V Semiconductors Optics and Optoelectronics I

V-1 States of Matter: Order

The physical matter can be characterized as solid, liquid and gaseous as shown in figure V-1-1.

Figure V-1-1: The different states of matter and the classification of materials according to the ordered nature of the arrangement of atoms.

One way to characterize these states of matter is by the density and the hardness of the material and the ability of the matter to flow. Thus, a solid is usually dense with atomic spacing of a few Angstroms (10^-8 cm). A solid is also hard and has his its own shape, in contrast to a liquid or a gas, which takes the shape of the container it is in.

For our purpose, a more useful way of characterizing materials is by the presence of order in the collection of atoms. While order can be defined precisely through mathematical expressions for the correlation functions for the arrangements of atoms, we will simply use a physical and intuitive definition of order. Let us imagine we can actually look at the atoms inside a material. We may find that in some cases the atoms are arranged in complete precision so that by knowing the position and species of a few atoms we can predict the position and chemical nature of all the atoms in the sample. When this occurs, we say that the structure has long range order. Such a long range order is only found in solids and such solids are called crystalline.

In some materials we may find that the precise arrangement of atoms exists over thousands of atoms, but these ordered regions are separated by a boundary without an order. Such solids are called polycrystalline materials. The size of the region across which order persists is called the grain size and, typically, the grain size is a few microns.

There is yet another class of solids in which, if we examine a particular atom, we find that the neighboring atoms (nearest neighbors or second nearest neighbors) are precisely arranged, but as one moves further out, the arrangement becomes less and less predictable. This kind of order is called short ranged order. Such materials are called amorphous materials. Often the term “glass” is used for such non-crystalline materials.

Matter that falls in the category of liquid or gaseous phase has no short (or long) order. The atoms have an average spacing between them determined by the density of the material, but there is no precise arrangement of the atoms.

Another important phase of matter is the liquid crystal, which is an important material system for display technology. The liquid crystals have the ability to flow and take the shape of the container just like any other liquid. However, atoms or molecules do have a certain long range order, at least in some direction. The attraction of liquid crystals in optoelectronics is due to the fact that this long range order can be broken by an electric field. Since the optical properties are dependent on the order of the atom arrangement, one can alter the optical properties by an electric field.

V-2 Crystalline Materials

Most of the high performance optoelectronic and electronic devices are made from crystalline materials, where long range order is present among the atoms. Semiconductors form the most important class of materials used in optoelectronics and, as shown in figure V-2-1, essentially all types of device can be fabricated from semiconductors. We will discuss the important electronic and optical properties of these materials in later chapters. Certain kinds of materials are called ferroelectrics and some dielectrics are also used in modern optoelectronics. We will now discuss some general properties of crystalline materials and will then focus on the specific structural properties of important optoelectronic crystals.

Figure V-2-1: In solid state optoelectronics crystalline semiconductors, ferroelectrics and dielectrics are widely used.

V-3 Periodicity of a crystal

Crystals are made up of identical building blocks being an atom or a group of atoms. The underlying periodicity of crystals is the key, which controls the properties of the electrons inside the material. Thus, by altering crystalline structure artificially, one is able to alter the electronic properties.

To understand and define the crystal structure, two important concepts are introduced. The lattice represents a set of points in space forming a periodic structure. The lattice is by itself a mathematical abstraction. A building block of atoms called the basis is then attached to each lattice point yielding the crystal structure.

An important property of a lattice is the ability to define three vectors a₁, a₂, a₃, such that any lattice point r’ can be obtained from any other lattice point r by the translation

r’=r+m₁a₁+m₂a₂+m₃a₃ (V-3-1)

where m₁, m₂, m₃ are integers and T= m₁a₁+m₂a₂+m₃a₃ is the translation vector. Such a lattice is called Bravais lattice. Figure V-2-2 visualizes equation (V-3-1).

Figure V-2-2: Visualization of equation (V-3-1). The arrangement of the atoms in the crystal looks the same for an observer at r’ and r. In the example T=-a₁+3a₂.

The entire lattice can be generated by choosing all possible combinations of the integers m₁, m₂, m₃. The crystalline structure is now produced by attaching the basis to each of these lattice points: lattice+basis=crystal structure. The translation vectors a₁, a₂, a₃ are called primitive if the volume of the cell formed by them is the smallest possible. There is no unique way to choose the primitive vectors. One choice is to pick a₁ to be the shortest period of the lattice, a₂ to be the shortest period not parallel to a₁, and a₃ to be the shortest period not coplanar with a₁ and a₂. It is possible to define more than one set of primitive vectors for a given lattice, and often the choice depends upon convenience. The volume cell enclosed by the primitive vectors is called primitive unit cell.

