39.2 / F. Castles

39.2: Fast-Switching Flexoelectric Display Device with High Contrast Flynn Castles, Stephen M. Morris, and Harry J. Coles University of Cambridge, Cambridge, United Kingdom

Abstract The flexoelectro-optic effect provides a fast switching mechanism (τ90-10 ~ 10-100 μs) suitable for use in field-sequential-color, full motion video displays. An in-plane field is applied to a short-pitch chiral nematic liquid crystal aligned in the Uniform Standing Helix (or Grandjean) texture. The display performance is presented as a function of device parameters.



Liquid crystals (LCs) are useful in a wide range of display devices because their optical properties may be manipulated by an applied electric field. The speed of the response of the LC to the electric field will be governed by a number of factors, including the viscosity, the elastic restoring forces, and the specific orientation of the LC molecules. Improving the response time of Liquid Crystal Displays (LCDs) is a primary technical challenge in their development. A faster switching speed will lead to superior quality of moving images, and a suitably fast switch will enable the use of Field Sequential Color (FSC) generation. FSC has a number of associated benefits, such as increased resolution, greater brightness, and a potential reduction in cost, due to the fact that a color filter is no longer required. The implementation of FSC has been hindered by the restrictively slow response times of conventional display modes. The flexoelectro-optic effect is a naturally fast switching mechanism seen in chiral nematic (N*) LCs, whereby the optic axis is rotated by an electric field applied perpendicular to it [1]. Response times are typically of the order of 10-100 μs, which is 2-3 orders of magnitude faster than conventional display modes. It was first discovered in, and is usually investigated in, the Uniform Lying Helix (ULH) configuration, in which the helical axis of the N* lies in the plane of the device. The drawback to this configuration is that the ULH structure must be formed under the application of an electric field, and is inherently unstable. An alternative device geometry is generated when the N* is aligned in the Uniform Standing Helix (or ‘Grandjean’) configuration, in which the helical axis of the N* stands normal to the plane of the device. The rotation of the optic axis is now created by an inplane electric field. Between crossed polarizers an electrically driven optical switch is produced. This was first proposed by Broughton et al. in the context of a polarization controller at optical communications wavelengths [2,3]. We investigate this device in the context of a fast-switching display without motion blur. We will call this the ‘Uniform Standing Helix Flexoelectric’, or ‘USHF’ display mode. In addition to the fast response in this mode, the LC structure is stable using existing, robust, alignment techniques. Further, because the short-pitch N* is optically isotropic at normal incidence, an excellent dark state is achievable, leading to a high contrast ratio (typically ≈ 2000:1). We use well-established theoretical methods to investigate primary aspects of the device performance as a function of material and device parameters. Specifically, we investigate the switching speed, the quality of the dark state, the electro-optic curves, the driving voltage, and the isocontrast curves with and without a compensation film. We use existing, experimentally

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determined, material parameters to generate realistic predictions of these device characteristics. The structure of the device is shown in Fig. 1. The N* is aligned in the USH configuration using anti-planar alignment on the glass surfaces. With no field applied, the helical structure is undistorted, and the device is non-transmissive between crossed polarizers. When an in-plane electric field is applied, the optic axis of the N* rotates, and a birefringence is induced. For a sufficient tilt angle (which depends on the birefringence of the LC) the device becomes transmissive.

Figure 1. The chiral nematic liquid crystal is aligned in the Uniform Standing Helix configuration. An in-plane electric field is applied, in the same direction as the surface alignment, which rotates the optic axis via the flexoelectrooptic effect. Crossed polarizers are positioned at ±45° to the direction of the electric field.


The Flexoelectro-optic Effect

Conventionally, the switching mechanisms in LCDs exploit dielectric coupling, where, due to the dielectric anisotropy Δε, the LC molecules will tend to align in the applied electric field. Flexoelectro-optic switching may be considered as a separate switching mechanism to dielectric coupling, in that it can be seen in LCs even if Δε = 0. The flexoelectro-optic effect in N* LCs is a result of flexoelectricity [4], which is caused by a linear coupling between the distortion of the LC and the applied electric field. It is analogous to piezoelectricity in normal, solid, crystals. The strength of the flexoelectric response is characterized by the splay and bend flexoelectric coefficients es and eb, which appear in the free energy expression as an additional, flexoelectric, term

f flexo = −E ⋅ [esn(∇ ⋅ n ) + eb n × ∇ × n ] where E is the applied electric field, and n is the director. In a short-pitch N* under the application of an electric field perpendicular to the helical axis, the flexoelectric effect manifests itself as a fast rotation of the optic axis. This is known as the flexoelectro-optic effect. The average of the flexoelectric coefficients ē = (es+eb)/2 dictates the strength of the flexoelectrooptic response. Whilst all LCs are, to some degree, flexoelectric, the effect is usually small and the response of the LC is dominated by the dielectric coupling in most circumstances. However, under certain circumstances the flexoelectric effect may dominate, particularly if the dielectric anisotropy is small Δε→0, and ē is large. “Bimesogenic” LCs have been developed specifically to have

