Derivation of the high field semiconductor equations

Conference · Tue Jan 01 00:00:00 EST 1991

OSTI ID:5217784

Hagan, P S ^[1]; Cox, R W ^[2]; Wagner, B A ^[3]

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1. Los Alamos National Lab., NM (United States)
2. Department of Computer Information Science, Indiana University, Purdue University, Indianapolis (USA)
3. Arizona Univ., Tucson, AZ (United States). Dept. of Mathematics

Electron and hole densities evolve in x-z phase space according to Boltzmann equations. When the mean free path of the particles is short and electric force on the particles is weak, a well-known expansion can be used to solve the Boltzmann equation. This asymptotic solution shows that the spatial density of electrons and holes evolves according to diffusion-drift equations. As devices become smaller, electric fields become stronger, which renders the Basic Semiconductor Equations increasingly inaccurate. To remedy this problem, we use singular perturbation techniques to obtain a new asymptotic expansion for the Boltzmann equation. Like the Hilbert expansion, the new expansion requires the mean free path to be short compared to all macroscopic length scales. However, it does not require the electric forces to be weak. The new expansion shows that spatial densities obey diffusion-drift equations as before, but the diffusivity D and mobility {mu} turn out to be nonlinear functions of the electric field. In particular, our analysis determines the field-dependent mobilities {mu}(E) and diffusivities D(E) directly from the scattering operator. By carrying out this asymptotic expansion to higher order, we obtain the high frequency corrections to the drift velocity and diffusivity, and also the corrections due to gradients in the electric field. Remarkably, we find that Einsteins's relation is still satisfied, even with these corrections. The new diffusion-drift equations, together with Poissons' equation for the electric field, form the high-field semiconductor equations, which can be expected to be accurate regardless of the strength of the electric fields within the semiconductor. In addition, our analysis determines the entire momentum distribution of the particles, so we derive a very accurate first moment model for semi-conductors by substituting the asymptotically-correct distribution back into the Boltzmann equation and taking moments.

• Research Organization: Los Alamos National Lab., NM (United States)
• Sponsoring Organization: USDOE; USDOE, Washington, DC (United States)
• DOE Contract Number: W-7405-ENG-36
• OSTI ID: 5217784
• Report Number(s): LA-UR-91-2951; CONF-9107176-1; ON: DE92000249
• Resource Relation: Conference: 1991 IMA semiconductor workshop, Minneapolis, MN (United States), 15 Jul - 9 Aug 1991
• Country of Publication: United States
• Language: English

Related Subjects

75 CONDENSED MATTER PHYSICS
SUPERCONDUCTIVITY AND SUPERFLUIDITY
SEMICONDUCTOR MATERIALS
BOLTZMANN EQUATION
DIFFUSION
EIGENFUNCTIONS
EIGENVALUES
MEAN FREE PATH
PHASE SPACE
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
MATERIALS
MATHEMATICAL SPACE
PARTIAL DIFFERENTIAL EQUATIONS
SPACE
656002* - Condensed Matter Physics- General Techniques in Condensed Matter- (1987-)