study of bandgap narrowing in the space-charge region of heavily doped silicon mos capacitors

Solid-State Electronics Vol. 32, No. 5, pp. 339-344, 1989 0038-1101/89 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright © 1989 Pergamon Press pie

S T U D Y OF B A N D G A P N A R R O W I N G IN THE SPACE-CHARGE REGION OF HEAVILY DOPED

SILICON MOS CAPACITORS

H. C. CI-rEN I, SHENG S. LI 1 and K. W. TENG 2

I Department of Electrical Engineering, University of Florida, Gainesville, FL 32611, U.S.A. and 2APRDL, Motorola Inc., Austin, TX 78721, U.S.A.

(Received 27 April 1987; in revised form 24 October 1988)

Al~traet--A new approach for analyzing the bandgap narrowing in the space-charge region of heavily doped silicon MOS capacitors is presented in this paper. Using high-frequency capacitance-voltage (C-V) measurements on the ultra-thin MOS capacitor structure, the bandgap narrowing versus dopant density in the space-charge region was determined by fitting the measured capacitance minimum, C~n, with our theoretical calculations using a modified C-V theory which considered the heavy doping effects. The results showed a significant bandgap reduction (e.g. AEg = 0.49 eV at N B = 1.6 x l019 cm -3) for dopant densities greater than l018 cm -3.

i. INTRODUCTION

The bandgap-narrowing effect plays an important role in semiconductor device modeling, particularly for minority-carrier transport in heavily doped silicon. Slotboom and de Graft[l] used n - p - n transis- tors to measure the bandgap narrowing from the product of p~n~e, where #~ is the minority carrier mobility in the p-base region. A similar approach was used by del Alamo et a/.[2] to extract rigid bandgap narrowing data in n-type silicon. Infrared absorption studies of Vol'fson and Subashiev[3] and Schmid[4] also found bandgap reduction, but the effect was smaller compared to those observed in Ref.[l,2,5]. The discrepanices between the results from electrical and optical measurements were studied in Ref.[6]. The bandgap narrowing effect observed by these authors was mainly in the quasi-neutral region of silicon p - n junction devices.

In a recent study Lowney and Thurber[7] measured the differential capacitance vs the reverse bias voltage in heavily-doped silicon p - n junction diodes, and found a significant discrepancy between the theoretical and measured values of the gradient voltages. They attributed this to the larger bandgap narrowing effect in the space-charge region of the heavily doped p - n diodes. In fact, values of AEg reported by them are considerably larger than those deduced from the quasi-neutral region of the p - n diodes[l-5]. This discrepancy may be attributed to the lack of the classical Debye-Huckel screening in the space-charge region which causes the bandgap narrowing. The reduction of the gradient voltage is due to the large potential drop across the Debye-tail region of the p - n diodes[8]. To analyze the C - V measurements of the built-in potential Lowney[8] calculated the density of states for both the valence and conduction bands in the space charge region of a heavily doped linearly

S.S.E. 32/5--A

graded p - n diode. He attributed the emergence of band tail states into the forbidden gap as the main reason for the large bandgap narrowing observed in the heavily-doped silicon p - n diodes.

The importance of the bandgap narrowing effect in the space charge region of a heavily-doped silicon p - n diode or MOS capacitor is obvious in view of the increasing interests in using heavily-doped silicon for VLSI and submicron device applications. In this paper, we present a new approach for analyzing the bandgap narrowing in the space-charge region of boron-implanted silicon MOS capacitors using high- frequency capacitance-voltage ( C - V ) measurements. Values of AEg were determined by comparing the measured and calculated values of Cmin using a modified C - V model which takes into account the bandgap-narrowing and other heavy-doping effects. Analytical and numerical models for computing C~i~ are described in Section 2 and in the Appendix, respectively. The deionization effect of boron[9] and the heavy-doping effects such as carrier degeneracy[10] and bandgap narrowing are included in our modified C - V formula. Experimental details are presented in Section 3. Section 4 presents the results and discussion. Summary and conclusions are given in Section 5.

