Semiclassical equations of motion in solid-state physics

Semiclassical equations of motion in solid-state physics describe the dynamics of electrons in a crystalline solid, when external perturbations vary slowly compared to the lattice spacing and the electronic wave packet remains well localized in both real and momentum space. In this regime, the electron behaves neither as a purely classical particle nor as a fully delocalized quantum state, but as a quantum wave packet whose center follows effective classical-like equations of motion.

These equations provide the foundation of transport theory in solids, including the description of electrical conduction, cyclotron motion, and Bloch oscillations. They emerge from the Schrödinger equation under the assumption that the electron occupies a single energy band and that interband transitions are negligible. Within this approximation, the crystal momentum evolves under external forces in a manner directly analogous to Newton's second law for a classical particle, while the electron velocity is given by the gradient of the band energy rather than by the momentum itself.

In this analogy, the semiclassical variables ${\displaystyle (\mathbf {r} ,\mathbf {k} )}$ play roles similar to the classical phase-space variables ${\displaystyle (\mathbf {r} ,\mathbf {p} )}$, with ${\displaystyle \hbar \mathbf {k} }$ replacing the canonical momentum. The force determines the evolution of ${\displaystyle \mathbf {k} }$, while the band dispersion ${\displaystyle \varepsilon _{n}(\mathbf {k} )}$ determines the velocity. Departures from this classical analogy arise from the geometric properties of the Bloch states.

Extensions of the semiclassical formalism incorporate geometric properties of the Bloch states, such as the Berry curvature and the quantum metric. These quantities describe not only how the energy of an electronic mode varies along a band, but also how the corresponding quantum eigenstate itself evolves in momentum space. They give rise to additional transport effects including the anomalous Hall effect and coherent interband dynamics. Semiclassical equations thus form a central bridge between quantum band theory and observable macroscopic transport phenomena in condensed-matter physics.

Basic semiclassical equations

Electrons in a crystalline solid can be described by semiclassical equations of motion when external perturbations vary slowly compared to the lattice period. In this approach, an electron is treated as a wave packet localized in a single energy band, and its center-of-mass motion follows equations formally analogous to Hamilton's equations for a classical particle.^[1][2]

If ${\displaystyle \varepsilon _{n}(\mathbf {k} ,\mathbf {r} )}$ is the energy of the electron in band ${\displaystyle n}$ as a function of the crystal momentum ${\displaystyle \mathbf {k} }$ and position ${\displaystyle \mathbf {r} }$, the semiclassical equations of motion are

${\displaystyle {\dot {\mathbf {r} }}={\frac {1}{\hbar }}\nabla _{\mathbf {k} }\varepsilon _{n}(\mathbf {k} ),}$

${\displaystyle {\dot {\mathbf {k} }}=-{\frac {1}{\hbar }}\nabla _{\mathbf {r} }\varepsilon _{n}(\mathbf {k} ).}$

The first equation relates the velocity of the wave-packet center to the slope of the electronic band dispersion. The second equation describes the effect of external forces on the crystal momentum and is directly analogous to Newton's second law ${\displaystyle {\dot {\mathbf {p} }}=\mathbf {F} }$ for a free particle, with the correspondence ${\displaystyle \mathbf {p} \leftrightarrow \hbar \mathbf {k} }$.

In a periodic potential without external forces, the crystal momentum ${\displaystyle \mathbf {k} }$ remains constant and the electron moves at the group velocity

${\displaystyle \mathbf {v} _{n}(\mathbf {k} )={\frac {1}{\hbar }}\nabla _{\mathbf {k} }\varepsilon _{n}(\mathbf {k} ).}$

When a uniform electric field ${\displaystyle \mathbf {E} }$ is applied, the momentum evolves as

${\displaystyle \hbar {\dot {\mathbf {k} }}=-e\mathbf {E} .}$

Derivation and validity of the semiclassical approximation

The semiclassical equations of motion are obtained by constructing a wave packet from Bloch states belonging to a single isolated energy band. The wave packet is assumed to be narrowly localized both in real space and in reciprocal space, allowing the external fields to be treated as slowly varying over its spatial extent. Under these conditions, the center of mass of the wave packet follows effective equations of motion derived from the time evolution of its expectation values.^[3][2]

The approximation is valid when interband transitions are negligible, which requires that the external perturbations vary slowly in time and space compared to the characteristic band gaps and lattice scale.^{[citation needed]} In this adiabatic regime, the electron remains confined to a single band and acquires only geometric phase corrections due to the Berry curvature.^{[citation needed]}

When this separation of energy and length scales breaks down, such as in strong electric fields or in the presence of significant interband coherence, the standard semiclassical description must be extended to include multiband and quantum-geometric effects.^{[citation needed]}

Bloch oscillations

In the presence of a uniform electric field, the semiclassical equation

${\displaystyle \hbar {\dot {\mathbf {k} }}=-e\mathbf {E} }$

implies a uniform drift of the crystal momentum through the Brillouin zone. Because the band energy ${\displaystyle \varepsilon _{n}(\mathbf {k} )}$ is periodic in momentum space as a consequence of the underlying lattice periodicity, the real-space velocity

${\displaystyle \mathbf {v} _{n}(\mathbf {k} )={\frac {1}{\hbar }}\nabla _{\mathbf {k} }\varepsilon _{n}(\mathbf {k} )}$

becomes a periodic function of time. As a result, the electron undergoes an oscillatory motion in real space known as a Bloch oscillation.

