Run this chapter’s code in Google Colab (no installation).
Slides 10–17 and 25: electron and hole concentrations, effective density of states, mass action law, intrinsic Fermi level, average electron energy, the meaning of \(E_F\), Table 3.1.
The electron concentration in the CB is the area under \(g_{cb}(E)f(E)\) (slide 10). When \(E_c-E_F\gg k_BT\) (a non-degenerate semiconductor) the Fermi–Dirac function reduces to a Boltzmann tail and the integral is analytic (slides 11–13):
\[ n=N_c\exp\!\left[-\frac{E_c-E_F}{k_BT}\right],\qquad p=N_v\exp\!\left[-\frac{E_F-E_v}{k_BT}\right],\qquad N_{c,v}=2\left(\frac{2\pi m_{e,h}^*k_BT}{h^2}\right)^{3/2}. \]
Their product does not depend on \(E_F\). This is the mass action law (slide 14), valid in equilibrium and in the dark:
\[ np=n_i^2=N_cN_v\exp\!\left(-\frac{E_g}{k_BT}\right). \]
In an intrinsic semiconductor \(n=p=n_i\), which places the Fermi level slightly off mid-gap (slide 15): \(E_{Fi}=E_v+\tfrac12E_g-\tfrac12k_BT\ln(N_c/N_v)=E_v+\tfrac12E_g-\tfrac34k_BT\ln(m_e^*/m_h^*)\). Electrons in the CB have an average energy \(E_c+\tfrac32k_BT\) above the band edge, like the atoms of an ideal gas (slide 16).
Choose a material, a temperature and a Fermi-level position. The panel integrates \(g(E)f(E)\) numerically (no Boltzmann approximation) and compares the result with the formulas above. Move \(E_F\) inside a band to see the Boltzmann formula fail: that is a degenerate semiconductor (Section 2.6).
rows = {}
for m, p in sl.MATERIALS.items():
mass = lambda N: (N/2)**(2/3)*sl.h**2/(2*np.pi*sl.kB*300)/sl.me
rows[m] = {"a (nm)": p["a"]*1e9, "Eg (eV)": p["Eg"], "χ (eV)": p["chi"],
"Nc (cm⁻³)": f"{p['Nc']/CM3:.2e}", "Nv (cm⁻³)": f"{p['Nv']/CM3:.2e}",
"ni table (cm⁻³)": f"{p['ni']/CM3:.1e}",
"ni = √(NcNv)e^(−Eg/2kT)": f"{np.sqrt(p['Nc']*p['Nv'])*np.exp(-p['Eg']/(2*sl.kT()))/CM3:.1e}",
"ε_r": p["eps_r"], "μe (cm²/Vs)": p["mu_e"]/sl.CM2, "μh (cm²/Vs)": p["mu_h"]/sl.CM2,
"me*/me": round(mass(p["Nc"]), 3), "mh*/me": round(mass(p["Nv"]), 3)}
pd.DataFrame(rows).T| a (nm) | Eg (eV) | χ (eV) | Nc (cm⁻³) | Nv (cm⁻³) | ni table (cm⁻³) | ni = √(NcNv)e^(−Eg/2kT) | ε_r | μe (cm²/Vs) | μh (cm²/Vs) | me*/me | mh*/me | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ge | 0.565 | 0.66 | 4.13 | 1.04e+19 | 6.00e+18 | 2.3e+13 | 2.3e+13 | 16.0 | 3900.0 | 1900.0 | 0.556 | 0.385 |
| Si | 0.5431 | 1.11 | 4.05 | 2.80e+19 | 1.20e+19 | 1.0e+10 | 8.7e+09 | 11.8 | 1450.0 | 490.0 | 1.076 | 0.612 |
| InP | 0.5868 | 1.35 | 4.5 | 5.20e+17 | 1.10e+19 | 3.0e+07 | 1.1e+07 | 12.6 | 4600.0 | 150.0 | 0.075 | 0.577 |
| GaAs | 0.5653 | 1.42 | 4.07 | 4.70e+17 | 7.00e+18 | 2.1e+06 | 2.1e+06 | 13.0 | 8500.0 | 400.0 | 0.071 | 0.427 |
| AlAs | 0.5661 | 2.17 | 3.5 | 1.50e+19 | 1.70e+19 | 1.0e+01 | 9.5e+00 | 10.1 | 200.0 | 100.0 | 0.71 | 0.771 |
The recomputed \(n_i\) agrees with the table to within the uncertainty of the data (the slide notes that \(n_i\) of Si varies between books; \(9.7\times10^9\) cm\(^{-3}\) is the modern value). One correction: the slide prints \(N_v=6.0\times10^{19}\) cm\(^{-3}\) for Ge, but the textbook and the mass action law both require \(6.0\times10^{18}\) cm\(^{-3}\), which is used here.
