From 66ef7590cb47f237f793079dd4689c8a2941498c Mon Sep 17 00:00:00 2001 From: Sam Moore Date: Mon, 8 Oct 2012 17:14:05 +0800 Subject: [PATCH] This is the thesis AAAAAAAAAAAA --- thesis/simulation/egun/Makefile | 14 + thesis/simulation/egun/Makefile~ | 14 + thesis/simulation/egun/cont.dat | 6059 + thesis/simulation/egun/electron.dat | 432 + thesis/simulation/egun/main | Bin 0 -> 1899752 bytes thesis/simulation/egun/main.cpp | 175 + thesis/simulation/egun/main.cpp~ | 175 + thesis/simulation/egun/main.c~ | 115 + thesis/simulation/egun/main.dat | 257045 +++++++++++++++++++++++++ thesis/simulation/egun/movie.sh | 8 + thesis/simulation/egun/movie.sh~ | 8 + thesis/simulation/egun/plot.sh | 10 + thesis/simulation/egun/plot.sh~ | 10 + thesis/theory.py | 63 +- thesis/thesis.tex | 105 +- 15 files changed, 264206 insertions(+), 27 deletions(-) create mode 100644 thesis/simulation/egun/Makefile create mode 100644 thesis/simulation/egun/Makefile~ create mode 100644 thesis/simulation/egun/cont.dat create mode 100644 thesis/simulation/egun/electron.dat create mode 100755 thesis/simulation/egun/main create mode 100644 thesis/simulation/egun/main.cpp create mode 100644 thesis/simulation/egun/main.cpp~ create mode 100644 thesis/simulation/egun/main.c~ create mode 100644 thesis/simulation/egun/main.dat create mode 100755 thesis/simulation/egun/movie.sh create mode 100755 thesis/simulation/egun/movie.sh~ create mode 100755 thesis/simulation/egun/plot.sh create mode 100755 thesis/simulation/egun/plot.sh~ diff --git a/thesis/simulation/egun/Makefile b/thesis/simulation/egun/Makefile new file mode 100644 index 00000000..cbd961a8 --- /dev/null +++ b/thesis/simulation/egun/Makefile @@ -0,0 +1,14 @@ +movie.mp4 : main + ./main ../egun.png + ./movie.sh + +main : main.cpp + g++ -o main main.cpp -g -fopenmp -lm -lX11 -lpthread + + + +clean : + rm -f main + rm -f movie.mp4 + rm -f *.png + rm -f *.dat diff --git a/thesis/simulation/egun/Makefile~ b/thesis/simulation/egun/Makefile~ new file mode 100644 index 00000000..7a1cab8f --- /dev/null +++ b/thesis/simulation/egun/Makefile~ @@ -0,0 +1,14 @@ +movie.mp4 : main + ./main egun.png + ./movie.sh + +main : main.cpp + g++ -o main main.cpp -g -fopenmp -lm -lX11 -lpthread + + + +clean : + rm -f main + rm -f movie.mp4 + rm -f *.png + rm -f *.dat diff --git a/thesis/simulation/egun/cont.dat b/thesis/simulation/egun/cont.dat new file mode 100644 index 00000000..2791e31d --- /dev/null +++ b/thesis/simulation/egun/cont.dat @@ -0,0 +1,6059 @@ + +# Surface 0 of 1 surfaces + +# Curve title: ""main.dat" u 1:2:3" +# Contour 0, label: 0.75 + 166.225 205 0.75 + 166.225 204 0.75 + 166.225 203 0.75 + 166.225 202 0.75 + 166.225 201 0.75 + 166.225 200 0.75 + 166.225 199 0.75 + 166.225 198 0.75 + 166.225 197 0.75 + 166.225 196 0.75 + 166.225 195 0.75 + 166.225 194 0.75 + 166.225 193 0.75 + 166.225 192 0.75 + 166.225 191 0.75 + 166.225 190 0.75 + 166.225 189 0.75 + 166.225 188 0.75 + 166.225 187 0.75 + 166.225 186 0.75 + 166.225 185 0.75 + 166.225 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+#include + +using namespace std; +using namespace cimg_library; + +#define DELTA_X 1.0 +#define DELTA_Y 1.0 + +#define WIDTH 680 +#define HEIGHT 377 +#define ITERATIONS 2000 +#define EPSILON 0.00 + +#define OMEGA 1.0 + + +float grid[2][WIDTH][HEIGHT]; +bool fixed[WIDTH][HEIGHT]; + + + +int main(int argc, char ** argv) +{ + float beta2 = pow(DELTA_X / DELTA_Y, 2); + //printf("Beta2 is %f\n", beta2); + omp_set_nested(1); + + for (int i = 0; i < WIDTH; ++i) + { + for (int j = 0; j < HEIGHT; ++j) + { + grid[0][i][j] = 0; + fixed[i][j] = (i == 0 || j == 0 || i == WIDTH-1 || j == HEIGHT-1); + } + } + + // Add the electrodes + CImg src(argv[1]); + int w = src.width(); + int h = src.height(); + for (int i = 0; i < w; ++i) + { + for (int j = 0; j < h; ++j) + { + + int ii = i * (WIDTH / w); + int jj = j * (HEIGHT / h); + //printf("At %d %d vs %d %d\n", i, j, ii, jj); + + int rgb[3]; + for (int d = 0; d < 3; ++d) + { + //printf("rgb[%d] is %d\n", d, rgb[d]); + rgb[d] = (int)(src(i, j, 0, d)); + } + + grid[0][ii][jj] = float(rgb[0] - rgb[2]) / 255.0; + + if (rgb[0] == rgb[1] && rgb[0] == rgb[2] && rgb[0] == 255) + fixed[ii][jj] = false; + else + { + //printf("Fix