2. THEORY

For high-frequency C - V measurements, the expression of Cr~in for a heavily-doped MOS capacitor can be derived by using either the conventional depletion approximation or the exact numerical integration of Poisson's equation taking into account the heavy-doping effects. The modified C - V theory which considers the heavy-doping effect is described in this Section, and the numerical integration of Cm~n from Poisson's equation is given in the Appendix.

339

340 H.C. CrlE~ et al.

In an MOS capacitor, if the surface of the silicon becomes strongly inverted, then the depletion-layer width in the silicon substrate will remain relatively constant at a certain value of wm. Under this condition the maximum depletion-layer width is obtained, and the capacitance of the MOS capacitor reaches its minimum value, namely, Cram, independent of the applied gate voltage. At the onset of strong inversion ~ ~ 2q~B, where ~b~ is the surface potential, and ~b B is the bulk potential given by (for p-type):

(Er - Ei) ~ - (I) q

and

Cox Es E0/Xm Cmi n = Cox -[- (Esg0/Xm), (2)

where Xm = (4EsEO[CksI/qNA) ~': is the maximum depletion layer width obtained from the conventional depletion approximation. Cox = Eox~o/Xox is the oxide capacitance; Xox is the oxide thickness; E~ and Eox denote the dielectric constants of silicon and silicon dioxide, respectively, and Eo is the permittivity of free space. If the spatial rearrangement for a fixed number of minority carriers within the silicon inversion layer is taken into account, then the capacitance behavior will be altered. A comparison of the results calculated from the improved high-frequency MOS capacitance formula by Brews[11] and that of the Sah-Pierret-Tole model (which neglects the inversion-layer-polarization effect), showed a good agreement (within 7%) for the lightly-doped silicon MOS capacitors. As the dopant density increases towards the doping-density range considered in this study, the discrepancy between these two models becomes negligible, and the error introduced by this effect in the bandgap-narrowing calculations is negligible.

To derive an expression for the bulk potential 4~B, the heavy-doping effects such as carrier degeneracy and bandgap narrowing are included. From the generalized expression of the pono product for the heavily-doped silicon one can write[12]:

n~ = n~exp(AE~/ka T)~rl,:(q *) (3) e q °

where ni is the intrinsic carrier density for the nondegenerate silicon, and AEg is the bandgap narrowing. Note that ,,ar'l.2(r/*) denotes the Fermi integral of order 1/2, and q * = ( E ~ - E F ) / k B T is the reduced Fermi energy. The bulk potential 4~a can be expressed in terms of the intrinsic Fermi level E~ by relating it to q* via eqn (3), which reads:

(Ep - Ei) ( ~ B - - - - -- q

- , * - i n ~ +In ~ - 2 k . r _ j (4)

An excellent empirical expression for r/* was given in Ref.[10] for the parabolic band case, which is

assumed valid for the present analysis[13], is used in eqn (4). Therefore, the first two terms on the right- hand side of eqn (4) can be approximated by (Po/N~.)/[64 + 3.6(P0/N,.)] '4, and the error introduced by this formula is less than 10 -2 for (Po/N,.) <~ 10.6. Now, solving eqns (2)-(4) yields:

Cmin

Cox 1

{1+ Co~{ 4kBT[q*- ln(p°/Nv)+~0qZp01n(p°/ni)]-2AEgll : 1

(5)

It is noted that the expression for Cram given above was derived from the depletion approximation. The validity of this approximation in the heavy-doping regime was verified by comparing the values of Cmln calculated from eqn (5) and the exact numerical integration of Poisson's equation given by eqn (A.7) in the Appendix. In fact, a comparison of the results calculated from eqn (5) and eqn (A.7) for (-'rain revealed that the agreement was found within 3% between values calculated from these two equations for NB < 2 x 10~gcm 3, provided that the same surface band-bending was assumed. It is interesting to note that results of our calculations showed only a very small (less than 5%) discrepancy among several different models considered, namely, (1) the depletion approximation and assuming ~bs(inv.) = 24~B, (2) depletion approximation and 4~(inv.) equal to band bending given by Lindner[14], and (3) numerical integration of eqn (A.7), which has the same nature of integrating an accurate depletion layer width given by eqn (7) in Ref.[15]. Therefore, the depletion approximation used in calculating the Cmm from high frequency C - V curves for the heavy doping case is adequate for the dopant densities studied in this paper.