kT = sl.kT(300)
E = np.linspace(0, 12*kT, 3000)
nE = np.sqrt(E)*np.exp(-E/kT) # g(E) f(E) up to a constant
mean = np.trapezoid(E*nE, E)/np.trapezoid(nE, E)
fig, ax = plt.subplots(figsize=(7.5, 3.4))
ax.fill_between(E*1e3, nE/nE.max(), alpha=0.25, lw=0); ax.plot(E*1e3, nE/nE.max())
ax.axvline(kT/2*1e3, color=sl.INK2, ls=":", lw=1.2); ax.axvline(mean*1e3, color=sl.PALETTE[1], ls="--", lw=1.5)
ax.text(kT/2*1e3, 1.02, " peak kT/2", fontsize=8, color=sl.INK2); ax.text(mean*1e3, 0.9, f" mean = {mean/kT:.3f} kT", fontsize=8)
ax.set(xlabel="E − E_c (meV)", ylabel="n_E (normalised)"); plt.show()
\(n_i\propto T^{3/2}\exp(-E_g/2k_BT)\). The exponential dominates, so \(\ln n_i\) against \(1/T\) is nearly a straight line with slope \(-E_g/2k_B\). The bandgap itself shrinks as \(T\) rises (Varshni law, Chapter 9), which is included below.
T = np.linspace(200, 800, 400)
fig, ax = plt.subplots()
for m in ("Ge", "Si", "GaAs"):
ni = sl.n_i(m, T)/CM3
ax.semilogy(1000/T, ni, label=m)
Tx = T[np.argmin(abs(np.log(ni/1e15)))]
if ni.max() > 1e15: ax.plot(1000/Tx, 1e15, "o", color=ax.lines[-1].get_color(), ms=6)
ax.axhline(1e15, color=sl.INK2, ls="--", lw=1)
sec = ax.secondary_xaxis("top", functions=(lambda x: 1000/x, lambda T: 1000/T)); sec.set_xlabel("T (K)")
ax.set(xlabel="1000/T (K⁻¹)", ylabel="n_i (cm⁻³)"); ax.legend(); plt.show()
out = {}
for m in ("Ge", "Si", "GaAs"):
p = sl.MATERIALS[m]
for T in (250, 300, 400):
Nc, Nv = p["Nc"]*(T/300)**1.5, p["Nv"]*(T/300)**1.5
out[(m, T)] = {"n_i (cm⁻³)": f"{sl.n_i(m, T)/CM3:.2e}",
"E_Fi − midgap (meV)": round(-0.5*sl.kT(T)*np.log(Nc/Nv)*1e3, 1)}
pd.DataFrame(out).T| n_i (cm⁻³) | E_Fi − midgap (meV) | ||
|---|---|---|---|
| Ge | 250 | 8.81e+11 | -5.9 |
| 300 | 2.30e+13 | -7.1 | |
| 400 | 1.54e+15 | -9.5 | |
| Si | 250 | 7.84e+07 | -9.1 |
| 300 | 1.00e+10 | -11.0 | |
| 400 | 4.92e+12 | -14.6 | |
| GaAs | 250 | 3.93e+03 | 29.1 |
| 300 | 2.10e+06 | 34.9 | |
| 400 | 6.11e+09 | 46.5 |
In GaAs the light electron mass (\(N_c\ll N_v\)) pushes \(E_{Fi}\) about 40 meV above mid-gap at room temperature.
The exact result is \(n=N_c\,\mathcal F_{1/2}(\eta)\) with \(\eta=(E_F-E_c)/k_BT\) and the Fermi–Dirac integral \(\mathcal F_{1/2}\). The Boltzmann formula is the limit \(\mathcal F_{1/2}(\eta)\to e^\eta\):
eta = np.linspace(-6, 6, 241)
F = sl.fermi_half(eta)
fig, ax = plt.subplots(figsize=(7.5, 3.6))
ax.semilogy(eta, F, label="exact F₁/₂(η)"); ax.semilogy(eta, np.exp(eta), "--", label="Boltzmann e^η")
ax.axvspan(-3, 6, color=sl.PALETTE[7], alpha=0.06); ax.text(-2.9, 200, "within 3kT of the band edge", fontsize=8, color=sl.INK2)
ax.set(xlabel="η = (E_F − E_c)/k_BT", ylabel="n / N_c"); ax.legend(); plt.show()
print("error of Boltzmann at η = -3, -1, 0, +2:", [f"{(np.exp(e)/sl.fermi_half(e) - 1)*100:.0f} %" for e in (-3, -1, 0, 2)])
error of Boltzmann at η = -3, -1, 0, +2: ['2 %', '12 %', '31 %', '162 %']\(E_F\) is the electrochemical potential of electrons. In equilibrium, in the dark and without applied voltage, it is flat across any system of materials in contact. An applied voltage \(V\) separates the Fermi levels at the two ends by \(\Delta E_F=eV\). This single rule builds every band diagram in the following chapters: pn junctions (Chapter 4), heterojunctions (Chapter 8), and the quasi-Fermi levels of a forward-biased LED.
scipy.optimize.brentq on sl.fermi_half).