Ion Transport and Switching Currents in Smectic Liquid Crystal Devices

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Ferroelectrics, 344:255–266, 2006 ... Keywords Ion transport; liquid crystal; switching voltage. 1. ..... of surface-stabilized ferro-electric liquid crystal devices.
Ferroelectrics, 344:255–266, 2006 Copyright © Taylor & Francis Group, LLC ISSN: 0015-0193 print / 1563-5112 online DOI: 10.1080/00150190600968405

Ion Transport and Switching Currents in Smectic Liquid Crystal Devices Downloaded By: [Universiteit Gent] At: 17:01 5 March 2007

KRISTIAAN NEYTS∗ AND FILIP BEUNIS ELIS Department, Universiteit Gent, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium In SSFLC and AFLC devices, the voltage over the liquid crystal is not only determined by the applied voltage, but also by the separation of ions and the spontaneous polarization. The motion of ions and the switching of the spontaneous polarization are therefore interfering phenomena. A simplified model for ion transport and switching of polarization is introduced which is able to explain the shift in apparent threshold voltage and the variation of the hysteresis width. Keywords Ion transport; liquid crystal; switching voltage

1. Introduction All liquid crystal devices contain some concentration of ionic species. Even if the original material is very pure, ions may appear in the liquid crystal due to the alignment layers, rubbing the glue, voltage operation or UV illumination. Ions drift under influence of the electric field and diffuse in the liquid crystal due to their thermal motion. As long as the ion concentration is sufficiently small, the charge density they represent can only cause a small variation in the electric field. In nematic liquid crystal devices, the charge densities have been reduced thanks to technological advances and the effect of ions on the transmission is now limited. However, in a number of applications ions still cause important problems for the image quality. In chiral smectic liquid crystals the ion concentrations are usually larger, because of the spontaneous polarization of the material and the resulting internal electric fields. The behavior of ions in nematic liquid crystals has been studied extensively in literature [1–4]. The presence of ions has been detected by current measurements under sine wave, square wave or triangular voltages. Matching these measurements with numerical simulations has led to the determination of different mechanisms: drift in the electric field, diffusion, trapping at the interfaces [5], generation of new ions [6], recombination and leakage through the alignment layers. The importance of ions in smectic materials has been recognized early on [7–9] and there are many reports in the literature about experimental work and theoretical simulations of ion transport [7–11]. In smectic materials the electric field generated by the ions can play an important role in the switching and the bistability of smectic devices. More recently, the influence of ions in so-called V-shaped switching has been investigated [12, 13]. Received September 12, 2005. ∗ Corresponding author. E-mail: [email protected]

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The aim of this paper is to elucidate the role of a number of parameters in relation to ion transport and switching in SSFLC and AFLC devices. Combination of detailed descriptions for the different mechanisms leads to complex numerical simulations which sometimes give relatively little insight. In this paper we will simplify the description of switching to a simple threshold behavior. This simplification makes it possible to understand the ion transport and how it influences the switching of the spontaneous polarization. It is also shown that the mechanisms of drift and diffusion can be described in a good approximation by a conductivity in combination with a limitation for the ion separation. With this approximation, an analytical model is obtained in which the influence of different device parameters is readily visible.

2. LC Device Definition Figure 1 illustrates the definition of various parameters in a one-dimensional structure with area S. The bottom electrode is grounded and an external voltage Ve (t) is applied to the top electrode. Each electrode is covered with an alignment layer with thickness 12 dal and dielectric constant εal . The liquid crystal layer has a thickness dlc and the dielectric tensor ε¯ (z, t) varies as a function of the z-coordinate in the liquid crystal layer. Different types of ions can be present, but we will only consider the case of positive and negative ions carrying the elementary charge e. The concentrations of the ions n + (z, t) and n − (z, t) are functions of the z-coordinate. In a smectic liquid crystal, there may also be a (z-dependent) spontaneous polarization P¯ s (z, t) related with the ordering of the molecules. The electrode with potential Ve (t) carries a charge Q e (t); the grounded electrode an opposite charge. The electric field in the LC-layer can be written as a function of the charges using Gauss law [8, 10]:   z+  Q e (t) ∂ Ps,z   εzz (z, t)E(z, t) = + ρ(z , t) − (z , t) dz  , (1.1) S ∂z  0− with the charge density given by: ρ(z, t) = e(n + (z, t) − n − (z, t)).

(1.2)

The charge at the electrode Q e (t) can be found by integrating the field over the entire device and setting this value equal to Ve (t). To close the system of equations, the dynamic behavior for the ions and the spontaneous polarization are needed.

Figure 1. Structure of the liquid crystal device, with indication of the liquid crystal layer, alignment layer, average spontaneous Ps and ionic polarization Pi , external voltage Ve and charge Q e .

