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Schrodinger's equation in rectangular coordinates inside the well for electrons having constant total energy becomes 2. Since the right-hand side of E q. We assume for t h e sake of simplicity that each of the sides of the cubic well has length a.
The solution to each of t h e equations in C. Possible solutions, wherein the cosine terms d r o p out and, subject to t h e b o u n d a r y conditions in E q. Using the above expressions in E q. In the three coordinate system that contains , , and n , we assume that their m a x i m u m values are N , , and N. That section is o n e for which t h e values of n , , n are all positive. This section forms one eighth of the sphere, shown in Fig.
By substituting Eq. E a c h point inside the section of Fig. I, corresponding to integer values of n , n and n represents an energy state that contains a pair of electrons. Thus the total volume inside this section represents t h e total n u m b e r of states. We can then write x. Gil The question is, what is t h e distribution of the density of the electrons as a function of energy?
We first introduce the following terms defined as N E. Volume The term can be written as C. T h e r a t e of change of t h e density of states becomes s. Thus E q. This is what we defined as N E so that G14 A t absolute zero, electrons occupy the lowest levels possible. Thus one can d e t e r m i n e the m a x i m u m energy that electrons can have at absolute zero from E q.
We label that energy t h e Fermi energy E. O n c e E is fixed, the highest q u a n t u m states at absolute zero are fixed and E is defined as the m a x i m u m energy that an electron in a certain material can have at absolute zero. This commonality exists b e t w e e n the mobility and t h e diffusion constant. To determine the relation b e t w e e n t h e mobility and the diffusion constant, we consider a slab of semiconductor that has b e e n d o p e d to a nonuniform distribution of electrons in a region, as shown in D.
In accordance with E q. For the total electron current to b e zero we have qO dn a. We show, in Fig. F l b the potential distribution in the slab, and in Fig. F l c the energy b a n d diagram. The electron density in Fig. F1 can be expressed as. The electric field intensity and the potential energy of an electron to the potential by. F1 Distribution of electrons, potential and energy versus distance.
BJT: D C ratio of collector to emitter current with collector shorted to base in active region BJT: D C ratio of collector to emitter current with emitter shorted to base in inverse active region. BJT: D C ratio of collector to base current with emitter shorted to base in inverse action region. These advances have b e e n largely responsible for the information revolution, b o t h in the processing and transmission of intelligence.
Materials used in t h e fabrication of large-scale circuits may be classified as: good conductors, insulators, and semiconductors. T h e basic difference b e t w e e n the three is their resistance to current flow, defined in terms of t h e resistivity of the material. Insulators that have resistivities greater than 10 ohm-cm are used as isolators of circuits and devices, and in the formation of capacitors.
B e t w e e n these two limits of resistivities lie the semiconductor materials: the p u r e elemental semiconductors, silicon and germanium, and some II-VI, IV-VI, and III-V compounds, t h e most i m p o r t a n t of which is gallium arsenide. Their mid-range resistivity, or conductivity, is not the direct reason for the importance of semiconductors. Rather, it is the extent to which their properties are influenced by light, temperature, and m o r e importantly, by the addition of minute amounts of special impurities.
Extensive changes take place in the resistivity of silicon when one part of an impurity is added to a million parts of silicon.
Silicon is abundant in nature in the form of sand silica and clay. However, before silicon can b e used in devices, major purification of the material is required. A n application that illustrates the use of the three types of materials in the formation of integrated circuits follows. Integrated circuits are fabricated on 8 to 20cm diameter circular sections of silicon k n o w n as wafers.
T h e major c o m p o n e n t s in integrated circuits are transistors, 3. In the fabrication process, controlled a m o u n t s of particular impurities are a d d e d to the silicon in order to form the various regions of a transis tor. Certain sections of the wafers are oxidized to form silicon dioxide, which serves as an insulator and as an isolator of different circuit parts. After the various devices are formed, a good conductor, such as aluminum, is deposited on the surface to interconnect t h e various parts of the circuits.
Finally, the complete circuit, with external leads attached, is encapsulated in a plastic or ceramic package. O u r objective in this book is to study, first, some of the properties of pure semiconductors and those semiconductors to which impurities are added.
Having studied these properties, we then investigate the operation and the current-voltage characteristics of semiconductor devices, such as diodes and transistors. To do that, we n e e d to consider the internal structure of semiconductors, t h e types of current carriers, and the modes of transport of these carriers. In crystalline solids, the atoms are arranged in an orderly three-dimensional array that is r e p e a t e d throughout t h e structure.
