An introduction to solid state physics.

Logistics

  • Location: 307 Pupin
  • Time: 2:40-3:55 PM Tuesdays and Thursdays
  • Textbook: The Oxford Solid State Basics
  • Grading Scheme
    • 30% problem sets (5 in total. 1 per 2 weeks)
      • Due on Saturdays at noon
    • 20% midterm
      • Closed book
    • 20% research project presentation
    • 30% final exam

Heat Capacity of Solids

  • Heat capacity is defined as $C = \frac{\partial Q}{\partial T}$
    • For solids, heat capacity at constant pressure and at constant volume are essentially the same, since the atoms are so closely packed
  • For a gas, we have that $\frac{C_{v}}{N} = \frac{3}{2} k_{b}$
  • For (most) solids, we know that $\frac{C}{N} = 3k_{b}$. Called the law of Dulong-Petit
  • For these derive the above (like Boltzmann did), pretend that we have N atoms trapped in a box.
    • Assume that each atom is trapped within a harmonic potential well of the form $U = \frac{k|\vec{r}|^{2}}{2}$
      • This arises due to electrostatic forces from neighboring atoms
    • From the equipartition theorem, we have that for every quadratic degree of freedom in the Hamiltonian adds $\frac{k_{b}}{2}$ to the heat capacity
      • For a monoatomic gas, we have 3 quadratic degrees of freedom from the 3 momentum components (position is not a d.o.f. since the particles don’t interact with each other)
      • For solids, we have 6 degrees of freedom, which implies that $\frac{C}{N} = 3k_{b}$
  • As the tempurature starts to get lower, the tempurature starts to fall of rapidly

Einstein Model of Heat Capacity of Solids

  • The main idea is that Einstein used the same model of the N particles in a box, but changed the energy model of the confining potential
    • Instead of $E = \frac{kx^{2}}{2}$, we utilize $E = \hbar \omega(n+\frac{1}{2})$
      • The original derivation omitted the $\frac{\hbar \omega}{2}$ (ie. the zero point energy), since the Schrodinger equation hadn’t been invented yet
    • $\omega$ is called the Einstein frequency, and is a free parameter of the system
  • Recall the partition function is $Z = \Sigma_{n} exp(-\beta E_{n})$ where $\beta = \frac{1}{kT}$
    • $<E> = -\frac{1}{Z}\frac{\partial Z}{\partial \beta}$
    • Plug in energy of quantum harmonic oscillator, turn the crack, and then take derivative w.r.t. temperature yields the specific heat
    • Can define Einstein temperature as $\frac{\hbar\omega}{k_{b}}$, which defines the “freeze out” temperature of the system
  • Problem with this is that at lower potentials, the head capacity goes as $T^{3}$ experimentally, while Einstein is exponential decays

