Modeling of the microdischarges in plasma addressed liquid
crystal displays
Citation for published version (APA):
Hagelaar, G. J. M., Kroesen, G. M. W., Slooten, van, U., & Schreuders, H. (2000). Modeling of the
microdischarges in plasma addressed liquid crystal displays. Journal of Applied Physics, 88(5), 2252-2262. https://doi.org/10.1063/1.1287529
DOI:
10.1063/1.1287529 Document status and date: Published: 01/01/2000 Document Version:
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Modeling of the microdischarges in plasma addressed
liquid crystal displays
G. J. M. Hagelaara)and G. M. W. Kroesen
Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands U. van Slooten and H. Schreuders
Philips Electronics Nederland B.V., Prof. Holstlaan 4, 5656 AA Eindhoven, The Netherlands 共Received 20 March 2000; accepted for publication 31 May 2000兲
Plasma addressed liquid crystal共PALC兲 is a promising technology for large size flat display devices, which uses gas discharges as electrical switches for the addressing of a liquid crystal共LC兲 layer. This work presents a comprehensive two-dimensional fluid model, that we developed for the simulation of the microdischarges occurring in PALC displays. The model comprises continuity equations and drift-diffusion equations for plasma particle species, a balance equation for the electron energy, and Poisson’s equation for the electric field. Using this model, we succeeded in simulating the full PALC operation, reproducing a series of discharge pulses and afterglows in three consecutive PALC discharge channels. Results are presented for helium and helium–hydrogen mixtures. The results include: calculated particle densities, current–voltage curves, plasma decay times, surface charges, and LC transmission profiles. The influence of electrical crosstalk between adjacent channels is demonstrated. © 2000 American Institute of Physics.
关S0021-8979共00兲06417-3兴
I. INTRODUCTION
For decades, display researchers have been looking for alternatives to the conventional bulky cathode ray tube dis-plays. Recently, two alternative technologies have emerged for large size television monitors, which both make use of plasma. The first and most common is the plasma display panel technology, using the ultraviolet emission from a
plasma to excite phosphors.1 The other is the plasma
ad-dressed liquid crystal 共PALC兲 technology.2–4 This
technol-ogy, invented at Tektronics, uses microdischarges as electri-cal switches for the addressing of a liquid crystal layer.
Liquid crystal 共LC兲 is an electro-optic material: It can alter the state of polarization of light passing through it. The polarization changing effect depends on the electric field ap-plied to the LC. In liquid crystal displays共LCDs兲, light from a uniform backlight passes first through a polarizing layer, then through a layer of LC, and finally through a second polarizing layer. The percentage of light transmitted by the second polarizer depends on the electric field in the LC, which is controlled for every display pixel independently, by a matrix of electrical switching elements. Conventional LCDs, as applied in laptop computers, use thin film transis-tors 共TFTs兲 as switching devices. These TFTs are manufac-tured by submicron semiconductor technology. Due to the low tolerances of this technology, the yield expectations on larger screen sizes are very low. In PALC displays however, the pixels are addressed by plasma switching devices, which are much more easily manufactured, permitting the produc-tion of large size displays.
Figure 1 schematically represents a PALC panel. The LC and a protective microsheet are sandwiched between two
glass plates. The rear glass plate contains parallel channels filled with a discharge gas, typically helium or helium-based binary mixtures at a pressure of a few hundred Torrs. These channels correspond to the picture rows of the display. At the bottom of each channel two thin parallel electrodes run all along the channel. The front glass plate is patterned with so-called data electrodes: transparent, conductive stripes of indium–tin–oxide, which correspond to the picture columns of the display.
The image on the display is written row by row, i.e., all pixels of one picture row are addressed at the same time. In order to address a certain row, a discharge is created in the corresponding channel by applying a short dc voltage pulse to the channel electrodes. During the afterglow of the dis-charge, small voltages—representing the data to be written on the pixels of the row—are applied to the data electrodes with respect to the channel electrodes. The decaying plasma in the channel screens itself from the resulting electric fields by depositing surface charge on top of the microsheet. It continues to build up this surface charge until the fields in the channel have vanished and the data voltages stand en-tirely across the LC layer and the microsheet. After the plasma has completely decayed, the surface charge is fixed and unaffected by any change in the data voltages. The elec-tric fields in the LC 共and the resulting transmission through
the second polarizer兲 of the picture row will now remain
nearly unchanged until the next discharge pulse is applied to the channel.
