4  The pn junction at equilibrium

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

Slides 40–47 (space charge layer, Poisson’s equation, built-in field and voltage, depletion widths, law of the junction) and 82–85 (band diagrams of the junction, built-in voltage from the band diagram).

4.1 Key relations

When p- and n-type regions touch, electrons and holes diffuse across the junction and recombine. They leave behind ionised acceptors (\(-eN_a\)) on the p side and ionised donors (\(+eN_d\)) on the n side: the space charge layer (SCL) or depletion region (slide 40). Its charge creates a field that stops further diffusion. Poisson’s equation links charge, field and potential (slide 41):

\[ \frac{d^2V}{dx^2}=-\frac{dE}{dx}=-\frac{\rho_\text{net}(x)}{\varepsilon},\qquad \varepsilon=\varepsilon_0\varepsilon_r . \]

For an abrupt junction in the depletion approximation (slides 43–44):

\[ N_aW_p=N_dW_n,\qquad E_o=-\frac{eN_dW_n}{\varepsilon}=-\frac{eN_aW_p}{\varepsilon},\qquad V_o=-\tfrac12E_oW_o, \] \[ V_o=\frac{k_BT}{e}\ln\frac{N_aN_d}{n_i^2},\qquad W=\left[\frac{2\varepsilon(N_a+N_d)(V_o-V)}{eN_aN_d}\right]^{1/2}. \]

\(V\) is the applied voltage (positive for forward bias). In equilibrium the Fermi level is flat (slide 45) and the minority and majority concentrations on the two sides obey the law of the junction (slides 46–47), Boltzmann statistics applied across the barrier \(eV_o\):

\[ \frac{p_{no}}{p_{po}}=\exp\!\left(-\frac{eV_o}{k_BT}\right),\qquad \frac{n_{po}}{n_{no}}=\exp\!\left(-\frac{eV_o}{k_BT}\right). \]

4.2 Junction explorer

In the band diagram the energy axis is electron energy, referenced to the p-side Fermi level. Forward bias (\(V>0\)) raises the n-side Fermi level by \(eV\) and lowers the barrier. Reverse bias does the opposite and widens the SCL. The depletion region extends mostly into the lightly doped side: try \(N_a=10^{19}\), \(N_d=10^{15}\) cm⁻³ (a p⁺n junction).

4.3 Worked numbers, recomputed

Show the Python 
cases = {"GaAs, Na = Nd = 1e17, V = 0 (slide 78)": ("GaAs", 1e17, 1e17, 0.0),
         "GaAs, Na = Nd = 1e17, V = 0.80 V (slide 79)": ("GaAs", 1e17, 1e17, 0.80),
         "Si p⁺n, Na = 1e18, Nd = 1e16, V = 0": ("Si", 1e18, 1e16, 0.0),
         "Si p⁺n, same, V = −10 V": ("Si", 1e18, 1e16, -10.0),
         "Ge, Na = 1e17, Nd = 1e16, V = 0": ("Ge", 1e17, 1e16, 0.0)}
rows = {}
for k, (m, Na, Nd, V) in cases.items():
    p = sl.MATERIALS[m]
    Vo = sl.built_in_voltage(Na*CM3, Nd*CM3, p["ni"])
    d = sl.depletion(Na*CM3, Nd*CM3, p["eps_r"], Vo, V)
    rows[k] = {"V_o (V)": round(Vo, 3), "W (µm)": round(d["W"]*1e6, 4), "W_p (µm)": round(d["Wp"]*1e6, 4),
               "W_n (µm)": round(d["Wn"]*1e6, 4), "|E_o| (kV/cm)": round(d["Eo"]/1e5, 1)}
pd.DataFrame(rows).T
Table 4.1: Built-in voltage, depletion widths and peak field for three abrupt junctions at 300 K. The GaAs row is the device of slides 75–79 (V_o = 1.27 V and W = 0.116 µm at 0.8 V forward bias on the slides).
V_o (V) W (µm) W_p (µm) W_n (µm) |E_o| (kV/cm)
GaAs, Na = Nd = 1e17, V = 0 (slide 78) 1.271 0.1911 0.0956 0.0956 133.0
GaAs, Na = Nd = 1e17, V = 0.80 V (slide 79) 1.271 0.1164 0.0582 0.0582 81.0
Si p⁺n, Na = 1e18, Nd = 1e16, V = 0 0.833 0.3313 0.0033 0.3280 50.3
Si p⁺n, same, V = −10 V 0.833 1.1946 0.0118 1.1828 181.4
Ge, Na = 1e17, Nd = 1e16, V = 0 0.374 0.2696 0.0245 0.2451 27.7

