Two-photon photovoltaic effect

TPA is typically several orders of magnitude weaker than linear absorption at low light intensities. It differs from linear absorption in that the optical transition rate due to TPA depends on the square of the light intensity, thus it is a nonlinear optical process and can dominate over linear absorption at high intensities. Therefore, the power dissipation from the TPA and the resulting free carrier scattering are harmful problems in semiconductor devices which operate based on the nonlinear optical interactions such as the Kerr and Raman effects, when dealing with high intensities. The TPP effect is studied as a possible solution to this double crisis on energy efficiency.

Although some improvements and theoretical investigation on the field have been done in the past, the concrete application of the effect was numerically and experimentally analysed for the first time by Bahram Jalali and colleagues in 2006 in Silicon.[1]

TPP effect devices are based on waveguides with lateral p–n junction diodes, in which the pump power is nonlinearly lost due to TPA and free-carrier absorption (FCA) along the z-direction, perpendicular to the junction x-y cross-section.

Coupled optical intensity is governed by the following equation:

${\displaystyle {\frac {dI_{p}(z)}{dz}}=\underbrace {-(\alpha +\alpha _{FCA})I_{p}(z)} _{\text{Linear absorption}}-\underbrace {\beta I_{p}^{2}(z)} _{\text{TPA}}}$

(1)

where:

α is the linear absorption coefficient;

β the TPA coefficient;

and α_FCA is called the FCA coefficient which is given by Soref´s expression.

Carrier photogeneration rate is defined by:

${\displaystyle G={\frac {dN}{dt}}=-{\frac {dI_{TPA}}{dz}}\cdot {\frac {1}{2E_{p}}}={\frac {\beta I_{p}^{2}}{2E_{p}}}}$

where E_p is the energy of the photon and the factor ${\displaystyle {\tfrac {1}{2}}}$ is due to the fact that there are two photons involved in the process.

Photocurrent per unit length: ${\textstyle I_{G}=q\cdot A_{eff}\cdot G}$, where ${\textstyle A_{eff}}$is the effective area of the waveguide and q is the electron charge. For a waveguide of length L, we have

$${\displaystyle I_{G}={\frac {\beta qA_{eff}}{2E_{p}}}\int _{0}^{L}I_{p}^{2}(z)dz}$$

We define ${\textstyle I_{p0}}$ as the coupled pump intensity at ${\displaystyle z=0}$. Therefore, we obtain the following expression:

$${\displaystyle I_{G}={\frac {\beta qA_{eff}I_{p0}^{2}}{2E_{p}}}\int _{0}^{L}{\frac {I_{p}^{2}(z)}{I_{p0}^{2}}}dz}$$

(2)

${\displaystyle L_{NL}=\int _{0}^{L}{\frac {I_{p}^{2}(z)}{I_{p0}^{2}}}dz}$

This last expression is called the effective length which is the nonlinear equivalent to the interaction length defined in optical fibers. Contribution to carrier injection and recombination to the total current need to be considered as well so that the total photodiode current is expressed as:[2]

${\displaystyle I_{T}=I_{G}+I_{D}+I_{R}}$

(3)

The Shockley equation gives I–V (current-voltage) characteristic of an idealized diode:[3]

${\displaystyle I_{D}=I_{s}(e^{\frac {qV_{s}}{k_{B}T}}-1)}$

(4)

The value of ${\displaystyle I_{s}}$ is called the reverse bias saturation current and is defined by:[3]

${\displaystyle I_{s}=qhL\cdot ({\frac {D_{n}n_{p0}}{L_{n}}}+{\frac {D_{p}p_{n0}}{L_{p}}})}$

where h and L are defined in Fig. 1 and the remaining parameters have the usual meaning defined in reference Sze's Physics of semiconductor devices.[3]

The Shockley equation is valid since photogeneration in the N and P doped regions is negligible in the p–n diode. This contrasts the conventional solar cell theory, where photogeneration predominantly occurs in the N and P doped regions[4] as shown in Fig. 2.

Due to PIN structure (Figure 2) we have to take into account the recombination current which we approximate by Shockley–Read–Hall recombination given by :

${\displaystyle I_{R}={\frac {qh(W+2d)LN_{eff}}{\tau _{n}+\tau _{p}}}}$

(5)

where ${\displaystyle W+2d}$ is defined in Fig.1, ${\displaystyle N_{eff}}$ is the effective carrier density along ${\displaystyle {\hat {z}}}$ and ${\displaystyle \tau _{n}}$ and ${\displaystyle \tau _{p}}$ are the electron and hole bulk recombination lifetimes, respectively.

In a circuit, power dissipation refers to the rate at which energy is lost due to resistive elements and is defined traditionally as follows:

${\displaystyle P=-I_{G}\cdot V_{m}}$

(6)

We now define collection efficiency, which is number of carriers/photons consumed by TPA:[4]

${\displaystyle \eta _{c}={\frac {I_{T}}{2I_{G}}}}$

(7)

This is appropriate for such devices as amplifiers and wavelength converters where energy harvesting is a useful byproduct but not the main functionality of the device itself. If the TPP effect is intended to be used in a photovoltaic cell, then the power efficiency ${\displaystyle \eta _{p}}$ should be considered.

First, external quantum efficiency is given by ${\displaystyle \eta _{ext}=\eta _{coupling}\cdot \eta _{q}}$, where ${\displaystyle n_{coupling}}$ refers to the coupling efficiency of the light into the waveguide and

${\displaystyle \eta _{q}={\frac {E_{p}I_{T}}{qI_{p0}A_{eff}}}}$

which can be approximated to:

${\displaystyle \eta _{q}(V_{m})\approx {\frac {\beta L_{NL}I_{p0}}{2}}}$

Finally, power efficiency is given by:

${\displaystyle \eta _{p}=\eta _{ext}\cdot {\frac {qV}{E_{p}}}\approx {\frac {\eta _{coupling}q\beta }{2E_{p}}}V_{m}L_{NL}I_{p0}}$

(8)

Semiconductor materials are so relevant due to the fact that their conducting properties can be altered in useful ways by introducing impurities ("doping") into the crystal structure. Where two differently-doped regions exist in the same crystal, a semiconductor junction is created. The development of these junctions is the basis of diodes, transistors and all modern electronics. Examples of semiconductors are silicon, germanium, gallium arsenide. After silicon, gallium arsenide is the second most common semiconductor.[3]

A potential application of the two-photon photovoltaic effect is remote power delivery to physical sensors installed in critical environments where electrical sparks are dangerous and copper cables must be avoided.