Near-field heat transfer measurement

The monolayer graphene used in the work is a commercial product made by chemical vapor phase deposition (CVD) (7440-44-0, XFNANO Mater Tech Co., Ltd, China) originally grown on a copper foil. Standard technique has been employed to transfer the macroscopic Gr sheets onto silicon substrates which are adhered together by the Van der Waals force34. The sample is cut into 2×2 cm squares for experiment. All these processes including the measurement are conducted in a cleaning environment. The details are addressed in the Methods section. Silicon hosting substrates are employed here primarily for three reasons. First, their controllable infrared properties through doping can be tailored to identify the Gr plasmon’s role in the near-field heat transfer. Second, silicon wafer with an ultra-small bow is achievable (in this work, a 500 °C post-annealing in vacuum is carried out to release the cutting-caused wafer bending)35. And lastly, the Gr–Si heterostructure may be a potential assembly for the important TPV or photothermoelectric applications. They have a roughness about 1–2 nm and a maximum bow value less than 20 nm determined by a Laser interferometer (ZYGO OMP-035/M) with an area of 2×2 cm. Figure 1a sketches our home-made measurement setup for the near-field thermal radiation. The two graphene layers are separated by a vacuum gap (experimentally with a pressure <5 × 10−6 Torr) assisted by four AZ photoresist (PR) posts (diameter = 50 μm, height = 500 nm, edge distance = 0.15 mm and conductivity = 0.18 W m−1 K−1). The PR posts have good physical stability below 150 °C and their height decides the gap value. In our setup, the top Gr/Si assembly (temperature T e ) behaves as the emitter sourced by a same sized heating plate through a copper spreader and the bottom Gr/Si assembly (temperature T r ) is the receiver sunk by a temperature electric controller (TEC) (1-12705, Realplay, China) through another copper spreader. Silicone grease is used to improve the boundary thermal contact between the stacks. A 200-nm thick titanium film has been deposited on the top surface of the silicon substrate in order to block the photons emitted from the grease layer. Two temperature sensors are inserted into the center of the top and bottom copper spreaders through the drilled slots to monitor T e and T r . A pair of magnets consisting of a 5×5×3 cm3 NdFeB hard magnet (H-magnet, bottom) and a 2×2×1 cm3 iron soft magnet (S-magnet, top) are used to strengthen the contact and mechanical stability of the system by forming a clamping force of about 10 N. The exposed surface of the top half unit is coated by a 200-nm thick aluminum film to minimize the background radiation. In the measurement, we have assumed the Gr/Si assembly has the same temperature with the copper spreader since the contacting thermal resistance is negligibly small compared with the radiation one. The Gr/Si/Cu triplets at the receiver side are thermally maintained at a stable temperature around 30 ± 0.30 °C (equal to the environment temperature T b ) assisted by the TEC cell. The maximum temperature difference (ΔT = T e − T r ) we obtain is ~50 °C at a vacuum gap around 500 nm and ~70 °C as the gap is larger than 1 µm.

Fig. 1 Measurement setup. a A schematic of the home-made measurement setup. From top to bottom, the emitter side consists of S-magnet, heater, copper spreader, and Si/Gr and the receiver side consisting of Gr/Si, copper spreader, TEC layer, copper spreader, and H-magnet. Four photoresist posts are used to separate the emitter and the receiver. b The equivalent thermal circuit of our measurement setup. The input power P in is split into the background branch with power P b and the near-field radiation with power P r . c Schematic of the background heat loss measurement. The receiver is maintained at 30 °C and displaced from the emitter at a macroscopic vacuum distance of 2 mm. The input power (heat) will be dissipated along the metal wires and via radiation. d The relation of input power with the temperature increment of the emitter. The experimental data points are denoted by the dots and the solid/dashed line represents the linear/non-linear fitting, from which we obtain the background heat loss P b (ΔT). The error bar for the temperature is 2.3 °C, derived from the five times repeating measurement. Figure 1 by Yang J et al. Full size image

