Supporting Online Material for

Transcription

1 Supporting Online Material for Hot Carrier Assisted Intrinsic Photoresponse in Graphene Nathaniel M. Gabor, Justin C. W. Song, Qiong Ma, Nityan L. Nair, Thiti Taychatanapat, Kenji Watanabe, Takashi Taniguchi, Leonid S. Levitov, Pablo Jarillo-Herrero* *To whom correspondence should be addressed. Published 6 October 2011 on Science Express DOI: /science This PDF file includes: Materials and Methods SOM Text Figs. S1 to S4 References (31 38)

2 Materials and Methods Graphene dual-gated devices are fabricated using standard e-beam lithography techniques. Graphene is mechanically exfoliated onto degenerately doped silicon substrate with 285 nm layer of silicon oxide (31). Single and double layer flakes are identified by G-values obtained using high color contrast optical microscope images (32), and later confirmed by Raman spectroscopy. Chromium-gold electrodes (0.6 nm Cr, 60 nm Au) are deposited by electron beam evaporation to form electrical contacts to the graphene flakes. For the top-gate electrode of the MLG p-n junction device, we first mechanically exfoliate high crystallinequality boron nitride flakes to a poly(methyl methacrylate) (PMMA) substrate (33). High quality flakes (~ nm thick over a large coverage area) are isolated and subsequently transferred from the PMMA to cover the entire graphene device. Hexagonal boron nitride acts as an insulating layer over which a chromium-gold (0.6 nm Cr, nm Au) top-gate electrode is deposited using e-beam lithography and thermal evaporation. For the top-gated BLG devices, similar fabrication steps are used (32, 34). In place of boron nitride, however, aluminum oxide (Al 2 O 3 ) is used as a top-gate insulator (3 nm of Al 2 O 3 deposited by e-beam evaporator and 20 nm of Al 2 O 3 deposited using atomic layer deposition at 250C), followed by e-beam deposition of chromium-gold (1 nm Cr, 50 nm Au) top-gate electrodes. 2

3 SOM Text Scanning photocurrent microscopy Devices are placed in an optical scanning microscope setup (Fig. S1) that combines electronic transport measurements with spatially scanned laser (λ = 850 nm) illumination (3, 12). The collimated excitation beam is expanded (beam expander BE) to fill the back aperture of the microscope (MIC) objective, while a CCD camera is used to image and locate the device. A diffraction-limited beam spot is scanned (using a two axis piezo-controlled scanning mirror PSM) over the graphene device and the current is recorded to form a spatial map of photocurrent. The photocurrent image resolution is limited by the wavelength of light used. Reflected light from the sample is collected and the reflected intensity is monitored to form a simultaneous image of the device. The absolute location of the photo-induced signal is found by comparing the photocurrent map to the reflection image. In scanning photocurrent microscopy, heating of the substrate and contacts can heat up graphene (both lattice and carriers), however the resulting change of the electron temperature will be very small compared to direct heating due to photoexcitation (6). In our measurements, the wavelength of light (λ = 850 nm) was chosen to minimize light absorption in the thin gold top-gate electrode. The optical power that reaches the graphene will be reduced by reflection and absorption in the gold film (in the top-gated region), the hexagonal boron nitride film (in both regions), and at the silicon-silicon dioxide interface. In addition to 5% absorption in the gold film (35), the absorption of thin boron nitride (thickness ~ 20 nm) reduces the incident power in all regions by ~ 1-2% (36). The optical power absorbed in the graphene sheet is then ~2.3% of the remaining incident power (37). From the reflected intensity recorded in the reflection images, we 3

