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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

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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 simpli?ed 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 in?uence of the electric ?eld and diffuse in the liquid crystal due to their thermal motion. As long as the ion concentration is suf?ciently small, the charge density they represent can only cause a small variation in the electric ?eld. 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 ?elds. 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 ?eld, 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 ?eld generated by the ions can play an important role in the switching and the bistability of smectic devices. More recently, the in?uence of ions in so-called V-shaped switching has been investigated [12, 13].

Received September 12, 2005. ? Corresponding author. E-mail: neyts@elis.ugent.be

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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 simpli?cation makes it possible to understand the ion transport and how it in?uences 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 in?uence of different device parameters is readily visible.

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2. LC Device De?nition

Figure 1 illustrates the de?nition 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 1 dal and 2 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 Ps (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 ?eld in the LC-layer can be written as a function of the charges using Gauss law [8, 10]: εzz (z, t)E(z, t) = Q e (t) + S

z+ 0?

ρ(z , t) ?

? Ps,z (z , t) dz , ?z

(1.1)

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 ?eld 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 ?ux F ± in the liquid crystal [14] is described with a drift term containing the ± ?± ? mobility tensor ? and a diffusion term containing the diffusion tensor D : ?n ± (z, t) ?± ? ? ? F ± (z, t) = ±n ± (z, t)?± (z, t) E(z, t) ? D (z, t) . ?z (1.3)

Note that due to the anisotropy of the liquid crystal, the ion ?ux 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 ?ux 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 ± ?± ? D = ? . e (1.4)

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The ion concentration is modi?ed when the ?ux 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 de?ne the following capactitances: Cal = and the ratio [8]: α= Cal , Cal + Clc (1.7) εal S dal Clc = εlc S dlc Ce = Cal .Clc , Cal + Clc

(1.6)

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 ?eld over the entire device or by using Ramo’s theorem:

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

(1.8)

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

0 d

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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. Simpli?ed 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 simpli?ed model. We propose the following simpli?ed model, in which diffusion is neglected, two types of ions have opposite charge and the same mobility, and the electric ?eld 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 + ?+ + n ? ?? ) 0 0 dt dlc dlc (1.12)

with n ± the initial density of positive and negative ions. This means that the ion transport 0 can be described by a conductivity [1] σ = e(n + ?+ + n ? ?? ), with n + = n ? = n 0 if only 0 0 0 0 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 : Ve (t) = 4Vem T T t ? <t < T 4 4 4Vem T T 3T = ?t <t < , T 2 4 4

(1.15)

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These waveforms are not often used as driving voltage in applications, but are quite common in experimental studies. It is often convenient to look at the polarization as a function of the applied voltage by eliminating the time dependency. Using the expression in the ?rst part of the triangular waveform (1.15), it is possible to eliminate the time in the differential equation (1.14) and obtain: Vi with: Vi = 4Vem dlc Cal , T ασ S (1.17) d Pi Cal + Pi = Ve ? Ps , d Ve S (1.16)

which can be interpreted as the voltage over the electrodes for which the ion current equals the external current supplied to the electrodes, when Pi and Ps are equal to zero. Typical values [1] in the case of an FLC device are: dlc = 2 · 10?6 m dal = 10?7 m Clc /S = 2.22 · 10?5 Fm ?2 Cal /S = 2 · 10?4 Fm ?2 α = 0.9 n 0 = 1020 m ?3 ? = 10?10 m 2 V ?1 s ?1 σ = 3.2 · 10?9 Psm = 10?4 Cm ?2 Using these values and we ?nd for a slope of the applied voltage of 10 V/s a critical voltage for ion transport: Vi = 0.14 V . First we investigate the ion transport behavior in the absence of spontaneous polarization. Using the described simpli?ed model, we can obtain analytical results, which are then compared with the simulation results of the rigorous equations. As long as Pi does not reach the extremal values ±Pim , the general solution can be determined from the differential equation (1.16): Pi (Ve ) = V Cal (Ve ? Vi ) + A exp ? S Vi . (1.19)

?1

(1.18)

m ?1

Pim = 3.2 · 10?5 Cm ?2

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The constant A can be found by requiring an anti-symmetric solution with Pi (Vem ) = ?Pi (?Vem ). This leads to: Pi (Ve ) = Cal Cal Vi Ve + S S exp (?Ve /Vi ) ?1 . cosh (Vem /Vi ) (1.20)

This is the bottom branch of the solution; the top branch is anti-symmetrical: ?Pi (?Ve ). This solution corresponds to an equivalent circuit of two capacitors Cal and Clc in series, with a resistor Rlc = dlc /σ S in parallel with Clc . It is valid as long as the ions do not reach the interfaces and the polarization does not reach the maximum value Pim : Cal Vi Vem ln cosh S Vi ≤ Pim . (1.21)

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If this condition is not ful?lled, the solution is somewhat more complicated, but can still be found analytically: Pim For ? SCal < Ve < Vem : Pi (Ve ) = Min Pim , Cal Cal Vi S Pim Ve Ve + exp ? ? S S Vi Cal Vi ?1 , (1.22)

Pim and for ?Vem < Ve < ? SCal we have:

Pi (Ve ) = Max ? Pim , · exp ?

