Optoelectronics formulas

Holds for a photon of free-space wavelength $\lambda$. Numeric form: $E(\text{eV})=\frac{1240}{\lambda (\text{nm})}=\frac{1.24}{\lambda (\mu \text{m})}$.

Number of photons per second carried by monochromatic light of optical power $P_{opt}$ (W). With $h\nu$ in eV, $\Phi =P_{opt}/(qh\nu [\text{eV}])$.

Band-to-band absorption occurs only for $\lambda \le {\lambda }_g$ (i.e. $h\nu \ge E_g$). For $\lambda >{\lambda }_g$ there is no band-to-band absorption; the material is nearly transparent apart from weaker impurity and free-carrier absorption.

$\alpha$ is the absorption coefficient (cm${}^{-1}$), which depends on wavelength; $1/\alpha$ is the penetration depth. The fraction of the light entering the material that is absorbed within a thickness $W$ is $1-e^{-\alpha W}$.

For band-to-band emission in a direct-gap material. The emitted photon energy is close to, and slightly above, $E_g$ because the injected carriers are spread over a few $kT$ above the band edges; this also gives the emission a finite spectral width.

Fraction of recombination events that emit a photon. $R_r$, $R_{nr}$ are the radiative and non-radiative recombination rates; ${\tau }_r$, ${\tau }_{nr}$ the corresponding lifetimes, with total lifetime $1/\tau =1/{\tau }_r+1/{\tau }_{nr}$.

$I$ is the forward current and $h\nu$ the photon energy (J). ${\eta }_{ext}$ (external quantum efficiency) is photons leaving the device per electron passing through it; ${\eta }_{inj}$ is the fraction of the current that recombines in the active region, and ${\eta }_{extr}$ the fraction of generated photons that escape. With $h\nu$ in eV: $P_{out}={\eta }_{ext}Ih\nu [\text{eV}]$ (W, with $I$ in A).

Light striking the top surface at more than ${\theta }_c$ from the normal is totally internally reflected. For a typical III-V semiconductor ($n_s\approx 3.5$), ${\theta }_c\approx 17^{\circ }$, which is why only a small fraction of the generated light escapes a planar LED. At normal incidence the reflected fraction is ${\left(\frac{n_s-1}{n_s+1}\right)}^2$.

$V$ is positive for forward bias; $I$ is the current flowing into the p terminal from the external circuit (the usual diode convention). $I_0$ is the reverse saturation (dark) current, $n$ the ideality factor ($n=1$ ideal) and $V_T\approx 25.9$ mV at 300 K. The photocurrent $I_{ph}$ flows in the reverse direction and simply adds to the dark characteristic, which shifts the whole I-V curve down by $I_{ph}$. Assumes $I_{ph}$ does not depend on $V$ and neglects series and shunt resistance.

$P_{opt}$ is the incident optical power (W) and $\eta$ the quantum efficiency. Assumes no internal gain (each collected pair gives one electron of external current), i.e. not an avalanche photodiode.

Electron-hole pairs collected per second divided by photons incident per second; $0\le \eta \le 1$ without internal gain. It is reduced by surface reflection, by light passing through without being absorbed, and by carriers recombining before they are collected. Ignoring collection losses, $\eta \approx (1-R_f)\left(1-e^{-\alpha W}\right)$, where $R_f$ is the surface reflectance and $W$ the absorbing (depletion) width.

Holds for $\lambda \le {\lambda }_g$; beyond the cutoff wavelength $\eta$, and so $R$, falls to nearly zero. At constant $\eta$ the responsivity rises linearly with $\lambda$ (a longer-wavelength photon carries less energy, so a watt contains more photons), then drops sharply near ${\lambda }_g$. The maximum possible value, at $\eta =1$, is $\lambda (\mu \text{m})/1.24$ A/W.

An APD is reverse biased close to breakdown so that impact ionisation multiplies each photogenerated carrier; $M$ is the average multiplication (gain) factor. Its responsivity can therefore exceed the $\eta =1$ limit of an ordinary photodiode.

$V$ is the terminal voltage (positive, since the cell forward biases itself), $I_{out}=-I$ is the current delivered to the external load, positive for $0\le V\le V_{oc}$. $I_0$ is the reverse saturation (dark) current, $n$ the ideality factor ($n=1$ ideal) and $V_T=kT/q\approx 25.9$ mV at 300 K. Ideal model: series and shunt resistance neglected, $I_L$ independent of $V$.

For a fixed spectrum, $I_L$ is proportional to the incident optical power (photon flux), so $I_{sc}$ is too. The ideal model is assumed (no series resistance).

Obtained by setting $I_{out}=0$. $V_{oc}$ grows only logarithmically with light intensity: multiplying the intensity by $k$ raises $V_{oc}$ by $n{V}_T\ln k$. $V_{oc}$ falls as temperature rises, because $I_0$ (proportional to $n_i^2$) rises steeply with temperature.

From $\frac{dP}{dV}=0$ with $P=V{I}_{out}$. The maximum power point lies on the curve between $(0,I_{sc})$ and $(V_{oc},0)$, with $V_{mp}<V_{oc}$ and $I_{mp}<I_{sc}$.

Ratio of the maximum-power rectangle to the rectangle $V_{oc}\times I_{sc}$. Always $FF<1$, since $V_{mp}<V_{oc}$ and $I_{mp}<I_{sc}$. It is reduced by series resistance and by low shunt resistance.

$P_{in}$ is the total optical power incident on the cell, under the same illumination at which $I_{sc}$ and $V_{oc}$ are measured.