Thermopower of Atomic-Size Contacts at Low Temperature

Low Temperature

Dissertation submitted for the Degree of Doctor of Natural Sciences (Dr. rer. nat.)

presented by

Bastian Kopp

at the

Faculty of Mathematics and Natural Sciences Department of Physics

Date of the oral examination: 04.08.2016 First referee: Prof. Dr. Elke Scheer Second referee: Prof. Dr. Paul Leiderer

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-0-361403

1 Introduction 9

2 Theoretical Background 11

2.1 Electrical transport in bulk metals . . . 12

2.1.1 Bloch theory . . . 12

2.1.2 Matthiessen’s rule . . . 13

2.1.3 Electron-phonon scattering . . . 13

2.1.4 Electron-electron scattering . . . 14

2.2 Resistivity of thin films . . . 14

2.2.1 Fuchs-Sondheimer model . . . 15

2.2.2 Mayadas-Shatzkes model . . . 16

2.3 Charge transport through mesoscopic systems . . . 19

2.3.1 Mesoscopic systems . . . 19

2.3.2 Historical background of charge transport theories . . . 19

2.3.3 Theory of mesoscopic charge transport . . . 20

2.3.4 Conductance quantization . . . 21

2.3.5 Atomic-size contacts . . . 22

2.4 Thermoelectricity in metals . . . 23

2.4.1 Seebeck effect . . . 23

2.4.2 Peltier effect . . . 25

2.4.3 Figure of merit . . . 25

3 State of the Art 27 3.1 Publications: Thermopower of atomic-scale Au-Au contacts . . . 28

5

3.2 Publications: Techniques for determining the thermopower of molecular

junctions and thermoplasmonic . . . 43

3.3 Summary . . . 49

4 Experimental Background 51 4.1 Sample preparation . . . 53

4.1.1 MCBJ . . . 53

4.1.2 NThC . . . 55

4.2 Experimental setup: MCBJ samples . . . 58

4.2.1 MCBJ principle and setup . . . 58

4.2.2 Cryogenic system . . . 61

4.2.3 Optical setup . . . 61

4.2.4 Electronic setup . . . 63

4.3 Experimental setup: NThC . . . 65

4.3.1 Optical setup . . . 65

4.3.2 Electronic setup . . . 65

5 Results and Discussion on Nano-Thermocouples at Room Temperature 67 5.1 Previous results . . . 68

5.2 Results . . . 70

5.2.1 Cu-Au-Cu nano-thermocouples . . . 71

5.2.2 Ag-Ni-Ag nano-thermocouples . . . 74

5.3 Discussion . . . 76

5.3.1 Thermovoltage . . . 76

5.3.2 Temperature-dependent resistance change . . . 78

5.4 Summary NThC measurements . . . 81

6 Results and Discussion on MCBJ at Low Temperature 83 6.1 Characterization measurements . . . 84

6.1.1 Sample . . . 85

6.1.2 Setup characterization . . . 86

6.1.3 Data acquisition concepts of the electronic measurements . . . 91

6.2 Results and discussion of the temperature determination . . . 101

6.2.1 Resistivity measurements . . . 102

6.2.2 Spatially resolved resistance change measurements . . . 106

6.2.3 Simulation . . . 109

6.3 Results of thermovoltage measurements . . . 118

6.3.1 Spatially resolved thermovoltages . . . 119

6.3.2 Influence of the laser intensity on the thermovoltage . . . 121

6.3.3 Conductance dependence of the thermovoltage . . . 122

6.3.4 Influence of the laser intensity on the thermovoltage of atomic-size contacts . . . 127

6.4 Discussion of thermopower results . . . 129

6.4.1 Results . . . 129

6.4.2 Discussion . . . 130

7 Outlook 135 8 Summary and Conclusion 137 9 Zusammenfassung 141 10 Acknowledgements - Danksagungen 145 A Cryostat 147 B Simulation 149 B.1 Material properties and parameters of the simulation . . . 149

C Further measurements with the temperature circuit 151 C.1 Cross section at room temperature . . . 151

C.2 Lock-in measurements . . . 151

Introduction

Thermoelectric effects allow the conversion of thermal energy into electricity without any mechanical intermediate step. Other motivations contain that thermoelectric devices would be an important step toward a more environmentally friendly use of energy [1],are considered as one of the ideal systems for harvesting that can directly convert heat energy into electricity [2] or even an ideal route for power generations being a greenhouse gas emission-free technology [3].

This work may represent a piece of the puzzle to save the worlds energy problems?!

