Department of Physical Electronics
Faculty of Science, Masaryk University, Brno
Physical laboratory 3
Task G Band gap width
Tasks
1. Determine the energy band gap of silicon and germanium using the photoelectric effect.
Introduction
Condensed or solid state materials can be generally divided into conductors and insulators. Conductors
can be further divided into metals and semiconductors. The width of the electron band
gap is considered one of the criteria for such division.
To understand the importance of semiconductors, we need to be aware of the relations between
their fundamental properties and their real-world usage. The milestone for the description of the
physical properties of semiconductors is the energy band model described by A. H. Wilson in 1931
[1]. The dependence of the electrical conductivity on the purity of the material with the possibility
of doping the material with small concentrations of dopants, together with being able to influence
their properties by external conditions (such as electric or magnetic field, temperature or radiation)
lead to a large number of applications. Besides the electrical resistance’s negative temperature
coefficient, semiconductors are also characterised by exhibiting the photovoltaic effect, the diode
effect, Zenner effect, or the photoelectric effect. The last-named effect is studied in this laboratory
task.
Energy bands
A single isolated atom can only have discrete energy levels. However, as multiple atoms forming a
condensed matter close together, their energy levels merge together forming the material’s energy
bands (figure 1). These energy bands are mutually divided by energy gaps called the energy band
gaps. These correspond to energy intervals, where no electron of the corresponding energy can
exist. Each energy band consists of many very close energy levels. Every state corresponding to
these individual energy levels can be occupied by only a single electron as the Pauli exclusion
principle describes it.
Insulators
In an insulator, the highest populated energy band is fully occupied, and it is divided from an
unoccupied band by an energy band gap so wide that electrons can not be sufficiently thermally
activated to jump from the full band to the unoccupied band.
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Figure 1: Band structure of a condensed matter. The magnified view shows that each band is
formed from many close-lying energy levels. [2]
Metals
In metals, the highest occupied energy level falls somewhere near the middle of an energy band.
It is said that the valence and conduction bands overlap. The band gap width is zero.
Semiconductors
The band structure of a semiconductor is similar to an insulator with the difference that the
energy band gap (Eg) is significantly lower. A small number of electrons is thermally activated
in a semiconductor such as silicon at the room temperature, and these electrons jump to the
conduction band. In this way, the same number of holes is created in the valence band as is the
number of electrons that jumped to the conduction band. Electrons and holes act as the charge
carriers.
The number of electrons in the conduction band can be significantly increased by doping the
semiconductor by e.g. small concentration of phosphorus, arsenic or antimony. This creates the
n type doped semiconductor. The number of holes can be analogically increased by doping of the
original semiconductor by e.g. boron. This creates the p type doped semiconductor.
p-n junction
The p-n junction is a semiconductor monocrystal with one part doped to form the p type material
and the other part doped to form the n type material. The transition from one region to the
other is perfectly sharp, occurring at a single junction plane. The following processes occur at
this plane under thermal equilibrium:
1. The majority carriers (electrons on the n side and holes on the p side) diffuse through the
junction plane. The motions of both the electrons and the holes contribute to a diffusive
current.
2. The minority carriers (holes on the n side and electrons on the p side) are carried by the
potential fall over the junction place and contribute to the drift current. At equilibrium,
the average diffusion current that moves through the junction plane from the p side to the
n side is balanced by an average drift current that moves in the opposite direction. These
two currents cancel themselves out and the net current through the junction plane would
be zero.
3. A charge carrier-depleted zone containing the immobile donor or acceptor ions forms in the
area of the junction plane.
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Figure 2: A schematic sketch of the band structure of an insulator is shown on the left. The
levels occupied by electrons are shown in red. The highest occupied energy is lying on the top
of a band, and the next free energy level is separated by a relatively large energy gap - the band
gap Eg. The band structure of a metal is shown in the middle. The highest occupied energy level
– the Fermi level EF – is lying within a band. As free energy levels are located within the same
energy band, electrons can easily change their energy levels within the whole band, leading to the
electrical conduction. The energy structure of a semiconductor shown on the right is similar to
that of an insulator, the band gap is, however, much narrower. The electrons are able to jump
from the occupied to the unoccupied band through the thermal activation. [2]
Figure 3: The band gap width and the lattice parameter of the most common elemental and
compound semiconductors at room temperature. [3].
4. Contact voltage establishes over the depleted zone.
Photoelectric effect
The photoelectric effect is a physical effect where electrons are released (emitted) from the material
because of the absorption of electromagnetic radiation (such as visible light). If these electrons
are completely released from the material, the effect its called the external photoelectric effect.
Suppose the electrons are not completely released from the material and instead remain as free
conduction electrons in a higher energy band. In that case, we are talking about the internal
photoelectric effect.
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Generation of free conduction electrons by the electromagnetic radiation takes place if the
photon energy is equal to or higher than the band gap width. If this condition is fulfilled, electrons
are released from the valence band into the conduction band. If this occurs in the vicinity of the
p-n junction, these excessive (from the point of thermodynamic equilibrium) charge carriers are
divided by the inner electric field. Electrons flow towards the n type material and holes towards the
p type material. This leads to the alteration of the charge in the volume of the material in the p-n
junction’s vicinity compared to the case without the excessive charge carriers’ external generation.
