Hall Effect Devices, Second Edition (Series in Sensors)

Hall Effect Devices, Second Edition......Page 1
Contents......Page 5
Preface......Page 11
Acknowledgments......Page 12
1.1 The basic physical effects......Page 15
1.1.1 The Hall effect......Page 16
1.1.2 The magnetoresistance effect......Page 18
1.2 Hall effect devices and their applications......Page 19
1.2.1 Hall plates......Page 20
1.2.2 Other Hall effect devices......Page 21
1.2.3 Applications of Hall effect devices......Page 22
References......Page 23
2.1.1 Conductor, semiconductor, insulator......Page 25
2.1.2 Permitted and forbidden energy bands......Page 26
2.1.3 Electrons in solids......Page 28
2.1.4 Electrons in the conduction band......Page 29
2.1.5 Holes in the valence band......Page 31
2.2.1 Density of states......Page 33
2.2.2 Distribution function......Page 35
2.2.3 Carrier concentration......Page 37
2.2.4 Intrinsic semiconductors......Page 38
Dopants......Page 39
Carrier concentration......Page 40
Saturation range......Page 42
Freeze-out range......Page 43
Example......Page 45
2.2.6 Heavy doping effects......Page 46
2.3 Scattering of carriers......Page 47
2.3.1 Free transit time......Page 48
2.3.2 Relaxation time......Page 51
2.4 Charge transport......Page 52
2.4.1 Drift in a weak electric field......Page 53
Drift velocity......Page 55
Mobility......Page 56
Current density......Page 60
2.4.2 High electric fields......Page 62
Saturation of drift velocity......Page 63
2.4.3 Diffusion current......Page 64
2.4.4 Non-stationary transport......Page 67
2.5 Other relevant phenomena......Page 68
2.5.1 Recombination and generation......Page 69
2.5.2 PN junction......Page 70
2.5.3 Noise......Page 71
2.5.4 Piezoresistance......Page 73
References......Page 75
3: Galvanomagnetic effects......Page 77
Smooth-drift approximation......Page 78
3.1.1 The (original) Hall effect......Page 79
Hall electric field......Page 80
Hall voltage......Page 81
Hall coefficient......Page 82
Example......Page 83
3.1.2 The current deflection effect......Page 85
Corbino disc......Page 86
Hall angle......Page 87
3.1.3 The magnetoresistance effect......Page 88
3.1.4 Galvanomagnetic current components......Page 90
3.1.5 Charge carrier transport in very high magnetic fields......Page 95
Free charged particle......Page 96
Charge carrier in a solid......Page 97
3.1.6 The role of collisions......Page 98
3.1.7 Summary and further questions......Page 100
3.2 Kinetic equation approach......Page 102
3.2.1 Kinetic equation......Page 103
3.2.2 Relaxation time......Page 105
3.2.3 Solution for the steady state......Page 106
Small disturbance of the distribution function......Page 107
Correction of the distribution function......Page 108
3.2.4 Non-equilibrium distribution function......Page 109
3.2.5 Galvanomagnetic transport of carriers. Current density......Page 111
Integrals determining the carrier transport......Page 112
Galvanomagnetic current density......Page 114
3.2.6 Kinetic coefficients......Page 115
Non-degenerate carriers......Page 116
3.2.7 Magnetic field dependence of the kinetic coefficients......Page 118
3.3.1 Basic equations......Page 120
3.3.2 Drift current density at B= 0......Page 122
3.3.3 Current density in a plate under perpendicular magnetic field......Page 123
3.3.4 Boundary conditions for B ? E......Page 125
Current density at an ohmic contact......Page 126
Insulating boundary......Page 127
3.3.5 Infinitely short plate with B ? E. Corbino conductivity and Hall mobility......Page 128
Hall mobility......Page 129
3.3.6 Infinitely long plate with B ? E. Physical magnetoresistance and Hall effect......Page 131
