P-N JUNCTION CAPACITANCES PART I: THE … Bound... · P-N junction capacitances,Part L'the depletion capacitance In fig. 2 the perfect agreement with numerical TRAP values is shown

Philips J. Res. 40, 88-102, 1985 Rll04

P-N JUNCTION CAPACITANCESPART I: THE DEPLETION CAPACITANCE

by J. J. H. VAN DEN BIESENPhilips Research Laboratories, 5600 JA Eindhoven, The Netherlands

AbstractAccording to the depletion approximation the depletion capacitance be-comes infinite when the applied voltage equals the built-in or diffusionvoltage. However, experiments and computer simulations show a finitemaximum. It is important to investigate this discrepancy since the high-frequency behaviour of a bipolar transistor is very sensitive to the forward-biased emitter-base depletion capacitance. Moreover circuit analysis pro-grams need a well behaved model for depletion capacitances. In this paperthe exact analytical equation for the depletion capacitance of a symmetricalabrupt junction is presented. It is obtained from a direct solution of thePoisson equation. The Debye length turns out to play an important role. Inthe case of linear or asymmetrical abrupt junctions such exact solutions arenot feasible. On the basis of computer simulations, however, expressionsfor the asymptotic behaviour of the depletion capacitance could be estab-lished at high reverse as well as at high forward bias. For both junctiontypes the classical value of the built-in voltage as obtained from the deple-tion approximation turned out to be incorrect.The depletion capacitance models proposed by Poon and Gummei and byde Graaff have been evaluated on the basis of the present results.PACS numbers: 7340 Lq; 8530 DeEECS numbers: 2530 B; 2560 B.

88 PhllipsJournalof Research Vol.40 No. 2 1985

1. Introduetion

During the past three decades considerable effort 1-28) has been made todevelop equations for the depletion capacitance of (a)symmetrical abrupt andof linear junctions which are more accurate than the classical expressions pro:vided by the depletion approximation 2~-31). The depletion approximation iseasily seen to be not self-consistent 32): the potential distribution derived fromthe depletion assumption can be used to calculate the mobile carrier coneen-trations again. Especially at forward bias the so-called depletion region turnsout to be far from depleted of mobile carriers! Accordingly, it is obvious thatthe classical expressions for the depletion capacitance lose their validity at for-ward bias. Whereas the classical depletion capacitance becomes infinite when

P-N junction capacitances, Part I: the depletion capacitance

"the applied voltage equals the built-in or diffusion voltage, numerical devicesimulation programs 3,33-36) find a maximum slightly below this voltage. Ex-perimental evidence for the existence of such a maximum has been obtainedindirectly fromh measurements37) and, more directly, from high-frequencycapacitance measurements 4). It is important to investigate this failure of the'depletion approximation, since the high-frequency behaviour of a bipolartransistor is very sensitive to the forward-biased emitter-base depletion capaci-tance. Moreover, circuit analysis programs need a well behaved model fordepletion capacitances.After some general remarks in sec. 2 the exact expression for the depletion

capacitance of a symmetrical step junction will be derived in sec. 3. Asym-metrical step junctions will be discussed in sec. 4 and linear junctions in sec. 5.In sec. 6 the two most sophisticated depletion capacitance models proposed inthe literature will be evaluated.

2. Definition of capacitanees

In this section we will recall the definition of the junction capacitance andits subdivision into depletion, diffusion and neutral capacitance. Moreover thespatial distribution of charges and associated capacitances will be illustrated.

In the following a pn-diode will be considered with contacts at x = Wp onthe p-side and at x = Wn on the n-side. The metallurgical junction is situated atx = 0 (Wp < 0). The total capacitance per unit area, Ctot, is given by

Id Wn I

ClOt = q dV ~pdx (1)

or, equivalently, by

Id Wn ICtot = q dV ~ n dx , (2)

where pand n are the mobile hole and electron concentrations, respectively,and Vis the applied voltage. According to Kwok 36) the location Xo of the elec-trical junction relevant for this problem is situated at the point wheredp/dV = dn/dV, which only for a symmetrical junction coincides with themetallurgiçal junction at x = O. Denoting the positions of the edges on the p-and on the n-side of the classical depletion region by Xp and xn, respectively,the total capacitance can be split into 36)

(3)

with

Phlllps Journal of Research Vol. 40 No. 2 1985 89

J. J. H. van den Biesen

Id w. ICdepI = q dV L (n - p) dx , (4)

. Id{XP W'}ICdiff = q dV ~ n dx +!p dx ,

I d{XO x'}1CN = q dV L n dx +Lp dx .

