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UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 INTERNAL REPORT NO. 6h 
 
 FUNDAMENTALS OF 
 JUNCTION TRANSISTOR PHYSICS 
 
 By 
 W. J. Poppelbaum 
 
 August 30, 1955 
 
 This work has been supported "by 
 Contract N6ori-71 Task XXIV 
 United States Navy 
 ONR NR 048 094 
 
TABLE OF CONTENTS 
 
 1. The Number of Possible States of a System of Particles 1 
 
 2. Number of Occupied States. Fermi Statistics 2 
 
 3. The Physical Meaning of the Fermi- level J4. 
 k. Electrons and Holes. Energy Bands. Electron and Hole Densities 5 
 
 5. Origin of Energy. Intrinsic Densities. Equation for the Electro- 
 static Potential 7 
 
 6. P- regions and n-regions. and y in an Equilibrium Junction. 9 
 Quasi Fermi- levels 
 
 7. Mobilities, Diffusion Constants, Einstein Relationship. Current 
 Densities expressed as Gradients of Quasi Fermi-levels 11 
 
 8. Generation and Recombination of Holes and Electrons. Continuity 
 Equation 13 
 
 9. Variation of , and "^ in a Non-equilibrium pn Junction lk 
 
 10. The Voltage- current Characteristic of a pn Junction 17 
 
 11. Variation of , and y/in a Non-equilibrium pnp Transistor 19 
 
 12. The Voltage- current Characteristics of a pnp Transistor 22 
 
 13. Behavior of the Base Region at High Injection Levels 2k 
 lk. Carrier Delay in the Base Region as a Function of Injection Level 25 
 
 15. The Current Amplification a of a Transistor 28 
 
 16. The a Cutoff Frequency as a Function of Injection Level 31 
 
 17. The Transition Region Capacitance 55 
 
 18. The Collector Space Charge Layer 5° 
 
 Ac know ledge me nt 
 
 The deepest gratitude is due to Professor J. Bardeen 
 for his helpful criticism and the suggestions he has 
 made for the improvement of many sections. 
 
INTRODUCTION 
 
 The following sections are an introduction to the theories used in 
 the discussion of transistor "behavior. To simplify explanations -we only con- 
 sider pnp transistors: all results can easily be transformed for npn units. 
 The text is in no way original except for details of exposition and certain 
 conclusions reached in the latter sections of the report. The exposition 
 leans on diverse treatises on solid state physics and a number of articles 
 which have become classic in the field. The principal sources used are: 
 
 J. Bardeen: 
 R. C. Tolman: 
 
 F. Seitz: 
 
 W. Shockley: 
 
 W. Shockley, 
 M. Sparks , 
 
 G. K. Teal: 
 
 W. Shockley: 
 
 C . Herring 8 
 
 E. So Rittner: 
 T. Misawa: 
 
 J. M. Early: 
 
 Fundamentals of Transistor Devices. Lectures at 
 the Graduate Summer School in Semiconductors (1953) 
 
 The Principles of Statistical Mechanics. Oxford 
 University Press (1950). 
 
 Modern Theory of Solids, McGraw-Hill (1940). 
 
 Electrons and Holes in Semiconductors, Van Nostrand 
 (1950). 
 
 PN Junction Transistors, Physical Review, vol. 83 , 
 No. 1, pp. 151-162 (July 1951). 
 
 The Theory of pn Junctions in Semi-conductors and 
 pn Junction Transistors, BSTJ, vol. 28, pp. 435-489, 
 (July 19^9). 
 
 Theory of Transient Phenomena in the Transport of 
 Holes in an Excess Semiconductor, BSTJ, vol. 28 , 
 pp. 402-469 (July 1949). 
 
 Extension of the Theory of the Junction Transistor, 
 Physical Review, vol. 94, No. 5, pp. Il6l-1171 (June 1954) 
 
 Diffusion Capacitances and High Injection Level Operation 
 of Junction Transistors, Proceedings I.R.E. p. 749 
 (June 1955)- 
 
 Effects of Space- charge Layer Widening in Junction 
 Transistors, Proceedings I.R.E. , p. l401 (November 1952). 
 
1. The Number of Possible States of a System of Particles 
 
 Let V be a volume in free space containing 
 
 N particles (for example electrons). Let x be 
 
 the radius vector from an arbitrary origin and 
 
 —> — > 
 
 let a- = mv be the momentum vector of a particle. 
 
 It is convenient to represent the succession of 
 
 states of movement (x,c-) in a 6 dimensional space 
 
 called phase space and having coordinates x x x 
 
 Figure 1 a ~ cr °z. The fundamental assumptions of quantum 
 
 Coordinate Vector and mechanics can then be summarized as follows : 
 Momentum Vector 
 
 — » — > 
 If by any apparatus we try to determine the components of x and o~ 
 
 simultaneously, we will encounter uncertainties Ax. and A&~. such 
 
 J ' 11 
 
 that for every value of i = 1, 2, 3 Ax .As". = h, where h is Planck's 
 
 constant. /„ . , . „ . . , \ 
 
 (Heisenberg's Principle; 
 
 We deduce from Heisenberg's Principle that if we divide phase space into cells 
 of size Ax 1 Ax_Ax : A < 3- Ao^ioo = h , we cannot distinguish the state of movement of 
 two particles in the same cell. Statistics applied to electrons are based upon 
 the following hypothesis: 
 
 3 
 Each cell of size h in phase space can only accomodate 2 particles 
 
 (of opposite magnetic momenta or "spins") at the most. 
 
 (Pauli's Principle) 
 
 Remark : We can apply the above theory even in the case where the electrons are 
 
 inside a crystal. Then the "crystal momentum"^ is no longer equal to mv but 
 
 h -> -* 
 still equals — r— k where k is the wave vector of the associated wave, 
 2rt 
 
 Notice that each cell corresponds to a definite energy E = E(x,cr) but that for 
 a given energy there may be several cells . 
 
 Let us now choose a volume V = Ax..Ax^Ax small enough to neglect the variation 
 of x; then E = E(s-) and the size of each cell in cr-space ("momentum space") becomes 
 
 This cell contains again at most 2 particles . 
 
 We can now calculate the number of possible states having energies between 
 E and E + dE if we know E as a function of o-„ It turns out that very often 
 
2 
 
 where E = potential energy of particle 
 
 (zero of energy to be fixed later) j 
 
 2 
 
 %r- = kinetic energy. 
 2m 
 
 Now the volume of a spherical shell between the radii (T and <r + do - is kgf* d<f. 
 The number of states N(cr)dcr corresponding to this shell is therefore 
 
 N(<r)dc-= =* W^<5\ (1.2) 
 
 IT 
 
 Let E and E + dE he the energies corresponding to crand o~+ do - by (l): 
 differentiating 
 
 dE = 2"do-. (1.3) 
 
 m * 
 
 Substituting (3) and (l) into (2) we obtain for the number of possible states 
 N(E)dE between E and E + dE 
 
 N(E)dE = i£ (2m) 3/2 (E - E ) l/2 . (lA) 
 
 IT n 
 
 2. Number of Occupied States. Fermi Statistics . 
 
 The total number of possible states can be much larger than the number N 
 
 of particles in V. We shall now try to determine the percentage of the possible 
 
 states which are occupied. For this purpose we divide the states (corresponding 
 
 to a small volume V) into groups 1, 2, ..., v, ... such that the r group has 
 
 an energy between E and E + dE . Suppose that there are S possible states 
 
 in the group, N of which are occupied. Let us count the number of ways in 
 
 which the distribution N,N^...N ... can be set up: This number is called the 
 
 1 2 r 
 
 probability W of the distribution (notice that W is not normalized). By the 
 laws of permutations 
 
 r r r r 
 
 It is evident that the physically realized distribution will render W 
 maximum: small perturbations satisfying 
 
En ■ Constant N (constant number of particles ) (2 „2 ) 
 ZEN = Constant E (constant total energy) (2.3) 
 
 must therefore leave W (or InW) unchanged . Let 1 and 2 he two perturbations 
 such that 1 changes E "by 5E. and InW hy (5 InW), while 2_ changes E "by 5E p and 
 InW by (5 lnW) 2 » Then 
 
 (5 InW) -I- (6 InW) = 5 InW = 
 
 and 5E X + 5E p =0 (hy 2.3) 
 
 simultaneously. Since the perturbations are arbitrary, this means that 
 
 5 InW = /6>oE (2 ok) 
 
 where g is a quantity independent of E and W. By comparing the result of 
 statistical calculations to classical thermodynamics, it can be shown that 
 
 ^-4r (2 - 5) 
 
 where k is Boltzmann's constant and T the absolute temperature. 
 
