Run this chapter’s code in Google Colab (no installation).
Slides 70–74: depletion capacitance, dynamic resistance, diffusion capacitance (long and short diodes), small-signal equivalent circuit.
The SCL stores charge \(\pm Q\) separated by \(W\), like a parallel-plate capacitor whose gap changes with the voltage. Its incremental capacitance is the depletion capacitance (slides 70 and 74):
\[ C_\text{dep}=\left|\frac{dQ}{dV}\right|=\frac{\varepsilon A}{W}=\frac{A}{(V_o-V)^{1/2}}\left[\frac{e\varepsilon N_aN_d}{2(N_a+N_d)}\right]^{1/2} \ \xrightarrow{\ p^+n\ }\ A\left[\frac{e\varepsilon N_d}{2(V_o-V)}\right]^{1/2}. \]
Under forward bias the injected minority charge \(Q=\tau_hI\) also changes with \(V\). This gives a dynamic resistance and a diffusion capacitance (slides 71–73):
\[ r_d=\frac{dV}{dI}=\frac{\eta V_\text{th}}{I},\qquad C_\text{diff}=\frac{\tau_hI}{2V_\text{th}}=\frac{\tau_h}{2r_d}\ \text{(long diode, ac)},\qquad C_\text{diff}=\frac{\tau_tI}{V_\text{th}}\ \text{(short diode, } \tau_t=\ell_n^2/2D_h), \]
with \(V_\text{th}=k_BT/e\). The small-signal equivalent circuit is a series resistance \(r_s\) followed by \(r_d\), \(C_\text{diff}\) and \(C_\text{dep}\) in parallel (slide 71c). \(C_\text{dep}\) dominates under reverse bias (photodiodes, varactors) and \(C_\text{diff}\) under forward bias (LEDs).
The dashed grey curves are the three junctions of slide 70(b). Your junction is the solid line.
The capacitance is set by the lightly doped side: the 5×10¹⁶/5×10¹⁴ junction is almost identical to the 5×10¹⁴/5×10¹⁴ one. The curve ends near \(V_o\), where the depletion approximation breaks down (and \(C_\text{diff}\) takes over).
fig, ax = plt.subplots(1, 2, figsize=(10, 4))
V = np.linspace(-10, 0.5, 300)
p = sl.MATERIALS["Si"]
for Na, Nd in ((5e16, 5e16), (5e16, 5e14), (5e14, 5e14)):
Vo = sl.built_in_voltage(Na*CM3, Nd*CM3, p["ni"])
Vv = V[V < Vo - 0.05]
ax[0].semilogy(Vv, sl.c_dep(1e-6, Na*CM3, Nd*CM3, p["eps_r"], Vo, Vv)*1e12, label=f"N_a = {Na:.0e}, N_d = {Nd:.0e}")
ax[0].set(xlabel="V (V)", ylabel="C_dep (pF mm⁻²)"); ax[0].legend(fontsize=7)
# synthetic measurement on a p⁺n junction, then fit
rng = np.random.default_rng(1)
Na, Nd, A = 1e19*CM3, 3e15*CM3, 1e-6
Vo = sl.built_in_voltage(Na, Nd, p["ni"])
Vm = np.linspace(-8, 0.3, 30)
Cm = sl.c_dep(A, Na, Nd, p["eps_r"], Vo, Vm)*(1 + 0.01*rng.standard_normal(Vm.size))
slope, icpt = np.polyfit(Vm, 1/Cm**2, 1)
Nd_fit = -2/(sl.qe*p["eps_r"]*sl.eps0*A**2*slope)
ax[1].plot(Vm, 1/Cm**2/1e22, "o", ms=5, label="'measured'")
ax[1].plot(Vm, np.polyval([slope, icpt], Vm)/1e22, label="linear fit")
ax[1].set(xlabel="V (V)", ylabel="1/C² (10²² F⁻²)"); ax[1].legend(fontsize=8)
plt.tight_layout(); plt.show()
print(f"true N_d = {Nd/CM3:.2e} cm⁻³, fitted N_d = {Nd_fit/CM3:.2e} cm⁻³; true V_o = {Vo:.3f} V, fitted V_o = {-icpt/slope:.3f} V")
true N_d = 3.00e+15 cm⁻³, fitted N_d = 3.02e+15 cm⁻³; true V_o = 0.862 V, fitted V_o = 0.894 VFor a p⁺n junction \(1/C_\text{dep}^2=2(V_o-V)/(e\varepsilon N_dA^2)\), so \(N_d=-2/(e\varepsilon A^2\,d(1/C^2)/dV)\). Applied point by point, with \(W=\varepsilon A/C\) as the depth, the same formula gives the doping profile \(N_d(W)\). This is the standard C–V characterisation of LED and photodiode wafers.
