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6.2: Semiconductors and Energy Level Diagrams

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    Semiconductor Definitions

    Some semiconductors are made up of atoms of a single type like pure Si or pure Ge. Others contain a combination of elements in column 13 and column 15 of the periodic table. Semiconductors of this type include AlAs, AlSb, GaAs, and InP. Other semiconductors contain a combination of elements in columns 12 and 16 of the periodic table. Examples of this type include ZnTe, CdSe, and ZnS [9]. Most semiconductors involve elements located somewhere near silicon on the periodic table, but more complicated compositions and structures are also possible. Materials made from three different elements of the periodic table are called ternary compounds, and materials made from four elements are called quaternary compounds.

    To understand the operation of devices like solar cells, photodetectors, and LEDs, we need to study the flow of charges in semiconductors. Electrical properties in semiconductors are determined by the flow of both valence electrons and holes. Valence electrons, as opposed to inner shell electrons, are the electrons most easily ripped off an atom. A hole is an absence of an electron. Valence electrons and holes are known as charge carriers because they are charged and they move through the semiconductor when an external voltage is applied. At a finite temperature, electrons are continuously in motion, and some electron-hole pairs may form an exciton. These electron-hole pairs naturally combine, also called decay, within a short time. However, at any time, some charge carriers are present in semiconductors at temperatures above absolute zero due to the motion of charges.

    Crystalline semiconductors can be classified as intrinsic or extrinsic [9, p. 65]. An intrinsic semiconductor crystal is a crystal with no lattice defects or impurities. At absolute zero, \(T = 0\) K, an intrinsic semiconductor has no free electrons or holes. All valence electrons are involved in chemical bonds, and there are no holes. At finite temperature, some charge carriers are present due to the motion of electrons at finite temperature. The concentration of these charge carriers is measured in units \(\frac{electrons}{m^3}\), \(\frac{holes}{m^3}\), \(\frac{electrons}{cm^3}\) or \(\frac{holes}{cm^3}\). The intrinsic carrier concentration is the density of electrons in a pure semiconductor, and it is a function of the temperature \(T\). At higher temperatures, more charge carriers will be present even if there are no impurities or defects in the crystalline semiconductor due to more motion of charges. If we apply a voltage across an intrinsic semiconductor at \(T = 0\) K, no charges flow. When the equilibrium concentration of electrons \(n\) or holes \(p\) is different from the intrinsic carrier concentration \(n_i\) then we say that the semiconductor is extrinsic. If either impurities or crystal defects are present, the material will be extrinsic. If a voltage is applied across an extrinsic semiconductor at \(T = 0\) K, charges will flow. If a voltage is applied across either an extrinsic or intrinsic semiconductor at temperatures above absolute zero, charge carriers will be present and will flow.

    The process of introducing more electrons or holes into a semiconductor is called doping. A semiconductor with an excess of electrons compared to an intrinsic semiconductor is called n-type. A semiconductor with an excess of holes is called p-type. Silicon typically has four valence electrons which are involved in bonding. Phosphorous has five valence electrons, and aluminum has three. When a phosphorous atom replaces a silicon atom in a silicon crystal, it is called a donor because it donates an electron. When an aluminum atom replaces a silicon atom, it is called an acceptor. Column 15 elements are donors to silicon and column 13 elements are acceptors. If silicon is an impurity in AlP, it may act as a donor or acceptor. If it replaces an aluminum atom, it acts as a donor. If it replaces a phosphorous atom, it acts as an acceptor.

    How can we dope a piece of silicon? More specifically, how can we dope a semiconductor with boron? Boron is sold at some hardware stores. It is sometimes used as an ingredient in soap. Start with a silicon wafer, and remove any oxide which has formed on the surface. Each silicon atoms forms bonds with four nearest neighbors. At the surface though, there is no fourth neighbor, so silicon atoms bond with oxygen from the air. Smear some boron onto the wafer, or place a chunk of boron on top of the wafer. Place it in a furnace at slightly less than silicon's melting temperature, around \(1000 ^{\circ}C\). Some boron will diffuse in and replace silicon atoms. Remove the excess boron. The same procedure can be used to dope with other donors or acceptors. What is the most dangerous part of the process? Etching the oxide off the silicon because hydrofluoric acid HF, a dangerous acid, is used [69].