For successful application of the PALC discharges, it is essential to understand the physics underlying their behavior. The discharges in question are in the glow regime: they are characterized by a very low ionization degree (⬍10⫺6) and space charge densities governing the electric field. Although a兲Electronic mail: hagelaar@discharge.phys.tue.nl
2252
the physical principles of glow discharges are nowadays quite well understood, their exact implications for the PALC operation are unclear. Issues of special interest are the charg-ing of the microsheet, which must be accurate and homoge-neous, and the decay time of the plasma in the channels, which must be as short as possible. In view of the small size of the discharges, experimental results are usually coarse and hard to interpret, so that discharge modeling is an indispens-able tool in the development of the PALC technology. Mod-eling work on PALC has been reported previously in Ref. 5. In this article we present a two-dimensional discharge model, which is suitable for the simulation of the PALC operation. The article is organized as follows: Sec. II describes the physical and numerical principles of the model, as well as the input data that have been used in the calculations. In Sec. III we present several examples of PALC simulation results, illustrating the usefulness of the model.
II. DESCRIPTION OF THE MODEL
This section discusses the physical and numerical prin-ciples of the model. Input data are presented for pure helium and helium–hydrogen mixtures.
A. Fluid model
We pursue the well-known fluid approach and describe the behavior of reactive particle species by the first few mo-ments of the Boltzmann equation: the continuity equation, the momentum transport equation, and the energy transport equation. Since this approach has been described previously in numerous articles,6–17we will only outline it briefly here. Given the low ionization degree, the density and tem-perature of the gas particles are assumed to be constant and unaffected by the plasma. For every plasma particle species p the density results from a continuity equation
np
t ⫹ⵜ•⌫p⫽Sp, 共1兲
where np is the density,⌫p the flux, Sp the source term, and
the index p can indicate: electrons共p⫽e兲, an ion species, or a
neutral species. The flux is given by the momentum transport equation, which we approximate by the drift–diffusion equa-tion
⌫p⫽sgn共qp兲pEnp⫺Dpⵜnp. 共2兲
E is the electric field, qpthe particle charge,pthe mobility,
and Dp the diffusion coefficient. The first term gives the flux
due to the electric field共drift兲 and the second term represents the flux due to concentration gradients 共diffusion兲. Particle inertia is neglected.
The source term Sp is determined by the reactions
oc-curring in the plasma. It consists of positive contributions from the reactions in which a particle of species p is created and negative contributions from those in which such a par-ticle is lost:
Sp⫽
兺
r
cp,rRr. 共3兲
The index r refers to a reaction; cp,r is the net number of
particles of species p created in one reaction of type r, and it can be negative as well as positive. The reaction rate R is proportional to the densities of the reacting particles:
R⫽kn1n2 共4兲
for two-body reactions, and
R⫽kn1n2n3 共5兲
for three-body reactions. The proportionality constant k is the reaction rate coefficient. Similarly, the rate of spontaneous decay processes is
R⫽kn. 共6兲
where k is the decay frequency.
Transport equations 共1兲 and 共2兲 require the input of re-action rate coefficients k and transport coefficients and D. In general these quantities depend on the energy distribution of the considered particles. We use the following approxima-tions concerning these dependencies:
We assume that the charged particle transport coeffi-cients satisfy the Einstein relation
D⫽kBT
e , 共7兲
where kB is the Boltzmann constant, e is the elementary
charge, and T is the particle temperature, corresponding to the energy of the random particle motion.
For ions we use the local field approximation, which assumes a direct relation between the particle energy distri-bution and the electric field. Transport and rate coefficients are regarded as functions of the electric field:
⫽共E兲, D⫽D共E兲, k⫽k共E兲. 共8兲
These relations can be found in the literature as results of experiments and classical theories. In particular, the ion dif-fusion coefficients are found from the mobilities by the Ein-stein relation 共7兲, in which the ion temperature is related to the electric field by18,19
kBT⫽kBTg⫹
m⫹mg
5m⫹3mg
mg共E兲2, 共9兲
where Tg is the gas temperature and m and mg are the ion
and gas particle mass, respectively.
For electrons however, the local field approximation of-ten leads to unsatisfactory modeling results, as a result of the poor energy transfer in electron-neutral collisions共due to the huge mass difference兲. Rather than using the relations 共8兲, we therefore assume the electron transport coefficients and the rate coefficients of electron impact reactions to be func-tions of the electron mean energy, as in Refs. 9 and 11:
e⫽e共¯⑀兲, De⫽De共¯⑀兲, k⫽k共¯⑀兲, 共10兲
where the subscript e refers to electrons, and the electron mean energy¯ is calculated as a function of time and space⑀ from an energy balance equation
n⑀ t ⫹ⵜ"
冉
5
3¯⑀⌫e⫹q
冊
⫽S⑀. 共11兲In this equation n⑀is the electron energy density
n⑀⫽ne¯ ,⑀ 共12兲
and q is the heat flux, which we assume to be proportional to the electron mean energy gradient according to
q⫽⫺53neDeⵜ¯ .⑀ 共13兲
The source term for electron energy is given by S⫽⫺e⌫e"E⫺ne
兺
r ⑀
¯
rkrnr, 共14兲
where the two terms represent heating by the electric field and energy loss in collisions, respectively. The summation in the loss term is only over the electron impact reactions, with nr the density of the target particles and ¯⑀r the threshold energy.
