2.4 Thermoelectricity in metals
2.4.3 Figure of merit
The efficiency of power generation as a function of ZT is given by η=
where TC is the temperature on the cold side and TH on the hot side. The mean tem-perature is ¯T = (TC+TH)/2. As already mentioned in the introduction, the efficiency of thermoelectric devices can be described by the thermoelectric figure of merit,ZT [42–45].
ZT =S2σT /κ (2.22)
withS the thermopower,σ the conductivity,κ=κe+κph+κmthe thermal conductivity and T the absolute temperature, whereκedenotes the contributions of the electrons, κph of the phonons and κm of the magnons. To improve ZT, either one increases the power factor (S2σ) [46–51], or reduces the thermal conductivity κ, for example by materials which reduce the fraction of the phonons [52–56].
ObvisouslyZT is enhanced by increasing S2σ and decreasing byκ. No theoretical max-imum value for ZT exists, but so far the highest observed was 2.6 for SnSe single crys-tals [57] at very high temperatures, while a value of at least 4 is necessary to exhibit a higher efficiency than existing power conversion technologies [43].
State of the Art
Much previous research engaged with nano-thermoelectric and several successful ther-mopower experiments were performed [2, 3, 58–66]. Here, these works are discussed to provide an overview of the state of the art. Two standard experimental setups are used in this thematic field to create atomic-sized contacts, namely the scanning tunneling micro-scope (STM) and the mechanically controlled break-junction (MCBJ). These techniques allow the investigation of transport properties of atomic-scale devices with different ad-vantages and disadad-vantages.
This chapter is divided into two parts, Publications: Thermopower of atomic-scaled Au-Au contacts and Publications: Techniques to determine the thermopower of molecular junctions and thermoplasmonics.
First, the experimental results of Ludoph et al., Tsutsui and Morikawa et al., Evangeli et al., and Lee et al. regarding their findings about thermopower of atomic contacts are discussed. Furthermore, techniques of creating atomic-scale contacts and generating temperature differences across these contacts, as well as methods to determinate the tem-perature differences will be described.
Publications about thermoelectricity of molecular junctions will be discussed briefly in the second part, mainly focusing on the techniques and setups since these kinds of con-tacts are not the major topic of this thesis.
In addition, the work of Herzog et al. [67] measuring the optically-induced change in resistance of nanowires is described.
Finally, a summary and a discussion about the advantages and disadvantages of the different setups and techniques will be given.
27
3.1 Publications: Thermopower of atomic-scale Au-Au contacts
Ludoph et al.
a) b)
c)
Figure 3.1: a) Sample with measurement setup. A notched-wire MCBJ is placed on a bronze substrate and a piezo element controls the conductance precisely. RuO2 heaters and RuO2 thermometers apply and measure the temperature across the junction. b) Simultaneous measurement of thermopower S and conductance Gversus the elongation of the contact that is proportional to the piezo-voltage. The vertical grey lines show that changes in conductance are leading to jumps of the thermopower caused by rearrangements of the atomic-scale contact. c) The plot thermopower versus conductance reveals huge fluctuations especially for lower conductanceG.1 G0. Taken from Ref. [58].
One of the pioneering works in the field of thermopower of atomic-sized contacts was performed by Ludoph et al. (1999) [58] by simultaneously measuring the conductance and thermovoltage in a modified MCBJ setup at low temperature. For this purpose, a notched 100µm thick gold wire was glued on a phosphor bronze substrate, as illustrated in Fig. 3.1a). The fine adjustment of the conductance was achieved by a piezo-element underneath the substrate.
A temperature gradient was generated by two RuO2 heaters on each side of the junction.
To determine the temperature, two RuO2thermometers were located next to the heaters.
With this setup a temperature difference in the range of 4 K to 6 K at a base temperature
of 12 K could be reached.
The traces of typical conductance plateaus were observed (see Fig. 3.1b)) and linked to the measured thermopower S [68, 69]. Changes of conductance on the one hand lead to a simultaneous jump in thermopower. Further fluctuations on the thermopower signal (with positive and negative values) are caused by atomic rearrangements of the contact, even when the conductance seems to be stable.
