--- title: Microscopic BCS Theory module: Superconductivity moduleNumber: 10 lessonNumber: 4 order: 1004 summary: > A phonon-mediated attraction, however weak, binds two electrons above the Fermi sea — the Cooper problem shows the sea is unstable. The BCS variational ground state pairs all electrons near the Fermi surface and, through a self-consistent gap equation, opens an energy gap. Weak-coupling solution gives the exponential T_c and the universal ratios 2 Delta(0) = 3.53 k_B T_c and Delta C / C_n = 1.43. topics: [Superconductivity] draft: false sources: - book: Kittel ref: "Ch. 10 — Superconductivity (BCS theory)" - book: Ashcroft & Mermin ref: "Ch. 34 — Superconductivity" - book: Hook & Hall ref: "Ch. 12 — Superconductivity" --- [Ginzburg–Landau theory](/condensed-matter/superconductivity/ginzburg-landau-theory) leaves the coefficients $\alpha$ and $\beta$ and the pairing charge $2e$ as inputs. Bardeen, Cooper, and Schrieffer (1957) derived them from the electron–phonon interaction. Two steps carry the argument: Cooper's demonstration that the Fermi sea is unstable against pairing, and the BCS variational state that pairs every electron near the Fermi surface self-consistently. The weak-coupling solution predicts the energy gap, the critical temperature, and their universal ratio. ## The phonon-mediated attraction An electron distorts the lattice it moves through, drawing the positive ions inward and leaving a trailing region of excess positive charge. A second electron is attracted to that region. In second-order perturbation theory the exchange of a virtual phonon produces an effective electron–electron interaction that is **attractive** when the energy transfer is less than the phonon energy. BCS model this by a constant attractive matrix element within a shell of width $\hbar\omega_D$ (the Debye energy) about the Fermi surface, $$ V_{\vec k\vec k'} = \begin{cases} -V & |\xi_{\vec k}|,\ |\xi_{\vec k'}| < \hbar\omega_D \\ 0 & \text{otherwise,} \end{cases} $$ where $\xi_{\vec k} = \varepsilon_{\vec k} - E_F$ is the single-electron energy measured from the Fermi level and $V > 0$. The cutoff $\hbar\omega_D$ carries the ion mass through $\omega_D \propto M^{-1/2}$, which is the origin of the isotope effect. ## The Cooper problem Add two electrons to a filled Fermi sea at $T = 0$, letting them interact through $V_{\vec k\vec k'}$ while the sea electrons stay inert and merely block the states below $k_F$. A pair of zero total momentum and opposite spins has the wave function $$ |\psi\rangle = \sum_{k > k_F} g_{\vec k}\,c^\dagger_{\vec k\uparrow}c^\dagger_{-\vec k\downarrow}|\mathrm{FS}\rangle . $$ $$ % caption: The Cooper pair. Two electrons in a thin shell just outside the filled % Fermi sphere, with equal and opposite momenta and opposite spins, bind through % the phonon attraction into a state below the sea's energy. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} % Fermi sphere \draw[black, thick, fill=black!6] (0,0) circle (1.6); \node[black, font=\scriptsize] at (0,0) {Fermi sea}; \draw[black, dashed] (0,0) circle (1.9); \node[black, anchor=west, font=\scriptsize] at (1.95,1.35) {Debye shell}; % two electrons at opposite points just outside \fill[acc] (1.75,0.35) circle (3pt); \draw[acc, ->, thick] (1.75,0.35) -- (2.5,0.5); \node[acc, anchor=west, font=\scriptsize] at (2.15,0.05) {electron 1}; \fill[acc] (-1.75,-0.35) circle (3pt); \draw[acc, ->, thick] (-1.75,-0.35) -- (-2.5,-0.5); \node[acc, anchor=east, font=\scriptsize] at (-2.15,-0.05) {electron 2}; % pairing link \draw[acc, dotted, thick] (1.75,0.35) .. controls (0,1.2) and (0,-1.2) .. (-1.75,-0.35); \end{tikzpicture} $$ The two-body Schrödinger equation, with the pair energy $E$, reads $$ (2\xi_{\vec k} + 2E_F - E)\,g_{\vec k} = -\sum_{\vec k'} V_{\vec k\vec k'}\,g_{\vec k'} = V\sum_{\vec k'} g_{\vec k'} \equiv V\,C , $$ so $g_{\vec k} = V C/(2\xi_{\vec k} + 2E_F - E)$. Summing over $\vec k$ in the shell and cancelling $C$ gives the consistency condition, converted to an integral over the density of states $N(0)$ at the Fermi surface, $$ 1 = V\sum_{\vec k}\frac{1}{2\xi_{\vec k} - (E - 2E_F)} = N(0)V\int_0^{\hbar\omega_D}\frac{\d\xi}{2\xi + |E - 2E_F|} . $$ Writing the binding energy $E_b = 2E_F - E > 0$ and integrating, $$ 1 = \frac{N(0)V}{2}\ln\!\frac{2\hbar\omega_D + E_b}{E_b} \quad\Longrightarrow\quad E_b = \frac{2\hbar\omega_D}{e^{2/N(0)V} - 1} \approx 2\hbar\omega_D\,e^{-2/N(0)V} . $$ The last form holds in the weak-coupling limit $N(0)V \ll 1$. The binding energy is positive for any $V > 0$: **however weak the attraction, the pair binds**. The result is non-analytic in $V$ — no order of perturbation theory reproduces $e^{-2/N(0)V}$ — so the normal Fermi sea is unstable, and a new ground state is needed.[^cooper] [^cooper]: Kittel, Ch. 10, "Cooper pairs"; the pairing instability of the filled Fermi sea. ## The BCS ground state and the gap equation Because the instability involves every electron near $E_F$, the ground state pairs them all at once. BCS proposed the variational wave function $$ |\Psi_{\mathrm{BCS}}\rangle = \prod_{\vec k}\left(u_{\vec k} + v_{\vec k}\,c^\dagger_{\vec k\uparrow}c^\dagger_{-\vec k\downarrow}\right)|0\rangle , \qquad |u_{\vec k}|^2 + |v_{\vec k}|^2 = 1 , $$ where $v_{\vec k}^2$ is the probability that the pair $(\vec k\uparrow, -\vec k\downarrow)$ is occupied. Minimizing the expectation of the reduced Hamiltonian $H = \sum_{\vec k}\xi_{\vec k}c^\dagger c - V\sum_{\vec k\vec k'}c^\dagger_{\vec k\uparrow}c^\dagger_{-\vec k\downarrow}c_{-\vec k'\downarrow}c_{\vec k'\uparrow}$ introduces the **gap parameter** $$ \Delta = V\sum_{\vec k} u_{\vec k}v_{\vec k} = V\sum_{\vec k}\langle c_{-\vec k\downarrow}c_{\vec k\uparrow}\rangle . $$ The minimization yields the excitation energy and the coherence factors $$ E_{\vec k} = \sqrt{\xi_{\vec k}^2 + \Delta^2}, \qquad v_{\vec k}^2 = \frac{1}{2}\left(1 - \frac{\xi_{\vec k}}{E_{\vec k}}\right), \qquad u_{\vec k}v_{\vec k} = \frac{\Delta}{2E_{\vec k}} . $$ The minimum energy to add a single excitation (a Bogoliubov quasiparticle) is $E_{\vec k} \geq \Delta$: there is an energy gap $\Delta$ between the paired ground state and the lowest excitation. Substituting $u_{\vec k}v_{\vec k}$ into the definition of $\Delta$ gives the **BCS gap equation**, $$ 1 = V\sum_{\vec k}\frac{1}{2E_{\vec k}} = N(0)V\int_0^{\hbar\omega_D}\frac{\d\xi}{\sqrt{\xi^2 + \Delta^2}} . $$ At $T = 0$ the integral is $\sinh^{-1}(\hbar\omega_D/\Delta)$, so $$ \Delta(0) = \frac{\hbar\omega_D}{\sinh\!\big(1/N(0)V\big)} \approx 2\hbar\omega_D\,e^{-1/N(0)V} \quad (N(0)V \ll 1) . $$ The gap equation is solved graphically by intersecting the decreasing curve $N(0)V\sinh^{-1}(\hbar\omega_D/\Delta)$ with the horizontal line at $1$; a stronger coupling $N(0)V$ raises the curve and pushes the intersection to a larger gap. $$ % caption: Graphical solution of the gap equation. The right-hand side falls with % increasing Delta; its intersection with the value 1 fixes Delta(0). A stronger % coupling (upper curve) gives a larger gap. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.2,0) node[below] {gap}; \draw[->, black] (0,0) -- (0,3.2) node[left] {gap-equation value}; % weaker coupling: k1 * asinh(w/Delta), w=5 \draw[very thick, domain=0.35:5.6, samples=90, variable=\x] plot ({\x},{0.62*ln(5/\x + sqrt((5/\x)*(5/\x)+1))}); % stronger coupling \draw[black, very thick, dashed, domain=0.35:5.6, samples=90, variable=\x] plot ({\x},{0.92*ln(5/\x + sqrt((5/\x)*(5/\x)+1))}); % line at 1 \draw[black, dashed] (0,1.5) -- (6.0,1.5); \node[anchor=east, font=\scriptsize] at (-0.05,1.5) {$1$}; % intersection markers \fill[acc] (2.35,1.5) circle (1.8pt); \draw[black, dashed] (2.35,0) -- (2.35,1.5); \node[anchor=north, acc, font=\scriptsize] at (2.35,0) {solution}; \node[anchor=west, font=\scriptsize] at (3.4,1.05) {weaker}; \node[black, anchor=west, font=\scriptsize] at (3.4,2.15) {stronger}; \end{tikzpicture} $$ ## Density of states and the gap in the spectrum The pairing removes all single-electron states within $\Delta$ of $E_F$ and piles them into sharp coherence peaks at the gap edges. Conserving states, $N_s(E)\,\d E = N(0)\,\d\xi$, gives the superconducting density of states $$ \frac{N_s(E)}{N(0)} = \begin{cases} \dfrac{|E|}{\sqrt{E^2 - \Delta^2}} & |E| > \Delta \\[4pt] 0 & |E| < \Delta , \end{cases} $$ with $E$ measured from $E_F$. The gap $2\Delta$ is directly visible in single-particle tunnelling, and the coherence peaks are its signature. $$ % caption: The BCS density of states. States within Delta of the Fermi level are % swept out, leaving a gap of width 2 Delta and divergent coherence peaks at the % two gap edges. The normal density of states N(0) is shown flat for comparison. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (-3.6,0) -- (3.6,0) node[below] {$E$}; \draw[->, black] (0,0) -- (0,3.2) node[left] {$N_s(E)$}; % normal DOS flat \draw[black, dashed] (-3.4,1.2) -- (3.4,1.2); \node[black, anchor=west, font=\scriptsize] at (2.4,1.35) {$N(0)$}; % right branch: E/sqrt(E^2 - 1), Delta=1 at x=1 \draw[acc, very thick, domain=1.06:3.3, samples=90, variable=\x] plot ({\x},{1.2*\x/sqrt(\x*\x - 1)}); % left branch mirror \draw[acc, very thick, domain=1.06:3.3, samples=90, variable=\x] plot ({-\x},{1.2*\x/sqrt(\x*\x - 1)}); % gap markers \draw[black, dashed] (1,0) -- (1,2.9); \draw[black, dashed] (-1,0) -- (-1,2.9); \draw[black, <->] (-1,0.4) -- (1,0.4); \node[anchor=south, font=\scriptsize] at (0,0.45) {gap}; \end{tikzpicture} $$ ## Critical temperature and the universal ratios At finite temperature the quasiparticle states are thermally populated with the Fermi function $f(E_{\vec k})$, which weakens the pairing. The factor $1 - 2f(E_{\vec k}) = \tanh(E_{\vec k}/2k_B T)$ enters the gap equation, $$ 1 = N(0)V\int_0^{\hbar\omega_D}\frac{\d\xi}{\sqrt{\xi^2 + \Delta^2}}\, \tanh\!\frac{\sqrt{\xi^2 + \Delta^2}}{2k_B T} . $$ The gap shrinks as $T$ rises and vanishes at $T_c$. Setting $\Delta = 0$ there, $$ 1 = N(0)V\int_0^{\hbar\omega_D}\frac{\d\xi}{\xi}\tanh\!\frac{\xi}{2k_B T_c} \quad\Longrightarrow\quad k_B T_c = \frac{2e^{\gamma}}{\pi}\,\hbar\omega_D\,e^{-1/N(0)V} \approx 1.13\,\hbar\omega_D\,e^{-1/N(0)V} , $$ with $\gamma = 0.577$ the Euler constant. Both $\Delta(0)$ and $k_B T_c$ carry the same exponential $e^{-1/N(0)V}$, so their ratio is a pure number independent of the material, $$ \frac{2\Delta(0)}{k_B T_c} = \frac{2\times 2}{1.13} = 3.53 . $$ The energy gap at zero temperature is $E_g(0) = 2\Delta(0) = 3.53\,k_B T_c$, matching the phenomenological value $E_g \approx 3.5\,k_B T_c$. Because $\omega_D \propto M^{-1/2}$, the theory reproduces the isotope effect $T_c \propto M^{-1/2}$ with exponent $\alpha = \tfrac{1}{2}$. > **Worked example.