Beyond BCS: d-wave, Multiband, Topological, and Frontier Research
While BCS theory brilliantly explains conventional superconductors like aluminum and lead, many materials discovered since the 1980s exhibit superconductivity that cannot be fully described within the BCS framework. These unconventional superconductors challenge our understanding and point toward new physics, exotic pairing mechanisms, and potential applications ranging from quantum computing to room-temperature superconductivity.
This chapter explores the rich landscape of unconventional superconductivity, from the d-wave pairing in cuprates to topological superconductors hosting Majorana fermions, and concludes with current research frontiers that may revolutionize the field.
A superconductor is considered unconventional if it violates one or more key assumptions of BCS theory:
In conventional superconductors, phonons mediate Cooper pairing. Unconventional superconductors may involve:
The pairing interaction $V(\mathbf{k}, \mathbf{k}')$ is no longer isotropic and phonon-mediated, leading to momentum-dependent gap structures.
BCS predicts s-wave pairing with angular momentum $L = 0$ and a momentum-independent gap $\Delta(\mathbf{k}) = \Delta_0$. Unconventional superconductors can have higher angular momentum:
For non-s-wave symmetry, the gap function $\Delta(\mathbf{k})$ vanishes along certain directions or points on the Fermi surface, creating nodes. These nodes lead to:
| Property | s-wave (nodeless) | d-wave (nodal) |
|---|---|---|
| Low-T specific heat | $C_v \propto e^{-\Delta_0/k_B T}$ | $C_v \propto T^2$ |
| Penetration depth | $\lambda(T) - \lambda(0) \propto e^{-\Delta_0/k_B T}$ | $\Delta\lambda \propto T$ |
| Thermal conductivity | Exponentially small | Finite at $T \to 0$ |
| NMR relaxation | $T_1^{-1} \propto e^{-\Delta_0/k_B T}$ | $T_1^{-1} \propto T^3$ |
The high-temperature superconducting cuprates (discovered 1986, $T_c$ up to 133 K at ambient pressure) are the most prominent examples of d-wave superconductivity.
The gap symmetry for cuprates follows the $d_{x^2-y^2}$ representation:
where $a$ is the lattice constant. This function has key properties:
In polar coordinates on a circular Fermi surface, the d-wave gap can be written as: $$ \Delta(\theta) = \Delta_0 \cos(2\theta) $$ where $\theta$ is the angle from the $k_x$ axis. The gap changes sign four times as one traverses the Fermi surface.
Multiple experimental techniques confirm d-wave symmetry in cuprates:
ARPES directly measures the electronic band structure and gap as a function of momentum $\mathbf{k}$. Experiments on Bi₂Sr₂CaCu₂O₈ clearly show:
Since the gap changes sign, Josephson junctions formed at different crystal orientations show distinctive phase relationships. The famous tricrystal experiment (Tsuei and Kirtley, 2000) measured the phase around a loop with three grain boundaries and observed a spontaneous half-flux quantum, proving the sign change.
The power-law behavior predicted for nodal superconductors is observed:
Philip Anderson proposed the Resonating Valence Bond (RVB) theory to explain high-$T_c$ superconductivity. The key ideas:
While RVB captures important physics, a complete microscopic theory of cuprate superconductivity remains an active research area.
