THEORETICAL GAS ADSORPTION ISOTHERMS ON SEMICONDUCTOR SURFACES BASED ON CHARGE TRANSFER MECHANISMS VO J. Lee ABSTRACT Theoretical gas adsorption isotherms of neutral and charged particles on semiconductor surfaces have been formulated, based on a generalized model. It is assumed in this model that the temperature is moderately high so that the particles can be ionized with or without dissociation. The combirked-thermal and electrostatic effects will make the adsorbed particles rxbiln Om.;..-Te:sen.fconu;uctor surfaces.

THEORETICAL GAS ADSORPTION ISOTHERMS ON SEMICONDUCTOR SURFACES BASED ON CHARGE TRANSFER MECHANISMS V. Jo Lee MODEL It is postulated in this model that the gas phase is in equilibrium with several adsorbed phases on the surface of a semiconductor. The various adsorbed phases can include the molecular state, the atomic state, the singly charged ionic state, and the doubly charged state. The relative number of particles in each phase is a function of temperature and the properties of the semiconductor This model is represented by the following generalized equilibrium equations. For a donor gas, D (gas)- D (ads) nD(ads) DD+ ads) nnD (ads) (1) n "Z n For an acceptor gas, A (gas) An (ads)= nA(ads) =nA (ads) - nA (ads) (2) It is further assumed that the system is such that there is at least one phase in the charged ionic state. As a result of the ionic state or states a static space charge region is created beneath the surface of the semiconductor. The static charge in the semiconductor surface distributes itself so that potential peaks or valleys are neutralized, and a substantially equipotential surface is created. Therefore, in the absence of potential wells on the surface which can trap individual ions, it is further postulated that the particles in the adsorbed state are mobile and can be treated approximately as a perfect two-dimensional gas. This also means that these adsorbed species are not covalently bonded to the surface. This assumption is not a gross restriction, however, since covalent bonds would not create an appreciable space charge region beneath the semiconductor surface~ These assumptions mean first that the variations in the surface potential are actually less than kT, so that the kinetic energy of the -1

-2species on the surface is large enough to surmount the variation in the surface potential. In addition it means that the ionization energy required to transfer charge between the surface atom and the semiconductor is not excessive with respect to kTo The periodic potential along the surface is commonly called the van der Waals' potential and is represented by Hill by the relationship __ ~2l7Tx Vo 2x V(x,y) = Voo + V (-cos ) + -- (1 - cos (3) 2 a 2 a where V is the depth of the potential wells on the surface. When there 0 are no static charges V is the order of 0. 3 to 1 0 K cal per mole (1). No attempt will be made to verify this postulate theoretically. It will be considered true, if the results derived from this postulate agree with the result of experiments. BASIC THERMODYNAMIC RELATIONS Let 6 U J. = (, N ) = the partial chemical potential of the jth phase, in icj (Njs V 6 N' S energy units/unit particle N! = the generalized partial external field potential at the jth phase, such as gravitational potential, van der Waals' potential, or electrostatic potential, in energy units/unit particle U = internal energy F = free energy = U-TS T = absolute temperature, in K0 S = entropy W = work Nj = number of particles in phase j N = total number of particles When there is an external potential, js, the first law of thermodynamics takes the following form: dU + Nd 2 = dQ - dW (4)

-3From the second law of thermodynamics, TdS 7/ dQ (5) therefore, TdS - ( dU + Nd r + dW')7/ 0 (6) Under isothermal equilibrium conditions, and when there is no work performed either on or by the system, equation (6) becomes, d (TS - U -'4) = 0 (7) or, dF + dT = 0 (8) Let us consider that there are two phases, j and k, a species, i, which exists in both phases j and k, and transfers from the j-phase into the k-phase, but the total number of particles of species i remains constant. The following relationships then exist d (Ni)j + d (Ni) = 0 (9) ij k dF = ijdNi - ikdNi (10) dt = (fij, - ifik) dN (11) 1' 1J ik 1 Combining equations (8), (10), and (11), one obtains, (ij Vij) (ik + Vik) (12) Having established equation (12), the author: will outline a procedure to evaluate the number of particles per unit volume or area in each phase under equilibrium conditions. Since in this model, the phases exchange particles among themselves Gibbs' distribution for a variable number of particles should be used. This distribution will lead to Fermi-Dirac as well as Bose-Einstein statistics as has been shown by Landau-Lifshitz. Under the condition that exp (a - E ) / kT <C, (13) both Fermi-Dirac and Bose-Einstein distributions reduce to the Boltzman distribution. Fowler and Guggenheim have discussed this problem in detail and have given it the name, Classical Distribution~

