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Week 8 LNs

R S C Solid State Year 4 Inglés 51 fichas

Can you recall lecture 15s Magnetism points

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1Can you recall lecture 15s Magnetism points
2State the relation for Pauli Spin Magnetisation For paramagnetism in conduction electrons in metals, describe the energy balance that takes place
3Describe ferromagnetism in metals
4Describe the stoner model for ferromagnetsFerromagnetism ultimately stems from Pauli exclusion. The simplified model of a solid which is nowadays usually called the Stoner model, can be formulated in terms of dispersion relations for spin up and spin down electrons.
5Sketch; A schematic band structure for the Stoner model of ferromagnetism
6State the stoner criterion YB
7For Copper, sketch the 4s, 3d shells showing fermi energy. Which are filled Is redistribution possible Is there ferromagnetism
8For Nickel, sketch the 3d up and down shells showing fermi energy for T greater and lower than Tc. Which are filled Is redistribution possible Size of stoner parameter Is there ferromagnetism
9Describe the magnetisation properties of a FMAnisotropic
Easy and hard directions of magnetisation
Heisenberg Hamiltonian is isotropic
10Describe ferromagnetic domainspreferred magnetisation directions,
linked to crystal directions
local magnetisation saturated within each domain;
but overall magnetisation can be very small
domains expand or shrink under applied field
magnetisation rotates only in very strong applied fields
11Describe the origin of Magnetic Domains and Anisotropy; Two types of interactions
12Describe the following FM domains; single, two, closure, numerous small ones, overall optimum
13Describe the length scales of exchange and dipolar interactionsact on different length scales.
14How are domain walls movedDomain walls can be moved easily by applying a magnetic field,
even by a rather weak field
15Decribe domain walls thicknesshave finite thickness with gradual transition of the
direction of the moments (Bloch walls)
16For bloch walls state the boundary energy relation, and what happens for extended domain walls. What limits thickness
17YB; for the susceptiblity of an antiferromagnet, state the relation for Chi for T > TN (Neel Temp),
18In the context of Susceptibility of an Antiferromagnet, what does the magnetic response depend ondepends on the
direction of the applied field Ba
relative to the spin direction
19Can you recall lecture 16s points?
20Describe briefly the origin of the term critical temperatureIn 1911, Kamerlingh Onnes reported the observation that the resistance of mercury dropped to immeasurably small values when
cooled below 4.2 K. The transition temperature between the
normal and this new, superconducting state is generally known
as the critical temperature, Tc
21Describe Kamerlingh Onnes zero resistance experimentIt is impossible to prove scientifically that the
resistance is zero. All we can do is to show that the resistance
is smaller than a certain upper limit. In this spirit, Kamerlingh
Onnes performed resistance measurements on long coiled wires
and concluded that the resistance in superconducting mercury
is at least ten orders of magnitude lower than at room temperature (in the sense of an upper limit).
Kamerlingh Onnes et al. also devised an ingenious method for
demonstrating that the resistance is in fact zero for all practical
purposes. They cooled a closed metal ring in the presence of
a magnetic field to below the superconducting transition temperature. Then they removed the field and induced thereby
an electrical current in the closed superconducting ring. By
measuring the magnetic field of this ring current, they could
monitor the ring current itself, without introducing a normalconducting ammeter into the circuit. Kamerlingh Onnes et
al. could not detect any change over several hours
22Describe the failures of Kamerlingh Onnes when using high electrical currentsMany of Kamerlingh Onnes’ attempts to measure the electrical
resistance in superconducting wires failed when high electrical
currents were used (so as to become more sensitive). Eventually,
it became clear that the current or current density as such was
not the issue, but rather the high magnetic field produced by a
large current passing through a coil. The figure above shows how
the application of a moderately strong magnetic field of order
0.1 T (1000 Gauss) destroyed superconductivity in Pb (lead)
wires, even though they remained below the critical temperature
of 7.2 K (at 2 and 4.25 K). It can also be seen that the sharp
transition from zero to the finite resistance of the normal state
occurred at a higher critical magnetic field when the sample was
at a lower temperature.