Because of the periodicity of a lattice, it is useful to define the symmetry of the structure. The symmetry is defined via a set of point group operations, which involve a set of operations applied around a point, such as rotation, reflection and inversion. The symmetry plays a very important role in the electronic properties of the crystal. For example, the inversion symmetry is extremely important and many physical properties of semiconductors are tied to the absence of this symmetry. In the diamond structure (Si, Ge, C, etc.), inversion symmetry is present, while in the zincblende structure (GaAs, AlAs, InAs, etc.), it is absent. Because of this lack of inversion symmetry these semiconductors are piezoelectric, i.e., when they are strained an electric potential is developed across the opposite faces of the crystal.

V-4 Basic Lattice Types

There are 14 types of lattices in three dimensions. These lattice classes are defined by the relationships between the primitive vectors a₁, a₂, and a₃, and the angles a, b, and g. The general lattice is triclinic (a¹b¹g and a₁¹a₂¹a₃) and there are 13 special lattices. Table V-4-1 provides the basic properties of these three dimensional lattices.

Table V-4-1: The 14 Bravais lattices and their properties.

There are three kinds of cubic lattices, simple cubic, body centered cubic (bcc), and face centered cubic (fcc).

Simple cubic: The simple cubic lattice shown in figure V-4-1 is generated by the primitive vectors ax, ay, az, where x, y, and z are the unit vectors.

Figure V-4-1: Simple cubic lattice showing the primitive vectors. Repeating the cubic cell through space provides the crystal.

Body centered cubic: The bcc lattice shown in figure V-4-2 can be generated from the simple cubic structure by palcing a lattice point at the center of the cube. If x, y, and z are three orthogonal unit vectors, then a set of primitive vectors for the bbc could be a₁=ax, a₂=ay, and a₃=(a/2)(x+y+z).

Figure V-4-2: The body centered cubic lattice along with the choice of primitive vectors.

Face Centered Cubic: Another equally important lattice for semiconductors is the fcc Bravais lattice. To construct the face centered cubic Bravais lattice add to the simple cubic lattice an additional point in the center of each square face (figure V-4-3). A symmetric set of primitive vectors for the fcc cubic lattice is a₁=(a/2)(y+z), a₂=(a/2)(z+x), a₃=(a/2)(x+y).

Figure V-4-3: Primitive basis vectors for the face centered cubic lattice.

The bcc and fcc Bravais lattices are of great importance, since an enormous variety of solids crystallize in these forms with an atom (or ion) at each lattice site. Essentially most of the semiconductors of interest for electronics and optoelectronic have fcc structure.

The diamond and zincblende structures: Figure V-4-4 visualizes the lattices of the most important semiconductors, silicon (Si) and Gallium arsenide (GaAs), which crystallize in the diamond and zincblende lattice, respectively. They have two atoms per basis. The coordinates of the basis atoms are (0, 0, 0) and (a/4, a/4, a/4). The lattice belongs to the tetrahedral phases; that is, each atom is surrounded by four equidistant nearest neighbors, which lie at the corners of a tetrahedron. The diamond and the zincblende lattices can be considered as two interpenetrating fcc lattices.

Figure V-4-4: Diamond and zincblende structure. For the diamond lattice, such as Si, all the atoms are Si; in a zincblende lattice, such as GaAs, one sublattice is gallium and the other is arsenic.

GaAs is a III-V compound, since it is formed from elements of groups III and V of the periodic table. Most III-V compounds crystallize in the zincblende structure; however, many semiconductors (including some III-V compounds) crystallize in the wurtzite or rock-alt structure.

Wurtzite (hexagonal) structure: Figure V-4-5 shows the wurtzite lattice, which can be considered as two interpenetrating close-packed lattices (e.g. the sublattices of cadmium and sulfur). The wurtzite structure has a tetrahedral arrangement of four equidistant nearest neighbors, similar to a zincblende structure. Note that some semiconductor compounds, such as ZnS and CdS, can crystallize in both zincblende and wurtzite structure. We should stress that the wurtzite structure exhibits, in contrast to the diamond structure, an optic axis (c-axis), i.e. semiconductors with wurtzite structure show birefringence and dichroism.