The atoms in polycrystalline solids are so arranged that, within certain sections, s o m e sort of a p a t t e r n of the atoms exists but the various sections are randomly arranged with respect to each other. Most semiconductors are crystalline in nature. Let us look into the internal a r r a n g e m e n t of the atoms of semiconductors in the basic building block k n o w n as t h e unit cell. D i a m o n d is a form of carbon, which is an element in Column I V of the peri odic table.
In the diamond lattice, shown in Fig. The unit cell for t h e d i a m o n d lattice has an a t o m in each corner of the cube 8 , one at the center of each of the six faces 6 , and four 4 internal to the cube located along the diagonals. F r o m this total of 18 atoms, the a t o m at each corner is shared by eight cells and each face a t o m is shared by two cells. We will use the above information in the following examples to calculate the density of atoms and the mass density of silicon.
Figure 1. Reprinted by permission of PrenticeHall, Inc. In this structure, G a and A s atoms are found at alternate locations inside the unit cell. O n close study of the unit cell lattice of Fig. This attachment of atoms, in fact, represents the force that holds the lattice atoms together and is known as covalent bonding. E a c h a t o m of C o l u m n I V elements has four electrons in its outermost shell.
T h e covalent bonding results from the sharing of electrons b e t w e e n atoms. W h e n each atom, say A, is shown b o n d e d to four neighboring atoms, each of t h e four neighboring atoms contributes one electron to the b o n d with A.
A t o m A, therefore, contributes one electron to each bond, so that two electrons are shared by a t o m A with each of the four atoms. This sort of bonding accounts for some physical proper ties of these solids. Eventually, we are interested in determining the current in a semiconductor device in response to the application of a source of energy, such as an electric source or light source.
To d o that, we n e e d to k n o w t h e types and densities of t h e current carri ers and their masses. In a semiconductor, there are b a n d s of energy in which electrons can exist and other b a n d s in which they cannot. We n e e d to establish the bases for the formation of these bands. The existence of b a n d s results from t h e allocation of specific energy levels to an electron in an a t o m and the consequent displacement of these energy levels by introducing t h e effect of t h e forces that atoms exert on each other.
A s a result, each electron in t h e solid has a specific energy level; the combinations of these levels form bands. Before we study the formation of bands, we will present a perspective of the theories that have evolved and have a t t e m p t e d to explain experimentally observed properties of h e a t e d materials.
Classical mechanics' concepts fail to explain these properties as well as all other p h e n o m e n a that take place at the atomic levels. Such explanations required the evolution of the science of Wave Mechanics, which treats electrons as particles that have wave-like properties. These were: blackbody radiation and the sharp discrete spectral lines emitted by h e a t e d gases. First, we will consider blackbody radiations. A blackbody is a h e a t e d solid labeled as an ideal radiator of electromagnetic waves.
W h e n a solid is heated, it emits radiation over a certain b a n d of frequencies. The h e a t e d atoms vibrate so that the amplitudes and frequencies of the vibrations seem to resemble the radiating a n t e n n a of a broadcast station. Classical mechanics predict that this h e a t e d solid emits radiation over a continuous b a n d of frequencies.
M e a s u r e d responses, however, indicate that this is not true. A s a m a t t e r of fact, radi ation takes place only over a certain b a n d of frequencies. A similar p h e n o m e n o n occurs w h e n certain gases are heated.
It was experi mentally observed that h e a t e d gases emit radiation in small discrete quantities at certain discrete spectral wavelengths. Again, Classical mechanics had n o explana tion for this. The conclusion that was reached confirmed that Classical mechanics could not predict p h e n o m e n a that occur at microscopic scales or at atomic levels. Planck's hypothe sis, presented in , was that light is emitted or absorbed in discrete units of energy called photons.
From E. Uiga, Optoelectronics, Prentice Hall
Semiconductor Devices by Kanaan Kano
Schrodinger's equation in rectangular coordinates inside the well for electrons having constant total energy becomes 2. Since the right-hand side of E q. We assume for t h e sake of simplicity that each of the sides of the cubic well has length a. The solution to each of t h e equations in C. Possible solutions, wherein the cosine terms d r o p out and, subject to t h e b o u n d a r y conditions in E q. Using the above expressions in E q. In the three coordinate system that contains , , and n , we assume that their m a x i m u m values are N , , and N.
A mainstream, solid state book that represents a balance in orientation between a strong physics and a strong applications approach. This book has a unique chapter on the fabrication of devices; integrates a wealth of illustrations and pictorial representations; and contains an extensive discussion and analysis of the MOSFET because of its universal application in digita. This book has a unique chapter on the fabrication of devices; integrates a wealth of illustrations and pictorial representations; and contains an extensive discussion and analysis of the MOSFET because of its universal application in digital circuits. Goodreads helps you keep track of books you want to read. Want to Read saving….