Solid State Physics Flashcards
What is a lattice and a basis?
Lattice: Describes the underlying periodicity of the structure.
Basis: Group of atoms being repeated.
What is a Bravais Lattice?
A periodic array of points that look identical from every point.
What is the primitive unit cell?
The minimum volume that fills all space when translated by all lattice vectors.
What is the volume of the primitive unit cell in lattice vectors?
V = |a1 * (a2 x a3)|
How many lattice points does a primitive unit cell contain?
One.
How is the volume of the Weigner-Seitz cell defined?
The volume enclosed by planes that perpendicularly bisect the lines to the nearest lattice points.
How does Bragg reflection work?
Lattice planes reflect x-rays specularly, with constuctive interference when paths drift by n*2pi.
State Bragg’s Law.
2dsin(theta) = n*lamda
For Bragg’s Law, what is the use of using a neutron beam over X-rays?
Neutrons are scattered only by nuclei, not electrons, and so are more useful in studying light elements.
How do you find the Miller indices of a crystal plane?
- Take any lattice plane and find its intercepts, m1, m2, and m3, relative to the crystallographic axis.
- Find reciprocals 1/m1, 1/m2, 1/m3
- Scale all reciprocals by a common factor to find the smallest set of integers; h,k,l.
How is the reciprocal lattice defined?
The reciprocal lattice is the set of all wavevectors k that yield plane waves with the periodicity of a Bravais lattice.
How do reciprocal wavevectors G relate to lattice vectors R?
exp( i(G*R)) = 1
or GR = 2pin
How do lattice vectors a1, a2 and a3 relate to reciprocal lattice vectors b1, b2 and b3?
- b1 = 2pi/V * (a2 x a3)
- b2 = 2pi/V * (a3 x a1)
- b3 = 2pi/V * (a1 x a2)
where volume V = a1 * (a2 x a3)
What is the formula for the volume of a reciprocal unit cell?
V = b1 * (b2 x b3)
What is the formula for distance between reciprocal crystal planes?
d_hkl = 2pi / | G_hkl |
What is the formula for distance between reciprocal planes of a cubic crystal?
d = a / sqrt( h^2 + k^2 + l^2 )
where a = lattice constant
How is momentum transfer Q defined in Laue diffraction, with regard to wave vectors?
Q = k’ - k
What is the Laue condition for diffraction?
Q = G
How is the structure factor S_G defined?
S_G = sum[ f_j * exp( i(-G * r_j)) ]
where f_j = integral [ n_j (rho) * exp( -G * rho) ], the atomic form factor in the amplitude scattered by the electron.
How is the Brillouin zone defined?
The Brillouin zone is the Weigner-Seitz cell of the reciprocal lattice; i.e. it is formed by the volume enclosed of the perpendicular bisectors of lines to nearest atoms.
It has the same volume as the primitive unit cell of the reciprocal lattice.
How does the Brillouin zone play into the Laue condition for diffraction?
When G = Q = k’ - k is satisfied, k terminates of the Brillouin zone boundary. This is a sufficient condition for diffraction; i.e if a wavevector terminates on the BZ boundary, the wave will be diffracted.
What is the dispersion relation for monatomic atoms on a 1D chain?
w^2 = 4C/m * sin^2(ka/2)
where C is the spring constant, m is the atomic mass, and a is the lattice constant.
What does the dispersion relation describe?
It describes how frequency of an elastic wave (w) changes with wavevector k (where k = 2pi/wavelength).
In regards to atomic displacement, why is the first Brillouin zone boundary unique?
The first BZ, i.e -pi/a
To find solutions of 1D chain problems, what steps are needed?
- Use equations of motion (F=ma, F=kx) on displacements of atoms, u_n and v_n.
- Try travelling wave solutions: u_n = u_0exp[ i(kna - wt)]
- Substitute.
- If necessary, express equations in the form of a matrix, and find the determinant for solutions.
What is a phonon?
