Impedance of semiconductor-insulator-metal capacitors

Solid-State Electronics Pergamon Press 1964. Vol. 7, pp. 59-79.

Printed in Great Britain

IMPEDANCE OF S E M I C O N D U C T O R - I N S U L A T O R - M E T A L CAPACITORS K. LEHOVEC a n d A. SLOBODSKOY* Sprague Electric Company Research Laboratories, North Adams, Massachusetts

(Received 15 April 1963) A b s t r a c t - - T h e a.c. behavior of a semiconductor-insulator-metal capacitor is analyzed as a function of the frequency of a small a.c. voltage and of a d.c. bias voltage. Our analysis differs from previous work by GAP.~TT, BERZ and YUNOVICN, respectively, which dealt mainly with field effect conductivity, in the following respects: (1) Both (I), the extreme case of negligible recombination rate in the space-charge layer, and (II) the opposite extreme case of infinite recombination rate in the space-charge layer, are treated. (2) Deviations from the Boltzmann distributions of carriers in the space-charge layer at current flow are taken into account. (3) Equivalent circuits are derived f r o m which the frequency dependence and the loss angle can be more readily appreciated than from the involved analytic expressions for the impedance. (4) Graphs are provided for the determination of the impedance as function of bias, frequency and of resistivity of the semiconductor for a semiconductor free of surface states. R 6 s u m 6 - - L e comportement ~t courant alternatif d'un condensateur semiconducteur-isolantm6tal est analys6 en fonction de la fr6quence d'une petite tension h courant alternatif et d'une polarisation ~t courant continu. Cette analyse diff~re de3 travaux pr6c6dents de GAm~TT, Blmz et YUNOVlCH respectivement qui consid6raient la conductivit6 h effet de champ dans les aspects suivants : (1) Les deux cas, (I) le cas extreme d'un taux de recombinaison n6gligeable duns la couche de charge d'espace et (II) le cas oppos6 extreme comprenant un tattx de recombinaison infini duns la couche de charge d'espace sont trait6s. (2) Les d6viatlons des distributions Boltzmann de porteurs dans la couche de charge d'espace au courant sont prises en consid6ration. (3) Des circuits 6quivalents sont d6riv6s dans lesquels la d6pendance de fr6quence et l'angle de perte peuvent ~tre plus facilement compris que dans les expressions analytiques compliqu6es de l'imp6dance. (4) Des graphiques sont fournis pour d6terminer 1'impEdance d'un semiconducteur exempt d'6tats de surface en fonction de la polarisation, de la fr6quence et de la r6sistivit6 du semiconducteur. Z u s a m m e n f a s s u n g - - D a s Verhalten einer Halbleiter-Isolator-Metall-Kapazit~it bei Wechselstrom wird als Funktion einer kleinen Wechselspannung tend einer Gleichstrom-Vorspannung untersucht. Die vorliegende Untersuchung unterscheidet sich yon der Arbeit yon GAm~TT, BEP.Zund YUNOVlCH, die hauptsiichlich Feldeffekt-LeitfShigkeit behandeln, in folgenden Punkten: (1) Sowohl der ~iusserste Fall einer vernachl~issigbaren Rekombinationsrate in der Raumladungsschicht (I) und der entgegengesetzte ~iusserste Fall einer unendlichen Rekombinationsrate in der Raumladungsschicht (II) werden behandelt. (2) Abweichungen v o n d e r Boltzmannverteilung der Tr~iger in der Raumladungsschicht bei Stromfluss werden beriicksichtigt. (3) Durch Ableitung yon Ersatzschaltbildern lassen sich Frequenzabhiingigkeit und Verlustwinkel leichter feststellen als aus den komplizierten analytischen Impedanzausdriicken. (4) Die Abh~ingigkeit der Impedanz von Vorspannung, Frequenz und spezifischem Widerstand des Halbleiters wird fiir einen Halbleiter ohne Oberfl~ichenzust~inde dutch Kurven dargestellt. 1. I N T R O D U C T I O N THE c o m b i n a t i o n s e m i c o n d u c t o r - i n s u l a t o r - m e t a l

voltage d e p e n d e n c e o f t h e c a p a c i t a n c e o f a s p a c e c h a r g e layer in t h e s e m i c o n d u c t o r is utilized, w h i l e

has r e c e n t l y b e c o m e o f c o n s i d e r a b l e i n t e r e s t as a p a r a m e t r i c capacitor.(1,2) I n this application, t h e

* Now at: California. 59

Fairchild Semiconductor, Palo Alto,

60

K. L E H O V E C

a n d A. S L O B O D S K O Y

the insulator serves essentially to block any d.c. current. The combination semiconductor-insulator-metal can also be used as a bias-independent capacitor, at least in certain bias ranges: T h e silicon-silicon oxide-metal system provides a highly stable, temperature and bias-independent capacitor,(3) if appropriately designed to suppress the bias-dependent contributions from the spacecharge layer mentioned previously. Furthermore, the combination semiconductor-insulator has potential interest for electroluminescent applications; e.g. in the case of a film prepared by imbedding luminescent semiconductor particles in an insulating matrix with an a.c. field applied. Finally, electrical measurements on a semiconductor-insulator-metal capacitor should become an important research tool to explore the electrical structure of surface states,(4-7) similar to the well-known method of the field effect conductivity modulation. In view of these various applications, a detailed analysis of the a.c. impedance of the combination semiconductor-insulator-metal as function of frequency and of d.c. bias appears warranted. Our analysis is similar to that of GARRETT,(8) BERZ(9,10) and YUNOVICH(11,12) o n the frequency dependence of the field-effect conductivity modulation; i.e. of the conduction parallel to the semiconducting surface as a function of an a.c. field applied perpendicular to that surface. As a matter of fact, some of GARRETT'S equations (equations 15b and 15c of Ref. 8) immediately provide the impedance of the space-charge layer in the semiconductor for current flow perpendicular to the surface. Unfortunately, these equations are so involved that their usefulness for a detailed discussion is highly impaired. It appeared desirable, therefore, to associate these equations with circuit elements of an equivalent circuit having physical significance. YUNOVICH(11) has already derived such an equivalent circuit for the very special case where only majority carriers are involved. The authors mentioned above (8-~2) made the special assumptions of (a) negligible recombination rate in the space-charge layer; and (b) Boltzmann distributions of electrons and holes in the space-charge layer at current flow. Some of these assumptions may not always be justified in practical cases.

In this paper, the impedance of the metal-semiconductor-insulator capacitor will be described by equivalent circuits for various bias ranges; the effect of current flow on the electron and hole distribution in the space-charge layer will be taken into account, and the two extreme cases, (I) of negligible recombination in the space-charge layer and (II) of an infinite recombination rate, will be discussed. 2. N O T A T I O N [] around a symbol indicates a complex quantity; the bracket is replaced in Figs 1-5 and 7 by a circle around the symbol position co-ordinate x = 0 interface semiconductor-insulator x > 0 position in the semiconductor x = d boundary between the space-charge layer and the bulk of the semiconductor x* position at which on = ¢rp = o* charge per unit area 0 current density i is current at x = 0 (horizontal flow in the energy-level diagram) i s flow from the conduction-or valenceband into surface states (vertical flow at x = 0 in the energy-level diagram) (i is counted positive: for electrons flowing from the bulk toward the surface of the semiconductor entering the surface states and for holes flowing from the surface toward the bulk of the semiconductor or leaving the surface states) V potential in the space-charge layer V-- Oatx = d V s P = V s - V a space charge potential

f~ = ( k T / e ) -1 inverse voltage equivalent of temperature reduced potential in the space-charge layer y* potential at the position x* potential defined by equation (49) YD reduced space-charge potential Y = ~Vsv YL reduced potential separating the bias ranges of strong inversion and weak inversion v a.c. potential a.c. potential across the insulator 'I)o a.c. potential across the space-charge 7)Sp layer a.c. voltage equivalents of electron and Vn's } hole variations defined by the equations (9), (8) and (10) ~p,g space-independent a.c. field in the bulk of the semiconductor y = flV

i o n,p

applied a.c.e.m.f. conductivity e* conductivity defined by equation (1.28) electron and hole concentrations

IMPEDANCE e tz D s E co co r rn, rp o~ ~, y i Lo Ct P C R

OF SEMICONDUCTOR-INSULATOR-METAL

absolute value of the electron charge mobility diffusion constant surface recombination velocity dielectric constant of the semiconductor dielectric constant of the insulator 8"854 x 10-14 A-sec/V-cm bulk lifetime of holes time constants pertaining to the transitions of electrons and holes into surface states angular frequency of the a.c. voltages a reduced angular frequency defined by equation (II.9) real and imaginary parts of (1 +jf~)l/2 V-1 Debye-length defined by equation (20) a characteristic capacitance defined by equation (11.14) a dimensionless parameter defined by equation (11.13) a capacitance per unit area resistance per unit area

The following capacitances are defined by the equations indicated in parentheses behind the symboh [CI (2); Co (5); [Cap] (6); [Csl (7); Cs (23); [Csl (II.23) ; [Col (24) ; [Czl (25); [Ca] (26) ; Cz (27) ; CA (28) ; CDo (32); CDo° (33); C O (II.16); C~ (II.14). The following resistances or impedances are defined, respectively, given by the equations listed in parentheses behind the symbol: Rn,D (14, 17); [Ru,n] (15, 16); Rp,no (11.12); Rp,A (18); RD (19); Rn,s, (21); Rp,S (22). Subscripts S P space-charge layer .4 accumulation layer D depletion layer I inversion layer O insulator M metal electrode at the insulator S surface states s the position x = 0 ("surface"; i.e. semiconductorinsulator interface) d the position x = d in the semiconductor o the bulk of the semiconductor (x = o0) z the intrinsic (undoped) semiconductor n electrons p holes Superscripts a.c. quantityt * the position x* 0 frequency oJ = 0 frequency to = oo 3. GENERAL CHARGE CONSIDERATIONS I n analyzing a capacitor it is c o n v e n i e n t to start

? Used only where confusion is possible; the voltages yap, etc., are used only as a.c. quantities and ~ is omitted.