    Sometimes it is possible to grow one layer of a semiconductor material on top of a layer of a different type of material. A stack of different semiconductors on top of each other is called a heterostructure. Not all materials can be made into heterostructures. GaAs and AlAs have almost the same atomic spacings, so heterostructures of these materials can be formed. The spacing between atoms, also called lattice constant, in AlAs is 0.546 nm, and the spacing between atoms in GaAs is 0.545 nm [9]. If the atomic spacing in the two materials is too different, mechanical strain in the resulting material will pull it apart. Even moderate mechanical strain can negatively impact optical properties of a device because defects may be introduced at the interface between the materials. These defects can introduce additional energy levels which can trap charge carriers.

    Energy Levels in Isolated Atoms and in Semiconductors

    In a solar cell, light shining on a semiconductor causes electrons to flow which allows the device to convert light to electricity. How much energy does it take to cause an electron in a semiconductor to flow? To answer this question, we will look at energy levels of:

    • An isolated Al atom at \(T = 0\) K
    • An isolated P atom at \(T = 0\) K
    • Isolated Al atom and P atoms at \(T > 0\)
    • An AlP crystal at \(T = 0\) K
    • An AlP crystal at \(T > 0\) K

    Aluminum has an electron configuration of \(1s^22s^22p^63s^23p^1\). It has 13 total electrons, and it has 3 valence electrons. More specifically, it has two valence electrons in the \(3s\) subshell and one in the \(3p\) subshell. Phosphorous has an electron configuration of \(1s^22s^22p^63s^23p^3\), so it has 5 valence electrons. Ideas in this section apply to materials regardless of whether they are crystalline, amorphous, or polycrystalline.

    6.2.1.png
    Figure \(\PageIndex{1}\): Energy level diagram of an isolated aluminum atom at \(T = 0\) K plotted using data from [70].
    6.2.2.png
    Figure \(\PageIndex{2}\): Energy level diagram of isolated aluminum and phosphorous atoms at \(T = 0\) K plotted using data from [70].

    Energy Levels of Electrons of Isolated Al and Isolated P Atoms at \(T = 0\) K

    To understand the interaction of light and a semiconductor, start by considering an isolated Al atom and an isolated P atom at absolute zero, \(T = 0\) K. How much energy does it take to rip off electrons of Al? It takes significantly less energy to rip off a valence electron than an electron from an inner shell. In fact, when we say an electron is a valence electron, or an electron is in a valence shell, we mean that the electron is in the shell for which it takes the least energy to rip off an electron. We do not mean that the electron is further from the nucleus, although often it is. When we say an electron is in an inner shell, we mean the electron is in a shell for which it takes more energy to rip off an electron. This text focuses on energy conversion devices which operate at moderate energies, so all of the devices discussed involve interactions of only valence electrons. Inner shell electrons will not be involved. It is also possible to excite, but not rip off, an electron. When an electron is excited, its internal momentum changes and its quantum numbers change. The terms valence electron and quantum number were both defined in Sec. 1.5.2. Less energy is required to excite than rip off an electron. The energy required to excite or rip off electrons can be supplied by thermal energy, an external voltage, an external optical field, or other external sources.

