9  The LED emission spectrum

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

Slides 98–109: energy distribution of carriers, emission spectrum, temperature dependence, worked examples on the linewidth and on the Varshni shift, degenerately doped junctions.

9.1 Key relations

Electrons in the CB are spread over a few \(k_BT\) above \(E_c\), holes a few \(k_BT\) below \(E_v\) (slide 99). The photon energy is \(h\nu=E_g+(\text{electron KE})+(\text{hole KE})\), so the spectrum follows the joint density of states times the Boltzmann tail:

\[ I(h\nu)\propto(h\nu-E_g)^{1/2}\exp\!\left[-\frac{h\nu-E_g}{k_BT}\right], \qquad h\nu_\text{peak}=E_g+\tfrac12k_BT,\qquad \Delta(h\nu)\approx1.8\,k_BT\text{ to }3\,k_BT . \]

Converting the energy width to wavelength with \(|d\lambda/d(h\nu)|=\lambda^2/hc\) gives the LED linewidth (slides 103–104):

\[ \Delta\lambda\approx\lambda_o^2\,\frac{3k_BT}{hc}. \]

The bandgap shrinks with temperature (Varshni law, slide 107): \(E_g(T)=E_{go}-AT^2/(B+T)\). The peak moves to longer wavelengths and the line broadens as the LED warms up.

9.2 Spectrum explorer

Dashed: the same LED at 25 °C. Without extra broadening the ideal spectrum has a FWHM of \(1.8k_BT\), narrower than the \(3k_BT\) rule of the slides. Real LEDs are broader because of alloy fluctuations, band tails and heating. Add 10–20 meV of broadening to reach the measured widths.

9.3 The ideal spectrum: peak and width

Show the Python 
x = np.linspace(0, 8, 80001)
I = np.sqrt(x)*np.exp(-x)
xp = x[np.argmax(I)]; above = x[I >= I.max()/2]
fig, ax = plt.subplots(figsize=(7.5, 3.3))
ax.plot(x, I/I.max()); ax.axvline(xp, ls=":", color=sl.INK2, lw=1.2)
ax.hlines(0.5, above[0], above[-1], color=sl.PALETTE[1], lw=2); ax.text(above[-1] + 0.1, 0.5, f"FWHM = {above[-1] - above[0]:.2f} kT", va="center", fontsize=9)
ax.set(xlabel="(hν − E_g)/k_BT", ylabel="relative intensity"); plt.show()
print(f"peak at {xp:.3f} kT, half-maximum points {above[0]:.3f} and {above[-1]:.3f} kT, FWHM {above[-1] - above[0]:.3f} kT")
Figure 9.1: Ideal LED spectrum in reduced units x = (hν − E_g)/k_BT. Peak at x = 1/2, half-intensity points at x = 0.05 and 1.85, so FWHM = 1.80 k_BT (slide 99). 
peak at 0.500 kT, half-maximum points 0.051 and 1.846 kT, FWHM 1.795 kT

9.4 Worked examples (slides 103–108), recomputed

Slides 103–104. Linewidth of a 1310 nm LED at 300 K with \(\Delta(h\nu)=3k_BT\). Slides 105–106. An AlGaAs LED has \(\lambda_o\) = 804, 822, 837 nm and \(\Delta\lambda\) = 30, 40, 48 nm at −40, 25 and 85 °C. Plot \(\Delta\lambda/\lambda_o^2\) against \(T\) and find \(m\) in \(\Delta(h\nu)=mk_BT\). Slides 107–108. Shift of the peak of a GaAs LED cooled from 25 °C to −25 °C (Varshni: \(E_{go}=1.519\) eV, \(A=5.41\times10^{-4}\) eV/K, \(B=204\) K).