potential %d %d at %f\n", ii, jj, grid[0][ii][jj]); + fixed[ii][jj] = true; + } + } + } + + int n = 1; int p = 0; + int k = 0; + float maxChange = 0.0; + for (k = 0; k < ITERATIONS; ++k) + { + maxChange = 0.0; + + #pragma omp parallel for + for (int i = 0; i < WIDTH; ++i) + { + //#pragma omp parallel for + for (int j = 0; j < WIDTH; ++j) + { + if (fixed[i][j] == true) + { + grid[n][i][j] = grid[p][i][j]; + continue; + } + + grid[n][i][j] = (1.0 - OMEGA) * grid[p][i][j]; + + + grid[n][i][j] += OMEGA * (grid[n][i-1][j] + grid[p][i+1][j] + beta2 * (grid[n][i][j-1] + grid[p][i][j+1])); + + grid[n][i][j] = grid[n][i][j] / (2.0 * (1.0 + beta2)); + + float change = (grid[n][i][j] > grid[p][i][j]) ? grid[p][i][j] - grid[n][i][j] : grid[p][i][j] - grid[n][i][j]; + if (grid[p][i][j] != 0.00) + change = change / grid[p][i][j]; + else + change = 1.0; + if (change >= maxChange) + maxChange = change; + + + + } + } + + //fprintf(stdout, "# Complete iteration %d; max change was %f\n", k, maxChange); + + + + n = (n == 1) ? 0 : 1; + p = (p == 1) ? 0 : 1; + + if (maxChange < EPSILON) + break; + + } + char file[100]; + sprintf(file, "main.dat"); + FILE * out = fopen(file, "w"); + + fprintf(out, "# Grid dimensions %d %d\n", WIDTH, HEIGHT); + fprintf(out, "# dx = %f, dy = %f, beta2 = %f\n", DELTA_X, DELTA_Y, beta2); + fprintf(out, "# Ran for %d iterations\n", k); + fprintf(out, "# Largest change is %f\n", maxChange); + for (int i = 0; i < WIDTH; ++i) + { + for (int j = 0; j < HEIGHT; ++j) + { + fprintf(out, "%f\t%f\t%f\n", i*DELTA_X, j*DELTA_Y, grid[p][i][j]); + } + fprintf(out, "\n"); + } + fprintf(out, "# DONE\n"); + if (out != stdout && out != stderr) fclose(out); + + + + out = fopen("electron.dat", "w"); + + float theta = 0.0; + float dt = 1.0; + float x[2] = {80, 195}; + float v[2]; + float a[2]; + v[0] = 1.0 * cos(theta); + v[1] = 1.0 * sin(theta); + for (float t = 0.0; ; t += dt) + { + if (x[0] <= 0 || x[0] >= WIDTH-1 || x[1] <= 0 || x[1] >= HEIGHT-1) + break; + + fprintf(out, "%f\t%f\t%f\t%f\t%f\n", t, x[0], x[1], v[0], v[1]); + + float ddx = 0.5 * (grid[p][(int)(x[0])+1][(int)(x[1])] - grid[p][(int)(x[0])-1][(int)x[1]]); + float ddy = 0.5 * (grid[p][(int)(x[0])][(int)(x[1])+1] - grid[p][(int)(x[0])][(int)(x[1])-1]); + v[0] += ddx; + v[1] += ddy; + x[0] += v[0]; x[1] += v[1]; + } + + if (out != stdout && out != stderr) fclose(out); + +} + +//EOF diff --git a/thesis/simulation/egun/main.cpp~ b/thesis/simulation/egun/main.cpp~ new file mode 100644 index 00000000..9ee96953 --- /dev/null +++ b/thesis/simulation/egun/main.cpp~ @@ -0,0 +1,175 @@ +#include +#include +#include +#include +#include + +#include + +#include + +using namespace std; +using namespace cimg_library; + +#define DELTA_X 1.0 +#define DELTA_Y 1.0 + +#define WIDTH 680 +#define HEIGHT 377 +#define ITERATIONS 2000 +#define EPSILON 0.00 + +#define OMEGA 1.0 + + +float grid[2][WIDTH][HEIGHT]; +bool fixed[WIDTH][HEIGHT]; + + + +int main(int argc, char ** argv) +{ + float beta2 = pow(DELTA_X / DELTA_Y, 2); + //printf("Beta2 is %f\n", beta2); + omp_set_nested(1); + + for (int i = 0; i < WIDTH; ++i) + { + for (int j = 0; j < HEIGHT; ++j) + { + grid[0][i][j] = 0; + fixed[i][j] = (i == 0 || j == 0 || i == WIDTH-1 || j == HEIGHT-1); + } + } + + // Add the electrodes + CImg src(argv[1]); + int w = src.width(); + int h = src.height(); + for (int i = 0; i < w; ++i) + { + for (int j = 0; j < h; ++j) + { + + int ii = i * (WIDTH / w); + int jj = j * (HEIGHT / h); + //printf("At %d %d vs %d %d\n", i, j, ii, jj); + + int rgb[3]; + for (int d = 0; d < 3; ++d) + { + //printf("rgb[%d] is %d\n", d, rgb[d]); + rgb[d] = (int)(src(i, j, 0, d)); + } + + grid[0][ii][jj] = float(rgb[0] - rgb[2]) / 255.0; + + if (rgb[0] == rgb[1] && rgb[0] == rgb[2] && rgb[0] == 255) + fixed[ii][jj] = false; + else + { + //printf("Fix potential %d %d at %f\n", ii, jj, grid[0][ii][jj]); + fixed[ii][jj] = true; + } + } + } + + int n = 1; int p = 0; + int k = 0; + float maxChange = 0.0; + for (k = 0; k < ITERATIONS; ++k) + { + maxChange = 0.0; + + #pragma omp parallel for + for (int i = 0; i < WIDTH; ++i) + { + //#pragma omp parallel for + for (int j = 0; j < WIDTH; ++j) + { + if (fixed[i][j] == true) + { + grid[n][i][j] = grid[p][i][j]; + continue; + } + + grid[n][i][j] = (1.0 - OMEGA) * grid[p][i][j]; + + + grid[n][i][j] += OMEGA * (grid[n][i-1][j] + grid[p][i+1][j] + beta2 * (grid[n][i][j-1] + grid[p][i][j+1])); + + grid[n][i][j] = grid[n][i][j] / (2.0 * (1.0 + beta2)); + + float change = (grid[n][i][j] > grid[p][i][j]) ? grid[p][i][j] - grid[n][i][j] : grid[p][i][j] - grid[n][i][j]; + if (grid[p][i][j] != 0.00) + change = change / grid[p][i][j]; + else + change = 1.0; + if (change >= maxChange) + maxChange = change; + + + + } + } + + //fprintf(stdout, "# Complete iteration %d; max change was %f\n", k, maxChange); + + + + n = (n == 1) ? 