3  Doping, Fermi level and conductivity

Open in Colab  Run this chapter’s code in Google Colab (no installation).

Slides 18–32: drift mobility and conductivity, n- and p-type doping, band diagrams, compensation, degenerate semiconductors, worked examples, bands in an applied field.

3.1 Key relations

A group-V atom (As, P) in Si gives a donor level a few tens of meV below \(E_c\); at room temperature it is ionised and \(n\approx N_d\). A group-III atom (B) gives an acceptor just above \(E_v\) and \(p\approx N_a\) (slides 19–23). The minority concentration follows from the mass action law. With both dopants present (compensation, slides 26–27):

\[ N_d>N_a:\ \ n=N_d-N_a,\ \ p=\frac{n_i^2}{N_d-N_a};\qquad N_a>N_d:\ \ p=N_a-N_d,\ \ n=\frac{n_i^2}{N_a-N_d}. \]

The Fermi level moves away from \(E_{Fi}\) by (slides 29–30)

\[ E_{Fn}-E_{Fi}=k_BT\ln\frac{n}{n_i},\qquad E_{Fp}-E_{Fi}=-k_BT\ln\frac{p}{n_i}. \]

Carriers drift in a field with \(v_{de}=\mu_eE_x\), \(v_{dh}=\mu_hE_x\), and the conductivity is (slide 18)

\[ \sigma=en\mu_e+ep\mu_h . \]

The code solves the exact neutrality condition \(n+N_a=p+N_d\) with \(np=n_i^2\) (sl.carriers). It reduces to the formulas above whenever the net doping is much larger than \(n_i\), and it also handles the intermediate cases.

3.2 Doping explorer

3.3 Worked examples (slides 29–31), recomputed

Slides 29–30. n-type Si with \(N_d=10^{16}\) cm\(^{-3}\) P. Find \(E_{Fn}-E_{Fi}\). Then add \(2\times10^{17}\) cm\(^{-3}\) B and find \(E_{Fp}-E_{Fi}\). Slide 31. Intrinsic conductivity of Si, then \(n\), \(p\) and \(\sigma\) for \(N_d=10^{16}\) cm\(^{-3}\) with \(\mu_e=1200\) cm² V⁻¹ s⁻¹.

Show the Python 
ni = Si["ni"]
rows = {}
for label, Nd, Na, mue in (("intrinsic Si", 0, 0, Si["mu_e"]),
                           ("Nd = 1e16 (slides 29, 31)", 1e16*CM3, 0, 1200*CM2),
                           ("Nd = 1e16 + Na = 2e17 (slide 30)", 1e16*CM3, 2e17*CM3, Si["mu_e"])):
    n, p = sl.carriers(Nd, Na, ni)
    rows[label] = {"n (cm⁻³)": f"{n/CM3:.3g}", "p (cm⁻³)": f"{p/CM3:.3g}",
                   "E_F − E_Fi (eV)": round(float(sl.fermi_level(n, ni)), 3),
                   "σ (S/cm)": f"{sl.conductivity(n, p, mue, Si['mu_h'])/100:.3g}"}
pd.DataFrame(rows).T
Table 3.1: Slides 29–31 recomputed (n_i = 10¹⁰ cm⁻³, kT = 25.85 meV).
n (cm⁻³) p (cm⁻³) E_F − E_Fi (eV) σ (S/cm)
intrinsic Si 1e+10 1e+10 0.0 3.11e-06
Nd = 1e16 (slides 29, 31) 1e+16 1e+04 0.357 1.92
Nd = 1e16 + Na = 2e17 (slide 30) 526 1.9e+17 -0.433 14.9

The slides give \(+0.358\) eV, \(-0.434\) eV, \(\sigma_i=3.1\times10^{-6}\) Ω⁻¹ cm⁻¹ and \(\sigma=1.92\) Ω⁻¹ cm⁻¹. The compensated row uses the Table 3.1 hole mobility of 490 cm² V⁻¹ s⁻¹, which is too optimistic at \(2\times10^{17}\) cm⁻³ of ionised impurities (see the mobility section below). Note also that slide 22 is titled “n-Type” and writes \(p\approx N_d\): it describes p-type doping, with \(p\approx N_a\).