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The ion flux F¯ ± in the liquid crystal [14] is described with a drift term containing the ¯ ±: ¯ ± and a diffusion term containing the diffusion tensor D mobility tensor µ ± ¯ ± (z, t) ∂n (z, t) . ¯ ¯ ± (z, t) E(z, F¯ ± (z, t) = ±n ± (z, t)µ t) − D ∂z

(1.3)

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Note that due to the anisotropy of the liquid crystal, the ion flux is usually not along the z-axis, even if the model considered is one-dimensional (parameters do not vary along the x and y axes). In some cases this anisotropy may lead to lateral transport of ions over a distance of mm [14–16]. The ion flux in the alignment layer is usually set to zero. The diffusion tensor is linked to the mobility tensor by the Einstein diffusion-mobility relation: ¯ ± = kT µ ¯ ±. D e

(1.4)

The ion concentration is modified when the flux is inhomogeneous, according to [8]: ∂n ± (z, t) ∂ F ± (z, t) =− z , ∂t ∂z

(1.5)

In order to reach agreement with experimental conditions, often other mechanisms are also involved related to ions, such as: trapping at the interfaces [5], ion generation [6], recombination, leakage through the alignment layers [17]. The dynamic behavior of nematic liquid crystal can be quite accurately described using the Oseen Frank elastic energy density, the electric energy density and the viscosity. For smectic liquid crystals, the description of the dynamic behavior is usually based on a onedimensional description [7, 8, 10, 13]. However, the real behavior is more complicated, because switching is often inhomogeneous, through domain wall motion. In a standard SS-FLC device the spontaneous polarization is more or less uniform across the thickness of the LC layer and switches roughly between ±Psm , with Psm equal to (or slightly lower then) the spontaneous polarization of the material. The transitions −Psm → Psm and Psm → −Psm occur approximately if the voltage over the LC layer reaches a certain threshold value ±Vs . In an AFLC device, there are four transitions for the spontaneous polarization: −Psm → 0, 0 → Psm , Psm → 0 and 0 → −Psm occurring respectively at: −Vs1 , Vs2 , Vs1 , and −Vs2 . In this paper we will assume that the dielectric constant, mobilities and diffusion constants in the liquid crystal are homogeneous and constant in time, e.g. εzz (z, t) = εlc . This approximation is good for nematic LC devices below the switching threshold and also acceptable for SSFLC devices when the tilt is limited. We can then define the following capactitances: Cal =

εal S dal

Clc =

εlc S dlc

Ce =

Cal .Clc , Cal + Clc

(1.6)

and the ratio [8]: α=

Cal , Cal + Clc

(1.7)

which is usually close to one and indicates the relative contribution of the liquid crystal layer in the impedance. We further assume the total charge in the layer is zero.

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The charge at the electrode Q e (t) can then be found by integrating the field over the entire device or by using Ramo’s theorem:

Q e (t) = Ce Ve (t) + αS (Pi (t) + Ps (t)) ,

(1.8)

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with Pi (t) and Ps (t) the average polarization in the liquid crystal layer due to the displacement of ions or spontaneous polarization: 

d

Pi (t) = 0

z ρ(z, t)dz Ps (t) = − dlc



d 0

1 z d Ps (z, t) dz = dlc dz dlc



d

Ps (z, t)dz, (1.9) 0

The voltage over the liquid crystal layer is given by: Vlc (t) = αVe (t) −

αS (Pi (t) + Ps (t)) . Cal

(1.10)

The supplied to the external electrodes is given by: Ie (t) = Ce

d Ve + αS dt



d Ps d Pi + dt dt

 .

(1.11)

3. Simplified Model for Ion Transport The detailed drift and diffusion behavior of several ion types can be rather complex and therefore it is interesting to determine which features of the behavior can be explained with a simplified model. We propose the following simplified model, in which diffusion is neglected, two types of ions have opposite charge and the same mobility, and the electric field is assumed to be homogeneous: Elc (z, t) = Vlc (t)/dlc . As long as all ions participate in the transport, the average current in the liquid crystal layer is given by: Ji =

dPi Vlc − − Vlc + =σ , = e(n + 0 µ + n0 µ ) dt dlc dlc

(1.12)

with n ± 0 the initial density of positive and negative ions. This means that the ion transport − − + − + can be described by a conductivity [1] σ = e(n + 0 µ + n 0 µ ), with n 0 = n 0 = n 0 if only two types of ions are present. The expression for the variation in Pi is only acceptable as long as none of the ions have reached the edge of the liquid crystal layer. When all ions have arrived at the lc/al interface, the average polarization due to ions obtains the maximum value ±Pim : Pi = ±en 0 dlc = ±Pim ,

(1.13)

with the + sign in the case a positive voltage is applied. Combining equations (1.12) and (1.10) leads to the following differential equation for Pi : dlc Cal dPi Cal = Ve (t) − Ps (t) − Pi (t). ασ S dt S

(1.14)

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In this paper we will limit the discussion to the case of a triangular voltage waveform with period T and amplitude Vem : 4Vem T T t −