4 observe a reflected intensity contrast of ~ 6-8% between the gold top-gate contact and the boron nitride film, indicating that the incident power on the graphene membrane is similar (within 5%) across the entire area of the device. The devices are mounted in a Janis ST-500 helium optical cryostat (CRYO) that can be cooled to 4 K. Incident laser power is measured at the output of the microscope objective using a calibrated photo-detector. Electrical feed-throughs allow electronic transport measurements while the device is being illuminated at various temperatures. All measurements are taken in the linear optical power response regime. The photovoltage V PH is the product of the measured photocurrent I PH and the measured device resistance R, both of which depend on gate voltage. Thermoelectric transport measurements For thermoelectric measurements, we heat the gold contact far from the graphene-metal junction (marked by a triangle in Fig. S2A and 1D in main text) and measure the resulting thermocurrent. By heating the contact, we introduce an un-calibrated, but fixed, temperature ΔT C between the source electrode (labeled V SD in Fig. S2A) and the drain electrode (labeled I). While ΔT C is unknown, the thermovoltage V T measured across the entire device has gate voltage dependence originating from Seebeck coefficients. The thermovoltage will have a dominant gate voltage-dependent contribution from the Seebeck coefficient in the bottom-gated region S(V BG ) and a small contribution from the p-n interface. As shown below, the observed contribution due to the p-n interface is negligible, and so the thermovoltage can be written as V T = S(V BG )ΔT C (18-21). 4

5 In Fig. S2B, we show the laser-induced thermocurrent measured as a function of the electrode length far from the graphene-metal (G-M) contact on the V SD side. While the thermocurrent is large near the G-M contact (likely including photocurrent generation effects there), the current decays with distance, yet persists to the end of the electrode ~ 6 microns away. We fit the thermocurrent at the V SD contact with an exponential decay (red line) and compare this to the Gaussian fit of the photocurrent at the opposite contact I (blue line). The Gaussian fit gives the photocurrent imaging resolution with full width at half maximum ~ 1 micron, while the exponential decay results from the cooling length of laser-induced heating on the electronic contact. Importantly, while both contacts show a photocurrent signal near the graphene, only the V SD contact shows a thermocurrent signal far from the G-M contact, consistent with laserinduced heating leading to a thermovoltage V T across the device. For the measurements shown in Fig. 2C, we fix the laser at the location of the triangle and measure the thermocurrent and thermovoltage as a function of V BG and V TG. The thermovoltage as a function of V BG and V TG (Fig. S2C) exhibits strong dependence on V BG with little dependence on V TG, indicating that the thermoelectric voltage changes primarily with the charge density in the bottom-gated region. Using Fourier transform to extract the contributions of different spatial regions to photovoltage Fourier analysis is a convenient mathematical tool for analyzing signals that arise from mixing contributions with different spatial structure (as in tomographic imaging). Here we generalize these ideas to analyze the measured photovoltage dependence on the gate voltages V BG and V TG. According to the thermoelectric model, the photovoltage is a sum of contributions 5

6 arising from two spatial regions, which are gated separately (see Eq. (1) of the main text). Each contribution is a complex nonmonotonic function of the density on the corresponding region, giving rise to a 6-fold photovoltage pattern as a functon of V BG and V TG. However, each of the two contributions is a function of a single variable, the density in the corresponding region, n BG =a V BG and n TG =a V BG + a' V TG, where a and a' are capacitance coefficients. This property can be used to extract these contributions individually, as described below. We consider photovoltage V(V BG, V TG ), which is a linear sum of two not-yet-known functions V = f 1 ( n 1 v ) + f 2 ( n 2 v ) (S1) where v = (V BG, V TG ) is a vector in the 2-dimensional V BG vs. V TG coordinates. Each of the functions f 1 and f 2 is a simple scalar function that depends on two separate combinations of gate voltages, V BG and V TG, and varies along two directions, n 1 and n 2 in the 2D map. Taking the Fourier transform of the first term gives V 1 = d 2 v f 1 ( n 1 v )e i k v. This integral decomposes the function f into its constituent frequencies k. The vector v in f 1 ( n 1 v ) can be separated into components that are parallel v 1 and perpendicular v 1 to the direction n 1 along which the function varies: V 1 = d 2 v f 1 ( n 1 v )e i k v = dv 1 1 dv f 1 ( n 1 v 1 )e i k v 1 ik 1 v. (S3) The integral yields a delta function of the perpendicular component of k, which isolates the components of f 1 along only n 1 (S2) 6

