Microscopic Superconductivity & Quantum Tunneling: BCS Theory, Josephson & SQUIDs

1. The Attractive Electron-Phonon Interaction

In 1950, Fröhlich pointed out that two electrons can experience an effective attractive interaction mediated by the polarizable positive ion lattice.

As an electron moves through the crystal, its negative charge attracts the positive ionic cores, creating a trailing localized concentration of positive charge (lattice polarization). Because the massive ions respond slowly on the timescale of the Debye frequency $\omega_D \sim 10^{13}\text{ s}^{-1}$, this positive wake persists long after the first electron has passed ($t \sim 2\pi/\omega_D \sim 10^{-13}\text{ s}$, during which an electron at the Fermi velocity $v_F \sim 10^6\text{ m/s}$ travels $\sim 100\text{ nm}$).

A second electron can then be attracted to this net positive wake. In second-order perturbation theory, the effective interaction between electrons with initial states $\vec{k}$ and $\vec{k}'$ exchanging a phonon $\vec{q}$ is:

$$V_{\text{eff}}(\vec{q}, \omega) = \frac{e^2}{\epsilon_0 q^2} + \frac{2 |M_{\vec{q}}|^2 \hbar\omega_{\vec{q}}}{(\epsilon_{\vec{k}'} - \epsilon_{\vec{k}})^2 - (\hbar\omega_{\vec{q}})^2}$$

Whenever $|\epsilon_{\vec{k}'} - \epsilon_{\vec{k}}| < \hbar\omega_D$, the phonon term becomes negative and overcomes the screened Coulomb repulsion, producing a net attractive pairing interaction!

2. The Cooper Pair Problem (1956)

Leon Cooper proved that the filled Fermi sea is fundamentally unstable against arbitrarily weak attraction.

Consider two electrons added to an inert, filled Fermi sea with Fermi momentum $k_F$. The pair wavefunction with zero total center-of-mass momentum ($\vec{k}_1 = \vec{k}, \vec{k}_2 = -\vec{k}$) and singlet spin is:

$$\psi(\vec{r}_1, \vec{r}_2) = \sum_{k > k_F} g(\vec{k}) e^{i \vec{k} \cdot (\vec{r}_1 - \vec{r}_2)} \chi_0^0$$

The Schrödinger equation is:

$$(2\epsilon_k - E) g(\vec{k}) + \sum_{k' > k_F} V_{\vec{k}\vec{k}'} g(\vec{k}') = 0$$

Assuming an attractive potential $-V$ within an energy shell $\hbar\omega_D$ above $E_F$:

$$V_{\vec{k}\vec{k}'} = \begin{cases} -V, & E_F < \epsilon_k, \epsilon_{k'} < E_F + \hbar\omega_D \\ 0, & \text{otherwise} \end{cases}$$

Setting $C \equiv \sum_{k'} g(\vec{k}') = \text{const}$:

$$g(\vec{k}) = \frac{V C}{2\epsilon_k - E} \implies \frac{1}{V} = \sum_{k > k_F} \frac{1}{2\epsilon_k - E} = \int_{E_F}^{E_F + \hbar\omega_D} \frac{N(0)}{2\epsilon - E} d\epsilon$$

Evaluating the integral:

$$\frac{1}{N(0)V} = \frac{1}{2} \ln\left( \frac{2E_F + 2\hbar\omega_D - E}{2E_F - E} \right)$$

Defining the binding energy $\Delta E \equiv 2E_F - E > 0$:

$$\Delta E \approx 2\hbar\omega_D \exp\left( -\frac{2}{N(0)V} \right)$$

Crucially, because $\Delta E > 0$ for any $V > 0$, electrons at the Fermi surface always bind into Cooper pairs! The non-analytic exponential factor $e^{-2/N(0)V}$ proves that superconductivity can never be derived by finite-order perturbation theory.

1. The BCS Many-Body Hamiltonian

In 1957, John Bardeen, Leon Cooper, and J. Robert Schrieffer published the microscopic BCS Theory. The reduced pairing Hamiltonian in second quantization is:

$$\hat{H}_{\text{BCS}} = \sum_{\vec{k}\sigma} \epsilon_k c_{\vec{k}\sigma}^\dagger c_{\vec{k}\sigma} - V \sum_{\vec{k},\vec{k}'} c_{\vec{k}\uparrow}^\dagger c_{-\vec{k}\downarrow}^\dagger c_{-\vec{k}'\downarrow} c_{\vec{k}'\uparrow}$$

Because Cooper pairs are bosons formed of fermions, the macroscopic ground state is not a simple Bose-Einstein condensate of independent molecules, but a coherent quantum superposition spanning the entire Fermi sea.

