How to calculate Fermi energy metals semiconductors

Calculate Fermi energy E_F = (hbar^2/(2m))*(3*pi^2*n)^(2/3) degenerate electron gas metals semiconductors. Cu Au Si GaAs white dwarf. Enrico Fermi Italy-USA 1901-1954 Nobel Prize 1938 + 1926. Intel + TSMC + Princeton.

Calculation parameters
Was this calculator useful?

How to calculate the Fermi energy (E_F) maximum energy occupied by electrons at T=0 K in metals and semiconductors according to Fermi-Dirac statistics of fermions: E_F = (hbar^2 / (2*m_eff)) * (3*pi^2*n)^(2/3). hbar = 1.0546e-34 J*s reduced Planck constant + m_eff effective mass electron (m_e = 9.109e-31 kg metals; 0.26*m_e silicon; 0.067*m_e GaAs); n electron density m^-3 (copper Cu 8.45e28; silver Ag 5.85e28; gold Au 5.9e28; aluminum Al 1.81e29; sodium Na 2.65e28; doped silicon 1e22-1e26; doped GaAs 1e21-1e25; white dwarf 1e36; neutron star 1e44).

Fermi temperature T_F = E_F/k_B: copper T_F=82000 K + sodium T_F=37500 K + doped silicon 100-1000 K.

Fermi wave vector k_F = (3*pi^2*n)^(1/3): copper 1.36e10 m^-1 + lambda_F = 2*pi/k_F = 0.46 nm.

Fermi velocity v_F = hbar*k_F/m_eff: copper 1.57e6 m/s + 0.5% speed light + classical reference.

Fermi-Dirac statistics: f(E) = 1/(exp((E-mu)/k_B*T) + 1) + mu chemical potential ~ E_F for T<<T_F + degenerate electron gas + classical reference Sommerfeld 1928.

Density of states DOS: g(E) = (1/(2*pi^2)) * (2*m_eff/hbar^2)^(3/2) * sqrt(E) for free electron 3D + g(E_F) Sommerfeld expansion T<<T_F + classical reference.

Electronic heat capacity: C_e = (pi^2/3) * g(E_F) * k_B^2 * T = gamma*T + Sommerfeld coefficient gamma + linear in T + classical reference + vs C_phonons = (12*pi^4/5)*N*k_B*(T/Theta_D)^3 Debye T^3 low.

Regime classification: E_F > 1 eV degenerate metal + alkali Na K + noble Cu Ag Au Pt + transition; kT < E_F < eV degenerate semiconductor + n+ p+ doped 1e25-1e26 m^-3 + tunnel diodes; E_F < kT non-degenerate semiconductor + Boltzmann distribution + classical reference + Si GaAs intrinsic 1e16; E_F << kT classical Boltzmann electron gas.

Related numbers: thermal kT energy 0.025 eV @ 298 K; Rydberg energy 13.6 eV hydrogen; gap energy E_g Si 1.12 + GaAs 1.42 + diamond 5.5; effective masses Si 0.26 + Ge 0.12 + GaAs 0.067 + InSb 0.014; mobility mu; conductivity sigma = n*e*mu.

Drude-Sommerfeld 1928 model: sigma = n*e^2*tau/m_eff + tau collision time + Wiedemann-Franz law L = K/(sigma*T) = pi^2/3*(k_B/e)^2 = 2.44e-8 W*ohm/K^2 + classical reference Wiedemann 1853 Franz Lorenz number. History: Enrico Fermi (Rome Italy 1901 - Chicago USA 1954) Sapienza University Rome 1923-1938 + Columbia 1939-1942 + Univ Chicago 1942-1954 + Manhattan Project Los Alamos 1943-1945 + 1942 first nuclear chain reaction Chicago Pile-1; 1926 paper Z Phys 36:902 'Sulla quantizzazione del gas perfetto monoatomico' classic Fermi statistics + 1934 beta decay theory neutrino + classical reference + Nobel Prize Physics 1938 'demonstrations of the existence of new radioactive elements produced by neutron irradiation' + Fermi unit 1 fm + Fermi level eponym + Member US National Academy + classical reference + Italian-American physicist + Chicago Pile-1 1942 first sustained nuclear chain reaction + ENRICO FERMI INSTITUTE Chicago 1955 eponym annual + classical reference comprehensive.

