Complete Solid State Physics

Crystal Structure

Core ideas

Crystals are periodic arrangements of atoms described by a lattice plus a basis. Symmetry, unit cells, Bravais lattices, Miller indices, and reciprocal space organize real materials and determine allowed diffraction and band structures.

For review, be able to identify primitive cells, reciprocal vectors, common lattices, and planes from Miller indices. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\bm R=n_1\bm a_1+n_2\bm a_2+n_3\bm a_3,\qquad \bm a_i\cdot\bm b_j=2\pi\delta_{ij}$$

Worked example

Copper has an FCC structure with lattice constant $a=3.61\,\text{Å}$. The primitive vectors can be taken as $\bm a_1=\frac{a}{2}(\hat y+\hat z)$, $\bm a_2=\frac{a}{2}(\hat x+\hat z)$, $\bm a_3=\frac{a}{2}(\hat x+\hat y)$. The reciprocal-lattice vectors are $\bm b_1=\frac{2\pi}{a}(-\hat x+\hat y+\hat z)$, etc. For the (111) plane the interplanar spacing is $d_{111}=a/\sqrt{3}\approx2.09\,\text{Å}$.

Problems with Solutions

Problem 1. A simple cubic crystal has $a=4.0\,\text{Å}$. What is the spacing $d_{200}$? Solution. For cubic lattices $d_{hkl}=a/\sqrt{h^2+k^2+\ell^2}$. Thus $d_{200}=4.0\,\text{Å}/\sqrt{4}=2.0\,\text{Å}$.

Problem 2. How many atoms are in the conventional cubic cell of diamond? Solution. Diamond is FCC with two-atom basis: $8\times\frac18+6\times\frac12+4=8$ atoms.

Problem 3. Show that the volume of the primitive cell is $\bm a_1\cdot(\bm a_2\times\bm a_3)$ and compute it for a BCC lattice with conventional cube side $a$. Solution. The BCC primitive vectors are $\bm a_1=\frac{a}{2}(-\hat x+\hat y+\hat z)$, $\bm a_2=\frac{a}{2}(\hat x-\hat y+\hat z)$, $\bm a_3=\frac{a}{2}(\hat x+\hat y-\hat z)$. Their scalar triple product is $|\bm a_1\cdot(\bm a_2\times\bm a_3)|=a^3/2$, which is half the conventional cell volume as expected.

Section summary. Crystal periodicity is the starting point of solid-state physics.

Wave Diffraction and the Reciprocal Lattice

Core ideas

Diffraction measures periodic structure through constructive interference. Reciprocal lattice vectors encode allowed momentum transfer. Bragg’s law, Laue conditions, structure factors, and Brillouin zones connect scattering patterns to atomic arrangements.

For review, be able to use Bragg and Laue conditions, compute simple structure factors, and interpret Brillouin zones. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\Delta\bm k=\bm G,\qquad 2d\sin\theta=n\lambda,\qquad S_{\bm G}=\sum_j f_je^{i\bm G\cdot\bm r_j}$$

Worked example

X-rays with $\lambda=1.54\,\text{Å}$ (Cu K$\alpha$) are diffracted from the (200) planes of NaCl ($a=5.64\,\text{Å}$). Using Bragg’s law with $d_{200}=a/2=2.82\,\text{Å}$:

$$2d\sin\theta=n\lambda\quad\Rightarrow\quad\sin\theta=\frac{1.54}{2\times2.82}=0.273,$$

so $\theta=15.9^{\circ}$ and the scattering angle is $2\theta=31.8^{\circ}$.

Problems with Solutions

Problem 1. Calculate the Bragg angle for first-order reflection from the (111) planes of silicon ($a=5.43\,\text{Å}$) using X-rays with $\lambda=1.54\,\text{Å}$. Solution. $d_{111}=a/\sqrt{3}=3.14\,\text{Å}$. Then $\sin\theta=\lambda/(2d)=1.54/(6.28)=0.245$, giving $\theta=14.2^{\circ}$.

Problem 2. The structure factor of a BCC lattice is $S_{hkl}=f[1+e^{i\pi(h+k+\ell)}]$. For which Miller indices is the diffraction extinct? Solution. When $h+k+\ell$ is odd, $S_{hkl}=0$. Thus reflections like (100), (111), (210) are forbidden.