3.4 Where is the Fermi level?

Show the Python 
N = np.logspace(10, 20, 300)*CM3
kT = sl.kT()
EFi = Si["Eg"]/2 - 0.5*kT*np.log(Si["Nc"]/Si["Nv"])
# exact (Fermi–Dirac) position: solve Nc F1/2((EF−Ec)/kT) − Nv F1/2((Ev−EF)/kT) = Nd − Na
def EF_exact(Nnet):
    f = lambda EF: (Si["Nc"]*sl.fermi_half((EF - Si["Eg"])/kT) - Si["Nv"]*sl.fermi_half(-EF/kT) - Nnet)/abs(Nnet)
    return brentq(f, -0.4, Si["Eg"] + 0.4)
Nn = np.logspace(14, 20, 25)*CM3
fig, ax = plt.subplots()
ax.semilogx(N/CM3, EFi + kT*np.log(sl.carriers(N, 0, Si["ni"])[0]/Si["ni"]), "--", color=sl.PALETTE[0], lw=1.4, label="n-type, Boltzmann")
ax.semilogx(N/CM3, EFi - kT*np.log(sl.carriers(0, N, Si["ni"])[1]/Si["ni"]), "--", color=sl.PALETTE[1], lw=1.4, label="p-type, Boltzmann")
ax.semilogx(Nn/CM3, [EF_exact(x) for x in Nn], color=sl.PALETTE[0], label="n-type, Fermi–Dirac")
ax.semilogx(Nn/CM3, [EF_exact(-x) for x in Nn], color=sl.PALETTE[1], label="p-type, Fermi–Dirac")
ax.axhspan(Si["Eg"] - 3*kT, Si["Eg"] + 0.1, color=sl.PALETTE[0], alpha=0.08); ax.axhspan(-0.1, 3*kT, color=sl.PALETTE[1], alpha=0.08)
for y, t in ((Si["Eg"], "E_c"), (0, "E_v"), (EFi, "E_Fi")):
    ax.axhline(y, color=sl.INK if t != "E_Fi" else sl.INK2, lw=1 if t != "E_Fi" else 0.8, ls="-" if t != "E_Fi" else ":")
    ax.text(1.2e10, y + 0.015, t, fontsize=8)
ax.set(xlabel="dopant concentration (cm⁻³)", ylabel="E_F − E_v (eV)", ylim=(-0.1, 1.2)); ax.legend(fontsize=8, loc="center left")
plt.show()
Figure 3.1: Fermi level position in Si at 300 K versus net doping. Dashed: Boltzmann formula E_F − E_Fi = ±kT ln(N/n_i). The shaded bands mark where E_F comes within 3kT of a band edge: the material becomes degenerate (slide 28) and the formula overestimates the shift.

3.5 Mobility depends on doping

The constant mobilities of Table 3.1 hold for lightly doped crystals. Ionised dopants scatter carriers, and the empirical Caughey–Thomas law describes the drop:

\[ \mu(N)=\mu_\text{min}+\frac{\mu_\text{max}-\mu_\text{min}}{1+(N/N_\text{ref})^\alpha}. \]

Show the Python 
def mu_ct(N, carrier):
    mmin, mmax, Nref, a = (68.5, 1414, 9.2e16, 0.711) if carrier == "e" else (44.9, 470.5, 2.23e17, 0.719)
    return mmin + (mmax - mmin)/(1 + (N/Nref)**a)      # N in cm^-3, result in cm²/Vs
N = np.logspace(13, 20, 300)
fig, ax = plt.subplots(1, 2, figsize=(9.5, 3.8))
ax[0].semilogx(N, mu_ct(N, "e"), label="electrons"); ax[0].semilogx(N, mu_ct(N, "h"), label="holes")
ax[0].plot(1e16, 1200, "o", color=sl.PALETTE[7], ms=7); ax[0].text(1.3e16, 1230, "slide 31", fontsize=8)
ax[0].set(xlabel="dopant concentration (cm⁻³)", ylabel="μ (cm² V⁻¹ s⁻¹)"); ax[0].legend()
ax[1].loglog(N, 1/(sl.qe*N*mu_ct(N, "e")), label="n-type"); ax[1].loglog(N, 1/(sl.qe*N*mu_ct(N, "h")), label="p-type")
ax[1].set(xlabel="dopant concentration (cm⁻³)", ylabel="resistivity ρ (Ω cm)"); ax[1].legend()
plt.tight_layout(); plt.show()
print(f"μh at 2e17 cm⁻³: {mu_ct(2e17, 'h'):.0f} cm²/Vs (Table 3.1: 490); μe at 1e16: {mu_ct(1e16, 'e'):.0f} cm²/Vs")
Figure 3.2: Si at 300 K: drift mobility (left) and resistivity (right) versus doping (Caughey–Thomas parameters of Masetti/Arora). The slide-31 value μe ≈ 1200 cm²/Vs at 10¹⁶ cm⁻³ is marked. 
μh at 2e17 cm⁻³: 266 cm²/Vs (Table 3.1: 490); μe at 1e16: 1184 cm²/Vs

