⚛️Solid State Physics Unit 5 Review

The Fermi surface is the constant-energy surface in reciprocal space (k-space) that separates occupied from unoccupied electron states at absolute zero temperature. Every point on this surface corresponds to an electron state with energy equal to the Fermi energy. The geometry of this surface controls a material's electrical, thermal, and magnetic behavior, so understanding it is central to solid state physics.

Fermi Energy

The Fermi energy ( $E_F$ ) is the energy of the highest occupied electron state at absolute zero. All states with energy below $E_F$ are filled; all states above are empty. Its value depends on the electron density and the density of states of the material. In copper, for example, $E_F \approx 7 \text{ eV}$ .

Fermi Wave Vector

The Fermi wave vector ( $k_F$ ) is the magnitude of the wave vector for states sitting on the Fermi surface. For a free-electron gas (where the Fermi surface is a perfect sphere), $k_F$ is simply the sphere's radius in k-space.

It relates to the electron density $n$ by:

$k_F = (3\pi^2 n)^{1/3}$

This expression comes directly from counting the number of allowed k-states inside a sphere of radius $k_F$ and setting that equal to the total number of electrons.

Fermi Velocity

The Fermi velocity ( $v_F$ ) is the group velocity of electrons at the Fermi surface:

$v_F = \frac{\hbar k_F}{m^*}$

Here $\hbar$ is the reduced Planck constant and $m^*$ is the effective mass. Only electrons near the Fermi surface participate in transport (deeper electrons can't scatter into nearby empty states), so $v_F$ directly governs electrical conductivity and the plasma frequency. Typical values in metals are on the order of $10^6$ m/s.

Fermi Surface Properties

The shape, connectivity, and nesting characteristics of the Fermi surface encode how electrons respond to external fields and interactions. Two metals with the same number of conduction electrons can behave very differently if their Fermi surfaces have different geometries.

Fermi Surface Shape

In the simplest case (free-electron model), the Fermi surface is a sphere. Real materials deviate from this because the periodic crystal potential distorts the energy bands, especially near Brillouin zone boundaries.

Simple metals (Na, K): nearly spherical Fermi surfaces.
Polyvalent metals (Al, Pb): complex shapes with multiple sheets, pockets, and necks arising from bands crossing zone boundaries.
Transition metals (Cu, Fe): intricate, multi-sheeted surfaces due to overlapping s- and d-bands.

Fermi Surface Topology

Topology here refers to the connectivity and genus (number of holes) of the surface.

Closed surfaces form isolated pockets. Electron pockets enclose occupied states; hole pockets enclose unoccupied states.
Open surfaces extend continuously across multiple Brillouin zones, leading to open electron orbits in a magnetic field.

A Lifshitz transition is a topological change in the Fermi surface triggered by varying temperature, pressure, or chemical composition. At a Lifshitz transition, a pocket may appear, vanish, or merge with another, producing anomalies in thermodynamic and transport quantities.

Fermi Surface Nesting

Nesting occurs when large, roughly parallel segments of the Fermi surface can be connected by a single wave vector $q$ . When nesting is strong, the electronic susceptibility $\chi(q)$ diverges, making the system unstable toward forming a new ordered state.

Strong nesting is common in quasi-1D and quasi-2D systems because their Fermi surfaces contain extended flat portions.
Nesting drives charge density waves (CDWs) and spin density waves (SDWs), discussed further below.

Experimental Determination of Fermi Surfaces

Several techniques probe the Fermi surface, each with different strengths. Quantum oscillation methods give extremal cross-sectional areas with high precision, while ARPES provides a direct image of the surface in k-space.

De Haas–van Alphen Effect

When a magnetic field is applied, electron orbits in k-space become quantized into Landau levels. As the field strength changes, successive Landau levels cross the Fermi energy, causing oscillations in the magnetization.

The oscillation frequency $F$ is related to the extremal cross-sectional area $A$ of the Fermi surface perpendicular to the field by the Onsager relation:

$F = \frac{\hbar}{2\pi e} A$

By rotating the sample relative to the field, you can map out the full 3D shape of the Fermi surface from the angular dependence of $F$ .