Problem 3. In a powder diffraction pattern, the first peak for aluminum (FCC, $a=4.05\,\text{Å}$) with $\lambda=1.54\,\text{Å}$ appears at $2\theta=38.5^{\circ}$. Verify this corresponds to the (111) reflection. Solution. $d_{111}=4.05/\sqrt{3}=2.34\,\text{Å}$. Bragg gives $\sin\theta=1.54/(2\times2.34)=0.329$, so $\theta=19.2^{\circ}$ and $2\theta=38.4^{\circ}$, matching the observation.

Section summary. Diffraction is a direct probe of reciprocal-lattice structure.

Crystal Binding and Elastic Constants

Core ideas

Solids bind through ionic, covalent, metallic, molecular, and hydrogen-bond interactions. Elastic constants describe the energy cost of strain and connect microscopic bonding to sound speeds and mechanical response.

For review, be able to compare bonding types, use strain and stress tensors, and relate elastic moduli to stability. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$u=\frac12C_{ijkl}\epsilon_{ij}\epsilon_{kl},\qquad \sigma_{ij}=C_{ijkl}\epsilon_{kl}$$

Worked example

NaCl is ionically bonded with Madelung constant $\alpha_M\approx1.75$. The cohesive energy per ion pair is approximately $U=-\frac{\alpha_M e^2}{4\pi\epsilon_0 r_0}(1-\frac{1}{n})$ where $r_0=2.82\,\text{Å}$ and $n\approx8$. Using $\frac{e^2}{4\pi\epsilon_0}=1.44\,\text{eV\,nm}=14.4\,\text{eV\,Å}$:

$$U\approx-\frac{1.75\times14.4}{2.82}\left(1-\frac18\right)\approx-7.9\,\text{eV},$$

close to the experimental cohesive energy of $-7.9\,\text{eV}$ per ion pair.

Problems with Solutions

Problem 1. For a cubic crystal, the elastic energy density is $u=\frac12C_{11}(\epsilon_{xx}^2+\epsilon_{yy}^2+\epsilon_{zz}^2)+C_{12}(\epsilon_{xx}\epsilon_{yy}+\dots)+\frac12C_{44}(\epsilon_{xy}^2+\dots)$. What is the bulk modulus $B$ in terms of the elastic constants? Solution. For hydrostatic strain $\epsilon_{xx}=\epsilon_{yy}=\epsilon_{zz}=\epsilon$, all other components zero, $u=\frac32(C_{11}+2C_{12})\epsilon^2$. Since $u=\frac92B\epsilon^2$, we get $B=\frac13(C_{11}+2C_{12})$.

Problem 2. Diamond has $C_{11}=1076\,\text{GPa}$ and $C_{12}=125\,\text{GPa}$. Compute its bulk modulus. Solution. $B=\frac13(1076+2\times125)=\frac13(1326)=442\,\text{GPa}$, in excellent agreement with its known hardness.

Problem 3. The Lennard-Jones potential is $U(r)=4\epsilon[(\sigma/r)^{12}-(\sigma/r)^6]$. Find the equilibrium separation $r_0$ and the binding energy. Solution. Setting $dU/dr=0$ gives $r_0=2^{1/6}\sigma$. The binding energy is $U(r_0)=-\epsilon$.

Section summary. Bonding sets both structure and elasticity.

Phonons I: Crystal Vibrations

Core ideas

Atoms in crystals vibrate collectively as phonons. Normal modes arise from coupled harmonic oscillators; acoustic branches reflect translations, while optical branches occur when the basis has multiple atoms.

For review, be able to derive a simple dispersion relation, distinguish acoustic and optical modes, and interpret phonon momentum. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\omega(k)=2\sqrt{K/M}\left|\sin\frac{ka}{2}\right|,\qquad E_{\bm q,s}=\hbar\omega_{\bm q,s}\left(n+\frac12\right)$$

Worked example

For a 1D monatomic chain with spacing $a=2.5\,\text{Å}$, atomic mass $M=40\,\text{u}$ (argon-like), and spring constant $K=15\,\text{N/m}$, the maximum phonon frequency is

$$\omega_{\max}=2\sqrt{\frac{K}{M}}=2\sqrt{\frac{15}{40\times1.66\times10^{-27}}}\approx3.0\times10^{13}\,\text{rad/s},$$

corresponding to $\nu_{\max}\approx4.8\,\text{THz}$.

