How to calculate the Hall effect (V_H, R_H) fundamental solid state semiconductor physics phenomenon generating transverse Hall voltage V_H when an electric current I traverses a conductor in presence of perpendicular magnetic field B due to Lorentz force F = q*v x B on charge carriers: V_H = (I * B) / (n * q * t) = R_H * I * B / t.

Hall coefficient R_H: R_H = 1 / (n * q). I current A (typical 1 A metals; 1 mA semiconductors; 10 mA Hall sensors); B magnetic field T (lab 0.1-2; permanent magnet 0.05-1; Earth 5e-5; superconducting 1-15); t sample thickness m (sensors 1-100 um; lab 0.1-1 mm; thin film 1-100 nm); n carrier density m^-3 (copper 8.45e28; doped silicon 1e22-1e26; GaAs 1e23; InSb 1e22; bismuth semimetal 3e25; intrinsic 1e16); q charge e = 1.602e-19 C (positive +e holes; negative -e electrons; sign R_H discriminates carrier type).

Carrier type from R_H sign: R_H<0 electrons n-type semiconductors and most metals; R_H>0 holes p-type semiconductors + alkali metals + Be Bi superconductors.

Hall mobility mu_H: mu_H = R_H * sigma = R_H/rho + sigma conductivity + rho resistivity + classical reference + Hall mobility ~ Hall scattering factor 1.18 vs drift mobility.

Classical Drude R_H: R_H_Drude = -1/(n*e) single-band + classical reference; R_H_band-structure: multi-band + Lorentz force + classical Ashcroft-Mermin 1976.

Advanced Hall effects: quantum Hall effect QHE 1980 Klitzing discovery + R_xy = h/(e^2*nu) integer plateau + classical reference Nobel Prize 1985 + von Klitzing constant R_K = 25812.8 ohm primary SI 2019 standard; fractional QHE Tsui-Stormer-Laughlin 1982 + Nobel Prize 1998 + nu = 1/3 1/5 fractional quasi-particles + classical reference; spin Hall effect SHE; anomalous Hall AHE; topological Hall THE; quantum anomalous Hall QAHE; quantum spin Hall QSHE Bernevig-Zhang 2005 + Konig 2007.

Commercial Hall sensors: Hall integrated circuits IC + 1-20 mV/T sensitivity + linear ratiometric + automotive Tesla EV BMW + position rotation speed + Allegro Microsystems Melexis Honeywell Infineon NXP.

Hall thrusters spacecraft propulsion: Hall-effect thruster HET + xenon ionized + radial magnetic field + azimuthal Hall current + thrust 50-500 mN + Isp 1500-3000 s + Aerojet XR-5 + classical reference.

Related semiconductor numbers: conductivity sigma = n*e*mu; resistivity rho = 1/sigma; mobility mu; drift velocity v_d = mu*E; carrier density n; Hall mobility mu_H; Hall angle theta_H = atan(mu*B).

4-probe van der Pauw 1958 measurements: standard thin film characterization technique + classical reference IEEE standard + Ashcroft-Mermin. History: Edwin Herbert Hall (Gorham USA 1855 - Cambridge MA 1938) Johns Hopkins University 1879 PhD + Harvard University 1881-1921 emeritus + Member US National Academy; 1879 paper Am J Math 2:287 'On a New Action of the Magnet on Electric Currents' classic Hall effect discovery under Henry Augustus Rowland supervision Johns Hopkins + classical reference Hall coefficient eponym + 1880 Phil Mag 9:225 + 1881 Phys Rev measurement of Hall coefficient various metals + classical reference solid state physics + Henry Draper Medal 1888 National Academy USA + Rumford Medal 1895 + classical reference + Edwin H Hall Building Harvard.

Klaus von Klitzing (Schroda Germany 1943-) Univ Wurzburg Diplom 1969 PhD 1972 + ETH Zurich + Max-Planck-Institut Festkorperforschung Stuttgart 1985-2011 emeritus; 1980 paper Phys Rev Lett 45:494 classic discovery quantum Hall effect QHE + R_K = 25812.8 ohm von Klitzing constant + Nobel Prize Physics 1985 'for the discovery of the quantized Hall effect' + classical reference + Member US National Academy + Royal Society 1996 + classical reference modern quantum metrology + SI-2019 redefinition kilogram primary standard.

