Inversion charge redistribution model of the high-frequency MOS capacitance

Solid-State Electronics, 1974, Vol. 17, pp. 735-742.

Pergamon Press.

Printed in Great Britain

INVERSION CHARGE REDISTRIBUTION MODEL OF THE H I G H - F R E Q U E N C Y MOS CAPACITANCE ARNOLD BERMAN and DONALD R. KERR IBM System Products Division, East Fishkill Development Center, Hopewell Junction, New York 12533, U.S.A. (Received 30 August 1973; in revised form 29 November 1973)

Abstract--The high-frequency semiconductor capacitance in an MOS structure is ordinarily calculated by a depletion-charge analysis approach assuming that there is no response of the inversion layer charge to the a.c. signal. The more realistic model, in which the inversion charge is allowed to be spatially redistributed at the high frequency, is treated here by the solution of the Poisson-Boltzmann equation incorporating an appropriate quasi-Fermi level for the inversion layer carriers. The inclusion of this effect leads to a much faster saturation of the capacitance and increases the value at strong inversion by 2-5 per cent for silicon at room temperature. It also predicts a shallow minimum less than 1 per cent below the asymptotic value. Agreement with experiment is shown to be excellent. An analytic expression for the asymptotic value is given.

1. I N T R O D U C T I O N

The metal-oxide-semiconductor (MOS) capacitor has been extensively used for studies of the oxide, interface, and semiconductor properties[I-6]. The most common measurement on this structure is the small-signal high-frequency capacitance vs d.c. voltage ( C - V ) trace. From this can be deduced, for example, information on the interface states, oxide charge, ion transport through the oxide, and the silicon impurity profile. For many of these applications it is necessary to be able to accurately calculate the theoretical C - V characteristic for a uniformly doped semiconductor in the absence of interface states and oxide charge. The high-frequency C - V analysis for such a case has been carried out by many authors[2, 3, 7-9]. The most commonly accepted approach for accurate calculations is a depletion-charge analysis [3, 8, 9]. In this approach the semiconductor depletion charge is exactly calculated from a solution of the Poisson equation. This charge is then treated as if it were a step-function and the semiconductor capacitance is calculated as the ratio of the dielectric constant, E, to the width, Xu, of this depletion layer.* In this scheme the capacitance in inversion and depletion (Fermi level beyond midgap at surface) is given by

*All subsequent uses of the term "charge analysis" in this paper refer to this depletion-charge analysis, rather than a more general meaning of the term.

Cs = e/Xu - e q N AQ'

(I)

where q is the electron charge, N is the semiconductor doping density, and AQ is the charge per unit area due to the depletion of majority carriers. AQ is obtained by a numerical integration at any desired surface potential[10]. Once strong inversion is reached, AQ saturates and the highfrequency capacitance approaches a minimum value asymptotically. Experimentally we have consistently observed in our laboratory a more rapid saturation of capacitance to its minimum than is predicted by charge analysis. This would be expected if minority carrier redistribution were taken into account. Since the inversion layer is not infinitesimally thin, redistribution of the carriers within the inversion layer at the a.c. measurement frequency will, in effect, short-circuit a portion of the depletion layer. The effective width of this shorted region will increase with d.c. bias and therefore tend to cancel the increase of Xd, causing capacitance to saturate more abruptly. As a further consequence, the saturation capacitance will be larger than predicted by charge analysis. This redistribution effect has been discussed by Lehovec et al. [2], but has been generally ignored in the literature. In this paper we present in sections 2 and 3 an exact high-frequency analysis of the MOS capacitor which allows for the redistribution of the inversion layer carriers. This is done by defining a quasi-

735

736

A. BERMANand D. R. KERR

Fermi level which is spatially uniform within the inversion region. This uniformity ensures internal equilibrium of the inversion layer carriers, while the displacement of this quasi-Fermi level in phase with the a.c. signal permits the total number of carriers to be held constant. The resulting capacitance expression requires a numerical integration of the same order of difficulty as that required by charge analysis. C - V curves thus calculated are shown in section 4 to be in much better agreement with precision measurements than those calculated by charge analysis. The exact calculations for the asymptotic value of the high-frequency silicon capacitance are compared in section 3 to the commonly used approximations. In addition, a useful analytical expression is given which closely approximates this exact asymptotic value. 2. T H E

