Single Electron in Quantum Well Structures

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Single Electron in Quantum Well Structures

SchrÖdingers equation in QW structures
where the bulk Hamiltonian is defined as
Although the bulk Hamiltonian is periodic, the QW
potential V is neither periodic nor small, so we
can not treat it as a perturbation.
What to do?
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Single Electron in Quantum Well Structures

An analogy between bulk and QW
Bulk free space periodic lattice QW free
space periodic lattice non-periodic QW
potential
Bulk global motion (plane wave) local motion
(lattice wave)
QW global motion (wave packet) local motion
(lattice wave)
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Single Electron in Quantum Well Structures

Plug
into the SchrÖdingers equation
to obtain
where
Therefore
or
where
and
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Single Electron in Quantum Well Structures

For conduction bands, we have
For valence bands, we have
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Single Electron in Quantum Well Structures

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Single Electron in Quantum Well Structures

Again, by ignoring spin-orbit split bands and the
small anisotropy in x-y plane, we may reduce the
6 ? 6 Hamiltonian to 4 ? 4. Hence we obtain
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Single Electron in Quantum Well Structures

Following the convention to use the term heavy
and light holes according to their masses in z
direction, we define
The valence band effective mass becomes
anisotropy and mass reversal happens since the
heavy hole has a smaller in-plane mass than the
light hole.

• Unlike in bulk semiconductors, the heavy hole and
  light hole bands in QW structures become
  degenerate even at k0.
• Band mixing effect

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Single Electron in Strained Layer Structures

Strain changes the semiconductor lattice
structure. If we adopt r as the new coordinate
system for the strained lattice, SchrÖdingers
equation takes the same form under r
All material and structural parameters in the
Hamiltonians (for both bulk semiconductors and QW
structures) are still given in the original
(unstrained) lattice structure under
coordinate system r. Therefore, all we need to
do is to project the new spatial variables (x,
y, z) in the equation to the original
space coordinate system (x, y, z). As such, the
governing equation for strained layer structures
will be obtained.
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Single Electron in Strained Layer Structures

Coordinate projection from r to r
Dimensionless relative shift in the space
domain due to the lattice deformation
Since the lattice deformation must be small to
avoid relaxation through defects generation
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Single Electron in Strained Layer Structures

Or
Hence we find
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Single Electron in Strained Layer Structures

Consequently, the Hamiltonian for the strained
layer structure becomes
where
Since the extra term introduced in the
Hamiltonian is a Perturbation, following the same
approach in dealing with the k-p term in the L-K
model, we can readily find the solution directly
through a mapping from kaß to eaß, and from
the Luttinger constants to the deformation
(hydrostatic and shear) energies.
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Single Electron in Strained Layer Structures

Strained bulk semiconductor
Conduction bands
Valence bands in the same (unstrained bulk) L-K
Hamiltonian form, with extra terms added to its
parameters
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Single Electron in Strained Layer Structures

Strained QW structure
Conduction bands
Valence bands in the same (unstrained QW
structure) L-K Hamiltonian form, with extra terms
added to its parameters
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Single Electron in Strained Layer Structures

For zinc blende structure
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Single Electron in Strained Layer Structures

Effect of strain for bulk semiconductors
For compressive strain e0lt0, Pcelt0, Pvegt0, Qegt0

For tensile strain e0gt0, Pcegt0, Pvelt0, Qelt0
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Ec
EcEg
Elh
Ec
Qe
Qe
Ehh
EhhElh0
Ehh
Qe
Qe
Elh
Unstrained bulk semiconductor
Tensile strained bulk semiconductor
Compressive strained bulk semiconductor
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Single Electron in Strained Layer Structures

Effect of strain for QW structures similar to
the effect for bulk semiconductors, with more
features due to the non-degeneracy of the heavy
hole and light hole energies at k0 and the band
mixings.
For compressive strain HH and LH further
separate apart at k0, which reduces the mixing
and bring in smaller effective mass in the
neighborhood of k0 consequently, the density of
states reduces which will increase the
differential gain and lower the transparent
carrier density
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Single Electron in Strained Layer Structures

For tensile strain HH and LH get closer or even
reversed at k0 if not reversed, the tensile
strain enhances the band mixing and bring in
larger effective mass, which may not be desired
however, if HH and LH get reversed with LH on
top, we may stillobtain higher differential gain
and lower transparent carrier density, due to the
larger overlap between the conduction
band electron and valence band light hole wave
functions (i.e., larger dipole matrix element).
This effect is often more significant than the
effective mass reduction.
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Ec
EcEg
Elh
Ec
Ehh0 Elhlt0
Qe
Qe
Ehh
Ehh
Qe
Qe
Elh
Unstrained QW structure
Tensile strained QW structure
Compressive strained QW structure