The functions共10兲 are obtained from cross sections, as-suming a Maxwellian electron energy distribution function, or—which usually gives better results—using the electron energy distribution function resulting from uniform-field Monte Carlo or Boltzmann calculations. In the above equa-tions, the electron mean energy is supposed to result mainly from random motion, so that it is consistent to use
Te⫽
2
3¯⑀ 共15兲
with the Einstein relation 共7兲 to find the electron diffusion coefficient.
Finally, the electric field depends on the space charge density according to Poisson’s equation
ⵜ"共⑀E兲⫽⫺ⵜ"共⑀ⵜV兲⫽, 共16兲
where⑀ is the dielectric permittivity, V the electrostatic po-tential, and the space charge density
⫽
兺
p
qpnp. 共17兲
B. Boundary conditions
Since the boundary conditions we apply are slightly dif-ferent from the boundary conditions found in most articles, we will discuss them somewhat more elaborately. A full derivation and discussion of our boundary conditions for par-ticle transport is given in Ref. 20.
The transport equations for heavy particles are solved for the boundary condition of zero particle influx. A microscopic analysis yields the following expression for the particle flux toward the wall:21,22
⌫"n⫽a sgn共q兲E"nn⫹14vthn⫺
1
2Dⵜn"n, 共18兲 where n is the normal vector pointing outward andvthis the
thermal velocity vth⫽
冑
8kBT
m . 共19兲
The number a is set to one if the drift velocity is directed toward the all and to zero otherwise:
a⫽
再
1, sgn共q兲E"n⬎0
0, sgn共q兲E"n⭐0. 共20兲
The first term on the right-hand side of Eq. 共18兲 represents the flux due to the electric field; the last two terms are the diffusion flux, due to the random motion of the particles. Most authors8–10,15,17apply Eq. 共18兲 without the last term, which they wrongly ignore; this term reflects the fact that the random motion flux involves all particles within a certain mean free path from the wall, not just the local particles at the wall. In order to incorporate the last term without running into numerical difficulties in accurately evaluating the den-sity gradient it contains, we rewrite Eq.共18兲 as follows:
Imposing Eq.共18兲 as a boundary condition for the drift–
diffusion Eq. 共2兲 implies that the following equation must
hold at the boundary:
sgn共q兲E"nn⫺Dⵜn"n⫽a sgn共q兲E"nn
⫹1
4vthn⫺ 12Dⵜn"n. 共21兲 Note that although both members of this equation contain similar terms, their nature is very different: the left member is a continuum expression, which in principle can be used anywhere in space, but only has physical meaning inside the plasma volume, whereas the right member is a kinetic ex-pression for the flux at the boundary. From Eq.共21兲 we find an expression to replace the last term in共18兲. Substitution of that expression gives
⌫"n⫽共2a⫺1兲sgn共q兲E"nn⫹1
2vthn, 共22兲
which is an appropriate boundary condition for heavy species.
The boundary condition for electrons is similar, but in-cludes influx by secondary electron emission. It is important to realize that the electrons emitted from the surface all have velocities directed away from the wall, and are described rather poorly by the drift–diffusion approximation. The most serious artifact arising from the poor description of these electrons is an unrealistic flux back to the surface by random
motion. In order to find an appropriate boundary condition, which avoids this undesired effect, we distinguish between two electron groups at the wall:␥electrons, emitted from the
surface, and␣electrons, coming from the bulk. Both groups
are treated equally and indistinguishably with the drift– diffusion equation, but have different boundary conditions.
To the␣ electrons we apply the boundary condition共22兲:
⌫␣"n⫽⫺共2ae⫺1兲eE"nn␣⫹ 12vth,en␣, 共23兲
where⌫␣and n␣ are the flux and density of the␣ electrons. In contrast, the␥ electrons do not flow共back兲 to the wall;
⌫␥"n⫽⫺共1⫺ae兲
兺
p ␥p⌫p•n, 共24兲
where⌫␥is the flux of␥ electrons, and the factor (1-ae兲 is included to cancel the flux in case the electric field is
di-rected away from the wall. Using Eq. 共24兲 as a boundary
condition for the drift-diffusion equation for ␥ electrons
gives
⫺eE"nn␥⫺Deⵜn"n␥⫽⫺共1⫺ae兲
兺
p ␥p⌫p"n. 共25兲
For the regions where secondary electron emission is the most important, it is justified to neglect the diffusion term in Eq. 共25兲 in any case, which yields the following density for the ␥ electrons:
n␥⫽共1⫺ae兲
兺p␥p⌫p"n
eE"n
. 共26兲
Adding the fluxes 共23兲 and 共24兲, and using Eq. 共26兲, now
leads to a suitable boundary condition for the total electron flux ⌫e"n⫽⫺共2ae⫺1兲eE"nne⫹ 1 2vth,e共ne⫺n␥兲 ⫺2共1⫺ae兲
兺
p ␥p⌫p"n, 共27兲where n␥ is once again given by Eq.共26兲. For the sake of
numerical stability, it is useful to approximate n␥共26兲 in this equation by n␥⬇共1⫺ae兲 1 e
兺
p ␥p冋
共2ap⫺1兲sgn共qp兲 ⫹1 2冑
16共mp⫹mg兲mg 3共5mp⫹3mg兲mp册
p np. 共28兲For this approximation we assumed the electric field to be perpendicular to the wall, and used Eqs.共22兲, 共19兲, and 共9兲. The boundary condition for the electron energy transport is very similar to the one for electron transport 共27兲:
⌫⑀"n⫽⫺共2ae⫺1兲53eE"nn⑀⫹ 23vth,e共n⑀⫺n⑀,␥兲 ⫺2共1⫺ae兲
兺
p ␥p⑀ ¯ p⌫p"n, 共29兲 where n⑀,␥⫽共1⫺ae兲 兺p␥p¯⑀p⌫p"n 5 3eE"n , 共30兲and ¯⑀p is the mean initial energy of electrons emitted by
incidence of species p. Expression共30兲 can then be
approxi-mated analogous to共28兲.