In other words, for CdS, the direction of the polarization of the impinging light beam is important for the energy position of the absorption edge. The absorption edge of GaAs, on the other hand, does not show a dependence on the polarization. We say GaAs is an isotropic material whereas CdS is an anisotropic material.

Figure V-4-5: The wurtzite structure.

V-5 The Reciprocal Lattice

If one deals with the real lattice, the interaction of waves (light) and matter is expressed in terms of wavelength and the lattice constant (space). It is useful to replace these parameters by wave vector (see equation I-1-2 and I-1-3) and reciprocal lattice for the theoretical treatment of many problems in solid state physics. The x-ray patterns of a solid, for example, correspond to the reciprocal lattice of the material.

The reciprocal vector is given by b₁=1/a₁. Multiplying the right side with the vector product a₂´a₃, one gets the reciprocal lattice vectors

, (V-5-1)

so that a₁b₁=2p and a₁b₂=0 and so on. The factors 2p are not used by crystallographers but are convenient in solid state physics. If a₁, a₂, and a₃ are primitive vectors of the lattice, then b₁, b₂, and b₃ are the primitive vectors of the reciprocal lattice. The general reciprocal lattice vector is given by

G=hb₁+kb₂+lb₃, (V-5-2)

where, h, k, and l are integers.

A simple scheme is used to describe lattice planes, directions and points. For a plane we use the following procedure. (1) Define the x, y, z primitive vectors. (2) Take the intercepts of the plane along the axes in units of lattice constants. (3) Take the reciprocal of the intercepts and reduce them to the smallest integers having the same ratio. The result, enclosed in parentheses (hkl), is called the index of the plane and may denote a single plane or a set of parallel planes. In a cubic system, the Miller indices of a plane are the same as the directions perpendicular to the plane. The notation [] is for a set of parallel directions; < > is for a set of equivalent directions. Figure V-5-1 shows how some important planes in the cubic system along with their Miller indices.

Figure V-5-1: Miller indices of some important planes in the cubic crystal.

The unit cell of a reciprocal lattice can be represented by a Wigner-Seitz cell (Brillouin zone). The cell is constructed by drawing perpendicular bisector planes in the reciprocal lattice from the chosen center to the nearest equivalent reciprocal lattice sites. Figure V-5-2 shows a typical example for an fcc structure. If one first draws lines from the center point (G) the eight corners of the cube, then forms the bisector planes, the result is the truncated octahedron with in the cube – a Wigner-Seitz cell.

Figure V-5-2: Brillouin zone for diamond and zincblende lattice.

The first Brillouin zone of one dimension is shown in figure V-5-3. The zone boundaries of the linear lattice are k=±p/a, where a is the primitive axis of the crystal lattice.

Figure V-5-3: Crystal and reciprocal lattices in one dimension.

Example: The lattice constant of Si is 5.43´10-10 m. Calculate the number of of Si atoms in a cubic centimeter.

Solution: Si has a diamond structure, which is made up of the fcc lattice with two atoms on each lattice point. The fcc unit cube has a volume of a³. The cube has eight lattice sides at the cube edges. However, each of this point is shared with eight other cubes. In addition, there are six lattice points on the cube face centers. Two adjacent cubes share each of these points. Thus, the number of lattice points per cube is of volume a³ are

.

We have two atoms per lattice point. Therefore, the number of atoms in a cubic centimeter is

.

V-6 Technology Issues

a) Bulk Crystal Growth

Bulk crystal growth techniques are used mainly to produce substrates on which devises are eventually fabricated. While for some semiconductors like Si, GaAs and InP (and lately also GaN) the growth technologies are highly matured, for most other semiconductors it is difficult to obtain high quality, large area substrates. The aim of the bulk crystal growth techniques is to produce single crystal boules with as large a diameter as possible and with a few defects as possible. Common defects in a semiconductor are shown schematically in figure V-6-1. In Si the boule diameters have reached 30 cm with boule length approaching 100 cm. Large size substrates ensure low cost device production.

For the growth of boules from which substrates are obtained, one starts out with a purified form of elements that are to make up the crystal. One important technique that is used is the Czochralski (CZ) technique. In the CZ technique shown in figure V-6-2, the melt of the charge (i.e., the polycrystalline material) is held in a vertical position crucible. The top surface of the melt is just barely above the melting temperature. A seed crystal is then lowered into the melt and slowly withdrawn. As the heat from the melt flows up to the seed, the melt surface cools and the crystal begins to grow.