Cal Cal Vi Ve + S S ?1 .

2 ? exp ?

Vem S Pim ? Vi Cal Vi (1.23)

Vem ? Ve Vi

Figure 2 illustrates the variation of Pi as a function of Ve for different values of Vi if Pim is not reached (this happens at high frequencies). Large values of Vi compared to Vem lead to a small effect of the conductivity, small values correspond with a strong effect of the conductivity in the LC layer. Figure 3 illustrates the variation of Pi as a function of Ve for different values of Pim . As the applied voltage is proportional with time, the analytical expressions for the current supplied to the electrodes can be obtained using equation (1.11) and the solution for Pi (Ve ). Usually current measurements are given in the literature. However, plotting the polarization Pi versus Ve is more interesting, because the voltage over the liquid crystal layer can be directly read from this diagram using (1.10).

4. Ion Transport with Ferro-Electric Switching

For the spontaneous polarization, we assume that Ps remains constant as long as the voltage over the liquid crystal layer does not reach the value αVs . Ion transport is continuing under in?uence of the voltage over the LC layer according to (1.14). Once the voltage over the LC layer reaches the value αVs , the spontaneous polarization changes at a rate which keeps the voltage Vlc clamped to αVs . The variation of Ps (switching) according to this equation stops when either the voltage Ve starts to decrease or the maximum value for Ps is reached: Ps = Psm . When one of these conditions is reached, the spontaneous polarization remains constant again and only the ion transport continues.

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Figure 2. Average polarization in the liquid crystal layer due to ions Pi versus applied voltage Ve for a triangular voltage waveform, with Pim suf?ciently large, for (a) Vi = 10 · Vem , (b) Vi = Vem and (c) Vi = 0.1 · Vem . The horizontal dashed line represents insulating LC, the diagonal dashed line perfectly conducting LC.

We start again with the simpli?ed model for ion transport with a triangular voltage applied. This is combined with the model outlined above for switching of the spontaneous polarization. As both the ionic polarization or the spontaneous polarization can be constant or not, four different regions can be distinguished:

r Case 1: Ps constant and Pi constant

Figure 3. Average polarization in the liquid crystal layer due to ions Pi versus applied voltage Ve for a triangular voltage waveform, with limitation due to Pim . Parameters are Vi = Vem and (a) Pim = 0.1 · Cal Vem /S, (b) Pim = 0.2 · Cal Vem /S and (c) Pim = 0.3 · Cal Vem /S.

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r Case 2: Ps constant and Pi variable:

The variation of Pi is determined by the differential equation (1.16) and the solution is of the form: Pi (Ve ) =

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Ve Cal (Ve ? Vi ) + A exp ? S Vi

? Ps ,

(1.24)

with A a constant which can be determined from boundary conditions.

r Case 3: Ps variable and Pi constant:

In this case Ps is determined by the requirement that Vlc = αVs . With equation (1.10) this leads to: Ps (Ve ) = Cal Cal Ve ? Vs ? Pi . S S (1.25)

r Case 4: Ps variable and Pi variable:

In this case, the voltage over the LC layer is equal to the threshold voltage and the variation of Pi is simply given by (1.12): d Pi σ T αVs = , d Ve 4dlc Vem and the solution is of the form: Pi (Ve ) = σ T αVs Ve + B, 4dlc Vem (1.27) (1.26)

with B an integration constant. The spontaneous polarization is then: Ps (Ve ) = Cal Cal σ T αVs Ve ? ? Vs ? B, S 4dlc Vem S (1.28)

For a given set of parameters, the solution for Pi and Ps of this model is a combination of different sections. The transition from one region to another is governed by the limiting conditions for Pi , Ps , Ve and Vlc . During such a transition these parameters remain continuous. As the device is symmetric we set as boundary condition for the steady state situation: Pi (?Vem ) = ?Pi (Vem ) Ps (?Vem ) = ?Ps (Vem ). (1.29)

Figure 4 illustrates the simpli?ed model, when the ion concentration is suf?ciently large, Pim is not reached and the ion transport is described by a conductivity. Under these circumstances only cases 2 and 4 occur. The importance of the ion transport is then again determined by Vi . Figure 5c illustrates the simpli?ed model, for the situation in which all ions reach the interfaces before the switching of Ps starts, because the switching voltage Vs is high. Under these circumstances only cases 1, 2 and 3 occur. In order to evaluate the accuracy of the results obtained with the simpli?ed model, Pi and Ps have also been calculated according to formula (1.9) for the detailed ion transport

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Figure 4. Average total polarization in the liquid crystal layer due to Pi and Ps versus applied voltage Ve for a triangular voltage waveform, with switching. Parameters are Psm = 0.2 · Cal Vem /S, Vs = 0.2 · Vem and (a) Vi = 20 · Vem , (b) Vi = Vem . The dashed lines indicate the conditions for switching: Vlc = ±αVs .