4.4 Charge, field, potential and bands

Show the Python 
Na, Nd = 1e17*CM3, 1e16*CM3
ni = sl.MATERIALS["Si"]["ni"]
Vo = sl.built_in_voltage(Na, Nd, ni)
x, rho, E, V = sl.junction_profile(Na, Nd, 11.8, Vo)
kT = sl.kT(); Eg = 1.11
EcP = Eg/2 + kT*np.log(Na/ni)                     # Ec − EF on the neutral p side (E_Fi at mid-gap)
fig, ax = plt.subplots(4, 1, figsize=(8, 8.5), sharex=True)
ax[0].fill_between(x*1e6, rho/sl.qe/CM3, alpha=0.3, lw=0); ax[0].plot(x*1e6, rho/sl.qe/CM3); ax[0].set_ylabel("ρ/e (cm⁻³)")
ax[1].fill_between(x*1e6, E/1e5, alpha=0.3, lw=0, color=sl.PALETTE[1]); ax[1].plot(x*1e6, E/1e5, color=sl.PALETTE[1]); ax[1].set_ylabel("E (kV/cm)")
ax[2].plot(x*1e6, V, color=sl.PALETTE[2]); ax[2].set_ylabel("V (V)")
ax[3].plot(x*1e6, EcP - V, label="E_c"); ax[3].plot(x*1e6, EcP - Eg - V, label="E_v", color=sl.PALETTE[1])
ax[3].axhline(0, color=sl.PALETTE[6], ls="--", lw=1.4, label="E_F"); ax[3].set(ylabel="energy (eV)", xlabel="x (µm)")
ax[3].legend(fontsize=8, ncol=3, loc="upper right")
plt.tight_layout(); plt.show()
Figure 4.1: Si junction with N_a = 10¹⁷ and N_d = 10¹⁶ cm⁻³ at equilibrium (slides 40 and 42): net space charge, field (triangular), electrostatic potential (two parabolas) and the resulting band diagram, where E_c(x) = E_c(p) − eV(x). The field peaks at the metallurgical junction.

4.5 Beyond the depletion approximation: solving Poisson exactly

The depletion approximation assumes a sharp edge between “fully depleted” and “neutral”. In reality the mobile carriers follow Boltzmann statistics in the local potential, \(n=n_ie^{\psi/V_T}\), \(p=n_ie^{-\psi/V_T}\), and Poisson’s equation becomes nonlinear:

\[ \frac{d^2\psi}{dx^2}=-\frac{e}{\varepsilon}\left[n_ie^{-\psi/V_T}-n_ie^{\psi/V_T}+N_d(x)-N_a(x)\right]. \]

A few Newton iterations on a finite-difference grid solve it (this is the core of every device simulator):

Show the Python 
def poisson_pn(Na, Nd, eps_r, ni, T=300.0, L=None, N=2001, tol=1e-10):
    Vt, eps = sl.kT(T), eps_r*sl.eps0
    Vo = sl.built_in_voltage(Na, Nd, ni, T)
    L = L or 2*sl.depletion(Na, Nd, eps_r, Vo)["W"]
    x = np.linspace(-L, L, N); dx = x[1] - x[0]
    Nnet = np.where(x < 0, -Na, Nd)
    u = np.where(x < 0, -np.log(Na/ni), np.log(Nd/ni))          # ψ/Vt, start from neutral values
    c = sl.qe*ni/(eps*Vt)*dx**2
    for _ in range(200):
        lap = np.zeros(N); lap[1:-1] = u[2:] - 2*u[1:-1] + u[:-2]
        R = lap + c*(np.exp(-u) - np.exp(u) + Nnet/ni); R[0] = R[-1] = 0
        main = -2 - c*(np.exp(-u) + np.exp(u)); main[0] = main[-1] = 1
        J = sp.diags([np.r_[np.ones(N-2), 0], main, np.r_[0, np.ones(N-2)]], [-1, 0, 1], format="csc")
        du = spsolve(J, -R); du = np.clip(du, -1, 1)              # damped Newton step
        u += du
        if np.max(abs(du)) < tol: break
    psi = u*Vt
    return x, psi - psi[0], -np.gradient(psi, x), ni*np.exp(u), ni*np.exp(-u)

