The Scope of this Thesis

Chapter 1 Introduction

1.4. The Scope of this Thesis

The main purpose of this thesis is to obtain of a two-dimensional fluid model to help

understand the behavior of the plasma backlight source, and the model can be used as a tool for optimizing the micro-discharge operating conditions, geometry, and gas composition.

Chapter 2 presents the numerical methods that can be used for the simulation of plasma backlight. It discusses the basis of the global and fluid models. We use the models to investigate the discharge phenomena. Chapter 3 provides all the details of the parameters used in this simulation. In the Chapter 4, we present the results of global and fluid modeling studies of plasma backlight discharge, respectively. The modeling results presented in this chapter will help us understand the mechanism of discharge of plasma backlight.

There are large numbers of variables in discharge experiments, so numerical simulation is a good useful tool for understanding the discharge mechanism and designing new cell structure.

Chapter 2 Numerical Methods

2.1. Global Model

The Global model was developed by Lieberman and Gottscho in 1994. It is based on the steady state model for a cylindrical plasma discharge of radius R and length L. The model assumes uniform spatial distribution of plasma parameters over the volume of bulk plasma, with the plasma density

n

^e in the bulk dropping sharply to edge values

n and

_sL

n at the

^sR thin sheaths close to the axial and circumferential walls. Electron-ion pairs are assumed to be created by electron-impact ionization of background gas and are lost by diffusive flow to the walls. The plasma is assumed electrically neutral and that the ion and electron fluxes toward the walls balance at all time.

2.1.1. Modeling Equations

Two main sets of equations are used in the global model: power balance and particle balance for all species of interest. For a monatomic gas, the equations are straightforward, as discussed by Lieberman and Gottscho.

The electron temperature ( , in units of eV) is simply a function of the pressure and geometry of the system, the plasma density is proportional to the input power, and the collisional energy loss per electron–ion pair created ( , in units of eV) is a function of only. For molecular gases, the situation is more complicated. As we will see, the plasma composition, i.e., ion and neutral densities, plays an important role in determining the electron temperature and .

T

E

T

E

The total power balance has the general form of

abs ev iw ew

P

P ( ) ¹

where is the power absorbed by the system, is the electron energy loss due to all electron–neutral collision processes in the volume, is the ion energy loss to the walls, and is the electron energy loss to the walls.

P

abs

P

_ev

P

The total surface particle loss to the total volume ionization,

(

)

0 _B 2 _L 2 _R _{iz g} 0

n u π R h

π RLh

K n n R L π ( ) ²

2.1.2. Derivation of Rate Constants

The rate coefficient for an electron impact collision is obtained by integrating the cross sections over an assumed Maxwellian distribution

( ) ⁴

₀

( )

is the Maxwellian velocity distribution, is the electron mass,

e

is the electron charge and is the electron temperature. Using the relationship for the velocity and kinetic energy of a particle,

( )

to be the normalized Maxwellian energy distribution. Thus

( )

^{3 2} ^{1 2}

is the rate coefficient.

2.1.3. Solution Procedures

The most important rate constants for electron collisions are , , for electron-neutral ionization, excitation and momentum transfer. There are given as a function of electron temperature. The collisional energy loss per electron-ion pair created,

K

_ex

K

_el coefficient for excited state and is the elastic scattering rate coefficient. The terms of the RHS account for the loss of electron energy due to ionization, excitation, and elastic (polarization) scattering against neutral atoms. The quantity

E

K

_iz

K

_ex

K

( ³ )

m M T

E

is the mean

energy lost per electron for a polarization scattering. is a function of only, depending on the electron-neutral species collisional energy loss process in the gas.

E

T

The total electron energy lost per ion lost from the system:

T = + +c e

E E E E

( ) ¹³

The quantity is the mean kinetic energy lost per electron lost. represents the mean kinetic energy lost per ion when it bombards a wall surface. It is the sum of the ion energy entering the sheath and the energy gained as it traverse the sheath.

e 2

T

E

At pressures for which the ion loss velocity is the Bohm velocity , the overall discharge power balance for a cylindrical plasma having radius R and length L can be written in terms of as edge, and A is the area for particle loss. The Bohm velocity is relatively constant for a given ion mass and for the typical limited range of

T s

_e' of 2-5 V. Hence

n is controlled by , A,

_s and .

E

P

abs

Godyak and Maximov have given heuristic equations for the axial sheath density

n

_sL and for the radial sheath density

n in the terms of dimensionless quantities

_sR and define as

at the axial sheath edge and

1 2

at the radial sheath edge, assuming cylindrical reactor geometry having a radius R and length L.