 Now consider a perturbation more in detail. Suppose it consists 
 in the transfer of 5N particles from group r to group s « Then 8N = -5N 
 and SE = fa - EjSN . Further (2.1) gives 
 
 5 InW = -5 ln(N i) - 5 ln[(S - N )l] 
 ~S ln(N g O a. 5 ln[ (S - N )'.]. 
 
 Since lnxi = x lnx, we have 
 
 5 InW = -8N InN + 5N ln(S - N ) - SN In N + 5N ln(S - N ) 
 r r r N r r s s ss s 
 
 5(S - N ) = -5N etc.'.) and hy using (2.k) and (2.5) 
 
 [-InN + ln(S - N ) + InN - ln(S - N )] 
 r x r r s s s' 
 
 E N 
 
 r s s 
 
 1 
 
 o_ 
 
 InW 
 
 1 
 
 kT 
 
 
 5E 
 
 E * E 
 r s 
 
 gives 
 
 
 
 
 
 
 E 
 r 
 
 N 
 
 1 "■ n ' 
 
 
 + J1 S m N 
 
This being true for all groups r and s, we snail set each side equal to E_/lff 
 and call EL, the Fermi-level. Then for any r 
 
 N 
 In s _ r N = (Eg, - E r )AT. (2.7) 
 
 r r 
 
 Now introduce the distribution function (value for group r) by 
 
 N 
 
 r 
 
 Then 
 
 f 1 
 
 r " (E -E )/KT 
 1 + e ' 
 
 i.e., for any given energy range E, E + dE the quotient (possible states)/ 
 (occupied states ) is 
 
 (2.9) 
 
 1 + e (E-V/M 
 
 ^. The Physical Meaning of the Fermi -Level 
 
 As can be seen from (2-9) f = l/2 when E = E^, i.e. 50$ of the states are 
 
 r 
 
 occupied for an energy equal to the Fermi-level. 
 
 We are now going to prove that if we transfer 5N electrons (of charge-q) 
 
 from a conducter in equilibrium at level EL, and electrostatic potential T, 
 
 to a conductor 2 in equilibrium at level E and potential V in a reversible 
 
 J? 2 2 
 
 fashion, 
 
 -*C*i - V 2 ) = E F1 - E F2 (3-D 
 
 or, for any origin of E^ and V 
 
 V = + constant (3-2) 
 
 It is possible to show by irreversible thermodynamics that in the most general 
 case of flow of electric and thermal currents (3'2) is still true. 
 
 To prove (3-1), we transfer 5N electrons from conductor 1 to a conductor 
 2 at the same temperature in a reversible fashion (total 8S = and therefore 
 5F = 5E - T6S = 5E, where F is the free energy). Now 5F = 5F + 5F and by a 
 
well-known theorem of thermodynamics S = k InW (in the union of 2 systems we 
 have to add entropies and multiply probabilities'.). Therefore 
 
 5F 2 = 5E 2 - T5S 2 = 5E 2 - kT(S lnW) 2 (3-3) 
 
 Suppose that we add the electrons to states in group r in conductor 2 . Then by 
 the preceding section 
 
 (5 lnW) 2 = - lJ—{— 
 
 It follows that 
 
 / N 
 5? 2 = 5E 2 + kT 8N ln/g— §- ^ 
 
 \ r ' r / 2 
 
 = E^ SN (3 A) 
 
 by utilizing (2.6). We see that 8F is independent of the group of states to 
 which the electrons are added. 
 
 Using 8F = SF + SF we find that 
 
 5F = (E^ - E F1 )5N (3-5) 
 
 while 5E = q.(V - V g )5N (3*6) 
 
 As mentioned above, we have in this particular case BF = BE. Equation (3«l) 
 thus follows immediately. 
 
 h. Electrons and Soles. Energy Bands. Electron and Hole Density 
 
 By a more detailed examination of the quantum mechanics involved in the 
 movement of electrons inside a crystal, it can be shown that these electrons 
 can only have energies in certain permitted ranges called bands. In the most 
 common case there is a nearly empty band in which the electrons behave more or 
 less like free particles % this is the so-called, conduction band. Then there 
 is a nearly filled band; the valency band. Electrons in the valency band are 
 better described by the behavior of the vacant energy levels or holes. 
 Practically a hole behaves like a particle of charge +q having approximately 
 the electronic mass m. 
 
We shall now calculate the electron density in the conduction hand. For 
 this we set V = 1 in (l.k), obtaining the number of possible states per unit 
 volume : 
 
 N(E)dE = i| (2m) 3/2 (E - Ej 1 / 2 
 h 
 
 The fraction f by which we must multiply to find the occupied states is given 
 by (2.9): 
 
 f _ 1 
 
 (E-1L, J/kT 
 1 + e * 
 
 The number of occupied states between the energy limits EL and E of the con- 
 duction band is thus 
 
 •E 
 
 
 2 N(E)dE_ 
 1 1 + e 
 
 (E-EpVl- (U - 1} 
 
 Now this is obviously the electron density. We shall calculate (^.l) under 
 the hypothesis that E > > E_ 3 then 
 
 (E-E J7S 
 1 + e ^ 
 
 1 -(E-E F )/kT 
 
 = e 
 
 (^.2) 
 
 and 
 
 'E, 
 
 = e V«r M^t' 2 f 2 (E . a )i/2 ,-Vh 
 
 dE 
 
 E i 
 
 Putting E* = E - E and taking account of the rapid convergence (which permits 
 to extend the upper limit to +°°), we may write 
 
 n - jEF-EnV^ M2m) 3 / 2 (^ ^&/^ & 
 
 Y? Jo 
 
 _ . e -(En-E F )/kT (k.3) 
 
 n 
 
where 
 
 o ^/p 
 N = 2(2jtmkT/h ) ' = "effective density of electrons in the 
 
 conduction hand." (^••^■) 
 
 By reasoning on the holes in the valency band, the potential energy E^ 
 
 r n€ 
 
 density 
 
 being necessarily measured from the same origin as E , we find for the hole 
 
 where 
 
 p = N e"^ "EpV^ (^5) 
 
 P 
 
 N = "effective density of holes in the valency band." 
 
 5. Origin of Energy. Intrinsic Densities. Equation for Electrostatic 
 Potential 
 
 We shall now proceed to choose the origin of the energies E, E , E , E_, 
 
 n p s 
 
 in such a way that there will be some simplification in the formalism. Let 
 
 y= electrostatic potential (measured from infinity) 
 
 E = energy gap between the top of the valency band and 
 the bottom of the conduction band. 
 
 The energy origin will then be chosen in such a way that 
 
 ] /M E r 
 E n = -qy+^KE Hr)'T- (5 ' 1} 
 
 leading im mediately to 
 
 /N \ E 
 E = KV+ikT ln(Jl--^ . (5.3) 
 
 V p/ 
 
 Further let us introduce a Fermi -potential f> such that 
 
 E F = -q0 (5-3) 
 
 Then (4-3) and (4.5 ) can be written 
 
 ; q(^-0)/kT 
 
 ,q(0-yVfcr 
 
 n = n. e 
 
 1 
 
 (5.10 
 
 n. e 
 
 1 
 
where 
 
 n i = / i n i p e " EG/kT = i^~' (5-5) 
 
 It can be seen that f or V' = 0, p = n = n.: n. is called the intrinsic 
 carrier density. (^.^0 shows that n. is independent of the location in the 
 crystal as long as E is constant: This is usually assumed to be the case. 
 Formulae (5«*0 are therefore perfectly general, n. having the same value every- 
 where; only and "y vary with the location. 
 