Residual heat is produced in countless processes around the modern world, e.g. in car engines, personal computers and refrigerators, and escapes into the environment. Even more, some of the devices have to be cooled under further (electric) energy consumption.

The big issue of the thermoelectricity is its inefficiency. For metal combinations only a few µV per K temperature difference are generated. To be realistic, such small voltages cannot solve the energy problems of the world. To increase the figure of merit, ZT, which describes the efficiency of thermoelectric heat conversion, fundamental research is stringently required. First approaches with atomic-size and molecular junctions reveal a promising progress in this topic. Hence, the miniaturization in this field yields not only to a higher integration density, it has also positive influence on the efficiency. The understanding of the thermoelectricity on the atomic scale is essential to enhance the energy output.

The aim of this work is to contribute to this process by the development of a novel setup for investigations at low temperature. To do so, a cryogenic system with a vertical mechanically controllable breakjunction mechanism was established to create atomic-size

9

contacts at low temperature.

The temperature difference across the contact was achieved by a moveable laser spot on the sample. Compared to other nano-thermoelectricity realizations, which use the Joule heating effect of fixed wires on the samples for the creation of a temperature difference, this setup allows a spatially variable heating on the sample.

This thesis presents the development of this system and the results of the thermopower of atomic-size gold contacts at low temperature. Therefore, the next chapter 2 offers an overview over theTheoretical Background which is required for the understanding of this work, followed by a more detailed study of theState of the Art (chapter 3) of the recent publications in the field of nano thermoelectricity.

Chapter 4 reports the Experiment Methods, hence the sample fabrication as well as the cryogenic, electronic and optical setups. Moreover, the results of this work are divided into two chapters, namely in Results and Discussion on Nano-Thermocouples at Room Temperature (chapter 5), and Results and Discussion on MCBJ at Low Temperature (chapter 6), where the results of the low temperature measurements are further devided into the sections Characterization measurements (section 6.1), Results and discussion of the temperature determination (section 6.2), Results of thermovoltage measurements (section 6.3) and Discussion on thermopower results (section 6.4). Moreover, an Outlook (chapter 7) is provided that describes open questions and further possibilities with the developed system. This thesis closes with a Summary and Conclusion in chapter 8.

As to applications, every technology had to pass different stages of development until it was sold on the mass market. The thermoelectricity has the potential for such a development and hopefully this work can support this process.

Theoretical Background

This chapter offers the reader the theoretical background of this work and should help to understand the presented effects in later sections.

First, a brief introduction is given which covers general ascepts of electronic transport and the resistivity of thin films. Furthermore, the models of Fuchs-Sondheimer and Mayadas- Shatzkes that describe the resistivity in thin films are introduced.

In addition, the charge transport through mesoscopic systems will be explained to un- derstand the behavior of atomic-size contacts.

Finally, thermoelectricity effects, in particular the observations from Seebeck and Peltier, will be considered.

This chapter is based on the manuscripts Refs. [4–6] with contributions by Refs. [7, 8].

11

2.1 Electrical transport in bulk metals

Based on the classical kinetic theory of gases, Paul Drude posited 1900 a model (free electron gas model) for the charge transport of free electrons in metals. There, the elec- trons are considered to be point-like objects, which do not interact with each other. With adjustments regarding the quantum behavior, this model serves very well as description for the charge transport of simple metals.

The conductivityσ, which is defined as the constant between an external electric field E and the electric current density (J = σE), is limited by the scattering of the electrons.

Drude’s model describes this scattering at lattice ions of the crystal, but it does not match with the reality completely, since at low temperatures mean free paths of up to 1 cm can occur for very pure metals. According to Drude, the distances between lattice ions would lead to mean free paths of 1 ˚A.

The electronic conductivity can also be described by Boltzmann transport theory. Refs.

[9, 10] provide a detailed insight, whereas here only a brief summary is given. The con- ductivity can be described by

σ = N e²τ me

= N e²l_e mevF

(2.1) withN the charge carrier density,τ the mean time between two scattering events, m_e the electron mass,l_e=v_F·τ the electron mean free path, andv_F the velocity of the electrons.

To calculate the Fermi velocity with v_F = ~k_F/m_e only a limited amount of electrons can contribute to the scattering event. The wave vectors have to be similar to the Fermi wave vector k_F =√

2·m_eE_F/~with the Fermi energy E_F as shown in Fig. 2.1. There it is illustrated that scattering events can happen if the energy of the electrons are in the (green) interval of the thermally spread out range (kBT, with kB Boltzmann constant) aroundEF due to Pauli’s exclusion principle. In the (orange) region, in which the energies of the electrons are much lower thanE_F, scattering events of electrons are disabled since there are no free states available.

