Matriz de Triangulación

This can be solved numerically to obtain:

∆(T = 0) = 1.764kbTc (2.48)

where Tcis the transition temperature. The gap is a critical property of BCS superconductivity. At

low temperatures T < Tc, the electron gas is unstable to the formation of a coherent state of Cooper

pairs, stabilised by an energy gap of order a few kelvin. Tunnelling measurements can accurately measure ∆, and provide an experimental confirmation of the accuracy of BCS theory.

BCS theory also produces a value for the transition temperature, at which ∆ → 0:

kbTc= 1.13~ωDe−1/V N (EF) (2.49)

BCS theory has proved fairly successful at describing weak-coupling superconductivity in the ele- ments and some compounds. It explains why superconductivity occurs, and gives an estimate of the transition temperatures:Tc ∼ ΘDe−1/N (EF)V where V = Vkk0_q is the strength of the attractive

electronic interaction and ΘDis the Debye temperature.

In the BCS limit of a weak electron-electron interaction, we have that V N (EF)  1. This implies

that Tc should in general be fairly low (some small fraction of the Debye temperature, which is

typically of the order of a few hundred kelvin) - and this is observed experimentally for many BCS superconductors.

2.4.3 Strong coupling superconductivity

The above theory is valid for weak coupling, where N (EF)V  1. It does not hold for systems

Theoretical overview 2.4 Superconductivity

transition temperatures, this is arguably the more interesting region - even the simple BCS theory captures the essential physics that an increase in λ gives an increase in Tc.

A more sophisticated expression for the dependence of Tcon λ was obtained by McMillan [15].

He showed that λ is a critical parameter for determining transition temperatures. McMillan used the phonon spectrum of niobium to numerically solve the strong-coupling phonon superconductivity equations worked out by Eliashberg, Migdal and others, thereby obtaining an expression for Tcset

by only three parameters: the Debye temperature ΘD, the electron-phonon coupling strength λ and

the renormalised Coulomb pseudopotential µ∗_:

Tc =

ΘD

1.45e

− 1.04(1+λ)

λ−µ∗ (1+0.62λ)_. (2.50)

Rather conveniently, the renormalised Coulomb pseudopotential always has approximately the same value ∼ 0.10, because it is reduced from the bare Coulomb energy µ = N (EF)VCoulombby:

µ∗= µ 1 + µ lnΩp

ωD

(2.51)

and Ωp ωDfor typical metals.

McMillan’s expression is widely used and found to fit the data for quite a large number of mate- rials. These include the elemental superconductors niobium (λ ≈ 1.1) and lead (λ ≈ 1.5), as well as many superconducting compounds such as the A15 materials V3Si and Nb3Sn. The Debye tempera-

ture can be obtained fairly accurately from heat capacity measurements; λ is trickier to measure but can be found from superconducting tunnelling measurements, or the high-temperature resistivity [16, 17].

Experimental methods 3.1 Resistivity measurements

3

Experimental methods

In this section, we provide an overview of the experimental techniques used to investigate our samples. We first consider the three cryostats used for resistivity and DC magnetisation measurements, and then provide a description of the high-pressure apparatus used.

3.1

Resistivity measurements

Resistivity is one of very few material properties than can be measured as well at high pressure as at ambient pressure. Resistivity measurements provide a comparatively straightforward diagnosis of significant transitions in the sample, although a detailed analysis can be complex (see Section 2.3). We measure resistivity with a four-point technique: two wires attached to the sample apply cur- rent, and two measure voltage. See Fig. 3.1 for an example. Because the input impedance of the voltmeter is enormous (> 10 MΩ), the voltage wires drain very little current, and the voltage mea- surement therefore does not pick up a significant contribution from the contact resistance between the wires and the sample (which can be several orders of magnitude higher than the resistance of the sample). The resistivity is then given by:

ρ = wt l

V

I (3.1)

where w, t are the sample’s width and thickness, l is the separation between voltage contacts, V is the measured voltage, and I the applied current. Typically currents I ≈ 1 mA are applied, a level at which the resistive heating of the sample is insignificant. For our resistive measurements we use 25 μm 99.999% pure Au wires. These are attached to the samples by two techniques. Spot-welding traps the wire between a sharp tungsten tip and the sample itself, which is positioned on a grounded plate. A sharp current discharge through the tip and sample then heats the wire, bonding it to the sample. Alternatively, contacts are made with conducting epoxy. We use either DuPont 4929 epoxy diluted with 2-n-butoxyethyl acetate (which dries in ∼ 10 minutes at room temperature), or DuPont 6838 epoxy (which requires heat treatment; we typically use 2 hours at 160 °C).

We have found that spot-welding to bismuth works straightforwardly, and makes contacts with a resistance of ∼ 1 Ω at room temperature. However, there is some concern that the significant pres- sure that must be applied by the tungsten tip could damage the crystal structure or cause twinning; the resulting contacts are also quite weak. Contacts made with 6838 epoxy are generally preferred; these are much quicker to prepare (it takes only ∼ 10 minutes to attach four wires to a sample, rather than ∼ 1 hour), but have a higher resistance (∼ 5 − 12 Ω). However, this does not appear to be a problem: in bismuth, features as delicate as quantum oscillations are so large that they can easily be seen even with quite high-resistance contacts. Contacts made with 4929 have a still higher resis- tance (∼ 20 Ω), but also yield perfectly sensible results. The 4929 epoxy makes rather weak bonds, which are prone to breaking when the wires are bent to connect to measurement electronics, so is avoided if possible.

Superconductivity corresponds to the total vanishing of ρ(T ) below T = Tc. Applied magnetic

Experimental methods 3.1 Resistivity measurements

Figure 3.1: A sample of bismuth contacted for resistivity measurements using DuPont 6838 con- ducting epoxy.