# Recent Advances in Photodetectors Based on Two-Dimensional Material/Si Heterojunctions
## 2 Figures of Merit for Photodetectors
### Quantum Efficiency and Gain  
Quantum efficiency is typically divided into two categories, namely, external quantum efficiency (EQE) and internal quantum efficiency (IQE), with EQE being the more commonly used parameter. The quantum efficiency serves as an indicator of a photodetector's sensitivity. The EQE (\(\eta_{\text{ext}}\)) is the ratio of the number of charge carriers generated in the circuit to the number of photons incident on the device [51; 52]:  
\[\eta_{ext}=\frac{I_{ph}/e}{P/h\upsilon}=\frac{R_{I}h\upsilon}{e\lambda} \tag{2}\]  
where \(e\) is the elementary charge value; \(h\) is the Plank constant; \(\upsilon\) and \(\lambda\) are the frequency and wavelength of the incident light, respectively; and \(c\) is the speed of light in a vacuum. It is important to note that a photodetector cannot fully absorb incident light due to the reflection and transmittance. The ratio of the number of photons absorbed by the photodetector to the number of incident photons is denoted as \(\eta_{\text{a}}\). Additionally, a portion of the photogenerated electron-hole pairs recombine, which cannot be collected as electric current in the circuit. The collection efficiency \(\eta_{\text{c}}\) is used to account for the collected carriers. The IQE (\(\eta_{\text{i}}\)) is the ratio of the number of electron-hole pairs generated to the number of photons absorbed by the device. If there is no gain in the device, the relationship between EQE and IQE can be expressed as follows:  
\[\eta_{ext}=\eta_{\text{i}}\eta_{\text{a}}\eta_{c} \tag{3}\]  
According to the definition, \(\eta_{\text{i}}\), \(\eta_{\text{a}}\) and \(\eta_{\text{c}}\) are smaller than 1, and \(\eta_{\text{ext}}\) is smaller than \(\eta_{\text{i}}\), as stated in Equation (3). For high-quality photodiodes, IQE can approach 1. Gain (\(G\)) is the number of charge carriers collected in the circuit, due to light illumination, divided by the number of photons absorbed by the device. If there is no recombination loss of photogenerated electron-hole pairs (\(\eta_{\rm c}=1\)), the gain is equivalent to the IQE. The gain can be expressed as follows [52; 53]:  
\[G=\frac{\tau_{\rm e}}{t_{\rm{Ire}}}+\frac{\tau_{\rm p}}{t_{\rm{Irp}}} \tag{4}\]  
where \(\tau_{\rm e}\) and \(\tau_{\rm p}\) are the lifetimes of photogenerated electrons and holes, respectively, while \(t_{\rm{Ire}}\) and \(t_{\rm{Irp}}\) are the transit times between electrodes for electrons and holes, respectively. In the case of a photodiode, the lifetime of the photogenerated carriers is equal to their transit time, and the gain is 1 (note that it is not 2 [51]). The relation between gain and EQE is given by  
\[\eta_{ext}=\eta_{\rm a}G \tag{5}\]  
The EQE for diode-like photodetectors is typically less than 100% when no carrier multiplication is occurring [54]. However, for photoconductive detectors, the EQE can exceed 100%. In the case of photodetectors with an EQE of 100% (i.e., an ideal photodiode), the photocurrent responsivity is expressed as follows [52]:  
\[R_{I}=\frac{\lambda}{1.24} \tag{6}\]  
where \(\lambda\) is the wavelength of the incident light with a unit of \(\upmu\)m. Here, the unit of \(R_{\rm I}\) is A/W. A simple model for the gain in a photoconductor with a perfect semiconductor can be described as follows: The mobility of electrons is higher than that of holes, so electrons move faster than holes under an external field. When an electron-hole pair is generated at the center of the device, the photogenerated electron is collected by the drain electrode before the hole reaches the source electrode. To maintain charge balance in the device, an electron is injected from the source electrode. This process continues until the hole reaches the source electrode or recombines with an electron, causing multiple electrons to be collected in the circuit and resulting in gain. Additionally, gain can also occur due to charge traps in an imperfect semiconductor.