9.3 BCS theory

BCS theory explains superconductivity at the microscopic level. It describes how electrons form Cooper pairs through phonon-mediated interactions, leading to a coherent quantum state with zero electrical resistance and perfect diamagnetism.

The theory predicts key features of superconductors, including the energy gap, critical temperature, and coherence length. It provides a framework for understanding experimental observations like the isotope effect and tunneling measurements, while also having limitations for unconventional superconductors.

Origins of BCS theory

• Developed in 1957 by John Bardeen, Leon Cooper, and John Robert Schrieffer to explain the microscopic mechanism of superconductivity
• Built upon the earlier phenomenological theories, such as the London equations and the Ginzburg-Landau theory, which described superconductivity without providing a microscopic understanding
• Provides a comprehensive framework for understanding the key features of superconductors, including the absence of electrical resistance, the Meissner effect, and the existence of an energy gap in the electronic excitation spectrum

Phonon-mediated electron interactions

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• Proposes that the attractive interaction between electrons, which leads to the formation of Cooper pairs, is mediated by the exchange of virtual phonons (quantized lattice vibrations)
• Electrons interact with the lattice, causing a local positive charge concentration, which in turn attracts another electron
• This indirect electron-electron interaction overcomes the Coulomb repulsion between electrons, resulting in a net attractive force

Cooper pairs

• Two electrons with opposite spins and momenta form a bound state called a Cooper pair due to the attractive interaction mediated by phonons
• Cooper pairs have a lower energy than individual electrons and are responsible for the superconducting properties
• The formation of Cooper pairs is a many-body effect, involving a large number of electrons simultaneously

BCS ground state

• The superconducting state is described as a coherent superposition of Cooper pairs, all occupying the same quantum state
• This coherent state is separated from the excited states by an energy gap, which is a characteristic feature of superconductors
• The BCS ground state is a macroscopic quantum state, exhibiting long-range order and phase coherence

Coherent state of Cooper pairs

• In the BCS ground state, all Cooper pairs are in the same quantum state, described by a single macroscopic wavefunction
• The coherence of the Cooper pairs leads to the superconducting properties, such as zero electrical resistance and the Meissner effect
• The macroscopic wavefunction has a well-defined phase, which is responsible for the phase coherence in superconductors

Energy gap

• The formation of Cooper pairs leads to the opening of an energy gap in the electronic excitation spectrum
• The energy gap separates the superconducting ground state from the excited states, which consist of broken Cooper pairs (quasiparticles)
• The magnitude of the energy gap is related to the binding energy of the Cooper pairs and is typically on the order of 1 meV (much smaller than the Fermi energy)

Electron-phonon coupling

• The strength of the electron-phonon interaction determines the magnitude of the attractive potential between electrons and, consequently, the properties of the superconducting state
• The electron-phonon coupling is characterized by a dimensionless parameter, usually denoted as $\lambda$, which depends on the material properties and the phonon spectrum

Attractive interaction potential

• The electron-phonon interaction leads to an attractive potential between electrons, which is responsible for the formation of Cooper pairs
• The attractive potential is often approximated by a simple model, such as the square-well potential or the delta-function potential
• The range of the attractive potential is determined by the phonon wavelength, which is typically much larger than the interatomic spacing

Coupling strength

• The strength of the electron-phonon coupling, $\lambda$, determines the magnitude of the attractive potential and the superconducting properties
• Stronger coupling (larger $\lambda$) leads to a higher critical temperature, a larger energy gap, and a shorter coherence length
• The coupling strength can be estimated from experimental data, such as the isotope effect or tunneling measurements

Critical temperature

• The critical temperature, $T_c$, is the temperature below which a material becomes superconducting
• BCS theory provides a microscopic expression for the critical temperature in terms of the electron-phonon coupling strength and the phonon spectrum

Calculation of Tc

• In the weak-coupling limit, the BCS expression for the critical temperature is: $T_c \approx 1.13 \hbar \omega_D \exp(-1/\lambda)$

  where $\hbar \omega_D$ is the Debye energy (related to the phonon spectrum) and $\lambda$ is the electron-phonon coupling strength

• The exponential dependence on $\lambda$ implies that small changes in the coupling strength can lead to significant changes in the critical temperature

Factors affecting Tc

• The critical temperature depends on several factors, including the electron-phonon coupling strength, the phonon spectrum, and the electronic density of states
• Materials with higher phonon frequencies (e.g., lighter atoms) and stronger electron-phonon coupling tend to have higher critical temperatures
• Other factors, such as pressure, impurities, and dimensionality, can also influence the critical temperature

Coherence length

• The coherence length, $\xi$, is a characteristic length scale in superconductors that describes the spatial extent of the Cooper pairs
• It represents the distance over which the macroscopic wavefunction of the superconducting state varies significantly

Spatial extent of Cooper pairs

• The coherence length determines the size of the Cooper pairs and the length scale over which the superconducting properties are maintained
• In conventional superconductors, the coherence length is typically much larger than the interatomic spacing (hundreds to thousands of angstroms)
• The large coherence length is a consequence of the weak binding energy of the Cooper pairs compared to the Fermi energy

Temperature dependence

• The coherence length depends on temperature and diverges as the temperature approaches the critical temperature