p = sl.MATERIALS["GaAs"]
Na = Nd = 1e23; A = 1e-6; tau = 50e-9
Vo = sl.built_in_voltage(Na, Nd, p["ni"])
gaas = dict(A=A, Na=Na, Nd=Nd, mat="GaAs", B=2e-16, mu_e=5000*CM2, mu_h=250*CM2)
rows = {}
for V in (0.8, 1.0, 1.05, 1.1):
d = sl.diode_currents(V, **gaas)
I = float(d["I"]); dI = float(sl.diode_currents(V + 1e-4, **gaas)["I"]) - I
rd = 1e-4/dI
Cd = sl.c_diff(float(d["I_diff"]), tau) # only the diffusion part stores minority charge
Cj = sl.c_dep(A, Na, Nd, p["eps_r"], Vo, V)
rows[f"V = {V} V"] = {"I (mA)": I*1e3, "r_d (Ω)": rd, "C_dep (pF)": Cj*1e12, "C_diff (pF)": Cd*1e12,
"r_d(C_dep + C_diff) (ns)": rd*(Cj + Cd)*1e9}
pd.DataFrame(rows).T.map(lambda v: f"{v:.3g}")| I (mA) | r_d (Ω) | C_dep (pF) | C_diff (pF) | r_d(C_dep + C_diff) (ns) | |
|---|---|---|---|---|---|
| V = 0.8 V | 0.00217 | 2.37e+04 | 989 | 0.117 | 2.34e+04 |
| V = 1.0 V | 0.351 | 83.1 | 1.3e+03 | 268 | 131 |
| V = 1.05 V | 2.09 | 12.9 | 1.44e+03 | 1.85e+03 | 42.7 |
| V = 1.1 V | 13.7 | 1.92 | 1.64e+03 | 1.28e+04 | 27.8 |
At low bias the recombination current has no stored minority charge, \(\eta=2\), and the RC product is set by \(C_\text{dep}\). At LED operating currents \(r_dC_\text{diff}\to\tau/2\): the diode cannot respond faster than its minority carrier lifetime. This is the electrical side of the LED modulation bandwidth of Chapter 12, \(f_{3\text{dB}}\propto1/\tau\).
f = np.logspace(3, 10, 400); w = 2*np.pi*f; rs = 2.0
d = sl.diode_currents(1.05, **gaas); I = float(d["I_diff"] + d["I_rec"])
rd = sl.r_d(I); Cf = sl.c_diff(float(d["I_diff"]), tau) + sl.c_dep(A, Na, Nd, p["eps_r"], Vo, 1.05)
Cr = sl.c_dep(A, Na, Nd, p["eps_r"], Vo, -5)
Zf = rs + rd/(1 + 1j*w*rd*Cf); Zr = rs + 1/(1j*w*Cr)
fig, ax = plt.subplots(figsize=(7.5, 3.6))
ax.loglog(f, abs(Zf), label=f"forward, I = {I*1e3:.1f} mA (r_d = {rd:.1f} Ω, C = {Cf*1e9:.1f} nF)")
ax.loglog(f, abs(Zr), label=f"reverse −5 V (C_dep = {Cr*1e12:.0f} pF)")
ax.set(xlabel="frequency (Hz)", ylabel="|Z| (Ω)"); ax.legend(fontsize=8); plt.show()