A phonon is a quantum of energy of a mechanical excitation, similar to photons.
They are bosons with spin 0, and so obey Bose-Einstein statistics: when a mode is excited to quantum number n, it is said to be occupied by n phonons:
n(w) = 1 / [exp( hbar * w / k_b * T) - 1]
Phonons carry energy hbarw and momentum hbark, known as crystal momentum.
How would you express the components of total heat capacity of a generic crystal?
Cv = Cv_phonons + Cv_electrons + Cv_magnetic
Cv_phonons is found in all solids.
Cv_electrons is only found in metals.
Cv_magnetic is only found in magnets.
How does heat capacity Cv relate to energy E and temperature T?
Cv = dE/dT
For a crystal of N atoms, what is its classical heat capacity?
Cv = 3Nk_b
where k_b is the Boltzmann constant. This formula breaks down at low temperatures.
In Debye Theory, what is the formula for total lattice energy E?
E = integral [0 -> w_D] [hbar*w * g(w) * u(w)] dw per branch.
Where g(w) is the density of states and u(w) is the bose-einstein factor, 1/(e^(hbar*w/kt)).
What assumptions does Debye Theory make?
- The crystal is harmonic (independent modes)
- Elastic waves are non-dispersive (w = k*v_s)
- Crystal is isotropic
- There is a high-frequency cutoff, w_D.
In Debye theory, how many unit cells are the there in a crystal?
N = integral [0 -> w_D] {g(w)*dw]
What happens in a ‘Normal’ and an ‘Umklapp’ event? How does it affect heat flow?
- Phonons interact and scatter off each other (k1 + k2 = k3), which defines the thermal conductivity of a crystal.
- In normal events, k3 is within the 1st BZB.
- In Umklapp events, k3 lies outside the 1st BZB, but can be translated into it by reciprocal lattice vector G, producing a phonon sometimes going the opposite direction.
- Heat flow carried by phonons is unaffect by normal (N) events, but is impeded by Umklapp scattering, so U events contribute to thermal resistivity.
What assumptions does the Free Electron Theory make?
- A fixed background of fixed positive charges doe to nuclei
- Valence electrons propagate freely without interacting with ion cores or other electrons.
- Zero potential energy.
How is the speed of sound v_s (i.e. group velocity v_g) related to frequency w and wavevector k? What about phase velocity v_p?
v_s = v_g = dw/dk
v_p = w/k
In regards to interatomic potentials, what is anharmonicity? What are its consequences?
- Anharmonicity is the idea that force exerted on an atom is not directly proportional to its displacement, due to either core repulsion or Coulomb attraction.
- Assuming this, there is thermal expansion: > a, as it costs less energy to expand than contract.
- Phonons also interact with each other, and causes momentary expansion or contraction of a lattice. Phonons scattering off each other determine the thermal conductivity of a solid.
How is thermal current density j_u related to thermal conductivity K and temperature gradient?
j_u = -K * dT/dx
What is the formula for thermal conductivity K, based on kinetic theory of gases?
K = 1/3 * v * l * Cv
where v = average velocity of sound, l = mean free path, and Cv = heat capacity per unit volume
What is mean free path l proportional to? What happens to l as T varies?
- l is proportional to 1 / # phonons
- At high T, (T > Debye temp. T_D) all phonon modes up to hbar*w_D are excited, and so there are many photons with large enough |k| to produce Umklapp events. Energy is proportional to T, so # phonons is proportional to T, and so l is proportional to 1/T.
- At intermediate T (T ~ T_D), U-events start to stop as the average energy of phonons decreases. # phonons ~ exp( -T_D/T), so l is proportional to exp(T_D/T).
- At low T (T
What is the equation for density of states (wrt phonon energy e) in the Free Electron Model?
g(e) = V/2pi^2 * (2m/hbar^2)^(3/2) * e^1/2
This comes from calculating the density of states w.r.t. k, multiplying by 2 to account for spin, and substituting the formula for energy of a phonon.
