Data acquisition concepts of the electronic measurements

To measure the thermovoltages and the resistance changes for temperature determina-tion, different measurement techniques and analysis methods were applied which will be explained here.

Analysis of measurements with the temperature circuit

One of the two electrical circuits of the MCBJ setup is designed for the determination of the temperature difference across the junction. The illumination causes resistance changes in the sensor leads. By measuring the difference of the resistance changes ∆R_Sens =

∆RSens, R −∆RSens, L, between the left sensor lead ∆RSens, L and the right sensor lead

∆RSens, R, information about the position of the illumination and heat input can be drawn from this data. Later, these results will be used to calibrate simulations, which yield a temperature distribution on the sample and so to a temperature difference ∆T across the contact.

The way how the resistance changes ∆R_Sens are measured will be explained based on Fig.

6.8.

First, the voltage is recorded versus the time for different applied currents (see Fig. 6.8b) and c)). For every curve the ∆V =V_on−V_off (laser on and laser off) was determined.

These values lead to ∆V−Icurves, see Fig. 6.8 d) and e). By fitting theses curves linearly with ∆V = ∆RSens·I+V0, the slope leads to a resistance change of ∆RSens, 1=−0.56 Ω for the left-hand side measurement and a slope of ∆R_{Sens, 2} = 0.55 Ω for the right-hand side measurement. The signs can be explained because the signal of the left sensor lead is subtracted from the right one.

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Figure 6.8: Explanation of the analysis concept of the measurements with the temperature circuit based on the example of two different illumination spots on the sensor leads. The voltage-time plots in b) and c) show the time resolved signal of the heating position illustrated in a) (the left plots belongs to the left spot, the right ones to the right spot, respectively). 5 different currents were applied (from

−3.77µA (red curves) to 3.77µA (black curves). The signal of the laser is drawn in blue and has a length of4.6 msand an off-time of25 ms. In d) and e), the voltage changes∆V are plotted versus the applied current I. Fitting the curves leads to the equations ∆V =−0.56 Ω·I−0.82µV for the left plot and∆V = 0.55 Ω·I−0.65µVfor the right plot with different sign of the slope. So the resistance change from the left sensor lead was subtracted from the right one. The error bars in d) and e) denote the standard deviation of the time traces in b) and c) calculated between10 msand25 msand multiplied by a factor of√

2due to error propagation.

Principle of simultaneous determination of thermovoltage and conductance

Another aspect of this work was to measure the thermovoltage and the conductance si-multaneously, hence the dependence of the thermovoltage concerning the conductance can be studied.

Thermovoltage measurement

The procedure of the data acquisition will be explained based on Fig. 6.9.

A typical measuring sequence had a duration of up to 200 s, whereas the laser pulses had a length of about 4 ms and an off-time of 30 ms, altogether a frequency of around 30 Hz with overall 6000 heating events per sequence.

The recorded voltage is plotted in Fig. 6.9a), in which around every second the applied voltage (see Fig. 6.9b)) is changed by 1 mV in loops from −5 mV to 5 mV. The whole record set is then divided into smaller parts, where every voltage from −5 mV to 5 mV occurs once, see Fig. 6.9c) and d).

They serve as base for ∆V −I curves, wherein the voltage changes between laser on and laser off were averaged for each applied voltage. The voltage change is defined as the difference of the averaged second part of voltage while the laser is heating and the averaged 1 ms before the laser pulse starts, illustrated in Fig. 6.9e) and f).

Between the changes of the applied currents, parts in the datasets were cut out (see Fig.

6.9a)-d)). This was necessary, because the system needs about 0.5 s to relax due to elec-tronic filters.

Fitting the measured ∆V against the current of the 11 different applied voltages leads to the thermovoltage VTh = ∆V(0µA) gained by the intercept.

The results of a stable contact are depicted in Fig. 6.10, where in a) the linear fit of the ∆V − I curve has the equation ∆V = ∆R_Sens · I +V_Th = (347.1±0.7) Ω ·I + (−0.68±0.22)µV. So, the thermovoltage is determined toV_Th =−(0.68±0.22)µV (see zoom-in in Fig. 6.10a)).

