Ion Transport and Switching Currents in Smectic Liquid Crystal Devices

Transcription

1 Ferroelectrics, 344: , 2006 Copyright Taylor & Francis Group, LLC ISSN: print / online DOI: / Ion Transport and Switching Currents in Smectic Liquid Crystal Devices 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 1. Introduction Ion transport; liquid crystal; switching voltage 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, Corresponding author. neyts@elis.ugent.be [499]/255

2 256/[500] K. Neyts and F. Beunis 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 V e (t) isapplied to the top electrode. Each electrode is covered with an alignment layer with thickness 1 2 d al and dielectric constant ε al. The liquid crystal layer has a thickness d lc 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 V e (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]: ε zz (z, t)e(z, t) = Q e(t) S + z+ 0 ( ρ(z, t) P ) s,z (z, t) dz, (1.1) z 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 V e (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 P s and ionic polarization P i,external voltage V e and charge Q e.

3 Ion Transport and Switching Currents [501]/257 The ion flux F ± in the liquid crystal [14] is described with a drift term containing the mobility tensor µ ± and a diffusion term containing the diffusion tensor D ± : F ± (z, t) =±n ± (z, t) µ ± (z, t)ē(z, t) D ± (z, t) n± (z, t). (1.3) z 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: D ± = kt e µ ±. (1.4) The ion concentration is modified when the flux is inhomogeneous, according to [8]: n ± (z, t) = F z ± (z, t), (1.5) t z 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 ±P sm, with P sm equal to (or slightly lower then) the spontaneous polarization of the material. The transitions P sm P sm and P sm P sm occur approximately if the voltage over the LC layer reaches a certain threshold value ±V s.inanaflc device, there are four transitions for the spontaneous polarization: P sm 0, 0 P sm, P sm 0 and 0 P sm occurring respectively at: V s1, V s2, V s1, and V s2. 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. ε zz (z, t) = ε lc. 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: and the ratio [8]: C al = ε al S d al C lc = ε lcs d lc C e = C al.c lc C al + C lc, (1.6) α = C al C al + C lc, (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.

4 258/[502] K. Neyts and F. Beunis 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) = C e V e (t) + αs (P i (t) + P s (t)), (1.8) with P i (t) and P s (t) the average polarization in the liquid crystal layer due to the displacement of ions or spontaneous polarization: P i (t) = d 0 z d ρ(z, t)dz P s (t) = d lc 0 z dp s (z, t) d lc dz The voltage over the liquid crystal layer is given by: dz = 1 d P s (z, t)dz, (1.9) d lc 0 V lc (t) = αv e (t) αs C al (P i (t) + P s (t)). (1.10) The supplied to the external electrodes is given by: I e (t) = C e dv e dt 3. Simplified Model for Ion Transport ( dpi + αs dt + dp ) s. (1.11) dt 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: E lc (z, t) = V lc (t)/d lc.aslong as all ions participate in the transport, the average current in the liquid crystal layer is given by: J i = dp i dt = e(n + 0 µ+ + n 0 µ ) V lc d lc = σ V lc d lc, (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 P i 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 ±P im : P i =±en 0 d lc =±P im, (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 P i : d lc C al ασ S dp i dt = C al S V e(t) P s (t) P i (t). (1.14)