    Figure \(\PageIndex{1}\) is a plot of the energy required to excite or remove electrons from an isolated neutral Al atom at \(T = 0\) K. The figure was plotted using data from [70]. While energy levels are drawn using actual data, the thickness of the lines is not drawn to scale. Energy is on the vertical axis. Allowed energy levels are shown by horizontal lines. Each electron can only have energy corresponding to one of these discrete possible energy levels. At \(T = 0\) K, electrons occupy the lowest possible energy levels. One electron can occupy each line, so the lowest 13 energy levels are occupied by electrons. While not shown due to the resolution of the figure, the density of allowed energy levels increases as energy approaches zero at the top of the figure. Since we are considering the case of absolute zero temperature, these upper energy levels are not occupied by electrons.

    The left side of Fig. \(\PageIndex{2}\) replots the allowed energy levels of the electrons in an isolated Al atom at \(T = 0\) K. The energy levels are also labeled. The right side of the figure plots the allowed energy levels of electrons in an isolated P atom also at \(T = 0\) K. Data on phosphorous energy levels also comes from [70]. As with the Al atom, the electrons of the P atom can only occupy certain specific discrete energy levels. Since the atoms are at absolute zero, the electrons occupy the lowest energy levels possible. Figure \(\PageIndex{3}\) contains the same information, but is zoomed in vertically to show the valence electron levels more clearly.

    The P atom has two more electrons than the Al atom. Phosphorous atoms have more protons, so the electrons are a bit more tightly bound to the nucleus. For this reason, it takes a bit more energy to rip the electrons off, and the allowed energy levels are a bit different than for Al.

    The amount of energy required to rip a \(3p\) electron off the atom is the vertical distance from the \(3p\) level to the ground line at the top of the figure. The amount of energy required to rip a \(2p\) electron off is the vertical distance from the \(2p\) level to the ground line. As expected, these figures show that it requires more energy to rip off the inner shell \(2p\) electron than the valence shell \(3p\) electron. If enough energy is supplied, an electron will be ripped off, and the electron will flow freely through the material. If some energy is supplied but not enough to rip off the electron, the electron can get excited to a higher energy level. The energy required to excite an electron is given by the vertical distance in the figure from an occupied to an unoccupied energy level. In either case, we say that an electron-hole pair forms. If the amount of energy supplied is too small to excite an electron from a filled to unfilled state, the external energy will not be absorbed.

    Energy Levels of Electrons of Isolated Al and Isolated P Atoms at \(T > 0\) K

    How do the energy levels change when the Al and P atoms are at temperatures above absolute zero, where electrons are continuously vibrating and moving? First, the energy levels broaden. The electrons can still only take certain energy levels, but there is a wider range to the allowed energy levels. Second, occasionally, electrons spontaneously get excited into higher states. For example, a \(3p\) electron may get excited into the \(4s\) state temporarily. If it does, it will quickly return to the ground state.

    Energy Levels of AlP at \(T = 0\) K

    How much energy does it take to rip an electron off an AlP crystal at \(T = 0\) K? The three valence electrons of each Al atom and the five valence electrons of each P atom form chemical bonds. The energy required to rip off these electrons is slightly different than the energy required to rip off the equivalent electrons of isolated Al and isolated P atoms. Figure \(\PageIndex{4}\) illustrates the energy levels of the valence electrons of AlP. Unlike in the previous figures, these energy levels do not come from actual data. Instead, they are meant as a rough illustration of the effect. The amount of energy required to rip off an electron is represented on the energy level diagram by the vertical distance from that level to the ground level at the top of the diagram. The energies needed to remove inner shell electrons do not significantly change from the energy levels of isolated atoms.

    6.2.3.png
    Figure \(\PageIndex{3}\): Zoomed in version of the energy level diagram of isolated aluminum and phosphorous atoms at \(T = 0\) K plotted using data from [70].
    6.2.4.png
    Figure \(\PageIndex{4}\): Energy level diagram at \(T = 0\) K of an isolated aluminum atom, AlP crystal, and isolated phosphorous atom. Energy levels for the isolated atoms are from [70]. Energy levels for AlP are a rough illustration and not from actual data.
    6.2.5.png
    Figure \(\PageIndex{5}\): Energy level diagram of a semiconductor zoomed in to show only the conduction and valence band.

