Part 1 The Applications
1. Introduction ................................................. 3
1.1 Scope of X-ray Metrology (XRM) ........................... 3
1.2 Specular X-ray Reflectivity (XRR) ........................ 7
1.3 Diffuse Scatter ......................................... 11
1.4 X-ray Diffraction ....................................... 16
1.5 High-Resolution X-ray Diffraction ....................... 21
1.6 Diffraction Imaging and Defect Mapping .................. 24
1.7 X-ray Fluorescence ...................................... 27
1.8 Summary ................................................. 29
2. Thickness Metrology ......................................... 31
2.1 Introduction ............................................ 31
2.2 Dielectrics and Metals .................................. 32
2.2.1 Interferometric Methods ........................... 32
2.2.2 Intensity Methods ................................. 35
2.3 Multiple Layers ......................................... 39
2.4 Epitaxial Layers ........................................ 42
2.4.1 Interferometric Methods ........................... 42
2.4.2 Intensity Methods ................................. 44
2.4.3 Small Measurement Spots ........................... 44
2.4.4 Comparison of XRR and XRD for Epitaxial
Thickness Metrology ............................... 45
2.5 Summary ................................................. 46
3. Composition and Phase Metrology ............................. 47
3.1 Introduction ............................................ 47
3.2 Amorphous Films ......................................... 48
3.3 Polycrystalline Films ................................... 52
3.4 Wafers and Epitaxial Films .............................. 53
3.4.1 Variation of Lattice Parameter with Composition:
Vegard's Law ...................................... 53
3.4.2 Coherency Distortion in Epilayers ................. 54
3.4.3 Absolute Lattice Parameter Measurements ........... 55
3.4.4 Relative Lattice Parameter Measurements ........... 57
3.5 Summary ................................................. 59
References .................................................. 60
4. Strain and Stress Metrology ................................. 61
4.1 Introduction ............................................ 61
4.2 Strain and Stress in Polycrystalline Layers ............. 62
4.2.1 sin2ψ Analysis .................................... 62
4.2.2 GIIXD Analysis .................................... 62
4.3 Relaxation of Epitaxial Layers .......................... 67
4.3.1 Relaxation in SiGe ................................ 70
4.3.2 Relaxation in Compound Semiconductors ............. 71
4.3.3 Relaxation in Compounds Based on GaN .............. 72
4.4 Thin Strained Silicon Layers ............................ 73
4.5 Whole Wafer Defect Metrology ............................ 75
4.6 Summary ................................................. 76
References .................................................. 77
5. Mosaic Metrology ............................................ 79
5.1 Grain Size Measurement .................................. 79
5.2 Mosaic Structure in Substrate Wafers .................... 81
5.3 Mosaic Structure in Epilayers ........................... 82
5.4 Summary ................................................. 86
References .................................................. 86
6. Interface Roughness Metrology ............................... 87
6.1 Interface Width and Roughness ........................... 87
6.2 Distinction of Roughness and Grading .................... 90
6.2.1 Measurement by Grazing Incidence Rocking Curves ... 90
6.2.2 Measurement by Off-Specular Specimen Detector
Scans ............................................. 92
6.3 Roughness Determination in Semiconductors ............... 92
6.4 Roughness Determination in Metallic Films ............... 94
6.5 Roughness Determination in Dielectrics .................. 96
6.6 Summary ................................................. 97
References .................................................. 97
7. Porosity Metrology .......................................... 99
7.1 Determination of Porosity ............................... 99
7.2 Determination of Pore Size and Distribution ............ 100
7.3 Pores in Single Crystals ............................... 106
7.4 Summary ................................................ 106
References ................................................. 107
Part 2 The Science
8. Specular X-ray Reflectivity ................................ 111
8.1 Principles ............................................. 111
8.2 Specular Reflectivity from a Single Ideal Interface .... 115
8.3 Specular Reflectivity from a Single Graded or Rough
Interface .............................................. 116
8.4 Specular Reflectivity from a Single Thin Film on a
Substrate .............................................. 119
8.5 Specular Reflectivity from Multiple Layers on a
Substrate .............................................. 122
8.5.1 Reflectivity from a Bilayer ...................... 124
8.5.2 Reflectivity from a Periodic Multilayer .......... 125
8.6 Summary ................................................ 127
References ................................................. 127
9. X-ray Diffuse Scattering ................................... 129
9.1 Origin of Diffuse Scatter from Surfaces and
Interfaces ............................................. 129
9.2 The Born Approximation ................................. 130
9.2.1 Interface Modeling within the Born
Approximation .................................... 133
9.3 The Distorted-Wave Born Approximation .................. 135
9.3.1 Separation of Topological Roughness and
Compositional Grading within the DWBA .................. 137
9.4 Effect of Interface Parameters on Diffuse Scatter ...... 140
9.5 Multiple-Layer Structures .............................. 141
9.6 Diffuse Scatter Represented in Reciprocal Space ........ 145