The physical magnetoresistance effect......Page 132
The Hall effect......Page 135
3.3.7 The galvanomagnetic effects in a mixed conductor......Page 136
Infinitely short samples. Average Corbino conductivity and mobility......Page 137
Infinitely long samples. Average magnetoresistivity and Hall coefficient......Page 138
3.3.8 The Hall effect in a many-valley semiconductor......Page 139
Mobility, Hall factor, Hall coefficient......Page 140
Modelling anisotropy......Page 141
Physical insight......Page 145
3.3.9 The planar Hall effect......Page 146
The planar Hall coefficient......Page 147
Infinitely long samples......Page 148
3.3.11 Relation between the magnetoresistance effect and the planar Hall effect. Magnetoresistivity for an arbitrary direction of B......Page 150
3.4.1 Scattering factors. Hall factor......Page 152
The Hall factor......Page 153
Non-degenerated carriers......Page 158
3.4.3 Hall mobility......Page 159
Non-degenerated carriers......Page 160
Strongly degenerated carriers......Page 161
3.4.5 Magnetoresistivity......Page 162
3.4.6 Hall coefficient......Page 163
Non-degenerated carriers......Page 164
Strongly degenerated carriers......Page 165
Experimental results......Page 167
3.4.7 Planar Hall coefficient......Page 169
3.5.1 Diffusion current. Complete galvanomagnetic equation......Page 170
3.5.2 Magnetoconcentration effect......Page 173
3.5.3 Galvanothermomagnetic effects......Page 175
Example......Page 176
3.5.4 The Hall effect at a high electric field......Page 177
3.5.5 The piezo-Hall effect......Page 180
References......Page 182
4.1 Galvanomagnetic effects: integral quantities......Page 185
4.1.1 From the basic equation to integral quantities......Page 186
4.1.2 Hall plate as a two-port device......Page 188
4.2 Influence of the plate geometry......Page 189
4.2.1 Plates neither long nor short. Geometrical factors......Page 190
4.2.2 Conformal mapping: method and results......Page 191
Example 1......Page 197
Example 2......Page 199
4.2.4 Numerical analysis using circuit simulation method......Page 202
Hall plates......Page 207
Magnetoresistors......Page 209
4.2.6 Geometrical factor of Hall voltage......Page 211
4.2.7 Geometrical factor of magnetoresistance......Page 216
4.3 The Hall voltage mode of operation......Page 218
4.3.1 Input and output resistances......Page 219
4.3.2 Output voltage......Page 222
4.3.3 Hall voltage......Page 223
Example......Page 225
4.3.4 Offset voltage......Page 226
4.3.5 Noise voltage......Page 228
4.3.6 Signal-to-noise ratio......Page 229
4.3.7 Self magnetic fields......Page 231
4.4.1 Large-contact Hall devices......Page 232
4.4.2 Dual Hall devices. Hall current......Page 233
4.5 The magnetoresistance mode of operation......Page 236
4.5.1 Magnetoresistance ratio......Page 237
Relaxation time frequency limit......Page 238
Dielectric relaxation frequency limit......Page 239
4.6.2 Global capacitive effects......Page 240
Eddy currents......Page 244
References......Page 247
5: Hall magnetic sensors......Page 250
5.1.1 Sensitivity......Page 251
5.1.2 Offset-equivalent magnetic field......Page 253
5.1.3 Noise-equivalent magnetic field......Page 254
5.1.4 Cross-sensitivity......Page 255
5.1.5 Non-linearity......Page 256
5.1.6 Frequency response......Page 260
5.1.7 Stability......Page 261
5.2 Material and technology......Page 263
5.2.1 Choice of material......Page 264
5.3 Examples of Hall plates......Page 266
Thin-film indium-antimonide Hall plates......Page 267
Silicon Hall plates......Page 268
5.3.3 2DEG Hall plates......Page 271
5.4.1 Generalizing the criteria for a good Hall device......Page 273
5.4.2 Vertical Hall devices......Page 275