2.1.1 Bloch theory

The model of the free electrons in a periodic background potential (like lattice ions) is generalized by Bloch’s theory. In contrast to Drude’s model, the electric resistivity is not generated by the scattering of electrons with the lattice ions. Bloch’s theory shows that the wave vector of an electron in a defect-free and infinite crystal is preserved.

Reasons of this resistivity are deviations of the perfect periodicity of the lattice. On the one hand they can arise from defects and impurities of the crystal, on the other hand from thermal vibrations of the ion lattice. These vibrations are quantized quasi-particles

k_y

k_x k_F

k_BT << E_F

Figure 2.1: Illustration of possible energiy regions for electron scattering. Due to the Pauli’s exclusion principle the scattering process is limited to the green region aroundEF within the thermally softened interval ofkBT.

and are called phonons.

2.1.2 Matthiessen’s rule

The inverse of the conductivity is the resistivity withρ=σ⁻¹. According to Matthiessen’s rule, the contributions to the resistivity due to the different kinds of scattering (electron- phonon, electron-defect, electron-impurities and electron-boundary) can be summed up to a total resistivity, if the various mechanisms do not influence each other.

ρ=ρ_ph+ρ_defect+ρ_impurities+ρ_boundary (2.2) whereinρphis the contribution of the electron-phonon,ρdefectthe electron-defect,ρimpurities

the electron-impurity and ρ_boundary the boundary scattering. Here the first part, ρ_ph vanishes at T for 0 K, whereas the other parts (ρ_res = ρ_defect+ρ_impurities+ρ_boundary) are independent of the temperature.

2.1.3 Electron-phonon scattering

The most prominent consequence of the electron-phonon scattering is the temperature- dependent resistivity of metals. For example, gold has a resistivity of 22 nΩ m at room temperature, but only 4.8 nΩ m at 80 K [11]. This is caused by the electron-phonon scat- tering. The higher the temperature, the more phonons are excited and more scattering events take place, which increases the resistivity. Above the Debye temperature the num- ber of thermal phonons is roughly proportional to the absolute temperature. Within this temperature range, the resistivity increases linearly with the temperature. Well below the Debye temperature, the resistivity behaves like T⁵, which is depicted in Fig. 2.2.

0 K 0.1Θ_D

ρ

ρ

~ T

ρ

~ T

Temperatur

Reistivity

phonon scattering

Figure 2.2: Illustration of the temperature dependence of the electrical resistivity of metals. For temperatures far below the Debye temperatureΘ_D[9] a T⁵ behavior occurs, above a linear one.

2.1.4 Electron-electron scattering

Another scattering mechanism is caused by the interaction of the electrons among each other. The probability for such a scattering event is very small compared to the electron- phonon scattering. The Pauli exclusion principle has to be taken in account for the incident electron and the scattered electron [12].

At higher temperatures, theT² behavior of the electron-electron scattering is not visible since the electron-phonon scattering dominates. Even at lower temperatures, regions in which the electron-phonon scattering accords to T⁵ and should decrease very fast, the electron-electron scattering cannot be observed due to the scattering off impurities.

2.2 Resistivity of thin films

Bulk materials are primarily classified by the influence of the sample boundaries. Based on the ratio of the boundaries compared to the dimensions of the system, the influence of the boundaries is negligible. The resistivity is mainly given by the mean free path of the electrons. Due to the occurrence of a further limitation of the mean free path the resistivity increases, besides the already presented scattering opportunities.

Depending on the temperature and the purity of the crystals, the mean free path can adopt dimensions in the order of the thin film thickness and therefore further scattering mechanisms occur which are negligible in bulk. Therefore, the scattering of electrons on grain boundaries and on internal surfaces have to be taken into account. A model to describe the scattering on grain boundaries is described by Mayadas and Shatzkes [13,14],

whereas the Fuchs-Sondheimer model explains the scattering at the surface [15, 16].

2.2.1 Fuchs-Sondheimer model

In 1901, Thomson [17] had already the idea that the dimensions of a system could in- fluence the resistivity. Based on the Boltzmann transport equation, Fuchs postulated 1938 [15] a size-effect for a free-electron model which can be simplified by [18]:

• disorder of the film is independent from the film thickness

• clean and parallel surfaces of the film

• a spherical Fermi surface

• isotropic scattering events

p = 0 p = 1

d

Figure 2.3: Description of parameter pof the Fuchs-Sondheimer model, which describes the fraction of the electrons which are scattered specularly.