CAPACITORS

61

from considerations of charge storage. T h e charge QM at the metal b o r d e r i n g the insulator is balanced b y equal charges of opposite sign at the semiconductor, either at the s e m i c o n d u c t o r - i n s u l a t o r interface as a surface charge Q s , or as a space charge Q s P , associated with an excess (or deficit) of electrons and holes: QM

=

- Qs - Psi.

(1)

I n analyzing small signal a.c. conditions, it is c u s t o m a r y to compose each t i m e - d e p e n d e n t variable of a d.c. portion and of an a.c. portion varying periodically with time, and to neglect terms of higher t h a n first order; i.e. p r o d u c t s of a.c. portions. Designating the amplitudes of the a.c. portions b y a wave above the t e r m symbol, we have b y definition:

[c] =

(z)

where ~ is the applied a.c.e.m.f, which is the s u m of the a.c. voltage across the dielectric vo, a n d that across the space-charge layer in the semiconductor yaps P. = v o + v s e

(3)

and, therefore, from (2) with (3) and (1): 1

1

[cl

Co

1

[Cse]+[Cs]'

where

Co = ¢ /vo

(5)

is the insulator capacitance; [Csp] = - O.sp/vsp

(6)

is the space-charge capacitance (which can be complex), a n d

[cs] = - Os/,'s

(7)

is the surface-state capacitance (which also can be complex). E q u a t i o n (4) indicates that the insulator capacitance is in series with a parallel c o m b i n a t i o n of two complex capacitances, associated with charge storage in the space-charge layer a n d in surface states, respectively. Since changes of the charge in surface states with time involve flow of electrons or holes + The symbol ~ is omitted because there is no danger of confusion with d.c. quantities.

62

K. L E H O V E C and A. S L O B O D S K O Y

through the space-charge layer, the rate of these changes will be limited by the a.c. voltage across the space-charge layer, by its conductance and by the transition rates of carriers from the conduction band and valence band to the surface states. We may conclude, therefore, that joJ [Cs] remains finite with increasing frequency. Thus, at sufficiently high frequencies the equivalent circuit should approach the series combination of the insulator capacitance and the space-charge capacitance. Electrical measurements at such frequencies provide then [Csp] in series with Co from which the d.c. charge in the space-charge layer, QsP, may be deduced by standard semiconductor theory.t13) From this charge and the d.c. charge on the metal, the d.c. charge in surface states may be derived by means of equation (1).t4-7) Analysis of the d.c. charge in surface states as a function of bias permits conclusions on concentration and energy levels of surface states. While the description of the capacitance of the metal-insulator-semiconductor as expressed by (4) with (5), (6) and (7) is general and formally simple, it does not lead immediately to an analysis of the relation between electrical properties and physical structure, since the "capacitances" [Csp] and [Cs] are rather complicated expressions. Furthermore, since [Csp] and [Cs] are generally complex quantities, equation (4) in its present form does not even permit an electrical circuit analysis; e.g. a discussion of the frequency dependence or of dielectric losses. We shall, therefore, present in the next Section an approach leading to a more detailed equivalent circuit whose components have physical significance in terms of the structural elements of the semiconductor-insulator interface. 4. E Q U I V A L E N T

CIRCUITS

T o arrive at an equivalent circuit one may follow step by step the flow of a.c. charges, electrons and holes, from the bulk of the semiconductor to the interface semiconductor-insulator. Each step can then be associated with a resistor. Accumulation of a.c. charges can be associated with capacitors. T o be explicit, we shall consider an n-type semiconductor and for simplicity we shall consider unit area. First we shall discuss the bias range pertaining to a depletion-type space-charge layer in the semi-

conductor, Vsp < 0; there is a region, 0 < x < d, in which the majority carriers, electrons, are substantially depleted. Later in this Section we shall consider the bias range pertaining to an accumulation space-charge layer of majority carriers, Vsp > O. Storage of a.c. charges in a depletion spacecharge layer may occur: _ (a) in surface states at x = 0 (surface charges

Qs), (b) as free holes (minority carriers) close to the surface near x ~ 0 t (inversion charge 0 I ) , and (c) as free electrons (majority carriers) near the boundary x = d of the depletion layer (depletion charge QD). These charge accumulations are associated with the capacitors Cs, [Cz] and [Co], respectively, which will be defined mathematically later. Let us first discuss the resistances through which the current is fed into these capacitors. T h e surfacestate capacitor Cs is fed by the electron and hole currents in,s and ip,s in the case of surface states of the SHOCKLEY-READ type.(14) The inversion charge capacitor [C;] is fed by the hole-current, ip,s. The depletion capacitor [CD] is fed by the bulk electron current. T h e current ip,s feeding holes into surface states is not the same as the hole current ip,s~ to the interface, since a portion of the current ires is stored as the inversion charge. In addition to the three capacitors Cs, [CI] and [C9], the equivalent circuit contains, of course, the insulator capacitance Co and resistances that characterize the following physical processes: (1) The transition of electrons from the conduction band into surface states (Rn,s). (2) The transition of holes from the valence band into surface states (Rp,s). (3) The flow of holes through the depletion space-charge layer (Rp,D). (4) The flow of electrons through the depletion space-charge layer (Rn,o). (5) The generation (or recombination) of holes ~rThe extension of the inversion charge is much smaller than the width of the depletion layer, and the inversion charge may thus be considered to be located at the interface rather than in the space-charge layer. :~The subscript s pertains to the flow at x ~ 0 (horizontal flow in the energy-level diagram), while the subscript S pertains to flow from the conduction or valence band into the surface states at the interface x = 0 (vertical flow in the energy-level diagram).

I M P E D A N C E OF S E M I C O N D U C T O R - I N S U L A T O R - M E T A L and their flow rate in the region of quasi-neutrality in the bulk adjacent to the space-charge layert

[Rp,B]. (6) The flow of electrons and holes through the bulk of the semiconductor (Ro). T h e arrangement of these resistances in the equivalent circuit differs for the two cases of (I) negligible recombination rate in the space-charge layer and (II) infinite recombination rate in the space-charge layer. In the case (I) the hole current and the electron current are each independent of position within the space-charge layer,:~ and we have thus a parallel arrangement of the holeimpedance of the space-charge layer and of the electron-impedance of the space-charge layer. This leads to the equivalent circuit of Fig. 1.

J-co

CAPACITORS

63

In the case (II) there can be a transition of the current in the space-charge layer from a dominant electron current at the bulk side (i ~_ in,a at x ___ d) to a dominant hole current near the surface (i ~_ ip,, at x x 0). It is thus impossible to represent the current flow through the depletion layer by an impedance for holes in parallel with an impedance for electrons; rather, a single impedance RD must be introduced that is in part due to electron flow and in part due to hole flow, and replaces the resistances Rn,o and Rp,D of Fig. 1. The equivalent circuit for the combination semiconductor-insulator-metal at an infinite recombination rate in the semiconductor and for bias voltages corresponding to a depletion space-charge layer is shown in Fig. 2.

=~-Co

iR o

Ro FIG. 1. Equivalent circuit of the combination: n-type semiconductor-insulator-metal assuming negligible recombination of electrons and holes in the space-charge layer (bias range corresponding to depletion-inversion). The left-hand side of the diagram represents the equivalent circuit for the flow of electrons (majority carriers), and the right-hand side that for the flow of holes (minority carriers). t This is not a simple resistance; it has a eapacitive component. A.C. charge storage within the space-charge layer is insignificant.

FIo. 2. Equivalent circuit for the combination: n-type semiconductor-insulator-metal assuming an infinite recombination rate of electrons and holes in the spacecharge layer (bias range corresponding to depletioninversion). In ordinary circuitry, an electric potential difference is the "driving force" for current flow. In the case of the network consisting of Cs, Rn,s and Rp,s the "driving force" cannot be an ordinary potential difference, since the currents in,s and ip,s represent transitions of electron

64

K. L E H O V E C

and A. S L O B O D S K O Y

and holes between different energy states at the same position. T h e "driving forces" in this case are the concentration variations of electrons and holes at the surface, gs and ibs, from which potentials may be defined b y : ~,,s

= -~,/(bs/~)

(8)

v.,8

=

~8/(~s~),

(9)

and

where fl = (kT/e) -1 is the reciprocal of the voltage equivalent of temperature. Similarly the "driving force" for the excesshole current,~" which results from diffusion in the region of quasi-neutrality x > d, is not an ordinary potential, but the concentration potential:

v,,a = ['a/(Po~).