23For Tl, Hg, Pb, sketch the critical magnetic field vs temperature in K What is this curve?
24Define and describe the Meissner effectThe Meissner effect (or Meißner–Ochsenfeld effect) is the expulsion of a magnetic field from a superconductor during its transition to the superconducting state when it is cooled below the critical temperature. This expulsion will repel a nearby magnet.
A magnetic
field gets always expelled from the interior of a superconductor,
no matter whether the material is cooled from its normal- to
its superconducting state with the magnetic field applied or
without it (and then applied later). This behaviour makes a real
superconductor distinct from a hypothetical perfect conductor
with zero electrical resistance as described by Maxwell’s equations. For the latter, the order in which the sample is made
zero-resistive (by cooling) and the magnetic field is applied
matters.
25Describe the 4 cases in the diagram3 and 4; For a superconductor, the situation is much simpler:
While in its superconducting state, the magnetic field
inside is always zero. It does not matter in which order
the sample was cooled and the external magnetic field
changed. This, however, cannot be understood on the
basis of Maxwell’s equations
26Explain magnetic levitation and suspensionIt is the expulsion of a magnetic field from the interior of a
superconductor that produces the magnetic levitation shown
in the photo above. When a magnet is brought near a superconductor, the field of the magnet induces an equally strong
magnetisation in the superconductor (M = −Ha thanks to the
perfectly diamagnetic susceptibility of χ = −1), which creates
a repulsion between magnet and superconductor. No matter
how the magnet might move and tumble, the magnetisation
of the superconductor will constantly adjust, which can then
create stable levitation. Once the magnet has been brought into
the vicinity of the superconductor and a stable magnetic configuration been achieved, the superconductor actually resists any
change. Not only can the magnet not sink any further down,
one also needs to do some work to lift it off the superconductor. This explains why both magnetic levitation and suspension can
be achieved.
27Define 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.
28State the differences between type 1 and type 2 SC from graphhe 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.
29Describe the benefits of type 2 SC in applicationsType-II superconductors are of interest for practical application because superconductivity in these can be retained up to
much higher magnetic fields than for type-I superconductors.
The above slide illustrates how the type-I superconductor lead
turns into a type-II superconductor after alloying with moderate
amounts of indium
30Sketch C/T vs T^2K^2 showing heat capacity of normal metal and of a superconductor and describe itMeasurements of the heat capacity C(T) yield important insight
on the evolution of the internal energy U of a superconductor
with temperature. The slide above shows the familiar linear
variation of C(T)/T vs. T^2
for gallium metal in its normalconducting state. The slope relates to the Debye/phonon contribution, while the intercept at T = 0 gives the Sommerfeld
constant γ that describes the electronic contribution to the heat
capacity. (Superconductivity was suppressed here by the application of a magnetic field.) Without a magnetic field applied,
the transition from the normal to the superconducting state at
T^2 = 1.18 K2
(T ≈ 1.1 K) can be seen in the form of a jump in
C/T (and thus the heat capacity itself). The more important
observation is that the heat capacity vanishes for T → 0 — the
electronic contribution to the heat capacity becomes zero! The
simplest explanation of this is that there is an excitation gap in
the electronic excitation spectrum of a superconductor, with a
magnitude of the order of meV
31What are the consequences of the excitation gap in superconductorsSuch an excitation gap prevents thermal excitation of the electrons from their ground state as T → 0. This bears some resemblance to the electronic excitation gap or band gap in semiconductors, albeit on a very different energy scale and for very
different reasons. The superconducting excitation gap varies
with temperature with the gap being the largest at zero kelvin
and a reduction to zero as the critical temperature is approached.
32State the relation for Tc varying with average atomic mass in isotope mixtures
33Describe the parts of the periodic table to form superconductorsMany chemical elements are superconductors (at sufficiently
low temperature, the usual caveat). Many elements in the middle
of the periodic table are superconductors already at normal
pressure, whereas the main-group elements on the right side of
the periodic table need to be subjected to high pressure. This
turns these semiconductors into metals, which then become
superconducting at sufficiently low temperatures. Magnetism
and superconductivity are competing forms of electronic order
and generally mutually exclusive. This is nicely demonstrated
by iron, which is ferromagnetic and not superconducting at
normal and low pressures. At 21 GPa, it undergoes a structural
phase transition, where it loses its ferromagnetism and becomes
superconducting
34Describe the classes of superconducting compounds I1. Most commonly used in superconducting magnets are
simple binary alloys and intermetallic compounds such
as Nb3
Sn, V3
Si and Nb3Ge, whose Tc
’s are mostly below 30 K, but which can be turned into wires and coiled.
MgB2 mentioned in text context of DFT also belongs to
this group and is the member with the highest Tc of 39 K.
2. The highest critical temperatures at normal pressure are
observed in the cuprate superconductors, also known as
the high-Tc superconductors, even if there are now noncuprate superconductors with similarly high Tc
’s. They
are attractive for technical applications because cooling
with liquid nitrogen (77 K) would suffice, which is far
less expensive than the still-standard cooling with liquid
helium, but turning these brittle ceramic materials into
wires and coils remains a challenge.
35Describe the classes of superconducting compounds II3. Superconductors containing lanthanide and actinides elements such as uranium and cerium form the group of
heavy-fermion superconductors. Even though they generally have rather low Tc
’s, they are of interest because
many of these are close to magnetic phases, and superconductivity in some of these is understood to be driven
by magnetic interactions.
4. Iron pnictides are a fairly recent addition. Initially considered another iron-containing curiosity, critical temperatures of up to ∼100 K have now been achieved in this
group.
36Describe the classes of superconducting compounds III5. The most recent discoveries concern superconductivity
in hydrogen-rich compounds such as hydrogen sulfide,
lanthanum hydride and lutetium hydride. Hydrogen sulfide and lanthanum hydride were found to be superconducting with critical temperatures of ∼200 K and ∼250 K,
respectively, but only under extremely high pressures of
around 2 megabar (200 GPa). This makes experimental
investigations of these materials extremely challenging,
and there are only few researchers worldwide who have
the experience, skills and instrumentation to perform
them.
37Can you recall Ch 16s main points
38Define the London equationsThey are constitutive relations for a superconductor relating its superconducting current density to electromagnetic fields in and around it.
As a material drops below its superconducting critical temperature, magnetic fields within the material are expelled via the Meissner effect. The London equations give a quantitative explanation of this effect.
39Exam; State the London equation and explain which experimental observation necessitated this addition to Maxwell’s equations.The experimentally observed
magnetic properties of a superconductor differ markedly from
what one would expect for a (hypothetical) perfect electrical conductor on the basis of classical electromagnetism, i.e., Maxwell’s
equations. The crucial point was that superconductors exhibit
the Meissner effect, i.e., the repulsion of any magnetic field from
the interior of a superconductor, whereas a hypothetical perfect conductor would show more complicated behaviour that
depends on the sample history. Maxwell’s equations do not describe the Meissner effect. Apart
from showing their limitations, this suggests that superconductivity originates from physics beyond classical electromagnetism.
The London Equation describes the Meissner effect of superconductors.
Symbols;
j; superconducting current density
n; charge carrier density
B; magnetic field,
lamL; london penetration depth,
uo; permeability of free space
40What is the result of the London equation to define the London Penetration depthone
finds that a magnetic field does not drop abruptly to zero inside a
superconductor, but rather decreases to zero exponentially. We
will see that the characteristic length scale of this reduction is
given by the London penetration depth λL
41Explain the physics of the following relation from the London equation and Meissner effectThis shows that a uniform or constant magnetic field cannot
exist inside a superconductor (unless it is zero everywhere, trivially). An exponentially increasing magnetic field would yield
a mathematical solution to (1) but is physically not plausible.
What is left is an exponentially damped magnetic field.