 We can now examine the case in which the crystal contains electrons, holes 
 and ionized atoms. Let N, be the density of positively ionized atoms (called 
 donors) and N the density of negatively ionized atoms (called acceptors). The 
 
 8 
 
 distribution of donors and acceptors is determined during the growing of the 
 crystal. We thus obtain for the charge density 
 
 f>= q(p - n + N d - N a ). (5-6) 
 
 Substituting for n and p the values (5«*0 we have 
 
 /> = q[N d (x) - N a (x) + 2n. sinhq^(x) -y(x)|/kT] , (5-7) 
 
 If we specify 0(x) we can calculate ~yu from Poisson's equation 
 
 *) 
 
 -div E = lap y> = - ■£ (5.8) 
 
 r o 
 
 (MKS-system, e = dielectric constant of vacuum 
 
 € = relative dielectric constant of crystal) 
 
 In the treatment of transistor problems two cases are usually considered: 
 as a first approximation a zero space charge solution (/°= 0) at low current 
 densities and then a zero space charge solution at high current densities. Only 
 in exceptional cases is one able to treat the case p f 0: we shall not attempt 
 to do so until section 18. 
 
 :2 
 
 *) In the one dimensional case lap = — gf 
 
 9x 
 
6. p-regions and n-regions. and y/ in an Equilibrium Junction. Quasi 
 
 Fermi -levels 
 
 p 
 
 n 
 
 N N, 
 ap dp 
 
 N N, 
 an dn 
 
 P n 
 P P 
 
 P n 
 n n 
 
 Figure 2 
 
 Notation for p-regions 
 and n-regions 
 
 (5.6 ) gives 
 
 Consider two adjacent regions distinguished by 
 the indices p and n such that in the left hand region 
 and in the right hand region respectively we have 
 
 ap dp 
 
 = P o > ° ) 
 
 N - N^ = -n < 
 an dn o 
 
 (6.1) 
 
 Let us consider the zero space charge case 5 then 
 
 p = p + n 
 P o p 
 
 n = n 
 n o 
 
 (6.2) 
 
 n 
 
 We shall suppose that 
 
 p > > n 
 
 P P 
 
 n > > p 
 
 n 
 
 . 3 
 
 (6-3) 
 
 The p-region then practically only contains holes, while the n-region practically 
 
 only contains electrons . Electrons in the p-region or holes in the n-region are 
 
 called minority carriers . 
 
 Let us try to gain some insight into the equilibrium behavior of and 
 
 "y^when we go from a p-region into an n-region across a "pn junction" i.e. 
 
 the transition region. The important thing to notice is that must have the 
 
 same value throughout the sample: if it were not so, energy changes according 
 
 to section 3 would accompany the transfer of electrons (or holes ) from one 
 
 region to the other and this contradicts the hypothesis of equilibrium. This 
 
 does not imply by any means that /-'is the same everywhere. But we can see 
 
 that Y' must be constant at all locations where the concentration of carriers 
 
 p n and p n is constant, i.e. inside the p-region and inside the n-region. 
 *p p *n n ' * & to 
 
 This is evident because carriers can only drift as a consequence of concentra- 
 tion gradients and electric fields E = -grady<*)s if there is equilibrium and 
 no concentration gradient, there can be no potential gradients either. 
 
 *; In the one dimensional case grad Y* = — — 
 
 a*- 
 
10 
 
 fl- reg 
 
 /0 5-7 
 
 
 r 
 
 72 - -r<2jt'er\ 
 
 Figure 3 
 and V i n an Equilibrium pn Junction 
 
 In the transistion zone however a big potential gradient must he present to 
 
 offset the influence of the rapidly varying concentration of holes and electrons: 
 
 and y> behave as indicated in Figure 3« 
 
 In equilibrium the computation of p and n (called p and n to indicate 
 
 * e n p n. p 
 
 equilibrium'.) offers no difficulty when p and n are given. (5«^) and (5«5) 
 
 are valid, therefore (p + n )n = n. and (n + p )p = n., giving by (6.3) 
 
 o p p 1 o n n 1 J v 
 
 o 
 1 
 P 
 
 = n,/p 
 
 2 
 
 n./n 
 
 (6A) 
 
 Let us now take the non-equilibrium case. Then we do not know anything 
 a priori about the Fermi-levels in the p- and n-regions . Worse stills when 
 current flows across a pn junction we are evidently no longer entitled to use 
 formulae (5«*0 which are a consequence of (^-3) and (^«5) and therefore valid 
 only under equilibrium conditions . 
 
 Nevertheless we can write relationships analogous to (5>*0 containing 
 "quasi Fermi-levels" (sometimes called "imrefs") and to be defined in 
 a suitable way: 
 
 n 
 
 n.e 
 
 1 
 
 q(/<-0 n )/kT 
 
 (6.5) 
 
 p = n.e 
 
 * 1 
 
 q(0 p -y)/kT 
 
 Solving (6.k) for and we then obtain formulae defining the quasi Fermi- 
 levels : 
 
n 
 
 r n ' q ' n i j 
 
 w,/.** (6>6) 
 
 In section 9 we shall look into the behavior of ys, and in a junction 
 when current flows across it, i.e. the behavior in the non-equilibrium case. 
 
 Mobilities, Diffusion Constants, Einstein Relationship and Current- 
 Densities Expressed as Gradients of Quasi Fermi -levels 
 
 The movement of electrically charged particles can come about in two ways; 
 firstly they can attain a certain average speed due to the electric field 
 present, the speed being limited because of collisions with the crystal lattice, 
 It turns out that often the average speed is proportional to the electric field: 
 one can then define a constant, called mobility u, which is the average speed 
 per unit electric field. 
 
 Secondly the movement can be caused by concentration gradients 2 the 
 
 (random-) thermal Telocity then produces a general drift to the regions of 
 
 lower carrier density. The number of carriers drifting across unit area in 
 
 unit time often turns out to be proportional to the concentration gradient; 
 
 one can then define a constant, called diffusion constant D, which is the 
 
 particle current density per unit concentration gradient. 
 
 Take for example holes having a mobility u and a diffusion constant D . 
 
 P P 
 
 Let p be their volume density and q their charge as before. Call E = - grad y* 
 
 = - —*■ the electric field. Then the total particle current density due t,o E 
 and grad p is u- pE - D grad p, i.e. the current density I for holes is given 
 
 fey 
 
 I = qG-yE - D p grad p) . (7-1) 
 
 In the same way we would have for electrons 
 
 I = q(jj. nE + D grad n) . (7-2) 
 
 n n n *' ' 
 
 Let us take an inhomogeneous p-region in equilibrium. Then we can write 
 
 P = Ae-^/ M (7.3) 
 
q0 
 where A = n.e^ has the same value throughout the region as discussed in section 6 
 
 but Y= y(x). Using (7-3) we obtain 
 
 arad T,-^-- B i^-H E 
 grad P - 3^ - P Hdx - M ^ 
 
 Since we have equilibrium, I =0 and (7*1 ) gives 
 
 IT 
 
 . pqEfc - D p £) 
 
 or 
 
 |i = rt" D (Einstein relationship) (7«^) 
 
 In a similar way we can prove that 
 
 We can now establish a relationship between the gradients of and 
 and I and I in the general non-equilibrium case. We then have by equations 
 
 (6.5) 
 
 n = n e q(v-0n)/kT 
 
 (7.6) 
 
 p = n r «l(0p-/)/KP 
 
 r i 
 
 Let us now use (7-1) and (7'2), remembering (7^) and (7-5 )• Calculating' 
 grad p from (7-6) we have 
 
 grad p = p ^r (grad - grad 7^), 
 
 so that (7'l) gives 
 
 I = qp. pE - q — up^r- (grad + E) 
 p p q p^ kT r -p 
 
 = -wjp g rad 0-. (7*7) 
 
 x^ Jr 
 
 In the same way we can prove that 
 
 I = -qu n grad (7 «8) 
 
 n * p r n 
 
 *) It is sufficient to consider the one -dimensional case in which grad p = 5g£ 
 
13 
 
 8. Generation and Recombination of Holes and Electrons. Continuity 
 
 Equation 
 
 If we isolate unit volume inside a semiconductor (not necessarily in equili- 
 brium), the number of holes (or electrons) in this volume varies for several 
 causes. First of all holes and electrons recombine (i.e. electrons "fall" into 
 the vacant energy levels called holes) at a rate r particle pairs/unit time. 
 Then holes are generated due to thermal excitation, light, etc. at a rate g 
 particles/unit time. Let us try to estimate the rate of change g-r. 
 
