Karteikarten
Final Week LNS
Karten in diesem Stapel (33)
- 1What does the formation of cooper pair implyThe formation of Cooper pairs implies that the spatial distribution of the electrons in the superconducting state deviates from
that in the normal electron gas. The formation of pairs leads to the occupation of higher kinetic energy states, the total energy lowered by pair binding energy. - 2Explain the 2∆(0) relationIts the relation is the energy required to break a Cooper pair. All Cooper pairs are in the same ground
state, which has a distinct energy. A Cooper pair can be broken
up by adding an energy of 2∆ (i.e., ∆ per electron in the pair).
The free electrons will then occupy a continuum of states at
higher energies. The excitation gap is temperature dependent
as noted in the previous lecture. It has the largest value at zero
kelvin and decreases to zero at the superconducting transition
temperature, Tc
. BCS theory relates the zero-temperature gap
to Tc through this relation. - 3For an electron fermi gas at 0K, describe energy when removing an electron from an occupied state and adding an electron to an unoccupied state. Sketch energy and quasiparticles energies as a function of wave vector showing quasi holes and quasi electrons
- 4Describe the superconducting quasiparticle dispersionIn the superconducting state, the addition of an electron or hole
quasiparticle requires at least an energy ∆ per particle because
the continuum of electron states starts at an energy ∆ above
the Cooper pair ground state. This modifies the quasiparticle
dispersion near EF . Without going through the theory, the
resulting dispersion for the superconducting quasiparticles is
given by relation shown. e En(k) is the original electron dispersion relation. Correspondingly, one obtains a modified density of states, Ds(E), for
the superconducting quasiparticles. It deviates from the normal
electron density of states, Dn(E), only within a few ∆ (typically
a few meV) around the Fermi energy (several eV). This is a
representation of the superconducting excitation gap (and the
pair breaking energy) in the energy spectrum of the electron
quasiparticles. The electron states that were within ±∆ of EF are
now redistributed to lower and higher energies, creating the gap
and two peaks in the density of states.
- 5Describe scattering of Cooper pairsCooper pairs are
electrically charged and carry an electrical current. However,
as all Cooper pairs form a collective state, they are not affected
by scattering events in the same way as individual electrons.
The wave vectors of individual electrons or phonons are much
smaller than the wave vector of the macroscopic boson state, so
that the scattering does not affect a superconducting current. A
complete lack of effective scattering processes leads naturally
to zero electrical resistance. Note that scattering events may
break individual Cooper pairs, but this does not destroy the
superconducting state as a whole. - 6Describe the following relationWithin BCS theory, the superconducting transition temperature
Tc can be expressed (approximately) in terms of three material
properties. The Debye temperature θD represents the average
phonon frequency of the system and characterises how quickly
the crystal lattice can distort around the moving electrons. The
electron density of states (of the normal state!) is a measure of
how many electrons can contribute to the superconducting state.
Finally, the strength of the electron-phonon (or electron-lattice)
coupling is quantified by the parameter V. From the expression
for Tc
, it becomes clear that the isotope effect originates indeed
from the mass dependence of the phonon frequencies.
- 7Describe the setup of a metal superconductor tunnelling experimentFor such a measurement, a thin film of either the metal or the
superconductor is deposited on a substrate in the form of a
strip that connects two of the four electrodes. A thin oxide
surface oxide layer is then formed by exposing the strip to a low
concentration of oxygen. Subsequently, the other material is
deposited on top, forming a strip that connects the other two
electrodes. This creates the metal-insulator-superconductor
structure where the two strips overlap (at the centre of the X). A
regulated current I is then passed through the sandwich while
the voltage drop U across the structure is measured. Measuring
U(I) rather than I(U) is done for experimental reasons, to
achieve better accuracy. In the end, a voltage is applied, which
produces a current through the structure. - 8Explain the metal insulator curve of the tunnelling curve; current increasing linearly with applied voltageThe origin of the linearI(U) relation for a metal-insulator-metal
contact can be understood by considering the electronic densities of states in the metals on either side of the insulating barrier.