7 V 1 = f 1 k 1 n 2πδ(1) ( k 1 v 1 ) 1 (S4) where f 1 is the Fourier transform of the function f 1. Using the linearity of Fourier transforms we can add the contributions of two regions to obtain the Fourier transform of photovoltage V V = f 1 k 1 2πδ (1) ( k 1 v 1 ) + 2 k f 2 2πδ (1) ( k 2 v 2 ) n 1 n 2 (S5) where f 1 and f 2 correspond to the two regions. The two terms are streaks in Fourier space pointing in different directions, which can be separately masked and analyzed to extract individual contributions f 1 and f 2. We can now apply this to the photovoltage in the graphene p-n junction. The PTE model of photoresponse predicts that the photovoltage can be written as V PH = ( S( n 1 V GATE ) S( n 2 V GATE ))ΔT (S6) where V GATE is the general gate-voltage dependence of S (Equation (2) in main text). Applying the Fourier transform to Eq. (S6), we have V PH S 1 k 2πδ (1) ( k 1 v 1 ) + 2 k S 2πδ (1) ( k 2 v 2 ). n 1 n 2 This suggests that the Fourier transform should appear as two streaks: one along n 1 and one along n 2. If we mask one of the streaks, we obtain the scalar function for the other streak. An inverse Fourier transform yields S scaled by a multiplicative pre-factor. We perform the Fourier analysis using the built-in Matlab utility. Indeed, two such streaks appear as shown in Fig. 3A in the main text. We mask the horizontal streak using a onedimensional Gaussian (with variance 2σ 2 = 1 / (0.07 V)) and then perform an inverse Fourier (S7) 7

8 transform, shifting the origin back to the sides (resulting in V PH [ ˆ N B ] of Fig. 3B). To isolate the diagonal streak, we next take the masked horizontal streak and subtract it from the total Fourier transform. This acts to remove the overlapping elements near the origin. We take a diagonal mask using the same procedure with a slightly larger variance (2σ 2 = 1 / (0.16 V)), resulting in the V PH [ ˆ N T ] shown in Fig. 3B. By adding the two inverse transforms, we regain the six-fold photovoltage pattern. As discussed in the main text and described by Eq. (S6), line cuts from the two inverse transforms show gate voltage-dependence consistent with the expected density dependence of the thermoelectric coefficient S. Optoelectronic characteristics of bilayer graphene In the bi-layer graphene (BLG) p-n-p device, electronic transport and photocurrent measurements show similar results to the MLG p-n junction (Fig. S3). Fig. S3A and S3B show a schematic and scanning photocurrent image (also shown in main text) of the top-gated BLG device. In Fig. S3C, we show the resistance R as a function of V BG and V TG for the BLG device. Similar to the MLG p-n junction, we observe two intersecting lines of maximum resistance that intersect at the charge neutrality point CNP (shown as dashed lines in the main text Fig. 4C). In addition to measuring photocurrent vs. V BG and V TG in the BLG device, we can also measure the spatial dependence of photocurrent as a function of V TG. For the data shown in Fig. S3D, we measure the photocurrent along the solid white line in Fig. S3B as a function of V TG. At V BG = 14.8 V, the bottom gated region is doped p-type, and we tune the top-gated region from n- type to p-type (shown schematically in Fig. S3E). Along the dashed line in Fig. S3D, photocurrent starts out negative, tends toward zero at the CNP, and then becomes negative again, 8

9 though with a smaller magnitude. This behavior re-iterates that the photocurrent behavior is not consistent with the photovoltaic (PV) effect. In the PV effect, the electric field, which scales as the difference in chemical potential between p- and n- regions, should change directions resulting in a single photocurrent sign change. Additionally, while the photocurrent at the n-p and p-n junctions changes significantly, the photocurrent at the contacts changes only slightly and follows the top gate-dependence of resistance R. Estimate of ΔT and the Seebeck coefficient Equation (1), ΔT = V PTE / (S 1 - S 2 ) allows us to estimate the electronic temperature ΔT. We first estimate the maximum Seebeck coefficient difference in the p-n junction, which is a figure of merit for the performance of our device. Using the phenomenological formula for conductivity in graphene (6) σ(µ) = σ MIN 1+ µ2, Δ 2 (S10) we obtain the maximum Seebeck coefficient difference as (S 1 S 2 ) MAX = π 2 3 k B 2 T eδ. (S11) At T = 40 K and using an estimate for the charge neutrality width Δ = 70 mev, we arrive at (S 1 - S 2 ) MAX = 24 µv/k where (S 1 - S 2 ) MAX is twice the Seebeck coefficient ~12 µv/k. Thermoelectric measurements on graphene give a similar value of S ~ 10 µv/k at T= 40 K (20). This is larger than the Seebeck coefficients of other metals at higher temperatures, such as gold, copper, palladium, and silver, all of which have Seebeck coefficients below 5 µv/k at room temperature 9