The electrode potentials are boundary conditions for Poisson’s equation. Dielectric materials surrounding the
TABLE I. Reactions in helium–hydrogen mixtures. The second column gives the mean electron energy lost in the reaction.
No. Reaction Energy共eV兲 Rate coefficient Ref.
1 e⫹He→e⫹He* 20.215 Fig. 3共a兲 29
2 e⫹He*→2e⫹He⫹ 4.365 Fig. 3共b兲 29
3 e⫹He→2e⫹He⫹ 24.58 Fig. 3共c兲 29
4 e⫹He*→e(20.2 eV)⫹He ⫺20.215 2.9⫻10⫺9 cm3/s 30
5 He⫹⫹2He→He2⫹⫹He 1.1⫻10⫺31 cm6/s 31
6 He*⫹He*→e共15.0 eV兲⫹He⫹⫹He ⫺15.0 8.7⫻10⫺10 cm3/s 30
7 He*⫹He*→e共17.4 eV兲⫹He2⫹ ⫺17.4 2.03⫻10⫺9 cm3/s 30
8 e⫹H2→e⫹2H 11.7 Fig. 3共f兲 29 9 e⫹H2→2e⫹H2⫹ 15.4 Fig. 3共d兲 29 10 e⫹H3⫹→3H 0 Fig. 3共g兲 29 11 e⫹H3⫹→H2(v⬙)⫹H 0 Fig. 3共h兲 29 12 H2⫹⫹H2→H3⫹⫹H 2.1⫻10⫺9 cm 3/s 32 13 H⫹⫹2H2→H3⫹⫹H2 3.1⫻10⫺29 cm 6/s 32 14 He*⫹H2→e⫹H2⫹⫹He ⫺4.815 3.18⫻10⫺11 cm 3/s 33 15 He2⫹⫹H2→H2⫹⫹2He 4.1⫻10⫺10 cm 3 /s 34 16 He*⫹H→e⫹H⫹⫹He ⫺6.615 1.1⫻10⫺9 cm3/s 35
17 e⫹H2→e⫹H2(v⬙) 12.0 Fig. 3共e兲 29
18 e⫹H2(v⬙)→H⫺⫹H 0 Fig. 3共i兲 29
19 H⫺⫹H→e⫹H2 0 7.6⫻10⫺11 cm3/s 32
20 H⫺⫹H3⫹→2H2 2⫻10⫺7 cm
3
plasma do not naturally impose boundary conditions for the electric field, so that it is necessary to solve Poisson’s equa-tion in the entire dielectric surroundings of the plasma. The effect of surface charge on top of a dielectric wall is de-scribed by Gauss’s law
⑀wallEwall"n⫺⑀0Eplasma"n⫽, 共31兲
where Ewall and Eplasma are the electric fields at the surface,
respectively, inside and outside the dielectric material. The surface charge density results from plasma currents strik-ing the wall. We assume that this wall charge does not dif-fuse along the surface, but stays in the very spot it is depos-ited by the plasma:
⫽
冕
j"ndt, 共32兲where j is the plasma current density j⫽
兺
p
qp⌫p. 共33兲
The current through an electrode is given by I⫽
冕冕
electrodesurface
冉
j"n⫺⑀ tE"n
冊
d2S. 共34兲
The second term of this equation is the displacement current; it corresponds to changes in total amount of charge present at the surface of the electrode. An external electric circuit, in-volving back coupling from the discharge current to the elec-trode voltage, is not included in the model: The elecelec-trode potentials are imposed at all times.
