--- title: "Josephson Effects and Unconventional Superconductors" module: Superconductivity moduleNumber: 10 lessonNumber: 5 order: 1005 summary: > Two superconductors joined by a thin barrier carry a supercurrent set by their phase difference — the dc Josephson effect — and oscillate at 2eV/h under a voltage. A two-junction loop turns flux quantization into a magnetometer of single-quantum sensitivity. The cuprates superconduct in CuO2 planes with a doping-dependent dome, d-wave pairing, and a pseudogap that lie outside the phonon picture. topics: [Superconductivity] draft: false sources: - book: Kittel ref: "Ch. 10 — Superconductivity (Josephson effects, high-temperature superconductors)" - book: Tipler & Llewellyn ref: "Ch. 10 — Solid State Physics; §10-9 Superconductivity, Josephson Junction" - book: Hook & Hall ref: "Ch. 12 — Superconductivity" --- The macroscopic condensate wave function has a definite phase, and a phase is observable only through interference. A thin barrier between two superconductors lets their condensates interfere, and the resulting Josephson effects turn the phase into measurable current, voltage, and flux. The same interference, plus a survey of the cuprate superconductors, closes the module by pointing past the [phonon BCS mechanism](/condensed-matter/superconductivity/bcs-theory) to pairing that is still not fully understood. ## The dc and ac Josephson effects Two superconductors separated by an insulating barrier a few nanometres thick form a **Josephson junction**. Label the condensate wave functions $\psi_1 = \sqrt{n_1}\,e^{i\theta_1}$ and $\psi_2 = \sqrt{n_2}\,e^{i\theta_2}$. Across the barrier they are weakly coupled, and the two-state Schrödinger equations read $$ i\hbar\frac{\partial\psi_1}{\partial t} = \mu_1\psi_1 + K\psi_2 , \qquad i\hbar\frac{\partial\psi_2}{\partial t} = \mu_2\psi_2 + K\psi_1 , $$ with coupling energy $K$ and chemical potentials $\mu_1, \mu_2$. Separating each equation into modulus and phase and writing the phase difference $\delta = \theta_2 - \theta_1$, the rate of pair transfer across the barrier is $$ I = I_c\sin\delta , \qquad I_c = \frac{2K\sqrt{n_1 n_2}}{\hbar} , $$ the **dc Josephson effect**: a supercurrent up to $I_c$ flows with no voltage, fixed entirely by the phase difference of the two condensates. The phase evolves according to $$ \frac{\d\delta}{\d t} = \frac{\mu_2 - \mu_1}{\hbar} = \frac{2eV}{\hbar} , $$ since a voltage $V$ across the junction shifts the pair energy by $2eV$. With a constant voltage the phase winds linearly and the current oscillates — the **ac Josephson effect**, $$ I = I_c\sin\!\left(\delta_0 + \frac{2eV}{\hbar}t\right), \qquad f = \frac{2eV}{h} = 483.6\ \text{MHz}/\mu\text{V} . $$ The frequency depends only on $2e/h$, so a measured frequency fixes a voltage to the precision of a frequency standard; the ac Josephson effect defines the volt.[^josephson] [^josephson]: Kittel, Ch. 10, "Josephson tunneling"; Tipler & Llewellyn, §10-9. $$ % caption: A Josephson junction and its current-phase relation. Cooper pairs % tunnel through the barrier carrying a supercurrent I = I_c sin(delta) set by the % phase difference delta of the two condensates; the maximum is the critical % current I_c. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % junction schematic \draw[acc, thick, fill=acc!10] (0,0.4) rectangle (1.7,2.0); \node[acc, align=center, font=\scriptsize] at (0.85,1.2) {super 1}; \draw[black, thick, fill=black!6] (1.7,0.4) rectangle (2.1,2.0); \node[black, font=\scriptsize, rotate=90] at (1.9,1.2) {barrier}; \draw[acc, thick, fill=acc!10] (2.1,0.4) rectangle (3.8,2.0); \node[acc, align=center, font=\scriptsize] at (2.95,1.2) {super 2}; \draw[acc, ->, thick] (1.55,2.3) .. controls (1.9,2.6) .. (2.25,2.3); \node[acc, anchor=south, font=\scriptsize] at (1.9,2.5) {pairs tunnel}; % current-phase plot \begin{scope}[xshift=5.2cm, yshift=0.3cm] \draw[->, black] (0,0) -- (4.0,0) node[below] {phase}; \draw[->, black] (0,-1.4) -- (0,1.6) node[left, font=\scriptsize] {$I$}; \draw[acc, very thick, domain=0:3.6, samples=90, variable=\x] plot ({\x},{1.2*sin(\x r)}); \draw[black, dashed] (0,1.2) -- (0.15,1.2); \node[anchor=east, font=\scriptsize] at (-0.05,1.2) {$I_c$}; \end{scope} \end{tikzpicture} $$ ## SQUID magnetometry Put two junctions in parallel on a superconducting loop that encloses a magnetic flux $\Phi$. Flux quantization ties the two phase differences together: going once around the loop, the phase must change by a multiple of $2\pi$, so $$ \delta_2 - \delta_1 = \frac{2\pi\Phi}{\Phi_0} , \qquad \Phi_0 = \frac{h}{2e} . $$ The total supercurrent is the sum through the two arms, and with $\delta_{1,2} = \delta_0 \mp \pi\Phi/\Phi_0$ it becomes $$ I = I_c(\sin\delta_1 + \sin\delta_2) = 2I_c\cos\!