 Take for instance holes in the n-region; then for equilibrium p = p 
 given by (6.k). Let g and r be the values of g and r in the equilibrium 
 cases g = r . Thus g-r = when p - p =0. We shall now make the funda- 
 mental assumption that g is constantly equal to g , i.e. that the minority 
 carrier concentration does not affect the creation of holes. Since the rate 
 of recombination r must be proportional to p (for the probability of a re- 
 combination is proportional to p ) we must have 
 
 g - r = g o - X? n 
 
 Taking the equilibrium case we find that g = X.p . Further we deduce from the 
 
 definition that X has the dimensions (time) j we shall therefore introduce 
 
 _, -l 
 a constant c = \ called lifetime of holes in n-type material and write 
 
 P 
 
 ^ o 
 
 P - P 
 
 (g - r), , _ . . — s± • (8*1) 
 
 vo hoxes m n- region 7~ 
 
 -p 
 
 In the same way we can treat electrons in p-type material by introducing 
 
 a lifetime 2" and writing 
 n 
 
 o 
 n - n 
 
 ^° ' electrons in p-region _r* v • / 
 
 n 
 
 Notice that (8.1) and (8.2) have been established for the non-equilibrium 
 case, i.e. the case where current flows. We shall now express that in this 
 case the change of the number of holes (or electrons) per unit volume and unit 
 time is due to g-r and the holes carried into the volume by the currents this 
 is the continuity equation. 
 
 Take again the case of holes in an n-regiom I , the current density of 
 holes, is given by (7«1 )• The total charge of the holes going out of unit 
 volume is given by the divergence of I ; the number of holes going into unit 
 . _____ 3x 
 
 *) In the one dimensional case div I = 
 
 P 9x 
 
Ik 
 
 volume in unit time is thus div I . The continuity equation thus "becomes 
 
 q. p 
 
 3p P - P -, 
 
 t=— ; = =: dlV I 
 
 5^ c q p 
 
 (8.3) 
 
 In exactly the same way we have for electrons in the p-region 
 
 o 
 
 <?n n - n 
 
 -£ = ? r B + 1 div I 
 <?r ^ q. n 
 
 (8.^) 
 
 Remark : The ahove reasoning can be extended to holes in the p-region and 
 
 ro / o 
 Calling p (= p or 
 
 n 
 
 p ) and n (= n or n ) the equilibrium concentrations we have the general 
 equations 
 
 |E = E^L^ - I div I 
 3^ Z^ q p 
 
 (8.5) 
 
 dn n-n 1,. 
 
 jrt=u= — ■==. + — div I 
 
 "9^ 7T q n 
 
 (8.6) 
 
 n 
 
 9« Variation of , and ~y^ in a Non-equilibriun pn Junction 
 
 Let us go back to the problem of section 6, but supposing this time that 
 the equilibrium is disturbed by modifying y- at the left hand side of the p- 
 region by by, leaving -)^ at the right hand side of the n-region unchanged. 
 
 p-region 
 
 an$ 
 Re 
 
 it Lon 
 
 loci 
 
 1 
 
 n-region 
 
 1 2 3 h 5 
 
 Figure in- 
 significant Points in a pn Junction 
 
 We shall make 3 hypothesis: 
 (l) We shall assume that far from the junction, in 1 and 5* equilibrium con- 
 ditions obtain. The carrier densitites will then be given by (6.U): p(l) = p , 
 n(l) = n , etc. This means that almost the total current density I is produced 
 by holes alone in 1 and by electrons alone in 5 J 
 
 I - I p (D = I n (5) 
 
 (9-1) 
 
15 
 
 (2) We shall assume that the transition region 2,k is so narrow that we 
 can neglect recombination in its 
 
 i p (2) = yio ) 
 
 (9-2) 
 
 I (2) = I (k) J 
 n v n J 
 
 (3) We shall assume that n and p > > p and n i this is the so-called 
 w ' o o n p 
 
 "low" injection level case." We shall partially abandon this hypothesis in 
 section 13' 
 
 Using (7-7) ve find that 
 
 T 
 
 |grad I < — - — in the p-region 
 
 i 1 
 
 p 1 
 
 < 
 
 I 
 
 m- p q. 
 p 
 
 1 1 
 
 < 
 
 1 
 
 n ' 
 
 u p q 
 p n 
 
 grad < — — — in the n-region 
 p n 
 
 Since p > > p by hypothesis 3 , 
 
 I grad < < Igrad 
 
 P P r p'n 
 
 and it is reasonable to put 
 
 |grad p | p = (9-3) 
 
 i.e. is constant in the p-region. 
 In the same way we can find that 
 
 Igrad =0 (9.^) 
 
 i.e. is constant in the n-region. 
 r n 
 
 As regards the transition region, we have to distinguish two cases. For 
 
 forward and low reverse bias the carrier densities of either type will be 
 
 sufficient to apply simultaneously (9 «3) aud (9.^). For high reverse bias there 
 
 will be a very low carrier density and then and will vary in the transition 
 
 region. Figure 5 gives the shape of and in the low bias case. Notice that 
 
 the variation of in the p-region and that of in the n-region is arbitrary? 
 
 it has no influence whatsoever on the other reasonings in the present section. 
 
 By hypothesis (l) and must of course coincide at points 1 and 5s going 
 
 n i 
 back to (6.k) p(l) = and p(l) = p . Using (6.6) we obtain the announced 
 
 result . 
 
16 
 
 
 o o 
 
 '/'S%>. 
 
 r>o* 
 
 Figure 5 
 and in a Non- equilibrium Junction 
 
 Let us now turn our attention to )^. First of all (6.6) shows that"/' is 
 constant between 1 and 2, for is constant and p ~ p everywhere. In the 
 same way we can prove that ~v^ is constant between k and 5* The variation of y 
 is therefore as represented in Figure 6: 
 
 
 ■n 
 
 S H\ X 
 
 £?&/'/ y 
 
 
 n 
 
 ►> Cor>sfdr?7 v'j/i/e 
 
 > c//3 7&/7CQ 
 
 I -2 3 * 
 
 Figure 6 
 ■y-in a Non-equilibrium Junction 
 
 Lastly we can see that (l) - (5 ) = (l) - (5) = 5/* The reason is 
 
 kT o 
 again (6.6): -")£ = — In — at point 1 whether we consider equilibrium or 
 
 not. If we therefore increase yby 5y$ (and = ) will increase by by. 
 
 The same reasoning applied to point 5 shows that here and keep their 
 
 initial equilibrium case value. Summing up Figure 7 gives 0,0 and"/-' for 
 
 the equilibrium case in dashed lines while the heavy lines refer to a biased 
 
 junction with 0,0 and")^: 
 
 p. e/'-s7bnc^ 
 
 y, and in a Non- equilibrium Junction 
 
17 
 
 10. The Voltage Current Characteristic of a p-n Junction 
 
 Let us again consider the junction of section 9- In point 4 (of coordinates 
 x, ) we have from (6.5) 
 
 v (k) - n e titj>M-Y(VVw (10-D 
 
 y n K i 
 
 n (k) = n elMV-tnMl/m (10-2) 
 
 n i 
 
 where p and n carry an index n to indicate the fact that we are in an n-region. 
 Now by section 6 n = n = (N, - N ) in an n-region because we neglect the space 
 charge. (10.2) therefore gives 
 
 ^i e -qy*nOOAT m e -q)W/Kr . 
 
 n 
 o 
 
 Substituting into (10. l) we find that 
 
 (10 = !i JLtpfrhtrnfM/* 
 
 p 
 
 r n n 
 o 
 
 2 
 
 But by Figure 7 (4) - (4) = by and by (6.9) n./n = p = normal density 
 of minority carriers in the n-region. The last equation therefore becomes 
 
 P n (*)=P° e* 8 tf ffi . (10.5) 
 
 3p n 
 Let us now try to determine p (x) in the static case ( ~z-^ = 0) for 
 
 n v ^ c 
 
 x > x, . The continuity equation (8.3) then becomes 
 
 p - p (x) . p - p (x) 31 
 
 n *n n 1 , . _ ^n *n 1 n /, A i, \ 
 
 - — — div I = =s — 1 — ■* • (10.4) 
 
 T q p IT q 3 x v 
 
 .f x^ 
 
 To this we must join the equation giving I for a given grad p (x) and a given 
 
 E; Js being constant to the right of 4, E = and (7-1 ) gives 
 
 3p (x) 
 I = -qD grad p (x) = -qD * (10. 5) 
 
 Inserting (10.5) into (10.4) 
 
 P - p (x) 9 p (x) 
 
 = *V n + D —^— (10.6) 
 
 P 9* 
 
18 
 
 P being a constant, we can write the solution to (10.6) which satisfies the 
 boundary condition p finite for x -»<*= in the form 
 
 m o <x4-x)/Lp (10.7) 
 
 *n n 
 
 vhere L =JdF (10.8) 
 
 p V p p * ' 
 
 is called the diffusion length of holes in n-type material. 
 