Fig. 2.3 illustrates the phenomenological specularity parameterp. For p= 0, a complete diffusive scattering of the electrons takes place, whereas for p = 1 the electrons were scattered according Snell’s Law, so-called specular reflection.

A way to calculate the conductivity of the film σ_f is σ_f

σ₀ = 1− 3 2κ₀

Z ∞

1

1 t³ − 1

t⁵

1−e^κ0t

1−pe^κ0t dt (2.3)

with σ0 the bulk conductivity and κ0 = _l^d

0, the ratio between film thickness d and the mean free path l0 adopted from a corresponding bulk system (for example density of impurities or lattice defects).

A further modification simplifies the formula since Eq. 2.3 is only numerically solvable.

For thick films (dl₀) one obtains [16]:

σ_f σ₀ = ρ₀

ρ_f = 1 + 3 8

l₀

d(1−p) (2.4)

and for thin films (dl₀) σ_f σ0

= ρ₀ ρf

= 1 + 4 3

(1−p) (1 +p)

l₀

dln^l_d⁰ (2.5)

In summary, the Fuchs-Sondheimer theory describes the scattering of electrons on parallel and smooth surfaces with a scattering parameter p. The scattering on a rough surface cannot be explained with this model. However, inspired hereby, further works integrated the roughness into their models [12, 19–22]. Also in some models a second parameter q was introduced due to the fact that the second surfaces of a film can differ from the other one [23–25].

2.2.2 Mayadas-Shatzkes model

Mayadas and Shatzkes take another approach to describe the resistivity of thin films [13].

They considered the scattering at grain boundaries of polycrystalline metals, which occur between two crystals. The ground boundaries were treated as potential barriers, where the electron can either be transmitted or reflected. Therefore, a grain boundary reflection coefficient R_gb was introduced, whereby an increase of the resistivity for increasing R_gb was observed. In this model the differences between specular and diffusive scattering is negligible, compared to the approach of Fuchs and Sondheimer. Another length scale has to be introduced to describe the resistivity, the density of the grain boundaries, here denoted with the average grain size D.

Since Mayadas and Shatzkes developed a 1D model only, some assumptions and sim-

D

R

Figure 2.4: Sketch of the Mayadas scattering model withDthe distance between two grain boundaries andRgbthe grain boundary reflection coefficient.

plifications [26] had to be done. Only those grain boundaries were taken into account which are aligned perpendicular to the electric field. The contributions of all other grain boundaries have negligible effect on the total resistivity. In addition, the remaining perpendicular grain boundaries were described by a Gaussian distribution of planar scat- terers, with the averaged distance ¯Dand the standard deviations. Potentials of the form V_x =Sδ(x−x_n) describe these scatterers at the position x_n with S the strength of the

scatterer andδ the Delta Dirac function.

The relation between strength of the scattererS and the grain boundary reflection coef- ficient R_gb is given by

S² = ~³

2m_ev_Fk_F R_gb

1−R_gb (2.6)

with Fermi velocity v_F and Fermi wave vector k_F.

The conductivityσ_g for polycrystalline metals can be derived from the Boltzmann equa- tion. Thereby, the conductivity inside a crystalσ₀ is multiplied by the scattering rate.

σ_g =σ₀·f(α) (2.7)

l0 is the mean free path inside the crystal, according to σg. f(α) = 1−3

2α+ 3α²−3α³ln

1 + 1 α

(2.8)

α= l0

D Rgb

1−R_gb (2.9)

Since this first model of Mayadas-Shatzkes describes the scattering on grain boundaries only, they extended it for thin films according to the model of Fuchs-Sondheimer. In addition, film surfaces were also considered and lead to the expression for the film con- ductivity σf [14]:

σ_f =ρ⁻¹_f = 1

ρ_g − 6 πκ₀ρ₀

Z π/2

0

dφ Z ∞

1

dt cos²φ H²(t, φ)

1 t³ − 1

t⁵

1−e^{−κ0tH(t,φ)}

1−pe^{−κ0tH(t,φ)} (2.10) with σ_g =ρ⁻¹_g , σ₀ =ρ⁻¹₀ , κ₀ =d/l₀ and the definition:

H(t, φ) = 1 + α cosφp

(1−1/t²) (2.11)

