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Macroscopic models for semiconductor devices

A hierarchy of
semiconductor
models
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:
mean electron velocity
q – elementary charge, h – Planck const,
ε (k) – energy wave vector function
Using parabolic band approximation:
V = V (x,t) – electric potential
Ec – conduction band min energy,
m* - effective electron mass
- electron density
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
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:
Energy-transport models
Using ε as a parameter in the diffusion time scaling:
letting ε -> 0 in 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
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:
5
Equations 1-5 are called the bipolar
isentropic drift-diffusion model
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:
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 quantum Boltzmann equation (called Vlasov equation) becomes:
equivalent to a system of countable Schroedinger equations for the wave functions:
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:
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:
References
Macroscopic models for semiconductor devices: a review, Juengel, Ansgar