- OSUM=OSUM-EV+DV1- TRANSFER TO (SUM)- FD ESUM=ESUM-DV+EV1- SUM CONTINUE- EPO= (ESUM*2+0SUM**+ )*H/3- 72 WRITE OUTPUT,1,(CJ, YE (K) ,Y,EN,HM,EPO)- SOLVED CONT3MJE- END PROGRAM (START)- EPO gives the voltage drop in the base region corresponding to the current density CJ. There should be a small correction to this voltage according to (44). But if the number of steps 2M is large, this correction can be safely neglected. The following are some computed values for the case Sp=10“15, N=3sd0ll+, a=20, n ^ l O ^ 3 , z=l.962x10^, d=0.08 in e.g.s. unit: *i x3■j Ja (xi) Jb (Xj ) d d 0.22 1.616xl0"3 0.75 1.291xlQ"3 0.23 2.003xl0“3 0.76 1.696xl0“3 0.24 2.469xl0“3 0.77 2.217xl0~3 0.25 3.024xl0-3 0.78 2.884xl0“3 The interpolated value for the current density is 2.096x10 ampere/cm2 , at xj/d=.232 and Xj/ds.768. BIBLIOGRAPHY 1. M. Lax, "Cascade Capture of Electrons in Solids," Fhys. Rev., vol. 119, 1502; September, 1960. 2. W. Shockley and W. T. Read, Jr., "Statistics of the Recombina tions of Holes and Electrons," Fhys. Rev., vol. 87, 835; September, 1952. 3. A. Rose, "Recombination Processes in Insulators and Semicon ductors," Fhys. Rev., vol. 97, 322; January, 1955. 4. N. Holcnyak, Jr., "Double Injection Diodes and Related DI Phenomena in Semiconductors," Proc. IRE, vol. 50, 2421; December, 1962. 5. R. H. Bube, Photoconductivity of Solids, p. 313, John Wiley and Sons, Inc., New York, N. Y.; 1960. 6. W. W. Tyler, "Injection Breakdown in Iron-Doped Germanium Diode," Fhys. Rev,, vol. 96, 226; October, 1954. 7. C. T. Sah, R. N. Noyce and W. Shockley, "Carrier Generation and Recombination in PN Junctions and PN Junction Characteristics," Proc. IRE, vol. 45, 1228; September, 1957. 8. W. E. Milne, Numerical Solution of Differential Equations, p. 72, John Wiley and Sons, Inc., New York, N. Y.; 1953. 73 9. M. J. Romanelli, " Runge-Kutta Method for the Solution of Ordinary Differential Equation," Matheratical Methods far Digital Computer, pp. 110-120, John Wiley and Sons, Inc., New York, N. Y.; 1960. 10. W. E. Milne, op. cit., p.*+8. 11. R. D. MLddlebrook, An Introduction to Junction Transistor Theory, p. 289, John Wiley and Sons, Inc., New York, N. Y.; 1957. 12. A. K. Jonscher, Principles of Semiconductor Devioe Operation, p. 85, John Wiley and Sons, Inc., New York, N. Y.; 1960, 13. L, B. Valdes, The Fhysical Theory of Transistors, p. 188, McGraw-Hill Book Co., New York, N.Y.; 1961. 14. M. Lampert, "Double Injection in Insulators," Fhys. Rev., vol. 125, 763; January, 1962. 15. W. Shockley and R. C. Prim, "Space-Charge Limited Emission in Semiconductors," Fhys. Rev., vol. 90, 753; June, 1953. 16. V. I. Stafeev, "Modulation of Diffusion Length as A New Principle of Operation of Semiconductor Devices," Soviet Fhys. — Solid State, vol. 1, 763; Deoember, 1959. 17. 709/7090 SCATRAN Primer, Numerical Computation Laboratory, The Chio State Ihiversity, Columbus, Chio; 1962. AUTOBIOGRAPHY I, Tet Chong Pang, b o m in North Borneo, attended high school in Hong Kong. I received the B.S. degree in Engineering Physics from the Uhiversity of Illinois, Qiampagne-Ufrbana, Illinois in 1954, and the M.Sc. degree in Electrical Engineering from The Chio State Uhiversity, Columbus, Ohio in 1957. I worked in the capacity of a design and development engineer at Admiral Corporation, Chicago, Illinois from 1954 to 1955; and was associated with the Electron Device Laboratory of The Ohio State University as a research assistant from 1956 to 1961. Since 1962 I have been associated with F. W. Bell, Inc. and Continental Electronics, Inc., Columbus, Ohio, engaging in research and develop ment in semiconductor devices. I am a member of the Institute of Electric and Electronics Engineers, and Tau Beta Pi. 75