Daniel Chee Tsui (Henan China-USA 1939-) Bell Labs 1968-1982 + Princeton 1982-2010 emeritus; 1982 paper Phys Rev Lett 48:1559 with Horst Stormer Arthur Gossard 'Two-Dimensional Magnetotransport in the Extreme Quantum Limit' classic fractional quantum Hall effect FQHE + classical reference + Nobel Prize Physics 1998 shared with Stormer Laughlin + classical reference + Member US National Academy + Franklin Medal 1981.

Robert Betts Laughlin (Visalia USA 1950-) MIT + Stanford; 1983 paper Phys Rev Lett 50:1395 'Anomalous quantum Hall effect An incompressible quantum fluid with fractionally charged excitations' classic Laughlin wavefunction quasi-particles fractional charge + 2005 'A Different Universe' Basic Books + classical reference + Nobel Prize Physics 1998 shared with Tsui-Stormer + Member US National Academy + Franklin Medal 1998.

Horst Ludwig Stormer (Frankfurt Germany 1949-) Bell Labs 1977-1998 + Columbia 1997-2014 emeritus; 1982 with Tsui-Gossard fractional QHE classic Bell Labs era discovery + classical reference + Nobel Prize Physics 1998 + Member US National Academy. Legacy: Datta 'Electronic Transport in Mesoscopic Systems' 1995 Cambridge 393 pages classical + Yu-Cardona 'Fundamentals of Semiconductors' 4th 2010 Springer 778 pages classical + Sze 'Physics of Semiconductor Devices' 3rd 2007 Wiley 832 pages classic + Ashcroft-Mermin 'Solid State Physics' 1976 Saunders 826 pages classical + Kittel 'Introduction to Solid State Physics' 8th 2004 Wiley 704 pages + Bernevig-Hughes 'Topological Insulators and Topological Superconductors' 2013 Princeton 247 pages classical + Hasan-Kane 2010 Rev Mod Phys 82:3045 classical topological.

USA research solid state + semiconductor + Hall: NIST Gaithersburg Boulder 4800 + quantum Hall metrology von Klitzing standard SI 2019; NSF Materials 12000; DOE national labs Argonne ORNL LBNL LANL Sandia 35000; Lawrence Berkeley LBL 4500; SLAC Stanford 1500; Brookhaven BNL 2700; NREL Boulder 3700; NASA Jet Propulsion Lab JPL 6000 + Hall thruster Deep Space + Dawn mission + Psyche mission.

USA universities solid state + semiconductor: MIT 11500 + Mildred Dresselhaus carbon legacy + Karl Berggren physics; Stanford 11000 + James Harris semiconductor + Robert Laughlin Nobel 1998; UC Berkeley 8500 + Eli Yablonovitch photonic; Caltech 8800 + Harry Atwater materials; Princeton 5400 + Daniel Tsui Nobel 1998 legacy; Harvard 22000; Cornell 5200; Univ Illinois UIUC 7000 + Bardeen legacy; Wisconsin Madison 7000; Northwestern 7100; Georgia Tech 36000; UCSB UC Santa Barbara 23000 + Herbert Kromer Nobel 2000.

USA industry semiconductor + Hall sensors: Allegro MicroSystems Manchester NH 4500 + Hall sensor IC world-leading + automotive position; Honeywell Charlotte NC 95000 + Hall sensors aerospace defense; Texas Instruments TI Dallas 33000 + Hall TMR + automotive; Analog Devices Wilmington MA 25000 + integrated Hall; Melexis USA 200; Diodes Inc Plano TX 9000.

USA industry automotive Hall + EV: Tesla Palo Alto 127000 + Hall current battery + position sensor; Ford Detroit 173000; GM Detroit 167000; Stellantis North America 80000; Lucid Motors Newark CA 7000 + EV; Rivian Plymouth MI 14000 + EV.

USA industry Hall thrusters spacecraft: NASA JPL Pasadena 6000 + Hall thruster Dawn + Psyche; Aerojet Rocketdyne Sacramento 5000 + XR-5 Hall thruster; SpaceX Hawthorne 13000 + Starlink ion thruster; L3Harris Melbourne FL 47000 + propulsion; Northrop Grumman Falls Church 95000; Lockheed Martin 122000.

