This part combines the results of the two previous chapters, where the applied temper-ature difference across the contact (section 6.2) was determined and the thermovoltage was measured (section 6.3).
These findings result in the thermopower of atomic-scale Au contacts at low temperature.
Temperature differences across the contact for various laser intensities and positions will be listed and used to calculate the thermopower for the measured contacts depicted in Fig. 6.28 to Fig. 6.31.
Finally, at the end of this chapter the results will be discussed and compared with recent publications [3, 58, 66] of other groups.
6.4.1 Results
Temperature differences across the contact
Various heat distributions of the sample were simulated using the calibrated parameter PSim = 23·PLaserin section 6.2.3 for the different illumination positions and laser intensities PLaser in Fig. 6.28, Fig. 6.29, Fig. 6.30 and Fig. 6.31. As a consequence, cross sections at y= 0µm along the contact as shown in Fig. 6.23d), exhibit the temperature differences, which are listed for the different positions and laser intensities in Tab. 6.2.
Position
PLaser [mW]
0.9 1.5 2.4 2.7 3.0
purple −2.0 K −3.3 K −5.0 K -
-black - 5.0 K - - 9.1 K
red 1.5 K 2.4 K 3.7 K - 4.6 K
green 1.5 K 2.5 K - -
-(dark) blue - 3.7 K - -
-light blue 1.8 K 2.9 K 4.5 K 5.0 K 5.5 K
yellow 3.7 K 5.9 K 8.8 K -
-Table 6.2: Temperature differences across the contact for different illumination positions and laser intensitiesPLaser. The positions are denoted in Fig. 6.28, Fig. 6.29, Fig. 6.30 and Fig. 6.31.
Thermopower results
Fig. 6.33 presents the thermopower plotted against the conductance, calculated from the thermovoltages in the previous section 6.3.3 and the temperature differences listed in Tab. 6.2.
Most of the recorded thermopower values are in the range between 1 G0 to 4 G0 and
0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 0
- 2 - 1
012Thermopower [µV/K]
C o n d u c t a n c e [ G 0 ]
Figure 6.33: Collected thermopowers in the range up to 120 G0 for different heating inputs and positions. The mean value of S =−0.30µV K−1 is marked as red dotted line. The blue dotted lines and the shaded region symbolizing the standard deviationσ= 0.44µV K−1. The error bars were taken over from the thermovoltages, the uncertainty due to the temperature determination is not taken into account.
−1.5µV K−1 to 1.5µV K−1 and reveal a negative averaged value of S = −0.30µV K−1, which is marked as red dotted line. However, the standard deviation ofσ= 0.44µV K−1, which is symbolized by the shaded region and bounded by the blue dotted lines, is rela-tively high due to the huge intrinsic fluctuations.
Furthermore, the error bars denote the standard errors of the linear fits from which the thermovoltage values were determined. The uncertainty of the temperature determina-tion is not taken into account.
6.4.2 Discussion
In the following section the presented findings will be compared with recent literature, whereby the averaged value, the fluctuations, and the standard deviation of the ther-mopower will be considered and discussed.
On the one hand, the measurements of Ludoph et al. (presented in section 3.1) with a MCBJ setup at 12 K reveals a vanishing averaged thermovoltage for low conductances (see Fig. 3.1c)).
On the other hand, Evangeli et al. found a small negative value of S =−0.75µV K−1 at room temperature with their STM realization. They claim the different base tempera-ture of the measurements as a reason for the deviation of both experiments, which was considered in Ref. [1]. In these calculations, the averaged thermopower at low temper-ature should be 30× smaller than at room temperature. Therefore, a recorded value of S =−0.30µV K−1 for measurements in this work at 77 K is reasonable.
Additionally, as depicted in Fig. 6.33, the standard deviation ofσ= 0.44µV K−1 is larger than the averaged values. Hence, these measurements would still be consistent with a vanishing thermopower.
Furthermore, due to the fluctuations, positive thermopower values appear. This is also reported in the other publications [3, 58, 66]. A trend like in Ref. [58], which exhibits higher fluctuations for lower conductances, is not visible.