PHYSICAL

AND MATHEMATICAL

(2) internal equilibrium of the carriers in the inversion region. This is the carrier redistribution effect which has not been properly taken into account in previous calculations. (3) no response of the magnitude of the inversion charge to the a.c. voltage: the high-frequency condition. The calculation proceeds with a solution of the Poisson-Boltzmann equation in which the distribution of the inversion charge carriers is governed by a quasi-Fermi level. In accordance with condition (2) this quasi-Fermi level is spatially uniform in the inversion region. The energy band diagram under these conditions is illustrated in Fig. 1 for a p-type semiconductor. Deep in the semiconductor the electron and hole Fermi levels must, of course, coalesce. This coalescence will take place rapidly where the minority carrier density is very small, i.e., outside the region of significant inversion. The detailed shape of this merging of levels will not affect our results since the total number of minority carriers in this region is negligibly small, In the inversion region the displacement, a, of the Fermi levels (which varies in synchronism with the high-frequency surface potential) is determined by the conservation of inversion charge as dictated by condition (3). The capacitance is calculated from

MODEL

The high-frequency semiconductor capacitance is calculated under the following physical assumptions. (1) d.c. equilibrium. For an MOS capacitance-voltage measurement this is equivalent to a voltage ramp rate slow enough to avoid deep depletion.

EC

E.

/ /

J_

A

/ ~ / - "

/

~

- ...........

!

EFp

EFn

/ Oxide

Semiconductor X

Fig. l. Energy band diagram for a p-type semiconductor. E, and E~. are the conduction and valence band edges, respectively, and E, is the intrinsic Fermi level. All potentials are measured in units of kT/q. The solid lines show the equilibrium values, and the dashed lines correspond to a high-frequency perturbation (6us) of the surface potential. E.~-p and E~,, are the quasi-Fermi levels for holes and electrons, respectively.

737

Redistribution model of the MOS capacitance the derivative of the surface field with respect to the surface potential in the limit of vanishing ot. This ensures compliance with condition (I). The value of a will depend upon the high frequency shift of the surface potential. It will clearly be some average value of this potential shift over the inversion region, therefore ot < 6Us, where 6Us is the instantaneous high frequency displacement of the surface potential Us. Insofar as the inversion charge resides very near to the surface, it may be anticipated that ot ~ 6Us and ot ~ ~Us for strong inversion. 3. C A L C U L A T I O N O F T H E H I G H - F R E Q U E N C Y CAPACITANCE

with the discussion of section II, it is assumed that the dependence of K upon a is negligible. The high-frequency semiconductor capacitance is given by C s H v = - e l i m~o OUs Us

(2)

where n, is the intrinsic carrier concentration and u is the semiconductor potential measured as the displacement of the intrinsic Fermi level from the bulk Fermi level and normalized to kT/q. Note that for the p - t y p e semiconductor u is negative in the bulk. The electron concentration, on the other hand, is given by n = n,e "-° ~ (1 - ot)n,e u,

(3)

as determined by the quasi-Fermi level. The expansion for small ot is made since only the limit ot ~ 0 will be of interest. The Poisson-Boltzmann equation then becomes 1

1 da

u"=~---~(sinhu-sinhuB-~ote),

2(

,

(u')2=~-~ c o s h u - u s i n h u B - ~ a e " +

,

(5)

where K is a constant of integration. In accordance SSE Vol. 17, No. 7--G

Qo=q

(6)

f

ndx=-n~(1-ot)jo

[ u s ~Tau, eu.

(7)

where again the minority carrier charge outside the inversion region is neglected. Setting to zero the derivative of this expression with respect to Us and taking the limit ot ~ 0 : _e,~ o-~-jo

[,s e" {Ou']

dot [ " s e " d u

(-ffr~ky~u~/,du-T~-U~Jo ~

.

