pspice electronic model of a ferroelectric liquid crystal cell

PSpice electronic model of a ferroelectric liquid crystal cell

J.R.Moore and A.R.L.Travis

Abstract: A model that predicts how the optical transmission and charge acceptance of a ferroelectric liquid crystal cell vary over time with an arbitrary drive voltage is presented. The model is written using PSpice, such that it can be directly integrated with the normal PSpice components used to simulate drive circuitry. The model agrees to within 10% of experimental results, over an applied electric field of 1 to 20 V/pm. It only requires simple measurements of a sample cell to provide the model parameters. The applications are in optimising shutter design, switching and decay of ferroelectric liquid-crystal on silicon, and effects of drive line and component impedances. With additions to the model, it is expected to be able to simulate Joers/ Alvey passive multiplexing.

1 Introduction

Ferroelectric liquid crystals switch in less than 10 micro- seconds with contrast ratios of less than 0.01 over a wide range of ray angles and wavelengths, so ferroelectrics make excellent shutters and high bandwidth microdisplays. The speed of ferroelectrics, combined with the switching charge taken by the high electric polarisation needed to get that speed, result in a strong interaction with the electronic drive components which can lead to unexpectedly slow switching or poor contrast ratios. Ferroelectric liquid crystals are nonlinear, so their performance is usually predicted by a computer simulation [l-31 such as might be written in C+ +, Matlab, Mathematica or such like, but the great majority of transistor-based components are now defined by libraries written in Spice (or PSpice, its PC manifestation. Spice and PSpice are registered trade marks of MicroSim Corporation) It makes sense to specify the behaviour of ferroelectric liquid crystals also using Spice so that they can be added to these libraries, even though the original electronic parameters used by Spice need to be stretched into the optical domain to do so.

This paper presents the physics and the equations that the simulation program needs to reproduce, describes the modelling of a unit cell and the parameter measurements required, and gives comparisons of the simulated and measured response of ferroelectric liquid crystal test cells.

2 Liquid crystal physics

The liquid crystals in ferroelectric shutters are rod-like molecules which tumble and move, but tend to dwell in a particular direction called the molecular director. Groups of molecules dwell in layers which, within a conventional

0 IEE, 1999

IEE Proceedings online no. 19990841 Dol: 10.1049/ip-opt:19990841 Paper first received 10th May and in revised form 1 1 th October 1999 J.R. Moore is with JMEC Ltd., Cambridge CB3 7AZ, UK, A.R.L. Travis is with the Engineering Department, University of Cambridge, Cambridge CB2 lPZ, UK

IEE Proc.-Optoelectron., Vol. 146, No. 5, October 1999

cell, are approximately perpendicular to the glass walls of the cell, and the molecules' directors adopt an angle to the layer so that the set of positions which each molecule is free to adopt constitutes a cone whose axis is orthogonal to the layer. If each molecule has a permanent transverse dipole, the molecules within a layer combine to give a spontaneous polarisation Ps, and if the liquid crystal molecules inside a thin cell are made to align, the result, as demonstrated by Clark and Lagerwall, has macroscopic polarisation.

If an electric field 'E is placed across such a cell, according to a simple model [4] the rate of change of 4, the angle of the director round the cone, is limited by the rotational viscosity y and a restoring torque K ( 4 ) to:

(1) d 4 y - = P,E cos 4 + K ( 4 ) dt

Ps, y, IC(+), and 4 all vary with the material, the cell separation, and the temperature.

Layers are usually bent into a chevron structure, as shown in Fig. 1. When projected onto the cell surface, two quasistable states of the molecule director result (Fig. 2 ) . If 6 is the angle at which the layer is tilted due to the chevron structure, the rate of change of 4 changes to:

(2 ) d 4 ~ - = P S E C O S ~ C O S ~ + K ( ~ ) dt

Fig. 1 Alignment of directors is continuous through chevron surface

23 1

tilt angle 6 director angle y, on the plane of cell

VZ

Fig. 2 alignment surface intersects tilted cone

Chevron structure bistable director positions are where plane of

A "3

Applying a DC electric field in the appropriate direction rotates the director further round the cone and $, the projection of the director onto the plane of the cell, can be pulled out towards the maximum of the cone angle, although the director will return to the chevron stable position once the field is removed. By inspection of Fig. 2 , trigonometry shows that $ > $s is given by:

A v 5

(3) sin e sin 4

tan$ = cos cos 6 + sin 0 cos 4 sin 6

The material exhibits a birefringence A,; the thickness of the cell d is chosen so that at the operating wavelength /z the molecules combine to act as a half-wave retarder, or pi- cell:

d = A/2A, (4)

If linearly polarised light of any wavelength is shone parallel to the director, the light will pass through unchanged and can be completely extinguished by an analysing polariser at right angles to the plane of polarisation of the light. However, if the director is rotated by an angle $, the incident light will be resolved into two components: one parallel to the director which passes through the cell unchanged, and one orthogonal to the director which is delayed by 180" (equivalent to being made negative).

The transmitted components sum to a linearly polarised output, which is equivalent to the incident light rotated through some angle. Ideally, the ferroelectric material will have a cone angle (3 of 22.5" so that, when the cell is switched, the director moves through 45", and the effect of the cell is equivalent to rotating the incident light through 90". In practice, the director will move through an angle $, the cell will have the effect of rotating the plane of

polarisation of incident light through an angle 2 I,!,, and the transmission T through the cell and analyser will be:

T = sin2 2$ (5)

3 Model

A ferroelectric cell behaves in an electronic circuit as a capacitor with some leakage or series resistance, in parallel with a variable current sink, which draws current during the rotation of the ferroelectric director. The current drawn in a unit area cell equals the rate of change of the resolved part of the spontaneous polarisation in the plane of the cell, which itself is determined by the director rotation angle, so the basic Spice model is given by Fig. 3.

In advanced versions of Spice the equations for I , c)), and T, can be programmed directly, but the authors intended that their programme should be compatible with older and student versions of Spice, which do not explicitly permit integration and differentiation. Instead, integration was simulated by passing a controlled current into a perfect capacitor, as shown in Fig. 4 which is functionally ide~ntical to that of Fig. 3.

The voltage V, on the integrating capacitor Cps iepre- sents the angle of the director around the cone. If this is between the values equivalent to the bistable positions, it can be interpreted as indicating the ratio of areas switched in one direction or the other. A voltage of 0 volts indicates half the area is switched one way, half the other, and similarly for other proportions between the stable po'sition voltages. If the angle is greater than the bistable po'sition angle, it represents the observed average director angle: the complete area of the cell is switched uniformly to the director angle indicated. A further current-controlled1 stage can be added to calculate the light transmission froin V,. The restoring torque K(4) , which twists the director towards the nearest bistable angle if the director ex'ceeds

I I I I I

Fig. 3 PSpice model Director rotation angle Q, = s((V,/(12Ps cos 9 cos 6 +K(b))/y dt Optical power transmission T= sin Z(tan-' (sin@ sinQ/(cos 6 cos S + sin0 cos 9 sin 6)))

I, = P, (vg-V4) C O W , ) c o w Ik'K(V4) I" = Ps V3 I Y d R, in

Rs Ips=Vp cos (v4) cos (6) c,, = 1

Fig. 4 'd' = cell thickness '6' =tilt angle

232

PSpice model with rotation angle 4 simulated byV4


._ c 0 ' -

-5.0 -

rotation angle + deg

Fig. 5 at = f 25"

Restoring torque K (normalised to I ut 90") with stuhle positions

either bistable angle, is nonlinear so is simulated with a look-up table, which typically is of the form of Fig. 5.

4 Measurement of model parameters

The ferroelectric cell parameters required for the model of Fig. 4 are the cell area in square centimetres, the cell spacing in microns, the dielectric capacitance in nF/cm2, the spontaneous polarisation in nC/cm2, the viscosity in mPa.s (cP), the director cone angle, the director bistable angle from the rubbing direction (from which the tilt angle can be calculated), and the restoring force on the director as it twists outwards from the bistable position.