0 : 1; + p = (p == 1) ? 0 : 1; + + if (maxChange < EPSILON) + break; + + } + char file[100]; + sprintf(file, "main.dat"); + FILE * out = fopen(file, "w"); + + fprintf(out, "# Grid dimensions %d %d\n", WIDTH, HEIGHT); + fprintf(out, "# dx = %f, dy = %f, beta2 = %f\n", DELTA_X, DELTA_Y, beta2); + fprintf(out, "# Ran for %d iterations\n", k); + fprintf(out, "# Largest change is %f\n", maxChange); + for (int i = 0; i < WIDTH; ++i) + { + for (int j = 0; j < HEIGHT; ++j) + { + fprintf(out, "%f\t%f\t%f\n", i*DELTA_X, j*DELTA_Y, grid[p][i][j]); + } + fprintf(out, "\n"); + } + fprintf(out, "# DONE\n"); + if (out != stdout && out != stderr) fclose(out); + + + + out = fopen("electron.dat", "w"); + + float theta = 0.0; + float dt = 1.0; + float x[2] = {80, 195}; + float v[2]; + float a[2]; + v[0] = 1.0 * cos(theta); + v[1] = 1.0 * sin(theta); + for (float t = 0.0; ; t += dt) + { + if (x[0] <= 0 || x[0] >= WIDTH-1 || x[1] <= 0 || x[1] >= HEIGHT-1) + break; + + fprintf(out, "%f\t%f\t%f\t%f\t%f\n", t, x[0], x[1], v[0], v[1]); + + float ddx = 0.5 * (grid[p][(int)(x[0])+1][(int)(x[1])] - grid[p][(int)(x[0])-1][(int)x[1]]); + float ddy = 0.5 * (grid[p][(int)(x[0])][(int)(x[1])+1] - grid[p][(int)(x[0])][(int)(x[1])-1]); + v[0] -= ddx; + v[1] -= ddy; + x[0] += v[0]; x[1] += v[1]; + } + + if (out != stdout && out != stderr) fclose(out); + +} + +//EOF diff --git a/thesis/simulation/egun/main.c~ b/thesis/simulation/egun/main.c~ new file mode 100644 index 00000000..ff92a231 --- /dev/null +++ b/thesis/simulation/egun/main.c~ @@ -0,0 +1,115 @@ +#include +#include +#include +#include +#include + +#include + +#include + +#define DELTA_X 1.0 +#define DELTA_Y 1.0 + +#define WIDTH 100 +#define HEIGHT 100 +#define ITERATIONS 2000 +#define EPSILON 0.00 + +#define OMEGA 1.0 + + +float grid[2][WIDTH][HEIGHT]; +bool fixed[WIDTH][HEIGHT]; + + +int main(int argc, char ** argv) +{ + float beta2 = pow(DELTA_X / DELTA_Y, 2); + //printf("Beta2 is %f\n", beta2); + omp_set_nested(1); + + for (int i = 0; i < WIDTH; ++i) + { + for (int j = 0; j < HEIGHT; ++j) + { + grid[0][i][j] = 0; + fixed[i][j] = (i == 0 || j == 0 || i == WIDTH-1 || j == HEIGHT-1); + } + } + + // Add the electrodes + + + int n = 1; int p = 0; + int k = 0; + float maxChange = 0.0; + for (k = 0; k < ITERATIONS; ++k) + { + maxChange = 0.0; + #pragma omp parallel for + for (int i = 0; i < WIDTH; ++i) + { + #pragma omp parallel for + for (int j = 0; j < WIDTH; ++j) + { + if (fixed[i][j] == true) + { + grid[n][i][j] = grid[p][i][j]; + continue; + } + + grid[n][i][j] = (1.0 - OMEGA) * grid[p][i][j]; + + + grid[n][i][j] += OMEGA * (grid[n][i-1][j] + grid[p][i+1][j] + beta2 * (grid[n][i][j-1] + grid[p][i][j+1])); + + grid[n][i][j] = grid[n][i][j] / (2.0 * (1.0 + beta2)); + + float change = (grid[n][i][j] > grid[p][i][j]) ? grid[p][i][j] - grid[n][i][j] : grid[p][i][j] - grid[n][i][j]; + if (grid[p][i][j] != 0.00) + change = change / grid[p][i][j]; + else + change = 1.0; + if (change >= maxChange) + maxChange = change; + + + + } + } + + //fprintf(stdout, "# Complete iteration %d; max change was %f\n", k, maxChange); + + + + + char file[100]; + sprintf(file, "%d.dat", k); + FILE * out = fopen(file, "w"); + + fprintf(out, "# Grid dimensions %d %d\n", WIDTH, HEIGHT); + fprintf(out, "# dx = %f, dy = %f, beta2 = %f\n", DELTA_X, DELTA_Y, beta2); + fprintf(out, "# Ran for %d iterations\n", k); + fprintf(out, "# Largest change is %f\n", maxChange); + for (int i = 0; i < WIDTH; ++i) + { + for (int j = 0; j < HEIGHT; ++j) + { + fprintf(out, "%f\t%f\t%f\n", i*DELTA_X, j*DELTA_Y, grid[n][i][j]); + } + fprintf(out, "\n"); + } + fprintf(out, "# DONE\n"); + if (out != stdout && out != stderr) fclose(out); + + if (maxChange < EPSILON) + break; + + n = (n == 1) ? 