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Table of contents
1 General introduction
1.1 Tunneling spectroscopy
1.2 Spatially resolved measurement of the LDOS with a combined AFM-STM.
1.3 An AFM-STM in a table-top dilution refrigerator
1.4 Benchmarking tunneling spectroscopy
1.5 An experiment on the proximity effect in S-N-S structures
1.6 Perspectives
1.6.1 Proximity effect in ballistic 1D systems
1.6.2 Proximity effect in a 2D system
1.6.3 Spin injection and relaxation in superconductors
1.6.4 Energy relaxation in quasi-ballistic (“Superdiffusive”) structures
2 Design, Fabrication and Operation of the Microscope
2.1 The microscope structure
2.1.1 General description
2.1.2 Materials and construction
2.1.2.1 Material selection
2.1.2.2 Mechanical structure
2.1.2.3 CAD Design
2.1.2.4 Assembly
2.1.2.5 Plating of titanium
2.2 Piezoelectric actuators
2.2.1 The piezoelectric effect
2.2.1.1 General mathematical description of piezoelectricity
2.2.1.2 Practical equations for piezoelectric devices
2.2.2 Practical piezoelectric materials
2.2.2.2 Quartz crystal
2.2.2.3 PZT piezoelectric material
2.2.2.3.a Description
2.2.2.3.b PZT ceramics: poling and depoling
2.2.2.3.c PZT actuators shapes and functioning
2.2.2.3.d Imperfections of PZT ceramics: drift and hysteresis
2.2.2.3.e PZT ceramics at low temperature
2.2.3 The piezoelectric tube
2.2.3.1 Poling procedure
2.2.3.2 Calibration at high and low temperatures
2.3 Coarse positioning and indexing
2.3.1 Design & principle
2.3.1.1 Stick-slip motion
2.3.1.2 Electrical requirements
2.3.1.3 Mechanical performance of the motors
2.3.1.4 Inertial vs. non-inertial stick-slip motion
2.3.2 Position indexing
2.3.2.1 Design
2.3.2.2 Resolution and accuracy of the capacitive position sensors
2.3.2.3 Practical implementation
2.3.2.4 Testing actuators and sensors.
2.4 The atomic force sensor
2.4.1 implementing AFM sensors at low temperature
2.4.1.1 Optical sensors
2.4.1.2 Electrical sensors
2.4.2 Quartz Tuning Fork (TF)
2.4.2.1 A high Q harmonic oscillator
2.4.2.2 Tuning forks for atomic force microscopes
2.4.2.3 Oscillating modes of the tuning fork
2.4.2.4 Basic analysis of a tuning fork: 1 Degree-of-Freedom harmonic oscillator model
2.4.2.4.b Mechanical analysis
2.4.2.4.c Tip-sample interaction
2.4.2.5 Discussion of the 1-DOF model
2.4.2.6 Model of a tuning fork as three coupled masses and springs
2.4.2.6.b Free dynamics of the 3-DOF model
2.4.2.6.c Balanced tuning fork: the unperturbed modes
2.4.2.6.d Broken symmetry: a perturbative analysis
2.4.2.6.e Discussion
2.4.2.7 Estimating characteristics of the connecting wire for preserving the tuning fork balance .
2.4.2.8 Measuring the oscillator parameters
2.4.2.8.b Resonance curves during the setup
2.4.2.8.c Capacitance compensation
2.4.2.8.d Extracting the resonator parameters
2.4.3 Force measurement
2.4.3.1 Typical forces on the atomic scale
2.4.3.2 “Force” measurements: sensor stability and sensitivity
2.4.3.3 “Force” measurements: detection schemes
2.4.3.4 Phased locked loop detection
2.4.3.5 Feedback loop in imaging mode
2.4.3.5.b Optimizing AFM imaging
3 Experimental techniques
3.1 A dilution refrigerator well adapted for a local probe microscope.
3.1.1 Functioning of the inverted dilution refrigerator
3.1.2 (not so good) Vibrations in dilution refrigerator
3.1.3 Living with vibrations
3.1.4 Possible improvements
3.2 Wiring a microscope for very low temperature experiments.
3.2.1 Thermal load
3.2.2 Low level signals and low noise requirements
3.2.3 Filtering
3.2.3.1 Necessity of filtering
3.2.3.2 Why filtering all the lines?
3.2.3.3 Propagation of thermal noise in networks