The cell spontaneous polarisation and rotational viscosity vary with cell spacing for a given material [5], so data sheet values, while useful as indicators to the real values, cannot be used. Surface effects can appreciably extend into the cell thickness [6], and the strength of alignment also varies with the cell processing [7], so restoring torque

treatment. Direct measurements of the cell to be modelled have to be taken, and the values for the parameters calculated from these.

values will also vary with cell separation and rubbing layer

4.7 Measurement of spontaneous polarisation If a symmetrical triangular voltage drive [SI is applied to the cell, the spontaneous polarisation charge can be calculated by integrating the current flow into the cell during one-half of the drive cycle.

A waveform generator imposes a voltage onto the test cell, and the current passing into the cell is measured as a voltage across a small current sense resistor.

Three measurements at different triangular wave repeti- tion rates were recorded for two test cells at three different periods (Fig. 6 shows a typical trace). The traces were photographed from the screen of an oscilloscope, which had calibrated voltage and time measurement cursors, and show several phenomena. The voltage pedestal is due to the dielectric capacitance of the cell: the capacitance differentiates the rising input voltage to absorb a constant current proportional to the rate of rise of the voltage and the cell dielectric capacitance. The step is not flat-topped, but has a voltage tilt due to cell leakage current.

Both cells at all drive waveforms showed a second peak in the current pulse. It was originally thought to be due to coupling of the rotation of the director to the bulk liquid crystal material called backflow [9], but is now believed to be due to the thickness and strong alignment of the surface

IEE Pjoc -Optoelectron, Vol 146, No 5, October I999

Fig. 6 Upper trace shows k I O volt triangular drive, lower trace shows input current to test cell, with zero current as horizontal line; pulse height is approximately 24pA; note slant on pulse due to leakage current, and surface layer second switching pulse; to compute switched charge, curve of response is traced out and line added below along slant of current leakage response; filled area (in white) is then calculated in pixels by the image processing program

Typical test cell, 39.8 ms slope time

layers of the ferroelectric material to the polyimide rubbing layer and at the chevron interface [6, 7, lo].

The charge transferred during the switching of the director can be measured from the area between the current curve and the basic slope of the resistive response to the drive voltage. This is not easy to measure using readily available instruments, so the oscilloscope traces were photographed, scanned into a computer, and the pixels between the curve and the trace slope counted using an image processing program [ 1 11.

The switching charge Q is proportional to the number of pixels, and the spontaneous polarisation is equal to Q divided by twice the area of the cell. Any slope on the dielectric current pedestal is attributable to ion or resistive leakage in the cell liquid crystal. The value of the cell dielectric capacitance C,, can be calculated from the height in pixels of the vertical pedestal at the triangular voltage peaks. The values of the spontaneous polarisation and dielectric capacitance for the liquid crystal cells at 18°C from the triangular drive were: (i) cell 1 spontaneous polarisation (ii) cell 1 capacitance (iii) cell 2 spontaneous polarisation (iv) cell 2 capacitance Measurements were repeatable to f 5 % and accurate to f 15%.

10.5 nC/cm2 3.6 nF/cm2 6.33 nC/cm2 4.9 nF/cm2

4.2 Measurement of restoring torque K(4) The director angle $ was found by applying a symmetrical square wave voltage across the cell, and recording the positions of crossed polarisers on either side of the cell for maximum opacity for the positive and negative voltage levels. Fig. 7 plots twice the projection of the director onto the cell plane 2 $ against peak drive voltage for cell 1, and shows that the stable chevron angle $s is given by 2 $s = 16".

What is required for the model is 4, the rotation position of the director around the cone, and the restoring torque. Trigonometry on Fig. 2 shows that:

233

*O r ?

I I 0 40 50

rotation angle y ~ , deg

Fig. 7 cell plane -*- data -.- fit

Cell 1: applied cell voltage against projected director angle in

but neither cone angle, 0 nor tilt angle, 6 are easily measurable. However trigonometry on Fig. 2 also shows that the chevron angles I)$ are given by:

cos 0 cos 6

cos = ~ (7)

and the chevron angles are measurable. Various combina- tions of cone and tilt angle can be calculated to find the correct bistable cone angles, shown in Fig. 8 for cell 1

Using the data from Fig. 7, a voltage K (4) proportional to the restoring torque where IC ( 4 ) = K (q5)d/Ps can be plotted for differing values of 0 (Fig. 9). The restoring torque must have a finite value as 4 approaches 90" (i.e. not increase without limit, or reduce to zero). Fig. 9 may be used to estimate the cone angle for the cell by selecting the value of 0, which appears to show a finite value for the torque as the rotation angle approaches 90". Values of 22.25" for old cell 1, and 22.16' for cell 2 are indicated with respective chevron tilt angles 6 of approximately 20.8" and 2 1.5".

