Chapter 1: What is Superconductivity?

Learning Objectives

• Understand the historical context of superconductivity discovery
• Explain the phenomenon of zero electrical resistance
• Describe the Meissner effect and its significance
• Distinguish between Type I and Type II superconductors
• Visualize superconducting behavior using Python

1.1 The Discovery of Superconductivity

The Race to Absolute Zero

In the early 20th century, physicists were racing to reach ever lower temperatures. In 1908, Dutch physicist Heike Kamerlingh Onnes at Leiden University became the first to liquefy helium, reaching temperatures below 4.2 K (-269°C). This breakthrough opened the door to studying matter at temperatures never before achieved in a laboratory.

Onnes was particularly interested in how electrical resistance behaves as temperature approaches absolute zero. The prevailing theories at the time were divided:

• Lord Kelvin's prediction: Electrons would freeze in place, causing resistance to become infinite
• Other theories: Resistance would decrease smoothly to zero as thermal vibrations diminished

The Unexpected Discovery (1911)

On April 8, 1911, Onnes and his team were measuring the resistance of mercury as they cooled it with liquid helium. What they observed was completely unexpected:

This was the discovery of superconductivity—a state where a material conducts electricity with absolutely zero resistance. Onnes received the Nobel Prize in Physics in 1913 for this groundbreaking work.

Visualizing the Discovery

Let's recreate Onnes' historic observation using Python:

import numpy as np
import matplotlib.pyplot as plt

# Temperature range (Kelvin)
T = np.linspace(0, 10, 1000)

# Critical temperature of mercury
Tc = 4.2  # K

# Normal state resistance (simplified model)
# R(T) = R0 * (1 + alpha * T) for T > Tc
R0 = 1.0  # Normalized resistance
alpha = 0.01

# Resistance behavior
R = np.where(T > Tc, R0 * (1 + alpha * (T - Tc)), 0)

# Plotting
plt.figure(figsize=(10, 6))
plt.plot(T, R, 'b-', linewidth=2)
plt.axvline(x=Tc, color='r', linestyle='--', label=f'Tc = {Tc} K')
plt.fill_between(T, 0, R, where=(T <= Tc), alpha=0.3, color='cyan',
                  label='Superconducting state')

plt.xlabel('Temperature (K)', fontsize=12)
plt.ylabel('Resistance (normalized)', fontsize=12)
plt.title('Superconducting Transition in Mercury (Hg)', fontsize=14)
plt.legend(fontsize=11)
plt.grid(True, alpha=0.3)
plt.xlim(0, 10)
plt.ylim(-0.05, 1.2)

# Add annotation
plt.annotate('Abrupt drop\nto zero!', xy=(Tc, 0.02), xytext=(5.5, 0.3),
            fontsize=11, arrowprops=dict(arrowstyle='->', color='red'))

plt.tight_layout()
plt.show()

1.2 Zero Electrical Resistance

What Does "Zero Resistance" Really Mean?

In a normal conductor like copper, electrical resistance is caused by electrons colliding with lattice vibrations (phonons), impurities, and defects. Even at very low temperatures, some resistance always remains.

In a superconductor, something fundamentally different happens:

Persistent Currents

One of the most dramatic demonstrations of zero resistance is the persistent current experiment:

1. Create a closed loop of superconducting wire
2. Induce a current in the loop
3. Remove the power source
4. The current continues to flow indefinitely!

Experiments have measured persistent currents that show no decay over periods of several years. This is only possible if the resistance is truly zero.

Ohm's Law in Superconductors

Recall Ohm's Law: V = I × R

If R = 0 (superconductor), then:

• V = 0 regardless of the current I
• No voltage drop across the superconductor
• No energy dissipation (no heat generated)

import numpy as np
import matplotlib.pyplot as plt

# Compare voltage drop in normal vs superconductor
current = np.linspace(0, 10, 100)  # Amperes

# Normal conductor (copper wire)
R_normal = 0.1  # Ohms
V_normal = current * R_normal

# Superconductor
R_super = 0
V_super = current * R_super

# Power dissipation
P_normal = current**2 * R_normal
P_super = current**2 * R_super

fig, axes = plt.subplots(1, 2, figsize=(12, 5))