As sketched in Fig. 1b, the input electric power P in after subtracted by the resistive heating loss of electrode wires is majorly dissipated through two channels: one we called is the background power loss (P b ) including one part conducted along the electrode wires of source and temperature sensors and the other part radiated toward the surrounding space and the second is the net power (P r ) radiated via the vacuum gap. As the total input power is known, it is crucial to correctly evaluate the background transfer branch P b , which in our case is done in a separate experiment measuring the temperature rising in a far-field configuration. As schematically shown in Fig. 1c, for the background measurement, the top half of the device is displaced from the bottom one at a gap of ~2 mm supported by the four rigid metal wires (belonging to the source and sensors). The other ends of these wires are screw fixed with the bottom steel sink platform. By this structure, P b at different T e is estimated through a simple relation: P b (T e ) = P in (T e ) − P r Far(T e ), where P r Far = εσ(T e 4 − T b 4) (ε-emissivity and σ-coefficient) is the power emitted via the Gr/Si surface. As T e is relatively small, the radiation via other Al-film coated surfaces is neglected, i.e., assuming that P b is primarily caused by the heat flowing out along the electrode wires. Figure 1d plots the measured temperature rising curve as a function of the input power. The experimental data could be well fitted by a binomial expression P in (ΔT) = a × ΔT + b × ΔT4 with the fitting coefficients a = 0.014 W K−1 and b = 5.2 × 10−9 W K−4. Note here we use ΔT = T e − T e 0 and T e 0 = T b = 30 °C. When ΔT < 40 °C, the temperature is linearly dependent on the input power, indicating that the background radiation could be neglected at small temperature changes. Thereafter, in the following, the near-field radiated heat is evaluated by the equation P r = P in (ΔT)−a × ΔT − P AZ (ΔT), where P AZ is the heat conducted via the four posts.

Graphene on intrinsic insulating silicon

Our first experimental sample utilizes an insulating silicon (i-Si) substrate (resistivity > 20,000 ω cm) without doping so that the substrate contribution to the near-field heat flux could be minimized and the role of the Gr plasmons can be more clearly examined. The maximum temperature change for the emitter is less than 100 °C. The variation of infrared properties due to temperature change for both silicon and graphene can be practically neglected. Figure 2a gives a typical three-dimensional (3D) atomic force microscopy (AFM) morphology picture of the Gr–Si heterostructure in an area of 20×20 μm2. Statistical measurement shows the surface roughness of our sample is less than ~50 nm, which is relatively small when considering the multiple physical and chemical processes involved in the sample preparation. As a consequence, in the experiment, we set the height of the photoresist posts or gap size at values larger than 400 nm. Figure 2b gives the Raman spectrum of the Gr–Si heterostructure. The typical peaks from the D-mode and G-mode resonances are observed at 1149.98 and 1591.46 cm−1, respectively. Their small peak intensity ratio indicates the high quality of the CVD graphene36. According to the G-mode position, we can estimate the Fermi level of graphene at E F = 0.27 eV due to the electron doping from the absorbed molecules, which is close to the previous reported value (0.32 eV) also for a CVD graphene37. In addition, the strong single 2D peak identifies the dominant monolayer nature of our graphene sample38. Figure 2c gives the absorption spectrum of the Gr–Si heterostructure measured by a Fourier transform infrared (FTIR) spectrometer, where T Gr and T i-Si denote the transmittance with and without graphene, respectively. There is a broad peak at 4832.8 cm−1, associated with doping inhomogeneity, temperature effect and other broadening mechanisms. The spectrum has no obvious change as we raise the temperature from 30 to 70 °C, indicating the thermal stability of the heterostructure. The middle-IR broad absorption curve can be phenomenologically fitted by a Gaussian step function37. From the full width at half-maximum (FWHM), we can estimate the “2E F onset” of the inter-band transition, as indicated by the arrow in Fig. 2c, which corresponds to a Fermi level of 0.27 ± 0.03 eV, highly consistent with that estimated from the Raman spectrum. The gap distance d of the silicon substrates is mainly decided by the height of the photoresist posts. The real value may have a certain distribution due to the bow of wafer or roughness, etc. As shown in Fig. 2d, we use the measured FTIR spectra (dashed lines) and the theoretical ones (solid lines) to evaluate the gap value. The red and blue lines calculated at different d represent the edges of the experimental lines measured at various spatial points. Figure 2e shows there is a good linear correspondence between the measured and the target gap distances when it is within few micron meters. The uncertainty is about 25 nm at a central distance of 430 nm and gets larger when the distance increases. The measured values have been used in the following numerical calculations for the heat transfer efficiency.

