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Piezoelectric actuators
Piezoelectric materials are the standard actuators used in local probe microscopy. They allow controlling electrically very small displacements, down to subatomic levels. In the following, I first review the basics of these actuators. Then, I detail how we use them in our microscope:
· for accurately scanning the tip over the sample.
· for performing coarse positioning over the sample.
· in the AFM sensor.
The piezoelectric effect
Piezoelectricity was first predicted and explained by the Curie brothers in 1880. It is the property of certain crystals to generate charge when stressed. This effect is due to the intrinsic polarization of the crystalline unit cell (e.g. pyroelectric materials), and more generally to a non-centro-symmetric charge distribution.
When a crystalline cell with non uniform charge distribution is mechanically deformed, the geometric centers of positive and negative charges move by a different amount resulting in electric polarization. Conversely, the application of an electrical field results in different motions of the charges geometric centers in the cell, that finally induce macroscopic deformation of the material, either shear or longitudinal. The converse piezoelectric effect was mathematically deduced by Lippmann in 1881 and was immediately confirmed by the Curie brothers, proving the complete reversibility of electroelasto- mechanical strains.
General mathematical description of piezoelectricity
In a microscopic (local) view, strain (S) develops within any material, due to an applied stress (T), and the link between those quantities is the mechanical compliance1 (s). Mechanical compliance is a local tensor (6×6), when describing continuous media with a large number of degrees of freedom (DOF)2. Identically, from the electrical point of view, the electric displacement (D) is related to the electric field (E) by the permittivity (e), where D an E are 3-dimensional vectors, and e is a rank 3 diagonal matrix.
Practical equations for piezoelectric devices
Piezoelectric devices are objects made of piezoelectric materials. Their shape is cut relative to the polarization axis of the material and they are covered with metallic electrodes to apply an electric field to the material or to collect the piezoelectric charges, whether it is used in reverse or direct piezoelectric mode.
The tensorial relation (2.1) is a local microscopic description of the piezoelectric effect. By integration over the spatial coordinates, taking into account the shape of the electrodes, one establishes a link between the charge q on the electrodes (from ∫∫D), the voltage U applied (∫E), the displacement z of the object (∫S), and the force F applied (∫∫T), of the same form as (2.1), but with macroscopic parameters.
For example, in the case of a one dimensional actuator (i.e. designed to move an extremity in a single given direction), this yields a linear relationship between electrical and mechanical quantities: z k s F q s c U .
PZT ceramics: poling and depoling
For most applications3 the material consists of a ceramic made of small sintered grains. As fabricated, all the grains are randomly oriented, so that the ceramic element has no global dipolar moment, and no net piezoelectric effect.
In order to make them usable, a net macroscopic piezoelectric effect is obtained in these ceramics by poling (i.e. polarizing) them. The poling process consists in using the ferroelectric property of the material: under a sufficiently large electric field (typically in the MV/m range at room temperature), the microscopic individual crystalline cell dipoles can be permanently rotated. This process is often assisted thermally as the ferroelectric coercive field vanishes at the Curie temperature, which typically ranges from 150 to 350°C depending on the type of PZT. After this process, the grains that were earlier randomly oriented are globally aligned along the direction of the poling field, and a permanent polarization remains.
However, care must be exerted with these ceramics, as they may be “depoled” by applying a too large reverse electric field or by raising temperature above the Curie temperature.
Imperfections of PZT ceramics: drift and hysteresis
Since the macroscopic piezoelectric effect of PZT ceramic devices results from the addition of microscopic contributions, it is subject to the motion of screening charges in the material and to thermodynamic fluctuations. As a result, the material has large drift and hysteresis (typically a sizeable fraction of the total range), as well as noise and energy losses. In practice, no bi-univocal relationship exists between the position of an actuator and the voltage applied to it: an actual position depends on the history before reaching this point, and, for a fixed applied voltage, the position slowly relaxes (drifts) towards an equilibrium position. These imperfections are particularly pronounced when performing finite frequency motion of the actuator: the amplitude rapidly decreases with frequency and one must take care to avoid having too much internal dissipation which could lead to damaging the device (overheating the glue in stacks, for instance), or partly depoling it.
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