# Voltage comparison
axes[0].plot(current, V_normal, 'r-', linewidth=2, label='Normal conductor (R=0.1Ω)')
axes[0].plot(current, V_super, 'b-', linewidth=2, label='Superconductor (R=0)')
axes[0].set_xlabel('Current (A)', fontsize=12)
axes[0].set_ylabel('Voltage Drop (V)', fontsize=12)
axes[0].set_title('Voltage Drop: V = IR', fontsize=14)
axes[0].legend(fontsize=11)
axes[0].grid(True, alpha=0.3)

# Power dissipation
axes[1].plot(current, P_normal, 'r-', linewidth=2, label='Normal conductor')
axes[1].plot(current, P_super, 'b-', linewidth=2, label='Superconductor')
axes[1].set_xlabel('Current (A)', fontsize=12)
axes[1].set_ylabel('Power Loss (W)', fontsize=12)
axes[1].set_title('Power Dissipation: P = I²R', fontsize=14)
axes[1].legend(fontsize=11)
axes[1].grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

print("At 10 A current:")
print(f"Normal conductor power loss: {10**2 * R_normal:.1f} W")
print(f"Superconductor power loss: {10**2 * R_super:.1f} W")

1.3 The Meissner Effect

More Than Just Zero Resistance

In 1933, German physicists Walther Meissner and Robert Ochsenfeld discovered that superconductivity involves more than just zero resistance. They found that superconductors actively expel magnetic fields from their interior.

Why is the Meissner Effect Important?

A "perfect conductor" (hypothetical material with zero resistance but not a superconductor) would behave differently:

The Meissner effect proves that superconductivity is a true thermodynamic phase, not just a state of zero resistance.

Magnetic Levitation

The Meissner effect enables one of the most visually striking demonstrations of superconductivity: magnetic levitation. A magnet placed above a superconductor will float in mid-air because the superconductor expels the magnetic field, creating a repulsive force.

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Circle, Rectangle, FancyArrowPatch

fig, axes = plt.subplots(1, 2, figsize=(14, 6))

# Left plot: Normal conductor (field penetrates)
ax1 = axes[0]
ax1.set_xlim(-3, 3)
ax1.set_ylim(-2, 4)
ax1.set_aspect('equal')
ax1.set_title('Normal Conductor\n(Field Penetrates)', fontsize=14)

# Draw conductor
conductor1 = Rectangle((-2, -1.5), 4, 1.5, color='gray', alpha=0.7)
ax1.add_patch(conductor1)
ax1.text(0, -0.75, 'Normal\nMetal', ha='center', va='center', fontsize=11)

# Draw magnet
magnet1 = Rectangle((-0.5, 1.5), 1, 1, color='red', alpha=0.8)
ax1.add_patch(magnet1)
ax1.text(0, 2, 'N', ha='center', va='center', color='white', fontsize=12, fontweight='bold')

# Draw field lines (penetrating)
for x in [-1.5, -0.75, 0, 0.75, 1.5]:
    # Lines go through the conductor
    y_points = np.linspace(3.5, -1.5, 50)
    ax1.plot([x]*len(y_points), y_points, 'b-', alpha=0.5, linewidth=1.5)
    ax1.annotate('', xy=(x, -1.5), xytext=(x, 3.5),
                arrowprops=dict(arrowstyle='->', color='blue', alpha=0.5))

ax1.axis('off')

# Right plot: Superconductor (Meissner effect)
ax2 = axes[1]
ax2.set_xlim(-3, 3)
ax2.set_ylim(-2, 4)
ax2.set_aspect('equal')
ax2.set_title('Superconductor\n(Meissner Effect)', fontsize=14)

# Draw superconductor
super1 = Rectangle((-2, -1.5), 4, 1.5, color='cyan', alpha=0.7)
ax2.add_patch(super1)
ax2.text(0, -0.75, 'Super-\nconductor', ha='center', va='center', fontsize=11)

# Draw levitating magnet
magnet2 = Rectangle((-0.5, 2), 1, 1, color='red', alpha=0.8)
ax2.add_patch(magnet2)
ax2.text(0, 2.5, 'N', ha='center', va='center', color='white', fontsize=12, fontweight='bold')

# Draw expelled field lines (going around)
theta = np.linspace(0, np.pi, 50)
for r_offset in [0.8, 1.3, 1.8]:
    x_curve = r_offset * np.cos(theta)
    y_curve = r_offset * np.sin(theta) + 0
    ax2.plot(x_curve, y_curve, 'b-', alpha=0.5, linewidth=1.5)

