Semiconductors and Light A non-standard way to deal with it Assumes confidence with Basic Solid State Physics: Crystals Electrons, Holes, direct-indirect Band Gap Their foundation on the Bloch’s Theorem pn junction DC characteristics Basic light-matter interaction theory Quantum-mechanic perturbation theory Spontaneous and stimulated emission, absorption

Semiconductors and Light Does NOT aim to be complete Aims to build up a self-consistent view of: Electron, Hole and Photon densities Their link with measurable quantities: I, V, P Their call for technological solutions

Semiconductors and Light It gives a new insight on: Threshold current Gain and loss coefficients The theoretical foundation of known empirical formulas It should be compared with literature. Coldren LA, Corzine SW, Mašanović ML. Diode lasers and photonic integrated circuits, Wiley series in microwave and optical engineering. Jhon Wiley & Sons, Inc., Hoboken, New Jersey; Second Edition 2012. J.T. Verdeyen. Laser Electronics. Third Edition, Prentice Hall, 1995.

Semiconductors and Light Absorption Emission Partial reflection Transparency Refraction Eg n Easily explained by Eg: Absorbption Transparency Refraction Partial Reflection

Diamond has the closest lattice spacing It is Mechanically Hard Electrically Insulating Optically Transparent Optically Refractive

… but Light Emission is another tale Fermi Golden Rule (holds for any system) Momentum Selection Rule (specific for crystals) Si excluded from light emitting devices Recombination rate is no more It must obey the selection rules High probability Low probability

Something may be anticipated about light emission in direct gap semiconductors Few electrons Few holes Few photons Many electrons Many holes Many photons No electrons No holes No photons h𝜈 Eg 𝜙𝜈 = photon density Typical ≈kT Typical 40kT

How to get many electrons and many holes together? By injecting a forward current into a pn junction But in an ideal diode, they meet inside the depletion layer without recombining. Then recombine, separately, entering the neutral regions

This is a Light Emitting Diode = LED The practical solution is to force e-h recombination inside the depletion layer introducing a thin layer at smaller bandap between the p and n regions. In a well designed device, a forward bias V will cause a current I to flow and an optical power POUT to leave the structure. This is a Light Emitting Diode = LED

Possibility for population inversion But something more may be anticipated about light emission in direct gap semiconductors More empty than filled states More filled than empty states More filled than empty states More empty than filled states 𝜙n Possibility for population inversion 𝜙p More top-down (emitting) than bottom-up (absorbing) transitions stimulated by light. GAIN = Light Amplification The LED achieves Light Amplification by Stimulated Emission of Radiation This is a LASER DIODE

Form qualitative to quantitative: we should: List all mechanisms involving photons and electrons and holes Balance them Bring optical properties properties inside the world of diodes: Spectrum 𝜙𝜈, gain g, loss α, optical power POUT Bring diode properties properties inside the world of lasers: Current I, Voltage V Include concurring phenomena non involving photons Find lumped equations for V, I and POUT The starting point will be a Rate Equation for photons

Our program: Consider a Double Heterostructure ideal diode made of direct gap semiconductor where all recombinations are radiative and happens only inside the central active layer qV p n depletion layer active layer Inside that layer all electron, holes and photon densities are uniform We will look first for the electro-optical characteristics of such ideal diode Non-radiative recombination inside the layer and later on, we will include Other recombinations and currents outside the layer

A quick recall of the Einstein’s treatment of Black Body radiation (1905) Search for the spectral power density u𝜈 at equilibrium Fermi’s Golden rule not yet discovered No Fermi-Dirac or Bose-Einstein statistics available: only Boltzmann. No exclusion principle No quanta. Planck (1901) not sure of their existence. Einstein going to explain the phototelectric effect on the same year (1905) Classical results from Thermodynamics Stefan’s Law Wien’s (Displacement) Law Rayleigh and Jeans’ Ultraviolet Catastrophe . But good for low 𝜈.