C. Numerical implementation
The numerical method used for solving the system of equations is described elaborately in Ref. 23, and will there-fore only be outlined here. A unique feature of this numerical method is the time integration scheme: When integrating the system of equations numerically with respect to time, the explicit evaluation of coupled quantities necessitates strong
time step restrictions, resulting in an enormous slowdown of the calculation. The strongest time step restriction is caused by an explicit treatment of the coupling between the charged particle transport and the space charge field.24 In order to circumvent this constraint, we use a semi-implicit technique, similar to the method reported in Refs. 25 and 26. When using this technique however, the explicit handling of the
dependence of the electron energy source term 共14兲 on the
mean electron energy becomes limiting for the time step. Therefore, we have developed a new technique for the im-plicit treatment of the electron energy source term, based on linearization with respect to electron mean energy. This ap-proach makes it possible to increase the time step by several
FIG. 2. Electron mobility in 97% helium–3% hydrogen. The data come from Ref. 21 (⬍1 eV兲 and aBOLSIG共Ref. 29兲 calculation (⬎1 eV兲.
FIG. 3. Rate coefficients of electron impact reactions in 97% helium–3% hydrogen, as calculated withBOLSIG共Ref. 29兲. 共a兲 total excitation He(23S)
and He(21S)共I-1兲, 共b兲 ionization from He*共I-2兲, 共c兲 direct ionization 共I-3兲,
共d兲 direct ionization H2 共I-9兲, 共e兲 20%⫻total excitation H2(C1⌸u) and H2(B1⌺u⫹)共I-17兲, 共f兲 dissociation H2共I-8兲, 共g兲 recombination H3⫹共I-10兲, 共h兲
recombination H3⫹共I-11兲, 共i兲 dissociative attachment 共I-18兲.
FIG. 4. Ion mobilities in helium. The data are taken from Ref. 18共for He⫹, He2⫹, H2⫹, H3⫹, and H⫺), Ref. 19共for H⫹), and Ref. 37共for He2⫹at high
orders of magnitude, thus giving a tremendous speedup of the calculation. Details and a discussion of the implicit tech-nique can be found in Ref. 23.
The spatial numerical grid is two-dimensional, equidis-tant, and Cartesian. The discretization of the equations is
based on the widely used Sharfetter–Gummel scheme,27 as
in Ref. 7. This scheme, which is derived from the analytical solution for a constant drift–diffusion flux between two grid points, supports large density gradients, as opposed to the more straightforward central difference scheme. All dis-cretized equations are solved by the modified strongly im-plicit technique published in Ref. 28, which we slightly modified in order to ensure convergence for all cases. Be-sides being simple to implement, we found this method to be extremely efficient for solving the plasma equations.
The model is flexible with respect to the discharge ge-ometry. Material properties are defined in every separate grid cell: a cell can be filled with discharge gas, with electrode
material at a certain voltage, or with dielectric material with a certain permittivity. In this way, arbitrarily shaped dis-charge channels, surrounded by an arbitrary configuration of electrodes and dielectric materials, can be defined. The plasma equations共1兲, 共2兲, and 共11兲 are solved only within the
gas area’s. Poisson’s equation 共16兲 is solved on the entire
grid, taking into account the effect of possible surface charge on the channel walls. At the edges of the simulation domain, the electric field inside the dielectrics is taken to be parallel to the edges.
D. Input data
The simulations presented in this article concern pure helium and helium hydrogen mixtures. The reactions taken into account are listed in Table I; this reaction scheme has been optimized for PALC discharges, and is only valid for
mixtures with a small percentage of hydrogen (⬍5%兲. Since
all the important metastable helium states—He共23S), He共2 1S), and He
2(2s3⌺u)—participate in similar reactions with
very similar rate coefficients, we have grouped them together into one species: He*. Highly vibrationally excited hydrogen molecules共v⬎6兲 are treated as one separate species: H2(v
⬙
兲.The electron mobility and the rate coefficients of the electron impact reactions are obtained with the Boltzmann codeBOLSIG,29which calculates the electron energy distribu-tion in uniform electric fields. Figures 2 and 3 show these coefficients for a mixture of 97% helium and 3% hydrogen; for other mixing ratios they are slightly different. The mo-bilities and diffusion coefficients of all heavy species are FIG. 5. Standard geometry used for the calculations. The dielectric constant
of the glass plate with interchannel walls is 6.0, the LC layer with mi-crosheet has an effective dielectric constant of 5.0.
FIG. 6. Electric potential, particle densities, electron mean energy, and electron impact reaction rates in a dc discharge in pure helium. These plots correspond to the middle channel of the geometry shown in Fig. 5. The gas pressure is 150 Torr. The cathode共2兲 is set to ⫺260 V, all other electrodes are grounded. The resulting current is 3.0 mA/cm. The increment of the contours is 1/10 times the maximum value indicated in the top right corner of each plot. The darkest regions correspond to this maximum value.
assumed to be determined merely by the helium gas. Ion mobilities are shown in Fig. 4. The diffusion coefficients of
the neutral species are all taken from Ref. 38: D⫻p
⫽450 cm2Torr/s for He*, D⫻p⫽2102.9 cm2Torr/s for
H, and D⫻p⫽1156.7 cm2Torr/s for H2(v
⬙
), where p is thegas pressure. We use constant secondary electron emission coefficients: 0.2 for helium ions and 0.01 for hydrogen ions.21Secondary electron emission by neutral species is ne-glected. The mean initial energy is taken to be 5 eV for the
secondary electrons emitted by helium ions21 and 1 eV for
those emitted by hydrogen ions.