USA industry semiconductor manufacturing: Intel Santa Clara 132000 + leading-edge logic; TSMC USA 8000 Phoenix; GlobalFoundries 13000; Texas Instruments 33000; Applied Materials 35000 + Hall measurement equipment; Lam Research 19000.

USA industry MRI: GE Healthcare Chicago 47000 + MRI scanner + Hall coil; Siemens Healthineers USA 65000; Philips Healthcare USA 6000.

Hall formula

Hall voltage: V_H = (I * B) / (n * q * t) = R_H * I * B / t. I current A + B magnetic field T + n carrier density m^-3 + q charge + t thickness m.

Hall coefficient: R_H = 1 / (n * q) + sign discriminates carrier type (positive holes p-type; negative electrons n-type).

Hall mobility: mu_H = R_H * sigma = R_H/rho + Hall scattering factor 1.18 typical.

Typical values: copper Cu n=8.45e28 + R_H=-7.4e-11 m^3/C; aluminum Al R_H=-3.4e-11; bismuth Bi R_H=-7e-7 semimetal; doped silicon n=1e22 + R_H=-6.2e-4; GaAs n=1e23 + R_H=-6.2e-5; InSb 1e22 + R_H=-6.2e-4 high mobility; intrinsic Si 1e16 + R_H=-625 m^3/C very high.

Carrier type from R_H sign: R_H0 holes p-type + alkali metals + Be Bi.

History and context

Edwin Herbert Hall (Gorham USA 1855 - Cambridge MA 1938) Johns Hopkins 1879 PhD + Harvard 1881-1921 emeritus; 1879 paper Am J Math 2:287 'On a New Action of the Magnet on Electric Currents' classic Hall effect discovery under Henry Rowland supervision + classical reference + 1880 Phil Mag 9:225 + 1881 Phys Rev measurement Hall coefficient various metals + classical reference + Henry Draper Medal 1888 + Rumford Medal 1895 + Member US National Academy + Edwin H Hall Building Harvard.

Klaus von Klitzing (Schroda Germany 1943-) Wurzburg + ETH Zurich + Max-Planck-Institut FKF Stuttgart 1985-2011; 1980 Phys Rev Lett 45:494 quantum Hall effect QHE + von Klitzing constant + Nobel Prize Physics 1985 + Member US National Academy + Royal Society 1996 + SI-2019 kilogram standard.

Daniel Tsui (1939-) Bell Labs + Princeton 1982-2010; 1982 Phys Rev Lett 48:1559 with Stormer Gossard fractional QHE + Nobel Prize Physics 1998 + Franklin Medal 1981.

Robert Laughlin (1950-) Stanford; 1983 Phys Rev Lett 50:1395 Laughlin wavefunction + 2005 'A Different Universe' Basic Books + Nobel Prize Physics 1998 + Franklin Medal 1998.

Horst Stormer (Germany 1949-) Bell Labs + Columbia; 1982 with Tsui-Gossard FQHE + Nobel Prize Physics 1998 + Member US National Academy. Legacy: Datta 1995 Cambridge 393 + Yu-Cardona 4th 2010 Springer 778 + Sze 3rd 2007 Wiley 832 + Ashcroft-Mermin 1976 Saunders 826 + Kittel 8th 2004 Wiley 704 + Bernevig-Hughes 2013 Princeton 247 + Hasan-Kane 2010 Rev Mod Phys classical topological.

USA research: NIST 4800 + quantum Hall metrology von Klitzing SI 2019; NSF Materials 12000; DOE labs 35000; LBL 4500; SLAC 1500; BNL 2700; NREL 3700; NASA JPL 6000 + Hall thruster Dawn Psyche.

Universities: MIT 11500; Stanford 11000 + Laughlin Nobel; UC Berkeley 8500; Caltech 8800; Princeton 5400 + Tsui Nobel; Harvard 22000; Cornell 5200; UIUC 7000; Wisconsin 7000; Northwestern 7100; UCSB 23000 + Kromer Nobel.

Industry Hall sensors: Allegro MicroSystems 4500 world-leading; Honeywell 95000; Texas Instruments 33000; Analog Devices 25000; Melexis USA 200; Diodes Inc 9000.

