Superconductivity

10

Superconductivity: phenomenology

Some metals have the remarkable property that their electrical resistivity ρ vanishes below a critical temperature Tc . Understanding this phenomenon has been one of the main goals of solid state physics since its origins in the beginning of the 20th century. Superconductivity is now well understood for elemental metals and intermetallics; an entirely successful microscopic theory (the BCS theory) has been worked out for these systems. The more recently discovered ”high temperature superconductors” (”high” T c >77K) are still mysterious on this level, however. In this section, I would like to cover only those aspects of superconductivity which follow from the observation of the phenomenon, not saying anything about its microscopic origin. Two key experimental characteristics of superconductors, the absence of resitivity ρ, and the perfect diamagnetism χm = −1, have some interesting implications. If we use only classical E&M theories combined with thermodynamics, two experimentally observed characteristics follow: the existence of a penetration depth for magnetic fields, and the existence of a critical magnetic field above which superconductivity disappears. Superconductivity is not a tremendously useful materials property–not yet. Visions of superconductor-enabled technologies, such as magnetically levitated trains, or lossless power transmission, have not yet materialized in any real sense. Niche applications exist, such as construction of superconducting magnets and use in very high quality cellular telephone base station filters. For the most part, however, I am including a discussion of superconductivity since it has always been an active area in the study of solid state phenomena and is likely to remain so.

10.1

Perfect conductors or not?

Superconductors are not merely perfect conductors. A perfect conductor is one in which there is vanishing resistivity, ρ = 0, and no other salient feature. Something more is involved in superconductivity, manifested in the path independence of transitions to the superconducting state. This something more is the Meissner effect: the expulsion of magnetic field. The magnetic field is constant: B = Bo Let’s examine the consequences of ρ = 0. Remember that we can write Ohm’s law as E = ρJ 1

(1)

where J is the current density. For a perfect conductor, ρ = 0, so we have E = 0 ∇×E = 0 ∂B ∇×E = ∂t ∂B = 0 ∂t

(2) (3) (4) (5)

which implies that the magnetic field inside a perfect conductor never changes: B = Bo

(6)

We have already learned something: the magnetic field inside a perfect conductor must always be the same as it was at the time the perfectly conducting state was set up. Meissner effect: B = 0 The condition 6 is true for superconductors, but not restrictive enough. It turns out to be the case that superconductors expel magnetic fields. This is found experimentally in the reversibility of the transition to the superconducting state. To see how reversibility comes in, let’s think about the following 2x2 matrix of possibilities. Physical situations correspond to 1) predicted behavior for a perfect conductor and 2) observed behavior for superconductors; experimental configurations count a) cooling through Tc first, applying H second, and b) applying H first, cooling through Tc second. If the phase transition is reversible, it should be path-independent, with no difference in the final state depending on whether we take path a) or b). The possibilities are shown in Figure 1. The experimental sequence proceeds from top to bottom, with columns corresponding to situations 1a, 1b, 2a, and 2b. In the first column, we expect that a perfect conductor will have B = 0 inside in its final state: we start with no field applied, cool through Tc , and apply a magnetic field Hext , which is expelled since B = Bo at the birth of the superconducting state (=0). Surface currents are set up on the sample to create M = −H; the perturbed B observed outside the sample results from the sum of Hef f and Hd , the demagnetizing field. Removing Hef f causes the surface currents to die off, and we have again M = 0 and B = 0. 2

superconductor (observed)

perfect conductor (hypothetical) H=0 T >Tc

key:

Step #1 normal

Step #2

superconductor outcome

H=0 path independence

hysteresis

Figure 1: Superconductors are not merely perfect conductors: the Meissner effect makes the transition to superconductivity path-independent. Field lines plotted are B. Left: prediction for a perfect conductor; right: observed phenomena for superconductors. See text for details.

3

If instead, for a perfect conductor, we reverse the order of applying the fields and cooling, so that we apply the field first and then cool, we must always have Bo = µo Happ inside the sample. This was the field present at the birth of the superconducting state. Removing the field then means that B = Bo = µo M , and we would measure a demagnetizing field from the sample, as pictured. Perfect conductors would thus exhibit hysteresis in the transition to the superconducting state: the final state depends on the sample history. For superconductors as they exist in nature, there is no hysteresis. All experimental facts are the same except in the sequence b). Here, it turns out that when cooling the material through TC in the presence of a magnetic field, the field is expelled from the SC! We have B = 0 satisfied if M = −H. This is the expression for perfect diamagnetism. Recalling the definition of χM , we thus have. B = 0

(7)

M = −H

(8)

χM

= −1

(9)

The presence of perfect diamagnetism is called the Meissner-Ochsenfeld effect.

