Metals, Insulators and Semiconductors

Question 1

Realistically, what happens to the density of states in some materials?

Answer 1

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)

Question 2

How can we tell if a material is insulating or not from an n(E) - E graph?

Answer 2

If E(F) lies in a gap, the material is insulating.

Question 3

When is a material transparent to light?

Answer 3

With ћω = hc/λ < Δ, where Δ is the length of the gap on the n(E) - E graph.

Question 4

For what values of Δ is a material an insulator?

Answer 4

Δ > 2eV = insulator, < 2eV is a semiconductor

Question 5

What is the origin of the gaps on an n(E) - E graph?

Answer 5

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.

Question 6

What is a second insight into the origin of the gaps in the n(E) - E graph?

Answer 6

• 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

Question 7

What is the Kronig-Penney model diagram?

Answer 7

Periodic array of square potentials with spacing a, height v0 and width b.

Question 8

In the Kronig-Penney model, what form does the solution of the Schrodinger equation take?

Answer 8

Ψ(x) = exp(ikx)*u(x), where u(x) = u(x+a) = u(x+2a)=… this ensures the probability distribution is periodic

Question 9

What is Block’s theorem?

Answer 9

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.

Question 10

What does using Block’s theorem tell us in the Kronig model?

Answer 10

We need only find a solution for a single period and ensure that it is continuous and smooth.

Question 11

In the Kronig-Penney model, what does the schrodinger equation equal to 0 <= x <= a-b (width of the gap between square potentials)?

Answer 11

-ћ^2/2m d^2Ψ(x)/dx^2 = EΨ(x), where Ψ(x) = Aexp(iαx)+Bexp(-iαx), and α = sqrt(2mE/ћ^2)

Question 12

In the Kronig-Penney model, what does the schrodinger equation equal to -b <= x <= 0 (the width of the square potential)?

Answer 12

-ћ^2/2m d^2Ψ/dx^2 + V0Ψ= E*Ψ, where Ψ(x) = Cexp(βx)+Dexp(-βx), β=sqrt(2m(V0-E)/ћ^2)

Question 13

What does u(x)I equal for the 0<=x<=a-b?

Answer 13

ΨI(x) = exp(ikx)*(Aexp(i(x-k)x) + Bexp(-i(x+k)x)), part in the brackets is u(x)I.

Question 14

What does u(x)II equal for the -b<=x<=0?

Answer 14

ΨII(x) = exp(ikx)*(Cexp((β-ik)x) + Dexp(-(B+ik)x)), part in the brackets is u(x)II.

Question 15

How do we check to see if Ψ is continuous and smooth?

Answer 15

ΨI(0) = ΨII(0), dΨI/dx at x= 0 = dΨII/dx at x = 0

Question 16

How do we check that u(x) is periodic?

Answer 16

uI(a-b) = uII(-b), duI/dx at (a-b) = duII/dx at -b

Question 17

What equation can we get from the 4 equations checking for periodicity?

Answer 17

cos(ka) = cos(α(a-b))*cosh(βb)+(β^2-α^2)/2αβ sin(α(a-b))sinh(βb)

Question 18

What can we get from this cos equation? What does this show?

Answer 18

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.

Question 19

What is the Fermi-dirac distribution?

Answer 19

F(E, μ, T) = 1/(exp((E-μ)/k(B)T) + 1)

Question 20

What is the equation for N the number of electrons in terms of the fermi-dirac distribution?

Answer 20

N = integral from 0 to inf of n(E)*F(E, μ, T) dE

Question 21

What does each atom in Group IV semiconductors have?

Answer 21

A core of electrons tightly bound to the nuclei and 4 more loosely bound valence electrons.

Question 22

Why can semiconductors conduct sometimes?

Answer 22

At finite T, a small proportion of electrons will be thermally excited from the valence to the conduction band and therefore it can conduct.

Question 23

What is the equation for E(F) in terms of Ev and Δ?

Answer 23

E(F) = Ev + Δ/2

Question 24

What are n-type and p-type semiconductors?

Answer 24

Semiconductors like Si, Ge deliberately doped with very low concentrations of impurities which have either more or less valence electrons.