Eq. 2.10 reveals that the specularity parameter p (Fuchs-Sondheimer) and the grain boundary reflection coefficient quantities R_gb (Mayadas-Shatzkes) of the models can be derived simultaneously. Furthermore, it is shown that p and Rgb are not independent which leads to the finding that Matthiessen’s rule is not valid for this boundary consid- eration. Nevertheless, for thick films (dl₀) Matthiessen’s rule is valid since pand R_gb are independent, so the resistivity of a thin film can be written as:

ρ_f =ρ₀+ρ_ss+ρ_gb (2.12)

The contribution of the film surface (Fuchs-Sondheimer) amounts to ρ_ss = 3

8(1−p)ρ₀l₀

d (2.13)

whereas grain boundary scattering leads to ρ_gb = 3

2ρ₀ R_gb 1−Rgb

l₀

D (2.14)

Other groups worked on simplifications and statistical models of these findings, as re- ported in Refs. [27–32].

Influence of the film thickness

Figure 2.5: Resistivity as a function of the film thickness for the Fuchs-Sondheimer (in a)) and the Mayadas-Shatzkes model (in b)). Two kinds of grains must be considered in the model of Mayadas- Shatzkes. The columnar grains are thickness independent, whereas the spherical grains reveal a behavior like the Fuchs-Sondheimer model. Taken from Ref. [4].

Fig. 2.5 illustrates the resistivity as a function of the film thickness for both models.

The Fuchs-Sondheimer model reveals a reciprocal behavior for the resistivity (the thinner the film thickness, the higher the resistivity), while in the Mayadas-Shatzkes model two kinds of grains have to be considered. The first kind is the columnar grain, which yields a resistivity independent of the film thickness. However, for the second kind, the spherical grains are proportional to the film thickness. Like in the model of Fuchs-Sondheimer, a reciprocal behavior of the resistivity versus the film thickness can be observed. Since for metal films the grains grow constantly with the film thickness, the thickness dependence of both models are very similar, although two types of scattering were considered.

2.3 Charge transport through mesoscopic systems

2.3.1 Mesoscopic systems

The charge transport through mesoscopic systems is one of the fields in physics which cannot be described by a simple downscaling of the macro-scale physics. Some phenomena occur in the range between macro and micro, the mesoscopic scale.

Responsible for these phenomena are different quantities which play no role in macro- scale physics, since their sizes are negligible compared to the dimensions of the system.

In decreased dimension the phase coherence length (l_φ ∼ 1µm), the mean free path of electrons (l_e ∼ 100 nm) and the Fermi wavelength of electrons (λ_F ∼ 5 ˚A) influence the charge transport. Fig. 2.6 shows three types of transport through a conductor which occur in this size scale.

l

w

diffusive conductor (quasi-) ballistic conductor quantum point contact

Figure 2.6: Three types of charge transport through mesoscopic conductors. Left-hand side: Diffusive transport with l_φ > w, l l_e, center: ballistic transport with l, w, < l_e and on the right-hand side quantum point contact withl, w.λ_F

2.3.2 Historical background of charge transport theories

So far the basics of the electronic transport were explained, this part provides a short historical background.

Inspired by the discovery of the wave-particle duality in 1927 [33], Felix Bloch developed in 1928 [34] a theory about the free metal electrons, which could explain the influence of the underlying crystals structure.

Another milestone in the history of the charge transport is the microscopic theory of Rolf Landauer. His theory merges the established methods of the quantum theory and atomic physics. The scattering theory and the description of the electronic eigenstates enabled Landauer’s theory to explain electronic conductors with a restricted geometry.

Markus B¨uttiker extended this theory 1986 [35] from an effective two-point contact to a four-point contact. The Landauer-B¨uttiker formula allows the description of mesoscopic systems as well as system following Ohm’s Law.

2.3.3 Theory of mesoscopic charge transport

The theory of mesoscopic charge transport will be explained on the basis of Fig. 2.7, which is taken from Ref. [5]. There, a model of the charge transport in mesoscopic conductors is shown. By applying a voltage U between the left-hand and the right-hand side of the constriction, the chemical potential difference amounts to −eU. The band edge of the electrons in the leads is represented by the Fermi function, which is thermally rounded by about 2k_BT. The occupied and unoccupied states are located on the parabolic dispersion relation. In mesoscopic conductors only a few discrete states are available as transport modes. This assumption applies for systems much larger than atomic scale, when the eigen states are well-described by Bloch waves.

