Simulation

Simulations are necessary to derive the temperature difference ∆T across the contact from the measured ∆R_Sens. Hence, the commercial software package COMSOL Multiphysics [95] was used to perform finite element simulations which were done by Thomas M¨oller.

Material properties and parameters of the simulation are listed in the appendix B.1.

To do so, the geometry and proportion of the relevant part of sample was implemented into the software (see Fig. 6.21a)). The constriction was chosen to have a contact area of 63 nm×100 nm and was modeled as a freestanding bridge (see Fig. 6.21b)). This contact was constructed so that no heat and no electric current can pass, since the measurement next to the contact was performed at a relatively low conductance of about 4 G₀.

A 500µm thick Polyimide tape (Kapton HN) served as substrate with an additional 2.5µm thick polyimide layer on top. The properties were taken from COMSOL MEMS module: Polyimid [95].

The laser source with a wavelength of 500 nm was replaced in the simulation by a

b) a)

4 µm 50 µm

Figure 6.21: View on top of simulated structure in a) and in b) on the constriction.

Gaussian heat source [96] with variable intensity PSim and a diameter of 12µm. All results of the simulation reveal the status after 4 ms of heating adopting the laser pulse length of 4 ms in the experiments. It was assumed that gold absorbs 50 % of the heat,

the PI layer 0 % and the Kapton surface 100 %. Furthermore, the bottom of the Kapton substrate was held at a temperature of 77 K.

Calibrating experiments

In order to calibrate the simulation, three different measurements were performed with various parameters like the laser intensity P_Laser, the position and the resistance of the contact. The experimental results will be described in this section and later compared with the simulation.

Figure 6.22: a) Raw data of the voltage plotted against the time while the laser (indicated by the blue line) was pulsing on the substrate for different applied currents from−4µA to4µA. The curves have an arbitrary offset to enhance the illustration. b) Calculated voltage changes from a) versus the applied current. The linear fit (black line) corresponds to the equation∆VSens= 0.145 Ω·I+ 0.10µVand error bars denote the standard deviation during the laser was in the time between10 msto30 ms multiplied by a factor of√

2.

Position 1: Next to the constriction Fig. 6.22 shows the first ∆RSens measurement which is used to calibrate the simulation. In a) the voltage is plotted against the time for different applied currents from −4µA to 4µA which result in b) to the ∆V_Sens−I plot. The linear fit (black line) reveals the equation ∆V_Sens = 0.145 Ω·I + 0.10µV and so a ∆R_Sens = 0.145 Ω. It was performed at a conductance of 4 G₀, thus the transport leads contribute to the resistance changes, see explanation in section 6.2.2. The inset demonstrates the position of the laser spot, which had an intensity of 1.5 mW.

Position 2 and 3: On the sensor leads Two further calibrating measurements were already presented in section 6.1.3, Fig. 6.8. There, the sensor leads were illuminated directly with a laser intensity of P_Laser = 3 mW, which causes resistance changes ∆R_Sens

of −0.56 Ω and 0.55 Ω. In this case, the sample had a resistance of 70 Ω, so that the resistance changes of the transport leads played was negligible.

The slightly different absolute values can be explained by the fact that the illumination on the right sensor lead was closer to the widened lead which acts as heat sink. This behavior is confirmed later by the simulations.

Calibrated simulation

This part presents the simulations of the experiments in section 6.2.3, whereby different heating positions, laser intensities and resistances of the contact were considered.

For each position (see Fig. 6.23, Fig. 6.24 and Fig. 6.25) the following results of the simulations are shown: In a) the resistance changes against the time with ∆RSim =

∆R_{Sim, L}−∆R_{Sim, R}, in b) the heat distribution on the sample, in c) cross section along the different leads (whereby the red curves belong to the left leads and the black curves to the right leads, respectively), as well as cross sections with y = 0µm in d). Tab. 6.1 provides an overview on the parameters of the experimental results and simulations.

