7  Inside a forward-biased LED: a drift–diffusion simulator

Complements slides 48–66 (injection, recombination current, ideality factor), 86–87 (LED principle, quasi-Fermi levels) and 124–126 (internal quantum efficiency). The panel below solves the full semiconductor equations of a GaAs pn junction in your browser. It is a JavaScript port of pn_led_sim.py, which you can also run offline (see the end of the chapter).

7.1 Key relations

The diode model of Chapter 5 splits the current into a diffusion term (η = 1) and a recombination term (η = 2), each with its own approximations. A drift–diffusion simulation needs neither. It solves Poisson’s equation and the two continuity equations together,

\[ \frac{d}{dx}\!\left(\varepsilon\frac{d\psi}{dx}\right)=-e\,(p-n+N_d-N_a),\qquad \frac{1}{e}\frac{dJ_n}{dx}=R,\qquad -\frac{1}{e}\frac{dJ_p}{dx}=R, \]

with \(n=n_ie^{(\psi-\phi_n)/V_t}\) and \(p=n_ie^{(\phi_p-\psi)/V_t}\) (\(V_t=k_BT/e\)). The quasi-Fermi levels \(E_{Fn}=-e\phi_n\) and \(E_{Fp}=-e\phi_p\) (slide 87c) give the product

\[ np=n_i^2\exp\!\left(\frac{E_{Fn}-E_{Fp}}{k_BT}\right)\;\approx\;n_i^2\exp\!\left(\frac{eV}{k_BT}\right)\ \text{in the SCL.} \]

Three recombination channels compete. All of them vanish at equilibrium, when \(np=n_i^2\):

\[ R=\underbrace{B\,(np-n_i^2)}_{\text{radiative}} +\underbrace{(C_nn+C_pp)(np-n_i^2)}_{\text{Auger}} +\underbrace{\frac{np-n_i^2}{\tau_p(n+n_1)+\tau_n(p+p_1)}}_{\text{SRH (traps)}}, \qquad n_1=n_ie^{(E_t-E_i)/k_BT},\; p_1=n_ie^{-(E_t-E_i)/k_BT}. \]

  • Radiative. The rate goes as \(B\,n\,p\), so it is the same everywhere in the SCL, where \(np\) is constant, and decays into the neutral regions with the minority carriers. The only channel that makes light.
  • Shockley–Read–Hall (SRH). Recombination through a defect level \(E_t\). Its rate peaks where \(\tau_pn\approx\tau_np\), that is near the middle of the SCL, and is then \(\approx n_ie^{eV/2k_BT}/(\tau_n+\tau_p)\) for a mid-gap trap. This peak gives the \(\eta=2\) current of slide 62.
  • Auger. The energy goes to a third carrier instead of a photon. The \(C_nn+C_pp\) factor makes it grow faster than the other two channels, so it matters at high injection (the “droop” of high-power LEDs).

Integrating each rate over the device gives its share of the current, \(J_\text{rad}=e\int R_\text{rad}\,dx\) and so on. The internal quantum efficiency is \(\eta_\text{IQE}=J_\text{rad}/J\) (Chapter 11).

ImportantCarriers recombine everywhere, not only in the SCL

Every electron that crosses the junction recombines somewhere, so the whole current is recombination: \(J=e\int R\,dx\) over the device, plus what reaches the contacts. The two terms of the diode equation are two parts of this integral:

  • the diffusion current (Shockley, η = 1) is the recombination in the neutral regions. The injected minority carriers diffuse away from the SCL and recombine within a diffusion length. The name describes how they get there, not what happens to them;
  • the recombination current (η = 2) is the SRH recombination inside the SCL.

The SCL gets its own term because its rate depends differently on the voltage, not because it is the only place where carriers recombine. In a neutral region the majority density is fixed and the minority density goes as \(e^{eV/k_BT}\), so any rate linear in it (radiative \(Bn N_a\) or SRH \(n/\tau_n\)) gives η = 1. In the SCL, SRH peaks where \(n\approx p\approx n_ie^{eV/2k_BT}\), which gives η = 2. Radiative recombination in the SCL goes as \(np\propto e^{eV/k_BT}\) (η = 1) and is negligible over a few tens of nm, so it has no term of its own. The panel below shows all six contributions separately.

