Motivos de caída: reflexiones sobre las necesidades del tomador y

In the hot electron regime, the electron-electron scattering length le−eis much shorter than the device length L and therefore many inelastic scattering events take place while an electron travels from N1 to N2. This leads to an effective electron temperature that follows the temperature profile along the device as described in Eq. 1.55 and shown in Fig. 4.4. The dominating cooling me-chanism in this regime is due to hot electron out-diffusion. It follows that the temperature in the middle of the sample is linearly dependent on bias with the Lorenz number L0 as the proportionality factor. This holds for U²/L0 T0, where T0is the temperature of the reservoirs.

3.0

2.0 1.0

0.0 Te (K)

1.0 0.8

0.6 0.4

0.2 0.0

x/L Σep = 0 WK^-3.6m^-2 Σep = 0.098 WK^-3.6m^-2 Σep = 1 WK^-3.6m^-2 U = 1 mV

R = 4.1 kΩ

Figure 4.4. Temperature profile in the hot electron regime: Te(x) is obtained by numerical solving the heat transfer equation 1.53. The de-vice dimensions of dede-vice A are used with different electron-phonon coupling strengths as indicated. A power law of 3.6 is used as was observed in device C, see section4.5. The phonon and reservoir temperature is assumed to be 100 mK.

Fig.4.5(a) shows the tunnelling differential conductance for several values of heating bias U. An increased U leads to a smearing of the sharp supercon-ducting gap and the middle of the gap is shifted by U/2 since the tunnel probe is located in the middle of the sample. The extracted electron temperature is shown in Fig. 4.5 (b) as function of heating bias U for several values of back gate voltage VBG. It can be seen that Te depends linearly on U. The inset in (b) shows the gate dependence of the graphene conductance measured from N1 to N2. While the graphene resistance is tuned by roughly a factor of two (by changing the charge carrier density by ∼ 7 × 10¹²cm⁻²), the

re-80

4.4. Hot electron regime

sulting electron temperature is independent of the graphene resistance. If a large contact resistance would be present, a substantial part of the heating bias would drop on this resistor and a bias smaller than U would drive the graphene out-of-equilibrium. The ratio of the voltage dropping on the grap-hene and on the contact resistance would depend on the gate voltage as the graphene resistance is tunable. Therefore, different temperatures for the same U would be expected for different gate voltages. However, the measurements show that neither the electrical contact nor the charge carrier density plays a significant role.

A linear dependence of Te on U is expected in the hot electron regime, where the cooling is given by the Wiedemann-Franz law that relates electrical conductivity to thermal conductivity. As shown in section 1.5 the electron temperature profile can be calculated analytically by Eq.1.55and the expected Te at the location of the superconducting probe electrode is shown as solid black line in Fig.4.5(b). All extracted values for Te fall below the expected value. Possible deviations are discussed later in section4.7. Increasing L0 by 24 % leads to a lower Te as shown by the solid blue line in Fig. 4.5 (b). A similar effect is observed by including the phonon cooling, see the two solid purple lines. The heat diffusion equation is numerically solved taking into account the electron-phonon coupling extracted in section4.5. The two lines originate from the largest and smallest device resistance as this influences the total cooling power through the phonons.

Similar results have been obtained for device B, which are shown in Fig.4.6.

The extracted electron temperatures of device B are also independent of the gate voltage. Here, the graphene resistance changes by a factor of three while changing the charge carrier density by ∼ 7 × 10¹²cm⁻². This again confirms the negligible role of electrical contact resistance. Here, the dependence of the temperature on heating bias U can be divided into two qualitatively different regimes. For U ≤ 1 mV, a linear dependence similar to device A is observed.

Again, the extracted values for Teare smaller than calculated by Eq.1.55(solid black line). An increase of 48 % in L0results in lower values of Te, close to the experimentally observed ones as shown by the solid blue line. For U > 1 mV, the extracted electron temperature is much lower than expected (even for the larger L0). This "kink" is well explained by the onset of electron-phonon cooling which reduces the electron temperature below the expected value. This is visualized by the two solid purple lines that represent the expected electron temperature including phonon cooling. The two lines originate from the largest and smallest device resistance as this influences the total cooling power through the phonons. Further possible explanations are discussed in section4.7.

4. Non-equilibrium properties of graphene probed by superconducting tunnel

Figure 4.5. Device A in the hot electron regime: (a)shows the dif-ferential conductance measured through the superconducting electrode to the graphene for different values of heating bias U at a gate voltage of −7.5 V. (b) shows the extracted electron temperature from fitting a Fermi-Dirac distribu-tion to the numerically extracted distribudistribu-tion funcdistribu-tion for several values of VBG. The electron temperature increases linear with applied bias as expected by a dominating cooling mechanism due to electron out diffusion. The solid black lines represents the expected temperature based on the Wiedemann-Franz law, whereas the solid blue line represents the Wiedemann-Wiedemann-Franz law for an increased Lorenz number. The solid purple line shows a numerical es-timate of Te(using Eq.1.53) including the influence of the phonon cooling by using Σep extracted in section4.5. The two lines originate from the largest and smallest device resistance. The inset shows the two-terminal conductance through the graphene from N1 to N2 as a function of gate voltage VBG.

82

4.4. Hot electron regime

6 5 4 3 2 1 0 Te (K)

2.0 1.5

1.0 0.5

0.0

U (mV) 15

10

5 G (e2 /h)

-40 0 40

V_BG (V)

V_BG = -47.5 V VBG = +47.5 V TWF

L = 1.3 µm W = 4 µm

TWF + Σep

TWF, 1.48*L0

Figure 4.6. Device B in the hot electron regime: The extracted elec-tron temperature from fitting a Fermi-Dirac distribution to the numerically extracted distribution function for two different gate voltages is shown. Ty-pical differential tunnel conductance traces are shown in Fig.4.2(b) and (c).

The electron temperature increases nearly linear with applied bias as expected by a dominating cooling mechanism due to electron out diffusion. The solid black lines represents the expected temperature based on the Wiedemann-Franz law, whereas the solid blue line represents the Wiedemann-Wiedemann-Franz law for an increased Lorenz number. The solid purple line shows a numerical es-timate of Te(using Eq.1.53) including the influence of the phonon cooling by using the Σepextracted in section4.5. The two lines originate from the largest and smallest device resistance. The inset shows the two-terminal conductance through the graphene from N1 to N2 as a function of gate voltage VBG.

4. Non-equilibrium properties of graphene probed by superconducting tunnel spectroscopy