Paul Dirac (UK 1902-1984) Cambridge Lucasian Chair 1932-1969 + Florida State 1971-1984; 1926 Proc Roy Soc A 112:661 'On the theory of quantum mechanics' classic Fermi-Dirac statistics + 1928 Dirac equation relativistic electron + 1928 prediction positron + 1930 'Principles of Quantum Mechanics' Oxford classical textbook + Nobel Prize Physics 1933 shared with Schrodinger + Royal Society 1930.

Arnold Sommerfeld (Konigsberg 1868 - Munich 1951) Univ Munich 1906-1940; 1928 Z Phys 47:1 'Zur Elektronentheorie der Metalle' classic Sommerfeld theory metals applying Fermi-Dirac + 1916 Sommerfeld fine-structure constant alpha + 1919 'Atombau und Spektrallinien' Vieweg classical textbook + Sommerfeld nominated Nobel 84 times never received + Lorentz Medal 1939 + Max-Planck-Medaille 1931.

Felix Bloch (Zurich 1905-1983) ETH + Stanford 1934-1971; 1928 PhD thesis 'Uber die Quantenmechanik der Elektronen in Kristallgittern' classic Bloch theorem + 1946 NMR + Nobel Prize Physics 1952 NMR shared with Purcell + classical reference solid state.

John Bardeen (1908-1991) Princeton + Bell Labs 1945-1951 + Univ Illinois 1951-1991; 1957 BCS Bardeen-Cooper-Schrieffer superconductivity classic + 1947 transistor with Shockley-Brattain + Double Nobel Prize Physics 1956 transistor + 1972 superconductivity + only person 2 Nobel Physics. Legacy: Ashcroft-Mermin 'Solid State Physics' 1976 Saunders 826 pages classical textbook + Kittel 'Introduction to Solid State Physics' 8th 2004 Wiley 704 pages classical + Marder 'Condensed Matter Physics' 2010 Wiley 1004 pages + Pines-Nozieres 'Theory of Quantum Liquids' 1966 Benjamin classical + Mahan 'Many-Particle Physics' 2000 Springer 800 + Fetter-Walecka 1971 McGraw-Hill 600 classical + Sze 'Physics of Semiconductor Devices' 3rd 2007 Wiley 832 pages classic semiconductor + Yu-Cardona 'Fundamentals of Semiconductors' 4th 2010 Springer 778 pages.

USA research solid state + semiconductor + nuclear: NIST Gaithersburg Boulder 4800 + materials standards SRM; NSF Materials 12000; DOE national labs Argonne ORNL LBNL LANL Sandia 35000 + materials science Fermi legacy; Argonne National Laboratory ANL 3500 + Fermi-named after Enrico Fermi original site + Chicago Pile-1 1942 site + Material Science world-leading; Fermilab Batavia IL 1900 + particle accelerator Tevatron + DOE + Fermi legacy world-leading; SLAC National Accelerator Stanford 1500; Brookhaven BNL Long Island 2700; NREL National Renewable Energy Lab Boulder 3700 + photovoltaic; NASA JPL 17000.

USA universities solid state + semiconductor: MIT 11500 + Karl Berggren physics nanoelectronics + Marc Baldo OLED + nuclear + Mildred Dresselhaus carbon nanotubes legacy world-leading; Stanford 11000 + James Harris semiconductor; UC Berkeley 8500 + Eli Yablonovitch photonic + Tom Richmond physics; Caltech 8800 + Harry Atwater materials photovoltaic; Princeton 5400 + Bardeen legacy; Harvard 22000 + Charles Lieber nanowires; Cornell 5200; Univ Chicago 9000 + Fermi alma mater + James Cronin Nobel 1980 + Robert Wald GR; Univ Illinois Urbana-Champaign UIUC 7000 + John Bardeen legacy + Anthony Leggett Nobel 2003 + classical reference; Wisconsin Madison 7000; Northwestern 7100; Texas Austin 50000 + Steven Weinberg legacy.

