Superconductivity

Meissner effect

Meissner effect is the complete expulsion of magnetic flux from the interior of a material when it is cooled below its critical temperature $T_c$ in a weak magnetic field ($H < H_c$). The superconductor becomes a perfect diamagnet.

Inside the superconductor:

$$\begin{aligned} B &= \mu_0 (H + M) = 0 \\ \Rightarrow M &= -H, \quad \chi_m = \frac{M}{H} = -1 \end{aligned}$$

Screening currents flow in a thin surface layer (the London penetration depth, $\lambda_L \approx 10$ to $100$ nm) and create a field that exactly cancels the applied field inside.

   T > Tc (normal)        T < Tc (superconducting)
   | | | | | | |           \ \         / /
   | |+-----+| |            \ \ +---+ / /
   | ||     || |   cool      ) )|   |( (
   | ||     || |   ---->    / / +---+ \ \
   | |+-----+| |           / /         \ \
   | | | | | | |      flux lines bend around it

The effect is independent of history: whether the field is applied before or after cooling, the flux is expelled. This shows a superconductor is not just a perfect conductor (a perfect conductor would trap the flux present at cooling).

Type-I and type-II superconductors

Superconductors are classified by how they respond to a rising magnetic field.

• Type-I: the material stays a perfect diamagnet (full flux expulsion) until the field reaches $H_c$, then it suddenly becomes fully normal and flux enters completely.
• Type-II: below $H_{c1}$ flux is fully expelled. Between $H_{c1}$ and $H_{c2}$ flux enters as thin quantized tubes called vortices (each carries $\Phi_0 = h/2e = 2.07\times10^{-15}$ Wb). The material around them stays superconducting (mixed state). Above $H_{c2}$ it becomes normal.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

A superconductor stays superconducting only while the magnetic field at every point is below the critical field $H_c$. A strong field destroys superconductivity, and the critical field depends on temperature:

$$H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

where $H_0$ is the critical field at 0 K.

Effect of a strong magnetic field

• For $H < H_c$: flux is expelled (Meissner effect), resistance is zero.
• For $H \ge H_c$ (type-I): flux penetrates completely and the material becomes normal, with normal resistance.
• For type-II: flux starts entering at $H_{c1}$ (mixed state), and superconductivity is fully lost at $H_{c2}$.
• A stronger field means a lower temperature is needed to stay superconducting, so $T_c$ falls as $H$ rises.

  H
 H0 |*
    |   *
    |  Super-   *      Normal
    |  conducting  *
    |                *
    +-----------------*---- T
    0                 Tc

Critical current (Silsbee's rule)

A current in the wire itself produces a magnetic field. If this self-field at the surface reaches $H_c$, superconductivity is destroyed. The current at which this happens is the critical current $I_c$.

Consider a long straight superconducting wire of radius $r$ carrying current $I$:

        H (circles around wire)
      .-~~~~~-.
     /  .---.  \
    |  ( I ⊙ )  |   radius r
     \  '---'  /
      '-~~~~~-'

By Ampere's circuital law, the field at the surface is

$$\oint H\,dl = I \;\Rightarrow\; H \cdot 2\pi r = I \;\Rightarrow\; H = \frac{I}{2\pi r}$$

Superconductivity is lost when $H = H_c$, so

$$\begin{aligned} I_c &= 2\pi r H_c \\ &= 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \end{aligned}$$

The critical current density is

$$J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r}$$

If an external field $H_a$ is also applied (along the same direction as the self-field at the surface), the total must stay below $H_c$:

$$H_a + \frac{I}{2\pi r} \le H_c \;\Rightarrow\; I_c = 2\pi r\,(H_c - H_a)$$

So $I_c$ falls linearly as the applied field increases, and falls as temperature rises. This sets the maximum current a superconducting cable or magnet winding can carry.

Below $T_c$, a superconductor expels all magnetic flux from its interior. This is the Meissner effect, and it makes the material a perfect diamagnet ($B = 0$ inside, $\chi_m = -1$).

How the flux is expelled

1. When the material is cooled below $T_c$, electrons form Cooper pairs (BCS theory). The pairs move together as one coherent quantum state, without scattering.
2. When a magnetic field is present, the pairs set up persistent surface (screening) currents in a thin layer of thickness about the London penetration depth $\lambda_L$ (10 to 100 nm).
3. These loss-free currents produce a magnetic field equal and opposite to the applied field inside the body, so the net field inside is zero:

$$B = \mu_0(H + M) = 0 \;\Rightarrow\; M = -H$$

4. The field does not stop sharply at the surface. By London's equation it decays exponentially inside:

$$B(x) = B_0\,e^{-x/\lambda_L}$$

So beyond a few $\lambda_L$ from the surface, $B \approx 0$.