(All capacitances in this paper are per unit area; a wide variety of other defini-tions and names for parts of the total capacitance have been proposed in theliterature.)

This paper deals with the depletion capacitance, CdepI • Cdiff is the diffusioncapacitance stemming from the storage of neutral charge (i.e. compensatedcarrier charges) in the neutral regions of the diode. CN is the neutral capaci-tance associated with storage of neutral charge in the depletion layer. In part 11of this work 38) an approximation for CN will be presented. Each of the threecomponents of the total capacitance has a different dependence on the appliedvoltage.

For a better understanding the illustration presented in fig. lis very en-lightening. Here the quantities ql dp/dVI and ql dn/dVI have been plotted,so that the different components of the total capacitance can easily be distin-guished. The curves have been obtained from the numerical device simulation

(5)

and

(6)

(Ol

E::J,u..s,

.g.1~50---0:--

c::1::.."0 "0

o-

Fig. 1. The three components of the total capacitance of a symmetrical step junction with NA =ND = 1X1017 cm-so W.= - Wp = 0.25 urn and V = 0.75 Volts (cf. eqs (3)-(6)). -- qldpldVI.--- ql dnldVI·

90 Phllips Journol of Research Vol.40 No. 2 1985


program TRAP 34). The peaks in ql dp/dVI and ql dn/dVI correspond to theedges of the depletion region. (In the depletion approximation delta functionpeaks appear at Xp and xn.)

It is remarked at this point that neither in the analytical equations nor in thecomputer simulations has the effect of bandgap narrowing at high dopinglevels been taken into account. All.examples in this paper concern silicondiodes at a temperature T = 298 K with an intrinsic carrier concentrationn. = 1.033 x 1010/cm3•

3. Symmetrical step junctions

For a symmetrical step junction with doping concentrations NA = ND thePoisson equation will be solved at the n-side of the junction. Note that in thiscase Xo = O. It is assumed that the quasi-Fermi potentials (/Jp and (/Jn are con-stant throughout the device. The consequences of this assumption will be dis-cussed at the end of this section. Introducing the potential w so that 1fI(0) = 0,the quasi-Fermi potentials in the case of an applied voltage V over the junctionare given by

(/Jp = ~V (7)

and(/Jn = -~ V, (8)

yielding for the carrier concentrations

(9)

and

(10)

where VT = k'Tlq is the thermal voltage.The built-in or diffusion voltage of the junction, VBh is defined as

VB! = 2 VT In (~~ ). (11)

At the contact at Wn neutrality applies: ND - n(Wn) + p(Wn) = O. For thecarrier concentrations at the contact this condition yields

n(Wn) = ND [~ + ~VI + 4 exp ( V ~ VBI) ] (12)

. T

andp(Wn) = n(Wn) - ND• (13)

Phllips Journalof Research Vol. 40 No. 2 1985 91


The potential IJI(Wn) at the contact is then given by

(W.) - VBI- V v: I n(Wn)Ijl n i r= 2 + T n No . (14)

Wn Wa

Qdepl= q f (No - n +p) dx = f g dx,o 0

(15)

It is convenient to introduce at this point the depletion charge per unit area,Qdephdefined as

where g is the space charge density. Since the boundaries of the integral do notdepend on the voltage the depletion capacitance can be written as

C I dQdepl Idepl= dV . (16)

Using the Poisson equation

(17)

(8 is the permittivity) it is easily seen that

Qdepl= 8 [E(Wn) - E(O)), (18) .

where E = - dlJlldx is the electric field. From the neutrality condition at theÎ contact it follows that E(Wn) = O. The electric field at the junction, E(O), can\ be obtained from the Poisson equation following the method of Garrett and

Brattain 2). Multiplying both sides of eq. (17) by 2dlJl/dx and integrating over xone finds

2 \II(Wn)

E(0)2 = - f g dlJl.e 0 (19)

Here again use has been made of the fact that E(Wn) = O. Substitution of eqs(9) and (10) in g, integration over Ijl and subsequent substitution of eq. (14)for IJI(Wn) lead to

r--------------------------===~--_=====~--Qdepl= qN08(VBI- V + 2VT[ln n~n) - {V n~n) - VP~n)} 2J) (20)

and, after differentiation, to

c = {V n(W,,) - VP(Wn)}2 Xdepl No N

D

!qNo8

92 Philips Journalof Research Vol.40 No. 2 1985

P-N junction capacitances, Part L' the depletion capacitance

In fig. 2 the perfect agreement with numerical TRAP values is shown. Acomparison with the classical equation

V~qNDeCdeplcl = VB! - V ' (22)

shows that the exact solution contains two extra terms involving the carrierconcentrations at the neutral contact.