-4In any phase the mean number of particles at an energy level E is given by the relationship nr 1/ xP (E rj - rj )/kT] ~} exp( rj - E /kT (14) Total number of particles in the j phase is given by, Nj = exp (l j/kT) r exp (- E rj/kT) (15) = (x f).r where X= exp ([1 /kT) = the absolute activity as defined by Fowler and Guggenheim~ f =. exp (- E /kT) = partition function of the particles in phase j j r rj FORMULATION OF ADSORPTION ISOTHERMS For convenience, the phases in equations (1) and (2) are labelled as follows: j = 1, = gas phase j = 2, = Dn (Ads), or An (Ads) phases j = 3, = nD (Ads) or nA (Ads) phases; = 4, = nD (Ads) or nA (Ads) phases j = 5, = nD (Ads) or nA (Ads) phases I. Equilibrium Relationship Between Phase 1 and Phase 2. From equation (12), for component i one obtains +11 = [ 2+N2 (12a) That is P l +2l exp exp k T kT From equation (15) it follows that 2 = 1 exp t( l - ~ fz)/kT} Let W = 1 - Z = the van der Waals' energy difference between the lowest free state and the lowest adsorbed state without

-5dissociation. (It is noted that if, in the range considered, the van der Waals' force is an attractive force, W will be positive and heat will be evolved upon adsorption. ) Therefore 2 = 1 exp W/kT (16) From equation (15) by definition N. f. N exp W/kT (17) 2 1 f The partition function of three dimensional and two dimensional gas are given in any standard textbook of statistical physicso For the partition function of particles in a volume'T n M kT 3/2 f. = I V (18) Correspondingly, the partition function for particles on a surface is Z 12Tn M kT f2 = I, S (19) -:~J J where M = atomic weight of D, or A I, = partition function of internal degree of freedom in the particle n = number of atoms per molecule V = total gas volume in the system, cm3 S = total adsorbing semiconductor area in the system, cm To simplify the writing let = ( )(in cm ) (20) Then f = (n)3/2 l 1V (18a)

-6f2 = (n3) I S (19a) Combining equations (17), (18a) and (19a) one obtains 2N2S 1 i(X 1 )()/2 exp W/kTJ (21) S V ii For ideal gas approximation PV = N kT 1 (-) =(P) (22) For the sake of avoiding confusion in units it is defined 2 3 that P is in dynes/cm, kT in erg, and volume in cm Then (P/kT) is in rcm- 3 Combining equations (21) and (22), and writing [D or [A for N2 one has [D l ( ( l exp (W/kT) (23a) [A" = (k) ( -7)( )1 exp (W/kT) (23b) II. Equilibrium Relationships Between Phase 1 and Phase 3. In entering phase 3, the molecule D or A dissociates into n n n atoms, and the total energy of the particles in phase 1 and phase 2 now is comprised of a chemical energy associated with each particle, 39 a generalized external field energy per atom,'3( and an energy of dissociation per atom, L This can be expressed as 11 1 -+ 1 112 + * 2 n nB3 + n'l3 + n(12b where = dissociation energy per atom which is also a form of potential energy~ FollowTing the procedure as in Section I, define nW' - ~ - nXK, (24)

-7Now substitution of equation (24) into equations (12b) and (15) gives the relationship 1 exp ___(I____ =X eXP W J (16a) Substitution for the X's gives N3 N1 __ W - 3 f (y) ~ exp WkT (25) 3 fS I n kT Now, defining the partition functions for the atoms on the surface in phase 3 f3 = 3 3 3 9 k)I S = p(IrS (19b) Substitution of equations (19b), (22) and (18a) into equation (25) gives 2n 3 N 3 2n S [3 2/3 T exp k k 3 (26) But since N3 /S is the surface concentration of atoms in phase 3, then for donor atoms, equation (26) can be written as CA] " =n B) (- P g3 n3D3 nf) 3 exp k k (2 a) For acceptor atoms on the surface in phase 3 the result is [Al = exp (27 b) Equations (27a) and (27b) relate the concentration of dissociated atoms on the surface coming from an n-atom molecule, to the dissociation energy, the partial pressure of the molecular gas, and other factors as indicated,