  B
 B0|\
   | \
   |  `.
   |    `-._
   |        `--.____
   +--|-------------- x (into sample)
   0  λL

5. Because the screening currents never decay (zero resistance), the expulsion is permanent as long as $T < T_c$ and $H < H_c$.

The flux lines therefore bend around the sample instead of passing through it. This expulsion happens whether the field was applied before or after cooling, which shows that superconductivity is a true thermodynamic state and not just perfect conduction.

The Meissner effect (expulsion of flux, $B = 0$ inside, $M = -H$) behaves differently in the two classes of superconductors. Plotting magnetization $-M$ against applied field $H$ separates them clearly.

Type-I (soft) superconductors

• Show a complete Meissner effect right up to the critical field $H_c$: $-M$ rises linearly with $H$ (perfect diamagnetism).
• At $H = H_c$ the flux enters all at once, $M$ drops sharply to zero and the material becomes normal.
• $H_c$ is low (about 0.01 to 0.2 T). Examples: Pb, Hg, Sn, Al.

Type-II (hard) superconductors

• Show a complete Meissner effect only up to the lower critical field $H_{c1}$.
• Between $H_{c1}$ and $H_{c2}$ flux partly enters as quantized vortices, so the Meissner effect is incomplete (mixed or vortex state). $-M$ falls gradually.
• At the upper critical field $H_{c2}$, $M = 0$ and the material is normal. $H_{c2}$ can be tens of tesla. Examples: NbTi, Nb$_3$Sn, YBCO.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

So by measuring how $M$ varies with $H$, a sharp drop identifies type-I and a gradual fall over a range identifies type-II.

Superconductor

A superconductor is a material whose electrical resistance falls abruptly to zero when it is cooled below a certain critical temperature $T_c$, and which expels magnetic flux from its interior (Meissner effect). Example: mercury becomes superconducting at 4.2 K (discovered by Kamerlingh Onnes, 1911); YBa$_2$Cu$_3$O$_7$ at about 92 K.

Superconductivity exists only while temperature, magnetic field and current density are all below their critical values ($T_c$, $H_c$, $J_c$). It is explained by BCS theory: electrons bind into Cooper pairs through lattice vibrations and move without scattering.

  R
    |        ___----
    |   ___--
    |  |   normal metal
    |  |
    |__|__________ T
       Tc

Type-I vs type-II superconductors

• Type-I: perfect diamagnet up to $H_c$, then suddenly normal.
• Type-II: perfect diamagnet up to $H_{c1}$, mixed (vortex) state between $H_{c1}$ and $H_{c2}$, normal above $H_{c2}$.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

Critical magnetic field

The critical magnetic field $H_c$ is the minimum field that destroys superconductivity at a given temperature. It is largest at 0 K and becomes zero at $T_c$:

$$H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

Critical current density

The critical current density $J_c$ is the maximum current density a superconductor can carry without losing superconductivity. The current's own magnetic field must stay below $H_c$. For a wire of radius $r$ (Silsbee's rule):

$$I_c = 2\pi r H_c, \qquad J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r}$$

Soft vs hard superconductors

Soft superconductors are type-I; hard superconductors are type-II.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

Hard superconductors are "hard" because flux vortices get pinned at defects, so they resist flux movement, show hysteresis like hard magnetic materials, and can carry large currents in strong fields.

How a superconductor expels the magnetic field

When a superconductor is cooled below $T_c$ in a field $H < H_c$, it pushes all flux out of its interior. This is the Meissner effect, and it makes the material a perfect diamagnet.

1. Below $T_c$ electrons form Cooper pairs, which move without scattering.
2. The applied field induces persistent screening currents on the surface, in a layer about the London penetration depth $\lambda_L$ (10 to 100 nm) thick.
3. These currents create a field that exactly cancels the applied field inside:

$$B = \mu_0(H + M) = 0 \;\Rightarrow\; M = -H, \quad \chi_m = -1$$

4. Inside, the field decays as $B(x) = B_0 e^{-x/\lambda_L}$, so it is essentially zero beyond a few $\lambda_L$.

     normal (T > Tc)          super (T < Tc)
     | | | | | |              \  \     /  /
     | |+---+| |               \  +---+  /
     | ||   || |    ---->       ) |   | (
     | |+---+| |               /  +---+  \
     | | | | | |              /  /     \  \

The expulsion happens whether the field is applied before or after cooling, so it is a property of the superconducting state, not just of zero resistance.

