Electrical detection of graphene plasmons for mid-infrared photodetection and chemical sensing: A computational study

Surface plasmon polaritons (SPPs) in single layer graphene (SLG)^1–5 are a promising platform for light detection,^6–13 optical modulation,^14–16 and sensing applications.^17–22 SLG-SPPs are gate-tunable throughout the infrared (IR),^23–25 sensitive to chemical doping,²⁶ and exhibit high spatial field confinement.^1,17,27,28 The latter is a result of significant momentum mismatch between SLG-SPP and free-space light,^2,4 which, however, also renders the light in-coupling to SLG-SPPs inefficient due to momentum conservation.^2,4 Techniques to match momenta and increase in-coupling efficiency include SNOM assisted scattering,^24,25 scattering from metallic-nanoparticles,^29–31 and nanostructured SLG, e.g., in the form of nanodisks^32–37 or nanoribbons.^23,38–42

Detection of SLG-SPPs is typically done optically and is indirect, i.e., by spectral measurements on the reflected and transmitted light,^{15,17–19,23,35–37,39} followed by post-processing and simulations to identify and subtract background absorption and peaks coming from other device elements, such as metallic antennas and Fabry–Pérot cavities. An electrical measurement technique yielding direct information on SLG-SPP excitation, however, would be ideal for reasons both fundamental (understanding of the excitation and decay mechanisms) and practical (clear distinction between SLG-SPPs and other device resonances, avoidance of bulky optics). There have been several studies addressing electrical detection of SLG-SPPs. In all of them, the elevated SLG electronic temperature at plasmonic resonance is utilized to induce and harvest a photocurrent.^7–13 One approach involves the photo-thermoelectric effect (PTE).⁷ In Ref. 7, an external responsivity (R_ext) of 20 mA/W was measured at zero bias, with a calculated noise equivalent power (NEP) of 400 pW/Hz^0.5. Another approach involves the temperature-induced changes in SLG conductivity.^8–10 However, this requires electrically biasing SLG, which introduces significant dark current noise (SLG is a conductor). A further bottleneck is also related to the weak temperature dependence of SLG conductivity,⁴³ especially in chemical vapor deposition (CVD) graphene where carrier transport is dominated by defects and grain boundaries.^44,45 As a result, the measured R_ext in Ref. 10 was found in the order of $Rext≈10 μA/W$⁠. An improvement, utilizing the conductivity change due to elevated SLG carrier temperature mechanism, was reported in Ref. 9, using graphene disk plasmonic resonators (GDPRs)^8,9 connected with graphene nanoribbons⁹ (GNRs), whereby the GDPRs generate hot carriers population via plasmon resonant absorption and thermalization, while carrier transport is thermally activated along the GNRs. This work resulted in measuring $Rext≈16$ mA/W,⁹ with a measured NEP $≈1.3$ nW/Hz^0.5, and a theoretical lower limit⁹ of NEP calculated at $≈460$ pW/Hz^0.5. An improved configuration was reported by Safaei et al.,¹¹ with a high voltage responsivity of $∼2500$ V/W and low NEP of $∼7$ pW/Hz^0.5, exploiting the PTE effect in an asymmetrically patterned graphene configuration, where plasmons are excited and absorbed at one side of an electrically biased graphene layer. Using similar configurations, Shabir et al. also reported similar performances,^12,13 reaching responsivities in the order of $∼105$ V/W and NEP as low as $∼10$ fW/Hz^0.5. The latter approaches, however, also rely on applying a source–drain bias across the SLG channel.^8,9,11–13

Here, we propose and theoretically investigate an unbiased SLG/Si Schottky junction configuration, operating in the thermionic regime,^46,47 as a platform for electrical detection of SLG-SPPs. We assume a periodic array of SLG-GNR on top of a p-doped silicon substrate (p-Si) forming a series of SLG/Si Schottky contacts,^46,48,49 where graphene plasmon resonance frequency scales with nanoribbon width^27,41 and can be tuned by gating⁵⁰ and/or reverse voltage bias.^51,52 The absorbed optical power in GNR, promoted by SPPs excitation, leads to a thermalized hot carrier gas in SLG,^7–9,53 which in turn creates photocurrent via thermionic emission across the SLG/Si junction.^46,47 We extend our theoretical framework, which was developed in Ref. 46 for inter-band optical excitation in SLG with photons energy $Eph>2EF$⁠, to the intra-band absorption process at longer wavelengths (i.e., at Pauli blocking regime $Eph<2EF$⁠), and show that GNR-SPPs based Schottky junctions can produce significant photocurrent. Specifically, using optimized device geometry and realistic operational parameters, we calculate an external responsivity $Rext≈110$ mA/W, with a NEP $≈190$ pW/Hz^0.5. We demonstrate this electrical detection scheme in a chemical sensing application, utilizing the surface enhanced infrared absorption technique (SEIRA)^{17–21,54–57} and assuming toluene⁵⁸ as an analyte paradigm. We optimize the sensor device architecture and operational parameters and explore its detection and sensitivity limits.