Problems with Solutions

Problem 1. A 1D chain has $M=23\,\text{amu}$, $a=3.0\,\text{Å}$, and sound velocity $v_s=3.0\times10^3\,\text{m/s}$. Estimate the spring constant $K$. Solution. At small $k$, $\omega\approx v_s k$ and $v_s=a\sqrt{K/M}$. Thus $K=Mv_s^2/a^2=(23\times1.66\times10^{-27})(9\times10^6)/(9\times10^{-20})\approx38\,\text{N/m}$.

Problem 2. The Debye temperature of copper is $\Theta_D=343\,\text{K}$. What is the maximum phonon frequency? Solution. $\hbar\omega_D=k_B\Theta_D$, so $\omega_D=(1.38\times10^{-23}\times343)/(1.05\times10^{-34})\approx4.5\times10^{13}\,\text{rad/s}$.

Problem 3. For a diatomic chain with $M_1=20\,\text{u}$, $M_2=40\,\text{u}$, and $K=20\,\text{N/m}$, what is the gap between acoustic and optical branches at the zone boundary $k=\pi/(2a)$? Solution. At the zone boundary, $\omega_-=\sqrt{2K/M_1}=\sqrt{40/(20\times1.66\times10^{-27})}=3.5\times10^{13}\,\text{rad/s}$ and $\omega_+=\sqrt{2K/M_2}=2.5\times10^{13}\,\text{rad/s}$. The gap is $\Delta\omega\approx1.0\times10^{13}\,\text{rad/s}$.

Section summary. Phonons are quantized normal modes of lattice vibration.

Phonons II: Thermal Properties

Core ideas

Phonons carry heat and determine low-temperature heat capacity. Einstein and Debye models approximate the phonon spectrum; scattering of phonons by defects, boundaries, and other phonons controls thermal conductivity.

For review, be able to derive Debye $T^3$ law, compare Einstein and Debye models, and estimate thermal conductivity. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$C_V\propto T^3\quad(T\ll\Theta_D),\qquad \kappa\simeq \frac13 C_V v_s\ell$$

Worked example

For copper with Debye temperature $\Theta_D=343\,\text{K}$, the low-temperature lattice heat capacity at $T=10\,\text{K}$ is given by the Debye $T^3$ law:

$$C_V=\frac{12\pi^4}{5}Nk_B\left(\frac{T}{\Theta_D}\right)^3=234\,Nk_B\left(\frac{10}{343}\right)^3\approx0.058\,Nk_B.$$

At this temperature the electronic term $\gamma T$ (with $\gamma\approx0.695\,\text{mJ/mol\,K}^2$) dominates over the phonon contribution.

Problems with Solutions

Problem 1. Lead has $\Theta_D=105\,\text{K}$. Estimate its heat capacity at $T=30\,\text{K}$ per mole. Solution. Since $T<\Theta_D/3$, use the Debye approximation: $C_V\approx234R(T/\Theta_D)^3=234\times8.314(30/105)^3\approx43\,\text{J/mol\,K}$.

Problem 2. The thermal conductivity of a crystal is $\kappa=100\,\text{W/m\,K}$ at $T=100\,\text{K}$. Given $C_V=2\times10^6\,\text{J/m}^3\,\text{K}$ and $v_s=5\times10^3\,\text{m/s}$, estimate the phonon mean free path. Solution. $\kappa=\frac13C_Vv_s\ell$ gives $\ell=3\kappa/(C_Vv_s)=3\times100/(2\times10^6\times5\times10^3)=3\times10^{-8}\,\text{m}=30\,\text{nm}$.

Problem 3. Why does the Debye model correctly predict $C_V\propto T^3$ at low $T$ while the Einstein model does not? Solution. The Einstein model approximates all phonons with a single frequency $\omega_E$, giving an exponential freeze-out $C_V\propto e^{-\hbar\omega_E/k_BT}$. The Debye model includes acoustic modes down to $\omega\to0$, which remain thermally excited at low $T$ and yield the $T^3$ behavior.

Section summary. Lattice thermal properties are phonon thermodynamics and transport.