(8)

The term (Ou'/OUs), can be obtained from equation (5): '] = 1 e. dot OUs/ . - 2--L-f-~ du--~"

(9)

Using this expression in equation (8):

(4)

where L is the intrinsic Debye length (EkT/2q2nl) ~/2 and uB is the bulk potential ( - I n N / n ~ ) . This expression differs from the equilibrium case only in the last term containing the factor a. Because of this term, it should be noted, this equation differs in character from the equilibrium case in that the right hand side is a function not only of u but of the surface potential, Us, as well. In the inversion region, where a is constant, the first integral is

)

where dot/dus must be evaluated under the constraint that the total electron charge is independent of Us. This charge is to be evaluated from the expression

da

u

(sinh us - sinh uB 2 dus e"s '

The calculation is here carried out for a p - t y p e semiconductor. The hole concentration, using Boltzmann statistics, is given by p = nie ",

~ •

eUs -

f"sre"

1

h--~u~- u ; J o L ~

e 2u ]

2L-2 (..-77Z3" u ) J du"

(lO)

As it stands equation (10) is not convenient for calculation because the integrand is a small difference between two large numbers. This can be simplified by a partial integration to the more convenient form da dus

1 =

u'sexp(-(Us+U,)) 1+ 2L 2

("~ e" . J0 ( - ~ d u

(11)

This expression is plotted against Us in Fig. 2 for several values of impurity concentration in silicon at room temperature. As anticipated it approaches unity for strong inversion. This approach to unity is especially marked for Us - > - UB, i.e., beyond the threshold for strong inversion.

738

A. BERMAN and D. R. KERR

.ooI 0.99 !

0.9B I dUS

1018~

0.970.96 1

:

// ~/

/"

/

0

1

6

~

i014

0.950

I

I

5

10

15

L

Z

20

25

.

30

US Fig. 2. Calculated values da/du~ (equation (11)) for silicon at 25°C with impurity concentrations of 10 '~, 10'* and 10'~/cm ~. The vertical lines on the curves are the strong inversion threshold points,

U s i n g the f o r m of equation (11) in equation (6), we h a v e for Us > 0,

capacitances calculated by the two models differ significantly only in the region of strong inversion.

e e "~

1

C~,~F = Cs,~ + 2L2u} 14 u } e x p ( - ( u s + u o ) ) 2L 2 w h e r e CsL~ is the low f r e q u e n c y s e m i c o n d u c t o r capacitance. Since the limit ~ ~ 0 has b e e n taken, the integral in equation (12) is to be evaluated with u' depending only upon u, as in the K i n g s t o n N e u s t a d t e r [10] relationship, i.e., from equation (5) with a set to zero and K = cosh uB + u, sinh uR. The asymptotic limit of the high-frequency capacitance is f o u n d by taking the limit of equation (12) as us goes to infinity. This gives ee""f

~

("~ e" J,, ~ 7 ~ o u

As expected, the redistribution-model curves saturate more rapidly than do the charge-analysis curves. The difference b e t w e e n the asymptotic values of the capacitance for the two models varies from 2.6 to 5 per cent o v e r this doping range. An u n e x p e c t e d result is the shallow minimum shown by all the redistribution-model curves, after which the capacitance rises slightly (0.4-0.8 per cent) to Cs~. This suggests that the effective shunting width e"du

Cs-~ = 2V2L Jr [cosh u - c o s h u~ 22 (--u - uB) sinh uB] 3/~" Capacitance vs us curves using charge analysis and those using the redistribution model (equation (12)) are shown in Fig. 3 for silicon at 25°C (n~ = 1.186 x 10 t° cm -3, E = 1.04 x 10-n F / c m ) with three doping concentrations spanning the range of interest for typical M O S devices. Although our use of B o l t z m a n n statistics is not appropriate for a part of the range of Us s h o w n (Us ~ 2 0 ) , the use of F e r m i - D i r a c statistics would not be e x p e c t e d to change the qualitative b e h a v i o r of the curves. In addition, oxide b r e a k d o w n would limit the accessible experimental range to Us ~ 24. N o t e that the

(12)

(13)

of the inversion layer saturates more slowly than does the extension of Xd. T h e m i n i m u m occurs at about 5kT/q units b e y o n d the threshold of us = - uB. Since total surface charge is increasing rapdily with Us in this region, the asymptotic capacitance is difficult to achieve experimentally. The fixed oxide capacitance in an M O S structure will, of course, reduce the observable m i n i m u m - a s y m p t o t e difference e v e n further. This minimum should not be confused with the minimum c o m m o n l y observed in M O S capacitors due to channeling[11]. Table 1 gives the silicon capacitance at 25°C

