Hot-Carrier Seebeck Effect: Diffusion and Remote Detection of Hot Carriers in Graphene

We investigate hot carrier propagation across graphene using an electrical nonlocal injection/detection method. The device consists of a monolayer graphene flake contacted by multiple metal leads. Using two remote leads for electrical heating, we generate a carrier temperature gradient that results in a measurable thermoelectric voltage VNL across the remaining (detector) leads. Due to the nonlocal character of the measurement, VNL is exclusively due to the Seebeck effect. Remarkably, a departure from the ordinary relationship between Joule power P and VNL, VNL ~ P, becomes readily apparent at low temperatures, representing a fingerprint of hot-carrier dominated thermoelectricity. By studying VNL as a function of bias, we directly determine the carrier temperature and the characteristic cooling length for hot-carrier propagation, which are key parameters for a variety of new applications that rely on hot-carrier transport.

determine the carrier temperature and the characteristic cooling length for hot-carrier propagation, which are key parameters for a variety of new applications that rely on hot-carrier transport.
Understanding the heating, energy flow and relaxation of two-dimensional carriers in graphene is essential for the design of graphene electronic devices. 1 In contrast to conventional metals with large Fermi surfaces, thermal decoupling of electrons from the crystal lattice leads to very slow electron-lattice cooling rates. [2][3][4] Electrons can be easily pushed out of thermal equilibrium with the lattice, even under weak electrical driving, and the generated hot carriers can propagate over extended distances. [5][6][7] Remarkably, such hot-carrier transport regime can occur at room temperature, resulting in novel thermoelectric and optoelectronic phenomena. 6 In combination with its large mobility and fast electrical response, these phenomena make graphene a material with great potential for a variety of applications, including bolometry, calorimetry and THz detectors. [8][9][10] The inefficiency of carrier cooling originates from the intrinsic properties of graphene.
Because of the large optical phonon energy ℏΩ ~ 200 meV, the most efficient mechanism available for hot-carrier cooling at low energies is the emission of acoustic phonons. [2][3][4] However, a small Fermi surface and momentum conservation severely restrict the acoustic phonons that can scatter off electrons. This leads to the observation of unconventional high-order cooling pathways assisted by disorder ("supercollisions", SC), which become dominant by relaxing the restrictions in phase space for acoustic phonon scattering. [11][12][13] Experimentally, the temperature of hot carriers T C has been determined by means of spontaneous optical emission 14 and Johnson-noise thermometry measurements, 13,15 whereas their dynamics has been investigated using ultra-fast pump-probe spectroscopy. [16][17][18][19][20] However, the propagation of hot carriers injected by electrical driving has yet to be investigated. This is critical for the understanding of energy flow at the nanoscale and its control in high-speed devices.
In this work, we report hot-carrier propagation across monolayer graphene (MLG). Hot carriers are generated locally by a bias current, then diffuse away from the injection point, and are detected electrically with voltage probes in a region where no current circulates. The carrier temperature T C is deduced from the detected thermoelectric voltage V NL and the Seebeck coefficient S of the sample. The temperature and current dependence of V NL enable us to identify when the carriers reach the detectors before thermalizing with the lattice, which remains at a lower temperature T L . We demonstrate that the presence of hot carriers results in a strong increase of V NL at low bath temperatures T bath and a clear departure from the ordinary linear relationship between Joule power P at the injector and V NL , V NL ~ P, which is typically found in conventional thermoelectric experiments for which T C ~ T L (see Refs. 21,22 and 23).
Additionally, we observe that the bias dependence of T C is consistent with the energy relaxation rate predicted by the SC mechanism. 11 By measuring V NL as a function of the distance from the injection point, we can determine the characteristic cooling length ξ for the electrically injected hot carriers. The measured voltage approaches 1 mV in our devices but we expect it to be much larger for high-quality graphene, resulting in a response that needs to be taken into account in high-frequency graphene transistors.