Fig. 3.1c shows the correlation between thermopowers and conductance. The ther-mopower generally fluctuates around zero, at very small averaged values which show an increasing spread for smaller contacts. Ludoph et al. explain this behavior with interfer-ence effects of backscattered electrons at impurities next to the contact.
Furthermore it is shown that the measured standard deviation (Fig. 3.2) is in agreement
Figure 3.2: Standard deviation of thermopower plotted against conductance G. The theoretical predicted curve is presented in solid lines, whereas the black squares show the measured data. Taken from Ref. [58].
with the model for the standard deviation where the modes contributing to the conduc-tance open one by one. By analyzing the amplitude they estimated the mean free path tole= (5±1) nm.
Positive peaks expected for thermopower oscillation at the values of (n + 12)G0, n = 0,1,2, . . . could not be observed in these measurements since the appearing fluctuations are much larger than the expected oscillations.
Tsutsui and Morikawa et al.
Several publications about the thermoelectricity of atomic and molecular junctions were released by the group of Taniguchi. Tsutsui [3, 63] et al. and Morikawa [2] et al. re-ported experiments conducted at room temperature with thin-film mechanically control-lable break junctions (MCBJ). In this setup, a microheater was embedded in the sample, which consists of a 500µm thick phosphor bronze substrate with a 4µm polyimide layer and a 20 nm Al2O3 layer on top (Fig. 3.3a)-c)). The structures above the Pt coils (300 nm line-space geometry) serving as microheater and the 100 nm thick1 nano-junction made of gold were fabricated by electron beam lithography.
Finally, the sample was etched by reactive ion etching to produce a freestanding junction.
The creation of atomic-scale contacts was realized by a three-point bending
configura-a) b)
c) d)
Figure 3.3: a) and b) SEM images of the nano-junction and the Pt microheater embedded in the sample.
c) Illustration of the three-point breaking mechanism. The red arrow indicates the piezo-element which enables the precise control and stabilization of the atomic contact. Furthermore the electrical circuit of the setup is sketched. d) Top: ConductanceGversus time tfor different heat inputs under Vh= 0.5 V andVh= 2.0 V. Predicted stepwise conductance changes occur in both cases. Bottom: Fluctuations of the thermovoltage, whereby the signal underVh= 0.5 V is much lower compared toVh= 2.0 V. Taken from Ref. [3].
1 1 nm chromium was desposited as an adhesion layer underneath the gold
tion. Additionally, a piezo-element was used for fine adjustment and stabilization of the contact by a feedback control. The typical conductance plateaus predicted by the Lan-dauer theory appeared (see section 2.3.3 and Fig. 3.3d) top), as well as the characteristic conductance histogram (see Fig. 3.4a)). By applying a voltage Vh at the microheater it was possible to control the local temperatures. The simultaneously recorded thermovolt-age ∆V is depicted in Fig. 3.3d) bottom for two different heat input voltages, whereby forVh= 0.5 V the thermoelectric signal is very small compared to the Vh = 2.0 V. While the conductance decreases stepwise, both curves show large fluctuations also on the con-ductance plateaus.
Tsutsui et al. summarize that the thermopower of Au atomic-size contacts is mainly dependent on geometry and can adopt both signs, positive and negative. Fluctuations of geometric contact changes were minimized by data averaging so that conductance-dependent thermopower oscillations can be observed. In the experiments of Ludoph et al. (see Ref. [58]) no averaging took place and the oscillations were superposed by the fluctuations.
Additionally they explain the negative sign of the thermopower with a theory for a bal-listic one-dimensional system that predicts thermopower quantizations at SC = e(n+kBln 21
2) ≈
−(n+601
2)µV K−1 for the nth sub-band, withkB the Boltzmann constant ande the electron charge.
