Influence of high currents on atomic contacts

Dissertation

Christian Schirm

Influence of high currents on atomic contacts

Experimental Physics

Dissertation

Christian Schirm

Influence of high currents on atomic contacts1

Dissertation for obtaining the academic degree Doctor of Natural Sciences (Dr. rer. nat.) at the University of Konstanz, Department of Physics, submitted by Christian Schirm. Day of the oral examination: 2009-10-08 Referees: Prof. Dr. Elke Scheer Prof. Dr. Juan Carlos Cuevas 1 The original work was published in German language with the title “Einfluss hoher Ströme auf atomare Kontakte”.

In this English version some corrections have been included: lines 22-26 in listing 3.1, p. 39, swapped values in left and right figures 3.5, 3.6 and 3.8, p. 32-33, corrected timing in figure 5.16, p. 92, additional footnote 5, p. 42.

Contents 1. Introduction

9

Overview of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. Basics

11

2.1. Mesoscopic transport . . . . . . . . . . . . . . . . . . 2.1.1. Mesoscopic systems . . . . . . . . . . . . . . . 2.1.2. Short historical context of the theory . . . . . 2.1.3. Theory of mesoscopic transport . . . . . . . . Multiple leads . . . . . . . . . . . . . . . . . . . 2.1.4. Conductance quantisation . . . . . . . . . . . 2.1.5. Atomic point contact . . . . . . . . . . . . . . . 2.2. Superconducting SNS junctions . . . . . . . . . . . . 2.2.1. Multiple Andreev reflections . . . . . . . . . . 2.2.2. IV curves . . . . . . . . . . . . . . . . . . . . . . 2.3. Conductancs histograms of break junctions . . . . . 2.3.1. Conductance quantisation . . . . . . . . . . . 2.3.2. Shell effects . . . . . . . . . . . . . . . . . . . . 2.3.3. Theoretical histograms . . . . . . . . . . . . . . 2.4. Voltage-dependent conductance of atomic contacts 2.4.1. Conductance fluctuations . . . . . . . . . . . . 2.4.2. Excitation of vibrational modes or phonons . 2.4.3. Lattice distortion . . . . . . . . . . . . . . . . . 2.4.4. Influence of band structure . . . . . . . . . . . 2.4.5. Temperature dependence . . . . . . . . . . . . 2.5. Electromigration . . . . . . . . . . . . . . . . . . . . .

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3. Numerical Methods 3.1. Numerical channel analysis . . . . . . . . . . 3.1.1. Problem . . . . . . . . . . . . . . . . . 3.1.2. Symmetry considerations . . . . . . . Symmetry in Φ . . . . . . . . . . . . . Degeneration of the global minimum Consequence . . . . . . . . . . . . . . 3.1.3. Number of theory curves . . . . . . . 3.1.4. Bandgap . . . . . . . . . . . . . . . . .

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5

C ONTENTS

3.1.5. Fit algorithm . . . . . . . . . . . . . . . . . . . . . Systematical scanning . . . . . . . . . . . . . . . G AUSSian regression . . . . . . . . . . . . . . . . Simulated annealing . . . . . . . . . . . . . . . . Automatic detection of the number of channels 3.1.6. Efficient calculation of the fit error Φ . . . . . . 3.2. Smoothing algorithm . . . . . . . . . . . . . . . . . . . . 3.2.1. Known smoothing algorithms . . . . . . . . . . Real smoothing . . . . . . . . . . . . . . . . . . . Comparison . . . . . . . . . . . . . . . . . . . . . 3.2.2. New method for smoothing . . . . . . . . . . . . The algorithm . . . . . . . . . . . . . . . . . . . . Fast calculation of the weights . . . . . . . . . . Derivation of the Iteration . . . . . . . . . . . . . Variant with D IRACs delta functions . . . . . . . Improved matching to the G AUSS-function. . .

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4. Experimental Techniuqes 4.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1. Known and new techniques . . . . . . . . . . . . . . 4.1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 4.1.3. Procedure of the experiments . . . . . . . . . . . . . 4.2. Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 4.3. Breaking mechanics . . . . . . . . . . . . . . . . . . . . . . 4.4. Cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Electrical wiring of the cryostat . . . . . . . . . . . . . . . . 4.5.1. Switchable wiring . . . . . . . . . . . . . . . . . . . . Solution idea . . . . . . . . . . . . . . . . . . . . . . Failed strategies . . . . . . . . . . . . . . . . . . . . . Best strategies . . . . . . . . . . . . . . . . . . . . . . Demonstration measurements for both strategies . Details of the relay . . . . . . . . . . . . . . . . . . . Details of the cables . . . . . . . . . . . . . . . . . . 4.5.2. Copper powder filters . . . . . . . . . . . . . . . . . Filter A . . . . . . . . . . . . . . . . . . . . . . . . . . Filter B . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3. Grounding and shielding . . . . . . . . . . . . . . . 4.6. Equipment and control . . . . . . . . . . . . . . . . . . . . 4.6.1. Measuring electronics . . . . . . . . . . . . . . . . . 4.6.2. Computer control . . . . . . . . . . . . . . . . . . . . 4.6.3. Scripting language Python . . . . . . . . . . . . . . Application examples of Python . . . . . . . . . . .

6

34 34 35 35 37 40 41 42 42 44 44 45 46 47 48 49

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51 51 52 52 53 54 55 55 55 56 56 58 58 59 59 59 60 61 61 62 62 64 65 65

C ONTENTS

4.7. Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1. Mechanical optimization of the substrate . . . . . . . . . . . . . . . . . . . 4.7.2. Optimization of sample designs . . . . . . . . . . . . . . . . . . . . . . . .

5. Measurement Results 5.1. Atomic stability at high currents . . . . . . . . . . . . . . . . 5.1.1. Measurement abrupt atomic rearrangements . . . . 5.1.2. Statistics on many measurements . . . . . . . . . . . Control algorithm and evaluation of the experiment 5.1.3. Result and interpretation . . . . . . . . . . . . . . . . Histogram of step heights . . . . . . . . . . . . . . . . Explanation for the return to mechanical histogram Origin of the Preferred conductances . . . . . . . . . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4. Local temperature . . . . . . . . . . . . . . . . . . . . Calculation . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Bistable switching . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of many hysteresis . . . . . . . . . . . . Training Phase . . . . . . . . . . . . . . . . . . . . . . . Atomic switches in other works . . . . . . . . . . . . . Suitability as a memory? . . . . . . . . . . . . . . . . . 5.3. Material quality and superconductivity . . . . . . . . . . . . Lead resistance and RRR . . . . . . . . . . . . . . . . . Energy spectrum . . . . . . . . . . . . . . . . . . . . . Critical values I c , Bc , T c . . . . . . . . . . . . . . . . . 5.4. Channel analysis . . . . . . . . . . . . . . . . . . . . . . . . . 5.5. Conductance histograms . . . . . . . . . . . . . . . . . . . . 5.6. Hysteresis shapes . . . . . . . . . . . . . . . . . . . . . . . . . Origin of the forms . . . . . . . . . . . . . . . . . . . . Similarities of the shape . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1. Individual observations . . . . . . . . . . . . . . . . . 5.7. Overview of all measurements . . . . . . . . . . . . . . . . .

A. Graphical Illustration

67 67 70

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111

A.1. Simulation to illustrate a break junction . . . . . . . . . . . . . . . . . . . . . . . . 111 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

Bibliography

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Summary

125

7

» Where the frontier of science once was is now the centre. « — Georg Christoph Lichtenberg1

1

C HAPTER 1.

Introduction

No other physical phenomenon can be used as universally as the electrical current. It enables the focused and uncomplicated transport of energy and information, two basic elements of modern technology. The knowledge of its fundamental physical laws allows it to improve its ability to transform itself into other forms of energy and information, as well as to further develop its applications. In the future, technological advances down to the atomic scale in particular will give rise to hopes for new applications of electrical current in data processing. In the present study physical relations between the electrical transport behavior and the structure of a few atoms have been investigated. The technique of mechanically controllable break junctions [RAP+ 96] have been used to produce atomic point contacts, which can be modified in its geometry in more or less controlled way. Since no microscopic method is known to determine the exact atomic geometry of the contacts, it is only possible to deduce certain properties of the geometry by observing the conductance. The focus of this work was on influencing the geometry as specifically as possible and, for comparison, measuring the resulting changes in conductance. The atomic contacts were investigated using conductance histograms, channel spectrum analysis by means of superconductivity and the measurement of current-voltage characteristics. In recent years, these methods have yielded the most important experimental results on fundamental questions of atomic conductivity. In this way, effects of the conductance quantization as well as the geometrical preference of certain atomic configurations (which in part can be perceived as apparent conductance quantization) could be found. In addition, at the microscopic limit of a single atom, the quantum mechanical conductance channels predicted by R. L ANDAUER [Lan57], could be assigned to individual orbitals of the central contact atoms [SJE+ 97]. A certain classification of many properties can be seen in the distinction between monovalent and multivalent metal atoms. The differences can be seen in the channel spectra of single atom contacts [SAC+ 98], the behavior of conductance fluctuations [Lud99] and (more in exceptional cases) in the occurrence of conductance quantization [KRF+ 95, YR97, XNOS04, THNA05]. In this thesis the investigations were focused on the trivalent metal aluminum. 1 1742–1799, first Geman professor for experimental physics, citation from “Sudelbuch” H, written 1784–1788.

9

1. I NTRODUCTION

Overview of the chapters Chapter 2 conveys the theoretical basics and the important results of previous experiments on atomic conductivity. The L ANDAUERtheory of the conductance channels will be shown, superconductivity effects of atomic contacts (multiple Andreev reflections) and some theoretical calculations on the conductivity and atomic structure. Furthermore, quantum interference (conductance fluctuations), geometrically preferred shell effects, vibrational excitation and electromigration forces will be treated. Chapter 3 shows numerical methods that have been used and further developed for the evaluation of measurement data. The first part deals with improved methods for channel analysis to fit theoretical characteristics of superconducting contacts on experimental curves. The aim of the further development was to protect against numerical artifacts and fit errors. At the same time, a significant increase in calculation speed was achieved. In the second part, a smoothing algorithm is presented, which was originally intended to serve as a background treatment for conductance fluctuations, but was also applied several times in this work. Chapter 4 describes the experimental setup. A few improvements compared to previously known superstructures are presented. In particular, a special cabling of the cryostat is discussed, which had to be developed for the experiments carried out here. Chapter 5 shows the measurement results. First of all, measurements on the current-driven reassignment of atomic contacts are described and the results and statistical analyses are discussed. In order to limit the changes of the atomic influence more strongly, a reversible local switching behaviour is forced by alternating current directions. The bistable switching states are examined by means of channel analysis. Histograms are generated via conductance values and channel transmission and compared with each other. The forms of IV curves are also discussed in more detail. The question of where the origin of the effects lie, whether locally at the atomic contact or in the immediate vicinity, is investigated. Finally, individual observations of bistable switching hystereses are shown and interpreted.

10

» I will call these electric particles, following more recent names, [...] electrons. « — Paul Drude, 19001

2

C HAPTER 2.

Basics

2.1. Mesoscopic transport 2.1.1. Mesoscopic systems For many physical systems, a microscopic atomic model and knowledge of macroscopic geometry is sufficient for a complete theoretical description of the physical processes. In electrical transport, on the other hand, some phenomena take place on a medium length scale, which can strongly influence the overall behavior. For systems of this size the name mesoscopic systems (Greek: meso = medium) has been introduced. Responsible for the phenomena of this size are various length scales, such as e.g the Fermi wavelength of the electrons (λ f ∼ 5 Å), the elastic mean free path (l e ∼ 100 nm) and the phase coherence length (l Φ ∼ 1 µm). In Figure 2.1 three classes of such mesoscopic conductors are shown, in which these length scales are crutial for the behavior.

2.1.2. Short historical context of the theory The development of theories about the electronic current began in 1900, three years after the electron [Fal97] was discovered, with the particle model proposed by PAUL D RUDE [Dru00]. The second milestone followed just as shortly after the beginning of modern quantum mechanics (1926, Schrödinger equation) and the discovery of the wave character of the electron (1927, [DG27]), when F ELIX B LOCH could, in 1928, apply the new findings to a new theory of free metal electrons [Blo29], which also allowed the influence of the underlying crystal structure to be explained. Finally, in 1957, R OLF L ANDAUER developed a microscopic theory that used the now mature methods of atomic physics and quantum theory, such as scattering theory and the quantum mechanical description of electronic eigenstates. For the first time, the theory was able to correctly describe electrical conductors with very small dimensions. It was initially limited to effective two-point contacts and was extended to true four-point 1 Cited from: Zur Elektronentheorie der Metalle [Dru00].

11

2. B ASICS

L

W diffusive conductor

(quasi-) ballistic conductor

quantum point contact

Figure 2.1.: Various extreme cases in mesoscopic conductors. Diffusiv: l Φ > W, L À l e , ballistic: L,W < l e , Quantum point contact: W, L > λ f

-U

μL

μR

E

E

E

μL

f(E)

0V

E

eU

0

0

k

0

k

μR

0

k

0

f(E)

Figure 2.2.: Model of electronic transport through a ballistic mesoscopic conductor at an applied voltage U and finite temperature.

contacts in 1986 by M ARKUS B ÜTTIKER [Bü86]. The resulting L ANDAUER-B ÜTTIKER formula can be used to describe mesoscopic systems as an extension or substitute for Ohm’s law. It will be briefly explained in the following.

2.1.3. Theory of mesoscopic transport Figure 2.2 shows a model of the electronic transport in the mesoscopic conductor with an imbalance of the chemical potential around eU , corresponding to an applied external voltage U . The band edge of the electrons in the leads is shown on the left and right by the Fermi function smeared at finite temperature around 2k B T . The occupied and unoccupied states often lie on parabolic dispersion relations, shown here for both k-directions parallel to the conductor. In the mesoscopic conductor itself, only a few discrete states are available as transport modes. Transport of electrons through the conductor can only take place from an occupied state of the left to an unoccupied state of the right reservoir. Within the conductor itself, free states, called channels, must be available in the energy interval in between. To calculate the total current through the conductor, let’s first look at such a single channel. Its contribution to the total conductance depends mainly on the overlap of the wave functions of the states in the electrodes with the channel. However, it turns out that even in the most

12

2.1. M ESOSCOPIC TRANSPORT

favorable case, there is an upper limit of the conductance for a single channel, which can not be exceeded and which represents a universal constant. This result is provided by the following somewhat rough estimate. For this the conductance is written as G=

I ∆Q 1 = U ∆t U

(2.1)

first for the transport of individual electrons with ∆Q = e und ∆E = eU as G=

e2 . ∆E ∆t

(2.2)

In [Bat02] it is now argued with the energy-time uncertainty ∆E ∆t ≥ h that the transport process, taking place in the energy interval ∆E = µ2 − µ1 , requires the electron to have a minimum residence time ∆t ≥ h/∆E . For subsequent electrons, the channel is blocked for the appropriate amount of time due of the Paulian exclusion principle. Therefore the conductance

G≤

e2 h

(2.3)

of a single channel has an upper limit. Twice this maximum conductance G0 =

2e 2 h

(2.4)

for transport through both spin states is called conductance quantum and is abbreviated by G 0 . The sum over the conductance of all available channels G=

2e 2 X τi h i

(2.5)

is called the L ANDAUER-equation. The τi are values between 0 and 1, which represent the coupling strengths of the outer modes overlapping with the channel. They are referred to as transmissions and represent the probability that an electron passes through the channel without being reflected back. In the original [Lan57] and in later works and textbooks the derivation of the formula was carried out in more detail and more carefully. In this short derivation or estimation shown here, it “might appear at first sight that the uncertainty principle derived value is just some sort of an accident”2 . However, it provides a plausible idea of how a universal limit for the conductivity of individual channels can also be set within perfect ballistic conductors.

2 Citation from the publication [Bat02]. Here also an alternative derivation is shown with the uncertainty principle

for position and momentum.

13

2. B ASICS

Multiple leads When the leads of the excitation and voltage measurement are brought together close to the mesoscopic conductor within the ballistic range, the conductance measurement provides different values. B ÜTTIKER set up a corresponding formula that assumes four (or any number of ) leads. All leads are treated the same regardless of their function. The result is the L ANDAUER-B ÜTTIKER-Equation Ip =

X q

G pq [U p −U q ],

(2.6)

which specifies the current I p through the lead with number p. U q is the voltage on another lead q and

G pq ≡

N N ¯2 2e 2 2e 2 Xp Xq ¯¯ τp←q = S mp,nq ¯ h h m=1 n=1

(2.7)

are the coefficients of the conductance matrix G. The scattering matrix S contains the transition amplitudes of the state n of the lead q to the state m of the lead p. The two equations replace the well-known ohmic law I = GU

mit

A G =σ , L

(2.8)

which is in the end also included as a limit. 3

2.1.4. Conductance quantisation In 1988, VAN W EES ET AL . produced a mesoscopic cunductor with only channels of transmission 1. In a 2D electron gas system of a semiconductor, depletion zones were spatially varied with two variable voltage gate electrodes such that a conductive channel in between could be constricted as desired. The edges of this conductor seemed to be sufficiently smooth so that as the width was continually changed, the number of waveguide modes accommodated therein regularly increased, as shown in Fig. 2.3. The temperature here was small enough (0.6 K), and the Fermi edge accordingly sharp, so that the modes appeared at a certain thresholds. The conductance, which increases by 1G 0 per mode, is effectively quantized here. However, the conductance quantization must not be understood in the strict sense, as e. g. the quantization of the electric charge by charge quanta, the electrons, which are fundamentally not divisible. Principally, the fact that conductivities can occur in arbitrary non-integer G 0 , especially for contacts with only one channel, is shown in particular in the example in the next section. 3 For further details on mesoscopic transport theory, it is referred to the textbook of P. D ATTA [Dat97].

14

Conductance (2e²/h)

2.1. M ESOSCOPIC TRANSPORT

10 8 6 4 2 0 -2

-1.8 -1.6 -1.4 -1.2 Gate Voltage (V)

-1

Figure 2.3.: Conductance quantization with perfect ballistic conductor with variable width [WHB+ 88].

E tight binding

p-orbitals

EF

s-orbitals

0

DOS

Figure 2.4.: Atomic point contact made of aluminum

LDOS

Transmission 1.0 Al

Al 0.4

0.8

spz

spz

0.6 p

0.2

xy

pxy

0.4 0.2

0.0

-5

0 E(eV)

5

0.0

-5

0 E(eV)

5

Figure 2.5.: Theoretical calculation of the local density of states and the transmission of an atomic point contact of aluminum as shown schematically in Fig. 2.4, plotted against the position of the Fermi energy (from [Cue99]). Recent calculations show that conductances up to 2G 0 can occur on single-atom contacts [PVH+ 08].

15

2. B ASICS

2.1.5. Atomic point contact Transport modes such as those found in Fig. 2.3 require a corresponding waveguide potential. Such modes are not possible in the spherical symmetric core potential of a single atom. In atomic point contacts, as shown schematically left in Fig. 2.4, the electrons are forced to propagate through the narrow potential of a single atom. In this case, it can be assumed that the electron wave functions predominantly take the form of the atomic orbitals rather than plane wave transport modes when passing through the constriction. Figure 2.4 shows in the middle the level scheme of a free aluminum atom. The valence electrons appear in club-shaped p- and spherically symmetric s-orbitals. Coupled to the contacts, the energy eigenfunctions of the orbitals are changed and thereby energetically expanded in the density of states (figure left). The originally discrete levels may thereby overlap, and in particular come into contact with the Fermi energy E F . This allows the electrons of the feed line to occupy the orbitals for a short time in order to get to the other side. The occupation probability is proportional to the density of states at the Fermi energy. Some orbitals are correspondingly stronger and others less conductive. A quantization of the conductance to integer G 0 is therefore not expected for atomic contacts. Figure 2.5 shows a theoretical calculation of the expanded levels schematically shown in Fig. 2.4 [Cue99]. In addition, the transmission was calculated here. The s-orbital in aluminum hybridizes with one of the three p-orbitals, which lies in the transport direction. The antibonding state of spz is non-conductive, leaving one spz and two p-orbitals left as channels for transport. In the calculation, idealized geometry was calculated (pyramidal peaks in (111) growth direction). The spz orbital conducts much better than the two similar p-orbitals. To confirm this signature of a single aluminum atom, measurements are shown in the chapter 5. For the first time such measurements were published in [SJE+ 97]. It was possible to determine the transmission values of the individual channels using superconducting effects, in particular for several channels involved in current transport at the same time.4 2.6 shows an opening curve of an atomic break junction, with the simultaneous transmissions of the individual channels. The measurements confirm the atomic model quite well, since at the lowest plateaus the three measured transmission values can be convincingly assigned to the individual atomic levels of the aluminum atom. Visible are namely two similar channels and one behaving differently with higher transmission, which agrees with the calculated local density of states. Also, the increase of the sp-channel with increasing distance has been confirmed by theoretical calculations. In addition to aluminum, other metals were also investigated [SAC+ 98]. In each case it could be shown that at the last plateau of the atomic contact the number of channels is less than or equal to the number of free valence electrons of the metal atom. In the case of point contacts that are connected by more than one atom in the break junction, it is unclear whether quantization is to be expected here, or whether the local atomic states determine the conductance, in view of this and the previous section. In fact, there is usually only a quantization tendency, as shown below. 4 The method is explained in more detail in the sections 2.2 and 3.1.