** Tin has $T_c = 3.72\ \text{K}$. The predicted zero-temperature > gap is > > $$ > 2\Delta(0) = 3.53\,k_B T_c = 3.53\,(8.62\times10^{-5}\ \text{eV/K})(3.72\ \text{K}) > = 1.13\times10^{-3}\ \text{eV} , > $$ > > so $\Delta(0) = 0.57\ \text{meV}$. Tunnelling measurements on tin give > $2\Delta(0) \approx 1.15\ \text{meV}$, a ratio of $3.6$, close to the BCS value. The gap closes with a square-root law near $T_c$, $\Delta(T) \approx 1.74\,\Delta(0) \sqrt{1 - T/T_c}$, and is nearly flat at low temperature. $$ % caption: Temperature dependence of the BCS gap. Delta(T) is flat at low % temperature and closes as the square root of (1 minus T over T_c) at the % transition. The universal ratio 2 Delta(0) = 3.53 k_B T_c fixes the vertical % scale. \begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize] \definecolor{acc}{HTML}{4A6FA5} \draw[->, black] (0,0) -- (6.4,0) node[below] {$T/T_c$}; \draw[->, black] (0,0) -- (0,3.4) node[left] {scaled gap}; % curve sqrt(1 - (T/Tc)^4): flat at low T, sqrt near Tc; Tc at x=5.4, top y=3 \draw[acc, very thick, domain=0:5.4, samples=140, variable=\x] plot ({\x},{3.0*sqrt(1 - (\x/5.4)^4)}); \draw[black, dashed] (5.4,0) -- (5.4,0.1); \node[anchor=north, font=\scriptsize] at (5.4,0) {$1$}; \draw[black, dashed] (0,3.0) -- (0.12,3.0); \node[anchor=east, font=\scriptsize] at (-0.05,3.0) {$1$}; \node[acc, anchor=west, font=\scriptsize] at (0.3,1.2) {universal ratio}; \end{tikzpicture} $$ ## Thermodynamics The gap in the excitation spectrum controls the thermal properties. At low temperature the quasiparticle population is set by the Boltzmann factor across the gap, so the electronic heat capacity and the thermal conductivity fall off exponentially, $$ C_s \sim e^{-\Delta(0)/k_B T} \quad (T \ll T_c) , $$ in contrast to the normal-state $\gamma T$. This exponential activation is direct evidence for a gap. At $T_c$ the heat capacity jumps, and BCS predicts the ratio of the jump to the normal electronic value as a second universal number, $$ \frac{\Delta C}{C_n}\bigg|_{T_c} = \frac{\Delta C}{\gamma T_c} = 1.43 , $$ independent of the material within weak coupling. Measured jumps in tin, aluminium, and other classic superconductors cluster near this value; deviations upward (lead, mercury) mark strong-coupling materials where $N(0)V$ is not small. | Quantity | BCS weak-coupling prediction | | --- | --- | | Gap ratio | $2\Delta(0)/k_B T_c = 3.53$ | | Critical temperature | $k_B T_c = 1.13\,\hbar\omega_D\,e^{-1/N(0)V}$ | | Isotope exponent | $\alpha = \tfrac{1}{2}$ | | Heat-capacity jump | $\Delta C/\gamma T_c = 1.43$ | | Low-$T$ heat capacity | $\propto e^{-\Delta(0)/k_B T}$ | BCS theory accounts for the gap, the exponential $T_c$, the isotope effect, and the universal ratios from one attractive interaction. It also justifies Ginzburg–Landau theory near $T_c$, fixing the pairing charge at $2e$ and giving $\alpha$ and $\beta$ in terms of $N(0)$ and $T_c$. What it does not explain are the cuprates, whose critical temperatures and pairing symmetry lie outside the phonon weak-coupling picture, taken up in the [final lesson](/condensed-matter/superconductivity/josephson-and-high-tc).