Drude Model Calculator
Enter carrier density and relaxation time to compute metal conductivity σ=ne²τ/m and mobility μ=eτ/m. Copper n=8.5e28, τ=2.5e-14 → σ≈5.99e7 S/m.
輸入資料
計算結果
重點速覽：The Drude model (1900) treats metal electrons as a classical free-electron gas (ideal gas) that accelerates under an electric field and scatters off ions, re-thermalising after a relaxation time τ. Conductivity σ = n·e²·τ/m, with n carrier density, e = 1.602176634×10⁻¹⁹ C, τ mean collision time and m electron mass (or effective mass m*). Mobility μ = e·τ/m, so σ = n·e·μ. Classical example: copper n=8.5e28 m⁻³, τ=2.5e-14 s → σ ≈ 5.99e7 S/m (measured 5.96e7, excellent agreement). Historically Drude (1900) explained the Wiedemann–Franz law (κ = LσT) but predicted electron heat capacity ~100× too large — fixed by Sommerfeld (1928) with quantum Fermi–Dirac statistics. Applications: (1) metal conductivity theory; (2) semiconductor mobility; (3) Hall-effect carrier measurement; (4) thin-film and nanowire conduction; (5) high-field (differential mobility) effects.
計算公式
Conductivity: σ = n·e²·τ/m (S/m).
Mobility: μ = e·τ/m (m²/(V·s)).
Conductivity: σ = n·e·μ.
Mean free path: λ = v_F·τ, v_F the Fermi velocity.
Microscopic Ohm's law: J = n·e·v_d = σ·E.
$$\sigma = \frac{n e^2 \tau}{m}, \quad \mu = \frac{e\tau}{m}, \quad \sigma = ne\mu$$使用說明
- Enter carrier density n (m⁻³), relaxation time τ (s) and effective mass m* (kg).
- The tool computes σ=ne²τ/m, μ=eτ/m, λ=v_F·τ (v_F≈1e6 m/s).
- Typical: copper n=8.5e28, τ=2.5e-14 → σ=5.99e7 S/m, μ=44 cm²/(V·s), λ=25nm.
Room-temperature metal conductivity and mobility (T=300K)
| Metal | Carrier density n (m⁻³) | Relaxation time τ (s) | Conductivity σ (S/m) | Mobility μ (cm²/(V·s)) |
|---|---|---|---|---|
| Silver | 5.86e28 | 3.8e-14 | 6.30e7 | 67 |
| Copper | 8.50e28 | 2.5e-14 | 5.99e7 | 44 |
| Gold | 5.91e28 | 3.0e-14 | 4.10e7 | 53 |
| Aluminium | 18.1e28 | 8.0e-15 | 3.55e7 | 14 |
| Tungsten | 1.20e29 | 1.5e-14 | 1.79e7 | 26 |
| Iron | 1.70e29 | 2.2e-15 | 1.00e7 | 4 |
Metal electrons v_F≈1e6 m/s, τ~1e-14 s, so mean free path λ~10–50nm (far larger than lattice spacing 0.3nm).
理財情境案例
Copper wire conductivity
Copper n=8.5e28 m⁻³ (one valence electron per atom); room-temperature τ=2.5e-14 s.
σ=8.5e28×(1.6e-19)²×2.5e-14/9.11e-31 = 5.99e7 S/m (measured 5.96e7, 0.5% off).
Mobility μ=1.6e-19×2.5e-14/9.11e-31 = 4.4e-3 m²/(V·s) = 44 cm²/(V·s); mean free path λ=1e6×2.5e-14=25nm, far larger than the 0.3nm lattice spacing.
GaAs electron mobility
GaAs n=1e23 m⁻³ (doped), τ=2e-13 s, m*=0.067m_e=6.1e-32 kg.
μ=1.6e-19×2e-13/6.1e-32 = 5.2e-1 m²/(V·s) = 5200 cm²/(V·s) (high-purity measured ~8500).
GaAs mobility is ~5× that of Si (μ=1500), used in high-speed transistors (HEMT, mm-wave). InSb m*=0.014m_e reaches μ 80000, used in infrared detectors.
常見問題
Why does the Drude model overestimate electron heat capacity?
Drude assumed all electrons follow Boltzmann statistics and contribute to heat capacity, predicting c_e=3nk_B/2 — 100× the experiment. Sommerfeld corrected with quantum Fermi–Dirac statistics: only electrons within kT of the Fermi surface can be excited, c_e=(π²/2)·n·k_B·(k_B·T/E_F), smaller by T/E_F≈T/T_F≈0.01, matching experiment.
How is the relaxation time τ measured?
(1) Invert from conductivity: τ=σ·m/(n·e²), needing σ and n; (2) Hall effect for n; (3) SdH quantum oscillation period for the Fermi surface; (4) ARPES directly for electron lifetime; (5) anomalous skin effect at high frequency. Metal τ~1e-14 s corresponds to mean free path 10–50nm.
Why use effective mass, not electron mass, for semiconductors?
In a semiconductor electrons move in the periodic lattice potential, effectively as an effective mass m*. Si electron m*=0.26m_e, GaAs 0.067m_e, InSb 0.014m_e. Smaller m* gives larger mobility μ=eτ/m*, so GaAs (μ=8500) far exceeds Si (μ=1500).
Why does metal conductivity rise sharply at low temperature?
At room temperature electrons are mainly scattered by phonons, τ∝T⁻⁵. As T→0 phonons freeze out and τ grows rapidly, so σ can rise 10⁴–10⁶×. Residual impurity scattering sets the limit (residual resistivity). Ultra-pure copper at 4K can reach σ~10¹² S/m; entering superconductivity makes σ→∞.
How does Drude differ from band theory?
Drude treats electrons as free particles (no periodic field); band theory includes the periodic lattice, electrons obey Bloch's theorem and energy forms bands. Metals have overlapping conduction and valence bands, semiconductors have a band gap Eg. Band theory explains conductor/semiconductor/insulator differences; Drude applies only to metal conduction-band electrons.
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參考資料
Content reviewed by the Hong Kong Calculator science team. σ=ne²τ/m; copper n=8.5e28, τ=2.5e-14 → σ=5.99e7 S/m (measured 5.96e7). Mobility μ=eτ/m=44 cm²/(V·s). Mean free path λ=v_F·τ=25nm.