In a pn junction LED, consider a point source in the...

In a pn junction LED, consider a point source in the semiconductor at the junction and assume that photons are emitted uniformly in all directions. Show that (neglecting photon absorption) the external quantum efficiency of the LED is given by $$ \eta_{\mathrm{ext}}=\frac{2 \bar{n}_{1} \bar{n}_{2}}{\left(\bar{n}_{1}+\bar{n}_{2}\right)^{2}}\left(1-\cos \theta_{c}\right) $$ where $\bar{n}_{1}$ and $\bar{n}_{2}$ are the index of refraction parameters for the air and semiconductor, respectively, and $\theta_{c}$ is the critical angle.

In a pn junction LED, consider a point source in the semiconductor at the junction and assume that photons are emitted uniformly in all directions. Show that (neglecting photon absorption) the external quantum efficiency of the LED is given by
$$
\eta_{\mathrm{ext}}=\frac{2 \bar{n}_{1} \bar{n}_{2}}{\left(\bar{n}_{1}+\bar{n}_{2}\right)^{2}}\left(1-\cos \theta_{c}\right)
$$
where $\bar{n}_{1}$ and $\bar{n}_{2}$ are the index of refraction parameters for the air and semiconductor, respectively, and $\theta_{c}$ is the critical angle.

Key Concepts

External Quantum Efficiency (EQE)

EQE is a measure of the fraction of electron-hole recombination events in an LED that result in photons escaping from the device. It takes into account both the internal conversion efficiency and the optical extraction efficiency, the latter being significantly influenced by the geometry and material properties at the semiconductor/air interface.

Refractive Index and Snell's Law

The refractive index is a material property that affects how light propagates through it. Snell's law describes the relationship between the angles of incidence and refraction when light passes from one medium to another. In the context of LEDs, the refractive indices of the semiconductor and the surrounding medium determine the bending of light rays at the interface, influencing which photons can escape.

Critical Angle and Total Internal Reflection

The critical angle is defined as the angle of incidence above which total internal reflection occurs at an interface between two media. For photons inside a semiconductor, only those hitting the interface at angles smaller than the critical angle will refract out into the surrounding medium, while others are reflected back, limiting the external quantum efficiency.

Isotropic Photon Emission

Assuming the LED behaves as a point source, the emitted photons are distributed uniformly in all directions. This isotropic distribution, when combined with the angular limitation imposed by the critical angle, forms the basis for calculating the fraction of photons that can escape, thereby influencing the external quantum efficiency.

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