3. EXPERIMENTAL

The MOS capacitors used in this study were fabri- cated by boron implantation through a 400A sacrificial oxide-on-(100)-oriented p-type silicon sub- strates using 18 to 21 keV implant energy and 1-8 x 10 ~4 cm- : implant doses. In order to observe a maximum change of the capacitance with dopant density in the heavily doped silicon MOS capacitors, an ultra-thin gate oxide of 75-115/~, was grown on silicon at 900°C for 1 h to form an MOS capacitor structure by using a dry-oxide process. Secondary Ion Mass Spectroscopy (SIMS) analysis [16] was em- ployed to obtain the depth profile of the boron density. In fact, SIMS data taken from our boron implanted silicon wafers showed that after annealing at 900°C for 3 h the boron-impurity-density profile near the surface region of the implant layer (a few hundred A thick) was quite uniform. It is worth noting that SIMS depth profile resolution for the

Bandgap narrowing in the space-charge region 341

boron implant is around 100 A and the boron density resolution is about 10 '6 cm -3. Thus, measurements of boron impurity density by the SIMS technique can be quite accurate for doping densities greater than 10~Bcm -3 (i.e. 1% accuracy). Li[17] has shown that for boron densities greater than 3 x 1018 cm -3, the boron atoms are 100% ionized in silicon at room temperature. Therefore, the total boron density determined by SIMS and the ionized boron density determined by the C - V measurements should be equal for boron dopant densities greater than 3 x 10 'a cm -3. In fact, since the depletion layer width for our heavily- doped MOS capacitors is very small and close to the surface (i.e. less than 0.05/~m at high doping), the mean dopant densities determined by the C - V measurements were found in good agreement with the SIMS data obtained at high doping densities. Thus the boron densities used in our calculations are quite accurate, based on the results discussed above. The net boron densities for the MOS capacitors studied in this work vary from 10 ~5 to 1.6 x 1019cm -3.

The oxide thickness determined by ellipsometer was found in good agreement with that determined by the C - V measurements. The thickness of oxide films varies from 75 to 115 A for the MOS capacitors used in this study. Using an ultra-thin oxide- capacitor structure allows us to observe a more radical change of Cmi~ ~Co, with gate bias voltage, and thus enhance the accuracy of the measured C,~i ~ values. Since the breakdown electric field for those MOS capacitors is around 9 x 106V/cm, the C - V measurements were performed for electric field strengths less than this critical field strength.

The C - V measurements were made by using a HP4280A capacitance meter and a HP4274A LCR meter, with frequencies varying from 1 kHz to 1 MHz. Note that the minority-carrier response for these MOS capacitors is fast enough to allow the observation of typical high-frequency C - V curves among these samples. The variation of Cm~,/Cox from sample to sample was found to be less than 3% over the above measured frequency range. The oxide capacitance is determined directly from eoxEo/Xo, (per unit area). When the MOS capacitor was biased into the accumulation region, the asymptotic capacitance Cma, was found in good agreement with Co,. This was also confirmed by our quasi-static C - V measurements in both accumulation and strong inversion regions. In the 1 MHz C - V measurement the gate- bias sweep rate was chosen so that it was slow enough to avoid deep depletion from occurring[18]. The C-V measurements were taken by using a mathematical function called "percent(%) function" which calcu- lates the ratio between the measured and the pre- viously stored reference value.