    Energy levels due to electrons shared amongst atoms in a solid semiconductor are called energy bands. The filled energy level closest to the top of an energy level diagram for a semiconductor is called the valence band. The energy level above it is called the conduction band. The energy gap \(E_g\), also called the bandgap, is the energy difference from the top of the valence band to the bottom of the conduction band. The term valence electron refers to an outer shell electron while the term valence band refers to a possible energy level it may occupy. At \(T = 0\) K, the valence band is typically filled, and the conduction band may be empty or partially empty. We often are only interested in the valence and conduction bands because we are interested in energy conversion processes involving small amounts of energy. For this reason, we often plot energy level diagrams zoomed in vertically to just show these two energy levels as shown in Fig. \(\PageIndex{5}\).

    If the AlP crystal has defects or impurities, the energy levels broaden a bit because the electrical potential (in volts) seen by each Al and each P atom is slightly different from the potential seen by other Al and P atoms in the crystal. Thus, it takes slightly different amounts of energy to rip off each electron. For this reason, energy levels in amorphous materials are quite a bit broader than energy levels in crystals of the same composition [10]. If the AlP crystal has defects or impurities, additional allowed energy levels may be present. Some of these energy levels may even fall within the energy gap.

    Energy Levels of AlP at \(T > 0\) K

    As with isolated atoms, there are two differences between energy levels for crystals such as AlP at \(T > 0\) K compared to at \(T = 0\) K. First, energy levels broaden. Second, some electrons get excited to higher energy levels and quickly, perhaps in a few microseconds, decay back down.

    Definitions of Conductors, Dielectrics, and Semiconductors

    Conductors, dielectrics, and semiconductors were defined in section 1.5.1. Now that we have seen example energy level diagrams, we should revisit these definitions as well as define the term semimetal. In the presence of an applied external voltage, electric field, optical field, or other energy source, valence electrons flow easily in a conductor [10, p. 429] [11, ch. 4]. In a conductor, the conduction band is partially filled with electrons, so there are many available energy states for electrons remaining in the conduction band. With just a little bit of external energy, possibly even from vibrations that naturally occur at \(T > 0\) K, valence electrons flow easily. Inner shell electrons can be ripped off their atoms and flow, but significantly more energy is needed to rip off inner shell than valence electrons.

    In the presence of an applied external voltage, electric field, optical field, or other energy source, electrons do not flow easily in an insulator [10, p. 429] [11, ch. 4]. The valence band is filled and the conduction band is empty. The energy gap between valence band and conduction band in an insulator is typically above 3 eV. A little heat or energy from vibrations is not enough to excite an electron from one allowed energy state to another. If a large enough external source of energy is applied, though, an electron can be excited or ripped off of an insulator.

    In Sec. 3.3, electro-optic materials were discussed. Some insulators are electro-optic which means that in the presence of an external electric or optical field, the spatial distribution of electrons changes slightly which cause a material polarization to build up. Photons of the external electric or optical field in this case do not have enough energy to excite electrons in the insulator, so the internal momentum of electrons in the material does not change. The electro-optic effect occurs in insulators and involves external energies too small to excite electrons from one allowed energy state to another while the affects discussed in Sec. 6.2 involve semiconductors and external energies large enough to excite electrons from one energy level to another.

    At \(T = 0\) K in a semiconductor, the valence band is full, and the conduction band is empty. The energy gap of a semiconductor is small, in the range \(0.5 eV \lesssim E_g \lesssim 3 eV\). In the presence of a small applied voltage, electric field, or optical field, a semiconductor acts as an insulator. In the presence of a large applied voltage or other energy source, a semiconductor acts as a conductor, and electrons flow. Photodiodes and solar cells are made from semiconductors. If enough energy is supplied to a photodiode, for example from an optical beam, the valence electrons will flow. More specifically, the photons of the external optical beam must have more energy than the energy gap of the semiconductor for the valence electrons to flow.