Type-I vs type-II superconductors

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

A normal conductor (e.g. copper) always has some resistance that falls gradually with cooling, while a superconductor loses all resistance abruptly below its critical temperature $T_c$ and also expels magnetic flux.

  R |        /  normal
    |      /
    |    /
    |  _/  residual R
    |  |
    |  |  superconductor
    |__|________ T
       Tc

A material stays superconducting only while three quantities are below their critical values. Exceeding any one of them returns it to the normal state.

Limiting factors

1. Critical temperature $T_c$: above $T_c$ thermal energy breaks the Cooper pairs (e.g. Hg 4.2 K, Nb$_3$Sn 18 K, YBCO 92 K).
2. Critical magnetic field $H_c$: a field above $H_c$ destroys superconductivity. It depends on temperature: $H_c(T) = H_0[1 - (T/T_c)^2]$.
3. Critical current density $J_c$: a current produces its own field; when this field at the surface reaches $H_c$, superconductivity is lost. For a wire of radius $r$: $I_c = 2\pi r H_c$.

The three form a critical surface; the material is superconducting only inside it.

Applications

• MRI and NMR magnets: NbTi coils give strong, stable fields with no power loss.
• Maglev trains: superconducting magnets for levitation (e.g. Japan's SCMaglev).
• Particle accelerators: bending magnets in the LHC.
• SQUIDs: extremely sensitive magnetometers (brain and heart signals).
• Power cables and fault current limiters: low-loss transmission and fast fault limiting.
• SMES: superconducting magnetic energy storage.
• Josephson junction devices: fast switching and quantum computing qubits.

Superconductor

A superconductor is a material whose resistance becomes exactly zero below a critical temperature $T_c$ and which expels magnetic flux from its interior (Meissner effect, perfect diamagnetism). Examples: Hg ($T_c$ = 4.2 K), Pb (7.2 K), Nb$_3$Sn (18 K), YBCO (92 K).

Critical magnetic field

The critical magnetic field $H_c$ is the minimum magnetic field that destroys superconductivity at a given temperature. It is maximum ($H_0$) at 0 K and zero at $T_c$:

$$H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

  Hc
  H0|*
    |    *    normal
    | super  *
    |         *
    +----------*--- T
               Tc

Critical current density

The critical current density $J_c$ is the largest current density a superconductor can carry without becoming normal. The current's own magnetic field at the surface must stay below $H_c$. For a wire of radius $r$ (Silsbee's rule):

$$I_c = 2\pi r H_c, \qquad J_c = \frac{I_c}{\pi r^2} = \frac{2H_c}{r}$$

Meissner effect

Meissner effect is the complete expulsion of magnetic flux from the interior of a material when it is cooled below its critical temperature $T_c$ in a weak magnetic field ($H < H_c$). The superconductor becomes a perfect diamagnet.

Inside the superconductor:

$$\begin{aligned} B &= \mu_0 (H + M) = 0 \\ \Rightarrow M &= -H, \quad \chi_m = \frac{M}{H} = -1 \end{aligned}$$

Screening currents flow in a thin surface layer (the London penetration depth, $\lambda_L \approx 10$ to $100$ nm) and create a field that exactly cancels the applied field inside.

   T > Tc (normal)        T < Tc (superconducting)
   | | | | | | |           \ \         / /
   | |+-----+| |            \ \ +---+ / /
   | ||     || |   cool      ) )|   |( (
   | ||     || |   ---->    / / +---+ \ \
   | |+-----+| |           / /         \ \
   | | | | | | |      flux lines bend around it

The effect is independent of history: whether the field is applied before or after cooling, the flux is expelled. This shows a superconductor is not just a perfect conductor (a perfect conductor would trap the flux present at cooling).

Type-I vs type-II superconductors

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

Why type-II is called a hard superconductor

• It stays superconducting up to a very high field $H_{c2}$, so superconductivity is hard to destroy.
• In the mixed state, flux vortices get pinned at defects, dislocations and impurities. This makes the magnetization irreversible with a large hysteresis loop, just like a hard magnetic material.
• Because of pinning, it can carry a large critical current density in strong fields.
• Most are alloys or compounds that are mechanically hard and brittle (e.g. Nb$_3$Sn).

The critical field varies with temperature as

$$H_c(T) = H_0\left[1 - \frac{T^2}{T_c^2}\right]$$

Given: $H_{c1} = 0.176$ T at $T_1 = 14$ K, $H_{c2} = 0.528$ T at $T_2 = 13$ K.