The transport of electrons through a conductor is only possible, if it happens between

E

k 0

k 0

eU

E E

f(E) f(E)

E

k μ_L

μ_R

μ_L μ_R

0

0 0

-U 0V

Figure 2.7: Model of the charge transport through a mesoscopic ballistic conductor with an applied voltage ofU at a finite temperature. Taken from Ref. [5].

occupied state of one, let’s say the left, reservoir to an unoccupied state in the other, let’s say right, reservoir. It is required that the conductor reveals free states, so-called channels, within this energy interval.

For the calculation of the overall current through the mesoscopic conductor, one has to consider first a single channel. Its contributions mainly depend on the overlap of the wave functions of the states in the electrodes. However, even for a single channel an upper limit appears, which cannot be exceeded, and which symbolizes a universal constant.

This approach results in the following rough approximation for the conductance G= I

U = ∆Q

∆t · 1

U . (2.15)

For the transport of an electron with ∆Q=e and ∆E =e·U one obtains:

G= e²

∆E·∆t (2.16)

Using the energy-time uncertainty relation ∆E·∆t ≥has argued in Ref. [36], a transport process within the energy interval of ∆E =µ₂−µ₁ would result in a dwell time of at least

∆t = h/∆E for the electron. Due to Pauli’s exclusion principle, the channel is blocked for this time span. Therefore, the conductance for one channel has an upper limit:

G= e²

h (2.17)

Since two spin states are allowed, the twice the maximum conductance is called conduc- tance quantum:

G₀ = 2e²

h (2.18)

The sum of all available channels lead to the equation G= 2e²

h X

i

τ_i (2.19)

which is called the Landauer formula. Thereby, τ_i can adopt values between 0 and 1.

The so-called transmissions denote the probability of an electron passing the constriction without being backscattered.

The derivation is a simplification of the original publication of Ref. [37], but nevertheless it shows the conductance limitation of individual channels.

2.3.4 Conductance quantization

In 1988, a mesoscopic conductor was realized in an experiment of Wees et al. [38]. In a 2D electron gas of a semiconductor it was observed that only channels with transmission of 1 contribute to the charge transport in this system. The system was realized by two gate electrodes whereby the variation of the voltage yields a constriction of the depletion region and thus a limitation of the conducting channels. Fig. 2.8a) shows an opening curve with well-defined steps, with small rounding Fermi edges caused by the base tem- perature of 0.6 K.

In this experiment the conductance changed always by steps of 1 G0, but in other ex- periments conductances with a non-integer number of conductance quanta G₀ occurred.

Therefore, a quantization of the conductance is not a general feature.

a) b) c)

Figure 2.8: a) Conductance quantization of a ballistic conductor with variable width. b) Sketch of the experiment. c) Transport modes formed by Bloch waves and clasified by their transverse momentum.

Taken from Ref. [5].

2.3.5 Atomic-size contacts

Classifying transport modes by their transverse wave vector as shown in Fig. 2.8c) implies a certain potential of the waveguide. For spherical core potentials of single atoms such modes are not possible. Fig. 2.9a) illustrates the potentials of an atom. Therefore, it is assumed that the electron wave functions adopt the shapes of the atom orbitals by passing the contact.

Fig. 2.9b) shows the atomic orbitals schematics of a free aluminum atom. The valence electrons appear in dumb bell shaped p-orbitals and in spherical s-orbitals. Is an atom linked to both leads, the energy eigenstates of the atomic orbitals change and the density of states is widened (see Fig. 2.9c)). In addition, an overlap of initially discrete levels can happen, especially at the Fermi energy E_F. As a consequence, electrons of the leads can occupy the orbitals temporarily and reach the other side of the contact. The probability of occupancy is proportional to the density of states at the Fermi energy. Therefore, some orbitals are more conductive than others. A quantization in multiples of G₀ is not expected for atomic-size contacts.

The number of modes, which contribute to the charge transport of single atom contacts, depend on the number of valence orbitals. This behavior was observed by Scheer et al. [39, 40] for superconducting aluminum contacts. Furthermore, the transmission values of individual channels were determined.

a) b) c)

p-orbitals s-orbitals

Figure 2.9: Atomic-size aluminum contact. Taken from Ref. [5].

2.4 Thermoelectricity in metals

In this section, thermoelectric effects will be considered. The two prominent effects are the Seebeck effect which generates a thermovoltage due to an applied temperature and the inverse observation (Peltier effect), in which a temperature gradient occurs by applying a current. Furthermore, the figure of merit, ZT, will be considered that describes the efficiency of thermoelectric heat converters.