    The term semimetal is used to describe conductors with low electron concentration. Similar to conductors, in a semimetal at \(T = 0\) K, there is no energy gap because the conduction band is partially filled with electrons, and there are plenty of available energy states. The concentration of electrons for semimetals, however, is in the range \(n < 10^{22} \frac{electrons}{cm^3}\) while \(n\) is greater for conductors [26, p. 304].

    Why Are Solar Cells and Photodetectors Made from Semiconductors?

    Energy level diagrams for AlP were illustrated above. The energy gap of AlP is \(E_g = 2.45 eV\), so it is a semiconductor [9] [10, p. 432,543]. If a beam of light with photons of energy \(E < 2.45 eV\) is applied to a piece of AlP, the photons will not be absorbed, and no electrons will be excited. If a beam of light with photons of energy \(E \geq 2.45 eV\) is applied to a piece of AlP, some of those photons may be absorbed. When a photon is absorbed, an electron will be excited from the valence band to the conduction band. A blue photon with energy \(E = 3.1 eV\) will be absorbed by AlP, for example, but a red photon with energy \(E = 1.9 eV\) will not. When the electron is excited, the internal momentum of the electron necessarily changes. The excited electron quickly spontaneously decays back to its lowest energy state, and it may emit a photon or a phonon in the process. If a beam of light with photons of significantly higher energy is applied to a piece of AlP, it is possible to rip off electrons entirely from their atom.

    Why are solar cells and optical photodetectors made from semiconductors instead of insulators? Sunlight is composed of light at multiple wavelengths, and it is most intense at wavelengths that correspond to yellow and green light. Green photons have energies near \(E \approx 2.2 eV\), and visible photons have energies in the range \(1.9 eV < E < 3.1 eV\). Solar cells are made from materials with an energy gap less than the energy of most of the photons from sunlight. Semiconductors are used because the energy of each photon is large enough to excite the electrons in the material. Insulators are not used because most of the photons of visible light do not have enough energy to excite electrons in the material. The material should not have an energy gap that is too large otherwise photons will not be absorbed.

    6.2.1T.png

    Table \(\PageIndex{1}\): Energy gap of various semiconductors.

    Why are solar cells and optical photodetectors made from semiconductors instead of conductors? When light shines on a solar cell or photodetector, photons of light are absorbed by the material. If the photon absorbed has energy greater than the energy gap of the material, the electron quickly decays to the top of the conduction band. With some more time, it decays back to the lowest energy state. In a solar cell or photodetector, a pn junction is used to cause the electrons to flow before decaying back to the ground state. The amount of energy converted to electricity per excited electron depends on the energy gap of the material, not the energy of the incoming photon. Only energy \(E_g\) per photon absorbed is converted to electricity regardless of the original energy of the photon. Thus, the energy gap of the material used to make a solar cell or photodetector should be large so that as much energy per excited electron is converted to electricity as possible. The material should not have an energy gap that is too small otherwise very little of the energy will be converted to electricity. The electron and hole will release the excess energy, \(hf - E_g\), quickly in the form of heat or lattice vibrations called phonons.

    Each semiconductor has a different energy gap \(E_g\). Many solar cells and photodetectors are made from silicon, which is a semiconductor with \(E_g = 1.1 eV\). Predicting the energy gap of a material is quite difficult. However, all else equal, if an element of a semiconductor is replaced with one below it in the periodic table, the energy gap tends to get smaller. This trend is illustrated in Table \(\PageIndex{1}\). Data for the table comes from [9]. This trend is also illustrated in Fig. \(\PageIndex{6}\), which plots the energy gap and lattice constant for various semiconductors. Figure \(\PageIndex{6}\) is taken from reference [71]. The horizontal axis represents the interatomic spacing in units of angstroms, where one angstrom equals \(10^{10}\) meters. The vertical axis represents the energy gap in eV. This figure illustrates energy gaps and lattice constants for materials of a wide range of compositions. For example, the energy gap for aluminum phosphide can be found from the point labeled AlP, and the energy gap of aluminum arsenide can be found from the point labeled AlAs. Energy gap for semiconductors of composition \(\text{AlAs}_xP_{1−x}\) can be found from the line between these points.