Bloch oscillations therefore originate from the periodic structure of the energy bands imposed by the crystal lattice, and their dynamics is conveniently described within the semiclassical equations of motion. The oscillation frequency is given by

${\displaystyle \omega _{B}={\frac {eEa}{\hbar }},}$

where ${\displaystyle a}$ is the lattice constant in the direction of the field.

Bloch oscillations were first predicted by Felix Bloch in 1929^[4] and later analyzed by Jones and Zener^[5] and Wannier,^[6] and are reviewed in standard solid-state physics textbooks.^[2]

Bloch oscillations in a periodic lattice under a constant electric field.

Semiclassical equations with Berry curvature

In 1999, Sundaram and Niu introduced geometric corrections to the semiclassical equations by incorporating the Berry curvature of the energy band.^[7] The Berry curvature ${\displaystyle {\boldsymbol {\Omega }}_{n}(\mathbf {k} )}$ is a vector field in momentum space defined from the Bloch eigenstates.

Physically, the Berry curvature characterizes how the quantum eigenstate associated with a given energy band twists and rotates as the crystal momentum is varied through the Brillouin zone. While the band dispersion ${\displaystyle \varepsilon _{n}(\mathbf {k} )}$ determines how the energy of the electron changes with momentum, the Berry curvature describes how the internal phase structure of the Bloch state evolves in momentum space.

The modified velocity becomes

${\displaystyle {\dot {\mathbf {r} }}={\frac {1}{\hbar }}\nabla _{\mathbf {k} }\varepsilon _{n}(\mathbf {k} )-{\dot {\mathbf {k} }}\times {\boldsymbol {\Omega }}_{n}(\mathbf {k} ).}$

For an electron in a uniform electric field,

${\displaystyle \hbar {\dot {\mathbf {k} }}=-e\mathbf {E} ,}$

which yields the anomalous velocity

${\displaystyle \mathbf {v} _{\mathrm {anom} }={\frac {e}{\hbar }}\mathbf {E} \times {\boldsymbol {\Omega }}_{n}(\mathbf {k} ).}$

This transverse contribution is responsible for the anomalous Hall effect even in the absence of an external magnetic field.^[3]

Semiclassical equations with quantum metric

A generalized semiclassical framework beyond the single-band adiabatic limit was developed for coherent two-band systems, where interband superpositions play an essential dynamical role.^[8] In this regime, the electron wave packet cannot be described solely by its position and crystal momentum, but must also include an internal degree of freedom associated with the coherent occupation of the two bands.

In this formulation, the wave packet is characterized by its center-of-mass position ${\displaystyle \mathbf {r} }$, crystal momentum ${\displaystyle \mathbf {k} }$, and a pseudospin vector ${\displaystyle \mathbf {S} }$ defined on the Bloch sphere. The pseudospin encodes the internal quantum state of the wave packet and directly represents the relative weight and phase coherence between the two band components. A pseudospin aligned with one pole of the Bloch sphere corresponds to a state entirely in one band, while intermediate orientations describe coherent interband superpositions.

Bloch sphere representation of a two-level quantum state (pseudospin).

The equations of motion take the Hamiltonian form

${\displaystyle \hbar {\dot {\mathbf {k} }}=-\nabla _{\mathbf {r} }E(\mathbf {k} ,\mathbf {S} ),\qquad {\dot {\mathbf {r} }}={\frac {1}{\hbar }}\nabla _{\mathbf {k} }E(\mathbf {k} ,\mathbf {S} ),\qquad {\dot {\mathbf {S} }}=\mathbf {S} \times {\boldsymbol {\Omega }}(\mathbf {k} ),}$

where ${\displaystyle E(\mathbf {k} ,\mathbf {S} )}$ is the expectation value of the energy of the two-band wave packet and ${\displaystyle {\boldsymbol {\Omega }}(\mathbf {k} )}$ is an effective pseudomagnetic field that governs the precession of the pseudospin. The third equation describes coherent interband dynamics in direct analogy with the precession of a spin in a magnetic field.

The effective pseudomagnetic field ${\displaystyle {\boldsymbol {\Omega }}(\mathbf {k} )}$ arises from the momentum dependence of the two-level band Hamiltonian. It represents the local coupling between the two bands in momentum space and governs the precession of the pseudospin in direct analogy with spin dynamics in a magnetic field, although it is not associated with any real magnetic field in physical space.

The Bloch-sphere representation of ${\displaystyle \mathbf {S} }$ is mathematically equivalent to the description of a generic two-level quantum system and corresponds to the geometrical picture introduced by Feynman in his analysis of two-state dynamics.^[9] In this picture, the motion of the pseudospin directly visualizes the quantum evolution of the internal band superposition.

In this two-band semiclassical regime, not only the band energy dispersion ${\displaystyle \varepsilon _{n}(\mathbf {k} )}$ but also the evolution of the associated quantum eigenstate contributes to the real-space dynamics. The geometry of these internal-state variations is encoded in the quantum geometric tensor, whose symmetric part defines the quantum metric

${\displaystyle g_{ij}(\mathbf {k} )=\operatorname {Re} \left[\langle \partial _{i}u|\partial _{j}u\rangle -\langle \partial _{i}u|u\rangle \langle u|\partial _{j}u\rangle \right].}$

The quantum metric measures the infinitesimal distance between neighboring Bloch eigenstates in momentum space and quantifies how rapidly the internal structure of the wave packet changes as ${\displaystyle \mathbf {k} }$ varies. Physically, it sets the characteristic spatial scale and amplitude of coherent interband motion, and governs phenomena such as spatial oscillations and Zitterbewegung-like dynamics in two-band systems.^[10][8]

See also

• Bloch oscillations
• Berry curvature
• Anomalous Hall effect
• Semiclassical physics