- 9Explain the metal insulator curve of the tunnelling curve; what is considered process to enable tunnelling from small applied voltage result
- 10For tunnelling, explain what happens at finite T but below TcAt finite temperatures, T, but below Tc
, the I(U) characteristic
becomes smoother. This originates from the more gradual transition from occupied to unoccupied states in the metal, over
energies of order kBT, and correspondingly the existence of
some occupied (unoccupied) states above (below) the gap in
the superconductor. A small tunnelling current can then occur
already at voltages well below ∆(T)/e, but the pronounced increase in tunnelling remains at ∆(T)/e. This provides a means
of measuring the temperature dependence of the gap, ∆(T),
accurately. - 11What observation can be made of the curve for single particle tunnelling experimentsgap is essentially constant
for temperatures below half of Tc
, and it becomes zero at T
- 12Explain that the magnetic flux Φ through the hole in a superconducting ring can only be changed in discrete stepsIt is important to bear in mind that the interior of the superconducting ring is field-free because of the Meissner effect. (For
simplicity, we consider a type-I superconductor.) The external
magnetic field may induce a current, but it will be confined to
a thin surface layer of the material. “Thin” means of the order
of the London penetration depth. The basic idea is now to follow the superconducting ground state around the ring, on a
path inside the material and away from the surface layer. The
phase of the wave function will change along the path. When
we return to the starting point, the total phase change must be a
multiple of 2π so that the wave function is unambiguous. This
condition leads to the quantisation of the flux. - 13State the quantisation of the magnetic flux for a superconducting ring What does it predict?n; integer
h; planks constant
2e; charge of a cooper pair
Φ0; magnetic flux quantum
result predicts that the magnetic
flux through the ring changes in discrete steps even if one
changes the applied magnetic field continuously
- 14Can you recall Ch 18s points?
- 15Define type-I superconductorType I; The interior of a bulk superconductor cannot be penetrated by a weak magnetic field, a phenomenon known as the Meissner effect. When the applied magnetic field becomes too large, superconductivity breaks down. Superconductors can be divided into two types according to how this breakdown occurs. In type-I superconductors, superconductivity is abruptly destroyed via a first order phase transition when the strength of the applied field rises above a critical value Hc. This type of superconductivity is normally exhibited by pure metals, e.g. aluminium, lead, and mercury.
- 16Define type II superconductorIn superconductivity, a type-II superconductor is a superconductor that exhibits an intermediate phase of mixed ordinary and superconducting properties at intermediate temperature and fields above the superconducting phases. It also features the formation of magnetic field vortices with an applied external magnetic field. This occurs above a certain critical field strength Hc1. The vortex density increases with increasing field strength. At a higher critical field Hc2, superconductivity is destroyed. Type-II superconductors do not exhibit a complete Meissner effect
- 17With a sketch and description outline the differences between type I and II superconductorsThe main difference can be seen in the magnetisation of the two
types of superconductor as a function of the applied field. As
discussed above, the magnetisation of a type-I superconductor
increases linearly with the applied field up to the critical field
Hc
, at which point superconductivity breaks down, and the
magnetisation becomes essentially zero (from weak para- or
diamagnetism).
A type-II superconductor behaves similarly for low applied
fields up to its first critical field, Hc1
, but the magnetisation then
decreases gradually upon increasing the applied field further.
The superconductor retains zero electrical resistance in this
intermediate region, and superconductivity completely disappears only above the second critical field, Hc2. Between Hc1 and
Hc2, a type-II superconductor is in a mixed state in which both
super- and normal-conducting regions are present, the latter in
the form of channels or tubes embedded in a superconducting
matrix. The magnetic field penetrates the superconductor only
through these normal-state channels.
- 18How is the mixed state in type II superconductors understoodFlux Quantisation
- 19Describe the flux lines in type II SCThey tend to form a regular hexagonal lattice
- 20Describe the concept of flux quantisation in superconductors. Describe an aspect of the superconducting state that is revealed through flux quantisation experiments.The origin of flux quantisation has been shown through experiments where magnetic flux through a hole in a superconducting ring can only be changed in discrete steps. Experiments managed to quantify and trap flux in the ring reaching expectations for particles carrying two elementary charges in the superconducting state. These experiments showed evidence for the macroscopic quantum ground state and Cooper pairs having two electrons each.