It should be pointed out that all the C - V data taken from MOS capacitors scribed from the same wafer (4") were quite reproducible (less than I% variation in the measured capacitance values from device to device), which is essential in order to ensure

1.0

0.9

0 .8

0.7"

0.6

Boron density (1) 1.5 x 1019 cm -3 (2 ) 4 .5 x 1018 c m - 3

x 1015 cm -3

(1)

- - :2(1)

- - - ( 2 )

0.11 ~ - (3) 0 I I I I I I - 1 2 - 8 - 4 0 4 8 12

VoLtoge (v) Fig. 1. C- V curves for three boron implanted silicon MOS capacitors. Curves (1) and (2) are for the heavily doped samples. The solid lines are the experimental data, while the dashed lines are the calculated values using eqn (5) and assuming AE 8 = 0. Curve (3) is for the lightly doped sample.

a reliable and accurate determination of Cmi ~. Al- though the presence of oxide charges is unavoidable in a practical MOS capacitor, we have assumed that the oxide charges and oxide-silicon interface charges would only cause the C - V curves to shift along the V s axis and not affect the values of the Cm~,/Cox ratio.

4. RESULTS AND DISCUSSION

Figure 1 shows the C - V plots for three boron- implant MOS capacitors with different implant doses. Curve (3) is for the lightly-doped controlled sample; curves (1) and (2) are for the higher doped samples. The solid lines are the measured C - V curves, while the dashed lines are the calculated values using eqn (5) without considering the bandgap-narrowing effect. It is noted that the discrepancy between the measured and the calculated Cmi . at high dopant densities are so large that it can not be properly accounted for by other effects cited in this paper (e.g. depletion approximation, deionization and carrier degeneracy effects) except the bandgap narrowing. effect. Therefore, by using the measured Cm~n ~Co, as shown in Fig. 1 and the modified C - V formula given by eqn (5), the bandgap narrowing AEg can be determined.

Figure 2 shows the plot of Craig~Co, vs dopant density for p-type silicon along with the experimental data taken at 300 K for the boron implanted silicon MOS capacitors. The solid line was calculated from eqn (2) using the depletion approximation, and the dashed line was calculated when the deionization effect[9] and carrier degeneracy [10] were included. It is worth noting that Fermi-Dirac statistics and bandgap narrowing have an opposite influence on the value of bulk potential OB and hence on the value of

342 H.C. CrlEN et al.

1.o

• Boron doped Si

o., . , / / - ; /

~ 0 . 4 - -

0.2

@ • 0 _ _ L L ~ t i _1 I __ .1 I _ t I

1015 1016 1017 1018 1019 1020

Doping density (crn -3)

Fig. 2. A comparison of the calculated and measured Cmin/Cox v s dopant density for the boron implanted silicon MOS capacitors. Solid line is from the conventional C- V theory, and the dash line takes into account the deionization[9] and carrier degeneracy effects[10]. The solid circles were obtained from our C - V measurements on the

boron implanted silicon MOS capacitors.

Cram. Using Fermi-Dirac statistics will lower the values of n~e at high dopant densities, whereas bandgap narrowing will increase n~. However, the latter has a much stronger effect on n], than the former.

In determining AEg from eqn (5), we first incorporated the data for bandgap narrowing in the space-charge region as reported in Ref. [7], with and without considering the deionization effect. This will

enable us to see the deionization effect on the value

of Cmi n. Figure 3 shows a compar ison of the Cmia/Cox VS dopan t density curves calculated from the conventional and the modified C - V formula given in the text. The dashed line was calculated from eqn (2) using the depletion approximat ion, the solid and dot-dash lines were calculated from the modified equat ion [eqn (5)] with and without including the deionization effect. Table 1 summarized the calculated and measured values of Cmin, and AEg obtained from this work and those from Ref. [7]. Note that the increase of CmjCox with dopant density in our C - V measurement can be best described by the increase of

"-. 0.6 c

L)

O.B

. ~ / ~ ~ . ~ / " / j / /

0.7 " ~ J ~

0.5 / /

0 . 4

0.3 I _ _ 1 I I J 0 4 8 12 16 20

Doping density (x 101Scrn -3)

Fig. 3. A comparison of Cml ~ calculated from the conventional and the modified C V theory by including heavy doping[7] and degeneracy effects[10]. The dashed line is the conventional C-V theory given by eqn (2); the solid line and the dash-dot line are obtained from the modified formula [eqn (5)] with and without including the deionization

effect[9].