42—With the given geometry, the vector relation (1) simplifies to
a one-dimensional problem along x. The general solution to
this simple second-order differential is given by the sum of an
exponentially increasing and an exponentially decaying term.
The former is physically not plausible — the magnetic field
cannot increase towards infinity inside the superconductor. The
constants a and α for the exponentially damped solution can be determined from the boundary condition at the surface. The final result allows one to see that the London penetration depth λL is indeed the
characteristic distance over which the magnetic field decreases
inside the superconductor. Unsurprisingly, this length scales
with the inverse of the electron density n
43What is another length that determines properties of a superconductor, what is its originCoherence Length; The starting point is that the London equation is a local equation,
which implies that the current density j (and thus charge density n) follows any applied magnetic field exactly, even if it had
abrupt, step-like changes. However, the standard kinetic energy
operator from quantum mechanics, Tˆ = −
ħ
2
2m ∇
2
, suggests that
abrupt changes in the charge density (or the underlying wave
function) are associated with additional kinetic energy. Consequently, superconductors have an intrinsic, minimum length
scale — the coherence length — over which the charge and
current densities can vary without breaking superconductivity.
44When does the superconducting state break downbreaks down if the kinetic energy of the electrons
increases by more than the magnitude of the energy gap
45Why was Ginzburg Landau theory introducedIt was developed as a
phenomenological theory of electrons in the superconducting
state, well before the microscopic origin of superconductivity
was understood. It remains the theory of choice for describing
the macroscopic properties of superconductors
46What determines whether a superconductor is type I or type IIThe ratio of two characteristic lengths, London penetration
depth λL and intrinsic coherence length ξ0
47Explain the origins or the critical field and include a sketchAt low field, the superconducting
state must have a lower energy than the normal state, or else
the former could not be observed. As derived above, the energy
of the superconducting state increases quadratically with the
applied field, whereas the energy of the normal states remains
essentially constant. Consequently, there must be a critical field,
Bac, at which both states have the same energy. Above the critical field, the normal state becomes the stable state as it has
the lower free energy. In other words, an applied magnetic field
induces a transition from the superconducting to the normal
state if it exceeds the critical field Bac. Its value is determined by
the condensation energy Fc
, which is difference in free energy
between the normal and the superconducting states at zero field,
and thus the energy gained by adopting the superconducting
state at zero field.
48Explain BCS theoryThe BCS theory, named after the authors Bardeen, Cooper and
Schrieffer of the original formulation, is a microscopic quantum
theory of a certain class of superconductors.
The most important points that a theory of superconductivity must explain are (i) the origin of zero electrical resistance,
(ii) the origin of the energy gap in the excitation spectrum and
(iii) the origin of the isotope effect. A key ingredient for all
known types of superconductivity are electron pairs — the socalled Cooper pairs.
Basis; One can imagine how
the two electrons move through the crystal together as a pair. A
more careful consideration shows that the coupling is stronger if
the two electrons move in opposite directions, i.e., through each
other’s lattice distortions, which gives the result that electrons
of opposite wave vectors form a Cooper pair. At this point, a
model based on classical particle starts to fail. The BCS theory
is the quantum-mechanical treatment based on this idea.
49Define a Cooper PairA Cooper
pair is a quantum-mechanical superposition of two electron
states. Before coupling, the two states have opposite wave vectors,
±k, and opposite spins. The pair binding energy (analogous to
the binding energy in a molecule, but for two electrons instead
of atoms) is what creates the excitation gap: a certain amount
of energy must be added to the system to break up a Cooper
pair
50Define the intrinsic coherence lengthThe intrinsic coherence length ξ0 introduced earlier is a measure of the correlation length or typical distance between the
electrons in a Cooper pair. A stronger interaction leads to a
smaller correlation length (“smaller Cooper pairs”), and the resulting electron pairs require more energy to be broken up than
for a weaker interaction. This is equivalent to a larger energy
gap Egap, and one can now understand the physical meaning of
the expression for the coherence length ξ0
51Can you recall Chapter 17s points