Show the Python 
T = np.array([-40, 25, 85]) + 273.15
lam0 = np.array([804, 822, 837])*1e-9; dlam = np.array([30, 40, 48])*1e-9
y = dlam/lam0**2
slope = np.sum(y*T)/np.sum(T**2)                       # least squares through the origin
m = slope*sl.h*sl.c0/sl.kB
fig, ax = plt.subplots(figsize=(7, 3.4))
ax.plot(T, y*1e-4, "o", ms=8, label="slide 106 data")
Tl = np.linspace(0, 400, 10); ax.plot(Tl, slope*Tl*1e-4, label=f"slope {slope:.0f} m⁻¹ K⁻¹  →  m = {m:.2f}")
ax.set(xlabel="T (K)", ylabel="10⁻⁴ × Δλ/λo² (m⁻¹)", xlim=(0, 400), ylim=(0, 8)); ax.legend(fontsize=8); plt.show()
Figure 9.2: Slides 105–106: Δλ/λo² against T for the AlGaAs LED, with the best line forced through the origin. The slope gives m ≈ 2.8.
Show the Python 
Eg298, Eg248 = sl.varshni(298, *sl.VARSHNI["GaAs"]), sl.varshni(248, *sl.VARSHNI["GaAs"])
l298 = sl.wavelength_from_eV(Eg298 + sl.kT(298)/2); l248 = sl.wavelength_from_eV(Eg248 + sl.kT(248)/2)
hc_slide = 3e8*6.626e-34/1.602e-19                     # constants used on slide 107
pd.DataFrame({"value": {
    "Δλ of a 1310 nm LED at 300 K, m = 3 (slide 104)": f"{sl.led_linewidth(1310e-9)*1e9:.1f} nm",
    "same with the ideal width m = 1.8": f"{sl.led_linewidth(1310e-9, m=1.8)*1e9:.1f} nm",
    "m from the AlGaAs data (slide 105)": f"{m:.2f}",
    "E_g(298 K) of GaAs": f"{Eg298:.4f} eV",
    "λo at 298 K, hc/(E_g + kT/2)": f"{l298*1e9:.1f} nm  (slide constants: {hc_slide/(Eg298 + sl.kT(298)/2)*1e9:.1f} nm)",
    "E_g(248 K)": f"{Eg248:.4f} eV",
    "λo at 248 K": f"{l248*1e9:.1f} nm  (slide constants: {hc_slide/(Eg248 + sl.kT(248)/2)*1e9:.1f} nm)",
    "shift for −50 K": f"{(l298 - l248)*1e9:.1f} nm = {(l298 - l248)*1e9/50:.2f} nm/K"}})
Table 9.1: Slides 103–108 recomputed with CODATA constants. The slide values are 107 nm, m = 2.81, 864.2 nm, 852.4 nm and 11.8 nm.
value
Δλ of a 1310 nm LED at 300 K, m = 3 (slide 104) 107.3 nm
same with the ideal width m = 1.8 64.4 nm
m from the AlGaAs data (slide 105) 2.81
E_g(298 K) of GaAs 1.4233 eV
λo at 298 K, hc/(E_g + kT/2) 863.3 nm (slide constants: 864.0 nm)
E_g(248 K) 1.4454 eV
λo at 248 K 851.5 nm (slide constants: 852.2 nm)
shift for −50 K 11.8 nm = 0.24 nm/K

The 0.9 nm offset in the absolute wavelengths comes from the rounded constants on slide 107 (\(c=3\times10^8\) m/s and \(h=6.626\times10^{-34}\) J s give \(hc/e=1.2409\) eV µm instead of 1.2398). The shift of 0.24 nm/K is unaffected. Measuring this shift is a practical way to monitor the junction temperature of a power LED.