<;'E::J..LL--

V/Volts

Fig. 2. The depletion capacitance Cd,pl as a function of applied voltage V for a symmetrical stepjunction withNA = No = 1X 1017 cm:" and zero current densities. --. -exact solution (eq. (21»,• TRAP simulations, -- depletion approximation (eq. (22», --- reverse bias asymptoticbehaviour (eq. (24», -' - high forward bias asymptotic behaviour (eq. (25».,

It will appear useful to introduce the extrinsic Debye length, LD' defined as

LD = V e VT• (23)

qND

It is the characteristic length associated with the shielding of impurity chargesby mobile carriers and, as a consequence, with spatial variations of the poten-tial.

Phllips Journalof Research Vol. 40 No. 2 1985 93

(24)


In the limit of high reverse bias one obtains from eq. (21)

so that the intercept voltage in a plot of Cdepl-2 vs. Vis found to be VBI - 2VT,

consistent with earlier findings by many authors 2,6,6,9,11,13,16,16,20,22,24-27). Thebehaviour of the depletion capacitance at high forward bias can easily befound from eq. (21) (see fig. 2)

Cdepl = vliLo exp (VB;~V) =No V 32~:ni exp (;::). (25)

A maximum of the depletion capacitance occurs at VB!- 3.45 VT and hasthe value 0.243 elLo. Comparison with the parallel plate capacitor equationC = elw tells us that the minimum effective separation w between the two'capacitor plates' is approximately equal to 4Lo. Considering that any poten-tial variation in a semiconductor has a minimum spatial extension of a fewDebye lengths, the depletion approximation, which assumes an abrupt transi-tion of the carrier concentrations, is bound to fail when x; :5LD 39). It is there-fore not surprising that in fig. 2 the depletion approximation yields reasonablevalues for Cdepl up to the maximum of the exact curve, where x; :::::2Lo. Fromthe foregoing it is obvious that eq. (21) is only valid if the device is not toosmall (Wn ~ LD) and not completely depleted (w;, ~ xn).

In the derivation of eq. (21) the assumption of constant quasi-Fermi poten-tials (i.e. zero current densities) has been made. TRAP simulations with. non-zero current densities «({Jp and ({Jn not constant) never show any significant de-viations from eq. (21) up to about VBI• The dependence of the depletion capa-citance on the current density that is found at higher voltages is clearly relatedto the change of the quasi-Fermi levels across the device. In the case of non-zero current densities the quasi-Fermi levels in the neutral regions can be shownto be complicated functions of the distance, the length of the device, the diffu-sion constant and the recombination in the bulk and at the contacts. In the de-pletion region the extrinsic Debye length also plays a role. Since so manyvariables are involved it is not possible to specify the deviations from eq. (21)at high current densities.

4. Asymmetrical step junctions

For an asymmetrical step junction Xo in eq. (4) no longer coincides with themetallurgical junction, but becomes voltage-dependent. This gives rise to anextra term - qND Idxo/dVI for Cdepl in eq. (16) 13). As a consequence, an exactanalytical solution of the Poisson equation is not feasible. Numerical resultsobtained from TRAP for NA = 2 X 1019 cm:" and No = 2 X 1016 cm-3 are pre-

94 PhllIpsJournalof Research Vol.40 No. 2 1985

~E::J.,LL....<,

ë...(.)"0 0.1


100r-------~--.--->T._------~

0.01l- _i __ .L-__è[_J..._~ __ _J

o 0.5 1.0V/Volis .

Fig. 3. The depletion capacitance Cd,pl as a function of applied voltage V for various step junctionsat zero current densities. a) NA = No = 2x 1016cm:", b) NA = 2 X 10'9 cm:", No = 2x 10'6 cm",c) NA =No = 2x 1019 cm-so ---- TRAP simulations, -- depletion approximation for b)(eq. (26», --- reverse bias asymptotic behaviour for b) (eq, (27», -'- high forward biasasymptotic behaviour for b) (eq. (28».

