11.2 Electrons in metals

Properties of metals

Statistical mechanics gives you the tools to understand how electrons collectively behave inside metals. Metals have unique electrical, thermal, and optical properties that trace back to their electronic structure and the quantum mechanical nature of their electrons. This topic connects classical transport ideas with quantum statistics, and those connections show up everywhere in modern technology.

Crystalline structure

Metals form periodic lattice structures where atoms sit in repeating geometric patterns. The most common arrangements are body-centered cubic (BCC) (like iron), face-centered cubic (FCC) (like copper and aluminum), and hexagonal close-packed (HCP) (like zinc).

• Phonons are quantized lattice vibrations that propagate through these structures. They're central to understanding thermal properties and how electrons scatter as they move through the metal.
• Real crystals always contain imperfections: dislocations, vacancies, and grain boundaries. These defects scatter electrons and affect both electrical resistivity and mechanical strength.

Free electron model

The free electron model treats conduction electrons as a gas of non-interacting particles that move freely inside the metal. Despite its simplicity, it explains a surprising amount: electrical conductivity, heat capacity trends, and the basic optical response of metals.

The key assumptions are:

1. The potential inside the metal is constant (electrons don't "feel" individual ions).
2. The potential at the boundaries is infinite (electrons are confined to the metal).
3. Electron-electron interactions are ignored.

Under these assumptions, the energy-momentum relationship is parabolic:

$E = \frac{\hbar^2 k^2}{2m}$

where $k$ is the electron wavevector and $m$ is the electron mass.

The model's main limitations: it ignores the periodic potential of the lattice (which gives rise to band gaps) and neglects electron-electron correlations. The nearly-free electron model and band theory address these shortcomings.

Conduction vs. valence electrons

• Valence electrons are the outermost electrons of an atom and participate in chemical bonding.
• Conduction electrons occupy states near the Fermi level and are free to move through the lattice, carrying electrical current.

In metals, the distinction between these two often blurs because the valence band and conduction band overlap. The number of conduction electrons per atom varies: copper contributes 1, aluminum contributes 3. This directly affects properties like the Hall coefficient and the plasma frequency.

Fermi-Dirac statistics

Because electrons are fermions (spin-1/2 particles), no two can occupy the same quantum state. This constraint means you need Fermi-Dirac statistics rather than classical Maxwell-Boltzmann statistics to describe their energy distribution.

Fermi energy

The Fermi energy $E_F$ is the energy of the highest occupied electron state at absolute zero ($T = 0$ K). Below $E_F$, every state is filled; above it, every state is empty.

For a 3D free electron gas:

$E_F = \frac{\hbar^2}{2m}(3\pi^2 n)^{2/3}$

where $n$ is the number density of conduction electrons. For typical metals, $E_F$ falls in the range of 2 to 12 eV. Copper, for example, has $E_F \approx 7.0$ eV. This is a huge energy scale compared to thermal energy at room temperature ($k_B T \approx 0.025$ eV), which is why most electrons sit deep below the Fermi level and don't participate in thermal processes.

Density of states

The density of states $g(E)$ tells you how many electron states are available per unit energy interval. For a 3D free electron gas:

$g(E) = \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2} \sqrt{E}$

The $\sqrt{E}$ dependence means there are more available states at higher energies. This function is critical for calculating thermodynamic quantities because any integral over electron properties (specific heat, magnetic susceptibility, etc.) weights the integrand by $g(E)$.

Temperature dependence

At finite temperature, the sharp step function at $E_F$ softens into the Fermi-Dirac distribution:

$f(E) = \frac{1}{e^{(E - E_F)/k_B T} + 1}$

At $T = 0$, this is exactly 1 for $E < E_F$ and 0 for $E > E_F$. As temperature increases, electrons within roughly $k_B T$ of the Fermi level get thermally excited to states just above it. Only this thin shell of electrons near $E_F$ participates in thermal and transport processes. That's why the electronic specific heat is so much smaller than classical predictions.

Electron energy bands

Band theory basics

When atoms come together to form a crystal, their discrete atomic energy levels broaden into continuous energy bands separated by band gaps (forbidden energy ranges). This happens because the periodic potential of the lattice splits what would otherwise be free-electron states.

• The tight-binding model starts from isolated atomic orbitals and shows how their overlap creates bands.
• The nearly-free electron model starts from free electrons and shows how the periodic lattice potential opens gaps at Brillouin zone boundaries.