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
# d-wave gap on a 2D Fermi surface
def d_wave_gap(kx, ky, Delta0=1.0, a=1.0):
"""
d-wave gap function: Delta(k) = Delta0/2 * (cos(kx*a) - cos(ky*a))
"""
return Delta0 / 2 * (np.cos(kx * a) - np.cos(ky * a))
# Create k-space grid
k = np.linspace(-np.pi, np.pi, 200)
KX, KY = np.meshgrid(k, k)
Delta_k = d_wave_gap(KX, KY)
# Plot 1: Heatmap of gap function
fig, axes = plt.subplots(1, 3, figsize=(15, 4))
# Heatmap
im1 = axes[0].contourf(KX, KY, Delta_k, levels=50, cmap='RdBu_r')
axes[0].contour(KX, KY, Delta_k, levels=[0], colors='black', linewidths=2)
axes[0].set_xlabel(r'$k_x a$', fontsize=12)
axes[0].set_ylabel(r'$k_y a$', fontsize=12)
axes[0].set_title('d-wave Gap $\Delta(k)$', fontsize=13)
axes[0].set_aspect('equal')
plt.colorbar(im1, ax=axes[0], label=r'$\Delta(k)/\Delta_0$')
# Angular dependence on circular Fermi surface
theta = np.linspace(0, 2*np.pi, 300)
Delta_theta = np.cos(2 * theta)
axes[1].plot(theta, Delta_theta, 'b-', linewidth=2)
axes[1].axhline(0, color='black', linestyle='--', linewidth=1)
axes[1].fill_between(theta, 0, Delta_theta, where=(Delta_theta > 0),
color='red', alpha=0.3, label='Positive')
axes[1].fill_between(theta, 0, Delta_theta, where=(Delta_theta < 0),
color='blue', alpha=0.3, label='Negative')
axes[1].set_xlabel(r'Angle $\theta$ (rad)', fontsize=12)
axes[1].set_ylabel(r'$\Delta(\theta)/\Delta_0$', fontsize=12)
axes[1].set_title(r'Angular Dependence: $\cos(2\theta)$', fontsize=13)
axes[1].legend()
axes[1].grid(True, alpha=0.3)
# Polar plot
ax_polar = plt.subplot(133, projection='polar')
r = np.abs(Delta_theta)
colors = np.where(Delta_theta > 0, 'red', 'blue')
for i in range(len(theta)-1):
ax_polar.plot([theta[i], theta[i+1]], [r[i], r[i+1]],
color=colors[i], linewidth=2)
ax_polar.set_title('Polar Representation\n(Red: +, Blue: -)', fontsize=13, pad=20)
plt.tight_layout()
plt.savefig('d_wave_gap_cuprate.png', dpi=150, bbox_inches='tight')
plt.show()
print("d-wave gap visualization:")
print(f" - Gap maximum: ±{1.0:.2f} Δ₀")
print(f" - Nodal lines: kₓ = ±kᵧ")
print(f" - Four-fold sign change around Fermi surface")
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import quad
def dos_d_wave(E, Delta0=1.0, broadening=0.05):
"""
Density of states for d-wave superconductor using angle-averaged BCS DOS.
For d-wave, Delta(theta) = Delta0 * cos(2*theta).
N(E) = (1/2π) ∫ dθ N_BCS(E, Delta(θ))
where N_BCS(E, Delta) = |E| / sqrt(E^2 - Delta^2) for |E| > Delta, else 0
"""
def integrand(theta):
Delta_theta = Delta0 * np.abs(np.cos(2 * theta))
if abs(E) > Delta_theta:
# Add small broadening to avoid singularities
return abs(E) / np.sqrt(E**2 - Delta_theta**2 + broadening**2)
else:
return 0.0
result, _ = quad(integrand, 0, 2*np.pi)
return result / (2 * np.pi)
# Calculate DOS for range of energies
E_range = np.linspace(-2.5, 2.5, 500)
dos_normal = np.ones_like(E_range) # Normalized to 1
dos_s_wave = []
dos_d_wave = []
Delta0 = 1.0
for E in E_range:
# s-wave DOS
if abs(E) > Delta0:
dos_s = abs(E) / np.sqrt(E**2 - Delta0**2)
else:
dos_s = 0.0
dos_s_wave.append(dos_s)
# d-wave DOS (angle-averaged)
dos_d = dos_d_wave(E, Delta0)
dos_d_wave.append(dos_d)