We consider a periodic GNR array of width w and period L on top of a p-Si substrate (⁠$NA,Si=1017 cm−3$⁠), forming a Schottky junction,^46,48 as depicted in Fig. 1(a). We assume the GNRs to be p-doped, as is common in CVD fabricated graphene,^59,60 and a fixed $w/L=0.5$ ratio for simplicity. The structure is supplemented by an Au backmirror at spacer distance $d=1 μm$⁠, to form a Fabry–Pérot cavity and enhance light absorption.^15,17,54 The lowest energy GNR-SPP resonant wavelength can be calculated by^27,41,42 $λres=α [(1+ϵr,Si)ϵ0 w/EF]1/2$⁠, where $α=η (2πℏc/e), ℏ$ is the reduced Planck constant, c is the speed of light, e is the electron charge, E_F is the SLG Fermi level, ϵ₀ and $ϵr,Si=11.7$ are the free space and relative permittivity⁶¹ of the p-Si substrate, respectively. The prefactor η accounts for the hybridization and redshift⁴² of plasmons in neighboring GNRs, where $η→1$ for large (⁠$w/L→0$⁠) GNR separations. η can also be affected by SPP-cavity interactions with the backmirror.⁵⁴ Here, the value $η=1.25$ is extracted by fitting the relationship for λ_res to the finite difference time domain⁶² (FDTD) calculations of the proposed structure (see the supplementary material). By adjusting E_F and/or w, the GNR-SPP resonance can be tuned within the application desired wavelength range (e.g., for toluene sensing we target $λ≈13−15 μm$⁠). Assuming w = 40 nm (more w values will also be explored), the required SNR doping level is $EF≈−0.36$ eV. The free carriers relaxation time in SLG, $τopt$⁠, depends on E_F and carrier mobility μ_q via^63,64 $τopt=μqEF/eυF2$⁠, where $υF=106 m/s$ is the SLG Fermi velocity⁴⁸ and is typically in the order of $∼100 fs$⁶⁴ for polycristalline CVD graphene. For GNR, we assume $τopt$ to be w-limited,⁶⁵ i.e., $τopt=w/υF=40 fs$⁠. This yields mobility $μq≈1000 cm2 V−1 s−1$⁠, similar to what expected for defected and/or nanostructured SLG.^54,65 The optical conductivity of the GNRs is modeled by the Kubo formula,^66,67 and the device optical response is simulated by the FDTD method⁶² using the Lumerical FDTD solver.⁶⁸ Under IR illumination, the calculated GNR-SPP absorption coefficient is plotted in Fig. 1(b), reaching a maximum of $∼60%$⁠.

Regarding the sensing application, toluene has two main absorption lines at $λ1=13.7 μm$ and $λ2=14.4 μm$⁠.⁵⁸ These spectral features are poorly probed directly by light, due to the large mismatch between resonant wavelength (⁠$>10 μm$⁠) and molecular size (⁠$<nm$⁠).^18,55 The principle of SEIRA is to use the deep sub-wavelength electric field (E-field) confinement and intensity enhancement of the SPP mode in GNR^{17–21,54–57} to probe these vibrational fingerprints. If a thin $dtol=8 nm$ toluene layer (more d_tol values will also be explored) is placed on top of GNR, the latter's absorption will be reduced at the spectral lines of toluene absorption peaks, as shown in Fig. 1(b). This also involves a small redshift of the overall response, due to the different dielectric environment introduced by the toluene layer.¹⁸ We next evaluate the thermionic photocurrents, induced by the GNR-SPP light absorption.

