Formula sheet · PU-303

Condensed Matter Physics — formula sheet

Every boxed result from the 17 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.

D-247

Reciprocal Lattice from the Bravais Lattice

\[ \vec{b}_i = 2\pi\,\frac{\vec{a}_j\times\vec{a}_k}{\vec{a}_1\cdot(\vec{a}_2\times\vec{a}_3)}\ \ (ijk\ \text{cyclic}),\qquad \vec{a}_i\cdot\vec{b}_j = 2\pi\,\delta_{ij} \]

The three reciprocal primitive vectors are the "biorthogonal dual" of the direct primitive vectors: \(\vec{b}_i\) points perpendicular to the plane spanned by the other two direct vectors, and its length is scaled so that its projection onto \(\vec{a}_i\) equals \(2\pi\). The integer combinations of the \(\vec{b}_i\) are exactly the wavevectors whose plane waves are invisible to (periodic on) the lattice. The factor \(2\pi\) is a convention of solid-state physics; crystallographers often drop it, replacing \(e^{i\vec{G}\cdot\vec{R}}=1\) by \(\vec{G}\cdot\vec{R}\in\mathbb{Z}\).

D-248

Laue and Bragg Diffraction Conditions

\[ \vec{q}=\vec{k}'-\vec{k}=\vec{G}_{hkl} \;\;\Longleftrightarrow\;\; 2d_{hkl}\sin\theta=n\lambda, \qquad I \propto \Big|\sum_\alpha f_\alpha\, e^{-i\vec G\cdot\vec d_\alpha}\Big|^2 \]

A crystal diffracts only when the change in the photon's wavevector lands exactly on a reciprocal-lattice point. Equivalently, the classical Bragg picture of mirror-like reflection from lattice planes at the special angle \(\theta\) is nothing but this momentum-matching condition rewritten in real space. Which reflections actually appear — and how bright they are — is decided by the structure factor, the phased sum of atomic form factors over the basis; when that sum cancels, the reflection is systematically absent regardless of geometry.

D-249

Bloch's Theorem from Translational Symmetry

\[ \boxed{\ \psi_{n\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}\,u_{n\vec{k}}(\vec{r}),\qquad u_{n\vec{k}}(\vec{r}+\vec{R})=u_{n\vec{k}}(\vec{r}),\qquad \vec{k}\in\text{1st BZ}\ } \]

Every eigenstate of a periodic Hamiltonian is a running plane wave \(e^{i\vec{k}\cdot\vec{r}}\) whose amplitude is modulated by a function \(u_{n\vec{k}}\) that repeats with the lattice. The plane wave carries the long-range coherence — it is the same at equivalent points of every cell up to a fixed phase \(e^{i\vec{k}\cdot\vec{R}}\) — while \(u_{n\vec{k}}\) carries the atomic-scale structure inside a cell. The wavevector \(\vec{k}\) is a conserved crystal momentum, unique within the first Brillouin zone; the discrete index \(n\) counts the bands, i.e. the tower of solutions of the cell problem at fixed \(\vec{k}\).

D-250

Band Gaps in the Nearly-Free-Electron Model

\[ \boxed{\,E_{\mathrm{gap}}=2\,|V_{\mathbf{G}}|\,} \]

A weak periodic potential mixes the two counter-propagating plane waves \(e^{i\mathbf{k}\cdot\mathbf{r}}\) and \(e^{i(\mathbf{k}-\mathbf{G})\cdot\mathbf{r}}\) that Bragg-reflect into one another at the zone boundary. The mixing produces standing waves: one piling charge on the ion cores (lower energy, \(E_-\)) and one piling charge between them (higher energy, \(E_+\)). The energy separation of these two standing waves is \(2|V_{\mathbf{G}}|\), a range of energies no propagating Bloch state can occupy — the band gap. Its size is set purely by the strength of the Fourier component of the lattice potential at the reciprocal vector that connects the two states.