USA industry semiconductor: Intel Santa Clara 132000 + leading-edge logic 2nm 18A node world-leading; TSMC USA Phoenix 8000 + Arizona 4nm 3nm fab; GlobalFoundries Malta NY 13000 + 12nm 14nm specialty; Texas Instruments TI Dallas 33000 + analog + automotive; Applied Materials Santa Clara 35000 + plasma etch CVD world-leading; Lam Research Fremont CA 19000 + plasma etch deposition world-leading; KLA Milpitas 14000 + inspection metrology; Micron Boise ID 49000 + DRAM NAND memory; Western Digital San Jose 64000 + storage; Nvidia Santa Clara 30000 + GPU H100; AMD Santa Clara 26000 + EPYC Ryzen; Qualcomm San Diego 51000 + Snapdragon.

USA industry solar PV: First Solar Tempe AZ 7000 + thin-film CdTe; Sunpower Maxeon 1500; SunRun San Francisco 12000; Tesla Solar Palo Alto 127000 + solar roof.

USA industry nuclear + accelerators: Westinghouse Cranberry 9000 + AP1000; GE Hitachi Wilmington NC 3000 + PRISM sodium; Holtec International Camden NJ 1000; SLAC Stanford 1500; Argonne 3500; Fermilab 1900; Brookhaven 2700.

Fermi formula

Definition: E_F = (hbar^2 / (2*m_eff)) * (3*pi^2*n)^(2/3). hbar = 1.0546e-34 J*s + m_eff effective mass + n density m^-3.

Practical metals form: E_F[eV] = (3.65e-19) * (n[m^-3])^(2/3) / m_eff_ratio + Ashcroft-Mermin 1976.

Fermi temperature: T_F = E_F/k_B + copper 82000 K + sodium 37500 K.

Fermi wave vector: k_F = (3*pi^2*n)^(1/3) + lambda_F = 2*pi/k_F + copper 0.46 nm.

Typical values: copper Cu n=8.45e28 + E_F=7.0 eV + T_F=82000 K + v_F=1.57e6 m/s; silver Ag E_F=5.5 eV; gold 5.5 eV; aluminum 11.7 eV; sodium 3.2 eV; doped silicon 0.18 eV; GaAs 0.011 eV; white dwarf 1 MeV relativistic.

History and context

Enrico Fermi (Rome Italy 1901 - Chicago 1954) Sapienza Rome 1923-1938 + Columbia 1939-1942 + Univ Chicago 1942-1954 + Manhattan Project Los Alamos 1943-1945 + 1942 first nuclear chain reaction Chicago Pile-1; 1926 paper Z Phys 36:902 Fermi statistics + 1934 beta decay theory neutrino + classical reference + Nobel Prize Physics 1938 + Fermi unit 1 fm + Fermi level eponym + Member US National Academy + Italian-American physicist + Chicago Pile-1 1942 + ENRICO FERMI INSTITUTE Chicago 1955 eponym annual + classical reference.

Paul Dirac (UK 1902-1984) Cambridge Lucasian Chair 1932-1969; 1926 Proc Roy Soc A 112:661 Fermi-Dirac statistics + 1928 Dirac equation + 1930 'Principles of Quantum Mechanics' Oxford classical + Nobel Prize Physics 1933 + Royal Society 1930.

Arnold Sommerfeld (1868-1951) Munich 1906-1940; 1928 Z Phys 47:1 Sommerfeld theory metals + 1916 fine-structure alpha + nominated Nobel 84 times never received + Lorentz Medal 1939 + Max-Planck-Medaille 1931.

Felix Bloch (1905-1983) Stanford 1934-1971; 1928 Bloch theorem + 1946 NMR + Nobel Prize Physics 1952 NMR shared with Purcell.

John Bardeen (1908-1991) Princeton + Bell Labs + Univ Illinois 1951-1991; 1957 BCS superconductivity + 1947 transistor + Double Nobel Prize Physics 1956 + 1972 + only person 2 Nobel Physics. Legacy: Ashcroft-Mermin 1976 Saunders 826 + Kittel 8th 2004 Wiley 704 + Marder 2010 Wiley 1004 + Pines-Nozieres 1966 + Mahan 2000 Springer 800 + Fetter-Walecka 1971 + Sze 2007 Wiley 832 + Yu-Cardona 2010 Springer 778.

USA research: NIST 4800; NSF Materials 12000; DOE labs 35000 Fermi legacy; Argonne 3500 Fermi original site + Chicago Pile-1 1942 + world-leading; Fermilab 1900 + particle accelerator + Fermi legacy world-leading; SLAC 1500; Brookhaven 2700; NREL 3700; NASA JPL 17000.

Universities: MIT 11500 + Berggren + Dresselhaus legacy world-leading; Stanford 11000 + Harris; UC Berkeley 8500 + Yablonovitch; Caltech 8800 + Atwater; Princeton 5400 + Bardeen legacy; Harvard 22000 + Lieber; Cornell 5200; Chicago 9000 + Fermi alma mater + Cronin + Wald; UIUC 7000 + Bardeen legacy + Leggett 2003; Wisconsin 7000; Northwestern 7100.

Industry semiconductor: Intel 132000 world-leading 2nm 18A; TSMC USA 8000 Phoenix; GlobalFoundries 13000; Texas Instruments 33000; Applied Materials 35000 world-leading; Lam Research 19000 world-leading; KLA 14000; Micron 49000 DRAM NAND; Western Digital 64000; Nvidia 30000 GPU; AMD 26000; Qualcomm 51000 Snapdragon.

Solar PV: First Solar 7000 CdTe; Tesla Solar 127000.

Nuclear accelerators: Westinghouse 9000; GE Hitachi 3000 PRISM; SLAC 1500; Argonne 3500; Fermilab 1900; Brookhaven 2700.

Fermi values table

  • E_F = (hbar^2/(2m))*(3*pi^2*n)^(2/3): definition
  • T_F = E_F/k_B: temperature
  • k_F = (3*pi^2*n)^(1/3): vector
  • v_F = hbar*k_F/m: velocity
  • g(E_F): density of states
  • C_e = gamma*T: linear heat capacity
  • Wiedemann-Franz: 2.44e-8 W*ohm/K^2
  • Cu: E_F=7.0 eV T_F=82000 K
  • Si doped: E_F=0.18 eV
  • White dwarf: 1 MeV relativistic

5 numerical examples

  1. Copper Cu metal: n=8.45e28 m^-3 + m_eff=1 + E_F=7.0 eV + T_F=82000 K + v_F=1.57e6 m/s + classical conductor.
  2. Silver Ag: n=5.85e28 + E_F=5.5 eV + T_F=64000 K + Wiedemann-Franz law.
  3. Doped silicon n+ degenerate: n=1e25 + m_eff=0.26 + E_F=0.18 eV + Intel TSMC 12-inch wafer.
  4. GaAs heterostructures HEMT: n=1e23 + m_eff=0.067 + E_F=0.011 eV + classical reference.
  5. White dwarf core: n=1e36 + m_eff=1 + E_F=1 MeV + v_F~c relativistic + Chandrasekhar limit.

Limitations

Free electron approximation: ignore lattice + Bloch theorem extension band structure + Ashcroft-Mermin 1976.

Effective mass: parabolic dispersion + non-parabolic GaAs InSb + Yu-Cardona 2010.

Spin degeneracy: factor 2 included in formula + classical reference.

3D vs 2D 1D 0D: dimension affects DOS + 2D constant + 1D 1/sqrt(E) + 0D delta + Sze 2007.

Relativistic correction: white dwarf neutron star ultra-relativistic + Chandrasekhar.

Common mistakes

  • Confuse E_F and chemical potential mu: E_F at T=0 K + mu(T) Sommerfeld expansion + T<<T_F + classical.
  • Effective mass error: m_e free vs m_eff conduction band + parabolic vs heavy/light hole + Yu-Cardona 2010.
  • Density n units: m^-3 SI vs cm^-3 cgs + 1e6 conversion.
  • Degeneracy criterion: T<>T_F classical Boltzmann + Ashcroft-Mermin.
  • Ignore band structure: insulator gap + multi-band + Sze 2007.