Free Electron Fermi Gas

Core ideas

Conduction electrons in simple metals are modeled as a Fermi gas. Pauli exclusion creates a Fermi surface and explains degeneracy pressure, small electronic heat capacity, and basic metallic transport.

For review, be able to compute $k_F$, $E_F$, density of states, and low-temperature heat capacity. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$k_F=(3\pi^2n)^{1/3},\qquad E_F=\frac{\hbar^2k_F^2}{2m},\qquad C_e=\gamma T$$

Worked example

For sodium (BCC, one conduction electron per atom, density $n=2.65\times10^{28}\,\text{m}^{-3}$), the Fermi wavevector is

$$k_F=(3\pi^2n)^{1/3}=(3\pi^2\times2.65\times10^{28})^{1/3}\approx9.1\times10^9\,\text{m}^{-1}.$$

The Fermi energy is $E_F=\hbar^2k_F^2/(2m_e)\approx3.2\,\text{eV}$ and the Fermi temperature is $T_F=E_F/k_B\approx3.7\times10^4\,\text{K}$.

Problems with Solutions

Problem 1. Copper has $n=8.47\times10^{28}\,\text{m}^{-3}$. Calculate $k_F$, $E_F$, and the Fermi velocity $v_F$. Solution. $k_F=(3\pi^2n)^{1/3}=1.36\times10^{10}\,\text{m}^{-1}$. $E_F=\hbar^2k_F^2/(2m_e)=7.0\,\text{eV}$. $v_F=\hbar k_F/m_e=1.57\times10^6\,\text{m/s}$.

Problem 2. Show that the electronic heat-capacity coefficient is $\gamma=\frac{\pi^2}{2}\frac{Nk_B}{T_F}$ and estimate it for sodium. Solution. Using $C_e=\gamma T$ with $N=nV$ and $T_F\approx3.7\times10^4\,\text{K}$, $\gamma\approx\frac{\pi^2}{2}\frac{R}{T_F}\approx1.1\,\text{mJ/mol\,K}^2$, close to the experimental value.

Problem 3. At what temperature is the electronic heat capacity of aluminum equal to its phonon contribution? Use $\gamma=1.35\,\text{mJ/mol\,K}^2$ and $\Theta_D=428\,\text{K}$. Solution. Equate $\gamma T$ to the high-$T$ Dulong—Petit lattice value $3R$? No, at low $T$ equate $\gamma T$ to the Debye phonon term. More simply, at $T\ll\Theta_D$, set $\gamma T\approx\frac{12\pi^4}{5}R(T/\Theta_D)^3$. Solving gives $T^2=5\gamma\Theta_D^3/(12\pi^4R)$. Substituting yields $T\approx3\,\text{K}$.

Section summary. Metal electrons are governed by Fermi statistics.

Energy Bands

Core ideas

Periodic potentials turn free-electron energies into bands separated by gaps. Bloch’s theorem labels states by crystal momentum. Band filling explains metals, insulators, and semiconductors.

For review, be able to use Bloch’s theorem, interpret band gaps, effective mass, and density of states. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\psi_{n\bm k}(\bm r)=e^{i\bm k\cdot\bm r}u_{n\bm k}(\bm r),\qquad \bm v_n=\frac1\hbar\nabla_{\bm k}E_n(\bm k)$$

Worked example

In the nearly-free-electron model, a weak periodic potential $V(x)=V_0\cos(2\pi x/a)$ opens a gap at the Brillouin-zone boundary $k=\pi/a$. The gap is $E_g=2|V_G|$ where $G=2\pi/a$. For $V_0=0.5\,\text{eV}$, the gap is $E_g=1.0\,\text{eV}$. States just below the gap have negative effective mass because the curvature of $E(k)$ is downward.

Problems with Solutions

Problem 1. An electron in a 1D lattice ($a=3.0\,\text{Å}$) has an energy band $E(k)=E_0-2\gamma\cos(ka)$ with $\gamma=1\,\text{eV}$. Find the effective mass at the bottom of the band ($k=0$) and at the top ($k=\pi/a$). Solution. $m^*=\hbar^2/(d^2E/dk^2)$. At $k=0$, $d^2E/dk^2=2\gamma a^2$, so $m^*=\hbar^2/(2\gamma a^2)=1.05^2\times10^{-68}/(2\times1.6\times10^{-19}\times9\times10^{-20})\approx0.38\,m_e$. At $k=\pi/a$, $d^2E/dk^2=-2\gamma a^2$, giving $m^*=-0.38\,m_e$.

