Solid State Physics Essentials

Cheatsheet Content

Crystal Structure Lattice: An infinite array of points in space, with each point having identical surroundings. Basis: An atom or group of atoms associated with each lattice point. Crystal Structure: Lattice + Basis. Unit Cell: Smallest component of the crystal that repeats to form the entire crystal. Primitive Unit Cell: Smallest unit cell containing only one lattice point. Bravais Lattices: 14 unique types of lattices in 3D. Common Lattices: Simple Cubic (SC): Atoms at corners. Coordination number = 6. Body-Centered Cubic (BCC): Atoms at corners and body center. Coordination number = 8. Face-Centered Cubic (FCC): Atoms at corners and face centers. Coordination number = 12. Hexagonal Close-Packed (HCP): ABAB... stacking of close-packed layers. Coordination number = 12. Coordination Number: Number of nearest neighbors to an atom. Atomic Packing Factor (APF): Volume of atoms in unit cell / Volume of unit cell. SC: $0.52$ BCC: $0.68$ FCC: $0.74$ HCP: $0.74$ Miller Indices: $(hkl)$ for planes, $ $ for directions. Find intercepts on axes ($x, y, z$) in terms of lattice constants. Take reciprocals. Clear fractions to get smallest integers. Enclose in parentheses $(hkl)$. Negative indices denoted by a bar over the number. Reciprocal Lattice: Fourier transform of the real lattice. Basis vectors: $\vec{b}_1 = 2\pi \frac{\vec{a}_2 \times \vec{a}_3}{\vec{a}_1 \cdot (\vec{a}_2 \times \vec{a}_3)}$, etc. Any reciprocal lattice vector $\vec{G} = h\vec{b}_1 + k\vec{b}_2 + l\vec{b}_3$ is normal to the $(hkl)$ plane in real space. X-ray Diffraction Bragg's Law: $2d \sin\theta = n\lambda$ $d$: interplanar spacing $\theta$: Bragg angle $n$: integer (order of diffraction) $\lambda$: wavelength of X-rays Scattering by a crystal: Constructive interference occurs when $\Delta\vec{k} = \vec{G}$, where $\vec{G}$ is a reciprocal lattice vector. Structure Factor ($F_{hkl}$): Amplitude of scattering from a unit cell. $$ F_{hkl} = \sum_j f_j e^{-i (\vec{G} \cdot \vec{r}_j)} $$ where $f_j$ is atomic scattering factor, $\vec{r}_j$ is position of $j$-th atom in unit cell. Intensity: $I \propto |F_{hkl}|^2$. Systematic Absences: For certain lattices, some reflections are absent due to destructive interference within the unit cell (e.g., FCC: $h,k,l$ all odd or all even; BCC: $h+k+l$ is even). Lattice Vibrations (Phonons) Diatomic Chain: Two types of atoms with masses $M_1, M_2$. Leads to acoustic and optical branches. Acoustic Branch: Atoms in unit cell move in phase. Dispersion relation $\omega \to 0$ as $k \to 0$. Optical Branch: Atoms in unit cell move out of phase. Dispersion relation $\omega \ne 0$ as $k \to 0$. Quantization of Lattice Waves: Phonons are quanta of lattice vibrations. Energy of a phonon: $E = \hbar\omega$. Momentum of a phonon: $\vec{p} = \hbar\vec{k}$. Density of States (DOS): $g(\omega) d\omega$ is the number of vibrational modes between $\omega$ and $\omega+d\omega$. For 1D: $g(\omega) \propto 1/\frac{d\omega}{dk}$. For 3D: $g(\omega) \propto \omega^2$ for small $\omega$ (Debye approximation). Specific Heat of Solids: Dulong-Petit Law: $C_V = 3R$ at high temperatures (classical limit). Einstein Model: Assumes all atoms oscillate at a single frequency $\omega_E$. $$ C_V = 3Nk_B \left(\frac{\hbar\omega_E}{k_BT}\right)^2 \frac{e^{\hbar\omega_E/k_BT}}{(e^{\hbar\omega_E/k_BT}-1)^2} $$ Underestimates $C_V$ at low T. Debye Model: Treats phonons as elastic waves, valid at low T. $$ C_V = 9Nk_B \left(\frac{T}{\Theta_D}\right)^3 \int_0^{\Theta_D/T} \frac{x^4 e^x}{(e^x-1)^2} dx $$ At low T ($T \ll \Theta_D$): $C_V \propto T^3$ (Debye $T^3$ Law). $\Theta_D$: Debye temperature. Free Electron Model Assumes valence electrons are free to move within the crystal, forming a "Fermi gas". Fermi Energy ($E_F$): Maximum energy of an electron at $T=0$ K. $$ E_F = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3} $$ where $n$ is electron density. Fermi Sphere: In k-space, states