🔬Condensed Matter Physics Unit 6 – Superconductivity

Introduction to Superconductivity

• Superconductivity phenomenon where certain materials exhibit zero electrical resistance and expel magnetic fields below a characteristic critical temperature ($T_c$)
• Discovered by Dutch physicist Heike Kamerlingh Onnes in 1911 while studying the resistance of solid mercury at cryogenic temperatures
• Requires extremely low temperatures, typically below 30 K ($-243.2°C$ or $-405.7°F$)
• Characterized by the Meissner effect, where a superconductor expels magnetic fields from its interior, demonstrating perfect diamagnetism
• Enables the flow of electric current without dissipation, leading to potential applications in power transmission, energy storage, and high-performance electronics
• Quantum mechanical phenomenon that involves the formation of Cooper pairs, where electrons with opposite spins and momenta become bound together
• Requires materials to be cooled below their critical temperature ($T_c$), critical magnetic field ($H_c$), and critical current density ($J_c$) to achieve and maintain the superconducting state

Basic Principles and Properties

• Zero electrical resistance allows superconductors to conduct electricity without energy loss, enabling persistent currents
• Meissner effect perfect diamagnetism where a superconductor expels magnetic fields from its interior
  • Occurs when the material transitions into the superconducting state
  • Differs from perfect conductivity, as it actively excludes magnetic fields
• Critical temperature ($T_c$) the temperature below which a material becomes superconducting
  • Varies among different superconducting materials (e.g., lead: 7.2 K, niobium: 9.3 K)
• Critical magnetic field ($H_c$) the maximum external magnetic field a superconductor can withstand before losing its superconducting properties
  • Type I superconductors have a single critical field, while Type II superconductors have lower and upper critical fields
• Critical current density ($J_c$) the maximum current density a superconductor can carry without losing its superconducting state
• Coherence length ($\xi$) the characteristic length scale over which the superconducting order parameter varies
  • Determines the size of Cooper pairs and the thickness of superconductor-normal interfaces
• London penetration depth ($\lambda$) the distance over which an external magnetic field penetrates into a superconductor before being expelled
  • Relates to the Meissner effect and the screening of magnetic fields

Types of Superconductors

• Type I superconductors exhibit a complete Meissner effect and have a single critical magnetic field ($H_c$)
  • Examples include pure metals like mercury, lead, and aluminum
  • Characterized by a sharp transition from the superconducting to the normal state when the critical field is exceeded
• Type II superconductors have two critical magnetic fields: lower critical field ($H_{c1}$) and upper critical field ($H_{c2}$)
  • Examples include alloys and compounds like niobium-titanium (NbTi) and yttrium barium copper oxide (YBCO)
  • Exhibit a mixed state between $H_{c1}$ and $H_{c2}$, where magnetic flux partially penetrates the material in the form of quantized vortices
  • Can withstand much higher magnetic fields than Type I superconductors
• Conventional superconductors explained by the Bardeen-Cooper-Schrieffer (BCS) theory, which describes the formation of Cooper pairs through electron-phonon interactions
  • Examples include most elemental superconductors and some alloys
• Unconventional superconductors have properties that cannot be fully explained by the BCS theory
  • Examples include high-temperature superconductors (cuprates) and iron-based superconductors
  • May involve different pairing mechanisms, such as electron-electron interactions or spin fluctuations
• High-temperature superconductors materials with critical temperatures above 30 K, often containing layered copper oxide (cuprate) compounds
  • Examples: yttrium barium copper oxide (YBCO) with $T_c \approx 93$ K and bismuth strontium calcium copper oxide (BSCCO) with $T_c \approx 110$ K
  • Potential for applications at liquid nitrogen temperatures (77 K)