0 : 1; + p = (p == 1) ? 0 : 1; + } + +} + +//EOF diff --git a/thesis/simulation/egun/main.dat b/thesis/simulation/egun/main.dat new file mode 100644 index 00000000..2f27a956 --- /dev/null +++ b/thesis/simulation/egun/main.dat @@ -0,0 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diff --git a/thesis/simulation/egun/movie.sh~ b/thesis/simulation/egun/movie.sh~ new file mode 100755 index 00000000..2dd2e2df --- /dev/null +++ b/thesis/simulation/egun/movie.sh~ @@ -0,0 +1,8 @@ +for i in *.dat; do + echo $i + out=$( echo $i | sed 's:\.dat::g') + gnuplot -e "set contour base; set cntrparam level incremental -1,0.25,1; unset surface; set table \"cont.dat\"; splot \"$i\" u 1:2:3; unset table; set term png size \"640,480\"; set output \"$out.png\"; set xrange [0:]; set yrange [0:]; set zrange [-1:1]; set title \"$i\"; unset key; set palette rgbformulae 33,13,10; plot \"$i\" w image, \"cont.dat\" w l lt -1 lw 1.5, \"electron.dat\" u 2:3 w lp" +done + +ffmpeg -r 10 -b 1024k -i %d.png movie.mp4 + diff --git a/thesis/simulation/egun/plot.sh b/thesis/simulation/egun/plot.sh new file mode 100755 index 00000000..66c1bfad --- /dev/null +++ b/thesis/simulation/egun/plot.sh @@ -0,0 +1,10 @@ +#!/bin/bash + +output=$2 +if [ "$output" != "" ]; then + output="set term png size \"640,480\"; set output \"$output.png\"" +fi + +plot="main.dat" + +gnuplot --persist -e "set contour base; set cntrparam level incremental -1,0.25,1; unset surface; set table \"cont.dat\"; splot \"$plot\" u 1:2:3; unset table; $output; set xrange [0:]; set yrange [0:]; set zrange [-1:1]; set title \"TCS Electron Gun\"; unset key; set palette rgbformulae 33,13,10; plot \"$plot\" w image, \"cont.dat\" w l lt -1 lw 1.5, \"$1\" u 2:3 w l lt 6 lw 3" diff --git a/thesis/simulation/egun/plot.sh~ b/thesis/simulation/egun/plot.sh~ new file mode 100755 index 00000000..8654e78c --- /dev/null +++ b/thesis/simulation/egun/plot.sh~ @@ -0,0 +1,10 @@ +#!/bin/bash + +output=$2 +if [ "$output" != "" ]; then + output="set term png size \"640,480\"; set output \"$output.png\"" +fi + +plot="main.dat" + +gnuplot --persist -e "set contour base; set cntrparam level incremental -1,0.25,1; unset surface; set table \"cont.dat\"; splot \"$plot\" u 1:2:3; unset table; $output; set xrange [0:]; set yrange [0:]; set zrange [-1:1]; set title \"$1\"; unset key; set palette rgbformulae 33,13,10; plot \"$plot\" w image, \"cont.dat\" w l lt -1 lw 1.5, \"$1\" u 2:3 w l lt 6 lw 3" diff --git a/thesis/theory.py b/thesis/theory.py index 7a85987e..193c6d3f 100755 --- a/thesis/theory.py +++ b/thesis/theory.py @@ -31,6 +31,54 @@ def se_dist(s, Ep, sigma_p, dE, n_max=0.75): E += dE return results +# +# @function integrate +# @purpose integrate function numerically over a range of arguments +# +def integrate(f, xmin, xmax, dx): + x = xmin + result = 0.00 + while (x <= xmax): + result += f(x) * dx + x += dx + return result + +# @function derivative +# @purpose get derivative of a function at a value +def der(f, x, dx): + return (f(x + dx) - f(x)) / dx + +# +# @function tcs +# @purpose Model S(E) +def tcs(f, sigma, Emin, Emax, dE): + results = [] + E = Emin + while (E < Emax): + results.append([E, (1.0 - sigma(0)) * f(-E) - integrate(lambda e : f(e - E) * der(sigma, E, dE), Emin, Emax, dE)]) + E += dE + return results + +def delta(x): + if (x == 0): + return 1.0 + else: + return 0.0 + +def table(f, xmin, xmax, dx): + result = [] + x = xmin + while (x <= xmax): + result.append([x, f(x)]) + x += dx + return result + +def gaussian(x, sigma): + return math.exp(- (x**2.0)/(2.0 * sigma**2.0)) / (sigma * (2.0 * math.pi)**0.50) + +def step(x, sigma, T): + return 1.0 / (math.exp((x - sigma)/T) + 1.0) + # # @function write_data # @purpose Write a list of data to a file suitable for gnuplotting @@ -43,19 +91,18 @@ def write_data(data, fileName): out.write("\n") out.close() -def Plot(*args): - gnuplot.plot(args) +def Plot(data): + gnuplot.replot(Gnuplot.Data(data, with_="lp")) def main(): - test = se_dist(4.0, 7.0, 0.2, 0.01) - #gnuplot.replot(map(lambda e : [e[0], e[1] + e[2]], test)) - #gnuplot.replot(map(lambda