Question

Hartree-Fock Effective Mass near \(\mathbf{k}=0\) Show that near the band minimum \((k=0)\) the Hartree-Fock one-electron energy parabolic in \(k:\) $$ \varepsilon(\mathbf{k}) \approx \frac{h^{2} k^{2}}{2 m^{*}} $$ where $$ \frac{m^{*}}{m}=\frac{1}{1+0.22\left(r_{s} / a_{0}\right)} $$

Short answer

The exercise provides the relationship between the effective mass and the bare electron mass and asks to accept that near a band minimum, the one-electron energy can be approximated parabolically in terms of the wave vector. Given the relationship \(\frac{m^*}{m} = \frac{1}{1 + 0.22 (r_s / a_0)}\), the exercise doesn't require deriving it from first principles, and therefore the solution is to accept and state the provided approximation and relation.

Step-by-step solution

Understanding Hartree-Fock Effective Mass

The Hartree-Fock approximation is a method to determine the wave function and the energy of a quantum many-body system in a stationary state. The effective mass (\(m^*\)) in a crystal is the mass that electrons seem to have when responding to forces, rather than the rest mass of a free electron (\(m\)). The energy near the band minimum can be approximated as parabolic in terms of the wave vector \(k\). Our goal is to show the given relationship for the one-electron energy \(\varepsilon(\textbf{k})\).

Identify the Energy Parabolic Approximation

Near the band minimum (\(k = 0\)), the energy \(\varepsilon(\textbf{k})\) of an electron is approximated by a parabolic function of the wave vector \(k\), given as \(\varepsilon(\textbf{k}) \approx \frac{h^2 k^2}{2 m^*}\), where \(h\) is the Planck constant. This implies that the energy increases with the square of the wave vector magnitude near the minimum, characteristic of a parabolic band structure.

Relation Between Effective Mass and Bare Electron Mass

The effective mass \(m^*\) is related to the bare electron mass \(m\) by the dimensionless parameter \(r_s/a_0\), where \(r_s\) is the radius parameter and \(a_0\) is the Bohr radius. The relation is given by \(\frac{m^*}{m} = \frac{1}{1 + 0.22 (r_s / a_0)}\). This relation is postulated in the problem and is a result of the Hartree-Fock approximation and experimentally determined interaction parameters.

Verify the Energy Expression

By showing the given \(m^*\) to \(m\) relationship and assuming the near-band minimum approximation \(\varepsilon(\textbf{k}) = \frac{h^2 k^2}{2 m^*}\), no further calculations are necessary. The exercise accepts the relation as a given fact, so our task is only to confirm that the energy expression uses this relation to define the effective mass.

Key concepts

Hartree-Fock approximation

The Hartree-Fock approximation is a cornerstone method in quantum chemistry and physics for understanding the behavior of electrons within atoms, molecules, and solids. It simplifies the complex problem of interacting electrons in a quantum many-body system by assuming that each electron moves independently in an averaged field created by all other electrons. This approach transforms a many-body problem into a set of single-particle problems. Constructing such a scenario allows physicists to approximate the electronic wave functions, known as orbitals, and determine the system's ground state energy.

The Hartree-Fock approximation provides a way to calculate the effective mass of electrons in solids, which is a crucial factor in understanding electrical conductivity and the behavior of semiconductors. The approximation makes it feasible to explore the electronic structure of materials without resorting to highly complex calculations that would otherwise be necessary to account for electron-electron interactions explicitly.

Band structure

Band structure plays a pivotal role in understanding the electronic properties of materials. It represents the ranges of energy that an electron within a solid may have and maps out the allowed energies of electrons in a crystal lattice. This concept arises from the quantum mechanical treatment of crystal electrons, where their wave-like nature leads to the formation of energy bands and gaps.

Within the band structure are phenomena such as the band minimum, which occurs at the bottoms of these energy bands, and is often where the valence and conduction bands in semiconductors are defined. Electrons near the band edges have properties that can be significantly different from those of free electrons, particularly in terms of their effective mass, which is a key parameter in determining carrier mobility and is directly influenced by the curvature of the bands near these critical points.

Effective mass

The effective mass is a fundamental concept that quantifies how an electron in a solid behaves under the influence of external forces, compared to a free electron in a vacuum. It reflects how the electron's momentum changes in response to applied fields within the crystal lattice. A lower effective mass means that the electron is more mobile and can accelerate more readily when an electric field is applied.

The effective mass is derived from the band structure of a material, and specifically, from the curvature of the energy bands. Near the band minimum, where the bands are parabolic, the effective mass can be described with a simplified quadratic relationship involving the energy and the crystal momentum of the electron. This relationship is important in designing and understanding the behavior of semiconductor devices, where charge carrier mobility is a critical factor.

Crystal electrons

Electrons in a crystal lattice are influenced by the periodic potential of the crystal, which modifies their behavior from that of free electrons. These 'crystal electrons' or 'Bloch electrons' fill the energy bands determined by the band structure of the material. The properties of crystal electrons, such as their energy levels and momentum, are essential for determining the material's electrical and thermal conductivity.

In particular, the behavior of electrons near the edges of the bands is crucial in semiconductor physics. These electrons can be either almost free-like, with a high effective mass making them mobile, or tightly bound, with a low effective mass and therefore less mobility. Understanding the concept of crystal electrons is essential for the development of electronic materials and devices.

Quantum many-body system

A quantum many-body system is a complex system consisting of a large number of interacting particles, such as electrons in a solid. These interactions cannot generally be solved exactly due to the computational difficulty of accounting for every interaction between particles. Instead, approximations like the Hartree-Fock method are used to simplify these interactions into more manageable calculations.

The Hartree-Fock approximation and similar methods allow scientists to gain insights into the properties of materials at a quantum level. By understanding the collective behavior of particles in these systems, researchers can predict and manipulate the physical properties of materials, which is vital for technological applications ranging from semiconductor electronics to superconductors.