5.4.3 Cylindrical Hall device......Page 278
5.5 Very small Hall devices......Page 280
Linear drift range......Page 281
Saturated velocity range......Page 282
Velocity overshoot range......Page 284
Ballistic transport range......Page 285
5.5.2 Examples of micrometre and submicrometre Hall devices......Page 288
5.6.1 Circuit model of a Hall device......Page 289
5.6.2 Biasing a Hall device. Temperature cross-sensitivity and non-linearity......Page 294
5.6.3 Reducing offset and 1/f noise......Page 296
5.6.4 Amplifying the output signal......Page 300
5.6.5 Integrated Hall magnetic sensors......Page 303
5.7 Hall sensors with magnetic concentrators......Page 306
Magnetic gain......Page 307
Remanent field......Page 309
5.7.2 Discrete and hybrid magnetic concentrators......Page 310
5.7.3 Integrated magnetic concentrators......Page 311
Twin integrated magnetic concentrator......Page 312
Single integrated magnetic concentrator......Page 314
Application of the single IMC: two-axis Hall magnetic sensor......Page 316
5.8.1 Packaging stress......Page 319
5.8.2 Reducing packaging stress......Page 321
5.9.1 Measuring magnetic fields......Page 322
5.9.2 Mechanical displacement transducers......Page 323
Angular position measurement......Page 325
Linear position sensor......Page 326
5.9.3 Measuring electric current, power and energy......Page 328
Current sensors......Page 329
Electric power sensor......Page 331
References......Page 332
6.1.1 Homogeneous plates and layers......Page 338
6.1.2 Inhomogenous layers......Page 341
6.1.3 Measurement of profiles......Page 345
6.2.1 Measurements on a long slab......Page 346
6.2.2 Van der Pauw method......Page 348
Resistivity......Page 352
Hall measurements......Page 353
6.2.4 Measurements on a large-contact sample......Page 356
6.2.5 The Hall mobility of minority carriers......Page 358
6.3 Measuring packaging stress......Page 360
6.3.1 Measurement method......Page 361
6.3.2 Examples of the measurement results......Page 362
References......Page 364
7: Magnetic-sensitive field-effect and bipolar devices......Page 366
7.1.1 MOS Hall plates......Page 367
7.1.2 Split-drain MOS devices......Page 370
7.2.1 The magnetodiode effect......Page 376
7.2.2 Structures and technology......Page 378
7.3 Magnetotransistors......Page 379
7.3.1 The current deflection mechanism......Page 380
7.3.2 Injection modulation......Page 385
7.3.3 The magnetodiode effect in magnetotransistors......Page 391
Sensitivity......Page 392
Offset-equivalent magnetic field......Page 393
Noise-equivalent magnetic field......Page 394
Cross-sensitivity......Page 395
7.3.5 Magnetotransistor circuits......Page 397
7.4 Carrier-domain magnetic-sensitive devices......Page 400
7.4.1 Three-layer carrier-domain devices......Page 401
7.4.2 Four-layer carrier-domain devices......Page 404
References......Page 408
8.1 Hall elements versus other Hall effect devices......Page 411
8.1.1 Hall elements: Hall voltage mode versus Hall current mode and magnetoresistance mode of operation......Page 412
8.1.2 Hall Elements versus magnetodiodes, MagFETs, MagBJTs, . . .......Page 413
8.2.2 Performance of ferromagnetic magnetoresistors......Page 414
8.2.3 Integrated Hall sensors versus AMRs and GMRs......Page 417
8.2.4 Looking into a crystal ball......Page 419
References......Page 420
Appendix A: International system of units......Page 421
Appendix B: Physical constants......Page 422
Appendix C: Properties of some semiconductors......Page 423
Appendix D: Magnetic quantities, units, relations and values......Page 424
II2. Let us find the inverse function ( y = (x)) of the function......Page 425