model based on drift and diffusion and the variation of ion distributions in the liquid crystal. In Fig. 5, curves (a) and (b) are calculated with respectively the simpli?ed model and the detailed model for the case without switching, when Pim is not reached. To obtain curve (b) the parameters dlc , Cal /S, α, ? in (1.18) were used, with n 0 = 1.25 · 1021 m?3 ,

Figure 5. Average total polarization in the liquid crystal layer due to Pi and Ps versus applied voltage Ve for a triangular voltage waveform, with switching. Comparison between the simpli?ed model (a,c) and the detailed model (b,d) for ion transport. Parameters for (a,b): Vi = 2 · Vem , Psm = 0 and for (c,d) Vi = 0.1 · Vem , Psm = 0.2 · Cal Vem /S, Vs = 0.38 · Vem . The dashed line indicates the condition for switching: Vlc = ±αVs .

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T = 0.022 · s and Vem = 10 · V to obtain Vi /Vem = 2. The comparison illustrates that the simpli?ed model overestimates the ion motion. This is due to the fact that, on average, more charges are located in the regions with a smaller electric ?eld. Similarly curves (c) and (d) in Fig. 5 compare the total polarization for the case with switching and limitation to Pim . In this case the ion transport is more spread for the detailed model, because the ?eld is inhomogeneous, but otherwise the correspondence is quite good.

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5. Discussion of Results

First we discuss the validity of the simpli?ed ion transport model. From curves (a) and (b) in Fig. 5 it is obvious that the results are not completely equivalent. Because of the inhomogeneity of the electric ?eld over the LC layer, the actual current is somewhat lower than the current estimated from the simpli?ed model. This deviation is more important for larger ion concentrations. The simpli?ed model describes both the ion transport in the LC material and the limitation when the ions have reached the interfaces. In this respect the agreement between curves (c) and (d) in Fig. 5 is very good. Another interesting comparison can be made between curves 4b and 5c. In both simulations, there is ion transport and switching of spontaneous polarization. For curve 4b, the ion transport occurs mainly after the switching, because the voltage Vi is larger than Vs . For curve 5c, the situation is exactly opposite: the ion transport is now mainly before the switching because Vi is now smaller than Vs . This difference has important implications for the voltage Ve at which the spontaneous polarization reverses. In curve 4b, the switching towards +Ps happens at a lower value of Pi + Ps . Because switching occurs on the dashed line, corresponding to Vlc = αVs this means it appears at a lower voltage Ve . In curve 5c, the ion transport has exactly the opposite effect, it shifts the switching to a higher voltage Ve .

Figure 6. Average total polarization in the liquid crystal layer due to Pi and Ps and Transmission versus applied voltage Ve for a triangular voltage waveform, with SSFLC switching. Schematic illustration for two cases: (a) ion transport occurs mainly before switching, (b) ion transport occurs mainly after switching. The dashed line indicates the threshold for switching. (See Color Plate XLVI)

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Figure 7. Average total polarization in the liquid crystal layer due to Pi and Ps and Transmission versus applied voltage Ve for a triangular voltage waveform, with AFLC tri-state switching. Schematic illustration for two cases: (a) without ions (b) ion transport after switching reverses the hysteresis loop.

As a conclusion we can state that ion motion after switching (as in 4b) destabilizes the switched state and leads to switching at a lower voltage during the next voltage pulse. Ion motion before switching (as in 5c) stabilizes the spontaneous polarization state and requires a higher voltage to switch. The magnitude of the shift in voltage is increased for larger values of Pim or smaller values of Ca . The variation in the apparent switching voltage in the transmission/voltage curve is illustrated in Fig. 6 for the case of SS-FLC switching with polarizers parallel with the director of one of the two stable states. In case (a) the hysteresis curve becomes wider due to ions, in case (b) it becomes narrower. A similar variation in the apparent switching voltages is given in Fig. 7 for the case of tri-state switching in AFLC devices. If the ion concentration is suf?ciently large and transport occurs mainly after switching, the hysteresis order may be reversed as in Fig. 7b. This mechanism can also lead to V-shaped switching as has been explained elsewhere [12, 13].

6. Conclusion

An electrical model has been proposed for the behavior of ion transport and switching of spontaneous polarization. The model is based on a number of assumptions: conduction and limited polarization in the liquid crystal layer and switching of spontaneous polarization at a threshold voltage. The simulation results compare well with results obtained with a detailed drift-and-diffusion ion transport model. The simpli?ed model makes it possible to visualize the relation between applied voltage, average polarization in the liquid crystal layer and voltage over the liquid crystal layer. The model can also explain the apparent shift in threshold voltages and changes in the hysteresis width, due to ion transport before or after the switching of spontaneous polarization.

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Acknowledgments

The authors want to thank the Belgian Research Project IAP 5/18 on Photonics and the European Research Training Network SAMPA.

References

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