Each band has a dispersion relation $E(k)$ that relates electron energy to crystal momentum $\hbar k$. The shape of $E(k)$ determines the effective mass of electrons and their response to external fields.

Metals vs. insulators vs. semiconductors

The band structure determines whether a material conducts:

• Metals have a partially filled band or overlapping bands, so electrons can easily move to nearby empty states and carry current.
• Insulators have a completely filled valence band separated from an empty conduction band by a large gap (typically > 4 eV). Electrons can't reach the conduction band at ordinary temperatures.
• Semiconductors have a small band gap (typically < 4 eV), so thermal excitation or doping can promote electrons into the conduction band. Silicon has a gap of about 1.1 eV.

Brillouin zones

Brillouin zones are the fundamental cells of the reciprocal lattice in $k$-space.

• The first Brillouin zone contains all unique wavevectors needed to describe electron states. Any wavevector outside it can be mapped back inside by adding a reciprocal lattice vector.
• Zone boundaries correspond to the condition for Bragg diffraction of electron waves by the lattice. This is exactly where band gaps open up.
• Higher-order zones are periodic repetitions of the first zone.

Understanding Brillouin zones is essential for interpreting band structures and Fermi surfaces in real metals.

Electrical conductivity

Drude model

The Drude model is the classical starting point for electron transport. It pictures conduction electrons as a gas of particles that accelerate under an electric field and lose momentum through collisions with ion cores.

The resulting conductivity is:

$\sigma = \frac{ne^2\tau}{m}$

where $n$ is the electron density, $e$ is the electron charge, $\tau$ is the mean free time between collisions, and $m$ is the electron mass. This directly gives you Ohm's law: $\mathbf{J} = \sigma \mathbf{E}$.

The Drude model also predicts the frequency-dependent (AC) conductivity, which explains why metals are reflective at optical frequencies. Its main failure is predicting the wrong electronic specific heat (too large by a factor of ~100), which Fermi-Dirac statistics fixes.

Matthiessen's rule

Different scattering mechanisms contribute independently to the total resistivity:

$\rho_{\text{total}} = \rho_{\text{impurity}} + \rho_{\text{phonon}} + \rho_{\text{defect}}$

• $\rho_{\text{impurity}}$ and $\rho_{\text{defect}}$ are roughly temperature-independent (residual resistivity).
• $\rho_{\text{phonon}}$ is strongly temperature-dependent.

This additive rule lets you separate intrinsic scattering (phonons) from extrinsic scattering (impurities, defects), which is useful for characterizing material purity.

Temperature effects on conductivity

In metals, resistivity generally increases with temperature because higher temperatures mean more phonons and more electron-phonon scattering.

• Low temperatures ($T \ll \Theta_D$, the Debye temperature): resistivity follows the Bloch-Grüneisen law, $\rho(T) \propto T^5$. The steep power law reflects the freezing out of phonon modes.
• High temperatures ($T \gg \Theta_D$): resistivity grows linearly, $\rho(T) \propto T$, because all phonon modes are excited and their population grows linearly with $T$.
• The residual resistivity ratio (RRR), defined as $\rho(300\,\text{K})/\rho(4.2\,\text{K})$, is a common measure of sample purity. High-purity copper can have RRR values exceeding 1000.

Thermal properties

Electronic specific heat

Classical theory (equipartition) predicts each electron contributes $\frac{3}{2}k_B$ to the specific heat. Experiments show the electronic contribution is far smaller. Fermi-Dirac statistics explains why: only electrons within $\sim k_B T$ of the Fermi level can be thermally excited.

At low temperatures, the electronic specific heat is:

$C_e = \gamma T$

The coefficient $\gamma$ is called the Sommerfeld coefficient and is proportional to the density of states at the Fermi level, $g(E_F)$. Below about 1 K, this linear electronic term dominates over the lattice contribution (which goes as $T^3$). Measuring $\gamma$ gives direct information about the effective mass of electrons and the strength of electron-electron interactions.

Wiedemann-Franz law

The Wiedemann-Franz law states that the ratio of thermal conductivity $\kappa$ to electrical conductivity $\sigma$ is proportional to temperature:

$\frac{\kappa}{\sigma T} = L$

where $L$ is the Lorenz number. The theoretical value from Fermi-Dirac statistics is:

$L = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2 \approx 2.44 \times 10^{-8}\;\text{W}\,\Omega\,\text{K}^{-2}$

This works because the same electrons carry both charge and heat. The law holds well at very low and very high temperatures but can break down at intermediate temperatures where inelastic scattering processes become important.