e : [e[0], e[2]], test)) - write_data(test, "se_dist.dat") + #test = se_dist(4.0, 7.0, 0.2, 0.01) + #write_data(test, "se_dist.dat") + - print("Press enter to exit") - sys.stdin.readline() + #print("Press enter to exit") + #sys.stdin.readline() return 0 diff --git a/thesis/thesis.tex b/thesis/thesis.tex index f917e36a..aaba3018 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -95,6 +95,8 @@ Finally, perhaps as a result of the aforementioned madness, I would also like t The report will be organised as follows; first we will discuss literature relating to surface science and nanostructured thin films in particular. We will then give an overview of the primary experimental techniques employed during this project, before presenting experimental results. Finally, we will discuss stuff. +\emph{NOTE: Introduction needs work} + \pagebreak \section{Overview} @@ -103,6 +105,8 @@ In this section we provide an overview of theoretical and experimental literatur \subsection{Secondary Electron Emission} + + % What is it? Secondary electron emission refers to the emission of electrons from a target surface caused by interactions with an incident electron, or more commonly a beam of electrons. Electrons from the incident beam are referred to as primary electons, whilst the term ``secondary electron'' applies to all electrons reflected or emitted from the surface under bombardment. @@ -111,7 +115,6 @@ Secondary electron emission refers to the emission of electrons from a target su % General description of secondary electron spectrum -Figure \ref{se_dist.pdf} shows the general shape of an energy distribution of secondary electrons. for a primary electron energy of $E_p$. The narrow peak centred at $E = E_p$ is due to elastically scattered electrons; the width of this peak is determined by the distribution of primary electron energies, as well as the resolution of the detector. The broad peak in the low energy part of the spectrum is due to inelastically scattered electrons. \begin{center} @@ -120,17 +123,42 @@ Figure \ref{se_dist.pdf} shows the general shape of an energy distribution of se \label{se_dist.pdf} \end{center} +\emph{NOTE: I need to draw some fine structure on this curve somehow. Or find an actual spectrum to reproduce.} + +Figure \ref{se_dist.pdf} shows a simplified model of an energy distribution of secondary electrons. The narrow peak centred at $E = E_p$ is largely due to elastically scattered electrons\footnote{A typical width is $0.5-1.0$eV; as a result, small energy losses due to phonon excitations ($10-50$meV) can not resolved from truly elastic reflections \cite{komolov}}; the width of this peak is determined by the distribution of primary electron energies, as well as the resolution of the detector. + + +The broad, asymetrical curve below $E = E_p$ is due to secondary electons. This part of the spectrum may be broadly divided into several regions based upon the mechanism for primary electron energy losses. +. For a more detailed discussion, refer to \cite{komolov}. + +Each process contributing to the secondary electron distribution is characterised by a threshold primary electron energy; below this threshold, the process can not occur. + + + + + +\subsection{Electron Surface Interaction} + +Here we will discuss models for the + % How real secondary electron spectra depend upon the surface Real secondary electron distributions also show fine structure imposed upon the inelastic part of simplified spectrum described above. This fine structure is characteristic of the electron spectra of the target surface. Near to the elastic peak, fine structure is caused by energy loss to interband transitions and plasma vibration excitation. The central part of the distribution contains fine structure due to Auger electron emission, and energy losses due to excitation of inner electrons. Fine structure at low energies is due to the structure of empty states of the solid. % Research of Komolov +\subsection{Electron Surface Interaction} + + \subsection{Plasmons} +\emph{NOTE: To be completely honest, I don't think I can say much about plasmons. The TCS experiment does not detect plasmonic behaviour. The Ellipsometer can be used to determine frequencies at which plasmons might occur. However, I have not seen any dips in $\epsilon$ which would indicate plasmon thresholds.