Fig. 3.4 shows measured thermovoltage values ∆V. In b) the quadratic behavior of the peak height ∆Vp (see inset) versus heat input voltage Vh that indicates the Joule heat-ing of the mircoheater is shown. Oscillations occur by averagheat-ing thermovoltages ∆Vave which are within a window of 0.02 G0 (see Fig. 3.4c)). Close inspection of the maxima at the positions Gm,1, Gm,2, Gm,3 shows that they are independent of the input heat Vh (Fig. 3.4d)). The standard deviation Qof ∆Vave shows also oscillations but for multiple integers of G0, as predicted by the coherent backscattering model (see Fig. 3.4e)). Again, the maxima of e) are irrespective of Vh (f).
Additional measurements performed with the same setup, published in Ref. [2] confirm the thermopower oscillations, see Fig. 3.5. In Fig. 3.5a) the black circles represent the raw data and the red circles the values of the thermovoltage which are again averaged over a window of 0.02 G0 plotted against the conductanceGand Gaussian fits emphasize the oscillations Fig. 3.5b).
In order to calculate thermopowers, the temperature is estimated from the average con-tact lifetimeτ =f0−1exp(−EB/kBTC) withf0the attempt frequency,EBthe critical bond strength in the contact, kB the Boltzmann constant as well as the effective temperature at the breakpoint TC . In this measurement an averaged lifetime of τ = 6.2 s for a data base of 50 conductance traces was observed at (1.0±0.2) G0. To do so, a temperature
a) d)
b) e)
c) f)
Figure 3.4: a) Conductance histogram of 50 traces under a heat input ofVh= 2.0 V. b) Peak heights
∆Vpof the thermovoltage (see inset) plotted against the applied heat voltage Vh. The dotted blue line is a fit which indicates the quadratic increase. c) Averaged thermovoltages∆Vave versus conductance G. ∆V within a window of0.02 G0 were summed up to values∆Vave and show so-called thermopower oscillation. Three peaks occur at the positionsGm,1, Gm,2, Gm,3which were fitted by a Gaussian function and drawn with solid lines. d) The three positionsGm,1, Gm,2, Gm,3are independent ofVh. e) Standard deviation Q of the thermovoltage ∆V versus conductance G. The blue lines present the theoretically predicted coherent backscattering model. f) Again, the positions of local maxima in e) are independent ofVh. Taken from Ref. [3].
a) b)
Figure 3.5: a) Black circles show the thermovoltage versus the conductanceG. The red dots symbolize the signals averaged over a window of0.02 G0. b) Gaussian fits emphasize the oscillations. Taken from Ref. [2]
increase from ambientT0 = 293 K to 312 K is assumed. The temperature profile of a heat input voltage of Vh = 2.2 V across the sample is drawn in Fig. 3.6 accounting for the temperature of the microheater of T0 = 402 K determined by the change in resistance.
More detailed calculations regarding the technique of temperature determination by an-alyzing the average lifetime τ of the contacts are reported in Ref. [70].
Moreover, Tsutsui et al. [3] analyzed the geometry-dependent thermovoltage
fluctua-Figure 3.6: Temperature profile (red line) of a heat input voltage ofVh= 2.2 Vacross the sample. The maximumTh was determined by the change of resistance of the Pt coil. TC= 312 Kand T0 = 293 K are the temperatures on both sides of the junction. Taken from Ref. [2].
tions of single-atom contacts. To this end a conductance trace was divided into different regions (see Fig. 3.7a)) and compared with transmission curves and their derivative ∆V. For region I at 1 G0, where a transmission channel is fully opened, the transmission peak is close to the Fermi energy and therefore ∆V is very small. An elongation of the contact in region II leads to a shift of the transmission and increased thermovoltage. The
step-wise decrease of conductance in region III causes an additional shift of the transmission resulting in monotonic decreasing ∆V.
In Ref. [63] properties of BDT single molecular junctions were investigated with the same MCBJ setup and techniques as described above and a thermopower of 15µV K−1 at a con-ductance 0.01 G0 was observed for the suggested electrical transport through the highest occupied molecular orbital (HOMO).
a)
b)
Figure 3.7: a) Conductance versus time at the top, below the simultaneously measured thermovoltage
∆V atVh= 2.0 V. The elongation process is divided into 3 regions and compared to the corresponding transmission curves and the deduced∆V, both plotted againstE−EFin b). In region I the transmission peak is close to the Fermi energy and ∆V is very small. In region II the transmission peak is shifted due to the elongation of the contact which results in an increased signal of∆V. The stepwise decrease in conductance (region III) leads to a monotonic decrease in conductance that can be explained by the additional shift of the transmission away fromEF.