16

2.2. S UPERCONDUCTING SNS JUNCTIONS

Figure 2.6.: Opening curve of an aluminum break junction with determination of the transmissions of the individual conductivity channels (from [SJE+ 97]).

2.2. Superconducting SNS junctions 2.2.1. Multiple Andreev reflections Superconductivity 5 also occurs in mesoscopic conductors, while the critical current is quickly exceeded on very narrow contacts. Atomic point contacts no longer have a crystalline lattice, which is a prerequisite for superconductivity. In superconductor-normal-conductorsuperconductor contact (short SNS contact), however, with sufficient small spatial extent of the N-region, an overlap of the Cooper pair wave functions of the two superconducting sides can occur (J OSEPHSON contact). Here Cooper pairs and normal conducting electrons, depending on the voltage difference, can pass the barrier, more or less directly, in certain many-body processes [And64]. Fig. 2.7 above shows the electronic states of the left and right superconducting leads. The intermediate point contact is indicated by the dashed line. With a potential difference of at least eU ≥ 2∆, normally conducting electrons can pass directly from the occupied to the unoccupied band region of the other side above the band gap. At lower voltages, this transition is not possible, but two such electrons could combine to form a Cooper pair, which requires a voltage of at least eU ≥ ∆. Below this voltage, at least a 3-particle process with a virtual intermediate level is possible, as illustrated graphically. It turns out that n particle processes occur from a voltage of eU ≥ 2δ/n. Alternatively, the decreasing potential difference was not plotted in the lower part of the figure, accordingly the electrons move upwards by eU . In the n particle process, the particle transition can be alternately drawn as an electron and as a hole propagating in the opposite direction, until finally the band gap has been overcome. From each of two virtual levels, symmetrical about the Cooper pair level, a 5 The basics of superconductivity are not discussed here. A suitable overview is provided by the history of

superconductivity [ST99], summarized in 4 pages by S CHRIEFFER and T INKHAM.

17

2. B ASICS

1-particle process eU ≥ 2Δ/1




1Δ 1Δ

P

P

τ

P

τ²

τ³

P

h

h e

e

I

e

I

2Δ U

e

I

Δ

τ⁴

U

I

2Δ/3

U

2Δ/4

U

Figure 2.7.: Top and center: Mechanisms of charge transport in SNS contact. Every time a potential difference of eU = 2∆/n is exceeded, in addition to the previous transitions with n +1, n +2, . . . particles, another transition with only n particles is possible. We call the transitions quasiparticle transition (n=1), andreev reflection (n=2) and multiple andreev reflections (n ≥ 3), Bottom: Simple idea explaining the current-voltage characteristics.

Cooper pair can form at the end, or with odd n a normal conducting electron can be lifted into the unoccupied band. In a certain auxiliary concept, electrons are repeatedly reflected back as holes and holes as electrons, which led to the name multiple Andreev reflections (MAR).

2.2.2. IV curves Although many-body processes generally carry more charge for large numbers of n, the current contribution is lower because the process is less frequent. The transmission probability of an electron in the normal conducting state is given by τ, therefore in n particle processes the single probabilities of simultaneous occurrence have to be multiplied, whereby a probability P ∝ τn is decisive for the current contribution. The current resulting from the transitions therefore decreases with increasing n. Fig. 2.7 below shows the contribution of each transition to the current-voltage diagram, which falls below a threshold of eU = 2∆/n down to zero. The sum of all contributions here gives a step-shaped curve. Due to the different powers of τ, the ratios of the individual contributions and thus the relative step heights are shifted so that different curve shapes for different τ are to be expected. This proves to be a stroke of luck

18

2.3. C ONDUCTANCS HISTOGRAMS OF BREAK JUNCTIONS

Current (G 0 V∆ /τ)

4 3

Transmission τ

1.0 0.9 0.8 0.7 0.5 0.3 0.0

5

2 1 0

0

1 2 Voltage (V∆ )

3

Figure 2.8.: From C. C UEVAS calculated IV-curves [CMRY96], normalized with τ (see non-normalized curves in Fig. 3.1, page 27). For each transmission value, the characteristic has an individual shape that is linearly independent of the others.

because the associated linear independence can be exploited to analyze the measurement data. The actual waveforms differ significantly from the idealized model (Fig. 2.7 below). However, the exact characteristics can be calculated numerically. The result of such an invoice shows Fig. 2.8. The thresholds of the many-particle processes are not recognizable here as steps, but more as small kinks. The agreement of these calculations with experiments, however, proves to be almost perfect. Fig. 2.8 is not shwon with absolute SI units (A and V), but instead units of the energy gap. In this way, the result becomes universally valid for different metals, as this is the only and also linear material parameter. Different transport channels that run in parallel (not in the spatial sense, but electronically as an effective parallel connection) lead to a sum of such IV characteristics. Apart from the transmission values τi and the bandgap, no further parameters are used to emulate the IV characteristic of a specific multi-channel contact. By means of a suitable fit procedure (see section 3.1), the IV characteristic obtained in the experiment can also be decomposed into the individual IV characteristics in order to determine the individual transmissions τi .

2.3. Conductancs histograms of break junctions 2.3.1. Conductance quantisation Mechanically controlled break junctions, which were also investigated in this work, can be opened and closed in constantly new geometries, as if the sample would be permanently replaced. Through this ergodic assumption, statistics on millions of samples, that is atomic contacts, can be formed in one experiment. Conductance histograms, ie occurrence probability distributions of the occurring conductance values, are particularly suitable for the

19

2. B ASICS

Normalized numbers of counts

0.40 0.35

Aluminum, T = 4.2 K

0.30 0.25 0.20 0.15 0.10 0.05 0.00 0

1

2

3

4

5

6

7

8

9

10

Conductance [2e²/h]

Figure 2.9.: Histogram of 30 000 opening operations of an aluminum break junction at 10 mV (from [YR97, AYR03]).

evaluation. Thus, in [KRF+ 95], a tendency to conductance quantization in sodium and copper contacts was demonstrated, which can be explained by cylindrical modes. In [YR97] (see Fig. 2.9) a histogram of 30 000 opening processes was also formed for the material aluminum investigated here. Although, as shown in the previous section, the individual conductance channels do not show any indication of conductance quantization, at first sight, preferred values can be seen in the sum of the channels (ie in the overall conductance). These are even close to the conductance quantum. However, in [YR97] it is emphasized that the maxima are also somewhat higher than the integer multiples of the conductance quantum. Overall, this indicates that no pure quantization of the electric modes as a cause comes into question. Rather, a geometric preference could shape the shape of the histogram. This was also clearly demonstrated by later measurements on shell effects. Similarly, the ergodic assumption had to be partially discarded because the history of the metal (the crystalline structure) can reproducibly influence the conductance histograms.

2.3.2. Shell effects Experiments showing a pure conductance quantization can be explained by eigenstates of the electron waves [KRF+ 95]. However, if preferred geometric configurations occur when the contact is formed, this is also visible as an accumulation of associated conductance in the histogram. The accumulations are in some cases on a regular scale around G 0 , which makes it difficult to distinguish from the conductance quantization. The preferred geometries are similar to metallic clusters. These form on the surface of the roundish particles shell structures, partly geometric with monoatomic layers, and partly electronic type, similar to the shells in the atomic model. Clusters with closed shells are called magic and are particularly stable and correspondingly common. Geometric and electronic shells are usually not magical for the same number of particles. If so, it is called double magic.

20

2.3. C ONDUCTANCS HISTOGRAMS OF BREAK JUNCTIONS

Figure 2.10.: Left: Theoretical predictions of shell structures in aluminum (from [GET98]). Right: Measured histograms on hardened aluminum at very regular intervals with ∆G > 1 (from [YSRS08]).

In nanowires similar shells occur in the cross section of the wire [YYR99, Mar06]. Fig. 2.10 left shows some theoretical predictions of such preferred geometries for aluminum wires [GET98]. Experimentally, the shell structures can be detected by break junction experiments with voltages À 10 mV and conductance histograms [YYR99]. Especially with aluminum, electronic or geometrical shells can dominate under certain conditions [Mar06, MUB+ 07]. Or, as in Fig. 2.10 on the right, hardened6 aluminum can have very regular maxima in the distance 1.15G 0 [YSRS08]. A pure conductance quantization [KRF+ 95] is completely suppressed here by the geometric quantization.

2.3.3. Theoretical histograms To clarify these and other questions, a comprehensive simulation of M. D REHER was performed [DPH+ 05, Dre08]. Molecular-dynamic simulations were used to calculate complete opening processes of break junctions including the conductance values. In the evaluation, the respective cross-sectional radii of the contact were determined, which are not accessible in the experiment. Although clear peaks are visible in the histogram of the cross-section radii (Fig. 2.11 left), the same cross sections in the conductance histogram are distributed over larger areas. In particular, they are not correlated with multiples of the conductance quantum. The 6 The difference between soft and cold tempered material is much more sigificant in gold [YSC+ 05, HWHS08].

21

2. B ASICS

Figure 2.11.: Molecular dynamics simulation of several opening curves of a gold break junction of M. D REHER [DPH+ 05]. Left: Histogram of the cross-section radii. Right: Conductivity histogram with color-coded allocations to the cross-section radii.

deviation of the quantization of gold observed in the experiment is ascribed to the influences of the relatively narrow initial geometry in [Dre08]. Unfortunately, due to certain complicated properties of the metal, aluminum could not be analyzed in the same quantity and accuracy as gold.

2.4. Voltage-dependent conductance of atomic contacts 2.4.1. Conductance fluctuations In mesoscopic conductors, in contrast to ohmic resistances, the conductance is usually voltagedependent. In the diffusive regime, interference between different electron paths can occur (see scheme in Fig. 2.12). Considering the electron waves as F EYNMAN pathways which divide as they enter the conductor and converge again at the end and within the phase coherence length, the magnitude of the quantum mechanical amplitude can be calculated with the sum over all path integrals along the paths. The probability of getting an electron from a location A to a location B can be calculated by ¯ ¯2 ¯ ¯ ¯ X ¯ ¯ P A→B ∼ ¯ Ψi ¯¯ ¯ alle Pfade ¯ A→B

1 with Ψi = ΨA,i exp(i ϕi ) and ϕi = h

Z

tB tA

L(~ x ,~ x˙ , t ) dt .

(2.9)

The Lagrange function, which is set as L = T − V , increases with increasing voltage gradient. As a result, sinus periodic variations of the amplitude as a function of the voltage are to be expected between pairs of paths with different path lengths. The total conductance is composed of a sum of different frequencies, which can provide information on the path length differences and indirectly on the path lengths. Conductance fluctuations are also expected in atomic point contact. Fig. 2.13 shows

22

2.4. V OLTAGE - DEPENDENT CONDUCTANCE OF ATOMIC CONTACTS

¯ ¯2 ¯ ¯ ¯ X ¯ ¯ P A→B ∼ ¯ Ψi ¯¯ ¯ all path ¯

e-

eA

A→B

B

Path 1 Path 2 elastic scatter scenter

Figure 2.12.: Conductance fluctuations in the diffusive mesoscopic conductor

Ballistic point contact

diffusive bank

diffusive bank

Figure 2.13.: Left: Scheme of conductance fluctuations at atomic point contacts. The ballistic scattering provides two passes through contact with different parts of the electron’s wave function. Right: Measurement of conductance fluctuations at different conductance values. From [Lud99, LDE+ 99].

schematically the prerequisite for it. In the ballistic regime, there must be scattering centers which allow parts of the electron wave to make contact, e. g. a second time out of phase to go through. On the right are measurements of conductivity fluctuations with different amplitude. Close to the conductance 1G 0 , the fluctuation amplitude is very small, since it is usually a full channel in which the reflected component is small compared to the transmitted amplitude.

2.4.2. Excitation of vibrational modes or phonons Further voltage dependencies of the conductance can occur if, starting from certain threshold voltages, vibration modes of individual atoms or phonons in the supply lines are excited. The excitations occur symmetrically in the positive and negative current direction. Above the threshold, the excitation leads to a change in the conductance. Vibration modes could be clearly demonstrated for a single H2 molecule between two

23

2. B ASICS

atomic platinum electrodes (and for more massive isotopic molecules) [SNU+ 02]. However, (transversal) modes of vibration could also be identified on a purely metallic aluminum contact [BES09, Bö08]. The effect can not be partially distinguished from conductance fluctuations. Characteristics of vibrational excitations and phonons are one peak at a positive and one dip at the same negative voltage in the second derivative from I (V ) to V . This corresponds to a (usually poorly visible) kink in the curve I (V ).

2.4.3. Lattice distortion The voltage- or current-dependent distortions of the local atomic geometry can also lead to significant conductance changes in IV characteristics. The measurements of this work provide clues to this, as will be shown later. Physical reasons for the responsible forces on single atoms are mentioned in section 2.5 for electromigration. The forces that increase with the voltage deflect individual atoms further and further out of their rest position, which in turn causes a voltage-dependent conductance.

2.4.4. Influence of band structure As can be seen from the theoretical local density of states in Fig. 2.5, p. 15, the density of states and thus the transmission can depend on the chemical potential of the electrons. When the voltage is varied, the chemical potential can change locally, and thus also the conductance. However, this effect is negligible in the experimentally accessible voltages.

2.4.5. Temperature dependence The temperature can rise at one-atom contact with increasing current. The heat dissipated there can already be detected with the thermometer, although the heat, when it arrives at the thermometer, is already very thinned out. At the point of origin around the break junction, however, it is strongly concentrated. The local temperature can exert an influence on the conductance via phonons or vibrations.

2.5. Electromigration The previous topics of the first chapter showed influences of the conductor on the transport behavior of the electrons. What was announced by the title of this work is the reverse case in which the electric current causes changes in the conductor. The movement of individual metal atoms by electric current is caused by two forces pointing in opposite directions. In [Lod05], the historically separate forces were combined into a unified theory describing the interplay of both forces. The two forces are named direct force and wind force. For an atom, the force arises in the metal F = Fdirect + Fwind = (Zdirect + Zwind ) eE = (Zi + Z scr + Zwind ) eE

24

(2.10)

The Amount of calculatet scr screening Z

2.5. E LECTROMIGRATION

0

Coulomb potential Square well with width 1/l Square well with width 2/l Average

-0.05 -0.1 -0.15 -0.2 -0.25 -0.3

Sodium

Aluminum

0.5 0.6 0.7 0.8 0.9 Fermi wave vector kF of the metal

1

Figure 2.14.: Theoretical calculation of the shielding effect, depending on the Fermi wave vector and the nature of the surrounding potential (from [Lod05]).

in an electric field E. The direct force is the force of the field on the positive nucleus with the effective charge Zdirect , which is composed of the ion valence Zi (positive) and an effective shielding of the ion charge by the valence electrons Z scr (negative). Wind power, on the other hand, is an indirect force on the core in the opposite direction. The negative electron shell senses the repulsive force of the incoming conduction electrons, which in turn keep their distance from the nucleus by repulsive Pauli forces. The wind force is thus a force due to the pressure of the flowing electric current, while the direct force is caused by the electric voltage. It is in the end crucial which of the forces outweighs, that is, how strong the shielding is. Fig. 2.14 shows an account of how the shielding may depend on the metal and the exact potential form in which the particle resides. At experimental works especially those of S TAHLMECKE ET AL . are to be mentioned [SHC+ 06, SCHD06, SD07a, SD07b, Sta08]. Here, in-situ observations of electromigration processes were performed with a scanning electron microscope. Among other things, it was shown how material can be transported by the electromigration from one side to the other and back, depending on the current direction. For different conductors (for example, polycrystalline gold and single-crystal silver, or differently contaminated surfaces), even opposite directions of electromigration have been observed [Sta08], meaning that in one case the direct force and in the other the wind force exerts stronger influence on the atoms.

25

» In some ways, technical systems [computers] are similar to the sorcerers and wizards of fairy tales. They will satisfy your wishes, but they won’t tell you what to wish. « — Norbert Wiener1

3

C HAPTER 3.

Numerical Methods

This chapter describes some numerical data analysis methods used and developed in this work.

3.1. Numerical channel analysis 3.1.1. Problem

4 3

0.7

2 1 0

0

1 2 Voltage (V∆ )

3

0.6 0.5 0.4 0.3 0.2 0.1 0.0

Sum over 3 channels: τ1 = 0.8 τ2 = 0.54 τ3 = 0.46

6 Current (V∆G 0 )

Current (V∆G 0 )

5

Transmission

1.0 0.95 0.9 0.85 0.8

4

2

0

0

1 2 Voltage (V∆ )

3

Figure 3.1.: MAR theory curves, calculated by C. C UEVAS and example of a sum of three specific channels with transmissions 0.8, 0.54 and 0.46.

As already mentioned in section 2.2, the transport channels of an atomic point contact can be determined experimentally by numerically dividing the measured current-voltage characteristic into the contributions of individual channels. The characteristics of individual such 1 US-American mathematician, 1894–1964, (developed the W IENER -filter). Source of quote unknown.

27

3. N UMERICAL M ETHODS

channels calculated by C. C UEVAS are shown in Fig. 3.1 and were available for the analysis of the measured data of this work. Each of these theoretical curves is uniquely linked to a transmission 2 of the channel. By breaking it down into the single-channel curves, the transmission value of each contributing channel is determined. This also determines the number of channels present in the experiment by considering channels with transmission near zero as non-existent channels. To determine the channels, n theory curves with different transmissions (τ1 , τ2 , . . . , τn ) are combined so that the mean error v !2 uZ Ã n u X t Φ(τ1 , τ2 , τ3 , . . . , τn ) = (3.1) I Theo (τi ,U ) − I Mess (U ) dU i =1

becomes minimal. n must be greater than or equal to the actual number of channels. The different curves turn out to be linearly independent, so that there is exactly one minimum and therefore the decomposition should unambiguously and correctly be possible. Practically, however, with increasing measurement noise and a large number of channels, the information contained in the channels is increasingly lost. The following section discusses the occurrence of an artifact that causes similar channels to merge into seemingly identical channels.

3.1.2. Symmetry considerations Symmetry in Φ The error Φ forms a N -dimensional potential landscape. From the interchangeability of the summands follows for Φ a symmetry with respect to pairwise interchange of two arbitrary coordinates τi and τ j . Thus, for each value in Φ with unequal coordinates, there exist N ! identical copies. The potential landscape therefore decays in N ! identical, partially mirrored subregions. The inner interfaces ¯ © ª S p,q := (τ1 , τ2 , . . . , τp , . . . , τq , . . . , τN ) ¯ τi ∈ [0, 1], τp = τq

(3.2)

are N2!! linear subspaces of dimension N − 1 for all pairs p and q with p 6= q. They form mirror p planes, since for every point in the distance 2d to such a plane S p,q the potential Φ(τ1 , τ2 , . . . , τp − d , . . . , τq + d , . . . , τN )

(3.3)

Φ(τ1 , τ2 , . . . , τp + d , . . . , τq − d , . . . , τN )

(3.4)

is identical to because of τp = τq and the symmetry requirement . The distance vectors d+ = (0, . . . , d , . . . , −d , . . . , 0) and d− = (0, . . . , −d , . . . , d , . . . , 0) to the center and symmetry point are orthogonal to the N − 12 The transmission (the probability of an electron traversing the channel) can be converted directly into the 2 1 channel’s conductance by multiplication with G 0 = 2eh ≈ 12906 Ω.

28

3.1. N UMERICAL CHANNEL ANALYSIS

1

0.6

0.8

0.55

0.5

τ3

τ3

0.6 0.4 0.45

0.2 0

0.4 0

0.25

0.5 τ2

0.75

1

0.4

0.45

0.5 τ2

0.55

0.6

Figure 3.2.: Left: Fitting error with three parameters plotted against two of the parameters in different sections. Shown is the respective minimum error with respect to the third parameter in colors from dark (small) to bright (large) in a strongly non-linear color scale. Here, a theoretical curve with the channels 0.8, 0.54 and 0.46 is fitted. The found minimum (black cross) is as expected in one of the 3! = 6 mirror images with τ1 = 0.8000, τ2 = 0.5400 and τ3 = 0.4600. Right: enlarged detail.

0.04

75

0.02

50

0.01

FWHM (µV)

Error (V∆G 0 )

0.03

25

0

0 0.4

0.45

0.5 τ2

0.55

0.6

Figure 3.3.: Diagram of the minimum along the τ2 direction with the global minima, represented by black dots. From bottom to top, the curve to be fitted was increasingly rounded. The right scale assigns the respective Gaussian width to the numerical rounding of the individual curves. At a rounding of 45 µV, the minima in this example coincide incorrectly.