This results in a change of the electric field in the said area and the so-called photovoltage arises
over the p-n junction. The generated photovoltage depends on the way of its generation and the
radiation intensity.
Measurement method principle
We will measure the photoelectric effect on a silicon and germanium photodiode. A photodiode is
a semiconductor device containing the p-n junction, and it is used to transform an optical signal
to an electric signal. For this, it uses the photoelectric effect. A cross-section of one type of a
photodiode is schematically shown in figure 4a. A p type area is made by a dopant diffusion in the
base n type silicon material. Therefore, a p-n junction is created. Electrical contacts are present
in both the p type and the n type areas. The diode surface is protected by a SiO2 coating. The
whole device is usually encased in resin forming the outer case of the diode.
Figure 4: (a) A photodiode schematic; (b) The dependence of the photovoltage on the photon
energy.
Radiation of different wavelength is absorbed in different thickness of the semiconductor.
Therefore, the p type material’s thickness needs to be such that the used radiation is absorbed in
the p-n junction’s vicinity. The intensity of the radiation intensity in the material is mathematically
described as
I(x) = I0R e−αx
, (1)
where I(x) is the intensity in depth x under the surface, R is the optical reflectivity of the surface
and α is the absorption coefficient. You can see that the intensity of the radiation exponentially
decreases with depth.
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Suppose photons with a too high wavelength not having high enough energy for excitation of
electrons into the conduction band (i.e. creation of conduction electron and a hole in the valence
band) are used. In that case, the radiation passes through the material with a low absorption
coefficient. The internal photoeffect can occur as the wavelength is sufficiently decreased to correspond
the energy Eg.
The absorption coefficient increases with the decreasing wavelength, and the number of photons
that can be absorbed in the vicinity of the p-n junction increases as well. Therefore, the photovoltage
first increases with the increasing absorption coefficient. However, it reaches its maximum,
and it then decreases as more and more electron-hole pairs are created just under the surface of
the photodiode (therefore far from the p-n junction) – see figure 4b.
The band gap width Eg is determined from the spectral dependence of the photovoltage per
one photon S(λ); that is the relation of the ratio of the measured photovoltage U(λ) and the
number of incoming photons N(λ) on the wavelength:
S(λ) =
U(λ)
N(λ)
(2)
The band gap width can then be determined from the photon energy for which the photodiode
starts to be sensitive, i.e. for which the photovoltage U(λ) on the p-n junction is detected (see
fig. 4b). It needs to be considered that photons with high wavelength cannot excite conduction
electrons and do not contribute to the photoeffect.
Experimental setup
The experimental configuration (obr. 5) consists of two main parts. These are the optical setup
and the device for the measurement of low voltages. In the optical setup, light from a halogen lamp
passes through an optical route through an input slit to a monochromator. A monochromatic ray
then passes through an output slit to the photodiode. Narrower slits lead to a lower intensity
light, but they also provide higher spectral resolution. On the other hand, wide-opened slits
provide high intensity but worse monochromaticity of the light. The individual semiconductor
photodiodes (silicon and germanium) are interchangeable. It is necessary to pay attention to the
light being aimed at the photodiode’s active area, where the voltage, measured by a sensitive
voltmeter, is generated.
Figure 5: Experimental setup: 1 - Halogen lamp, 2 - Convex lens, 3 - Monochromator, 4 Semiconductor
photodiode, 5 - Voltmeter.
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1
5
3 4
2
Figure 6: Measurement setup: 1 - Voltmeter, 2 - Monochromator, 3 - Lamp, 4 - Lamp voltage
source, 5 - Photodiode cover.
Experimental procedure
1. Familiarise yourselves with the measurement setup as per figure 5.
2. Use both the silicon and germanium photodiodes for the measurements.
3. Aim the light on the active area of the photodiode during the measurement.
4. Set the monochromator slits and the voltmeter range in a way to achieve the highest possible
monochromaticity of the light while using the whole range of the voltmeter.
5. For each diode measure the dependence U = f(λ).
(a) Using the table at the end of the manual find the values of D(λ) ∝ N(λ) for those
wavelengths used for the measuring of U(λ).
(b) For each wavelength λ determine the ralative value of S(λ) according to equation (2).
6. For each wavelength calculate the photon energy.
7. Plot the dependence S = f(E) into a graph and determine the band gap width Eg for the
first non-zero value.
References
[1] Frank H.: Fyzika a technika polovodičů, SNTL, 1990
[2] Jearl Walker, David Halliday, Robert Resnick: Fundamentals of physics. 10th edition. Wiley,
2014, ISBN 978-1-118-23072-5
[3] Ibach H. a Lueth H.: Solid-State Physics, Springer Verlag, 2003
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Appendix
Spectral radiation of the halogen lamp.
Halogen lamp spectral radiation intensity
D = f (λ)
λ D λ D
(nm) (nm)
1000 158 1500 110
1100 140 1600 95
1150 135 1700 70
1250 125 1800 46
1350 120 1850 35
1400 115 1900 15
Relative intensity of the photodetector signal (D) used for the measurement of the spectral distribution
of the radiation of the halogen lamp. This value is directly proportional to the number
of photons passing through the monochromator and therefore to the number of photons arriving
at the photodiode (N).