10.2

The London equations

So far we know two experimental phenomena about superconductors: 1) ρ = 0, they have no resistance; 2) they are perfect diamagnets. What do these two facts imply? The London brothers figured out one important implication of these facts: that there is a finite penetration depth of magnetic fields into the superconductor. B = 0 cannot be satisfied right at the surface; it takes about 100 atomic layers to satisfy this condition. We’ll see why. They begin with the assumption that there are two types of electrons in a superconductor, normal electrons (n) and superconducting electrons (s). This is sometimes known as a two fluid model; we will see this language reappear in the discussion of ”spintronics” where two fluids refer to electrons with opposite spins. Assuming that these electron types have two separate concentrations, n s and nn , we can rewrite Ohm’s law:

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Jn = nevn = σn E

(10) (11)

Recall that in the Drude model, there is a relaxation time τ between scattering events, during which the electron is accelerated, but after which its velocity is returned to zero. For a superconductor, since there is no scattering, the electrons are continuously accelerated: me

∂v = eE ∂t

(12)

We can write for the flux: J s = n s e vs ∂vs ∂Js = ns e ∂t ∂t

(13) (14)

and substituting in from 12 n s e2 E ∂Js = ∂t me

(15)

This is the first London equation. In itself, it is not useful, but it brings us to the second London equation. If we take the curl of the above, we have:

∇×

∂Js ∂t

n s e2 ∇×E me ns e2 ∂B = − me ∂t =

(16) (17)

or, rewriting:   n s e2 ∂ ∇ × Js + B =0 ∂t me

(18)

So the quantity in brackets needs to be a constant. To be consistent with the Meissner effect, however, B = 0, and the whole term in brackets is set to zero: ∇ × Js = − 5

n s e2 B me

(19)

This is the second London equation. To go further, we can assume that in the superconducting state, the normal contribution to the total current Jn , J = Js + Jn is negligible. Rewriting the third Maxwell’s equation: ∇ × B = µo J

(20)

∇ × (∇ × B) = µo (∇ × Js ) µo n s e2 = − B me

(21) (22)

Introducing 1 µo n s e2 ≡ 2 me λL

(23)

we rewrite as ∇ × (∇ × B) = −

1 B λ2L

(24)

and, revisiting our vector calculus identity, ∇ × ∇ × v = ∇(∇ · v) − ∇2 v, through the fact that ∇ · B = 0, we simplify the LHS as 1 B λ2 Similarly, from the second London equation, we have ∇2 B =

n s e2 ∇×B me µo n s e2 = − Js me 1 ∇ × (∇ × Js ) = − 2 Js λL 1 ∇2 J s = Js λ2 ∇ × (∇ × Js ) = −

λL is the London penetration depth. Checking units, we have

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(25)

(26) (27) (28) (29)

λL

=

r

[=]

s

=

r

= =

me µo n s e2 kg N m−3 C 2 A2

kg m3 2 sN s  kg m 1 2 m s2 N

m

(30) (31) (32) (33) (34)

Let’s consider how these equations work out for a magnetic field applied parallel to the surface of a superconductor. We’ll have B = Bo x ˆ, and the distance into the superconductor measured in the z direction. The london equations can be simplified in one dimension (z) as:

∇2 B = ∂ 2 Bx (z) ∂z 2

=

1 B λ2 1 Bx (z) λ2L

(35) (36)

and solved as: Bx (z) = B0x e−z/λL

(37)

similarly, for the superconducting current (supercurrent) density, Jsy (z) = Js0y e−z/λL since ∇ × B = µo Js , and ∇ × B = we have Jsy = −B0x /µo λL . Jsy (z) = −

∂Bx ∂z

(38)

y ˆ = µo Jsy y ˆ, evaluated at x = 0,

B0x −z/λL e µo λL

(39)

Thus the magnetic field can penetrate the superconductor to a characteristic depth given by λL , typically 500˚ A in conventional superconductors such as Pb and Sn. The surface supercurrent Js acts to screen the rest of the sample from this applied field. 7