7.2 Drift–diffusion explorer

The GaAs device is 30 µm long, with the junction at \(x=0\) and ohmic contacts at \(x=\pm15\) µm (\(\mu_e=4000\), \(\mu_h=200\) cm² V⁻¹ s⁻¹, \(n_i=2.1\times10^6\) cm⁻³, 300 K). Changing a device parameter re-runs the voltage sweep from 0 to 1.3 V (about half a second). The voltage slider only selects a step of the stored sweep, so it responds instantly. Set a coefficient to its minimum or untick a channel to switch that channel off.

7.3 What to look for

Start with the default device (\(N_a=3\times10^{18}\), \(N_d=3\times10^{17}\) cm⁻³, \(B=10^{-10}\) cm³ s⁻¹, \(\tau_n=\tau_p=10\) ns, mid-gap trap, \(C_n=C_p=10^{-30}\) cm⁶ s⁻¹).

  1. The B·n·p plateau. On the log axis, the radiative rate is flat across the SCL: \(n\) falls and \(p\) rises, but their product does not change, because the quasi-Fermi levels are flat there. Its value is \(Bn_i^2e^{eV/k_BT}\), so it grows by a decade every 60 mV. Outside the SCL the rate follows the minority carriers, electrons on the p side and holes on the n side. It decays only over a diffusion length: at 1 V it is 3× lower at −5 µm but already 12× lower at +5 µm, because holes on the n side diffuse less far than electrons on the p side. Narrow the x window to ±0.3 µm to see the plateau in the SCL.
  2. The rate peaks in the SCL, but the light comes from the neutral regions. The SCL is only about 60 nm wide, while the injected electrons spread over several µm of the p side. The neutral regions therefore produce about 99 % of the photons. Switch to the linear axis to see the area under the green curve.
  3. SRH peaks where \(\tau_pn\approx\tau_np\). At 0.5–0.8 V the SRH rate in the SCL is orders of magnitude above B·n·p, and the current follows \(e^{eV/2k_BT}\) (η ≈ 2, IQE close to zero). Above about 1 V the injected carriers take over and η → 1.
  4. Trap level. Move \(E_t-E_i\) toward a band edge. \(n_1\) or \(p_1\) becomes large, the SRH channel weakens at low bias and the η = 2 region shrinks. Mid-gap traps are the most harmful recombination centres.
  5. Auger. Raise \(C_n\) and \(C_p\) toward \(10^{-28}\) cm⁶ s⁻¹. Auger is negligible at low bias but grows as \(n^2p\) and takes a growing share of the current at high bias. Real LEDs show this as efficiency droop.
  6. Switch channels off. Without SRH the IQE is constant and η = 1 over the whole sweep. It reaches about 88 %, and the rest of the current goes to the contacts: the electron diffusion length on the p side (≈ 6 µm) is not much shorter than the 15 µm p region.

7.4 Why a homojunction is a poor LED

Radiative recombination happens everywhere, yet homojunction LEDs are inefficient. The simulation shows three of the reasons:

  • The carriers are diluted. The injected electrons spread over a diffusion length of several µm. At 10 A/cm² their density on the p side is only about \(10^{15}\) cm⁻³, so \(np\) is small and the photon generation rate \(B\,n\,p\) is low per unit volume.
  • The IQE is fixed by the doping. In the neutral p region each electron recombines radiatively at the rate \(BN_a\) and through traps at \(1/\tau_n\). Both are linear in \(n\), so \(\eta_\text{IQE}\approx BN_a\tau_n/(1+BN_a\tau_n)\) does not improve when you drive the LED harder. In the explorer the IQE stays near 40 % from 1 to 100 A/cm². A short radiative lifetime needs heavy doping, and heavy doping brings defects, so in practice \(\tau_n\) falls too. The model keeps \(\tau_n\) fixed and is optimistic.
  • Carriers are lost on both sides. Holes are injected into the n side, where \(BN_d\) is ten times smaller. Electrons reach the contact, and at low bias SRH in the SCL takes most of the current.