As described in section 3.1 the model set up by Tsutsui et al. [3] describes ballistic quan-tum point contacts by a simple free-electron model. Unfortunately, this model cannot explain the positive values for the thermopowers found here, as it assumes only negative signs.
Ludoph et al. report that the fluctuations are caused by the coherent backscattered elec-trons at impurities close to the contact. However, Evangeli et al. neglected the influence of the backscattered electrons, since their simulations reveal that the thermopower fluc-tuations can be described only by local disorders at the contact. As a consequence, the detailed knowledge about the atomic configuration of the contact is important. Therefore, the average behavior and the sign of the thermopower is given by the intrinsic electronic structure of the contact.
A trend towards the positive bulk thermopower value of 1µV K−1 like in Evangeli et al.
is not visible in this data since here only conductances up to 120 G0 were measured which cannot be considered as bulk contacts.
In addition, the amount of data points is too small to make general statements.
A method to consider the fluctuations is given by the standard deviation. These were cal-culated for the thermopowers within a conductance window of 0.1 G0 and plotted against the conductance, see Fig. 6.34. The red dots are the findings of this work, whereas the background with black squares and the solid line were taken from [58], see also Fig. 3.2.
In summary, the 22 data points do not reveal any clear correlation. The limited statis-tical data might also explain the suppression of the expected oscillation of the standard deviation, which Ludoph et al. could show (see black squares and line in Fig. 6.34). It is
0 1 2 3 4 5 0
1 2 3
Standarddeviation[µV/K]
Conductance [G0]
Figure 6.34: Standard deviation of the thermopower plotted against the conductance. The ther-mopowers within0.1 G0 were combined and the standard deviation calculated. The red dots show the results of this work, whereas the black squares and the black lines represent the findings and the model of Ludoph et al. [58].
expected that the fluctuations vanish for integer number of G0 since in this regime only fully opened transmission channels contribute to the charge transport but not combina-tions of several. Another difference between Ludoph’s measurements and the analysis of this work is that the standard deviation here was derived of altogether 220 contacts in-stead of just one contact in Ludoph’s study. It is expected that the values of the standard deviation should be on or above the solid line, since the contribution of more conductance modes can only cause higher fluctuations. However, the findings here reveal that nearly all values are below the theoretical calculations, which can be explained by the limited statistical data. Furthermore, too tight restriction concerning the discard of supposedly unstable contacts limit the fluctuations.
Conclusion Summarized, the 220 data points measured at 77 K confirm previous ex-perimental findings of the thermopower of atomic-size gold contacts.
A negative average value of −0.30µV K−1 for the thermopower of atomic-size contacts with a standard deviation 0.44µV K−1 agree very well with the observation of Ludoph et al. [58] and Evangeli et al. [66], who found a vanishing thermopower at 12 K base temper-ature (Ludoph) and a negative value of −0.75µV K−1 at room temperature (Evangeli).
Furthermore, the observed fluctuations are also in good agreement with the previous findings of Ludoph et al. [58] and Evangeli et al. [66]. On the basis of the simulations of Evangeli et al., they can explain the averaged negative value of the thermopower as well as the fluctuations by impurities close to the contact.
Nevertheless, an oscillating behavior of the standard deviations cannot be detected as shown in Ref. [58]. This can be explained by the limited statistical data and too tight restrictions on the discard of supposedly unstable contacts.
In conclusion, despite the missing statistical data points, the measurements presented here match the common theories and experimental findings about the thermopower of atomic-size gold contacts.
Outlook
This PhD thesis started from scratch building up a complete new measurement technique, which was not established before in Konstanz. Now at the end of this project an operat-ing experiment exists, but obviously improvements are possible and a variety of scientific questions can be addressed, for example variable temperature difference, variable contact size and variable heating position. This part provides an overview of further possibilities concerning the results of thermopower measurements and the potential application of the developed optically-accessible, cryogenic MCBJ setup.