up to $k_F$ are occupied. $$ k_F = (3\pi^2 n)^{1/3} $$ Density of States (DOS) for Free Electrons: $$ g(E) = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} E^{1/2} $$ Specific Heat of Electrons: At low temperatures, $C_{el} \propto T$. $$ C_{el} = \frac{\pi^2}{2} n k_B \frac{T}{T_F} $$ where $T_F = E_F/k_B$ is the Fermi temperature. This is much smaller than phonon contribution at room temperature. Electrical Conductivity: $\sigma = ne^2\tau/m$ $n$: electron density $e$: electron charge $\tau$: relaxation time $m$: electron mass Thermal Conductivity: $K = \frac{\pi^2}{3} \frac{n k_B^2 T \tau}{m}$ Wiedemann-Franz Law: $K/\sigma T = L = \frac{\pi^2}{3} (k_B/e)^2$ (Lorenz number). Nearly Free Electron Model Considers weak periodic potential of the lattice. Energy Gaps: Occur at Brillouin zone boundaries (where $\vec{k} = \pm \vec{G}/2$). $$ E_{\pm} = \frac{\hbar^2 k_G^2}{2m} \pm |V_G| $$ where $k_G = |\vec{G}|/2$ and $V_G$ is the Fourier component of the potential. Brillouin Zones: Regions in reciprocal space defined by planes that bisect reciprocal lattice vectors. Effective Mass ($m^*$): Describes how electrons respond to external forces in a periodic potential. $$ \frac{1}{m^*} = \frac{1}{\hbar^2} \frac{d^2E}{dk^2} $$ Can be negative near band edges. Holes: Vacancies in an otherwise filled band. Behave as positive charge carriers with positive effective mass. Band Theory of Solids Bloch's Theorem: Electron wavefunctions in a periodic potential are of the form $\psi_{\vec{k}}(\vec{r}) = e^{i\vec{k}\cdot\vec{r}} u_{\vec{k}}(\vec{r})$, where $u_{\vec{k}}(\vec{r})$ has the periodicity of the lattice. Energy Bands: Allowed energy levels for electrons, separated by forbidden energy gaps. Classifications of Materials: Conductors (Metals): Partially filled band or overlapping bands. Electrons can easily move. Insulators: Full valence band, large energy gap to conduction band ($E_g > 3$ eV). Semiconductors: Full valence band, small energy gap to conduction band ($E_g Fermi-Dirac Distribution: Probability of an electron occupying a state with energy $E$. $$ f(E) = \frac{1}{e^{(E-E_F)/k_BT} + 1} $$ Semiconductors Intrinsic Semiconductors: Pure semiconductors (e.g., Si, Ge). Electron concentration in conduction band: $n = N_c e^{-(E_c-E_F)/k_BT}$ Hole concentration in valence band: $p = N_v e^{-(E_F-E_v)/k_BT}$ For intrinsic: $n=p=n_i$. $n_i^2 = N_c N_v e^{-E_g/k_BT}$. Intrinsic Fermi level: $E_F = \frac{E_c+E_v}{2} + \frac{3}{4} k_BT \ln(m_h^*/m_e^*)$. Extrinsic Semiconductors: Doped with impurities. n-type: Doped with donors (e.g., P in Si). $E_F$ shifts closer to conduction band. $n \approx N_D$. p-type: Doped with acceptors (e.g., B in Si). $E_F$ shifts closer to valence band. $p \approx N_A$. Hall Effect: Measures carrier concentration and type. $$ R_H = \frac{1}{nq} $$ where $R_H$ is the Hall coefficient, $n$ is carrier concentration, $q$ is charge. pn-Junction: Formed by joining p-type and n-type semiconductors. Depletion Region: Region near the junction depleted of free carriers, creating an electric field. Built-in Potential ($V_{bi}$): Potential difference across the depletion region. $$ V_{bi} = \frac{k_BT}{e} \ln\left(\frac{N_A N_D}{n_i^2}\right) $$ Forward Bias: Reduces barrier, current flows. Reverse Bias: Increases barrier, small leakage current. Magnetism Magnetic Susceptibility ($\chi$): $\vec{M} = \chi \vec{H}$ ($\vec{M}$ magnetization, $\vec{H}$ magnetic field strength). Diamagnetism: Weak, negative susceptibility. Induced by orbital motion of electrons opposing external field. Present in all materials. Paramagnetism: Weak, positive susceptibility. Due to permanent magnetic moments (e.g., unpaired electrons) that align with external field. Curie's Law: $\chi = C/T$ (for paramagnets). Ferromagnetism: Strong, positive susceptibility. Permanent magnetic moments align spontaneously even without external field (below Curie temperature $T_C$). Due to exchange interaction. Exhibits