2. The BCS Ground State Wavefunction

The BCS trial ground state wavefunction is:

$$|\Psi_{\text{BCS}}\rangle = \prod_{\vec{k}} \left( u_{\vec{k}} + v_{\vec{k}} c_{\vec{k}\uparrow}^\dagger c_{-\vec{k}\downarrow}^\dagger \right) |0\rangle$$

where:

• $v_{\vec{k}}^2$ is the probability that the Cooper pair state $(\vec{k}\uparrow, -\vec{k}\downarrow)$ is occupied.
• $u_{\vec{k}}^2$ is the probability that the state is empty.
• Normalization requires $u_{\vec{k}}^2 + v_{\vec{k}}^2 = 1$.

Minimizing the expectation energy $\langle \Psi_{\text{BCS}} | \hat{H} - \mu \hat{N} | \Psi_{\text{BCS}} \rangle$ with respect to $v_{\vec{k}}$ yields:

$$u_{\vec{k}}^2 = \frac{1}{2} \left( 1 + \frac{\xi_k}{E_k} \right), \quad v_{\vec{k}}^2 = \frac{1}{2} \left( 1 - \frac{\xi_k}{E_k} \right)$$

where $\xi_k = \epsilon_k - \mu$ is the normal single-electron energy relative to the Fermi level, and $E_k$ is the Bogoliubov quasiparticle excitation energy:

$$E_k = \sqrt{\xi_k^2 + \Delta^2}$$

The minimum energy required to create a quasiparticle excitation is $E_{\min} = \Delta$. Breaking a Cooper pair requires minimum energy $2\Delta$, opening an energy gap $2\Delta$ centered symmetrically at the Fermi level!

1. The BCS Gap Equation

At finite temperature $T$, thermal excitations create quasiparticles that block pair states. The self-consistent BCS gap equation is:

$$\frac{1}{N(0)V} = \int_0^{\hbar\omega_D} \frac{\tanh\left( \frac{\sqrt{\xi^2 + \Delta^2(T)}}{2k_B T} \right)}{\sqrt{\xi^2 + \Delta^2(T)}} d\xi$$

• At $T = 0\text{ K}$ ($\tanh \to 1$): $$\frac{1}{N(0)V} = \int_0^{\hbar\omega_D} \frac{d\xi}{\sqrt{\xi^2 + \Delta_0^2}} = \operatorname{arcsinh}\left(\frac{\hbar\omega_D}{\Delta_0}\right) \approx \ln\left(\frac{2\hbar\omega_D}{\Delta_0}\right)$$ $$\Delta(0) = 2\hbar\omega_D \exp\left( -\frac{1}{N(0)V} \right)$$
• At $T = T_c$ ($\Delta \to 0$): $$\frac{1}{N(0)V} = \int_0^{\hbar\omega_D} \frac{\tanh(\xi / 2k_B T_c)}{\xi} d\xi = \ln\left( \frac{1.134 \hbar\omega_D}{k_B T_c} \right)$$ $$k_B T_c = 1.134 \hbar\omega_D \exp\left( -\frac{1}{N(0)V} \right)$$

Taking the ratio eliminates the interaction parameters $N(0)V$ and $\omega_D$, yielding the universal BCS ratio:

$$\frac{2\Delta(0)}{k_B T_c} = \frac{2 \times 2}{1.134} \approx 3.528$$

For weakly coupled conventional superconductors, $2\Delta(0)/k_B T_c \approx 3.53$ is an exact universal constant!

2. The Isotope Effect

Because the Debye frequency $\omega_D \propto \sqrt{K/M} \propto M^{-1/2}$ depends inversely on the square root of the isotopic nuclear mass $M$, BCS theory predicts:

$$T_c \propto \omega_D \propto M^{-1/2} \implies T_c M^\alpha = \text{const}, \quad \alpha = 0.5$$

This isotope effect was experimentally discovered in mercury isotopes ($\text{Hg}^{198}$ to $\text{Hg}^{204}$) by Maxwell and Reynolds in 1950, providing definitive proof that electron-phonon interactions drive conventional superconductivity.

1. Superconducting Density of States

From the Bogoliubov quasiparticle dispersion $E = \sqrt{\xi^2 + \Delta^2}$, the quasiparticle density of states $N_s(E)$ is related to the normal density of states $N(0)$ by particle conservation $N_s(E) dE = N(0) d\xi$:

$$N_s(E) = N(0) \frac{d\xi}{dE} = N(0) \frac{E}{\sqrt{E^2 - \Delta^2}}, \quad |E| > \Delta$$ $$N_s(E) = 0, \quad |E| < \Delta$$

The density of states vanishes completely inside the gap $|E| < \Delta$ and exhibits an inverse square-root square-integrable singularity at the gap edges $E = \pm\Delta$.