2.4.1 Seebeck effect

The Seebeck effect is the most prominent effect in the field of thermoelectricity. It describes the generation of a voltage (for an opened electronic circuit) in absence of an applied electric field, when two different materials are connected and their ends are kept at different temperatures. Fig. 2.10 illustrates the heat distribution of a conductor, when the ends of the conductor are kept on different temperatures. The Fermi distribution of the hot side is thermally more softened, which yields higher energies of the electrons on this side compared to the cold one. As a result, above E_F a diffusive electron current J_e from the hot side to the cold side occurs. For energies below E_F a current J_h from the cold to the hot side is generated. A thermovoltage ∆V =−S∆T is generated by an applied ∆T, where S is the Seebeck coefficient, also-called thermopower, see Fig. 2.11.

It is required that the scattering of the oppositely diffusing hotter and colder electrons is energy dependent (electron-hole asymmetry) to obtain a non vanishing S.

The thermopower can also be expressed for a degenerate diffusive system withEF kBT by the Mott formula [41] to relate S and σ:

S =−eL₀T

δlnσ(E) δE

E=EF

(2.20) with L₀ = 2.44×10⁻⁸V²/K² (Lorenz number) and σ(E) the energy-dependent conduc- tivity.

Figure 2.10: Illustration of the temperature distribution of a conductor which ends are on different temperatures. The Fermi edge of the hot side is thermally more smeared out than the cold side due to the increased temperature. Therefore, a diffusive electron currentJe from the hot side to the cold side aboveE_Foccurs, whereas belowE_Fthe electron flow (hole currentJ_e) takes place from the cold to the hot side. Taken from Ref. [6].

a) b)

Figure 2.11: a) Sketch of the Seebeck effect, where a temperature difference ∆T is applied and a thermovoltage ∆V is generated. b) Illustration of the Peltier effect. An applied current leads to a temperature difference between the contact points. Taken from Ref. [6].

2.4.2 Peltier effect

The Peltier effect is illustrated in Fig. 2.11b), in which two different materials are con- nected to a circuit. By applying a current density j, a heat current Q occurs, which results in a temperature difference between both connection points. Analogous to the Seebeck effect, the Peltier coefficient Π is correlated to the other quantities withQ= Πj.

However, the Peltier coefficient is not directly accessible by an experiment like the See- beck coefficient due to the Joule heating which occurs in this case. Nevertheless, the Thomson-Onsager relates the thermopower S and the Peltier coefficient by Π = ST₀ at any temperature T₀.

2.4.3 Figure of merit

The efficiency of power generation as a function of ZT is given by η=

√

1 +ZT¯−1

√1 +ZT¯+_T^TC

H

, (2.21)

where TC is the temperature on the cold side and TH on the hot side. The mean tem- perature is ¯T = (T_C+T_H)/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=κ_e+κ_ph+κ_mthe thermal conductivity and T the absolute temperature, whereκ_edenotes the contributions of the electrons, κ_ph of the phonons and κ_m 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 disadvantages.

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 tol_e= (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 V_h= 0.5 V andV_h= 2.0 V. Predicted stepwise conductance changes occur in both cases. Bottom: Fluctuations of the thermovoltage, whereby the signal underV_h= 0.5 V is much lower compared toV_h= 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 V_h 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 forV_h= 0.5 V the thermoelectric signal is very small compared to the V_h = 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 S_C = _e(n+^{kBln 2}1

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 ∆V_p (see inset) versus heat input voltage V_h that indicates the Joule heat- ing of the mircoheater is shown. Oscillations occur by averaging thermovoltages ∆V_ave which are within a window of 0.02 G₀ (see Fig. 3.4c)). Close inspection of the maxima at the positions G_m,1, G_m,2, G_m,3 shows that they are independent of the input heat V_h (Fig. 3.4d)). The standard deviation Qof ∆V_ave 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(−E_B/k_BT_C) withf₀the attempt frequency,E_Bthe critical bond strength in the contact, k_B the Boltzmann constant as well as the effective temperature at the breakpoint T_C . 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

∆V_pof the thermovoltage (see inset) plotted against the applied heat voltage V_h. The dotted blue line is a fit which indicates the quadratic increase. c) Averaged thermovoltages∆V_ave versus conductance G. ∆V within a window of0.02 G₀ were summed up to values∆V_ave and show so-called thermopower oscillation. Three peaks occur at the positionsG_m,1, G_m,2, G_m,3which were fitted by a Gaussian function and drawn with solid lines. d) The three positionsG_m,1, G_m,2, G_m,3are independent ofV_h. 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 V_h = 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 andp_z (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 =

