Preface ........................................................... xiii Acknowledgments .................................................... xiv 1 Introduction ....................................................... 1 1.1 The basic physical effects ..................................... 1 1.1.1 The Hall effect .......................................... 2 1.1.2 The magnetoresistance effect ............................. 4 1.2 Hall effect devices and their applications ..................... 5 1.2.1 Hall plates .............................................. 6 1.2.2 Other Hall effect devices ................................ 7 1.2.3 Applications of Hall effect devices ...................... 8 References ......................................................... 9 2 Semiconductor physics: a summary .................................. 11 2.1 Energy bands .................................................. 11 2.1.1 Conductor, semiconductor, insulator ..................... 11 2.1.2 Permitted and forbidden energy bands .................... 12 2.1.3 Electrons in solids ..................................... 14 2.1.4 Electrons in the conduction band ........................ 15 2.1.5 Holes in the valence band ............................... 17 2.2 Distribution and concentration of carriers .................... 19 2.2.1 Density of states ....................................... 19 2.2.2 Distribution function ................................... 21 2.2.3 Carrier concentration ................................... 23 2.2.4 Intrinsic semiconductors ................................ 24 2.2.5 Extrinsic semiconductors ................................ 25 2.2.6 Heavy doping effects .................................... 32 2.3 Scattering of carriers ........................................ 33 2.3.1 Free transit time ....................................... 34 2.3.2 Relaxation time ......................................... 37 2.4 Charge transport .............................................. 38 2.4.1 Drift in a weak electric field .......................... 39 2.4.2 High electric fields .................................... 48 2.4.3 Diffusion current ....................................... 50 2.4.4 Non-stationary transport ................................ 53 2.5 Other relevant phenomena ...................................... 54 2.5.1 Recombination and generation ............................ 55 2.5.2 PN junction ............................................. 56 2.5.3 Noise ................................................... 57 2.5.4 Piezoresistance ......................................... 59 References ........................................................ 61 3 Galvanomagnetic effects ........................................... 63 3.1 Approximate analysis .......................................... 64 3.1.1 The (original) Hall effect .............................. 65 3.1.2 The current deflection effect ........................... 71 3.1.3 The magnetoresistance effect ............................ 74 3.1.4 Galvanomagnetic current components ...................... 76 3.1.5 Charge carrier transport in very high magnetic fields ... 81 3.1.6 The role of collisions .................................. 84 3.1.7 Summary and further questions ........................... 86 3.2 Kinetic equation approach ..................................... 88 3.2.1 Kinetic equation ........................................ 89 3.2.2 Relaxation time ......................................... 91 3.2.3 Solution for the steady state ........................... 92 3.2.4 Non-equilibrium distribution function ................... 95 3.2.5 Galvanomagnetic transport of carriers. Current density ................................................. 97 3.2.6 Kinetic coefficients ................................... 101 3.2.7 Magnetic field dependence of the kinetic coefficients ........................................... 104 3.3 An accurate analysis ......................................... 106 3.3.1 Basic equations ....................................... 106 3.3.2 Drift current density at B = 0 ........................ 108 3.3.3 Current density in a plate under perpendicular magnetic field ........................................ 109 3.3.4 Boundary conditions for B ^ E ......................... 111 3.3.5 Infinitely short plate with B ^ E. Corbino conductivity and Hall mobility ........................ 114 3.3.6 Infinitely long plate with B ^ E. Physical magnetoresistance and Hall effect ..................... 117 3.3.7 The galvanomagnetic effects in a mixed conductor ...... 122 3.3.8 The Hall effect in a many-valley semiconductor ........ 125 3.3.9 The planar Hall effect ................................ 132 3.3.10 Alternative forms of the basic equations .............. 134 3.3.11 Relation between the magnetoresistance effect and the planar Hall effect. Magnetoresistivity for an arbitrary direction of B .............................. 136 3.4 Galvanomagnetic coefficients ................................. 138 3.4.1 Scattering factors. Hall factor ........................ 138 3.4.2 Corbino conductivity ................................... 144 3.4.3 Hall mobility .......................................... 145 3.4.4 Planar Hall-current coefficient ........................ 148 3.4.5 Magnetoresistivity ..................................... 148 3.4.6 Hall coefficient ....................................... 149 3.4.7 Planar Hall coefficient ................................ 155 3.5 Related effects .............................................. 156 3.5.1 Diffusion current. Complete galvanomagnetic equation ... 156 3.5.2 Magnetoconcentration effect ............................ 159 3.5.3 Galvanothermomagnetic effects .......................... 161 3.5.4 The Hall effect at a high electric field ............... 163 3.5.5 The piezo-Hall effect .................................. 166 References ....................................................... 168 4 Hall plates and magnetoresistors. Basic properties ............... 171 4.1 Galvanomagnetic effects: integral quantities ................. 171 4.1.1 From the basic equation to integral quantities ......... 172 4.1.2 Hall plate as a two-port device ........................ 174 4.2 Influence of the plate geometry .............................. 175 4.2.1 Plates neither long nor short. Geometrical factors ..... 176 4.2.2 Conformal mapping: method and results .................. 177 4.2.3 Conformal mapping: examples ............................ 183 4.2.4 Numerical analysis using circuit simulation method ..... 188 4.2.5 Common shapes of Hall plates and magnetoresistors ...... 193 4.2.6 Geometrical factor of Hall voltage ..................... 197 4.2.7 Geometrical factor of magnetoresistance ................ 202 4.3 The Hall voltage mode of operation ........................... 204 4.3.1 Input and output resistances ........................... 205 4.3.2 Output voltage ......................................... 208 4.3.3 Hall voltage ........................................... 209 4.3.4 Offset voltage ......................................... 212 4.3.5 Noise voltage .......................................... 214 4.3.6 Signal-to-noise ratio .................................. 215 4.3.7 Self magnetic fields ................................... 217 4.4 The Hall current mode of operation ........................... 218 4.4.1 Large-contact Hall devices ............................. 218 4.4.2 Dual Hall devices. Hall current ........................ 219 4.5 The magnetoresistance mode of operation ...................... 222 4.5.1 Magnetoresistance ratio ................................ 223 4.6 High-frequency effects ....................................... 224 4.6.1 Local high-frequency effects ........................... 224 4.6.2 Global capacitive effects .............................. 226 4.6.3 Inductive effects ...................................... 230 References ....................................................... 233 5 Hall magnetic sensors ............................................ 236 5.1 Basic characteristics ........................................ 237 5.1.1 Sensitivity ............................................ 237 5.1.2 Offset-equivalent magnetic field ....................... 239 5.1.3 Noise-equivalent magnetic field ........................ 240 5.1.4 Cross-sensitivity ...................................... 241 5.1.5 Non-linearity .......................................... 242 5.1.6 Frequency response ..................................... 246 5.1.7 Stability .............................................. 247 5.2 Material and technology ...................................... 249 5.2.1 Choice of material ..................................... 250 5.2.2 Technology ............................................. 252 5.3 Examples of Hall plates ...................................... 252 5.3.1 High mobility thin-film Hall plates .................... 253 5.3.2 Silicon and GaAs Hall plates ........................... 254 5.3.3 2DEG Hall plates ....................................... 257 5.4 Non-plate-like Hall magnetic sensors ......................... 259 5.4.1 Generalizing the criteria for a good Hall device ....... 259 5.4.2 Vertical Hall devices .................................. 261 5.4.3 Cylindrical Hall device ................................ 264 5.4.4 Three-axis Hall magnetic sensor ........................ 266 5.5 Very small Hall devices ...................................... 266 5.5.1 Scaling down a Hall plate .............................. 