• Near $T_c$, the temperature dependence of the coherence length is given by: $\xi(T) \propto (1 - T/T_c)^{-1/2}$

• The divergence of the coherence length near $T_c$ is related to the disappearance of the superconducting state and the onset of phase fluctuations

Density of states

• The density of states (DOS) describes the number of electronic states available per unit energy interval
• In superconductors, the formation of the energy gap leads to a modification of the DOS compared to the normal state

Electron energy distribution

• In the normal state, the DOS is typically a smooth function of energy, with a finite value at the Fermi level
• In the superconducting state, the DOS is modified by the presence of the energy gap, which opens up around the Fermi level
• The DOS in the superconducting state is zero within the energy gap and exhibits sharp peaks at the gap edges

Divergence at gap edges

• The DOS in the superconducting state exhibits a divergence at the gap edges, known as the coherence peaks
• The divergence is a consequence of the singularity in the quasiparticle excitation spectrum at the gap edges
• The presence of the coherence peaks in the DOS is a characteristic feature of superconductors and can be observed in tunneling experiments

Thermodynamic properties

• BCS theory provides a framework for calculating various thermodynamic properties of superconductors, such as the specific heat and thermal conductivity
• The thermodynamic properties are determined by the electronic excitations (quasiparticles) and the presence of the energy gap

Specific heat

• The specific heat of a superconductor exhibits a characteristic jump at the critical temperature, known as the specific heat jump
• Below $T_c$, the specific heat decreases exponentially with temperature, reflecting the presence of the energy gap and the reduced number of available electronic excitations
• The magnitude of the specific heat jump and the low-temperature behavior provide information about the strength of the electron-phonon coupling and the size of the energy gap

Thermal conductivity

• The thermal conductivity of a superconductor is strongly suppressed compared to the normal state due to the absence of electronic excitations within the energy gap
• At low temperatures, the thermal conductivity is dominated by phonons, as the electronic contribution is exponentially suppressed
• The temperature dependence of the thermal conductivity provides information about the scattering processes and the mean free path of the phonons in the superconducting state

Magnetic properties

• Superconductors exhibit unique magnetic properties, such as perfect diamagnetism (the Meissner effect) and the ability to sustain persistent currents
• BCS theory provides a microscopic understanding of these properties in terms of the coherent state of Cooper pairs and the energy gap

Meissner effect

• The Meissner effect is the complete expulsion of magnetic fields from the interior of a superconductor, leading to perfect diamagnetism
• It occurs because the superconducting state minimizes its free energy by screening out the external magnetic field
• The Meissner effect is a consequence of the coherence of the Cooper pairs and the existence of a well-defined macroscopic wavefunction

Type I vs type II superconductors

• Superconductors can be classified into two types based on their response to an external magnetic field
• Type I superconductors exhibit a complete Meissner effect up to a critical field, above which superconductivity is destroyed abruptly
• Type II superconductors allow partial penetration of the magnetic field in the form of quantized flux tubes (vortices) above a lower critical field, and superconductivity persists up to a higher upper critical field
• The distinction between type I and type II superconductors is determined by the ratio of the coherence length to the magnetic penetration depth (the Ginzburg-Landau parameter)

Experimental evidence

• BCS theory has been extensively tested and confirmed through various experimental observations
• Key experimental evidence supporting BCS theory includes the isotope effect, tunneling measurements, and the observation of the energy gap

Isotope effect

• The isotope effect refers to the dependence of the critical temperature on the mass of the lattice ions

• BCS theory predicts that $T_c$ should be inversely proportional to the square root of the isotope mass, $M$: $T_c \propto M^{-\alpha}$, with $\alpha \approx 0.5$

• The experimental observation of the isotope effect with the predicted exponent provided strong support for the phonon-mediated pairing mechanism

Tunneling measurements

• Tunneling experiments, such as scanning tunneling microscopy (STM) and planar junction tunneling, provide a direct probe of the electronic density of states in superconductors
• The presence of the energy gap and the coherence peaks in the tunneling spectra is a key prediction of BCS theory
• Tunneling measurements have been used to determine the size of the energy gap, the strength of the electron-phonon coupling, and the symmetry of the superconducting order parameter

Extensions and limitations

• While BCS theory successfully describes the properties of conventional superconductors, it has some limitations and has been extended to account for various experimental observations

Strong coupling corrections

• BCS theory is based on a weak-coupling approximation, which assumes that the electron-phonon coupling is much smaller than the Fermi energy
• In some materials, the electron-phonon coupling is strong, leading to deviations from the predictions of the weak-coupling BCS theory
• Strong coupling corrections, such as the Eliashberg theory, have been developed to describe the properties of strongly coupled superconductors, including higher critical temperatures and larger energy gaps

Unconventional superconductors

• Some superconductors, such as high-temperature cuprates and heavy fermion compounds, exhibit properties that cannot be explained by the conventional BCS theory
• These unconventional superconductors often have a complex electronic structure, competing interactions, and a pairing mechanism that may not be solely phonon-mediated
• Extensions of BCS theory, such as the resonating valence bond (RVB) theory and the spin fluctuation mechanism, have been proposed to describe the properties of unconventional superconductors
• The study of unconventional superconductors is an active area of research, aiming to uncover new pairing mechanisms and develop a unified theory of superconductivity