Debye Model of Heat Capacity of Solids

  • Basic idea: individual atomic displacements are coupled, which creates a collective excitation/mode which acts like a sound wave
  • The particles which arise from quantizing sound waves are called phonons
    • acts like bosons
  • For sound waves, the dispersion relationship is $\omega = v_{s}|k|$
  • The average energy is now
    • $<E> =3\Sigma_{k}\hbar \omega(k)(n+\frac{1}{2})$
      • Namely, now the frequency depends on the wavevector (ie. the dispersion relationship)
  • Can replace sum with integral: $\Sigma = \int 4\pi k^{2} dk$ where k is the magnitude of the wavenumber
    • This comes from assuming periodic boundary conditions, deducing that we have 1 state per every $\frac{(2\pi)^{3}}{L_{x}L_{y}L_{z}} = \frac{(2\pi)^{3}}{V}$ volume in k-space, then converting to spherical coordinates and integrating out solid angle
  • $<E> = 3 \frac{V}{(2\pi)^{3}}4\pi \int_{0}^{\infty} \omega^{2} (\frac{1}{v_{s}})^{3}\hbar\omega(n+\frac{1}{2}) = \int_{0}^{\infty} d\omega g(\omega) \hbar\omega(n+\frac{1}{2})$
    • Made change of variables from k to $\omega$
    • $g(\omega) = \frac{12\pi \omega^{2}}{(2\pi)^{3}v_{s}^{3}}V = N\frac{9\omega^{2}}{\omega_{d}^{2}}$ is the density of state
      • Can think of this as the number of modes that exist in the frequency band of $\omega$ and $\omega+d\omega$
      • $\omega_{d}^{3} = 6\pi^{2}\frac{N}{V}v_{s}^{3}$
      • Do the tedious integral. The following is useful: $\int_{0}^{\infty} dx \frac{x^{3}}{e^{x}-1} = \frac{\pi^{4}}{15}$
  • Taking T derivative yields the experimentally observed low tempurature power-law scaling
    • $C_{v} = Nk_{b}(\frac{k_{b}}{\hbar \omega})^{3} \frac{12\pi^{4}}{15} T^{3}$
    • $k_{b}T_{D} = \hbar \omega_{d}$
  • Currently, the model blows up at higher tempuratures. Problem is that by integrating to infinity, we are assuming an infinite number of modes. But we are limited by the number of modes present in the system
  • The physical assumption is that $\int_{0}^{\omega_{d}} g(\omega) d\omega = 3N$ ie. there exists a cut-off frequency $\omega_{d}$ that includes all possible modes of the system
    • The integral becomes intractable (ie. must be numerically solved)
  • Problems with model
    • cutoff is a bit ad. hoc.
    • the dispersion relationship is a bit different than sound waves
    • not an exact match
    • The heat capacity of metals follows a different pattern: $\alpha T^{3}+\gamma T$

Electrical Properties (Drude Model)

  • Apply classical kinetic theory to electrons in materials
  • Assumptions
    • Having a scattering time $\tau$ between collisions implies that the probability of a scatter occuring in some dt is $\frac{dt}{\tau}$
    • After scattering, the averaged momentum of the ensemble is 0 (ie. $<\vec{p}> = 0$)
    • Between scattering events, electrons feel E&M forces since they are charged
  • Consider an electron with some $p(t)$ at time t. We want to know p(t_dt)
    • $(1-\frac{dt}{\tau})(p(t)+Fdt)+\frac{dt}{\tau}*0 = p(t)+Fdt-p(t)\frac{dt}{\tau}\rightarrow \frac{d\vec{p}}{dt} = \vec{F}-\frac{\vec{p}}{\tau}$

Constant E&M Force

  • Let $\vec{F} = q(\vec{E}+\vec{v}\times\vec{B})$
  • Define current density $\vec{j} = ne\vec{v} \rightarrow \vec{v} = \frac{-\vec{j}}{ne}$
  • $\vec{E} = \frac{1}{ne}\vec{j}\times \vec{B}+\frac{m}{ne^2\tau}\vec{j}$
    • First term is an $E_{\perp}$ (Hall field) and second term is $E_{\parallel}$, where we take parallel to be in the flow of the current
  • If B=0, then $\vec{j} = \frac{ne^{2}\tau}{m}\vec{E} = \sigma \vec{E}$ where $\sigma$ is the conductivity (ie. $\sigma = \frac{1}{\rho}$ where $\rho$ is resistivity)
  • $E_{Hall} = =R_{H} \vec{B}\times \vec{j}$
    • $R_{H} = \frac{-1}{ne}$

Thermal Conductivity

  • $j_{q} = R \nabla T$ is the heat current density
  • For monoatomic gases, $R = \frac{1}{3}nc_{v}<v>\lambda$
    • $v = \sqrt{\frac{8k_{b}T}{m\pi}}$
  • Assuming that R of monoatomic gases works for solids, replace v with the appropriate expression
  • Taking the ratio of thermal conductivity to electrical conductivity (called the Lorenz number), we get something on te order of $10^{8}$
    • called the Wiedemann-Franz law