D-251

Tight-Binding Band Dispersion

\[ E(\mathbf{k})=\sum_{\mathbf{S}}e^{i\mathbf{k}\cdot\mathbf{S}}\,h(\mathbf{S}) \;\;\xrightarrow[\text{1D chain}]{\text{NN}}\;\; E(k)=\epsilon-2t\cos(ka) \]

A single atomic level of energy \(\epsilon\) broadens into a continuous band as orbitals on neighbouring sites overlap and let the electron hop with amplitude \(t\). The band is a cosine of total width \(W=2zt\) (\(=4t\) for the chain, \(z=2\); \(=12t\) for simple cubic, \(z=6\)): larger overlap means wider bands and lighter carriers. Near the band bottom the cosine is parabolic, \(E\simeq(\epsilon-2t)+t a^2 k^2\), so the electron behaves as a free particle with effective mass \(m^{*}=\hbar^{2}/(2ta^{2})\) — the lattice has replaced the bare mass with one set by the hopping.

D-252

Effective Mass and Semiclassical Electron Dynamics

\[ \mathbf{v}_n=\frac{1}{\hbar}\nabla_{\mathbf{k}}E_n,\qquad \hbar\dot{\mathbf{k}}=\mathbf{F}_{\text{ext}},\qquad (M^{-1})_{ij}=\frac{1}{\hbar^2}\frac{\partial^2 E_n}{\partial k_i\,\partial k_j} \]

A Bloch electron drifts at the band's group velocity and responds to a force by having its crystal momentum \(\hbar\mathbf{k}\) pushed at exactly the free-particle rate; the entire influence of the lattice is compressed into the curvature of \(E_n(\mathbf{k})\). Where the band curves up (band bottom) the mass is positive; where it curves down (band top) the mass is negative — the origin of holes. Off-diagonal curvature makes the mass a tensor, so acceleration need not be parallel to force.

D-253

Density of States and Van Hove Singularities

\[ g(\varepsilon)=\frac{1}{(2\pi)^d}\oint_{\varepsilon_n(\mathbf{k})=\varepsilon}\frac{dS_k}{\left|\nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k})\right|}, \qquad \nabla_{\mathbf{k}}\varepsilon_n(\mathbf{k}_c)=\mathbf{0}\;\Rightarrow\; \text{Van Hove singularity} \]

The density of states is the area of a constant-energy surface weighted by the inverse group speed: where electrons move slowly (flat bands, small \(\left|\nabla_{\mathbf{k}}\varepsilon\right|\)) they pile up in energy. Band extrema and saddle points, where the group velocity vanishes, are the sources of the non-analytic features. Their exact form — step, kink, log, or inverse-root — is fixed only by the spatial dimension \(d\) and the Morse index \(p\), independent of microscopic detail. Restore a factor of 2 for spin: \(g\to 2g\).

D-254

Sommerfeld Free-Electron Heat Capacity

\[ \boxed{\,C_{\mathrm{el}} = \gamma\, T,\qquad \gamma = \frac{\pi^2}{3}k_{\mathrm B}^2\, g(\varepsilon_{\mathrm F}) = \frac{\pi^2}{2}\frac{N k_{\mathrm B}}{T_{\mathrm F}}\,} \]

The electronic heat capacity rises linearly from zero as the metal is warmed. Its slope \(\gamma\) (the Sommerfeld coefficient) is a direct measure of the density of single-particle states at the Fermi energy. Physically, only electrons within an energy shell \(\sim k_{\mathrm B}T\) of \(\varepsilon_{\mathrm F}\) — a fraction \(\sim T/T_{\mathrm F}\) of the total — are thermally active, and each carries \(\sim k_{\mathrm B}T\) of excess energy, so \(U-U_0\sim g(\varepsilon_{\mathrm F})(k_{\mathrm B}T)^2\) and \(C\sim k_{\mathrm B}^2 g(\varepsilon_{\mathrm F})T\).

