IEEE/OSA/IAPR International Conference on Informatics, Electronics & Vision

A New Technique for Finding Sub-threshold Current of MOSFETs

Taufiq Ahmed Department of Electrical and Electronics Engineering

Bangladesh University of Engineering and Technology Dhaka, Bangladesh

topitww@gmail.com

Abstract- A new technique for finding sub-threshold current of

MOSFETs is proposed. In this approach, first I-D Poisson's

equation for potential within the depletion region formed beneath

the SiOz-Si interface is solved. The solution gives potential as a

function of the distance along the width the of the depletion

region. The intersection of the tangent of the potential curve at

the interface with the distance axis gives the effective thickness of the channel. The new effective channel thickness is applied for finding new expression of sub-threshold current. The result of

this model is compared with previous models and found smaller sub-threshold current.

Keywords-sub-threshold current, electrostatic potential, surface

potential, space charge density, effective channel thickness.

I. INTRODUCTION

The prediction of sub-threshold current characteristics in MOSFETs is important in low-voltage digital logic and memory circuits. Therefore, it is of great importance to obtain an accurate, yet relatively simple analytical current model for design and analysis purposes. Numerous papers regarding sub­threshold current have been published [1-6]. A new current model with reduced sub-threshold leakage current is proposed in this paper.

In previous works, effective channel thickness has been

assumed to the distance where potentiallfJ ' decreases by kT q

from its surface value [1-2]. For the exponential dependence of electron density on the potentiallfJ ' this assumption has

been made. In another approach [4], an expression for surface charge, Qn has been obtained from charge-sheet model. It has been assumed that, weak inversion occurs before the onset of gate voltage at which surface potentialljl s, is double of the

bulk potential. These approach also use predetermined potential value for finding sub-threshold current.

In this paper the analytical expression for sub-threshold current is developed using comparatively large effective channel thickness. The thickness is computed as the region where potential is non-zero. So the thickness is in perfect agreement with the potential. The electron density extending to this deeper level has been assumed constant. The new drain

978-1-4673-1154-0112/$3l.00 ©20 12 IEEE

Nazmul Hasan Department of Electrical and Electronics Engineering

Bangladesh University of Engineering and Technology Dhaka, Bangladesh

nazmulhasan772@gmail.com

current expression gives accurate result. It also accurately describes the relationship between 1D and VD.

The paper starts with brief description of MOSFET geometry. It is followed by mathematical derivations along with necessary explanations. The paper concludes with results obtained from the model along comparison with other models.

II. THE PROPOSED METHOD OF SUB-THRESHOLD CURRENT

In weak inversion state of MOSFET, there is significant amount of electron in the surface of a p-type substrate. The actual dependence of sub-threshold current 1D is an exponential function of potential 'P, resulting from the Si-Si02 interface. It is dominated by diffusion [1-3]. So it can be written as

dn 1 D = -qADn dx '

1D = qADn nCO) - n(L) .

L

(1)

(2)

Where q is the charge of an electron, A is the cross section of current flow, Dn is the diffusion coefficient for electron, n(O) and n(L) is the electron concentration at the edge of the source and drain respectively. These are given by:

nCO) = npoexp(qljls), (3)

kT q(ljIs - Vd)

n(L) = npoexp( . kT

(4)

The area A is given by the width W of the device and the effective channel thickness normal to the semiconductor­insulator interface. In this paper the main focus is determining a suitable channel thickness. First, it is assumed that the effective channel thickness is Xd. It implies that, the cross­sectional area is given by: A=WXd. (5) Using equations (3), (4) and (5), (2) can be written as,

1 =wx llkTn ef3If1'(l_ e-f3vD) (6) D L d ,..n po .

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IEEE/OSA/IAPR International Conference on Informatics, Electronics & Vision

x

Gate G

p-type

Substrate

Fig. I MOSFET geometry and corresponding X-Y direction

In Fig.l Y direction indicates the channel length and x direction indicates the axis for effective channel thickness. For finding the relationship between channel thickness and potential, first one dimensional Poisson's equation is solved at the surface charge region of drain.

(7)

The integrated result can be rearranged as

dlJl = 2q [(IJI-1JI )(n _ p )+eP'I'(npo _ ppo)_ npo eP'I" + Ppo e-fJ'I" ].

dx Ss s

po po f3 f3 f3 f3

(14) We Rearrange equation (14) and then integrate again

=4��(1+�

'I' )�� +eP'I'+k-��

(I+�'I")

�� +eP'I"+k=x.