-8ll. Equilibrium Relatinship of Phases 3, 4 and 5. With neutral atoms adsorbed on the semiconductor surface, the possibility exists for electronic interactions between the adsorbed atoms and the semiconductor. Furthermore, for the sake of completeness in the development ard in order to make comparisons with actual situations, it is assumed that there are singly charged ions as well as doubly charged ions on the surface. The total charge on the surface is given belowo For donors, the surface charge density is related to the ion concentrations by the relationship T = L[D + 2 D J (28) For acceptors, the relationship is M3 = A- + Z A-] (29) For a donor gas, it is assumed that the neutral adsorbed donor atom such as a hydrogen atom has an uncompensated electron spin. Therefore in going from phase 3 to phase 4 the following reactions can occur, D ) D+ + e~~~+ ~~~~~~~~(30) D ({) - +(3 e Since the electrons initially are in a surface state swith energy level EB then from the distribution function it follows that CD ] = 2 [D+a exp (fF -ED)/kT] (31) For the transition from phase 4 to phase 5, with the removal of another electron from the donor atom, the following reaction can occur. D tP D++(i) + e- (32a) D — D () + e (32b) Obviously, these reactions canrnot occur when hydrogen is the donor

-9atom, but this case is included for completeness of the development of the theory~ Therefore, from the distribution function, the relationship between the singly ionsized donor concentration and the doubly ionized donor concentration can be obtained. D = [D+ exp E - Ed kT3 (33) where Ed is the donor level of the first electron Ed-, that of the second electron. The factor of 1/2 arises because of the assumption that the electrons leaving the donor atom now combine with the original electrons and form a compensated pair,.E1 Using the bottom of the conduction band as reference, EF = E - F EC n Ed E - Y - 1 ~~1 Ed = E Y - c s d2 Therefore, EF - Ed = Y + d - n (34) EF EdY + d - (3 5) F d2ns

-10Substitute (34) into (3l1) [DI = 2 [D: exp Y /kT exp (36) kT Substitute (35) into (33) +Di = 2 rm exp T d exp d n (37) = { For the acceptor gas, it is assumed that the A atom has all electron spins compensated. When these atoms ionize, then ions with uncomplicated spins can be formed. A A- gt) + e (38) A A ($} + e where e is a hole in the semiconductor Jurfaceo Therefore from the distribution function it follo ws that eA = [A exp EF EA) / kT (39) where A is the total singly charged acceptor ion concentration. For the second ionization of the acceptor atom,'the reactions are A (f) A + e (40a) A ()=_ _ A + e (40b) The total singly ionized acceptor ion concentration then is related to the doubly ionized acceptor concentration through the distribution function by the relationship [A- = 2 [A exp F E) / T (41) where EA and E A are the acceptor levels associated with the ions on the surfaceo

-11fb I. This case can be looked upon as the transferring of a hole from the acceptor gas atom or ion to the valence band. Choosing the top of the valence band as the energy reference (which is arbitrary since ultimately only energy differences enter into the final equations) the following definition can be made. EF =E -4 F v p E =E - Y A v s al E E - Y - A v s a2 Therefore E -E = Y + E -~ (42) F A s a p EF - EA = Y + E - (43) ~sF A s p Substitution of equations (42) and (43) into equations (39) and (41), respectively, gives CA = 2 EAI- exp exp f(E ) /kT] (44) CA-] = 2 CA= exp {-4 exp (E ) /kT (45) Having formulated equations (36), (37), (44), (45), one is ready to find the equilibrium relationships between phase 3, phase 4

-12and phase 5. These relationships will be established by using the relationships between Y and o which were previously derived. These s 0 relationships can be regrouped into classes. The first class of cases is associated with accumulation layers on intrinsic or doped semiconductors, and with highly inverted surface layers on nearly intrinsic semiconductors. The second class of cases is associated with depletion region or slightly inversion space charge region. The Class I cases are discussed first and the adsorption isotherms are derived completely before the Class II cases are considered. A. Class I Cases: Accumulation Space Charge Regions. In the next section the relationships between the surface potential and the surface ion concentration which have been previously derived are summarized. A. Accumulation Layers Produced by Surface Donors. Case 1. Surface Donors on an Intrinsic Semiconductor. Y L 2 kT In [1 C 1 (46) kT n, Case 9. Surface Donors on a Nearly_Intrinsic -te Semiconductor n n (47) Y L kT n B. Accumulation Layers Produced by Surface Accepftors. Case 2. Surface Accettors on an Intrinsic Semiconductor.

