Resistivity measurements

As the temperature dependence of the resistivity of thin gold films is crucial for the sim-ulations, it was further studied in the work of Isele [80] for the temperature range from 64 K to 296 K.

Polished substrates (Kapton and bronze) were covered with a 2 nm to 3 nm thick layer of PI, and structures of 1 cm length and 1 mm width were evaporated through a mask. The preparation of the samples is described in more detail in Ref. [80]. 4-point measurements were performed following the procedure in section 4.2.4.

The possible effect of film thickness and substrate on a) the absolute value of resistivity and b) their temperature dependences was investigated by testing three different thick-nesses (30 nm, 65 nm and 170 nm, electron-beam evaporation, evaporation pressure of 1×10⁻⁷mbar, evaporation speed around 1 ˚A s⁻¹) and two different substrates (Kapton and bronze). The impact of the substrate’s thermal expansion on the same two param-eters was tested by evaporating gold films of 65 nm on Kapton and bronze substrates at 143 K. All other samples were evaporated at room temperature.

Fig. 6.15 shows four examples of temperature-dependent resistance measurements, in which the resistivity is plotted versus temperature for a film thickness of 65 nm evaporated at a base temperature of 143 K in a) and of 170 nm in b) evaporated at room temperature.

All curves exhibit a linear behavior with fit parameters in a) ρ = 80.4 pΩ m K⁻¹ ·T + 7.12 nΩ m (65 nm on Kapton), ρ= 79.2 pΩ m K⁻¹·T + 8.95 nΩ m (65 nm on bronze) and in b) (170 nm on Kapton) ρ= 91.7 pΩ m K⁻¹·T + 6.38 nΩ m (170 nm on Kapton) as well as ρ= 84.0 pΩ m K⁻¹·T + 2.86 pΩ m (170 nm on bronze).

A small deviation of the linear behavior can be observed in Fig. 6.15b), in which the black curve shows non-linearity between 150 K and 200 K. Since this is the only irregular-ity in the linearirregular-ity among all samples, a temporary non-equilibrium of the temperature between the sample and the PT100 resistor that was used to determine the temperature is likely at the origin of this deviation.

A comparison of the resistivity at temperatures of 77 K (a)) and 296 K (b)) is shown in Fig. 6.16, including results of this study as well as from Sambles et al. [94] for gold film thicknesses of 35 nm, 80 nm and 507 nm and gold bulk values of Matula [11]. The values can be described by Mathiessen’s rule and the models of Fuchs-Sondheimer and Mayadas-Shatzkes, see section 2.2. As expected, the bulk value has the lowest resistivity.

With decreasing film thicknesses, the resistivity increases due to the bigger influence of the boundaries, and higher concentration of impurities and defects of the films.

Between 77 K and 296 K the resistivity increases/decreases by a factor between 1.7 and 2.4 for all films.

50 100 150 200 250 300

Figure 6.15: Temperature-dependent resistivity of thin gold films from64 Kto 296 K. a)65 nm on Kapton (black curve) and bronze (red curve) substrates, evaporated at 143 K. The linear fit yields the equationρ= 80.4 pΩ m K⁻¹·T+7.12 nΩ mfor Kapton andρ= 79.2 pΩ m K⁻¹·T+8.95 nΩ mfor bronze.

b)170 nmon Kapton (black curve) and bronze (red curve) substrates, evaporated at room temperature.

The linear fit in the range from64 K to296 Kgives the equation ρ= 91.7 pΩ m K⁻¹·T + 6.38 nΩ m

Figure 6.16: Absolute values of resistivity in a) at77 Kand in b) at296 Kversus the film thickness.

Fig. 6.17 summarizes the temperature dependence of resistivity in gold films. It is shown that all gold films, including the analyses reported by Sambles et al. [94] and Matula [11], exhibit a slope of about 81 pΩ m K⁻¹ (dark grey region symbolizes the standard devia-tion) independent of their thickness.

The following equation shows that the thickness of the film is directly related to the slope dρ

dT = dR dT

W

L ·d (6.2)

with thicknessd, widthW, lengthLand ^dR_dT the temperature-dependent resistance of the gold film.

Although the films on bronze and Kapton were evaporated at the same time to ensure a similar film thickness, the 65 nm gold film on bronze reveals a 1.4×smaller temperature dependence than the same film thickness on Kapton, which has a similar temperature dependence as the other samples. This might be related to differences in the adhesion of the gold to the substrate. However, this effect should be eliminated by the PI layer on top of the substrate, like for the other film thicknesses. The ratios of the width to the length of the film might also affect the resistance but these quantities are facilely accessible and their error is negligible. Therefore, this deviation cannot be explained easily.

The high value of the temperature dependence for 170 nm on Kapton might be related to an overestimation of the film thickness, however the value is still within the expected error range.

Figure 6.17: Temperature dependence of resistivity in the range from 64 Kto 296 Kversus the film thickness. The grey region symbolizes the standard deviation around the average of all data points (81 pΩ m K⁻¹) except the value for one of the65 nmgold films on bronze

Conclusion resistivity measurements The resistivity measurements of thin gold films on Kapton and bronze substrates reveal a consistent behavior and are in good agreement with values from literature [11, 94].

In summary, it can be said that thinner films cause higher resistivity and resistivity in-creases by a factor of 1.7 to 2.4 for the different films between liquid nitrogen and room temperature. The temperature dependence follows a linear trend, all with a slope of about (81±5) pΩ m K⁻¹.

No significant and/or systematic influence of the thermal expansion of the substrates on the resistivity is found within the limits of measurement accuracy, also not for the temperature dependence studies.

Finally, the linear behavior of the resistivity of thin gold films between room temperature and 64 K allows for their application as thermometers.