Automotive EV: Tesla 127000; Ford 173000; GM 167000; Lucid 7000; Rivian 14000.

Hall thrusters: NASA JPL 6000; Aerojet Rocketdyne 5000; SpaceX 13000; L3Harris 47000; Northrop Grumman 95000; Lockheed Martin 122000.

Semiconductor manufacturing: Intel 132000; TSMC USA 8000; GlobalFoundries 13000; Texas Instruments 33000; Applied Materials 35000; Lam Research 19000.

MRI: GE Healthcare 47000; Siemens Healthineers 65000; Philips Healthcare 6000.

Hall values table

• V_H = I*B/(n*q*t) = R_H*I*B/t: Hall voltage
• R_H = 1/(n*q): Hall coefficient
• mu_H = R_H*sigma: Hall mobility
• R_H<0: electrons n-type
• R_H>0: holes p-type
• Cu: R_H=-7.4e-11 m^3/C
• Si doped: R_H~-6e-4
• Bi semimetal: R_H=-7e-7
• QHE von Klitzing: R_K=25812.8 ohm
• Hall thruster: thrust 50-500 mN Isp 1500-3000 s

5 numerical examples

1. Copper Cu metal: I=1 A + B=1 T + t=1 mm + n=8.45e28 + R_H=-7.4e-11 m^3/C + V_H=-74 nV small.
2. Doped silicon n+: I=1 mA + B=0.1 T + t=0.5 mm + n=1e22 + R_H=-6.2e-4 + V_H=-12 mV easily measurable.
3. GaAs HEMT 2DEG: I=1 mA + B=0.5 T + t=1 um + n=1e23 + R_H=-6.2e-5 + V_H=-31 mV high mobility classical.
4. InSb high mobility sensor: I=10 mA + B=0.1 T + t=10 um + n=1e22 + R_H=-6.2e-4 + V_H=-62 mV Hall sensor InSb classic.
5. Bismuth Bi semimetal: I=100 mA + B=1 T + t=0.1 mm + n=3e25 + R_H=-7e-7 + V_H=-7e-4 V research material.

Limitations

Single carrier band: real semiconductor multi-band + Hall scattering factor + Sze 2007 + Yu-Cardona 2010.

Drift Hall mobility: mu_H = mu_d * scattering factor 1.18 typical + Ashcroft-Mermin.

Strong field: omega_c*tau >> 1 saturated Hall + magneto-resistance.

Quantum Hall regime: 2DEG GaAs + low T + high B + von Klitzing 1980 + integer plateau + classical reference.

Boundary effects: contact resistance + finite geometry + Van der Pauw 1958 method correction + IEEE.

Common mistakes

• Confuse R_H and V_H: R_H = 1/(n*q) coefficient + V_H = R_H*I*B/t voltage + classical distinct.
• Sign convention: R_H0 holes p-type + Sze 2007 + sign error common.
• Drift vs Hall mobility: mu_H = mu_d * scattering factor 1.18 + Ashcroft-Mermin.
• Multi-band: real semiconductor heavy/light hole + Yu-Cardona 2010.
• Quantum vs classical: low T high B QHE plateau vs classical Hall + classical reference.

Applications

1. Hall sensors automotive: Allegro MicroSystems world-leading + Honeywell + position rotation speed + Tesla EV Ford GM.
2. EV battery current sensors: Tesla 4680 + Lucid + Rivian + Hall current sensors battery management.
3. Semiconductor characterization: Intel TSMC GlobalFoundries + 12-inch wafer + n p-type identification + Van der Pauw.
4. Hall thrusters spacecraft: NASA JPL Dawn Psyche + Aerojet Rocketdyne XR-5 + SpaceX Starlink + xenon Hall current.
5. Quantum Hall metrology: NIST + von Klitzing constant R_K standard SI 2019 + classical reference.
6. MRI scanner: GE Healthcare + Siemens Healthineers + Philips + Hall coil magnetic field.
7. Spintronics MRAM: Magnetic tunnel junction + spin Hall effect + classical reference.
8. Topological insulators: classical reference + Bi2Te3 quantum spin Hall.
9. Smart meters: utility + Hall current sensors + EPA.
10. Industrial control: Honeywell + Allegro + position rotation.

Updated July 2026