Fundamentals of Semiconductor Physics: Problem Set 1

1. The Delta function atom:

Consider a particle of mass m in one dimension subject to an attractive δ-functionpotential (Fig. 1):

V(x)= −λδ(x) Figure 1.

where the delta function is defined so

that δ(x) = {0

∞

x ≠ 0

x = 0, with

−∞

∞∫ δ(x)dx = 1.

(a) Write down the Schrodinger Equation for the particle, and by integrating theSchrodinger equation across the δ-function show that the wavefunction has aslope discontinuity at the potential site given by

ε → 0lim )(0m2

dxd

dxd

2xx

ψλψψεε h

−=

−

−=+=

(b) By writing down the wavefunction with an energy E < 0 for x < 0, and x > 0 withacceptable behavior at x = ± ∞, and matching at x = 0, show that the delta

function has only one bound state with energy E = 2

2

2mh

λ− .

2. The Delta function molecule:

Now consider a particle of mass m subject to two delta functions at x = ± a, (Fig. 2), so

V (x) = −λ [ ])()( axax −++ δδ .

Figure 2.

x0

V(x)

0

0

V(x)

- a + ax

)( x−

⇒+− )( ax (x-a) −⇐

(a) Write down the form of the wavefunction in the three regions for a particle ofenergy E < 0 with proper behavior in the x→ ± ∞ limits for a bound state. Bymatching at x = −a and x = +a, derive the transcendental equation:

Ka422

e1m

K −=

−λh

where 2Em2K h/−=

(b) Show graphically that no matter what a is, there is always one bound state with

energy Eo given by m2K

E2

o2

0h−= where a

oK2o

2

e1m

K −=−λh

. Show that this

corresponds to the symmetric (bonding) solution )()( xx oo ψψ =− , and sketch

)(xoψ .

(c) Show also, that for a > λm2

2h , there exists a second bound state with energy

E1 = m2K 2

12h− where 1 aK21

21e

mK −=− λ

h. Show that )()( xx 11 ψψ −=

(antisymmetric, antibonding solution) and sketch ψ1 (x).

(d) Sketch E0(a) and E1(a) qualitatively.

3. The Delta function solid:

As a final example using delta functions, we consider an electron of mass m movingin a periodic lattice of delta function potentials (Fig. 1):

)( V(x) nax −∑−= δλn

Figure . 3.

-3a -2a -a 0 a 2a 3a 4ax

(a) Assume a solution:

)(xψ = AeKx

+Be-Kx

for 0 < x < a with E = m2K 22 / h− for E < 0. Similarly

for E > 0 assume a solution

)(xψ = Aeiqx

+Be-iqx

with E = m2q22 /h . By using )()( xeax ikaψψ =+ and matching boundaryconditions show that the attached values of energy satisfy:

cos ka = cosh Ka K

m2h

λ− sinh Ka for E < 0

and coska = cos qa 2q

mh

λ− sinqa for E > 0.

(b) By plotting the right hand side versus Ka or qa, exhibit the energy bands andgaps.

(c) Sketch qualitatively the energy band diagram resulting from (b), explainingsalient features.

4. Simple model for Germanium

Germanium has four anisotropic conduction band minima with longitudinal andtransverse masses given by ml = 1.6 and mt = 0.08 in units of the free electron mass.The density of states of each minimum can be shown to be the same as that for an

isotropic mass *em = (mlmt

2)1/3. We simplify the true valence bands by a pair of

degenerate bands with effective mass *hm

(a) Given that intrinsic Ge has a band gap of 0.67 eV and a carrier density of 2.4 x1013 cm-3 at room temperature (300 K), what must the value of m*

h be?

(b) The true valence band structure consists of two bands with masses 0.04 and0.28 (in units of the free electron mass). Which of the two bands do youexpect to give rise to more carriers and what does that imply about theeffective mass m*

h?

5. Fermi Level of Doped Silicon

Silicon has six conduction band minima with ml = 0.9 and mt = 0.2, and at roomtemperature intrinsic Si has a carrier density around 1010 cm-3.

(a) Given a donor binding energy of 0.05 eV, calculate the position of the Fermilevel relative to the conduction bound edge in Si doped with ND=2x1016 donoratoms/cm3 at room temperature (300 K), assuming complete ionization.

(b) Check if the assumption of complete ionization is justified for ND=2x1018 cm-3.Compute the Fermi energy in this case.

Problem 6:Consider a free electron gas contained in a rigid 1-D container of length L.

(a) Compute the density of states g(ε)dε for electrons in this system. Plot qualitativelyg ε( ) .

(b) One puts N electrons in this container. Find an expression for the Fermi energy of thesystem ε F(0) at T= 0 K.

(c) At T=0 K, find an expression for average energy per electron ε , in terms of ε F(0).

(d) Plot (qualitatively) N(ε)dε, the number of electrons with energy in the range ofε,ε + dε[ ], at T=0 K. Calculate ε F(0) for n/L = 0.5 electrons/Å.

(e) What is the qualitative behavior of ε F with temperature T as T increases from zero?Explain.

Problem 7:

(a) Calculate the density of states factor g(ε)dε for a 2-D system of free particlescontained within a rigid container of area A and dimensions xo and yo. Plot g(ε).

(b) Suppose the particles obey Fermi-Dirac statistics. At T=0 K, find the Fermi energy,εF(0), for this system. The system has N particles.

(c) Find an expression for εF(T), the Fermi energy as a function of temperature, in termsof εF(0).

(d) Now suppose the particles in system of part (a) obey Maxwell-Boltzmann statistics.Find the chemical potential µ(T).

8. Three Dimensional Crystal

A three dimensional cubic crystal with lattice constant a (see figure) has an energy-wavevector relation:

E )(→k = E0 ( sin2

2akx + sin2

2

ak y + sin2

2akz )

(a) Calculate the group velocity ),, , v(i.e. ).( E )( v k zyxikEkk

ii =∂

∂=∇≡→→→→

at an

arbitrary point )(→k .

(b) Is )( v→→k along the direction of

→k ? If not, for what directions

→k is it true?

(c) What are the components of the inverse mass tensor

? 1 2

*ji

2ij

kkE

m1

∂∂∂=

h

(d) Assume there is one electron for every two atoms in the lattice. Do youexpect the Fermi surface to be spherical? Explain.

(Hint: Do the surfaces of constant E )(→k correspond to fixed values of k?).

Sketch in what way you expect the Fermi surface to distort from a sphericalshape (i.e. where are the bulges and where are the troughs).

(e) Now, calculate the Fermi energy EF in eV assuming the arguments of the sine

functions in E )(→k are small, so xx sin ≅ . Take E0 = 2.0 eV, a = 2.5Å and one

electron/atom. Do you expect the true EF to be higher or lower? Explain.

Bonus Problem: