Chapter 3: Superconducting Materials

Learning Objectives

• Survey the major classes of superconducting materials
• Understand the evolution from low-Tc to high-Tc superconductors
• Compare properties of different superconductor families
• Learn about material selection criteria for applications
• Visualize superconductor properties using Python

3.1 Elemental Superconductors

Many pure elements become superconducting at low temperatures. Of the 118 known elements, about 30 are superconductors under normal pressure, and more become superconducting under high pressure.

Periodic Table of Superconducting Elements

import numpy as np
import matplotlib.pyplot as plt

# Elemental superconductors data
elements = {
    'Nb': {'Tc': 9.3, 'type': 'II', 'atomic_num': 41},
    'Tc': {'Tc': 7.8, 'type': 'II', 'atomic_num': 43},
    'Pb': {'Tc': 7.2, 'type': 'I', 'atomic_num': 82},
    'La': {'Tc': 6.0, 'type': 'II', 'atomic_num': 57},
    'V': {'Tc': 5.4, 'type': 'II', 'atomic_num': 23},
    'Ta': {'Tc': 4.5, 'type': 'II', 'atomic_num': 73},
    'Hg': {'Tc': 4.2, 'type': 'I', 'atomic_num': 80},
    'Sn': {'Tc': 3.7, 'type': 'I', 'atomic_num': 50},
    'In': {'Tc': 3.4, 'type': 'I', 'atomic_num': 49},
    'Tl': {'Tc': 2.4, 'type': 'I', 'atomic_num': 81},
    'Re': {'Tc': 1.7, 'type': 'II', 'atomic_num': 75},
    'Al': {'Tc': 1.2, 'type': 'I', 'atomic_num': 13},
    'Zn': {'Tc': 0.85, 'type': 'I', 'atomic_num': 30},
}

# Prepare data for plotting
names = list(elements.keys())
Tc_values = [elements[e]['Tc'] for e in names]
types = [elements[e]['type'] for e in names]
colors = ['blue' if t == 'I' else 'red' for t in types]

# Create bar chart
fig, ax = plt.subplots(figsize=(12, 6))
bars = ax.bar(names, Tc_values, color=colors, alpha=0.7, edgecolor='black')

# Add horizontal lines for reference
ax.axhline(y=4.2, color='green', linestyle='--', alpha=0.5, label='Liquid He (4.2 K)')

ax.set_xlabel('Element', fontsize=12)
ax.set_ylabel('Critical Temperature Tc (K)', fontsize=12)
ax.set_title('Critical Temperatures of Elemental Superconductors', fontsize=14)

# Create legend
from matplotlib.patches import Patch
legend_elements = [
    Patch(facecolor='blue', alpha=0.7, label='Type I'),
    Patch(facecolor='red', alpha=0.7, label='Type II'),
]
ax.legend(handles=legend_elements, fontsize=11)

plt.xticks(rotation=45)
plt.grid(True, alpha=0.3, axis='y')
plt.tight_layout()
plt.show()

3.2 Alloy and Compound Superconductors

A15 Compounds

The A15 compounds (also called β-tungsten structure) have the formula A₃B and dominated superconductor technology before high-Tc materials:

NbTi: The Workhorse Superconductor

Niobium-titanium alloy (NbTi) is the most widely used superconductor despite its modest Tc:

MgB₂: A Surprising Discovery

In 2001, magnesium diboride (MgB₂) was discovered to be superconducting at 39 K—remarkably high for a simple binary compound:

• Conventional BCS superconductor (phonon-mediated)
• Two-gap superconductor (unique physics)
• Inexpensive raw materials
• Can operate with cryocoolers (no liquid helium needed)

3.3 High-Temperature Cuprate Superconductors

The 1986 Revolution

In 1986, Bednorz and Müller at IBM Zurich discovered superconductivity at 35 K in La-Ba-Cu-O, a copper oxide ceramic. This was shocking because:

• Ceramics were considered insulators, not superconductors
• The Tc was far higher than any known superconductor
• It didn't fit BCS theory predictions

They won the Nobel Prize in 1987—one of the fastest recognitions in physics history.

Breaking the Liquid Nitrogen Barrier

In 1987, Wu, Chu, and colleagues discovered YBCO (YBa₂Cu₃O₇) with Tc = 93 K. This was revolutionary because:

Cuprate Family

Structure of Cuprates

All cuprate superconductors share a common structural feature: CuO₂ planes. Superconductivity occurs within these copper-oxygen layers.

import numpy as np
import matplotlib.pyplot as plt

# Timeline of Tc discoveries
discoveries = [
    ('Hg', 1911, 4.2, 'Element'),
    ('Pb', 1913, 7.2, 'Element'),
    ('Nb', 1930, 9.3, 'Element'),
    ('NbN', 1941, 16, 'Compound'),
    ('Nb₃Sn', 1954, 18.3, 'A15'),
    ('Nb₃Ge', 1973, 23.2, 'A15'),
    ('LBCO', 1986, 35, 'Cuprate'),
    ('YBCO', 1987, 93, 'Cuprate'),
    ('BSCCO', 1988, 110, 'Cuprate'),
    ('Tl-cuprate', 1988, 125, 'Cuprate'),
    ('Hg-cuprate', 1993, 133, 'Cuprate'),
    ('MgB₂', 2001, 39, 'Compound'),
    ('Fe-pnictide', 2008, 56, 'Iron-based'),
    ('H₃S (high P)', 2015, 203, 'Hydride'),
]

years = [d[1] for d in discoveries]
Tc = [d[2] for d in discoveries]
names = [d[0] for d in discoveries]
categories = [d[3] for d in discoveries]

