Resonant-cavity-enhanced photo detector

The RCE photodetectors are expected to have higher quantum efficiency η than compared to conventional photodiodes. The formulation of η for RCE devices gives insight to the design criteria.

A generalized RCE photodetector schematic as given in Figure 1 can give the required theoretical model of photodetection. A thin absorption region of thickness d is sandwiched between two relatively less absorbing region, substrate, of thickness L₁ and L₂. The optical cavity is formed by a period of λ/4 distributed Bragg reflector (DBR), made of non-absorbing larger bandgap materials, at the end of the substrate. The front mirror has a transmittance of t₁ and generally has lower reflectivity than compared to the mirror at back (R₁ < R₂ ). Transmittance t₁ allows light to enter into the cavity, and reflectivity R₁ (=r₁²) and R₂ (=r₂²) provides the optical confinement in the cavity.

The active region and the substrate region have absorption coefficient α and α_ex respectively. The field reflection coefficients of the front and the back mirrors are ${\displaystyle r_{1}e^{-j\phi _{1}}}$ and ${\displaystyle r_{2}e^{-j\phi _{2}}}$ respectively, where ф₁ and ф₂ are the phase shifts due to the light penetration (see Penetration Depth) into the mirrors.

The optical microcavity allows building up an optical field inside the optical cavity. In compared to conventional detector, where light is absorbed in a single pass through the absorption region, for RCE detectors trapped light is absorbed each time it traverses through the absorption region.

The Quantum efficiency ${\displaystyle \eta }$ for a RCE detector is given by:

${\displaystyle \eta =(1-R_{1})(1-e^{-\alpha d})\left[{\frac {(e^{-\alpha _{ex}L_{1}}+r_{2}^{2}e^{-\alpha _{ex}L_{2}-\alpha _{c}L})}{1-2r_{1}r_{2}e^{-\alpha _{c}L}\cos(2\beta L+\phi _{1}+\phi _{2})+(r_{1}r_{2})^{2}e^{-\alpha _{c}L}}}\right]}$

Here ${\displaystyle \alpha _{c}={\frac {\alpha _{\text{ex}}L_{1}+\alpha _{\text{ex}}L_{2}+\alpha d}{L}}}$. In practical detector design α_ex << α, so α_ex can be neglected and ${\displaystyle \eta }$ can be given as:

${\displaystyle \eta =(1-R_{1})(1-e^{-\alpha d})\left[{\frac {(1+R_{2}e^{-\alpha d})}{1-2{\sqrt {R_{1}R_{2}}}e^{-\alpha _{c}d}\cos(2\beta L+\phi _{1}+\phi _{2})+(R_{1}R_{2})e^{-\alpha _{c}d}}}\right]}$

The term inside the [] represents the cavity enhancement effect. This is a periodic function of ${\displaystyle 2\beta L+\phi _{1}+\phi _{2}}$, which has minima at ${\displaystyle 2\beta L+\phi _{1}+\phi _{2}=2m\pi }$. And η enhanced periodically at resonance wavelength that meets this condition. The spacing of the resonant wavelength is given by the Free Spectral Range of the cavity.

The peak value of η at resonant wavelength is given as:

${\displaystyle \eta =(1-R_{1})(1-e^{-\alpha d})\left[{\frac {(1+R_{2}e^{-\alpha d})}{(1-{\sqrt {R_{1}R_{2}}}e^{-\alpha _{c}d})^{2}}}\right]}$

for a thin active layer as αd<<1, η becomes:

${\displaystyle \eta =(1-R_{1})\alpha d\left[{\frac {(1+R_{2}e^{-\alpha d})}{(1-{\sqrt {R_{1}R_{2}}}e^{-\alpha _{c}d})^{2}}}\right]}$

This is a significant improvement from the quantum efficiency of a conventional photodetector which is given by:

${\displaystyle \eta =(1-R)\alpha L}$.

This shows that higher quantum efficiency can be achieved for smaller absorption regions.

The critical design requirements are a very high back mirror reflectivity and a moderate absorption layer thickness. At optical frequencies, metal mirrors have low reflectivity (94%) when used on materials like GaAs. This makes metal mirrors inefficient for RCE detection. Whereas distributed Bragg reflector (DBR) can provide reflectivity near unity and are ideal choice for RCE structures.

For a R2=0.99 and α=10⁴ cm-1 with a R1=0.2 a η of 0.99 or more can be achievable for d=0.7–0.95 µm. Similarly for different values of R1 very high η is possible to achieve. However, R1 =0 limits the length of thickness region, d>5 µm can achieve 0.99 η but at the cost of bandwidth.

The detection speed depends upon the drift velocities of the electrons and holes. And between these two holes have slower drift velocity than the electrons. The transit time limited bandwidth of conventional p-i-n photodiode is given by:

${\displaystyle f_{\text{transit}}=0.45{\frac {v_{h}}{L}}}$

However, the quantum efficiency is a function of L as:

${\displaystyle \eta =(1-R)\alpha L}$.

For a high-speed detector for a small value of L, as α is very small, η becomes very small (η<<1). This shows for an optimum value of quantum efficiency the bandwidth has to sacrifice.

A p-i-n RCE photodetector can reduce the absorption region to a much smaller scale. In this case the carriers need to traverse a smaller distance as well, L₁ (< L) and L₂ (< L) for electrons and holes respectively.

The length of L1 and L2 can also be optimized to match the delay between the hole and electron drift. And the transition bandwidth becomes:

${\displaystyle f_{\text{transit}}=0.45{\frac {v_{h}+v_{e}}{L+d}}}$

As in most of semiconductors ${\displaystyle v_{e}}$ is more than ${\displaystyle v_{h}}$ the bandwidth increases drastically.

It has been reported that for a large device of L=0.5μm 64 GHz of bandwidth can be achieved and a small device of L=0.25μm can give 120 GHz bandwidth, whereas conventional photodetectors have a bandwidth of 10–30 GHz.

An RCE structure can make the detector wavelength selective to an extent due to the resonance properties of the cavity. The resonance condition of the cavity is given as ${\displaystyle 2\beta L+\phi _{1}+\phi _{2}=2m\pi }$. For any other value the efficiency η reduces from its maximum value, and vanishes when ${\displaystyle 2\beta L+\phi _{1}+\phi _{2}=(2m+1)\pi }$. The wavelength spacing of the maxima of η is separated by the free spectral range of the cavity, given as:

${\displaystyle FSR={\frac {\lambda ^{2}}{2n_{\text{eff}}(L+L_{{\text{eff}},1}+L_{{\text{eff}},2})}}}$

Where n_eff is the effective refractive index and L_eff,i are the effective optical path lengths of the mirrors.

Finesse, the ratio of the FSR to the FWHM at the resonant wavelength, gives the wavelength selectivity of the cavity.

${\displaystyle {\text{Finesse}}={\frac {\pi (R_{1}R_{2})^{1/4}e^{-{\frac {\alpha d}{2}}}}{(1-{\sqrt {R_{1}R_{2}}}e^{-\alpha _{c}d})^{2}}}}$

This shows that the wavelength selectivity increases with higher reflectivity and smaller values of L.