Graphene devices used in this work were prepared by mechanical exfoliation of MLG onto p-doped Si/SiO 2 (440 nm) substrates. The devices were prepared in two steps using electron-beam lithography. First, we deposited an amorphous carbon layer in the contact area just after exfoliation using electron-beam induced deposition (EBID), as described elsewhere. 24 Second, we defined electrical contacts with a width of 100 nm. The contacts were made by electron beam evaporation of Ti (5 nm)/Pd (60 nm) in a chamber with a base pressure of 10 -8 Torr. The conducting p-Si substrate was used to apply a back-gate voltage V BG relative to the device to control the carrier density n of the MLG. The presence of the amorphous carbon layer results in a contact resistance of ~5 kOhm, and fulfills the purpose of reducing the influence of the contacts in the hot-carrier dynamics. We have recently found that amorphous carbon does not affect the charge transport properties of graphene, preserving its mobility, but notably helps electrically detach the MLG from the leads. 24  injection/detection technique that is commonly used to investigate spin transport. 25,26,27 A current I between contacts Iand I + generates hot carriers that diffuse away from the injection region, which are then detected with remote voltage probes. The measured voltage V NL is said to be nonlocal (NL) because no electrical current flows in the detection region. In our devices, we define a set of three detectors with associated potential V + d (d = 1, 2 and 3) that are located at specific distances (1, 1.5 and 2 µm respectively) from the injector contact I + (see Fig. 1(a)). The thermal and hot-carrier transport properties are then investigated by measuring the nonlocal Samples are placed in a liquid helium continuous-flow cryostat that allows us to precisely control T bath between 10 K and 296 K. Measurements are carried out as a function of T bath and the back-gate voltage V BG in the linear I-V regime, where scattering with interfacial SiO 2 substrate phonons and optical phonons is absent. [28][29][30] The three detectors allow us to determine the temperature of the carriers at different positions in the MLG. We have measured five devices that showed similar results. The data hereby presented were acquired with two of them (device 1 and 2). If not specified otherwise, all data shown correspond to detector 1.
We first realized a full electrical characterization of the devices. We carried out measurements where current is applied between the outermost contacts and the voltage measured between pairs of inner contacts. In device 1, graphene is slightly p-doped with the charge neutrality point (CNP) at V CNP = -2 V, the residual carrier density is n r = 10 12 cm -2 and the carrier mobility µ = 5,000 cm -2 /V·s at a carrier density of n = 10 12 cm -2 . Figure 1(b) shows the gatedependent square resistance R measured in the first detector. We found that R vs V BG presents a similar behavior independently of the voltage contacts selected, which demonstrates that the contact probes do not significantly modify the transport characteristics of the graphene underneath.
We then measured the non-local response V NL as a function of the applied current I for different V BG . Figure 2(a) shows measurements at room temperature (T bath = 296 K). At any given V BG , we observe a parabolic dependence of V NL with current I. The parabolic behavior is verified in Fig. S1 of the Supporting Information, where it is observed that the data in Fig. 2(a) can be linearized by plotting V NL (I) vs I 2 .The change from an upward (V NL > 0) to a downward parabola (V NL < 0) occurs progressively with n and correlates with the change from electron to hole conduction; in particular we observe that V NL (I) is zero for all I at the CNP (n = 0). Figure 2 (b) shows V NL vs n for specific applied currents (marked by vertical lines in Fig. 2(a)), where the progressive change from electrons to holes is clearly observed.
Notably, the parabolic dependence between V NL and I (i.e, V NL ∝ P) morphs into V NL ∝ ∝ with ν < 1 for temperatures T bath < 100 K (Fig. 2(c), see also Fig. S2). As for T bath = 296 K, V NL changes from positive to negative when passing from electron to hole conduction, with no signal at the CNP. However, V NL is larger and the change from upward to downward curvature is significantly more abrupt, as observed in V NL vs n cuts at fixed I ( Fig. 2(d)).
To interpret the results in Fig. 2, we first note that carrier-carrier scattering processes are much faster than the electron-phonon scattering pathways, 31 independently of T bath . As a consequence, a hot carrier population is established that can be described by a thermal distribution with a well-defined temperature T C > T L , 14,18,31 which decays away from the injector.
Therefore, V NL d can be expressed as (1) where T C d is the carrier temperature at the detector d and T C COM is the carrier temperature at the common detector. As discussed below, T C COM ~ T L ~T bath because the distance between the injector and the common detector (2.5 µm) is much larger than the cooling length ξ. We also note that, as shown in Refs. (21, 22 and 23), the thermopower in graphene can be predicted by the semiclassical Mott relation, 32 where e is the electron charge, k B is the Boltzmann constant, and E F is the Fermi energy. S Mott is therefore a good approximation for the graphene Seebeck coefficient S in Eq. (1). The calculated S Mott at 296 K is shown in Fig. 1(c). S Mott reaches a maximum value of 40 µV/K and changes from positive to negative at the CNP, indicating the nature of the majority charge carriers in the system.
