Abstract
Dynamic wavelength tunability has long been the holy grail of photodetector technology. Because of its atomic thickness and unique properties, graphene opens up new paradigms to realize this concept, but so far this has been elusive experimentally. Here we employ detailed quantum transport modeling of photocurrent in graphene fieldeffect transistors (including realistic electromagnetic fields) to show that wavelength tunability is possible by dynamically changing the gate voltage. We reveal the phenomena that govern the behavior of this type of device and show significant departure from the simple expectations based on vertical transitions. We find strong focusing of the electromagnetic fields at the contact edges over the same length scale as the bandbending. Both of these spatiallyvarying potentials lead to an enhancement of nonvertical optical transitions, which dominate even in the absence of phonon or impurity scattering. We also show that the vanishing density of states near the Dirac point leads to contact blocking and a gatedependent modulation of the photocurrent. Several of the effects discussed here should be applicable to a broad range of one and twodimensional materials and devices.
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Introduction
Photodetectors based on graphene have been extensively studied both experimentally^{1,2,3,4,5,6} and theoretically^{7,8}. A number of detector designs harnessing different photocurrent mechanisms have been realized and have shown promising performance. This success brings some interesting questions on whether graphene could provide novel modalities for photodetectors beyond the conventional photoresponse. One such modality is the ability to dynamically tune the spectral sensitivity of the detector. In existing technology, multicolor detection is achieved by using mechanical filters^{9}, multiple colocated pixels^{9}, or by using stacked layers^{10}, all of which have limitations: For example, mechanical filters are slow and introduce unwanted moving parts. In the multipixel approach, subpixels possessing differing spectral sensitivity are realized by changing material composition during fabrication. While providing hyperspectral response, resolution at a given wavelength is compromised. An alternative design is to stack multiple layers of materials possessing different absorption properties; while this approach is commonly used for twocolor detection, generalizing the concept beyond two colors is difficult due to the necessity of monolithically integrating multiple materials and electronically addressing each layer. Furthermore, these methods do not allow for postfabrication tuning of the spectral band. Thus, a photodetector concept with dynamic tuning of the intrinsic spectral range of a detector pixel composed of a single material possesses advantages over existing technology.
Graphene opens interesting possibilities to realize this concept, as recent work has shown that the optical properties of graphene can be modulated using a back gate^{11,12} to realize optical modulators. If this result can be translated into modulation of the photocurrent, it could be harnessed for tunable photodetector applications. Unfortunately the experimental realization of this concept remains elusive, and to our knowledge the only experimental report of spectral photocurrent tunability (by a few tens of nanometers) was realized by inducing conformal changes in graphene^{13}. In light of the experimental difficulties and nonidealities implicit in the fabrication of tunable graphene photodetectors, modeling can provide possible paths and identify key roadblocks. Previous modeling efforts have considered different possibilities for tunable graphene photodetectors based on multiple graphene layers, such as bilayer graphene photogating^{14} and tunneling between two separate graphene layers^{15} or nanoribbons^{16}. These exciting results, based on macroscopic transport equations appropriate for longchannel devices, show the promise of devices based on multiple graphene layers. However, since devices based on monolayer graphene are easier to fabricate, an open question is whether wavelength tunability is possible in that case.
In this paper, we address the challenges of identifying promising tunable detector designs and the necessity for more detailed quantum modeling by developing and implementing a nonequilibrium quantum transport approach for photocurrent calculations in graphene devices, including realistic electromagnetic fields. Our main result is that dynamic hyperspectral imaging should be possible in such devices, but a wealth of phenomena renders the system rather complex. For example, we find that translational symmetry breaking due to spatiallyvarying bandbending and electromagnetic fields leads to a dominance of nonvertical transitions that significantly change the behavior expected from pure vertical transitions and Fermi blocking^{17}. We also show that the vanishing density of states at the graphene Dirac point leads to a gatedependent modulation of the photocurrent due to a contact blocking effect.