 P can now be evaluated by the second boundary condition (10.3 ) 
 
 o 
 
 ^qS^/kT 
 
 u ^ r/ ^ (1+) « = 
 
 n n x ' -^n 4 
 
 giving 
 
 /■ \ ° /" q&y/KT i > ( x k-x)/Lp ,,. rt * 
 
 (x) = p + p (e 1 r/ - 1) e v 4 y/ * (10.9) 
 
 p , 
 
 r n n n 
 
 As a last step we evaluate I = -qD =r — at point 4. (10.9) gives 
 
 directly 
 
 I (k) = I (e^/ M - 1) (10.10) 
 
 p ps v ' 
 
 where 
 
 o^ 
 
 qp D 
 
 I = 5 P (10.11) 
 
 ps L v 
 
 P 
 
 is the saturation hole current density in the junction. We see that for Sy 
 
 big and negative I = -I 
 
 P P s 
 
 In a similar way we could discuss the electron current at point 2: 
 
 I (2) = I (e* 5 ^ - 1) (10.12) 
 
 ns 
 where 
 
 qn°D 
 
 I = H 2 - 2 - (10.13) 
 
 ns L v ' 
 
 n 
 
 is the saturation electron current density in the junction. Using hypothesis 
 (9.2) we thus find that 
 
 I . I (4) + I (4) = I (4) + I (2) 
 p v ' n v P n 
 
 = I (e^/^ - 1) (10.14) 
 
1=1 +1 
 
 s ps ns 
 
 19 
 (10.15) 
 
 "being the saturation current density. 
 
 Formula (10.1^) is the fundamental relationship used in rectifier theory. 
 We can of course multiply by the cross section A of the junction and call v 
 the applied bias 5"^. Then the current i = IA is given by 
 
 i = AI (e 
 
 s 
 
 ^/^ - 1) = i (e* v / H - 1) 
 s 
 
 (10.16) 
 
 11. Variation of 0_, and )o in a Non- equilibrium pnp Transistor 
 
 It will be sufficient, as in the last section, to solve for p in the con- 
 tinuity and the current density vs. field and concentration gradient equation, 
 to obtain a complete picture. But this time the boundary conditions are slightly 
 different. We shall deduce them by extending the arguments applied to a pn- 
 junction. Figure 8 gives the aspect of a pnp transistors an n-region (very 
 
 /'(/ric7/077 / 
 
 /1//7C 
 
 7/omZ 
 
 e 
 
 § 
 
 P- 
 
 A 
 
 n- 
 
 | 
 
 c 
 
 
 ^ 
 
 eroiTTc-r 
 
 6&SG 
 
 8 
 
 ee t/s-cTor 
 
 ;5 
 
 
 / 
 
 Schematic 
 
 Di 
 
 ?. z<zn<%% = 
 
 stor 
 
 h 
 
 t 
 
 
 Figure 8 
 agram of a Trans i 
 
 
 narrow in reality) is sandwiched between two p-regions . Contacts are made 
 to the 3 regions by an emitter electrode e, a base electrode b and a collector 
 electrode c. We shall again assume that we have equilibrium conditions in 1 
 and k, i.e. identity of and . Sometimes this assumption of equilibrium 
 at e and c is expressed by saying that e and c are ohmic constants, meaning 
 that no rectification occurs at these metal-crystal junctions. This implies 
 that y is identical on either side and that holes and electrons at e and c 
 have their equilibrium densities. 
 
20 
 
 The base contact b is an ohmic contact too, but it is not in direct 
 contact with the interior of the base region and therefore does not imply 
 equilibrium densities on the x-axis of the device. But we shall assume that 
 y* has the same value in points like A and B having the same x. Taking the base 
 electrode as reference electrode, the applied biasses by, and 5y, are then 
 simply equal to the potential differences v and v applied between b and e 
 or c respectively. 
 
 By virtue of these hypothesis we can again find the variation of and 
 
 and V i n "the transistor. We shall use the following principles discussed 
 P 
 in section 9: 
 
 (l) is constant in the p-regions . 
 
 (2)0 is constant in the n-regions . 
 
 (3) At the equilibrium points 1 and ^ =0 and the displacement 
 of =0 above the =0 line (for equilibrium throughout the transistor) 
 is equal to the applied bias v or v . 
 
 (h) yy in the n-region (base) has the same value as in the equilibrium 
 case, the base electrode being the reference electrode. 
 
 (5)0 =0 in the n-region, for y-' is the same as in the equilibrium 
 
 5° 
 
 n ' n 
 
 case and the density n = n : we can then use (6.5 )• 
 
 As in section 9 we must again distinguish two cases for the collector 
 
 junction: forward or low reverse bias and high reverse bias. In the latter 
 
 case and vary considerably in the junction. Figure 9a gives the 
 
 variation of and for the equilibrium case and for the case of low biasses 
 
 v and v . Figure 9b repeats the equilibrium case but also illustrates the 
 e c 
 
 case of a strongly reverse biassed collector junction. Luckily the calculations 
 of the next section, based upon Figure 9a > will still be valid when Figure 9b 
 holds, the reason being that the minority carrier density at the collector 
 junction becomes exceedingly small in either picture, i.e. the same boundary 
 conditions hold. 
 
21 
 
 V or ft 
 
 c o//oc /or £/->) 
 
 c//s7&ocq 
 
 origin of x 
 
 Figure 9a 
 
 y*x p and for a Transistor with Low (Forward) Collector Voltage 
 
 yorfi e m,7Ter Cp ) 
 
 6*se C-n ) Co//ecr r Cp) 
 
 Or/gin af X 
 
 Figure 9b 
 y^ s and p for a Transistor with High (Reverse) Collector Voltage 
 
22 
 
 12. The Voltage Current Characteristic of a pnp Transistor 
 
 As in section 10 we can calculate p(2) and p(3) "by taking the difference 
 
 between and . But (2 ) - (2 ) = v and (3) - (3) = v . Therefore 
 r p r n P & e r p mi c 
 
 P. 
 
 n 
 
 (2 ) = p° e qv e/ M 
 
 qv c /kT 
 
 (12.1) 
 
 P n (5) =P n e 
 
 The solution of the equation (10.6) (applicable because once again 
 E = in the base region), i.e. 
 
 = (12.2) 
 
 o 
 
 P- 
 
 " Pr>> 
 
 
 
 2 f 
 9 P„(x 
 
 :) 
 
 n 
 
 n 
 
 + 
 
 D 
 
 n 
 
 
 
 r 
 
 
 P 
 
 3x^ 
 
 
 
 p 
 
 
 
 
 
 which satisfies the boundary conditions (12. l) at points 2 and 3 of coordinates 
 x = and x = v is obviously 
 
 [p n (5) - p°] sinh jL + [p n ( 2 )-p°]sin^ 
 p(x)=p° + E E- (12.5) 
 
 sinh =— 
 P 
 
 where L =,/ D r as before. 
 
 P V p p 
 
 9P n (x) 
 
 This allows us to calculate I (3) = -q.D — | by (12.3 )• 
 
 P P v^ X = W 
 
 Introducing 
 
 j*( T ) = ^L£ (e^/M . 1} 
 P L P 
 
 = I (e^/^ - 1) (12 A) 
 
 ps v ' 
 
 we can then write 
 
 y 3 ) - cseh/ft W(r e ) - cottv'^-W ). 
 
 (12-5) 
 
 Now the total amount of current across the junction in 3, going from left to 
 right, is 
 
 1(3) = l p (3) + I n (3). (12-6) 
 
23 
 
 The evaluation of I (3) can be done along the same lines as that of I (3) by 
 considering the diffusion equation for electrons in the emitter and collector 
 region together with the boundary conditions discussed in section 11. It is, 
 however, easier to observe that the electron flow across each junction is 
 independent of that across the other one. The reason for this is that the 
 majority carrier density in the base, given by (6.5), is constant and equal 
 to that in the equilibrium case (y> - y* and = in the base by section 111 )« 
 Therefore the junction at 3 for example behaves like a pn junction biassed in 
 the reverse direction. Therefore by section 10 
 
 I (3) = -I = - T * n (12.7) 
 
 n x ns 
 
 if we consider the reverse bias big enough. 
 