29


dimensional mirror plane S p,q like the vanishing scalar product 〈d+ , s2 − s1 〉 = d (τp,2 − τp,1 ) − d (τq,2 − τq,1 ) = 0 mit s 1 , s 2 ∈ S p,q

(3.5)

shows. Figure 3.2 shows an example of a potential with a visible plane of symmetry and six identical minima, as well as a detail magnification.

Degeneration of the global minimum Another symmetry property of the problem is shown in the special form of the given curves. Here, the linear section in the higher voltage range has a particularly strong weight. In purely linear curves, the lack of linear independence would mean that no well-defined global minimum exists in Φ. Rather, a degeneration of the actual minima w occurs in the directions of the P same sum τi . The identical errors lie on a N − 1-dimensional plane X © ª K w := w + (δ1 , δ2 , . . . , δN ) | δi ∈ [0, 1], δi = 0 ,

(3.6)

whicht is perpendicular to the origin vector (1, 1, . . . , 1) due to 〈(δ1 , δ2 , . . . , δN ), (1, 1, . . . , 1)〉 =

X

δi = 0

(3.7)

and to the spatial diagonal contained in all mirror planes. Thus, all mirror planes S p,q intersect with the planes K w with identical errors. Since these are not identical but only similar properties, absolute minima nevertheless should exist, but they lie in elongated flat potential wells, as demonstrated in Fig. 3.2.

Consequence If two or more channels with similar transmissions are found in the experiment, the distance to the mirror plane and also to the mirror images is small. If, due to measurement inaccuracies, the depth of the potential well decreases around the global minimum, there is a risk that the well and its mirror image will no longer win against the concave form of the surrounding potential. As a result, both mirror-image well fuse together and the global minimum slips exactly onto the mirror plane. As a consequence, two or more similar channels seem to be identical. Fig. 3.3 shows in a cross-section of a potential how from a certain noise the unwanted case of two apparent identical channels occurs.

3.1.3. Number of theory curves The simulation and calculation of a theory curve with the program of C. C UEVAS requires some computation time, so it makes sense to calculate the curves once in a sufficiently small grid and to save them. For 5000 curves it took several days of computation time on multiple computers. 5000 steps in the transmission interval between 0 and 1 correspond to a step size of 0.0002. At first, this seems exaggeratedly accurate, since in the fit an absolute accuracy of the

30


0.8

0.8

0.6

0.6 τ4

1

τ4

1

0.4

0.4

0.2

0.2

0

0 0

0.2

0.4

a)

0.6 τ3

0.8

1

0

b)

0.2

0.4

0.6

0.8

1

τ3

Figure 3.4.: Fit error as color scale on a test curve with the channels 0.1, 0.5, 0.7 and 0.9, plotted against two of the four fit parameters τ3 and τ4 in a grid of 500 steps each. The other two channels τ1 and τ2 were rasterized in a) 1000 steps and b) 50 steps. Restricting the accuracy of channels τ1 and τ2 to a 0.02 raster has a much greater effect on the accuracy of τ3 and τ4 up to a minimum of around 0.1.

individual channels better than 0.05 can hardly be justified. But as it turns out, an inaccuracy of one channel affects the accuracy of other channels even more. In particular, this applies to deviations in channels with high transmissions, which have a strong impact on small channels. Fig. 3.4 shows the error as a color scale plotted against two of the four channels. While in a) the two unassigned channels τ1 and τ2 were rasterized with 1000 intermediate values, b) shows the result for only 50 intermediate values. The quantization artifacts lead to local minima at wrong positions. The roughness of the potential landscape also complicates the convergence of the Monte Carlo optimization method, which will be described below. 5000 appears to be a sufficient number of curves and transmission values. If the curves are in a coarser grid, intermediate values could be linearly interpolated. Here, however, experiments with channel histograms have shown that about 100 interpolation points are still marginal. In addition, the interpolation increases the computation time per iteration by a factor of 2 to 3.

3.1.4. Bandgap The theoretically calculated IV curves (see Fig. 3.1, p. 27) are universally valid for a large class of conducting nanostructures, independent of material parameters except for the band gap of superconductivity V∆ . However, this variable only enters into the IV characteristics as a linear scaling, so that with a suitable choice of the units, the material dependence is eliminated. When comparing with the experiment, however, the value plays a role, since the units must be converted into each other. V∆ has a value of approximately 180 µV for the used material aluminum. In addition to the material, a slight influence of the geometry of the contact on

31


0 µV

Ex:

0 µV

0.8 0.6 0.4

5

0.2 0 178

179

a)

180 V∆ (µV)

181

25

0 182

Th:

0 µV

Ex:

35 µV

24

1 0.8 0.6 0.4

23

0.2 22 178

179

b)

180 V∆ (µV)

181

Transmission

10

1 Error (10−3V∆G 0 )

Th:

Transmission

Error (10−3V∆G 0 )

15

0 182

Figure 3.5.: Fit a test curve with three channels (0.8, 0.54, 0.46) depending on V∆ . Minimum error (black line) with the resulting channels (blue dots) for a) non-smoothed and b) smoothed (35 µV) test curve) The vertical gray line is at the position of the minimum fit error The horizontal gray lines correspond to the true channel values.

V∆ of a maximum of 1 µV is imaginable, which the analyzes indicate. The extent to which a not exactly known value of V∆ has a negative effect on the determinability of the channels will now be examined by means of examples. First, test curves composed of theory curves are examined. The curves were constructed with a band gap of V∆ = 180 µV. Analyzing with values between 178 µV and 182 µV. Figure 3.5 a) shows investigations of test curves with channels identical to the examples in the previous chapter. As a black solid line the minimum error of the fit is drawn. At the actual band gap of 180 µV this error is zero. Deviating values lead to higher, approximately linearly increasing errors on both sides. The corresponding channels of the respective fit are represented by blue dots. They increasingly deviate from true values (gray horizontal lines). With realistic rounding of the constructed characteristic curves with 35 µV (Fig. 3.5 b), the minimum of the error remains at 180 µV, but the channels show small deviations at this value.

0 µV

Ex:

0 µV

0.8 0.6 0.4

5

0.2 0 178

a)

179

180 V∆ (µV)

181

0 182

28

Th:

0 µV

Ex:

35 µV

27

1 0.8 0.6 0.4

26

0.2 25 178

b)

179

180 V∆ (µV)

181

0 182

Figure 3.6.: Fit results as in Fig. 3.5, only with four channels (opposite Fig. 3.5 another channel at 0.25).

32

Transmission

10

1 Error (10−3V∆G 0 )

Th:

Transmission

Error (10−3V∆G 0 )

15


35 µV

Ex:

35 µV

0.8 0.6 0.4

5

0.2 0 178

179

a)

180 V∆ (µV)

181

12.5

0 182

Th:

30 µV

Ex:

1

35 µV

0.8

10

0.6

7.5

0.4

5

0.2

2.5 178

179

180 V∆ (µV)

b)

Transmission

10

1 Error (10−3V∆G 0 )

Th:

Transmission

Error (10−3V∆G 0 )

15

0 182

181

Figure 3.7.: Smoothed test curve (35 µV) as in Fig. 3.5 b) with smoothed theory curves with a) 35 µV and b) 30 µV. Thus, if the type of rounding is not modeled accurately, the Result of the fit are strongly influenced.

0 µV

Ex:

0 µV

0.6

10

0.4

5 0 178

a)

0.8

0.2 179

180 V∆ (µV)

181

0 182

15

Th:

35 µV

Ex:

35 µV

10

1 0.8 0.6 0.4

5

0.2 0 178

b)

179

180 V∆ (µV)

181

0 182

Figure 3.8.: Difficult set of channels with the values 0.15, 0.30, 0.45, 0.60 and 0.75 with a) none and b) 35 µV strong rounding. The band gap must be known here relatively accurately. Normally, the channel distances are much less balanced, which further favors the risk of merging two channels.

33

Transmission

15

1 Error (10−3V∆G 0 )

Th:

Transmission

Error (10−3V∆G 0 )

20


Much more sensitive is the result with an additional channel at 0.25 (see Fig. 3.6). Here, in the case of an unrounded test curve (a), the correct channels are determined as expected. For the rounded curve (b), however, the minimum value is slightly lower at 179.8 µV and second, the channels at this value as well as at exactly 180 µV are far off. At a certain larger value V∆ ≈ 181.3 µV the channels are reconstructed correctly. Such a value is often found in many other calculation examples. However, choosing a slightly larger value of V∆ is not a safe strategy for finding the right channels, because the distance to 180 µV or the minimum error varies too much. However, the deviations are generally slightly lower above 180 µV. As an alternative strategy, it is now attempted to better align the theory curves that are combined for the fit procedure by rounding them up as realistically as possible. For this purpose, a convolution with a G AUSS-bell function is performed as a model of the noise distribution present in the experiment. The purely electrical noise is seen as the main source of rounding and the finite temperature is classified as an effectively similar effect. The result shows Fig. 3.7. In picture a) the same rounding of 35 µV was applied to both the theory curves and the fitting curve. As expected, this does not improve the stability of the fit against unsmoothed curves. However, if V∆ is selected properly, the fit will provide the correct channels. In Figure b) the same curve smoothed with 35 µV was used, but weaker smoothed theory curves (30 µV) were used for fitting, which still leads to a gain in accuracy over the fit with unsmoothed theory curves (see Fig. 3.6 b) at both 180 µV, and at the position of the minimum error. However, the result also shows that the exact type of rounding has a strong influence. Although the approximately Gaussian noise distribution could be measured directly in the experiment, it was unclear what proportion of the noise was present on the sample and which, e. g. is induced by the amplifier afterwards. Thus, no accurate measured distribution was available for convolution. However, a G AUSS curve could be adjusted quite well in width by systematic fit attempts. A certain confirmation was also provided by the comparison of the fit error to the symmetry error. The analysis of real experimental data is shown in the section 5.

3.1.5. Fit algorithm As shown in the previous section, the problem of channel analysis reduces numerically to the determination of a global minimum of a multi-dimensional potential landscape, formed from the square deviation of the fit from the experiment. For this purpose, various strategies or algorithms come into question, of which the simulated annealing has proven to be the most advantageous. The G AUSS regression proved to be unstable and the systematic scanning of all values was too time-consuming.

Systematical scanning The complete scanning of all possible parameters required a number of steps proportional to the order O(N n ), where n is the number of channels to be expected and N the number of theoretical curves available (here is N = 5000). Figure 3.2 shows the result of such a scan on three channels, which takes a few minutes of computation time. With four channels, the calculation time would have to be expressed by days, with five channels by years. The 34


consideration of the symmetry accelerates the calculation by factor n!. A restriction to small ranges around the minimum also would reduce the computation time, but requires a manual selection or a previous coarser rasterization.

G AUSSian regression Of course, it would be attractive to be able to directly determine the global minimum analytically. This is possible by using the G AUSS regression method (see section 3.2.2) as long as the fit parameters are linear in the fit function. Although this is not the case here, one could provide all individual available theory curves with a linear factor as a free fit parameter. In this way, the problem would become linear, but with additional degrees of freedom. It is to be hoped that the additional degrees of freedom will not be occupied and that almost all factors will get the value zero, with the exception of a few, rounded to integers. The integer would then correspond to the number of channels to the associated transmission. Experiments showed, however, that this is only possible with exact (artificially created) curves. Even with barely visible noise, the number of channels increases dramatically and the factors take on a variety of non-integer and also negative values. The remaining error thus becomes very small, and even noisy measurement curves are perfectly approximated, but using physically nonsensical parameters, such as negative transmissions. The method is therefore unfortunately unsuitable. Nevertheless, a very efficient method for the calculation of Φ was found in the G AUSS equations (see chapter 3.1.6).

Simulated annealing The method simulated annealing from Madrid3 proved to be particularly suitable, a Monte Carlo optimization method that yields fast result with high probability. The algorithm allows finding a global minimum in a potential landscape if the local minima are neither very deep nor isolated. These two requirements satisfy the problem under investigation, as can be shown. The method first starts at a random starting position in the potential landscape and alters the coordinates by small steps in random directions. The step is discarded if it goes uphill and the condition exp(−Φneu /ΦRef ) > r is met at the same time, with the error square of the new position Φneu , a parameter ΦRef and a random number r ∈ [0, 1]. This corresponds to the movement of a particle in a potential landscape at a temperature ΦRef and the occupation probability of a higher lying potential according to the B OLTZMANN distribution. So the particle is very likely to move downhill. Nevertheless, it can escape from a flat local minimum with a certain probability. The temperature can be adapted to the problem depending on the depth of the local minima. The temperature can also be lowered slowly during the process. In this case, however, this does not bring any advantages, as the local minima only appear on a small scale. The position of the minimum potential is stored on the entire trajectory. Once the algorithm has approximately reached the depression of the global minimum, it remains there for all subsequent iteration steps and thoroughly scans the immediate environment due to the finite temperature. In order to additionally limit the residence time in small potential wells, 3 The idea for using the method originates from the group of G ABINO R UBIO -B OLLINGER

35


0.6

0.6

∆τi 2

0.8

pP

1

Distance

τ4

0.4

0.4

0.2

0.2 0 0

0.2

0.4

0.6

0.8

τ3

a)

0

1

0

10

b)

20 30 1000 iterations

40

50

Figure 3.9.: Example of 12 fits with simulated annealing of a curve composed of the values 0.1, 0.4, 0.7 and 0.9. a) Paths of the simulated particle trajectories starting from a random starting point along the first 50 000 iteration steps up to one of the 24 minima (here only 4!/2! = 12 are visible, since in this projection two lie on each other) , b) Minimum distance to one of the minima plotted against the first 50 000 iteration steps. 1 million iteration steps take about one second of computation time.

0.75 0.5 0.25 0

a)

1

5, 6, ≥ 7 channels with τ > 0 Transmission

Transmission

1

0

0.5 1 1.5 No of iterations (106 )

0.75 0.5 0.25 0

b)

5, 6, ≥ 7 channels with τ > 0

0

5 10 15 No of iterations (106 )

Figure 3.10.: Different fits depending on the number of iteration steps. All fits are independent and start with random values. a) Fit of four widely spaced channels (0.8, 0.6, 0.4, 0.2) with five free parameters. b) Five slightly denser channels (0.72, 0.68, 0.44, 0.3, 0.16) fitted with six free parameters. Here, more than ten times the number of iteration steps is required for a stable result.

36


Parameter Strength

ErrRef

KickOff

good value 0.001

10−7

0.0003

description Maximum amount of random change in each iteration. Since the theoretical curves are in grid 0.0002, this maximum step size should be sufficient. Higher values lead to slower convergence. Temperature for the occupation probability of higher potentials. This value is selected in such a way that in the case of large errors, it is practically only possible to move downwards and in the case of small errors, only occasionally degradations occur. Probability of change in each case. This parameter is effective in rare cases at saddle points or in small and deep wells.

Table 3.1.: Meaning of the parameters when fitting the IV characteristics with simulated annealing with the indication of suitable values.

a further parameter (Kickoff) specifies a probability for a movement step that is executed independently of the temperature criterion. Table 3.1 displays an overview of all parameters and specifies values that have been found to be suitable. To demonstrate the method, Fig. 3.9 shows the traces of a fit with the method of simulated annealing. In a) the minimum error with respect to two parameters was color-coded against the other two of the four free parameters. The black spots show the mirror-symmetrical global minima. As a starting point, four random values are selected and the trajectory drawn during the iteration steps. A dot marks the location reached after 500 000 iterations. b) shows the distance to the nearest minimum, plotted against the iteration steps. After 50 000 iterations the minimum was reached very well in the 12 examples. The trajectory first moves very fast, then slower and slower towards a minimum. Far from the minima, the potential usually seems to be very steep, as shown by Fig. 3.3. In rare cases the trajectory is temporarily stuck to a saddle point. A further example should give an idea of how many iteration steps are reasonable and sufficient. Slight fit problems with 4 channels as in 3.9 b) give quite a good result after 50 000 iterations. Fig. 3.10 a) shows fits of 5 free parameters at 4 given channels at a distance of 0.2. The fits have been calculated independently of each other with each random starting position. At least 500 000 iterations should be awaited here. In figure b) the channel spacing is only 0.14. Here 5 million iterations are required. The different channels can be fitted differently well. The uppermost channels generally converge particularly well. Channels with adjacent channels and those with very similar IV characteristics, such as between 0.2 and 0.5, converge poorly. Overall, the examples show that due to the asymptotic convergence, completely erroneous results are usually rare after a few iterations, if the global minimum is in the right position. The latter is likely to be the major source of error.

Automatic detection of the number of channels The number of channels is unknown before the fit. Therefore, you must start with a sufficiently large number of free parameters. Each degree of freedom, however, slows down the conver37


0.75 0.5 0.25 0

a)

1

5, 6, ≥ 7 channels with τ > 0 Transmission

Transmission

1

0


0.75 0.5 0.25 0

b)

5, 6, ≥ 7 channels with τ > 0

0


Figure 3.11.: Dependency of a fit on the number of iteration steps as in Fig. 3.10 for five given channels 0.8, 0.6, 0.5, 0.3, 0.1 if a) a maximum of 6 free parameters and b) a maximum of 20 free parameters are allowed. Despite the 20 free parameters, the convergence is not significantly slower. Only in a few cases will more than 6 of the initial 20 channels remain open at the end, as the color code shows. Often even the reserve channel is exactly zero.

gence of the fit. In simulated annealing, however, this only accounts for about a factor of 2 if the additional degree of freedom is not occupied. This is because the channels that are close to zero will drop to a negative value every second time they change randomly, and negative values will be set to zero. Thus, every second iteration step is identical to a fit with a channel fixed to zero or a smaller number of degrees of freedom. With two additional channels, this is the case on average every fourth time and so on. n additional free fit parameters only slow down the convergence by 2n , not by (N + n)!/N !. Nevertheless, it is worth using an extended algorithm to accelerate convergence by automatically adjusting the number of channels. To achieve this, once more than two channels have reached zero, a degree of freedom is removed. If no more channel has the value zero, on the other hand, a degree of freedom is added again, up to a given maximum number of degrees of freedom. Therefore, exactly one reserve channel is always left open which has the current value of zero. However, changes in the number of degrees of freedom should not be made in every iteration step, but only (to reduce the computational time) if a new best value is found. Figure 3.11 demonstrates the behavior at different degrees of freedom. In both cases, fits were started with different numbers of iterations. Initially, all values were set to random numbers. After a certain period of time, all but one of the superfluous degrees of freedom are usually closed and convergence is not slowed down. The following shows the source code of the algorithm in simplified notation (pseudo code). It was compiled in C++ and implemented as a dynamic library (DLL) in Python as a function. Basis for these codes was an algorithm of G ABINO RUBIO -B OLLINGER. The improvements are the elimination of interpolation, the more efficient calculation of the error and the extension, which automatically locks or reopens the channels by zero. All three improvements serve to speed up convergence by at least a factor of 20.