A fourth reason is not in the model, and it is the largest in practice. The photons are emitted with \(h\nu\approx E_g\) a few µm deep in a material that absorbs exactly these photons (\(1/\alpha\approx1\) µm near the gap). Most of them are reabsorbed before they reach the surface.

The double heterostructure solves all four problems at once: it confines both carriers in a thin layer, so \(n\) and \(p\) are high and the IQE grows with the current, and its wide-gap claddings are transparent. The same simulator with band offsets is in Section 8.5.1.

The unknowns are \(\psi\), \(\phi_n\) and \(\phi_p\) on a non-uniform mesh: 2 nm spacing near the junction, graded up to 0.1 µm toward the contacts, about 650 nodes. The currents use the Scharfetter–Gummel discretisation, which handles the exponential carrier profiles exactly between nodes. The equations are solved by Newton’s method with damping. The Jacobian is block tridiagonal and is built by finite differences in nine residual evaluations. The bias starts at 0 V and rises in 25 mV steps, each solution being the starting point of the next one.

The contacts are ideal ohmic contacts with infinite surface recombination. There is no series resistance, no heterostructure (see Section 8.5.1 for that) and no photon recycling, so at high bias the voltage on the junction is the applied one. The current there exceeds what a real chip can carry.

7.5 Running the Python version

pn_led_sim.py is the reference implementation. It uses the same equations on a mesh twice as fine, prints a summary table and saves a four-panel figure and the J–V breakdown as CSV:

python pn_led_sim.py                                     # default device, writes pn_led.png and pn_led_iv.csv
python pn_led_sim.py --Na 1e19 --Nd 1e18 --tau 1e-9      # heavier doping, shorter SRH lifetime
python pn_led_sim.py --tau-n 1e-6 --tau-p 1e-10 --Et 0.3 # asymmetric lifetimes, shallow trap
python pn_led_sim.py --Na 3e18 --Nd 3e17 --tau 1e-7 --dh-d 100 --xal 0.3 --nact 1e17 --vmax 1.4
                                                         # double heterostructure of @sec-dh-sim

The browser version reproduces its J–V curve to better than 0.1 %, and on the same mesh the two agree to all printed digits. Run python pn_led_sim.py --help for every option. When you import it, PNDiode(tau_n=..., tau_p=..., Et=..., Cn=..., Cp=..., dh=dict(d=..., dEc=..., dEv=..., Nact=...)) exposes the same parameters as the panels.

  • High injection. At which voltage does the electron density on the p side reach \(N_a/10\)? Check that η rises again above 1 there, as in Chapter 5.
  • Doping and IQE. The radiative lifetime on the p side is \(1/(BN_a)\). Compare it with \(\tau_n\) and predict the IQE of the neutral region. Check your prediction with \(N_a=10^{17}\) and \(10^{19}\) cm⁻³.
  • Asymmetric lifetimes. Set \(\tau_n=1\) µs and \(\tau_p=0.1\) ns. Where does the SRH peak move, and why?
  • Band parameters. The simulator handles heterostructures (Section 8.5.1) by adding \(\theta_n=-\Delta E_c/k_BT\) and \(\theta_p=-\Delta E_v/k_BT\) to the exponents of \(n\) and \(p\). Show that the Scharfetter–Gummel fluxes then use \(\psi+\theta_n\) for electrons and \(\psi-\theta_p\) for holes (fluxes in pn_led_sim.py). What would change with different effective masses in the claddings?

7.6 Try it yourself

  1. Read off \(J\) at 0.6 V and 0.7 V for the default device and compute η. Compare with the η = 2 prediction.
  2. Find the doping of the p side that maximises the IQE at 1.1 V. Why is there an optimum?
  3. With only the radiative channel on, the IQE stays near 88 %. Estimate \(L_n=\sqrt{D_e\tau_\text{rad}}\) with \(\tau_\text{rad}=1/(BN_a)\), compare it with the 15 µm p region, and explain where the remaining 12 % of the current goes.
  4. By how much must \(\tau_n=\tau_p\) increase to move the η = 2 → η = 1 transition down by 0.1 V?