We consider a simple cylindrical discharge model to estimate the plasma parameters and their variation with power, pressure, and source geometry. The electron temperature

T

_e, the

ion-bombarding energy , and the plasma density are the most significant quantities for plasma processing applications.

E

n

₀

We first determine by equating the total surface particle loss to the total volume ionization. Introducing an effective plasma size ,

T

the particle balance equation can be written as :

( ) ( )

iz e

B e g eff

K T

u T

n d ( ) ¹⁸

Using these parameters, can be deduced as a function of neutral gas pressure and geometry of the system solely. The ion-bombarding energy is the sum of the ion energy entering the sheath and the energy gained by the ion as traverse the sheath. The potential over the sheath is taken to be equal to

The ion kinetic energy lost at a surface is then 1

i =

V

s+2

T

E

( ) ²⁰

Finally, we estimate the plasma charged particles density is proportional to the power dissipated by the discharge electrons according to

n

where

A is an effective area defined by

_eff

( )

eff

2

L R

A = π R Rh + Lh ( ) ²²

2.2. Fluid Model

In fluid models the charged particle properties are characterized by macroscopic quantities such as density, mean velocity, and mean energy that are solutions of the first three moments of the Boltzmann equation in velocity space. Only two moments are used to describe electron and ion transport: continuity equations, and simplified momentum transfer equation, in the drift-diffusion approximation. The system is closed by assuming that the charged particle mobility and ionization and excitation coefficients depend only on the local reduced electric field (E (x,t) / N or E (x,t) / P ).

( ) , : electric firld E x t

: filled gas

N

: filled gas pressure

P

The model consists of a set of fluid equation for electron, ions, and excited species, Poisson equation for electric field, and the appropriate boundary conditions.

2.2.1. Modeling Equations 2.2.1.1. Continuity Equation

The governing equations for a plasma fluid model are based on Maxwell’s equations coupled with moments of the Boltzmann equations describing the transport of the ion and electron component. For every plasma particle species, the time evolution of the density is described by a continuity equation

n t S

∂ + ∇ ⋅Γ =

∂^l ^l ^l

( ) ²³

where , , and are the number density, flux, and source term including the creation and the destruction reaction of species , respectively.

n

_l Γ_l

S

The creations of electrons and ions include the ionization, step ionization, and Penning

ionization. We use the local field approximation (LFA) method to calculate the electron-driven rate coefficients. The LFA assumes that the electron energy distribution function at a given location and in time depends on E(x,t) / N or E(x,t) /P at this location. This is equivalent to assuming that the electron energy gain due to the field at a given location and time is exactly balanced by the collision loss at the same location and time. The model overestimates the rate in the sheath region near the peak of discharge because the electrons don’t accomplish equilibrium with the field in high electric field region.

2.2.1.2. Momentum Equation with Drift-Diffusion Approximation

For the time scales of interest, the electrons are inertia-free, eliminating the electron momentum transfer equation, allowing invoking the drift-diffusion approximation. The flux is given by the momentum balance equation. Simplified momentum equation is followed.

Γ = − ∇ +_p

D n

_p _p

µ

_{p p}

n E ( ) ²⁴

Γ = − ∇ +_e

D n

_e _e

µ

_{e e}

n E ( ) ²⁵

Γ = −_ex

D

_ex∇

n

_ex

( ) ²⁶

: mobility for charged particles

µ

2.2.1.3. Poisson’s Equation for Electrostatic Potential

The self-consistent treatment of the charged particle transport is ensured by coupling the above set of equations with Poisson’s equation. The electric field profile is calculated by

solving Poisson’s equation. The quantities obtained from Eqs. (23) and (24) are coupled with

: surface charge density on the dielecric surface

σ

Integrating in time the electron and ion current densities arriving at the dielectric surface obtain the charge density σ on the dielectric surface.

2.2.1.4. Radiation Transport Equation

The model includes 6 excited states of Xe [i.e. Xe*(³P1), Xe*(³P2), Xe2*(Ou+) Xe2*(¹Σu+), Xe2*(³Σu+), Xe**], and emission spectra from the radiative excited states (i.e. 147 nm, 150 nm, 173 nm, 828 nm). Collision between neutral species and Penning ionization included as well the electron impact ionizations and excitations.