Applications

  1. Semiconductor microelectronics: Intel TSMC GlobalFoundries world-leading + 12-inch wafer + CMOS MOSFET.
  2. Solar photovoltaic: First Solar CdTe + Tesla Solar + p-n junction Fermi level.
  3. Nuclear physics accelerators: Fermilab + SLAC + Argonne world-leading + Fermi legacy.
  4. Astrophysics white dwarf: Chandrasekhar limit + Fermi degenerate gas pressure.
  5. Superconductivity BCS: Bardeen-Cooper-Schrieffer + Cooper pairs at Fermi surface.
  6. Spintronics MRAM: Magnetic tunnel junction + spin valve + classical reference.
  7. Heavy-fermion materials: classical reference + many-body.
  8. Topological insulators: Bi2Te3 + classical reference modern.
  9. Graphene 2D Dirac fermion: linear dispersion + Andre Geim Nobel 2010.
  10. Quantum dots 0D semiconductor: discrete levels + Intel + classical reference.

Updated July 2026

Frequently Asked Questions

What does the Fermi energy E_F measure?

Fermi energy E_F = (hbar^2/(2m))*(3*pi^2*n)^(2/3) measures the maximum energy occupied by electrons at T=0 K in metals and semiconductors according to Fermi-Dirac statistics of fermions. Typical metals E_F = 5-12 eV (Cu 7.0 eV; Al 11.7 eV); doped semiconductors 0.01-0.5 eV; white dwarf 1 MeV relativistic. Enrico Fermi Italy-USA 1901-1954 1926 Z Phys 36:902 + Nobel Prize 1938.

What is Fermi-Dirac statistics?

Fermi-Dirac statistics f(E) = 1/(exp((E-mu)/k_B*T) + 1) describes distribution of fermions (electrons) at temperature T with chemical potential mu. T<>T_F classical Maxwell-Boltzmann gas. Enrico Fermi 1926 Z Phys 36:902 + Paul Dirac 1926 Proc Roy Soc A 112:661 independently + classical reference.

Difference between E_F and chemical potential mu?

Fermi energy E_F = mu(T=0) maximum occupied energy level at absolute zero temperature + material constant + T independent; chemical potential mu(T) = E_F*(1 - (pi^2/12)*(T/T_F)^2) Sommerfeld expansion + slightly T dependent for T<<T_F. For T<<T_F (typical room Cu T=298 K T_F=82000 K) mu ~ E_F. Classical reference Ashcroft-Mermin 1976 'Solid State Physics' Saunders 826 pages classical textbook.

What is the Fermi temperature T_F?

Fermi temperature T_F = E_F/k_B characterizes degenerate electron gas regime vs Boltzmann. T<>T_F classical Boltzmann + C_e = (3/2)*N*k_B constant. Typical metals T_F = 50000-200000 K + always degenerate up to melting; doped semiconductors T_F = 100-1000 K + transition for T>T_F.

What is the Fermi velocity v_F?

Fermi velocity v_F = hbar*k_F/m_eff = hbar*(3*pi^2*n)^(1/3)/m_eff velocity of electrons at Fermi level. Typical metals v_F = 1-2e6 m/s + 0.5% speed of light + classical reference Cu 1.57e6 m/s; doped semiconductors 1e5-1e6 m/s; relativistic white dwarf v_F~c + Chandrasekhar limit degenerate gas pressure. Important for resistivity transport scattering.

Drude-Sommerfeld vs classical model?

Drude 1900 classical electron gas model + collision tau + sigma = n*e^2*tau/m + Wiedemann-Franz fit but heat capacity wrong 3/2*N*k_B vs C_e=gamma*T linear in T metals. Sommerfeld 1928 Z Phys 47:1 corrects applying Fermi-Dirac statistics + explains linear heat capacity + classical Wiedemann-Franz L = pi^2/3*(k_B/e)^2 = 2.44e-8 W*ohm/K^2 exact + classical reference Ashcroft-Mermin.

Fermi energy industrial applications?

Applications: semiconductor microelectronics Intel TSMC GlobalFoundries world-leading + 12-inch wafer 2nm 18A CMOS MOSFET; solar photovoltaic First Solar CdTe + Tesla Solar + p-n junction Fermi level; nuclear physics Fermilab + SLAC + Argonne world-leading + Fermi legacy classical reference; astrophysics white dwarf Chandrasekhar limit degenerate gas pressure; superconductivity BCS + Cooper pairs at Fermi surface; spintronics MRAM; graphene 2D Dirac fermion; quantum dots.

Comments (0)

No comments yet. Be the first!