Evangeli et al.
d) c)
b) a)
Figure 3.8: Thermopower and conductance measurements of Evangeli et al.. a) Electronic setup of the modified STM. b-d) ConductanceGand thermopowerS versus distance between tip and substrate.
Changes in thermopower signal mainly due to atomic rearrangement of the contact. Taken from Ref. [66].
A recent publication by Evangeli et al. [66] reports simultaneous measurements of thermopower and conductance of Au-Au and Pt-Pt atomic-size contacts2. The STM technique was used at room temperature (see Fig. 3.8a)) generating the temperature difference by heating the STM tip with a resistive element to reach differences of 20 K and 40 K respectively between tip and substrate.
Fig. 3.8b)-d) show simultaneous measurements of conductance and thermopower during the breaking of the Au-Au contact. The typical stepwise change in conductance and changes of thermopower due to atomic rearrangements appear as in previous measure-ments of other groups [2, 3, 58]. Even for small changes of conductance (for example in Fig. 3.8c)), the thermovoltage fluctuates immensely adepting both positive and neg-ative values. For large conductance values G >104G0 the measured thermopowers of
≈1µV K−1 are in good agreement with bulk values in literature [71] (Fig. 3.9a)). By de-creasing the conductance, the sign changes to a negative value (Fig. 3.9b)-c)). Maxima of the thermovoltage occur at the positions of the conductance minima (Fig. 3.9c)), this behavior will be discussed later.
The measurements were compared with simulations of 100 stretching events. These
sim-2Here, primarily the results of the Au-Au contacts are discussed, since in this work the same kind of contact was used.
a)
b)
c)
d)
e)
f)
Figure 3.9: a) ThermopowerS versus conductanceGfor the range from10 G0 to 104G0. For large contacts the thermovoltage reaches the positive bulk value, whereas for atomic-scaled contacts the sign of the thermopower switches to a negative sign. b) Density plot ofSin the range of1 G0to10 G0where a small negative averaged value appears. c) The range between 1 G0 to 4 G0 exhibits thermovoltage oscillations. Simultaneously measured conductance reveals shifted oscillations. d) Simulations of the thermopower confirm the measured data in the range of1 G0 to10 G0, where a negative average value occurs, but as well positive and negative due to fluctuations. e) Computed conductance histogram with a prominent peak at1 G0. f) Largest transmission coefficients are plotted against the conductance. In the range close to1 G0 is dominated by a fully opened transmission channel. Taken from Ref. [66].
ulations show as well fluctuating values for the thermopower (see Fig. 3.9d)) with both, positive and negative sign due to the reconfiguration of the contact which is in good agreement with the experiment. Here, the conductance maxima (see Fig. 3.9e)) are also correlated with the minima of the thermopower. Analysis of the transmission coefficients (see Fig. 3.9f)) emphasizes that the elimination of the thermopower fluctuations close to 1 G0 is caused by the transport through a fully opened conductance channel. Therefore, the transmission curve has a maximum at the Fermi energy with a zero slope which leads to the suppression of the thermopower.
Evangeli et al. further suggest that theoretically the sign of the thermopower is de-a)
b)
Figure 3.10: Four different Au contact configurations lead to four transmission curves as a function of the energy. a) All four curves have a positive slope at the Fermi energy effecting a negative thermopower.
b) Local density of states as function ofE−EF. The transmission is dominated by thesandpz(transport direction) orbitals with a positive slope at the Fermi energy. Taken from Ref. [66].
termined by the electron-hole asymmetry in the transmission function since the slope around the Fermi energy is usually positive for gold leading to a negative thermopower (see Fig. 3.10a)). This was proven by simulating the transmission curves of four different atomic configurations of the contact. The local density of states (DOS) are depicted in Fig. 3.10b) revealing the transport through thes andpz (in transport direction) orbitals with a positive slope.