dos_s_wave = np.array(dos_s_wave)
dos_d_wave = np.array(dos_d_wave)
# Plot comparison
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
# Full DOS comparison
axes[0].plot(E_range, dos_normal, 'k--', linewidth=2, label='Normal metal')
axes[0].plot(E_range, dos_s_wave, 'b-', linewidth=2, label='s-wave')
axes[0].plot(E_range, dos_d_wave, 'r-', linewidth=2, label='d-wave')
axes[0].axvline(Delta0, color='gray', linestyle=':', alpha=0.5)
axes[0].axvline(-Delta0, color='gray', linestyle=':', alpha=0.5)
axes[0].set_xlabel(r'Energy $E/\Delta_0$', fontsize=12)
axes[0].set_ylabel(r'DOS $N(E)/N(0)$', fontsize=12)
axes[0].set_title('Density of States Comparison', fontsize=13)
axes[0].set_ylim([0, 3.5])
axes[0].legend(fontsize=11)
axes[0].grid(True, alpha=0.3)
# Low-energy region (zoom)
E_low = np.linspace(-0.5, 0.5, 200)
dos_d_low = [dos_d_wave(E, Delta0) for E in E_low]
axes[1].plot(E_low, dos_d_low, 'r-', linewidth=2)
axes[1].axvline(0, color='black', linestyle='--', linewidth=1)
axes[1].set_xlabel(r'Energy $E/\Delta_0$', fontsize=12)
axes[1].set_ylabel(r'DOS $N(E)/N(0)$', fontsize=12)
axes[1].set_title('d-wave: Finite DOS at Fermi Level', fontsize=13)
axes[1].grid(True, alpha=0.3)
axes[1].text(0.25, 0.5, 'V-shaped\nlow-E DOS', fontsize=10,
bbox=dict(boxstyle='round', facecolor='wheat', alpha=0.5))
plt.tight_layout()
plt.savefig('d_wave_dos.png', dpi=150, bbox_inches='tight')
plt.show()
print("\nKey differences:")
print("s-wave: Hard gap, N(E) = 0 for |E| < Δ₀")
print("d-wave: V-shaped DOS, N(E) ~ |E| near E = 0 (gapless)")
print(" => Power-law thermodynamics at low T")
Discovered in 2001, magnesium diboride (MgB₂) has $T_c = 39$ K, the highest among conventional phonon-mediated superconductors. It exhibits multiband superconductivity with two distinct gaps.
MgB₂ has a layered hexagonal structure with boron forming honeycomb sheets. The Fermi surface consists of:
These bands have different:
MgB₂ exhibits two superconducting gaps:
The ratio $\Delta_\sigma / \Delta_\pi \approx 2.8$ is confirmed by:
The two bands are coupled through interband scattering and pairing interactions. The gaps are determined by the coupled Eliashberg equations:
where $\lambda_{ij}$ are the intra- and inter-band coupling constants. The interband coupling $\lambda_{\sigma\pi}$ is crucial: it raises $T_c$ beyond what either band would have alone and causes the gaps to merge at $T_c$.
import numpy as np
import matplotlib.pyplot as plt
def two_gap_temperature_evolution(T_array, Tc=39.0,
Delta_sigma_0=7.0, Delta_pi_0=2.5):
"""
Simple model for temperature evolution of two gaps using
BCS-like temperature dependence with same Tc.
Delta(T) = Delta(0) * tanh(1.74 * sqrt((Tc/T) - 1))
"""
Delta_sigma = []
Delta_pi = []
for T in T_array:
if T >= Tc:
Delta_sigma.append(0.0)
Delta_pi.append(0.0)
else:
# BCS-like temperature dependence
t = T / Tc
factor = np.tanh(1.74 * np.sqrt(1/t - 1))
Delta_sigma.append(Delta_sigma_0 * factor)
Delta_pi.append(Delta_pi_0 * factor)
return np.array(Delta_sigma), np.array(Delta_pi)
def two_gap_dos(E, Delta_sigma, Delta_pi, w_sigma=0.6, w_pi=0.4):
"""
Density of states with two gaps.
Weighted sum of two BCS DOS functions.