3.2.3.4 Filtering technique
3.2.3.5 Fabrication
3.2.3.6 Electromagnetic simulations of the filters.
3.2.3.7 Filter attenuation
3.2.3.8 Analysing the full setup
3.2.3.9 Experimental validation of the filtering setup
3.2.3.10 Article reprint: microfabricated filters
3.2.4 Tunneling spectroscopy
3.2.4.1 Tunneling spectroscopy measurements
3.2.4.2 Data acquisition setup
3.2.4.3 Article reprint : Tunnel current pre-amplifier
3.3 Fabrication techniques
3.3.1 Tip fabrication
3.3.1.1 Electrochemical etching
3.3.1.2 Tungsten tips
3.3.1.3 Niobium tips
3.3.1.4 In-situ tip cleaning
3.3.1.5 Tip damages and tip reshaping
3.3.2 Sample preparation
3.3.2.1 Position encoding grid
3.3.2.2 Multiple angle evaporation
4 Mesoscopic superconductivity
4.1 Introduction to proximity effect
4.2 Theoretical description of Proximity Effect
4.2.1 Inhomogeneous superconductivity
4.2.2 The Bogolubov – de Gennes equations
4.2.3 Theoretical description of the proximity effect in diffusive systems at equilibrium.
4.2.3.1 Electronic Green functions
4.2.3.2 Green functions in the Nambu space
4.2.3.3 Quasiclassical Green functions in the dirty limit – Usadel equation
4.2.3.4 Properties of the Green function elements – Physical quantities
4.2.3.5 Boundary conditions
4.2.4 Parameterization of the Green functions
4.2.4.1 θ, φ parameterization
4.2.4.1.b Nazarov’s Andreev circuit theory
4.2.4.2 Ricatti parameterization
4.2.4.2.a Advantages of this parameterization
4.2.4.2.b Ricatti parameters in reservoirs
4.2.4.2.c (Dis-)Continuity of Ricatti parameters at interfaces
4.2.5 Spectral quantities in the Ricatti parameterization
4.2.6 A word on the non-equilibrium theory
4.3 Solving the Usadel equations numerically
4.3.1 Reduction of the problem dimensionality
4.3.2 Specifying the 1-D problem
4.3.3 Solving strategy – self-consistency
4.3.4 Implementation details
4.4 General results on proximity effect in SNS structures
4.4.1 “Minigap” in the DOS
4.4.2 Inverse proximity effect – Pairing amplitude and pairing potential .
4.4.2.2 Self-consistency
4.4.3 Role of interfaces
4.4.4 Dependence of the minigap on N size
4.4.5 Phase modulation of the proximity effect
4.5 SNS systems viewed as scattering structures
4.6 An STM experiment on the proximity effect
4.6.1 Sample geometry
4.6.2 Implementation of the experiment
4.6.2.1 Achieving a good phase bias
4.6.2.1.a Kinetic inductance correction
4.6.2.1.b Circulating currents correction
4.6.2.2 Sample fabrication
4.6.2.3 Overview of the measured SNS structures
4.6.3 What do we measure?
4.6.4 Density of states in a long SNS structure
4.6.5 Density of states in a short SNS structure
4.6.5.1 position dependence
4.6.5.2 phase dependence
4.6.5.2.b Phase calibration
4.6.5.2.c minigap as a function of phase
4.6.5.3 Is the Al electrode affected by the applied field?
4.6.6 Density of states in a medium-size SNS structure
4.6.6.1 Intermediate regime
4.6.7 Comparison with theory
4.6.7.1 Parameters entering the theory
4.6.7.2 Comparison with theoretical predictions
4.6.7.2.a Determination of the parameters
4.6.7.2.b Position dependence
4.6.7.2.c Phase dependence
4.6.7.2.d Length dependence
4.6.7.3 Summary of the fitting parameters
4.6.7.4 Experimental limitations
4.6.7.4.a Positioning accuracy
4.6.7.4.b Energy resolution
4.6.7.5 Model limitations
4.6.7.5.a Depairing
4.6.7.5.b Self consistent treatment of the proximity effect
4.6.7.5.c 1-D model of the NS interface
4.6.7.5.d Transverse extension, sample geometry
4.6.8 Article reprint: proximity effect in S-N-S structures
4.7 Conclusions
4.7.1 Further experiments
4.7.1.1 Out of equilibrium proximity effect
4.7.1.2 Gapless superconductivity