 Calling A the cross section of the two junctions, the current i flowing 
 into the collector is -l(3)A for continuity reasons (but I (3) is different 
 from l(2) because of recombination in the base'.). We can therefore write 
 
 i = -A cschf- T — |l*( v ) + A coth/=^\l*(v ) 4- AI (12.8) 
 
 [ L v p e (pJ P 
 
 and in a similar way we could establish that 
 
 1 = A <Wf] I*(v e ) - A esch/f ] I«(v ) ♦ AI [12.9) 
 
 This then solves the transistor problem completely, for we can, by differentiation 
 and elimination, calculate the fourpole quantities r., defined by 
 
 IK 
 
 dv = r.. ..di + r n ^di 
 e 11 e 12 c 
 
 dv = r_ n di + r 00 di 
 c 21 e 22 c 
 
 and these are the only quantities we must know in the small signal circuit 
 applications . 
 
 Remark? As can be seen from the definition of I in (12.7) and I* (v) in 
 
 , M — , ns \ p 
 
 (12 A), I can be neglected if p > > n . This is usually the cases the 
 ns n p 
 
 emitter and collector regions are more highly doped than the base. 
 
2k 
 
 13 « Behavior of the Base Region at High Injection Levels 
 
 The hypothesis 3 in section 9, i.e. n and p > > p and n . led to the 
 
 o ^o r n p 
 
 conclusion that = constant in a p-region and = constant in an n-region. 
 
 This fact, together with the nearly constant concentrations p and n of holes 
 
 00 
 
 and electrons then allowed lis to take y as a constant except in transition 
 regions: this means that we neglected electric fields. 
 
 We shall now discuss certain features of transistors in which the minority 
 carrier density p in the n-type base is actually much bigger than n . This can 
 come about when the density of hole current across the emitter-base junction is 
 high, i.e. for high injection levels. We can then still assume that the varia- 
 tion of and is given by Figure 9a (This means that p > > p , which does 
 not contradict p > > n l), while y/ would now vary in the base region on account 
 of the very variable electron density n = n + p (zero space chargel). Thus an 
 electric field appears in the base. 
 
 To gain some oversight, let us go back to the general equations (8.5), (8.6), 
 (7-1), (7.2), (5-6), (5-8) and I = I p + V 
 
 & = 2lz^ - i div 1 (13.1) 
 
 5r ^ 1 p 
 
 ^=^=^ + -divI (13.2) 
 
 I - q(u p pE - D p grad p) (13-3) 
 
 I n = q(u n nE + D n grad n) (13-^) 
 
 I =I p + I n (13.5) 
 
 div E = -S — (p - n + N, - N ) (13.6) 
 
 C € ** da' v ' 
 
 o r 
 
 Let us recall that p and n are the equilibrium concentrations of carriers 
 (in the p-region p = p in the n-region p = p etc.) We shall of course 
 retain the zero space charge condition: 
 
 p - n + N , - N =0 (13-7) 
 
 da 
 
 From (13.7) w e deduce 3 equations: 
 
25 
 
 div E = o (13-8) 
 
 £-!%. (13-9) 
 
 grad p = grad n (13-10) 
 
 Let us now define a dimensionless factor b by 
 
 H = bn (13.11) 
 
 n p 
 
 By Einstein's relationship (see 7-^ and 7-5) we then have 
 
 D = bD (13-12) 
 
 n p 
 
 Let us consider an n -region having an electron density of the form 
 discussed in (6.2): 
 
 n = n Q + p (13-13) 
 
 (we leave out the subscript n to simplify the formulae). Adding (13-3) and 
 (13 -M we then obtain 
 
 I - qD (b - l) grad p 
 
 E = r-7t n — C 1 — (13-1^) 
 
 qu Lp(b + 1) + bn J 
 
 P ° 
 
 Therefore, by (13.15) 
 
 pi - qbB (2p + n ) grad p 
 ±p ■ P(/ + 1WL ^) 
 
 which means that (13-1 ) can be written 
 
 - o pi - qbD (2p + n ) grad p 
 
 % + £-^- = - div 1 — Ij l-L— al . — (13.16) 
 
 d? -T p q[p(b + 1) + bn Q ] 
 
 This, then, is the differential equation for the concentration in an n-region 
 in the most general case, assuming zero space charge. Before integration, 
 one would have to discuss the boundary conditions. 
 
 lh. Carrier Delay in the Base Region as a Function of Injection Level 
 
 Lex us consider the mechanism by which signals are transmitted by a 
 
 sistor. 2$ and (for 1 
 r V 'n * 
 
 bias) are given by Figure 10, 
 
 transistor. and (for high or low injection level and strong collector 
 
 XT AA 
 
Vc 
 
 err>///er 
 
 fe e//'s7'c)/7C(? 
 
 Figure 10 
 
 Quasi Fermi-levels in Transistor with High 
 Reverse Collector Voltage 
 
 Modifying v means — because 
 e r p 
 
 is nearly horizontal in all 
 cases in the emitter region — 
 modifying the hole concentration 
 at the emitter-base junction; 
 p (2) is given by (12.1). This 
 modification is practically 
 instantaneous and there is no 
 signal delay in the emitter 
 region. In an analogous way 
 we could argue that the collector 
 region causes no delay. So we 
 are left with the time of pro- 
 pagation of a perturbation of 
 
 hole concentration from 2 to 3. This time dealy 5 is clearly the base width 
 divided by the average velocity v of the carriers*: 
 
 5 = 
 
 w 
 
 V 
 
 (l*.l) 
 
 Now it is clear that in a fixed point we can define an average velocity 
 v such that 
 
 I = vpq (2A.2) 
 
 P 
 
 In the low injection case there is no electric field in the base and 
 
 I = -<£D grad p (lA-3) 
 
 P P 
 
 It follows therefore that 
 
 D 
 v = - — ** grad p 
 
 l .e. 
 
 = m _ D /gradj> 
 
 p V p , 
 
 ~ _ D grad p 
 P P 
 Now for all practical purposed p (3) =0 and 
 
 (Ik.k) 
 
 *) v indicates an average over all the particles in a given point, averaged 
 over the whole base width. 
 
27 
 
 P n (3) - P_(2) -p (2) 
 
 w 
 
 while 
 
 p=\p n (2) (14.6) 
 
 where \ is a numerical factor of order unity. Thus it follows that 
 
 D 
 
 w 
 and 
 
 5 X =xf (14.7) 
 
 P 
 where the upper index 1 refers to the low injection level. 
 
 Let us now look at the high injection level case. For this let us go 
 
 back to equation (13.16), remembering that the square parenthesis is simply 
 
 I 
 
 -2. . From (13-4) we obtain, using (13.IO), (13.11 ) and (13.13) 
 
 I - qbD grad p 
 
 E = n "P -_ (14.8) 
 
 qbu (n Q + p) v 
 
 which in the high injection level case p > > n becomes 
 
 (14.9) 
 
 Substituting this into (13 »3) we have 
 
 I 
 I = *p - 2qD grad p = - 2qDp grad p (l4.10) 
 
 since I is negligible by the hypothesis p > > n°. 
 
 We see that (l4.3) is replaced by 
 
 I = -qD 1 grad p with 
 P P 
 
 D* = 2D . (14.11) 
 
 P P V ' 
 
 From there on all calculations remain the same and we obtain 
 . 2 
 
 n ~ W 
 
 p 
 
28 
 
 Notice that (ik.'j), (1^.12) and (1^.13) are in perfect agreement with 
 the results obtained in section l6 by more sophisticated raasonings . 
 
 15. The Current Amplification a of a Trans istor 
 
 In section 12 we found expressions f< 
 
 Positive Directions of Currents 
 in a Transistor 
 
 i and i as a function of the applied 
 c e 
 
 biases v and v . at least for the low 
 c e' 
 
 injection level case with zero electric 
 field in the base. Let us call 1 the 
 current flowing from the base electrode 
 into the transistor; then 
 
 i e ♦ ^ + i c = (15.1) 
 i.e. by using (12.8) and (12.9) with 
 
 I ~ 
 ns 
 
 2 sinh' 
 
 **-- 
 
 = -i - i = 
 
 2/ w 
 
 sinh/=^ 
 
 ifl 
 
 [I*(v ) + I*(v )] 
 p v e' p v c' 
 
 (15.2) 
 
 which shows clearly that i, f 0, or |i | > |i |. This base current clearly 
 
 corresponds (in the case where most current is due to holes) to recombination 
 
 of holes and electrons in the base region or flow of electrons across the 
 
 emitter junction. It is therefore sometimes called "recombination current." 
 
