EXPLORING SCANNING PROBE MICROSCOPY WITH MATHEMATICA Dror Sarid University of Arizona A WILEY-1NTERSCIENCE PUBLICATION JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto
CONTENTS PREFACE ^ 1 INTRODUCTION 1.1 General Comments 1.1.1 Style 1.1.2 Preparation 1.2 Units and Constants 1.2.1 Units 1.2.2 Physical Constants 1.2.3 Materials Constants 1.3 Figures 1.3.1 Figure Options 1.3.2 Figure Examples 1.4 Recommended Books 1.4.1 Mathematica Programming Lan 1.4.2 Scanning Probe Microscopy 2 UNIFORM CANTILEVERS 2.1 Highlights 2.2 Abstract 2.3 Introduction 2.4 Operating Conditions 2.4.1 Circular Cantilever 2.4.2 Rectangular Cantilever 2.5 Area Moments of Inertia 2.5.1 Circular Cantilever 2.5.2 Rectangular Cantilever 2.6 Force in the z Direction: Bending 2.6.1 Circular Cantilever 2.6.2 Rectangular Cantilever 2.7 Force in the x Direction: Bending 2.7.1 Circular Cantilever 2.7.2 Rectangular Cantilever
viii CONTENTS 2.8 Force in the j Direction: Twisting 12 2.8.1 Circular Cantilever 13 2.8.2 Rectangular Cantilever 13 2.9 Summary of Results 13 2.9.1 Circular Cantilever 13 2.9.2 Rectangular Cantilever 14 2.10 Numerical Examples 15 2.10.1 Spring Constant 15 2.10.2 Bending Resonance Frequencies 16 2.10.3 Characteristic Functions 18 2.11 Exercises 20 2.12 References 21 / 3 CANTILEVER CONVERSION TABLES 23 3.1 Highlights 23 3.2 Abstract 23 3.3 Introduction 23 3.4 Circular Cantilever 24 3.4.1 Conversion Table 24 3.4.2 Automatic Conversion 27 3.5 Square Cantilever 29 3.5.1 Conversion Table 29 3.5.2 Automatic Conversion 32 3.6 Exercises 33 3.7 References 34 4 V-SHAPED CANTILEVERS 35 4.1 Highlights 35 4.2 Abstract 35 4.3 Introduction 35 4.4 Operating Conditions 36 4.5 Spring Constant in the z Direction 36 4.5.1 General Solution 36 4.5.2 Special Cases 38 4.6 Spring Constant in the x Direction 41 4.6.1 General Solution 41 4.6.2 Special Cases 42 4.7 Resonance Frequencies 43 4.8 Characteristic Functions 45 4.9 Exercises 47 4.10 References 47
CONTENTS ix TIP-SAMPLE ADHESION 49 5.1 Highlights 49 5.2 Abstract 49 5.3 Introduction 49 5.4 Operating Conditions 50 5.4.1 Operating Conditions 50 5.4.2 Evaluation of Operating Constants 51 5.5 Identation as a Function of the Contact Force 53 5.5.1 Contact Radius and the Contact Force 53 5.5.2 Indentation and the Contact Radius 56 5.5.3 Indentation and the Contact Force 58 5.6 Inverted Functions 59 5.6.1 Contact Force and the Contact Radius 59 5.6.2 Contact Radius and the Indentation 61 5.6.3 Contact Force and the Indentation 62 5.6.4 Check of the Inversion Codes 63 5.7 Limits of Adhesion Parameters 64 5.8 Contact Pressure 65 5.8.1 Maximum Contact Pressure 65 5.8.2 Distribution of the Contact Pressure 66 5.9 Lennard-Jones Potential 67 5.10 Total Force as a Function of the Indentation 68 5.10.1 Push-in Region 69 5.10.2 Pull-out Region 71 5.10.3 Hysteresis Loop 71 5.11 Exercises 72 5.12 References 74 TIP-SAMPLE FORCE CURVE 75 6.1 Highlights 75 6.2 Abstract 75 6.3 Introduction 75 6.4 Operating Conditions 77 6.5 Tip-Sample Interaction 77 6.5.1 Lennard-Jones Potential 77 6.5.2 Lennard-Jones Force 79 6.5.3 Lennard-Jones Force Derivative 81 6.5.4 Morse Potential 82 6.6 Hysteresis Loop 83 6.6.1 Snap-in and Snap-out Points 83 6.6.2 Calculated Hysteresis Loop 85 6.6.3 Observed Hysteresis Loop 87
