EXPLORING SCANNING PROBE MICROSCOPY WITH MATHEMATICA

Transcription

1 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

2 CONTENTS PREFACE ^ 1 INTRODUCTION 1.1 General Comments Style Preparation 1.2 Units and Constants Units Physical Constants Materials Constants 1.3 Figures Figure Options Figure Examples 1.4 Recommended Books Mathematica Programming Lan Scanning Probe Microscopy 2 UNIFORM CANTILEVERS 2.1 Highlights 2.2 Abstract 2.3 Introduction 2.4 Operating Conditions Circular Cantilever Rectangular Cantilever 2.5 Area Moments of Inertia Circular Cantilever Rectangular Cantilever 2.6 Force in the z Direction: Bending Circular Cantilever Rectangular Cantilever 2.7 Force in the x Direction: Bending Circular Cantilever Rectangular Cantilever

3 viii CONTENTS 2.8 Force in the j Direction: Twisting Circular Cantilever Rectangular Cantilever Summary of Results Circular Cantilever Rectangular Cantilever Numerical Examples Spring Constant Bending Resonance Frequencies Characteristic Functions Exercises References 21 / 3 CANTILEVER CONVERSION TABLES Highlights Abstract Introduction Circular Cantilever Conversion Table Automatic Conversion Square Cantilever Conversion Table Automatic Conversion Exercises References 34 4 V-SHAPED CANTILEVERS Highlights Abstract Introduction Operating Conditions Spring Constant in the z Direction General Solution Special Cases Spring Constant in the x Direction General Solution Special Cases Resonance Frequencies Characteristic Functions Exercises References 47

4 CONTENTS ix TIP-SAMPLE ADHESION Highlights Abstract Introduction Operating Conditions Operating Conditions Evaluation of Operating Constants Identation as a Function of the Contact Force Contact Radius and the Contact Force Indentation and the Contact Radius Indentation and the Contact Force Inverted Functions Contact Force and the Contact Radius Contact Radius and the Indentation Contact Force and the Indentation Check of the Inversion Codes Limits of Adhesion Parameters Contact Pressure Maximum Contact Pressure Distribution of the Contact Pressure Lennard-Jones Potential Total Force as a Function of the Indentation Push-in Region Pull-out Region Hysteresis Loop Exercises References 74 TIP-SAMPLE FORCE CURVE Highlights Abstract Introduction Operating Conditions Tip-Sample Interaction Lennard-Jones Potential Lennard-Jones Force Lennard-Jones Force Derivative Morse Potential Hysteresis Loop Snap-in and Snap-out Points Calculated Hysteresis Loop Observed Hysteresis Loop 87

5 x CONTENTS 6.7 AR tip Product Product Obtained from the Snapping Points Product Obtained from the Area Total Tip-Sample Energy Total Energy Separate Energy Minima Total Energy Minima Animation Exercises References 99 " 7 FREE VIBRATIONS Highlights Abstract Introduction Operating Conditions Equation of Motion Numerical Solution Analytical Solution Exercises References NONCONTACT MODE Highlights Abstract Introduction Operating Conditions Tip-Sample Interaction Lennard-Jones Potential Numerical Solution Equation of Motion Numerical Solution Transient Regime Steady-State Regime Approximate Analytical Solution Equation of Motion Analytical Solution Steady-State Regime Exercises References 131

6 CONTENTS TAPPING MODE Highlights Abstract Introduction Operating Conditions Tip-Sample Interaction Lennard-Jones Potential Indentation Repulsive Force Total Tip-Sample Force Equation of Motion General Solution Transient Regime Steady-State Regime Summary of Results List of Key Results Four Figures Exercises References 149 METAL-INSULATOR-METAL TUNNELING Highlights Abstract Introduction Operating Conditions Tunneling Current Density General Solution Small Voltage Approximation Large Voltage Approximation The Image Potential Barrier with an Image Potential The Barrier Width Average Barrier Height Comparison of the Barriers Barrier without an Image Potential Barrier with an Image Potential Average Barrier Height The Plot of the Three Barriers Comparison of Tunneling Currents The General Solution with Image Potential 165

7 xii К CONTENTS Small Voltage Approximation without Image Potential Large Voltage Approximation without Image Potential Plot of the Three Tunneling Currents Apparent Barrier Height Exercises References 170 FOWLER-NORDHEIM TUNNELING Highlights Abstract Introduction Operating Conditions Fowler-Nordheim Current Density Numerical Example Oxide Field and Applied Field Oscillation Factor Averaged Oscillations Effective Tunneling Area Exercises References 182 \i SCANNING TUNNELING SPECTROSCOPY Highlights Abstract Introduction Scanning Tunneling Spectroscopy Fermi-Dirac Statistics Feenstra's Parameter File Reading Initialization File Characterization File Preparation Averaging Normalization Voltage Information Spectroscopic Data i(v) di\dv d\ni/d\nv Comparison of STS Results 195

8 CONTENTS xiii 12.7 Exercises References 197 COULOMB BLOCKADE Highlights Abstract Introduction Capacitance Sphere-Plane Capacitance Sphere-Sphere Capacitance Quantum Considerations Requirements and Approximations Coulomb Blockade and Coulomb Staircase Electrostatic Energy due to the Charging of the Center Electrode Electrostatic Energy due to the Applied Bias Total Electrostatic Energy Tunneling Rates Tunneling Current Examples Model Calculations Data Analysis Exercises References 210 DENSITY OF STATES Highlights Abstract Introduction Sphere in Arbitrary Dimensions Volume Area Solid Angle Density of States in Arbitrary Dimensions Volume in k-space Energy in the Parabolic Approximation Density of States Density of States in Confined Structures Quantum Well Quantum Wire Cubical Quantum Dot Spherical Quantum Dot 220

9 xiv CONTENTS 14.7 Exercises References ELECTROSTATICS Highlights Abstract Introduction Isolated Point-Charge Point-Charge and Plane Point-Charge and Sphere Isolated Sphere Sphere and Plane Position of Charges Inside the Sphere Magnitude of Charges Inside the Sphere Position of Charges Outside the Sphere Magnitude of Charges Outside the Sphere Table of Coefficients Potential and Field Potential along the Axis of Symmetry Capacitance Example Two Spheres Capacitance: Exact Solution Example Matrix Elements Capacitance: Approximate Solution Example Electrostatic Force Exercises References NEAR-FIELD OPTICS Highlights Abstract Introduction Operating Conditions Far-Field Solution Vector Potential Electric Field Magnetic Field Poynting Vector and Intensity 249

10 CONTENTS 16.6 Near-Field Solution Electric Field Magnetic Field Poynting Vector and Intensity Discussion of the Models Electric Field Intensity Exercises References 260 INDEX 261