Figure V-6-1: A schematic showing some point defects in a crystal.

Figure V-6-2: Czochralski-style grower used for substrate ingots. The technique is widely used for Si, GaAs and InP.

In the case of GaAs and InP the CZ technique has to fac the problem arising from the very high pressures of As and P at the melting temperature of the compounds. Not only does the chamber have to withstand the pressures, also the As and P leave the melt and condense on the sidewalls. To avoid the second problem one seals the melt by covering it with molten layer of a second material (e.g., boron oxide), which floats on the surface. The technique is then referred to as liquid encapsulated Czochralski (LEC).

A second bulk crystal growth method technique involves a charge of material loaded in a quartz container. The charge may be composed of either high quality polycrystalline material or carefully measured quantities of elements, which make up a compound crystal. The container called “boat” is heated till the charge melts and wets the seed crystal. The seed is then used to crystallize the melt by slowly lowering the boat temperature starting from the seed end. In the Bridgeman approach shown in figure V-6-3, the boat is kept stationary while the furnace temperature is temporally varied to form the crystal.

Figure V-6-3: Crystal growing from the melt in a crucible: Solidification from the end of the melt (Bridgeman method).

Table V-6-1 shows the representative impurity concentrations in as-grown Si and GaAs single crystals. Note that an impurity concentration of 5´10¹⁶ cm^-3 corresponds to about one impurity atom per 10⁶ atoms of the parent semiconductor.

Table V-6-1: Impurities in as-grown Si and GaAs single crystals.

b) Epitaxial Growth

The substrates grown by the aforementioned techniques are almost never used directly for devices. Invariably an epitaxial layer or epilayer is grown of a few mm thickness. The epitaxial growth techniques have a very slow growth rate (as low as a monolayer per second for some techniques), which allow one to control very accurately the dimension in the growth directions. In fact, techniques like molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD) achieve monolayer (3´10^-10 m) control in the growth direction. This level of control is essential for the variety of heterostructure devices that are being used in optoelectronics. Table V-6-2 gives a brief view of various epitaxial techniques used.

Table V-6-2: Various epitaxial crystal growth techniques and some of their positive and negative aspects.

In the following we briefly discuss the different techniques.

Liquid Phase Epitaxy (LPE):

LPE was an epitaxial technique of choice until the 70’s when it gradually started to be replaced by other techniques. LPE is still in use for the growth of crystals such as, HgCdTe for long wavelength detectors and AlGaAs for double heterostructure laser. Figure V-6-4 shows the principle of LPE. The

Figure V-6-4: Schematic of the LPE growth of AlGaAs and GaAs.

substrate is placed in a quartz or a graphite boat and covered by a liquid of the crystal to be grown. The slider moves the substrate, thus positioning itself to achieve contact with the different melts to grow heterostructures. By precise control of the liquid composition and temperature, the allow composition can be controlled. Because LPE is very close to equilibrium growth technique, it is difficult to grow allow systems, which are not miscible or even grow heterostructures whit atomically abrupt interfaces. Nevertheless for many applications, quality and interface extension of LPE heterostructures are sufficient and the technique is used widely in many commercial applications.

Vapor Phase Technique (VPE):

The VPE technique is used mainly for homoepitaxy and does not have the additional apparatus present in techniques as MOCVD for precise heteroepitaxy. As an example of the technique, consider VPE of Si. The Si containing reactant silane (SiH₄) or dichlorosilane (SiH₂Cl₂) or trichlorosilane (SiHCl₃) or silicon tetrachloride (SiCl₄) is diluted in hydrogen and introduced into a reactor in which heated substrates are placed as shown in figure V-6-5.

Figure V-6-5: Reactors for VPE growth. The substrate temperature must be maintained uniformly over the area. This is achieved better by lamp heating. A pyrometer is measured for temperature measurements.

The silane pyrolysis to yield silicon while the chlorine containing gases react to give SiCl₂, HCl and various other silicon-hydrogen-chlorine compounds. The reaction 2SiCl₂®Si+SiCl₄ then yields to Si.

An important consideration in VPE is safety related since hydrogen, which is produced in the deposition can explode in contact with any oxygen. Also, all the reactants are highly toxic.