9.6.1 Specular Scan .................................... 146
9.6.2 Off-Specular Coupled Scan ........................ 147
9.6.3 Transverse Scan .................................. 148
9.6.4 Radial Scan ...................................... 148
9.6.5 Transformation from Angular Coordinates to
Reciprocal Space Units ........................... 149
9.7 Summary ................................................ 150
References ................................................. 150
10. Theory of XRD on Polycrystals ............................. 151
10.1 Introduction ......................................... 151
10.1.1 Mathematical Health Warning ................... 152
10.2 Kinematical Theory of X-ray Diffraction .............. 152
10.2.1 Scattering from a Small Crystal ............... 155
10.2.2 The Reciprocal Lattice ........................ 158
10.2.3 Intensity Diffracted from a Thin Crystal ...... 159
10.3 Determination of Strain .............................. 162
10.4 Determination of Grain Size .......................... 164
10.3 Texture .............................................. 166
10.6 Reciprocal Space Geometry ............................ 167
10.7 Summary .............................................. 170
References ............................................... 171
11. High-Resolution XRD on Single Crystals .................... 173
11.1 Introduction ......................................... 173
11.2 Dynamical Theory of X-ray Diffraction ................ 174
11.2.1 The Takagi-Taupin Generalized Diffraction
Theory ........................................ 176
11.2.2 Thin-Layer and Substrate Solutions ............ 178
11.2.3 Calculation of Strains and Mismatches ......... 179
11.3 The Determination of Epilayer Parameters ............. 181
11.3.1 Selection of Experimental Conditions .......... 181
11.3.2 Measuring Composition ......................... 183
11.3.3 Measuring Thickness ........................... 185
11.3.4 Measuring Tilt ................................ 187
11.3.5 Measuring Curvature and Mosaic Spread ......... 187
11.3.6 Measuring Dislocation Content ................. 189
11.3.7 Measuring Relaxation .......................... 190
11.4 High-Resolution Diffraction in Real and Reciprocal
Space ................................................ 193
11.4.1 Triple-Axis Scattering ........................ 193
11.4.2 Setting up a Triple-Axis Measurement .......... 194
11.4.3 Separation of Lattice Tilts and Strains ....... 194
11.4.4 Reciprocal Space Mapping ...................... 197
11.4.5 The Relaxation Scan ........................... 201
11.4.6 Grazing Incidence In-Plane Diffraction ........ 204
11.5 Summary .............................................. 207
References ................................................ 207
12. Diffraction Imaging and Defect Mapping .................... 209
12.1 Introduction ......................................... 209
12.2 Contrast in X-ray Diffraction Imaging (XRDI) ......... 209
12.2.1 Images of Dislocations ........................ 212
12.3 Spatial Resolution in XRDI ........................... 216
12.3.1 Real-Time Image Detectors ..................... 217
12.4 X-ray Defect Imaging Methods ......................... 219
12.4.1 Lang Projection Topography .................... 219
12.4.2 The BedeScan Method ........................... 220
12.4.3 Section Topography ............................ 223
12.5 Example Applications ................................. 224
12.6 Summary .............................................. 228
References ................................................ 229
Part 3 The Technology
13. Modeling and Analysis ..................................... 233
13.1 What Has Been Measured? .............................. 233
13.2 Direct Methods ....................................... 234
13.3 Data-Fitting Methods ................................. 236
13.4 The Differential Evolution Method .................... 238
13.4.1 The Objective Function ........................ 241
13.4.2 Performance and Examples ...................... 242
13.5 Requirements for Automated Analysis .................. 246
13.6 Summary .............................................. 246
References ................................................ 247
14. Instrumentation ........................................... 249
14.1 Introduction ......................................... 249
14.2 X-ray Sources ........................................ 249
14.3 X-ray Optics ......................................... 251
14.4 Mechanical Technology ................................ 254
14.4.1 Angle Measurement and Calibration ............. 254
14.5 Detectors ............................................ 254
14.6 Practical Realizations ............................... 255
14.7 Summary .............................................. 255
References ................................................ 257
15. Accuracy and Precision of X-ray Metrology ................. 259
15.1 Introduction ......................................... 259
15.2 Design of X-ray Metrology ............................ 260
15.3 Repeatability and Reproducibility .................... 260
15.4 Accuracy and Trueness ................................ 261
15.4.1 X-ray Reflectivity ............................ 262
15.4.2 High-Resolution X-ray Diffraction ............. 262
15.5 Repeatability and Throughput ......................... 263
15.6 Absolute Tool Matching ............................... 265
15.7 Specimen-Induced Limitations ......................... 266
15.7.1 Effect of Layer Defects ....................... 266
15.7.2 Where Is the Surface? ......................... 266
15.7.3 Comparisons of XRM with Other Metrologies ..... 268
15.8 Summary .............................................. 269
References ................................................ 270
Index ......................................................... 271
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