AEg, which is known as the effective bandgap narrowing. The results obtained in Ref. [7] are in general agreement with our calculated values of AEg. It should be pointed out that one must simultaneously include both the deionization and bandgap narrowing effect because of the double counting. Either there is no bandgap narrowing, in which case deionization into an impurity band must be considered, or the impurity band has merged with the valence band, in which case bandgap narrowing and not the deionization must be considered. Generally deionization does not apply because the impurity band merges with the valence band prior to any measur- able effect occurring. The only reason to consider the deionization effect here is to show that it cannot account for the large change of the Cm~o values measured at high dopant densities.

The results of our study clearly showed that the bandgap narrowing in the space charge region of

Table 1. Values of Cm,n/Co~ and AEg for the heavily doped silicon MOS capacitors ~

Dopant Cmi./Co~ Cmin/Co, Craig/Co, b Cm~./Co~ AEg[eV] AE~[eV] Density (Classical) (Modified) (Modified) (Expt.) From Ref. [7] This work [cm 3]

0.760 0.812 0.812 0.812 - - 0.493 1.5E19 (p) - - - - - - 0.808 0.490 1.2El 9 (p)

0.760 0.809 0.809 -- 0.496 -- 1.4El 9 (n) 0.698 -- - - 0.760 -- 0.389 8.0E18 (p) 0.679 0.717 0.712 - - 0.321 -- 6.0E18(p) 0.648 0.711 0.701 0.701 - - 0.301 4.5E18 (p) 0.640 0.691 0.681 (0.710) 0.329 - - 4.5E18 (n) 0.611 0.643 0.625 -- 0.246 -- 3.2E 18 (p) 0.437 0.451 0.432 -- 0.102 -- 7.2E 17 (p) 0.336 0.345 0.327 -- 0.066 -- 2.9E 17 (p)

aOxide thickness is 105 ~. blncluding the deionization effect. cWith impurity gradient a = 3.7 × 1022 cm 4.

Bandgap narrowing in the space-charge region 343

heavily doped silicon MOS capacitors was much larger than in the quasi-neutral region [1-5]. This is a direct result and manifestation of negligible free- carrier screening effect in the space charge region. To explain the phenomenon being seen pertaining to the space charge region we have to look into the nature of the extraction of C - V data in such a system when the measurement frequency is high. Upon a change in gate bias voltage all of the incremental charge ap- pears at the edge of depletion region, the formula for the capacitance in the space charge region is given by Cs = -dQs/dcb~ = qEo/Xd, and the minimum capacitance value C~n occurs when Xd~Xm. It is the ionized boron impurities in the surface depletion layer and the surface potential that control the minimum capacitance. In fact, further increase in the bias voltage after strong inversion would result in shielding of the silicon from further penetration of the applied field by the inversion layer. The energy band is bent in the space-charge region with a spatial curvature which can be calculated by solving the one-dimensional Poisson's equation. The surface potential is then incorporated in this curvature variation, which is the electric field. Therefore, the bandgap narrowing in the space-charge region will reduce the surface potential necessary to reach strong inversion, which in turn makes the MOS structure easier to invert and hence to increase the value of C~n.

Finally, it is noted from Fig. 2 that the two calculated (dashed and solid) Cmin/Cox VS doping- density curves cross over at a dopant density around 3 x 10~Scm -3. This may be attributed to the deionization effect. It has been shown that at room temperature about 80% of the boron atoms are ionized at N A = 1018 cm-319] in silicon, and become fully ionized at dopant densities greater than 3 x 1018 cm -3. The deviation of Cmin/Co~ from the conventional C - V theory is dominated by the deionization effect for N A less than 3 x 1018 cm -3, and is otherwise dominated by the bandgap narrowing in the space-charge-region for N A exceeding this value. It is important to point out that the lack of screening both in the semiconductor and in the oxide will greatly enhance the perturbation due to dopant atoms near the interface of an MOS capacitor. One generally expects that the bandgap narrowing effect to be reduced near the Debye region.