The metastable–metastable ionization processes 共I-6兲
and 共I-7兲 produce electrons at 15 and 17.4 eV of energy, which is not enough to excite or ionize helium atoms, but at the same time too much to be efficiently lost in elastic col-lisions. As a result these electrons are not thermalized and treated incorrectly by the transport and rate coefficients
cal-culated with BOLSIG.29 In pure helium, where metastable–
metastable ionization plays an important role, we consider them as a separate species: monoenergetic electrons of 15 eV, which do not take part in any reaction. This approach is supported by Monte Carlo calculations.5,39 For the mobility
of these electrons we use ⫻p⫽6.8⫻105 cm2Torr/V/s,
the diffusion coefficient is taken to be D⫻p⫽6.8
⫻106 cm2Torr/s. In helium–hydrogen mixtures we use
only one electron group.
The importance of H⫺is unclear. The reactions共I-17,18兲 only represent the most probable mechanism for its produc-tion. Details and reliable values for rate coefficients are not
known. The rate coefficient for dissociative attachment
共I-18兲 increases by orders of magnitude with the increasing
vibrational level of H2,40 so that only states H2(v⬎6兲 may
contribute significantly to the production of H⫺. It has been suggested that these high vibrational levels are populated mainly through radiative decay of higher singlet electronic states excited by energetic electrons.32,36,41 Since the exact vibrational distribution resulting from these processes is un-known, the effective rate coefficient for reaction共I-18兲 is no more than a rough estimate. The rate coefficient of reaction
共I-17兲 is taken to be proportional to the excitation rates of
H2(C1⌸u) and H2(B1⌺u⫹);
36,41
the proportionality constant, however, is only a conjecture. In view of the large uncertain-ties concerning the production of H⫺, we choose to omit the species H⫺ and H共v
⬙
兲, as well as the last four reactions共I-17-20兲, from the reaction scheme, unless there is evidence
for the importance of H⫺in PALC discharges.
III. RESULTS
This section discusses examples of simulation results, illustrating the validity and usefulness of the model. All cal-culations presented here are based on the standard geometry shown in Fig. 5, which represents a cross section through three discharge channels of a PALC display. Three consecu-tive channels are taken into account in order to be able to study crosstalk effects: electrical influences of adjacent chan-nels on each other. In addition, a large region of the bottom glass plate is taken into account in order to obtain the correct electric field at the bottom wall of the discharge channels. Typical PALC simulations take 1 min—1 h of CPU time on a Pentium II PC.
FIG. 7. Particle densities and electron mean energy in a dc discharge in a mixture of 97% helium and 3% hydrogen. These plots correspond to the middle channel of the geometry shown in Figure 5. The gas pressure is 105 Torr. The cathode共2兲 is set to ⫺320 V, all other electrodes are grounded. The resulting current is 3.0 mA/cm. The increment of the contours is 1/10 times the maximum value indicated in the top right corner of each plot. The darkest regions correspond to this maximum value.
We first present the simulation of a discharge pulse共Sec. III A兲 and an afterglow 共Sec. III B兲 in a PALC channel. Then we present the calculation of the surface charge and the re-sulting transmission that develops during a series of dis-charge pulses and afterglows in three consecutive channels, corresponding to a realistic addressing procedure共Sec. III C兲. A. Discharge pulse
A PALC discharge is generated by a dc voltage pulse on the discharge electrodes. Figures 6 and 7 show the calculated steady state particle densities and electron mean energy un-der typical conditions at the end of the pulse, for helium and a mixture of 97% helium and 3% hydrogen.
The calculations clearly show a so-called cathode fall: a region in front of the cathode where the total positive ion density exceeds the electron density by orders of magnitude. The strong positive space charge density of the cathode fall screens the remainder of the channel largely from the applied electric field. This remaining region contains a quasineutral plasma. In the cathode fall region an extremely high electric field heats the electrons to a mean energy of about 50 eV, resulting in high electron impact ionization and excitation
rates. As a result particle species which are mainly created by electron impact reactions reach their maximum density around the cathode fall region, whereas species created in heavy particle reactions, such as He2⫹, are mainly present in the plasma region.
As can be seen comparing Fig. 6 to Fig. 7, the cathode fall in the helium–hydrogen mixture is very similar to the one in helium, but in the plasma region the role of the helium ions is taken over by the hydrogen ions.