ISSN/009-0966X/09/3902-0582-$1.00 © 2009 SID

39.2 / F. Castles large average flexoelectric coefficients ē~10 pC m-1 and very low dielectric anisotropy Δε ≈ 0 [5-7]. If Δε is sufficiently low, there will be no dielectric unwinding of the N* helical structure.


Response Time

The natural response time of the flexoelectro-optic effect τ can be shown to be given by the expression

τ 90−10 ≈

γP 2 , 2π 2 K

where P is the pitch of the N*, γ is an effective viscosity, and K is the average of the splay and bend elastic constants of the LC [8]. Typical, experimentally determined, material parameters for pure bimesogenic LCs are shown in Table I. Based on these, the predicted response time τ90-10 of the device as a function of the N* pitch is plotted in Fig. 2. It is seen that for N* pitch less than ≈360 nm, the response time is sub- 100 μs. Such fast response times have been confirmed experimentally in both the ULH and USH geometries [7,2]. For the specific value of the pitch P = 150 nm, a response time of τ90-10 = 17 μs is predicted.

polarization of light does not follow the helical rotation. As the pitch is reduced, the optical rotation falls; for sufficiently short pitch the N* may be considered optically isotropic at normal incidence, and thus provides an excellent dark state between crossed polarizers. An analogy may be drawn here with the VAN device (except that the short-pitch N* is negatively birefringent). So, how short must the pitch be to give a suitably dark off state? To answer this question we employ two methods; the Berreman numerical method, and an approximate analytic approach based on the de Vries equation, which give results in good agreement. We use the Berreman 4×4 method [9,10] to investigate the optical properties of the display in general cases. This method is frequently employed to model LC structures and has been found to give results in good agreement with experiment (see e.g. Ref. [11]). Polarizers are implemented using the ‘equivalent polarizer’ method [12]. Fig. 3 shows how, as the pitch P is 1.0

Symbol and value

Splay elastic constant

K1 = 6 pN

Twist elastic constant

K2 = 2 pN

Bend elastic constant

K3 = 7 pN

Viscoelastic ratio

γ/K = 1.5×1010 m-2s

100 90 response time, τ90−10 (μs)

80 60

P=1200 nm

0.5 1.0

P=400 nm

0.5 1.0

P =240 nm

0.5 1.0 0.5


P=150 nm


0.0 100 200 300 400 500 600 700 800 900 1000


wavelength, λ (nm)

40 30 20 10 0 50



200 250 pitch, P (nm)



Figure 2. The response time τ90-10 of the switching mechanism as a function of the chiral nematic pitch P. (The elastic constants and viscoelastic ratio are as given in Table I.)


1.0 relative transmitted intensity, I

Material Parameter

P=6000 nm


Table I. Throughout this work we use the following typical, experimentally determined, values for the material parameters of bimesogenic liquid crystals [7].

Field-off (Dark) State

The transmission in the field-off, dark, state will dominate the contrast ratio of the device. With no field applied, the LC structure is that of an undistorted N*. Under certain circumstances, the plane of polarization of light travelling along the helical axis of a N* will be rotated along the helical structure. This is the regime in which the Twisted Nematic (TN) and Super Twisted Nematic (STN) devices operate. However, if the pitch P is suitably smaller than the wavelength of light λ, the plane of

Figure 3. The transmitted intensity as a function of wavelength in the field-off state for various values of the chiral nematic pitch P. As P is decreased below ≈200 nm transmission in the visible region is suppressed and the device is in an effective dark state. Curves were calculated using the Berreman method. (d=6 μm, n||=1.7, and n⊥=1.5.) reduced below the wavelengths of visible light, the relative transmission in the visible falls, and the device becomes dark in the field-off state. As a specific example, consider the following, typical, device parameters: parallel and perpendicular component of the local refractive index n|| = 1.7, and n⊥ = 1.5, pitch P = 150 nm, and device thickness d = 6 μm. At the wavelength of green light, λ = 555 nm, the Berreman method predicts a transmitted intensity in the off state of Ioff = (2270)-1. This will give the display a contrast ratio of approximately CR ≈ 2000+ at normal incidence. In the limit of small pitch the de Vries equation [13] may be used to derive an analytic expression for Ioff, giving

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39.2 / F. Castles

for Pn|| < λ, where n = (n|| + n⊥)/2, and Δn = n|| - n⊥ [10]. For the above device parameters, this equation gives Ioff = (2244)-1, in approximate agreement with the Berreman method. This expression becomes exact in the limit Pn/λ→0.


