What is the unit of the energy gap

Band gap

As Band gap (English band gap), too Band gap or. forbidden zone, is the energetic distance between the valence band and the conduction band of a solid. Its electrical and optical properties are largely determined by the size of the band gap. The size of the band gap is usually given in electron volts (eV).

material Art Energy in eV
0 K 300 K
elements
C (as diamond) indirectly 5,4 5,46–5,6[1]
Si indirectly 1,17 1,12
Ge indirectly 0,75 0,67
Se directly 1,74
IV-IV connections
SiC 3C indirectly 2,36
SiC 4H indirectly 3,28
SiC 6H indirectly 3,03
III-V connections
InP directly 1,42 1,27
InAs directly 0,43 0,355
InSb directly 0,23 0,17
InN directly 0,7
InxGa1-xN directly 0,7–3,37
GaN directly 3,37
GaP 3C indirectly 2,26
GaSb directly 0,81 0,69
GaAs directly 1,52 1,42
AlxGa1-xAs x <0.4 direct,
x> 0.4 indirect
1,42–2,16
AlAs indirectly 2,16
As B indirectly 1,65 1,58
AlN directly 6,2
BN 5,8
II-VI compounds
TiO2 3,03 3,2
ZnO directly 3,436 3,37
ZnS 3,56
ZnSe directly 2,70
CdS 2,42
CdSe 1,74
CdTe 1,45

origin

According to the band model, bound states of the electrons are only permitted at certain intervals on the energy scale, the Ribbons. There can (but do not have to) be energetically forbidden areas between the bands. Each of these areas represents a gap between the bands, but for the physical properties of a solid only the possible gap between the highest band that is still completely occupied by electrons (valence band, VBM) and the next higher band (conduction band, CBM) is of decisive importance. Therefore is with the The band gap always means the one between the valence and conduction bands.

The occurrence of a band gap in some materials can be understood quantum mechanically through the behavior of the electrons in the periodic potential of a crystal structure. This Model of the quasi-free electrons provides the theoretical basis for the ribbon model.

If the valence band overlaps the conduction band, no band gap occurs. If the valence band is not completely occupied with electrons, the upper unfilled area takes over the function of the conduction band, consequently there is no band gap here either. In these cases, infinitesimal amounts of energy are sufficient to excite an electron.

Effects

Electric conductivity

Only excited electrons in the conduction band can move practically freely through a solid and contribute to electrical conductivity. At finite temperatures, there are always some electrons in the conduction band due to thermal excitation, but their number varies greatly with the size of the band gap. On the basis of this, the classification according to conductors, semiconductors and insulators is carried out. The exact limits are not clear, but the following limit values ​​can be used as a rule of thumb:

  • Conductors do not have a band gap.
  • Semiconductors have a band gap in the range from 0.1 to ≈ 4 eV.[2]
  • Insulators have a band gap greater than 4 eV.[2]

Optical properties

The ability of a solid to absorb light is linked to the condition that it absorbs the photon energy by exciting electrons. Since no electrons can be excited in the forbidden area between the valence and conduction bands, the energy $ E_p $ of a photon must exceed the energy $ E_g $ of the band gap

$ E_p> E_g, $

otherwise the photon cannot be absorbed.

The energy of a photon is coupled to the frequency $ \ nu $ (Ny) of the electromagnetic radiation via the formula

$ E_p = h \ nu $

with Planck's constant of action $ h. $

If a solid has a band gap, it is therefore transparent for radiation below a certain frequency / above a certain wavelength (in general, this statement is not entirely correct, as there are other ways of absorbing the photon energy). The following rules can be derived specifically for the permeability of visible light (photon energies around 2 eV):

  1. Metals can Not be transparent.
  2. Transparent solids are mostly insulators. But there are also electrically conductive materials with a comparatively high degree of transmission, e.g. B. transparent, electrically conductive oxides.

Since the absorption of a photon is linked to the excitation of an electron from the valence to the conduction band, there is a connection with the electrical conductivity. In particular, the electrical resistance of a semiconductor decreases with increasing light intensity, which z. B. can be used with brightness sensors, see also under photo line.

species

Direct band gap

More direct Band transition
(The transition to the right minor minimum would be indirectly, but would cost too much energy here.)

In the $ E (\ vec k) $ diagram, the minimum of the conduction band lies directly above the maximum of the valence band;
where $ \ vec k $ is the wave vector, which in photons is proportional to their vectorial momentum:

$ \ vec k = \ frac {1} {\ hbar} \ cdot \ vec p $

with Planck's reduced quantum of action $ \ hable. $

In the case of a direct transition from the valence band to the conduction band, the smallest distance between the bands is directly above the maximum of the valence band. Therefore the change is $ \ Delta \ vec k \ approx \ vec 0 $, whereby the momentum transfer of the photon is neglected because of its comparatively small size.