38


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

for count = 0 . . maxiter : //count iterations //−−−−−−−−−−−−−− Calculation of the new e r r o r −−−−−−−−−−−−−−−−−−−−−−−− ErrNew = c //Φ = c + ... for i = 0 . . channels −1: ErrNew += b [TNew[ i ] ] //+b i for j = 0 . . i −1: ErrNew += A[TNew[ i ] ,TNew[ j ] ] //+A i j //−−−−−−−−−−−−−− Improvement occurred ? −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− i f ErrNew < ErrNow : ErrNow = ErrNew //→ take over for i = 0 . . channels −1: TNow[ i ] = TNew[ i ] //→ take over //−−−−−−−−−−−−−− Degradation occurred ? −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− e l se : i f ( exp(−ErrNew/ ErrRef ) ) > (random ( 0 . . 1 ) ) : //Temperatur criterion ErrNow = ErrNew //→ take over for i = 0 . . channels −1: TNow[ i ] = TNew[ i ] //→ take over else : i f random ( 0 . . 1 ) < KickOff : //Kick-Off-criterion ErrNow = ErrNew //→ take over for i = 0 . . channels −1: TNow[ i ] = TNew[ i ] //→ take over //−−−−−−−−−−−−−− new best value found ? −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− i f ErrNew < ErrBest : ErrBest = ErrNew //→ save zerochannels = 0 for i = 0 . . channels −1: TBest [ i ] = TNew[ i ] //→ save i f TNow[ i ] = 0 : zerochannels += 1 //count zero channels //−−−−−−−−−−−−−− open or close channels ? −−−−−−−−−−−−−−−−−−−−−−−−−−−−− i f zerochannels 6= 1 : i f zerochannels > 1 : //close channels? i = 0 ; while TBest [ i ] 6= 0 : i += 1 //1st zero channel? TNow[ i ] = TBest [ i ] = TBest [ channels −1] //empty channel... TNow[ channels −1] = TBest [ channels −1] = 0 //...to the end i f channels > 1 : channels −= 1 //→ close channel else : i f channels < channelsMax : channels += 1 //→ open channel for i = 0 . . channels −1: //random change TNew[ i ] = min(TMax, max( 0 ,TNow[ i ]+ i nt ( Strength * ( random ( − 1 . . 1 ) ) ) ) ) //−−−−−−−−−−−−−−−−−− End −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− Listing 3.1: Simulated annealing with automatic channel reduction (pseudocode)

39


3.1.6. Efficient calculation of the fit error Φ The efficiency of calculating the fit error v !2 uZ Ã n u X t Φ(τ1 , τ2 , τ3 , . . . , τn ) = I Theo (τi ,U ) − I Mess (U ) dU

(3.8)

i =1

per iteration step initially seems to be limited by the integration. For discrete measured values, it corresponds to the summation via the number of interpolation points. Usually about 300 measuring points require 300(n + 2) arithmetic operations for n channels. However, this number can be reduced by appropriate preparation to only 12 n(n +1) additions. To accomplish this, it is taken advantage of the fact that only a finite number of (here N = 5000) theoretical curves ϕ j are present at discrete transmission values j /N . With f ≡ I Mess (U ),

g≡

n X

i =1

I Theo (τi ,U ) und ϕ j ≡ I Theo ( j /N ,U )

(3.9)

and with integer coefficients α j ∈ N (≥ 2 for several equal channels), so that holds g=

n X

i =1

ϕ(τi N ) =

N X

αj ϕj ,

(3.10)

j =1

the error can be reformulated to Φ2 (τ1 , τ2 , . . . , τn )

° °2 ® = °g − f °2 = g − f , g − f * + N N X X = αj ϕj − f , αk ϕk − f j =1

= =

(3.11b)

k=1

N N X N X ® ® X ® f , f −2 αj f ,ϕj + α j αk ϕ j , ϕk j =1

®

f,f +

= c+ =

(3.11a)

N X

j =1

c+

N X

j =1

Ã

αj

Ã

αj b j +

n X

j =1

Ã

(3.11c)

j =1 k=1

! jX −1 ® ® ® −2 f , ϕ j + ϕ j , ϕ j + 2 αk ϕ j , ϕk jX −1

αk A j k

k=1

b (τ j N ) +

jX −1

k=1

(3.11d)

k=1

!

(3.11e) !

A (τ j N )(τk N ) .

(3.11f)

® ® ® In 3.11e, A j k := 2 ϕ j , ϕk , b j := 21 A j j − 2 f , ϕ j and c := f , f were defined. Finally, the introduced integer α j are eliminated by irnoring the indexes or counting them single or multiple times. The scalar products, i. e. the computationally complex integrals, can now be

40

3.2. S MOOTHING ALGORITHM

calculated and stored once before the fit procedure. While b j and c have to be calculated once for each trace before the fit (approx. 50 ms computational time on a normal office computer), the matrix A j k can be calculated once for all measurement curves (approx. 5 minutes computational time) and saved for future fits (200 MByte file at 5000 × 5000 values). This means that the calculation time for Φ during fitting is now very short and independent of the number of sampling points of the curves. At e. g. n = 4 channels, the computational effort for the calculation of Φ per iteration step has dropped from originally 300(n+2) = 1800 arithmetic operations (in some cases multiplications) to only 12 n(n + 1) = 10 additions, which is at least a factor of 180 times faster. For scanning, a current processor (2.8 GHz quadcore) requires about 30 ns computational time per CPU core for the computation of Φ, i. e. 33 million values per second are obtained. An iteration step of the simulated annealing requires 0.7 µs. The calculation of Φ only takes up a very small part of the time. All together, the computation still results in 1.4 million iteration steps per second. A loss of computational accuracy is noticeable in the method only in the last 4–5 of a total of 15–16 decimal digits of the double-type, which is negligible.

3.2. Smoothing algorithm This section describes an algorithm for smoothing curves that can be found on local regression4 is based and improved in this work (accelerated to O(n) and extended for a numerically stable interpolation) and was applied several times. It was designed to be used for the underground treatment of conductance fluctuations, but has also proved to be a useful tool for the treatment of different measurement data and is superior to other smoothing methods.

Requirements Smoothing algorithms are often only used for better graphical representation of data to make certain tendencies easier for the reader to recognize. However, if the calculation results depend on the way of smoothing, the simple known algorithms are insufficient. This is the case, for example, with the background treatment of conductance fluctuations, if autocorrelation functions are to be calculated from them, or with the noise artifacts present in the experiment, which are to be modeled in theoretical current-voltage curves (see section (see Section 3.1.4). Smoothing algorithms should have two characteristics in particular: They should represent the curve as smoothly as possible and at the same time deviate as little as possible on average from the original curve. These two combined requirements define on the one hand the quality of the smoothing process and on the other hand they describe the information hidden in the measurement data, which is to be read out in the named application examples. Filtering certain frequency components would be an inappropriate description language of the problem. Often the terms filter and smoothening are hardly differentiated linguistically. Neither are 4 In the literature a detailed treatment is found e. g. in [Loa99]. There it is mentioned that the method was already

used in 1866 by the Italian astronomer S CHIAPARELLI (“discoverer” of the martian canals).

41


the different algorithms used in a targeted manner, which only have similar effects when viewed from afar. While in filters certain components are supposed to be separated, usually frequency components, smoothing, on the other hand, expresses the property of the result to be limited in curvature. Furthermore, a smoothing algorithm should also be able to be controlled precisely by means of understandable and problem-adapted, physically motivatable control parameters.

3.2.1. Known smoothing algorithms A variety of smoothing algorithms can be found in the literature. Widespread in secondary literature and numerical packages, however, are just those that do not always reliably smooth, as one would expect it qualitatively or would also draw by hand. Almost all of the known algorithms show an increase in curvature at the edge of a slant straight line. Very common and easy to program is the moving average, which averages over a fixed number of adjacent measurement points. However, in numerical reference works (e. g., Numerical Recipies [Pre02]) or in numerical software packages (e. g., Numpy / Scipy, Origin, NAG-library), more advanced algorithms such as e. g. a spline approximation [Die95], an FFT filterung, the gaussian blur, the optimal ( W IENER-)filtering [Wie64, Pre02] or the S AVITZKY-G OLAYsmoothing [SG64, Pre02]. In scientific publications, however, there is an unmanageably large number of filtering and smoothing algorithms, often for time series (i. e. asymmetric) or optimized for digital image processing.5

Real smoothing The less widely used W HITTAKER-H ENDERSON method6 most strictly satisfies [Whi23] the requirements of smoothing. Here, the mathematical optimum between the requirements for smoothness (small mean square n-th derivative) and accuracy (small mean square deviation from the original) is calculated. The control parameter is the ratio of both quantities. However, the parameter is somewhat unwieldy and a length scale must first be determined in an inconvenient way (see [Eil03]). The x values must also be in an equidistant grid, which limits the method. Similar results are provided by the local regression with a bell-shaped smoothing kernel. The smoothed curve is composed of locally evaluated regression polynomials obtained by local weighting of the respective local environment. As a control parameter, the width of the bell curve can be specified directly as a length scale of the smoothing, which can be physically motivated. In the case of very weak smoothing, the result is similar to the convolution with the bell curve; in the case of very strong smoothing, the result goes against the regression polynomial. Further advantages are: The x-values do not have to be equidistant and also 5 Another method should be mentioned: Gaussian process regression or kriging, a very mighty and versatile

interpolation and smoothing method. This comment was added in the translated english version of this thesis.

6 The W HITTAKER -H ENDERSON method [Whi23], also known in economics under H ODRICK -P RESCOTT filter

[HP97], according to [HP97] also goes to the Italian astronomer S CHIAPARELLI (1867).

42


40 Fast local regression

30

fwhm = 9, order = 1

20 10

G AUSS-Convolution fwhm = 10

0 10

FFT-Filter (F ERMI-function) f = 6, w = 0.8

0

y

10

Moving average d = 16

0 10

W IENER filter kernel = 15, noise = 50

0 10

S AVITZKY-G OLAY smoothing kernel = 15, order = 1

0 10

D IERCKX splinefit s = 1420

0 10 0 0

20

40

60

80

100

x Figure 3.12.: Comparison of local regression (with iterative bell convolution) with different common algorithms. Most of them have weaknesses at the edges. Many react to individual outliers with unfavorably overlapping rectangular discontinuities. In addition to the deficiencies seen here, there are further difficulties in applying some filters, such as e. g. a poor controllability of the behavior with the available parameters.

43


interpolation is possible. The local regression was therefore used as a basis for the algorithm described here.

Comparison Fig. 3.12 shows a qualitative comparison of the different smoothing methods with the algorithm presented here. The test curve was constructed from a doubly kinked 101 point line. On the left, peaks were added at two separate points, and on the right, noise up to the left end. 7 The obvious drawbacks and other drawbacks are summarized below: G AUSS-Convolution: Unintended curvature of the curve at the edges. If one extends the Gaussian convolution such that the area of the Gaussian curve is always normalized to exactly one in the overlap range, the edge effects improve (dashed curve), but then the edges always tend towards the mean value of the curve or not against the straight line. FFT-Filter (F ERMI): The deviation on the left edge unfavorably even depends on the curve on the right edge. This is due to the periodicity of F OURIER approximation. A cutoff function the F ERMI function was used. Moving average: Discrepancies always remain unstable even in the smoothed curve, which does not meet the expectations of smoothing. W IENER-Filter: The so-called optimal filtering produces artifacts similar to those of the moving average in the best setting. S AVITZKY-G OLAY-Smoothing: Also shows discontinuities, even if a higher polynomial order is chosen in the parameters. In addition, the control parameter for adjusting the amount of smoothing is not continuous. D IERCKX-Splinefit: Although splines appear smooth and show no edge artifacts, they are often unpredictable: some details are taken into account, others are ignored (see the two peaks in comparison). In addition, splines change abruptly with continuous change of the control parameter and with continuous change of function values of the curve. This is because the curve is described by a finite integer number of spline nodes. In the case of particularly strong smoothing, only very few nodes are present and thus the quantization artifacts are particularly conspicuous and disturbing.

3.2.2. New method for smoothing The new smoothing method is based on local regression with a bell-shaped smoothing kernel, which is described below. Instead of a Gauss curve, however, a approximated bell curve is 7 Values of sample for comparison: x = 0 . . . 100, y = 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 13.5, 14,

14.5, 40, 15.5, 16, 16.5, 17, 17.5, 18, 18.5, 19, 19.5, 35, 20.5, 21, 21.5, 22, 22.5, 23, 23.5, 24, 24.5, 25, 25.5, 26, 26.5, 27, 27.5, 28, 28.5, 29, 29.5, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 14, 9.75, 9.5, 10.25, 19, 9.75, 7.5, 17.25, 16, 25.75, 19.5, 11.25, 18, 26.75, 16.5, 19.25, 29, 24.75, 31.5, 22.25, 27, 25.75, 32.5, 23.25, 22, 27.75, 38.5, 28.25, 26, 29.75, 39.5.

44


used, which allows a faster calculation, so that the computational effort of the algorithm is scaled with O(n), i. e. linearly with the number of the given measuring points n and in particular is independent of the thickness of the smoothing or the number of points m = n of the convolution function. Even after intensive research, no such Gaussian filters with this rare “constant-time” property were found in the literature, as described, for example, found in a median filter (blur) [PH07]. Convolutions that scale with O(n) instead of O(n 2 ) (faster than FFT convolution with O(n log n)) are described in [Wer03] for a larger class of functions, but no bell curve is evidently included. This section introduces the O(n) iterative bell convolution found here. First, however, the already known local regression will be explained.

The algorithm To determine a smoothed value of a measurement curve, the coefficients for calculating a regression polynomial are weighted bell-shaped around the immediate vicinity. The regression polynomial obtained by this local weighting is evaluated at the respective position. The same procedure is performed for each value to be calculated and the bell curve is placed with its maximum at the position of this value. From these values a smoothed curve is formed, which locally corresponds to a regression polynomial, but is subject to a similar influence of the environment as in a convolution with the bell curve. The convolution and the calculation of a regression polynomial are in certain way combined into a single algorithm. The regression polynomial can be calculated very quickly. The “Least square method” roots back on C ARL F RIEDRICH G AUSS and A DRIEN -M ARIE L EGENDRE. This allows, for a measured function f with the measured values (x 1 , y 1 ) . . . (x n , y n ), to analytically calculate a fit of a k-th degree polynomial with minimum error square, or any other linear model, composed of arbitrary linear independent basis functions ϕi . The coefficients c i in front of the basis functions or the polynomial coefficients are obtained directly as a solution of the linear system of equations8 8 The derivation of the equation system is similar to the equation 3.11, p. 40, with the measurement function f ,

the fit function g , the basis functions ϕi , the coefficient c i and by the transformation of the squared error Φ2 (c 1 , c 2 , . . . , c k )

= = =

° ° ® °g − f °2 = g − f , g − f 2 * + k k X X c i ϕi − f , c j ϕj − f i =1

(3.12a) (3.12b)

j =1

D E k k X k X ® ® X f , f −2 c i f , ϕi + c i c j ϕi , ϕ j i =1

(3.12c)

i =1 j =1

and by the demand for the extreme value D E k ® X 1 ∂Φ = − f , ϕi + c j ϕi , ϕ j = 0, 2 ∂c i j =1

für i = 0, 1, . . . , k.

(3.13)

Furthermore, it can be shown that the second derivative (Hessematrix) is positive definite, so that the extremum found is a minimum [SK09, Gro69].

45


 ® ® ϕ0 , ϕ0 ϕ0 , ϕ1 ® ®  ϕ1 , ϕ1  ϕ1 , ϕ0  .. ..   . . ® ® ϕk , ϕ0 ϕk , ϕ1

··· ··· .. . ···

®    ® ϕ ,ϕ c f , ϕ0 0 k ®  0   ® ϕ1 , ϕk   c 1   f , ϕ1   .  =  .  ..  .   .   .   .  . ® ® ϕk , ϕk ck f , ϕk

(3.14)

As basis functions ϕp we choose the basis of the polynomials, thus the x-potentials ϕp = x p . The scalar products are defined in the local regression by means of bell-shaped weighting, depending on the x position x i to be calculated

f , ϕp

®(i )

:=

n X

j =1

f (x j ) ϕp (x j ) g (x j − x i )

(3.15)

with a bell function g (x), e. g. ¶ µ 1 1 ³ x ´2 . g σ (x) = p exp − 2 σ σ 2π

(3.16)

By evaluating the polynomial at the position x i , we now obtain the required smoothed i -th value from c p(i ) k X c p(i ) x i p . (3.17) si = p=0

The described procedure is performed for all positions x i , with the maximum of the bell function being at the same position x i . The x i in 3.15 and 3.17 do not necessarily coincide with the x j of the measurement. As a result, in the case of smoothing, interpolation and redistribution of the interpolation points can also be achieved.

Fast calculation of the weights In the algorithm described, for each position x i the scalar products that contain a convolution with a bell function must be calculated. The number of arithmetic operations for smoothing then grows overall with order O(n 2 ) or via FFT convolution with O(n log n). It was possible to reduce the computation time to O(n) by replacing the G AUSS bell-shaped curve 3.16 with the curve µ ¶ µ ¶ 1 |x| 1 |x| 21 geσ (x) = exp − − exp −γ mit γ = (3.18) σ σ γσ σ 20

, in the following named iterative bell function. This function is twice continuously differentiable and, as seen in Fig. 3.13, relatively bell-shaped. The similarity to the G AUSS form can be further enhanced, as will be shown later. The value γ should not be too close to one because of rounding errors, so that the shape does not get too sharp, but close enough to one. The effect of the iterative bell convolution becomes now visible when the function (3.18) is decomposed by geσ (x) = h σ (x) − 46

1 1 h σ/γ (x) − h σ (−x) + 2 h σ/γ (−x) 2 γ γ

(3.19)


1.5 G AUSS bell fast bell

1

0.5

0 −2

−1

0

1

2

Figure 3.13.: Comparison of the bell-shaped G AUSSian normal distribution (3.16) with the iterative bell convolution (3.18). Both functions are normalized to area 1 and maximum 1.

into four half e −x functions h σ (x) =

³ x´ 1 exp − Θ(x) mit Θ(x) = σ σ

½

0: 1:

x 900 Ω, the voltage drops to a greater extent over the sample so that no abrupt changes in conductance can cause any voltage drop in the conductors to jump over to the sample. Nor do unnecessary losses in the lines heat up the cryostat too much. The right variant is shown in the case of high-resolution measurements as introduced by the excitation currents interfering signals by a voltage divider in the amplitude of 1/37 can be reduced. The voltage divider is placed in the low temperature range to avoid thermal noise. However, it prevents the current through the sample from being measured with reliable accuracy, so that an additional reference resistor of known quantity is required to determine the current. It is attached in series to the sample and additionally reduces the excitation voltage and the interference signals introduced by the factorR Probe /(R Probe + R Ref ).

Failed strategies In order to combine the advantages of both cabling, various tests were carried out. For example, the sample was directly excited with a current source past the voltage divider by converting lines outside the cryotate. The problem was, however, the parasitic current through the 1 kΩ-resistance of the voltage divider the parallel to the sample and through the reference resistance. This led to the falsification of the measured value, to an additional ground loop and an associated responsibility to interference. In addition, the external relay switch box for changing the cable function and the non-coaxially shielded interrupts also produced noise. The alternative of adding more cables, each with its own fixed function and its own measuring device, would cause further interference. The reason for this are unavoidable ground loops 3 Incidentally, the resolution does not mean the low level of measurement noise in the signals. Rather, it is defined

by the smallest possible details that can be measured in non-linear current-voltage characteristics. In contrast, temporal noise of the measurement signal can be minimized by means of sufficiently slow measurement or, in the case of many data points, afterwards, in the evaluation in the computer. On the other hand, noise applied to the sample, which leads to smoothed current-voltage characteristics, irrevocably destroys information. Effectively, the characteristic to be measured is convolved with the noise distribution function.

56

4.5. E LECTRICAL WIRING OF THE CRYOSTAT

a)

Relay "off"

Relay "on"

c) U+ U-

U+ U- B+ B-

U+ U- B+B- I+ I-

I+ I - B+B-

UR

300 K

300 K 250 mK 250 mK 18k

18k

Filter A

1k

Filter B

56k

Sample b)

Sample RRef

PC Current source

output

output

A/D-D/A-convert. input

10x, 100x 1000x, 10000x

U+ U-

I+ I-

1M

10x, 100x 1000x, 10000x

1M

Preamp. Preamp.

1k

latching relay

output

18k

18k

Voltage source

latching relay

digital out

monost. relay

UR B+ B-

Filter B Sample

56k

RRef

Figure 4.6.: Switchable cables: a) Effective wiring between which can be switched back and forth. b) Wiring outside the cryostat with relays for switching between current and voltage source. c) Wiring within the cryostat with relays for activating and deactivating a voltage divider and a serial reference resistor.

through the two permanently connected sources (current and voltage source). Omitting the voltage divider, it was not possible to achieve the required low-noise and measuring solution. A further alternative would be to store the resistors outside the cryostat in liquid nitrogen, which was partly implemented in the work of P. KONRAD [Kon03, KBS+ 05], where an adequate voltage measurement could be achieved with an external voltage divider. Based on this, the external voltage divider could be deactivated via a switch. The internal series resistance of 100 kΩ could then be bypassed by re-working the cables. For this purpose, however, no further attempts were made, since the reproducibility of this designed as a last resort variant appeared very uncertain.

57

4. E XPERIMENTAL T ECHNIUQES

R (Ω)

30

Relay “off” (high current) R N = 31 Ω, T3 He ≤ 248 mK

1 I (nA)

40

20

50 Relay “on” (low noise) 0 R N = 9 kΩ −50 T3 He ≤ 244 mK −400

0

400

0

10

a)

0 −400

−200

0 I (µA)

200

400

b)

−1 −40

6 µV −20

0 U (µV)

20

40

Figure 4.7.: Demonstration of cabling performance in the two relay positions: a) critical current measurement experiment, sometimes up to 1 mA b) High resolution multiple Andreev reflection measurement, detail around zero and larger view in the inset.

Best strategies Figure 4.6 b) and c) show the solution of the problem realized in this work. The centerpiece is an electromechanical relay placed at low temperatures that can switch the voltage divider and reference resistor on and off. This effectively switches between the two strategies shown in Fig. 4.6 a). The test leads U retain the fixed function of voltage measurement over the sample. Lines I are only required when using the voltage divider. Lines U and I can both remain connected to the external meter without interruption. Only the excitation lines B must be changed when switching. However, these are less problematic because introduced interference in the high-resolution measurement are minimized by the voltage divider. The test leads U and I do not require such protective resistors. They are connected to very highimpedance (1 GΩ) preamplifiers. This means that there is hardly any current flowing through it, which in turn produces few external interference signals. The interference generated by the antenna effect is minimized by twisted-pair cables. Figure 4.6 b) shows the wiring outside the cryostat. Using the outer switch box, the power source (Keithley 6221a) can be replaced with the power source (Yokogawa). During the switching process, 1 MΩ-resistances are interposed to protect the sample from short current peaks in case of slight potential differences.