Redistribution model of the MOS capacitance 295

CSDA=

739

rL 2 k T ( U s

-- US --

1 ''

1)J "

(14)

RM

290

N

A commonly used approximation for the highfrequency asymptote is obtained from equation (14) by letting Us = - uB. Figure 4 shows that this approximation gives a capacitance value ranging from 4 to 6 per cent above the exact asymptotic value calculated from the redistribution model. The difference between the redistribution-model and chargeanalysis asymptotes is also plotted in Fig. 4. We have found an analytic expression which gives excellent agreement (within 0.02 per cent) with C,~ over a wide range of uB. It is obtained by replacing Us in equation (14) with - uB + I n 1.15 ( - uB - 1). This gives

IOI8¢m "3

28o ] 34 ~

33

N = 1016cm -3

32

4.2

CsAp =

RM

4.0

Eq 2 N ] ~/2 {- 2uA - 1 + l n 1"15 ( - uB - 1)}J " (15)

N = 1014era °3

4. EXPERIMENTAL WORK

3.9

3.8

2kT

I 10

__1 15

I 20

__1 25

I 30

I 35

US

Fig. 3. Calculated semiconductor capacitance vs us for charge analysis (CA) and the redistribution model (RM) of this paper. Calculations are for silicon at 25°C and the us-values corresponding to the doping concentrations of 10TM, 10~ and 10'a/cm~, which are -9.04, -13-64 and - 18.25, respectively. calculated to four significant figures at various doping densities for the redistribution-model minimum, C, m~,, and asymptote, Cs~, as well as for the charge-analysis asymptote, CscA. Percentage differences between C~ and Cs~. and between C,~ and Csca are also given. The depletion approximation for semiconductor surface capacitance, which agrees with the exact calculation before the onset of strong inversion, is (for uB < O)

It has been shown that the redistribution model does, indeed, predict a C - V curve which saturates rapidly in strong inversion as is commonly observed experimentally. Careful measurements have been made on MOS capacitors in order to compare in detail the experimental and theoretical curve shapes. Of all the relevant variables of the device, the most difficult to measure with precision is the semiconductor impurity concentration. This variable is, therefore, taken as a parameter whose value is chosen to fit theory to experiment at strong inversion. The self-consistency of the value chosen can then be examined by an analysis of the C - V shape in the region where the mobile carrier density is negligible compared with the ionized impurity density (the depletion approximation). An MOS fabrication process was chosen to avoid channeling, to minimize the redistribution of impurities from the substrate, and to give a low faststate density. Wafers of n-type silicon, (100) orientation, nominally l l L c m , were oxidized in an 02-

Table 1. Capacitance calculations for silicon at 25°C N

--

UB

C m,°

C~.

(10-gF/cmz) (10-9F/cm2) (10-gF/cm2)

(cm-3) 10" 10'5 10~6 I0 '~ 10'8

C ~

9-04 11.34 13.64 15"95 18"25

4.098 11"58 33"46 98.04 290.3

4"066 11-51 33.28 97.59 289-2

3'889 11.11 32.32 95-16 282.8

C ~ - Csm i n

Cx ~ --

CSCA

(%)

(%)

0.78 0.60 0-54 0"46 0.38

5-10 4.06 3.41 2.94 2.58

740

A. BERMAN and D. R. KERR 7 6 5 i

4 / 3

C

CSD,S,]

~E~ t 4 US

- 3B

2 C - C~

I

CS ~ {%)

0 -1 -2

C

CSC,~

-3 -4 -5

, i!

i

-6

io 14

1o 15

io j6

1o 17

N ( c m -s )

Fig. 4. Comparison of an approximation (equation (14) with u~ = u~) and the charge analysis asymptote to the exact (redistribution-model) asymptote for silicon capacitance, C~, at 25°C.