Co

I©

(1 O)

T h e location of these concentration potentials and of the ordinary a.c. potential vsp across the space-charge layer has been indicated in Figs 1 and 2. In the case ( I I ) of an infinite recombination rate one has:

Vp,s ---- Vn,s

where an accumulation of electrons occurs toward the surface. A.C. charge storage occurs either in surface states or in the accumulation layer of electrons, the former corresponding to the capacitance Cs and the latter to the capacitance [CA]. Figs. 3 and 4 show that the capacitance Cs is fed through the resistances Rn,s and R p , s corresponding to the transitions of electrons and holes

VSp

wT

T1 Vp,d ! -

11

(11)

as an immediate consequence of the small signal formulation of:

np = n~2.

(12)

Equation (12) results from the fact that deviations from thermal equilibrium are the "driving force" for a net generation or recombination, and that such deviations must remain negligibly small at an infinite generation or recombination rate coefficient. Thus, equation (12) should be valid in our case (II) not only at x = 0, b u t throughout the entire semiconductor. As a further consequence of the small-signal formulation of (12) it follows that: vv,a = 0 (13) in the region of quasi-neutrality where ~ = j5 - 0, in the case (II). W e shall now consider the bias range V s e > O, 1"The "excess hole current" designates that part of the hole current at x = d that is in excess of the spaceindependent a.c. hole current, ip,o = r~p.oFB, throughout the bulk of the semiconductor.

FIG. 3. Equivalent circuit for the combination n-type semiconductor-insulator-metal assuming negligible recombination rate of electrons and holes in the spacecharge layer (bias range corresponding to accumulation). from the semiconductor into surface states. These resistances were already mentioned when discussing the bias range of depletion (Figs 1 and 2). T h e accumulation capacitance [CA] is fed through a resistance Rn,n resulting from the electron flow through the accumulation space-charge layer. T h e resistances Rp,a and [RIo,B] result from the hole flow through the accumulation layer and through the region of quasi-neutrality adjacent to that layer. Since the conductivity of electrons increases toward the surface in the case of an accumulation layer, one may surmise that the resistance Rn,A is small corresponding to the bulk resistance Ro. T h e condition of flat band position, Vsp = O,

IMPEDANCE

OF SEMICONDUCTOR-INSULATOR-METAL

CAPACITORS

65

(b) through the bulk of the semiconductor adjacent to the space-charge layer, and (c) through the space-charge layer. Such calculations are contained in A p p e n d i x I, and the results will be summarized in what follows. One hag:

[Rp,B] = (Dp'r)l/2/[ap,o(1 +joJ~)l/2],

© : Rr,s

RA :Ro

l

(16)

where r is the bulk life time of holes, Dp is the diffusion coefficient of holes, and ap, o is the bulk hole conductivity. It will be shown in A p p e n d i x I that the resistances Rn,D, Rp,D and RD can be obtained b y integration over the profiles of the specific resistivitiest of electrons pn, holes pp, or of both p; i.e. d Rn,D = f. pndx ~-- O,/(n2e21~n) (17) o d

Rp,A = ~ ppdx ~- (~Al(n~e21~p)

(1 8)

o

and:~ d

FIG. 4. Equivalent circuit for the combination: n-type semiconductor-insulator-metal assuming an infinite recombination of electrons and holes in the space-charge layer (bias range corresponding to accumulation).

R o = S Pdx~-TrLo/(2n*noe2tLn#p ln~n,o/Op,o) 1/2, (19) o

where

LD = [Eeo/(2en~fl) ]1/2, is intermediate between the two bias ranges we have treated; i.e. that of an accumulation layer and that of a depletion (or inversion) layer. T h e case of V s p = 0 is of particular interest because it permits an exact solution for any (space-independent) recombination rate.~" 5. RESISTANCES A N D CAPACITANCES OF THE EQUIVALENT CIRCUITS

T h e resistances occurring in the Figs 1-4 are defined as ratios of voltages v and currents i:

Rn, o = ( v s e - - Vn,s)/in Rp,B = v , , a / i , ,

(14)

(15)

etc. T h e magnitudes of these resistances can be expressed b y a detailed analysis of the electron and hole flows: (a) from the semiconductor into surface states, t A mathematical analysis of the case of flat-band condition will be given in a forthcoming paper.

(20)

al, n~ are the intrinsic conductivity and electron concentration of the semiconductor;/~n,/~p are the electron and hole mobilities; and n o" crn, o' crp, o are quantities pertinent to the electrically neutral bulk of the semiconductor. T h e resistances R ~ , 9 and Rn,A are quite small and generally negligible being associated with carrier concentrations that are larger than the corresponding bulk concentrations. T h e resistances:

Rn,s = Cs/~n

(21)

Rp,s = Cs/~'p

(22)

and

"~This relation ceases to be valid if there is a finite d.c. current of electrons or holes causing deviations of the d.c. distributions of these carriers from the Boltzmann type. :~ The last expression for RD applies only at sufficiently large bias voltages in the depletion region, satisfying Y
K. L E H O V E C and A. S L O B O D S K O Y

66

are associated with transitions of electrons and of holes into surface states having the time constants rn and ~-p, respectively, which can be expressed in terms of the SHOCKLEY-RrAD parameters(14) of the surface states. The capacitance of the surface states is:

flNsef(1--f),

Cs =

(23)

where f is the Fermi factor for the surface states (it is assumed here that all surface states have the same energy level) and N s is the concentration of surface states. The other capacitances occurring in the equivalent circuits are defined as the following ratios of a.c. charges to a.e. potentials:t

[c.]

=

-

~D/V~,

(24)

[c,]

=

-#,/v.,~

(25)

[ca]

=

- 0A/v.,s.

(26)

and

In general, the charges QD, Qx and QA are functions of three independent variables; e.g., Vse, Pa and i~. These functions are obtained by integration of Poisson's equation with the hole and electron distributions appropriate to the instantaneous current flow through the space-charge layer. However, it can be shown (Appendix II) that quite good approximations to [CI] and [CA] are the real and frequency-independent expressions:

CI = -dQl/dVsp

(27)

CA = --dQA/dVsp.

(28)

and

The justification of these expressions may be seen in the fact that the principal variables in Qx and QA are Ps and ns, respectively, while the effect of Vs~ is a comparatively minor one, except that P8 and n8 are functions of Vs~ on account of:

Ps = pa" exp (--fl Vsp)

(29)

The negative signs result from the fact that the physical charges Qo, Qx and QA correspond to those at the bulk side of the capacitances [CD], [(21] and [Ca]; i.e. at the negative sides of the potentials vsP, vp,, and v.,s.

and ns = na exp(fl Vse).

(30)

Thus,

[CI] ~- - (dQi/dp,) (ps/vp,8) = (dpz/dps)fl#s, (31) which becomes (27) on account of (29), assuming again that QI depends only on Ps but not on Vsp (except through Ps). There is a small effect on Vs~, in addition to that through Ps, whose magnitude will be estimated in equation (II.3). If Qo would be considered only as a function of Vsp, one would obtain:

C° = -dQD/dVsv.

(32)

However, in the bias range of strong inversion, a change in Vse arises not only from a change in QD, but also from a change in QI; i.e. a change in Ps. The dependence of QI on Vse has a complicated frequency behavior due to the impedances [R~,B] and R~,D in Fig. 1. Accordingly, Co becomes also dependent on frequency, equation (32) being the low-frequency limit. At sufficiently high frequencies, the capacitance Cn tends toward the limit:

C D = -dQn/dVsp

+(dQddVse)" (dQoldPa)l(dQzldpa). ( 3 3 ) The second term in (33) is significant only in the inversion range. By using the high-frequency limit at all frequencies, no significant error in the impedance of the entire circuit is made, since in the lower-frequency range where CD differs signicantly from C~ the impedance of the" branch containing Co can be shown to be large (and is thus of little significance) compared to that containing CI, considering biases in the inversion range. Outside of the inversion range C~ and C ° become practically identical. We have thus expressed the resistances Rn,o and R~,A and the capacitances CD, Cz and CA in terms of the d.c. charges Qo, Q1 and QA and of their derivatives with respect to Vsp and to the boundary concentrations of carriers. These are well-known expressions in semiconductor theory.(13A5,16) Integration of Poisson's equation provides:

QsP = 2nleLDF(Y, pa),

(3 4)

IMPEDANCE OF SEMICONDUCTOR-INSULATOR-METALCAPACITORS

67

where

i n,O

1) +

no+pa-po ( - ¥ + e Y -

n,

~1/2

1)j

;

[

(36)

In the depletion-inversion range:

Pt = n,eLD G(Y,pa),

l

CD

(37)

where

G ( Y , p a ) - n~LD Pa f (e-U-- 1) dx. o

Rp,S

(38)

tRo

T h e depletion space charge is: QD = Q s P - QI = meLn(2F- G).

(39)

T h e functions F(Y,pa) and G(Y,pa) have been presented in graphical form.(15,16) AND

,S

Rn,D

d

6. D I S C U S S I O N

ip,O

(35/

In the accumulation range:

Qa ~- 9sP.