What is the Fermi Energy? What is the formula for it in the Free Electron Model?
- Fermi Energy is the energy at which states are filled up to at T=0.
- e_F = hbar^2 / 2m * ( 3 pi^2 N / V)^(2/3)
- This corresponds to the filling of all k-states inside a sphere of radius k_F, the Fermi Radius.
What is the formula for the Fermi Radius?
- k_F = (3pi^2 N / V) ^ (2/3)
How is total heat capacity Cv related to T?
Cv_total = Cv_phonon + Cv_electron
= bT^3 + gT
where g = pi^2/2 * N/e_F * k_b * T
What happens to the Fermi Surface in the presence of an electric field?
The Fermi Surface is shifted in the direction of the field, as the electric field applies force to all electrons, via the Lorentz force.
State the Wiedemann-Franz law.
K/(sigma * T) = L
Where K =thermal conductivity, sigma = electric conductivity, and L = Lorentz number (constant).
What is the Hall effect? What is its formula?
- When a metal is placed in magnetic field B and a current j flows through it, a transverse electric field E appears:
- E = R * (B x j)
where R is the Hall coefficient.
What is the formula for the Hall coefficient?
E = - 1/n_e,
where n_e is the electron concentration.
What is the tight binding model?
Electron bands emerge due to overlap of the wavefunctions of isolated atoms as they are brought closer to form a solid.
What is Bloch’s Theorem?
Wave functions in a periodic potential differ from a plane wave of a free electron only by a periodic modulation. I.e, if V(r) = V(r+R) where R is any lattice vector, then irrespective of the form of V(r) the solutions of the Schrodingher equation is of the form:
phi(r) = u(r)exp(ikr)
where u(r) = u(r+R), i.e. u(r) has the periodicity of the lattice.
An alternative statement:
phi(r+R) = exp(ik*R) * phi(r)
What is the Meissner effect?
The exclusion of magnetic flux for the interior of a superconductor.
i.e. Susceptibility chi = -1
What is the critical field?
The magnetic field strength at which superconductivity is destroyed.
What are screening currents?
If the interior magnetic field of a superconductor is 0, then there must be electric currents created to counteract the field, found on the surface of the superconductor to penetration depth lamda.
What is the demagnetisation factor of a long thin rod, a sphere, and an infinite disc? What is the formula for demagnetization?
0, 1/3, 1
H_inside = D * H_applied
What is persistent current in a superconductor?
It is trapped flux that causes a current to flow ‘forever’ around a superconductor, with resistance very very close to 0.
What is the critical current?
Similar to critical field, critical current is the current strength around a superconductor at which superconductivity is destroyed. Also known as the ‘Silsbee Hypothesis’.
What is the temperature dependence of the critical current, Bc?
It is an approximation:
Bc (T) ~ Bc (0) (1 - (T/Tc)^2)
i.e Bc becomes quadratically smaller as T increases, until it reaches critical temperature, Tc.
How is the Gibbs energy of the superconducting state, Gs, and the Gibbs energy of the normal state, Gn, related to critical field, Bc?
Gn (0, T) - Gs (0, T) = Bc^2 / (2* mu_0)
For type-2 superconductors, what happens above the first critical field?
Above Bc, the type-2 superconductor is in a ‘mixed state’, where magnetic flux penetrates the material but doesn’t destroy the superconductivity.
What is the London equation?
j = - (ne^2 / m) A
Where n is the number of electrons per unit volume, e is electron charge, m is electron mass, and A is vector potential.
What relates penetration depth L to magnetic field B?
del^2 B = (1 / L^2) B
What is the equation for magnetic flux Phi in a superconductor?
Phi = n Phi (0),
Where n = integer and Phi (0) = h / 2e
Provides evidence of ‘Cooper Pairs’
What formula determines between type-1 and type-2 superconductors?
- If S/L > 1, it is type-1
- If S/L
What is the difference between direct and indirect band gaps?