"""
dos = 0.0
# σ-band contribution
if abs(E) > Delta_sigma:
dos += w_sigma * abs(E) / np.sqrt(E**2 - Delta_sigma**2)
# π-band contribution
if abs(E) > Delta_pi:
dos += w_pi * abs(E) / np.sqrt(E**2 - Delta_pi**2)
return dos
# Temperature evolution
T_array = np.linspace(0, 45, 200)
Delta_sigma, Delta_pi = two_gap_temperature_evolution(T_array)
# DOS at T=0
E_range = np.linspace(-12, 12, 500)
dos = [two_gap_dos(E, 7.0, 2.5) for E in E_range]
# Plotting
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
# Temperature dependence
axes[0].plot(T_array, Delta_sigma, 'b-', linewidth=2.5, label=r'$\Delta_\sigma$ (σ-band)')
axes[0].plot(T_array, Delta_pi, 'r-', linewidth=2.5, label=r'$\Delta_\pi$ (π-band)')
axes[0].axvline(39, color='gray', linestyle='--', linewidth=1, alpha=0.7, label=r'$T_c = 39$ K')
axes[0].set_xlabel('Temperature (K)', fontsize=12)
axes[0].set_ylabel('Gap (meV)', fontsize=12)
axes[0].set_title('Two-Gap Temperature Evolution in MgB₂', fontsize=13)
axes[0].legend(fontsize=11)
axes[0].grid(True, alpha=0.3)
axes[0].set_xlim([0, 45])
axes[0].set_ylim([0, 8])
# DOS
axes[1].plot(E_range, dos, 'g-', linewidth=2)
axes[1].axvline(7.0, color='blue', linestyle='--', alpha=0.7, label=r'$\Delta_\sigma = 7.0$ meV')
axes[1].axvline(-7.0, color='blue', linestyle='--', alpha=0.7)
axes[1].axvline(2.5, color='red', linestyle='--', alpha=0.7, label=r'$\Delta_\pi = 2.5$ meV')
axes[1].axvline(-2.5, color='red', linestyle='--', alpha=0.7)
axes[1].set_xlabel('Energy (meV)', fontsize=12)
axes[1].set_ylabel('DOS (arb. units)', fontsize=12)
axes[1].set_title('Two-Gap Density of States at T=0', fontsize=13)
axes[1].legend(fontsize=11)
axes[1].grid(True, alpha=0.3)
axes[1].set_ylim([0, 4])
plt.tight_layout()
plt.savefig('mgb2_two_gap.png', dpi=150, bbox_inches='tight')
plt.show()
print("MgB₂ Two-Gap Structure:")
print(f" σ-band gap: Δ_σ(0) = {7.0} meV")
print(f" π-band gap: Δ_π(0) = {2.5} meV")
print(f" Gap ratio: Δ_σ/Δ_π = {7.0/2.5:.2f}")
print(f" Both gaps vanish at Tc = 39 K (interband coupling)")
Discovered in 2008, iron-based superconductors (FeSC) form the second family of high-$T_c$ materials after cuprates, with $T_c$ up to 55 K. They exhibit exotic pairing symmetry and compete with magnetic order.
FeSC have layered structures with FeAs or FeSe layers:
The Fe atoms form a square lattice (rotated 45° from the unit cell) with As or Se above/below.
Unlike cuprates (one large Fermi surface), FeSC have multiple disconnected Fermi surfaces:
The nesting between hole and electron pockets at wavevector $\mathbf{Q} = (\pi, 0)$ is crucial for both magnetism and superconductivity.
The most widely accepted pairing state is s± symmetry:
This sign change is crucial:
The parent compounds are antiferromagnetic metals with stripe-type order. Superconductivity emerges upon:
The phase diagram shows an interplay between magnetism and superconductivity, with SC often appearing where magnetic order is suppressed. This suggests that spin fluctuations (remnants of magnetic order) mediate Cooper pairing.