The governing equation of the resonant state density is the modified Holstein equation [Holstein, 1947 and 1951],

( ) ( ) ( ) ( ) ( ) ( )

includes the production rate by electron impact excitation, the depopulation of the excited states by step ionization, diffusion, and other collisions. The kernel function,

^G ( ^{r r} ^, ^′ )

^{, is the}

probability of radiation emitted at the position ′

r

being absorbed at the position . It is given by the expression,

r

(

)

₃

(

)

traversing a distance R without being absorbed.

For uniform ground state density, the transmission factor depends only on the distance between two positions and , and therefore it is shift invariant and isotropic in Cartesian coordinate.

: wavelength at the line center

, : degeneracies of lower and upper states respectively : frequency at the line center

: line width defined as the FWHM of the lineshape

The peak absorption coefficient

k

₀ at the line center is defined as

2.2.2. Discretization of Modeling Equations

2.2.2.1. Continuity Equation with Drift-Diffusion Approximation

As the gas pressure in the plasma backlight cells is usually high (~100 Torr), it was not found necessary to solve the conservation momentum equation per se. We instead used the drift-diffusion approximation, which considerably reduces the simulation time. The simulation geometry for two-dimensional plasma backlight cell is shown in Figure 2.1. We

calculate the electron density at the center of cell and potential at the grid point.

In order to calculate the density implicitly, we have to substitute a discretized expression for the drift-diffusion flux. We employ the Scharfetter-Gummel exponential [Scharfetter and Gummel, 1969] representation of the charged particle fluxes. The scheme supports large density gradients.

A standard different scheme of Eq. (25) would be lead to numerical instability whenever the voltage difference between grid point is of the order or larger than the characteristic energy

D µ

. The basic idea of the Scharfetter-Gummel method is to assume that the flux is constant between half grid points and is calculated at the grid point. Flux is interpolated with adjacent two half-grid points shown in Figure 2.2. Consider one direction (X-direction). Integration is followed:

(sign): signature denoting 0 (neutral species), -1 (electron), and 1 (ions) then analytic integration between m and m+1 lead to

¹²

m m

n

m m

n

∆ x

: the distance between mesh points.

The continuity equation is discriminated as follow:

, , 1/ 2, 1/ 2, , 1/ 2 , 1/ 2

We use the Alternating Direction Implicit (ADI) method [Reale, 1995] to integrate the continuity equation. Two time steps are used in two dimensions to update the quantities between and

t t + ∆ t

. The continuity equation in X direction is replaced with Eq. (34) as

The left side of Eq. (38) is calculated at time

t

^k^{% %}⁺¹ and the right side is obtained from present time . The two adjacent grid points (m+1 and m-1) are needed to update the densities at the grid point m. Matrix representation of Eq. (38) forms a tridiagonal matrix that has non-zero value in diagonal and its adjacent two elements. The density can be obtained from the inversion of tridiagonal matrix. The update from to

t

k accomplished with Y direction integration similar to previous matrix solver.

2.2.2.2. Poisson’s Equation

Poisson’s equation is solved with a Successive Over-Relaxation (SOR) [Kinder and Kushner, 2001] method. The electric field is taken at time t when the continuity equations are integrated between t and t + ∆t. The electric field in the integration of the continuity equation between t and t + ∆t is not the field at time t, but rather a prediction of the electric field at time

t + ∆t. The semi-implicit integration of Poisson’s equation is followed as:

( ^{εε φ}

⁰

) ^{e n} (

ⁿ

)

The continuity equations and momentum transfer equations for ion and electron is shown

( )

Continuity equations and momentum equations are coupled with Poisson’s equation to obtain the quantities.

( ) ( )

( )

The Poisson’s equation is discriminated x and y directions,

( )

We define the dielectric constant ε , number n , and mobility µ at the center of cell in our code (Figure 2.3).

, , 1 , , 1 , ,

For the calculation of the electric field, we make use of the electric potential and substitute the difference.

φ

is defined at the grid point.

1, ,

On this substitution Poisson’s equation becomes a five-point equation for the potential

1 1 1 1 1

, k1, , k1, , k, 1 , k, 1 , k, k, k, ,

i j i j i j i j i j i j i j i j i j i j i j i j i j

a φ

₊⁺

+ b φ

₋⁺

+ c φ

⁺₊

+ d φ

⁺₋

+ e φ

⁺

= ∆ tS + φ = f ( ) ⁴⁸

( ) ( )( )

: The surface charge density accumulating on intersection between plasma region and dielectric is obtained from boundary condition.

The classical SOR technique is often used in two-dimensional discharge modeling. We can calculate the plasma densities at time with the electric potential at a given location and time. Updated plasma densities have the electric potential redistributed using Poisson equation. The plasma densities are assumed updated in this method, the electric potential is frozen, and vice versa. There is a time limitation to valid the assumptions of the frozen plasma densities and electric potential.

