2 Charge Carrier Transport

Modelling the transport of charge carriers in a crystal, eg. silicon, containing artificially introduced impurities, ie. dopands, to modify the electrical conductance of the crystal is not an easy task. In order to fully describe the many-particle system, where r denotes the positions of the electrons and R denotes the positions of the ions, one needs to employ

	-∇_r² -∇_R² + H_ion-ion(R)
	+ H_el-ion(r,R) + H_el-el(r) + …Ψ(r,R) = iℏ∂_tΨ(r,R),	(2.1)

which is the time-dependent Schrödinger equation of the full many-particle system. In the above equation various interactions between the ions and electrons, such as the interactions between ions is given by the operator H_ion-ion(r,R), between electrons and ions is described by H_el-ion(r,R) and electron-electron scattering is described by H_el-el(r,R). At present it is practically impossible to solve the many-particle Schrödinger equation even for only a few atoms. Thus the usual approach to this is to decouple the electron-system and the system of lattice atoms by the separation Ansatz

Ψ (r,R ) = ψ (r )ψion(R ),

where the wave function ψ only describes the system of electrons and ψ_ion only describes the lattice. Inserting this into Equation (2.1) one obtains

ψ(r,R)	= E_elψ(r,R),
ψ_ion(R)	= E_ionψ_ion(R),

where V _ion(x_i) and V _el(x_i) are the ionic or electronic background potentials seen by the electrons or ions. To reduce this set of equations further, the ions are assumed to only slowly oscillate around their mean lattice position and the Schrödinger equation for the ions is not solved (jellium model). The main argument of this approach is based on the assumption that the mass of an ion M is much larger than the mass of an electron m, which means that the ions are much slower in their movement than the electrons. The approximation itself is often referred to as the ‘adiabatic approximation’ or Born-Oppenheimer-Approximation [10]. A thorough justification and investigation of the Born-Oppenheimer-Approximation is beyond the scope of this work and can be found in [11]. In order to reduce the many-particle Schrödinger equation for the electrons to a more tractable single particle equation the Hartree-Fock-Approximation and the Slater-Approximation are applied [12]. The Schrödinger equation under these approximations for a single electron state in a homogeneous semiconductor reads

( ) -ℏ2- 2 - 2m0 ∇ + V (x)+ φout(x,t)+ Hscatter(x,t) ψ(x,t) = iℏ∂tψ (x, t)

(2.2)

where ψ(x,t) is now a single electron wave function representing the whole ensemble of electrons, φout(x,t) is the external electrostatic potential and all other particle-particle interactions have been summarized in H_scatter. In order to obtain a semi-classical approximation in the special case of a periodic lattice Equation (2.2) has to be transformed such that the potential of the ions does not explicitly appear in the equation. This is done by solving Equation (2.2) without any scattering H_scatter and time dependence [13, 12],

( ) -ℏ2- 2 - 2m0 ∇ + V (x) ψ(x,k ) = E (k)ψ(x,k ),

(2.3)

and inserting the solution back into Equation (2.2), where the reciprocal wave vector k and the ionic potential V (x) (jellium-model for semiconductors) have been introduced [12, 14]. A solution ψ(x,k) to Equation (2.3) is easily obtained by using plane-waves and Bloch-functions [15] to account for the periodicity in the lattice. Additionally the solution to Equation (2.3) gives the relation Eν(k) for the νth band. Eν(k) is often referred to as the dispersion relation and is only dependent on the type of semiconductor. By introducing the operator E_ν(-i∇x), where

Eν(- i∇x )ψ(x,k ) = E ν(k)ψ(x,k )

according to [12], one can transform Equation (2.2) into

( E (- i∇ ) +φ (x,t)+ H (x,k))ψ (x, k,t) = iℏ∂ ψ(x,k, t). ◟-ν-◝◜-x◞ out scatter t -2ℏm2∇2+V (x) 0

(2.4)

The Hamilton operators used in Equation (2.4) form the basis of all transport models that are described in the course of this thesis. Often Equation (2.4) is arbitrarily separated into two equations, one for all bands (valence bands) below and one for all conduction bands above the center of the band gap. Next a new quasi-particle, the hole, is introduced in the equation for all valence bands. A hole represents an unoccupied state and thus has a positive elementary charge associated with it. This is done, since the valence bands contain many electrons occupying the bands. Thus it is easier to describe the flow of the empty states instead. For a proper definition of holes refer to [12].

2.1  Band structure
2.2  The Boltzmann Transport Equation
  2.2.1  Scattering
  2.2.2  Recombination and Generation
2.3  The Method of Moments
  2.3.1  The Drift Diffusion Model
  2.3.2  The Hydrodynamic Model
  2.3.3  Higher Order Models
2.4  The Monte Carlo Method
2.5  The Spherical Harmonics Expansion Method
2.6  Quantum Mechanical Effects - Confinement
2.7  Non-Equilibrium Greens Functions Approach

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