hysteresis. Antiferromagnetism: Neighboring magnetic moments align antiparallel, resulting in zero net magnetization. (below Néel temperature $T_N$). Ferrimagnetism: Like antiferromagnetism, but with unequal opposing moments, leading to a net magnetization. Superconductivity Zero Electrical Resistance: Below a critical temperature ($T_c$). Meissner Effect: Expulsion of magnetic flux from the interior of a superconductor (perfect diamagnetism). Perfect Diamagnetism: $\vec{B} = 0$ inside the superconductor. $\chi = -1$. Critical Magnetic Field ($H_c$): Above which superconductivity is destroyed. Type I Superconductors: Exhibit complete Meissner effect and single $H_c$. Usually pure metals. Type II Superconductors: Two critical fields ($H_{c1}, H_{c2}$). Magnetic flux penetrates in quantized vortices between $H_{c1}$ and $H_{c2}$. Usually alloys and ceramic materials (e.g., high-$T_c$ superconductors). BCS Theory (Bardeen, Cooper, Schrieffer): Explains conventional superconductivity. Cooper Pairs: Electrons form bound pairs due to electron-phonon interaction. These pairs condense into a macroscopic quantum state. Energy gap $2\Delta$ for breaking a Cooper pair. Flux Quantization: Magnetic flux through a superconducting loop is quantized in units of $\Phi_0 = h/2e$. Josephson Effect: Tunneling of Cooper pairs across a thin insulating barrier between two superconductors. Dielectrics and Ferroelectrics Dielectric Constant ($\epsilon_r$): $\epsilon = \epsilon_r \epsilon_0$. Measures material's ability to store electrical energy. Polarization ($\vec{P}$): Electric dipole moment per unit volume. $$ \vec{P} = \epsilon_0 (\epsilon_r - 1) \vec{E} $$ Types of Polarization: Electronic: Displacement of electron cloud relative to nucleus. Ionic: Displacement of positive and negative ions relative to each other. Orientational: Alignment of permanent dipoles. Local Field: Electric field experienced by an atom in a dielectric, which is generally different from the applied field. Clausius-Mossotti Relation: Relates dielectric constant to polarizability $\alpha$. $$ \frac{\epsilon_r - 1}{\epsilon_r + 2} = \frac{N\alpha}{3\epsilon_0} $$ Ferroelectricity: Materials exhibiting spontaneous electric polarization that can be reversed by an external electric field. Exhibit hysteresis in $\vec{P}$ vs. $\vec{E}$ curve. Occurs below a critical temperature (Curie temperature). Examples: BaTiO$_3$, Rochelle salt. Piezoelectricity: Generation of electric polarization by mechanical stress, and vice-versa. Pyroelectricity: Change in spontaneous polarization due to temperature change. Defects in Solids Point Defects: Vacancy: Missing atom from a lattice site. Interstitial: Atom occupying a non-lattice site. Substitutional Impurity: Foreign atom replacing a host atom. Frenkel Defect: Vacancy-interstitial pair. Schottky Defect: Pair of cation and anion vacancies to maintain charge neutrality. Line Defects (Dislocations): Edge Dislocation: Extra half-plane of atoms inserted into the crystal. Screw Dislocation: Lattice distorted into a helical path. Responsible for plastic deformation in materials. Planar Defects: Grain Boundaries: Interfaces between crystallites of different orientations. Stacking Faults: Errors in the stacking sequence of close-packed planes. Volume Defects: Pores, cracks, foreign inclusions. Transport Phenomena Drift Velocity: Average velocity of charge carriers under an electric field. $v_d = \mu E$. Mobility ($\mu$): Drift velocity per unit electric field. $\mu = e\tau/m^*$. Electrical Conductivity: $\sigma = ne\mu$. Diffusion: Movement of particles from high to low concentration. Diffusion Coefficient ($D$): $J = -D \frac{dn}{dx}$ (Fick's First Law). Einstein Relation: $D/\mu = k_BT/e$. Thermoelectric Effects: Seebeck Effect: Temperature difference creates an electric voltage. Peltier Effect: Electric current creates a temperature difference. Thomson Effect: Current in a conductor with a temperature gradient generates or absorbs heat.