=~ f.,

CONCLUSIONS

A. The equivalent circuit parameters as function of the bias voltage It has been shown that the equivalent circuit for the depletion-inversion range can be approximated by the network of Figs 5-8 with Co, CI and CA the real and frequency independent expressions? (33), (27) and (28), and the resistances [Rp,n], Rn,o, R~,A and Ro the expressions (16), (17), (18) and (19) respectively. There are several ranges of the bias voltage where the equivalent circuit parameters become rather simple expressions of the bias voltage. These ranges are characterized by strong inversion: Y < YL ; depletion with weak inversion: YL < Y < - 1 ; and strong accumulation : Y > 1. T h e bias voltage YL is defined by:

FIO. 5. Simplified equivalent circuit for the depletioninversion range at zero recombination in the spacecharge layer, in,o and ip,o indicate d.c. leakage current paths through the insulator. In absence of such leakage, real and frequency independent approximations to CI, Co and Rn,D are given by equations (27), (33) and (17), respectively. Ro bulk resistance; Rp,D see equation (16). The expressions for CI and CD have been derived for R~,s = ~, but are believed to be good approximations also for a finite Rp,s. At Y < YL the inversion charge exceeds the depletion charge, as seen readily from equation (34), considering that in (35) with Pa = Po the first term arises from the inversion charge while the second term arises from the depletion charge. In the bias range of depletion with weak inversion, YL < Y < --1, the space-charge potential is well approximated by the SCHOTTRY formula :(17)

Vsv = - no edZ/(2,,o).

(41)

PD = n o e d

(42)

Since: t Strictly, the values of these capacitances have been derived assuming that Rp.s = oo.

68

K. L E H O V E C and A. S L O B O D S K O Y

i0

I n,O

~3c°

Co 3

=CA -C s

Rn,s

p,O

-LCs -r. Rp,s Rp, A

CD-r-

R n,$

RD

i

RO

FIG. 6. Simplified equivalent circuit for the depletioninversion range at infinite recombination rate in the space-charge layer, io indicates the d.c. leakage path through the insulator. In absence of such leakage, CI is given by equation (27), Co by (33) and R9 by (19). OQD/bPa ~-- 0 and, using (33) or (32), C°~ = C ° = --(~,QD/Od) (~d/~Vsp) = ,,o/d. (43) T h e inversion charge can be calculated from the Boltzmann distribution of holes and using a linear development of the potential near the surface, Q1 z pse(kT/e)/(~V/ex)x=o.

(44)

In the bias range under consideration the field: (eV/ex)x-o ~ QD/e,o,

(45)

is practically independent of the inversion charge, but depends on the space-charge potential as shown by (42) with (41). T h e inversion charge depends exponentially on Vsp through the factor Ps; the dependence on Vsp through the factor ~V/~x)z=o is quite small by comparison, and we have, using (31), C1 ~ fiQl.

(46)

Ro

FIO. 7. Simplified equivalent circuit for the accumulation range at zero recombination in the space-charge layer. in,o and i~,o indicate d.c. leakage paths through the insulator. In absence of such leakage, real and frequency independent approximations of CA and Rp,a are given by equations (28) and (18); R0 bulk resistance; Rp,o see (16). T h e inversion capacitance increases as e x p ( - Y ) on account of (46) with (44) and (29). In the bias range of strong inversion, Y < YL, the field at the surface depends strongly on the inversion charge. Furthermore, the field varies so rapidly with distance that a linear development such as (44) is no more satisfactory for calculating QI. Instead, we obtain by integration of Poisson's equation, using only the space-charge density of holes (equation (34) with QsP ~ QI and (35) with (29) neglecting all terms under the square root of (35) except ps/n~): Qt = 2n~112eLD ps it2.

(47)

Since Qz depends only on Ps, we obtain from (31) : G = flQ,/2.

(48)

Note that, from (48) with (47) and (29), the inversion capacitance increases in this bias range as

I M P E D A N C E OF S E M I C O N D U C T O R - I N S U L A T O R - M E T A L

io Co Cs CA

"

CAPACITORS

frequencies (oJ-+oo) the depletion capacitance is given by (33), and for large negative bias voltages the second term becomes dominant. This term arises from the fact that the a.c. inversion charge Qx at very high frequencies tends toward zero.t Thus, at very high frequencies where Qz -~ 0, the a.c. potential across the space-charge layer yap must result entirely:~ from a change in the

tR0s

f Rn's

Ro FIG. 8. Simplified equivalent circuit for the accumulation range at infinite recombination rate in the space-charge layer, io indicates the d.c. leakage-current path through the insulator. In absence of such leakage, Ca is given by equation (28). e x p ( - Y / 2 ) . T h e change in slope on a semilogarithmic plot by a factor 2 in the Figs 10 and 11 results from the change in the voltage dependence of Qz between the ranges separated by YL. T h e contribution of the depletion charge to the space-charge potential is approximately: YD ~ In (no/Po),

(49)

since at this potential the space-charge density due to holes becomes equal to the depletion spacecharge density hoe on account of the Boltzmann distribution of holes. Closer to the surface, i.e. at potentials y < YD, the inversion space-charge density is dominant. T h e distance dD corresponding to y 9 by (41) characterizes the thickness of the space-charge layer at large negative bias potentials. While the position XD where y = yD shifts to larger distances from the surface with increasing negative Y, in all cases XD "~ dD, since the inversion charge is located very close to the surface. T h u s QD approaches the constant value noedD at large negative biases and equation (32) shows that the incremental depletion capacitance at zero frequency approaches zero. At very high

69

,o~

J~

1

10¢

r0

4 8

12

16 20

-y

24

28

FIG. 9. Bias dependence of the real part of the depletion space-charge capacitance at high frequencies with surface states absent and neglecting recombination in the depletion layer, using C OO= C~ of equation (33) and C~ of equation (II.14); parameter is A = n~/no. width of the depletion layer, and we obtain for Y ~ YL:

c~ -. ~o/d~.

(50)

A comparison of (50) using (41) and (49) agreeswell ~fff)I = Czvr,s = ClVSV/{I +joJCz

([Rp.B]+ R2~,D)}

and w ( [ R p , B ] + R p , D ) --> oo for ca -+ oo. We neglect here a change in the space-charge potential that results f r o m redistribution of ~)z with

Vsp at a constant total Qz. For a further discussion of this "redistribution effect", see Ref. 6.

70

K. LEHOVEC

and A. SLOBODSKOY

~

108

I

106

AIO-3 i0-2 lO-I

i06

t-i

_..I O)

104

0-2 U

t-n

!

tJ

r-~ O_

Y

0

H] -y

3-2 0

20

30

Fxo. 10. Bias dependence o f the differences at zero and at infinite frequency in the real parts of the depletion space-charge layer capacitance with surface states absent and neglecting recombination in the depletion layer. C o - C °° by equations (I 1:6) and (3 3), C~ by equation (II. 14); parameter is A = ni/no.

with the values of equation (33) at large negative biases taken from Fig. 9 confirming this interpretation. We are now in a position to discuss Figs 9-11, which show the bias and frequency dependences of the complex capacitance (6) in the case of absence of surface states and neglecting recombination in the space-charge layer. Furthermore, R~,o has been neglected vs. [R~,~], which is well justified at frequencies small compared to the bulk relaxation time (II.15). The capacitance Csv was obtained from equation (11.5). The real part of the capacitance Csv at high frequencies is shown in Fig. 9, and is C9 of equation (33), i.e. it results from the depletion capacitance. In the bias range

10 -¥

20

30~

FIG. 11. Bias dependence o f the parameter P of equation (II.13) with LD of equation (20) and A = re~no.

of weak inversion the (-Y)-l/2-dependence of a SCHOTTKyO7) depletion layer capacitance [equations (43) with (41)] is recognized while for strong inversion the limiting value (50) is approached. The difference between the low-frequency capacitance and the high-frequency capacitance, C o - c °°, is shown in Fig. 10. This difference results essentially from the inversion capacitance, Cz, which becomes ineffective at high frequencies due to the series impedancet [Rp,B]. Thus Fig. 10 reflects the bias dependence of CI showing the change in the slope by a factor 2 on the semilogarithmic plot, near Y ~ YL. Figure 11 shows the bias dependence of the factor P, defined by equation (II.13), which occurs in the frequency dependence of (Cap). The bias dependence of this factor reflects again the bias dependence of the inversion charge. The frequency dependences of the real and "~While [ R p , n ] - + O for (o--> co; [Rp,B]j~oCz approaches infinity.

I M P E D A N C E OF S E M I C O N D U C T O R - I N S U L A T O R - M E T A L imaginary parts of [Czp] are shown in Figs 12 and 13, and result mainly from the branch of Fig. 1 containing [Rv,n], RV,D ,.~ 0 and Cs. T h e essential features can be understood as the frequency dependences of a series RC combination, p.#3

0

° T I0-'

CAPACITORS

71

which is analogous to equation (47) with appropriate changes from holes to electrons. Since Qa depends only on ns we have from equation (28): (52)

CA = -- 3 QA/2.