2.2.2.3. Radiation Transport Equation

We use the piecewise constant approximation (PCA), the last term in Eq. (28) becomes

( ) ( ) ^{( )} ( )

The radiation transport matrix (Figure2.4) is defined as the spatial integral of the kernel function over the volume of the

; center of the

( ) ^{j k} ^,

_th

cell. The equation is simplified by analytic calculation,

( ) ( ) ( ) ( )

2.2.2.4. Boundary Condition

The boundary conditions for the above equations are an essential part of the description of the problem. The flux form at the left dielectric surface shown in Figure 2.5 is

1 ,

From the previous equation, we can find the coefficient

β

_{m n}_, . We calculated the flux at the grid point, is ahead

( ^∆ ^x ² )

than the position of densities. The m-1 (behind dielectric surface) and m (plasma region) are needed to calculate flux

( ^m ⁻ ^{1 2} )

at the boundary.

Because plasma densities are zero at m, only

β

_{m n}_, coefficient remains. With the same reason,

α

m n coefficient remains at the right dielectric surface.

The boundary condition of the surface charging equation at the interface between the dielectric surface and the plasma region is followed:

( ε

E

− ε

1 1

E ) ⋅ = n

σ ( ) ⁵⁴

: Electric field in the plasma region : Electric field at the dielectric surface

: Unit vector prependicular to the dielectric surface

: Time integration of charge densities entering into the diel

E E n

σ

ectric surface

Chapter 3 Simulation of Plasma Backlight

In this chapter we use two numerical models: the global model and fluid model presented in Chapter 2 to simulate the micro-discharge in a plasma backlight unit. The aim is to find out the actual improvements of the plasma backlight technology.

3.1 Global simulation

Here presents a simple analysis, for plasma applied to display panel backlight, by the global model which is introduced in Lieberman’s book. For the Xenon plasma, electron impact reaction rate from the integration of cross-section. The simulation cross sections for several reactions with respect to electron energy are shown in Figure 3.1.

is complex to calculate. There are some parameters for the analysis,

 Whole domain is a cylinder structure with radius 0.783 cm and height 0.28 cm

 Input power is 0.1 W, which is reduced by assuming panel size 32 x 30 x 0.28 cm³ and power 50W to a unit volume.

3.2 Fluid simulation

Table 3 outlines the characteristic properties of the micro-discharges considered in this section. We use the fluid model to simulate the micro-discharges in plasma backlight module.

All calculations presented here are based on the standard geometry in Figure 3.2. Figure 3.2 shows the electrode structure on bottom plate. In this approach, we solve the continuity equations for the species number density, using the drift-diffusion approximation for the species flux. The electron transport coefficients (Figure 3.5~3.8) (mobility,

µ

_e, diffusion

coefficient, , Electron energy, and Ionization coefficient) are from the BOLSIG, a user friendly code for the numerical solution of the Boltzmann equation for electrons in weakly ionized gases and in steady-state, uniform fields.

D

Table 3. Plasma backlight unit details and operating parameters.

Symbol (unit) Value

Protruded electrode length

l

(mm)

1.5 Protruded electrode width

w

(mm)

Electrode gap width

w

(mm)

5.4

Pulse frequency

f (kHz)

Applied pulse voltage amplitude

V

amp (V) 1740

Dielectric thickness

D

die (µm) 400

Dielectric relative permittivity ε 12

3.2.1. Applied Pulse Voltage

A plasma backlight discharge is generated by AC voltage pulses on the electrodes.

Figure3.7 is the waveform of the applied pulse voltage examined in this work. Va is the voltage amplitude. The duty ratio is defined as

r

d =

T

( T

on+

T

off

) , where

and are the periods of ‘on state’ and ‘off state’ of the pulse voltage. Figure 3.8 shows the pulse shapes of the applied voltages biased at the electrodes. Because the power supply is restrained unable to provide sufficient energy singly, so it uses two smaller voltages to build bigger one.

T

_off

Unipolar-pulsed excitation of Xenon excimer lamps has been proved to be able to improve dramatically the energy efficiency of DBDs in producing VUV excimer radiation [Liu and Neiger 2003]. The occurrence of the secondary discharge improves the performance of DBD technology. The VUV spectra of Xenon excimer radiation under unipolar pulse

excitation is identical to that under sine wave excitation. So we use unipolar-pulsed voltage in this simulation.