Demonstration measurements for both strategies In Figure 4.7 an example measurement is shown to demonstrate the properties of both switching states. a) shows the characteristic of the critical current with high currents in the variant without voltage divider. At 0.4 mA there is no noticeable increase in temperature on the 3 He thermometer. A maximum of about 50 mA is possible. B) shows the highest resolution achieved when the voltage divider is activated. Details of the characteristic of about 6 µV are still clearly recognizable.

58


Details of the relay The RAL-5 D W-K relay used in the cryostat is a bi-stable double relay. This means that the magnetic coil only needs to be supplied with power for a short time (for ≥ 6 ms).) The switched state is then permanently held by a permanent magnet, which is reversed with reversed polarity or alternatively via a second reversely wound coil. To avoid additional cable, the former option was chosen. Only with a bistable relay it is possible to switch at low temperatures, as in 250 mK, without heating up the cryostat. A relay of this kind has a power consumption of 5 V 150 mW. For a monostable relay, the cryostat would be overloaded with 40 µW cooling power at 300 mK, but the short switching time of 15 ms showed a very small temperature rise in 3 He only by 2 mK per switching operation for a few minutes. The plastic housing of the relay has been removed so that no air remains in the housing during pumping that could freeze on the contacts and worsen them. Instead, the relay was housed in a copperplate molded enclosure that provides electrical shielding and sufficient thermal contact with the cooling system and can be evacuated through a curved tube. In Fig. 4.6 c) the copper case is shown with a dashed line. The resistors of the voltage divider are in the same housing. As resistors SMD components were used, which were wrapped as far as possible with copper foil and thermally coupled, so that the resulting heat can flow away. As a disadvantage of the relay circuit is mentioned that no larger magnetic fields (& 50 mT) can be used in the experiment. The relay would otherwise switch unintentionally or cause mechanical damage in the case of stronger fields.

Details of the cables The cable pairs U, I and B consist of twisted manganin wires of length 1.5 m and of diameter 50 µm in a stainless steel capillary. In the intermediate area, the capillary was filled with dilute silicone (solvent toluene). The silicone holds the wires mechanically stable in the shielding and provides as electrolyte for a certain capacity. The resistance of manganin, which is quite high for a metal, the capacitance and inductivity together form an electrical lowpass with a cut-off frequency of 14 MHz and a damping of − 58 dB at 5 GHz (calculated for 1 m cable length). 5 GHz corresponds to mean thermal energy of excitations in the used cryostat with T = 250 mK converted into photon frequency ν = k B T /h. Table 4.1 summarizes some data about the cables.4

4.5.2. Copper powder filters In addition to the filter effect of the coaxial cable already mentioned, further filters were used to lower the electronic temperature and to cut off high frequencies from 100 MHz. The transfer of heat through the conductor is mainly due to electric fields (the “sound” of the electron gas propagating at almost the speed of light) and diffusion. Although the electrons also move very fast (v F ≈ 1600 km/s in Cu), the mean drift velocities are negligibly small for 4 Manufacturing and measurement of the cables was carried out by H.-F. P ERNAU

59


Properties of cable pairs U, I and B

Value

Cable material Cable diameter Cable length Shielding material Outer diameter of the shield Inner diameter of the shield Total resistance Specific restistance (manufacturer information) Specific capacity (at 5 kHz) Specific inductance Calculated damping at 5 GHz Calculated cutoff frequency

manganin 50 µm 1.5 m stainless steel 400 µm 200 µm 490 Ω 224.9 Ω/m 20 pF/m 140 nH/m −58 dB/m 14 MHz

Table 4.1.: Properties of the cables used.

the currents used (about 10 nm/s bis 1 µm/s)5 . Passing through the filter tube only 5 cm requires several days or months for one electron in average. The purpose of the filter is to cool the electrons and to dampen the frequencies of thermal noise in the electrical signal. Here, only the newly developed copper powder filters [MDC87, BGH+ 03] were used instead of conventional RC elements. The benefits are sharper cut-off frequencies and greater damping at high frequencies. Each of the room temperature pipelines passes through a copper powder filter as shown in Fig. 4.6, p. 57. Two types of these filters were installed:

Filter A Filter A refers to filters with insulated wires. The length of the wire must be sufficient for a filtering effect in the range of meters (here 2.4 m). Due to the skin effect, high frequency electrical transport takes place close to the surface of the conductor, as evanescent waves overlap through the insulation with the copper powder. Due to their electrically insulating oxide surface, the copper particles ensure very selective transmission and gradual damping of high-frequency signals, as well as good thermal conductivity for sufficient cooling of the heat generated. This is done with a special coiling with vanishing inductance, so that magnetic interference fields can induce no currents if possible. The damping of these filters at a frequency of 600 MHz is -70 dB. This means that no notable thermal noise can pass through the sample, which corresponds to temperatures of 250 mK and higher. 5 In a MAR measurement on a single-atom contact, e. g. I = 0 . . . 100 nA and v 2 Drift = I /(neπr ), with r = 50 µm, 22 −3 −19 n = 8.45 × 10 cm [Kit06] and e = 1.6 × 10 C.

60


2 cm Figure 4.8.: Filter B. Left: Image of the filter array in a compact copper package with eyelet for attachment and thermal coupling. Right: Filter curves with the transmitted power through two of the seven channels (Measurement: Scientific Workshop of University of Konstanz).

Filter B Filter B denotes a copper powder filter with a non-insulated wire in direct contact with the copper powder. Very fine grain copper powder, as used herein, is not percolating in conductivity due to the oxide surface at low frequencies and over long distances. Direct contact with the powder, therefore, does not cause a short to ground, but does significantly improve the filtering effect, so that much shorter wires in the filter result in the same filtering. Filter boxes can be built much more compact, since no coiling is necessary. At approximately 5 cm distance from the sample, such a compact (25 mm×21 mm×5 mm) sevenfold filter array could be well accommodated in the innermost shielding sleeve. Disadvantages are occasional conductive paths through the copper powder to the chassis ground or to adjacent wires. If such a case is present, the copper powder may be rearranged or loosened by slight shocks until no more shorts are detected. Figure 4.8 shows the measurement of two channels of the filter, both curves are very different depending on the arrangement. The filter curves were measured under changed conditions outside the cryostat at room temperature, but they show that despite the simplified design without a coaxial shielded wire, a filter effect can be seen. The filter serves as a second filter stage. An additional advantage is the cooling of the only 5 cm long wires that lead to the sample. Apart from the metals in the connector and in the solder joints, the sample is thereby coupled via a copper wire directly to the copper rod coming from the 3 He-Pot. The first 5 µm thick polyimide layer between the 100 nm thick aluminum structure and the bronze substrate thermally insulates the sample, so cooling through the leads plays a very important role. Table 4.2 summarizes the most important data of the two filter types.

4.5.3. Grounding and shielding To reduce the coupling of interference signals, a good shielding and a well-considered grounding is necessary. The latter means, in particular, that the grounding originates in a star-shaped branching of the cables from the special measurement ground, so that no ground loop and no

61


Properties Grain size of the copper powder Length of the copper wire Diameter of the copper wire Isolation Cut-off frequency (−3 dB)

Filter A

Filter B

< 40 µm 2.4 m 50 µm Kapton about 5 µm 100 MHz

< 40 µm 5 cm 100 µm none 1 to 3 GHz

Table 4.2.: Properties of the filters used.

contact with the supply line ground is formed. However, variations of this basic strategy had to be optimized by trial and error. It turned out to be successful in detaching the complete cryostat from any mass. Also the shielding of any test leads was only connected on one end. The connection to the ground was then made via a single shielding of the excitation line. A massless measurement with regard to the test leads themselves proved to be unfavourable. Thus the negative pole of the respective source has been grounded. This means that the sample itself was no longer on the ground potential of the shielding, but this could not be avoided due to the lead resistance up to the sample and the voltage divider.

4.6. Equipment and control 4.6.1. Measuring electronics When choosing the measuring devices, the tasks were distributed to specialized devices. In detail, these are a DC current source, DC voltage source, two measuring signal amplifiers, a D/A converter and, not to forget, the device, which should be specialized exclusively to the intelligence, the computer. So for each task the best available device was used in my opinion and at the same time a high flexibility was kept. When working with universal devices (SMUs = source and measurement units or digital oscilloscopes), one is often limited to the more or less well-thought-out range of functions and is sometimes faced with insurmountable limits for certain, experimentally compelling specifications.6

Requirements Due to the temperature-dependent resistances of the leads from the measurement device to the sample, a four-point measurement had to be performed, which requires separate sources and measurement devices. The measurement of the multiple Andreev reflections requires an extremely stable DC source and a good amplifier to detect a noise-free signal 6 Particularly treacherous in sources is the well-intentioned automatic change of the gain range, which, however,

can destroy or alter the atomic contact by means of voltage pulses. Pleasantly passive and transparent is e. g. the voltage source Yokogawa 7651.

62

4.6. E QUIPMENT AND CONTROL

of a few microvolts. Since the cable capacitances, depending on the sample resistance, can be experienced as time delays and the amplifiers have a limited bandwidth, AC voltage was dispensed with and measured correspondingly slowly, for example in the range from 0.5 to 1 s per sample. For some measurements, e. g. to assess the noise, it should still be possible to perform fast oscilloscopic measurements on the same setup.

Selection of the devices As a source, the choice fell on the very low-noise DC current source Keithley 6221 or equally low-noise DC voltage source Yokogawa 7651. On the measurement side the signal was shortly amplified after the vacuum feedthroughs with the very compact built, low-noise and low-drift FET voltage amplifiers DLPVA-100-FD from the company Femto. The subsequently insensitive signal between -10 V and 10 V is recorded digitally by the data acquisition system ADwin. As a further device for preparatory measurements, the Keithley 2182A nano-voltmeter was sometimes used instead of amplifiers and data acquisition. All devices were controlled and synchronized via GPIB or via Ethernet by the computer. The devices in detail: Current source Keithley 6221: Primarily a DC power source with some AC functions. A voltage limitation is possible in relatively small increments, which is important for the protection of the sample. Further function used: digital output, with which the outer relay box could be switched. A treacherous disadvantage should be mentioned: If the grounding in the circuit is too far away from the negative pole, the source sends out extreme interference signals. This is not explicitly stated in the manual. This, however, includes, for example, the possibility of placing the grounding of the circuit directly to the sample, since this point is already too far away from the negative pole by the lead resistors. Voltage source Yokogawa 7651: The DC source is extremely stable and low noise, and not equipped with unnecessarily complicated software. It offers the possibility of monitoring the voltage further away from the output with an additional feedback loop, thus compensating for the interference signals that occur along the line. However, the latter function was not used since no improvement could be achieved. Voltage amplifier Femto DLPVA-100-F-D: Two of these amplifiers were placed on the measuring side with only 20 cm short coaxial cables very close to the current feedthrough. The very small case could be easily attached to the cryostat with packet adhesive tape. The amplifier is sufficiently low-noise and has settings for the two bandwidths 1 kHz and 100 kHz as well as four voltage gain ranges 101 , 102 , 103 and 104 . The gain ranges can be set via a optically decoupled digital input from the computer. Data acquisition system ADwin: This handy device has two types of A/D converters: 16 bits at 200 kHz or 14 bits at 2 MHz. It also offers 16-bit D/A outputs and some digital outputs. Instead of extensive functions, the device provides a simple basic compiler to execute

63


arbitrary functions with an internal 40 MHz-Prozessor. For example, data can be transferred to the computer as averages, histogram or buffered and triggered oscilloscope image. For our experiments, an average was calculated. In the case of current ramps, a certain settling time was also waited after each current change. Each measurement point is a numerical average over a well-defined time interval with approximately 100 000 individual measurements. The load on the actual measuring computer and the data lines remains very low. The outputs of the device were used to switch the amplifiers and the relay in the cryostat. Nanovoltmeter Keithley 2182A: This device was originally used instead of the femto amplifiers and the ADwin data acquisition system. The advantage is a pulsed mode for differential measurements, in order to be able to read out a resistance when the cryostat is being cold-driven without drift and without thermal effects. A synchronization and data cable to the Keithley 6221 current source is used. The resistance is then displayed on the current source display. The disadvantage of the device is that no fast oscilloscope measurements are possible.

4.6.2. Computer control It turned out to be extremely advantageous that all devices of the experiment were completely designed for remote control operation by computer. The five electronic measurement devices, the control units for the cryostat and the magnet system, as well as the controller for the motor of the break junction mechanics could all be controlled or read out via digital data lines (GPIB, Ethernet, RS232) from the computer. The relays for switching the cables were switchable without difficulty on one of the other devices. Only then was it possible to carry out the experiments in high quantity. Due to the computer control, processes and rules in the form of Python scripts could be formulated once for the many individual processes and decisions. The following short excerpt from the log file 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24 2008-02-24

02:29:45 02:30:19 02:30:19 02:30:23 02:30:24 02:31:32 02:31:33 02:32:29 02:32:29 02:32:38 02:42:41 03:05:07 03:22:21 03:22:43 03:36:53 03:36:54 03:37:54 03:38:03 03:39:10 03:39:11

Current is rising Jump from 0.904 to 0.538 Current drops Jump from 0.563 to 0.904 Current drops Current is rising Switch off the field... Field is at 0mT Andreev-Reflexionen He3-Pot: wait for T 10). High values indicate that the resistance at room temperature is dominated by phonon scattering, at low temperatures the phonons are frozen and the resistance correspondingly lower. For small RRR, this reduction is minor, due to temperatureindependent impurities, such as grain boundaries or impurities, or reduced phonon state densities in constrained geometry. For pure bulk aluminum, RRR ≈ 20 000 [Lid04].

Energy spectrum If the sample is broken and tuned to a tunnel resistance of approximately 3 MΩ, the electronic density of states can be mapped by differential resistance by varying the voltage. In the range up to 1 mV the band gap of the superconductivity can be observed. Fig. 5.17 shows in a) the measured IV characteristic and in b) a numerical derivative using analytically differentiated spline functions. The measurement noise somewhat limits this numerical treatment, but the sharp band edges are clearly visible at ±2∆. The upper scale plots the units of these band gaps, assuming ∆ = 180 µeV. The peaks should be much sharper and higher at these temperatures of 250 mK. The only reason for the rounding is the noise of the excitation voltage.

92

5.3. M ATERIAL QUALITY AND SUPERCONDUCTIVITY

4

−0.5

0 U (mV)

−2

0.075

0

−0.25

a)

U (V∆ = 180µV) −2 0 2

dU /dI (G 0 )

I (nA)

0.25

−4

0.05 0.025 0

0.5

b)

U (V∆ = 180µV) 0 2

−0.5

0 U (mV)

0.5

Figure 5.17.: Band gap of superconductivity. a) measured IV characteristic, b) numerical derivative with the help of smoothing splines.

Critical values I c , Bc , T c The critical parameters of superconductivity provide additional information about the material. All three variables were only determined at the end of the measurements when the sample cs_b37 was already open for 4 months. Although, as shown above, the resistance was initially < 100 mΩ , the sample could only be closed afterwards up to a value of 5–10 Ω . This resistance is presumably caused by the shape of the break junction that changes over time (due to constant mechanical processing). Figure 4.11, p. 71 shows a REM image of the sample with number cs_b37. Here the break junction is clearly visible (the same picture is shown on the cover of this work somewhat larger, page 1). Both sides seem to be rounded, which can often (but not always) be observed on broken contacts. The actual contact is therefore very small, which could make up the 5–10 Ω. This confirms Fig. 5.18, where the critical current at the inner structure is exceeded very quickly. In comparison, Fig. 5.19 shows a measurement on a sample that was not previously broken. The geometry was chosen a little differently here, but this should have no influence on the characteristic curve. The inner section is a bit broader here. The step shape can be explained by the different widths of the conductor and can be identified by the resistors of the normal conductiing state. In both cases, the 3.5 mm long lead causes the jump at the highest current. The narrower parts jump much earlier. The hysteresis shows an effect caused by the band gap of superconductivity. When the current rises below the critical current, the complete voltage drops above the supply resistance. If the critical current is exceeded and as a result part of the conductor is conducting normally, the current source must apply additional voltage to the system in order to maintain the current. When the current is lowered, the voltage in the normally conducting area is first located outside the (doubled) band gap shown in Fig. 5.17. If the local voltage touches the peak of the band edge, the resistance rises rapidly. The current source must regulate the voltage accordingly. This causes the local voltage to rise again and the local voltage remains attached

93

20

4.2μm

a)

40μm

0 −400

−200

0 I (µA)

200

400

40μm

10

4μm

30

100nm

R (Ω)

5. M EASUREMENT R ESULTS

3.5μm

b)

Figure 5.18.: Critical current. The measurement took place after the bridge was closed again.

2.5

a)

40μm

−1000

0 I (µA)

1000

40μm

R (Ω)

4μm

5

0

4μm

100nm

7.5

3.5μm

b)

Figure 5.19.: Critical current of another, unbroken sample.

to the peak. This delays the entry into the superconducting area, which makes a hysteresis in the IV characteristic visible. Even at a critical B field and a critical temperature, the collapse of the superconductivity could be observed (Fig. 5.20). Since these measurements were also made after the break junction experiments had been completed and the remaining resistance had already risen to 10 Ω, the transitions may no longer correspond to those of pure aluminum. The critical temperature (measured with the 3 He-thermometer far from the sample) was clearly too high.

94

5.4. C HANNEL ANALYSIS

40

40

R (Ω)

R (Ω)

30 20

30 20

10 10

a)

−10

0 B (mT)

1

10

b)

1.25

1.5 T (K)

1.75

2

Figure 5.20.: a) critical B field and b) critical temperature.

5.4. Channel analysis The so far obtained results are all based on measurements of the total conductance of the atomic contact. Microscopically, however, the electrons pass through the contact in different quantum mechanical modes, the conductance channels described in 2.1.5, p. 16. These are experimentally accessible by measurements of the multiple Andreev reflections (MAR) (section 2.2, p. 17) and by numerical analysis (section 3.1, p. 27). The measurement is carried out by shutting down the magnetic field applied to suppress superconductivity and measuring the high-resolution IV characteristic up to 1 mV as described in chapter 4. Figure 5.21 shows the sequence of a measurement with bistable switching and measurement of the MAR. a) shows an opening curve on the left to generate an atomic contact as a function of the distance over which the substrate has been bent. Further to the right, the bistable switching in the conductance and the associated current ramp are shown as a function of time. In this example, except for a barely visible small rearrangement, the contact is in a bistable state almost from the very beginning, without a longer training phase. After the bistability has been detected, the measurement is interrupted at the zero crossing of the current ramp and a MAR measurement is performed. The time required for this (about 15 minutes each) was cut out of the time axis. b) shows all six measured MAR characteristics in one graph. Three characteristic curves are exactly superimposed on each other, which confirms that only two different states are actually taken. Figure 5.22 shows the result of the numerical channel analysis described in chapter 3.1, applied to a series of measurements. a) shows the first two MAR characteristics from b). The measuring points were displayed here in reduced numbers in order to make the continuous, fitted theory curve lying behind visible. The mean deviation between measurement and fit is about 40 pA. The fit parameters found now reflect the transmission probabilities of the individual channels and can be plotted directly as multiples of the conductance quantum G 0 together with the total conductance in b). The sums of the conductance channels were also entered as black triangles. They lie very precisely on the directly measured conductance and confirm the MAR measurements, which were carried out with a completely different

95


0 6

−20

Current Conductance MAR measurement

4

100

4 2

2

0 400 410 0 a) Bending (µm)

0.25 0.5 Time (h)

0.75

Current (nA)

Conductance (G 0 )

20

I (µA)

8

75 50 25 0

0

1 2 3 4 5 6

0

0.2 0.4 Voltage (mV)

b)

0.6

Figure 5.21.: a) Sequence of a measurement with bistable switching. b) Six measured MAR characteristics. In doing so, both the three even and the three odd ones completely overlap.

