Concepts in Condensed Matter Physics Flashcards
Define ‘equilibrium’.
Substance is in a uniform state.
Define ‘phase’.
Material of specific composition in a specific state.
Define ‘phase transition’.
Phases in equilibrium with each other.
What is the Lennard-Jones potential, V(r)?
V(r) = 4ε [(σ/r)^12 - (σ/r)^6]
Where:
r is the inter-atomic spacing
σ is the separation for V = 0 (~0.3nm)
ε is the minimum potential (at r(m) )
Describe a single crystal.
Single crystal:
* macroscopic specimin with uniform crystal orientation
* crystal growth > nucleation
Define ‘nucleation’.
The process of a substance changing from one state to another.
Describe a polycrystalline material.
Polycrystalline material:
* conglomerate of grains with different orientation
* nucleation > crystal growth
Define ‘conglomerate’.
Gather together into a compact mass.
Describe a ‘grain’.
Grain:
* Uniform composition
* Uniform crystal structure
* Uniform orientation
Describe a ‘grain boundary’.
Grain boundary:
* Interface between grains
* Disrupted long-range order
* Intersecting lattice planes
* Boundary disorder
Define ‘lattice’.
An array of points that represent identical locations in the crystal.
Define ‘unit cell’.
A repeat unit in 2D that generates the crystal.
Define ‘lattice vectors’.
Two vectors in 2D that define all points in the lattice,
r = n(a) 𝐚 + n(b) 𝐛
Define ‘primitive cell’.
A unit cell that contains only one lattice point. The area of the cell,
𝐀 = 𝐚 × 𝐛
Define ‘basis’.
Repeating shape at each lattice point.
Define ‘crystallography’.
Crystallography:
* Primitive lattice preferred
* Atoms (or groups of atoms or molecules) located at lattice points
* Unit cell as compact as possible
Define ‘Bravais Lattice’.
A lattice that fills space without gaps or overlaps.
The number of possible Bravais lattices is ______:
The number of possible Bravais lattices is finite:
* 5 in 2D
* 14 in 3D
What do crystals with the same Bravais lattice have?
- The same symmetry
- Different parameter values
What is a ‘symmetry element’?
An operation that maps each lattice point onto another.
Give the 7 crystal classes (lattice types).
- Cubic
- Tetragonal
- Ortho-rhombic
- Monoclinic
- Hexagonal
- Trigonal
- Triclinic
What is the FCC structure?
Face-Centred Cubic
What is the HCP structure?
Hexagonal Close-Packed
What is the BCC structure?
Body-Centred Cubic
What is a Wigner-Seitz cell?
It is an example of a primitive cell, which is a unit cell containing exactly one lattice point.
What is a phase diagram?
A phase diagram links any two state variables,
e.g. pressure and temperature,
and shows regions in which a substance in equilibrium is in a uniform state.
What is “short range order”?
Short range order refers to the
regular and predictable arrangement of atoms over a short distance,
usually with one or two atom spacings.
What does “amorphous” mean?
“without a clearly defined shape or form.”
“(of a solid) not crystalline, or not apparently crystalline”
What is “long range order”?
Long range order refers to a
regular pattern of arrangement of particles
which repeats itself periodically over the entire crystal.
Which Bravais lattice (in three dimensions) has three different lattice parameters, all angles identical, and four atoms in the unit cell?
Face-centred orthorhombic
What are the two major classes of semiconductors?
Intrinsic and extrinsic
What does intrinsic mean in regards to semiconductors?
Intrinsic, in which the electrical properties of the semiconductor are determined by the host material.
What does extrinsic mean in regards to semiconductors?
Extrinsic, in which the electrical properties are determined by chemical impurities, or dopants.
What is reciprocal space?
Reciprocal space is the Fourier Transform of the real space.
What is the Einstein model for crystal lattices?
It is a model for the heat capacity of a lattice.
It assumes that all the oscillators have the same frequency 𝜔0.
What is the Debye model for crystal lattices?
The Debye model is an improved Einstein model by
incorporating the long-wavelength acoustic phonons,
ignoring any zone boundary effects.
What are the differences between the Einstein and Debye models for crystal lattices?
Einstein model:
* All modes in a crystal have the same fundamental frequency
Debye model:
* Modes depend on crystal structure
* Standing waves in the crystal lattice
What is the dispersion relationship for 1D monatomic chain model?
𝜔(k) = √(4l/m) * |sin(ka/2)|
Explain the arrangement of
atoms in a crystalline solid.
- Orderly 3D array
- Long range order
- Uniform structure
Describe Bragg’s Law.
Bragg’s Law explains the relationship between an x-ray shooting into and its reflection off from a crystal surface.
What is Bragg’s Law?
nλ = 2dsinθ
Where:
λ = wavelength of the x-ray
d = spacing of the crystal layers
θ = is the incident angle (between incident ray and scatter plane)
n = integer
What is diamagnetism?
Diamagnetism is based on the interaction between electrons and the magnetic field.
Since all materials contain electrons, all materials are diamagnetic.
What is paramagnetism?
Paramagnetism is due to permanent magnetic moments of atoms.
Not all types of atom have magnetic moments, so this mechanism is not universal.
What is ferromagnetism?
Ferromagnetism arises when permanent magnetic moments are strong enough to interact and influence each other.
Describe Lenz’s Law.
● A magnetic field induces a current
● The resulting current induces a magnetic field
● Which opposes the original field
What is the Curie Law?