Fig. 2 Characterization of i-Si/Gr properties. a A typical 3D AFM picture of the i-Si/Gr sample. The largest roughness of the sample is about 50 nm. b The Raman spectrum of the i-Si/Gr sample. The monolayer feature of graphene is identified by the strong 2D resonance peak. c The absorption spectrum of graphene on intrinsic silicon. The red dots are measured results by FTIR and the blue line is a Gaussian function fitting. T Gr and T Si denote the transmittance with and without the graphene cover, respectively. The “2E F onset” of the inter-band transition is estimated at about 4832.8 cm−1, corresponding a Fermi level at E F = 0.27 eV. d Measured (dashed lines) and calculated (solid lines) FTIR spectra at three different target gap sizes of 3.7, 1.4, and 0.43 μm. The uncertainty of gap size is considered here so that the calculated lines (red and blue) could cover the experimental curves measured at various spatial points. The statistic value for the smallest gap we designed is about 430 ± 25 nm. e Correspondence between the measured and the target gap sizes. The gap uncertainty is shown by the error bar derived from the measurements at multiple points. The black curve is a guidance line for eye, which shows nearly a linear relationship when the gap value is within few micron meters. Figure 2 by Yang J et al. Full size image

Figure 3a gives the measured heat flux density h varied as a function of temperature difference ΔT for the samples with and without graphene. The measured gap size is 430 ± 25 nm and the receiver’s temperature is controlled at T r = 30 °C. For the bare Si–Si assembly, the measured heat flux density (blue dot) fluctuates below the line (black) representing the Blackbody radiation. Although the intrinsic silicon has an ultra-small imaginary permittivity (~10−4 derived from the FTIR absorption measurement), the thick substrate will still produce certain amount of thermal radiation and this effect will be enlarged in the near-field case with the involvement of evanescent wave tunneling23. The heat flux from the naturally oxidized ultrathin SiO 2 layer (typically ~1 nm39) on the silicon surface is negligibly small. The blue dashed line in Fig. 3a is the analytically calculated heat flux with the input of measured material parameters, which overall has a good agreement with the experimental data. The heat flux is substantially enhanced when covering the substrate with a graphene layer as shown (red triangles) in Fig. 3a. At ΔT = 36 °C, the heterostructure gives rise to a strong flux density at h = 1195 W m−2, nearly 4.5 times larger than the Blackbody radiation (h = 268 W m−2).The total radiation heat flow at this case is about 0.48 W for our sample, which is nearly four times larger than that (P AZ ~ 0.12 W) conducted along the four supporting posts. The red shaded region profiles the theoretical heat flux distribution calculated by considering the fluctuations: d = 430 ± 25 nm and E F = 0.27 ± 0.03 eV. The collision time factor for a CVD graphene used here is τ Gr = 100 fs40. The experiment data is generally well reproduced by the theory. Here we may overlook the radiation contribution from the surface residuals but they do have indirect influences by affecting the Fermi level of graphene and their near-field coupling strength.

Fig. 3 Heat flux density and transmission possibility between i-Si/Gr pair. a The measured heat flux density (symbols) compared with the analytical predictions (shaded region and dashed line) at different emitter temperature variation between i-Si substrates with or without graphene. The gap distance for both case is 230 nm. The blue dots are measured points for the bare i-Si pair, while the blue dashed line is the numerical prediction calculated using the measured material parameters. The red triangles are measured points between Gr-covered i-Si substrates, while the red shaded region denotes the corresponding theoretical heat flux density considering E F = 0.27 ± 0.03 eV and d = 430 ± 25 nm. The temperature uncertainty is 2 °C derived from the five times repeating measurement. The black line denotes the flux density of Blackbody radiation. b, c p-polarization transmission possibility τ p (ω, k || ) between bare i-Si pair and i-Si/Gr pair, respectively. The interference strip patterns in the transmission map within the free-space light cone is caused by the FP resonance in an ultra-low loss planar stack system. d The mode pattern denoted by the magnetic component between the i-Si/Gr pair at (2.5 × 1013 rad s−1, 2.3 × 104 rad cm−1) and d = 500 nm. Figure 3 by Yang J et al. Full size image