10 (38). Comparing the estimated Seebeck coefficient in graphene with the photovoltage in Fig. 2A and 2B we obtain the electronic temperature ΔT = 4.5 K. Cooling length To extract the cooling length resulting from the laser heating the graphene sample we consider the heat transport equation (see (6, 4) from main text): (κ T el ) + γc el T el = J LASER (S8) Where κ is the thermal conductivity of the sample, T el is the temperature of the electron subsystem, γ is the electron-lattice cooling rate, C el is the heat capacity of the electron subsystem and J LASER is the heat flux into the electron subsystem. The solutions of Equation (S8) can be found using the Green s function method. The finite system size Green s function for this is given by G(y > y') = ξ κ sinh( ( y L /2) /ξ)sinh y'+l /2 sinh L /ξ ( ) ( ) /ξ ( ) G(y < y') = ξ κ sinh( ( y + L /2) /ξ)sinh y' L /2 sinh L /ξ ( ) ( ) /ξ ( ) (S9) where y is the distance from the position of the laser, L is the full length of the device, and ξ = κ /γc el is the cooling length (6). We can obtain the temperature profile for a given cooling length, ξ, by taking the laser spot as a Gaussian and convoluting with the Green s function. We show in Fig. S4 various fits of the temperature profile obtained and can discern that ξ = 2, 3, all fit the measured curve very well. 10

11 When cooling lengths approach L/2 (or larger than L/2), the Green s functions approach a simple piece-wise linear expression G(y,y =0) = (1 2 y /L)ΔT, where the constant ΔT can be determined from direct substitution in Equation (S8) and produces Equation (3) of the main text. For excitation fixed at the p-n interface (y = 0), the differential temperature profile is then δt e (y,y =0) = (1-2 y /L)ΔT. This triangular form is clearly exhibited in Fig. 1D. Hence, we are limited by the device dimensions and can only establish a lower bound for the cooling length. Some asymmetry is present in the photocurrent line profile (Fig. 1E), likely resulting from the additional small photocurrent generated at the contacts and small differences between the net absorbed power in the top-gated region and bottom-gated region. As shown in the fits of cooling length (Fig. 1E and Fig. S4), we establish a lower bound for the cooling length of ξ > 2 µm. Using this estimate of cooling length, we can establish a similar lower bound for cooling time given by τ ~C el ξ 2 /κ. Plugging in the Einstein relation for the heat capacity, C el = (π 2 k 2 B T /3e)D(E F ), where D(E F ) is the density of states, the Wiedemann-Franz relation K = π 2 k 2 B T/3e 2 (1/R), and using a typical resistance of the device read from Fig. 1B of about 2 KΩ, we obtain cooling times of order 100 ps, which is consistent with recent measurements of slow cooling (23). Gate dependence of the ratio V PH [N B ]/V T ΔT Since the fits of the temperature profile indicate that the cooling length ξ L/2, by setting the second term in Equation (S8) to zero, we approximate the temperature at the p-n interface by ΔT = J LASER / (K 1 + K 2 ), where K is the thermal conductance. However, if ΔT is dominated by the temperature of the electron subsystem we can use the Wiedemann-Franz 11