 Its effect is to decrease the efficiency i /i of current transfer between 
 
 * c' e 
 
 emitter and collector. 
 
 However it is not very useful to calculate i /i since in practice we are 
 often interested in the response to small signals. In this case we define the 
 
 current amplification factor a by 
 
 di 
 
 a - 
 
 Si 
 
 v = constant 
 c 
 
 (15.3) 
 
 It may seem advantageous to derive a from (12.8) and (12.9) by differentiation. 
 But this would only be valid for very slow variations. Practically we are interested 
 in a at signal frequencies of considerable magnitude. We shall therefore take up 
 the problem from the beginning, i.e. solve equation (l3»l6) with its time dependent 
 
29 
 
 term S^» (left out in the steady-state discussion of section 12) and appropriate 
 boundary conditions (v variable'.) 
 
 e 
 Let us treat the low injection level case n > > p first. Then (l3°l6) 
 
 becomes 
 
 o o 
 
 ^£ + P " P = div D grad p = D lap p. (15-^) 
 
 P^ tr 
 
 P 
 
 Introducing as before L = , / D "C and restricting ourselves to the one 
 
 p V p p 
 
 dimensional case, we have 
 
 » P P - P 1 5P / ls o 
 
 rf -—2" - r ?? (15 * 5) 
 
 ax l p ^ 
 p 
 
 Suppose now that we apply an emitter bias 
 
 V = V + V , 
 
 e o e 
 
 leaving v constant according to the definition of a in (15«3)« Let x = 
 and x =w correspond to the position of the junctions as in section 12. Then 
 the boundary conditions (12. l) are replaced by 
 
 p( 2 )=p\'Me M )/ H 
 *n v -^n 
 
 Now assume that v > > v, then 
 
 o 3 
 
 qve Jj/t /kT ~ . jafc/wn 
 
 i ' = 1 + qve /KC 
 
 and 
 
 jo/C 
 
 where 
 
 P n (2) = p(0) = Pl + Pe J (15.6) 
 
 P 1= p°e^oM (15 . 7) 
 
 P = p^v/kT (15.8) 
 
 The boundary condition at x -w is the same as the second condition in 
 
 (12.1) because v is constant? 
 c 
 
 P n (5) = P(w) = p° e"c/H (15 . 9) 
 
30 
 
 Notice that in the last equation v < 0. 
 
 The solution of (15*5) satisfying (15.6) and (15«9) is 
 
 ^ P l " P n^ S±uh ( \~ / + ^ P ^ " P n^ s±Ilh ( jT ) 
 
 P = P. 
 
 sinn / — 
 L P 
 
 Pe J</U sinh 
 
 vgffigfj 
 
 sinh [^ l+jo/T (2L)] 
 
 (15.10) 
 
 We can now calculate 
 
 1 W - -qc ^| 
 
 (15.11) 
 
 To I (w) we must add I (w), hut again this turns out to be ~ -I and 
 id v n ns 
 
 therefore negligible if p > > n . Thus 
 
 i = -AI ( w ) 
 
 c p 11 ' 
 
 (15-12) 
 
 where A is the cross section of the junctions. But since we are only interested 
 in the alternating part i of i , it is sufficient to calculate (15.II) for the 
 second term in (15.IO). We then obtain 
 
 , J& \/l+3^ , 
 
 i = APe 
 c 
 
 y } 
 
 cB 4\/ 1+Jw p (sh]l 
 
 In a similar way we can compute the alternating part i of i : 
 
 i = APe c 
 e 
 
 -= * 1 coth 
 
 \l 1+ ^ r v 
 
 w 
 
 p /—I 
 
 and therefore (supposing everywhere v = constant) 
 
 (15.13) 
 
 (15.1*0 
 
 oC = 
 
 di i 
 
 c c 
 
 31 := 7 
 
 csch J 1+ J 
 
 ■s ( i , )] 
 
 coth \/l+j 
 
 "• (s)J 
 
i .e 
 
 31 
 
 oC = 
 
 Remark: For uj= Owe have 
 
 1 
 
 
 1 
 
 
 
 cosh 
 
 H + a* It 
 
 K P 
 
 oL r 
 
 w 
 
 :osh ( j — ) 
 
 (15-15) 
 
 (15-16) 
 
 Since w/L is quite small in general, a 
 ' p o 
 
 l6. The a Cutoff Frequency as a Function of the Injection Level 
 
 From (15.15) we deduce two facts: first of all there is a phase shift 
 between i and i at higher frequencies because 0. is complex. Secondly la] 
 decreases with ou: the current amplification of a transistor drops at higher 
 frequencies . 
 
 It is useful to define a a -cutoff frequency f = 00 /2n such that the 
 corresponding value |a | is "3db down." This means that 
 
 a 
 
 a 
 
 1 
 2 
 
 (16.1) 
 
 We shall first examine the low injection case of the preceding section. 
 
 (16.2) 
 
 In order to calculate uo we shall suppose that 
 
 uy V > > 1 
 
 c p 
 
 'hen 
 
 \ fTT H 
 
 ^=/5«-/?(w) 
 
 (16.3) 
 
 Let us introduce 
 
 >? 
 
 w_/t^5 
 
 L \ 
 P 
 
 = w 
 
 2D 
 
 (16.4) 
 
 by virtue of the definition of the diffusion length L . Then (15.15) gives 
 
32 
 
 a 
 
 1 _ 1 
 
 c ■ cosh^ + ">ij ) costnO cos>) + j sinh^sinvi 
 
 and 
 
 la 
 
 c 1 .2. 2 N . .2^ . 2, cos2n + cosh2>7 
 cosh n cos >7 + sinh ^9 sin V <. l 
 
 V/ cos >i + sinh ^sin "^ 
 Using (l6.l) this becomes 
 
 2 = cos2 ^ . ± cosh2 1 (16. 5 ) 
 
 Solving this transcendental equation we obtain 
 
 ■h = 1.103 
 
 which, using (l6.h) finally gives 
 
 D 
 ^ = 2.W-f (16.6) 
 
 w 
 
 where the upper index 1 draws attention to the fact that we are talking about 
 
 the low injection level case. 
 
 It seems at a first glance necessary to repeat the calculations of the 
 
 preceding section for high injection levels. Happily enough this is not 
 
 necessary; we shall show that everything behaves at high injection levels 
 
 as if D had been replaced by D' = 2D . We only have to repeat the argument 
 
 of section lk: using 
 
 I p = -2qD p grad p (16.7) 
 
 equation (13.16) becomes 
 
 §£ + ^^ = 2D lap p 
 
 ?r ^ P 
 
 = D p lap p (16.8) 
 
 This proves our contention for the differential equation. The boundary con- 
 ditions remain unchanged, so all we have to prove is that (l5-ll) is unchanged 
 
 if D' is substituted for D . But this is easy, for we have already shown at 
 
 P P 
 
 the beginning of this section that (130) — i.e. the equation for I taking 
 
 ir 
 
 account of electric fields --can be written in the form (16.7) in the high 
 injection level case. (i5.ll) therefore becomes: 
 
33 
 
 I = -qD' f*| 
 
 X = V 
 
 (16.9) 
 
 All this leads to the conclusion that the reasonings at the beginning of 
 
 2D 
 
 b 
 
 this section are valid if D ->D' 
 
 P P P 
 frequency (divided by 2rt ) for the high injection level case, we therefore have 
 
 Calling^ the value of the a cutoff 
 
 uO 
 
 2A3I+ D' 
 R 
 
 = 4.868 -g = 2<J 
 
 d C 
 
 W 
 
 (16.10) 
 
 1 7 • T he Trans ition Region Capacitance 
 
 Transition Region 
 
 'by 
 
 I k0fl-&n + eo»s7 
 
 -#!%#- - ^M ± —i 
 
 c/isT&'->CQ. 
 
 Figure 12 
 
 H> . and in a junction with Forward 
 Bias 
 
 Let us go back to the problem 
 
 of a single pn junction treated in 
 
 section 10. But this time we shall 
 
 take account of the fact that the 
 
 transition region between points 2 
 
 and h has a finite width, i.e. of 
 
 the fact that n and p do not abruptly 
 
 change at 3 but gradually between 
 
 2 and k. The width will however be 
 
 taken small enough to allow us to 
 
 assume that - in it is constant 
 r p r n 
 
 and equal to 5^ (see figure). 
 