267 5.5.2 Examples of micrometre and submicrometre Hall devices ................................................ 274 5.6 Incorporating a Hall device into a circuit ................... 275 5.6.1 Circuit model of a Hall device ......................... 275 5.6.2 Biasing a Hall device. Temperature cross-sensitivity and non-linearity ...................................... 280 5.6.3 Reducing offset and 1/f noise .......................... 282 5.6.4 Amplifying the output signal ........................... 286 5.6.5 Integrated Hall magnetic sensors ....................... 289 5.7 Hall sensors with magnetic concentrators ..................... 292 5.7.1 Basic characteristics of magnetic concentrators ........ 293 5.7.2 Discrete and hybrid magnetic concentrators ............. 296 5.7.3 Integrated magnetic concentrators ...................... 297 5.8 Packaging Hall magnetic sensors .............................. 305 5.8.1 Packaging stress ....................................... 305 5.8.2 Reducing packaging stress .............................. 307 5.9 Applications of Hall magnetic sensors ........................ 308 5.9.1 Measuring magnetic fields .............................. 308 5.9.2 Mechanical displacement transducers .................... 309 5.9.3 Measuring electric current, power and energy ........... 314 References ....................................................... 318 6 Hall devices as means for characterizing semiconductors .......... 324 6.1 Extraction of material characteristics ....................... 324 6.1.1 Homogeneous plates and layers .......................... 324 6.1.2 Inhomogenous layers .................................... 327 6.1.3 Measurement of profiles ................................ 331 6.2 Measurement methods for characterizing material .............. 332 6.2.1 Measurements on a long slab ............................ 332 6.2.2 Van der Pauw method .................................... 334 6.2.3 Four-point probe methods ............................... 338 6.2.4 Measurements on a large-contact sample ................. 342 6.2.5 The Hall mobility of minority carriers ................. 344 6.3 Measuring packaging stress ................................... 346 6.3.1 Measurement method ..................................... 347 6.3.2 Examples of the measurement results .................... 348 References ......................................................... 350 7 Magnetic-sensitive field-effect and bipolar devices .............. 352 7.1 MOS Hall effect devices ...................................... 353 7.1.1 MOS Hall plates ........................................ 353 7.1.2 Split-drain MOS devices ................................ 356 7.2 Magnetodiodes ................................................ 362 7.2.1 The magnetodiode effect ................................ 362 7.2.2 Structures and technology .............................. 364 7.3 Magnetotransistors ........................................... 365 7.3.1 The current deflection mechanism ....................... 366 7.3.2 Injection modulation ................................... 371 7.3.3 The magnetodiode effect in magnetotransistors .......... 377 7.3.4 Magnetotransistor figures of merit ..................... 378 7.3.5 Magnetotransistor circuits ............................. 383 7.4 Carrier-domain magnetic-sensitive devices .................... 386 7.4.1 Three-layer carrier-domain devices ..................... 387 7.4.2 Four-layer carrier-domain devices ...................... 390 References ....................................................... 394 8 Comparing Hall sensors with other galvanomagnetic sensors ........ 397 8.1 Hall elements versus other Hall effect devices ............... 397 8.1.1 Hall elements: Hall voltage mode versus Hall current mode and magnetoresistance mode of operation ........... 398 8.1.2 Hall Elements versus magnetodiodes, MagFETs, MagBJTs,... ............................................ 399 8.2 Hall sensors versus ferromagnetic magnetoresistors ........... 400 8.2.1 Performance of integrated Hall magnetic sensors ........ 400 8.2.2 Performance of ferromagnetic magnetoresistors .......... 400 8.2.3 Integrated Hall sensors versus AMRs and GMRs ........... 403 8.2.4 Looking into a crystal ball ............................ 405 References ....................................................... 406 Appendices A International system of units .................................. 407 B Physical constants ............................................. 408 C Properties of some semiconductors .............................. 409 D Magnetic quantities, units, relations and values ............... 410 E Mathematical relations ......................................... 411 Index .............................................................. 413