By performing steady state dc discharge calculations for different discharge voltages, it is possible to calculate
current–voltage 共I–V兲 curves, which can be compared to
experimental data. Exact quantitative agreement cannot be expected, in view of the fact that the I–V curves are ex-FIG. 8. Calculated and measured I–V curves of pure helium at 150 Torr, for
different values of the channel height. The simulated discharge configura-tions are all similar to the one shown in Fig. 5; the discharges are created only in the middle channel.
FIG. 9. Decay of the numbers of particles in the afterglow in pure helium, per centimeter of channel length. Two electron groups are shown separately in this figure: e are the共thermalized兲 electrons produced by electron impact ionization, e* are the 共nonthermalized兲 electrons originating from metastable–metastable ionization.共See Sec. II D兲 The discharge configura-tion is shown in Fig. 5; all electrodes are grounded. The gas pressure is 150 Torr. This afterglow belongs to the discharge shown in Fig. 6.
FIG. 10. Decay of the numbers of particles in the afterglow in a mixture of 97% helium and 3% hydrogen, per centimeter of channel length. The dis-charge configuration is shown in Fig. 5, all electrodes are grounded. The gas pressure is 105 Torr. This afterglow belongs to the discharge shown in Fig. 7.
tremely sensitive to many delicate parameters, such as sec-ondary emission coefficients. However, trends in calculated I–V curves turn out to be in very good agreement with trends in measured I–V curves. As an example, Fig. 8 presents a study of the influence of the channel height on the I–V curve of a pure helium discharge. The figure shows that channel height does not affect the I–V curve as long as it is well
above 200m. For smaller channel heights the cathode fall
region is somewhat squeezed by the channel walls, which leads to an increase in the discharge voltage.
B. Afterglow
During the period immediately following the discharge pulse—the so-called afterglow—the plasma in the channel decays. In the simulations we assume the electrode voltage to go down abruptly to 0 V at the beginning of the afterglow; this assumption seems to be a reasonable approximation of the reality. Figures 9 and 10 show the simulated decay of the different particle species in the afterglow in pure helium, and in a mixture of 97% helium and 3% hydrogen, respectively.
Initially, the densities of most charged species increase somewhat due to the sudden decrease in drift losses. During the first part of the afterglow the electrons and ions are coupled together by space charge fields, resulting in ambipo-lar diffusion. As the plasma density decreases, the space charge fields become weaker until the ambipolar diffusion breaks and the electrons run off leaving the ions behind.
In pure helium however, the decay is not determined by diffusion alone. In this gas the decay of charged species is enormously slowed down by metastable–metastable ioniza-tions 共I-6兲 and 共I-7兲, which continue far into the afterglow. As can be seen in Fig. 9, the electrons produced by these processes leave the channel very quickly, indifferent to the ambipolar coupling, due to their relatively high energies.
共See Sec. II D.兲 The produced ions then lead to enhanced
trapping of the less energetic thermalized electrons remain-ing from the discharge. In helium–hydrogen mixtures the helium metastables are strongly quenched by Penning reac-tions共I-14兲 and 共I-16兲 so that the metastable–metastable ion-izations have no influence.
C. Addressing
In true PALC operation, small voltages are applied to the data electrodes during the discharge pulse and the afterglow, which leads to the buildup of surface charge on the mi-crosheet. The surface charge invokes an electric field in the FIG. 11. Typical transmission–voltage curve. Note that all light is
transmit-ted if the microsheet is uncharged.
FIG. 12. Two charge coupled device camera images of the same part of a PALC display. The channels are directed horizontally; three and a half chan-nels are shown. The two images correspond to two different addressing procedures: in the left picture the channels were addressed from the top to the bottom, in the right one from the bottom to the top. The horizontal black stripes are the opaque channel electrodes. All the light stripes result from insufficient charging of the microsheet: the sharp horizontal light stripes correspond to the interchannel ribs, the vertical light stripes to the gaps in between the data electrodes. Additional horizontal vague light stripes can be seen inside the channels, close to the ribs, indicating charging inhomogeni-ties. In the left picture these effects mainly take place at the bottom sides of the channels, in the right picture at the top sides.
FIG. 13. Electrode potentials as a function of time in a realistic addressing procedure. This figure relates to the three channel geometry shown in Fig. 5. The last three discharge pulses共in the channels 1, 2, and 3, respectively兲 represent one picture frame, where the rows are addressed from left to right. The first pulse共in channel 3兲 belongs to the previous frame. Typical values for the addressing and pulse times are on the order of 10s.
FIG. 14. Calculated electric potential in the three channel geometry of Fig. 5, at the end of the addressing procedure represented by Fig. 13. The unit of the indicated potentials is volt. The increment of the contours is 4 V.