# Add arrows on curved lines
for angle in [np.pi/4, np.pi/2, 3*np.pi/4]:
    for r in [0.8, 1.3, 1.8]:
        ax2.annotate('', xy=(r*np.cos(angle-0.1), r*np.sin(angle-0.1)),
                    xytext=(r*np.cos(angle+0.1), r*np.sin(angle+0.1)),
                    arrowprops=dict(arrowstyle='->', color='blue', alpha=0.5))

# Add levitation gap indicator
ax2.annotate('', xy=(1.5, 0), xytext=(1.5, 2),
            arrowprops=dict(arrowstyle='<->', color='green', lw=2))
ax2.text(1.8, 1, 'Levitation\ngap', fontsize=10, color='green')

ax2.axis('off')

plt.tight_layout()
plt.show()

1.4 Type I and Type II Superconductors

Different Responses to Magnetic Fields

Not all superconductors behave the same way in magnetic fields. Based on their response, we classify them into two types:

Type I Superconductors

• Complete Meissner effect below critical field Hc
• Abrupt transition to normal state at Hc
• Usually pure elemental metals (Pb, Hg, Sn, Al)
• Low critical fields (typically < 0.1 T)

Type II Superconductors

• Two critical fields: Hc1 and Hc2
• Below Hc1: Complete Meissner effect
• Between Hc1 and Hc2: Mixed state (vortex state)
• Above Hc2: Normal state
• Alloys and compounds (NbTi, Nb3Sn, YBCO)
• Much higher critical fields (up to 100+ T)

import numpy as np
import matplotlib.pyplot as plt

# Magnetic field range
H = np.linspace(0, 2, 500)

# Type I superconductor
Hc = 0.5  # Critical field
M_type1 = np.where(H < Hc, -H, 0)  # Perfect diamagnetism below Hc

# Type II superconductor
Hc1 = 0.3   # Lower critical field
Hc2 = 1.5   # Upper critical field

M_type2 = np.zeros_like(H)
for i, h in enumerate(H):
    if h < Hc1:
        M_type2[i] = -h  # Perfect diamagnetism
    elif h < Hc2:
        # Mixed state: partial flux penetration
        M_type2[i] = -Hc1 * (1 - (h - Hc1)/(Hc2 - Hc1))
    else:
        M_type2[i] = 0  # Normal state

fig, axes = plt.subplots(1, 2, figsize=(14, 5))

# Type I
axes[0].plot(H, M_type1, 'b-', linewidth=2)
axes[0].axvline(x=Hc, color='r', linestyle='--', label=f'Hc = {Hc}')
axes[0].fill_between(H, M_type1, 0, where=(H < Hc), alpha=0.3, color='cyan',
                     label='Superconducting')
axes[0].set_xlabel('Applied Field H', fontsize=12)
axes[0].set_ylabel('Magnetization M', fontsize=12)
axes[0].set_title('Type I Superconductor', fontsize=14)
axes[0].legend(fontsize=11)
axes[0].grid(True, alpha=0.3)
axes[0].set_xlim(0, 2)
axes[0].set_ylim(-0.6, 0.1)

# Type II
axes[1].plot(H, M_type2, 'b-', linewidth=2)
axes[1].axvline(x=Hc1, color='orange', linestyle='--', label=f'Hc1 = {Hc1}')
axes[1].axvline(x=Hc2, color='r', linestyle='--', label=f'Hc2 = {Hc2}')
axes[1].fill_between(H, M_type2, 0, where=(H < Hc1), alpha=0.3, color='cyan',
                     label='Meissner state')
axes[1].fill_between(H, M_type2, 0, where=(H >= Hc1) & (H < Hc2), alpha=0.3,
                     color='yellow', label='Mixed state')
axes[1].set_xlabel('Applied Field H', fontsize=12)
axes[1].set_ylabel('Magnetization M', fontsize=12)
axes[1].set_title('Type II Superconductor', fontsize=14)
axes[1].legend(fontsize=11, loc='lower right')
axes[1].grid(True, alpha=0.3)
axes[1].set_xlim(0, 2)
axes[1].set_ylim(-0.6, 0.1)

plt.tight_layout()
plt.show()

1.5 Timeline of Superconductivity Discovery

Let's visualize the major milestones in superconductivity research:

Summary

Practice Problems