Macroscopic models for semiconductor devices

1. Macroscopic models for semiconductor devices

2. A hierarchy of semiconductor models

3. The Boltzmann equation Models the flow of charge carriers (electrons) in semiconductor crystals f – phase space distribution function x – position variable, k – wave vector physical constants: q – elementary charge, h – Planck const, ε (k) – energy wave vector function V = V (x,t) – electric potential mean electron velocity Using parabolic band approximation: Ec – conduction band min energy, m* - effective electron mass - electron density

4. Hydrodynamic models System of hyperbolic-parabolic equations(for the first moments of the distribution function) particle density current density energy tensor function used for this method after inserting the function f If Cu, CT(relaxation terms)omitted - Euler equations of gas dynamics for a gas of charged particles in an el. field

5. Hydrodynamic models (cont.) Using: the system of hydrodynamic equations can be rewritten in terms of n, Jn, E as: the relaxation terms are given by: Isentropic model consists of the first two equations (for n, Jn) with the assumption, that the temperatures depend only on the density:

6. Energy-transport models Using εas a parameter in the diffusion time scaling: letting ε -> 0in the equations for n and Jn above and assuming the Wiedemann-Franz law: the equations for the energy-transport model are derived: is the internal energy is the energy relaxation term The diffusion matrix is given by: with mobility constant

7. Drift-diffusion models In gas dynamics an ideal gas satisfies the law r = nT, where r is the pressure of the gas. For the isentropic case T(n) = T0n2/3 and using the diffusion matrix above, the electron current density is rewritten as: or 1 Together with the continuity and Poisson equation the isentropic drift-diffusion model is derived: 2 The motion of holes also contributes to the current flow in the crystal: 3 p - hole density 4 Jp - hole current density and the Poisson becomes: Equations 1-5 are called the bipolar isentropic drift-diffusion model 5

8. Drift-diffusion models (cont.) Another derivation starts from the transport equations: The carrier densities are given by Boltzmann statistics: So, from the standard drift-diffusion model is derived, consisting of the continuity equations , the Poisson equation and the current relations: where the diffusivity is related to the mobility by the Einstein relation:

9. Quantum models For ultra-small electronic devices in which quantum effects are present The quantum Boltzmann equation: w – Wigner function, Qh(w) – collision operator, V – effective potential, given by The Poisson equation If h -> 0, the collisionless quantumBoltzmann equation (called Vlasov equation) becomes: equivalent to a system of countable Schroedinger equations for the wave functions:

10. Quantum models (cont.) From the quantum Boltzmann equation above the (full) quantum hydrodynamic model can be derived: where the energy density is: The term below is the Bohm quantum correction and CE the energy relaxation term, defined:

11. Quantum models (cont.) Using a diffusion scaling in the full quantum hydrodynamic equations, similar as in the full hydrodynamic model, the so-called quantum energy-transport model: and the energy density is:

12. References Macroscopic models for semiconductor devices: a review, Juengel, Ansgar