The comparison with the experiment reveals that not the complete laser power P_Laser Position x/y PLaser PSim ∆RSens ∆RSim ∆TSim

1 11µm/−13µm 1.5 mW 1 mW 0.14 Ω 0.14 Ω 7.5 K 2 −19µm/60µm 3 mW 2 mW −0.56 Ω −0.66 Ω −1.8 K

3 23µm/72µm 3 mW 2 mW 0.55 Ω 0.61 Ω 1.1 K

Table 6.1: Overview on the parameters on the experimental results and simulations.

contributes to the heating, because much larger resistance changes would be expected.

As a consequence, the intensity of the heat source P_Sim used in the simulation was re-duced by a factor of ²₃ compared to the experimental laser intensities P_Laser, due to the measurement at position 1.

The calibrated value of the heat sourceP_Sim = ²₃·P_Laserwas adjusted by the experimental results at position 1. This reduction of the intensity also yields similar values of ∆R_Sens and ∆R_Sim for the illuminated sensor lead at position 2 and 3. The difference can be explained by a not perfectly shaped Gaussian laser spot in the experiment as well as an exact illumination position on the sensor leads.

The illumination at position 1 (11µm/−13µm) results in a small resistance change be-tween the leads ∆R_Sim = 0.14 Ω, whereby the resistance changes of the transport leads were taken into account (see Fig. 6.20). There, the mean temperature of the right lead is at least 5 K higher than the mean temperature of the left lead, see Fig. 6.23d).

In addition, the illumination position next to the contact gives a temperature difference across the contact of ∆T_Sim =T_{Sim, R}−T_{Sim, L} = 7.5 K.

The influence of the resistance change of the unheated sensor on the total resistance change ∆R_Sim = ∆R_{Sim, L}−∆R_{Sim, R} is obviously smaller in the simulations of heating spots 2 and 3 compared to spot 1, see. Fig. 6.24a) and Fig. 6.25a). This behavior also illustrates the heat distributions in Fig. 6.24b) and in Fig. 6.25b), where it is shown that the temperature of the second sensor lead increases only marginally because of the distance to the heat position.

The simulated resistance changes ∆R_{Sim, 2} =−0.66 Ω and ∆R_{Sim, 3} = 0.61 Ω show a small difference of the absolute values between both illumination spots. This finding can be explained by the closer position of spot 3 towards the widened leads, which acts as heat sink. This behavior was already observed in the experiments, see Fig. 6.8.

The temperature differences result in ∆T_{Sim, 2} =−1.8 K and ∆T_{Sim, 3} = 1.1 K for spot 2 and 3.

c) d) of the sample due to the illumination at the position (11µm/−13µm). c) Cross sections of the heat distribution along the leads and in d) across the contact with y = 0µm. The simulation results in a resistance change of∆RSim=0.14 Ωand a temperature difference across the contact of∆TSim= 7.5 K.

-40 -20 0 20 40 of the sample due to the illumination at the position (−19µm/60µm). c) Cross sections of the heat distribution along the leads and in d) across the contact with y = 0µm. The simulation results in a resistance change of ∆R_Sim = −0.66 Ω and a temperature difference across the contact of ∆T_Sim =

−1.8 K.

0 50 100 150 200 250 of the sample due to the illumination at the position (23µm/72µm). c) Cross sections of the heat distribution along the leads and in d) across the contact with y = 0µm. The simulation results in a resistance change of∆R_Sim=0.61 Ωand a temperature difference across the contact of∆T_Sim= 1.1 K.

Conclusion

In summary, the calibrated simulations are a promising tool to determine the heat distri-bution in the sample and therefore the derivation of the temperature difference across the contact. To achieve this, three different measurements were compared with simulations in which parameters like laser intensity, heating positions and resistance of the contact were varied.

The calibration of the heat source intensity with P_Sim = ²₃ ·P_Laser leads to resistance changes in the simulation which are in good agreement with the experimental findings.

Although the simulated resistance changes result in a slightly higher value than the ex-periment, the simulation can be used for the determination of the temperature difference

∆T_Sim across the contact.

As a result, cross sections along the x-axis at y = 0µm reveal the ∆T_Sim, which can be used to calculate the thermopower.