40

20

0

a)

1 Theory fit Experiment

Conductance (G 0 )

Current (nA)

60

0

0.2 0.4 Voltage (mV)

0.75 0.5 0.25 0

0.6

b)

G(t ) τ Pi τi

0.2

0.4 Time (h)

0.6

Figure 5.22.: a) Measured MAR characteristic and fit of two states. b) Time conductance curve with the entered conductance of the determined individual channels.

measuring electronics (plus superconducting state). Here, the directly measured conductance would have a completely different and also voltage-dependent value. This confirms both the measurement, the MAR theory and the numerical analysis. These channel spectra are only valid for the zero crossing of the current ramps. The conductance continuously changing with the current must also be noticeable in the individual channels. However, they can only be measured around zero, since the determination itself is based on a voltage ramp around zero. The theoretical fit contains as further parameters the width w (full width at half maximum, FWHM) of the noise and the temperature (which rounds the MAR characteristic) and the exact value of the band gap V∆ of the superconductivity. These values could not be measured directly and had to be determined first. For this purpose fits with different values w and V∆ were performed and the minimum of the error was determined. Fig. 5.23 a) shows such a variation of the different parameters. The solid lines indicate the minimum error as a function of the bandgap. The curve showing the smallest error in terms of noise w is shown in black. In

96

5.4. C HANNEL ANALYSIS

4

0.75 0.5

2

0.25 179

180 V∆ (µV)

181

Error (10− 3 V∆ G 0 )

1

0 178

a)

20

6 Transmission

Error (10−3 V∆ G 0 )

Φmin = 2.6, V∆ =179.7 µV, w =20 µV

0 182

10 0 − 10 1

b)

2

3 U (V∆ )

4

5

Figure 5.23.: Fit error. a) Dependency of the fit error on the assumed band gap V∆ and additional on the variation of the Gaussian width w of the rounding. Black: Fit error on the best w, light red/light blue: larger/smaller values of w, blue: found channels matching the best w. b) Remaining difference to the theory curve, positive and negative edge of the IV characteristic. The unit 10−3 V∆G 0 of the error axis corresponds to about 14 pA.

red, the corresponding curves are shown for higher values of w and in blue for smaller values. In all investigations, the smallest errors in noise were found to be w=20 µV and a bandgap of V∆ =180. 0 µV. This value pair was used for all following fit procedures. Fig. 5.23 b) shows the difference between the measurement and the fit for the positive and negative slope of the MAR characteristic to the found minimum. While in this example fit to variations is relatively stable, Fig. 5.23 shows two measurements in which the found transmissions (blue) are much stronger, or much less, dependent on the assumed bandgap. The remaining mean errors of the fit are usually around 5 · 10−3V∆G 0 ≈ 70 pA and are barely visible. Nevertheless, deviations of this magnitude can strongly influence the results (due to the high number of degrees of freedom). Before each fit, the exact point of symmetry between the positive and negative edge is first determined in order to compensate for the usually low offset of the source and the amplifiers. The difference of the mirrored flanks is on average slightly lower at about 3 · 10−3V∆G 0 . Fig. 5.25 shows the dependence of the symmetry and fit error on the conductance of the contact. The symmetry error grows approximately linearly with the conductance, which can be interpreted as the current noise of the source (or as a voltage noise above the reference resistance) or as a voltage noise of the amplifier. The fit error also depends on the same noise as the symmetry error, and in addition on the inaccurately modelable noise distribution function. It is therefore always larger. Figure 5.26 (Pages 99 and 100) shows 14 more examples of a total of 190 similar measurements. The selection shows cases with particularly clear reproducibility in all details. The acquired channel spectra also show the same channels. Most of the measurements not shown here have a somewhat poorer reproducibility of the numerically obtained channel spectra, although the associated MAR curves, judged by the naked eye, appear exactly the same. In an earlier evaluation without the modeling of the noise curve, the reproducibility of the spectra

97


0.8 7.5

0.4

2.5

0.2 179

180 V∆ (µV)

181

7.5 1

5

0.75 0.5

2.5

0 182

0.25 0 178

179

b)

180 V∆ (µV)

181

0 182

Error (10−3 V∆G 0 )

Figure 5.24.: a) Extremely sensitive fit with four very small similar channels. b) Very robust result with three channels that are not too small and far apart.

Symmetry error Fit error

10

5

0

0

1 2 3 P Conductance (G 0 ) or τi

4

Figure 5.25.: Symmetry error (mean deviation to the point-mirrored IV characteristic) and fit error (mean deviation to the theory) as a function of the total conductance. Shown are average fit results of 1580 measured MAR curves.

under the related states was a lot better. At the same time, however, groups of channels also appeared, which (as shown in the chapter 3.1) indicate bad fits and which are related to the seemingly very high stability of the fit. The current values can be expected to be more reliable, although they fluctuate more. Error bars were omitted here, since a reliable quantitative quantity for individual measurements could only be obtained with high computational effort. As a guideline, an error of about ±0.03 can be assumed in the typical compositions for high transmissions, and ±0.1 for small transmissions, whereby these values are strongly influenced by the exact number and constellation of the channels.

98

Transmission

5

0 178

a)

0.6

Transmission

Error (10−3 V∆ G 0 )

10

Φmin = 5.4, V∆ =180.3 µV, w =21 µV

1 Error (10−3 V∆ G 0 )

Φmin = 3.8, V∆ =179.9 µV, w =20 µV

5.5. C ONDUCTANCE HISTOGRAMS

1 Conductance (G 0 )

Conductance (G 0 )

1 0.75 0.5 0.25

0.4

0.5 0.6 Time (h)

1.5 1 0.5 0

0.2

0.4 Time (h)

0.6

0.5 0.25 0

0.7

Conductance (G 0 )

Conductance (G 0 )

0

G(t ) τ Pi τi

0.75

0.5

0.75

1 Time (h)

1.25

0.5

1

1.5 Time (h)

2

1.5

1 0.75 0.5 0.25 0

Conductance (G 0 )

1.5 1 0.5 0

0.2

0.4 Time (h)

1

0.5

0

0.2

0.4 Time (h)

0.6

2 1.5 1 0.5 0

0.6

Conductance (G 0 )

Conductance (G 0 )

Conductance (G 0 )

2

0.2

0.3 0.4 Time (h)

0.5

1.5 1 0.5 0

0

0.2

0.4 Time (h)

0.6

99

2

Conductance (G 0 )

Conductance (G 0 )


1.5 1 0.5 0

0.6

0.8 1 Time (h)

1 0.75 0.5 0.25 0

1.2

1.5 1 0.5 0.2

0.4 0.6 Time (h)

2 1.5 1 0.5 0

0.8

0.25

0.5 0.75 Time (h)

1

1.25

1.5 1 0.5 0

0.8

Conductance (G 0 )

Conductance (G 0 )

0

0.6 Time (h)


Conductance (G 0 )

2

0.4

0.25

0.5 Time (h)

0.75

1

0.5

0

0.1

0.2 Time (h)

0.3

Figure 5.26.: A selection of 14 more of the 190 similar measurements. Only channels were plotted τ > 0.05. In the Fit, all channels below the transmission τ < 0.01 were closed except for a reserve channel.

100

5.5. C ONDUCTANCE HISTOGRAMS

mech. formed bistable (×1.47)

200

a)

40 20

100 0

mech. formed bistable

60 Counts

Counts

300

0

0.25 0.5 0.75 Transmission

0

1

b)

0

0.2

0.4 0.6 0.8 1 Conductance (G 0 )

1.2

Figure 5.27.: a) Histogram over 2018 (mech.) and 1367 (bistab.) channels from 713 and 409 channel spectra up to the conductance 1.2G 0 . b) Histogram over the respective sums of the channels or over the conductance values.

Channel histogram 0.85G 0 ≤ G ≤ 1.1G 0

Counts

100

50

0

a)

mech. formed bistable (×1.52)

0


1

b)

Figure 5.28.: a) Channel histogram over 288 or 156 contacts with 0.85G 0 ≤ G ≤ 1.1G 0 . b) Theoretical calculation of the histogram of C. C UEVAS [CYMR98].

5.5. Conductance histograms To study the composition of the single-atom contacts from individual channels, those channel spectra with total conductance < 1.2G 0 were selected from all 190 measured bistable switching states (409 spectra) and shown in a histogram (Fig. 5.27 a). For comparison, an experiment was performed in which spectra were measured after a purely mechanical formation of the contacts (713 spectra). After every 5–10 measurements on only slightly changed contacts, the contact was closed and reshaped up to a conductance of 50G 0 , so that the good statistical mixing leads to a reproducible statement. There is no significant difference between bistable and mechanically generated contacts. Only the peak above 0.5 seems to be weakened for bistable states in Fig. 5.27 a). However, this effect can be explained. Looking at the histogram of the sums in Fig. 5.27 b), one can see even in the first aluminum

101


2 · 105

Channel histogram G > 1G 0 Counts

Counts

600 400 200 0

a)

1.5 · 10

Conductance histogram

5

1 · 105 5 · 104 0

0


1

b)

0

1 2 3 Conduction (G 0 )

4

Figure 5.29.: a) Channel histogram over 7426 channels from 1380 channel spectra with conductance values > 1G 0 . b) Histogram of all bistable and non bistable current-driven rearrangement experiments of sample cs_b37 to verify purity.

peak at 0.85 a flattening or even a minimum for bistable contacts 8 . It is possible that contacts are so stable around 0.85 that bistable switching is very rare. As shown in the section 5.1.3, these values can withstand very high voltages. However, the partner state of this bistable contact would also have to be able to withstand this high voltage in order not to be destroyed immediately after switching. However, such partner states at other conductances do not or only rarely exist, so that only a few of the bistable pairs contain a value around 0.85. Of course, the attenuated 0.85 peak is also noticeable in the channel histogram, and apparently affects especially the higher channels, presumably the spz orbitals. The club-shaped px and py -orbitals, which reflect back about 80 % of the incoming electrons in each case, place themselves more as an obstacle in the way of arriving electrons and are therefore easier to switch. So it is not surprising to find them more often in bistable contacts. Channels with high transmission, on the other hand, act more like "lightning conductors"and are more stable. The assignment of the two peaks of the channel histogram to the spz , px and py orbitals, provides a comparison to the theory. Fig. 5.28 b) shows a calculation of a channel histogram with assignment of the orbitals. In this calculation, pyramidal (111) contacts were assumed and averaged over some atomic position disorder. The peaks are more distinct than in the experiment. Fig. 5.28 a) shows in comparison a channel histogram of purely mechanically generated contacts, which (limited to 0.85G 0 ≤ G ≤ 1.1G 0 ) has a good agreement. In more recent calculations total contact values up to 2G 0 were calculated in contacts of this type [PVH+ 08]. Further calculations of this method, published by M. H ÄFNER and published in [Sch08], show how the channels can change significantly by the angles or distances between the dimer contacts. Accordingly, this accumulation of contacts does not show ideal, but mainly tensed contacts. The evaluation of all channel spectra with sums > 1 (Fig. 5.29 a) provides a very balanced distribution of high transmissions, but still a preference for smaller values. However, in the 8 The steep flank in the histogram of the mechanical contacts is conditioned by the specification of the control

parameters for the break junction mechanics in the experiment and is not a regularity.

102

5.6. H YSTERESIS SHAPES

Channels with τ > 0.975

Counts

100

50

0

0

1 2 3 Conductance G (G 0 )

Figure 5.30.: Number of channels with transmission near 1G 0 , summed over all contacts whose conductance does not exceed G, plotted against G. About 1000 bistable and 1000 mechanically generated contacts were investigated here.

case of the many channels with high conductance values, the significance of the numerical evaluation of very small channels is diminished, so that the height and shape of the peaks around 0.05 and 0.2 are not guaranteed. Figure 5.30 shows from which total conductance the highest histogram bar in Fig. 5.29 starts to rise. A few contacts consist of a single fully transmitting channel, as predicted in the calculations of M. H ÄFNER in [Sch08] for dimer contacts, without voltage. Otherwise, the number of full channels will start to increase linearly with the number of contacts added only after a total conductance of about 1.7G 0 . Fig. 5.29 b) shows a conductance histogram of all rearrangement experiments. This is mainly composed of non-bistable current-driven conductance changes, but with changing current ramps after each reallocation (i. e. the failed attempts to generate bistable contacts). In the experiment, an interval of 0 to maximum 3G 0 was allowed, which limits the histogram upwards. The two peaks have a typical width and sit in usual places, confirming the integrity of the sample, which has been opened for almost five months. Although the temperature was continuously below 1.5 K, foreign atoms or molecules could migrate into the atomic contact when they are frequently opened and closed again. This could noticeably change the histogram, e. g. by an additional peak.

5.6. Hysteresis shapes Origin of the forms in Fig. 5.31 most of the bistable switching hystereses are shown again. The hystereses were normalized in height and width and sorted according to the standard deviation of the characteristic curves. Conspicuous among the many forms are many with tips at the zero crossing. This contradicts the assumption that the variation of the conductance can be explained alone with conductance fluctuations, i. e. quantum interference between different electron paths. These were e. g. observed in [Lud99, LDE+ 99] on gold mostly without kinking at the zero crossing (see, for example, Fig. 2.13, p. 23). Further effects (see also section 2.4, p. 22) such as vibration excitations also do not explain a kink at zero. Only electromigration forces are 103


Figure 5.31.: Hystereses of bistable contacts. For normalization, the IV characteristics of the upper and lower sides of each hysteresis were shifted vertically to each other so as to give a square. For further normalization, the squares in the x and y-directions were scaled uniformly. By doing so, the gradients keep comparable units to each other. The sorting follows in ascending order the standard deviation to the bottom and to the right.

possible, which displace individual particles continuously out of the rest position until the edge of the potential well is exceeded and the jump takes place. Depending on the situation, this displacement can only depend on the magnitude of the current and be independent of the direction, so that a pointed shape at the zero crossing can be explained. Although various conceivable zero-bias effects could also make a tip visible, they only point in one direction. The peaks occur in both directions, which speaks for the electromigration forces.

Similarities of the shape The standard deviation of the hysteresis in Fig. 5.31 is usually similar at the top and bottom. There are also some hystereses, which even resemble the shape of the upper and lower side. This correspondence despite different conductance shows that the cause of the form is not local to individual atoms but is determined by properties of the more distant environment. Since the channel spectrum was measured for each of the hystereses, these cases can be examined more closely. Figure 5.33 shows such a choice of similar conductance changes. Among the hystereses, the detected channels are plotted for the respective upper and lower levels. Channel spectra with similar channels have been sorted to the right. In the left two, an additional channel appears to be in the upper state, while the other channels still resemble each other. In the next few cases, one channel disappears and another is added. In other cases, no similarity can be seen. However, the similarity cases do not differ significantly from those of random similarity. Similarities between one hysteresis and another can be found just as often.

104


1.4 d2 I /dV 2 (G 0 /Ω)

Conductance G (G 0 )

5

1.3 1.2 1.1 −10

a)

0 Current I (µA)

2.5 0 −2.5 −5

10

b)

−100

0 V (mV)

100

Figure 5.32.: Hysteresis with very clear kink. a) Absolute conductance against the current. b) Double differential conductance versus voltage (numerical derivative).

Transmission

1 0.8 0.6 0.4 0.2 0

0.6 1.6

1.6 2.1

1.4 1.8

2.6 2.9

0.6 0.8

0.6 0.9

0.6 1.5

1.7 2.0

1.5 1.9

1.8 2.0

Figure 5.33.: Hystereses with strong similarities between the upper and lower characteristic curve with the associated channels. The hystereses normalized to unit size are at different absolute conductivities, as indicated by the numbers given below in G 0 .

Conclusion These results and further considerations of other hystereses show that the minimal local atomic rearrangements of bistable contacts strongly change the channel distribution. In many cases, however, the forms of hysteresis appear to be caused by more distant geometries.

5.6.1. Individual observations In the following some selected hystereses are considered, which have certain special features. Fig. 5.34 shows a hysteresis with a seemingly continuous transition. In the zoom b) one recognizes an increased noise at this point. This is interpreted as a temporally fluctuating two-level system. As the flow continuously changes, the likelihood of residence increasingly shifts from one position to another. Each measured value averaged over 500 ms therefore gives a seemingly intermediate conductance, which gradually changes from one to the other

105

2.5

Conductance (G 0 )

Conductance (G 0 )


2 1.5 −10

a)

0

10 20 Current (µA)

30

2 1.75 1.5

40

4

b)

5 Current (µA)

6

Figure 5.34.: Hysteresis with seemingly steep continuous transition showing only the mean of fast two-level temporal fluctuation.


Conductance (G 0 )

2

1.8

1.6 − 20

a)

− 10

0 10 20 Current (µA)

1.5 1 0.5 0

30

b)

0.4

0.6 0.8 Time (h)

1

Figure 5.35.: a) Unexplained individual case of a slow steady change of contact. The coloration from red to blue reflects the time course. b) Time-based switching behavior with the measured channel spectra.

conductance. Such observations have been made at least two times. An yet uncleared individual case shows Fig. 5.35 a). Here, a gradual continuous change of the hysteresis occurs, which manifests itself clearly in the upper form of the conductance variation. The coloring corresponds to the chronological order (from red to blue). Since the measurement takes place over a period of about 40 minutes, a relaxation of the metal or substrate is unlikely. The temperature on the 3 He thermometer drops from 0.3 K to 0.249 K. It is possible that this temperature profile is much stronger at the sample and also influences a temporal fluctuation between two positions, which is very fast here, and shows no amplified noise. Although the channel spectra in b) are not very reproducible for the small channels, there is also no tendency for a systematic change. Fig. 5.36 shows that the geometry changes are concentrated in a small space. The chronological order (from red to blue) shows that many geometric states are repeated over and over again. In particular, the second lowest level is taken far in the beginning (red) and far at the end

106


Conductance (G 0 )

1.75 1.5 1.25 1 −10

−5 0 5 Current (µA)

10

a)

0.8

Conductance (G 0 )

Conductance (G 0 )

Figure 5.36.: Example of the restricted degrees of freedom of the rearrangements. The chronological order is shown by the colors red to blue. At the beginning and at the very end, the contact seems to be in the same configuration.

0.7

0.6 −10

0 10 Current (µA)

20

b)

0.75 0.7 0.65 0.6 −5

0

5 10 Current (µA)

15

Figure 5.37.: Small rearrangements that are traversed several times in the same order.

(blue) of the measurement. This limitation to recurrent states can only mean that a spatially severely restricted area is responsible for the changes, in which only a few possible geometries can be taken. Smaller reproducible multiple rearrangements can also be seen in the two hystereses in Fig. 5.37. In both cases, the lower side of the hysteresis shows additional changes in conductance as the current increases, which would be difficult to explain by a rearrangement with a large number of atoms involved. In both examples, it is possible to switch between more than two atomic geometries in a reproducible manner. Such situations can be targeted by special experiments by reversing the current ramp, e. g. after the second change in conductance. In this case, the conductance change was too small to be detected as a jump by the experiment controller. In Fig. 5.38 the local small (actually unintentional) changes of the conductance variation cause changes which leave the rough course of the signal unaffected. This again shows that such forms are not determined by local geometry alone or dominated by chaotic behavior.

107


a)

2 1.8 1 1.6

1.4 −20

1 1.5 Time (h) 0

20 Current (µA)

Conductance (G 0 )

Conductance (G0)

2 0.9 0.8 0.7 0.6

40

b)

−20

0 Current (µA)

20

Figure 5.38.: Small random changes that lead to very similar conduction curves driven by the current.

The examples in Fig. 5.38 show that small local changes in geometry cause only minor changes in the shape of the conductance variations. Although [Lud99] has already shown how the fluctuations in the conductance of gold change continuously with the opening of the contact, a complete half of all atoms of the contact are in motion compared to the other half. Even if the mechanical changes in this technique can be chosen to be as small as desired, interference effects could possibly cause a significant change by pulling the contact apart (e. g. in a Fabry-Perot interferometer). Local changes due to bistable rearrangements are small in a sense that affects the spatial extent of the changed area.

5.7. Overview of all measurements In the following table 5.2 all important measurements are listed and assigned to the shown results.

108

5.7. O VERVIEW OF ALL MEASUREMENTS

time span

Sample Type Number of measurements and reference to the result

16.03.2003

cs_61

Al

12.04.2006–16.04.2006 22.04.2006–30.04.2006 24.06.2006–03.07.2006 19.03.2007–10.04.2007 15.03.2007–23.03.2007 05.12.2007–14.12.2007

cs_a35 cs_a38 cs_b2 cs_b14 cs_b14 cs_b33

Au Au Al Al Al Al

14.12.2007–18.12.2007

cs_b33 Al

01.02.2008–23.04.2008

cs_b37 Al

08.02.2008–03.06.2008

cs_b37 Al

04.06.2008–13.06.2008

cs_b37 Al

single experiments on current-driven rearrangements → Fig. 5.1 46 current-driven rearrangements → Fig. 5.2 123 current-driven rearrangements → Fig. 5.2 479 current-driven rearr. → Fig. 5.3a, 5.4,5.5, 5.7 906 current-driven rearr. → Fig. 5.3a, 5.4,5.5, 5.7 90 attempts for bistable switching → Fig. 5.11 67 current-driven rearrangements with one MAR measurement per reorder → so far unevaluated 70 attempts for bistable switching with MAR measurements→ Fig. 5.16 2449 attempts for bistable switching, 190 where bistable, with 1583 MAR measurements → all figures from 5.21 to 5.37 (except 5.28) 662 Contacts around 0.85G 0 with MAR measurements and current-driven rearrangement → Fig. 5.27, 5.5, 5.28 440 MAR measurements aat mechanical generated contacts → Fig. 5.27, 5.28, 5.29

Table 5.2.: Overview of all important measurements. In the nomenclature of the samples means a = Apical®-substrat (Polyimid), b = Bronce substrat. “Current-diven rearrangement” here designates experiments with an increase of the current in only one current direction until the interruption of the contact.

