10.3 Tunneling Spectroscopy and Point-Contact Andreev Reflection

Tunneling spectroscopy and point-contact Andreev reflection are powerful tools for studying superconductors. These techniques probe the electronic structure and properties of materials by measuring current flow across tiny junctions or contacts.

By analyzing the resulting data, scientists can uncover key information about superconducting energy gaps, density of states, and pairing mechanisms. This helps reveal the fundamental physics behind superconductivity in different materials.

Tunneling Spectroscopy for Superconductors

Principles and Applications

• Tunneling spectroscopy utilizes the quantum tunneling effect to probe the electronic structure and properties of superconductors by placing a thin insulating barrier between a normal metal and a superconductor, forming a metal-insulator-superconductor (MIS) junction
• Applying a bias voltage across the MIS junction and measuring the resulting tunneling current allows for the direct probing of the density of states (DOS) of the superconductor, as the tunneling current is proportional to the convolution of the DOS of the normal metal and the superconductor
• Tunneling spectroscopy provides direct access to the superconducting energy gap, a fundamental parameter characterizing the superconducting state, and is sensitive to the presence of quasiparticle states within the gap, enabling the study of phenomena such as impurity-induced states (Shiba states), vortex cores, and unconventional superconductivity
• The technique has been extensively used to investigate pairing symmetry (s-wave, d-wave), gap structure (isotropic, anisotropic), and electronic inhomogeneity in a wide range of superconducting materials, including conventional superconductors (Nb, Pb) and unconventional superconductors (cuprates, heavy fermions)

Experimental Setup and Measurement

• The experimental setup for tunneling spectroscopy typically consists of a low-temperature scanning tunneling microscope (STM) or a mechanically controllable break junction (MCBJ) to form the MIS junction
• The sample is cooled to cryogenic temperatures (below the superconducting transition temperature) to minimize thermal broadening effects and enhance the energy resolution
• A bias voltage is applied across the MIS junction, and the resulting tunneling current is measured as a function of the bias voltage, yielding the tunneling current-voltage (I-V) characteristics
• The differential conductance (dI/dV) is obtained by numerically differentiating the I-V characteristics or by using a lock-in amplifier technique, which directly measures the differential conductance by superimposing a small AC modulation on the DC bias voltage

Interpreting Tunneling Spectroscopy Data

Superconducting Energy Gap and Density of States

• The tunneling conductance (dI/dV) provides a direct measure of the superconducting DOS, with pronounced peaks at the superconducting energy gap edges known as coherence peaks, whose position directly corresponds to the superconducting energy gap magnitude
• The shape and width of the coherence peaks provide information about the sharpness of the superconducting DOS and the presence of broadening effects, such as thermal smearing or lifetime effects due to inelastic scattering processes (electron-phonon interactions)
• In the subgap region between the coherence peaks, the tunneling conductance is suppressed due to the absence of quasiparticle states within the superconducting energy gap, while additional features or structures can indicate the existence of impurity-induced states (Yu-Shiba-Rusinov states), multiple gaps, or unconventional pairing symmetries

Theoretical Modeling and Data Analysis

• Fitting the experimental tunneling conductance data with theoretical models, such as the Bardeen-Cooper-Schrieffer (BCS) theory or its extensions (Dynes model, Maki model), allows for the extraction of quantitative information about the superconducting gap, DOS, and other relevant parameters
• The BCS theory predicts a characteristic DOS with a square-root singularity at the gap edges for a conventional s-wave superconductor, while modifications to the BCS theory (Eliashberg theory) incorporate strong-coupling effects and energy-dependent gap functions
• Deviations from the BCS behavior, such as the presence of in-gap states or asymmetric coherence peaks, can provide insights into the underlying pairing mechanism, impurity effects, or the presence of unconventional superconductivity (d-wave, p-wave)
• Advanced data analysis techniques, such as the maximum entropy method or the Kramers-Kronig transformation, can be employed to extract the underlying DOS from the tunneling conductance data, accounting for the effects of the normal metal DOS and the tunneling matrix elements

Andreev Reflection in Superconductivity

Concept and Mechanism

• Andreev reflection is a unique scattering process that occurs at the interface between a normal metal and a superconductor, where an incident electron from the normal metal with energy below the superconducting gap is reflected as a hole, while a Cooper pair is simultaneously created in the superconductor
• The reflected hole has opposite spin and momentum compared to the incident electron, leading to a charge transfer of 2e across the interface, which is a consequence of the coherent nature of the superconducting state and the presence of the superconducting energy gap
• The probability of Andreev reflection depends on the transparency of the interface, determined by factors such as the Fermi velocity mismatch and the interface barrier strength, and the energy of the incident electron relative to the superconducting gap

Significance and Applications

• Andreev reflection plays a crucial role in the proximity effect, where superconducting correlations are induced in a normal metal layer in contact with a superconductor, leading to phenomena such as the superconducting proximity effect and the Josephson effect in superconductor-normal metal-superconductor (SNS) junctions
• The study of Andreev reflection provides valuable insights into the pairing mechanism, coherence length, and gap structure of superconductors, as well as the spin-polarization of ferromagnetic materials in superconductor-ferromagnet (SF) junctions
• Andreev reflection forms the basis for various experimental techniques, such as point-contact spectroscopy, where the differential conductance across a nanoscale contact between a normal metal and a superconductor is measured to probe the superconducting order parameter, and the study of Andreev bound states in superconducting junctions, which are sensitive to the phase difference across the junction and the pairing symmetry of the superconductor

Point-Contact Andreev Reflection Analysis

Experimental Technique and Data Acquisition

• Point-contact Andreev reflection (PCAR) spectroscopy utilizes Andreev reflection to probe the superconducting order parameter at the nanoscale by forming a nanoscale point contact between a sharp metallic tip and a superconducting sample
• The differential conductance (dI/dV) measured across the point contact provides information about the energy-dependent Andreev reflection probability and the superconducting order parameter, with the shape and features of the PCAR conductance spectrum depending on the transparency of the point contact, the superconducting gap magnitude, and the symmetry of the order parameter
• Experimental factors such as the contact size, stability, and cleanliness play a crucial role in obtaining reliable and reproducible PCAR data, with techniques such as the needle-anvil method or the mechanically controllable break junction (MCBJ) method being commonly employed to form stable point contacts

Data Analysis and Interpretation

• For a conventional s-wave superconductor, the PCAR spectrum exhibits a double-peak structure at the gap edges, similar to the tunneling conductance in tunneling spectroscopy, while the presence of a zero-bias conductance peak (ZBCP) can indicate the existence of unconventional pairing symmetries (p-wave, d-wave) or the presence of surface Andreev bound states
• The width and height of the ZBCP provide information about the strength and nature of the unconventional pairing, with a sharp and tall ZBCP being indicative of a fully gapped unconventional superconductor, while a broad and shallow ZBCP may suggest the presence of nodes in the gap structure
• Fitting the PCAR data with theoretical models, such as the Blonder-Tinkham-Klapwijk (BTK) theory or its extensions, allows for the extraction of quantitative information about the superconducting gap, barrier strength (Z parameter), and spin-polarization, with the BTK theory being particularly useful for analyzing the conductance spectra of conventional superconductors and the extended BTK models incorporating the effects of spin-polarization, Fermi surface mismatch, and unconventional pairing symmetries
• PCAR spectroscopy has been widely used to study the order parameter symmetry, gap structure, and spin-polarization in various superconducting materials, including unconventional superconductors (heavy fermions, organic superconductors), topological superconductors (Majorana bound states), and spin-triplet superconductors (Sr2RuO4)