direct and indirect band gaps

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Direct and indirect band gaps 1

Direct and indirect band gaps

In semiconductor physics, the band gap of a semiconductor is always one of two types, a direct band gap or an

indirect band gap. The band gap is called "direct" if the momentum of electrons and holes is the same in both the

conduction band and the valence band; an electron can directly emit a photon. In an "indirect" gap, a photon cannot

be emitted because the electron must pass through an intermediate state and transfer momentum to the crystal lattice.

The minimal-energy state in the conduction band and the maximal-energy state in the valence band are each

characterized by a certain crystal momentum (k-vector) in the Brillouin zone. If the k-vectors are the same, it is

called a "direct gap". If they are different, it is called an "indirect gap".

Energy vs. crystal momentum for a semiconductor with an indirect band gap, showing

that an electron cannot shift from the lowest-energy state in the conduction band (green)

to the highest-energy state in the valence band (red) without a change in momentum.

Here, almost all of the energy comes from a photon (vertical arrow), while almost all of

the momentum comes from a phonon (horizontal arrow).

Implications for radiative

recombination

Interactions among electrons, holes,

phonons, photons, and other particles

are required to satisfy conservation of

energy and crystal momentum (i.e.,

conservation of total k-vector). A

photon with an energy near a

semiconductor band gap has almost

zero momentum. One important

process is called radiative

recombination, where an electron in

the conduction band annihilates a hole

in the valence band, releasing the

excess energy as a photon. This is

possible in a direct band gap

semiconductor if the electron has a

k-vector near the conduction band

minima (the hole will share the same

k-vector), but not possible in an

indirect band gap semiconductor, as

photons cannot carry crystal momentum, and thus conservation of crystal momentum would be violated. For

radiative recombination to occur in an indirect band gap material, the process must also involve the absorption or

emission of a phonon, where the phonon momentum equals the difference between the electron and hole momentum.

(It can also, instead, involve a crystallographic defect, which performs essentially the same role.) The involvement of

the phonon
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Energy vs. crystal momentum for a semiconductor with a direct band gap, showing that

an electron can shift from the lowest-energy state in the conduction band (green) to the

highest-energy state in the valence band (red) without a change in crystal momentum.

Depicted is a transition in which a photon excites an electron from the valence band to the

conduction band.

makes this process much less likely to

occur in a given span of time, which is

why radiative recombination is far

slower in indirect band gap materials

than direct band gap ones. This is why

light-emitting and laser diodes arealmost always made of direct band gap

materials, and not indirect band gap

ones like silicon.

The fact that radiative recombination is

slow in indirect band gap materials

also means that, under most

circumstances, radiative

recombinations will be a small

proportion of total recombinations,

with most recombinations being

non-radiative, taking place at point

defects or at grain boundaries.

However, if the excited electrons are

prevented from reaching these

recombination places, they have no

choice but to eventually fall back into the valence band by radiative recombination. This can be done by creating a

dislocation loop in the material. At the edge of the loop, the planes above and beneath the "dislocation disk" are

pulled apart, creating a negative pressure, which raises the energy of the conduction band substantially, with the

result that the electrons cannot pass this edge. Provided that the area directly above the dislocation loop is defect-free

(no non-radiative recombination possible), the electrons will fall back into the valence shell by radiative

recombination and thus emitting light. This is the principle on which "DELEDs" (Dislocation Engineered LEDs) are

based.

Implications for light absorption

The exact reverse of radiative recombination is light absorption. For the same reason as above, light with a photon

energy close to the band gap can penetrate much farther before being absorbed in an indirect band gap material than

a direct band gap one (at least insofar as the light absorption is due to exciting electrons across the band gap).

This fact is very important for photovoltaics (solar cells). Silicon is the most common solar-cell material, despite thefact that it is indirect-gap and therefore does not absorb light very well. Silicon solar cells are typically hundreds of

micrometres thick; if it was much thinner, much of the light (particularly in the infrared) would simply pass through.

On the other hand, thin-film solar cells are made of direct band gap materials (such as CdTe, CIGS or CZTS), which

absorb the light in a much thinner region, and consequently can be made with a very thin active layer (often less than

1 micrometre thick).

The absorption spectrum of an indirect band gap material usually depends more on temperature than that of a direct

material, because at low temperatures there are fewer phonons, and therefore it is less likely that a photon and

phonon can be simultaneously absorbed to create an indirect transition. For example, silicon is opaque to visible light

at room temperature, but transparent to red light at liquid helium temperatures, because red photons can only be

absorbed in an indirect transition.
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Formulas for absorption

A common and simple method for determining whether a band gap is direct or indirect uses absorption spectroscopy.

By plotting certain powers of the absorption coefficient against photon energy, one can normally tell both what value

the band gap has, and whether or not it is direct.

For a direct band gap, the absorption coefficient is related to light frequency according to the following

formula:[1][2]

, with

where:

is the absorption coefficient, a function of light frequency

is light frequency

is Planck's constant ( is the energy of a photon with frequency )

is reduced Planck's constant ( )

is the band gap energy

is a certain frequency-independent constant, with formula above

, where and are the effective masses of the electron and hole, respectively ( is

called a "reduced mass") is the elementary charge

is the (real) index of refraction

is the vacuum permittivity

is a "matrix element", with units of length and typical value the same order of magnitude as the lattice

constant.

This formula is valid only for light with photon energy larger, but not too much larger, than the band gap (more

specifically, this formula assumes the bands are approximately parabolic), and ignores all other sources of absorption

other than the band-to-band absorption in question, as well as the electrical attraction between the newly created

electron and hole (see exciton). It is also invalid in the case that the direct transition is forbidden, or in the case that

many of the valence band states are empty or conduction band states are full.[3]

On the other hand, for an indirect band gap, the formula is:[3]

where:

is the energy of the phonon that assists in the transition

is Boltzmann's constant

is the thermodynamic temperature

(This formula involves the same approximations mentioned above.)

Therefore, if a plot of versus forms a straight line, it can normally be inferred that there is a direct band gap,

measurable by extrapolating the straight line to the axis. On the other hand, if a plot of versus

forms a straight line, it can normally be inferred that there is an indirect band gap, measurable by extrapolating the

straight line to the axis (assuming ).
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Article Sources and Contributors 5

Article Sources and ContributorsDirect and indirect band gaps Source: http://en.wikipedia.org/w/index.php?oldid=478644616 Contributors: Ankur Banerjee, Chris the speller, Heroszeros, Hgrosser, Keenan Pepper,

Lightmouse, Quarky2001, Sbyrnes321, Shaddack, TDogg310, Wtshymanski, 15 anonymous edits

Image Sources, Licenses and ContributorsImage:Indirect Bandgap.svg Source: http://en.wikipedia.org/w/index.php?title=File:Indirect_Bandgap.svg License: Public Domain Contributors: Matanbz, Ricky81682, 2 anonymous edits

Image:Direct.svg Source: http://en.wikipedia.org/w/index.php?title=File:Direct.svg License: Creative Commons Attribution-ShareAlike 3.0 Unported Contributors: Profjohn

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direct and indirect band gaps

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