2-level system with N0 particles Energy Density of states Population E2 g2 Level 2 E1 g1 Level 1 Rate of spontaneous 2→1 transitions Rate of stimulated 2→1 transitions Rate of stimulated 1→2 transitions Coefficients A and B can depend on 𝜈 but not on T At equilibrium 2→1 = 1 →2

At high temperature: Increase with T4 Do not change with T Go to unity Wien’s (Displacement) Law Do not include T

Must be (Rayleigh and Jeans): Planck’s Law

Top-down optical transitions proportional to Bottom-up optical transitions proportional to

The diode at equilibrium must give back the Black Body formula Conduction-to-valence spontaneous transitions (e-h recombination, photon emission) Conduction-to-valence stimulated transitions(e-h recombination, photon emission) Valence-to-conduction stimulated transitions (e-h generation, photon absorption) At equilibrium V=0 As for Einstein’s treatment:

Out of equilibrium, rates can be not constant, V≠0 and an escape term must be introduced, that must vanish at equilibrium In the steady state: Using the previous forms we can solve for

B may be and 𝜙0𝜈 is a slow function of h𝜈. We can safely assume for both: B and 𝜙0𝜈 are constant where h𝜈 Eg 𝜙𝜈 = photon density 𝜙0𝜈 Is the joint density of states for electrons and holes. It is null for and is linear in for thick layers and constant for thin layers (the normal case) 𝜏C is the average permanence time of radiation inside the active layer. It is a function of 𝜈 because of refraction and resonances. But we start keeping it constant.

Let us suppose Eg= 1 eV and qV=0.5 eV Thick layer case. Linear vertical scale. Pay attention to the values in the abscissa calculated Linear scale h𝜈 Eg 𝜙𝜈 = photon density Expected (qualitative)

Let us now change qV: Thick layer case. Log vertical scale. Pay attention to the values of qV Logarithmic scale

Thin layer case. Log vertical scale. Pay attention to the values of qV Logarithmic scale At qV=1.02 eV Something happens when qV approaches Eg

For possible emission, we have exponentials at the denominator are extremely small

But the denominator vanishes as The total optical power increases unlimited Infinite energy is required to further increase qV. Voltage clamps at a threshold value that is the minimum among the ones allowed For a thin layer this is

What is it the happening? Resc Rsp Rst Rabs qVth Eg Spontaneous emission and absorption dominate over stimulated emission. It is the LED regime. Stimulated emission balances absorption. It the the transparency condition. Stimulated emission overcomes absorption. Light starts to be amplified (super-radiance) Spontaneous emission is blocked. Voltage is calmped. Stimulated emission dominates. It is the LASER regime.

The more the bathtube is filled, the higher is the output flux qV PTOT But when the edge is reached, the flux can increase without increasing the water level

Current in the ideal laser diode The current is q times the net recombination rate. For the ideal LED/Laser diode, where recombination is always radiative Integration factor. h𝜈 Eg 𝜙𝜈 = photon density

This is the DC transfer function I(V) of an ideal laser diode. Equation of an ideal diode with saturation current = V<Vth: LED range Laser region V=Vth: Laser range Shockley region (LED) This is the DC transfer function I(V) of an ideal laser diode.

Optical Power in the ideal laser diode Optical Power=Photon Density x Photon Energy x Volume / Lifetime But: Iph POUT Collection efficiency (coupling+conversion+…)

Optical Power in the real laser diode Iph is not the only current measured For V<Vth non-radiative recombination dominates A non-radiative current Inr exists qVth Ith The total current is made of I= Inr + Iph As qV clamps at qVth, Inr stops, and Iph grows alone Quantum Efficiency: A threshold current Ith defines the transiton In practise:

Threshold current

What do textbooks tell? for any V,I for V> Vth (that is I> Ith ) Collected escaping power Total escaping power What do textbooks tell? Conversion efficiency (often omitted) My opinion: a big mistake. It must go to unity for I> Ith

Lumped equations for the DC regime I(V), POUT(V) → direct substitution POUT(I) → eliminate is analytic under the form I(POUT) calculates and embeds the threshold condition deals with the sub-thresholdregime It comes out a huge formula, but it