5 Ion Transport and Switching Currents [503]/259 In this paper we will limit the discussion to the case of a triangular voltage waveform with period T and amplitude V em : V e (t) = 4V em T t T 4 < t < T 4 = 4V ( ) em T T 2 t T 4 < t < 3T 4, (1.15) 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 first part of the triangular waveform (1.15), it is possible to eliminate the time in the differential equation (1.14) and obtain: with: V i dp i dv e + P i = C al S V e P s, (1.16) V i = 4V em T d lc C al ασ S, (1.17) which can be interpreted as the voltage over the electrodes for which the ion current equals the external current supplied to the electrodes, when P i and P s are equal to zero. Typical values [1] in the case of an FLC device are: d lc = m d al = 10 7 m C lc /S = Fm 2 C al /S = Fm 2 α = 0.9 (1.18) n 0 = m 3 µ = m 2 V 1 s 1 σ = m 1 P im = Cm 2 P sm = 10 4 Cm 2 Using these values and we find for a slope of the applied voltage of 10 V/s a critical voltage for ion transport: V i = 0.14 V. First we investigate the ion transport behavior in the absence of spontaneous polarization. Using the described simplified model, we can obtain analytical results, which are then compared with the simulation results of the rigorous equations. As long as P i does not reach the extremal values ±P im, the general solution can be determined from the differential equation (1.16): P i (V e ) = C ) al S (V e V i ) + A exp ( VVi. (1.19)

6 260/[504] K. Neyts and F. Beunis The constant A can be found by requiring an anti-symmetric solution with P i (V em ) = P i ( V em ). This leads to: P i (V e ) = C al S V e + C [ ] alv i exp ( Ve /V i ) S cosh (V em /V i ) 1. (1.20) This is the bottom branch of the solution; the top branch is anti-symmetrical: P i ( V e ). This solution corresponds to an equivalent circuit of two capacitors C al and C lc in series, with a resistor R lc = d lc /σ S in parallel with C lc.itisvalid as long as the ions do not reach the interfaces and the polarization does not reach the maximum value P im : C al V i S ( ln cosh V ) em P im. (1.21) V i If this condition is not fulfilled, the solution is somewhat more complicated, but can still be found analytically: For SP im C al < V e < V em : P i (V e ) = Min and for V em < V e < SP im C al we have: { P i (V e ) = Max P im, C al { P im, C al S V e + C [ ( alv i exp V e SP ) ]} im 1, (1.22) S V i C al V i ( exp V em V e V i S V e + C (( ( alv i 2 exp V em SP )) im S V i C al V i ) )} 1. (1.23) Figure 2 illustrates the variation of P i as a function of V e for different values of V i if P im is not reached (this happens at high frequencies). Large values of V i compared to V em 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 P i as a function of V e for different values of P im. 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 P i (V e ). Usually current measurements are given in the literature. However, plotting the polarization P i versus V e 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 P s remains constant as long as the voltage over the liquid crystal layer does not reach the value αv s. Ion transport is continuing under influence of the voltage over the LC layer according to (1.14). Once the voltage over the LC layer reaches the value αv s, the spontaneous polarization changes at a rate which keeps the voltage V lc clamped to αv s. The variation of P s (switching) according to this equation stops when either the voltage V e starts to decrease or the maximum value for P s is reached: P s = P sm. When one of these conditions is reached, the spontaneous polarization remains constant again and only the ion transport continues.

7 Ion Transport and Switching Currents [505]/261 Figure 2. Average polarization in the liquid crystal layer due to ions P i versus applied voltage V e for a triangular voltage waveform, with P im sufficiently large, for (a) V i = 10 V em, (b) V i = V em and (c) V i = 0.1 V em. The horizontal dashed line represents insulating LC, the diagonal dashed line perfectly conducting LC. We start again with the simplified 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: Case 1: Ps constant and P i constant Figure 3. Average polarization in the liquid crystal layer due to ions P i versus applied voltage V e for a triangular voltage waveform, with limitation due to P im.parameters are V i = V em and (a) P im = 0.1 C al V em /S, (b) P im = 0.2 C al V em /S and (c) P im = 0.3 C al V em /S.