−∆V_TC/S_TC of the thermocouple occurs and the heat dissipation Q_P can be derived by Q_P = ∆T_TC/R_P, with R_P the thermal resistance of the NTISTP and S_TC the Seebeck coefficient of the thermocouple. Both values, R_P and S_TC 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 ∆T_{TC, Avg} were recorded and following the time-averaged power dissipation in the tip Q_{P, 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 ∆T_{TC, Avg} and Q_{Total, 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 1,4-benzenediisonitrile (BDNC, Fig. 3.13) and 1,4-benzenediamine (BDA, Fig. 3.14).

Both molecules exhibit a nonlinearI−V curve (Fig. 3.13c) and Fig. 3.14c)) but differ in the ∆T_{TC, Avg}/Q_{P, Avg}−Q_{Total, Avg} plot (Fig. 3.13b) and Fig. 3.14b)). There it has been figured out that for Au-BDNC-Au contacts the power dissipation in the probe is lower un- der a positive bias than at negative bias and inversely for Au-BDA-Au junctions. In both cases the dissipation is asymmetric. This can be explained with the hole-dominated elec- trical transport of the Au-BDA-Au junctions. The computed transmission functions (Fig.

3.14b)) reveal a negative slope of the transmission at the Fermi energy (E =EF, V = 0) leading to a positive Seebeck coefficient, whereas in Au-BDNC-Au junctions the Seebeck coefficient is negative due to the positive transmission slope at the Fermi energy.

Furthermore, DFT simulations were performed to verify the heat dissipation in the probe, which is drawn also in Fig. 3.13b) and Fig. 3.14b) (solid lines).

a) b)

c) d)

Figure 3.13: a) Conductance histogram of Au-BDNC-Au junctions with examples of conductance traces. b) Asymmetric heating of the system exhibit the measured time-averaged temperature rise

∆TTC, Avg and time-averaged power dissipation in the probeQP,Avg) as function of the time-averaged total power dissipation in the junction. The computed power dissipation are depict with solid lines and correspond to the measured values which result in a smaller power dissipation in the substrate under a negative bias. c)I−V characteristics, the solid line represents the average over 100 individual curves and the green area the standard deviation. d) Transmission function versus energy, computed by DFT. Around the Fermi energy the slope of the transmission is positive which leads to negative Seebeck coefficient.

Taken from Ref. [72].

a) b)

c) d)

Figure 3.14: Equivalent to Fig. 3.13, but here, the hole-dominant electrical transport of the Au-BDA- Au junction causes a heat dissipation in the probe larger for positive than for negative biases. This results in a negative slope in transmission near to the Fermi energy and thereby a positive Seebeck coefficient.

Taken from Ref. [72].

3.2 Publications: Techniques for determining the thermopower of molecular junctions and thermoplasmonic

This section sums up several leading works regarding thermopower of molecular junctions.

The realizations of the molecular contacts as well as of applying a temperature difference will be presented here. Furthermore, the results of the measurements are discussed briefly.

Finally, the publication of Herzog et al. [67] is presented, where the optically induced change in resistivity is used for temperature determination.

Kaneko et al.

Kaneko et al. [65] performed simultaneous thermopower and conductance measurements with a notched-wire Au MCBJ setup at low temperatures of about 50 K. The ben- zenedithiol molecular junction were characterized by IETS measurements with a standard lock-in technique.

To apply a temperature gradient, they placed a Pt heater sheet under the metallic

Figure 3.15: Notched-wire MCBJ setup of Kaneko et al. which enables the simultaneous measurement of thermopower and conductance at low temperatures. Taken from Ref. [65].

electrodes and resistive thermometers made of RuO₂ on top to measure the temperature difference (see Fig. 3.15). Polyimide tapes provided the electrical insulation between the layers. A temperature difference across the junction of ∆T = 10 K was achieved.

The simultaneous measurements of thermopower and conductance were realized by cy- cling the bias voltage 0 mV, 50 mV, 0 mV and −50 mV within 0.3 s. The values for the thermopower were averaged.

The measurements resulted in the observation of an anticorrelation between the electri- cal conductance and the thermovoltage of the BDT junction. The positive sign of the thermopower indicates that electrical transport is dominated by the HOMO. Further- more, the fluctuations in thermopower and conductance are explained by changes in the orientation of the molecule in the junction.