The effective mass Conductivity effective mass – determines mobility. Density of states effective mass – determines NC Cyclotron effective mass – can be measured directly
Electron in a periodic potential Why does the semiconductor industry use single crystal material (when possible) ?
Electrons are not scattered by a periodic potential – move with a constant velocity as in vacuum !
Electron in vacuum
Dispersion
Electron in vacuum
Electron in a periodic potential
Electron in a 3D periodic potential
primative reciprocal lattice basis vector respectively. and .
Expansion of E(kx,ky,kz) near a minimum value E0= E(kx0,ky0,kz0)
Expansion of E(kx,ky,kz) near a minimum value E0= E(kx0,ky0,kz0)
In the coordinate system in which the effective mass tensor is diagonal
Acceleration due to an electric filed (F)
Effective mass tensor – valid near E(k) minima and maxima only
Constant energy surfaces in crystal momentum space cookies
Constant energy surfaces near a minimum are ellipsoids
Constant energy surfaces in Si and Ge near a minimum are ellipsoids of revolution
Acceleration of an electron near an energy minimum in silicon
Electron transport effective mass in silicon and germanium
Homework competition – find a shorter way to prove this equation for germanium than given in last year’s home exam
The spherical case - Electrons in GaAs Holes in Si, Ge, GaAs
Cyclotron resonance effective mass
Cyclotron resonance effective mass –to be shown in the tutorial
Cyclotron resonance effective mass –to be shown in the tutorial
Density of states effective mass
Density of states
Example: density of states of hydrogen gas
Density of states of solids g(E) E
Density of electrons in an energy band g(E) E E f
Density of holes in an energy band g(E) E E f
Approximation for the Fermi Dirac distribution for E-Ef>3KT
Density of electrons in an energy band g(E) E E f
Density of states of solids in K space
The density of states in an energy interval is proportional to the volume in K space between two constant energy surfaces
Constant energy surfaces in crystal momentum space cookies
Volume of an ellipsoid
Volume of a constant energy (E’) ellipsoid
Density of states near a conduction band minimum or valence band maximum, and the definition of the density of states effective mass
Conduction band density of states effective mass
Valence band density of states effective mass
Summary Conductivity effective mass – determines mobility. Density of states effective mass – determines NC Cyclotron effective mass – can be measured directly
Elective home exercise Derive conductivity and density of states effective mass for holes. k E hh lh