Field-on (Transmissive) State


Tilt as a function of applied electric field

The director profile of the LC is determined using a onedimensional continuum model with strong planar alignment [14,10]. The figure of merit is θmax, the rotation of the optic axis. In the limit of zero dielectric anisotropy Δε→0, the tilt of the optic axis is given as a function of the electric field E by tan θ max =

eP K − 2K 2 + K3 E- 1 sin θ max 2πK 2 2K2


(In this situation, the tilt as a function of field is identical to that found in the ULH case [10].) The tilt angle is thus dominated by the combination of the average flexoelectric coefficient and the pitch ēP. Using Eq. (1), the rotation of the optic axis as a function of electric field is plotted in Fig. 4 for various values of ēP.

80 70





50 -18








5 -1 electric field E (V μm )



n||=1.7 λ=532 nm K1=6 pN


K2=2 pN K3=7 pN




10 15 -1 electric field, E (V μm )


Figure 5. Electro-optic curves for typical device parameters at various values of the average flexoelectric coefficient ē. V may be determined from the electro-optic curve by multiplying the electric field required for full intensity modulation EFI by the spacing of the in-plane electrodes V = EFI×de.


Viewing angle dependence

We investigate the viewing angle dependence of the display by considering the contrast ratio CR. The CR is defined as the ratio of the luminance of the device in the on state, to the luminance in the off state. To determine the CR, the spectral power density of transmitted light is integrated over the visible range 380 nm to 780 nm, weighted by the standard color matching function of the green component of light [15]. We use Standard Illuminant A as the light source. To reduce the expense of the computation we approximate the short-pitch N* to be a uniaxial, negatively birefringent structure. (This approximation can be shown to be accurate at normal incidence for the values of pitch P used here [10]. However, the accuracy of the approximation may degrade as the polar viewing angle becomes large.)


It is noted that, in the limit of small θmax, Eq. (1) may be approximated by

θ max ≈

eP E 2πK

which describes the linear part of the plots at sufficiently low E.

Electro-optic curves

We investigate the transmission of the device as a function of the applied electric field, again using the Berreman 4×4 method. For typical device parameters, we plot the electro-optic curves for various values of the average flexoelectric coefficient ē in Fig. 5. It is seen that as ē increases, the field value required for full intensity modulation decreases. The driving voltage of the display

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Figure 4. The tilt of the optic axis of the chiral nematic is plotted as a function of the applied, in-plane, electric field for various values of the combined quantity ēP (where ē is the average flexoelectric coefficient and P is the pitch).


P=150 nm d=6 μm n⊥=1.5


e =40 pCm -1 e =30 pCm -1 e =20 pCm -1 e =10 pCm

Fig. 6 plots the CR as a function of viewing angle - the isocontrast





polar viewing angle (degrees)

maximum tilt angle θmax (deg)


relative transmitted intensity, I

I off


⎛ ⎞ ⎜ ⎟ 2 ⎜ π (Δn ) Pd ⎟ 2 ≈ sin ⎜ ⎟ 2 ⎜ 2 ⎛⎜ ⎛ λ ⎞ ⎞⎟ ⎟ λ 4 1 − ⎜ ⎜ ⎜⎝ nP ⎟⎠ ⎟ ⎟ ⎠⎠ ⎝ ⎝



60 40



20 0


90 1000 100 50

20 40



60 80


150 180

Figure 6. Contrast ratio (CR) as a function of viewing angle, with no optical compensation. Lines denote equal CR contours. Polarizers are at 0° and 90° azimuthal angles, and the electric field E is applied at 45°.

39.2 / F. Castles curves - with no optical compensation. The device parameters used for this plot are the same as for Fig. 5, with the average flexoelectric coefficient set to ē =10 pC m-1.

Our theoretical results suggest the USHF display is a promising candidate for future display devices, and further theoretical and experimental work is being undertaken.

It may be expected that the viewing angle will be widened using optical compensation films. As a simple example of this we implement a c-plate compensation film. The film is chosen such that its phase retardation cancels that of the LC. In this way, the light leakage in the dark state is greatly reduced. The film is required to be a uniaxial, positively birefringent structure, with optic axis normal to the plane of the device. For simplicity we set the thickness of the film to be equal to that of the LC layer, in which case the required birefringence of the film Δnfilm is given by Δnfilm = -ΔnN*, where ΔnN* is the effective birefringence of the N*. The c-plate compensated isocontrast curves are plotted in Fig. 7. The viewing angle is significantly widened. It is noted that the viewing angle dependence may be further improved using a multidomain electrode structure.


polar viewing angle (degrees)

330 300

of Chiral Nematic Bimesogens for the Flexoelectro-optic Effect,” Liquid Crystals 26, 1235 (1999).


