Superconductivity
In 1911, Heike Kamerlingh Onnes cooled mercury to 4.2 K and watched its electrical resistance vanish. Not decrease, not become very small, but drop to exactly zero. This discovery of superconductivity launched one of the most remarkable chapters in physics: a macroscopic quantum phenomenon in which billions of electrons act as a single coherent entity, producing effects that seem to defy classical intuition. Understanding superconductivity required decades of theoretical effort and ultimately revealed a profound connection between quantum mechanics, symmetry breaking, and the nature of the vacuum itself.
This lesson builds on quantum mechanics (lessons 6-11), statistical physics (lessons 21-25), and band theory (lesson 68). The BCS theory of superconductivity is one of the great triumphs of many-body quantum physics, and the concepts it introduced, including spontaneous symmetry breaking of a gauge symmetry, were later adapted by particle physicists to construct the Higgs mechanism.
What Makes Superconductivity Remarkable?
Superconductivity is not merely "very good" conductivity. It is a qualitatively different state of matter: a macroscopic quantum state in which electrons form bound pairs (Cooper pairs) that condense into a single quantum state described by a single wave function with a well-defined phase. This coherence extends over macroscopic distances, producing zero resistance and perfect diamagnetism, two properties that have no classical explanation.
Discovery and Phenomenology
Zero Resistance
In a normal metal, resistance arises because electrons scatter off lattice vibrations (phonons) and impurities. As temperature decreases, the resistance of a metal decreases but remains finite, limited by impurity scattering. A superconductor behaves completely differently: below a critical temperature $T_c$, the resistance drops abruptly to zero.
This is not a measurement limitation. Persistent currents have been observed to flow in superconducting rings for years with no measurable decay. The upper bound on the resistivity of a superconductor is $\rho < 10^{-25}$ $\Omega$m, at least 17 orders of magnitude smaller than the best normal conductor (copper at low temperature).
The Meissner Effect
In 1933, Meissner and Ochsenfeld discovered something even more striking than zero resistance: a superconductor actively expels magnetic fields from its interior. When a material is cooled below $T_c$ in the presence of a magnetic field, the field is ejected. This is the Meissner effect.
Beyond Perfect Conductivity
The Meissner effect proves that a superconductor is not merely a perfect conductor ($\rho = 0$). A perfect conductor would trap whatever magnetic flux was present when it became conducting; it would resist changes in flux. A superconductor goes further: it actively expels flux that was already there. This means the superconducting state is a true thermodynamic equilibrium state, not just a metastable state with zero resistance.
Mathematically, the Meissner effect means $\mathbf{B} = 0$ inside a superconductor. The magnetic field penetrates only a thin surface layer of thickness $\lambda_L$ (the London penetration depth), typically 20-200 nm.
The London Equations
Fritz and Heinz London proposed the first phenomenological description of superconductivity in 1935. Their equations modify classical electrodynamics to capture zero resistance and the Meissner effect.
The London Equations
First London equation (zero resistance): The supercurrent $\mathbf{J}_s$ accelerates without friction in an electric field:
$$\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m}\mathbf{E}$$This replaces Ohm's law $\mathbf{J} = \sigma\mathbf{E}$: instead of a resistive current proportional to $E$, the current accelerates under $E$ (like free particles with no scattering).
Second London equation (Meissner effect): Taking the curl and using Faraday's law:
$$\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m}\mathbf{B}$$Magnetic field screening: Combining with Ampere's law $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s$ gives:
$$\nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2}\mathbf{B}, \quad \lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}$$Exponential decay: At a flat surface, the solution is $B(x) = B_0 e^{-x/\lambda_L}$. The field decays exponentially into the superconductor with characteristic length $\lambda_L$, the London penetration depth.
Cooper Pairs and BCS Theory
The Cooper Instability
The microscopic theory of superconductivity was developed by Bardeen, Cooper, and Schrieffer (BCS) in 1957, earning them the Nobel Prize in 1972. The key insight came from Leon Cooper, who showed that the Fermi sea of electrons is unstable against the formation of bound pairs when there is any attractive interaction, no matter how weak.
Where Does the Attraction Come From?
Electrons repel each other through the Coulomb interaction. But in a crystal, there is a second, indirect interaction mediated by phonons (lattice vibrations). An electron moving through the lattice attracts nearby positive ions, creating a local region of positive charge density. A second electron is attracted to this region. The net effect is an attractive interaction between the two electrons, but it is retarded: the lattice distortion persists long enough for the first electron to move away before the second arrives. The time scale for the lattice to respond ($\sim 10^{-13}$ s) is much slower than the electron transit time ($\sim 10^{-16}$ s).