(15)

Let, (16)

N D + and NA -are the densities of ionized donors and acceptors. So, equation (15) becomes,

Equilibrium surface densities are:

N; = npo' (8)

N� = Ppo' (9)

p = ppoe-fJ'I', (1 0) n = npoefJ'I'-fJVD . (11)

By Substituting equations (8),(9),( 1 0),( 11) in equation(7), it is found

(�+efJ'I' + k) 2ss f3

---'-------;;--f3q (l+efJ'I' f

(17)

This is the relationship between potential and channel thickness, which is exponential. For getting the slope of this curve, equation (13) is differentiated with respect tox,

dlJl dx

(18) :� = -: [npo {l_efJ('I'-VD)} -Ppo (1- e-fJ'I')]. (12) Atx=O,1JI =lJIs' s

Multiplying equation (8) by 2

(13)

After Integrating equation (13), it is found,

894

dVI I b(1 + eP"' )' fJq dx ,.0 Ii, [( 1+ eP'" )' (i + fJeP'" ) - 2 ( V� + eP'" + k } 1+ eP'" ) fJeP"-]

(19) Using this slope a straight line is constructed. The equation of straight line is:

dlJl 1JI-lJIs =d;(x-O),

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IEEE/OSA/IAPR International Conference on Informatics, Electronics & Vision

(20)

Xd x

Fig. 2 Calculation of channel depth

The channel thickness is up to the distance where potential becomes zero. So, putting IJI =0 in equation (20) yields,

-lJIs Xd = dlJl

dx

If/,S, [2(� + eP'" + k}1 + eP'" ) peP,., - (I +eP"' ), (i+ peP,., J] =>x =----=.---'-'--------'-------:,--.,.-----"-----'-=-d b(1 + eP,., )4 pq

(21 ) This is the new expression for effective channel thickness Xd'

It is shown in Fig. 2. Using new effective channel thickness in

equation (6), the new analytical expression for sub-threshold

current iD is found,

(22)

III. COMPARISON OF SUB-THRESHOLD CURRENT USING

DIFFERENT METHODS

Using equation (22) the sub-threshold current, iD vs. VD curve is plotted in Fig. 3.

895

-12 1.4'10

1.2

0.8

0.6

0.4

0.2

0.1

Subthreshold Current computed from equation(22)

0.2 0.3 0.4 0.5 V0lf6 0.7 0.8

Fig. 3 Sub-threshold current for proposed method

0.9

An alternative form of sub-threshold current is proposed in previous papers [1-2].

W (kTJ n2 [ qvD ] q'P, i D = J.l" L

q q ;; A

1- e kT e kT (23)

Another equation for sub-threshold current by calculating charge under source using charge-sheet approximation [4] is,

_ W 1 (-!3Vosl ID --J..I.n -(-QI source )(I-e ). L f3 .

Where Q = yCox e(If/,-21f1B-VDBl!3 I,sollrce 2f3fo; .

(24)

A comparison of current curves using equation (22), (23) and (24) is plotted in Fig. 4.

1 X 10-10

0.9 0.8 0.7 0.6

�05 0.4 0.3 02V 0.1

00 0.1

subthreshotd current comparison between different methods

0.2

I current using equation(24) I

0.3

I current using equation(23) I

I current using equation(22) I 0.4 0.5 0.6

V(V) 0.7 0.8 0.9

Fig.4 Comparison between different methods of sub-threshold current.

It appears that our proposed method produce the lowest sub­threshold current. In calculating sub-threshold current using

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IEEE/OSAIIAPR International Conference on Informatics, Electronics & Vision

equations (22), (23) and (24), the used values of different parameters are:

Ppo =1016cm-3 ni =1. 5 x1010cm-3 If/, =1BkT/q w/L =15

Since a larger depletion width has been considered, smaller sub-threshold current is found. The surface charge (Qs) in weak inversion is the sum of inversion charge (Ql) and depletion layer charge (QB)' QB increases with increased depletion layer area. As there is no change in gate voltage, Qs remains constant so Ql attains a smaller value. This results in reduced sub-threshold current.

IV. CONCLUSION

The main purpose of this work is to present a new technique for calculating sub-threshold current of MOSFETs. The previous models used some assumptions and predetermined values .The model proposed in this paper gives desired lower current and also in perfect agreement with device physics.

ACKNOWLEDGEMENT We are very much grateful to Dr. M.M. Shahidul Hasan,

Professor, Department of EEE, BUET, who has supervised our work. We are also indebted to Mr. AsifZaman, Lecturer, Department of EEE, BUET for providing valuable references. In this paper, both authors contributed equally.

896

REFERENCES

[1] D. J. Wouters, J. P. Collinge and H. E. Maes, "Subthreshold Slope in Thin Film SOl," in IEEE Transactions on Electronic Devices, vo1.37, no. 9, pp. 2022-2031, September 1990.

[2] S. M. Sze, Physics of Semiconductor Device, 2nd ed., New York: John Wiley & Sons, 1981, p. 440.

[3] G.W. Taylor, "Subthreshold Conduction in MOSFET's," in IEEE transactions on Electron Device, Vol. ED-25, No.3, p. 344, March 1978.

[4] Y. Tsividis, Operation and Modeling of MOS Transistor, 2nd ed. , New Delhi, India: Tata McGraw Hill, pp. 59-65, January 1999.

[5] L. Dunlop, "An Efficient MOSFET Current Model for Analog Circuit Simulation- Subthreshold to Strong Inversion," in IEEE Journal Of

Solid State Circuits, vo1.25, no.2, pp. 616-619, April 1990.

[6] E. Vittoz and J. Fellrath, "CMOS Analog Integrated Circuits Based On Weak Inversion Operation", in IEEE JOURNAL OF SOLID STATE CIRCUITS vol. sc-12, no3, pp. 224-231, June 1977.

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