To have a deeper understanding about the physical picture, Fig. 3b, c plot the calculated p-polarization photon transmission probability τ p (ω, k || ) across the vacuum gap without and with graphene at ΔT = 50 °C and d = 500 nm, respectively41. The metallic ground is described by a Drude model with the data input from ref. 42. Usually, the p-polarized contribution is dominant in the near-field flux43. Without graphene, only traveling waves inside the free-space light cone (from y-axis to the line ω = ck) has a high transmissivity, in particular for those near the edge of the light cone that have longer light paths in the substrate. The inhomogeneous transmission pattern reflects the Fabry–Perot (FP) resonance features of our low loss planar system. The tunneling possibility for evanescent waves inside the silicon light cone (defined by the line ω = v/k with v the light velocity in silicon) is rather small, which will become obvious only using much thicker substrates or at smaller gaps (<100 nm). When a graphene layer is introduced, as shown in Fig. 3c, the transmission feature for traveling waves has no obvious change, which accords to the nature of graphene that has low far and mid-IR response due to Pauli block44. In addition, there appears a new and p-polarized evanescent wave band below the silicon light cone arising from the near-field EM coupling in an asymmetric super-mode pattern (Fig. 3d). These bounded surface modes carry high density of state of photons and can largely raise the evanescent wave tunneling and thus overall thermal radiation efficiency, as found in the measurement. This effect will be more obvious at smaller gaps or replacing Si with a low-index substrate that will enhance the bandwidth and intensity of the bounded modes45.

Graphene on doped silicon

In the second experiment, we will repeat the experiment using doped (~1018 cm−3) n-type silicon substrates that can strongly modify the dielectric properties of the background and grapheme layers. Especially, the doped silicon (d-Si)/Gr heterostructure is interesting for the potential applications in TPV cells or thermophotoelectric sensors due to the formation of a Schottky junction at their interface33,46, which will modulate the infrared response of graphene as we observed before31. The dielectric properties of the heterostructure are derived from the measured FTIR spectrum. Further information is given in the Methods section. Figure 4a plots the measured reflectance curves of the d-Si substrate with (shifted upward by 0.05) and without grapheme at different substrate temperatures. Note in this case the substrate is totally opaque in the IR region and has no transmission. The reflectance of both samples has almost no variation when the temperature varies in 30–70 °C, implying the stability of charge densities in both substrate and graphene layer under the agitation of moderate temperature changes. This result is expected for a highly doped semiconductor47. Then, a Drude model fitting (solid line in Fig. 4a) is applied here to retrieving the dielectric constant of the substrate. The results are plotted in Fig. 4b. Our doped silicon substrate will have a metallic response below the plasmon frequency of 5.6 × 1014 rad s−1 (3.3-μm wavelength) accompanied with a dramatic change on both real and imaginary permittivity. Figure 4c gives the real (blue dots) and imaginary (red squares) parts of the spectral sheet conductivity of graphene directly retrieved from the measured reflectance with the parameters of silicon input from Fig. 4b48. From terahertz to far-IR, the monolayer graphene will also electromagnetically behave as an optical metal and can be described by a Drude model conductivity resulted from the intra-band electron scattering. In Fig. 4c with the shaded regions, the measured conductivity data is well fitted by the Drude model with a slight fluctuation of Fermi level at E F = 0.15 ± 0.03 eV. The E F value on the d-Si substrate is nearly half of that (0.27 eV) of the intrinsic silicon, indicating the occurrence of electron transfer from the d-Si substrate to the p-type graphene (a general property for a CVD-grew graphene) assisted by the Schottky heterojunction at the Gr–Si interface37,49.

Fig. 4 Characterization of d-Si/Gr properties. a The measured reflection spectra of the d-Si and d-Si/Gr samples at different temperatures. We have intentionally shifted up the value of d-Si/Gr by +0.05 for better vision. The black solid/dashed line is the fitting curve by considering a Drude conductivity model for both Si and graphene in the far-mid-IR range. b The retrieved real (blue) and imaginary (red) parts of permittivity for the doped silicon substrate. c The retrieved real (blue dots) and imaginary (red squares) parts of sheet conductivity for the graphene layer. The shaded regions represent the theoretical conductivity distribution calculated using E F = 0.15 ± 0.03 eV. Figure 4 by Yang J et al. Full size image

Figure 5a plots the measured heat flux density (symbols) between two d-Si substrates with and without graphene at various temperature differences. The receiver side is maintained at T r = 30 °C. For both samples, we design two different gaps at d = 430 and 1150 nm. Their measured uncertainties are ±25 nm and ±50 nm, respectively. First, a large heat flux density exceeding the Blackbody radiation is experimentally observed between the bare silicon substrates with a 1150 nm gap, indicating the participation of thermally excited evanescent waves. The heat flux density is substantially strengthened when the gap size is reduced to 430 nm. These results agree with the theoretical calculations quite well by considering the fluctuation of gap. Similar super-Planckian near-field thermal radiation between doped silicon wafers was also recently reported by Watjen et al.50. Introducing a graphene layer will bring no obvious change to the radiation efficiency at a large gap (1150 nm). The increment becomes obvious when the gap is shrunk to 430 nm. At an example point of ΔT = 37 °C, the heat flux density increases about 11% after introducing graphene. The measured results are consistent with the analytical predictions (shaded regions) calculated using the measured material and structural parameters. In addition, it is noted that with the same configurations, the d-Si/Gr heterostructure pair will give a much larger heat flux than the i-Si/Gr pair, primarily due to the loss-induced additional channel for the generation and transmission of thermal photons using doped silicon.