Cooper showed that two electrons with opposite momenta ($\mathbf{k}$ and $-\mathbf{k}$) and opposite spins ($\uparrow$ and $\downarrow$) near the Fermi surface always form a bound state, regardless of how weak the attraction is. This is the Cooper pair.
The BCS Ground State
The BCS ground state is a coherent superposition of Cooper pair states. It can be written as:
where $|u_{\mathbf{k}}|^2 + |v_{\mathbf{k}}|^2 = 1$, and $|v_{\mathbf{k}}|^2$ is the probability that the pair state $(\mathbf{k}\uparrow, -\mathbf{k}\downarrow)$ is occupied. This is not a simple product of independent pair states: it is a highly entangled state in which the number of pairs is not fixed. The BCS state spontaneously breaks the $U(1)$ gauge symmetry associated with charge conservation (more precisely, it breaks the symmetry of particle number conservation).
The Energy Gap
BCS theory predicts an energy gap $\Delta$ in the excitation spectrum: breaking a Cooper pair costs at least $2\Delta$ of energy. At zero temperature:
where $\omega_D$ is the Debye frequency, $N(0)$ is the density of states at the Fermi level, and $V$ is the pairing interaction strength. The gap decreases with increasing temperature and vanishes at $T_c$:
BCS theory also gives the famous relation between the gap and the critical temperature:
This prediction is well confirmed experimentally for conventional superconductors.
Connection: Symmetry Breaking and the Higgs Mechanism
The BCS ground state breaks a $U(1)$ gauge symmetry: the phase of the superconducting order parameter $\Delta = |\Delta|e^{i\phi}$ is spontaneously chosen. This is precisely the mechanism that Nambu, Anderson, and Higgs later adapted to particle physics. In the Standard Model (lesson 37), the Higgs field plays the role of the superconducting order parameter, and the mass of the W and Z bosons is analogous to the photon acquiring an effective mass inside a superconductor (which is what produces the Meissner effect and the London penetration depth).
Type I and Type II Superconductors
The response of a superconductor to an applied magnetic field defines two fundamentally different types.
Two Types of Superconductors
Type I: Superconductivity is destroyed abruptly at a single critical field $H_c$. Below $H_c$, the Meissner effect is complete ($B = 0$ inside). Above $H_c$, the material reverts entirely to the normal state. Most elemental superconductors (mercury, lead, tin) are Type I. Their $T_c$ values are low (below 10 K).
Type II: There are two critical fields. Below $H_{c1}$, the Meissner effect is complete. Between $H_{c1}$ and $H_{c2}$, magnetic flux penetrates in quantized vortices (the "mixed state" or "vortex state"). Above $H_{c2}$, superconductivity is destroyed. Type II superconductors include most alloys, all high-$T_c$ materials, and all practical superconductors used in technology. $H_{c2}$ can be extremely large (100 T or more).
The distinction is determined by the Ginzburg-Landau parameter $\kappa = \lambda_L / \xi$, where $\xi$ is the coherence length (the size of a Cooper pair). Type I: $\kappa < 1/\sqrt{2}$; Type II: $\kappa > 1/\sqrt{2}$.
Flux Quantization and the Josephson Effect
Flux Quantization
The magnetic flux through a superconducting ring is quantized in units of the flux quantum:
The factor of $2e$ (not $e$) in the denominator was one of the earliest experimental confirmations that the superconducting carriers have charge $2e$, i.e., they are Cooper pairs. Each vortex in a Type II superconductor carries exactly one flux quantum.
The Josephson Effect
When two superconductors are separated by a thin insulating barrier (a Josephson junction), Cooper pairs can tunnel through the barrier. Brian Josephson predicted this in 1962 (Nobel Prize 1973).
Josephson Effects
DC Josephson effect: A supercurrent flows through the junction even with zero voltage, proportional to the sine of the phase difference:
$$I = I_c \sin(\phi_1 - \phi_2)$$AC Josephson effect: When a constant voltage $V$ is applied across the junction, the phase difference evolves linearly in time, producing an oscillating current at frequency:
$$f = \frac{2eV}{h}$$This frequency-voltage relation is so precise that it is used to define the volt in terms of fundamental constants.
High-Temperature Superconductors
In 1986, Bednorz and Mueller discovered superconductivity at 35 K in a copper-oxide (cuprate) ceramic, shattering the prevailing belief that BCS theory set an upper limit of about 30 K for $T_c$. Within a year, $T_c$ had been pushed above 90 K in YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO), above the boiling point of liquid nitrogen (77 K), transforming the practical prospects for superconducting technology.