# Color mapping
color_map = {
    'Element': 'blue',
    'Compound': 'green',
    'A15': 'orange',
    'Cuprate': 'red',
    'Iron-based': 'purple',
    'Hydride': 'magenta'
}
colors = [color_map[c] for c in categories]

fig, ax = plt.subplots(figsize=(14, 8))

# Plot points
for i, (year, tc, name, cat) in enumerate(zip(years, Tc, names, categories)):
    ax.scatter(year, tc, c=color_map[cat], s=100, zorder=5)
    offset = 5 if i % 2 == 0 else -15
    ax.annotate(name, (year, tc), textcoords="offset points",
                xytext=(5, offset), fontsize=9)

# Connect with lines
ax.plot(years, Tc, 'k-', alpha=0.3, linewidth=1)

# Add reference lines
ax.axhline(y=77, color='cyan', linestyle='--', alpha=0.7, linewidth=2,
          label='Liquid N₂ (77 K)')
ax.axhline(y=4.2, color='blue', linestyle='--', alpha=0.5, linewidth=2,
          label='Liquid He (4.2 K)')

# Highlight 1986 revolution
ax.axvspan(1986, 1995, alpha=0.1, color='red', label='High-Tc era')

ax.set_xlabel('Year', fontsize=12)
ax.set_ylabel('Critical Temperature Tc (K)', fontsize=12)
ax.set_title('Evolution of Superconductor Critical Temperatures', fontsize=14)
ax.set_xlim(1900, 2025)
ax.set_ylim(0, 220)

# Create legend for categories
from matplotlib.patches import Patch
legend_elements = [Patch(facecolor=c, label=cat) for cat, c in color_map.items()]
ax.legend(handles=legend_elements, loc='upper left', fontsize=10)

plt.grid(True, alpha=0.3)
plt.tight_layout()
plt.show()

3.4 Iron-Based Superconductors

In 2008, Hosono and colleagues in Japan discovered superconductivity in iron-arsenide compounds. This was surprising because iron is magnetic and usually suppresses superconductivity.

Iron Pnictides and Chalcogenides

• LaFeAsO₁₋ₓFₓ: Tc = 26 K (first discovery)
• SmFeAsO₁₋ₓFₓ: Tc = 55 K
• FeSe: Tc = 8 K (bulk), up to 65 K (thin film on SrTiO₃)

3.5 Comparing Superconductor Classes

import numpy as np
import matplotlib.pyplot as plt

# Comprehensive comparison data
materials = {
    # Low-Tc
    'NbTi': {'Tc': 10, 'Hc2': 15, 'Jc': 3000, 'class': 'Low-Tc', 'practical': True},
    'Nb₃Sn': {'Tc': 18, 'Hc2': 24, 'Jc': 2000, 'class': 'Low-Tc', 'practical': True},
    'MgB₂': {'Tc': 39, 'Hc2': 16, 'Jc': 1000, 'class': 'Low-Tc', 'practical': True},
    # High-Tc
    'YBCO': {'Tc': 93, 'Hc2': 100, 'Jc': 5000, 'class': 'High-Tc', 'practical': True},
    'BSCCO': {'Tc': 110, 'Hc2': 60, 'Jc': 500, 'class': 'High-Tc', 'practical': True},
    'Hg-1223': {'Tc': 133, 'Hc2': 100, 'Jc': 100, 'class': 'High-Tc', 'practical': False},
    # Iron-based
    'Fe-pnictide': {'Tc': 55, 'Hc2': 80, 'Jc': 500, 'class': 'Fe-based', 'practical': False},
}

fig, axes = plt.subplots(1, 3, figsize=(15, 5))

# Color by class
class_colors = {'Low-Tc': 'blue', 'High-Tc': 'red', 'Fe-based': 'green'}

# Plot 1: Tc comparison
ax1 = axes[0]
names = list(materials.keys())
Tc_vals = [materials[m]['Tc'] for m in names]
colors = [class_colors[materials[m]['class']] for m in names]
bars = ax1.barh(names, Tc_vals, color=colors, alpha=0.7)
ax1.axvline(x=77, color='cyan', linestyle='--', label='LN₂ (77K)')
ax1.set_xlabel('Tc (K)', fontsize=12)
ax1.set_title('Critical Temperature', fontsize=14)
ax1.legend()

# Plot 2: Hc2 comparison
ax2 = axes[1]
Hc2_vals = [materials[m]['Hc2'] for m in names]
bars = ax2.barh(names, Hc2_vals, color=colors, alpha=0.7)
ax2.set_xlabel('Hc2 (T)', fontsize=12)
ax2.set_title('Upper Critical Field', fontsize=14)

# Plot 3: Practical status
ax3 = axes[2]
practical = ['Yes' if materials[m]['practical'] else 'No' for m in names]
practical_colors = ['green' if p == 'Yes' else 'gray' for p in practical]
ax3.barh(names, [1]*len(names), color=practical_colors, alpha=0.7)
ax3.set_xlabel('', fontsize=12)
ax3.set_title('Practical Applications', fontsize=14)
ax3.set_xlim(0, 1.5)
ax3.set_xticks([])
for i, (name, prac) in enumerate(zip(names, practical)):
    ax3.text(0.5, i, prac, ha='center', va='center', fontsize=11, fontweight='bold')

# Add legend
from matplotlib.patches import Patch
legend_elements = [Patch(facecolor=c, label=cat, alpha=0.7)
                   for cat, c in class_colors.items()]
fig.legend(handles=legend_elements, loc='upper center', ncol=3,
          bbox_to_anchor=(0.5, 1.02), fontsize=11)

plt.tight_layout()
plt.subplots_adjust(top=0.88)
plt.show()

3.6 Material Selection for Applications

Summary

Practice Problems