D-255

Phonon Dispersion of the Diatomic Chain

\[ \omega_\pm^2(q) = K\left(\frac{1}{M}+\frac{1}{m}\right) \pm K\sqrt{\left(\frac{1}{M}+\frac{1}{m}\right)^2 - \frac{4\sin^2(qa/2)}{Mm}} \]

The lower root \(\omega_-\) is the acoustic branch: it starts at \(\omega=0\) at \(q=0\), where both sublattices oscillate in phase (\(A=B\)) like a long-wavelength sound wave, and rises to \(\sqrt{2K/M}\) at the zone boundary. The upper root \(\omega_+\) is the optical branch: at \(q=0\) it sits at \(\omega=\sqrt{2K(1/M+1/m)}=\sqrt{2K/\mu}\) (with reduced mass \(\mu=Mm/(M+m)\)), where the two sublattices move exactly out of phase (\(MA=-mB\), centre of mass fixed), and it falls to \(\sqrt{2K/m}\) at the boundary. Between \(\sqrt{2K/M}\) and \(\sqrt{2K/m}\) no propagating mode exists — the zone-boundary gap, which is finite precisely when \(M\neq m\).

D-256

Quantization of Lattice Vibrations

\[ \boxed{\;\hat H=\sum_{\mathbf{k},s}\hbar\omega_{\mathbf{k}s}\left(\hat a_{\mathbf{k}s}^{\dagger}\hat a_{\mathbf{k}s}+\tfrac12\right),\qquad [\hat a_{\mathbf{k}s},\hat a_{\mathbf{k}'s'}^{\dagger}]=\delta_{\mathbf{k}\mathbf{k}'}\delta_{ss'}\;} \]

The crystal is a gas of non-interacting bosons. Each mode \((\mathbf{k},s)\) can hold any number \(n_{\mathbf{k}s}=0,1,2,\dots\) of phonons, each carrying energy \(\hbar\omega_{\mathbf{k}s}\); \(\hat a_{\mathbf{k}s}^{\dagger}\) adds one phonon, \(\hat a_{\mathbf{k}s}\) removes one. Eigenstates are occupation-number (Fock) states \(|\{n_{\mathbf{k}s}\}\rangle=\prod_{\mathbf{k},s}\frac{(\hat a_{\mathbf{k}s}^{\dagger})^{n_{\mathbf{k}s}}}{\sqrt{n_{\mathbf{k}s}!}}|0\rangle\) with energy \(\sum_{\mathbf{k},s}\hbar\omega_{\mathbf{k}s}(n_{\mathbf{k}s}+\tfrac12)\). The \(\tfrac12\) is the zero-point energy that survives even at \(T=0\).

D-258

Intrinsic Carrier Concentration in Semiconductors

\[ np=N_cN_v\,e^{-E_g/k_BT},\qquad n_i=\sqrt{N_cN_v}\,e^{-E_g/2k_BT}=2\left(\frac{k_BT}{2\pi\hbar^2}\right)^{3/2}(m_e^*m_h^*)^{3/4}e^{-E_g/2k_BT} \]

In equilibrium the product of electron and hole densities depends only on the gap and temperature, not on doping or the Fermi level's position — add donors and \(n\) rises exactly as \(p\) falls so that \(np\) is pinned. The intrinsic density carries the tell-tale \(T^{3/2}\) prefactor from the phase-space of thermally accessible band states and an Arrhenius factor with activation energy \(E_g/2\), half the gap, because a single thermal event creates one electron and one hole sharing the cost \(E_g\).

D-259

Electrical Conductivity from the Boltzmann Equation

\[ \sigma_{ij}=e^2\int\frac{2\,d^3k}{(2\pi)^3}\,\tau\,v_i v_j\left(-\frac{\partial f_0}{\partial\varepsilon}\right)\;\xrightarrow[\text{constant }\tau]{\text{isotropic}}\;\sigma=\frac{ne^2\tau}{m} \]

The conductivity is not a property of "all \(n\) electrons" but of the thin Fermi-surface shell, weighted by how long each state survives (\(\tau\)) and how fast it moves along the field (\(v_iv_j\)). The kernel \(-\partial f_0/\partial\varepsilon\) is a sharply peaked window that selects states within \(k_BT\) of \(\varepsilon_F\). The Drude form emerges only because, for a spherical band, the large Fermi velocity that makes each carrier effective is exactly cancelled by the small density of states, leaving the naive count \(n\). The tensor is symmetric and positive: current never flows against the field's projection.