The model assumes a symmetric chevron with the same restoring forces in either direction. An asymmetric or monostable chevron could be measured and simulated using the same techniques, by plotting the director angle for separate positive and negative voltage excursions. The maximum positive and negative excursions would allow the position of the asymmetric stable positions to be inferred.

= f S o ) and cell 2 (I)s = f 5.5")

4.3 Measurement of rotational viscosity Measurements described so far provided values of spontaneous polarisation for the two test cells, and a restoring

Cell 1 cone angle (0) 21.7" 22.0" 22.25" 22.50"

Tilt angle (6) 20.3" 20.6" 20.8" 21.1"

Cell 2 cone angle (0) 21.20" 21 50" 21.67" 22.16"

Tilt angle (8) 20.5" 20.8" 21.0" 21.5"

Fig. 8 (Ys = i P) and cell 2 (Y5 = f 5.54,

234

Tilt angle for various cone angles, calculated for cell I

3

1 100

rotation angle around cone, deg Fig. 9 cone angle H -4- 21.75 deg -.- 22.0 deg -A- 22.25 deg - x - 22.5 deg

Cell I ; restoring torque K plotted against 4 for diffrrent values of

torque curve for the forces apparently acting on the macroscopic liquid crystal director. There is now enough information to simulate the cell performance with various values chosen for the rotational viscosity, and to check the computed input current shape and position of the model with the measurements.

A look-up table was used to mimic the shape of the restoring torque, against cone angle, which for cell 1 was the curve of Fig. 9 with 8 = 22.25".

5 Results

A good test (and the original need for the simulation) is to reproduce the light transmission switching curve with a square wave drive. This was simulated for a range of va.lues for the rotational viscosity, and the simulated waveforms compared with the actual photographic trace. Figs. 10-15 compare the photographed cell switching with the results of simulations for cell 2, with a best-fit value of 65 mPa.s for the rotational viscosity.

Figs. 10 and 11 show the drive voltage and light transmission curve for f 19 volt drive. The simulated light curve shape and delay and fall time is within 10% of the original, and the shape of the real light curve is well reproduced. The simulated peak light transmission is 129% of the theoretical maximum transmission through par.alle1 polarisers.

The start of the light curve is slow as the director has rotated to the point where the voltage providing the torque is decreased by the cos 4 term, but there is not a long delay since the chevron restoring torque is aiding the rotation.

Figs. 12 and 13 show the drive voltage for f 2 . 3 5 volts drive. The director is no longer switching through the full cone angle, so the light transmission is reduced to 83% of the theoretical maximum (the simulated polarisers have been rotated for minimum light transmission in the off state). When the drive voltage is reversed to close the shutter, the director is already pointing at an angle where a large torque is applied by the electric field in the same direction as the chevron restoring torque, so the light curve has a sharp start to it, which is correctly simulated in the model, and is an important indication of its accuracy. As


Simulation with 65 mPa.s viscosity and 6.4 nC/cmZ polarisation Fig. 13 Decreased light transmission

Cell 2: f 2 . 3 5 volts square wave drive

the director rotates to its final Dosition. the chevron restor-

Fig. 14 Cell 2: f 0.6 volts square wuve drive

Fig. 12 Sharp start to light curve decay


Cell 2: f 2 . 3 5 volts square wuve drive Fig. 15 Long ‘tail’ on photographed light curve

Cell 2: f 0.6 volts square wave drive

235

curve look-up table approaches zero with a finite slope rather than asymptotically, so the tail is not simulated. Because of this, the long settling time is not reproduced, and the switching speed is too fast. This might be improved by generating the restoring torque from a continuous function rather than from straight line sections, but PSpice does not easily permit this with the necessary ‘dead band’ between the bistable positions. For the purposes of the drive circuit for which this simulation was developed, this degree of accuracy at low drive voltages was unnecessary, and if the simulation had used a straight line fit similar to Fig. 6, it would have been almost as accurate.