109

» I paint objects as I think them, not as I see them « — Pablo Picasso

A

A PPENDIX A.

Graphical Illustration

A.1. Simulation to illustrate a break junction The simulation, which is shown below, pursues the goal of producing a realistic-looking graphic illustration of the break junction as a spherical model with a raytracing program. It is not about a physically correct model, but only about an image for didactic purposes. The manual positioning of the balls in the computer, which initially seemed obvious, turned out to be much too time-consuming and from theoretical physics only images with very few atoms could be obtained. In contrast, these simulations generate images with over one million particles. After the incorporation of some heuristic terms and parameters, some effects could even be observed in the simulation that were not explicitly incorporated, but nevertheless appear plausible. First, a brief overview of how it works.

Algorithm The potential of the atoms becomes the L ENNARD -J ONES potential V =∼

³ σ ´12

r

−

³ σ ´6

r

(A.1)

used. However, the dynamics of the system is not described by an equation of motion of the second, but first order. That is, the particles do not experience acceleration through the potential, but instead directly receive the velocity from the gradient of potential. This is the derivative µ ³ ´ ¶ σ 13 ³ σ ´7 ∂V 0 V ≡ =∼ − 2 − (A.2) ∂r r r from the potential V and this value is evaluated for each neighbor atom per iteration step and added to the coordinate. The basic equation of the simulation is the iteration formula with the ¡ ¢ distances to the neighboring atoms di j = x j − xi and d i j = |di j | for the i -th atom xi = xi +

X di j j

di j

V 0 (d i j ).

(A.3)

111

A. G RAPHICAL I LLUSTRATION

Figure A.1.: Simulation of the opening process of a break junction with a wire comprising 100×100×200 atoms, with 1.2 million atoms being truly mobile.

112

A.1. S IMULATION TO ILLUSTRATE A BREAK JUNCTION

To increase the computational speed, the interaction is not calculated with each atom. Instead, every atom gets one limited number of nearest neighbors (here 14) with which an interaction takes place. As the atoms mix with each other over time, the 14 atoms with the smallest distance are stored as new neighbors at regular intervals from the set of neighbors as well as neighbors of the neighbors. In this way the calculation time only scales with ∼ O(N ), ie linear with the particle number N . Further acceleration was achieved by calculating V 0 (d i j )/d i j in an initialization step for 5000 values and storing them in an array. In order to be able to keep the number of iteration steps small, the equation A.3 is applied to the coordinate immediately, and not just at the end of the iteration step. As a result, the following particles can count on even more recent positions of the neighbors. In this way, within a step of the intervention, some momentum can be carried far through the system. The order of the particles in the array must be chosen to match the initial crystal order and every other iteration step must be done in reverse order. This strategy allows the spontaneous recrystallization at the beginning even with few iterations. Overall, this algorithm can be used to calculate simulations of over one million particles in one night on a common office computer. In addition to the basic equation, a few heuristic additional terms have been added, such as a slowing down of particles approximately at the bond distance to neighboring atoms or a parameter for an additional surface tension. The pulling motion is forced by some nonmoving particles on the left and right concave concave edges. In addition, the moving atoms may belong to either the left or the right side and accordingly move through an additional term at the same speed as the associated edge. The membership is a continuous parameter between −1 and 1, which may change with time, depending on the affiliation of the atoms in the neighborhood. In this way, a composite of atoms can adhere to one of the two broken halves and follow the constant motion, similar to a second-order equation of motion. The system does not contain temperature effects due to the first-order equation of motion.

Result Figure A.1 shows the result of a simulation of a 100 × 100 × 200 atom-containing wire with 1.2 million truly mobile atoms, shown with a ray tracing program 1 . Since it is a nonspecific spherical model that is not motivated by quantum mechanical processes, no specific material is described. Nevertheless, it is obviously subject to some basic material properties on a larger scale, which together apply to different systems of atoms, colloids, foam or similar adhesive objects. After an initial slight narrowing, some areas in the example begin to change their crystal orientation under pulling strain. At one point the crystal structure dissolves locally. The first cavities are formed there and soon a crevice is formed. The surface tension (e. g. following the example of gold) is set here by means of parameters and partly filamentous bridges are formed, which deform into peaks shortly before tearing off. In the example shown, the last contact between the two halves forms a dimer contact. Figure A.2 shows such a tip that has formed by itself. The surface tension, which helps in the formation, makes the two tips after tearing again slightly round. Tips like these can also be found in the physically correct atomic simulations of theoretical physics, as in [Dre08, 1 Here the freeware software Povray was used [pov].

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Figure A.2.: Atomic tip that has formed itself. The influence of the given surface tension contributes to a more pointed shape, but in turn causes a slight rounding after tearing.

Figure A.3.: Grain boundaries between different crystal regions, which were locally excited by the tensile stress to recrystallize.

DPH+ 05]. In Fig. A.3, the transitions between the grain boundaries are highlighted by lines. The crystals can take very different angles to each other. The originally chosen crystal direction, on which hexagon shapes are seen, appears to be the least stable under tension and is never found around the fracture. These effects are actually conceivable. Overall, despite its very simple assumptions, the simulation generates realistic structures. It therefore serves its purpose of providing illustrative pictorial material to stimulate imagination and fantasy, while the manifold processes in the “black box” of the experiment are only “visible” by means of one single scalar quantity, the conductance.

114

Bibliography [And64]

A NDREEV, A. F.: The Thermal Conductivity of the Intermediate State in Superconductors. In: Soviet physics JETP-USSR 19 (1964), Nr. 5, S. 1228–1231 17

[AYR03]

A GRAÏT, N. ; Y EYATI, Alfredo L. ; RUITENBEEK, Jan M.: Quantum properties of atomic-sized conductors. In: Physics Reports 377 (2003), April, Nr. 2-3, S. 81–279. http://dx.doi.org/10.1016/S0370-1573(02)00633-6. – DOI 10.1016/S0370–1573(02)00633–6. – ISSN 03701573 20

[Bü86]

B ÜTTIKER, M.: Four-Terminal Phase-Coherent Conductance. In: Phys. Rev. Lett. 57 (1986), Oct, Nr. 14, S. 1761–1764. http://dx.doi.org/10.1103/ PhysRevLett.57.1761. – DOI 10.1103/PhysRevLett.57.1761 12

[Bö08]

B ÖHLER, Tobias: Elektronischer Transport durch einzelne C60-Moleküle. Universitätsstr. 10, 78457 Konstanz, Universität Konstanz, Diss., 2008. http: //kops.ub.uni-konstanz.de/volltexte/2008/5829 24

[Bas]

B ASS, J.: 1.5 Bloch-Grüneisen resistivity function. http://dx.doi.org/10. 1007/10307022_30. In: Landolt-Börnstein - Group III Condensed Matter Bd. 15a. Hellwege, K.-H., Olsen, J. L. (ed.). SpringerMaterials - The Landolt-Börnstein Database. – DOI 10.1007/10307022_30 84

[Bat02]

B ATRA, Inder P.: From uncertainty to certainty in quantum conductance of nanowires. 124 (2002), Nr. 12, S. 463–467. http://dx.doi.org/10.1016/ S0038-1098(02)00550-1. – DOI 10.1016/S0038–1098(02)00550–1 13

[BES09]

B ÖHLER, T. ; E DTBAUER, A. ; S CHEER, E.: Point-contact spectroscopy on aluminium atomic-size contacts: longitudinal and transverse vibronic excitations. In: New Journal of Physics 11 (2009), Nr. 013036, S. 013036. http://dx.doi.org/10.1088/1367-2630/11/1/013036. – DOI 10.1088/1367– 2630/11/1/013036 24

[BGH+ 03]

B LADH, K. ; G UNNARSSON, D. ; H ÜRFELD, E. ; D EVI, S. ; K RISTOFFERSSON, C. ; LANDER, B. S. ; P EHRSON, S. ; C LAESON, T. ; D ELSING, P. ; TASLAKOV, M.: Comparison of cryogenic filters for use in single electronics experiments. In: Review of Scientific Instruments 74 (2003), Nr. 3, S. 1323–1327. http://dx.doi. org/10.1063/1.1540721. – DOI 10.1063/1.1540721 60

115

B IBLIOGRAPHY

[Blo29]

B LOCH, Felix: Über die Quantenmechanik der Elektronen in Kristallgittern. In: Zeitschrift für Physik A Hadrons and Nuclei 52 (1929), Nr. 7-8, S. 555–600. http://dx.doi.org/10.1007/BF01339455. – DOI 10.1007/BF01339455 11

[CMRY96]

C UEVAS ; M ARTÍN -R ODERO ; Y EYATI: Hamiltonian approach to the transport properties of superconducting quantum point contacts. In: Phys Rev B Condens Matter 54 (1996), Sep, Nr. 10, S. 7366–7379. http://dx.doi.org/10.1103/ PhysRevB.54.7366. – DOI 10.1103/PhysRevB.54.7366 19

[COL+ 03]

C HEN, Yong ; O HLBERG, Douglas A. A. ; L I, Xuema ; S TEWART, Duncan R. ; W ILLIAMS, R. S. ; J EPPESEN, Jan O. ; N IELSEN, Kent A. ; S TODDART, J. F. ; O LYNICK, Deirdre L. ; A NDERSON, Erik: Nanoscale molecular-switch devices fabricated by imprint lithography. In: Applied Physics Letters 82 (2003), Nr. 10, S. 1610–1612. http://dx.doi.org/10.1063/1.1559439. – DOI 10.1063/1.1559439 91

[Cue99]

C UEVAS, Juan C.: Electronic Transport in Normal and Superconducting Nanocontacts, Universidad Autónoma de Madrid, Diss., 1999 15, 16

[CYMR98]

C UEVAS, J. C. ; Y EYATI, A. L. ; M ARTÍN -R ODERO, A.: Microscopic Origin of Conducting Channels in Metallic Atomic-Size Contacts. In: Phys. Rev. Lett. 80 (1998), Feb, Nr. 5, S. 1066–1069. http://dx.doi.org/10.1103/PhysRevLett. 80.1066. – DOI 10.1103/PhysRevLett.80.1066 101

[Dat97]

D ATTA, Supriyo: Electronic Transport in Mesoscopic Systems. Cambridge : Cambridge University Press, 1997. – ISBN 9780521599436 14

[DG27]

D AVISSON, C. ; G ERMER, L. H.: The Scattering of Electrons by a Single Crystal of Nickel. In: Nature 119 (1927), S. 558–560. http://dx.doi.org/10.1038/ 119558a0. – DOI 10.1038/119558a0 11

[Die95]

D IERCKX, Paul: Curve and Surface Fitting with Splines. 9780198534402 42

[DPH+ 05]

D REHER, Markus ; PAULY, Fabian ; H EURICH, Jan ; C UEVAS, Juan C. ; S CHEER, Elke ; N IELABA, Peter: Structure and conductance histogram of atomicsized Au contacts. In: First publ. in: Physical Review B 72 (2005), 7, Article 075435 (2005). http://dx.doi.org/10.1103/PhysRevB.72.075435. – DOI 10.1103/PhysRevB.72.075435 21, 22, 114

[Dre08]

D REHER, Markus: Untersuchung elektronischer Eigenschaften komplexer Materialien mittels Computer-Simulationen. Universitätsstr. 10, 78457 Konstanz, Universität Konstanz, Diss., 2008. http://kops.ub.uni-konstanz.de/ volltexte/2008/6094 21, 22, 114

[Dru00]

D RUDE, Paul: Zur Elektronentheorie der Metalle. In: Annalen der Physik 306 (1900), Nr. 3, S. 566 – 613. http://dx.doi.org/10.1002/andp.19003060312. – DOI 10.1002/andp.19003060312 11

116

(1995).

ISBN

B IBLIOGRAPHY

[DSDV06]

D’A GOSTA, Roberto ; S AI, Na ; D I V ENTRA, Massimiliano: Local Electron Heating in Nanoscale Conductors. In: Nano Letters 6 (2006), dec, Nr. 12, S. 2935– 2938. http://dx.doi.org/10.1021/nl062316w. – DOI 10.1021/nl062316w. – ISSN 1530–6984 80, 85

[Eil03]

E ILERS, Paul H C.: A perfect smoother. In: Anal Chem 75 (2003), Jul, Nr. 14, S. 3631–3636. http://dx.doi.org/10.1021/ac034173t. – DOI 10.1021/ac034173t 42

[ES05]

E INSTEIN, A. ; S EELIG, C.: Mein Weltbild. Bd. 30. Ullstein, 2005 73

[Fal97]

FALCONER, Isobel: J J Thomson and the discovery of the electron. In: Physics Education 32 (1997), Nr. 4, S. 226–231. http://dx.doi.org/10.1088/0031-9120/ 32/4/015. – DOI 10.1088/0031–9120/32/4/015 11

[GET98]

G ÜLSEREN, Oguz ; E RCOLESSI, Furio ; T OSATTI, Erio: Noncrystalline Structures of Ultrathin Unsupported Nanowires. In: Phys. Rev. Lett. 80 (1998), Apr, Nr. 17, S. 3775–3778. http://dx.doi.org/10.1103/PhysRevLett.80.3775. – DOI 10.1103/PhysRevLett.80.3775 21

[GML09]

G OMEZ -M ARLASCA, F. ; L EVY, P.: Resistance switching in silver – manganite contacts. In: Journal of Physics: Conference Series 167 (2009), 012036 (6pp). http://dx.doi.org/10.1088/1742-6596/167/1/012036. – DOI 10.1088/1742– 6596/167/1/012036 91

[Gro69]

G ROSSMANN, Walter: Grundzüge der Ausgleichsrechnung nach der Methode der kleinsten Quadrate nebst Anwendung in der Geodäsie. 3., erw. A. SpringerVerlag GmbH, 1969. – ISBN 9783540044956 45

[HP97]

H ODRICK, R. J. ; P RESCOTT, E.C.: Postwar US Business Cycles: An Empirical Investigation. In: Journal of Money, Credit & Banking 29 (1997), Feb, Nr. 1, S. 1–16 42

[HWHS08]

H OFFMANN, R. ; W EISSENBERGER, D. ; H AWECKER, J. ; S TÖFFLER, D.: Conductance of gold nanojunctions thinned by electromigration. In: Applied Physics Letters 93 (2008), Nr. 4, S. 043118. http://dx.doi.org/10.1063/1.2965121. – DOI 10.1063/1.2965121 21

[JWB+ 07]

J OOSS, C. ; W U, L. ; B EETZ, T. ; K LIE, RF ; B ELEGGIA, M. ; S CHOFIELD, MA ; S CHRAMM, S. ; H OFFMANN, J. ; Z HU, Y.: Polaron melting and ordering as key mechanisms for colossal resistance effects in manganites. In: Proceedings of the National Academy of Sciences 104 (2007), Nr. 34, S. 13597. http://dx.doi. org/10.1073/pnas.0702748104. – DOI 10.1073/pnas.0702748104 91

[KBS+ 05]

KONRAD, Patrick ; B ACCA, Cecile ; S CHEER, Elke ; B RENNER, Patrice ; M AYERG INDNER, Andreas ; L ÖHNEYSEN, Hilbert: Stable single-atom contacts of

117

B IBLIOGRAPHY

zinc whiskers. In: First publ. in: Applied Physics Letters 86 (2005), 213115. http://dx.doi.org/10.1063/1.1926408. – DOI 10.1063/1.1926408 57 [Kit06]

K ITTEL, Charles: Einführung in die Festkörperphysik. 14. Auflage. Oldenbourg Wissenschaftsverlag, 2006. – ISBN 3–486–57723–9. – Daten aus den Tabellen 6.1 und 6.3. 60, 84

[Kon03]

KONRAD, Patrick: Herstellung und elektronischer Transport von Kontakten atomarer Größe aus Zink. Universitätsstr. 10, 78457 Konstanz, Universität Konstanz, Diplomarbeit, 2003 57

[KRF+ 95]

K RANS, J. M. ; RUITENBEEK, J. M. ; F ISUN, V. V. ; YANSON, I. K. ; J ONGH, L. J.: The signature of conductance quantization in metallic point contacts. In: Nature 375 (1995), S. 767 – 769. http://dx.doi.org/10.1038/375767a0. – DOI 10.1038/375767a0 9, 20, 21

[Lan57]

L ANDAUER, Rolf: Spatial variation of currents and fields due to localized scatterers in metallic conduction. In: IBM Journal of Research and Development 1 (1957), Nr. 3, 223-231. http://researchweb.watson.ibm.com/journal/rd/ 013/ibmrd0103D.pdf 9, 13

[LDE+ 99]

L UDOPH, B. ; D EVORET, M. H. ; E STEVE, D. ; U RBINA, C. ; RUITENBEEK, J. M.: Evidence for Saturation of Channel Transmission from Conductance Fluctuations in Atomic-Size Point Contacts. In: Phys. Rev. Lett. 82 (1999), Feb, Nr. 7, S. 1530–1533. http://dx.doi.org/10.1103/PhysRevLett.82.1530. – DOI 10.1103/PhysRevLett.82.1530 23, 103

[Lid04]

L IDE, David: Crc Handbook of Chemistry and Physics: a Ready Reference Book of Chemical and Physical Data: Special Student Ed. City : CRC Press Inc, 2004. – ISBN 0849305977 85, 92

[LKDK08]

L I, Fushan ; K IM, Tae W. ; D ONG, Wenguo ; K IM, Young-Ho: Formation and electrical bistability properties of ZnO nanoparticles embedded in polyimide nanocomposites sandwiched between two C60 layers. In: Applied Physics Letters 92 (2008), Nr. 1, S. 011906. http://dx.doi.org/10.1063/1.2830617. – DOI 10.1063/1.2830617 91

[Loa99]

L OADER, Clive: Local Regression and Likelihood. Berlin : Springer, 1999. – ISBN 9780387987750 41

[Lod05]

L ODDER, A.: Electromigration theory unified. In: Europhysics Letters 72 (2005), Nr. 5, S. 774–780. http://dx.doi.org/10.1209/epl/i2005-10306-9. – DOI 10.1209/epl/i2005–10306–9 24, 25

[LSC02]

L Ü, X. ; S HEN, WZ ; C HU, JH: Size effect on the thermal conductivity of nanowires. In: Journal of Applied Physics 91 (2002), S. 1542. http://dx.doi. org/10.1063/1.1427134. – DOI 10.1063/1.1427134 81, 83

118

B IBLIOGRAPHY

[Lud]

L UDOPH, Bas: Private communication with Elke Scheer 75

[Lud99]

L UDOPH, Bas: Quantum conductance properties of atomic-size contacts, Universiteit Leiden, Diss., 1999 9, 23, 103, 108

[Mar06]

M ARES, Ancuta I.: Search for shell effects in metallic nanowires, Universiteit Leiden, Diss., 2006 21

[MDC87]

M ARTINIS, John M. ; D EVORET, Michel H. ; C LARKE, John: Experimental tests for the quantum behavior of a macroscopic degree of freedom: The phase difference across a Josephson junction. In: Phys. Rev. B 35 (1987), Apr, Nr. 10, S. 4682–4698. http://dx.doi.org/10.1103/PhysRevB.35.4682. – DOI 10.1103/PhysRevB.35.4682 60

[Mei08]

M EIJER, G. I.: MATERIALS SCIENCE: Who Wins the Nonvolatile Memory Race? In: Science 319 (2008), Nr. 5870, 1625-1626. http://dx.doi.org/10. 1126/science.1153909. – DOI 10.1126/science.1153909 91

[MUB+ 07]

M ARES, A I. ; U RBAN, D F. ; B URKI, J ; G RABERT, H ; S TAFFORD, C A. ; RUITENBEEK, J M.: Electronic and atomic shell structure in aluminium nanowires. In: Nanotechnology 18 (2007), Nr. 26, S. 265403 (9pp). http://dx.doi.org/10. 1088/0957-4484/18/26/265403. – DOI 10.1088/0957–4484/18/26/265403 21

[Mus06]

M USTAFA, Jakob: Design and Analysis of Future Memories Based on Switchable Resistive Elements, Rheinisch-Westfälische Technische Hochschule Aachen, Diss., 2006. http://darwin.bth.rwth-aachen.de/opus3/volltexte/2006/ 1605/ 91

[NNWA01]

N AKAMURA, Jun ; N AKAYAMA, Tomonobu ; WATANABE, Satoshi ; A ONO, Masakazu: Structural and Cohesive Properties of a C60 Monolayer. In: Phys. Rev. Lett. 87 (2001), Jul, Nr. 4, S. 048301. http://dx.doi.org/10.1103/PhysRevLett. 87.048301. – DOI 10.1103/PhysRevLett.87.048301 91

[PH07]

P ERREAULT, Simon ; H ÉBERT, Patrick: Median Filtering in Constant Time. In: IEEE Transactions on Image Processing 16 (2007), Sept., Nr. 9, S. 2389–2394. http://dx.doi.org/10.1109/TIP.2007.902329. – DOI 10.1109/TIP.2007.902329 45

[pov]

Povray. Freeware, Vision Raytracer Pty. Ltd. Version: 3.7.0.beta33, Freeware 113

[Pre02]

P RESS, William: Numerical Recipes in C++. Cambridge : Cambridge University Press, 2002 http://www.nr.com/. – ISBN 0521750334 42

[PVH+ 08]

PAULY, F ; V ILJAS, J K. ; H UNIAR, U ; H AFNER, M ; W OHLTHAT, S ; B URKLE, M ; C UEVAS, J C. ; S CHON, G: Cluster-based density-functional approach to

http://www.povray.org/.