Conductance measurement The conductance was simultaneously determined by recording the current, by measuring the voltage drop, across the pre-resistor and, with another offset-stable pre-amplifier, the voltage drop across the sample.

Furthermore, in Fig. 6.10b) an averaged conductance of G = 1.42µV was observed.

There, the conductance deviates around zero from the average value because of the off-sets of the amplifiers. They cause an increasing error the smaller the currents are.

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Figure 6.9: Determination principle of the thermovoltage. a) shows a whole dataset of recorded voltage drop across the sample and b) the applied current. Divided into sequences of the applied voltages from

−5 mV to 5 mV (c) and d)) they are the base of ∆V −I curves (later shown in Fig. 6.11a) and 6.10a)). How the voltage changes are determined is illustrated in e) and f). In f) appear the peaks at the beginning and end of the laser pulse which are caused by the EOM, but they play no role for the∆V determination.

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Figure 6.10: a) Example of a∆V −I curve with the simultaneous detected conductance in b) for a contact which was stable during measurement. The determination of the conductance at low currents leads to huge errors, so these points were removed. A linear fits (red and blue lines) reveal in a) the equation ∆V = (347.1±0.7) Ω·I+ (−0.68±0.22)µV. In b) the averaged conductance amounts to 1.42 G0.

Crucial changes in the atomic configuration of the contact during recording a ∆V −I curve can be observed by a non-linear shape shown in Fig. 6.11a) and further jumps by the conductance in b). Here, between −0.35µA and−0.1µA such a reconfiguration took place. The conductance jumped from 1.2 G₀ to 1.7 G₀. Often, a rearrangement of the contact is not obvious like here, an indication of the stability of the contact is the calcu-lated standard error of the intercept by the linear fit. Here, the linear fit (red line) yields the equation ∆V = ∆R·I+V_Th = (−305.7±45.5) Ω·I+ (−42.5±13.9)µV. By dividing the ∆V −I curve into two parts, the regions from −0.2µA to 0.4µA and from −0.6µA to−0.3µA, the linear fits (green lines) expose the equations of ∆V = (−516.5±17.4) Ω· I+ (−2.65±4.00)µV and ∆V = (−257.7±94.0) Ω·I+ (−56.9±42.4)µV. In general, sequences with a standard error above 1µV were discarded for further analysis because the contact was not stable during the measurement, like in this case with a huge error of 13.9µV. The criterion of a maximum standard error of 1µV was selected to ensure a stable contact during the measurement. Summarized, the recorded signals first were averaged for certain applied currents, then plotted as ∆V −I curves for every sequence from −5 mV to 5 mV. The intercepts of linear fits of these curves then lead to thermo-voltages at certain conductance values with corresponding standard errors.

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Voltagechange[µV]

Current [µA]

a) b)

Figure 6.11: a) Example of a ∆V −I curve with the simultaneously detected conductance in b) for a contact which changed during measurement. Between −0.35µA and −0.1µA a reconfiguration of the atomic contact takes place, which can be observed in the ∆V −I curve as well as in a jump of the conductance. A linear fit to the ∆V −I curveleads to the equation ∆V = ∆R·I+VTh =

−(305.7±45.5) Ω·I−(42.5±13.9)µV. The green lines denote the linear fits for the parts from−0.2µA to 0.4µA and from −0.6µA to−0.3µA and lead to the equations of ∆V = (−516.5±17.4) Ω·I+ (−2.65±4.00)µV and∆V = (−257.7±94.0) Ω·I+ (−56.9±42.4)µV.

Interpretation of the slope of the ∆V −I curves

Of interest is not only the intercept (thermovoltage V_Th) of ∆V −I curves with ∆V =

∆R·I+V_Th, but also the slope ∆R.

The slopes in Fig. 6.10 and Fig. 6.11 show not only a huge difference in magnitude, but also a different sign.

In previous results of the NThC (see section 5.1) and the resistance change measurements of the sensor leads (see section 6.2) reveal also an non-zero slope in the ∆V −I curves.