The parabolic behavior found in Fig. 2(a) can be understood from the Joule dissipation at the current injector and the cooling rates involved. At large enough bias, heat diffusion into the leads and into the unbiased graphene regions can be neglected and thus the cooling of the carriers is mediated by the electron-lattice coupling. 15 In addition, at room temperature, we expect that the carrier cooling time for our MLG is ~10 ps, 11 and the typical cooling length ξ < 100 nm, which is much smaller than the electrode separation. Therefore, the carriers are thermalized with the lattice (T C = T L ) at the detector region all the way to within 100 nm of the injector.
Under these conditions, 15 the steady state solution of the heat equation shows that the temperature gradient ∇T C at the detectors is we obtain, using Eq. (1), Therefore, at sufficiently high T bath , the thermoelectric response of our devices is similar to that found using an external heater, [21][22][23] despite the fact that, in our case, graphene is part of said heater.
In contrast, at low enough temperatures (below ~100 K), the hot-carrier lattice thermalization length can exceed a few hundred nanometres 11 and, therefore, the conditions T C = T L and T C d -T bath << T bath are no longer satisfied at the detectors. As illustrated in Fig. 2(c) for T bath = 10 K, this leads to a remarkable increase in V NL and to a dramatic departure from the linear relationship between V NL and P. Such behavior contrasts to that observed in conventional thermoelectric experiments, where the relationship V NL ∝ P is valid at all temperatures. [21][22][23] This is verified in Fig. S3, where we show thermoelectric measurements using an external heater. In those conventional experiments, the temperature pre-factor in Eq. (2) reduces the thermoelectric voltage by a factor ~30 from room temperature to 10 K. In our hot-carrier Seebeck measurements, we observe that V NL is actually larger at T bath = 10 K than at T bath = 296 K. By inspecting Eq. (1) and (2), this is possible for T C d ≈ 100 K.
The presence of hot carriers and the main relaxation by disorder-mediated scattering, or supercollisions (SC), accounts for the change in the functional dependence at low temperatures.
Recent experiments using Johnson-noise thermometry in two-terminal devices 13 have demonstrated that T C can reach 400-700 K with a Joule power P = 0.2 mW/µm 2 . Therefore, assuming ≫ ; 0123 and 9 0123 , we integrate Eq. (1). By taking into account Eq.
The increase that we observe in V NL at low temperatures is also a signature of SCs, as the dependence expected from momentum-conserving scattering by acoustic phonons predicts the opposite behavior. 11 This phenomenon can be understood by considering the temperature dependence of the cooling rate γ for hot carriers in the regime T L ≥ T BG , where T BG is the Bloch-Grüneisen temperature, which in graphene sets the boundary between direct electron-phonon scattering (T L < T BG ) and the regime which is dominated by SCs (T L ≥ T BG ). Direct emissions of acoustic phonons are rare leading to a cooling rate γ e-p ∝ 1 = that would produce a decrease of the non-local response with decreasing temperature. 2,3 However, SCs lead to a cooling rate > ? ∝ where the proportionality constant is related with the amount of disorder in the system. 11 The decrease of the cooling rate with decreasing temperature in the latter model explains the increase in the nonlocal response.
The nonlocal voltage decreases monotonically with temperature, indicating that we do not achieve the regime for direct emissions of phonons. The reason is twofold. First, the high disorder concentration in our devices hinders the direct emission of phonons, and second, the injection of current likely results in @ AB , even for 0123 C AB . Indeed, the expression for currents. This argument is supported by the data in Fig. 2c, which appears to be somewhat more parabolic at low I for large gate voltages, although the results are not conclusive (see Fig. S4).
The abrupt change in the V NL vs n cuts at fixed I (Fig. 2(d)) is a consequence of a larger rise in T C close to the CNP. This is partly due to the fact that the Joule power dissipated in the MLG is distributed over a relatively small number of carriers close to the CNP. Therefore, even though the overall shape of the Seebeck coefficient vs back-gate voltage is unchanged with temperature, the temperature gradient at the detector is gate dependent at low temperatures. This does not occur at room temperature because the dissipation, and temperature gradient, is dominated by the high-resistance contacts.