-13Case 4. Surface Acceptors on a p-type Semiconductor. Y L 2 S = I n p (50) Highly Inverted Layer Produced by Surface Acceptors. Case 10. Surface Acceptors on a Nearly Intrinsic n-type Semiconductor. Y L s = in (51) kT The mathematical expressions for these cases are similar. After a more complete development of these cases the remaining four cases will be considered in a later section. A. Accumulation Layers Produced by Surface Donors. Case 1. For Donors on an Intrinsic Semiconductor. Substitution of equation (46) into equation (36) gives, LEd 2n n k where d - rD o++D Furthermore, since IT=2 ( kT (m m3/4 exp i n= pi = 2 2 TkT)3/2(m exp i T kT = (N N) /exp' Therefore L 2 ED:( N )l/2 exp {dl??ED [D++2D JJ (52) c~l 2N / expk T cv where 1b= n (2 Substitution of equation (46) into equation (37) gives

LD. 2 n exp t dT I D [D +2D 1 Lm FEdZ I ~'ID1 rD2+ 2D1 2 m 21.1', 2(N N )/2 exp kT c v For simplicity, let L C1 - N N exp l/kT (54) {1d/kT) c v 2 N e 2 = N N exp | Z / kT (55) c v Then [D] = Z C1 [DJ D + 2D I (52a) [DJ = 1 C2 [D+ ED + 2D 2 (53 a) 2 2 Equation (53a) can be rearranged into y = { 2 cr ]D i - 4 C2 CD[+ZlJ L~~~~~~~~~~~~~D =~~(54) One must choose the negative sign before ~ t1 - 4 C2D + 1/L so that when LD J - 0, then rD J - O alsoo Since [D+] must be real and positive number, one also must have 1 - 4 C2 [D+22 -0 In order to obtain a minimum value for the singly ionized donor concentration consider that the radical in equation (54) is zero. That is 1 -4 C2 D = 0 Under these circumstances, then [D+ ma C 1/2 (55)

-15.Substitution of this result into equation (54) gives nln= - 1/2 [rD in C2 (56) A relationship between LD++ max and 1D+3 n then follows directly. LD1 = 2L mr m (57) max 2 min Now consider the situation wherein the radical in equation (54) is positive and [D+I is below its maximum value~ Therefore 1 - 4 C2 D++ 2 0 Under this condition it follows that 0 <4 2 CI D 1 One can expand the expression, - 4 C2 LD+ 2 J into an infinite power series for the situation wherein 0 < z < 1. This series is 1/2 1 1 2 1 3 2z m (m-2).. 4m-1 m=2 Therefore, applied to this situation, the result is )I - 40 C.4 1C -D++ m m (m-2), ni (58) m-=2 Substitution of equation (58) into equation (54) with the negative sign as indicated gives the result 2m-2 CD+] = 4 C2 D ( m -- 3) 2 [D+. 2 (59) m2M (M - )o 2 There are two nore points that are needed to ponder. (i) If =D - CD,

is it generally true that 0 - = 1/2 (ii) If 0 y ~< 1/2 how fast does the series defined in equation (59) converge? Is it converging more rapidly as y decreases? To investigate these questions one substitutes the definition [LDI = [ ED+ into equation (54), and one obtains C [D+*:2'y] 2(60) 02 D+ = (1 + 2 y )2 Substitute equation (60) into equation (58) m 1 - 2y D2 m (m-3) 2 D++ 1 + y 1 I - 2.... m. (m-)'. DC2+ D (61) m=-2 Since, when 4 C2 D+ 2 < 1, the right-hand side of equation (61) is greater than zero. Therefore 70 1 + 2 That is, <y. 1/2. When this result is combined with equation (57) and the fact that y cannot be negative is observed, then we have the conclusion that 0 - y < 1/2 (62) As to question (ii), an examination of equation (60), will show that as y decreases, the series defined by equation (59) converges more rapidly. For adsorption isotherms, we can substitute equation (53a) into equation (52a) to give [D~3 = c'D_ J +O 2 + D+ 1 (53 b) CD9 =, C1Z rl aC LD z+ z C 23 (532D) Furthermore, using the definition for y and its restriction imposed by equation (62), equation (52a) becomes