\left(\frac{\pi\Phi}{\Phi_0}\right)\sin\delta_0 . $$ The maximum supercurrent the loop can carry is $$ I_{\max}(\Phi) = 2I_c\left|\cos\!\frac{\pi\Phi}{\Phi_0}\right| , $$ an interference pattern periodic in the enclosed flux with period $\Phi_0$. A **SQUID** (superconducting quantum interference device) tracks $I_{\max}$ and so resolves flux changes far smaller than one quantum, giving field sensitivities near $10^{-15}\ \text{T}$ — the most sensitive magnetometer known, used from geophysics to brain imaging.[^squid] [^squid]: Kittel, Ch. 10, "superconducting quantum interference." Flux quantum $\Phi_0 = h/2e$, NIST, . $$ % caption: The dc SQUID. Two junctions on a loop enclosing flux Phi; the maximum % supercurrent oscillates as the modulus of cos(pi Phi over Phi_0), with each % period corresponding to one added flux quantum. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % loop with two junctions \begin{scope}[yshift=0.2cm] \draw[acc, very thick] (0,0) -- (0,1.6); \draw[acc, very thick] (2.0,0) -- (2.0,1.6); \draw[acc, very thick] (0,1.6) -- (0.8,1.6); \draw[acc, very thick] (1.2,1.6) -- (2.0,1.6); \draw[acc, very thick] (0,0) -- (0.8,0); \draw[acc, very thick] (1.2,0) -- (2.0,0); % junctions (crosses) \draw[black, thick] (0.8,1.5) rectangle (1.2,1.7); \draw[black, thick] (0.8,-0.1) rectangle (1.2,0.1); \node[black, font=\scriptsize] at (1.0,0.8) {enclosed}; \draw[acc, ->, thick] (1.0,2.1) -- (1.0,1.75); \draw[acc, ->, thick] (1.0,-0.5) -- (1.0,-0.15); \node[acc, anchor=south, font=\scriptsize] at (0.3,2.15) {current}; \end{scope} % interference pattern \begin{scope}[xshift=4.2cm] \draw[->, black] (0,0) -- (4.6,0) node[below, font=\scriptsize] {quanta}; \draw[->, black] (0,0) -- (0,2.0) node[left, font=\scriptsize] {$I_{\max}$}; \draw[acc, very thick, domain=0:4.2, samples=120, variable=\x] plot ({\x},{1.6*abs(cos(180*\x))}); \foreach \x in {1,2,3,4} \node[anchor=north, font=\scriptsize] at (\x,0) {\x}; \end{scope} \end{tikzpicture} $$ ## The cuprate superconductors Until 1986 the highest known $T_c$ was $23\ \text{K}$. Bednorz and Müller then found superconductivity near $35\ \text{K}$ in a layered lanthanum copper oxide, and within two years $\text{YBa}_2\text{Cu}_3\text{O}_7$ reached $92\ \text{K}$, above the boiling point of liquid nitrogen; the mercury cuprates reach $133\ \text{K}$. These **cuprates** share a layered perovskite structure in which the current flows in $\text{CuO}_2$ planes separated by charge-reservoir layers that dope the planes with carriers. The undoped parent compound is not a metal but a **Mott insulator**: strong on-site Coulomb repulsion localizes one hole per copper, and the spins order antiferromagnetically. Adding holes to the $\text{CuO}_2$ planes destroys the antiferromagnetism and, past a threshold doping, produces superconductivity. The critical temperature traces a **dome** as a function of doping, peaking at optimal doping near $0.16$ holes per copper and falling on both the underdoped and overdoped sides. $$ % caption: Schematic cuprate phase diagram. The undoped parent is an % antiferromagnetic Mott insulator; doping holes yields a superconducting dome % peaking at optimal doping, with a pseudogap region above the dome on the % underdoped side. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.6,0) node[below] {hole doping}; \draw[->, black] (0,0) -- (0,3.6) node[left] {$T$}; % antiferromagnet: narrow tall region at low doping \fill[black] (0,0) -- (0,3.2) .. controls (0.5,2.0) and (0.9,0.6) .. (1.1,0) -- cycle; \node[black, anchor=south, font=\scriptsize, rotate=64] at (0.42,1.2) {antiferro}; % superconducting dome \draw[acc, very thick] (1.5,0) .. controls (2.6,2.0) and (4.0,2.0) .. (5.2,0); \fill[acc!12] (1.5,0) .. controls (2.6,2.0) and (4.0,2.0) .. (5.2,0) -- cycle; \node[acc, anchor=south, font=\scriptsize] at (3.35,0.55) {superconducting}; % pseudogap line \draw[black, thick, dashed] (1.5,3.2) -- (3.5,0.9); \node[black, anchor=west, font=\scriptsize] at (1.7,2.6) {pseudogap}; % strange metal \node[black, anchor=west, font=\scriptsize] at (3.9,2.5) {strange metal}; \end{tikzpicture} $$ ## d-wave pairing and the pseudogap BCS superconductors have an isotropic **s-wave** gap: $\Delta(\vec k)$ is the same sign and magnitude everywhere on the Fermi surface. The cuprates instead pair in a $d_{x^2-y^2}$ channel, $$ \Delta(\vec k) = \Delta_0\,\frac{\cos k_x a - \cos k_y a}{2} , $$ which is largest along the copper–oxygen bond directions, changes sign between them, and has **nodes** — points where the gap vanishes — along the diagonals. Phase-sensitive experiments, using corner junctions and tricrystal geometries to detect the sign change directly, confirmed the d-wave symmetry. The nodes leave quasiparticles available at arbitrarily low energy, so the heat capacity and penetration depth follow power laws in $T$ rather than the exponential BCS activation. $$ % caption: Gap symmetry on the Fermi surface. The s-wave gap is uniform around % the circle; the d-wave gap bulges outward along the bond axes, changes sign, and % vanishes at four nodes on the diagonals. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % --- s-wave --- \begin{scope} \draw[black, thick] (0,0) circle (1.0); \draw[acc, very thick] (0,0) circle (1.35); \node[anchor=north] at (0,-1.7) {s-wave}; \node[black, font=\scriptsize] at (0,0) {Fermi}; \end{scope} % --- d-wave --- \begin{scope}[xshift=5.2cm] \draw[black, thick] (0,0) circle (1.0); % gap magnitude R + A cos(2t): lobes along x, dents along y \draw[acc, very thick, domain=0:360, samples=160, variable=\t] plot ({(1.0 + 0.45*cos(2*\t))*cos(\t)},{(1.0 + 0.45*cos(2*\t))*sin(\t)}); \node[acc, font=\scriptsize] at (1.75,0) {$+$}; \node[acc, font=\scriptsize] at (-1.75,0) {$+$}; % nodes on diagonals \foreach \a in {45,135,225,315} \draw[black, fill=white] ({1.0*cos(\a)},{1.0*sin(\a)}) circle (2.2pt); \node[black, anchor=south west, font=\scriptsize] at (0.75,0.75) {node}; \node[anchor=north] at (0,-1.7) {d-wave}; \end{scope} \end{tikzpicture} $$ Above the dome on the underdoped side lies the **pseudogap**: below a temperature $T^\ast$ well above $T_c$, part of the density of states at the Fermi surface is suppressed without full superconductivity. Whether the pseudogap is a precursor to pairing or a distinct competing order remains unsettled, and it is the central open problem of cuprate physics. The pairing glue is widely attributed to antiferromagnetic spin fluctuations rather than phonons, which is consistent with the proximity of the superconducting dome to the antiferromagnetic parent. ## Other unconventional families Superconductivity beyond the phonon s-wave picture now spans several classes. - **Iron-based superconductors** (2008), such as the oxypnictide $\text{LaFeAsO}$ and the selenides, reach $T_c$ up to $55\ \text{K}$ and pair in an $s_{\pm}$ state whose sign reverses between electron and hole Fermi pockets. - **Heavy-fermion and organic superconductors** pair near magnetic quantum critical points, again suggesting a magnetic rather than phononic mechanism. - **Hydrogen-rich compounds under pressure**, $\text{H}_3\text{S}$ and $\text{LaH}_{10}$ at megabar pressures, superconduct above $200\ \text{K}$ by a conventional but extreme phonon mechanism, the light hydrogen mass pushing the Debye energy and the BCS $T_c$ upward. - $\text{MgB}_2$ ($39\ \text{K}$) is a phonon superconductor with two distinct gaps on different sheets of its Fermi surface. | Family | Example | $T_c$ | Pairing | | --- | --- | --- | --- | | Elemental (BCS) | Nb | $9.3\ \text{K}$ | s-wave, phonon | | A15 alloy | Nb$_3$Sn | $18\ \text{K}$ | s-wave, phonon | | Two-gap | MgB$_2$ | $39\ \text{K}$ | s-wave, phonon | | Cuprate | YBa$_2$Cu$_3$O$_7$ | $92\ \text{K}$ | d-wave, unconventional | | Iron-based | LaFeAsO(F) | $55\ \text{K}$ | $s_{\pm}$, unconventional | | Hydride (high $P$) | LaH$_{10}$ | $250\ \text{K}$ | s-wave, phonon | The Josephson effects show that the condensate phase is a real, measurable quantity, and flux quantization through $\Phi_0 = h/2e$ recurs in every measurement as the fingerprint of pairing. The cuprates and their relatives keep the pairing state — its symmetry and its glue — an active question, one that the weak-coupling BCS theory frames but does not close.