Sommerfield Model

Assumptions

  • Assume the free electron model: electrons don’t interact with the nuclei
  • assume independent electrons: ie. electrons don’t interact with each other
    • Combining the above two means that there is only kinetic energy
  • Assume some phonomenological relaxation time $\tau$
  • The Pauli exclusion principle holds
  • Construct an LxLxL periodic box
  • Solutions of Schrodinger equation for free particles are plane waves (ie. $\phi \propto e^{i\vec{k}\cdot\vec{r}}$)
    • $\vec{k} = \frac{2\pi}{L}<n_{x},n_{y},n_{z}>$
  • Energy of free particles is $\epsilon(\vec{k}) = \frac{\hbar^{2}\vec{k}^{2}}{2m}$

Fermi-Dirac Statistics

  • Assume a number occupancy of $n_{F} = \frac{1}{e^{\beta(\epsilon-\mu)}+1}$
    • Probability that an eigenstate is occupied
    • $\beta = \frac{1}{kT}$
  • At T=0, the probability of being above $\mu$ is 0
    • Called the Fermi energy at this tempurature
  • As higher tempuratures, the tempurature distribution smears out around $\mu$ (mu remains at the same level regardless though)
  • The spread around $\mu$ is roughly $k_{b}T$

Sommerfield Model (cont.)

  • $N = 2(\frac{L}{2\pi})^{3}\int n_{F}d^3k$
    • 2 from Pauli exclusion principle
  • $<E> = 2(\frac{L}{2\pi})^{3}\int \frac{\hbar^{2}k^{2}}{2m}n_{F}d^3k$
  • At T=0, we know that there is an exact cut off fermi momentum:
    • $N(T=0) = \frac{2V}{(2\pi)^{3}}\int_{0}^{k_{F}} d^{3}k$
  • Integrating over Fermi sphere at $k_{F}$ in spherical coordinates yields that
    • $E_{F} = \frac{\hbar^{2}}{2m}(3\pi^{2}n)^{\frac{2}{3}}$
  • Can also calculate density of states: $g(\epsilon) = \frac{dN}{dE}$
    • $g(\epsilon) = \frac{Vm^{\frac{3}{2}}\sqrt{2\epsilon}}{\pi^{2}\hbar^{3}} \propto \sqrt{\epsilon}$
  • Occasionally, we can treat higher temperatures like they are T=0, since the Fermi tempurature of many metals is much larger than room tempurature

LCAO (Linear Combination of Atomic Orbitals)/ Tight Binding Theory

Chemistry Review

Shell Theory

  • Electrons fill shells of lowest energies (Aufbau’s principle), as defined by $E_{n} = \frac{-R_{y}}{n^{2}}Z^{2}$
    • There are additional corrections from fine, hyperfine etc.
  • The exact shells that get filled are described by Madelung’s rule (the wrapping snake thing)
    • Places 3d orbital below 4s orbital. Arises from electron electron interactions which are normally not relavent

Ionic Bonds

  • Electrons getting transferred from one atom to another
  • $E_{ionization}$ is the energy required to remove an electron from an atom
  • $E_{e-aff}$ is the energy liberated by attaching an electron to an atom
  • $E_{ionization}-E_{e-aff}$ is the energy needed to transfer an electron from one atom to another (assuming that they are very far away)
  • There is also $E_{cohesive}$, which is the energy liberated by putting two ions togther
    • Say you have one sodium and clorine ion. Putting them together will release energy into the environment
  • if $\Delta E = E_{ion}-E_{aff}-E_{coh} < 0$, then the atoms react and bonding occurs

Covalent Bonds

  • By letting wave function be more spread out (due to electron spending time near both atoms), we reduce the average momentum of the electron. Smaller energy means atoms wants to be in that state
    • Eventually, exclusion principle forbids atoms from being too close
  • Explains why $H_{2}$ is abundant (electons of opposite spin pair up) but $He_{2}$ is not (electrons already paired up, so higher energy anti-bonding pair is formed, which discourages formation)

