Page 126 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Free and Bound Electrons, Dimensionality Effects
3.2.6 The Hydrogen Atom
The paradigm for the electronic states of a central symmetric potential is
the hydrogen atom since it consists of one proton in the nucleus sur-
rounded by one electron. It helps in understanding the principle of elec-
tronic states of atoms which is similar to the electronic states of defects
and traps for electrons in the semiconductor or even helps to understand
the formation of a band structure. We shall not go into details of the deri-
vation but we shall have a closer look at the form of the electronic states.

Coulomb The Coulomb potential of the proton is spherically symmetric in three
Potential dimensions, which means that it depends only on the distance r of the
2
symmetry center U r() = – e ⁄ r (e is the unit charge). The best choice
to tackle the problem is spherical coordinates, with the radius vector r
θ
φ
given by its modulus , the polar angle and the azimutal angle . The
r
Hamilton operator thus reads
2 2
— 2 e
H = – ------∇ – ----- (3.68)
2µ r
To solve the Schrödinger equation we must write the ∇ 2 in spherical
coordinates, which reads
1 ∂  ∂  1  1 ∂ ∂ 1 ∂ 2 
2
∇ = ---- r 2 + ---- ----------- sin θ + -------------------- (3.69)
r ∂  r ∂  r 2 sin θ θ ∂ θ sin θ∂φ 2
∂
r
2
2
Inserting (3.69) together with (3.68) and solving the respective
Schrödinger equation is beyond the scope of this book. We restrict the
discussion to an interpretation of the electronic wavefunction and an
illustration of the atomic orbitals.

The common approach to calculate the bound electronic states of the
hydrogen atom (energy E < 0 ) starts with a product trial function for the
(
,
wavefunction ψ r() = Rr()Y θφ) . Then the Schrödinger equation sep-
arates into two eigenvalue problems, one for the radial motion (Rr() )
and one for the angular motion (Y θφ,( ) ). The solutions of the angular

Semiconductors for Micro and Nanosystem Technology 123