xs, Vex, Eex, n, p = poisson_pn(Na, Nd, 11.8, ni)
xd, _, Ed, Vd = sl.junction_profile(Na, Nd, 11.8, Vo, npts=2001, span=2)
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
ax[0].plot(xs*1e6, Vex, label="V, exact"); ax[0].plot(xd*1e6, Vd, "--", color=sl.PALETTE[0], lw=1.3, label="V, depletion approx.")
a2 = ax[0].twinx(); a2.plot(xs*1e6, Eex/1e5, color=sl.PALETTE[1]); a2.plot(xd*1e6, Ed/1e5, "--", color=sl.PALETTE[1], lw=1.3)
a2.set_ylabel("E (kV/cm)", color=sl.PALETTE[1]); a2.spines["right"].set_visible(True); a2.grid(False)
ax[0].set(xlabel="x (µm)", ylabel="V (V)"); ax[0].legend(fontsize=8, loc="upper left")
ax[1].semilogy(xs*1e6, n/CM3, label="n"); ax[1].semilogy(xs*1e6, p/CM3, label="p")
ax[1].axhline(ni/CM3, color=sl.INK2, ls=":", lw=1); ax[1].text(xs[0]*1e6, ni/CM3*2, "n_i", fontsize=8)
ax[1].set(xlabel="x (µm)", ylabel="concentration (cm⁻³)"); ax[1].legend(fontsize=8)
plt.tight_layout(); plt.show()
LD = np.sqrt(11.8*sl.eps0*sl.kT()/(sl.qe*Nd))
print(f"V_o: exact {Vex[-1]:.4f} V, formula {Vo:.4f} V;  Debye length on the n side L_D = {LD*1e9:.0f} nm")
print(f"law of the junction: p_n0/p_p0 = {p[-1]/p[0]:.3e},  exp(−eVo/kT) = {np.exp(-Vo/sl.kT()):.3e}")
Figure 4.2: Exact (Poisson–Boltzmann, solid) versus depletion-approximation (dashed) solution for the same Si junction. Left: potential and field agree except for soft edges a few Debye lengths wide. Right: carrier concentrations on a log scale (slide 40c): the majority carriers drop smoothly across the SCL and n·p = n_i² everywhere at equilibrium. 
V_o: exact 0.7738 V, formula 0.7738 V;  Debye length on the n side L_D = 41 nm
law of the junction: p_n0/p_p0 = 1.000e-13,  exp(−eVo/kT) = 1.000e-13

The potential step of the exact solution equals \(V_o\) to four digits, and the ratio of hole concentrations on the two sides reproduces the law of the junction: slides 44 and 46 describe the same physics. The depletion edges are smeared over a few Debye lengths \(L_D=\sqrt{\varepsilon k_BT/e^2N}\).

4.6 Built-in voltage from the band diagram (slides 82–83)

Before contact the two Fermi levels differ; after contact they align and the bands bend by exactly that difference: \(eV_o=E_{Fn}-E_{Fp}=\Phi_p-\Phi_n\). With the electron affinity \(\chi\) of Table 3.1 the work functions follow from the position of \(E_F\) in each region:

Show the Python 
p = sl.MATERIALS["Si"]
EcEFn = -sl.kT()*np.log(Nd/p["Nc"])                  # (3.9.1)
EcEFp = -sl.kT()*np.log(p["ni"]**2/(Na*p["Nc"]))     # (3.9.2)
pd.DataFrame({"n side": {"E_c − E_F (eV)": EcEFn, "Φ = χ + E_c − E_F (eV)": p["chi"] + EcEFn},
              "p side": {"E_c − E_F (eV)": EcEFp, "Φ = χ + E_c − E_F (eV)": p["chi"] + EcEFp},
              "eV_o = Φ_p − Φ_n": {"E_c − E_F (eV)": np.nan, "Φ = χ + E_c − E_F (eV)": EcEFp - EcEFn}}).T.round(3)
Table 4.2: Si, Na = 10¹⁷ and Nd = 10¹⁶ cm⁻³: work functions Φ = χ + (E_c − E_F) of the separated regions (slide 24) and the built-in voltage from their difference (slide 83).
E_c − E_F (eV) Φ = χ + E_c − E_F (eV)
n side 0.205 4.255
p side 0.979 5.029
eV_o = Φ_p − Φ_n NaN 0.774
  • Linearly graded junction. Implanted and diffused junctions are graded, \(N_d-N_a=ax\). Show that \(W\propto(V_o-V)^{1/3}\) and check it with poisson_pn by changing Nnet.
  • Breakdown. The peak field limits the reverse voltage. Si breaks down by avalanche near 3–5×10⁵ V/cm. Use the explorer to estimate the breakdown voltage of a p⁺n junction with \(N_d=10^{15}\) and \(10^{17}\) cm⁻³. Which one would you choose for an avalanche photodiode (Ch. 5)?
  • pin structure. Insert an intrinsic layer of width \(d\) between p and n. The field becomes nearly uniform. Modify poisson_pn and compare with the p–n case. This is the photodiode geometry of Chapter 5.
  • Under bias. Extend the Newton solver with two quasi-Fermi levels and the drift–diffusion equations (Scharfetter–Gummel): a 1D device simulator in 100 lines. Packages such as solcore and devsim do this.
  • Kasap, Optoelectronics and Photonics, sec. 3.3; Sze & Ng, ch. 2 (pn junctions).
  • S. Selberherr, Analysis and Simulation of Semiconductor Devices (Springer, 1984): numerical methods behind device simulators.
  • DEVSIM: open-source TCAD in Python, https://devsim.org.

4.7 Try it yourself

  1. A Si p⁺n junction has \(N_d=10^{15}\) cm⁻³. Find the reverse voltage at which \(W\) reaches 10 µm (useful for a photodiode).
  2. Why does the depletion layer of a p⁺n junction lie almost entirely on the n side? Show that \(W_n/W\to1\) when \(N_a\gg N_d\).
  3. Compute \(V_o\) for GaAs and Ge junctions with the same doping. Why does it scale roughly with \(E_g\)?
  4. In poisson_pn, double the temperature to 600 K. What happens to \(V_o\) and to the Debye tails?