This capacitance is real and frequency-independent. T h e associated series resistance Rn,a is negligibly small in comparison to the bulk resistance, since ~n,a >> crn,o. T h e lack of a frequency dependence of CA was to be expected in view of the

- -. "

~

~,o

2

"

,o i0 ~

--~

~-r,,,,,,,~ ~

_

~ ~ 1 0

I0~

x

-4-3-2-101 IogioQ

x

;06 7 oe

234

, li

,/~l°s

¢,///,/

,

FIG. 12. Frequency-dependent factor fl (t~, P) of equation (II.7) pertaining to the real part of the depletion space-charge layer capacitance in absence of surface states and neglecting recombination in the space charge layer. ~ = cot and P of Fig. 11. expressed as a capacitance shunted by an admittance. Figure 10 represents thus the function {I+(oJCR)2} -1. T h e decrease of the function plotted in Fig. 12 for £1 = oJ~- > 1 results from the fact that R ,,~ [R~,B] becomes frequency-dependent for ~ > 1 (equation 16). T h e frequency dependence of the imaginary part shown in Fig. 13 represents the function oJCRI{I+(oJCR)2}. T h e change in slope seen in Fig. 13 for ~ > 1 results again from the frequency dependence of [Rv,B]. For biases of the accumulation range, Y > 1, the space charge arises almost exclusively from electrons. We have then: Qa -~ - 2nf 1/2 eLDns112,

4/

(51)

IIII,

/// // I//

i,o' i

Jl j/

=4 -3 -2 -I 0

Iog~o0

i

I

I

I

ii

2

108 io9

3 4

•

F[o. 13. Frequency-dependent factor f2 (fl, P) of equation (I 1.8) pertaining to the imaginary part of the depletion space-charge layer capacitance in absence of surface states and neglecting recombination in the spacecharge layer. £1 = cot and P of Fig. 11. short relaxation times eEo/aa in the accumulation layer.

B. Comparison of the cases of negligible recombination in the space-charge layer and of an infinite recombination rate There is no difference in the capacitances CD,

72

K. LEHOVEC and A. SLOBODSKOY

CI and CA. However, there are significant differences in the bias dependences and in the magnitudes of the resistances in series with CI in the circuits of Figs 1 and 2. The impedance [Rp,B] (16) at a negligible recombination rate in the space-charge layer does not depend on bias but depends on frequency and has an imaginary component. At low frequencies [Rp,B] can be quite high, since (Dp~-)z/2 must be assumed larger than the width of the space-charge layer in order that recombination can be neglected, and since the equivalent conductivity of this layer of width (Dp'r)l/~ is ap,o. At sufficiently high frequencies, [R~,B] becomes independent of the bulk lifetime and decreases inversely with frequency. The resistance Rp,9 is negligible by comparison except at very high frequencies. (Appendix II.2.) In the case of negligible lifetime, the conduction in the depletion-inversion space-charge layer is due mainly to electrons near the bulk of the semiconductor and to holes near the surface. The transition from dominant electron conduction to dominant hole conduction takes place at the (reduced) potential, y*, given by (I.29) and the corresponding conductivity ~* (I.28) is close to the intrinsic conductivity. The resistance. RD, of Figs. 2 and 4 is the product of this conductivity and of the distance, (kT/e)/(~V/~x)u=u*, which is independent of bias and can be calculated from (I.32) and (I.29). The resistance RD in the case II of zero lifetime is much smaller than the resistance 9 R~,B of the case I of long lifetimes of minority carriers, since a* ~ a~,o and (kT/e)/(~V/~x)u-u* ~ (D~') i/~. Note further the frequency dependence and complex nature of [Rp,B] while Ro is frequency independent and real. In the accumulation range there is no significant difference between the resistance Rn,A and RA in the cases I and II. The resistance Rp,A is large compared to Rn,A. Significant differences in the space-charge impedance should be expected only if the surface-state capacitance and the resistance Rp,s play an important role in the equivalent circuit, the differences arising from the absence of Rp,A in Fig. 4 and from the presence of a shortcircuit connection between Rn,s and Rp,s. The space-charge impedance at intermediate

lifetimes between the extreme cases of negligible recombination in the space-charge layer {(Dp ~-)]/2 >> d} and infinite recombination rate (T = 0) cannot be derived from our calculations. We can only surmise that this impedance should be intermediate between those of the two extreme cases.

C. Evaluation of surface-stateparameters In evaluation experimental data one should attempt first to fit the data by a model based on the known resistances and capacitances, (Rp,B), Rn,o, RD, CD, CI and CA. The parameters entering these quantities are generally known, except perhaps the bulk lifetime ~- entering [Rp,B] which is, however, of importance only at low frequencies cot ~ 1. For the case of a negligible recombination rate in the space charge layer the graphs 9 to 13 should be helpful. Discrepancies between the experimental data and the expected behavior based on the resistances and capacitances mentioned above might result from the presence of surface states. Surface states should become particularly conspicuous at low frequencies and at moderate space-charge potentials in the depletion range, YL < Y < 1. At high frequencies the surfacestate capacitance [Cs] defined in equation (7) will be small compared to the depletion or accumulation capacitances because of the series resistances in the branch of Figs 1-8 containing C~s,. At biases Y ~ YL the inversion capacitance becomes so large, and at Y >> 1 the accumulation capacitance becomes so large, that the effects of Cs may become inconspicuous. Thus, the influence of Cs on the impedance will be comparatively strongest at biases near the minimum of C O, which can be obtained from Figs 9 and 10 and is in the range l
I M P E D A N C E OF S E M I C O N D U C T O R - I N S U L A T O R - M E T A L C A P A C I T O R S

73

obvious that by suitable choice of these sections (a) The fundamental difference concerns the a wide variety of frequency and bias dependences effect of current flow on the carrier distribution can be obtained. In order to investigate whether which is ignored by all the previous authors.(8-12) an interpretation of experimental data by a spec- Our resistances Rn,D, Ro and Rp,A are a consetrum of time constants of surface states is justified quence of the effect of current on the carrier in a given case, the energy spectrum of the sur- distribution. The possible importance of these face states can be ascertained from the d.c. analysis resistances is indicated by the fact that in the case described elsewhere.(4-7) This eliminates some of infinite recombination rate, the loss angle in of the arbitrariness of such an interpretation. absence of surface states arises only from Ro; Furthermore, one may attempt to correlate the i.e. from the influence of current flow on the carrier energy difference between a surface state and the distribution. top of the valence band, or the bottom of the The field-effect conductance rises from the inconduction band, with the time constants ~-v and version charge ~)z, the depletion charge Do, or Tn, respectively, assuming that the capture cross- from the accumulation charge QA. Assumption of sections of all surface states are approximately equal. Boltzmann distributions for electrons and holes Unfortunately, distributed network behaviour will have an effect on the field-effect conductance may also result from a variation of the concen- to the extent to which the a.c. charges stored in tration of surface states along the surface causing the capacitors CI and CD of Figs 5 and 6, or CA the d.c. space charge potential Y to vary with of Figs 7 and 8 are influenced by Rn,D, Ro or position along the surface. In such a case, the in- Rp,A. Hereby it must be kept in mind that the duced currents would not only flow in the direction voltage vsp itself is a function of all the comperpendicular to the surface but there would also ponents of the equivalent circuit, vsp being the be components of these currents parallel to the fraction of the applied a.c. voltage that extends surface. The impedance would then depend also across the space-charge layer, the remaining on the conductance along the surface (i.e. the field fraction extending across the insulator capacitance effect inversion-charge conductance). Networks of Co. this sort should be very difficult to analyze. It is deThe deviations from the Boltzmann distribution sirable to check the patchiness of the surface by can be of importance also for the surface recomphotoeffect measurements using a light-probe. bination velocity: A distributed network may also arise from ins = Dp/{a~,o(Rn,s+R~,s+Rn,D+Rp,D)), (53) homogeneities in the semiconductor. In extreme cases there may be "short circuit paths" through which is obtained from Fig. 1 at ¢o = 0, replacing the space-charge layer, perhaps along dislocations. [Rv,B] by : If we consider such a dislocation as a region of vp,a/(~aes) = Dp/(SOp,o) (54) infinite recombination rate, one could visualize a network consisting of the parallel combination of and considering that the electron and hole curthose of Figs 1 and 2 (or 3 and 4) with a suitable rents to the surface must be equal. While Rp,o area factor for the dislocation. Obviously a large can b e safely neglected, Rn,D may not be neglipart of the current would flow along the disloca- gible. The term Rn,o is missing in equation tion to the surface and would then spread along (22) of Ref. 8. the surface providing again a sort of distributed A Boltzmann distribution of electrons or holes network as in the case of patchiness of surface implies a horizontal quasi-Fermi level. In case I states. of negligible recombination in the space-charge region, and considering the depletion-inversion D. Comparison of the present results with those of range, the quasi-Fermi level for the (a.c.) hole disprevious authors tribution (minority carriers) is horizontal through Our results differ from those of previous the space-charge region but depends on position authors< s-12) (a) in a fundamental way, which will in the region of quasi-neutrality adjacent to the be discussed immediately, (b) in the basic assump- space-charge region approaching the quasi-Ferm tions, and (c) in the method of presentation. level of electrons in the bulk at distances of severa

