Metals, Insulators and Semiconductors Flashcards
Realistically, what happens to the density of states in some materials?
The DoS develops gaps where it goes to zero stays at zero for a bit then goes back up again (graph of n(E) against E)
How can we tell if a material is insulating or not from an n(E) - E graph?
If E(F) lies in a gap, the material is insulating.
When is a material transparent to light?
With ћω = hc/λ < Δ, where Δ is the length of the gap on the n(E) - E graph.
For what values of Δ is a material an insulator?
Δ > 2eV = insulator, < 2eV is a semiconductor
What is the origin of the gaps on an n(E) - E graph?
Free electron model altered when a lattice of potentials replaces the smeared out background uniform positive charge, the scattering from these potentials open up the gaps at Brillouin zone boundaries.
What is a second insight into the origin of the gaps in the n(E) - E graph?
- Large number of identical atoms: energy level diagram for whole system is wave over each atom
- Move atoms closer and get distorted and extended wave
- Outermost valence states now can accommodate N electrons in a band with N tightly spaced levels
- Between bands are gaps with no allowed energy levels
What is the Kronig-Penney model diagram?
Periodic array of square potentials with spacing a, height v0 and width b.
In the Kronig-Penney model, what form does the solution of the Schrodinger equation take?
Ψ(x) = exp(ikx)*u(x), where u(x) = u(x+a) = u(x+2a)=… this ensures the probability distribution is periodic
What is Block’s theorem?
The eigenstates of a Hamiltonian H(hat) = -ћ^2/2m *∇^2 + V(r), where V(r) = V(r+R(n1,n2.n3)) for all R(n1,n2,n3) lattice vectors of the crystal can be chosen to have the form of a plane wave exp(ikr) times the wavefunction u(r) with the periodicity of the lattice.
What does using Block’s theorem tell us in the Kronig model?
We need only find a solution for a single period and ensure that it is continuous and smooth.
In the Kronig-Penney model, what does the schrodinger equation equal to 0 <= x <= a-b (width of the gap between square potentials)?
-ћ^2/2m d^2Ψ(x)/dx^2 = EΨ(x), where Ψ(x) = Aexp(iαx)+Bexp(-iαx), and α = sqrt(2mE/ћ^2)
In the Kronig-Penney model, what does the schrodinger equation equal to -b <= x <= 0 (the width of the square potential)?
-ћ^2/2m d^2Ψ/dx^2 + V0Ψ= E*Ψ, where Ψ(x) = Cexp(βx)+Dexp(-βx), β=sqrt(2m(V0-E)/ћ^2)
What does u(x)I equal for the 0<=x<=a-b?
ΨI(x) = exp(ikx)*(Aexp(i(x-k)x) + Bexp(-i(x+k)x)), part in the brackets is u(x)I.
What does u(x)II equal for the -b<=x<=0?
ΨII(x) = exp(ikx)*(Cexp((β-ik)x) + Dexp(-(B+ik)x)), part in the brackets is u(x)II.
How do we check to see if Ψ is continuous and smooth?
ΨI(0) = ΨII(0), dΨI/dx at x= 0 = dΨII/dx at x = 0
How do we check that u(x) is periodic?
uI(a-b) = uII(-b), duI/dx at (a-b) = duII/dx at -b
What equation can we get from the 4 equations checking for periodicity?
cos(ka) = cos(α(a-b))*cosh(βb)+(β^2-α^2)/2αβ sin(α(a-b))sinh(βb)
What can we get from this cos equation? What does this show?
Sub in E, V0, b and a and obtain k i.e E vs k - energy bands E(k). These bands show the gaps that occur at the Brillouin zone boundaries.
What is the Fermi-dirac distribution?
F(E, μ, T) = 1/(exp((E-μ)/k(B)T) + 1)
What is the equation for N the number of electrons in terms of the fermi-dirac distribution?
N = integral from 0 to inf of n(E)*F(E, μ, T) dE
What does each atom in Group IV semiconductors have?
A core of electrons tightly bound to the nuclei and 4 more loosely bound valence electrons.
Why can semiconductors conduct sometimes?
At finite T, a small proportion of electrons will be thermally excited from the valence to the conduction band and therefore it can conduct.
What is the equation for E(F) in terms of Ev and Δ?
E(F) = Ev + Δ/2
What are n-type and p-type semiconductors?
Semiconductors like Si, Ge deliberately doped with very low concentrations of impurities which have either more or less valence electrons.
