PX262 - K6 - energy gaps

free electrons have the wave functions of the form: $ψ = A_{k} e^{i k x}$ , where $k \in (- \infty, \infty)$
the energies are given by:so

E_{k} = \frac{ℏ^{2} k^{2}}{2 m_{e}}

linear combinations: $ϕ_{k}^{+} = C_{k} \cos (k x)$ and $ϕ_{k}^{-} = D_{k} \sin (k x)$
these correspond to the same energy, $E_{k}$
a 1D lattice will have the separations of $a$ , and its reciprocal lattice, $2 π / a$
in the k-space, the first BZ: $- π / a \leq k \leq π / a$
looking at the free electron wavefunctions' intensities with $k = \pm π / a$ (values at the edges of the BZ): $| ϕ_{π / a}^{+} |^{2}$ and $| ϕ_{π / a}^{-} |^{2}$

when potentials are placed on each of the lattice points, the energy degeneracy is broken, since $| ϕ_{π / a}^{+} |^{2}$ is distributed more closely to the potentials than the $| ϕ_{π / a}^{-} |^{2}$ solution
energy gaps at $k = \pm π / a$ are produced at the BZ boundaries
if the free electron model (1D) is perturbed by a weak crystalline potential, the electronic states change
this can be plotted in the first BZ, with $k^{'} = k - G_{n}$ ie. folding back, $G_{n} = - n π / a$ or $n π / a$

a complementary insight can be obtained by starting with a model of a large number, $N$ , of identical atoms, which are well separated. eg: Li: $1 s^{2} 2 s^{1}$
each atom potential has a set of electronic energy levels for the valence electrons, eg: $2 s 2 p 3 s \dots$
the energy level diagram for the whole system is set by that of a single atom, but the pauli exclusion principle for the entire system permits each state to be occupied by up to $2 N$ (2 for spin) electrons

if the atoms are pushed closer together, the wavefunctions of the electrons distort and extend over more atoms
the outermost valence electronic states accommodate $2 N$ in a band of $N$ tightly spaced levels

the two figures above show how, in some materials, the DOS develops a gap at the fermi energy

such a material is insulating
there are no states near $E_{F}$ , no low-lying excitations and no conduction
the specific heat has no electronic contribution for $k_{B} T < Δ$
the material is also transparent to light, with $ℏ ω = h c / λ < Δ$ and only photons with $ℏ ω = h c / λ > Δ$ will be blocked
a material is termed as an insulator if $Δ > 2$ eV, and for $Δ \geq\sim 2$ eV, the material is a semiconductor