The effective pairing interaction mediated by spin fluctuations is: $$ V(\mathbf{q}) \propto -\frac{\chi(\mathbf{q})}{\chi_0} $$ where $\chi(\mathbf{q})$ is the magnetic susceptibility, peaked at $\mathbf{Q} = (\pi, 0)$. This interaction is repulsive at $\mathbf{Q}$, favoring sign changes in $\Delta(\mathbf{k})$ between pockets connected by $\mathbf{Q}$.
Topological superconductors are a class of materials where the superconducting state has nontrivial topological properties, leading to exotic boundary excitations protected by topology.
The fundamental principle: A bulk topological invariant (e.g., Chern number, $\mathbb{Z}_2$ invariant) dictates the existence of gapless boundary states. For topological superconductors:
These boundary states are topologically protected: they cannot be removed by local perturbations that preserve symmetries.
Majorana fermions are exotic quasiparticles that are their own antiparticles:
In condensed matter, Majorana zero modes can appear at boundaries (edges, vortex cores) of topological superconductors. Key properties:
The simplest model of a 1D topological superconductor is the Kitaev chain:
where:
The model has two phases:
The topological invariant is a $\mathbb{Z}_2$ winding number that can be computed from the Hamiltonian in momentum space.
Realizing topological superconductivity in real materials is challenging. Candidates include:
Strontium ruthenate is a leading candidate for intrinsic p-wave (spin-triplet) superconductivity:
More common approach: Induce superconductivity in a topological material:
import numpy as np
import matplotlib.pyplot as plt
def kitaev_chain_hamiltonian(N, mu, t, Delta):
"""
Construct the Bogoliubov-de Gennes Hamiltonian for the Kitaev chain.
H = sum_k Psi^dagger(k) H_BdG(k) Psi(k)
In real space for open boundary conditions (to see edge states).
"""
# BdG Hamiltonian is 2N x 2N (particle and hole sectors)
H = np.zeros((2*N, 2*N), dtype=complex)
for j in range(N):
# Particle sector: -mu and hopping -t
H[j, j] = -mu
if j < N-1:
H[j, j+1] = -t
H[j+1, j] = -t
# Hole sector: +mu and hopping +t
H[N+j, N+j] = mu
if j < N-1:
H[N+j, N+j+1] = t
H[N+j+1, N+j] = t
# Pairing: Delta * c_j c_{j+1}
if j < N-1:
H[j, N+j+1] = -Delta
H[N+j+1, j] = -Delta
H[j+1, N+j] = Delta
H[N+j, j+1] = Delta
return H
def compute_spectrum_vs_mu(N=50, t=1.0, Delta=1.0, mu_range=None):
"""
Compute energy spectrum as a function of chemical potential mu.
"""
if mu_range is None:
mu_range = np.linspace(-3*t, 3*t, 100)
energies = []
for mu in mu_range:
H = kitaev_chain_hamiltonian(N, mu, t, Delta)
eigvals = np.linalg.eigvalsh(H)
energies.append(eigvals)
return mu_range, np.array(energies)
# Compute spectrum
N = 50
t = 1.0
Delta = 1.0
mu_range, energies = compute_spectrum_vs_mu(N, t, Delta)
# Plot
fig, axes = plt.subplots(1, 2, figsize=(14, 5))
# Full spectrum
for i in range(energies.shape[1]):
axes[0].plot(mu_range, energies[:, i], 'b-', linewidth=0.5, alpha=0.6)
axes[0].axhline(0, color='red', linestyle='--', linewidth=2, label='E = 0 (Majorana)')
axes[0].axvline(-2*t, color='green', linestyle='--', linewidth=1.5, alpha=0.7, label=r'$\mu = -2t$ (phase boundary)')
axes[0].axvline(2*t, color='green', linestyle='--', linewidth=1.5, alpha=0.7)
axes[0].fill_betweenx([-4, 4], -2*t, 2*t, color='yellow', alpha=0.2, label='Topological phase')
axes[0].set_xlabel(r'Chemical potential $\mu/t$', fontsize=12)
axes[0].set_ylabel('Energy', fontsize=12)