Although first thermopower results of atomic-size gold contacts were presented in this work, there is room for further investigation. It seems self-evident that collecting more data which can be used for a statistically analysis has always been an issue. Therefore, some slight modifications of the measuring process would allow a faster data acquisition.
Then statistically robust statements about the thermopower fluctuations and their oscil-lations like reported in Ref. [58, 66] could be made.
Furthermore, a conductance-dependent measurement with a higher resolution is needed, since this work could show the results at only certain conductances. Therefore, the ther-mopower has to be recorded while the contact is opened very slowly.
Of great interest is also to verify the prediction of Evangeli’s simulation [66] that the averaged value of atomic-size gold contacts at 4 K is 30 times smaller than at room tem-perature, which can be realized with this setup. In general, studies at different base temperatures could help to understand the behavior of several systems like atomic or molecular contacts.
Additionally, information about the substrate could be collected by the temperature cir-135
cuit. For example, the thermal resistance of Kapton might be extracted by spatially resolved resistance change measurements of the sensor leads.
An experiment of the heat transfer through the contact is a further challenge but would be an interesting object of research.
Another point is the possibility of investigating the thermopower of different metals, like platinum or aluminum with the MBCJ mechanism since such samples are in Konstanz available. Evangeli et al. [66] already reported about their findings with atomic-size plat-inum contacts. Thereby, theoretical predictions of the behavior of the thermopower of different metal contacts at atomic scale can be tested.
As already mentioned in the introduction, not only atomic-size contacts are of interest, since molecular junctions exhibit a huge potential towards effective thermoelectric de-vices. Many approaches [2, 3, 58–66] report about the thermopower of different molecular contacts which reveal promising results of huge thermopowers for a later technical real-ization. In order to perform such measurements, the cryogenic setup was developed in a proper way.
However, not only thermoelectric effects can be investigated, but also light-induced effects of atomic-size and molecular junctions at low temperatures. Until now, most of those experiments [77, 83, 96–98] took place at room temperature in ambient conditions.
Furthermore, the optical setup allows the measurement of photochromic molecules, since laser sources with different wavelength can be used.
Summary and Conclusion
The goal of this work was to measure the thermopowerS =−∆V∆T of atomic-size contacts at low temperature. Since a suitable setup for this scope was not available, a big task of this work was to develop such a system. Further issues were the realization of a temper-ature difference across the contact generated by a laser source and the determination of this difference which was achieved using finite element simulations. Moreover, measure-ments of the thermovoltage and the final calculation of the thermopower were subjects of this thesis.
Since two kinds of samples, the shadow-evaporated nano-thermocouples (NThC) and mechanically controllable breakjunctions (MCBJ), were used in this work each sample preparation process as well as their electronic and optical setups were described in chap-ter 4. In addition, the electron beam lithography on Kapton was established
The results were divided into several parts, beginning with the findings of the NThC samples in chapter 5. In addition to the outcomes concerning the thermopower of this nano-thermocouples, resistance changes caused by the heating of the leads were detected.
All in all, the experiments offer a consistent overall picture in which nano–thermocouples can be described by macro-scaled physics.
Chapter 6.1 deals with the characterization measurements for low temperature setup de-veloped in this thesis. The breakjunction mechanism allows the precise adjustment of atomic-size contacts which could be demonstrated by a conductance histogram as well as opening and closing curves. Another development was the optical setup that allows a variable heating position on the sample by tilting a mirror, with a tunable laser as light source. It was proven that a controlled movement and positioning of the laser
137
spot on the sample with step sizes under 1µm was possible within a window of at least 500µm×500µm. Furthermore, the laser spot size was determined to be about 13µm by the knife-edge method. In addition, it was described how the data acquisition took place for the different measuring processes and a criterion to distinguish between stable and unstable contacts was introduced.
The determination of the temperature difference across the contact was described in chapter 6.2. First, the resistance changes due to heating of two additional sensor leads were measured. The sensor leads were placed on each side, next to the junctions and perpendicular to the leads of a standard breakjunction design. Furthermore, the relative resistance changes between the sensor leads were compared with the results of a finite element simulation (COMSOL). The calibrating experiments revealed that instead of the full laser intensity, only two thirds of the intensity act effectively as heat source, so that similar resistance changes in the experiment and simulation occurred. This calibration was performed at three different positions with different parameters like applied laser intensity or resistance of the contact. From the calibrated simulation derivated heat dis-tributions yield the temperature difference across the sample.