Theories of Superconductivity

• Bardeen-Cooper-Schrieffer (BCS) theory the first microscopic theory of superconductivity, proposed in 1957
  • Explains the formation of Cooper pairs through electron-phonon interactions
  • Describes the superconducting state as a coherent macroscopic quantum state with a single wavefunction
  • Predicts the existence of an energy gap ($\Delta$) in the excitation spectrum of superconductors
• Ginzburg-Landau (GL) theory a phenomenological theory that describes superconductivity using an order parameter ($\psi$)
  • Introduces the concept of coherence length ($\xi$) and penetration depth ($\lambda$)
  • Allows the classification of superconductors into Type I and Type II based on the ratio $\kappa = \lambda / \xi$
  • Provides a framework for understanding the behavior of superconductors in magnetic fields
• Eliashberg theory an extension of the BCS theory that includes strong-coupling effects and retardation
  • Accounts for the energy dependence of the electron-phonon interaction
  • Enables the calculation of superconducting properties from first principles using the electron-phonon spectral function ($\alpha^2F(\omega)$)
• Unconventional pairing mechanisms theories that go beyond the electron-phonon interaction to explain superconductivity in unconventional superconductors
  • Examples include spin fluctuation-mediated pairing, resonating valence bond (RVB) theory, and the t-J model
  • Aim to understand the high-temperature superconductivity in cuprates and other unconventional superconductors
• Josephson effect the phenomenon of supercurrent tunneling between two superconductors separated by a thin insulating barrier (Josephson junction)
  • Described by the Josephson equations, which relate the supercurrent and the phase difference across the junction
  • Forms the basis for superconducting quantum interference devices (SQUIDs) and other superconducting electronic devices

Experimental Techniques

• Resistivity measurements used to determine the critical temperature ($T_c$) and study the transition from the normal to the superconducting state
  • Four-point probe method eliminates contact resistance and provides accurate resistivity values
  • Temperature-dependent resistivity reveals the onset of superconductivity and the width of the transition
• Magnetic susceptibility measurements probe the Meissner effect and the expulsion of magnetic fields from superconductors
  • SQUID magnetometers offer high sensitivity and can detect small changes in magnetic flux
  • Allows the determination of the critical fields ($H_c$, $H_{c1}$, $H_{c2}$) and the superconducting volume fraction
• Specific heat measurements provide information about the electronic and phononic contributions to the heat capacity
  • Superconducting transition appears as a jump in the specific heat at $T_c$, reflecting the opening of the energy gap
  • Helps to distinguish between conventional and unconventional superconductors based on the shape of the specific heat curve
• Tunneling spectroscopy probes the density of states (DOS) and the energy gap ($\Delta$) in superconductors
  • Scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) offer high spatial resolution
  • Reveals the symmetry of the superconducting order parameter and the presence of quasiparticle excitations
• Neutron scattering investigates the crystal structure, magnetic order, and lattice dynamics of superconductors
  • Elastic neutron scattering determines the atomic positions and the presence of structural phase transitions
  • Inelastic neutron scattering measures the phonon dispersion and the electron-phonon coupling strength
• Muon spin rotation/relaxation ($\mu$SR) probes the local magnetic fields and the superconducting vortex lattice
  • Positive muons are implanted into the superconductor and their spin precession is measured
  • Provides information about the penetration depth ($\lambda$), the coherence length ($\xi$), and the vortex dynamics

Applications in Technology

• Superconducting magnets generate strong, stable magnetic fields for various applications
  • Magnetic resonance imaging (MRI) in medical diagnostics
  • Particle accelerators (e.g., Large Hadron Collider) for high-energy physics research
  • Magnetic levitation (Maglev) trains for high-speed transportation
• Superconducting quantum interference devices (SQUIDs) highly sensitive magnetometers based on Josephson junctions
  • Used in biomagnetism, geophysical exploration, and quantum computing
  • Enable the detection of extremely weak magnetic fields, such as those produced by the human brain or heart
• Superconducting power transmission efficient long-distance electrical power transmission without resistive losses
  • Potential to reduce energy waste and improve grid stability
  • Requires the development of high-temperature superconducting cables and fault current limiters
• Superconducting energy storage systems store electrical energy in the magnetic field of a superconducting coil
  • Provide a means for load leveling and power quality improvement in the electrical grid
  • Offer high energy density and fast response times compared to other storage technologies
• Superconducting electronics high-speed, low-power electronic devices based on Josephson junctions and superconducting circuits
  • Rapid single flux quantum (RSFQ) logic for high-performance computing
  • Superconducting quantum bits (qubits) for quantum computing and quantum information processing
  • Superconducting nanowire single-photon detectors (SNSPDs) for quantum optics and quantum communication

