Optimizing the Photothermoelectric Effect in Graphene

Among its many uses, graphene shows significant promise for optical and optoelectronic applications. In particular, devices based on the photothermoelectric effect (PTE) in graphene can offer a strong and fast photoresponse with a high signal-to-noise ratio while consuming minimal power. In this work, we discuss how to optimize the performance of graphene PTE photodetectors by tuning the light confinement, device geometry, and material quality. This study should prove useful for the design of devices using the PTE in graphene, with applications including optical sensing, data communications, multigas sensing, and others.

Figure 1

A schematic of the photothermoelectric effect in graphene. (a) Optical absorption generates an excited population of electrons and holes, (b) interparticle scattering thermalizes photoexcited carriers to an elevated temperature, and (c) electron-phonon scattering relaxes the electron temperature back to the lattice temperature. (d) A cross section of a graphene PTE detector, where split gates generate a $p$-$n$ junction in the graphene channel, resulting in a thermoelectric voltage between the source and drain contacts. (e) Examples of the temperature profile $T_{\mathrm{el}}(x)$ and the Seebeck coefficient $S(x)$ present in these devices.

Figure 2

The photovoltage as a function of the width of the incident light profile $\xi$, from Eq. (11). The light profile width is normalized by the electron-cooling length $L_c$ and $V_{\mathrm{PTE}}$ is normalized by its value at zero width. The inset shows the shape of the Gaussian optical profile.

Figure 3

Photodetector figures of merit as a function of the channel length $L_{\mathrm{ch}}$, from Eqs. (13)–(15). The channel length is normalized by the electron-cooling length $L_c$, the photovoltage and photocurrent are normalized by their maximum values, and the NEP is normalized by its minimum value.

Figure 4

The photovoltage $V_{\mathrm{PTE}}$ as a function of the light spot size $\xi$ and split-gate separation $W_g$, from Eq. (18): (a) for low native doping of the graphene layer; (b) for moderate doping; (c) for high doping, when Pauli blocking has set in. The dashed lines show the value of $\xi$ at which $V_{\mathrm{PTE}}$ is maximized for each value of $W_g$ and vice versa for the dotted line in panel (a). The thin solid lines are contours of $V_{\mathrm{PTE}}$, relative to the maximal value at $(\xi =0,W_g=0)$.

Figure 5

Photodetector figures of merit as a function of the channel width $W_{\mathrm{ch}}$, from Eqs. (20)–(23). The photovoltage and photocurrent are normalized by their maximum values and the NEP is normalized by its minimum value. Here, we assume a cooling length $L_c=1\mu \mathrm{m}$ and an optical absorption length $L_{\alpha }=50\mu \mathrm{m}$. The solid (dashed) lines are for the case of perfectly relaxing (reflecting) graphene edges.

Figure 6

The optimal photovoltage response as a function of the graphene mobility, for the cases of relaxing and nonrelaxing edges. For nonrelaxing edges, the optimal photovoltage is given by $R_V^{\mathrm{opt}}=\underset{E_F}{max}\{(S/\kappa )(L_c/L_{\alpha })\}$. For relaxing edges, the optimal photovoltage is $R_V^{\mathrm{opt}}=\underset{E_F,W_{\mathrm{ch}}}{max}\{2S{T}_{\mathrm{av}}/P_{\mathrm{in}}\}$, where $T_{\mathrm{av}}$ is defined in Eq. (21). The inset shows how the Seebeck coefficient and cooling length, at the point of optimal photoresponse, scale with the graphene mobility.

Figure 7

The optimal photocurrent response as a function of the graphene mobility, calculated as the maximum of Eq. (28) over $E_F$. The solid (dashed) lines show the photoresponse with (without) Peltier cooling.