Molecular Beam Epitaxy

MBE is one of the most important epitaxial techniques as far as heterostructure physics and devices are concerned. Almost every semiconductor has been grow by this technique. MBE is a high vacuum technique (»10^-11 torr) in which crucibles containing a variety of elemental charges are placed in the growth chamber. The elements contained in the crucibles make up the components of the crystal to be grown as well as the dopants that may be used. When a crucible is heated, atoms or molecules of the charge are evaporated and these travel in straight lines to impinge on a heated substrate.

The growth rate in MBE is about one monolayer per second. This slow rate coupled with shutters placed in front of the crucibles allows one to switch the composition of the growing crystal with monolayer control. Hence, the so-called delta doping is possible with MBE. A typical MBE machine is shown in figure V-6-6.

Figure V-6-6: A schematic of the MBE growth system.

Since no chemical reactions occur in MBE, the growth is the “simplest” of all epitaxial techniques and quite controllable. However, since the growth involves high vacuum, leaks can be a major problem. The growth chamber walls are usually cooled by liquid N₂ to ensure high vacuum and to prevent atoms/molecules to come off from the chamber walls.

The low background pressure in MBE allows one to uses electron beams to monitor the growing crystal. The reflection high energy electron diffraction (RHEED) techniques relies on electron diffraction to monitor both the quality of the growing layer.

MBE is a relatively safe technique and has become the technique of choice for the testing of almost all new ideas on heterostructure physics.

Metal Organic Chemical Vapor Deposition (MOCVD)

MOCVD is another important growth technique widely used for heteroepitaxy. Like MBE, it is also capable of producing monolayers and abrupt interfaces between semiconductors. A typical MOCVD system is shown in figure V-6-7.

Figure V-6-7: Schematic diagram of an MOCVD system.

Unlike MBE, the gases that are used in MOCVD are not made of single elements but are complex molecules, which contain elements like Ga or As to form the crystal. Hence, the growth depends upon the chemical reactions occurring at the heated substrate surface. For example, in the growth of GaAs one often uses Triethyl Gallium and Arsine and the crystal growth depends on the following reaction, Ga(CH₃)+AsH₃®GaAs+3CH₄.

One advantage of the growth occurring via a chemical reaction is that one can use lateral temperature control to carry out local area growth. Laser assisted local area growth is also possible for some materials and can be used to produce new kind of device structures. Such local area growth is difficult with MBE.

The use of MOCVD requires very serious safety precautions. The gases used are highly toxic and many safety features have to be incorporate to avoid accidents.

In addition to MBE and MOCVD one has hybrid epitaxial techniques often called MOMBE (metal organic MBE), which tries to combine the best of MBE and MOCVD. In MBE, the chamber has to be opened to load the charge for the materials to be grown while this is avoided in MOCVD where gas bottles can be easily changed. Additionally, in MBE one has occasional “spitting” of material in which small clumps of atoms are evaporated off on to the substrate. In MOCVD and MOMBE no spitting takes place.

V-7 Thin Film Production

So far we spoke about the preparation of “thin films” on semiconductor substrates, i.e., the production of epilayers. A very important branch of semiconductor technology is the preparation of thin films on amorphous substrates.

a) Vacuum evaporation

Evaporation is the simplest technique for preparing thin films with a few mm thickness. The evaporation takes place in a vacuum (»10^-5 torr) chamber, maintaining a few cm distance between the heated source (polycrystalline powder of the semiconductor to prepare) and the substrate. This method, however affects strongly the stoichiometry of the material. Therefore, films formed by evaporation are not very useful for optoelectronic applications, they are, however, extremely smooth and might be useful for applications where “only” optical features are important as for nonreflecting glasses.

b) Closed-Spaced Vapor Transport (CSVT)

CSVT is based on the physical transport of e.g. CdS in an Ar atmosphere from a CdS source to a substrate. For the source polycrystalline CdS powder is used. With this method one should expect extremely homogeneous, less disordered layers with much better stoichiometry than these films formed by evaporation. The CSVT setup is shown in figure V-7-1. The source and the target are located in graphite blocks keeping a distance of 6 mm between them. Each of the blocks is heated with 500 W quartz halogen lamps. The temperature of the source and substrate is 725^oC and 600^oC, respectively. Growth occurs in Ar atmosphere of 20 torr.