3.2.2. Parametric Studies

Simulation domain configuration is presented as Figure 2.1. The y-direction, w, between the cathode and anode is 9.4mm, the thickness of the dielectric lay

D

_die =400

µ m

and the relative permittivity of the dielectric is 12. At the center of electron 2 builds a tip with 4 mm height.

The grid dimensions used here were 165 points in the X-direction and 95 points in the Y-direction, for a total of 15,675. Applied voltage on electrode 1 is from -100 to 815 V while electrode 2 is from 300 to -525 V, with working frequency 55 kHz, duty ratio 35%, and two pulse voltages are the at same phase. Background gas pressure is 90 Torr. Gas and ions temperature are assumed 350 K.

Chapter 4 Results and Discussion

4.1 Global model

The variation of the plasma parameters, plasma density, sheath potential, electron temperature, mean free path, and debye length, with discharge pressure are shown in Figure 4.1~4.5. We assume a cylindrical stainless steel chamber with length L = 2.8 mm and radius R

= 7.83 mm with absorbed power 0.1W.

While pressure rising, the probability of ionization of Xenon will increase, so the electron density (Figure 4.1) increases gradually. Plasma density was varied between 6×10¹² and 5.5×10¹³ cm⁻³.

Sheath potential (Figure 4.2) is the voltage difference between bulk and boundary, which will affect material of edge. Figure 4.3 shows the electron temperature dependence on pressure for pure Xe discharges. The trends are with Te decreasing with increase pressure.

This variation in the electron temperature and sheath potential with pressure is due to the change in the electronegativity ratio with pressure. The electron energy reduced will influence the ionization rate. Mean free path, the average distance the particle travels between collisions with other particles, is shown with increasing pressure (Figure 4.4). Debye length, named after the Dutch physical chemist Peter Debye, is the scale over which mobile charge carriers (e.g. electrons) screen out electric fields in plasmas and other conductors. It is about 10^-4 cm.

4.2 Fluid model

The simulations of the discharge were performing with a two-dimensional fluid model to study the evolution of the plasma. The model is based on solutions of electron and ion transport equations coupled with Poisson’s equation for the electric field in a 2D Cartesian

geometry.

4.2.1 Typical Test Case

The gas and driving pulse are fixed as 90 Torr of Xe gas and amplitude 1740V with frequency 55 kHz and duty ratio 35 %. Applied voltage on electrode 1 is from -100 to 815 V while electrode 2 is from 300 to -525 V. Figure 4.6 shows the positions of the following results. Figure 4.7 is the contour plot of potential and related electric field distribution in different phase of pulse. Plasma potential stably sustains despite the changes of applied power.

Electric field, notability, is high in the sheath region, especially at tip edge, which has sharp shape and close to cathode. Figure 4.8~4.10 show the electron, ions, and excited species densities in different phase of pulse. During pulse rising, the electrons are mainly generated

在文檔中 TFT-LCD電漿背光源之二維流體模式模擬 (頁 20-0)

Chapter 1 Introduction

1.4. The Scope of this Thesis

Chapter 2

Numerical Methods

2.1. Global Model

n

n and

n at the

2.1.1. Modeling Equations

T

E

T

E

P

P

P

P ( ) 1

P

P

P

P

(

)

n u π R h

π RLh

K n n R L π ( ) 2

2.1.2. Derivation of Rate Constants

( ) 4

( )

( )

e

( )

( )

2.1.3. Solution Procedures

K

K

K

E

K

K

K

( 3 )

m M T

E

E

T

E E E E

( ) 13

T

E

E

T s

n is controlled by , A,

E

P

n

n in the terms of dimensionless quantities

T

E

n

T

( ) ( )

K T

u T

n d ( ) 18

V

T

E

( ) 20

n

A is an effective area defined by

( )

2

A = π R Rh + Lh ( ) 22

2.2. Fluid Model

( ) , : electric firld E x t

N

P

2.2.1. Modeling Equations 2.2.1.1. Continuity Equation

P ( ) ¹

K n n R L π ( ) ²

( ) ⁴

( ³ )

( ) ¹³

n d ( ) ¹⁸

( ) ²⁰

A = π R Rh + Lh ( ) ²²

( ) ²³

n E ( ) ²⁴

n E ( ) ²⁵

( ) ²⁶

^G ( ^{r r} ^, ^′ )

( ^{εε φ}

) ^{e n} (

ⁿ

= ∆ tS + φ = f ( ) ⁴⁸

( ) ( ) ^{( )} ( )

( ) ^{j k} ^,

( ^∆ ^x ² )

( ^m ⁻ ^{1 2} )

σ ( ) ⁵⁴