-17Therefore [DJ = 2 C1 (1 + 2Y) [ED+3 (63) Similarly, equation (53b) becomes [D - C 1C (2 + ) +5 (64) These equations can also be written in terms of the charged species as ( 1/3 ( 2/3 F 1/3:: = ( 1 ) ( )CD] (63a) and 4/5 1/5 V~~~~D ++~~~~~ [F~~DI (64a) Note that when y = 0, so that there is only [D+J, one has the relation (63a) only. Case 3. Surface Donors on an n-type Semiconductor. Substitution of equation (47) into equation (36) gives ] = 2 nBn exp k T+k D B kT where d, CD++ 2D++ Furthermore, since nB N exp k- bT/k then ~ 2 Nc2 exp kD ) J D+++ ZD (65) Substitute equation (47) into equation (37) CD<... Nm_-L- exp.. D' + Z DJ' (66) Now define C N wf- exp 3E'g(67)

2 N exp K T (68) Then [D2 = 2 C +] [D + D 2 (65a) C[D = z1C 2 D1 D D+ + DJ2 (66a) By comparing equations (65a) and (66a) with equations (52a), (53a), one sees that they differ only in constants C, GC/ and C C 1 Z 1 Z2 Therefore the remainder of the derivations and results should be the same. Therefore, equations similar to equations (63a) and (64a) are obtained. These new equations are CD (1 \/3 2/3 (1)2/3.+ 1 1/5 4/5 1/5 [D++] =) ([D) (66b) Case 9. Surface Donors on a Nea ntrinsic — te Semiconductor. The results for this case are exactly the same as are obtained for case 3 since equations (47) and (48) are identical and are represented by equations (63a), (64a). Accumulation Layers Produced by Surface Acceptors. Case 2 Surface ce tors on an Intrinsic Semiconductor. For the neutral atoms in phase 3, substitution of equation (49) into equation (44) gives [A] = - i a eXp [A- [A-+ ZA2 For the singly charged atoms in phase 4, substitution of equation (49) into equation (45) gives

- 19 - [CA 2 -Pi e x-p k A But p. can be expressed as P = (N N 1/ expkT and substituted into the above equations to give l, =. m( -)V/ exp {Im} -jAJ LA-+A2 A (67) V. C and A 2,_i e~xpP T2 c J 2_ A+A (68) (N N )/2 For simplicity, let K1 - TNNV 1/2; exp [kT (69) (N N) v/ c v K \T E\~ ~ e~exp (70) Then substitution of equations (69) and (70) into equation (67) and (68) gives l ] I K A] [ 2A j (67a) CAl 2 K2 +[A UA+ 2A (68a) The solution to equation (68a) then is 1 8 K 2 I 1 - 8 K2 L t3l 2 rEAI 1 K[- (71) 4 Kz [AIJ

-20From the reasoning similar to that given in connection with [D+3 and CD.| following equation (54) it follows directly that 16 K2 [A < 1 and 1 - 16 K2 [A2 1/2 = 1- 8 K2 CA=2 - n (m- 2) Z (72) m=2 By the same reasoning process as before, define y as the ratio of CAJ= / CA-3 That is =A = Y CA] (73) and the limits on y again are 0 - Y - 1/2 Therefore from equation (71) it follows that K2 LA: = - (74) 2 2 (1+ 2)2 From equations (67a) and (68a), one obtains [A] = K1K2 rAJ EA + 2 A J4 (68b) From equation (67a) and the relation (73) [A] = 1 K (1+2 y2 [CA) 3 (67b) From equations (68b) and (73) [A] K1K2 (2 + 1)4 CA- 5 (68c) Therefore, solving equation (67b) for [A-] gives IA u (6 )/3 I )2/3 (75) and solving equation (68c) for [A 3 gives