Spatially resolved thermovoltage measurements as well as the results of the thermovoltage as a function of the conductance were presented in chapter 6.3. The spatially resolved measurements on the gold structure show an expected anti-symmetric behavior of the thermovoltage amplitude concerning the contact, with thermovoltages up to 5µV. Most of the data for the conductance-dependent measurements were collected while the laser spot was positioned next to the contact but on the substrate to avoid unwanted opti-cal effects from the leads. Since a small variation of the position result in a significant change of the simulated temperature difference across the contact ∆TSim, the thermovolt-ages were plotted against the conductance for altogether seven positions.
The consideration of the thermopower results in chapter 6.4 completed the results. There, a mean thermovoltage of S = −0.30µV K−1 was observed with a standard deviation of σ = 0.44µV K−1. The averaged value at 77 K is in good agreement with the findings of Ludoph et al. and Evangeli et al. which reveal a vanishing thermopower at 12 K and a value of S = −0.75µV K−1 at room temperature, since the simulations of Evangeli et al. predicted a 30 times smaller thermopower at 12 K compared to 293 K. Moreover, the expected fluctuations were detected which caused some positive thermopower values.
Nevertheless, the oscillation of the fluctuation like Ludoph et al. could not be observed.
Reasons for that might be the limited statistical database.
Finally, an outlook was provided for further investigations, not only in the field of ther-mopower of atomic-size gold contacts, but also for possibilities which are offered by the cryogenic setup.
In conclusion, the developed cryogenic system enables the measurement of the ther-mopower of atomic-size contacts at low temperature. The contacts were formed by a vertical breaking mechanism thereby enabling optical access to the sample. The tem-perature difference across the contact was achieved by laser heating. In addition, this temperature difference was determined by finite element simulations, which were cali-brated by resistance change measurements of two additional sensor leads next to the contact. The results of the thermopower measurements were in good agreement with recent publications.
All in all, a versatile setup to address thermoelectric properties of atomic-size devices.
Parts of this thesis were already published, see Refs. [84, 85, 99].
Zusammenfassung
Das Ziel dieser Arbeit war, die ThermokraftS =−∆V∆T atomarer Kontakte bei tiefen Tem-peraturen zu messen und daf¨ur einen geeigneten experimenteller Aufbau zu entwickeln.
Weiterer Bestandteil dieser Arbeit war es eine Temperaturdifferenz entlang des atomaren Kontakts mit einer Laserquelle zu erzeugen und diese Differenz mit Hilfe von Finite El-emente Simulationen zu bestimmen. Zudem war die Messung von Thermospannungen, sowie die Berechnung der Thermokraft von atomaren Kontakten Bestandteil.
Da zwei verschiedene Arten von Proben, die schattenbedampften Nanothermoelemente (NThC) und die mechanisch kontrollierbaren Bruchkontakte (MCBJ) in dieser Arbeit untersucht wurden, wurde in Kapitel 4 deren Probenpr¨aparation, sowie deren optischen und elektronischen Aufbauten pr¨asentiert.
Die Ergebnisse wurden in verschiedene Segmente unterteilt, angefangen mit den Ergeb-nissen der NThC Proben in Kapitel 5. Zus¨atzlich zu den gemessenen Thermospan-nungen wurden temperaturabh¨angige Widerstands¨anderungen in den Zuleitungen detek-tiert. Zusammengefasst zeigten die Experimente ein konsistentes Gesamtbild, bei dem die Ergebnisse der Nanothermoelemente mit denen auf makroskopischer Skala ¨ubereinstimmen.
Kapitel 6.1 beschreibt die Charakterisierungsmessungen des entwickelten kryogenen Auf-baus. Die Bruchkontaktmechanik erlaubt die pr¨azise Einstellung von atomaren Kon-takten, was durch ein Leitwert-Histogramm sowie durch ¨Offnungs- und Schließkurven demonstriert werden konnte. Eine weitere Entwicklung war die des optischen Aufbaus.