D-260

Landau Levels and the Integer Quantum Hall Effect

\[ E_n=\hbar\omega_c\!\left(n+\tfrac12\right),\quad n_B=\frac{eB}{h},\quad \boxed{\ \sigma_{xy}=\nu\,\frac{e^2}{h}\ \xrightarrow{\ \nu=i\ }\ i\,\frac{e^2}{h}\ },\quad R_{xy}=\frac{h}{i e^2} \]

The kinetic energy of a 2D electron in a field is quantized into equally spaced Landau levels separated by \(\hbar\omega_c\), each holding \(eB/h\) states per unit area — one per flux quantum. When exactly \(i\) levels are full the Hall conductance is an integer multiple of the conductance quantum \(e^2/h\) and the longitudinal conductance vanishes: the system carries dissipationless edge current whose quantization is set only by fundamental constants and the integer \(i\).

D-261

Cooper Pair Instability

\[ \boxed{\;\Delta \;=\; 2\hbar\omega_D\,\exp\!\left(-\dfrac{2}{N(0)V}\right)\;>\;0\quad\text{for all }V>0\;} \]

The added pair sits an energy \( \Delta \) below the top of two free electrons at the Fermi surface, so the normal metal can always lower its energy by pairing — it is unstable. The binding is non-analytic in \( V \): every term of its Taylor series about \( V=0 \) vanishes, so no order of perturbation theory ever produces it. The Fermi sea is indispensable: without it the lower integration limit would go to zero and a weak attraction in 3D would not bind. Note \( \Delta\propto\hbar\omega_D \), tying pairing to the lattice — the origin of the isotope effect.

D-262

BCS Gap Equation and the Superconducting Gap

\[ 1=N(0)V\int_{0}^{\hbar\omega_{D}}\frac{\tanh\!\left(\dfrac{\sqrt{\xi^{2}+\Delta^{2}}}{2k_BT}\right)}{\sqrt{\xi^{2}+\Delta^{2}}}\,d\xi \]

The gap \(\Delta(T)\) is fixed self-consistently: an electron pair feels an attraction \(V\) but only pairs where a coherent gap \(\Delta\) already exists, weighted by how many quasiparticle states are thermally excited via \(\tanh\). At \(T=0\) the \(\tanh\to1\) and the integral gives \(\Delta_0=2\hbar\omega_D\,e^{-1/N(0)V}\); as \(T\to T_c\) the gap vanishes and \(k_BT_c=1.13\,\hbar\omega_D\,e^{-1/N(0)V}\).

D-263

Weiss Mean-Field Theory of Ferromagnetism

\[M=n g\mu_B S\,B_S\!\left(\frac{g\mu_B S(B_0+\Lambda M)}{k_B T}\right),\quad k_BT_c=\frac{zJS(S+1)}{3},\quad \chi=\frac{C}{T-T_c}\]

The magnetisation solves its own field equation: below \(T_c\) the self-consistency curve has a slope steeper than one at the origin, so a stable non-zero \(M\) branches off continuously from zero — a second-order transition. \(T_c\) rises linearly with the exchange \(J\) and the coordination \(z\), and with the "spin size" \(S(S+1)\). Above \(T_c\) the susceptibility diverges as \(1/(T-T_c)\): the Curie \(1/T\) law of independent moments, but shifted so the divergence occurs at \(T_c\) rather than \(0\), the shift measuring the strength of the molecular field.

D-264

Spin-Wave Dispersion and Magnons

\[ \boxed{\;\hbar\omega_{\mathbf k}=2JSz\,(1-\gamma_{\mathbf k})\xrightarrow{k\to0}Dk^2,\quad D=2JSa^2;\qquad \frac{\Delta M}{M_0}=\frac{\zeta(3/2)}{S}\left(\frac{k_BT}{4\pi D}\right)^{3/2}\propto T^{3/2}\;}\]

Low-energy excitations of a ferromagnet are gapless, quadratically dispersing magnons — collective spin precessions. Populating them thermally tilts spins away from full alignment, so the spontaneous magnetisation falls from its \(T=0\) value \(M_0=g\mu_B S/v_0\) as \(T^{3/2}\), not exponentially and not linearly. The prefactor is fixed by the spin stiffness \(D\), i.e. by the exchange \(J\).

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