axes[0].set_title('Kitaev Chain: Energy Spectrum vs. $\mu$', fontsize=13)
axes[0].set_ylim([-3, 3])
axes[0].legend(fontsize=10)
axes[0].grid(True, alpha=0.3)
# Zoom near E=0
axes[1].plot(mu_range, energies[:, N-1], 'r-', linewidth=2, label='Lowest positive E')
axes[1].plot(mu_range, energies[:, N], 'b-', linewidth=2, label='Highest negative E')
axes[1].axhline(0, color='black', linestyle='--', linewidth=1)
axes[1].axvline(-2*t, color='green', linestyle='--', linewidth=1.5, alpha=0.7)
axes[1].axvline(2*t, color='green', linestyle='--', linewidth=1.5, alpha=0.7)
axes[1].fill_betweenx([-0.5, 0.5], -2*t, 2*t, color='yellow', alpha=0.2)
axes[1].set_xlabel(r'Chemical potential $\mu/t$', fontsize=12)
axes[1].set_ylabel('Energy (near E=0)', fontsize=12)
axes[1].set_title('Zero-Energy Majorana Modes', fontsize=13)
axes[1].set_ylim([-0.3, 0.3])
axes[1].legend(fontsize=10)
axes[1].grid(True, alpha=0.3)
plt.tight_layout()
plt.savefig('kitaev_chain_spectrum.png', dpi=150, bbox_inches='tight')
plt.show()
print("Kitaev Chain Phase Diagram:")
print(f" Trivial phase: |μ| > 2t = {2*t}")
print(f" Topological phase: |μ| < 2t (Majorana zero modes at edges)")
print(f" Phase transition at μ = ±2t (bulk gap closes)")
The most exciting aspect of Majorana fermions is their non-Abelian statistics. When multiple Majoranas are present (e.g., in vortex cores of a 2D topological superconductor), exchanging (braiding) them implements unitary transformations on the ground state manifold.
For $2n$ Majoranas, the ground state is $2^{n-1}$-fold degenerate (exponentially large). Braiding operations:
This property makes Majorana systems ideal for topological quantum computation:
However, full universality requires additional non-topological gates.
Detecting Majorana zero modes experimentally is challenging but achievable:
Zero-bias peaks can arise from non-topological effects (disorder, Kondo resonances). Confirming true Majorana modes requires multiple complementary tests: quantized conductance, robustness to parameter variations, and ultimately braiding statistics.
The field of superconductivity continues to surprise with new discoveries and theoretical insights. Here are some of the most exciting frontiers:
The holy grail of superconductivity research. Recent progress:
These materials are conventional (phonon-mediated) but with extremely high phonon frequencies due to light hydrogen atoms. The challenge: achieving high $T_c$ at ambient pressure.
When two graphene layers are stacked with a small twist angle (~1.1°, the "magic angle"), they exhibit:
This system is a tunable platform for studying unconventional superconductivity and correlation physics, with ongoing debates about the pairing mechanism.
Infinite-layer nickelates (e.g., NdNiO₂) discovered in 2019:
AI/ML methods are accelerating discovery:
Example success: Prediction of new superconducting hydrides before experimental synthesis.
import numpy as np
import matplotlib.pyplot as plt
def s_wave_gap(theta):
"""s-wave: isotropic"""
return np.ones_like(theta)
def d_wave_gap(theta):
"""d-wave: cos(2θ)"""
return np.cos(2 * theta)
def p_wave_gap(theta):
"""p-wave: |cos(θ) + i sin(θ)| = 1 (magnitude)"""
# For plotting, show real part: cos(θ)
return np.cos(theta)
def s_plus_minus(theta, n_pockets=2):
"""
s± symmetry (simplified):
Sign change between pockets but s-wave on each pocket.
Represent as piecewise constant.