12 relation to link the thermal conductance with the device resistance, K = π 2 k 2 B T/3e 2 (1/R). Since the device resistance can be controlled via the top and bottom gates, a gate dependent ΔT provides evidence for an electronic ΔT. Conversely, a gate independent ΔT would show that it is the lattice temperature that dominates ΔT of Equation (1) from the main text. We are able to extract this dependence by analyzing the photovoltage. While the photovoltage already depends on a gate dependent quantity, S, we know the thermopower of the device from measurements indicated in Fig. 2C, up to a constant un-calibrated temperature ΔT C, and we can extract ΔT by taking the appropriate ratios described below. We take the thermovoltage trace along the line where the carrier densities in the two regions are the same. This occurs along the straight (white dashed) line defined by the node of zero photovoltage shown in Fig. 2A. This ensures that the thermopower depends on the product of ΔT C and S of a single carrier density. Furthermore, to eliminate complications arising from having 2 regions of resistance and Seebeck coefficient (arising from having both a top and bottom gate), we can use the photovoltage obtained after Fourier analysis (described above) and shown in Fig. 2B, described by V PH = SΔT. Taking the ratio V PH / V T along the trace indicated above eliminates any gate dependence that one might expect from the Seebeck coefficient leaving only the gate dependence of ΔT. Using the Wiedeman-Franz relation and Equation (3), we can see that ΔT should exhibit the same gate dependence as resistance. This is shown strikingly in Fig. 3C. Moreover, the peak of ΔT at the CNP can be understood from the vanishing density of states at the CNP, which suppresses the electronic heat capacity and hence requires a larger ΔT given the constant flux of heat into the electron system produced by the laser. 12

13 Fig. S1 Optoelectronic setup for measuring photocurrent in the graphene p-n junction. Components are labeled in text, except for simple optical elements such as mirrors and beamsplitters. 13

14 Fig. S2 Laser-induced heating of the electrode and thermoelectric measurements in the graphene p-n junction. (A) Spatially resolved photocurrent map at T = 40 K with continuous wave laser excitation λ = 850 nm and optical power = 200 µw for the same device (V BG =-5 V, V TG = 2 V, V SD = 0 V) as in main text Fig. 1. White lines indicate the location of the gold contact electrodes. Dashed white lines show edges of graphene. Scale bar 5 microns. (B) Thermocurrent measured as a function of distance along the black dashed line in (A) at optical power = 1 mw. The blue line is a Gaussian fit to the photocurrent signal measured at the electrode labeled I (blue dashed line) while the red line is an exponential fit to the thermocurrent at the electrode labeled V SD. The triangles in (A) and (B) mark the location of the laser when measuring the thermocurrent in Fig. 2C of main text. (C) Thermovoltage V T as a function of V BG and V TG. 14

15 Fig. S3 Electronic transport and photocurrent characteristics of the bi-layer graphene p-np device. (A) Experimental schematic of the top-gated p-n-p junction composed of bi-layer graphene (BLG) and incorporating aluminum oxide top gate dielectric. (B) Spatially resolved photocurrent map at T = 40 K with continuous wave laser excitation λ = 850 nm and optical power = 200 µw for the same device (VBG =15 V, VTG = 10 V, VSD = 0 V). Thin white lines indicate the location of the gold contact electrodes. Scale bar 4 microns. (C) Electronic resistance measured as a function of VBG and VTG at VSD = 1.4 mv and T = 40 K. The four regions are labeled according to the type of carrier doping, p- type or n-type, in the bottom gated and top gated regions, respectively. (D) Photocurrent line scan taken at solid white line of (B) as a function of VTG with VBG = 14.8 V and optical power = 200 µw. (E) Schematic of bi-layer graphene's band structure in the p-n-p junction showing hyperbolic electron (blue) and hole bands (red) that touch at the charge neutrality point. Dotted lines represent the electron Fermi energy in the p- and n- type regions of the device. 15

16 Fig. S4 Extracting the cooling length of hot carriers in the graphene p-n junction. Photocurrent line profile of the p-n configuration from Fig. 1 of the main text. Red (blue) line is the fit with ξ = 3 (ξ = ). The lower bound of the cooling length, ξ = 2, is shown in the main text. 16

17 Supporting references and notes 31. K. S. Novoselov, et al. Proceedings of the National Academies of Science 102, (2005). 32. T. Taychatanapat, P. Jarillo-Herrero. Physical Review Letters 105, (2010). 33. C. R. Dean, et al. Nature Nanotechnology 5, (2010). 34. S. Kim, et al. Applied Physics Letters 94, (2009). 35. S. A. Kovalenko. Semiconductor Physics, Quantum Electronics, and Optoelectronics 3, 514 (2000). 36. L. Ci, et al. Nature Materials 9, 430 (2010). 37. R. R. Nair, et al. Science 320, 1308 (2008). 38. N. Cusack, and P. Kendall. Proceedings of the Physical Society 72, 898 (1958). 17