 We shall simplify the problem 
 
 by assuming that relative to an 
 
 origin of x situated between 2 and k 
 
 (but not necessarily in the middle 
 
 if n * p ! . ) we have 
 o ' o 
 
 n - p = N , - N 
 da 
 
 ax 
 
 (17.1) 
 
 a and the exact situation of the origin being such that 
 
 n = ax (3) 
 o v ' 
 
 P. = ax (2) 
 
 (17.2) 
 
 *) The first half of the equation is a consequence of space charge neutrality. 
 
3k 
 
 Points 2 and h will "be chosen sufficiently far in the p- and n-regions res- 
 pectively to warrant the absence of electron flow in 2 and the absence of hole 
 flow in k. Further we shall neglect recombination between 2 and k and as 
 already mentioned, assume space charge neutrality. 
 
 Now modifying the applied bias by d5y =dv modifies 0,0 and the hole 
 densities between 2 and^. Between 1 and 2 p ~ p , between 4 and 5 p ~ p ; 
 i.e. the variation of the total charge of holes Q with a given variation of 
 bias is produced in the transition region alone. The same reasoning leads to 
 the conclusion that the change of the total electronic charge (-Q because of 
 the neutrality condition) is produced in the transition region. This means 
 then that dv produces a drift of charge to the right of dQ in 2 and a drift 
 of electrons to the left (-dQ) in k: the transition region behaves like a 
 capacitor of capacitance 
 
 o - i CW.3) 
 
 Using (6.5) we have in the general case 
 
 3 qfr-0 n )/kT 
 
 n = n. e 
 
 1 
 
 p = n. e 
 
 1 
 
 q(0p-y)/kT 
 
 (17.4) 
 
 Now by figure 12 we have throughout the (narrow) transition region 
 
 p - n = &/ = v. (IT- 5) 
 
 Introducing the quantities n and by 
 
 K + K 
 
 = e qv/ 2k T (l7-?) 
 
 we can then write 
 
 q(>-0l)KT u 
 
 n = n e^- v ' r ' = n e 
 
 ^Wi-Wkt _ „ -u 
 
 (17-8) 
 
 p = n -, e = n e 
 
 where 
 
 u = q ^AV M (17.9) 
 
35 
 
 We can then transform (17-1) into 
 
 2n.. coshu du = adx 
 
 (17-10) 
 
 Let P "be the total number of holes between 2 and 4; then 
 P = A l pdx = A / n n e coshu du 
 
 where A is once again the cross section of the junction and u. and u_ are the 
 values of u at points 2 and 4 respectively. Integrating 
 
 , 2 
 An, /u 
 
 An, 
 
 1 / 4 f . -2ux, 1 
 P = / (1 + e )du = 
 
 IX, 
 
 ." 2u 4 . o' 2u 2 
 
 K " u 2 } 
 
 (17.II) 
 
 Now at points 2 and 4 we have from (17.8) 
 
 U 2 
 
 In 
 
 P(2) 
 
 u ), = l 21 ~^ 
 
 4 ni 
 
 (17.12) 
 
 (i7.ll) can thus be written 
 
 P = 
 
 An, 
 
 2a~ 
 
 n. "^2 
 
 *\-v+m -^h 
 
 A 2 A 2 
 
 Ap An, 
 
 *o 1 
 
 2a 
 
 IT K - U 2 } 
 
 (17.13) 
 
 \} 2 n i 
 
 where we neglect y— 7rr ( = ~k because we can assume that the n-region is 
 
 ' o 
 sufficiently doped to have n > > n y p(2) is replaced p . We can now calculate 
 
 U 4 " V 
 
 . p(2)n(4) . p(2)n(4) . n i 
 u^ - u 2 = In *^ g * f = In ^ < 2 S ' + In — 
 
 n l n i n l 
 
 In 
 
 n P n 
 
 00 2qy 
 
 2 " 2kT 
 n. 
 
 1 
 
 (17-14) 
 
36 
 
 by using (17.7). Replacing n and u, - u by their value in (17.13) 
 
 2 . 2 
 
 Q = qP = 
 
 qAp qAn . /. _ 
 
 r o I qv/kT 
 
 — - — + e^ ' 
 
 2a a 
 
 In 
 
 n p 
 00 qv 
 
 2 ~ kT 
 n. 
 
 1 - 
 
 1 .e. 
 
 C = 
 
 2 A 2 
 q An. 
 
 _ 1 
 
 kTa 
 
 - n p 
 
 , 00 qv , 
 
 ln — " HP " 1 
 n. 
 
 1 
 
 (17.15) 
 
 Let us suppose that the n- and p-regions are strongly doped (n and 
 p > > n. ). Then at room temperature qv/kT is of order unity (v ~ 3Qniv'. ) 
 and 
 
 C = 
 
 2 « 2 
 q An. n p , 
 2 1 -, n 00 p av/kT 
 
 kTa 2 
 n. 
 
 (17.16) 
 
 Now by (10.1*0 we have 
 
 1 + 1 
 
 s qv/kT 
 
 Calling C the value of C for v = 0, we obtain 
 
 1 + 1 
 
 C = C 
 
 o 1 
 
 (17-17) 
 
 showing that the transition region capacitance increases with the junction 
 current . 
 
 It should be remarked that the reasonings in this section can only be 
 applied to the emitter junction, firstly because we suppose and constant 
 in the transition region (see section 9) and secondly because we assume space 
 charge neutrality. As explained in the next section this latter assumption is 
 not valid for the strongly reverse biased collector junction. 
 
 18. The Collector Space Charge Layer 
 
 Let us now consider the collector transition region, extending from -A 
 on the left of the junction 3 to +A at the right side of the junction, the 
 exact position of which is given below. 
 
37 
 
 - A 
 
 + A 
 
 3 
 
 t 
 
 Or/q/'n of X 
 
 Figure 13 
 Variation of y- in a Transistor 
 
 We have already explained that for sufficiently high reverse bias the 
 space charge neutrality condition can no longer be satisfied, the reason being 
 that the collector region forms a very good sink for the minority carriers in 
 the base: they are drained out at such a rate that a "depletion layer" is 
 formed in the transition region. In this depletion layer p ~ and n ~ 0, 
 meaning that 
 
 ^=q(N d -N a ) 
 As in section 17 we can then use the second half of (l7»l), giving 
 
 (18.1) 
 
 /<>= ax (18.2) 
 
 which fixes, as we have seen in section 17, the "midpoint" 3 (corresponding 
 to x = 0) of the junction. 
 
 We can now use (5 '8) in the one -dimensional case? 
 
 9_ 2L 
 
 a* 2 
 
 ax 
 
 € e 
 r o 
 
 (18-3) 
 
 By integrating twice we obtain 
 
 qax~ 
 
 t be £ 12 
 
 r o 
 
38 
 
 Now it is easy to find C, : 
 
 2 
 *#- m ^ ax . c 
 
 dx 2e e 1 
 
 r o 
 
 must be approximately zero for x = 4A when the "normal" p and n regions begin, 
 for ^is constant in these regions (see section 9)« Therefore 
 
 c i ■ frr ww> 
 
 r o 
 
 and if we put 
 
 u =y(+A) -y(-A) (18.5) 
 
 we obtain 
 
 
 
 
 u = 
 
 2a qA 5 
 
 3e e 
 r o 
 
 Now u = v , therefore 
 c 
 
 z 3e e v 
 a5 . -i£i (X8.6) 
 
 It is thus evident that the depletion layer extends further and further 
 into the base region as the collector voltage goes up, reducing the base width 
 w to an effective width 
 
 w eff = V-A. (18.7) 
 
 When w „„ becomes zero we talk about punch-through: the corresponding 
 collector voltage v is the punch-through voltage. Above this bias all 
 transistor action is lost. 
 
 Further (l8.6) and (18.7) show that the transistor equations (with w 
 replaced by w „„ ) become more complicated, in particular the collector swing 
 in an amplifier modifies V rr constantly and thereby introduces a feedback 
 effect . 
 
 Another consequence is a decrease of recombination of injected minority 
 carriers in the base layer since the average carrier diffuses across the 
 narrower base in a shorter time. This corresponds to an increase in a and 
 in the a cutoff frequency.