1.5

sented in fig. 3. The depletion capacitance shows a maximum and a shoulderwhich turn out to coincide with the maxima in the depletion capacitances ofsymmetrical step junctions withNA = No = 2 X 1019 cm:" and withNA = No =2 X 1016 cm", respectively.The classical depletion capacitance readsr--------------------

V !eqC - 2dep!e! - --:-1-----:1,.;:----=.--------,

(-N-A+ -N-o) (VBIcl- V)

where VBle!= VTln(NANo/ni2).Confirming the earlier findings of Smith 16) and of Kennedy 24) the depletion

capacitance at high reverse bias appears to be. rather well described by (seefig. 3)

(26)

1 1 '(- + -) (VBle! - V - 2VT)NA No .

~e q (27)

where VBI = 2VT In(No/ni}'

Phillps Journal of Research Va!. 40 No. 2 1985 95


Quite similar expressions, some involving extra corrections, have been derivedby several other authors 7,13,15,20,22). Hence the intercept voltage in a plot ofCdepl-2 vs. Vis approximately equal to 2VTln(ND/ni) - 2VT•

On the basis of TRAP calculations and the analogy with eq. (25) it has beenfound (without analytical proof) that at high forward bias the depletion capa-citance is given by (see fig. 3)

NA +ND lrr;8 (-V)Cdepl = 2 r ~exp 4V

T• (28)

The classical built-in voltage turns out to play no role at all in the depletioncapacitance.

(aLe)VBI = 2VT In ----;;: . (30)

5. Linear junctions

For the case of a linear junction with doping concentration N(x) = ax anexact analytical solution for the depletion capacitance is not possible either(the integration in eq. (19) cannot be performed). Again without analyticalproof mathematical similarities with the symmetrical step junction have never-theless led to expressions for the low- and high-voltage asymptotic behaviourof the depletion capacitance, which could be confirmed by TRAP calculations(see fig. 4).At high reverse bias the depletion capacitance turns out to be described by

(cf. eq. (24» re VTCdepl = 2Le VBI - V _ 2VT ' (29)

where

Here a characteristic length Le has been introduced defined by

L = f3eVTe 2qa . (31)

In many respects Le plays the role of an effective extrinsic Debye length.In this notation the classical depletion capacitance can be written as

rs VTCdeplel = 2L V; - V .

e BIel'(32)

VB!eI, the classical built-in voltage, follows from the transeendental equation 40)

VBIel = VB! + ~VTIn (V;;I ). (33)

96 Phillps Journal of Research Vol.40 No. 2 1985


1::J,IJ...--ö.'"t3'

Val VBlc1

VJVolts

Fig. 4. The depletion capacitance Cd,pl as a function ofapplied voltage Vfor a linear junction withdoping concentration gradient a = 2 X 1023 crn" and zero current densities. -- TRAP simula-tions, -- depletion approximation (eq, (32», --- reverse bias asymptotic behaviour (eq. (29»,_._ high forward bias asymptotic behaviour (eq. (34».

(It is remarked that the values presented by Grove 40) in his fig. 6.11 are about0.05 Volts lower than the actual solution ofhis eqs (6.31) and (6.33) or ofthepresent eq. (33).)From eq. (29) the intercept voltage in a plot of l/Cdepl-3 vs. Vis found to be

equal to VB! - 2VT• Other intercept voltages proposed in the literature aregiven in table I, and their values for a = 2 X 1023 cm+ are compared. Thenumerical accuracy of TRAP does not permit discrimination between thethree improved intercept voltages.

The depletion capacitance at high forward bias turns out to be described by(cf. eq. (25»

e (VBI-V) a e (-V)Cdepl = 4Le exp 2V

T= 4ni exp 2VT •

The TRAP calculations for linear junctions yield a maximum for Cdepl of0.327 ell.; at VBI - 3.4VT (note the similarity with the corresponding values forthe symmetrical step junction).

(34)

Phlllps JournnI of Research Vol. 40 No.2 1985 97


TABLE I

Values proposed for the intercept voltage of a linear junction.

VB!cl

VB!cl - ~ VT

VB1 2VT

VB! - (~lnI2)VT

value (Volts) fora = 2x 1023 cm-4

0.92780.80790.81540.8242

sourceequation

depl, approx. (eq. (33»Kuzmicz and Swit 18)present (eq. (30»Chawla and Gummel v')

Because of specific properties inherent in linear junctions 8) eqs (29) and (34)remain valid up to very high current densities.

