Re: what makes a semiconductor have a direct or indirect bandgap?

Greetings:

References:
Richard P. Feynman, Robert B. Leighton, Matthew Sands,
The Feynman Lectures on Physics,
Volume 3, Quantum Mechanics, Addison-Wesley, 1966.

S. M. Sze, Physics of Semiconductor Devices,
Second Edition, John Wiley & Sons, NY, 1981

Direct Band Gaps versus Indirect Band Gaps

To quote the late Nobel Laureate, Professor Richard Feynman: "Because
atomic behavior is so unlike ordinary experience, it is very difficult
to get used to, and it appears peculiar and mysterious to every one - both
to the novice and to the experienced physicist". " We cannot make the
mystery go away by explaining how it works. "We will just
tell you how it works". So in this discussion I will tell you
how it works in words that might help your understanding the problem.

Direct band gaps are particularly important for fabricating photonic
devices, such as semiconductor diode lasers, because they require very
high quantum efficiencies compared to semiconductor electronic devices
which can be fabricated from both direct and indirect band gap materials.

In low pressure gas lasers, such as helium neon lasers, the atoms and
molecules are essentially free from interaction with each other. The
energy levels of the atoms are very distinct and electrons transitioning
downward between energy levels generate photons that have a very precise
frequency. However, if the gas pressure in the laser is increased to high
pressures, such as in the carbon dioxide waveguide laser, the interaction
between the atoms leads to collisions and pressure broadening of
the lines. The broadened lines are still separated by energy gaps; however,
the photons transitioning downward between the broadened energy levels
can vary in frequency by as much as a gigahertz. This is very useful for
laser radar and communication applications because we can we can move the
spacing between the laser cavity mirrors to tune the laser's output
frequency to a number of different frequency channels within the pressure
broadened transitions.

When we uses a semiconductor material to fabricate a laser the atomic
spacing in the crystal lattice becomes so small that major and complex
quantum mechanical interactions occur between the atoms and energy level
broadening is carried to the extreme.

Let's conduct a thought experiment.
As atoms are brought closer together to form a semiconductor or an
insulator (not a metal), first the energy levels begin to broaden as we
discussed in the gas laser. We close the atomic spacing more and the gaps
between the higher energy levels disappear and the lower energy levels
broaden, however a gap remains between the lowest energy levels and the
merged higher energy levels. When the atoms become so close that they
begin to bond to each other, a semiconductor lattice is beginning to form.
Bonding causes extreme energy level broadening because now the Pauli
Exclusion Principle requires that the bonding electrons cannot occupy the
same quantum state at the same time and so the electrons must occupy many
higher and lower energy levels around the original free atom energy levels.
As the crystal lattice forms a new gap appears within the broadened
higher energy levels. In semiconductor physics we call the higher portion
of broadened energy levels the conduction band and the lower
portion of the broadened energy levels the valence band. The gap
between conduction band and the valence band we call the energy gap.

Most semiconductor physics books only show the two closest gaps between the
conduction band and the valence band; however, Sze's book shows that there
are many higher and lower energy bands within the semiconductor lattice and
that the band structure is dependent on the crystal orientation because the
atomic spacing within the lattice changes between the crystal's axes.

When powerful computers solve the Schrodinger wave equation for a periodic
crystal lattice to calculate the energy of the bottom of the conduction band
and the energy of the top of the valence band, the values for energy can be
plotted as a function of the electron wave number, which is commonly
referred to as momentum (k) in text books. The results are many complex
asymmetrical curves. Most books only show these E/k curves near a normalized
zero momentum point because this is the area of most interest for device
technology; however, Sze's book shows that at greater and lesser values of
energy and momentum there are other energy level bands that cross over what
we normally call the conduction and valance bands. Also zero momentum
is a normalized number which is usually, but not always, placed at the
highest point of the valence band.

The closest point between the top of the valance band curve and the bottom
of the conduction band is called the materials Energy Gap. For
insulating materials this gap can be greater than ten electron volts; however,
for semiconductor electronic devices operating a reasonable voltages we want
the gap to be a few electron volts. (Note: electron volts and wave numbers
are used by physicists, electrical engineers are used to using joules and
frequency which often is a confusing factor for engineers like myself).
Fortuitously nature has caused some crystalline materials such as GaAs to have
the highest energy point of the valence band to be directly below the lowest
energy point of the conduction band at the same value for momentum and to have
the gap be only a few electron volts. We call these direct band gap
materials. Crystalline materials such as silicon that have the highest
energy point of the valence band at a different value of momentum than the
lowest energy point of the conduction band, we call indirect band gap
materials.

Laser diodes require a material with a high quantum efficiency to overcome
optical losses within the laser cavity (between the mirror facets) to initiate
and sustain laser action. This means that a high number of the electrons
dropping in energy from the conduction band to the valance band must radiate
that energy loss in a photon. Very pure, defect free, direct band gap material
is optimum for high quantum efficiency because the electrons are at the same momentum
(wave number) and fall directly from the conduction band to the valence band.
Indirect band gap materials must loose some energy in momentum to reach the
high point of the valence band (side ways on the E/K plots). This requires
that energy be lost in the generation of an acoustic phonon, or giving of
energy as heat to a defect or impurity in the material. Few of the electrons
can loose momentum and still retain enough energy to generate a photon at the
desired frequency. Thus indirect band gap materials have poor quantum
efficiency. This material can be used to generate incoherent light in the form of light
emitting diodes (LED); however, they do not generate enough photons to sustain
coherent laser action.

Best regards, Your Mad Scientist
Adrian Popa