-21[_]M1 ]I( z Y' )5 ( d~ )/5 1/5 Case 4. Surface Acceptors on a p-type Semiconductor. Following the same procedure as in case 3, but defining L E'a K1 =N exp (77) 2 N k T LI E a2 i (77) K2 - N' exp T(78) The results which are obtained then are [A-] - [A] (79) K[ K / + 2 Case 10. Surface Acceptors on a Nearly Intrinsic n-type Semiconductor. The formulation is exactly like case 4 above, and the results can be represented by equations (79) and (80). IV. Equilibrium Relationship Between Phase 1 and Phases 4 and 5. Having established the relationship between phases 3 and 4, 3 and 5, on various semiconductor surfaces, the authors proceed to formulate the relations between phases 1 and 4, phases 1 and 5, i. e., the dependence of the surface ion concentrations on the partial pressure of the gas in phase 1. Case 1. Surface Donors on Intrinsic Semiconductor. The desired relationship can be found by combining equations (63a) and (27a) for singly charged donors. The result is + (OI 1/3 (Lz )2/3 ( 1 1/3 LDJ (z1/ yi+ 2nI kT(n p)3/

-22Since L Ed 1/2 exp Therefore ~D+] = L/ vl )} 1 / 31/3 ZLm I nk T(n) 3/2 J 1 W- - n E 1 dexp l p 3n (81) 3 nkT ) For doubly charged donors the result is obtained by combining equation (27a) and equation (64a). The result is ++=(__ 11/5___ ( 4/5 P ++ (1)/5 (~4/5 t 3 }~1/5 exp 5 n k T PD 5n S T n However, since 1/5 N Nd + d C C) L 2 exp 5 k T 1 2 L Then vL ci+4.).j?n. T.)3/i. 1/. + N'- ) 1 3 w a - n (E d a exp 5 n k T PD (82) For case 3 and case 9, the results are

-23[D3 1/3'- -n E ll/3 m(1+) 15 -exp- 3nkT PD 3m (83) DT(np) n N2 4 1/5 3 i 1/5' -A -n(E D++] =,2 4 exp 5kn PD 5nl (84) Lm (1+2-y) kT3/2n n Case 2. Surface Acceptors on Intrinsic Semiconductor. For surface acceptors on an intrinsic semiconductor equations (75) and (27b) can be combined to relate the concentration of singly ionized acceptors to the partial pressure of the acceptor gas. =13 )-/3 exp T p 3n EA ) ( ) (Y ) (np) kT I- nA Since L K (N N 1/2 exp k,1, v c Therefore [KJ....3(... exp PA3n (85) L(1+2 -y) L 3 T PAn r (n kT I T For doubly charged acceptor the result is obtained by combining equations (76) and (27b) ONN y4 1/5 /5 W'- -n/ ( Eal Wa _exp 5nkT a (86) Lm2 (+2y ))4 l(n) 3/2 PAn (86) For case 4 and case 10, the results are For case 4 and case 10, the results are

-24 - 1/3 [A3=?Vn)k exp kT PA3n (87) ______2~ (np) 3/2kTI 1 p Cases 5 and 6 as previously defined represent depletion layers on surface chargeI a al+ are summarized1 L 3 (1 + 2~) (nD M' kT I The Class II cases are associated with depletion regions or slight inversion space charge regions. A. Duepletion Layers. Cases 5 and 6 as previously defined represent depletion layers on non-intrinsic semiconductors. For these cases the relationships between surface charge and surface potential are summarized. Lm + 2 U - L d+ J + 1 for a p-type semiconductor Nd L 2 S u Cd~3 — + I for an n-type semiconductor For a donor gas the surface charge can be defined by equation (28) and for an acceptor gas the surface charge is defined by equation (29). By combining equations (36) and (37) the result is I~dl Ed -'+!__n D] CD++ exp (2 us) exp {l + (89) For a p-type semiconductor equation (89) can be modified by using the relations hip in = E- p(90)

-25where ~g = energy gap of the semiconductor. The energy 4p is related to the bulk hole concentration by the relationP ship. PB = N exp (-p /kT) v p Therefore +/ N e p/kT v (91) Equations (36), (90) and (91) can be combined to give d - (Eg - p) CD3 = 2 D+J exp (u) exp k T } E Ed E = 2 [D exp (u) exp L-d E exp kT N ~E d CD[ = 2 DJ exp u(s i) exp { d2 But since 2 L 2 u -+3 + 1 S N- + a Therefore substitution of the relationship above gives [DJ = 2 CD exp (I 2 + 3 exp L 2 N E (2e) LDI exp +2j N x p EgJ This relationship can also be written as CD] =(2 e) P\ L N- [D+ exp DZ+ exp jk- -T. Assume as in previous sections, that [D++= y D+I where y7// 0 and also PB Na