Critical temperature

Divide the two equations:

$$\begin{aligned} \frac{0.176}{0.528} &= \frac{T_c^2 - 14^2}{T_c^2 - 13^2} \\ \frac{1}{3} &= \frac{T_c^2 - 196}{T_c^2 - 169} \\ T_c^2 - 169 &= 3T_c^2 - 588 \\ 2T_c^2 &= 419 \\ T_c^2 &= 209.5 \\ T_c &= 14.47\ \text{K} \end{aligned}$$

Critical field at 0 K

$$\begin{aligned} H_0 &= \frac{H_c(14)}{1 - \frac{196}{209.5}} = \frac{0.176 \times 209.5}{13.5} \\ &= 2.731\ \text{T} \end{aligned}$$

Check with the 13 K value: $2.731 \times (1 - 169/209.5) = 2.731 \times 0.1933 = 0.528$ T, which matches.

Critical field at 4.2 K

$$\begin{aligned} H_c(4.2) &= 2.731\left[1 - \frac{4.2^2}{209.5}\right] \\ &= 2.731\,(1 - 0.0842) \\ &= 2.501\ \text{T} \end{aligned}$$

Answer: $T_c = 14.47$ K, $H_c(0) = 2.73$ T, $H_c(4.2\ \text{K}) = 2.50$ T.

Critical magnetic field

The critical magnetic field $H_c$ is the minimum magnetic field which, applied to a superconductor at a temperature below $T_c$, destroys its superconductivity and returns it to the normal state.

• It is largest ($H_0$) at 0 K and falls to zero at $T = T_c$.
• The temperature variation is nearly parabolic:

$$H_c(T) = H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right]$$

  Hc
  H0|*
    |    *
    | supercon-  *     normal
    | ducting      *
    |               *
    +----------------*--- T
    0                Tc

• Points under the curve are superconducting; points above it are normal.
• Type-I materials have a single $H_c$ (about 0.01 to 0.2 T). Type-II have $H_{c1}$ (flux starts entering) and $H_{c2}$ (fully normal), with $H_{c2}$ as high as tens of tesla.
• Example: lead has $H_0 \approx 0.08$ T and $T_c = 7.2$ K.

Critical current

The critical current $I_c$ is the maximum current a superconductor can carry without losing superconductivity. A current produces its own magnetic field; when this field at the surface reaches $H_c$, the material turns normal (Silsbee's rule).

For a long straight wire of radius $r$ carrying current $I$, Ampere's law gives the surface field:

$$H \cdot 2\pi r = I \;\Rightarrow\; H = \frac{I}{2\pi r}$$

Setting $H = H_c$:

$$\begin{aligned} I_c &= 2\pi r H_c \\ &= 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \end{aligned}$$

The critical current density is

$$J_c = \frac{I_c}{\pi r^2} = \frac{2 H_c}{r}$$

With an external field $H_a$ adding to the self-field, $I_c = 2\pi r (H_c - H_a)$.

So the critical current depends on wire size, temperature and applied field. It falls as $T$ rises towards $T_c$ and becomes zero at $T_c$. Thicker wires carry larger $I_c$, which is why superconducting cables use many fine filaments in a copper matrix for stability.

The critical current $I_c$ is the maximum current a superconductor can carry while remaining superconducting. Beyond it, the magnetic field produced by the current (plus any applied field) exceeds the critical field $H_c$ and the material becomes normal (Silsbee's rule).

Field produced by the current

Take a long superconducting wire of radius $r$ carrying current $I$.

         H_a (applied)
     --------------->
       .-~~~~~~-.
      /  .----.  \   H_I = I/(2πr)
     |  (  I ⊙ ) |   circles round
      \  '----'  /
       '-~~~~~~-'
     --------------->

By Ampere's circuital law, the self-field at the surface is

$$\oint H\,dl = I \;\Rightarrow\; H_I = \frac{I}{2\pi r}$$

Without an applied field, superconductivity is lost when $H_I = H_c$:

$$I_{c0} = 2\pi r H_c$$

With an applied field

Let a uniform external field $H_a$ act on the wire. At the point of the surface where the self-field and applied field point the same way, the total field is the sum:

$$H_{total} = H_a + \frac{I}{2\pi r}$$

The wire stays superconducting only while $H_{total} \le H_c$. The critical condition is

$$\begin{aligned} H_a + \frac{I_c}{2\pi r} &= H_c \\ I_c &= 2\pi r\,(H_c - H_a) \\ I_c &= I_{c0} - 2\pi r\,H_a \end{aligned}$$

This is a straight line in $H_a$ with negative slope $-2\pi r$. So the critical current decreases linearly with the applied field:

• at $H_a = 0$: $I_c = 2\pi r H_c$ (maximum),
• at $H_a = H_c$: $I_c = 0$.