74

K. L E H O V E C and A. S L O B O D S K O Y

(Dpv)l/2. T h e quasi-Fermi level of the (a.c.) electron distribution is practically space independent through the entire region of quasi-neutrality, but depends on position in the space-charge layer at current flow. In the case of an infinite recombination rate of electrons and holes, the quasiFermi levels of electrons and holes coincide and depend on position in the space-charge layer at current flow. (b) Our analysis differs from previous authors by including also the case of zero lifetime of minority carriers. This provides now an analysis of the two extreme cases bracketing the cases of intermediate lifetimes. It was surprising to us that the resistances associated with the space-charge layer can be calculated in both cases from "specific resistivities" of electrons and holes. Clearly this is the case only if there is zero current flow and the electrons and holes have d.c. Boltzmann distributions, therefore. If the insulator should carry a d.c. leakage current of electrons or holes, respectively, appropriate connections to the equivalent circuit can be made to accommodate these leakage currents. The connections have been marked in Figs 5-8. However, the d.c. electron and hole distributions in the space-charge layer are then no more of the Boltzmann type, and the resistances Rn,D, Rp,D, •., Rp,A and RD cannot be calculated from specific resistivities. The problem of finite d.c. leakage currents is of importance for capacitance measurements on semiconductor-electrolyte systems. (c) Our presentation has been directed toward equivalent circuits. The equivalent circuit of YUNOVlCH(10) who considered the electron flow only is identical with the electron branch of our equivalent circuit, Fig. 1, except that R n , h is replaced by a short circuit. This is a consequence of the assumption of a quasi-Boltzmann distribution of electrons. T h e fact that TERMAN'S measurements do not reach the bias range of dominant inversion might justify the omission of the inversion capacitance in his equivalent circuit.(7) However, he does not distinguish between surface states charged by electrons or by holes, respectively. If the surface states are charged by holes (minority carriers), the impedance (R~,B) should be in series with the surface-state capacitance; if the surface states are charged by electrons, the resistance R n , o should

be in series with the surface-state capacitance. Neither of these impedances is shown explicitly in Terman's equivalent circuit. Terman attributes the observed frequency dependence entirely to a distribution in the time constants, ~'n and 7p, of the surface states proper; however, there is the possibility that part of the frequency dependence may result from the impedances Rn,D or (Rp,B), or even from C1. APPENDIX

I

Mathematical Analysis of Electron and Hole Flow 1.1 Charge transfer into ShocMey-Read type (14) surface states By representing the electron transitions from the conduction band into surface states by a current density in,S we have:(s-l°) in, s =

Cs "on +'~p + jw~'n'C~

[Vn,s(1 + jw~'p ) - vp,8 ]

(I.1)

and similarly for holes, leaving surface states and entering the valence band:

i~,s =

Cs ~'n + l"p + jw'rn ~'~

[v.,8- v~,8(1 +jw..)]. (I.2)

The parameters in equations (I.1) and (I.2) are related to more familiar expressions (s a0) by equation (23); ip,s ~ --e~Tp; i,,,s = eOv; "rn = N s ( i - f ) / ( n l C n )

(I.3)

and rp = X s f / ( p , C ~ ) .

The equivalent circuit for the equations (I.1) and (I.2) consists of the capacitance Cs and of the resistances Rn,s and Rp,s given by (21) and (22) in the arrangement shown in Figs. 1 to 8. 1.2 Excess flow of minority carriers in the region of quasi-neutrality By solving the continuity equation for holes: jw15 = - ~/T + D~eZ~/Ox 2

with the boundary conditions p (x = d ) = /~a and fi(x = o~) = 0, oneobtains: : pa exp [-- (x-- d)(1 +jw'r)l/2/(Dp'r) 1/2] (I.4) from which ip,a = vp,a[Rp,B] + ~ , o 1 ~

(I.5)

IMPEDANCE

OF

SEMICONDUCTOR-INSULATOR-METAL

w i t h [Rv,B ] given b y (16) a n d /~B d e n o t i n g t h e spacei n d e p e n d e n t a.c. electric field in t h e bulk o f t h e s e m i conductor. B o u n d a r y c o n d i t i o n s appropriate to a s e m i c o n d u c t o r of finite w i d t h are discussed b y GARRETT{S}a n d by BERZ.(9,1°) I n this case the c u r r e n t iv,a d e p e n d s also on t h e variation of the hole concentration at b o t h surfaces of t h e s e m i c o n d u c t o r .

1.3 Electron and hole flow through the space-charge layer

CAPACITORS

w h e r e we have u s e d t h a t ~ = ~n = 0, t h a t t h e d.c. electron a n d hole c o n c e n t r a t i o n s are B o l t z m a n n d i s t r i b u t i o n s , a n d have neglected t h e t e r m s r e s u l t i n g f r o m t h e a.c. p o r t i o n s o f (plRp) a n d (tuRn) since t h e s e a.c. portions are m u l t i p l i e d b y i~ a n d in, respectively, a n d t h u s b e c o m e small quantities of h i g h e r order. Since t h e i n t e g r a n d s in (I.12) a n d (1.13) are B o l t z m a n n factors we m a y express t h e m in t e r m s of t h e d.c. d i s t r i b u t i o n s of electrons a n d holes to obtain : X~

T h e flow of electrons a n d holes will be described b y the equations :

ip = --petz; ~ V / ~ x - e p / e x eDu

(1.6)

in = --net*n eV/Ox+~n/~x eDn

(1.7)

75

R~ =

(I.17) Xl

and and X2

with mobilities un, up, a s s u m e d to be i n d e p e n d e n t of t h e electric field. T h e total c u r r e n t d e n s i t y is t h u s :

R n ~

i = in + ip = -- ~0 V/ex + e(DnOn/~x- D~ bp/~x).

(i.8) T h e potential d i s t r i b u t i o n o c c u r r i n g in these e q u a tions is d e t e r m i n e d b y P o i s s o n ' s e q u a t i o n :

02V/~x 2 = - ( e / e ¢ o ) ( - n + n o + p - p o ) .

Pl = P2 exp ( - y l + y z ) + f l i p p l R v and

nl = n2 e x p ( y l - y 2 ) - f l i n n l R n

(1.10) (I.11)

where 22

Rp(xl,x2)

= [exp(--yl)/(ple~,~)]j" exp y dx x~ (1.12) X2

Rn(Xl,X2) =

[exp(yl)/(nle~n)]f exp(-y)dx x~

(I.13)

and

y = flY(x).

(1.14)

--. ~n

(1.18)

I n t h e case I of negligible recombination in the space-charge layer we m a y p u t xt = 0 a n d x2 = d; the e q u a t i o n s (I.15) a n d (I.16) b e c o m e t h e n :

v s e = Vp,s+ Vp,a+ ip Rp,D

(1.9)

I n t h e region w h e r e t h e m a g n i t u d e s o f t h e electron a n d hole c u r r e n t s do n o t c h a n g e appreciably, t h e e q u a t i o n s (I.6) a n d (I.7) m a y be g i v e n t h e integral form:

fdx Xl

(I.19)

and

vsp = Vn,s-Vn,a+ in Rn,r,

(I.20)

w h e r e we have inserted R~,D for Rp(O,d) a n d Rn,D for Rn(O,d) h a v i n g in m i n d particularly t h e depletioni n v e r s i o n range. Because of t h e condition o f q u a s i neutrality at x = d, one has:

Vn,a = v~,a "po/no, (1.21) s h o w i n g that vn,a can be neglected if po/no ~ 1 as we m a y a s s u m e for an n - t y p e extrinsic s e m i c o n d u c t o r . T h e e q u a t i o n s (I.19) a n d (I.20) w i t h vn,a = 0 are t h e K i r e h h o f f e q u a t i o n s for t h e potentials a p p e a r i n g in t h e equivalent circuits o f t h e Figs 1 a n d 3. I n t h e depletion inversion region t h e hole c o n d u c t i v i t y increases toward t h e surfaces so t h a t R~,D b e c o m e s q u i t e small and is generally negligible. T h u s t h e t e r m in p r o p o r t i o n to ip in (I.19) can be safely neglected. O n t h e o t h e r h a n d , t h e electron concentration in t h e depletion i n v e r s i o n region is smaller t h a n that in t h e bulk a n d t h e resistance Rn,o m a y b e c o m e substantial. E x p r e s s i n g t h e electron concentration in t e r m s o f t h e hole concentration, u s i n g (12), one o b t a i n s for (I.18): d

P r o c e e d i n g to t h e small a.c. signal f o r m u l a t i o n s of (I.10) a n d (I.11), one obtains: ~1/Pl

=

p2/P2+Y2--yl'~-~ipRp

QI nl2e2tzn

poed n~2e2txn

0

(I.15)

and

~a/nl = ~2/n2+ y l - - y 2 - f l i n R n ,

R,D (n(%n)-lf pdx

(I.16)

(1.22) s h o w i n g t h a t t h e resistance Rn,D is proportional to t h e i n v e r s i o n charge, t h e second t e r m b e i n g quite negligible except w h e n closely a p p r o a c h i n g t h e flat b a n d condition, Y=0.