6. Modelling the depletion capacitanceCircuit analysis programs need a simple, well behaved model to describe

depletion capacitances. In this section the two most sophisticated models pro-

\\\\\\\\\\\\\

VI VoltsFig. 5. Comparison of model equations for the depletion capacitance of.a symmetrical step junc-tion with NA =ND = 1X 1017 cm:" and zero current densities. -- exact (eq. (21», -- Poonand Gummei (eq. (35»; 0 = 0.314, --- de Graaff (eq. (36»; 0 = 0.00826.

98 PhllIps Journalof Research Vol.40 No. 2 1985


posed in the literature are evaluated on the basis of the results presented in theforegoing.Poon and Gummel ") suggested the model equation

Co ( m 0)CdeplPG = (u2 + ojm/2 1 + 1 _ m . il2 + 0 ' (35)

where Co, Vo, m and 0 are free parameters and u = 1 - VIVo.A modification was proposed by de Graaff "), which, after correction of a

typographical error, readsCo { u + (u2 + 0)1/2 } I-m

CdepldG = (U2 + 0)1/2 2 . (36)

In practice the four free parameters have to be determined from a fit ofexperimental data. However, for the purpose of testing the models on thepresent results for symmetrical step and linear junctions, Co, Vo and in aregiven their theoretical values: Co = Cdepl (V = 0), Vo = VBI - 2VT, m = ~ for asymmetrical step junction and m = ~ for a linear junction. With these values

0.1 L.._---'-___J'---'-_'---'----L_-'--"'---'-_J0.0 0.5 1.0

V/Volts

Fig. 6. Comparison of model equations for the depletion capacitance of a linear junction witha = 2 X 1023 cm:" and zero current densities. -- TRAP simulations, -- Poon and GummeI(eq, (35»; ö = 0.143, --- de Graalf (eq. (36»; ö = 0.00335.

Phlllps Journalor Research Vol. 40 No.2 1985 99


both models perfectly describe the depletion capacitance in the limit of highreverse bias. Adjusting the parameter 0 so as to obtain the proper maximumvalue of the depletion capacitance, the results shown' in figs 5 and 6 are ob-tained. Supported by additional tests for double-diffused junctions, it may beconcluded that the model proposed by de Graaff is by far superior to themodel of Poon and GummeI.

7. ConclusionIn this paper the depletion capacitance of abrupt and linear junctions has

been analysed in detail. The exact expression for the depletion capacitance ofa symmetrical step junction has been derived from the Poisson equation. It isvalid at low and moderate current densities in devices which are not too smallcompared with the Debye length and not completely depleted.

For asymmetrical step junctions and for linear junctions such exact solu-tions are not feasible. Nevertheless, for both junction types it proved possibleby means of computer simulations to establish expressions for the behaviourof the depletion capacitance at high reverse bias and at high forward bias.On the basis of the present results it may be concluded that the simple model

equation developed by de Graaff for circuit simulation purposes provides amost satisfactory description of the depletion capacitance in the voltage rangeof practical interest. -

Notationa gradient of doping concentration in a linear junctionCdepl depletion capacitance per unit areaCdiff diffusion capacitance per unit areaCN neutral capacitance per unit areaCtot total junction capacitance per unit areak Boltzmann constantLe characteristic length for linear junctionLD extrinsic Debye lengthNA acceptor ion concentrationND donor ion concentrationN doping concentration for linear junctionn electron concentrationni intrinsic carrier concentrationp hole concentrationq elementary chargeQdepl depletion charge per unit areaT temperature

100 Phlllps Journalof Research Vol.40 No.2 1985

P-N junction capacitances, Part L' the depletion capacitance

VVB1

VB1cl

VT

'W

Wn

w;,X

XOXn

xpee(/Jn

(/Jp

cp

voltagebuilt-in voltageclassical built-in voltage from depletion approximationthermal voltage VT = kT/qcapacitor plate separationlocation of the n-side contactlocation of the p-side contactdistancelocation of electrical junctionlocation of depletion region edge on n-sidelocation of depletion region edge on p-sidepermittivityspace charge densityelectron quasi-Fermi potentialhole quasi-Fermi potentialpotential