  Ic
 2πrHc|\
      |  \
      |    \
      |      \
      +--------\---- Ha
      0        Hc

Since $H_c = H_0[1 - (T/T_c)^2]$, the whole line moves down as temperature rises. Critical current density is $J_c = I_c/\pi r^2 = 2(H_c - H_a)/r$.

Meissner effect

Meissner effect is the complete expulsion of magnetic flux from the interior of a material when it is cooled below its critical temperature $T_c$ in a weak magnetic field ($H < H_c$). The superconductor becomes a perfect diamagnet.

Inside the superconductor:

$$\begin{aligned} B &= \mu_0 (H + M) = 0 \\ \Rightarrow M &= -H, \quad \chi_m = \frac{M}{H} = -1 \end{aligned}$$

Screening currents flow in a thin surface layer (the London penetration depth, $\lambda_L \approx 10$ to $100$ nm) and create a field that exactly cancels the applied field inside.

   T > Tc (normal)        T < Tc (superconducting)
   | | | | | | |           \ \         / /
   | |+-----+| |            \ \ +---+ / /
   | ||     || |   cool      ) )|   |( (
   | ||     || |   ---->    / / +---+ \ \
   | |+-----+| |           / /         \ \
   | | | | | | |      flux lines bend around it

The effect is independent of history: whether the field is applied before or after cooling, the flux is expelled. This shows a superconductor is not just a perfect conductor (a perfect conductor would trap the flux present at cooling).

Complete Meissner effect in type-I

• In a type-I superconductor, $B = 0$ inside for all fields up to $H_c$. Magnetization rises linearly, $M = -H$.
• At $H = H_c$, flux enters all at once and the material becomes normal; $M$ drops to zero.
• There is no intermediate state in a long sample, so the expulsion is complete right up to $H_c$. Examples: Pb, Hg, Sn.

Incomplete Meissner effect in type-II

• Below the lower critical field $H_{c1}$, flux is fully expelled (complete Meissner effect).
• Between $H_{c1}$ and $H_{c2}$ it is energetically favourable for flux to enter as thin normal cores called vortices (fluxons), each carrying one flux quantum $\Phi_0 = h/2e = 2.07\times10^{-15}$ Wb, surrounded by circulating supercurrents.
• The material between vortices stays superconducting (zero resistance), but $B \ne 0$ inside on average. Hence the Meissner effect is incomplete in this mixed (vortex) state.
• As $H$ increases, vortex density rises; at $H_{c2}$ the cores overlap and the material becomes normal. Examples: NbTi, Nb$_3$Sn, YBCO.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

 Mixed state (top view of type-II slab)
   +-------------------+
   |  o    o    o    o |   o = vortex (normal
   |    o    o    o    |       core with flux)
   |  o    o    o    o |   rest = superconducting
   +-------------------+

Meissner effect: type-I vs type-II

The Meissner effect is the expulsion of flux ($B = 0$, $M = -H$) below $T_c$. The way it ends as $H$ increases separates the two types:

• Type-I: complete diamagnetism up to $H_c$, then an abrupt drop to $M = 0$ (normal). Single, low $H_c$. Examples: Pb, Hg.
• Type-II: complete only up to $H_{c1}$; between $H_{c1}$ and $H_{c2}$ flux partly enters as vortices (mixed state), $-M$ falls gradually; normal above $H_{c2}$. Examples: NbTi, Nb$_3$Sn.

 -M                         -M
  |    /|                    |   /\
  |   / |                    |  /  \
  |  /  |                    | /    \_
  | /   |                    |/       \__
  |/    |____ H              +----|------|--- H
  0    Hc                    0   Hc1    Hc2
   Type-I                     Type-II

Critical current

The critical current $I_c$ is the maximum current a superconductor can carry without becoming normal. The self-field of the current must remain below $H_c$ (Silsbee's rule).

For a wire of radius $r$, by Ampere's law the surface field is $H = I/(2\pi r)$. Setting $H = H_c$:

$$\begin{aligned} I_c &= 2\pi r H_c = 2\pi r H_0\left[1 - \left(\frac{T}{T_c}\right)^2\right] \\ J_c &= \frac{I_c}{\pi r^2} = \frac{2H_c}{r} \end{aligned}$$

        H = I/(2πr)
      .-~~~~~-.
     / .-----. \
    | (  I ⊙  ) |  wire radius r
     \ '-----' /
      '-~~~~~-'

With an applied field $H_a$, $I_c = 2\pi r (H_c - H_a)$, so $I_c$ falls with field and with temperature.