K. LEHOVEC

76

and

I n t h e a c c u m u l a t i o n range of biases, Rn,A can be neglected b u t :

Rp,A --

QA

+

poed -

-

n~Ze2l~p n/2e2~

A. SLOBODSKOY a n d c o n s i d e r i n g t h e B o l t z m a n n d i s t r i b u t i o n s of electrons a n d holes: -kO0

(I.23)

RD ~ ( a , ) _ l f c

d(x--x*)

_

-o~° s h [ ( ~ i x - x * ) ]

7r a*(~y/bx)x*

m a y b e c o m e substantial. T h e w i d t h d of t h e a c c u m u (I.31) lation space-charge layer is ill-defined, since t h e spacecharge e x t e n d s in a m a t h e m a t i c a l sense to infinity, and Since the c o n t r i b u t i o n s of holes to the space charge R p , a w o u l d t h u s include t h e infinite bulk resistance of an d e n s i t y are negligible in t h e region x 3. x*, because of infinitely thick layer. However, in practice t h e b u l k p < p * ~ n o , one o b t a i n s b y integration of 1.9 over resistance can be taken into a c c o u n t separately and t h e t h e range x* < x < d w h e r e n, p - p o ~ no: second t e r m of (I.23) can be ignored. I n t h e case II of an infinite recombination rate in the space-charge layer it is advisable to consider t h e total c u r r e n t (I.8). E l i m i n a t i n g t h e hole concentration b y m e a n s of (12) a n d i n t e g r a t i n g across t h e space-charge layer C o m b i n i n g (I.31), (I.32) a n d (I.29), t h e e q u a t i o n (19) one o b t a i n s : is obtained.

;dx

VSP

=

i

-

iDn+Dpn~2/n2 ~n dx, do n~n+n,Zt~/n ?x

(I.24)

o

II.1 General case at negligible recombination in the spacecharge layer T h e capacitances [CD], [C1] a n d [CA] are defined by

w h i c h can be written also in t h e f o r m :

Y = iflRD--ln(na/ns).

(I.25)

T h e a.c. portion of (1.25) is:

vsp = iflRv +five,s,

(I.26)

c o n s i d e r i n g that ha = 0 a n d u s i n g (11). T h i s is t h e K i r c h h o f f e q u a t i o n for t h e voltages o f t h e equivalent circuit of Fig. 2. T h e m i n i m u m conductivity occurs at the position x = x* (quantities p e r t i n e n t to x = x* will be m a r k e d b y *), w h e r e

n*e#n = p*elzp = n,e~/(t~n~p)

(I.27)

~r* = 2**C(t~ntZp)/(t~n+t~).

(I.28)

and

I n view of t h e B o l t z m a n n d i s t r i b u t i o n s of electrons a n d holes t h e r e d u c e d potential at t h e position x* is t h e negative q u a n t i t y :

y* = -½1n(..,0/a~,0).

(I.29)

O n l y if Y < y*, a m i n i m u m of t h e c o n d u c t i v i t y occurs w i t h i n t h e space-charge layer. F o r Y > y*, t h e c u r r e n t t h r o u g h t h e entire space-charge layer is practically carried b y electrons a n d we m a y replace t h e n RD b y Rn,D o f (I.22) for y* < Y < --1. T h u s we need to consider n o w only t h e case Y < y*. By developing t h e potential in t h e vicinity o f x* :

dy)

(x-x*)+ ...

Y ~ Y* + dx x*

A P P E N D I X II

The Capacitances [Co], [Cz] and [CA]

(I.30)

e q u a t i o n s (24-26). T h e charges QD, Q1 a n d QA d e p e n d in general on three variables, e.g. VsP, pa a n d ip. By developing QD with respect to t h e s e variables,~" one o b t a i n s f r o m (24):

\?VsP

~pcl

73Sp

~ip vsp

(II.1) Note that [CD] is c o m p l e x a n d d e p e n d s on f r e q u e n c y t h r o u g h t h e ratios vp,a/vsP a n d ip/vsp w h i c h can be expressed in t e r m s of t h e c o m p o n e n t s [Cr], [Rp,B], Rp,D, R~,s, Rn,s and Cs o f t h e equivalent circuit of Fig. 1. E q u a t i o n (II.1) and t h e c o r r e s p o n d i n g expression for [CI] [obtained b y replacing t h e s u b s c r i p t D in e q u a t i o n (II.1) b y t h e s u b s c r i p t 1] can be solved for [CD] and [Cz]. H o w e v e r , t h e s e capacitances are c o m plex a n d f r e q u e n c y - d e p e n d e n t a n d t h e f o r m a l solution is so involved that we shall n o t list it here. I n s t e a d we shall list t h e solution for t h e special case t h a t Rp.s is infinite and we shall t h e n d i s c u s s t h e validity of t h e real a n d f r e q u e n c y - i n d e p e n d e n t a p p r o x i m a t i o n s (27) a n d (28). 11.2 Negligible recombination in the space-charge layer; R p , s ~ 0(3 T h e a s s u m p t i o n that Rp,s = ~ i m p l i e s that t h e surface states are n o t c h a r g e d b y m i n o r i t y - c a r r i e r flow. T h i s includes, of course, t h e case of absence of surface states. T h e ratios vp,a/vse a n d ip/vsP can t h e n be t T o s i m p l i f y n o t a t i o n we have written OOD/~Vsp for OOD(VsP, Pd, ip)/~VsP w i t h pa, ip considered as constants d u r i n g t h e differentiation a n d pa = po, ip = 0 to be inserted after t h e differentiation.

IMPEDANCE

OF SEMICONDUCTOR-INSULATOR-METAL

expressed in terms of [Cx], [R2a,B] and [Rv,D]. Inserting these expressions into (II.1) and combining it with the corresponding expression for [C~] provides:

×{1- iI I +j~oflpodQi ~---~--([RvB]+ Rp,D ) -~1 \

dpa

'

(11.2) and

[C~]-

~Vs~

\

× [Rv,B]+Rp,D

~Vse

~Vsp

,

(II.3)

where we have used that:

(11.4)

to be proved later in this Section. By combining [CD], [CI], [Rp,B] and Rv,D and using (II.2), (I1.3) and (16) one obtains the following expression for the complex space-charge capacitance (6):

Cse = C°O+ (C ° - Coo){fi(O,P) +jfz(~,P)},

(II.5)

where

co =

_

~9.~/~ v s p - OQ~/ev s e = - ~Qsv/~ v s e (11.6) Coo = C ~ o f e q u a t i o n (33)

~z + ~(~ + ~ p ) fl(~,P)

~2 + (r + ~p)2

(11.7)

~p~2 f2(n,P) =

~2 + (7 + ~p)2 = o~"

(11.8) (II.9)

{(1 + ~22)1/2+ 1 }1/2

=

(IIAO) 2

{(1+ ~'~2)1/2--1 }1/2 =

2

R°,B = (D~)~/2/.~,0

C~ = fln~eL~.

(II.14)

T h e expressions f l and f2 have beenlisted forRv,D = 0, which is justified if 2~Rp,D/R~, B ~ 1, a condition that can be shown to be valid if:

m ~ noetzp/(zr,eO),

(11.15)

i.e. if oJ is small compared to the bulk relaxation time of the semiconductor (assuming that t*p/~r is of the order of ~n). Equation (II.4) is proved as follows. By considering xl, yl in equation (I.10) as variables and noting that the maximum contributions to Rp arise from positions close to the upper boundary x~, we may approximate Rv(x, x2) by Rv(0, x~) = Rv,D for xe = d. T h u s the hole distribution is again of the Boltzmann type (at least in the vicinity of the surface, where the significant contributions to QI arise): (11.16)

containing, however, a factor that depends on the current iv. Thus, the partial derivations of any function of p (such as the inversion charge) with respect to pa and ip are related by equation (II.4). It may appear objectionable to consider the hole current to be space-independent in the inversion region since the charge fluctuations in the inversion layer arise from a gradient in the hole current. However, by considering the factor iv in equation (I.10) as a suitable average value, replacing a factor iv(x)in the integrand of R~, it will be recognized that this average value i v is representative for the iv(x ) values close to the upper boundary where the principal contributions to Rv arise. T h u s iv indicates essentially the hole current outside of the inversion region where no significant charge fluctuations arise. It appears justified, therefore, to calculate the hole distribution in the inversion layer as if a space-independent current feeds a surface charge. 11.3 Justification of the approximations (27) and (33) Note that [C1] of equation (II.3) consists of the real and frequency-independent capacitance (27) in series with the impedance: -- {1 + flpo(OQx/@a)/(oQr/o Vsp)}{[Rv,B] + Rv,D}. In Fig. 1 with (Rv,s = co) the capacitance [CI] is already in series with the impedance ([Rv,B]+Rv,D) and adding the impedance above will not cause any significant change if:

(I1.11)

(11.12)

and

P = (~Qi/@a) (flpo)R°,B/r.

77

Graphs of C/C~; (C ° - C°O/CI; P(DvT)I/2/LD; fl and f j are shown in Figs. 9-13 with

p ~- (p~+Bpoi~ Rp,o) exp ( - y ) ,

= Rp,D" f ~ P o - - , @a

~iv

CAPACITORS

(11.13)

I1 +3po(aO,/@e)/(aOi/ev~,v)l

~ 1. (II.17)

T h u s , if this condition is fulfilled, the complex inversion capacitance can be approximated by the real and

78

K. LEHOVEC

and

f r e q u e n c y - i n d e p e n d e n t capacitance (27). N o t e t h a t t h e l e f t - h a n d side o f (II.17) w o u l d be zero if Q~ were a f u n c t i o n of p s only, in view o f (29). T h e a s s u m p t i o n QI = QI(ps) neglects t h e variation of t h e effective w i d t h o f t h e i n v e r s i o n layer with Vse, w h i c h is only a s e c o n d - o r d e r effect. I n t h e depletion range w i t h weak i n v e r s i o n one finds, b y c o m b i n i n g (44), (45), (42) a n d (29), t h a t t h e l e f t - h a n d side o f (II.17) is of t h e order o f (2 y ) - i i.e. small c o m p a r e d to unity. T h e inequality is even better satisfied in t h e range of s t r o n g i n v e r s i o n w h e r e Q t a p p r o a c h e s a f u n c t i o n o f p s only. T h e a d m i t t a n c e of t h e b r a n c h of t h e equivalent circuit c o n t a i n i n g C I of (27), [R~,B] a n d Rp,D contains a f r e q u e n c y - d e p e n d e n t p o r t i o n t h a t equals part of t h e a d m i t t a n c e of joJ[CD] u s i n g (II.2) except for the factor (SQD/Opa)/(aQz/Opa) b y w h i c h t h i s part of jw[CD] is multiplied. Since t h e two a d m i t t a n c e s are in parallel, t h e part of jco[CD] u n d e r d i s c u s s i o n can be o m i t t e d w i t h o u t c o m m i t t i n g a significant error provided that:

epa /

ap---a-~ 1.