Also, Flux quantisation is the key to understanding the mixed state
of type-II superconductors. - 21Describe the Vortex stateIn the mixed state of a type-II superconductor, normalconducting lines (or tubes or vortices) form in the otherwise
superconducting material. These vortices are parallel to the
applied magnetic field. The Meissner effect persists in the superconducting regions,
but not in the normal-conducting vortices, so that the overall magnetisation decreases in the mixed region. The gradual reduction of the magnetisation as the second critical field, Hc2,
is approached suggests that the density of vortices increases as
the applied magnetic field is increased. - 22Describe magnetic flux lines and how they are visualisedAs the non-superconducting vortices allow a magnetic field to
go through the otherwise superconducting sample, they are
also called (magnetic) flux lines. They can be visualised with
the Bitter decoration technique - 23For the vortex state describe how vortices repelling each other lead to a 2D hexagonal lattice
- 24What suggests the existence of an energy gap in superconductorsthe observation
of an exponential temperature dependence for the electronic
heat capacity in a superconductor suggests the existence of an
energy gap - 25State the DC Joesphson effectConsider two separated, independent superconductors. The
phases of the two wave functions are completely unrelated, so
that the phase difference can have any value once the superconductors are brought into contact. With this, the inconspicuous
result for the current becomes rather surprising. It holds for
any potential difference between the two sides, including zero
voltage. In other words, a constant superconducting current
passes through the barrier even if no voltage is applied. This is
known as the DC Josephson effect. - 26Describe AC Joesphson effecthow the superconductor-insulatorsuperconductor structure responds to a constant applied voltage.
Surprisingly, this gives rise to a high-frequency oscillating (AC)
current! This is know as the AC Josephson effect. t a constant applied voltage V causes the phase difference δ(t) between the
wave functions on either side of the barrier to change linearly
with time. From the result for the DC Josephson current it follows immediately that the current becomes now a sinusoidally
oscillating current. - 27For a loop of superconductor with two insulating layers, describe the diagram; Changing phase of SC wave function What effect is found for both junctions Describe current Resultthe phase of the superconducting wave function changes by (2e/ħ) Φ if the loop encloses a magnetic flux Φ.
The DC Josephson effect occurs for both junctions, giving rise
to DC currents Ja and Jb for the two junctions. The total current
Jtotal between points 1 and 2 through the loop is given by the
sum of Ja and Jb along the upper and lower part of the loop.
After plugging in the expression for the two phases changes and
combining the two sine terms, one obtain the magnetic flux
dependence of the total current through the loop. This shows that the combined DC Josephson current should
exhibit oscillations if the applied magnetic field (and thus the
flux Φ) is changed.
- 28What is used to measure SQUIDSThe maximum current, Jmax, exhibits the expected oscillatory
behaviour. More rapid oscillations are observed for larger loops.
The oscillation amplitude decreases with increasing field due
to some additional quantum interference effects inside the insulating layers of finite width. These current oscillations as a
function of the applied magnetic field are used in the most sensitive devices for measuring weak magnetic fields: SQUIDS —
Superconducting Quantum Interference Devices. - 29What is an advantage and disadvantage of SQUIDSThe sensitivity is high
enough to detect even the extremely weak magnetic fields that
are produced by electric currents during brain activity. With
a whole array of such sensors placed around a person’s head,
spatially resolved brain activity can be detected (at least in prototype systems). The drawback of SQUIDs is that they rely on
superconductivity, which requires such devices to be operated at
temperatures below the superconducting transition, which can
be anywhere below 100 K, depending on the superconducting
material used. - 30Describe heavy fermion superconductors
- 31Define the Jahn Teller effectThe Jahn–Teller effect (JT effect or JTE) is an important mechanism of spontaneous symmetry breaking in molecular and solid-state systems which has far-reaching consequences in different fields. This effect arises where transition metal oxides exhibit strong electron phonon coupling.
- 32Can you recall 19
- 33—