2. Giaever Tunneling Experiments

In 1960, Ivar Giaever fabricated thin-film planar junctions with an ultra-thin insulating oxide layer ($\sim 2\text{ nm}$ thick, e.g. $\text{Al}_2\text{O}_3$):

• Superconductor-Insulator-Normal Metal (S-I-N) Junction: Applying a bias voltage $V$ shifts the Fermi level of the normal metal by $-eV$. Quasiparticles cannot tunnel until $e|V| \ge \Delta$. The tunneling differential conductance at $T \to 0\text{ K}$ directly measures the BCS density of states: $$\frac{dI}{dV} \propto N_s(eV) = N(0) \frac{eV}{\sqrt{(eV)^2 - \Delta^2}}$$ The onset of current at $V = \Delta/e$ provides a direct spectroscopic measurement of the energy gap $\Delta(T)$.
• Superconductor-Insulator-Superconductor (S-I-S) Junction: With two identical superconductors having gap $\Delta$, tunneling is blocked until $e|V| = 2\Delta$. At $V = 2\Delta/e$, the sharp singularity in the filled states of one electrode aligns with the singularity in the empty states of the other, causing a discontinuous jump in current.

1. The DC Josephson Effect

In 1962, Brian Josephson predicted that Cooper pairs can tunnel quantum-mechanically across a thin insulating barrier separating two superconductors without any applied voltage.

Let $\psi_1 = \sqrt{n_1} e^{i\theta_1}$ and $\psi_2 = \sqrt{n_2} e^{i\theta_2}$ be the macroscopic wavefunctions of the two superconductors. Coupled Schrödinger equations across the barrier yield:

$$i\hbar \frac{\partial\psi_1}{\partial t} = E_1 \psi_1 + K \psi_2$$ $$i\hbar \frac{\partial\psi_2}{\partial t} = E_2 \psi_2 + K \psi_1$$

where $K$ is the coupling tunneling matrix element. Separating real and imaginary parts yields the DC Josephson Equation:

$$I = I_c \sin\phi$$

where $\phi \equiv \theta_2 - \theta_1$ is the gauge-invariant phase difference across the junction, and $I_c$ is the maximum zero-voltage critical supercurrent given by the Ambegaokar-Baratoff formula:

$$I_c = \frac{\pi \Delta(T)}{2e R_N} \tanh\left( \frac{\Delta(T)}{2k_B T} \right)$$

where $R_N$ is the normal-state junction resistance.

2. The AC Josephson Effect

When a constant DC voltage $V$ is applied across the junction, the energy difference between the Cooper pairs on opposite sides is $\Delta E = 2e V$. The phase evolves according to the AC Josephson Equation:

$$\frac{d\phi}{dt} = \frac{2e V}{\hbar} \implies \phi(t) = \phi_0 + \left( \frac{2e V}{\hbar} \right) t$$

Substituting into the current relation:

$$I(t) = I_c \sin\left( \phi_0 + \omega_J t \right)$$

A static DC voltage produces an alternating high-frequency supercurrent oscillating at the Josephson frequency:

$$\omega_J = \frac{2e V}{\hbar} \implies f_J = \frac{2e}{h} V = \frac{V}{\Phi_0}$$

Numerically, the frequency-to-voltage conversion ratio is:

$$\frac{f_J}{V} = \frac{2e}{h} \approx 483.5979\text{ GHz / mV} = 483.6\text{ MHz} / \mu\text{V}$$

Because frequency can be measured with atomic-clock accuracy, the AC Josephson effect provides the international metrological standard for the Volt.

1. Microwave Shapiro Current Steps

When a Josephson junction is irradiated with RF microwave radiation of frequency $\omega$ in addition to a DC bias $V_0$:

$$V(t) = V_0 + V_1 \cos(\omega t)$$

Integrating the phase evolution equation $\frac{d\phi}{dt} = \frac{2e}{\hbar} V(t)$:

$$\phi(t) = \phi_0 + \left(\frac{2e V_0}{\hbar}\right) t + \frac{2e V_1}{\hbar\omega} \sin(\omega t)$$

Using the Jacobi-Anger Bessel function expansion $\sin(\alpha + z\sin\theta) = \sum_{n=-\infty}^\infty J_n(z) \sin(\alpha + n\theta)$:

$$I(t) = I_c \sum_{n=-\infty}^\infty (-1)^n J_n\left(\frac{2e V_1}{\hbar\omega}\right) \sin\left[ \phi_0 + \left(\frac{2e V_0}{\hbar} - n\omega\right) t \right]$$

Whenever the Josephson frequency matches an integer harmonic of the microwave frequency:

$$\frac{2e V_0}{\hbar} = n\omega \implies V_n = n \frac{\hbar\omega}{2e} = n \left(\frac{h\nu}{2e}\right)$$

The time-dependent term vanishes, producing constant DC current spikes known as Shapiro steps.