REFERENCES1) H. U. Harten, W. Koch, H. L. Rath and W. Schultz, Z. Physik 138, 336 (1954).2) C. G. B. Garrett and W. H. Brattain, Phys. Rev. 99, 376 (1955).3) S. P. Morgan and F. M. Smits, Bell Syst. Techn. J. 39, 1573 (1960).4) C. T. Sah, Proc. IRE 49, 603 (1961).6) Y. F. Chang, J. Appl. Phys. 37,2337 (1966).6) Y. F. Chang, Solid-St. Electron. 10, 281 (1967).7) H. P. Kleinknecht, J. Appl. Phys. 38, 3034 (1967).8) D. P. Kennedy and R. R. O'Brien, IBM J. Res. Develop. 11, 252 (1967).9) H. K. Gummei and D. L. Scharfetter, J. Appl. Phys. 38, 2148 (1967).10) W. Nuyts and R. J. van Overstraeten, Electr. Lett. ~, 54 (1969).") A. B. Almazov and Ye. V. Kulikova, Radio Engineering and Electronic Physics 15, 737

(1970).12) F. van de Wiele and E. Demoulin, Solid-St. Electron. 13, 717 (1970).13) W. Nuyt s and R. van Overstraeten, J. Appl. Phys. 42, 5109 (1971).14) B. R. Chawla and H. K. Gummei, IEEE Trans. Electron Devices ED-18, 178 (1971).16) Z. Djurié, M. Smiljanié and D. Tj ap k in, Solid-St. Electron. 14,457 (1971).16) W. R. Smith, Int. J. Electronics 31, 201 (1971).17) D. L. Heald, P. F. Ordung, J. G. Skalnik and E. N. Nansen, Solid-St. Electron. 16,

1055 (1973).18) W. Kuárn icz and A. Swit, Solid-St. Electron. 17,457 (1974).19) J. E. Parrott and L. Ph. Leonidou, Phys, Stat. Sol. (a) 25, 231 (1974).20) J. E. Parrott, Phys. Stat. Sol. (I) 26,89 (1974).21) C. Shekhar, S. K. Sharma and S. Kumar, J. Instn Electronics and Telecom. Engrs. 21,

511 (1975).22) 1. E. Parrott and M. Tieng, Phys. Stat. Sol. (a) 34, K43 (1976).23) M. Katsuhata and S. Yoshida, Japan. J. Appl. Phys, 15, 565 (1976).24) D. P. Kennedy, Solid-St. Electron. 20, 311 (1977).26) G. Weinhausen, Phys, Stat. Sol. (a) 46,517 (1978).26) R. B. Shirts and R. G. Gordon, J. Appl. Phys. 50,2840 (1979).27) Ch. Kittel and H. Kroemer, Thermal Physics, 2nd Edn, Freeman, San Francisco, 1980; cf.

p.378.28) F. A. Lindholm, J. Appl. Phys. 53, 7606 (1982).

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29) w. Schottky, z. Physik 113, 367 (1939).30) W. Schottky, Z. Physik 118, 539 (1942).31) W. Sh o ck ley, Bell Syst, Techn. J. 28,435 (1949).32) D. P. Kennedy, IEEE Trans. Electron Devices ED-22, 988 (1975).33) A. De Mari, Solid-St. Electron. 11, 33 (1968).34) J. W. Slotboom, Thesis, Technische Hogeschool, Eindhoven, 1977.35) R. M. Sirsi and A. R. Boothroyd, IEEE Trans. Electron Devices ED-23, 348 (1976).36) c. Y. Kwok, J. Phys. D 16, L263 (1983).37) B. C. Bouma and A. C. Roelofs, Solid-St. Electron. 21, 833 (1978).38) J. J. H. van den Biesen, Philips J. Res. 40, 103 (1985).39) R. S. Muller and Th. I. Kamins, Device electronics for integrated circuits, Wiley, New

York, 1977; cf. p. 120.40) A. S. Grove, Physics and technology of semiconductor devices, Wiley, New York, 1967; cf.

p. 164.11) H. C. Poon and H. K. Gummei, Proc. IEEE 57,2181 (1969).42) H. C. de Graaff, Proc. Nato advanced study institute on process and device modeling for

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102 Phlllps Journalor Research Vol.40 No. 2 1985

Documents

P-N JUNCTION CAPACITANCES PART I: THE … Bound... · P-N junction capacitances,Part L'the depletion capacitance In fig. 2 the perfect agreement with numerical TRAP values is shown