There, the slope is caused by the resistance change of the leads due to heating and amounts to a few Ω and the sign is always positive.

Here, resistance changes were found between−500 Ω and 350 Ω. This cannot be explained by the resistance changes of the leads, because the leads have only an absolute value of some tens of Ω.

As a consequence, the resistance change must be caused by the contact. To study this effect, possible dependences of the resistance changes will be considered. Therefore, Fig.

6.12a) depicts the resistance changes of 220 different ∆V −I curves in a chronological order, whereby the unstable contacts are not shown. The corresponding thermovoltages will be shown later in Fig. 6.28 to Fig. 6.31.

Huge fluctuations can be observed, which are not necessarily related to conductance changes, since between two points time intervals from less than 14 s up to hours could be.

The zoom-in of Fig. 6.12a), shows a region in which the time between two measurements was 14 s or less.

To explore a possible correlation between the slope and the conductance, the resistance change is plotted in Fig 6.12b) against the conductance, but no significant correlation appears. The shaded region denotes the standard deviation of the resistance changes with 83.6 Ω and the blue line the mean value of −4.3 Ω.

Furthermore, the influence of position was studied in Fig. 6.13a). Therefore, the resistance changes of seven different positions, which are denoted on the right-hand side, are plotted against the conductance. A laser intensity of 1.5 mW was applied.

Since positive and negative resistance changes appear, Fig. 6.14 depicts the averaged ∆R of absolute values against the laser intensity to study if there is a correlation. There, a trend that a higher intensity causes higher resistance changes is visible. Since only 10 data points of only one measurement lead to the point at 2.7 mW, this value is limited in statistical certainty.

a)

b)

Figure 6.12: a) Slopes (resistance changes) of 220 different fitted ∆V −I curves are plotted in a chronological order, whereby between two points time intervals from less than14 sup to hours could be.

The zoom-in shows a part where the time between two points is always less than14 s. b)∆R versus the conductance exhibits no correlation. The shaded range symbolizes the standard deviation of the resistance changes with83.6 Ω and the blue line denote the mean value of−4.3 Ω. Errors gained from the standard error of fitted∆V −I are smaller than the dot size.

Figure 6.13: Resistance changes versus the conductances at P_Laser = 1.5 mW for seven different positions, which are denoted on the right-hand side.

Figure 6.14: Averaged resistance changes gained from absolute values plotted against the laser inten-sity.

Conclusion Except of one measurement, the results show no clear relations which can help to explain the slope of the ∆V −I curves, whereby ∆V is the difference of the voltage drops across the sample between laser on and laser off. Neither a conductance dependence nor a position dependence can be detected. By considering the absolute val-ues of the ∆R, a tendency that the higher the heat input the higher the resistance change occurs. This is reasonable since the probability to change the contact configuration is higher due to a higher energy in the system.

Since positive and negative resistance changes appear, many possible effect are excluded.

For example, resistance changes of the leads would always exhibit only positive signs.

The thermal expansion is expected to reveal either a positive or negative sign, depending on the fact whether the structure is expanded or the substrate. However, an alternating sign is unrealistic in this case.

Only two possibilities are left, which can explain the observation of different resistance changes. On the one hand, thermally induced elastic modifications of contact configura-tion may cause the resistance changes with both signs. Nevertheless, one would expect to see mainly positive changes with some possible negative exceptions. Since the mean value reveals a negative sign with a huge standard deviation, also this possibility has its limitations.

On the other hand, a reason for the resistance changes can be conductance fluctuations, which were observed by Ludoph et al. [92] and describe the voltage-dependent fluctua-tions of the conductance. They are caused by backscattered electrons close to the contact.

To adopt it to the results shown here, a linear behavior of the conductance against the voltage in the measured range is required. Anyhow, it is unclear if the conductance fluc-tuations have an influence on the resistance change due to laser on and off.

Summarized, the origin of slopes (resistance changes) in the ∆V −I curves cannot be answered finally. For both models, the possible reconfiguration of the contact due to heating and the influence of the conductance fluctuations, still questions remain and can-not convince all in all.