Thermoelectric effects

When charge flow and heat flow couple, you get thermoelectric effects:

• Seebeck effect: A temperature gradient across a conductor generates a voltage. The Seebeck coefficient $S = -\Delta V / \Delta T$ depends on the energy dependence of the conductivity near $E_F$.
• Peltier effect: Current flowing through a junction of two different conductors causes heat to be absorbed or released at the junction.
• Thomson effect: A single conductor carrying current in the presence of a temperature gradient absorbs or releases heat throughout its bulk.

The efficiency of thermoelectric devices is characterized by the dimensionless figure of merit $ZT = S^2 \sigma T / \kappa$. Higher $ZT$ means better conversion efficiency. Current commercial materials reach $ZT \approx 1$; values above 3 would make thermoelectric generators competitive with mechanical heat engines.

Quantum effects in metals

Landau levels

When you apply a uniform magnetic field to a metal, the continuous spectrum of electron energies perpendicular to the field collapses into discrete Landau levels:

$E_n = \left(n + \frac{1}{2}\right)\hbar\omega_c$

where $\omega_c = eB/m$ is the cyclotron frequency and $n = 0, 1, 2, \ldots$ These levels are highly degenerate (many electrons share the same energy). As the magnetic field changes, Landau levels sweep through the Fermi energy, causing oscillations in thermodynamic and transport properties.

Observing Landau quantization requires high magnetic fields and low temperatures so that $\hbar\omega_c \gg k_B T$ and the thermal broadening doesn't wash out the discrete levels.

de Haas-van Alphen effect

The de Haas-van Alphen (dHvA) effect refers to oscillations in the magnetization (or magnetic susceptibility) of a metal as a inverse magnetic field $1/B$ is varied. These oscillations arise because Landau levels periodically cross the Fermi energy.

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

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

By measuring dHvA oscillations for different field orientations, you can map out the full 3D shape of the Fermi surface. This technique also yields the effective mass of electrons through the temperature dependence of the oscillation amplitude.

Quantum Hall effect

In a two-dimensional electron gas (such as electrons confined at a semiconductor interface) subjected to a strong perpendicular magnetic field, the Hall conductance becomes quantized:

$\sigma_{xy} = \nu \frac{e^2}{h}$

where $\nu$ is an integer (integer quantum Hall effect) or a simple fraction (fractional quantum Hall effect). The quantization is extraordinarily precise and is now used as a resistance standard.

The integer effect can be understood from Landau level filling, while the fractional effect requires accounting for strong electron-electron interactions. Both are connected to the topological properties of the electron wavefunctions, a connection that has driven much of modern condensed matter research, including topological insulators.

Electron-phonon interactions

Cooper pairs

In a metal at low temperature, two electrons can form a bound state called a Cooper pair through an indirect attractive interaction mediated by phonons. Here's the mechanism:

1. An electron moving through the lattice attracts nearby positive ions, creating a slight concentration of positive charge.
2. A second electron (with opposite momentum and spin) is attracted to this positive charge concentration.
3. The net effect is an attractive interaction between the two electrons that overcomes their Coulomb repulsion.

The resulting pair has zero net momentum and zero net spin, making it a boson. Cooper pairs can therefore condense into a single macroscopic quantum state. The binding energy of a Cooper pair is tiny, typically on the order of meV (about $10^{-3}$ eV), which is why superconductivity only appears at low temperatures.

Superconductivity basics

Below a critical temperature $T_c$, certain metals and compounds enter a superconducting state with two defining features:

• Zero electrical resistance: Current flows without any energy dissipation.
• Meissner effect: Magnetic fields are expelled from the interior of the superconductor (perfect diamagnetism). This is not simply a consequence of zero resistance; it's a distinct thermodynamic property.

Superconductors come in two types:

• Type I: A single critical field $H_c$ above which superconductivity is destroyed abruptly. Most elemental superconductors (lead, mercury) are Type I.
• Type II: Two critical fields. Between $H_{c1}$ and $H_{c2}$, magnetic flux penetrates in quantized vortices (the mixed state). This allows Type II superconductors to sustain much higher fields, making them useful for magnets. Niobium-titanium and the cuprate high-$T_c$ materials are Type II.