Finally Evangeli et al. compare their results to previous works of Ludoph et al. (see Ref. [58] and above), Morikawa et al. and Tsutsui et al. (see Refs. [2, 3, 63] and above) showing similarities such as the jumps in the thermopower signal due to atomic rearrange-ments of the contact (see Fig. 3.8b)-d)) as well as the fluctuations by fewµV K−1 around zero (Fig. 3.9a)-b)). However, unlike in Ludoph’s work, the averaged thermopower is negative for small contacts. The vanished averaged thermopowers in the measurements of Ludoph can be explained by the low temperature base (12 K) causing the signal to be 30
times smaller. This assumption is confirmed by the theoretical result of Evangeli et al..
For larger contacts with G <104G0 (Fig. 3.9d)), the sign of the averaged thermopower is positive and the value corresponds to the bulk measurements [71].
Against Ludoph’s explanation of the fluctuations due to impurities and defects, the simu-lation of Evangeli show that the fluctuation can be traced back to atomic rearrangements at the contact. Further Tsutsui’s interpretation of the free electron model describing ballistic quantum point contacts is challenged since it can predict only negative thermo-voltages and does not explain the measured positive values (Evangeli et al.).
Lee et al.
Lee et al. [72] developed a so-called custom-fabricated nanoscale thermocouple integrated scanning tunneling probe (NTISTP) to detect heat dissipation in atomic-scale junctions.
There the relation between electronic transmission characteristics of molecular and atomic contacts and the properties of heat dissipation were studied. To explore this relation an STM tip was used, where a gold and chromium thermocouple is integrated to measure the temperature. The layout of the tip is illustrated in Fig. 3.11a)-b), where the metallic layers gold-chromium-gold are separated by an electrically insulating but thermal con-ducting 70 nm thick layer of silicon nitride.
In order to perform heat dissipation measurements, a gold substrate serves as second b)
a) c)
Figure 3.11: Taken from Ref. [72]: a) SEM image of the nanoscale thermocouple integrated into a STM tip. b) design layout of the tip with a tapped molecule between the two electrode. c) investigated contacts, Au-Au, Au-BDNC-Au and Au-BDA-Au junction.
electrode, where also molecules can be trapped in, here 1,4-benzenediisonitrile (BDNC) and 1,4-benzenediamine (BDA).
By applying a bias voltage along the tip and the substrate, a temperature rise ∆TTC =
−∆VTC/STC of the thermocouple occurs and the heat dissipation QP can be derived by QP = ∆TTC/RP, with RP the thermal resistance of the NTISTP and STC the Seebeck coefficient of the thermocouple. Both values, RP and STC were determined experimen-tally.
The heat dissipation of over 100 Au-Au contacts with (1.0±0.1) G0 at each bias was mea-sured. For negative and positive biases the time-averaged temperature rises ∆TTC, Avg were recorded and following the time-averaged power dissipation in the tip QP, Avg de-termined. The definition of positive and negative bias: For positive bias, the probe is grounded while the substrate is at a higher potential and vice versa for negative bias.
One can see in Fig. 3.12a) that there is a proportional behavior between ∆TTC, Avg and QTotal, Avg without any differences between positive and negative bias, so no
asymme-a) b)
Figure 3.12: a) Symmetric heating of Au-Au contacts, even for high power (inset). b) Computed transmission function of the Au-Au which shows a weak energy dependence in the range of the Fermi energyEF. Taken from Ref. [72].
try in the power dissipation can be observed. This result agrees with theory, where the symmetry is predicted due to the weak energy dependence of their transmission function (see Fig. 3.12b)), which also explains the vanishing average thermopowers in previous works [58]. The calculated zero-bias transmission for the contact in the upper left inset is depicted in Fig. 3.12b is almost energy independent over 1 eV nearby the Fermi energy.
By trapping molecules between the NTISTP and the substrate, the symmetry of positive and negative bias is broken. The following asymmetry depends on the molecule, here
By trapping molecules between the NTISTP and the substrate, the symmetry of positive and negative bias is broken. The following asymmetry depends on the molecule, here