Shubnikov–de Haas Effect

This is the resistivity counterpart of the de Haas–van Alphen effect. Oscillations appear in the electrical resistivity as a function of inverse magnetic field, with the same physical origin (Landau quantization). The oscillation frequencies yield the same extremal cross-sectional areas. The Shubnikov–de Haas effect is often easier to measure in thin films and 2D systems where magnetization signals are small.

Angle-Resolved Photoemission Spectroscopy (ARPES)

ARPES directly measures the band dispersion $E(\mathbf{k})$ by ejecting electrons from a sample with ultraviolet or soft X-ray photons and recording their kinetic energy and emission angle.

The emission angle maps to the in-plane crystal momentum $\mathbf{k}_\parallel$ .
By scanning energy and angle, you build up the full band structure.
The Fermi surface appears as the constant-energy contour at $E = E_F$ .

ARPES is especially powerful for layered and 2D materials where $k_z$ dispersion is weak.

Fermi Surface in Metals

Free Electron Model

This model treats conduction electrons as a non-interacting gas moving through a uniform positive background (no lattice potential). The dispersion is purely parabolic, $E = \hbar^2 k^2 / 2m$ , so the Fermi surface is a perfect sphere of radius $k_F$ .

The free-electron picture works surprisingly well for alkali metals (Li, Na, K), where the weak lattice potential barely distorts the sphere. It fails for transition metals and polyvalent metals where band interactions are strong.

Nearly Free Electron Model

Adding a weak periodic potential opens energy gaps at Brillouin zone boundaries. Near these boundaries, the Fermi surface bulges toward or away from the zone face, depending on whether the gap pushes states up or down in energy.

Key consequences:

The sphere gets "truncated" or "necked" where it approaches zone boundaries.
Electron pockets form where the Fermi surface bulges into a higher Brillouin zone.
Hole pockets form where the Fermi surface retreats, leaving unoccupied states inside the zone.
Fermi surface nesting can emerge from the flattened segments near zone boundaries.

This model explains the Fermi surfaces of metals like Al and Pb quite well.

Tight-Binding Model

The tight-binding approach starts from localized atomic orbitals and builds bands through inter-site hopping. It naturally captures narrow bands with large effective masses, which are characteristic of d- and f-electron systems.

Transition metals (Fe, Ni, Cu) have complex, multi-sheeted Fermi surfaces that tight-binding handles well.
The model is also the starting point for Hubbard-type treatments of strong electron correlations.
Band widths and Fermi surface shapes depend sensitively on the hopping integrals, which can be extracted from first-principles calculations.

Fermi Surface in Semiconductors

Strictly speaking, an intrinsic semiconductor at $T = 0$ has no Fermi surface because the Fermi level sits inside the band gap where no states exist. The concept becomes relevant when carriers are present (through doping or thermal excitation), creating small electron or hole pockets.

Intrinsic vs. Extrinsic Semiconductors

Intrinsic: pure material, equal numbers of thermally excited electrons and holes. Carrier concentrations are low at room temperature.
n-type: doped with donors (e.g., P in Si), introducing excess electrons. The Fermi level shifts toward the conduction band.
p-type: doped with acceptors (e.g., B in Si), introducing excess holes. The Fermi level shifts toward the valence band.

Fermi Level Position

In an intrinsic semiconductor, the Fermi level sits near the middle of the band gap (shifted slightly by the effective mass ratio of electrons to holes). Doping moves it:

Closer to the conduction band edge for n-type.
Closer to the valence band edge for p-type.

The carrier concentration depends exponentially on the distance between the Fermi level and the band edge, so even small shifts have large effects on conductivity.

Band Gap Effects on Fermi Surface

Because of the gap, the Fermi surface in a semiconductor is not a single connected sheet like in a metal. Instead, you get:

Small electron pockets at conduction band minima (e.g., the six equivalent ellipsoidal pockets in Si along the $\Delta$ directions).
Small hole pockets at valence band maxima (e.g., the heavy-hole and light-hole pockets at $\Gamma$ in Si).

The size of these pockets grows with doping or temperature. Their shape reflects the band curvature (effective mass tensor) at the relevant extrema.