BCS theory

BCS theory (Bardeen, Cooper, and Schrieffer, 1957) provides the microscopic explanation of conventional superconductivity. The central ideas:

• Electrons near the Fermi surface form Cooper pairs via phonon-mediated attraction.
• These pairs condense into a coherent ground state separated from excited states by an energy gap $\Delta$.
• The gap at zero temperature is related to the critical temperature by:

$2\Delta(0) \approx 3.5\, k_B T_c$

• The gap vanishes continuously as $T \to T_c$, consistent with a second-order phase transition.

BCS theory successfully predicts $T_c$, the specific heat jump at the transition, the coherence length, and the isotope effect ($T_c \propto M^{-1/2}$, where $M$ is the ion mass). It does not, however, explain high-temperature superconductors like the cuprates ($T_c > 100$ K), where the pairing mechanism remains an open question.

Experimental techniques

Photoemission spectroscopy

Photoemission spectroscopy measures the kinetic energy and emission angle of electrons ejected from a material by incident photons. From energy and momentum conservation, you can reconstruct the electronic band structure.

• ARPES (angle-resolved photoemission spectroscopy) maps out $E(k)$ and directly images the Fermi surface. It's one of the most powerful probes of electronic structure in metals and strongly correlated systems.
• XPS (X-ray photoemission spectroscopy) uses higher-energy photons to probe core-level electrons, giving chemical composition and bonding information.

Scanning tunneling microscopy

The scanning tunneling microscope (STM) brings a sharp conducting tip within a few angstroms of a surface. A bias voltage drives a tunneling current that depends exponentially on the tip-surface distance, giving atomic-scale spatial resolution.

• Scanning tunneling spectroscopy (STS) varies the bias voltage to measure the local density of states as a function of energy.
• STM/STS can image standing waves of electrons scattered by impurities, visualize the superconducting gap, and detect charge density waves.

Hall effect measurements

Applying a magnetic field perpendicular to a current-carrying conductor deflects charge carriers to one side, producing a transverse Hall voltage. The Hall coefficient for a simple metal is:

$R_H = -\frac{1}{ne}$

where $n$ is the carrier density and the sign tells you the carrier type (negative for electrons, positive for holes). This is one of the most straightforward ways to determine carrier density and mobility.

In ferromagnetic metals, the anomalous Hall effect adds a contribution proportional to the magnetization, which depends on spin-orbit coupling and the band structure.

Applications in technology

Semiconductor devices

While semiconductors aren't metals, understanding the free electron model and band theory for metals is the foundation for semiconductor physics. Controlled manipulation of band gaps through doping (adding impurity atoms) and band gap engineering (alloying or heterostructures) enables:

• Diodes: p-n junctions that allow current in one direction.
• Transistors: Switches and amplifiers that form the basis of all digital logic.
• Integrated circuits: Billions of transistors on a single chip.

Research into novel materials like graphene and transition metal dichalcogenides aims to push device performance beyond the limits of silicon.

Superconducting magnets

Superconducting coils carry persistent currents with zero resistive loss, generating strong, stable magnetic fields. Applications include:

• MRI machines: Require fields of 1.5 to 7 T, typically using NbTi coils cooled to 4.2 K.
• Particle accelerators: The LHC uses NbTi magnets producing fields up to 8.3 T to steer proton beams.
• Fusion reactors: Tokamak designs rely on superconducting magnets for plasma confinement.

The main practical challenge is cryogenic cooling. Research into high-temperature superconductors (like YBCO, $T_c \approx 93$ K) could eventually allow operation with cheaper liquid nitrogen cooling instead of liquid helium.

Thermoelectric materials

Thermoelectric devices convert heat directly to electricity (via the Seebeck effect) or pump heat using electrical current (via the Peltier effect). They have no moving parts, making them reliable for specialized applications:

• Waste heat recovery in automotive and industrial settings
• Radioisotope thermoelectric generators (RTGs) powering deep-space probes like Voyager
• Solid-state Peltier coolers for electronics and small-scale refrigeration

The efficiency bottleneck is the figure of merit $ZT$. Improving $ZT$ requires simultaneously high electrical conductivity, high Seebeck coefficient, and low thermal conductivity. Nanostructuring (superlattices, nanowires) has shown promise by reducing lattice thermal conductivity without degrading electrical properties.