Photodetectors (PD) play key role in high performance of optoelectronic and photonic systems. P-i-N photodetector is one type of photodetectors which converts optical signal into electrical response. This type of PD usually consists of heavily doped P and N regions, separated by an intrinsic layer. Usual P-i-N PD systems consist of semiconductor materials like, Germanium, Indium gallium arsenide, Lead sulphide and Silicon. The next generation flexible and wearable communication systems require efficient photodetector material which is compatible with the flexible fabrication process. In recent years, copious amount of research is going on to fabricate flexible PD systems using various types of flexible materials, like, Tin Monosulfide, CsPbBr3 microcrystals, two-dimensional (2D) layered materials, organic semiconductors and perovskite materials    . Among all existing flexible PD materials, graphene based PD systems are attractive research topic owing to its unique advantages. Graphene is a zero band gap hexagonal honeycomb carbon atomic layer  -  , which allows absorption in broad range of wavelengths. Along with other unique properties of graphene, this wonder material is extremely flexible and it has huge potential to be used in flexible device fabrication. In the past few years, graphene based photodetectors have been successfully reported     , they are mostly fabricated on rigid substrates, providing a limitation of its utilization is flexible electronic systems. Thus, though there are vast applications of photodetector mainly in the nano-metric dimensions with unique features, several areas require immediate attention to optimize the engineered properties of such devices. Implementing a useful model for graphene based photodetector is still under research. In this letter, we describe a model for graphene based p-i-n photodetector based on simple concepts, where all the layers of the device is made of graphene sheets allowing its applicability in flexible electronics applications.
The model is based on the previously reported study on photodetector devices      . We consider the graphene based p-i-n photodetector structure, as shown in Figure 1. For practical fabrication process, mechanically exfoliated graphene single layer or multilayer sheets can be utilized. In order to create doped graphene structures, chemical doping of the graphene sheets can be realized. P-type graphene sheets can be realized by intercalating halogen dopants (Cl, Br, I and F), alkali metal based dopants (K, Li, Na, etc.), acids (HCl, HNO3 and H2SO4) and some organic compounds     ; whereas, N-type graphene structures are usually created by intercalating metallic compounds like ZnMg  .
Here, the light is incident on the P side. The structure consists of a single layer of N+ graphene layer, an undoped multi-layer graphene with thickness l and finally a single layer graphene P+ layer. The nominal N+ and P+ region doping is taken of the order of 1 × 1012 cm−2, which is practically reliable value  .
In the theoretical model, we have to consider the effect of photogenerated carriers in the intrinsic layer as because the widths of the three regions are comparable to each other. Thus, the current continuity equations in the depletion layer is given by 
Figure 1. Schematic structure of a graphene based p-i-n photodetector.
where, g is the photo carrier generation rate, v is the velocity, n and p denote the electron and hole, respectively. Incident optical powers, absorption coefficient of graphene at the operating wavelength, reflectivity of the graphene surface are some of the important parameters which control the generation rate of the photo carriers.
Carrier distribution N(x,jω) for electrons and P(x,jω) for holes in the depletion region (in frequency domain) are calculated by solving the above current continuity equations simultaneously. In general, all the uppercase variables are used to indicate the Laplace transform of the corresponding lowercase variables. The entire depletion region is subdivided into equal energy spacing (Δx) for calculation. Along its path of motion, each carrier represents a specific position and energy state in the depletion region of the device. For this reason, each carrier is specifically represented as a function of two indices: one position index (i) and one energy index (j). So, we substitute N(x, jω) by N(i, j, jω) and P(x, jω) by P(i, j, jω).
To obtain the photo-current density, the carrier distribution N(i, j, jω) and P(i, j, jω) are multiplied by equal energy spacing (∆x) and then summed for all i and j. The current density J of the device is obtained using Equation (3)
where, L, q and v are the length of the PD, electronic charge and the carrier velocity respectively. We consider here that the carrier velocity is only function of the position. Suffix n and p is used for electrons and for holes respectively.
3. Results and Discussions
The material parameters for the graphene layer have been taken from the literature      . Using those parameters in this present model, 3-dB bandwidth, frequency response and responsivity of the device have been calculated. The calculated values are justified by the performance of the fabricated graphene based PD devices as described in literature     .
Figure 2 shows the 3-dB bandwidth variation i-layer thickness, where p and n layer consist of single layer graphene sheets (thickness ~1 nm). Areas of the PD devices are also varied in this case. Maximum bandwidth (~5 GHz) is obtained for the multilayer graphene i-layer (5 nm). Effect of device dimension on bandwidth can also be observed from the plot, where smaller sized devices have better bandwidth compare to the larger sized devices. Keeping i-layer thickness constant at 5 nm where maximum bandwidth is obtained, the P+ and N+ layer thicknesses are varied from 1 to 5 nm and area of the device is also varied, as seen from Figure 3. There is no significant variation of the 3-dB bandwidth when the P+ and N+ layer thicknesses are changed, which concludes that the
Figure 2. Variation of bandwidth with i-layer thickness keeping N+ and P+-layer thicknesses fixed at 1 nm.
Figure 3. Variation of bandwidth with N+ and P+-layer thicknesses keeping i-layer thickness fixed at 5 nm.
bandwidth of the device is dependent on the intrinsic layer thickness. This conclusion facilitates the applicability of the model in real fabrication of the device, since producing multilayer graphene sheets is comparatively easier than single layer graphene sheets.
Figure 4 shows the frequency response, keeping device area fixed at 50 µm2, which gave rise to the maximum 3-dB bandwidth as observed from Figure 2 and Figure 3. The i-layer thickness is varied from single layer graphene to few layers of graphene. Again we observe the multilayer graphene i-layer gives rise to better response. This is likely due to the fact that multilayer graphene sheets have poor electrical conductivity as compared to the single layer graphene sheets, giving rise to better performance of the intrinsic layer.
Keeping i-layer thickness as 5 nm and device area as 50 µm2, the responsivity of the system has been investigated, where p and n layer thicknesses are kept fixed at 1 nm. Figure 5 shows the change of responsivity of the PD system with
Figure 4. Normalized frequency response with i-layer thickness.
Figure 5. Responsivity is plotted as a function of wavelength for 5 nm i-layer (b) thickness in a Graphene-Graphene-Graphene n+-i-p+ Photodiode (PD). n+(a) and p+(c) Graphene layer thicknesses are taken to be 1 nm.
Figure 6. Responsivity is plotted as a function of bias at 633 nm for 5 nm i-layer (b) thickness in a Graphene-Graphene-Graphene n+-i-p+ Photodiode (PD). n+(a) and p+(c) Graphene layer thicknesses are taken to be 1 nm.
Table 1. Performance of p-i-n Photodetector (PD) based on Graphene: p = n = 1 nm, i = 5 nm, V = 1.5 V.
wavelength, which shows a linear response of the device for different applied bias voltages. We also investigate the responsivity of the PD at 633 nm wavelength, as a function of applied bias which is comparable with the reported values in the literature  as shown in Figure 6. Table 1 shows the performance of p-i-n Photodetector (PD) based on the optimized parameters discussed in this paper.
We have designed an effective graphene based flexible PD system whose performance matches well with the experimental values in the literature. The PD system can be effectively used in the next generation communication systems. Further investigations are under progress to fabricate graphene based PD devices on flexible polymer substrates.
The authors would like to thank Prof. S. Chaudhury, Director CEERI-Pilani and Prof. D. Bhattacharya, Director AOT for their support.