Fermi energy, Fermi-Energy Level for Extrinsic Semiconductor - Physics Stack Exchange

Fermi Surface in Superconductors

Superconductivity fundamentally modifies the electronic states near the Fermi surface. Cooper pairing gaps out portions of the Fermi surface, and the nature of this gap (its symmetry and magnitude) is one of the central questions in superconductivity research.

BCS Theory

The Bardeen-Cooper-Schrieffer (BCS) theory explains conventional superconductivity through three ingredients:

Electrons near the Fermi surface interact attractively via exchange of virtual phonons.
This attraction binds electrons into Cooper pairs.
The Cooper pairs condense into a macroscopic quantum state with zero resistance.

BCS theory predicts an isotropic (s-wave) gap opening symmetrically around the Fermi level in the density of states.

Cooper Pairs

A Cooper pair consists of two electrons with opposite momenta ( $\mathbf{k}$ and $-\mathbf{k}$ ) and opposite spins. The pair has zero net momentum and behaves as a composite boson. The binding energy is small (on the order of meV), so Cooper pairs are large objects, with a coherence length $\xi$ typically spanning hundreds of lattice spacings in conventional superconductors.

Superconducting Gap

The superconducting gap $\Delta$ is the energy cost to break a Cooper pair into two quasiparticle excitations. In BCS theory:

$\Delta(T=0) \approx 1.76 \, k_B T_c$

where $k_B$ is Boltzmann's constant and $T_c$ is the critical temperature. For Nb with $T_c \approx 9.3$ K, this gives $\Delta \approx 1.4$ meV.

In unconventional superconductors (e.g., cuprates), the gap can be anisotropic or have nodes on the Fermi surface, meaning it vanishes along certain directions. The gap symmetry is directly tied to the pairing mechanism.

Fermi Surface Applications

Electrical Conductivity

Only electrons near the Fermi surface contribute to conduction, because only they can scatter into nearby empty states. The conductivity depends on:

The Fermi surface area (more surface area means more conducting electrons).
The Fermi velocity (faster electrons carry more current).
The scattering rate from impurities, phonons, and electron-electron interactions.

Metals like Cu and Ag have large, relatively simple Fermi surfaces with high $v_F$ and long mean free paths, giving them excellent conductivity.

Thermal Conductivity

Heat in metals is carried by both electrons and phonons. The electronic thermal conductivity is linked to electrical conductivity through the Wiedemann-Franz law:

$\frac{\kappa}{\sigma T} = L_0 = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2$

where $L_0$ is the Lorenz number. A large Fermi surface and long electron mean free path enhance the electronic contribution to thermal conductivity.

Thermoelectric Effects

Thermoelectric effects (Seebeck, Peltier) arise from the energy dependence of the electronic transport near $E_F$ . The Seebeck coefficient $S$ is roughly proportional to:

$S \propto \frac{1}{N(E_F)} \frac{dN}{dE}\bigg|_{E_F}$

where $N(E_F)$ is the density of states at the Fermi level. Sharp features in the density of states near $E_F$ (from flat bands, van Hove singularities, or narrow pockets) enhance $S$ . Good thermoelectric materials combine a large Seebeck coefficient with low thermal conductivity and high electrical conductivity.

Fermi Surface in Reciprocal Space

Brillouin Zones

The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It contains all unique k-points needed to describe the electronic states. The Fermi surface is conventionally drawn within the first zone using the reduced zone scheme, but in the extended zone scheme it can span multiple zones.

Higher Brillouin zones matter when the Fermi sphere is large enough to cross zone boundaries. The nearly-free-electron construction of folding higher-zone pieces back into the first zone is a standard way to understand how complex Fermi surface sheets arise from a simple sphere.

Fermi Surface Mapping

Mapping the Fermi surface means determining its shape experimentally or computationally.

Experimental: ARPES for direct k-space imaging; de Haas–van Alphen and Shubnikov–de Haas for extremal areas as a function of field angle; positron annihilation for momentum-space density.
Computational: Density functional theory (DFT) band structure calculations, followed by interpolation onto a fine k-grid to extract the constant-energy contour at $E_F$ .

Comparing measured and calculated Fermi surfaces is one of the most stringent tests of electronic structure theory.