Upon GNR/p-Si contact, holes will flow from p-Si to GNRs due to their work function difference,^46,48,51 forming a depletion layer in p-Si and a Schottky junction with barrier height^46,48 (SBH) $ΦB=eV0+eϕSi$⁠, where V₀ is the built-in potential across the depletion layer^46,48 and $eϕSi$ is the energy difference between the Fermi level $EF,Si$ in Si and the valence band $EV,Si$⁠.^46,51 Using the framework developed in Ref. 46 (see also the supplementary material), we find that for $EF=−0.36$ eV, the GNRs must initially (before contact) be at $EF′$ = −0.35 eV, i.e., no significant E_F shift occurs. This is related to the increased density of states of SLG at elevated E_F values.⁴⁶ At the contact, GNR/p-Si Schottky junction gives rise to a Schottky barrier height (SBH) $ΦB∼0.31$ eV. In this work, we will not consider reverse bias, thus avoiding dark current and excess noise.⁴⁶ 

We assume that the incident IR light is continuous (or quasi-continuous, e.g., in long $≫ps$ pulses⁴⁶) and quasi-monochromatic, i.e., filtered by a grating or prism.⁶⁹ After light absorption, the excited GNR carriers thermalize within ∼20 fs^70,71 and eventually cool down through the emission of acoustical and optical phonons.^46,70–81 During this process, an elevated electronic temperature is achieved and additional fraction of charge carriers at the Fermi-Dirac distribution tail will overcome the SBH⁴⁶ and thermionically will be injected into p-Si. The latter will contribute to further cooling^46,82 as well as a net photocurrent across the junction.^46,47 To calculate the electronic temperature $Te,SLG$ of charge carriers in GNR under continuous illumination, we use the framework developed in Ref. 46 (see also the supplementary material) and self-consistently solve the equilibrium equation for $Te,SLG$⁠:

where P_in is the incident optical power focused in area S (here taken as the diffraction limited spot $S=λ2/π≈6×10−7 cm2$⁠, where $λ=14 μm$ is the average wavelength of interest) and $αSLG≡αSLG(EF,Te,SLG)$ is the absorption coefficient of the GNRs. $Je−ph≡Je−ph(EF,Te,SLG)$ is the thermal current density dissipated into the phonon bath via electron–phonon (e-ph) interactions,⁴⁶ and $Jth≡Jth(EF,Te,SLG)$ is the thermal current density across the junction,^46,82 for each GNR. For $Je−ph$⁠, we consider two main e–ph scattering processes: optical phonons^74–76 and acoustic phonons via disorder-assisted supercollisions.^77–80 For the former, we consider two branches,⁴⁶ at the K-point^83–85 (⁠$ΩK=161 meV$⁠) and at the doubly degenerate Γ-point^83–85 (⁠$ΩΓ=196 meV$⁠) of the SLG Brillouin zone. For the latter, we assume a concentration of short-range scatterers with a mean free path l.^46,77–80 For the GNR structure, we assume l to be w-limited, i.e., l = w. The combined contributions then lead to an effective e-ph relaxation time $τe−ph=cvΔTe,SLG/Je−ph∼400 fs$⁠, where $ΔTe,SLG$ is the electronic temperature rise at equilibrium (⁠$ΔTe,SLG≪T0=300 K$⁠) and $cv(EF,Te,SLG)$ is the electronic heat capacity.⁴⁶ This $τe−ph$ value is much lower compared to the one for a continuous SLG sheet (⁠$∼1.5$ ps)⁴⁶ due to the smaller mean free path l, which renders supercollision as the dominant cooling process for GNRs. We note that here we examine only the linear regime,⁴⁶ i.e., low enough P_in so that $ΔTe,SLG$ values are in the order of mK. Due to the much larger (⁠$×103$⁠) heat capacity of the SLG lattice compared to that of its carriers,⁸⁶ we assume the lattice temperature to be fixed at $Tl,SLG=300 K$⁠.

The thermionic electrical and thermal current densities across the Schottky junction are calculated using the Landauer transport formalism:^46–49,82

where $τinj$ is the effective injection time of charge carriers from SLG to Si,^46–49,$ν(ϵ)=2|ϵ|/(πℏ2uF2)$ is the graphene carrier density of states, and $Df(ϵ)=fFD(ϵ;μSLG,Te,SLG)−fFD(ϵ;μSi,Te,Si)$ with $fFD(ϵ;μ,T)={ exp [(ϵ−μ)/(kBT)]+1}−1$ being the Fermi–Dirac distribution at chemical potential μ and temperature T. $T(ϵ)$ is the probability of charge carrier transmission from SLG to Si over $ΦBCNP$⁠, where $ΦBCNP$ is the SBH with respect to the charge neutrality point⁴⁶ (for simplicity, we assume $T(ϵ)=1$ for $ϵ>ΦBCNP$ and zero otherwise^46–48). $τinj$ varies with SLG quality,^48,49 Schottky interface quality,^47,51,87 momentum conservation/relaxation at the junction,^88–90 and E_F and $Te,SLG$⁠.^88–90 A lower bound for $τinj$ can be estimated for the low temperature regime (⁠$kBTe,SLG≪EF$⁠) using⁸⁹ $τinj∼ℏ|EF|/(kBTe,SLG)2$ yielding $τinj≈350$ fs for $EF=−0.36 eV$ and $Te,SLG≈300 K$⁠.