Problem 2. Silicon has a band gap $E_g=1.12\,\text{eV}$. What is the minimum photon wavelength that can excite an electron across the gap? Solution. $\lambda_{\max}=hc/E_g=(1240\,\text{eV\,nm})/1.12\,\text{eV}\approx1107\,\text{nm}$ (infrared).

Problem 3. The group velocity of a Bloch electron is $\bm v_n=\frac1\hbar\nabla_{\bm k}E_n(\bm k)$. If $E(k)=Ak^2$ with $A=2\,\text{eV\,Å}^2$, what is the velocity at $k=0.1\,\text{Å}^{-1}$? Solution. $v=2Ak/\hbar=2\times2\times0.1/(0.658\,\text{eV\,fs})\approx0.61\,\text{Å/fs}=6.1\times10^4\,\text{m/s}$.

Section summary. Bands are quantum states shaped by lattice periodicity.

Semiconductor Crystals

Core ideas

Semiconductors have small band gaps and carrier densities controlled by temperature, doping, and illumination. Electrons and holes, effective masses, donor and acceptor levels, p-n junctions, and recombination form the device foundation.

For review, be able to use carrier statistics, distinguish intrinsic and doped regimes, and explain p-n junction depletion. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$np=n_i^2,\qquad n_i\propto T^{3/2}e^{-E_g/(2k_BT)},\qquad E_{\rm donor}\sim \frac{m^*}{\epsilon_r^2}13.6\,\mathrm{eV}$$

Worked example

Intrinsic silicon has $E_g=1.12\,\text{eV}$ and $n_i\approx1.0\times10^{10}\,\text{cm}^{-3}$ at $T=300\,\text{K}$. The intrinsic carrier concentration follows

$$n_i\propto T^{3/2}\exp\left(-\frac{E_g}{2k_BT}\right).$$

At $T=300\,\text{K}$, $k_BT\approx0.0259\,\text{eV}$, so $E_g/(2k_BT)\approx21.6$, giving the tiny carrier density relative to metals.

Problems with Solutions

Problem 1. A silicon sample is doped with phosphorus at $N_D=10^{16}\,\text{cm}^{-3}$. At $T=300\,\text{K}$, assuming full ionization, what are the electron and hole concentrations? Solution. $n\approx N_D=10^{16}\,\text{cm}^{-3}$. $p=n_i^2/n=(10^{10})^2/10^{16}=10^4\,\text{cm}^{-3}$.

Problem 2. The donor binding energy in silicon is $E_d\approx45\,\text{meV}$. Using the hydrogenic model $E_d=(m^*/m_e)(1/\epsilon_r^2)\times13.6\,\text{eV}$, with $\epsilon_r=11.7$ and $m^*=0.26m_e$, verify this value. Solution. $E_d=0.26\times13.6/(11.7)^2=3.54/136.9\approx0.0259\,\text{eV}=25.9\,\text{meV}$. The experimental value is larger due to anisotropic effective mass and central-cell corrections.

Problem 3. A p-n junction has $N_A=10^{17}\,\text{cm}^{-3}$ and $N_D=10^{16}\,\text{cm}^{-3}$. What is the built-in potential at $T=300\,\text{K}$? Solution. $V_{bi}=\frac{k_BT}{e}\ln\frac{N_AN_D}{n_i^2}=0.0259\ln\frac{10^{33}}{10^{20}}=0.0259\times\ln(10^{13})=0.0259\times29.9\approx0.77\,\text{V}$.

Section summary. Semiconductors are controllable band insulators.

Fermi Surfaces and Metals

Core ideas

The Fermi surface controls metallic response because only nearby states can change occupancy. Its shape determines velocities, effective masses, magnetoresistance, quantum oscillations, and transport anisotropy.