Fig. 5 Heat flux density and transmission possibility between d-Si/Gr pair. a The measured heat flux density (symbols) compared with the analytical predictions (shaded regions) at different temperature difference (T r is fixed at 30 °C). The solid circles and triangles represent the measured heat flux density between a pair of bare d-Si and d-Si/Gr, respectively, at the gap d = 1150 nm. These results can be well reproduced by the theory (solid royal and pink lines) using the measured material parameters. The empty circles and triangles represent the heat flux density measured at the gap of d = 430 nm for bare d-Si and d-Si/Gr pairs, respectively. The corresponding blue/red shaded region profiles the theoretical flux density distribution when considering the parametric fluctuations: E F = 0.15 ± 0.03 eV and d = 430 ± 25 nm. The black line denotes the flux density of Blackbody radiation. The temperature error bar is 2 °C derived from the five times repeating measurement. b, c p-polarization transmission possibility τ p (ω, k || ) between bare d-Si pair and d-Si/Gr pair, respectively. d, e The inter-layer mode patterns using magnetic component between bare d-Si pair and d-Si/Gr pair at (1.23 × 104 rad cm−1, 3.02 × 1013 rad s−1) and (1.94 × 104 rad cm−1, 3.02 × 1013 rad s−1), respectively. Figure 5 by Yang J et al. Full size image

Figure 5b, c gives the p-polarization transmission probability τ p (ω, k || ) between the d-Si pairs without and with graphene at ΔT = 50 °C and d = 500 nm, respectively. For the bare d-Si pair, compared with that shown in Fig. 3b for the i-Si pair, the transmission map becomes homogenous due to the disappearance of FP resonance and the evanescent portion (inside the silicon light cone but outside the free-space light cone) grows largely. The density of thermal source significantly increases in the lossy silicon substrates, which greatly raises the transmission possibility for both traveling and evanescent waves. As a result, it leads to the exceeding Blackbody’s heat flux. In addition, we also observe a low-frequency evanescent wave transmission band right below the line ω = v/k, arising from the plasmonic near-field coupling between the silicon (negative ε in this region) surfaces. A typical asymmetric field pattern for this super-mode is shown in Fig. 5d by magnetic field. With a top graphene layer, this band will be extended in the wavenumber domain. Compared with that on the i-Si substrate, the asymmetrical super-mode nature of the evanescent wave band does not change (Fig. 5e) but the dispersion diagram shifts to lower frequencies and in this case the inter-layer coupling will be weakened in particular at large gaps (e.g., d > 100 nm)45. On the other hand, the momentum and mode pattern matches will improve the near-field coupling of plasmonic polaritons between graphene and silicon substrate. As a consequence, more evanescent waves can be transferred to strengthen the heat flux, as shown in Fig. 5a.

In the above, the role of graphene plasmon polaritons in enhancing the near-field coupling51 and heat transfer efficiency has been experimentally observed at different substrates. The results not only validate the continuum theory of fluctuating electromagnetics in dealing with the single layer material, but also provide an indirect evidence about the existence of this highly bounded surface modes and their tunability in our macroscopic characterization. For this aim, it is crucial to have high measurement accuracy and good quality sample. The former is achieved by our custom-made setup as evidenced by the high agreement of the measured and calculated heat flux densities between the bare d-Si substrates (Fig. 5a). In the experiment, we have taken into account the influence of all the possible background heat loss, in particular including the heat conducted along the electrode wires which previously was not mentioned. For the sample quality, the CVD method is believed able to produce large-scale and high-quality graphene sheets. The transfer process needs to be expertly realized to avoid further damage or contamination. For a gap-size dependent heat flux characterization, the microsphere-plate measurement scheme will be more executable, e.g., by coating a SiO 2 microsphere with a graphene micro-disk.