The most critical simulation for the model would be for triangular voltage drive, since the polarisation current is changing at its fastest with the lowest applied field. From the results of Figs. 14 and 15 above, it would be expected that this would not be imitated well. Fig. 16 shows this simulation, scaled and superimposed on a real trace, for the test cell 1 for the range of rotational viscosity from 35 to 150 mPa.s. None of the simulated curves fits the actual curve well, and the one that is reasonably close (for 50

Fig. 16 Simirluted current waveform for =t 10 volt 1 4 . 6 ~ 1 ~ triangular wave Thick trace is photographed trace, thin traces are simulated currents for rotational viscosity of 35, 50, 100 and 150 mPa.s; closest fit is for approximately 50 mPa.s viscosity

*New cell, * 19 Volt Square wave PWL restoring torque * AC COUPLING .PARAM VSC =65 ; Viscosity in CP ( mPa.S) .PARAM D = 1.5 ; Cell separation in microns .PARAM PS =6.4 ; Spontaneous polarisation in nC/sq cm .PARAM CAP =4.9 ; Cell capacitance in nF/sq cm .PARAM SIZE = 10 ; Cell area in sq cms .PARAM INPOLAR = -24 .PARAM CONE =21.5 ; Cone angle in degrees .PARAM TILT = 18.6 ; Chevron Tilt angle in degrees .PARAM LEAK = 1000 .PARAM PO .PARAM SC ={SIN (3.14159*CONE/180) } .PARAM CC ={COS (3.14159*CONE/180) } .PARAM CT = {COS (3.14159*TILT/180) } .PARAM ST ={SIN (3.14159*TILT/180) ] .PARAM CN = {CC*CT} .PARAM SN = { SC*ST}

; Input polariser orientation in degrees

; CELL LEAKAGE RESISTANCE IN MEGOHMS/SQ CM = { 3.14159*INPOLAR/180 }

VIN CCOUP 1 2 1UF ; AC COUPLING CAPACITOR RDC 2 0 IMEG ; DC RESTORING RESISTOR RIN 2 3 100 ; CELL SERIES RESISTANCE CIN

R1 3 0 {LEAK*IMEG/SIZE} ; CELL LEAKAGE RESISTANCE R2 4 0 1 ; TORQUE SENSING RESISTOR R3 5 0 1G ; DC CONTINUITY RESISTOR R4 7 0 1G ; DC CONTINUITY RESISTOR

1 0 PULSE -19 19 1999u l u lu 1999u 4m ; INPUT SQUARE WAVE + - 19 V P-P ~

3 0 {CAP*SIZE*lN } IC = - 19 ; CELL DIELECTRIC CAPACITANCE WITH INITIAL VOLTAGE

GR 0 5 VALUE= { V(4,0)*1E-5*PS/(VSC*D) } ; VISCOSITY SETTING CURRENT SOURCE

COUT ROUT 5 6 1 ; CAPACITOR CURRENT SENSE

6 0 IN IC= -1 ; INTEGRATION CAPACITOR INITIAL SETTING 1 RADIAN

GPS 0 4 VALUE = {COS (V(6,0))*V(3,0)*CT

0 4 TABLE {V(5,0)} =(-1.57,5.2)(-1.41,4.5)(- 1.22,3.3)(- 1.05,2.4) ; SPONTANEOUS POLARISATION TORQUE

GK + (- .79,1.3)( - .58,.6)( - .34,. 1)( - .26,0)(.26,0)(.34, - . I ) + (.58,-.6)(.79,-1.3)(1.05,-2.4)(1.22,-3.3)(1.41,-4.5) + (1.57,-5.2) ; STATIC RESTORING TORQUE

GIN 3 0 VALUE = {COS(V(6,O))*V(5,6)*PS*CT*SlZE) ; FERROELECTRIC INPUT CURRENT

EOUT 7 0 VALUE={ SIN(2*(PO-ATAN( SIN(V( 6,O)) * SC/( CN+ SN*COS(V(6,0)))))) *SIN(2*(PO-ATAN(SIN(V(6,0))*SC/(CN+ SN*COS(V(6,0))))))}

; LIGHT TRANSMISSION THROUGH POLARISES .STEP PARAM TNPOLAR LIST -24 -21 - 18 - 15 .TRAN 10N 30m ON 1Ou UIC .PROBE .END

; STEP POLARISER ROTATION ; MTN TIME STEP l0NS MAX I O US

Fig. 17 Pspice programme 236 IEE Proc.-Oploelectron., Vol. 146, No. 5, Octob,zr 1999

mPa.s) is actually about half that of the best fit for high voltage square wave switching for cell 1. Almost the entire polarisation current flows as the applied voltage goes between - 1 and f 2 Vlmicron.