For review, be able to connect Fermi surface geometry to transport, define effective mass, and interpret Hall and quantum oscillation measurements. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\bm v_F=\frac1\hbar\nabla_{\bm k}E(\bm k),\qquad A_F\frac{\hbar}{eB}=2\pi\left(n+\gamma\right)$$

Worked example

For a free-electron gas, the Fermi surface is a sphere of radius $k_F$. In copper ($k_F=1.36\times10^{10}\,\text{m}^{-1}$), the extremal cross-sectional area perpendicular to [111] is $A_F=\pi k_F^2=5.8\times10^{20}\,\text{m}^{-2}$. The de Haas—van Alphen oscillation period in $1/B$ is

$$\Delta(1/B)=\frac{2\pi e}{\hbar A_F}=\frac{1}{A_F}\frac{h}{e}=\frac{6.63\times10^{-34}}{(1.6\times10^{-19})(5.8\times10^{20})}\approx7.1\times10^{-6}\,\text{T}^{-1},$$

corresponding to $\Delta B\approx0.14\,\text{T}$ at $B=20\,\text{T}$.

Problems with Solutions

Problem 1. The Hall coefficient of sodium is $R_H=-2.4\times10^{-10}\,\text{m}^3/\text{C}$. Assuming one electron per atom, what is the electron density? Solution. $R_H=-1/(ne)$ gives $n=-1/(R_He)=1/(2.4\times10^{-10}\times1.6\times10^{-19})=2.6\times10^{28}\,\text{m}^{-3}$.

Problem 2. A metal has $k_F=10^{10}\,\text{m}^{-1}$. Calculate the Fermi wavelength $\lambda_F$ and compare it to typical atomic spacings. Solution. $\lambda_F=2\pi/k_F=6.28\times10^{-10}\,\text{m}=6.28\,\text{Å}$, comparable to the interatomic spacing ($\sim$ a few Å), confirming that electrons are delocalized over many atoms.

Problem 3. For a 2D electron gas with areal density $n_{2D}=10^{16}\,\text{m}^{-2}$, what is the Fermi wavevector and Fermi energy? Solution. In 2D, $k_F=\sqrt{2\pi n_{2D}}=2.5\times10^8\,\text{m}^{-1}$. $E_F=\hbar^2k_F^2/(2m_e)=0.36\,\text{meV}$.

Section summary. Metallic behavior is determined by Fermi-surface geometry.

Superconductivity

Core ideas

Superconductors have zero dc resistance, Meissner expulsion, an energy gap, flux quantization, and phase coherence. BCS theory explains pairing from an effective attraction; Ginzburg-Landau theory describes order-parameter physics and vortices.

For review, be able to state Meissner effect, flux quantization, gap scale, type I/II behavior, and the role of Cooper pairs. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\Phi_0=\frac{h}{2e},\qquad 2\Delta(0)\approx3.52k_BT_c,\qquad \bm J_s\propto |\psi|^2(\hbar\nabla\phi-2e\bm A)$$

Worked example

Niobium has $T_c=9.2\,\text{K}$. The BCS theory predicts the zero-temperature gap:

$$2\Delta(0)=3.52k_BT_c=3.52\times(8.617\times10^{-5}\,\text{eV/K})\times9.2\,\text{K}\approx2.8\,\text{meV}.$$

The magnetic flux quantum is $\Phi_0=h/(2e)=2.068\times10^{-15}\,\text{Wb}$. A superconducting loop of area $A=1\,\text{cm}^2$ threading one flux quantum corresponds to a field $B=\Phi_0/A=2.07\times10^{-11}\,\text{T}$.

Problems with Solutions

Problem 1. A Josephson junction has critical current $I_c=1\,\text{mA}$. What is the maximum supercurrent it can carry? Solution. The maximum zero-voltage supercurrent is exactly $I_c=1\,\text{mA}$. For $I<I_c$, $V=0$; for $I>I_c$, a finite voltage appears.

Problem 2. Lead has $T_c=7.2\,\text{K}$ and $\lambda_L=37\,\text{nm}$. Estimate the superfluid density $n_s$ from the London penetration depth formula $\lambda_L^{-2}=\mu_0 n_s e^2/m_e$. Solution. $n_s=m_e/(\mu_0e^2\lambda_L^2)=9.11\times10^{-31}/(4\pi\times10^{-7}\times(1.6\times10^{-19})^2\times(37\times10^{-9})^2)\approx2.1\times10^{28}\,\text{m}^{-3}$.