Figure V-7-1: CSVT setup. Two separately tunable 500 W halogen lamps are used as heat sources. The substrate and the source are placed in the evaporation chamber where the growth takes place.

We should stress that with this technique it is possible to visualize the hexagonal lattice of CdS. Figure V-7-2 shows the surface image of thin film CdS grown by CSVT. The picture was done with a simple microscope. Nevertheless, hexagons are clearly seen.

Figure V-7-3: The surface of thin film CdS grown by CSVT.

CSVT suffers from mechanical problems. The films show the tendency to peel off and frequently cracks disturb the homogeneity of the films. The appearance of the latter are a problem for optical and optoelectronical applications.

c) Spray-Deposition

This technique uses a chemical reaction to form the film. In case of CdS the following reaction can be used, CdCl₂+SC(NH₂)₂+2H₂O ® 380^oC ®CdS(s)+CO₂(g)+2NH₄Cl(g), where (s) and (g) denote solid and gaseous. Figure V-7-4 shows the schematic of the spray-depositing setup.

Figure V-7-4: Schematic of the spray-depositing setup. The temperature is controlled with a Pt-PtRh thermocouple, providing feedback to the power supply. Thin film CdS is deposited on Pyrex through a pressure-regulated spray nozzle.

The optoelectronic features of the films are very good, though the crystallinity and smoothness of the surface is not the same as for films deposited by CSVT and evaporation, respectively. Films are useful for optical detectors and bistable devices.

d) Laser Ablation or Pulsed-laser Deposition (PLD)

PLD has emerged as a physical deposition technique delivering high quality thin films of chemically and structurally complex materials. A laser beam is used to evaporate the source material, which is, as in case of common vacuum evaporation, polycrystalline powder. Lately, considerable progress has been reported by demonstrating that thin film CdS formed by PLD shows outstanding emission and even laser features.

Figure V-7-5 shows laser ablation in action. The laser beam hits the target and causes a plume (highly ionized gas), which “transports” the particles to the substrate where form the thin film. PLD requires in many cases vacuum (»10^-4 torr), though it can be performed at ambient pressure. In this case the film might be doped by oxygen. The versatility of PLD exceeds by far that of all aforementioned preparation methods because the film features can be changed considerably by simply changing the laser wavelength or the so-called fluence (energy density J/cm²) of the laser beam used for the ablation process. In this context one speaks about ultraviolet (UV) and infrared (IR) PLD.

Figure V-7-6: PLD in action.

Figure V-7-7 shows the schematic of a laser ablation system. The laser beam is guided on the target (CdS) and a film is deposited on the substrate (glass). Various lasers are suitable for the ablation process, the choice depends on the material to be ablated.

Specifically one feature of PLD is highly interesting. It is possible by the variation of the fluence to control the orientation of thin film CdS. As shown in figure V-4-5, CdS exhibits a c-axis. The orientation of this axis rotates with respect to the substrate surface from a perpendicular to a parallel orientation if the fluence increases. Figure V-7-8 shows the x-ray patterns of CdS films formed at 355 nm (UV-PLD) with a fluence from 2-4 J cm^-2. From x-ray patterns the orientation of a solid can be determined. The pronounced appearance of the (002) peak indicates perpendicular orientation, whereas the (100) and (110) peaks indicate that the c-axis “lies” on the substrate.

Figure V-7-7: PLD setup.

Figure V-7-8: Orientation control of thin film CdS via fluence variations.

It was pointed out that in the ablated plume at energy densities ³1 J/cm2 the amount of fast particles (atoms and ions) decreases and that of slow particles (cluster and droplets) increases. The clusters coming to the surface stick with the smoothest and largest area, i.e., the cleavage, on the substrate. Since the cleavage is parallel to the c-axis, the film grows parallel to the surface at increased fluence.

V-8 Electronic States: Energy Bands (first approach)

Figure V-8-1 shows the band structure for a semiconductor in the real space. As mentioned in chapter III, the valence band is filled with electrons and the conduction band, separated by the gap energy w_g, is empty. The band diagram in real space is not useful since many important features of the semiconductor are not seen.

Figure V-8-1: Space band diagram of a semiconductor.