It needs to be pointed out that silicon may not be the best choice to host grapheme if one wants to maximize the heat flux and for this aim polar materials such as silica or BN may be used where the occurrence of plasmon–phonon hybridization can further strengthen the tunneling of evanescent waves20,45. However, the Si/Gr heterostructure proposed here can provide a promising framework to harvest thermal energy. A normal TPV cell needs layers of infrared semiconductors (such as InSb or InAs) with an interface p–n junction in order to excite and separate electron–hole charges52. From the material perspective, the Gr–Si heterostructure with the existence of Schottky junction could be more realistic compared with the p–n junction which for far-IR optics is not readily available yet. The tunability of Fermi level of graphene will also help to modulate the barrier height and improve the utilization of thermal photons, which becomes critical when the working temperature is relatively low. Direct integration with semiconductors is also advantageous for other device applications such as thermal rectification53 or thermoelectric sensor54,55. In addition, high level chemical doping using species like HNO 3 may be employed to further tune the Fermi level of graphene with larger modulation depth56 and the number of graphene layers may be optimized to strengthen the near-field coupling and further enhance the heat flux. These methods need more and deep investigations in future.

Si/Gr-based near-field TPV cell

In the following, we give a theoretical picture about the performance of the Si/Gr heterostructure based near-field TPV cell. As shown in Fig. 6a, the emitter and the cell (receiver) have the same structural configuration with temperatures of T 1 and T 2 (=30 °C), respectively. The TPV cell is specifically biased to have a potential difference V b in order to maximize the output power. A typical interface band diagram for the graphene-covered n-type silicon is drawn in Fig. 6b. Here, W Gr is the work function of graphene and \(\Phi_{\mathrm{b}}\) is the energy height of the Schottky barrier. χ is the electron affinity in silicon. E c (E v ) and \(E_{\mathrm{F}}^{\mathrm{Si}}\) are the edge of the conduction (valence) band and Fermi level of silicon, respectively. \({{\Phi }}_{\mathrm{b}}\) \(\left( { = W_{G{\mathrm{r}}} - \chi } \right)\) is dependent on the Fermi level of graphene and can be tuned by charge doping through the bias voltage. In our discussion, we assume a reasonable barrier value \({{\Phi }}_{\mathrm{b}} = 0.15\,{\mathrm{eV}}\). It means thermal photons with energy larger than this threshold absorbed by the graphene layer are able to cross the barrier and contribute to the photocurrent as indicated in Fig. 6c by the emission of the photons in the shaded region where \(\hbar \omega > \hbar \omega _0 \equiv {{\Phi }}_{\mathrm{b}}\). Here, the radiation power density spectra of both emitter and receiver are calculated at T 1 = 700 °C and d = 50 nm. The discontinuity of the curve for the cell side is caused by the excitation of non-thermal photons arising from the potential difference57, with more details addressed in the Methods section. Under the current device parameters, the main part of absorbed thermal photons are attenuated into heat. Smaller gaps or higher temperature will help to increase the efficiency of photon utilization but at the cost of practical challenge24,25. In Fig. 6d, we compare the total radiated power P rad received by the cell and the portion P Gr absorbed solely by the graphene layer on both insulating and doped silicon substrates as ΔT ranges from 100 to 700 °C. At this small gap (50 nm), lower loss for the substrate will be more favorable to strengthen the heat flux as the coupling of the graphene layers will be stronger. The thermal photons from the emitter are dominantly absorbed by the graphene layer, which is nearly affected by the loss degree of the substrate. In Fig. 6c, d, we plot the electric power density P PV and efficiency η ( = P PV /P rad ) of two Si/Gr TPV cells at different silicon doping. As a comparison, we also give the result of an InSb (band gap = 0.17 eV57) based semiconductor p–n junction at the same gap (50 nm). For both cases, we assume an ideal photocurrent responsivity of 100% (i.e., every absorbed photon satisfying \(\hbar \omega > {{\Phi }}_{\mathrm{b}}\) will generate one electron–hole pair) in order to examine the limit of the proposed devices. The power density of the Si–Gr junction can be a few times or even one order larger than the semiconductor dependent on the doping of silicon substrate. On the other side, the semiconductor junction has a larger power efficiency due to its relatively low absorption of undesired thermal photons at \(\hbar \omega < {{\Phi }}_{\mathrm{b}}\). However, for the Si–Gr junction, the power efficiency could be traded off with the cell power density by tuning the substrate doping degree. The results show that in the ideal case the Si–Gr junction has the cost and power density advantages competed with the conventional semiconductor scheme for TPV cells.