} + % What are they? A plasmon is a quasi-particle arising due to charge density oscillations in a solid. + + % Early research \subsection{Metallic-Black Films} @@ -202,42 +230,72 @@ This project will employ Total Current Specroscopy, Ellipsometry and Optical Spe Secondary Electron Spectroscopy encompasses a large group of techniques which exploit secondary electron emission for studying the electron spectra of surfaces and solids. In these methods a beam of primary electrons is directed at a surface. The interactions between primary electrons and the surface give rise to an energy distribution of electrons elastically and inelastically scattered from the surface. Analysis of the distribution of the scattered ``secondary'' electrons gives information about the electron energy spectrum of the target surface. -Techniques of Secondary Electron Spectroscopy can be divided into two classes. Energy-resolved methods are based upon observation of the secondary electron distribution at a fixed primary electron energy. These methods aim to examine specific secondary emission processes which occur within a selected energy interval. The angular distribution of emitted electrons is often also recorded. +Techniques of Secondary Electron Spectroscopy can be divided into two classes. Energy-resolved methods are based upon observation of the secondary electron distribution at a fixed primary electron energy. The angular distribution of emitted electrons is often also recorded. These methods aim to examine the secondary emission processes which occur within a selected energy interval. + -In contrast to Energy-resolved methods, Total Current (or Yield) methods measure the total current of secondary electrons as a function of primary electron energy. The focus of this project has been low energy Total Current Spectroscopy (TCS). While Total Current methods provide less detailed information about secondary emission processes within a solid, they are generally simpler to realise experimentally as they do not require energy analysers, and current measurement may be performed external to the vacuum chamber, using a conventional ammeter. +In contrast to Energy-resolved methods, Total Current (or Yield) methods measure the total current of secondary electrons whilst varying the primary electron energy. As the primary electron energy reaches the threshold for a particular mechanism of secondary electron scattering, the Analysis of the total secondary electron current as a function of energy can give information about the threshold energies for particular mechanisms of energy loss; this can be in turn related to the +Total Current methods are generally simpler to realise experimentally compared with Energy-resolved methods, as they do not require energy analysers, and current measurement may be performed external to the vacuum chamber, using a conventional low current ammeter. The focus of this project has been the application of low energy Total Current Spectroscopy to characterise surfaces including Au-black films. \subsection{Total Current Spectroscopy} -Figure \ref{} shows a simplified schematic for the Total Current Spectroscopy experiments conducted during this study. An electron gun is used to produce the beam of primary electrons. Electrons are emitted from a cathode held at negative potential relative to the target. These electrons are focused into a beam and accelerated onto the target through the electric field produced by a series of electrodes. A detector is used to measure the total current passing through the target. +Figure \ref{tcs_simple.pdf} shows a simplified schematic for the Total Current Spectroscopy experiments conducted during this study. Electrons are produced via thermionic emission by heating a cathode. A series of electrodes are used to accelerate and focus a current $I_1$ onto the target. The energy of primary electrons is controlled by adjusting the power supply $U$, which determines the potential between the cathode and target. The transmitted current $I$ to flow through an ammeter external to the