The second current pulse due to the surface layer switching is not simulated at all, but the effect this has on switching times at normal drive voltages is small in this case and can be ignored.

6 Conclusions

A Spice model for ferroelectric liquid crystal cells has been found to predict the response of cells to voltages ranging from 1 .O to 19 volts/micron with an accuracy of 10%. The model is less accurate at lower voltages and, as a conse- quence, gives poor prediction of response to triangular voltage drives. The model requires values for spontaneous polarisation, rotational viscosity, and restoring torque against cone rotation, each of which has to be measured because each varies with cell construction and temperature.

We have shown how spontaneous polarisation can be measured using computer integration of the switching current, how torque against cone rotation can be extra- polated from measurements of cell transmission against drive voltage, and how these measurements can be combined with switching times to infer rotational viscosity. It is anticipated that addition of the biaxial dielectric components of the voltage torque would allow simulation of JOERS-Alvey passive multiplexing, and breaking each pixel into a subpixel matrix to simulate the domains would allow simulation of spatial grey scale. Addition of an extra term for the alignment layer switching could also be incorporated.

7 Acknowledgment

John Moore acknowledges the help of A. B. Davey.

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References

GIESSELMANN, E, and ZUGENMAIER, P.: ‘Computer simulation of the electro-optic switching process in ferroelectric liquid crystal cells with bookshelf geometry’, Liquid Crystals, 1993, 14, (2) , pp. 389404 OTON, J.M., PENA, J.M.S., and SABATER, J.: ‘Modelling and char- acterisation of ferroelectric liquid crystal displays’, Opt. Quunt. Electr:, 1994, 26, pp. 571-581 MARREC, S., KILLWGER, M., and CAMBON, P.: ‘A simple electrical model of a ferroelectric liquid crystal cell’, Ferroelectrics, 1993, 149, pp. 305-318 DAHL, I., LAGERWALL, S.T., and SKARP, K.: ‘Simple model for the polarisation reversal current in a ferroelectric liquid crystal’, Phys. Rev. A, 1987,36, (9), pp. 43804389 GOODBY, J.W.: ‘Ferroelectric liquid crystals: principles, properties and applications’ Gordon and Breach, Philadelphia, 1991, pp. 190-194 MYRVOLD, B.O., KONDO, K., and OHARA, S.: ‘The thickness of the surface layer in polymer-aligned liquid crystal cells’, 1 SID, 1994,2, (3), pp. 129-134 KIMURA, M., NISHIKAWA, M., AKAHANE, T., and KOBAYASHI, S.: ‘Dynamic response of surface-stabilised ferroelectric liquid crystal: effect of alignment films’, Jup. 1 Appl. Phys., 1994, 33, (2 NO 7A),

MIYASATO, K., ABE, S., TAKEZOE, H., FUKUDA, A., and KUZE, E.: ‘Direct method with triangular waves for measuring spontaneous polarization in ferroelectric liquid crystals’, Jap. 1 Appl. Phys., 1983,22, pp. L661 SKARP, K.: ‘Rotational viscosities in ferroelectric smectic liquid crystals’, Ferroelectrics, 1988, 84, pp. 119-142 MIURA, S., KTMURA, M., and AKAHANE, T.: ‘Layer structure reformation of surface stabilised ferroelectric liquid crystal treated with electric field’, Jap. J Appl. Phys., 1994, 33, (1 IA), pp. 209-213 MOORE, J.R.: ‘Three dimensional television using liquid crystal shutters. PhD Thesis Cambridge University Engineering Department, 1996

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IEE Proc.-Optoelectron., Vol. 146, No. 5, October 1999 237

pspice electronic model of a ferroelectric liquid crystal cell

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