90 1000 100 50




150 180

Figure 7. Contrast ratio (CR) as a function of viewing angle, with positive c-plate compensation. Lines denote equal CR contours. Polarizers are at 0° and 90° azimuthal angles, and the electric field E is applied at 45°.


“Optimized Flexoelectric Response in a Chiral LiquidCrystal Phase Device,” J. Appl. Phys. 98, 34109 (2005).

[5] B. Musgrave, P. Lehmann, and H. J. Coles, “A New Series

60 80

[2] B. J. Broughton, M. J. Clarke, A. E. Blatch, and H. J. Coles,

Phys. Rev. Lett. 22, 918 (1969).


20 40

Cholesteric Liquid Crystal,” Phys. Rev. Lett. 58, 1538 (1987).

[4] R. B. Meyer, “Piezoelectric Effects in Liquid Crystals,”

20 0


[1] J. S. Patel and R. B. Meyer, “Flexoelectric Electro-optics of a

“Flexoelectro-optic Liquid Crystal Device,” patent No. WO/2006/003441 (2006).


60 40


[3] H. J. Coles, M. J. Coles, B. J. Broughton, and S. M. Morris,




We would like to thank the Engineering and Physical Sciences Research Council (EPSRC) for financial support. One of the authors (F.C.) gratefully acknowledges Merck for financial support in the form of a CASE studentship.


By applying an in-plane electric field to a short-pitch N* LC in the USH configuration, a fast optical switch may be generated, which operates using the flexoelectro-optic effect. We call this the ‘USHF’ display mode. Some properties of this mechanism have been theoretically investigated in the context of a fast-switching display device. Using typical, experimentally determined, material parameters we predict a response time of τ90-10 = 17 μs, together with a contrast ratio at normal incidence of CR ≈2000+. In this respect, the USHF device may be considered as an ultra-fast switching VAN device. Whilst the typical driving voltage for the display remains relatively high, further developments in materials with higher average flexoelectric coefficients ē may reduce this. The predicted isocontrast curves for the device were plotted and we demonstrated that the viewing angle may be improved by implementing a positively birefringent c-plate compensation film.

[6] H.J. Coles, M. J. Clarke, S. M. Morris, B. J. Broughton, and A. E. Blatch, “Strong Flexoelectric Behavior in Bimesogenic Liquid Crystals,” J. Appl. Phys. 99, 34104 (2006).

[7] S. M. Morris, M. J. Clarke, A. E. Blatch, and H. J. Coles, “Structure-flexoelastic Properties of Bimesogenic Liquid Crystals,” Phys. Rev. E 75, 041701 (2007).

[8] J. S. Patel and S. –D. Lee, “Fast Linear Electro-optic Effect Based on Cholesteric Liquid Crystals,” J. Appl. Phys. 66, 1879 (1989).

[9] D. W. Berreman, “Optics in Stratified and Anisotropic Media: 4×4 Matrix Formulation,” J. Opt. Soc. Am. 62, 502 (1972).

[10] F. Castles, S. M. Morris, and H. J. Coles, “The Flexoelectrooptic Properties of Chiral Nematic Liquid Crystals in the Uniform Standing Helix Configuration,” Phys. Rev. E (in submission).

[11] H. G. Yoon and H. F. Gleeson, “Accurate Modelling of Multilayer Chiral Nematic Devices through the Berreman 4×4 Matrix Methods,” J. Phys. D: Appl. Phys. 40 3579 (2007).

[12] H. A. van Sprang, “Angular Dependence of the Transmission of Nontwisted Liquid Crystal Displays,” J. Appl. Phys. 71, 4826 (1992).

[13] H. De Vries, “Rotary Power and Other Optical Properties of Certain Liquid Crystals,” Acta Cryst, 4, 219 (1951).

[14] A. J. Davidson, S. J. Elston, and E. P. Raynes, “Investigation Into Chiral Active Waveplates,” J. Appl. Phys. 99, 93109 (2006).

[15] J. Schanda, Colorimetry: Understanding the CIE System (Wiley, 2007).

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39.2: Fast-Switching Flexoelectric Display Device with ...

39.2: Fast-Switching Flexoelectric Display Device with High Contrast. Flynn Castles, Stephen M. Morris, and Harry J. Coles. University of Cambridge, Cambridge ...

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