Dieser erlaubt eine variable Beleuchtung- und Heizposition auf der Probe durch Verkippen eines Spiegels mit einer einstellbaren Laserquelle. Es wurde gezeigt, dass eine kontrollierte Bewegung des Laserspots auf der Probe in Schritten von kleiner als 1µm ¨uber einen
Bere-141
ich von 500µm×500µm m¨oglich ist. Des Weiteren konnte die Gr¨oße des Laserspots mit der Knife-Edge Methode auf 13µm bestimmt werden. Zus¨atzlich wurde beschrieben, wie die Daten f¨ur die einzelnen Messmethoden gesammelt und analysiert wurden. Außerdem wurde auch ein Kriterium aufgestellt, das zwischen stabilen und instabilen Kontakten unterscheidet.
Die Bestimmung der Temperaturdifferenz ¨uber dem Kontakt wird in Kapitel 6.2 erkl¨art.
Zun¨achst wurden die Widerstands¨anderungen der beiden Sensorzuleitungen aufgrund von Erw¨armung gemessen. Die Sensorzuleitungen waren jeweils auf einer Seite in der N¨ahe des Kontaktes senkrecht zu den Zuleitungen des Standard-Bruchkontaktdesigns angebracht.
Daraufhin konnte die relative Widerstands¨anderung zwischen den beiden Sensorzuleitun-gen mit den Ergebnissen von Finite Element Simulationen (COMSOL) verglichen werden.
Die Kalibrierungsmessungen offenbarten, dass 2/3 der im Experiment eingesetzten Laser-intensit¨at als W¨armequelle in der Simulation zu vergleichbaren Widerstands¨anderungen f¨uhrte. Die Kalibrierung wurde mit verschiedenen Parametern durchgef¨uhrt, wie der Po-sition, der angelegten Laserintensit¨at oder des Widerstandswertes des Kontakts. Aus den kalibrierten Simulationen gingen dann die Temperaturprofile entlang des Kontaktes hervor.
Ortsaufgel¨oste Thermospannungsmessungen sowie Ergebnisse der Thermospannung in Abh¨angigkeit des Leitwerts wurden in Kapitel 6.3 pr¨asentiert. Die ortsaufgel¨osten Mes-sungen auf der Goldstruktur zeigten das erwartete anti-symmetrische Verhalten bez¨uglich des Kontaktes von der Amplitude der Thermospannung, mit Werten bis zu 5µV. Bei den Messungen in Abh¨angigkeit vom Leitwert wurde bewusst das Kapton in der N¨ahe des Kontakts erw¨armt um ungewollte optische Effekte zu verhindern. Da kleine ¨Anderungen der Position signifikante Auswirkungen auf die simulierte Temperaturdifferenz ¨uber der Probe ∆TSimhatte, wurde die Thermospannungsergebnisse f¨ur die verschiedenen Positio-nen ¨uber den Leitwert aufgetragen.
Die Betrachtung der Thermokraft in Kapitel 6.4 vervollst¨andigt den Ergebnisteil. Dort wurde ein gemittelter Wert von S = −0.30µV K−1 mit einer Standardabweichung von σ = 0.44µV K−1 beobachtet. Damit ist dieser bei 77 K aufgenommener Wert in guter Uberstimmung mit den Ergebnissen von Ludoph et al.¨ und Evangeli et al.. Diese beobachteten zum einen einen verschwindenden Mittelwert der Thermokraft bei 12 K und zum anderen rndeckten sie einen Mittelwert von S = −0.75µV K−1 bei 293 K. Jedoch konnte keine Oszillation der Fluktuationen wie bei Ludoph et al. beobachtet werden.
Grund daf¨ur wird die begrenzte Anzahl von Messwerten sein. Der Ausblick stellte noch
Grund daf¨ur wird die begrenzte Anzahl von Messwerten sein. Der Ausblick stellte noch