2. The DC SQUID (Superconducting Quantum Interference Device)

A DC SQUID consists of two identical Josephson junctions connected in parallel on a superconducting loop enclosing magnetic flux $\Phi$.

The total current is the sum of currents through branches $a$ and $b$:

$$I_{\text{tot}} = I_c \sin\phi_a + I_c \sin\phi_b = 2I_c \cos\left(\frac{\phi_a - \phi_b}{2}\right) \sin\left(\frac{\phi_a + \phi_b}{2}\right)$$

By flux quantization around the loop, the phase difference across the two junctions is locked to the enclosed magnetic flux:

$$\phi_a - \phi_b = \frac{2e}{\hbar} \Phi = 2\pi \frac{\Phi}{\Phi_0}$$

The maximum zero-voltage critical current is modulated by the magnetic flux:

$$I_{\max}(\Phi) = 2I_c \left| \cos\left( \pi \frac{\Phi}{\Phi_0} \right) \right|$$

The SQUID exhibits a periodic quantum interference pattern analogous to Young's double-slit experiment. By monitoring the voltage across a current-biased SQUID, magnetic field changes as minute as $10^{-15}\text{ Tesla}$ ($10^{-6} \Phi_0 / \sqrt{\text{Hz}}$) can be detected, enabling magnetoencephalography (MEG) of neural brain activity.

1. Discovery of High-Temperature Cuprate Superconductors

In 1986, Georg Bednorz and K. Alex Müller discovered superconductivity in the copper oxide ceramic $\text{La}_{2-x}\text{Ba}_x\text{CuO}_4$ at $T_c = 35\text{ K}$, breaking the theoretical BCS phonon-mediated Eliashberg limit ($T_c \lesssim 30\text{ K}$).

Shortly thereafter in 1987, $\text{YBa}_2\text{Cu}_3\text{O}_{7-\delta}$ (YBCO) was discovered with $T_c = 93\text{ K}$, shattering the liquid nitrogen barrier ($77.3\text{ K}$). Subsequently, mercury cuprates reached $T_c = 138\text{ K}$ at ambient pressure and $164\text{ K}$ under $30\text{ GPa}$.

2. d-Wave Gap Symmetry & Electronic Pairing

High-$T_c$ cuprates possess quasi-2D layered perovskite crystal structures dominated by conducting $\text{CuO}_2$ planes separated by charge-reservoir oxide layers.

Key distinctions from conventional BCS superconductors:

• Parent State is a Mott Insulator: Undoped cuprates ($\text{La}_2\text{CuO}_4$) are antiferromagnetic Mott insulators due to massive on-site Coulomb repulsion $U \approx 8\text{ eV} \gg t$. Superconductivity emerges upon hole-doping ($x \sim 0.05-0.25$) into a strongly correlated pseudogap regime.
• Unconventional $d_{x^2-y^2}$ Order Parameter: While conventional superconductors have isotropic $s$-wave pairing ($\Delta = \text{const}$), cuprates exhibit $d$-wave pairing symmetry: $$\Delta(\vec{k}) = \Delta_0 (\cos k_x a - \cos k_y a)$$ The gap changes sign across the Brillouin zone diagonals ($k_x = \pm k_y$), creating four gapless nodes on the Fermi surface where $\Delta(\vec{k}) = 0$.
• Spin Fluctuation Pairing Mechanism: Instead of acoustic phonons, Cooper pairing is mediated by antiferromagnetic spin fluctuations (magnons) originating from the nearby Mott insulating phase.

In 2008, iron-based pnictide superconductors ($\text{LaO}_{1-x}\text{F}_x\text{FeAs}$, $T_c = 56\text{ K}$) were discovered, exhibiting multi-band $s_\pm$-wave pairing. In 2015-2020, hydrogen-rich polyhydrides under extreme megabar pressures reached record critical temperatures ($\text{H}_3\text{S}$ at $203\text{ K}$ at $150\text{ GPa}$, and $\text{LaH}_{10}$ at $250\text{ K}$ at $170\text{ GPa}$), representing phonon-driven conventional BCS pairing in ultra-dense hydrogen.