8 262/[506] K. Neyts and F. Beunis Case 2: Ps constant and P i variable: The variation of P i is determined by the differential equation (1.16) and the solution is of the form: P i (V e ) = C ( al S (V e V i ) + A exp V ) e P s, (1.24) V i with A a constant which can be determined from boundary conditions. Case 3: Ps variable and P i constant: In this case P s is determined by the requirement that V lc = αv s.with equation (1.10) this leads to: P s (V e ) = C al S V e C al S V s P i. (1.25) Case 4: Ps variable and P i variable: In this case, the voltage over the LC layer is equal to the threshold voltage and the variation of P i is simply given by (1.12): and the solution is of the form: dp i dv e = σ T αv s 4d lc V em, (1.26) P i (V e ) = σ T αv s V e + B, (1.27) 4d lc V em with B an integration constant. The spontaneous polarization is then: P s (V e ) = ( Cal S σ T αv s 4d lc V em ) V e C al S V s B, (1.28) For agiven set of parameters, the solution for P i and P s of this model is a combination of different sections. The transition from one region to another is governed by the limiting conditions for P i, P s, V e and V lc. During such a transition these parameters remain continuous. As the device is symmetric we set as boundary condition for the steady state situation: P i ( V em ) = P i (V em ) P s ( V em ) = P s (V em ). (1.29) Figure 4 illustrates the simplified model, when the ion concentration is sufficiently large, P im 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 V i. Figure 5c illustrates the simplified model, for the situation in which all ions reach the interfaces before the switching of P s starts, because the switching voltage V s is high. Under these circumstances only cases 1, 2 and 3 occur. In order to evaluate the accuracy of the results obtained with the simplified model, P i and P s have also been calculated according to formula (1.9) for the detailed ion transport

9 Ion Transport and Switching Currents [507]/263 Figure 4. Average total polarization in the liquid crystal layer due to P i and P s versus applied voltage V e for a triangular voltage waveform, with switching. Parameters are P sm = 0.2 C al V em /S, V s = 0.2 V em and (a) V i = 20 V em, (b) V i = V em. The dashed lines indicate the conditions for switching: V lc =±αv s. 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 simplified model and the detailed model for the case without switching, when P im is not reached. To obtain curve (b) the parameters d lc, C al /S,α,µin (1.18) were used, with n 0 = m 3, Figure 5. Average total polarization in the liquid crystal layer due to P i and P s versus applied voltage V e for a triangular voltage waveform, with switching. Comparison between the simplified model (a,c) and the detailed model (b,d) for ion transport. Parameters for (a,b): V i = 2 V em, P sm = 0 and for (c,d) V i = 0.1 V em, P sm = 0.2 C al V em /S, V s = 0.38 V em. The dashed line indicates the condition for switching: V lc =±αv s.

10 264/[508] K. Neyts and F. Beunis T = s and V em = 10 V to obtain V i /V em = 2. The comparison illustrates that the simplified 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 field. Similarly curves (c) and (d) in Fig. 5 compare the total polarization for the case with switching and limitation to P im. In this case the ion transport is more spread for the detailed model, because the field is inhomogeneous, but otherwise the correspondence is quite good. 5. Discussion of Results First we discuss the validity of the simplified 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 field over the LC layer, the actual current is somewhat lower than the current estimated from the simplified model. This deviation is more important for larger ion concentrations. The simplified 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 V i is larger than V s. For curve 5c, the situation is exactly opposite: the ion transport is now mainly before the switching because V i is now smaller than V s. This difference has important implications for the voltage V e at which the spontaneous polarization reverses. In curve 4b, the switching towards +P s happens at a lower value of P i + P s. Because switching occurs on the dashed line, corresponding to V lc = αv s this means it appears at a lower voltage V e.incurve 5c, the ion transport has exactly the opposite effect, it shifts the switching to a higher voltage V e. Figure 6. Average total polarization in the liquid crystal layer due to P i and P s and Transmission versus applied voltage V e 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)

11 Ion Transport and Switching Currents [509]/265 Figure 7. Average total polarization in the liquid crystal layer due to P i and P s and Transmission versus applied voltage V e 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 P im or smaller values of C a. 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 sufficiently 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 simplified 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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