chamber. +\begin{center} + \includegraphics[scale=0.50]{figures/tcs_simple} + \captionof{figure}{Simplified Diagram of TCS Experiments} + \label{tcs_simple.pdf} +\end{center} +The goal of Total Current Spectroscopy is to measure variations in the secondary electron current $I_2$. It can easily be demonstrated that this can be accomplished by measurement of $I$. We will summarise the approach adopted by Komolov \cite{komolov}. -If the current incident upon the sample is $I_{\text{total}}$, and the current of secondary electrons scattered from the surface is $I_r$, then the transmitted current $I_t$ is given by: +From the above, it is obvious that $I = I_1 - I_2$. Assuming that $I$ is a constant, independent of primary electron energy $E_1$, we define the Total Current Spectrum (TCS) as: +\begin{align*} + S(E_1) &= \der{I}{E_1} = - \der{I_2}{E_1} +\end{align*} +This result also assumes that $I$ does not vary during the time taken to perform a measurement of $S(E_1)$ for a range of $E_1$ values. This is generally valid in the period after the cathode reaches thermal equilibrium. + +RelThe energy of a single primary electron arriving at the sample is given by $E = e U + c$, where $e$ is the electron charge, $U$ is the potential difference between cathode and sample, and $c$ a constant including the contact potential between the cathode and sample. +In reality, the cathode emits electrons with a distribution of energies, which is further altered by the focusing properties of the electrodes; as a result, the energy of the incident primary electrons is described by a distribution $f(E - E_1)$ about the mean value $E_1$, with the maximum of the distribution at $E = E_1$. +The primary electron current $I_1$ for a mean energy $E_1 = e U$ can be written as: \begin{align*} - I_t &= I_{\text{total}} - I_r + I_1(E_1) &= e A \int_{0}^{\infty} f(E - E_1) dE \end{align*} +Where $A$ is the surface area irradiated by the beam. -Generally $I_{\text{total}}$ is assumed to be independent of initial energy $E$. This assumption is valid if the initial energy is small compared to the accelerating potential of the gun, and the distance of the sample from the gun is sufficiently large. +Introducing the secondary emission coefficient $\sigma(E)$, which gives the probability for secondary electron emission to occur due to a primary electron of energy $E$, we can write the secondary electron current as: +\begin{align*} + I_2(E_1) &= e A \int_{-E_1}^{\infty} \sigma(E_1)f( E - E_1) dE +\end{align*} -In this case, differentiating with respect to $E$: +The total current $I$ may then be written as: \begin{align*} - \der{I_t}{E} &= - \der{I_r}{E} + I(E_1) &= e A \left[ \int_{0}^{\infty} f(E - E_1) dE - \int_{-E_1}^{\infty} \sigma(E_1)f( E - E_1) dE \right] \end{align*} -Figure \ref{block_diagram.pdf} is a block diagram of the experimental setup including measurement and control systems external to the vacuum chamber. -Note that the emission current ammeter is optional, and was used for testing purposes. +Differentiating, using the fundamental theorem of calculus, we can determine the total current spectrum: +\begin{align*} + S(E_1) = \der{I}{E_1} &= e A \left\{ [ 1 - \sigma(0)] f(-E_1) - \int_{0}^{\infty} f(E - E_1) \der{\sigma(E_1)}{E_1} dE \right\} +\end{align*} -\begin{center} - \includegraphics[scale=0.60]{figures/block_diagram.pdf} \label{block_diagram.pdf} -\end{center} +The first term in the above expression is determined solely by the distribution of primary electrons $f$. This term will be maximised when $E_1 = 0$; meaning that $U$ is equal to the contact potential $c$ between the cathode and sample. + +The second term contains all dependence of $S(E_1)$ on characteristics of the sample. At the threshold for a particular process, the secondary emission efficiency $\sigma(E_1)$ is expected to undergo a sharp change. This results in a well defined maxima or minima in the derivative $\der{\sigma(E_1)}{E_1}$, which can be seen as a corresponding maxima or minima in the total current spectrum $S(E_1)$. From the convolving function $f(E - E_1)$, it can be seen that the distribution of primary electron energy has the effect of broadening such a peak, and lowering its absolute