5. CONCLUSIONS

A new approach for analyzing the bandgap- narrowing effect in the space charge region as a function of doping density in heavily-doped silicon MOS capacitors has been presented in this paper. A modified C - V formula for analyzing the high frequency C - V curves in heavily doped MOS capacitors has been proposed by taking into account the effects of bandgap narrowing[7], carrier degeneracy[10] and the deionization of shallow impurities[9]. Results of

our calculations of Cmi./Cox using the modified depletion approximation and the exact numerical integration of Poisson's equation were found in excellent agreement. A large discrepancy between the measured Cmi n and the calculated values from the conventional C - V theory given by eqn (5) and the numerical analysis given in the Appendix was observed when the effect of bandgap narrowing was not considered. To account for this discrepancy the bandgap narrowing effect is included in the present analysis, and the results are in good agreement with those reported in Refs[7,8]. Finally, it should be mentioned here that the theoretical calculations of the bandgap narrowing performed by Lowney[19] using o u r Cmi n

data showed that at NA=8 x 1018cm -3 and assuming a screening radius of 10nm the bandgap narrowing was 277 meV, which is about 70% of the value we found from our capacitance data. If a screening radius of 15 nm is used in his calculations, the results would agree well with our measured value.

Acknowledgements--The authors would like to acknowl- edge APRDL of Motorola Inc., for processing the ultra-thin boron implanted silicon MOS structures used in this study. One of the authors (Li) is most grateful to Dr Jeremiah Lowney of the National Institute of Standards and Tech- nologies for his many constructive comments and stimu- lating discussions during the revision of this paper. This research was supported by Semiconductor Research Corp. under contract no. 85-07-060.

REFERENCES

1. J. W. Slotboom and H. C. De Graaff, Solid-St Electron. 19, 857 (1976).

2. J. del Alamo, S. Swirhun and R. M. Swanson, Solid-St. Electron. 28, 47 (1985).

3. A. A. Volfson and V. K. Subashiev, Soy. Phys. Semicon. 1, 327 (1967).

4. P. E. Schmid, Phys. Rev. B 23, 5531 (1981). 5. H. S. Bennett, J. appl. Phys. 55, 3582 (1984). 6. S. T. Pantelides, A. Selloni and R. Car, Solid-St.

Electron. 28, 17 (1985). 7. J. R. Lowney and W. R. Thurber, Electron. Lett. 20, 142

(1984). 8. J. R. Lowney, Solid-St. Electron. 28, 187 (1985). 9. W. Kuzwicz, Solid-St. Electron. 29, 1223 (1986).

10. N. G. Nilsson, Appl. Phys. Lett. 33, (1977). 11. J. R. Brews, J. appl. Phys. 45, 1276 (1974). 12. R. J. Van Overstraeten, H. J. J. De Man and R. P.

Mertens, IEEE Trans. Electron. Devices ED-20, 290 (1973).

13. W. P. Dumke, Appl. Phys. Lett. 42, 196 (1983). 14. R. Lindner, Bell Syst. Tech. J. 41, 803 (1962). 15. J. R. Brews, IEEE Trans. Electron Devices EDo33, 182

(1986). 16. R. B. Marcus, Diagnostic Techniques, VLSI Technology

(S.M. Sze, Ed.). McGraw-Hill, New York (1983). 17. Sheng S. Li, Solid-State Electron. 21, 1109 (1978). 18. E. H. Nicollian and J. R. Brews, MOS Physics and

Technology. Wiley, New York (1982). 19. J. R. Lowney, private communications. 20. C. T. Sah, R. F. Pierret and A. B. Tole, Solid-St.

electron. 12, 681 (1969). 21. C. T. Sah R. F. Pierret and A. B. Tole, Solid-St.

electron. 12, 681 (1969). 22. J. R. Brews, Solid-st. Electron. 17, 447 (1974).