3.6 Beyond the slides: freeze-out and the intrinsic regime

“All donors are ionised” fails at low temperature, where electrons fall back onto the donors (freeze-out). It also fails at high temperature, where \(n_i\) overtakes \(N_d\). Solving neutrality with the ionised-donor fraction \(N_d^+=N_d/[1+2\exp((E_F-E_d)/k_BT)]\) shows the three regimes of every doped semiconductor:

Show the Python 
Nd, Ed = 1e15*CM3, 0.045
T = np.linspace(25, 900, 300)
n_out = []
for Tk in T:
    kt = sl.kT(Tk)
    Nc, Nv = Si["Nc"]*(Tk/300)**1.5, Si["Nv"]*(Tk/300)**1.5
    Eg = sl.varshni(Tk, *sl.VARSHNI["Si"]) - sl.varshni(300, *sl.VARSHNI["Si"]) + Si["Eg"]
    def neutral(x):                              # x = (Ec − EF)/kT
        n = Nc*np.exp(-x); p = Nv*np.exp(-(Eg/kt - x)); Ndp = Nd/(1 + 2*np.exp(Ed/kt - x))
        return np.log(n + 1e-300) - np.log(p + Ndp)
    x = brentq(neutral, -5, Eg/kt + 5)
    n_out.append(Nc*np.exp(-x))
n_out = np.array(n_out)
fig, ax = plt.subplots()
ax.semilogy(T, n_out/CM3, label="n (exact neutrality)")
ax.semilogy(T, sl.n_i("Si", T)/CM3, "--", color=sl.INK2, lw=1.3, label="n_i")
ax.axhline(Nd/CM3, color=sl.PALETTE[1], ls=":", lw=1.3, label="N_d")
ax.set(xlabel="T (K)", ylabel="n (cm⁻³)", ylim=(1e11, 1e17)); ax.legend(); plt.show()
Figure 3.3: Electron concentration of Si doped with 10¹⁵ cm⁻³ phosphorus (E_c − E_d = 45 meV) versus temperature. Ionisation (freeze-out), extrinsic plateau n ≈ N_d, and intrinsic regime n ≈ n_i. Devices are designed to work on the plateau.

3.7 Bands tilt in an applied field (slide 32)

A voltage \(V\) across a bar of length \(L\) adds the potential energy \(-eV(x)\) to every electron: the whole band diagram tilts with slope \(eE_x\). Electrons roll “downhill” in the CB, holes “uphill” in the VB, and the Fermi levels at the two contacts differ by \(eV\), exactly the rule of slide 17. A 1 cm Si bar with 1 V applied tilts by 1 eV over 1 cm. The same picture gives the band bending inside a pn junction (Chapter 4), where the field comes from the space charge.

  • Hall effect. Combine the drift equations with the Lorentz force to get \(R_H=(p\mu_h^2-n\mu_e^2)/[e(p\mu_h+n\mu_e)^2]\). At which \(p/n\) does the Hall coefficient of Si change sign?
  • Minimum conductivity. Show that \(\sigma\) is minimum when \(p=n_i\sqrt{\mu_e/\mu_h}\) (slightly p-type), not at intrinsic doping. Find \(\sigma_\text{min}\) for Si and GaAs.
  • Bandgap narrowing. Heavy doping (> 10¹⁸ cm⁻³) shrinks \(E_g\) and raises the effective \(n_i\). Look up the Slotboom model and add it to EF_exact. This matters for the emitter of an LED (Chapter 9, degenerate junction).
  • Velocity saturation. \(v=\mu E\) breaks down above ~10⁴ V/cm: \(v\) saturates near \(10^7\) cm/s. Plot \(v(E)\) with the Caughey–Thomas field model.
  • Kasap, Optoelectronics and Photonics, sec. 3.1; Sze & Ng, ch. 1 (mobility models, freeze-out).
  • G. Masetti, M. Severi, S. Solmi, IEEE Trans. Electron Devices 30, 764 (1983): doping-dependent mobility in Si.

3.8 Try it yourself

  1. A Si wafer has \(\rho=1\) Ω cm, n-type. Find \(N_d\) with the Caughey–Thomas mobility (use brentq).
  2. In the explorer, dope Si with \(N_d=10^{16}\) cm⁻³ and raise \(T\). At what temperature does \(p\) reach 10 % of \(n\)?
  3. A GaAs layer must be semi-insulating (\(\rho>10^7\) Ω cm). What is the maximum net doping allowed? Why is it easier to obtain with GaAs than with Si?
  4. Repeat the freeze-out figure with B acceptors (\(E_a-E_v=45\) meV) and with a deep level at 0.3 eV. What changes?