The photocurrent across the junction is calculated as $Iph=JelS/2$⁠. (the 1/2 factor is because of half-coverage, i.e., $w/L=0.5$⁠.) We assume incident power $Pin≈9.35 nW$⁠. This yields an external responsivity (calculated at $λ=14 μm$⁠) $Rext≈1$ mA/W with a noise equivalent power $NEP≈1.4$ nW/Hz^0.5. The specific detectivity, defined as⁶¹ $D*=S/NEP$⁠, is extracted having a value $D*=5.5≈105$ Jones. This configuration has a response time of $τresp≈5$ ns with a cutoff frequency $≈1.22$ GHz (see the supplementary material). Upon analyte placement, assuming analyte thickness d_tol = 8 nm, the chemical sensing signal I_S is determined by the photocurrent difference, $IS≡ΔIph=Iph,bare−Iph,toluene$⁠, which is plotted in Fig. 2(a), showing peaks at the toluene molecular resonances. The peak at $λ1=13.7 μm$ is higher due to this mode's stronger absorptivity.⁵⁸ Quantitative comparison between peaks can be done after subtraction of the baseline [red dashed line in Fig. 2(a)]. By operating the SLG/Si Schottky diode at zero bias, with eliminated dark current, the noise signal (normalized to a 1 Hz spectral band) is $⟨iS2⟩=⟨ish2⟩+⟨iJ2⟩$⁠, where $⟨ish2⟩=2e Iph$ is the shot noise contribution⁶¹ and $⟨iJ2⟩=4kB Te,SLG/Req$ is the Johnson (thermal) noise contribution,⁶¹ where $Req=dV/dIph$ is the diode shunt resistance.^46,61 The latter is calculated through the inverse of the derivative of the dark current with respect to applied bias, for low (⁠$∼mV$⁠) reverse bias values.⁴⁶ 

For chemical sensing, when measuring a signal I_S, the effective total noise is contributed by both $Iph,bare$ and $Iph,toluene$⁠, i.e., the total noise power is $⟨iS,tot2⟩=⟨iS,bare2⟩+⟨iS,toluene2⟩$⁠. To quantify a limit of detection associated with the photo-thermionic process, we calculate the minimum optical power P_min needed, i.e., signal to noise ratio $SNR=1$⁠, in order to detect the presence of an analyte. We define SNR, for each absorption line of the analyte, as $SNR=IS/iS,tot$⁠. In the proposed device, we find that the main contribution to $iS,tot$ is the Johnson noise, where the calculated $iJ≫ish$ is in the order of few pA and few fA, respectively. Since Johnson noise does not depend on P_in,^46,61 SNR can be approximated to scale linearly with the incident optical power, given the large (more than five decades) linear dynamical range (LDR) of SLG/Si Schottky junctions operating in the photo-thermionic regime.⁴⁶ Hence, $SNR=αPin$⁠, where α is a proportion coefficient. As a result, the limit of detection at $SNR=1$ is $Pmin=1/α$⁠. We calculated the power dependence of SNR for each absorption line of toluene, assuming d_tol = 8 nm (see the supplementary material), and found from the slope $α1≈0.14 nW−1$ and $α2≈0.10 nW−1$⁠, where the subscript denotes the $λ1=13.7 μm$ and $λ2=14.4 μm$ lines, respectively. The corresponding limits of detection are $Pmin,1≈7.1$ and $Pmin,2≈10$ nW, respectively.

Next, we study the effect of the various operational parameters on P_min. We first explore the effect of E_F. Besides changing the SBH, E_F variations also require the reevaluation of several other quantities, like the $τinj$ lower bound, the width w to keep the GNR-SPP frequency fixed, the free carrier relaxation time $τopt$⁠, and the mean free path l for supercollision scattering. These parameters are summarized in Table I for three different values of E_F.

