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ELEC 4510: Semiconductor Materials and Devices

创建日期：2024-05-24 18:09:34

所属分类：电气工程electric engineering

标签： ELEC 4510

Semiconductor Materials and Devices

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ELEC 4510: Semiconductor Materials and Devices

1. In Homework 3, you analyzed the reciprocal lattice and calculated the nearly free electron
band structure of a 2D hexagonal Bravais lattice that describes a graphene crystal. In the
figures below, a is the lattice constant of the direct lattice, ଵሬሬሬሬ⃗ and ଶሬሬሬሬ⃗ are the primitive
vectors of the direct lattice, and ଵሬሬሬ⃗ and ଶሬሬሬሬ⃗ are the primitive vectors of the reciprocal lattice.
Pairs of red and blue dots represent the two carbon atoms associated with each lattice point
(black dot) of graphene. The gray hexagons depicted in the direct and reciprocal lattices are
the Wigner–Seitz primitive cell of the direct lattice and the first Brillouin zone of the
reciprocal lattice, respectively.
Recall that the general reciprocal lattice vector is ⃗ = ଵଵሬሬሬ⃗ + ଶଶሬሬሬሬ⃗ , where ଵ and ଶ
are integers. Suppose the 2D periodic potential is:
(⃗) = ଵ cos൫ଵሬሬሬ⃗ ∙ ⃗൯ +ଶ cos൫ଶሬሬሬሬ⃗ ∙ ⃗൯ +ଷ cosൣ൫ଵሬሬሬ⃗ + ଶሬሬሬሬ⃗ ൯ ∙ ⃗൧ +ସ cosൣ൫ଵሬሬሬ⃗ − ଶሬሬሬሬ⃗ ൯ ∙ ൧
The periodic potential (⃗) lifts the degeneracies (i.e. opens energy gaps) of the two lowest
electron bands at the M-point ቀ ଶగ
√ଷ௔
, 0ቁ and the three lowest electron bands at the K-point
ቀ0, ସగ
ଷ௔
ቁ. The expected gap openings are marked by red circles in the nearly free electron
bandstructure above.
2
a) Due to () , the free-electron state at the M-point is coupled to one other
degenerate free-electron state at the zone boundary (recall that degenerate states
possess the same energy and hence the same หሬ⃗ ห ). Therefore, the electron wave-
function at the M-point can be written as a superposition of two free-electron states.
Coupling between two energy eigenstates ௞భ(⃗) and ௞మ(⃗) by () can be
described mathematically through a straightforward extension of the result derived
in Problem 1(b) of Homework 1:
 The strength of coupling is calculated by ∫ ௞భ
∗ ൣ()௞మ൧⃗.
 ∫ ௞భ
∗ ൣ(⃗)௞మ൧ ≠ 0 only if ଵሬሬሬሬ⃗ − ଶሬሬሬሬ⃗ = ±ଵሬሬሬ⃗ , ±ଶሬሬሬሬ⃗ , ±൫ଵሬሬሬ⃗ + ଶሬሬሬሬ⃗ ൯ , or
±൫ଵሬሬሬ⃗ − ଶሬሬሬሬ⃗ ൯. In other words, two states couple through () only if the
difference between their wavevectors corresponds to one of the Fourier
components of (⃗).
Based on the above information,
i. Identify the state which couples to the M-point. Mark the state in the
reciprocal lattice.
ii. Set up a 2×2 matrix energy eigenvalue equation following the steps outlined
in Problems 1(c) and 1(d) of Homework 1. Express the diagonal matrix
elements in terms of EM (the unperturbed electron energy at the M-point)
and the off-diagonal matrix elements in terms of ଵ, ଶ, ଷ, and ସ.
iii. If ଵ = ଶ = ଷ = ସ = −2଴, solve the matrix equation to obtain the new
(perturbed) energy eigenvalues at the M-point following the steps outlined
in Problem 1(e) of Homework 1.
b) Due to (), the free-electron state at the K-point is coupled to two other degenerate
free-electron states at the zone boundary. Therefore, the electron wavefunction at
the K-point can be written as a superposition of three free-electron states.
i. Identify the two states which couple to the K-point. Mark the states in the
reciprocal lattice.
ii. Set up a 3×3 matrix energy eigenvalue equation, which takes the form:
൥
ଵଵ ଵଶ ଵଷ
ଶଵ ଶଶ ଶଷ
ଷଵ ଷଶ ଷଷ
൩ ൥
ଵ
ଶ
ଷ
൩ = ൥
ଵ
ଶ
ଷ
൩,
The matrix elements ௡௠ = ∫ ௡∗ ൣ෡(⃗)௠ ൧⃗ represent coupling between
two distinct k-states, as defined in Homework 1. Express the diagonal matrix
elements in terms of EK (the unperturbed electron energy at the K-point) and
the off-diagonal elements in terms of ଵ, ଶ, ଷ, and ସ.
iii. If ଵ = ଶ = ଷ = ସ = −2଴, solve the matrix equation to obtain the new
(perturbed) energy eigenvalues at the K-point. You will notice that the three-
fold degeneracy at the K-point is only partially lifted; two bands remain
degenerate at a higher energy while the third one has a lower energy.
The K’-point is also coupled to two other degenerate free-electron states via (ሬ⃗ ). The
bandstructures at the K- and K’-points constitute a two-fold valley degeneracy in
graphene.
3
2. Consider a gallium arsenide crystal (a face-centered cubic crystal) with a lattice constant a
of 5.65 Å. Suppose an external DC electric field, ሬ⃗ = ௭̂, is applied to the material.
a) Assuming no scattering, what is the frequency of Bloch oscillations? To solve this
problem, you first need to determine the dimension of the first Brillouin zone along the
relevant direction in k-space, and that dimension is not ଶగ
௔
as with 1D crystals or simple
cubic crystals.
Hint: Refer to problem 2(c) of Homework 3.
b) Assuming a scattering time of 0.1 ps and a breakdown field of 0.4 MV/cm for gallium
arsenide, can electrons complete one period of Bloch oscillation without scattering?
c) Extra Credit Problem—Successful completion will double your homework score.
Electrons in typical semiconductors do not exhibit Bloch oscillations since they scatter
before reaching the zone edge. A possible solution is a superlattice structure, which is
a periodic potential obtained by a periodic variation of alloy composition, as illustrated
schematically by the figure below.

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