The highest confirmed $T_c$ at ambient pressure is about 135 K in mercury-based cuprates. Under extreme pressures, hydrogen-rich compounds (such as LaH$_{10}$) have shown superconductivity near room temperature (around 250 K at 170 GPa), though these require pressures that are impractical for applications.
The Great Unsolved Problem
Despite nearly four decades of intense research, there is no accepted microscopic theory of high-temperature superconductivity in cuprates. BCS theory in its standard form cannot explain $T_c$ values above about 30 K. The cuprates are strongly correlated electron systems where the standard tools of perturbation theory break down. Understanding these materials remains one of the greatest open problems in condensed matter physics. Proposed mechanisms include antiferromagnetic spin fluctuations, resonating valence bond states, and various forms of non-Fermi-liquid behavior.
Applications
Superconductivity has enabled technologies that would be impossible with normal conductors:
- MRI (Magnetic Resonance Imaging): The large, stable magnetic fields required for MRI (typically 1.5 to 7 T) are generated by superconducting magnets made of NbTi or Nb$_3$Sn wire. Without superconductors, the power consumption would be prohibitive.
- Particle accelerators: The LHC at CERN uses 1,232 superconducting dipole magnets cooled to 1.9 K to bend the proton beams around the 27 km ring.
- SQUIDs (Superconducting Quantum Interference Devices): Based on two Josephson junctions in a loop, SQUIDs are the most sensitive magnetometers ever built, capable of detecting fields as small as $10^{-15}$ T. They are used in brain imaging (magnetoencephalography), geological surveying, and fundamental physics experiments.
- Maglev trains: Superconducting magnets provide the strong fields needed for magnetic levitation, enabling trains that float above the track with virtually no friction.
- Quantum computing: Superconducting circuits, particularly transmon qubits based on Josephson junctions, are the leading platform for quantum computing. The companies building the most advanced quantum processors (IBM, Google) use superconducting technology.
Connection: Topological Superconductors
The intersection of superconductivity and topology (lesson 70) is one of the most active areas in modern physics. Topological superconductors are predicted to host Majorana fermions at their boundaries: exotic quasiparticles that are their own antiparticles. These Majorana modes could serve as the basis for topological quantum computing, which would be inherently protected from decoherence.
Exercises
- London penetration depth: Estimate $\lambda_L$ for aluminum, which has a superconducting carrier density $n_s \approx 1.8 \times 10^{29}$ m$^{-3}$. Use $m = 2m_e$ (Cooper pair mass) and $e \to 2e$ (Cooper pair charge). How does this compare to the wavelength of visible light?
- BCS gap equation: For aluminum, $T_c = 1.2$ K. Using the BCS relation $2\Delta(0) = 3.53 k_B T_c$, calculate the zero-temperature gap $\Delta(0)$ in eV. What photon wavelength would be needed to break a Cooper pair?
- Flux quantization: A superconducting ring has an area of 1 cm$^2$. What is the maximum magnetic field $B$ for which the ring contains exactly one flux quantum? Express your answer in teslas and compare it to the Earth's magnetic field ($\sim 50 \mu$T).
- AC Josephson effect: A Josephson junction is biased at $V = 10$ $\mu$V. Calculate the frequency of the AC supercurrent. In what part of the electromagnetic spectrum does this frequency lie?
- Critical field: A Type I superconductor has a critical field $H_c(0) = 0.08$ T at $T = 0$. The temperature dependence is approximately $H_c(T) = H_c(0)[1 - (T/T_c)^2]$, with $T_c = 7.2$ K. At what temperature does the critical field drop to half its zero-temperature value?
- Superconductors exhibit zero electrical resistance and the Meissner effect (expulsion of magnetic flux), the latter proving that superconductivity is a true thermodynamic state, not just perfect conduction.
- The London equations describe the electromagnetic response phenomenologically: magnetic fields decay exponentially inside a superconductor with penetration depth $\lambda_L$.
- BCS theory explains superconductivity through Cooper pairing: phonon-mediated attraction binds electrons into pairs that condense into a macroscopic quantum state with an energy gap $\Delta$.
- Type I superconductors have a single critical field; Type II superconductors have a vortex state between $H_{c1}$ and $H_{c2}$ where quantized flux tubes penetrate the material.
- Flux quantization ($\Phi_0 = h/2e$) and the Josephson effect are direct macroscopic manifestations of quantum coherence; they confirm the Cooper pair nature of the superconducting carriers.
- High-temperature superconductivity in cuprates remains unexplained by BCS theory and is one of the great open problems in physics; applications of superconductivity include MRI, particle accelerators, SQUIDs, maglev, and quantum computing.