LCAO Basics

  • Makes Born-Oppenheimer Approximation
    • Assume that nuclei don’t move very much, but electrons move around a lot. Makes some sense, since nuclei are heavy and thus slow, so electrons move relatively much faster in comparison
    • In math terms, we fix $R_{1}$ and $R_{2}$
  • $H = \frac{p^{2}}{2m}+V(r-R_{1})+V(r-R_{2})$
    • V follows a Coulomb potential
  • Imagine the solution to the above Hamiltonian with only one atom (ie. the atoms are very far apart):
    • $(K+V_{1})|1> = \epsilon_{0}|1>$
    • Similarly, for the other atom: $(K+V_{2})|2> = \epsilon_{0}|2>$
  • To solve the actual problem, assume that the true solution is some linear combination: $|\phi> = a|1>+b|2>$
    • Find the a and b that lower the energy of the system as much as possible
  • The effective Schrodinger equation for the system then becomes:
    • $\Sigma-{j}H_{ij}\phi_{j} = E \phi_{i}$
      • $H_{ij} = <i|H|j>$
  • $<1|H|1> = <1|K+V_{1}|1>+ <1|V_{2}|1> = \epsilon_{0}+V_{cross}$
    • $V_{cross}$ is the interaction and electron sitting at nucleus 1 interacting with nucleus 2
  • $<1|H|2> = <1|K+V_{2}|1>+ <1|V_{1}|2> = 0-t$
    • -t is called the hopping term and gives dynamics of electrons moving between atoms in time-dependent theory
  • If you include nuclei-nuclei interactions, you gain an additional term that cancels out $V_{cross}$
  • Can think of final eigenstates in chemistry terms as bonding (no relative minus) or antibonding (relative minus)

Vibrations

  • Can treat any local minima ina potential function as a SHO
  • Pretend that Coulomb force can be emulated by springs between adjacent molecules
  • Choose a random molecule in the center of the chain, and then compare spring forces of nearest neighbors
  • Assume wave solution: $\delta x_{n} = Aexp(i(\omega t -k x_{n}))$
  • Assume that $x_{n}$ is discretized like $x_{n} = i*a$ where a is the lattice spacing
  • For simplest model (same spring constant everywhere) we get $\omega = 2 \sqrt{\frac{k}{m}}|\sin(\frac{ka}{2})|$
  • 1st Brillouin zone is defined centered at k=0 and with width $\delta k = \frac{2\pi}{a}$

2 Types of Springs

  • If you model the next nearest heighbor as having a different spring constant, you can tease of two different regions in the dispersion relationship
    • Derived via finding normal modes of system of equations
    • Optical region is the higher band (only light dispersion relation can reach)
    • Acoustic is lower band (sound can feasibly reach this region)

Crystal Structure

  • Lattice: set of points defined as integer sums of primitive vectors
    • Use notation $[n_{1},n_{2},n_{3}]$ to denote what combination of scalars to do
  • Unit cell: region of space that when repeated and stacked together reconstructs the full periodic structure
  • Primitive unit cell: unit cell that contains exactly one lattice point
  • Conventional unit cell: a non primitive unit cell that is convenient (usually orthogonal axes)
  • Wigner-Seitz cell: unit cell around lattice point that is closer to that lattice point than any other
    • Construct by using perpendicular bisectors of nearest connections
  • Basis: Description of objects in unit cell w/ respect to a reference lattice point

Reciprocal Space

  • Reciprocal lattice space defines how solid distributed in k-space
    • Can think of this as the Fourier transform of the direct lattice
  • Definition of reciprocal lattice basis is $a_{i}\cdot b_{j} = 2\pi \delta_{ij}$
    • Satisfied in 3D by $b_{i} = 2\pi \frac{a_{j}\times a_{k}}{a_{1}\cdot (a_{2}\times a_{3})}$
  • Define $G = m_{1}b_{1}+m_{2}b_{2}+m_{3}b_{3}$ to define any point in the reciprocal lattice
  • Hello! How are you