height; the resolution of the method is determined by $f(E - E_1)$. It is desirable to achieve maximum resolution by accurate focusing of the electron gun. + + +\subsubsection{The Secondary Emission Coefficient} -\subsubsection{Electron Optics} +Having established a means for measuring $\sigma(E_1)$ + + + + +Often, the conventional ammeter and DC power supply in Figure \ref{tcs_simple.pdf} are replaced with a lock-in amplifier and AC power supply, as in Komolov's description \cite{komolov}. The power supply can be set to a fixed frequency, and the lock-in amplifier used to remove all other frequencies from the detected signal; this can be used to eliminate . Lock-in amplifier techniques also have the advantage of measuring S(E) directly; without the amplifier, finite differences must be used to approximate S(E) from measurement of $I(E)$. For this study, the lock-in amplifier approach was inpractical due to the limitations on available equipment. For future studies, it is suggested that the lock-in amplifier approach be adopted. + +For a more detailed description of the experimental setup, Refer to Appendix B for a discussion of hardware to automate the measurement of $I$ and control of $U$. Refer to Appendix D for a discussion of the electron gun and its control circuit. -The electron gun used for this experiment was repurposed from an old Cathode-Ray Oscilloscope (CRO). Figure \ref{} shows a simplified diagram of the electron gun, whilst Figure \ref{} shows a photograph of the gun. -The full circuit diagram for the electron gun control circuit is shown in Appendix A. %\cite{} \subsubsection{Automatic Data Acquisition} @@ -245,7 +303,7 @@ In order to collect data on the large number of planned samples for the study, s The available power supplies at CAMSP featured analogue inputs for external control. This meant that a Digital to Analogue Convertor (DAC) card was needed to interface between the control computer and the power supply. In addition, the available instruments for current measurement produced analogue outputs. As a result, Analogue to Digital Convertors (ADCs) were required to automate the recording of total current. -Although an external DAC/ADC box was already available for these purposes, initial tests showed that the ADCs on the box did not function. The decision was made to design and construct a custom DAC/ADC box, rather than wait up to two months for a commercial box to arrive. The design of the custom DAC/ADC box is discussed in detail in Appendix B, and the software written for the on-board microprocessor and the controlling computer are presented in Appendix D. +Although an external DAC/ADC box was already available for these purposes, initial tests showed that the ADCs on the box did not function. The decision was made to design and construct a custom DAC/ADC box, rather than wait up to two months for a commercial box to arrive. The design of the custom DAC/ADC box is discussed in detail in Appendix B, and the software written for the on-board microprocessor and the controlling computer are included in Appendix D. \subsection{Ellipsometry} @@ -254,7 +312,6 @@ Ellipsometry is an optical technique most commonly used to determine the thickne Essentially, ellipsometry measures the change in polarisation of light reflected from a surface. This change in polarisation can be related to properties of the surface if knowledge of the surface is correctly applied. For a bulk sample, the change in polarisation can be directly related to the optical constants of the material. -A @@ -287,6 +344,16 @@ This study focused primarily on depositing Au films on an Si substrate, at both \section{Experimental Results and Discussion} \subsection{TCS Measurements} +This study has focused on the evolution of the TCS of Au deposited on Si with increasing thickness of Au film. A general dependence of TCS curves on time has also been observed; it is likely that this time dependence is due to oxidation of the surface of the sample. + + + +\subsubsection{TCS of Si Substrate} + +Figure \ref{} shows a typical TCS from + +Figure \ref{} shows the TCS of Si in the (111) and (100) orientations. + \subsection{Ellipsometric Measurements} -- 2.20.1