119

B IBLIOGRAPHY

quantum transport through molecular and atomic contacts. In: New Journal of Physics 10 (2008), Nr. 12, 125019 (28pp). http://stacks.iop.org/1367-2630/ 10/125019 15, 102 [Pyt67]

P YTTE, E.: Contribution of the electron-phonon interaction to the effective mass, superconducting transition temperature, and the resistivity in aluminium. In: Journal of Physics and Chemistry of Solids 28 (1967), Nr. 1, S. 93–103. – ISSN 0022–3697 84

[RAP+ 96]

RUITENBEEK, J. M. ; A LVAREZ, A. ; P INEYRO, I. ; G RAHMANN, C. ; J OYEZ, P. ; D EVORET, M. H. ; E STEVE, D. ; U RBINA, C.: Adjustable nanofabricated atomic size contacts. In: Review of Scientific Instruments 67 (1996), Nr. 1, S. 108–111. http://dx.doi.org/10.1063/1.1146558. – DOI 10.1063/1.1146558 9, 51

[Ros]

R OSSUM, Guido van: Version: 2.5 65

[SAC+ 98]

S CHEER, Elke ; A GRAIT, Nicolas ; C UEVAS, Juan C. ; Y EYATI, Alfredo L. ; L UDOPH, Bas ; M ARTIN -R ODERO, Alvaro ; B OLLINGER, Gabino R. ; RUITENBEEK, Jan M. ; U RBINA, Cristian: The signature of chemical valence in the electrical conduction through a single-atom contact. In: Nature 394 (1998), Nr. 6689, S. 154–157. http://dx.doi.org/10.1038/28112. – DOI 10.1038/28112. – ISSN 0028–0836 9, 16

[SBMH09]

S TOJANOVIC, N. ; B ERG, JM ; M AITHRIPALA, DHS ; H OLTZ, M.: Direct measurement of thermal conductivity of aluminum nanowires. In: Applied Physics Letters 95 (2009), Nr. 9, S. 091905. http://dx.doi.org/10.1063/1.3216035. – DOI 10.1063/1.3216035. – ISSN 0003–6951 83, 84

[Sch02]

S CHIRM, Christian: Leitwertfluktuationen von Ein-Atom-Kontakten. Universitätsstr. 10, 78457 Konstanz, Universität Konstanz, Diplomarbeit, 2002. http://kops.ub.uni-konstanz.de/volltexte/2003/940 67

[Sch08]

S CHECKER, Olivier: Von Punktkontakten zu Nano-Elektro-Mechanischen Systemen – NEMS, Universität Konstanz, Diss., 2008 102, 103

[SCHD06]

S TAHLMECKE, B. ; C HELARU, L. I. ; H ERINGDORF, F.-J. M. ; D UMPICH, G.: Electromigration in Gold and Single Crystalline Silver Nanowires. In: AIP Conference Proceedings 817 (2006), Nr. 1, 65-70. http://dx.doi.org/10.1063/1.2173533. – DOI 10.1063/1.2173533 25

[SD07a]

S TAHLMECKE, B. ; D UMPICH, G.: Influence of the electron beam on electromigration measurements within a scanning electron microscope. In: Applied Physics Letters 90 (2007), Nr. 4, 043517. http://dx.doi.org/10.1063/1. 2432304. – DOI 10.1063/1.2432304 25

120

Python.

Open Source.

http://www.python.org/.

B IBLIOGRAPHY

[SD07b]

S TAHLMECKE, B. ; D UMPICH, G.: Resistance behaviour and morphological changes during electromigration in gold wires. In: Journal of Physics: Condensed Matter 19 (2007), Nr. 4, 046210 (11pp). http://dx.doi.org/10.1088/ 0953-8984/19/4/046210. – DOI 10.1088/0953–8984/19/4/046210 25

[SG64]

S AVITZKY, A. ; G OLAY, M. J. E.: Smoothing and Differentiation of Data by Simplified Least Squares Procedures. In: Analytical Chemistry 36 (1964), Nr. 8, S. 1627–1639 42

[SHC+ 06]

S TAHLMECKE, B. ; H ERINGDORF, F.-J. M. ; C HELARU, L. I. ; H OEGEN, M. H. ; D UMPICH, G. ; R OOS, K. R.: Electromigration in self-organized singlecrystalline silver nanowires. In: Applied Physics Letters 88 (2006), Nr. 5, 053122. http://dx.doi.org/10.1063/1.2172012. – DOI 10.1063/1.2172012 25

[SJE+ 97]

S CHEER, E. ; J OYEZ, P. ; E STEVE, D. ; U RBINA, C. ; D EVORET, M. H.: Conduction Channel Transmissions of Atomic-Size Aluminum Contacts. In: Physical Review Letters 78 (1997), Nr. 18, S. 3535–3538. http://dx.doi.org/10.1103/ PhysRevLett.78.3535. – DOI 10.1103/PhysRevLett.78.3535 9, 16, 17, 51

[SK09]

S CHWARZ, Hans R. ; KÖCKLER, Norbert: Numerische Mathematik. Teuber, 2009 (7.,überarb. Aufl.). http://dx.doi.org/10.1007/978-3-8348-9282-9. http: //dx.doi.org/10.1007/978-3-8348-9282-9. – ISBN 978–3–8348–9282–9 45

[SKB+ 06]

S CHEER, Elke ; KONRAD, Patrick ; B ACCA, Cecile ; M AYER-G INDNER, Andreas ; L ÖHNEYSEN, Hilbert ; H ÄFNER, Michael ; C UEVAS, Juan C.: Correlation between transport properties and atomic configuration of atomic contacts of zinc by low-temperature measurements. In: First publ. in: Physical Review B 74 (2006), 205430 (2006). http://dx.doi.org/10.1103/PhysRevB.74.205430. – DOI 10.1103/PhysRevB.74.205430 77

[SNU+ 02]

S MIT, R. H. M. ; N OAT, Y. ; U NTIEDT, C. ; L ANG, N. D. ; H EMERT, M. C. ; RUITENBEEK, J. M.: Measurement of the conductance of a hydrogen molecule. In: Nature 419 (2002), oct, Nr. 6910, 906–909. http://dx.doi.org/10.1038/ nature01103. – DOI 10.1038/nature01103. – ISSN 0028–0836 24

[SPS09]

S CHIRM, C. ; P ERNAU, H.-F. ; S CHEER, E.: Switchable wiring for high-resolution electronic measurements at very low temperatures. In: Review of Scientific Instruments 80 (2009), Nr. 2, 024704. http://dx.doi.org/10.1063/1.3073962. – DOI 10.1063/1.3073962. – weitere Quelle: http://kops.ub.uni-konstanz. de/volltexte/2009/7521 55, 125

[SSBW06]

S ZOT, Krzysztof ; S PEIER, Wolfgang ; B IHLMAYER, Gustav ; WASER, Rainer: Switching the electrical resistance of individual dislocations in singlecrystalline SrTiO3 . In: Nat Mater 5 (2006), apr, Nr. 4, 312–320. http://dx. doi.org/10.1038/nmat1614. – DOI 10.1038/nmat1614. – ISSN 1476–1122 91

121

B IBLIOGRAPHY

[ST99]

S CHRIEFFER, J. R. ; T INKHAM, M.: Superconductivity. In: Rev. Mod. Phys. 71 (1999), Mar, Nr. 2, S. S313–S317. http://dx.doi.org/10.1103/RevModPhys.71. S313. – DOI 10.1103/RevModPhys.71.S313 17

[Sta08]

S TAHLMECKE, Burkhard: Elektromigration in Gold und Silber Nanostrukturen, Universität Duisburg-Essen, Diss., 2008. http://nbn-resolving.de/ urn/resolver.pl?urn=urn:nbn:de:hbz:464-20080130-095740-3 25

[THNA05]

T ERABE, K. ; H ASEGAWA, T. ; N AKAYAMA, T. ; A ONO, M.: Quantized conductance atomic switch. In: Nature 433 (2005), Jan, Nr. 7021, 47–50. http://dx.doi. org/10.1038/nature03190. – DOI 10.1038/nature03190 9, 90, 91

[Tro09]

T ROUWBORST, M. L.: Electron Transport Through Single Gold Atoms an Hydrogen Molecules, University of Groningen, Diss., 2009 81, 85

[TVDMVW06] T ROUWBORST, ML ; VAN D ER M OLEN, SJ ; VAN W EES, BJ: The role of Joule heating in the formation of nanogaps by electromigration. In: Journal of Applied Physics 99 (2006), S. 114316. http://dx.doi.org/10.1063/1.2203410. – DOI 10.1063/1.2203410. – weitere Quelle: http://irs.ub.rug.nl/dbi/ 44f55f0d1fc1c 81, 85 [VNC]

TightVNC. Open Source. http://www.tightvnc.com/. – VNC = Virtual Network Computing, Protocol for password-protected transmission of screen content and keyboard and mouse events over the Internet 65

[VRD09]

VAN R OSSUM, G. ; D RAKE, F.L.: Python language reference. (2009) 65

[Wer03]

W ERMAN, Michael: Fast Convolution. In: Journal of WSCG Bd. 11, 2003. – http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.64.6978 45

[WGW02]

W ITCZAK, Zbigniew ; G ONCHAROVA, Valeria A. ; W ITCZAK, Przemyslaw: Elastic properties of a polycrystalline sample of the L12 Al5CrTi2 intermetallic compound under hydrostatic pressure up to 1 GPa at room temperature. In: Journal of Alloys and Compounds 337 (2002), Nr. 1-2, 58 - 63. http://dx.doi. org/10.1016/S0925-8388(01)01957-0. – DOI 10.1016/S0925–8388(01)01957– 0. – ISSN 0925–8388. – Zitiert nach http://www.springermaterials.com/docs/ VSP/datasheet/lpf-p/01103000/LPFP_1103887.html 84

[WHB+ 88]

W EES, B. J. ; H OUTEN, H. van ; B EENAKKER, C. W. J. ; W ILLIAMSON, J. G. ; KOUWENHOVEN, L. P. ; M AREL, D. van d. ; F OXON, C. T.: Quantized conductance of point contacts in a two-dimensional electron gas. In: Phys. Rev. Lett. 60 (1988), Feb, Nr. 9, S. 848–850. http://dx.doi.org/10.1103/PhysRevLett.60. 848. – DOI 10.1103/PhysRevLett.60.848 15

[Whi23]

W HITTAKER, E.T.: On a new method of graduation. In: Proceedings of the Edinburgh Mathematical Society 41 (1923), S. 63–75 42

122

B IBLIOGRAPHY

[Whi53]

W HITE, G K.: The Thermal Conductivity of Gold at Low Temperatures. In: Proceedings of the Physical Society. Section A 66 (1953), Nr. 6, 559. http://dx. doi.org/10.1088/0370-1298/66/6/307. – DOI 10.1088/0370–1298/66/6/307 81

[Wie64]

W IENER, Norbert: Extrapolation, Interpolation, and Smoothing of Stationary Time Series. Cambridge : The MIT Press, 1964. – ISBN 0262730057 42

[Woo05a]

W OODCRAFT, Adam L.: Predicting the thermal conductivity of aluminium alloys in the cryogenic to room temperature range. In: Cryogenics 45 (2005), Nr. 6, 421 - 431. http://dx.doi.org/10.1016/j.cryogenics.2005.02.003. – DOI 10.1016/j.cryogenics.2005.02.003. – ISSN 0011–2275 81, 83

[Woo05b]

W OODCRAFT, A.L.: Recommended values for the thermal conductivity of aluminium of different purities in the cryogenic to room temperature range, and a comparison with copper. In: Cryogenics 45 (2005), Nr. 9, S. 626–636. http://dx.doi.org/10.1016/j.cryogenics.2005.06.008. – DOI 10.1016/j.cryogenics.2005.06.008 81, 82, 83

[XNOS04]

X IE, F-Q. ; N ITTLER, L. ; O BERMAIR, Ch ; S CHIMMEL, Th: Gate-controlled atomic quantum switch. In: Phys Rev Lett 93 (2004), Sep, Nr. 12, S. 128303. http://dx.doi.org/10.1103/PhysRevLett.93.128303. – DOI 10.1103/PhysRevLett.93.128303 9, 90

[Yan]

YANSON, A.: Private communication with Elke Scheer 76

[YPX+ 08]

YANG, J. J. ; P ICKETT, Matthew D. ; X UEMA, Li ; O HLBERG, Douglas A. A. ; S TEWART, Duncan R. ; W ILLIAMS, R. S.: Memristive switching mechanism for metal/oxide/metal nanodevices. In: Nat Nano 3 (2008), jul, Nr. 7, 429–433. http://dx.doi.org/10.1038/nnano.2008.160. – DOI 10.1038/nnano.2008.160. – ISSN 1748–3387 91

[YR97]

YANSON, A. I. ; RUITENBEEK, J. M.: Do Histograms Constitute a Proof for Conductance Quantization? In: Physical Review Letters 79 (1997), Nr. 11, 2157-2157. http://dx.doi.org/10.1103/PhysRevLett.79.2157. – DOI 10.1103/PhysRevLett.79.2157 9, 20

[YSC+ 05]

YANSON, I. K. ; S HKLYAREVSKII, O. I. ; C SONKA, Sz. ; K EMPEN, H. van ; S PELLER, S. ; YANSON, A. I. ; RUITENBEEK, J. M.: Atomic-Size Oscillations in Conductance Histograms for Gold Nanowires and the Influence of Work Hardening. In: Phys. Rev. Lett. 95 (2005), Dec, Nr. 25, S. 256806. http://dx.doi.org/10.1103/ PhysRevLett.95.256806. – DOI 10.1103/PhysRevLett.95.256806 21

[YSRS08]

YANSON, I. K. ; S HKLYAREVSKII, O. I. ; RUITENBEEK, J. M. ; S PELLER, S.: Aluminum nanowires: Influence of work hardening on conductance histograms.

123

B IBLIOGRAPHY

In: Physical Review B (Condensed Matter and Materials Physics) 77 (2008), Nr. 3, 033411. http://dx.doi.org/10.1103/PhysRevB.77.033411. – DOI 10.1103/PhysRevB.77.033411 21 [YYR99]

YANSON, A. I. ; YANSON, I. K. ; RUITENBEEK, J. M.: Observation of shell structure in sodium nanowires. In: Nature 400 (1999), jul, Nr. 6740, S. 144–146. http: //dx.doi.org/10.1038/22074. – DOI 10.1038/22074. – ISSN 0028–0836 21

[YZNT08]

YAO, Jun ; Z HONG, Lin ; N ATELSON, Douglas ; T OUR, James M.: Etchingdependent reproducible memory switching in vertical SiO[sub 2] structures. In: Applied Physics Letters 93 (2008), Nr. 25, 253101. http://dx.doi.org/10. 1063/1.3045951. – DOI 10.1063/1.3045951 91

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» You know that I write slowly, [...] because I never like myself other than if in a small space is as much as possible, and to write short takes much more time than long. « — Carl Friedrich Gauss2

Summary Techniques Within the scope of this PhD thesis, conductance measurements were carried out on mechanically controllable single atomic contacts at low temperatures and their behaviour under small local changes of the atomic configuration were studied. In order to facilitate local manipulation, individual electrical migration events were stimulated with high currents and the corresponding jumps in the conductance were observed. For the contacts generated and modified in this way, the quantum-mechanical conductance channels could be determined for analysis with the aid of superconducting current-voltage characteristics, the so-called multiple Andreev reflections (MAR). In preparation, experimental techniques had to be developed in order to meet the high requirements of low noise for the MAR measurement and at the same time to obtain sufficient low-resistance lines for electrical migration excitation. The special wiring in the cryostat was published in Review of Scientific Instruments [SPS09]. Further experimental details that are important for reproducing the measurements have been described. A numerical method important for the analysis of MAR measurements and the determination of channel spectra (a special fit algorithm) had to be further refined, as the noise in the system (possibly also the finite temperature of 250 mK) has led to slightly smoothed MAR curves. Since there were many measurements available, the efficiency of the numerically complex optimization problem was also significantly improved, which reduced several weeks of computational time on an office computer to a few hours. In addition, a versatile smoothing algorithm based on the local regression has been developed for modeling noise and for other treatments of measurement data.

Measuring Results Individual examples have shown that high currents can be used to trigger atomic rearrangements near a single atomic contact in a way that leads to more stable configurations. Local melting is not considered the cause of atomic changes, as an estimation of the temperature shows. With a statistical analysis of many such measurements on gold and aluminium break junction contacts, preferred conductance values could be observed and manipulated by current-driven rearrangements. If local changes are investigated isolated from more global changes, preferred values are also shown, but with shifted conductance values. An interplay of local and nonlocal properties seems to be responsible for the form of the total conductance 2 1833, in a correspondence with H. C. Schumacher

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histograms. The histogram of the conductance jumps indicates that at higher voltages, the changes occur at a greater distance from the atomic contact. Analysis of the stability shows that one atomic contacts can withstand the highest voltage. This applies to gold with a conductance of 1G 0 , but also to aluminium, where the one atom contact is at a slightly lower conductance. By means of a special protocol of current ramps, reversible bistable contact geometries could be generated, which are interpreted as single atom switches. The switching hystereses were randomly distributed values and differed in shape and size. An analysis of many hystereses showed a reduced occurrence of conductance values that correspond to a single atom contact and were judged to be particularly stable in other experiments. The explanation is that a too stable configuration will not find a second partner state that meets the requirement to be more stable in one direction of current than in the other. Analysis of the channel spectra of bistable hysteresis shows a similar effect that channels with high transmission values occur less frequently than in purely mechanically generated contacts. In the case of high transmissions, the low backscattering probability means that the electromigration forces can exert very little pressure on the atom to cause it to move. This means that the prerequisite for bistable switching is less effectively fulfilled. A channel histogram of mechanically generated aluminum contacts around the single atomic contact shows a good correlation with a theoretical calculation of a pyramid-shaped single atomic contact in (111) crystal direction. For contacts with higher conductance values, the channel histogram increasingly shows an equal distribution. Full channels with transmission 1 only appear after a total conductance of about 1.7G 0 . Individual aluminium atoms, on the other hand, have several channels that are clearly below 1G 0 , with one of these channels usually dominating the others. Two observations can be made on the basis of the current-voltage characteristics of the bistable hystereses. On the one hand, small atomic changes of the contact can significantly change the channel distribution, while the shape of the characteristic curves is usually quite similar, sometimes even identical. This shows that switching is a very local phenomenon with few atoms involved. On the other hand, a symmetrical shape of the characteristic curves, which tends to become increasingly peaked at zero, indicates that a constant geometric distortion may occur as the voltage increases, which is caused by electromigration forces and has a similar effect in both directions of the current.

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Acknowledgement On this page I want to thank everyone who supports this work directly or indirectly: My doctoral supervisor Elke Scheer for her many years of support, the pioneering colleagues at the experiment, especially Hansi Pernau, all colleagues who have always provided very reliable support for the functioning of certain facilities (e. g. Vojko Kunej for the REM and Hansi Pernau, Michael Wolz, Stephan Egle for the "Dampfe", our metal evaporating facility), Ansgar Fischer for his precise metal works, Sabine Lukas for her friendly helpfulness, the staff of scientific workshops, my father and my brother for proofreading the manuscript and the external second referee Carlos Cuevas of the University of Madrid. Indirect assistance I received from Mr. Froböse, who introduced me to the general-purpose programming language Python. At this point one should also mention some invisible helpers from the internet, such as the open source developers of Python (Guido van Rossum et al.), the numerics packages Numpy / Scipy (T. Oliphant et al.), the typesetting programs LATEX (Knuth/Lamport et al.) and KOMA-Script (Markus Kohm et al.), the plot program Pyx (A. Wobst, P. Lehmann, M. Schindler), and the many nameless, who hide behind all the opensource projects.

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