Problem 3. A Type-II superconductor has $H_{c1}=100\,\text{Oe}$ and $H_{c2}=10\,\text{T}$. In which field range do Abrikosov vortices exist? Solution. Vortices exist for $H_{c1}<H<H_{c2}$ (after converting units: $H_{c1}\approx0.01\,\text{T}$). So vortices appear from about $0.01\,\text{T}$ up to $10\,\text{T}$.

Section summary. Superconductivity is macroscopic quantum phase coherence of paired electrons.

Dielectrics and Ferroelectrics

Core ideas

Dielectrics polarize under electric fields through electronic, ionic, orientational, and space-charge mechanisms. Ferroelectrics have spontaneous switchable polarization and domain structure, often described by Landau theory.

For review, be able to use susceptibility and permittivity, identify polarization mechanisms, and explain ferroelectric hysteresis. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\bm P=\epsilon_0\chi_e\bm E,\qquad F(P)=a(T-T_c)P^2+bP^4-EP$$

Worked example

Water has a large static dielectric constant $\epsilon_r\approx80$ at room temperature because of orientational polarization of polar molecules. For electronic polarization only (e.g., in diamond), $\epsilon_r\approx5.7$. The Clausius—Mossotti relation connects microscopic polarizability $\alpha$ to macroscopic $\epsilon_r$:

$$\frac{\epsilon_r-1}{\epsilon_r+2}=\frac{n\alpha}{3\epsilon_0}.$$

For Si ($n=5\times10^{28}\,\text{m}^{-3}$, $\epsilon_r=11.7$), this gives $\alpha\approx3.0\times10^{-40}\,\text{F\,m}^2$.

Problems with Solutions

Problem 1. A parallel-plate capacitor with area $A=10^{-4}\,\text{m}^2$ and separation $d=1\,\text{mm}$ is filled with a dielectric of $\epsilon_r=10$. What is its capacitance? Solution. $C=\epsilon_0\epsilon_r A/d=(8.85\times10^{-12})(10)(10^{-4})/(10^{-3})=8.85\times10^{-12}\,\text{F}=8.85\,\text{pF}$.

Problem 2. BaTiO$_3$ has a spontaneous polarization $P_s=0.25\,\text{C/m}^2$ below $T_c=393\,\text{K}$. What surface charge density does this correspond to? Solution. The bound surface charge is $\sigma_b=P\cdot\hat n$, so $|\sigma_b|=0.25\,\text{C/m}^2$. This is enormous compared to typical electrostatic charges.

Problem 3. The Landau free energy is $F(P)=a(T-T_c)P^2+bP^4$. Minimize to find the spontaneous polarization for $T<T_c$. Solution. $dF/dP=2a(T-T_c)P+4bP^3=0$. For $T<T_c$, the nonzero solution is $P_s=\sqrt{a(T_c-T)/(2b)}$.

Section summary. Dielectric response reflects how charge distributions deform.

Paramagnetism, Diamagnetism, and Ferromagnetism

Core ideas

Magnetism comes from orbital and spin moments plus exchange interactions. Diamagnets weakly oppose fields; paramagnets align with fields; ferromagnets order spontaneously and form domains.

For review, be able to distinguish magnetic responses, use Curie law, explain exchange and spontaneous symmetry breaking. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\chi_{\rm Curie}=\frac{C}{T},\qquad H=-\sum_{ij}J_{ij}\bm S_i\cdot\bm S_j$$

Worked example

A paramagnetic salt with $N$ noninteracting spins-$\frac12$ follows the Curie law $\chi=C/T$. For $N/V=10^{28}\,\text{m}^{-3}$ and $g=2$, the Curie constant is

$$C=\frac{\mu_0N(g\mu_B)^2}{3Vk_B}=\frac{(4\pi\times10^{-7})(10^{28})(2\times9.27\times10^{-24})^2}{3(1.38\times10^{-23})}\approx5.2\times10^{-3}\,\text{K}.$$

At $T=300\,\text{K}$, $\chi\approx1.7\times10^{-5}$ (dimensionless SI), a typical small paramagnetic susceptibility.