Figure 3(a) plots P_min calculations assuming different E_F, using the parameters as for Table I. Higher E_F (and w) values lead to lower P_min, i.e., detection capability at lower incident optical power. This is because increasing E_F brings to: (a) smaller SBH, which triggers an enhanced thermionic emission across SLG/Si Schottky interface and (b) larger w, which leads to longer $τopt$⁠, increased intraband SLG conductivity and higher light absorption, as well as larger supercollision mean free path l and, thus, less cooling, yielding higher $Te,SLG$ for the same P_in. A slight counterbalance comes from the longer $τinj$⁠, as shown in Table I. The overall trend, however, remains that of a lower P_min for larger E_F values. The performance differences, upon changing E_F, can also be understood by the R_ext and NEP values. At the central wavelength, the lower graphene doping (e.g., $EF=−0.265$ eV), which is attributed to higher SBH (Table I), results in $Rext≈40$ μA/W, $NEP≈7.4$ nW/$Hz0.5, D*≈105$ Jones, and response time $τresp≈4.9$ ns. For higher SLG doping (⁠$EF=−0.50$ eV, i.e., lower SBH), R_ext increases and reaches $Rext≈110$ mA/W with a lower $NEP≈190$ pW/$Hz0.5,D*≈4×106$ Jones, and response time $τresp≈12$ ns. There is, however, a lower limit of a SBH that can be achieved in the given configuration in the order of $ΦBmin≈eV0$ that depends on Si doping [see the supplementary material, Eq. (S3)]. The latter would allow achieving lower $ΦBmin$ using higher p-Si doping but would also lead to a fraction of light loss in Si (due to free carrier absorption), thus lowering the SLG carrier temperature increase and associated thermionic current across the Schottky junctions. Note also that a reduced ambient temperature will not improve performance. Because of the thermionic nature of the photocurrent generation, the responsivity will get strongly suppressed at lower temperatures⁴⁶ [see also Eq. 2(a)] and result in a NEP increase despite the reduction of the Johnson noise.

Up to now, we have assumed a toluene layer with thickness $dtol=8 nm$⁠. We next calculate the P_min as a function of d_tol, assuming $EF≈−0.36$ eV (and the corresponding operational parameters in Table I), and plot it in Fig. 3(b). For vanishing toluene thickness, P_min is increased, i.e., higher optical power is needed to detect a thinner layer of the analyte. On the contrary, P_min plateaus after $dtol∼35$ nm. This is due to SPP fields extending only a few tens of nm away from the GNRs.^{17–21,54–57} Further increase in d_tol does not change the overall I_ph yield, highlighting the significance and major impact of surface-enhanced fields of GNR SPPs to directly and electrically probe the toluene's (or other analytes) absorption lines.

In conclusion, we have theoretically studied the use of GNRs in a SLG/Si Schottky junction configuration, to electrically detect graphene plasmons. Using the computational framework developed in Ref. 46, and realistic considerations for all material and device parameters involved, our calculations suggest an increased I_ph yield at the SPP resonant wavelengths. Optimized devices can reach external responsivity $≈110$ mA/W with a corresponding NEP $≈190$ pW/$Hz0.5, D*≈4×106$ Jones, and response time $τresp≈12$ ns, corresponding to a cutoff frequency of $≈0.5$ GHz. This platform for electrical detection of SLG plasmons can operate either as a mid-IR photodetector or as a chemical sensor for analyte detection via the SEIRA technique. We studied the operation of such a sensor and extracted quantitative relations for its detection and sensitivity limits. The methods and conclusions derived in this work are applicable throughout the IR, where SLG SPPs can be realized. Limitations in reaching shorter wavelengths may arise due to fabrication constraints in reducing w and reaching large E_F values, as well as material availability in utilizing a semiconducting substrates with smaller dielectric constants. Assuming $wmin≈20$ nm and $EF,max≈0.55$ eV for p-Si with $NA,Si=1017 cm−3$⁠, there is a lower bound at $λ≈8 μm$⁠. There is no limitation, however, for longer wavelengths, since w can always be increased/tuned accordingly, and the proposed scheme is based on thermionic emission, which is not limited by the substrate's bandgap. This work facilitates the theoretical background for a unique tunable multi-band optics-free photodetector and/or chemical sensor based on graphene plasmons.

See the supplementary material for details in FDTD simulations, Schottky junction formation, absorbed power and carrier cooling calculations, temporal response of the proposed device, and SNR dependence on input power.

We acknowledge funding from the EU Graphene Flagship, the Israel Science Foundation (Grant No. 1732/18), and the Israel Innovation Authority (Grant No. 63350).