The same result is observed in the other measured devices. Figure 3 shows a comparison for device 2 between S Mott calculated at 296 K and at 77 K and V NL measured at the same temperatures. For this device, V NL has a similar magnitude at 77 K and at 296 K. However, the shape difference between them close to the CNP, due to the larger rise in T C , is evident; such a difference is not observed in S Mott (inset Fig. 3).
For a quantitative understanding of the thermoelectric response, we fit our results in Fig.   2 (a) to N O P , with N and ∑ as the fitting parameters. Here, ∑ ∝ S G -S l , with S G the graphene Seebeck coefficient; N and S l account for a small gate-independent thermoelectric voltage generated along the measurement lines. The resulting ∑ vs V BG is shown in Fig. 1(c) (open symbols). As discussed above, S Mott predicts the thermopower in graphene 21 (see Eq. (2)). We observe that β, which in a way represents the thermoelectric voltage due to the excess temperature of hot carriers, increases rapidly at low temperatures, especially below 100 K. For an additional test of the SC mechanism, we extract T C as a function of the dissipated power P in MLG. 33 We integrate Eq. (1) and take into account Eq. (2), obtaining, For T bath = 10 K, we estimate that the carrier temperature at the first voltage detector is as high as T C = 165 K for P∼ 0.65 mW/µm 2 and n ~ 2.5·10 11 cm -2 (see inset in Fig. 4(a)). Figure 4(a) illustrates $ / vs P. As predicted for the SC model, a plateau ∝ $ is evident for sufficient Joule power P where the condition T L ≥ T BG is fulfilled, whereas T C rises faster nearby the CNP due to the relatively small density of carriers, as discussed above. Finally, we use the three detectors ( Fig. 1 (a)) to estimate the cooling length ξ. At T bath = 10 K we measure the same sub-linear dependence of V NL vs P in the three detectors, indicating the presence of hot carriers in all of them. This is demonstrated in Fig. 4(b), where V NL ∝ P is shown for comparison. The temperature profile across the detectors is evaluated using Eq. (3) for P∼ 0.65 mW/µm 2 , presenting a characteristic cooling length on the order of 1 µm. Because of the large variations of T C , it is unclear if the profile obeys an exponential or a power law. In photothermoelectric experiments, 5,7 ξ was obtained using the heat diffusion equation whose general solution is ` ∝ sinh e f ` g = h, where D = 2.5 µm is the distance between the injector and the common voltage detector. By fitting our data with this model, we obtain ξ = 0.7-1.1 µm for n in the range of 1 × 10 12 cm -2 to 2.5 × 10 11 cm -2 (see inset of Fig. 4(b)).
Our devices show a rather fast relaxation of hot carriers. This is likely related to a relatively high concentration of impurities, which results in large SC rates. Extrapolation to x = 0 leads to hot-carrier temperatures of 370-450 K at the injector. These values are lower than those reported in Ref. 13, within a factor of 2, which might be related to a lower density of scattering centers in graphene on boron-nitride substrates. We demonstrate that the magnitude of the signal and its functional dependence with power are strong evidence of hot-carrier generated thermoelectric voltages and that the supercollision mechanism is the predominant cooling pathway for hot-carrier cooling. Future experiments could investigate the relative weight of the different electron-phonon scattering processes in multilayer graphene, in particular at high bias. 34 Beyond carrier-phonon physics and novel hot-carrier thermoelectricity phenomena, our work has important implications for the design of high-speed graphene-based devices. In the low temperature regime, the magnitude of the hot-carrier thermoelectric signal is as large as a few hundred microvolts but it can be much larger in high-quality graphene. Because the peak Seebeck coefficient scales with 1/ √ j k , a decrease in the residual carrier concentration n r by two orders of magnitude, which is typical for suspended graphene or graphene on boron nitride, will result in a tenfold increase in the signal. The signal will be further enhanced by an increase of the mean-free path, which will decrease the rate of supercollisions as well as strongly increase the carrier mobility and diffusion constant. Therefore, the hot carriers will be longer lived and diffuse much further than in our devices. Under these conditions, it is plausible that the signal can be as large as a few hundred mV, even at room temperature. It will thus strongly impact the performance of conventional graphene devices, and at the same time, create new opportunities for nanoscale bolometry and calorimetry.

Content
Linearized data in Fig. 2 Figure S4. V NL vs I for low I at 10 K, showing a parabolic-like behavior only at very low currents (I < 25 µA) and more clearly at negative back-gate voltages (negative V NL ). The results are nonconclusive due to the noise level of the measurements.