"""
gap = np.ones_like(theta)
# Sign change at theta = π/2, 3π/2
gap[(theta > np.pi/4) & (theta < 3*np.pi/4)] = -1
gap[(theta > 5*np.pi/4) & (theta < 7*np.pi/4)] = -1
return gap
# Angular grid
theta = np.linspace(0, 2*np.pi, 500)
# Calculate gaps
s_gap = s_wave_gap(theta)
d_gap = d_wave_gap(theta)
p_gap = p_wave_gap(theta)
s_pm_gap = s_plus_minus(theta)
# Plotting
fig, axes = plt.subplots(2, 2, figsize=(14, 12))
symmetries = [
('s-wave (Conventional)', s_gap, 'blue'),
(r'd-wave ($d_{x^2-y^2}$, Cuprates)', d_gap, 'red'),
('p-wave (Topological)', p_gap, 'green'),
(r's$\pm$ (Iron-based)', s_pm_gap, 'purple')
]
for idx, (title, gap, color) in enumerate(symmetries):
ax = axes[idx // 2, idx % 2]
# Polar plot
ax_polar = plt.subplot(2, 2, idx+1, projection='polar')
# Plot gap magnitude
r = np.abs(gap)
colors_sign = np.where(gap > 0, color, 'black')
for i in range(len(theta)-1):
alpha = 0.8 if gap[i] > 0 else 0.5
ax_polar.plot([theta[i], theta[i+1]], [r[i], r[i+1]],
color=colors_sign[i], linewidth=2, alpha=alpha)
ax_polar.set_title(title, fontsize=12, pad=20)
ax_polar.set_ylim([0, 1.2])
# Add legend for sign
from matplotlib.patches import Patch
legend_elements = [
Patch(facecolor=color, alpha=0.8, label='Positive'),
Patch(facecolor='black', alpha=0.5, label='Negative')
]
if idx == 0: # s-wave has no sign change
legend_elements = [Patch(facecolor=color, alpha=0.8, label='Isotropic')]
ax_polar.legend(handles=legend_elements, loc='upper right', fontsize=9)
plt.tight_layout()
plt.savefig('gap_symmetry_comparison.png', dpi=150, bbox_inches='tight')
plt.show()
# Print summary
print("\n=== Gap Symmetry Summary ===")
print("\ns-wave:")
print(" - Angular momentum: L = 0")
print(" - Gap: Δ(θ) = Δ₀ (isotropic)")
print(" - Materials: Al, Pb, Nb, MgB₂")
print("\nd-wave:")
print(" - Angular momentum: L = 2")
print(" - Gap: Δ(θ) = Δ₀ cos(2θ)")
print(" - Nodal lines at θ = π/4, 3π/4, 5π/4, 7π/4")
print(" - Materials: Cuprates (YBCO, BSCCO)")
print("\np-wave:")
print(" - Angular momentum: L = 1")
print(" - Gap: Δ(θ) = Δ₀(cos θ + i sin θ)")
print(" - Spin-triplet pairing")
print(" - Materials: ³He, Sr₂RuO₄ (candidate)")
print("\ns±:")
print(" - Angular momentum: L = 0 on each pocket")
print(" - Sign change between different Fermi surfaces")
print(" - Materials: Iron-based superconductors")
Unconventional superconductivity represents some of the most exciting and challenging physics in condensed matter. From the d-wave cuprates that revolutionized the field in the 1980s, to multiband systems like MgB₂, to the exotic physics of topological superconductors and Majorana fermions, these materials push the boundaries of our theoretical understanding and experimental capabilities.
The quest for room-temperature superconductivity continues, with hydride superconductors under extreme pressure reaching ever-higher critical temperatures. Meanwhile, twisted bilayer graphene and nickelates open new avenues for exploring strong correlations and unconventional pairing.
Perhaps most tantalizing is the promise of topological quantum computation using Majorana modes—a marriage of fundamental physics and transformative technology. As we develop better materials, theory, and experimental techniques, the next breakthrough in superconductivity may be just around the corner.