    6.2.6.png
    Figure \(\PageIndex{6}\): Energy gap versus interatomic spacing for multiple semiconductors. Used with permission from [71].

    Some solar cells are made from layered material with the largest energy gap material on the top. For example, a solar cell could be made from a top layer of ZnS, a middle layer of ZnSe, and a bottom layer of ZnTe. Photons with energy \(E > 3.6 eV\) would be absorbed in the ZnS layer. Photons with energy \(2.7 eV< E <3.6 eV\) would be absorbed by the ZnSe layer, and photons with energy \(2.25 eV< E <2.7 eV\) would be absorbed by the ZnTe layer. Each photon of energy absorbed by the ZnS layer and converted to electricity would have more energy than each photon absorbed by the ZnSe layer. Solar cells made from layers in this way can be more efficient at converting energy from optical energy to electricity than equivalent solar cells made of a single material.

    The photo in Fig. \(\PageIndex{7}\) shows naturally occurring zinc sulfide, also called sphalerite, collected near Sheffler's Rock shop near Alexandria, Missouri. The dark mineral embedded in the middle of the rock is the sphalerite.

    6.2.7.png
    Figure \(\PageIndex{7}\): The dark mineral embedded in the rock is naturally occurring zinc sulfide.

    Electron Energy Distribution

    The Fermi energy level of a semiconductor, denoted \(E_f\), represents the energy level at which the probability of finding an electron is one half [9] [10, p. 432,543]. The Fermi level depends on temperature, and it depends on the impurities in the semiconductor. Chemists sometime call the Fermi level by the name chemical potential, \(\mu_{chem}\).

    In a pure semiconductor at \(T = 0 \) K, all electrons occupy the lowest possible states. The valence band is completely filled, and the conduction band is completely empty. The Fermi level, \(E_f\), is the energy level at the middle of the energy gap. No electrons are found at energy \(E_f\) because no electrons can have an energy inside the energy gap. However, the Fermi level is a useful measure to describe the material.

    In a pure semiconductor at \(T > 0\) K, some electrons are excited into higher energy levels. As the temperature increases, more electrons are likely to be found at higher energy levels more often. The probability that an electron is in energy level \(E\) varies with temperature as \(e^{-E/k_BT}\) [9] [10]. The quantity \(kB\) is the Boltzmann constant.

    \[k_B = 1.381 \cdot 10^{-23}\frac{J}{K} = 8.617 \cdot 10^{-5} \frac{eV}{K} \nonumber \]

    The Fermi level for a material with \(T > 0\) K is slightly higher than the Fermi level for a material with \(T = 0\) K because more electrons are likely to be excited.

    The probability of finding an electron at energy level \(E\) at temperature \(T\) is

    \[F(E,T) = \frac{1}{1 + e^{(E-E_f)/k_BT}}. \label{6.2.2} \]

    Equation \ref{6.2.2} is called the Fermi Dirac distribution, and like any probability, it ranges \(0 \leq F \leq 1\). For energy levels far above the conduction band, (\(E − E_f\)) is large and positive, so electrons are quite unlikely to be found, \(F \approx 0\). For energy levels far below the valence band, (\(E − E_f\)) is large and negative, so electrons are quite likely to be found, \(F \approx 1\).

    The concentration and type of impurities influence the energy of the Fermi level. A p-type material has a lack of electrons. For this reason in a p-type material, \(E_f\) is closer to the valence band than the middle of the energy gap. An n-type material has an excess of electrons. For this reason in a n-type material, \(E_f\) is closer to the conduction band.


    This page titled 6.2: Semiconductors and Energy Level Diagrams is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.