Curie’s Law shows that the paramagnetic susceptibility, χpara, scales inversely with temperature.
χpara = C/T
Where C is the Curie constant.
Explain why there are only a finite number of Bravais lattices in 3D.
There are 14 Bravais lattices in 3D.
No more are needed because these describe all the possible ways of filling space using a lattice defined by two or three lattice vectors according to
R = n(1)a + n(2)b + n(3)c
How many atoms are there in a simple cubic lattice?
1 atom per simple cubic unit cell
How many atoms are there in a body-centred cubic lattice?
2 atoms per BCC unit cell
How many atoms are there in a face-centred cubic lattice?
4 atoms per FCC unit cell
Describe why an organic (polymer) semiconductor might be a poor choice for use in high-performance microelectronics.
These materials tend have amorphous microstructures,
and the resultant high scattering rates
and low carrier mobilities will preclude their use in such applications.
What is the formula for the Hall coefficient?
R(H) = 1 / nq = E(y) / J(x) B
Where:
* R(H) is the Hall coefficient
* n is the number of carriers
* q is the charge
* E(y) is the induced electric field
* J(x) is the current density
* B is the magnetic flux density
What is current density?
j = total current / total area
Units: A m^(-2)
What is the Hall voltage?
V(H) = IB / qnd
Where:
* V(H) is the Hall voltage
* I is the current
* B is the magnetic field strength
* q is the charge
* n is the number of charge carriers
* d is the separation
Describe applications for the Hall effect.
- Detection and measurement applications
Hall-effect sensors:
- Simple, inexpensive, electronic chips
- Positioning; e.g. disk-drive motors, hydraulics
- Velocity measurement; e.g. rotation of engine or crankshaft, anti-lock braking systems
Hall-effect probes:
- Expensive and sophisticated instruments
- Used in laboratories for measuring magnetic field strength with very high precision
What are the 5 types of centering for Bravais lattices in 3D?
- Body-centred
- Face-centred
- Side-centred
- Primitive
- No centring
What is meant by Brillouin zone?
The first Brillouin zone is a uniquely defined primitive cell in reciprocal space.
It is the set of points in k-space that can be reached from the origin without crossing any Bragg plane.
Imagine a simple cubic lattice with length a.
What is the (001) plane?
(001) Plane:
- Intersects the z-axis at z = a
- Parallel to the x-y plane
Imagine a simple cubic lattice with length a.
What is the (110) plane?
(110) Plane:
- Intersects the x-axis at x = a
- Intersects the y-axis at y = a
- Parallel to the z-axis
Imagine a simple cubic lattice with length a.
What is the (111) plane?
(111) Plane
- Intersects the x-axis at x = a
- Intersects the y-axis at y = a
- Intersects the z-axis at z = a
Describe longitudinal waves in 1D.
- Wave and displacement vectors are parallel
- Atoms move along the chain
- Analogous to beads in a necklace where the beads can slip along the string
Describe transverse waves in 1D.
- Wave and displacement vectors are perpendicular
- Atoms move sideways
- Analogous to beads in a necklace where the beads are tied to the string
Describe longitudinal waves in 3D.
- Wave and displacement vectors are parallel
- Planes move parallel to the wave vector
- Lattice spacing changes periodically
Describe transverse waves in 3D.
- Wave and displacement vectors are perpendicular
- Planes move sideways
- Lattice spacing does not change
In the context of extrinsic doping in semiconductors, explain what is meant by the term donors.
Donors are dopant atoms with an additional valence electron, which may be ionized to generate a free electron carrier.
In the context of extrinsic doping in semiconductors, explain what is meant by the term acceptors.
Acceptors are dopant atoms with fewer valence electrons,
which may produce vacant electron states (“holes”) by ionization.
In the context of extrinsic doping in semiconductors, explain what are meant by the term n-type semiconductors.
In an n-type semiconductor,
electrons are the majority carrier type
whilst holes are the minority carrier type.
In the context of extrinsic doping in semiconductors, explain what are meant by the term p-type semiconductors.
In a p-type semiconductor,
electrons are the minority carrier type
whilst holes are the majority carrier type.
In the context of extrinsic doping in semiconductors, explain what are meant by a compensated semiconductor.
In a compensated semiconductor, the number density of donor and acceptor dopants are closely matched.
Because of the principle of “mass action”, this results in the effects of extrinsic doping being cancelled out in the semiconductor as a whole,
producing properties similar to an intrinsic (undoped) system.
For an intrinsic semiconductor, every electron excited into the conduction band leaves a hole in the valence band, so that…
np =n(i)^2
Where:
n is the number of electrons per unit volume
p is the hole density
n(i) is the intrinsic carrier concentration
What are the two types of unit cell?
Primitive and non-primative.
What is a primitive unit cell?
Primitive unit cells contain only one lattice point,
which is made up from the lattice points at each of the corners.
What is a non-primitive unit cell?
Non-primitive unit cells contain additional lattice points,
either on a face of the unit cell or within the unit cell,
and so have more than one lattice point per unit cell.
Describe one method for measuring the atomic spacing in a 3-dimensional crystal using diffraction.
X-Ray or electron diffraction.
The wavelength of the radiation needs to be of the order of the lattice spacing, ~0.1nm.
How does the electrical conductivity of a semiconductor vary as a function of temperature?
Exponential increase
How does the electrical conductivity of a conductor (metal) vary as a function of temperature?
Linear decrease