Current Research and Future Prospects

• Room-temperature superconductivity the ultimate goal of superconductivity research, aiming to discover materials that remain superconducting at ambient conditions
  • Requires a deeper understanding of the mechanisms behind high-temperature superconductivity
  • Potential candidates include hydrogen-rich compounds under high pressure and novel unconventional superconductors
• Topological superconductors a new class of superconductors with non-trivial topological properties
  • Predicted to host Majorana fermions, which are their own antiparticles and have potential applications in topological quantum computing
  • Examples include certain semiconductor-superconductor heterostructures and iron-based superconductors
• Superconducting quantum computing harnessing the quantum properties of superconducting circuits for quantum information processing
  • Superconducting qubits (e.g., transmon, flux qubit) are among the leading platforms for scalable quantum computers
  • Challenges include improving qubit coherence times, reducing noise, and implementing error correction schemes
• Superconducting metamaterials artificial materials engineered to have unique electromagnetic properties by incorporating superconducting elements
  • Enable the realization of novel phenomena, such as negative refractive index and cloaking
  • Potential applications in superconducting sensors, antennas, and quantum metamaterials
• Superconductivity in two-dimensional materials exploring the emergence of superconductivity in atomically thin layers and heterostructures
  • Examples include twisted bilayer graphene, transition metal dichalcogenides (TMDs), and oxide interfaces
  • Offers a platform for studying the interplay between superconductivity, magnetism, and topology in low-dimensional systems
• Superconducting hydrides a new family of high-temperature superconductors based on hydrogen-rich compounds under high pressure
  • Examples include lanthanum hydride (LaH$_{10}$) with $T_c \approx 250$ K and yttrium hydride (YH$_6$) with $T_c \approx 220$ K
  • Potential for room-temperature superconductivity at extreme pressures, driving the search for stable ambient-pressure analogs

Key Equations and Formulas

• London equations describe the electrodynamics of superconductors and the Meissner effect
  • $\mathbf{E} = \frac{\partial \mathbf{J}_s}{\partial t}$ relates the electric field ($\mathbf{E}$) to the supercurrent density ($\mathbf{J}_s$)
  • $\mathbf{B} = -\mu_0 \lambda^2 \nabla \times \mathbf{J}_s$ relates the magnetic field ($\mathbf{B}$) to the supercurrent density ($\mathbf{J}_s$) and the penetration depth ($\lambda$)
• Ginzburg-Landau equations describe the spatial variation of the superconducting order parameter ($\psi$) and the supercurrent density ($\mathbf{J}_s$)
  • $\alpha \psi + \beta |\psi|^2 \psi + \frac{1}{2m^*} \left(-i\hbar \nabla - \frac{e^*}{c} \mathbf{A} \right)^2 \psi = 0$ is the first GL equation, governing the order parameter ($\psi$)
  • $\mathbf{J}_s = \frac{e^*\hbar}{2m^*i} (\psi^* \nabla \psi - \psi \nabla \psi^*) - \frac{{e^*}^2}{m^*c} |\psi|^2 \mathbf{A}$ is the second GL equation, expressing the supercurrent density ($\mathbf{J}_s$)
• BCS gap equation determines the superconducting energy gap ($\Delta$) in terms of the electron-phonon coupling strength ($V$) and the density of states ($N(0)$)
  • $\Delta = \frac{\hbar \omega_D}{\sinh(1/N(0)V)}$ where $\omega_D$ is the Debye frequency
• Josephson equations describe the supercurrent ($I_s$) and voltage ($V$) across a Josephson junction in terms of the phase difference ($\delta$)
  • $I_s = I_c \sin \delta$ is the first Josephson equation, relating the supercurrent ($I_s$) to the critical current ($I_c$) and the phase difference ($\delta$)
  • $\frac{d\delta}{dt} = \frac{2eV}{\hbar}$ is the second Josephson equation, relating the time derivative of the phase difference ($\delta$) to the voltage ($V$) across the junction
• Eliashberg equations describe the superconducting properties in the strong-coupling limit, taking into account the energy dependence of the electron-phonon interaction
  • $Z(i\omega_n) = 1 + \frac{\pi T}{\omega_n} \sum_{m} \lambda(i\omega_n - i\omega_m) \frac{\omega_m}{\sqrt{\omega_m^2 + \Delta^2(i\omega_m)}}$ is the mass renormalization function
  • $\Delta(i\omega_n) = \pi T \sum_{m} \left[ \lambda(i\omega_n - i\omega_m) - \mu^*(\omega_c) \right] \frac{\Delta(i\omega_m)}{\sqrt{\omega_m^2 + \Delta^2(i\omega_m)}}$ is the gap function