Polarization-stiffened ferroelectric liquid crystals: A thickness-independent bistable switching voltage with a lower limit of about 2 V and the transition to thresholdless switching

Typical ferroelectric $\text{Sm-}{C}^{*}$ liquid crystal (FLC) cells exhibit a voltage threshold for switching from one stable state to another despite the FLC's response being inherently continuous and thresholdless (free rotation of the director around the tilt cone). This switching threshold is due to FLC-surface interactions and to the chevron smectic structure commonly formed in cells. It is shown here that the FLC electrostatic energy contribution $\sim P_S{}^2$ responsible for thresholdless switching of high-$P_S$ FLCs also plays a key role in the bistable switching of lower-$P_S$ FLCs. Among the consequences are that it can be difficult to lower a cell's threshold below $V_{TB}\sim 3.4{(B/{\varepsilon }_F)}^{1/2}$ ($B$ and ${\varepsilon }_F$ are the FLC's elastic and dielectric constants), that a cell's threshold becomes independent of cell thickness once it substantially exceeds the characteristic length ${\xi }_P={\left({\varepsilon }_{F}B\right)}^{1/2}/P_S$, and that there are conditions under which alignment layer capacitance can decrease rather than increase the threshold (i.e., transition to thresholdless switching). A model that predicts and explains these behaviors is presented along with threshold measurements of representative FLC cells.

Figure 1

In this circuit model $C_A$ and $C_F$ represent alignment layer and FLC layer capacitances. The illustration [2] of bookshelf smectic layers within a cell shows the coordinate system used in the model.

Figure 2

Shown on the left is the model surface interaction energy $w\left({\phi }_S\right)=-\gamma \mathrm{co}{\mathrm{s}}^2{\phi }_S$ and torque $N=-dw/d{\phi }_S$. On the right the black dot represents the director position as it rotates around the FLC's tilt cone, lying on the surface at ${\phi }_S=0$.

Figure 3

The parameters are $L_F=1.6\mu \mathrm{m}$, $L_A=0$, $P_S=25\mathrm{nC}/\mathrm{c}{\mathrm{m}}^2$, $B=10\mathrm{pN}$, ${\xi }_P=75\mathrm{nm}$, ${\xi }_P/L_F=0.047$, ${\phi }_S={45}^{\circ }$, and ${\varepsilon }_F=4{\varepsilon }_0$. (a) Plot of $\phi \left(x\right)$ for voltages of 0–102 V. The heavy line is the $\phi \left(x\right)$ corresponding to an applied charge of $q=P_S$, occurring at voltage $V_P=2.4\mathrm{V}$. Below $V_P$ the voltage is varied in steps of $0.1V_P$. Above, it varies from $V_P$ to 102 V (where $q=10P_S$) through ten equal steps in ${(1/V)}^{1/2}$. See the text regarding the dashed line. (b) Plot of the electric field strength $E\left(x\right)$ for $V=0-5\mathrm{V}$ in steps of 0.5 V. (c) Plot of the midcell electric field strength $(x=0.5L_F)$ vs voltage. (d) Plot of $q$ vs voltage.

Figure 4

Computed FLC threshold voltages $V_{TF}$ vs surface torque density $N_M$, with $B=10\mathrm{pN}$ for (a)–(c): (a) three FLC layer thicknesses and three polarization values, (b) four peak torque angles ${\phi }_M$, (c) a larger range of $N_M$ at three thicknesses, and (d) three values of elastic constant $B$, where horizontal lines indicate corresponding values of $V_{TB}$.

Figure 5

(a) Numerically computed curves of $V_{TF}$ vs $P_S$ for surface strengths in the range $0.2-1.4\mathrm{erg}/\mathrm{c}{\mathrm{m}}^2$ for $L_A=0$, $B=10\mathrm{pN}$, $L_F=1.6\mu \mathrm{m}$, ${\phi }_M={45}^{\circ }$, and $\varepsilon =4{\varepsilon }_0$. (b) Same as (a) but showing detail at low voltage and additional $N_M$ curves; the dashed line marks $V_{TB}$.

Figure 6

(a) The $\phi \left(x\right)$ solutions for 11 FLC voltages ranging from 0 to 2.4 V in steps of 0.24 V. The inset illustrates the definition of the chevron angle  δ. See the text for FLC and cell parameters. (b) Computed curves of $V_{TF}$ vs $P_S$ for surface strengths $N_M$ in the range $0.1-0.7\mathrm{erg}/\mathrm{c}{\mathrm{m}}^2$. (c) Same as (b), except plotting $V_{Tcell}$ vs $P_S$ for 20 nm alignment layers $({\varepsilon }_A=4{\varepsilon }_0)$.

Figure 7

(a) Measured curves of memory angle vs pulse amplitude for the pulse widths shown in (b). Switching at lower voltage corresponds to a greater pulse width. (b) Measured threshold voltage vs pulse width for two cell thicknesses.

Figure 8

Same as Fig. 4 but with expanded scales to show detail at low voltage, for ${\phi }_M=0^{\circ },{20}^{\circ },{40}^{\circ },\mathrm{and}{60}^{\circ }$.