The simplest approach of the bandstructure is to write down the relation ship between energy and momentum of an electron introducing the k space,

, (V-8-1)

where m_e* is the so-called effective mass of the “free” electron, is Planck’s constant divided by 2p, and k is the wave number. Figure V-8-2 shows the correspondent band structures of a direct (a) and indirect (b) semiconductor. The transition – e.g. induced by an impinging photon - from an electron from the conduction band to the valence band in the direct semiconductors occurs with k conservation (E=w_g). On the other hand, in the indirect semiconductor, the transition conduction-to-valence band requires an additional energy W (E=w_g+W). The lattice vibrations called phonons provide this additional energy. The picture makes it plausible why indirect semiconductors are less suitable for applications to optoelectronic devices. Indirect semiconductors exhibit rather weak oscillator strength in comparison to a direct semiconductor between the conduction and valence band and exhibit therefore poor recombination features, which result in poor emission features. As a consequence, indirect semiconductors (Si) do not show lasing features.

Energy

Figure V-8-2: According to equation (V-8-1) the bandstructure is basically a parabola in the k space. (a) The lowest point of the conduction band occurs at the same value of k as the highest point of the valence band. (b) Te minimum of the conduction band and the maximum of the valence band do not occur at the same k.

Note that in figure V-8-2 an electron-hole pair is generated. Figure V-8-3 shows the every missing electron is associated with a hole of positive elementary charge.

Figure V-8-2: Diagram illustrating that every missing electron is associated with a hole.

Holes behave concerning transport features like electrons through the lattice though the exhibit a different effective mass (m_h*). The latter feature is discussed in the next point. Figure V-8-3 shows thee band structures of Si and GaAs. One sees formula V-8-1 describes the situation very well.

Figure V-8-3: Band structure of (a) Si and (b) GaAs.

V-9 Basic Considerations to the Carrier Transport in Semiconductors

Mass, like charge, is another very basic property of electrons and holes. Unlike charge however, the carrier mass is not a simple property. Seeking to obtain insight into the effective mass concept, let us first consider the motion of electrons in vacuum. If, as illustrated in figure V-9-1 (a), an electron of rest mass m0 is moving in a vacuum between two parallel plates under the influence of an electric field F_el, then, according to Newton’s second law, the force F on the electron will be

F=-qF_el=m₀dv/dt, (V-9-1)

where q is the elementary charge and v is the velocity of the electron. Does equation (V-9-1) describe the the overall motion of an electron in a semiconductor? The answer is clearly no!

Figure V-9-1: An electron moving in response to an applied electric field (a) within the vacuum (or air), and (b) within a semiconductor.

Strictly speaking, the motion of an electron in a solid can only be described using quantum mechanics. Fortunately, however, if one examines the carrier motion occurring between collisions, themathematically complex quantum-mechanical formulation simplifies to yield an equation of motion identical to equation (V-9-1), except mo is replaced by an effective carrier mass. In other words, for figure V-9-1 (b) we write

F=-qF_el=m_n*dv/dt. (V-9-1)

An equivalent formula can be written for holes. In each case the internal crystalline fields and quantum-mechanical effects are all suitably lumped into the effective mass factor.

V-10 Doping of Semiconductors

Besides chapter III-4, we spoke about intrinsic (without doping) semiconductors. However, semiconductor devices such as, diode and transistor make use of impurity semiconductors, which are created through the controlled addition of certain impurities to intrinsic semiconductors. This process is called doping. Figure V-10-1 is a schematic illustration of Si doped with arsenic such that arsenic atoms replace a few of the Si atoms. Arsenic has five valence electrons rather than the four of Si. Four of these electrons take part in bonds with the four neighboring Si atoms, and the fifth electron is very loosely bound to the atom. This extra electron occupies an energy level that is just slightly below the conduction band, and it is easily excited into the conduction band, where it can contribute to the electrical conduction. These levels just below the conduction bands are called donor levels because they donate electrons into the conduction band without leaving holes in the valence band. Such a semiconductor is called an n-type semiconductor.

Figure V-10-1: (a) Schematic illustration of Si doped with arsenic. (b) The extra electrons form a donor level below the conduction band.

Another type of impurity semiconductor can be made by replacing a silicon atom with a gallium atom, which has three valence electrons (see figure V-10-2). The gallium accepts electrons from the valence band to complete its four covalent bonds, thus creating a hole in the valence band. The levels formed are called acceptor levels since the impurities accept electrons to from the valence band.

Figure V-10-2: (a) Schematic illustration of Si doped with gallium. (b) The extra holes form a acceptor level below the conduction band.