2344 IEEE TRANSACTIONS ON ELECTRON DEVICES, DECEMBER 1976

Correspondence

Multiplexing and Contrast Ratio Optimization for Matrix Addressed Liquid Crystal Displays

P. F. ROBUSTO AND L. T. LIPTON

Abstract-Theoretical calculations have been made to allow maximizing the cpntrast and optimizing the multiplexing capabil 15y of rms responding liquid crystal displays. Exact equivalence of the results is shown, and equations for both the multiplexed voltage and the data voltage are written in terms of only the two variables: the OFF voltage (threshold voltage) and the number of multiplexed rows.

SUMMARY

In previous work on the multiplexing of a liquid crystal display derivations have been given both for optimizing the number of lines scanned [l] and for maximizing the contrast [2]. The present work is an expansion of the previous efforts which shows the generalization of those calculations, and then goes one step fur- ther by specifying the exact relation between the row and column voltage. Further, we show these voltages as functions of only two variables, the OFF voltage and number of multiplexed rows.

The matrix is addressed by multiplexing the N rows witE a voltage of magnitude Vs, and data information is presented '3y voltage signals of f VD. At the intersection of a row and column the voltage which appears across the liquid crystal element is given by the difference between the row and column signals. On the average, for a period T , an element will be ON if the voltaze during the time TIN is VS + Vo; similarly, an OFF element sees a potential difference of Vs - Vo. The rest of the time the voltage will be I VD I (see Fig. 1). In a fast-scan mode of operation, w r >> 1, where 7 is the characteristic response time of the liquid crysl.al and w is the smallest frequency component of the ac field.

Nematic liquid crystals have a characteristic time-integrati:sg response, and consequently the rms voltage must be investigahd during the entire scan time T. With the addressing scheme de- fined above, the rms voltages for an ON and OFF element are giwn by

V,,ff= [ ( v s - VD)'D + v D 2 ( 1 - D)]1'2

Van = [(Vs + V D ) ~ D + Vo2(1 - D)I1/' (1)

where D = 1/N = duty factor. The contrast ratio for the display is defined as C = R( Vor)/

R( Voff), where R ( V) is the optical response a t voltage V. It has been shown [a] that by maximizing the ratio Von/Voff, that is, setting d/dV~[(V,,)/(V,ff)] = 0, the contrast will be optimuln. This leads to the result

Vs = N 'I2 VD ( ;!)

which reduces to the result obtained in [2] when V, = 1. Tt rs shows that for optimized contrast the ratio of VS/VD is depe 1- dent only on N, the number of lines multiplexed.

Alt and Plesko [ l ] have shown that the number of scanned rows is maximized by solving (1) for N and setting dN/dVD = 0. The result is that N has a maximum when

v, = 2 (Von2 + V,ff2)1/'. 1

(3)

Manuscript received October 1, 1975; revised August 12, 1976. The authors are with Industrial Products Division, Hughes Aircrz ft

Company, Carlsbad, CA 92008.

Fig. 1.

N

vs-vo VD u 'OFF' CONDlTlON

O V 0 T,N T

v,+ v,

ov rI 'ON' CONDITION

-", 0 T/N

Typical voltage waveforms for an ON and OFF element.

Fig. 2. Plot of number of rows multiplexed N versus both Vs/Vt, and v D / v t h .

Upon substituting (1) into (3), (2) is again obtained. Conse- quently, (2) can be used as a general result which maximizes both the contrast ratio and the number of scanned lines. Substituting Vs = N112V~ into the V,ff equation in (1) and letting V,ff = Vthreshold, the following simple results are obtained:

and

vs = Nli2VD.

Thus the values of VD and VS are expressed only as functions of the liquid crystal threshold voltage and the number of multi- plexed lines N. Plots of VD/Vth and VS/Vth are shown in Fig. 2. I t should be noted that as N - m, VD = Vth /4 . Thus t he V D level changes little as N increases and is never lower than V t h l d .

CORRESPONDENCE 1345

REFERENCES TABLE I

[I] Paul M. Alt and Peter Plesko, “Scanning limitations of liquid crystal displays,” IEEE Trans. Electron Deuices, vol. ED-21, pp. 146-155, Feb. 1974.

[2] J. E. Bigelow, R. A. Kashnow, and C. R. Stein, “Contrast optimization in matrix-addressed liquid crystal displays,” IEEE Trans. Electron Deuices, vol. ED-22, pp. 22-24, Jan. 1975.

Modeling of Emitter-Base Bulk and Peripheral Space- Charge-Layer Recombination Currents in Bipolar

Transistors

N. G. CHAMBERLAIN AND D. J. ROULSTON

Abstract-The bulk and surface recombination currents of the E-B space-charge layer are related through a factor X F S having the dimensions of length. The factor XFS can be determined ex- perimentally and is an essential parameter for modeling ~ F E fall-off at low collector currents.

