Nearly free electron model

Last updated

In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual empty lattice approximation. The model enables understanding and calculation of the electronic band structures, especially of metals.


This model is an immediate improvement of the free electron model, in which the metal was considered as a non-interacting electron gas and the ions were neglected completely.

Mathematical formulation

Dispersion relation for the 2D nearly free electron model as a function of the underlying crystalline structure.

The nearly free electron model is a modification of the free-electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the ions in a crystalline solid. This model, like the free-electron model, does not take into account electron–electron interactions; that is, the independent electron approximation is still in effect.

As shown by Bloch's theorem, introducing a periodic potential into the Schrödinger equation results in a wave function of the form

where the function has the same periodicity as the lattice:

(where is a lattice translation vector.)

Because it is a nearly free electron approximation we can assume that

where denotes the volume of states of fixed radius (as described in Gibbs paradox).[ clarification needed ]

A solution of this form can be plugged into the Schrödinger equation, resulting in the central equation:

where is the total energy, and the kinetic energy is characterized by

which, after dividing by , reduces to

if we assume that is almost constant and

The reciprocal parameters and are the Fourier coefficients of the wave function and the screened potential energy , respectively:

The vectors are the reciprocal lattice vectors, and the discrete values of are determined by the boundary conditions of the lattice under consideration.

Before doing the perturbation analysis, let us first consider the base case to which the perturbation is applied. Here, the base case is , and therefore all the Fourier coefficients of the potential are also zero. In this case the central equation reduces to the form

This identity means that for each , one of the two following cases must hold:

  1. ,

If is a non-degenerate energy level, then the second case occurs for only one value of , while for the remaining , the Fourier expansion coefficient is zero. In this case, the standard free electron gas result is retrieved:

If is a degenerate energy level, there will be a set of lattice vectors with . Then there will be independent plane wave solutions of which any linear combination is also a solution:

Now let be nonzero and small. Non-degenerate and degenerate perturbation theory, respectively, can be applied in these two cases to solve for the Fourier coefficients of the wavefunction (correct to first order in ) and the energy eigenvalue (correct to second order in ). An important result of this derivation is that there is no first-order shift in the energy in the case of no degeneracy, while there is in the case of degeneracy (and near-degeneracy), implying that the latter case is more important in this analysis. Particularly, at the Brillouin zone boundary (or, equivalently, at any point on a Bragg plane), one finds a twofold energy degeneracy that results in a shift in energy given by:[ clarification needed ]


This energy gap between Brillouin zones is known as the band gap, with a magnitude of .


Introducing this weak perturbation has significant effects on the solution to the Schrödinger equation, most significantly resulting in a band gap between wave vectors in different Brillouin zones.


In this model, the assumption is made that the interaction between the conduction electrons and the ion cores can be modeled through the use of a "weak" perturbing potential. This may seem like a severe approximation, for the Coulomb attraction between these two particles of opposite charge can be quite significant at short distances. It can be partially justified, however, by noting two important properties of the quantum mechanical system:

  1. The force between the ions and the electrons is greatest at very small distances. However, the conduction electrons are not "allowed" to get this close to the ion cores due to the Pauli exclusion principle: the orbitals closest to the ion core are already occupied by the core electrons. Therefore, the conduction electrons never get close enough to the ion cores to feel their full force.
  2. Furthermore, the core electrons shield the ion charge magnitude "seen" by the conduction electrons. The result is an effective nuclear charge experienced by the conduction electrons which is significantly reduced from the actual nuclear charge.

See also

Related Research Articles

In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-12 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry. It is consistent with both the principles of quantum mechanics and the theory of special relativity, and was the first theory to account fully for special relativity in the context of quantum mechanics. It was validated by accounting for the fine structure of the hydrogen spectrum in a completely rigorous way.

<span class="mw-page-title-main">Ground state</span> Lowest energy level of a quantum system

The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. In quantum field theory, the ground state is usually called the vacuum state or the vacuum.

<span class="mw-page-title-main">Dirac sea</span> Theoretical model of the vacuum

The Dirac sea is a theoretical model of the electron vacuum as an infinite sea of electrons with negative energy, now called positrons. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-energy quantum states predicted by the Dirac equation for relativistic electrons. The positron, the antimatter counterpart of the electron, was originally conceived of as a hole in the Dirac sea, before its experimental discovery in 1932.

