ELEC 4510: Semiconductor Materials and Devices
Spring 2023
Homework 1 (Due in class on March 7th, 2023)
In this problem, you will calculate the energy gap opening in the electronic bandstructure of a
one-dimensional crystal of length L via a perturbative approach.
The energy eigenstates and corresponding energy eigenvalues of a free electron can be
obtained by solving the time-independent Schrodinger equation:
෡௞() = ௞௞(),
where ෡ = ௣ො
మ
ଶ௠೐
+ ෠() is the Hamiltonian (energy operator), ̂ = −ħ ௗ
ௗ௫
is the momentum
operator, and ௘ is the mass of a free electron. For a free electron, ෠() = 0. The eigenvalues and
eigenfunctions are indexed by their wavevector k.
a) Show that normalized plane-wave eigenfunctions of the form ௞() =
ଵ
√௅
௜௞௫ are
mutually orthonormal or linearly independent, i.e.
න ௞೘
∗ ௞೙
௅
଴
= ൜1, if = 0, if ≠
Here, ௠ =
ଶగ
௅
and ௡ =
ଶగ
௅
are the allowed wavevectors; = ±1, ±2, … and =
±1, ±2, … are quantum numbers. The asterisk denotes complex conjugation, i.e. ௞∗ () =
ଵ
√௅
ି௜௞௫.
Next, the electron is subjected to the periodic atomic potential of the crystal. This potential is
treated as a weak perturbation to the electron. The Hamiltonian is now modified to include a
nonzero ෠() = −2 cos ቀଶగ
௔
ቁ = − ቀ௜
మഏ
ೌ ௫ + ି௜
మഏ
ೌ ௫ቁ, where a is the interatomic spacing and
= for a total of N atoms in the crystal. The new Hamiltonian will lead to a set of new energy
eigenvalues and new energy eigenstates. The solution approach (matrix method) is outlined below,
and you will fill in the details.
b) The orthonormal, unperturbed energy eigenstates of the free electron are coupled by the
periodic potential. Every unperturbed state is connected to two other unperturbed states,
one separated by + ଶగ
௔
and the other separated by − ଶగ
௔
. The strength of the coupling is −.
Mathematically,
න ௞೘
∗ ൤−2 cos ൬
2

൰൨ ௞೙
௅
଴
= ൞
−, if ௡ − ௠ = ±
2
0, if ௡ − ௠ ≠ ±
2
Complete the steps to establish the above relationship.
2

The two states of interest in this problem are ±ே/ଶ =
ଶగ
௅
ቀ± ே
ଶ
ቁ = ± గ
௔
. These states are
degenerate (i.e. have the same energy) and both are Bragg-reflected by the periodic crystal
potential. According to the Expansion Principle of quantum mechanics, the new (perturbed) energy
eigenstates can be constructed as a linear superposition of the original (unperturbed) energy
eigenstates. Therefore, the Schrodinger equation for the new problem becomes

෡൫ାగ/௔ାగ/௔ + ିగ/௔ିగ/௔൯ = ±గ/௔ᇱ ൫ାగ/௔ାగ/௔ + ିగ/௔ିగ/௔൯

where ାగ/௔ and ିగ/௔ are the expansion coefficients and ±గ/௔ᇱ are the new (perturbed) energy
eigenvalues.

By applying the property of mutual orthogonality of the unperturbed energy eigenstates, the
following matrix equation can be derived from the new Schrodinger equation:

൤ଵଵ ଵଶଶଵ ଶଶ
൨ ቂ
ାగ/௔
ିగ/௔ቃ = ±గ/௔
ᇱ ቂ
ାగ/௔
ିగ/௔ቃ

The matrix elements are given by:
ଵଵ = න ାగ/௔∗ ൫෡ାగ/௔൯
௅
଴

ଶଶ = න ିగ/௔∗ ൫෡ିగ/௔൯
௅
଴

ଵଶ = න ାగ/௔∗ ൫෡ିగ/௔൯
௅
଴

ଶଵ = න ିగ/௔∗ ൫෡ାగ/௔൯
௅
଴


c) Show that ଵଵ = ଶଶ =
ħమ
ଶ௠೐
ቀగ
௔
ቁ
ଶ
, which is the energy of the original (unperturbed)
eigenstates at the Bragg points = ± గ
௔
. These diagonal elements are the expectation
values of the energy of the unperturbed eigenstates.

d) Show that ଵଶ = ଶଵ = −, which describes how strongly the two original (unperturbed)
eigenstates are coupled via the periodic potential.

e) Solve the matrix equation to obtain the new (perturbed) energy eigenvalues ±గ/௔ᇱ at the
Bragg points = ± గ
௔
. You need to first express the equation in the form

ቈ
ଵଵ − ±గ/௔ᇱ ଵଶ
ଶଵ ଶଶ − ±గ/௔ᇱ
቉ ቂ
ାగ/௔
ିగ/௔ቃ = 0.

Then, set the determinant of the 2×2 matrix to zero to obtain nontrivial solutions.
3

You should obtain ±గ/௔ᇱ =
ħమ
ଶ௠೐
ቀగ
௔
ቁ
ଶ
± as the two new energy eigenvalues
corresponding to each of the two states = ± గ
௔
. In other words, the perturbed electron
energies at the Bragg points are either higher or lower than the free-electron energy by .
The difference between the two values, 2, corresponds to an energy gap at = ± గ
௔
.

f) Solve the matrix equation to obtain the expansion coefficients ±గ/௔ at the Bragg points
= ± గ
௔
. Since we began with normalized wavefunctions, the expansion coefficients
satisfy: หାగ/௔ห
ଶ
+หିగ/௔ห
ଶ
= 1.

You should obtain two pairs of solutions: ቂ
ାగ/௔
ିగ/௔ቃ = ቈ
−1/√2
+1/√2
቉ or ቈ−1/√2
−1/√2
቉.
g) Construct the new (perturbed) energy eigenstates ൫ାగ/௔ାగ/௔ + ିగ/௔ିగ/௔൯ at the
Bragg points = ± గ
௔
. You will see that the new eigenstates are standing waves instead of
traveling waves. Figure from: C. Kittel, Introduction to Solid State Physics, 7th Ed.