Methods
We consider a subwavelength graphene fieldeffect transistor (FET) as illustrated in Fig. 1a,b. We focus on FETs with short channels for several reasons. First, the recombination time in graphene is on the order of picoseconds^{18} which, combined with the Fermi velocity, gives a recombination length of about one micron. Therefore, at low light intensity we can focus on single electronphoton scattering events without recombination. Second, the subwavelength channel leads to a nonuniform vector potential that introduces novel optical transition phenomena. Finally, because we consider uniform illumination of the full device, no temperature gradient exists between the two contacts and photothermoelectric effects^{19} can be neglected. While we present results for hot electron devices (i.e. purely ballistic transport), our results can be directly converted to the scattering case by multiplying the calculated photoresponse by the ratio l_{mfp}/L where l_{mfp} is the electron scattering mean free path and L is the channel length.
To calculate the photocurrent, we employ the nonequilibrium Green’s function (NEGF) approach, where we explicitly include the electronphoton interaction at the quantum level. We previously developed and applied this approach to carbon nanotube pn junctions^{20}; the graphene FET case differs significantly not only because of the material and the device geometry, but also because the FET does not possess a builtin field to generate a photocurrent at zero bias. Furthermore, to our knowledge, the only NEGF photocurrent calculation for graphene presented in the literature^{21} (for a model pn junction) did not include a nonuniform electromagnetic potential as considered in this paper.
Our calculations consist of two preliminary steps and a final step to obtain the photocurrent, as illustrated in more detail in the Supporting Information. The first preliminary step is a selfconsistent NEGF calculation of the potential and charge in the dark graphene FET using our previously developed approach for carbon nanotubes^{20,22,23} appropriately modified for graphene (specific details can be found in the Supporting Information). In the second preliminary step, we solve Maxwell’s equations in the geometry of Fig. 1a,b to obtain the vector potential on the graphene sheet. The final step is to combine the vector potential and the NEGFs from the dark simulations to calculate the photoresponse. Because the graphene FET does not possess a net builtin field, no photocurrent is generated for uniform illumination at zero bias. However, a small sourcedrain bias breaks the symmetry and leads to a net photocurrent; we therefore focus on the photoconductance in the small bias limit, defined as
where I_{ph} is the photocurrent at a sourcedrain bias V_{sd}.The full expression for σ^{ph} can be found in the Supporting Information.
In our simulations, we model the graphene using a tightbinding Hamiltonian with overlap integral of 2.5 eV and nearestneighbor bond length of 0.142 nm. The zigzag direction of the graphene is perpendicular to the electron transport direction. The source and drain electrodes are perfect metals vertically separated from the graphene sheet by a distance of 0.3 nm. Charge transfer between the metal and the graphene is naturally captured in our simulations by using a metal workfunction 0.5 eV larger than graphene. The electronphoton interaction is included by modifying the tightbinding Hamiltonian with the usual term where is the magnetic vector potential of the light field in the device geometry at the graphene location and is the graphene electron momentum. Details of the implementation can be found in the Supporting Information along with the graphene optical absorption calculated with this NEGF approach.
Results
Figure 1c shows the calculated selfconsistent bandbending potential for different gate voltages ranging from 0 V to 3 V (the bandbending potential is equivalent to the energy of the Dirac point). As expected for a FET, the potential in the middle of the channel is modulated by the gate; however, the metal contacts tend to pin the potential, leading to a bandbending over a distance of ~10 nm near the contacts. This length scale is determined by the screening environment of the device and the graphene. In this case, the bottom metal gate provides most of the screening, and the bandbending length scale is determined by the thickness and dielectric properties of the oxide. An additional effect is seen where the band potential under the contacts also changes with the gate bias. This gate modulation of the contacts has been previously discussed in the case of carbon nanotubes^{24} and we expect it to be relevant for other one and twodimensional materials and devices.
Figure 1d shows the small bias conductance in the dark based on the bandbending potential of Fig. 1c which displays an asymmetric shape and a minimum conductance of about 30 times the quantum of conductance of 4e^{2}/h. This result is in good agreement with experimental data for short channel devices^{25,26} which come within a factor of three of the minimum conductivity calculated here, and also show prominent asymmetry. In our case, the asymmetry is due to the contact modulation by the gate.
We obtained the vector potential by solving Maxwell’s equations in the device geometry of Fig. 1a assuming a periodic structure with perfect metals for the contacts and the bottom gate, for a monochromatic plane wave normally incident and polarized in the x direction (see Supporting Information for details of the calculations). Figure 2a shows that interaction of the plane wave with the device leads to the appearance of an additional component of the vector potential in the y direction. More importantly, the vector potential exhibits strong spatial dependence with significant enhancement near the contacts. Figure 2b shows the ratio of A_{x} at the contacts to that in the middle of the device, which is about 36 at the longer wavelengths and increases to more than 60 at a wavelength of 1 micron. These results suggest that the properties of photodetectors based on twodimensional materials should be particularly sensitive to nearcontact effects even under flood illumination.
The nearcontact focusing of the electromagnetic field is spatially colocated with the device bandbending. As discussed in the context of Fig. 1c, the bandbending near the contact occurs over a distance ~10 nm and is determined by the dielectric thickness. We find that the electromagnetic field also decays over this same length scale as illustrated in Fig. 2c. We believe this is also a consequence of the screening in the device geometry. For example, when solving Maxwell’s equations, the boundary conditions at the source and drain electrodes can be satisfied by introducing surface charges and currents, and these will in turn be screened by the bottom gate. We verified this idea by repeating our simulations for different oxide thicknesses, finding that the vector potential decay length (defined as the 1/5 decay distance from the contact) increases with increasing oxide thickness (Fig. 2d).
Having obtained the gatedependent dark bandbending and the vector potential resulting from illumination, we can calculate the photoconductance. Since the optical absorption in graphene for the outofplane contribution is very small^{27}, we can neglect the A_{y} component and consider only the spatiallyvarying A_{x} component. Figure 3 shows the results of such calculations for two photon energies of 0.1 eV (12.4 μm) and 0.25 eV (4.95 μm). Several conclusions emerge from the data in this figure. First, the photocurrent is gatetunable for each photon energy. Second, there are regions where clear discrimination between wavelengths is possible. Third, the curves do not follow the simple expectation based on Fermi blocking of vertical transitions. Indeed, if only vertical interband transitions are allowed, the symmetric graphene bands imply that the Fermi level must be less than above or below the Dirac point for transitions to occur. Using the Dirac point energy in the middle of the graphene FET to extract the gate voltage at which transitions should be blocked, we obtain the vertical dashed lines in Fig. 3, outside of which no photocurrent should be observed. The full calculations show a significant deviation from this simple model for both photon energies.
The above effects have their origin in two main factors: nonvertical transitions and gatedependent contact blocking. Nonvertical transitions can arise when the translational symmetry is broken; in the case of the graphene FET translational symmetry is broken by the spatiallyvarying electrostatic potential and spatiallyvarying magnetic vector potential. The missing momentum that allows nonvertical transitions arises from the infinite number of Fourier modes that make up these spatial variations. To illustrate the importance of nonvertical transitions, we plot in Fig. 4 the effective transmission probability for electrons as a function of their energy compared to the bandbending in the graphene FET at a gate voltage of 4.5 V. This transmission represents the probability that an electron injected from the source at energy absorbs a photon and is collected in the drain at energy E. If only vertical transitions were present, the transmission would show a peak at energy above the Dirac point. Note that because of the occupancy of the states, at zero temperature the transmission would be nonzero only in an energy window between E_{F} and . (Thermal broadening extends this window by a few K_{B}T above and below.)
