Pergamon

Solid-Stare Elecrronic~ Vol. 41, No. 3, pp. X17-509, 1997 0 1997 Ekvier Science Ltd

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NOTE

MOSFET INVERSION LAYER CAPACITANCE MODEL BASED ON FERMI-DIRAC STATISTICS FOR WIDE TEMPERATURE RANGE

(Received 12 April 1996; in revised form 13 June 1996)

1. INTRODUCIION

MOSFET inversion layer capacitance, C,,, has been extensively studied[l-31 because it has significant impact on both I-V characteristics of thin gate oxide MOSFET and channel mobility measurements. Based on Maxwell- Boltxmann statistics, a classical model proposed by Sze[l] and a charge sheet model proposed by Brews121 agree well with experimental data at room temperature (Fig. 1). However, both models show certain discrepancy from the experimental data at low temperature, as illustrated in Fig. 1. To model C,., more accurately, attempts including quantum mechanical analysis have been made[4]. However, iteratively and numerically solving the Schrodinger equation is not a desirable solution for many cases.

This note reports a new model based on Fermi-Dirac statistics. Measurement data taken from N-MOSFETs with gate oxide thickness as thin as 52 b; are compared with the model over a temperature range from 85 K to 300 K.

2. MODELLlNG AND RESULTS

Maxwell-Boltxmann statistics apply when the electron conduction band edge, &, is assumed to be much higher than the Fermi level, Er. The following equation of electron density in semiconductor conduction band

n.=N,exp(-w)

is used to approximate the more accurate Fermi-Dirac statistics solution:

(2)

where N, is the effective density-of-states at the conduction band edge, E,, and I(n) is the commonly used Fermi-Dirac integral[5]:

This integral reduces to the following equation in the limit 0fn< -1:

f(rt) z expftl ) (4)

When MOSFET is biased in strong inversion, con- duction band edge at the interface is pulled below the intrinsic Fermi level and Fermi-Dirac statistics should be used. Figure 2 illustrates the difference of charge distributions for a strongly inverted surface for two statistics at 77 K. Maxwell-Boltzmann statistics give a shorter average distance between the electrons and the oxide- silicon interface therefore it overestimates the dynamic capacitance.

For the case of N-channel MOSFETs, the hole concen- tration p,, and electron concentration nP can be expressed as follows, respectively:

and

=nti.exp(%$)-I(Er-~“) (6)

where subscripts 0 and b refer to the values in the bulk and Y(x) is the band bending or potential with respect to the bulk. It can be shown that the equation:

has the following solution:

Using these equations, the semiconductor capacitance C, can be obtained as:

.=_de,=d $E I ( > d’P, dY, ’ dx I

=& 1 - exp(-BY,) + ([expf-/?[& - &])] . I(j(Er - Eeb + Y,)) - 1) z 1 WY*) (9)

507

(7)

508 Note

Ql substrate.

16 4

0 0.4 0.8 1.2 1.6

E,,w~cm)

Fig. 1. Charge-sheet model (solid line) and Sze’s classical model (dashed line) based on Maxwell-Boltzmann statistics plotted with the measurement data for different tempera- tures. A large discrepancy appears at low temperature and high electric fields. The vertical scale on the right hand side is the effective average distance between the interface and electron, defined by the ratio between dielectric constant and

inversion layer capacitance.

where /I = l/H and Q, is the charge per unit area in the substrate and Yy, the surface potential. The inversion layer capacitance is equal to C, minus the depletion region capacitance and can be approximated as C,, since depletion region capacitance is much smaller than C,.,. The electric field inside the oxide is:

(10)

The inversion layer capacitance is measured using split C-V technique. The measurement data are taken from N-MOSFETs with P-type Boron substrate doping con- centration of 8 x 10’6cm-J. Figures 1 and 3 show the inversion layer capacitance versus the electric field in the oxide at temperatures at T = 85 K, 150 K and 300 K for three n-channel devices with the oxide thickness 52 A, 82 A and 142 A, respectively. The inversion layer capacitance is extracted from the measured gate-channel capacitance C,:

Fig. 2. Inversion layer charge distributions as a function of distance from silicon-oxide interface Dredicted bv models based on Maxwell-Boltzmann (solid line) and Fermi-Dirac (dashed line) statistics. Q, is the charge per unit area in the

16 4

0 0.4 0.8 1.2 1.8

E,x(~/cm)

Fig. 3. Comparison between the new model based on Fermi-Dirac statistics (solid line) and experimental data for wide range of temperature from 85 K to 300K. It shows that Fermi-Dirac statistics has to be used when evaluating the C,, at low temperature and high electric

fields.

and the electric field in the oxide E,, is given by

Eo. = 6 (QB + Qm., (12)

is the measured inversion charge and QB = where 8~ = (kT/q)ln(N&). The depletion

region capacitance

c*= f$g J B

= 0.079pF/cm2(85 K) 0.092~F/cm2(300 K) (13)

is much less than the extracted capacitance C,.,, normally on the order of several pF/cm* as shown in Figs. 1 and 3. This is commensurate with our previous treatment of C, as C,,,. Excellent match between the experimental data and the new model is shown over a wide range of temperature (Fig. 3). This is the first successful reconcilia- tion of measured C,., with theory.

When C. is calculated for temperatures 150 K and 85 K, because these two temperatures are below T1 = E. - E,/k ln(N,/8N.) z 250 K for Boron doped P-type Si with doning concentration of N. = 8 x 10’6cm-3, Pfl = JN,N,/Ze- & - ~~~~~ instead of Ppo = N. in eqn (9) is used when C, is evaluated. E. in above T, expression is impurity ionization energy.

3. CONCLUSION

A relatively simple (compared to applying quantum mechanical analysis otherwise) MOSFET inversion layer capacitance model based on Fermi-Dirac Statistics has been presented. The model is shown to match the measurement data well over a wide range of temperatures from 85 K to 300 K, oxide thicknesses from 52 A to 142 A and gate voltages (gate oxide field is up to 1.6 MV/cm). It is shown that Fermi-Dirac, instead of Boltzmarm statistics, is necessary to use for low temperature and high fields because the conduction band edge is well below the intrinsic Fermi level.

Ac/cnowledgemenrs-The author would like to thank Professor Peter Yu for his very helpful discussions. This work is partially supported by SRC contract 94-DC-324 and the Joint Servtces Electronics Program, F49620-94-C-0038 and AFOSR442427-25328.

Department of Elecrrieal Engineering and Computer Science

University of California, Berkeley, CA 94720-1772. USA

Intel Corp., Santa Clara, CA 95052, USA

BTA Inc.. CA, USA

Hong Kong Univ. of Science and Technology. Hong Kong

Note 509

Kai Chen REFERENCES

George Zhang Jon Duster

Chenming Hu

Jianhui Huang

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2. J. R. Brews, Solid-State Electronics 21, 345 (1978). 3. M. S. Liang, J.Y. Choi, P. K. Ko and C. Hu, IEEE

Trans. Electron Devices ED-33, 409 (1986). 4. S. A. Hareland, S. Krishnamuithy, s. Jaliepalli, C.-F.

Zhihong Liu Yeap, K. Hasnat, A. F. Tasch, Jr and C. M. Maziar, IEDM p. 933 (1995).

Ping K. Ko 5. C. Kittle and H. Kroemer, Thermal Physics, 2nd edition, p. 365. W. H. Freeman, San Francisco (1980).