If the minimal energy state in the conduction band and the maximal energy state of the valence band has the same k-vector, it is a ‘direct’ band gap, and can be crossed by just a photon of energy ~hbar*w.
An ‘indirect’ band gap has the min state of the conduction band and the max state of the valence band at different k-vectors, and so usually requires both a photon to cross the energy gap, and a phonon to cross the difference in k (i.e. the momentum).
See diagrams in notes.
What is the formula for effective mass of an electron, m_e*?
m_e* = hbar^2 / (d^2 e/ dk^2)
I.e. it depends on band curvature.
What is a ‘hole’?
It represents the empty state where a electron would be in a band, and is treated as a particle in its own right.
It has:
- positive effective mass m*
- positive charge +e
- ‘upwards’ dispersion
What is the law of mass action?
Regardless of doping, the product of electron and hole densities is constant at equilibrium. This constant depends on T:
n_p = N_c * N_v * exp(-e_G / (k_b * T))
Where n_p is carrier density, N_c is the number of electrons, N_v is the number of holes, and e_G is the ground energy (i.e energy at top of valence band).
What is the difference between intrinsic and extrinsic semiconductors?
Intrinsic semiconductors are undoped, while extrinsic semiconductors are doped with impurities.
What is charge neutrality in intrinsic semiconductors?
The number of holes equals the number of electrons, i.e. n=p
What is the formula for chemical potential in semiconductors, mu?
mu (T) = 1/2 e_G + 1/2 k_b T ln( N_v / N_c)
So if T=0 or N_c = N_v, mu = 1/2 e_G
What is the difference between n-type and p-type semiconductors?
Impurities in extrinsic semiconductors can act as either donors or acceptors of electrons.
If donors are the majority, it is n-type.
If acceptors are the majority, it is p-type.
In extrinsic semiconductors, what is carrier concentration, n_i, proportional to?
From the law of mass action,
n_i^2 = n*p is proportional to exp( -e_G / (k_b * T)).
What is the Diode equation?
Current I = I_0 * (exp( eV/kT) - 1)
Where I = current, I_0 = bias current of the diode.
I_0 is proportional to exp( -e_G/kT)
How is magnetism of a solid, M, related to magnetic intensity, H?
M = chi*H,
Where chi = magnetic susceptibility.
What is the difference between diamagnetic materials and paramagnetic materials?
In diamagnets, chi 0
What is the Curie Law?
The Curie Law is the thermodynamic equation of states for an ideal paramagnet:
chi = C/T
Where C is the ‘Curie constant’.
What is the equation for a magnetic dipole moment, mu?
mu = mu_b * l
Where mu_b is the Bohr Magneton.
What is the formula for Zeeman energy in a magnetic field? What is the formula for the magnetic moment operator?
E = -mu * B,
where mu in this case is the dipole moment operator:
mu = -gmu_bJ/hbar
What is the thermodynamic equation for magnetic work?
dW = B * dm
Where B is the applied field and m is magnetic moment.
In regards to angular momentum components L and S, what is the magnetic moment, mu?
mu = -mu_B * (L + 2S)
What is the formula for the thermally averaged magnetic moment?
In direction z:
= g * mu_B * S * tanh ( g * mu_B * S * B / kT)
What is the formula for the thermally averaged magnetic moment of a free ion in a low field?
= g_J^2 * mu_B^2 * J(J+1) * B / 3kT
What is the formula for Pauli paramagnetic susceptibility, chi? What is the free-electron diamagnetic contribution?
- chi_para = (3N * mu_0 * mu_B^2) / 2e_F
- chi_dia = - 1/3 chi_para
What is the formula for electromagnetic momentum?
p = mv + qA
Where q = particle charge, A = vector potential, v = velocity and m = mass.
What is the Curie-Weiss Law?
Chi = M / H = C / (T - theta)
Where C is the Curie constant, and theta is the ‘Curie - Weiss’ constant.
What is the formula for the Curie-Weiss constant, theta?
theta = 1/3 * z * J (S(S+1))