9.5 Temperature dependence (slides 101–102)

Show the Python 
E = np.linspace(1.35, 1.75, 3000)
def spectrum(TC, sigma=0.020, off=0.059):           # off: Al-induced gap increase, puts the 25 °C peak at 822 nm
    T = TC + 273.15
    Eg = sl.varshni(T, *sl.VARSHNI["GaAs"]) + off
    S = sl.led_spectrum(E, Eg, T)
    g = np.exp(-((E - E.mean())**2)/(2*sigma**2)); S = np.convolve(S, g/g.sum(), mode="same")
    return S*(1 - 0.005*(TC - 25))
ref = spectrum(25).max()
fig, ax = plt.subplots(figsize=(7.5, 3.8))
rows = {}
for TC in (-40, 25, 85):
    S = spectrum(TC)/ref; lam = sl.wavelength_from_eV(E)*1e9
    ax.plot(lam, S, label=f"{TC} °C")
    half = lam[S >= S.max()/2]
    rows[f"{TC} °C"] = {"λo (nm)": round(lam[np.argmax(S)], 1), "Δλ FWHM (nm)": round(half.max() - half.min(), 1)}
ax.set(xlabel="λ (nm)", ylabel="relative intensity", xlim=(760, 900)); ax.legend(); plt.show()
pd.DataFrame(rows)
(a) Model spectra of an Al₀.₀₅Ga₀.₉₅As LED at −40, 25 and 85 °C (ideal shape, Varshni gap, 20 meV Gaussian broadening, intensity scaled by a typical −0.5 %/K efficiency drop), normalised to the 25 °C peak as in slide 101.
-40 °C 25 °C 85 °C
λo (nm) 809.1 822.3 836.0
Δλ FWHM (nm) 33.7 38.3 43.3
(b)
Figure 9.3

The model reproduces the trend of slide 101: about +0.25 nm/K in peak wavelength and a width that grows roughly as \(T\) (the measured 30–48 nm also include alloy broadening).

9.6 Degenerately doped junctions (slide 109)

In a heavily doped n⁺ region \(E_{Fn}\) lies inside the CB. All states up to \(E_{Fn}\) can recombine, the spectrum widens to about \(\Delta E_{Fn}+1.5k_BT\), and the effective gap shrinks (band-gap narrowing):

Show the Python 
kT = sl.kT(300); x = np.linspace(0, 0.25, 3000)
fig, ax = plt.subplots(figsize=(7.5, 3.6))
for dEF in (-0.05, 0.0, 0.05, 0.10):
    I = np.sqrt(x)/(1 + np.exp((x - dEF)/kT))
    half = x[I >= I.max()/2]
    ax.plot(x*1e3, I/I.max(), label=f"ΔE_Fn = {dEF*1e3:+.0f} meV, FWHM = {(half.max() - half.min())*1e3:.0f} meV")
ax.set(xlabel="hν − E_g′ (meV)", ylabel="relative intensity"); ax.legend(fontsize=8); plt.show()
Figure 9.4: Emission from a degenerately doped n⁺ region (holes non-degenerate): I(hν) ∝ (hν − E_g′)^{1/2} f_n(E). As E_Fn moves into the band (ΔE_Fn = E_Fn − E_c′), the spectrum becomes a flat-topped band of width ≈ ΔE_Fn + 1.5kT.
  • Van Roosbroeck–Shockley. Derive the spontaneous emission spectrum from the absorption coefficient \(\alpha(h\nu)\) and detailed balance. With \(\alpha\propto\sqrt{h\nu-E_g}\) you recover the shape used here, multiplied by \(\nu^2\).
  • Coherence length. Combine the LED linewidth with Chapter 1: \(l_c=\lambda^2/\Delta\lambda\). Compute it for the 1310 nm LED and explain why LEDs are used in OCT (superluminescent diodes) but not in long-haul links.
  • Chromatic dispersion. A 1310 nm LED with \(\Delta\lambda=107\) nm launched into a fiber with \(D=2\) ps/(nm km) spreads by 0.2 ns/km. Use the Chapter 2 link simulator to find the maximum bit rate.
  • Urbach tail. Real absorption edges have an exponential tail \(\exp[(h\nu-E_g)/E_U]\). Add it to the model and fit the measured spectrum of a lab LED.
  • Kasap, Optoelectronics and Photonics, sec. 3.11; Schubert, Light-Emitting Diodes, ch. 5 (spontaneous emission), ch. 6 (spectrum).
  • Y. P. Varshni, “Temperature dependence of the energy gap in semiconductors”, Physica 34, 149 (1967).

9.7 Try it yourself

  1. What is the linewidth of a 460 nm InGaN LED at 300 K according to the \(3k_BT\) rule? Compare with a typical datasheet (~20 nm).
  2. Use the explorer to find the temperature rise that shifts a GaAs LED by 5 nm.
  3. Redo the fit of slides 105–106 without forcing the line through the origin. What does the intercept mean physically?
  4. Estimate the energy shift of the peak for a degenerate n⁺ region with \(\Delta E_{Fn}=50\) meV.