x CONTENTS 6.7 AR tip Product 91 6.7.1 Product Obtained from the Snapping Points 91 6.7.2 Product Obtained from the Area 92 6.8 Total Tip-Sample Energy 93 6.8.1 Total Energy 93 6.8.2 Separate Energy Minima 94 6.8.3 Total Energy Minima 95 6.9 Animation 97 6.10 Exercises 99 6.11 References 99 " 7 FREE VIBRATIONS 101 7.1 Highlights 101 7.2 Abstract 101 7.3 Introduction 101 7.4 Operating Conditions 102 7.5 Equation of Motion 103 7.5.1 Numerical Solution 104 7.5.2 Analytical Solution 108 7.6 Exercises 114 7.7 References 114 8 NONCONTACT MODE 115 8.1 Highlights 115 8.2 Abstract 115 8.3 Introduction 115 8.4 Operating Conditions 116 8.5 Tip-Sample Interaction 117 8.5.1 Lennard-Jones Potential 117 8.6 Numerical Solution 119 8.6.1 Equation of Motion 120 8.6.2 Numerical Solution 120 8.6.3 Transient Regime 121 8.6.4 Steady-State Regime 121 8.7 Approximate Analytical Solution 125 8.7.1 Equation of Motion 125 8.7.2 Analytical Solution 127 8.7.3 Steady-State Regime 127 8.8 Exercises 131 8.9 References 131
CONTENTS TAPPING MODE 133 9.1 Highlights 133 9.2 Abstract 133 9.3 Introduction 133 9.4 Operating Conditions 134 9.5 Tip-Sample Interaction 136 9.5.1 Lennard-Jones Potential 136 9.5.2 Indentation Repulsive Force 137 9.5.3 Total Tip-Sample Force 138 9.6 Equation of Motion 139 9.6.1 General Solution 140 9.6.2 Transient Regime 141 9.6.3 Steady-State Regime 142 9.7 Summary of Results 145 9.7.1 List of Key Results 145 9.7.2 Four Figures 147 9.8 Exercises 149 9.9 References 149 METAL-INSULATOR-METAL TUNNELING 151 10.1 Highlights 151 10.2 Abstract 151 10.3 Introduction 151 10.4 Operating Conditions 153 10.5 Tunneling Current Density 153 10.5.1 General Solution 153 10.5.2 Small Voltage Approximation 154 10.5.3 Large Voltage Approximation 156 10.6 The Image Potential 157 10.7 Barrier with an Image Potential 158 10.7.1 The Barrier Width 160 10.7.2 Average Barrier Height 162 10.8 Comparison of the Barriers 163 10.8.1 Barrier without an Image Potential 163 10.8.2 Barrier with an Image Potential 164 10.8.3 Average Barrier Height 164 10.8.4 The Plot of the Three Barriers 164 10.9 Comparison of Tunneling Currents 165 10.9.1 The General Solution with Image Potential 165
xii К CONTENTS 10.9.2 Small Voltage Approximation without Image Potential 166 10.9.3 Large Voltage Approximation without Image Potential 166 10.9.4 Plot of the Three Tunneling Currents 166 10.10 Apparent Barrier Height 167 10.11 Exercises 170 10.12 References 170 FOWLER-NORDHEIM TUNNELING 171 11.1 Highlights 171 11.2 Abstract 171 11.3 Introduction 171 11.4 Operating Conditions 172 11.5 Fowler-Nordheim Current Density 173 11.6 Numerical Example 175 11.7 Oxide Field and Applied Field 176 11.8 Oscillation Factor 177 11.9 Averaged Oscillations 180 11.10 Effective Tunneling Area 181 11.11 Exercises 182 11.12 References 182 \i SCANNING TUNNELING SPECTROSCOPY 185 12.1 