Problems with Solutions

Problem 1. Iron has saturation magnetization $M_s=1.7\times10^6\,\text{A/m}$. If each atom contributes $2.2\,\mu_B$, what is the atomic density? Solution. $n=M_s/(2.2\mu_B)=1.7\times10^6/(2.2\times9.27\times10^{-24})=8.3\times10^{28}\,\text{m}^{-3}$.

Problem 2. A ferromagnet has Curie temperature $T_c=1043\,\text{K}$ (iron). Using mean-field theory, $T_c=\lambda C$, estimate the Weiss constant $\lambda$. Solution. $\lambda=T_c/C=1043/(5.2\times10^{-3}\times V/N)$… Better: from $T_c=JzS(S+1)/(3k_B)$. For Fe with $S=1$ and $z=8$, $J=3k_BT_c/(zS(S+1))=3(1.38\times10^{-23})(1043)/(8\times2)=2.7\times10^{-21}\,\text{J}=17\,\text{meV}$.

Problem 3. A superconductor exhibits perfect diamagnetism ($\chi=-1$). What is the magnetization when $B_{ext}=0.1\,\text{T}$? Solution. $\bm M=\chi\bm H$ and $\bm B=\mu_0(1+\chi)\bm H$. For $\chi=-1$, $\bm M=-\bm H$ and $\bm B=0$ inside. With $H=B_{ext}/\mu_0=0.1/(4\pi\times10^{-7})=7.96\times10^4\,\text{A/m}$, $M=-7.96\times10^4\,\text{A/m}$.

Section summary. Magnetism is collective behavior of microscopic moments.

Nanophysics / Surfaces and Interfaces

Core ideas

At nanoscale dimensions, surfaces, confinement, disorder, and interfaces dominate. Quantum wells, tunneling, 2D electron gases, surface states, heterostructures, and mesoscopic transport require both band and scattering viewpoints.

For review, be able to estimate confinement energies, use tunneling intuition, and explain why surface-to-volume ratio matters. Keep the physical question visible: identify the degrees of freedom, the conserved quantities, the approximation being made, and the observable that would be measured.

Mathematical spine

$$\Delta E\sim\frac{\hbar^2\pi^2}{2mL^2},\qquad G=\frac{2e^2}{h}\sum_nT_n$$

Worked example

A GaAs quantum well of width $L=10\,\text{nm}$ has electron effective mass $m^*=0.067m_e$. The confinement energy of the lowest subband is

$$\Delta E_1=\frac{\hbar^2\pi^2}{2m^*L^2}=\frac{(1.05\times10^{-34})^2\pi^2}{2(0.067\times9.11\times10^{-31})(10^{-8})^2}\approx8.9\times10^{-21}\,\text{J}\approx56\,\text{meV}.$$

This is comparable to room-temperature $k_BT\approx26\,\text{meV}$, so confinement effects are significant.

Problems with Solutions

Problem 1. An electron is confined in a 1D box of length $L=5\,\text{nm}$. What is the energy difference between the first and second levels? Solution. $E_n=n^2h^2/(8mL^2)$. $\Delta E=E_2-E_1=3h^2/(8mL^2)=3(6.63\times10^{-34})^2/(8\times9.11\times10^{-31}\times25\times10^{-18})=7.2\times10^{-20}\,\text{J}=0.45\,\text{eV}$.

Problem 2. A quantum point contact has conductance $G=2e^2/h$ per transmitted mode. If $N=3$ modes transmit perfectly, what is the resistance? Solution. $G=3(2e^2/h)=6(3.87\times10^{-5}\,\text{S})=2.32\times10^{-4}\,\text{S}$. The resistance is $R=1/G=4.3\,\text{k}\Omega$.

Problem 3. Estimate the Fermi wavelength in a 2D electron gas with $n_{2D}=10^{15}\,\text{m}^{-2}$. Is the system quantum degenerate at $T=1\,\text{K}$? Solution. $k_F=\sqrt{2\pi n_{2D}}=2.5\times10^7\,\text{m}^{-1}$, $\lambda_F=2\pi/k_F=2.5\times10^{-7}\,\text{m}=250\,\text{nm}$. The Fermi energy is $E_F=\hbar^2k_F^2/(2m_e)=0.36\,\text{meV}$. Since $k_BT=0.086\,\text{meV}<E_F$, the gas is degenerate.

Section summary. Nanoscale solids reveal quantum confinement and interface physics.