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Encyclopedia of Complexity and Systems Science pp 1–21Cite as

Spin-Dependent Exchange and Correlation in Two-Dimensional Electron Layers

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Definition of the Subject

Since the advent of density functional theory (DFT), the exchange-correlation energy E xc of an interacting system has become a basic quantity in many-particle theory. Here, we study the E xc of two-dimensional (2D) electron layers. Such layers contain electrons confined in z and move only in the x and y directions. 2D layers are formed at insulator–semiconductor interfaces in heterojunctions and more particularly at metal–oxide–semiconductor (MOS) interfaces. These include two types of semiconductors (e.g., GaAs and the alloyed form Al x Ga1 − x As, containing a small fraction x of Al, and written as AlGaAs for brevity). The interface region defines a “confining potential” where an electron layer may form (Ando et al. 1982). SiO2 is an insulator with a large bandgap, while Si can be doped in a controlled manner to behave as a conductor. The Si/SiO2 interface supports the formation of an electron layer at the interface. The electron density nin such layers...

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Abbreviations

Atomic units a.u.:

The electron charge |e| and the mass m e are taken as unity. The unit of time is fixed by setting the Plank constant ℏ to unity. The Bohr radius \( {a}_0={\hslash}^2/\left({m}_e{e}^2\right) \) is one a.u. of length in the centimeter–gram–second(CGS) system which uses the “esu” (electrostatic unit of charge). The SI system uses the meter, kilogram, and second (with the ampere as the unit of current), \( {a}_0=\Big(4\pi {\varepsilon}_0{\hslash}^2/\left({m}_e{e}^2\right) \) where ε0 is the electric permittivity of the vacuum. The value of a 0 is 5.29177 × 10−9 cm. The unit of energy, the Hartree, is e 2/a 0 in the CGS system, being 27.2116 eV. In semiconductor physics, effective atomic units are used, with e 2/ε, replacing e 2, where ε is the dielectric constant. The band mass m b is used instead of m e, so that the effective Bohr radius a ∗0 = a 0(ε/m* ), where m* is the effective mass. Then the effective Hartree is of the order of millivolts.

Confining potential:

This potential keeps the electron in a given spatial region. It is due to the physical structure of the device, the applied gate voltages, etc.

Correlation energy:

The contribution to the total energy beyond the Hartree–Fock approximation, denoted by E c.

Correlation hole:

is the depletion of electron density near an electron due to the Coulomb repulsion effects.

Coupling constant:

is the ratio of the potential energy (PE) to the kinetic energy (KE). In a classical electron fluid, the KE (thermal energy) is T, and the PE is 1/r s (atomic units), and r s is the Wigner–Seitz radius. The coupling constant = PE / KE = Γ = 1/(r s T). In quantum systems, the Fermi energy E F is used instead of T for the KE, and Γ = r s.

Classical-map hyper-netted-chain (CHNC):

A method for using the classical hypernetted-chain equation to calculate the correlation functions of quantum systems.

Effective mass:

is denoted by m* and is the band mass m b in units of the electron mass m e.

Exchange energy:

The part of the Hartree–Fock energy due to electron exchange, i.e., the “Fock” part, denoted by E x. It is first order in the Coulomb interaction.

Fermi hole:

This denotes the reduction in the probability of finding a parallel-spin electron near another, due to Fermi statistics.

Hartree–Fock:

Hartree’s self-consistent one-body approximation for interacting electrons is based on a product wavefunction. Fock included exchange using an antisymmetrized product. “Hartree–Fock” is the label for calculations of the energy, wavefunctions, etc., where the electron moves in this mean potential generated by the electrostatics and the exchange effects. The Hartree term is zero in uniform systems.

HIGFET:

Heterojunction-insulated gate field-effect transistor.

Jellium:

A model “metal” where the positive ionic charges are replaced by a uniform static charge which neutralizes the free-electron charge

MOSFET:

Metal–oxide–semiconductor field-effect transistor

Heterojunction:

A semiconductor interface involving two dissimilar materials.

Hypernetted-chain:

(HNC) A classical integral equation due to van Leeuwen et al. (1959) which non-perturbatively sums “hypernetted-chain” diagrams, going beyond mean field theory.

Electron gas parameter rs :

See Wigner–Seitz radius.

Pair -correlation function:

Denoted by \( h\left(\overrightarrow{r}\right)=g\left(\overrightarrow{r}\right)-1 \), where g(r) is the pair distribution function (PDF).

Pair distribution function:

The pair distribution function (PDF), \( g\left(\overrightarrow{r}\right) \), is the probability of finding a particle at the location \( \overrightarrow{r} \), given a reference particle at the origin.

Plasma analogy:

A class of methods for approximately replacing a charged quantum fluid by an equivalent classical fluid at a finite temperature.

Pseudospin:

Discrete degrees of freedom beside the electron spin. The electrons in Si/SiO2 interfaces occupy two valleys. The valley index is a pseudospin.

Quantum Monte Carlo (QMC):

In molecular dynamics (MD), Newton’s equations of motion are integrated using a stochastic scheme based on the Metropolis algorithm. In QMC a trial wavefunction provides a probability measure for the Metropolis algorithm. The wavefunction is optimized in various ways, leading to “variational QMC,” where the nodes of the trial wavefunction are held fixed. In “diffusion QMC,” the nodes are also relaxed.

Random-phase approximation (RPA):

A time-dependent self-consistent field method where an electron with momentum k moves in an effective potential which contains the external potential and a k,ω-dependent screened potential. It reduces in the static k → 0 limit to Thomas–Fermi screening (or Debye–Hükel screening in classical systems). It is also called the “ring sum” or “bubble sum” and contains no exchange effects.

Subbands:

The electrons with the z-motion confined to a quantum well have discrete energy levels (index n). Each level carries with it a band of energies for the x, y motion. These are energy “subbands.”

Singwi–Tosi–Land–Sjölander (STLS):

A method due to Singwi et al. for determining the density–density correlation function of electrons (and other quantum systems) by truncating the equation of motion via an intuitive decoupling scheme involving the PDFs. STLS has been extended by Vashista, Ichimaru, and others.

Wigner–Seitz radius:

In 2D this is the radius, denoted by r s, of the circle containing, on the average, just one electron.

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Authors and Affiliations

  1. Institute of Microstructural Sciences, National Research Council of Canada, Ottawa, Canada

    M. W. Chandre Dharma-wardana

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  1. M. W. Chandre Dharma-wardana
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Correspondence to M. W. Chandre Dharma-wardana .

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  1. Ramtech Ltd., Larkspur, California, USA

    Robert A. Meyers

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Dharma-wardana, M.W.C. (2015). Spin-Dependent Exchange and Correlation in Two-Dimensional Electron Layers. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_512-3

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  • DOI: https://doi.org/10.1007/978-3-642-27737-5_512-3

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