Highlights 185 12.2 Abstract 185 12.3 Introduction 185 12.3.1 Scanning Tunneling Spectroscopy 186 12.3.2 Fermi-Dirac Statistics 186 12.3.3 Feenstra's Parameter 189 12.4 File Reading 189 12.4.1 Initialization 189 12.4.2 File Characterization 190 12.5 File Preparation 190 12.5.1 Averaging 190 12.5.2 Normalization 191 12.5.3 Voltage Information 192 12.6 Spectroscopic Data 192 12.6.1 i(v) 192 12.6.2 di\dv 193 12.6.3 d\ni/d\nv 194 12.6.4 Comparison of STS Results 195
CONTENTS xiii 12.7 Exercises 197 12.8 References 197 COULOMB BLOCKADE 199 13.1 Highlights 199 13.2 Abstract 199 13.3 Introduction 199 13.4 Capacitance 201 13.4.1 Sphere-Plane Capacitance 201 13.4.2 Sphere-Sphere Capacitance 202 13.5 Quantum Considerations 202 13.6 Requirements and Approximations 203 13.7 Coulomb Blockade and Coulomb Staircase 204 13.7.1 Electrostatic Energy due to the Charging of the Center Electrode 204 13.7.2 Electrostatic Energy due to the Applied Bias 205 13.7.3 Total Electrostatic Energy 205 13.8 Tunneling Rates 206 13.9 Tunneling Current 206 13.10 Examples 207 13.11 Model Calculations 208 13.12 Data Analysis 209 13.13 Exercises 210 13.14 References 210 DENSITY OF STATES 211 14.1 Highlights 211 14.2 Abstract 211 14.3 Introduction 211 14.4 Sphere in Arbitrary Dimensions 212 14.4.1 Volume 212 12.4.2 Area 213 12.4.3 Solid Angle 213 14.5 Density of States in Arbitrary Dimensions 214 14.5.1 Volume in k-space 214 14.5.2 Energy in the Parabolic Approximation 214 14.5.3 Density of States 215 14.6 Density of States in Confined Structures 217 14.6.1 Quantum Well 217 14.6.2 Quantum Wire 218 14.6.3 Cubical Quantum Dot 218 14.6.4 Spherical Quantum Dot 220
xiv CONTENTS 14.7 Exercises 222 14.8 References 222 15 ELECTROSTATICS 223 15.1 Highlights 223 15.2 Abstract 223 15.3 Introduction 223 15.4 Isolated Point-Charge 224 15.5 Point-Charge and Plane 226 15.6 Point-Charge and Sphere 227 15.7 Isolated Sphere 229 15.8 Sphere and Plane 230 15.8.1 Position of Charges Inside the Sphere 231 15.8.2 Magnitude of Charges Inside the Sphere 231 15.8.3 Position of Charges Outside the Sphere 232 15.8.4 Magnitude of Charges Outside the Sphere 232 15.8.5 Table of Coefficients 233 15.8.6 Potential and Field 233 15.8.7 Potential along the Axis of Symmetry 235 15.8.8 Capacitance 235 15.8.9 Example 236 15.9 Two Spheres 237 15.9.1 Capacitance: Exact Solution 237 15.9.2 Example 238 15.9.3 Matrix Elements 239 15.9.4 Capacitance: Approximate Solution 239 15.9.5 Example 239 15.10 Electrostatic Force 240 15.11 Exercises 241 15.12 References 241 16 NEAR-FIELD OPTICS 243 16.1 Highlights 243 16.2 Abstract 243 16.3 Introduction 243 16.4 Operating Conditions 244 16.5 Far-Field Solution 245 16.5.1 Vector Potential 245 16.5.2 Electric Field 246 16.5.3 Magnetic Field 249 16.5.4 Poynting Vector and Intensity 249
CONTENTS 16.6 Near-Field Solution 252 16.6.1 Electric Field 252 16.6.2 Magnetic Field 256 16.6.3 Poynting Vector and Intensity 257 16.7 Discussion of the Models 258 16.7.1 Electric Field 258 16.7.2 Intensity 259 16.8 Exercises 260 16.9 References 260 INDEX 261