steam-N,_ sequence at 950°C to a thickness of about 500/k. Circular aluminum electrodes (0-52 mm dia.) were evaporated through a mask from an electronbeam source. Aluminum was also evaporated onto the wafer backs after oxide removal. The penumbra of thin aluminum at the electrode perimeters was then removed by a brief aluminum etch. Finally, the wafers were annealed in forming-gas at 400°C for 60 mins. Precision capacitance vs voltage measurements were taken on a Boonton 75C bridge at 100 kHz with an rms signal of 20 mV, The oxide capacitance Cox was calculated from the measured capacitance C(V) in strong accumulation using the correction of Sah et a/.[12] giving

Cox

-

C(V) (1 4 A

2kT

qlV-Vv,

[)

'

(16)

where A is the electrode area and V~, is flat-band voltage. Electrode diameters were measured on a precision microscope stage driven by calibrated differential micrometers. For comparison with experiment, the theoretical MOS capacitance is given by C = A

CoxC~ Cox + C,'

(I 7)

where C, is the silicon capacitance, obtained from the low-frequency formula for us- < 0 and from the redistribution formula (equation (12)) for u, > 0 . The MOS voltage is given by

~[

,

~u;1

(18)

where the plus and minus signs are used for p- and n-type silicon, respectively, and u ~ is [10] u~-

X/2(u. - u~) Llu -"sl [(uB -

Us) sinh uB + c o s h Us - c o s h uB] '2.

(19)

A typical set of measurements is compared to a theoretical curve in Fig. 5. The theoretical curve is calculated by the redistribution model with N chosen to match the experimental minimum and is shifted along the voltage axis to match the experimental points at flat-band capacitance (100.8pF). We note excellent agreement between the data and the calculated curve. The improvement over charge-analysis theory can be seen clearly in Fig. 6, in which the data of Fig. 5 are shown on an expanded scale in the vicinity of the "corner" where the minimum capacitance is reached. Curve I is the

Redistribution model of the MOS capacitance

741

140

120

........

100

m I -5

13

-4

-

I

I

-2

-I

k

I

I

I

1

2

3

4

Aluminum B[os (Volts)

Fig. 5. Typical high-frequency MOS capacitance vs voltage measurements and a theoretical curve calculated from the redistribution model. The region enclosed in the dashed box is expanded in Fig. 6.

I

Somple

Q --

I

I

I

8496-4-9

M~sv,~ems Th~ry:

II

RedistributionModel N = 4.55x 1015cm-3 arg_~ • 4,~ al: sis. . . . . . -~ J12 ChN t3

ChorgeAnaly N = 4,86 x 10lScm'3

39

theoretical redistribution-model curve. Curve 2 is calculated by charge analysis using the same value of N (4.55 x 10~5/cm3). Curve 3 is a charge-analysis calculation with N chosen (4.86 × 10~5/cm3) to give an asymptote which matches the experimental minimum. Again, all theoretical curves are shifted along the voltage axis to match the measurements at flat-band capacitance. Note the excellent agreement of the shape of the redistribution-theory curve in Fig. 6 with measurement, in contrast with the charge-analysis curves, which saturate much more slowly. The capacitance minimum is not observable in Figs. 5 and 6 for the reasons given in section 3. Another method of obtaining the impurity density from the C - V data requires taking the slope of inverse-square-capacitance vs voltage. N is then given by[13] N =

t -3

,

l -2

~

2 _~ d(1/C2), qe~t ~

(20)

I -I

Al~m~.w B~os(Volt~)

Fig. 6. Capacitance-voltage data shown on an expanded scale plot and theoretical curves calculated from the redistribution model (curve 1) and charge analysis (curves 2 and 3).

if fast surface states are negligible. This expression is highly accurate for a depleted, but not inverted, surface. The bridge data of Fig. 5 show a very linear relationship between 1/C 2 and V over the range of -0.7 to - 1 . 3 V , and the resulting value of N

742

A. BERMAN and D. R. KERR

f r o m equation (20) is 4-62 × 10~5/cm 3. This is 1-5 per cent higher than the value (4.55 x 10~5/cm3) required to match the experimental minimum. If fast-states are present, the C - V curve will be broadened, but the minimum capacitance will be unchanged. Equation (20) will then give an N higher than the true value. Calculations with uniform fast state distributions show that a fast-state density of l x 101°/cm 2 eV in this structure would cause a 1.5 per cent error in N. Fast-state determination on this sample by c o m p a r i n g high-frequency and quasistatic capacitance[14, 15] indicates-densities in the range of 1-0-1.4 x l0 ~°states/cm 2 eV. Thus, the surface states can c o m p l e t e l y account for the small d i s c r e p e n c y in N as determined by the two methods. Alternately, this discrepancy could have been resolved by using the formula by Brews[16] which corrects equation (20) for interface-state effects. 5. SUMMARY