-26The final, result is 2eN L 21/Z L d1Eg CDJ -P -D +] exp(1+2) D exp(l (92) However, this expression is still rather intractable. In order to simplify it further, take the logarithm of both sides. This gives: C4 N22 (E -E 1 4e N d- g In L1 In —___ 2n 2 L m +' in:D+E + (1 + )2 m D+3 (93) 2 I3~68B P-B=-~ PB Substituting equation (27a) into equation (93) gives, t3 P(n 1 3/2 kT I Lm In PT d + WI E k T + n[ CD+] PB.. L 2 M 2 + (1+ t2 z y) p3 l (94) Equation (94) is the general isotherm for a donor gas on a p-type semiconductor. Now, If m +3 i// 13 so that the termn- In D+ can be considered as 10% or less of the total value, then equation (94) can be simplified and written as [D) 3J; p L In +"D H' (94a) D:,OJ1 n n p 2

' 27where H PB / Lm (1 + 2 y)2 (95a) p, 1 H = in HpF 2 n?=1 e.Ij31;T N i E d + n q + W' - n dl _ t (96a) For doubly charged donor gas, equation (89) is used. By similar substitutions,: one obtailns +E d2Eg+2 CD- p = e v exp p+ [+ (97) B JLmpJ B L By combining the relationship LDb+ = y [LD with equation (27a) and defining the coefficients IP (Ed +E d PB_ Z n Eq W' n _ d2 T - -n -, = np l Pm. (2 +2 + n k T (98a) (n3/2)I kT N e 1 V and J Y P /PB (99a) P, 2(2y+ l) L / Under the conditibn, 2(Lm) [D+ +2 D++ 2 I> In D(m) D++], elquation (97) gives:2 B P/ CD++ 1 (1D 0 (looa) B. Inversion Layers on Moderately Doped Semiconductors. For a donor gas associated with a slightly inverted space charge region, the s!irface potential is defi.ned withill the limits u: <'(2 i

-28This corresponds to cases 7 and 8 as previously defined. The surface potential is related to the surface charge through the relationship L m D] + 2 CD++] s PB By using the procedure which is exactly the same as that outlined in Sectioh V-A immediately above define H n= i m 3nfB +n Eg+WL dl = - (lOla) n 3/2 nkT (n) InkT For case 7 the result is 2 LD 1 - In PD + H (102a) p, n n p, 3 For doubly charged donors on the surface of a p-type semiconductor, define J =ln (3 B pL) PB 2n Eg +W' -n 1d d -(103a) r. +, (103a) p, 3 n /2 2 n kT (np)3/2kT N This result then is D++ P1In PD + J3 (1 04a) For case 8, an acceptor gas on an n-type semiconductor, the procedure of formulating the isotherms is similar to that used above. The various constants are defined as follows: nB H -- (95b) n, 1 + ) L (l+Zy)2 i _L 3 ) nE +W' n Ea a- -- n 2.... 1. 3 (96b)

-29B L I) nEg+W'-nE a1H n In (101b). n3/kT (n) I kT n 2 Y nB J 1 B (9 9b) n, 1 2(2~+ 1)L2 2(2+ 1) L m =- I3 - nB /nB Lm n, 2 n n, —------------ J2 m 3/2 2 2 (np) I kT Nc e + 2 n Eg + W' - n a I + a2 (98b) n k T 1 E 3 IIB L 2 n Eg + W'- n a 1 + a) - lfl( " Tn N nk+1 2(103b) n, 33-2 2nv;;=c nkT Ink T NJ+ The adsorption isotherms for an acceptor gas on an n-type semiconductor associated with a depletion space charge layer are then summarized in the following equations. For singly charged acceptors on the surface 2 =~. I C -Hni | ln (PAn ) + Hn,23 (94b) H n, 1 n n,H II 2 (94b) For doubly charged acceptors on the surface LA=3 = Jn 1 l( ( AnA + (100b) The adsorption isotherms for an acceptor gas on an n-type semiconductor associated with a slight inversion layer are also summarized. For singly ionized acceptors [A Hn, ]PA H 1 (102b) For doubly ionized acceptors