LC layer, resulting in a certain transmission of the共second兲 polarizing layer. The percentage of light transmitted by the polarizer as function of the total voltage across the LC layer and the microsheet is given by a so-called transmission– voltage curve. Using the transmission–voltage curve shown in Fig. 11, it is possible to calculate transmission profiles of simulated PALC channels, which can be compared to experi-mental results. Transmission measurements on real PALC panels, such as shown in Fig. 12, have raised the suspicion that the buildup of surface charge in a certain channel is influenced by the surface charge present in the adjacent channels 共electrical crosstalk兲. A proper calculation of the surface charge and the resulting transmission profile there-fore requires the full simulation of not only the channel in question, but also of the two neighboring channels.
We used the three channel geometry of Fig. 5. Following a realistic addressing procedure, the channels were addressed one by one from the left to the right, every time inverting the
sign of the voltage on the data electrodes 共row inversion兲.
The exact addressing scheme is represented by Fig. 13. We
applied a data voltage of⫾20 V, corresponding to the
maxi-mum slope of the transmission–voltage curve, where the transmission is most sensitive to charging errors. The calcu-lated electric potential at the end of the three channel ad-dressing scheme is plotted in Fig. 14. It can clearly be seen, that the surface charge field in the LC layer and microsheet is directed oppositely for the consecutive channels, due to the row inversion technique. Note that the outer sides of the first and last channel are not properly modeled, given the artificial boundary conditions for the electric field at the edges of the simulation domain; only the middle channel has been treated correctly. Figure 15 shows the final transmission profile of this middle channel. In the center of the channel a homoge-neous transmission has been achieved, but near the inter-channel walls共ribs兲, transmission inhomogenities occur. The charging is incorrect especially on the right-hand side of the channel. This asymmetric effect is in full agreement with the
experimental observations shown in Fig. 12. It results from electrical crosstalk: At the moment the surface charge in the middle channel is established, the right neighbor channel contains a repelling surface charge with the same polarity, whereas the left neighbor channel has an attracting surface charge with opposite polarity. In fact, the phenomenon de-pends on the addressing procedure; if the channels are
ad-dressed in reverse order 共so: from right to left兲, the main
charging errors occur on the other side of the channel. Figure 15 also illustrates that a minimum of addressing time is required for a proper charging: If the addressing time is too short for the plasma to decay, the row inversion will partially erase the surface charge, leading to an increase of
the transmission. According to Fig. 15, at least 30 s of
addressing time are needed for pure helium channels. The minimal required addressing time is determined by the decay rate of the plasma; one can define a plasma decay time as the addressing time corresponding to a 1% increase in transmis-sion in the center of the channel. Figure 16 shows a compari-son of calculated and measured decay times for helium at different pressures. The helium decay time goes up as the pressure increases, due to an increase of the metastable den-sity and a decrease of the diffusion coefficients. In Fig. 17 the calculated and measured decay times for different helium–hydrogen mixtures are compared. Adding hydrogen to helium strongly decreases the decay time due to the quenching of the helium metastables. For all calculations the agreement with experiments is excellent.
IV. CONCLUSIONS
We have developed a two-dimensional共2D兲 fluid model
of PALC discharges. The model comprises continuity equa-tions and drift–diffusion equaequa-tions for plasma particle spe-cies, a balance equation for the electron energy, and Pois-son’s equation for the electric potential. The boundary conditions for electron transport include a correction for the directed motion of the electrons emitted by secondary emis-sion. The implementation of the model allows for arbitrary FIG. 15. Calculated transmission profiles of a PALC channel filled with
pure helium, for different addressing times. This graph corresponds to the middle channel of the geometry shown in Fig. 5. The vertical lines indicate positions of the channel walls. The gas pressure is 150 Torr. The pulse voltage is 260 V, the addressing voltage⫺ 20V.
FIG. 16. Decay times for pure helium at different gas pressures. Here the decay time is defined as the addressing time corresponding to a transmission error of 1% 共see text兲. For all calculations the discharge current is 3.0 mA/cm and the data voltage⫺20 V.
2D geometries and arbitrary gas mixtures. Reaction schemes and data are presented for pure helium and helium–hydrogen mixtures.
Using this model, it is possible to simulate the full PALC operation. We have reproduced a series of discharge pulses and afterglows in three consecutive PALC channels, filled with pure helium or helium–hydrogen mixtures. The simulations show that at the end of a discharge pulse a cath-ode fall and a plasma region are present in the channel in question. Calculated I–V curves of the PALC discharges are in good agreement with measurements. The simulations re-produce nonuniformities in the charging of the microsheet, known from experiments, and related to electrical crosstalk between adjacent channels. Calculated plasma decay times are in good agreement with measured decay times. In pure helium, the decay is slowed down enormously by a continu-ous plasma production through metastable–metastable ion-ization; in helium–hydrogen mixtures the helium meta-stables are quenched so that this effect does not take place. ACKNOWLEDGMENT
This work was supported by the Philips Research Labo-ratories in Eindhoven, The Netherlands.
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defined as the addressing time corresponding to a transmission error of 1%
共see text兲. For all calculations the gas pressure is 105 Torr, the discharge