The M O S capacitance has b e e n analysed by use of a quasi-Fermi level a p p r o a c h which correctly accounts for the high-frequency redistribution of carriers within the inversion layer. Calculations based on this theory are in excellent a g r e e m e n t with experiments, in contrast to the conventional charge analysis. The numerical integration required in these calculations can be efficiently done on a digital computer. Specifically, an A P L program we have written for calculating a c o m p l e t e C - V curve, which evaluates equation (12) at 24 surface potentials b e t w e e n midgap (Us = 0) and the band-edge, requires only 0 . 4 s e c of C P U time on an I B M 360/85 computer. Sah et al.[9] h a v e found excellent a g r e e m e n t b e t w e e n charge analysis and an " e x a c t " analysis of the n o n u n i f o r m transmission line model. This a g r e e m e n t results f r o m the r e m o v a l of the equivalent circuit c o m p o n e n t s corresponding to the inversion layer conductance. If these c o m p o n e n t s are properly retained, a result in a g r e e m e n t with our analysis is obtained[17]. F o r a n o n u n i f o r m l y doped s e m i c o n d u c t o r the

P o i s s o n e q u a t i o n must be solved by numerical integration. The quasi-Fermi level analysis of the redistribution model can be straightforwardly incorporated in the numerical analysis. This will be the subject of a future publication. The charge-analysis approach, incidentally, b e c o m e s ambiguous in this case because the " q u a s i - n e u t r a l " charge must be subtracted f r o m AQ in equation (1). Since submission of this paper we have learned of two recent papers which take the inversion charge redistribution effect into account. Baccarani and Severi[18] d e v e l o p e d a model resulting in an integro-differential e q u a t i o n solved by iteration. B r e w s [19] used a quasi-Fermi level a p p r o a c h closely paralleling ours.

REFERENCES

1. L. M. Terman, Solid-St. Electron. 5, 285 (1962) 2. K. Lehovec, A. Slobodskoy and J. L. Sprague, Phys. Status Solidi 3, 447 (1963). 3. A. S. Grove, B. E. Deal, E. H. Snow and C. T. Sah, Solid-St. Electron. g, 145 0965). 4. E. H. Snow, A. S. Grove, B. E. Deal and C. T. Sah, jr. appl. Phys. 36, 1664 (1965). 5. K. H. Zaininger and G. Warfield, I E E E Trans. Electron Devices ED-12, 179 (1965). 6. K. H. Zaininger and F. P. Heiman, Solid State Tech. p. 49, May (1970). 7. R. Lindner, Bell Syst. Tech. J. 41, 803 (1962). 8. A. S. Grove, E. H. Snow, B. E. Deal, and C. T. Sah, J. appl Phys. 35, 2458 (1964). 9. C.T. Sah, R. F. Pierret and A. B. Tole, Solid-St. Electron. 12, 681 (1969). 10. R. H. Kingston and S. F. Neustadter, J. appl. Phys. 26, 718 (1955). 11. E. H. Nicollian and A. Goetzberger, I E E E Trans. Electron Devices, ED-12, 108 (1965). 12. C.T. Sah, A. B. Tole and R. F. Pierret, Solid-St. Electron. 12, 689 (1969). 13. W. van Gelder and E. H. Nicollian, J. Electrochem. Soc. 118, 138 (1971). 14. R. Castagne, C.R. Acad. Sci., Paris 267, 866 (1968). 15. M. Kuhn, Solid-St. Electron. 13, 873 (1970). 16. J. R. Brews, Jr. appl. Phys. 44, 3228 0973). 17. M. J. McNutt and C. T. Sah, Solid-St. Electron. 17, 377 (1974). 18. G. Baccarani and M. Severi, IEEE Trans. Electron Devices ED-21, 122 (1974). 19. J. R. Brews, J. appl. Phys. to be published.