For the case of Fig. 4, the Dirac point in the middle of the channel is more than 0.2 eV below the Fermi level. In principle the photoresponse should vanish unless the photon energy is larger than about 0.4 eV. However, we find that for photon energies of 0.1 eV and 0.25 eV, a significant photoresponse remains (Fig. 3). This can be explained from examination of Fig. 4b,d where the interband vertical transitions are Fermiblocked in both cases but significant transmission is observed at energies where only nonvertical transitions can arise. For example, at vertical interband transitions would give a peak at −0.17 eV based on the midchannel Dirac point, but these transitions are blocked because the Fermi level is at 0 eV and there are no empty states to excite into. However, nonvertical transitions can exist between ≈−0.1 eV and ≈0.2 eV (i.e. the Fermi window plus thermal broadening, and with appropriate momentum transfer from spatial symmetry breaking) in agreement with the transmission. By increasing the photon energy to we see the appearance of another effect: contact blocking. Indeed, at this photon energy, nonvertical transitions exist up to energies of 0.25 eV (plus thermal broadening). However, the transmission function has a strong dip to zero in the middle of this range at an energy that corresponds to the Dirac point in the contacts. This strong reduction in transmission is a consequence of the density of states of graphene that goes to zero at the Dirac point; thus, one can think of this effect as contact blocking of the photocurrent.
This effect is in part responsible for the shape of the photoresponse versus gate voltage observed in Fig. 3. To illustrate this effect in more detail, Fig. 5 plots the bandbending and transmission probabilities for a gate voltage of 1 V. In the case of a small peak at 0.061 eV is visible due to vertical transitions, but most of the photoresponse comes from nonvertical transitions. A similar situation arises for ; however, the Dirac point in the contact regions is now located in the Fermi window and blocks a significant portion of the carriers, leading to a reduced photoresponse and the unusual shape observed in Fig. 3.
To further determine the factors that lead to the unusual importance of nonvertical transitions, we calculated the transmission for a gate voltage of 1 V and illumination at . For this gate voltage, the Dirac point is 0.011 eV above the Fermi level in the middle of the channel and vertical interband transitions should give a peak at an electron energy of 0.061 eV. As shown in Fig. 6, a peak is observed at this energy, but is significantly broadened by nonvertical transitions. We then repeated this calculation for a constant electrostatic potential throughout the device (i.e. flat bands) with a value corresponding to the Dirac point in the middle of the channel for the full device simulations at V_{g} = 1V. Figure 6 shows that at low temperatures (30 K in this case) and for a uniform vector potential in the channel, there is a narrow peak due to vertical interband transitions; increasing the temperature to 300 K does not change this behavior and we therefore rule out temperature effects as the main source of the nonvertical transitions. However, introducing a nonuniform vector potential (but still with flat bands) gives a much broader peak, implying that the nonuniform light profile itself leads to significant nonvertical transitions. Finally, since the full device bandbending gives an even broader peak, nonvertical transitions are also enhanced by the spatiallyvarying bandbending that breaks the spatial symmetry and provides the momentum. (Additional details of how these effects emerge from the NEGF formalism can be found in the Supporting Information.)
Conclusion
In conclusion, we find that monolayer graphene FETs open a new avenue to realize photodetectors with dynamic spectral tunability, and display a number of novel underlying phenomena that should be general. For example, electromagnetic field focusing at the contacts plays a key role in governing photoresponse behavior, a phenomenon that should apply to a broad range of one and twodimensional materials. We also expect that nonvertical transitions induced by symmetry breaking due to the spatial variations of the bandbending and the vector potential should also become prominent in other lowdimensionality materials. Finally, additional studies are needed to assess the robustness of the wavelength tunability to nonideal factors such as substrate charged impurities. Nonetheless, our work establishes the first framework to guide and interpret experimental and theoretical efforts.
Additional Information
How to cite this article: Léonard, F. et al. Dynamic WavelengthTunable Photodetector Using Subwavelength Graphene FieldEffect Transistors. Sci. Rep. 7, 45873; doi: 10.1038/srep45873 (2017).
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Acknowledgements
Work supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DEAC0494AL85000. We thank Dr. Aron Cummings for help with the computational NEGF code.
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All authors contributed to development of the project and writing of the manuscript. F.L., C.S., and M.G. performed the computational work.
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Léonard, F., Spataru, C., Goldflam, M. et al. Dynamic WavelengthTunable Photodetector Using Subwavelength Graphene FieldEffect Transistors. Sci Rep 7, 45873 (2017). https://doi.org/10.1038/srep45873
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DOI: https://doi.org/10.1038/srep45873
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