344 H . C . CI-tEN et al.

A P P E N D I X

In this Appendix, a rigorous derivation and numerical calculation of high-frequency capacitance vs voltage re- lation for the heavily doped MOS capacitor is described. The calculation was carried out by numerically solving the nonlinear Poisson's equation for the high-frequency small signal capacitance for an MOS capacitor by considering the heavy doping effects and without using the depletion approximation.

An exact method for computing the high frequency small signal capacitance of an MOS capacitor was reported first by Sah et a/.[20], which is based upon direct solution of the small signal differential equation for the capacitance. Brews[21] obtained a similar equation as that of Sah[20] by deriving the capacitance of an MOS capacitor directly from the small signal analysis of Poisson's equation. Their results allow an accurate numerical calculation of the high frequency small signal capacitance of an MOS capacitor without the use of depletion approximation. To perform the error analysis of the depletion approximation for computing the Cm~, in the heavy doping regime, as described in the main text, we extend the high-frequency small signal capacitance analysis given by Brews[21] to the heavily doped regime. To take into account the heavy doping effects we consider: (1) modifying the small signal capacitance analysis of Poisson's equation by including the majority carrier degeneracy, (2) modifying the electric field by taking into account the free majority carrier concentration in the space charge region in the Poisson's equation, and (3) numerically solving the capacitance vs surface potential from Poisson's equation under conditions (1) and (2). Starting from Poisson's equation for a p-type MOS capacitor, under small signal condition we can write:

dC C 2 dp - - = + ( A . I ) dx EOE ~ q ~ '

where x is the distance from the semiconductor-oxide interface into the bulk, and p is the hole density.

Note that two basic assumptions were made in obtaining ¢qn (A.1): (1) the majority carriers (i.e. holes) respond instantaneously to the a.c. signal, and (2) the minority carrier generation does not follow the a.c. signal[20]. Thus, the hole density and its first derivative with respect to position for the degenerate case can be expressed respectively by:

p(x) = Nv~-lj2(t/) (A.2)

dx ~-P(X)[ ~ A (A.3)

where

r/ = [Ev(x ) -- Ev]/k . T

E - dx ~-D - ( ~ - q * ) + ~

1 2

x [o~1(--~ - A ) - - ~ - t ( - q * A)] (A.4)

where E is the normalized electric field at point x, and is obtained by integrating Poisson's equation once. ,~- ~2, ~-t/2, ,~l are the Fermi integral of order - 1/2, 1/2, and l, respectively, u = qdp/kBT is the dimension less potential. A = (Eg/2k B T) + ln(Nv/Nc) 1'2. L D = (~sEokB T/q2NA) 1'2 is the extrinsic Debye length for the nondegenerate silicon. Using the chain rule yields:

du = d.~ \ d x J = - p L ~ 3 IASt

Therefore, one can write:

Note that eqn (A.5) will reduce to the conventional expression as the carrier density approaches the nondegenerate regime [i.e. p = n, exp( -u) ] .

For a uniform doping profile, eqn (A. 1) can be simplified by changing the variable from x to u, and the result yields:

dC = ~ L - , ~ , ; i~ i JJ IA.7/ du LDHE,

where CFB H = ~sEo/DDH is the modified flat-band capacitance. LDH is the extrinsic Debye length for the degenerate case given by Ref. [22].

Runge-Kutta method of order four was used to solve eqn (A.7) which is a first-order nonlinear differential equation with dC/du = f ( C , u) and u~ ~< u ~< u s. The inital condition is given by C(uB)= CvB H. A comparison of the results calculated from eqn (5) and eqn (A.7) for Cram revealed that agreement of Cm, . was within 3% between these two equations for N B < 2 x 10 ~gcm 3.

study of bandgap narrowing in the space-charge region of heavily doped silicon mos capacitors

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