<span class="mw-page-title-main">Bloch's theorem</span> Fundamental theorem in condensed matter physics

In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, who discovered the theorem in 1929. Mathematically, they are written

<span class="mw-page-title-main">Fine structure</span> Details in the emission spectrum of an atom

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom by Albert A. Michelson and Edward W. Morley in 1887, laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant.

In solid-state physics, the electronic band structure of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have.

In quantum physics, Fermi's golden rule is a formula that describes the transition rate from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively independent of time and is proportional to the strength of the coupling between the initial and final states of the system as well as the density of states. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.

<span class="mw-page-title-main">Two-state quantum system</span> Simple quantum mechanical system

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

<span class="mw-page-title-main">Degenerate energy levels</span> Energy level of a quantum system that corresponds to two or more different measurable states

In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue. When this is the case, energy alone is not enough to characterize what state the system is in, and other quantum numbers are needed to characterize the exact state when distinction is desired. In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.

<span class="mw-page-title-main">Charge density</span> Electric charge per unit length, area or volume

In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m−3), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m−2), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m−1), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.

The Franz–Keldysh effect is a change in optical absorption by a semiconductor when an electric field is applied. The effect is named after the German physicist Walter Franz and Russian physicist Leonid Keldysh.

The Poisson–Boltzmann equation describes the distribution of the electric potential in solution in the direction normal to a charged surface. This distribution is important to determine how the electrostatic interactions will affect the molecules in solution. The Poisson–Boltzmann equation is derived via mean-field assumptions. From the Poisson–Boltzmann equation many other equations have been derived with a number of different assumptions.

In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function. Qualitatively, it corresponds to a potential which is zero everywhere, except at a single point, where it takes an infinite value. This can be used to simulate situations where a particle is free to move in two regions of space with a barrier between the two regions. For example, an electron can move almost freely in a conducting material, but if two conducting surfaces are put close together, the interface between them acts as a barrier for the electron that can be approximated by a delta potential.

In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure and optical properties of crystalline solids. It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the Luttinger–Kohn model, and of the Kane model.

Free carrier absorption occurs when a material absorbs a photon, and a carrier is excited from an already-excited state to another, unoccupied state in the same band. This intraband absorption is different from interband absorption because the excited carrier is already in an excited band, such as an electron in the conduction band or a hole in the valence band, where it is free to move. In interband absorption, the carrier starts in a fixed, nonconducting band and is excited to a conducting one.

<span class="mw-page-title-main">Envelope (waves)</span> Smooth curve outlining the extremes of an oscillating signal

In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

The Holstein–Herring method, also called the surface Integral method, or Smirnov's method is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energy becomes elusive at large internuclear systems, it is of prominent importance in theories of molecular binding and magnetism. This splitting results from the symmetry under exchange of identical nuclei. The basic idea pioneered by Theodore Holstein, Conyers Herring and Boris M. Smirnov in the 1950-1960.

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is thermal energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is different made (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

In quantum mechanics, the variational method is one way of finding approximations to the lowest energy eigenstate or ground state, and some excited states. This allows calculating approximate wavefunctions such as molecular orbitals. The basis for this method is the variational principle.

<span class="mw-page-title-main">Symmetry in quantum mechanics</span> Properties underlying modern physics

Symmetries in quantum mechanics describe features of spacetime and particles which are unchanged under some transformation, in the context of quantum mechanics, relativistic quantum mechanics and quantum field theory, and with applications in the mathematical formulation of the standard model and condensed matter physics. In general, symmetry in physics, invariance, and conservation laws, are fundamentally important constraints for formulating physical theories and models. In practice, they are powerful methods for solving problems and predicting what can happen. While conservation laws do not always give the answer to the problem directly, they form the correct constraints and the first steps to solving a multitude of problems. In application, understanding symmetries can also provide insights on the eigenstates that can be expected. For example, the existence of degenerate states can be inferred by the presence of non commuting symmetry operators or that the non degenerate states are also eigenvectors of symmetry operators.