I. INTRODUCTION

The separation of emitter-base bulk and surface recombina- tion currents and the effect on ~ F E fall-off at low collector cur- rents has been studied by several workers [1]-[5]. I t is shown here that the two components of base current (bulk and surface) can be related through a factor X F S which is constant for a given impurity profile and fabrication process. This factor X F S is de- termined experimentally and can be incorporated in existing models to predict ~ F E fall-off a t low current.

11. THEORY

If the carrier concentration p ( x ) and n ( x ) obey the Boltzmann relations in the E-B space-charge layer, thenp(x)n(x) = n% exp (YVEBIKT). I t can be shown [6] that for VEB > 50 mV, and for the abrupt junction approximation with a constant “average” electric field E = Em,,/2, the carrier recombination rate [ l ] for a single-level trap located at the intrinsic Fermi level gives rise to a space-charge layer current density given by

J S C L = Y ( ~ I ~ ) ( K T I Y E ~ , , ) ~ / ~ ~ , ( O ) ~ ~ ( O ) / . / ~ , O ~ , O (1) and to a recombination current in the bulk given by

ISCL = J s c L ( ~ L ) ( 2 ) where b is the total emitter stripe length and L is the emitter stripe width. The values of the minority-carrier lifetimes T,O and ~~0 are assumed to be constant and independent of the electric field. For a linearly graded p-n junction, a “constant” average electric field E can be taken as E = { vbi + V,) /W, and therefore, E,,, = (3/2) E, vbi and V , are the built-in and applied voltage, respectively; W is the space-charge layer width.

On the assumption that the same VEB is applied over the entire space-charge-layer region and further assuming the same values of T,O and ~~0 through this region, the recombination current in the peripheral region with a length approximately equal to (x /2)Xj1 is given by

Iperiph = JSCLB(b + L)(T/2)Xj lFp (3) where JSCL is defined in (1) and ( x /2 )X j l is the length of the

was supported by the National Research Council.

of Waterloo, Waterloo, Ont., N2L 3G1, Canada.

Manuscript received May 14, 1976; revised July 20, 1976. This work

The authors are with Department of Electrical Engineering, University

NO. 1 NO. 2 Average

IC mA h~~ h~~

cylindrical space-charge layer with radius Xj,. Fp is a factor which takes into account variations in JSCL, due to variations in doping level and thus built-in electric field and SCL width, from the bulk towards the semiconductor surface. Thle value of Fp was computed by numerical solution of Poisson’s equation in polar coordinates neglecting the free carriers, in the peripheral region, using a method similar to that used in [7]. For transistors with junction depths from 0.4 to 8 ym, the values of Fp were found to lie between 0.3 and 0.45.

The surface recombination current I , at the periphery of the SCL can be related to JSCL by

I , = 2(b + LIXFSJSCL (4) where 2(b + L ) is the periphery of the SCL and J S C L is defined in (1). Although X F S in effect lumps together the phenomena occuring at the surface, its relation to the real physical parameters is, a complex one. If for example, the enhanced surface density of recombination centers C, are located within a thin layer X,, near the surface in the SCL, the relation to the density of re- combination centers in the bulk C b will be given by x&$~ = XF&b where XFs will be much smaller than XFS. In physical terms the length XFS represents an effective equivalent extension of the E-B space-charge layer with density of recombination centers the same as that of the bulk. The total current due to carrier recombination in the whole of the E-B space-charge layer is then given by ( 2 ) , ( 3 ) , and (4) as

where ISCL is defined in (2 ) .

mainly due to carrier recombination in the SCL we obtain At low levels of collector current where the base current IB is

-- l C A - ITSCL

where I , is the collector current and ~ F E A Z,/IB. (The collec- tor-base reverse leakage current is assumed negligibly small.) For two transistors which are fabricated on the same chip and which differ only in their widths L , (6) can be used to determine the value of XFS. If the collector-current densities of both transistors are the same (Le., same VEB) , (5) and (6) give, for b >> L,

~ F E ~ ~ F E Z = [1 + ( 2 / L z ) ( ( ~ / 2 ) X j l F p + X F S ) ] /[I + (2/L1) ( ~ / ~ ) X J I J $ + X F S ) ] ( 7 )

and

XFS [(I - hFEZlhFE1)/(12/LZ)(hFEZIhFEl) -- 2/Ll )] - x j1Fp(x /2) . (8)

Measurements of h ~ ~ l and h ~ ~ 2 at the same collector-current density enable the factor X F S to be determined experimental- ly.

111. EXPERIMENTAL RESULTS

Table I shows the experimental values of X F S for devices fabricated on the same silicon chip for the purpose of determining XFS. A standard silicon planar technology was used with a final dry-nitrogen oxidation treatment at llOOo C for 12 min. For