Fermi Surface Reconstruction

The Fermi surface can be reshaped by perturbations that modify the electronic structure:

Magnetic ordering introduces new periodicity (via the magnetic unit cell), folding the Brillouin zone and reconstructing the Fermi surface. This is seen in antiferromagnetic metals like Cr.
Charge or spin density waves gap out nested portions of the Fermi surface, removing those segments and creating new, smaller pockets.
Pressure or strain changes lattice parameters and band overlaps, shifting where the Fermi surface intersects zone boundaries.

Fermi surface reconstruction often shows up as sudden changes in Hall coefficient, magnetoresistance, or quantum oscillation frequencies.

Fermi Surface Instabilities

When the electronic susceptibility diverges at some wave vector $q$ , the uniform metallic state becomes unstable and the system transitions into an ordered phase. Fermi surface geometry, particularly nesting, is the driving factor.

Charge Density Waves (CDWs)

A CDW is a periodic modulation of the electron charge density, accompanied by a static lattice distortion at the same wave vector $q$ . The mechanism:

Parallel (nested) segments of the Fermi surface are connected by $q$ .
The electronic susceptibility $\chi(q)$ is strongly enhanced.
Electron-phonon coupling softens the phonon at $q$ (Kohn anomaly).
Below a critical temperature, the lattice distorts and a gap opens on the nested portions of the Fermi surface.

Classic examples include NbSe $_2$ and the quasi-1D conductor K $_{0.3}$ MoO $_3$ (blue bronze).

Spin Density Waves (SDWs)

SDWs are the magnetic analog of CDWs: a periodic modulation of the spin density rather than the charge density. They require both good nesting and a significant exchange interaction.

Chromium is the textbook example, with an incommensurate SDW driven by nesting between electron and hole octahedra on its Fermi surface.
SDW formation splits the bands and gaps the nested regions, reducing the total electronic energy.

Peierls Instability

The Peierls instability is a special case of CDW formation in one-dimensional or quasi-1D systems. In a strictly 1D metal, the Fermi surface consists of just two points at $+k_F$ and $-k_F$ , giving perfect nesting at $q = 2k_F$ .

This perfect nesting means that any 1D metal is unstable at low enough temperature: the lattice dimerizes (or forms a longer-period distortion), a gap opens at $E_F$ , and the system becomes insulating. Polyacetylene is a classic example. In quasi-1D systems (e.g., organic conductors like TTF-TCNQ), the nesting is imperfect, so the Peierls transition occurs at a finite temperature rather than being inevitable.

Fermi Surface Engineering

Controlling the Fermi surface means controlling a material's electronic response. Three main knobs are available: doping, strain, and dimensionality.

Doping Effects on Fermi Surface

Adding or removing electrons shifts the Fermi level and changes the size of electron or hole pockets.

In semiconductors, doping moves $E_F$ toward a band edge, creating or enlarging carrier pockets.
In metals, substitutional doping (e.g., replacing Cu with Zn in brass) adds electrons, expanding the Fermi surface until it may contact new zone boundaries and change topology.
In correlated systems, doping can suppress or induce CDW/SDW order by degrading or enhancing nesting conditions.

Strain Effects on Fermi Surface

Strain modifies lattice parameters, which shifts band energies and changes orbital overlaps.

Tensile strain generally increases orbital overlap along the strain direction, broadening bands and potentially merging Fermi surface sheets.
Compressive strain reduces overlap, narrowing bands and possibly splitting the Fermi surface into disconnected pockets.
Epitaxial strain in thin films is a practical way to tune Fermi surface topology without changing composition.

Fermi Surface Tuning in 2D Materials

Two-dimensional materials like graphene and transition metal dichalcogenides (TMDs) are especially tunable because their electronic structure is sensitive to the environment.

Electrostatic gating shifts the Fermi level continuously, allowing you to move through van Hove singularities or trigger Lifshitz transitions in real time.
Substrate interactions and encapsulation modify the band structure through strain, dielectric screening, and hybridization.
Twist angle engineering (as in twisted bilayer graphene) creates moiré superlattices that reconstruct the Fermi surface and can produce flat bands associated with correlated phases and superconductivity.