Application examples: light emitting diode

Indirect band gap

More indirect Band transition
(in the band structure diagram)

In the case of an indirect band gap, the minimum of the conduction band is shifted relative to the maximum of the valence band on the $ \ vec k $ axis, i.e. H. the smallest distance between the bands is offset. The absorption of a photon is only effectively possible with a direct band gap, with an indirect band gap an additional quasi-pulse ($ \ vec k $) must be involved, whereby a suitable phonon is generated or destroyed. This process with one photon alone is much less likely due to the low momentum of the light, the material shows weaker absorption there.

The best-known semiconductor, silicon, has an indirect band transition.

Temperature dependence

The excitation of a semiconductor by thermal energy.

The energy $ E_ \ mathrm {g} $ of the band gap decreases with increasing temperature for many materials first quadratically, then linearly, starting from a maximum value $ E_ \ mathrm {g} (0) $ at $ T = 0 \ , \ mathrm {K} $. For some materials that crystallize in a diamond structure, the band gap can also increase with increasing temperature. The dependence can be described phenomenologically with the Varshni formula:[3]

$ E_ \ mathrm {g} (T) = E_ \ mathrm {g} (T = 0 \, \ mathrm {K}) - \ alpha \ cdot \ frac {T ^ 2} {T + \ beta} $

with the Debye temperature $ \ beta \ approx \ Theta_ \ mathrm {Debye}. $

The Varshni parameters can be specified for different semiconductors:

semiconductor E.G(T = 0 K)
in eV
$ \ alpha $
in 10−4 eV / K
$ \ beta $
in K
source
Si 1,170 4,73 636 [4]
Ge 0,744 4,774 235
GaAs 1,515 5,405 204 [4]
GaN 3,4 9,09 830 [5]
AlN 6,2 17,99 1462 [5]
InN 0,7 2,45 624 [5]

This temperature behavior is mainly due to the relative positional shift of the valence and conduction bands due to the temperature dependence of the electron-phonon interactions. A second effect, which leads to a negative $ \ alpha $ with diamond, for example, is the shift due to the thermal expansion of the lattice. In certain areas this can become non-linear and also negative, whereby negative $ \ alpha $ can be explained.[6]

Applications

There are applications above all in optics (e.g. different colored semiconductor lasers) and in all areas of electrical engineering, whereby the semiconductor or insulator properties of the systems and their great variability (e.g. by alloy). The systems with a band gap also include the so-called topological insulators, which have been in effect since around 2010 and in which (almost) superconducting surface currents occur in addition to the internal states that carry no current.

See also

  • Band structure
  • Schrödinger equation
  • Band Gap Reference
  • Solar cell
  • III-V compound semiconductors # Calculation of the ternary band transition energies
  • Wide bandgap semiconductors

literature

Web links

Individual evidence

  1. ↑ Jerry L. Hudgins: Wide and narrow band gap semiconductors for power electronics: A new valuation. In: Journal of Electronic Materials, June 2003, Volume 32, Issue 6. Springer, December 17, 2002, pp. 471–477, accessed on August 13, 2017 (english).
  2. 2,02,1A. F. Holleman, E. Wiberg, N. Wiberg: Inorganic Chemistry Textbook. 101st edition. de Gruyter, Berlin 1995, ISBN 3-11-012641-9, p. 1313 (limited preview in Google book search).
  3. ↑ Y. P. Varshni: Temperature dependence of the energy gap in semiconductors. In: Physica. Volume 34, No. 1, pp. 149-154, doi: 10.1016 / 0031-8914 (67) 90062-6.
  4. 4,04,1Hans-Günther Wagemann, Heinz Eschrich: Solar radiation and semiconductor properties, solar cell concepts and tasks. Vieweg + Teubner Verlag, 2007, ISBN 3-8351-0168-4, p. 75.
  5. 5,05,15,2Barbara Monika Neubert: GaInN / GaN LEDs on semipolar side facets using selective epitaxy produced GaN strips. Cuvillier Verlag, 2008, ISBN 978-3-86727-764-8, p. 10.
  6. ↑ Y. P. Varshni: Temperature dependence of the energy gap in semiconductors. In: Physica. Volume 34, No. 1, pp. 149-154, doi: 10.1016 / 0031-8914 (67) 90062-6.
fr: gang interdite