(II.18)

T h e r e m a i n i n g part of [Co] is the h i g h - f r e q u e n c y limit (33). T h e c o n d i t i o n (II.18) is o b v i o u s l y well satisfied, since an i n c r e m e n t in t h e depletion charge is located at t h e distance d, w h i c h is m u c h larger t h a n t h e (average) distance o f t h e i n v e r s i o n charge f r o m t h e surface. T h u s , at a c o n s t a n t potential Vsp a n d a given c h a n g e ~Qz/~pa, a m u c h smaller c h a n g e OQ/)/apa is required to c o m p e n s a t e for t h e potential change d u e to OQ1/Opa. I n j u s t i f y i n g t h e a p p r o x i m a t i o n s (27) and (33) we have u s e d t h e e q u a t i o n s (11.2) and (II.3), w h i c h were derived a s s u m i n g t h a t R~,8 = oo. A t finite R~,8, b u t at sufficiently h i g h f r e q u e n c i e s for w h i c h joJ Ct >~ Rp,s -1, t h e results o b t a i n e d for Rp,s = oo s h o u l d still be a good a p p r o x i m a t i o n . For sufficiently low f r e q u e n c i e s t h a t m o s t of t h e hole flow proceeds t h r o u g h R p . , a n d Cs rather t h a n Cz, even a large error in Cx s h o u l d be c o m paratively u n i m p o r t a n t for t h e i m p e d a n c e of t h e entire circuit, s u g g e s t i n g that t h e use of (27) m i g h t still be acceptable. 11.4 Negligible recombination in the space-charge layer; accumulation space-charge region T h e resistance Rn,A of Fig. 3 is quite small, since the electron c o n d u c t i v i t y increases toward t h e surface in an a c c u m u l a t i o n space-charge region. W e shall, therefore, p u t Rn,a = 0 a n d have t h e n also ~QA/~in ~ O. F u r t h e r n'lore :

II.5 Infinite recombination rate in the space-charge layer I n t h i s case, one h a s na = Pa = 0, a n d t h e charges Q I , QD and QA are f u n c t i o n s of only two variables: VsP a n d i. I n t h e d e p l e t i o n - i n v e r s i o n region we t h u s have :

~9~

[CD] =

b Vsp

a9~

i

8i

Vsp

= eOA/~ VSp -- (eOA/ ?na) fiPo v~,a/vse (II. 19) :b a because of t h e d. E q u a t i o n (28) t h a t vp,a/vsp is 3) and po/no ~ 1.

(11.20)

and

8QI

[CI] --

vsp

8QI

i

bi

vp,s

(11.21)

~V,s'p Vp,s

T h e ratios of t h e small-signal a.c. t e r m s can be expressed by t h e equivalent-circuit p a r a m e t e r s of Fig. 2, a n d we obtain t h e n f r o m (II.21):

[c,] ~Vsp

1+b,R~[Cs] {1 + bQdoi/(R~Q,/aVsv)} 1+jwRDSQ,/aVsp {1 + aQ,/ai/(RDeQZ/aVsp)) (11.22) where

[Cs] = Cs[1 +j~oCsRn,sR~s/(Rn,s + RAS)] -1

(11.23)

m u s t n o t be c o n f u s e d w i t h [Cs] of e q u a t i o n (7). A t zero f r e q u e n c y , Cx is given b y (27). T h i s expression r e m a i n s valid at finite a n d even at h i g h f r e q u e n c i e s if I+(8Q~/~i)/(RD 8QflOVsv) is sufficiently small. It is practical to express t h e derivation of Q1 w i t h respect to i by that w i t h respect to Pa'~ u s i n g t h e f u n c t i o n Qx(pa, Vsv) at i = 0. T h e d e p e n d e n c e of Q1 on t h e c u r r e n t arises only b y t h e influence of t h e c u r r e n t on t h e hole d i s t r i b u t i o n . U s i n g (I.24) w i t h t h e lower b o u n d a r y considered as t h e variable x i n s t e a d of o, RD (x, d) m a y be replaced b y t h e c o n s t a n t RD (o, d) for sufficiently small x ( ~ x*), w h i c h is certainly valid for t h e i n v e r s i o n region. W e obtain, t h u s , e q u a t i o n (25), with ns replaced b y n, a n d Y b y y. C h a n g i n g f r o m n to p b y m e a n s of (12), a n d for iflRo ~ 1,

p = (pa+iflRDPo) exp(--y).

(II.24)

T h u s , if Pa a n d i are i n d e p e n d e n t variables, t h e derivations of any f u n c t i o n o f p ( x ) are related by:

?/Si - R~flpoS/Spa.

[CA] = QA/Vn,s ~-~ --QA/7),SP

w h e r e we have u s e d t h e fact t h a t h a = condition of q u a s i - n e u t r a l i t y at x = follows b y u s i n g (51) a n d t h e fact o b v i o u s l y smaller t h a n u n i t y (of. Fig.

A. SLOBODSKOY

(II.25)

T h e e x p r e s s i o n in bracket in II.22 t h u s t r a n s f o r m s to t h e l e f t - h a n d side of (II.17). '~ T h e fact that, at an infinite r e c o m b i n a t i o n rate,

Pa = po is n o t a physical variable is n o t pertinent, since we deal here w i t h a purely m a t h e m a t i c a l t r a n s f o r m a t i o n .

IMPEDANCE

OF SEMICONDUCTOR-INSULATOR-METAL

In a similar manner we obtain from (II.20) :

~QD [co]

OQD

aV, s,p

eVsp

ja,{[Cs] + Cx}

ai 1 +joJRo{[Cs] + CI} ( I I ' 2 6 )

{,..o 0Qo/0 . t ~-~a/~-~sP!

(II.27)

which agrees with (33) on account of (II.17), which is always satisfied. We may thus write (II.26) in the form:

= c o + ( c ~ - Co)/{1 +jo, nn([Cs] + [C,])}. (II.28) Comparing the frequency-dependent portion with that of the equivalent capacitance to [Cs], [Cx] and RD, i.e. ([Cs]

+ C,)/{l+ i,,Ro([Cs] + [Eli)},

it is seen that CD can be used at all frequencies in a good approximation if:

gOD

~t,o ~

79

since Rn,a "~0, and since the current i has no significant effect on the accumulation charge.

showing that at zero frequency (32) is valid, while at very high frequencies:

C°°° -

CAPACITORS

g o t + [Cs].

~ gvs,,

(II.29)

Even for [Cs] = 0, this inequality is well satisfied on account of (II.17) and (II.18). Equation (28) is valid for the accumulation range,

REFERENCES

1. W. G. PFANN and C. G. B. GAP,RETT, Proc. I.R.E. 41, 2011 (1959). 2. J. L. MOLL, LR.E. Wescon Conf. Rec. Part 3, Electron Dev. p. 32 (1959). 3. J. A. MINAHAN, J. L. SVRAGUE and O. J. WIED, J. Electrochem. Soc. 109, 94 (1962). 4. K. LEHOVEC, J. MINAHAN, A. SLOBODSKOY and J. SPP,~GUE, Rep. 21st Ann. Conf. Phys. Electron., Mass. Inst. Techn., March, pp. 80-85 (1961). 5. K. LEHOVEC, A. SLOBODSKOYand J. SPRAGUE,LR.E. Trans. ED-8, 420 (1961). 6. K. LEHOVEC, A. SLOBODSKOYand J. SPRAGUE, Phys. Status Solidi 3, 447 (1963). 7. M . TERMAN, Solid-State Electron. 5, 285 (1962). 8. C. G. B. GARRETT, Phys. Rev. 107, 478 (1957). 9. F. BERZ,J. Electron. Control 6, 97 (1959). 10. F. BERZ, J . Phys. Chem. Solids 23, 1795 (1962). 11. A. E. YUNOVlCH, Soy. Phys., Tech. Phys. 3, 646 (1958). 12. A. E. YUNOVICH, Soy. Phys., Solid State 1, 998 (1960). 13. C. G. B. GAPmETT and W. H. BRATTAIN, Phys. Rev. 99, 376 (1955). 14. W. SHOCKLEY and W. T. READ, Phys. Rev. 87, 835 (1952). 15. R. H. KINGSTON and S. F. NEUSTAEDTER, J. Appl. Phys. 26, 718 (1955). 16. C. E. YOUNG, J . Appl. Phys. 32, 329 (1961). 17. W. SCHOTTKV, Z. Phys. 118, 534 (1942).