# Trends in Performance Limits of the HOT Infrared Photodetectors
## 5 2D Material Infrared Photodetectors
### 2D Material Photodetectors: Current Responsivity Versus Response Time  
The detector's current responsivity is given by equation  
\[R_{i}=\frac{\lambda\eta}{hc}qg, \tag{13}\]  
and is determined by the QE (\(\eta\)) and photoelectrical gain, \(g\). The QE is given by the number of electron-hole pairs generated per incident photon and shows how the detector is coupled to the impinging radiation. The second parameter, the photoelectrical gain describes the number of carriers reaching contacts per one generated pair and shows how well the generated carriers are used to increase photodetector current responsivity. Other symbols of Equation (13) mean: \(\lambda\)--Wavelength, \(h\)--Planck constant, \(q\)--Electron charge, and \(c\)--Light velocity. In general, the photoelectrical gain is given by  
\[g=\frac{\tau}{t_{t}}, \tag{14}\]  
where \(\tau\)--Carrier lifetime and \(t_{t}\)--Transit time of electrons between the device electrodes. If the drift length, L\({}_{\text{d}}\) = v\({}_{\text{d}}\tau\), is less or greater than the distance between electrodes, l, the photoelectrical gain can be less or greater than unity. The value of L\({}_{\text{d}}\) > l shows that a one carrier swept out by electrode is replaced directly by an equivalent carrier injected by the opposite contact. In this way, a carrier will circulate until it recombines. For the photodiode, the photoelectric gain usually =1, due to separation of minority carriers by the electrical field of depletion region. However, in a hybrid combination of 2D material photodetectors, the photogeneration and carrier transport occur in a separate regions: one for effective light absorption, and the second - to provide fast charge reticulation. In this way, high gain close to 10\({}^{8}\) electrons per one photon, and significant responsivities for SWIR photodetectors have been demonstrated [51]. The simple architecture of hybrid phototransistor, very popular in the 2D material photodetectors design with the fast transfer channel for carriers, is presented in Figure 14. 2D materials with atomic layer thickness are more vulnerable to local electric fields than conventional bulk materials and the photogating effect can strongly modulate the channel conductivity by external gate voltage, Vg. Improvement in the optical gain is particularly important since the QE is suppressed because of the weak absorption in 2D materials. This effect is especially seen in LWIR region, where the light absorption is weak. In the case of hybrid detector shown in Figure 14a, the holes are injected into transporting channel, whereas the electrons remain in the photoactive layer. The injected charges can reticulate even several thousand times before recombination, giving contribution to the gain under illumination. The photocarrier lifetime is enhanced through both the bandgap profile and defect engineering, and at the same time the trapping mechanisms limit the response time of photodetector even to several seconds. The trade-off between improvement in responsivity and response time must be considered during optimization process. The photocurrent versus photogating effect can be given by [57]  
\[I_{ph}=g_{m}\Delta V_{g}, \tag{15}\]  
where \(g_{m}\)--Transconductance, \(\Delta V_{g}\)--Equivalent photoinduced bias. Figure 14d indicates a shift of the \(I_{ds}(V_{g})\) trace after the light illumination. Generally, both positive and negative photoconductance effects are observed in hybrid 2D structures and operating points A and B, related to \(g_{m}\) and \(\Delta V_{g}\), perform opposite directions. Figure 15 compares the graphene-based detectors responsivity operating in visible and NIR with silicon and InGaAs photodiodes commercially available on the market [58,59]. The highest current responsivity, above \(10^{7}\) A/W, has been reached for hybrid Gr/quantum dot (QD) photodetectors with enhancement trapped charge lifetimes.