-30In order to summarize the. isotherms for the several cases, it is convenient to define various coefficients. Let I 121 AC"? exp W/k T (105) n (np) (k T) k —:- - exp nTd (106) B n B3/Zk.T 1 ", /s (W, ~ Edl ) N I3d _____ 3___ W -n d+I FCK.e4 enkTpj' (107a) Ci!ZL (1+3/2 1n3/ie3. m (nf) I kT p13 rW~~f1d3 d N e p 1 3 WI-..-n _dl_ C e (12Y 2 n/~3 exp k n ( T0b) exp L4 _I1/5 (E d+E / e - - F.= 1exp (108a) 1 L2 4l~zy~4 ~ n k T m (np) IkT 2 4 1/5 Ed + Ed C c 3- -n 1 2 2 n =kJTexp (108b) (+L2 )1+2~ )(4 3n/2 e L 2 (1+2N) 4 -n'5 n'k'T..' (n) 3/ZkT M ~~~~~~~II ~ I' ~1/3 (,n 2 4NN~ ~ 1/3e~xp (V- 1Z 3 nkT T. (1p9a TT C 1nP K= - ~~~~~~~s exp ~~~~~(109a) K,=(miZ).. i Ln(n+~,.Y ) 2 3nk3/ k 2N ~{33 ~1/3 p_,~n PI. "c (n'~) IlkT) ('k-nk-TTJ?)" (109b) K -E l- - exp Lm (1+2-y) kT7~ nk

-3 1G _ ______ exp2 } (liOa) L 2 (1+2y)4 5 n k T G (nP)D = n3 -n P G exp (Ir B b) e i2 4 5 n k T M (np) I kT n For a donor gas on any semiconductor substrate r:D] = B (PDn)n For ionized donors on an intrinsic semiconductor: D+ = C (PD'n 1 n LD++ = F (D ) For ionized donors on a highly p-type semiconductor, a depletion space charge region is formed, and the adsorption isotherms are p 1 (n LDJ -' H1 { In l(D ra+ =n (I ) 0 1 rnZ + For ionized donors on a moderately p-type semiconductor, a slightly inverted space charge region is formred. The adsorption isotherms are D+] = H r In (PD) + H

-3 2[D + = JP 1 in (n) p+ 3 P n p 1/n [AJ = B ( A1) n For ionized acceptors on an intrinsic semiconductor [A-] = K, (PA) /3n 1/5n [A = G. (A ) 1 n For ionized acceptors on a p:-type semiconductor or on slightly n-type semiconductor P /3n [A] = K (A) FAI = Ge (6A) /5 e n For ionized acceptors on a highly n-type semiconductor, a depletion space charge region is formed and the adsorption isotherms are ~A,2 = -H In ( A) + H n, 1 n n n, [A=l * n, 1 [n (n n 2 For ionized donors on a moderately p-type semiconductor a slightly inverted space charge region is formed. The adsorption isotherms areA- H I n A + H n, 1n n n 3 Conclusion Gas adsorption isotherms of ventral and charged particles based on a generalized. model have been formulated. Although the formulations

-33have been based on a molecule which yields n atoms upon dissociation, they can be applied to the cases wherein the molecules, such as D or A, can be ionized without dissociation. The functional n n relationship between [D.I or [A] and the respective pressures in the gas phases can be obtained by treating [Dn] or [An] as monatomic molecules. That is, by putting n equal to unity in the respective equations. Factual confirmations as well as practical applications of the formulations will be presented in subsequent publications. References 1. Hill, Terrell L., "An Introduction to Statistical Thermodynamics, " p. 126, Addison-Wesley Company, Reading, Mass. (1960). 2. Landau, L. D., and E. M. Lifshitz, "Statistical Physics," pp. 105-106; 152-154, Addison-Wesley Company, Reading, Mas s. (1960). 3. Fowler, R. H., and E. A. Guggenheim, "Statistical Thermodynamics, pp. 45-51; pp. 66-67, Macmillan Company, New York (1939). 4. Lee, V. J., and D. R. Mason, "The Relationships Between Surface Charge and Diffusion Potential Beneath Semiconductor Surfaces," Progress Report 04650-3-P, May 1962.